Emulsion Polymerization

Emulsion polymerization

@i; PROYECTOCONACYT

Contributors A. Berge D. C. Blackley A. S. Dunn T. Ellingsen Carlton G. Force Robert G. Gilbert A. E. Hamielec F. K. Hansen A. A. Khan Gottfried Lichti J. F. MacGregor P. C. M0rk Donald H. Napper Mamoru Nomura R. H. Ottewill Gary W. Poehlein Vivian T. Stannett J. Ugelst~d V. 1. Yeliseyeva

'.

EMULSION

POLYMERIZATION

Editad by

IRJA PIIRMA Institute of Polymer Science The Uníversity of Akron Akron, Ohía

ACADEMIC

PRESS

A Subsidiary of Harcourt Brace Jovanovich, Publishers

New York London -Toronto Sydney San Francisco 1982

----

COPYRIGHT @ 1982, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLlCATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGEANO RETRlEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLlSHER.

ACADEMIC PRESS, INC.

111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON)

24/28 Oval Road, London NWI

7DX

Ubrary of Congress cataloging Main entry under ti tle:

LTD.

in Publication .

Data

Emulsion polymerization. Includes bibliographies and index. l. Emulsion polymerization. l. Piirma, QD)B2.E48E48 660.2'8448 ISBN 0-12-556420-1

81-17626 AACR2

PRINTED IN THE UNITED STATES OF AMERICA 82 83 84 85

.

9 8 7 6 S 4 3 2 J

Irja,

Contents Contributors Preface

ix xi

1 The Stability and Instability of Polymer Latices R. H. Ottewi/l 1. Introduction 11. The Nature of Polymer Latex Particles 111. The Effect of Electrolytes on a Latex IV. The Theory of the Stability of Lyophobic Colloids V. Coagulation as a Kinetic Process VI. An Alternative Approach to the Critical Coagulation Concentration VII. The Determination of ccc Values VIII. The Effect of lons That Interact with Water IX. Secondary Minimum Effects X. The Effects of Organic lons: Added Surfactants XI. lonic Head Group with a Charge of the Same Sign as the Particle XII. lonic Head Group with a Charge of Opposite Sign to the Particle XIII. Nonionic Surfactants XIV. Mixed Electrolyte Systems xv. Heterocoagulation XVI. Surface Coagulation XVII. Peptization XVIII. The Effects of Adsorbed or Grafted Macromolecules XIX. Particle Stability in Emulsion Polymerization XX. Summary References

.

1 2 6 8 14 16 17 19 22 26 27 28 31 35 36 39 40 42 45 47 47

2 Particle Formation Mechanisms F. K. Hansen and J. Ugelstad

1.

Introduction

11. Micellar Nucleation: The Smith-Ewart 111. Radical Absorption Mechanisms IV. Micellar Nucleation: Newer Models V. Homogeneous Nucleation VI. VII.

Theory

Particle Coagulation during the Formation Period Nucleation in Monomer Droplets References

v

51 54 56 63 73 82 86' 91

Contents

vi

3 Theoretical Predictions of the Particle Size and Molecular Weight Distributions in Emulsion Polymerizations Gottfried Lichti, Robert G. Gilbert, and Donald H. Napper 1. 11.

Prediction of the PSD Molecular Weight Distributions 111. Separability of MWD and PSD IV. Conclusions References

4 Theory ofKinetics ofCompartmentalized Reactions

94 115 141 142 143

Free-Radical Polymerization

D. C. Blackley 1. Introduction 11. Reaction Model Assumed 111. The Time-Dependent Smith-Ewan Differential Difference Equations: Methods Available for Their Solution IV. Solution for the Steady State V. Solutions for the Nonsteady State VI. Predictions for Molecular Weight Distribution and Locus-Size Distribution VII. Theory for Generation of Radicals in Pairs within Loci List of Symbols References

5

146 149 156 164 167 183 185 187 189

Desorption and Reabsorption of Free Radicals in Emulsion Polymerization Mamoru Nomura

1. Introduction 11. Polymerization Rate Equations Involving Free-Radical Desorption 111. Derivation of Rate Coefficient for Radical Desorption from Panicles IV. Effect of Free-Radical Desorption on the Kinetics of Emulsion Polymerization List of Symbols References

191 192 199 210 217 219

6 Effects of the Choice of Emulsifier in Emulsion Polymerization A. S. Dunn 1. Introduction Monomer Emulsification 11. 111. Emulsion Polymerization with Nonionic Emulsifiers IV. Emulsion Polymerization with lonic Emulsifiers V. Latex Agglomeration Other Effects of Emulsifiers JVI. References

:¡

221 224 229 230 236 237 243

I

Contents

vii

7 Polyrnerization

of Polar Monorners

V. l. Yeliseyeva 1. Introduction 11. Interface Characteristics of Polymeric Dispersions 111. Relationship between Emulsifier Adsorption and the Difference in the Boundary-Phase Polarity IV. Mechanism of Particle Generation V. Colloidal Behavior of Polymerization Systems VI. Kinetics of Emulsifier Adsorption VII. Mechanism of Formation and Structure of Particles VIII. Polymerization Kinetics IX. Relationship between Polymerization Kinetics and Adsorption Characteristic of the Interface Nomenclature References

247 249 250 257 261 268 27q 278 283 286 287

8 Recent Developrnents and Trends in the Industrial Use of Latex Car/ton G. Force 1. 11. 111. IV. V. VI.

289 291 300 312 313 314 316

Introduction Factors in Adhesion Bonding Applications Construction Applications Rubber Goods Properties of Various Latexes References

9 Latex Reactor Principies:

Design, Oneration,

and Control

A. E. Hamie/ec and J. F. MacGregor 1. Introduction 11. Batch Reactors 111. Continuous Stirred-Tank Reactors: Steady-State Dperation IV. ContinuQus Stirred-Tank Reactors: Dynamic Behavior V. Dn-Une Control of Continuous Latex Reactors VI. Summary Nomenclature References

10

319 320 333 339 345 351 351 353

Ernulsion Polyrnerization in Continuous Reactors. Gary W. Poeh/ein

1. Introduction

. 11.

Smith-Ewart Case 2 Model for a CSTR 111. Deviations from Smith- Ewart Case 2 IV. Transient Behavior bf CSTR Systems V. Strategies for Process and Product Development VI. Summary References

357 361 367 375 378 381 381

Contents

viii

11

Effect of Additives on the Formation and Polymer Dispersions

of Monomer

Emulsions

J. Uglestad, P. C. MrjJrk, A. Berge, T. Ellingsen, and A. A.Khan 1. Introduction 11. ThermodynamicTreatment of Swelling and Phase Distributions 111. Rate of Interphase Transport IV. Preparation of Polymer Dispersions V. Effectof Additionof Water-InsolubleCompounds to the Monomer Phase VI. Emulsificationwith Mixed EmulsifierSystems Listof Symb91s References

383 384 392 396 401 408 411 412

12 Radiation-Induced Emulsion Polymerization Vivían T. Stannett 1. Introduction 11. Laboratory Results with Different Monomers 111. Copolymerizations IV. Radiation-Induced Emulsion Polymerization Using Electron Accelerators V. Pilot Plant and Related Studies References

lndex

415 418 433 436 437 447

451

Contributors Numbers in parentheses indicate the pages on which the authors' contributions begin.

A. Berge (383), SINTEF, Applied Chemistry Division, 7034-NTH, Trondheim, Norway D. C. Blackley (145), National College of Rubber Technology, The Polytechnic of North London, Holloway, London N7 8DB, England A. S. Dunn (221), Chemistry Department, University of Manchester Institute of Science and Technology, Manchester M60 lQD, England T. Ellingsen (383), SINTEF, Applied Chemistry Division, 7034-NTH, Trondheim, Norway Carlton G. Force (289), Westvaco Corporation, Research Center, North Charleston, South Carolina 29406 Robert G. Gilbert (93), Departments of Physical and Theoretical Chemistry, University of Sydney, New South Wales 2006, Australia A. E. Hamielec (319), Department.of Chemical Engineering, McMaster University, Hamilton, Ontario L8S 4L7, Canada F. K. Hansen (51), DYNO Industrier, Lillestr~m Fabrikker, N-2001 Lillestrs6m, Norway A. A. Khan (383), E. 1. du Pont de Nemours & Co., Polymer Products Department, Experimental Station, Wilmington, Delaware 19898 Gottfried Lichti (93), Australian Institute of Nuclear Science and Engineering, New South Wales, Australia J. F. MacGregor (319), Department of Chemical Engineering, McMaster University, Hamilton, Ontario L8S 4L7, Canada P. C. Mr)rk (383), Laboratory of Industrial Chemistry, The University of Trondheim, N-7034 Trondheim, Norway Donald H. Napper (93), Department ofPhysical and Theoretical Chemistry, University of Sydney, New South Wales 2006, Australia Mamoru Nomura (191), Department oflndustrial Chemistry, Fukui University, Fukui, Japan R. H. Ottewil/ (1), School of Chemistry, University of Bristol, Bristol BS8 1TS, England Gary W. Poehlein (357), School of Chemical Engineering, Georgia Institute ofTechnology, Atlanta, Georgia 30332 Vivian T. Stannett (415), Department of Chemical Engineering, North Carolina State University, Raleigh, North Carolina 27650 ix

x

Contributors

J. Uge/stad (51,383), Laboratory of Industrial Chemistry, The Norwegian Institute of Technology, The University of Trondheim, N-7034 Trondheim, Norway V. l. Ye/iseyeva (247), Institute ofPhysical Chemistry, Academy ofSciences USSR, Moscow, Union of Soviet Socialist Republics

.

x

Contributors

J. Ugelstad (51, 383), Laboratory of Industrial Chemistry, The Norwegian Institute of Techno1ogy, The University of Trondheim, N-7034 Trondheim, Norway V. l. Yeliseyeva (247), Institute ofPhysical Chemistry, Academy ofSciences USSR, Moscow, Union of Soviet Socialist Republics

.

1m

Preface Emulsion polymerization has been a very successful industrial process for four decades. In recent years it has undergone revitalization: while some old factories are closing, new and much more sophisticated ones using emulsion polymerization are sprouting. Historically, in an attempt to produce synthetic rubber, the polymerizations involving the use of an aqueous emulsifier solution resulted in a product that in physical appearance resembled nature's latex. It also gave a greatly improved synthetic rubber over that produced previously by other processes, e.g., the sodium-butadiene process. This industrial success was subsequently followed by a more theoretical approach to the problem during the 1940s that resulted in the publication of the first scientific papers in this field. Even then, since emulsion polymerization had been established first as an industrial process, the theories proposed by the scientists were primarily concerned with trying to put the industrial observations into a general and workable scientific framework. Unfortunately, the emulsion polymerization of styrene and the copolymerization of styrene-butadiene fit beautifully into a simple kinetic scheme proposed by Smith and Ewart, and during the two decades that followe9 almost all research efforts in the field attempted to make the other monomers fit into this framework. We now know that emulsion polymerization is not just another polymer synthesis method and that the complexity of the interactions, whether chemical or physical, must be considered before any control is possible over the outcome of the reaction. The creation and nucleation of particles, for example, is not necessarily and simply explained by the presence or or absence of micelles, but needs the understanding of interactions of all the ingredients present. Variables such as hydrophilic and hydrophobic associations or repulsions, polarity of the monomers, chemical structure of the surfactants, have to be taken into account. Research in the field is flourishing all over the world, and although numerous papers have been published and collections of papers have appeared recently, they have the disadvantage of presenting fragments of the subject and never the total picture. This book presents a collection of chapters, each written by scientists in their fields of expertise. These experts come from all over the world and thus sometimes represent different viewpoints of the same subject. It is the hope ofthe editor and the contributors that we have been successful in presentirig a total picture of the current understanding ofthe subject. xi

'"

1 xii

Preface

1 would like to thank Silvia Dolson for providing her time and talents to assist in the book's cover designo

1 !1 S ~ -

1 The Stability and Instabiliíy 01 Polymer Latices R. H. Ottewill

1. Introduction . 11. The Nature of Polymer Latex Particles 111. IV. V. VI. VII. VIII. IX. X. XI. XII. XIII. XIV. xv. XVI. XVII. XVIII. XIX. XX.

l.

.

The Effect of Electrolytes on a Latex . The Theory of the Stability of Lyophobic Colloids Coagulation as a Kinetic Process . An Alternative Approach to the Critical Coagulation Concentration The Determination of ccc Values . The Effects of lons that Interact with Water Secondary Minimum Effects . The Effects of Organic lons: Added Surfactants lonic Head Group with a Charge ofthe Same Sign as the Particle lonic Head Group with a Charge of Opposite Sign to the Particle Nonionic Surfactants Mixed Electrolyte Systems . Heterocoagulation . Surface Coagulation Peptization The Effects of Adsorbed or Grafted Macromolecules. Particle Stability in Emulsion Polymerization Summary. References

1 2 6 8 14 16 17 19 22 26 27 28 31 35 36 39 40 42 45 47 47

Introduction

Over the last two decades work on the formation and properties of polymer latices has developed extensively, and a very substantial amount of work has been devoted to the study of the processes of formation of polymer particles in a latex and to the characterization of the particles once formed (see for example, Fitch 1980). It is now generalIy recognized that in 1 EMULSION POLYMERIZATlON Copyright @ 1982 by Academic Press. Inc. AIl rights of reproduction in any form reserved. ISBN 0-12.556420-1

I

~

R. H. Ottewill

2

the majority of latices, the particles are within the size range 1 nm to 1 pm, which designates them as colloidal, and the name "polymer colloids" is becoming commonly used to describe this type of system. Consequently, the entire field of polymer latex manufacture and utilization is very dependent on an understanding of the basic principIes of colloid science in the widest sense whether it is to stabilize or coagulate the particles or to control the rheological properties of latex formation. In using the word stability in the colloidal sense we understand that the particles in the dispersion will remain for long periods of time, often years, dispersed as single entities in Brownian motion. The size range 1 nm to 1 pm is chosen to exclude at the lower end of the range small single ion s and molecules and at the higher end to exclude particles that settle under the influence of gravity and do not remain dispersed by Brownian motion. These are definitions of convenience rather than rigor and in considering polymer latices we shall frequently need to consider particles with a size. greater than 1 pm; in this case the colloidal arguments can be maintained but, in addition, the effects of gravity have to be included. When the la,tex loses its stability because the particles aggregate under the influence of chemical additives or mechanical action, the terms coagulation and flocculation are applied (see later). By the term rheology we describe the flow and deformation properties of the system, Le., whether it is viscous (either Newtonian or non-Newtonian) or elastic or whether it possesses both these properties and is viscoelastic.

ll.

The Nature oí Polymer Latex Particles

A typical polymer latex particle will be composed of a large number of polymer chains, with the individual chains having molecular weights in the range of about 105 to 107. According to the arrangement of the polymer chains within the particle, the latter can be amorphous, crystalline, rubbery, or glassy. Moreover, in many cases monomer is retained by the particle and hence the particles can also be, in cases where the polymer is soluble in the monomer, either extensively or minutely swollen. The physical state of the particle can be important in close-range interactions and in drying. For example, if the particles are soft, coalescence of the particles can occur to give continuous film formation, whereas with hard particles their individuality is retained in the dry state. In determining the colloidal behavior of a latex, the surface properties playa very important role, and these are directly related to the preparative . method employed. They frequently depend on (i) groupings arising from the initiator used; (ii) adsorbed or grafted surfactants; and (iii) adsorbed or grafted polymers, particularly, those soluble in the dispersion medium.

I j j

I j

j!

,

~

3

1. The Stability and Instability of Polymer Latices

In aqueous-based emulsion polymerizations using water-soluble initiators, the surface groupings formed are frequently determined by the nature of the initiator used (Ottewill et al., 1967; van den HuI et al., 1970; Goodwin et al., 1973) and the following have been reported: from hydrogen peroxide, persulfate, bisazocyanopentanoic acid

Weak acid

Strong acid

-0-S03

or

from persulfate

CH3 NH2 Base

Nonionic

I

,f'

-C-C~, I NH2 CH2 } -OH

+

from azobisisobutyramidine

from hydrogen peroxide or hydrolysis sulfate groups

of

In addition, latex particles with mixed anionic and cationic groups on the surface can be prepared (Bolt, 1978). In an ionizing medium of high relative permittivity (e.g., water) the acidic and basic groupings exist in the ionized form, depending on their pKa and pKb values and the pH, and consequently the surface of the particle becomes electrically charged. In addition the adsorption of other ionic species, such as surface-active ions, can also contribute to the surface charge. In physical terms the water is a good solvent for the ions and poor for the latex particle; that is, most of the polymers used for latex preparation are totally insoluble in water. A schematic illustration of this situation is shown in Fig. 1 where the particle surface is shown to be that of a smooth sphere with the charges evenly distributed over the spherical surface. The condition of electroneutrality is maintained by balancing the charge on the latex surface by the charges on small ions of opposite sign in the solution phase (counterions). This forms the so-called electrical double layer in which an equilibrium is set up between electrostatic forces and diffusion forces. As a consequence of its surface charge, the lat.ex particle surface has an electrostatic surface potential rjlswhich can be either positive or negative, depending on the nature of the surface groupings, relative to 'earth. This potential falls off exponentially with distance from the surface of the sphere according to the equation rjlr= rjls(a/r)exp[

-

K(r - a)]

(1)

where rjlris the surface potential at a distance (r - a) from the surface of the' sphere where a is the radius of the sphere. K is defined by (2) K2 = (8ne2NAl1000ekT)I

4

R. H. Ottewill

- -+ - -

,,/

o +

/ I I

"-

+ "-

+ \

+ \ \ I I

++

\ \

-

-

+

\

+ "-

"-

-

+

/

+

-- +

/

.,//

Fig. 1. SchematiciIlustration of a negatively charged spherical polymer latex particle with an electricaldouble layer. --- representsthe rangeof influenceof electrostaticforces.

and is dependent on the ionic .strength 1 of the solution phase, e the fundamental electronic charge, NA Avogadro's number, and ¡; the dielectric constant of the medium. It is K that determines how rapidly the electrostatic potential falls off with distance from the particle surface and consequently the range of electrostatiC interaction forces. The dashed line in Fig. 1 indicates that the range of electrostatic forces can extend well beyond the physical size of the particle. When latices are prepared in nonaqueous media such as hydrocarbons then charged-surface groups no longer pro vide a practical means of stabilizing the particles formed. Under these conditions polymer chains, soluble in the dispersion medium, can be grafted to the core polymer particle which remains insoluble in the dispersion medium. A typical example is the use of poly(12-hydroxystearic acid) chains to stabilize particles of poly(methyl methacrylate) in dodecane. This leads to what might be termed a "hairy particle" with the noncharged "hairs" extending into the solvent medium as shown in Fig. 2. Again the range to which the chains extend is important as it determines the distance at which one particle of this sort can start to interact sterically with another, giving the so-called sterically stabilized systems. Returning again to ionizing media, a combination of these two effects can be employed by grafting to a polymer core particle polyelectrolyte chains. This is illustrated schematically in Fig. 3. It pro vides a combinatorial effect of electrostatic and steric interactions. These will have different interactive ranges as illustrated in Fig. 3 by the dotted line for the electrostatic range and the dashed line for steric effects.

1.

5

The Stability and Instability of Polymer Latices

"-

\ .\ \ I ,

/

/

/ Fig. 2.

SchematiciIIustrationof a nonchargedpolymerlatex particlewith adsorbed or

grafted nonionic

polymer chains. ---

represents

the range of inftuence of steric forces.

The chains shown in Fig. 3 are those of long poly(ions) deliberately added. However, there is sorne evidence to indicate that even in conventional ernulsion polyrnerizations the particles forrned rnay not be as srnooth as those shown in Fig. 1, and the charged groups rnay be floating a short distance in the rnediurn as "rnicrohairs." In practice one should not be rnisled by the convenience of the srnooth sphere rnodel for theoretical rnodeling of colloidal phenornena.

. . . . . .'".. . +

.

'

+

.,

+

- .

. +

.+

+ +

. +

+

.

+

+ . . .. . . .

Fig. 3. Schematic iIIustration of a polymer latex core with grafted polyelectrolyte chains attached to the core surface. represents the range of electrostatic forces, and --represents the range of steric forces.

6

R. H. Ottewill

From this qualitative description of latex particIes we can immediately recognize the origins of three basic forces that ha ve to be considered in understanding the behavior of dispersions in both aqueous and nonaqueous media. These can be summarized as 1. Electrostatic effects: usualIy repulsive but opposite charges on particIes can lead to attraction. 2. Steric effects: arising from the geometry and conformation of adsorbed or grafted mo.1ecules. 3. Solvation effects: arising from the organization of solvent molecules near an interface or between the chains of adsorbed macromolecules. It will be noticed that these effects do not take into account to any great extent the bulk phase of the particIe. However, this also has to be taken into account and both the polarizability and the density are of importance in determining the attractive forces between particIes. Thus, we can add to the list

4. Attractive effects: which have their origin in molecular dispersive interactions (oftentermed van del Waals interactions).

m.

The Effect of Electrolytes00 a Latex

In an aqueous latex that has been cIeaned to remove various materials such as surface active agents one is dealing in many cases with a dispersion of charged spherical particIes. FrequentIy, the distribution of particIe sizes is very narrow and the term monodisperse is used to describe latices of this type. In an attempt to obtain a concise overview of the properties of aqueous latices in electrolyte solutions some of the essential features are summarized schematicalIy in Fig. 4. At intermediate electrolyte concentrations (-10-3 mol dm-3) and at low volume fractions of the dispersed phase, the charged particIes occupy random positions in the system and undergo continuous Brownian motion with transient repulsive contacts when the particIes approach each other. The range of the electrostatic repulsive forces is represented by the dashed circIe in Fig. 1, which implies that when a similar circIe on another particIe overlaps with it on a collision trajectory, a transient electrostatic repulsion occurs and the particIes move out of range. With most latices the particIes .have a real refractive index and their visual appearance is "milky" white. Ir, however, either the electrolyte concentration is reduced to about

10- 5 mol dm- 3, which increases the range of the eIectrostatic repulsive forces, or the concentration of the particIes is increased, a situation is

~

7

1. The Stability and Instability of Polymer latices Order Regular lattice Strong repulsion

Electrolyte 10- M 5

Remove electrolyte Imixed-bed ion exchang~

Dlsperslon Electrolyte

- sol 10-

3

resin)

Disorder Random arrangement of particles Brownian motion Repulsive contacts

M

Add electrolyte

Stable Unstable Electrolyte 0.2 M

Schulze

.

Hardy Flocculatlon Disorder Weak attraction

Fig. 4. partides

Coagulation Disorder Strong attraction

The effectof electrolyteand partide size on the properties of polyrner latex

in an aqueous

rnediurn. (Reproduced

with perrnission

of Chem. ¡nd. (London).

reached where the partic1es must maintain repulsive contacts over a long period of time. Con sequen tIy, an ordered arrangement.of the partic1es is set up so that the partic1es sit in lattice positions but remain well separated, Le., a "liquid crystal" arrangement is formed (see Fig. 4). When the interpartic1e spacing is of the order of the wavelength of light, Bragg diffraction effects become superimposed on the scattering from the partic1es and bright iridescent colors can be seen visually. This is well demonstrated with mono disperse latices which, in the right size and concentration range, show brilliant colors (Hiltner and Krieger, 1969; Hachisu, 1973; Goodwin et al., 1980).

R. H. Ottewill

8

In the lower part of Fig. 4 a schematic illustration is given of the transition from a stable dispersion, where all the particles exist essentially as single entities, to an unstable state where aggregation of the particles occur. This change from stability to instability has received a great deal of attention from several generations of colloid scientists. It is usually associated, when inorganic electrolytes are used, with the names of Schulze (1882, 1893) and Hardy (1900) who investigated this phenomenon at about the turn of the century. Today, our explanations of colloidal particle aggregation processes are largely based on the theories of interactions between particles that were first enunciated in quantitative form in the 1940s by Derjaguin and Landau (1941) and Verwey and Overbeek (1948), frequently now termed the DL VO theory in honor of the four authors.

IV.

The Theory of the Stability of Lyophobic CoUoids

The quintessence of the ideas put forward by DL VO was that the potential energy of electrostatic repulsion VRbetween the particles and the potential energy of the van der Waals attraction VAcould be added together to obtain the total potential energy of interaction VT. When the very shortrange Born repulsive energy was included, a potential energy against distance of surface separation curve of the form shown in Fig. 5 was obtained. This type of curve exhibits a number of characteristic features. At short distances, a deep potential energy minimum occurs, the position of which decides the distance of cIosest approach ho and is hence termed the. primary minimum. At intermediate distances, the electrostatic repulsion makes the largest contribution and hence a maximum occurs in potential energy of magnitud e Vm;this position is termed the primary maximum. At greater distances, the exponential decay of the electrical double layer term causes it to fall off more rapidly than the power law of the attractive term and another min!mum occurs in the curve, of depth VSM, termed the secondary minimum. In more quantitative terms VRfor weak interactions can be written in the form VR =

3.469 X 1019 B(kT)2ay2 exp( -

Kh)/V2

(3)

where Bis the dielectric constant of the dispersion medium, k is Bolzmann's constant, T is"the absolute temperature, v the magnitud e of the valency of the counterion, and y = [exp(velj¡s/2kT)- 1J/[exp(velj¡sl2kT)+ 1J with e as the fundamental electron charge. This expression given by Reerink and Overbeek (1954) is for the electrostatic interaction between two spheres of equal radius a of constant

1.

9

The Stability and Instability of Polymer Latices 10

Primary Maximum

. ...

.

l....

o Secondary Minim~m

Pri';;a~y- , Mínimum

h Fig. 5.

SchernaticiIIustration of a potential energy against distance of surface sepa-

ration curve to illustrate the rnain features used in discussing colloid stability. ~ Vr= energy barrier to coagulation;

~Vb

=

energy barrier to peptization;

i'd = dispersion

free energy of the

polyrner-water interface; Vm= height of the prirnary rnaxirnurn; VSM= depth of the secondary

rninirnurn.

surface potential and material 1 separated by a distance h in a medium 2 (Fig. 6). It is valid for the Ka range 3 to 10. For Ka < 3 and Ka > 10 useful expressions for small potentials are, respectively VR = [ea21jJ;/(h VR = tealjJ;

+ 2a)] exp( -'Kh)

ln[l + exp(- Kh)]

(4) (5)

The question of interaction at constant surface charge has also been discussed by several authors (Frens, 1968; Wiese and Healy, 1970; Gregory, 1975).

R. H. Ottewill

10

Fig. 6. Interaction betweentwo sphericalparticlesof material 1 in a liquid mediumof material2. a = radius of sphericalparticle;h = distanceof surfaceseparation.

For the attractive part of the interaction it was shown by Hamaker (1937) that VAcan be given for two spheres of the same radius and the same material by the expression V A-

-

A

1

1

12{ X2 + 2x+ X2+ 2x+ 1+

2l

X2

+ 2x

n(X2+ 2x+ 1)}

6)

(

where x = hl2a and A = the composite Hamaker constant for the particles in the mediumas givenby (7) TABLEI Values of Reported Hamaker Constants for Various Polymers

Material Poly(vinyl acetate) Poly(vinyl chloride) Poly(methyl methacrylate)

Styrene-butadiene Polytetrafluoroethylene

Polystyrene

AII/J (x 10-2°) 8.84 12.4 10.0 6.3 5.6 3.7 7.6

a Lifschitz, small separation distances. b Lifschitz, long separation distances.

AjJ (xlO-21) 5.4 5.5 7.2 5.5 3.0-3.8 4.0 2.9 3.6 3.5 9.0 3.2 6.5 7.0

Reference Dunn and Chong (1970) Evans and Napper (1973) Visser (1972) Visser (1972) Friends and Hunter (1971) Evans and Napper (1973) Force and Matijevié (1968a) Fowkes (1967) Rance (1976) Gingell and Parsegiana (1973) Visser (1972) Krupp et al. (1972) Gingell and Parsegiana (1973) Gingell and Parsegianb (1973) Evans and Napper (1973) Lichtenbelt et al. (1974)

11

1. The Stability and Instability of Polymer Latices

= the Hamaker constant of the particIes and A22 that of the medium. The Hamaker constant is directly related to the nature of the material by the expression

with All

(8) where Vj is the dispersion frequency of the material (c/A.o,with A.o= the dispersion wavelength), Ctjis the static polarizability and qj is the number of atoms/molecules per unit volume of the material. Some typical values of

A 11 for polymericmaterials are listed in Table I. For conditions such that x ~ 1 a useful approximation is obtained from Eq. (6),in the form (9)

VA = - Aa/12h

As the distance of separation between the particIe surfaces increases, particularly beyond 100 nm, a weakening of the attraction starts to occur,

150 11

\

\

\

., 100 t+--

\

\

.

\ \

\ \

\.,

:i.

11.

kT

50¡¡':'

\

.

\

.

\

\

\

'\ "-

\ \

o

...

.\

"-

"". -- .;-..,

.. .........

"........

,.,

- - =.= .'::..~,

10

...-.--.-20

h (nm) Fig. 7. Potential energy diagrams for the interaction between two spherical latex particles of radius 0.1 Jlm at a constant potential of 1/1,= 50 mV, with A = 7.0 X 10-21 J and T=298.2°K in a 1:1 electrolyte. -, 1O-3moldm-3; ---, 1O-2moldm-3; -'-', 0.075 mol dm-3; 0.5 mol dm-3.

R. H. Ottewill

12

100 VT kT

,......,

,

I!

~

r:

.,

"

.

'.

,

.'.

e.-- ..

'.

......-.

- .-.

k~..".-

O . ..-"

-8-

-

:: V..a.~.- . _8-

--

-

..- -¡ '8.~ I...:e-=."\:

f¡ h (nm)

Fig. 8.

Potential energy diagrams for the interaction between two spberical latex

partic1es of radius 0.1 Jlm at a constant

1:1 eIectrolyte

concentration

of 10-3 mol dm-3

A = 7.0 X 10-21 J. T = 298.2°K. and various values of "', as follows:

37.5mV;-u-o 25mV;

12.5mV;

5 mV.

-.

witb

50 mV'; -'-,

the so-called retardation effect, and this has to be allowed for in the calculation of VT. Equation (9) can then be rewritten in order to allow for retardation as (Kitchener and Schenkel, 1960) VA = -(Aa/h)(2.45/p

- 2.17/180p2 + 0.59/420p3)

(10)

with p equal to 21th/A.o.Usually, A.ocan be taken as -100 nm and Eq. (10) is a reasonable approximation for a ;p h and 0.5 < p < oo. An alternative treatment of attractive interactions between particles is that given by Lifshitz and co-workers (1961); it is beyond the scope of this article but is discussed by Ninham and Parsegian (1970, 1971) and Richmond (1975). From the various equations given, under the appropriate conditions of Ka, h. etc. the total potential energy of interaction between the particles can be calculated from the basic DL VO assumption that for lyophobic dispersions, (11)

"=""

13

1. The Stability and Instability of Polymer latices

" \

200H-'I I I

,

I I

\ \ \ \ \ \

100

O""'O\

'o

\

r! ,1

~,f o

!II-:! 8_'-..1 -=

~ -= .":--'::. 20

h

-- ~ _8-:--=30

(n m)

Fig. 9. Potential energydiagramsfor the interactionbetweensphericallatexparticlesin a 1:1 electrolyte at a concentration of 1O-2moldm-3 with A=7.0xl0-2IJ and T = 298.2°Kover a range of particle radii: -, a = 2 Jlm;---, a = 1Jlm;-'-, a = 0.5Jlm; , a = 0.1Jlm,t/J, = 25 mV. .

-

-. -

A number" of curves of VT against h are given in Figs. 7 to 9 to illustrate various points. For example, Fig. 7 shows the effect of electrolyte for spherical particles of radius 0.1 J1.mat a constant surface potential t/ls.For the same particle radius Fig. 8 shows the effect of changing t/ls at a constant electrolyte concentration of 10- 2 mol dm - 3. The effect of particle size for particles of the same t/ls at the same electrolyte concentration is demonstrated in Fig. 9. It is apparent from these curves that when the magnitude of Vmis greater than about 10kT conditions are favorable for the formation of a stable dispersion. Moreover, the form of the potential energy curve obtained by this approach indicates that the dispersion exists in a higher free energy state than that of the solid polymer with the depth of the primary rpinimum equal to twice the dispersion free energy of the polymer (Fowkes, 1964). The stability of the dispersion is kinetic in origin as a consequence of the large activation energy ~Jréwhich renders the forward transition into the primary minimum improbable. It also becomes apparent that because the activation

R. H. Ottewill

14

energy for the backward reaction /!:;. Vbis much larger than /!:;. Vr(see Fig. 5), and a decrease of free energy in the system has occurred once the primary minimum has been accessed, then spontaneous redispersion is unlikely to occur and mechanical work is needed to redisperse the aggregated particles. Móreover, time has to be considered in this context since prolonged contact of particles in the primary minimum can lead to welding of the particles, as interparticle diffusion of polymer chains occurs.

V. Coagulation as a Kinetic Process In a fundamental paper von Smoluchowski (1917) presented a theoretical model for the kinetics of the coagulation process. He showed that in the initial stages of coagulation the rate of disappearance of the primary particles, i.e., those present as single particles in the original dispersion, could be written as -dN/dt

.

= kN~

(12)

where No is number of primary particles per unit volume present initially and k is arate constant. For rapid coagulation, i.e., coagulation in the absence of an energy barrier, the process is diffusion controlled and k

= ko = 8nDR

where D is the diffusion coefficient of a single particle and R

the collision radius of the particle. In subsequent analyses it was shown that if diffusion in the presence of an energy barrier is considered (Fuchs, 1934; Overbeek, 1952) then the initial rate of disappearance of the primary particles could be written as -dN/dt

= koNUW

(13)

where W, termed the stability ratio, is related to VTby W

= 28 r'1) exp(VT/kT) Jo

dh

(14)

(h + 2a)2

In the absence of an energy barrier, i.e., setting VT = O,we find W = 1 and the equation for rapid coagulation is obtained. In the presence of an energy barrier VTbecomes positive and W becomes greater than unity and, clearly for these conditions, the rate of coagulation is slowed down; hence slow coagulation occurs. In practice at intermediate electrolyte concentrations

and medium potentials (ljIs'" 50 mV) W can attain values of the order of 107 so that coagulation is imperceptible on a reasonable time scale. The approach emphasizes the kinetic nature of the stability of lyophobic colloids.

15

1. The Stability and Instability of Polymer Latices

Measurements of the rate of coagulation of latex particles can be carried out by a number of techniques, two of those most commonly used being particle counting and light scattering (Ottewill and Shaw, 1966). These give values for the rate constant k and since a well-marked transition usually occurs between the slow and rapid coagulation regions the assumption is usually made that for rapid coagulation W = 1, whence for this region k = ko, and thus in the slow coagulation regíon, W can be obtained from the ratio ko/k. A typical example of the type of experimental data obtained is shown in Fig. 10 plotted in the form log Wagainst log C., where C. is the concentration of electrolyte in the dispersion after mixing has occurred at zero time. As can be seen, the transition between slow and rapid coagulation is clearly marked and a well-defined electrolyte concentration can be obtained from the graph at this point. This electrolyte concentration is termed the critícal coagulatíon concentratíon and will be abbreviated as cee. As a further development, it was shown by Reerink and Overbeek (1954)

:: C» o ...J

0.5

o

-2.5

-2.0 Log

-

--~

.

-1.5

ce

Fig. 10. Log W against log C. eurve for a polystyrene latex (a = 0.21 Jlm) in barium nitrate solutions (Ottewill and Shaw, 1966). indieates results obtained from light-seattering

measurements; O indieates resuIts obtained using a partic1e eounting teehnique; cee value.

i

indieates

R. H. Ottewill

16

that the gradient of the curve just before the cee was given as a first approximation, based on Eqs. (3) and (9), by (15) From this equation it is clear that the gradient should be directly proportional to the particle radius a and inversely proportional to the square of the valence of the counterion used. A number of experimental studies have been made tha~ do not seem to confirm these predictions (Ottewill and Shaw, 1966) and even more refined treatments of the kinetic process have not removed the discrepancy (Derjaguin and Muller, 1967; Honig et al., 1971). There is little doubt that as a kinetic process coagulation is rather complicated; further discussion on this point will be given later.

VI.

An Alternative Approach to the Critical Coagulation Concentration

From the previous discussion.it is apparent that provided the magnitude of Vmis substantial the probability of transition into the primary minimum is small. However, as shown above when VT = Vm = Othe transition is facile (Fig. 11) and the particles coagulate. Therefore, we can define conditions for the onset of instability as (Verwey et al., 1948) and

(16)

whence using Eqs. (3) and (9) we find Kcrit= 2.04 X 1O-Sy2/Av2

(17)

Distance >~

IJ) c: W

...

,

...

...

...

,

-

/

Fig. 11. Sehematie potential energy eurve to iIIustrate eonditions leading to a theoretical definition ofthe cee. -, VT;---, VAonly.

f

17

1. The Stability and Instability of Polymer Latices

and since for a symmetrical electrolyte 1( can be related directly to the concentration of the electrolyte C expressed in mol dm - 3 by 1(2

= 81tv2e2NAC/ekTx103

(18)

with NA = Avogadro'snumber, wethen find using Eqs. (17)and (18)that, CCril=

3.86 x 1O-25y4 -3 mol dm A 2v ~

(19)

where Ccritis the ccc and we note the inverse dependence on the sixth power of the valence of the counterion. For low surface po tentials, t/ls< 25 mV, a further simplification can be made to give mol dm-3

(20)

which gives an inverse dependence on the square of the valence, so that for univalent, divalent, and trivalent counterions, we obtain

The DL VO theory thus enabled theoretical significance to be given to the valence sequence in coagulation experiments that had been observed many years earlier by Schulze (1882) and Hardy (1900). Although expressions of this type are useful in a qualitative predictive sense, the implication that there is a simple rule applicable to alf systems must be treated with considerable caution. It must be borne in mind that coagulation is a complicated phenomenon involving quite a range of kinetic and specific ion effects.

VD.

Tbe Determination of eee Values

From an experimental viewpoint, however, there is no doubt that the ccc is a very important quantity to know for a polymer latex, since it represents the electrolyte concentration at which complete loss of stability occurs. Experimentally, the ccc can be obtained by a variety of methods and the use of light scattering and partic1e counting ha ve already been mentioned. Possibly the simplest method of all is visual observation in test tubes containing the same concentration of latex and different concentrations of electrolyte. A slightly more elaborate version of this method is to use a simple spectrophotometer to measure the optical density of the dispersion at specific time intervals after addition of electrolyte; the most convenient

R. H. Ottewill

18

time period having been determined from some preliminary experiments (OttewilI and Rance, 1977). An example is given in Fig. 12 of a plot of optical density against pH which shows the boundary between slow and rapid coagulation regions. Extrapolation of the rapidly descending portion of the curve to the abscissa yields a value for the cec. As can be seen the value obtained wilI be time dependent so that for comparative purposes, e.g., to evaluate the effectiveness of different electrolytes, standardization of the time period of the experiment is essential. The time dependence is not unexpected since each time corresponds to a different stage in the overall kinetic process. There is no doubt that the most precisely defined value is that obtained from the log Wagainst log Ce curve using initial rates, since this is always defined as t -+ O. We can anticipate that there will be some variation in the actual values of the ccc obtained by different methods and also between different workers. The general pattern of the results observed, however, should not change with either of the latter factors.

0.2

-

>-

VI c: CD

o

c. O 0.1

2.4

2.0

1.6 pH

Fig. 12. Optieal density against pH al various times after addition of hydroehlorie aeid. -0-,1 hr; -8-, 2 hr; -/;:,.-,26 hr; ---, cee 1 hr; -'-', cee 2 hr; , cee 26 hr.

1.

19

The Stability and Instability ot Polymer latiees TABlE

11

cee Values tor Various Polymer latiees Latex Polystyrene (Carboxylsurface) Polystyrene (Amidinesurface) Divinylstyrene Styrene-butadiene

Poly(vinyl chloride)

Counterion

cccjmmol dm - 3

H+ Na+ BaH La3+ (pH 4.6) CIBr1Na+ Na+ K+ MgH BaH La3+ (pH 3) Na+ MgH

1.3 160 14.3 0.3 150 90 43 160-560 200 320 6 6 0.5 50-200 2-10

Reference Ottewill and Walker (1968) Storer (1968) Ottewill and Shaw (1966) Ottewill and Walker (1968) Pelton (1976)

Neimann and Lyashenko (1962)

Force and Matijevié (1968a)

Bibeau and Matijevié (1973)

The cee values for a number of polymer latiees have been determined and some typical values are reported in Table II. The trends observed are qualitatively in agreement with those expected fram the theoretieal approach for particles with smooth surfaces, with 1/1.everywhere the same, using simple eleetrolytes, i.e., those which do not interact chemically with water to form new ionie species. These yalues should only be used for qualitative guidance since, in addition to the factors already mentioned, there can be variations of the cec with particle size, type and density of surface groupings, and the presence or absence of stabilizing materials such as surfactants. In practice it is advisable to determine the actual value for a particular latex system.

VIII.

Tbe Effects of IODSTbat Interact with Water

So far, the assumption has been made that the ions used in the coagulation experiments do not interact with water. In a number of cases, however, the ions do react with water under certain pH conditions to form hydrolyzed species. For example, in the case of aluminum, the AI3+ ion exists at pH values below about 3.3 as the hexaaquo ion, with six water molecules in the octahedral coordinate positions. As the pH is slowly increased, reaction occurs with water to form a sequence of species. The chemistry involved in these reactions is somewhat complex and has not

1-

l 20

R. H. Ottewill

been fulIy resolved but a plausible reaction scheme can be proposed, for the present purpose, as ~20

¡20 H20

¡ :

",

,

--.

'~'~I~+ ""

H;O

:

H20: .'.

Ji20

: '. ..

¡

/

H20: ". --.

''':~I~+ '.. '......

¡

..........

lI20

~20 OH

'. H20

H;O

OH

/i..... HO

'H20

¡

/

: : "AI+

H20

"H20

¡

H20

/ 1+

H'~ r~~",r~H "~I

/'~H/'" [ HO

.. ¡

~!

((

lI20

.. ¡

"'}{20

]

1120

1 Alx[OHy]"+

where Alx[OHy]n+ represents an inorganic polymer soluble in water. Polymeric species of this type can adsorb strongly onto negatively charged particles and reduce the effective surface potential on the particle to zero. As anticipated from Eq. (3) this situation leads immediately to coagulation. At higher concentrations of the aluminum species, superequivalent adsorption can take place, thus conferring a positive charge on the particle and leading to restabilization of the dispersion as one containing positively charged rather than the original negatively charged particles. In addition, it is also possible for the positive polymeric species to "bridge" two negatively charged particles. The exact nature of the polymeric species in solutions of aluminium salts at the pH conditions for charge reversal is not known with certainty. It is possible that several species coexist, depending on their stability constants, and that these also change with time with the ultimate product of hy-' drolysis being aluminum hydroxide particles. A number of species ha ve been pro posed in the literature and Matijevié et al. (1964) ha ve provided evidence for the existence of Als(OH)1t from coagulation studies. The higher valence of this type of species again reduces the concentration of ions

1.

21

The Stability and Instability of Polymer Latices

required to produce coagulation. The combined effects of high valence and reduction of the surface charge to zero makes aluminum salts very effective coagulants in the pH range of about 4 to 5.5. Coagulation can be achieved at very low salt concentrations and since most of the aluminum is adsorbed by the particles there is little salt left in the filtrate after remo val of the coagula. This factor is exploited in the use of aluminum salts for the treatment of potable water. The basic pattern of the coagulation of polymer latices with aluminum salts has been clearly demonstrated by the work of Matijevié and his collaborators (1968) using styrene-butadiene, poly(vinyl chloride) (1977), and PTFE (1976) latices. The results obtained by Matijevié and Force (1968b) for the coagulation of styrene-butadiene latices using alumirium nitrate are shown in Fig. 13. From these it can be seen that up to a pH of ",3.4 the ccc remains constant at 5 x 10-4 mol dm-3 and then decreases between pH 3.4 and 4.8 to reach a constant value of

'"

'";; O z

-2

~

c:

.2 -3 -;¡; ... ¡:

Q) () c: o Ü

RESTABILlZATlON

... -4 al o ~ ti' o ...J

STABlE

-5

REGION

2

3

5

..

6

7

pH Fig.13. Log[(AIN03h/mol dm-3J against pH showing the positions of the coagulation domains for a styrene-butadiene latex. Curves constructed from the data of Matijevié and Force (1968); reproduced with permission of Ko/loid Z.u.Z.fur Polymere.

.. ..

R. H. Ottewill

22 3.0

- 3.4

~

E

"O

-

~

R

Q)

c:

w

L-- ( a)

.

(b)

Distance

-------

Fig.27. repeptization.

AVb-kT

~

AVb-kT

.

L

R

R

(e)

(d)

Potential energy curves to iIIustrate the differences between coagulation and

42

R. H. Ottewill

One possibility is to provide the particles with a thin steric barrier such as the "micro hairs" which could occur on the surface of a latex by solvation of the polymer chain beyond the ionic end group. Indeed some evidence for this occurs with certain latices. For example, Smithan et al. (1973) have reported evidence of steric stabilization with polystyrene latices with a high content of carboxyl groups on the surface prepared by an essentially conventional emulsion polymerization method. Microsteric stabilization with latices could be an important factor and this is undoubtedly an area that needs more extensive investigation.

XVffi.

The Effects of Adsorbedor Grafted Macromolecules

Space prohibits a detailed discussion of this topic but a few general points can be made following the comments made in the previous section. With hairy particles of the type shown in Fig. 3, polyelectrolyte molecules can be chemically linked to the surface or adsorption can occur by several mechanisms, including ionic bonding-particularly via charges of opposite sign-hydrogen bonding, coupliñg with multivalent inorganic ions, and by hydrophobic bonding of the hydrophobic regions of the marcomolecule to the surface. The net result is shown schematically in Fig. 28. Instead of the array of surface charges leading to a well-defined surface charge density and surface potential there is now a distribution of charges in space which contri bu te to the electrical double layer surrounding the particle. At low electrolyte concentrations the latter will extend into the space beyond the polyelectrolyte layer so that VR will be significant. However, with increase in electrolyte concentration and compression of the electrical double layer, VR can become small or zero. However, under these conditions the particle will still be coated with an extensively hydrated layer of polymeric molecules which provide a steric barrier V.. Hence, this type of system provides a twotier mechanism of stabilization against electrolyte additions and the classical protective agents for colloidal particles such as gelatin, gum arabic, etc. almost certainly act in this way. A number of polymer latices falling into this category have been described in the literature but they are still relatively novel and have not received the extensive attention given to the more conven.tional latices. Probably the systems of this type most extensively characterized are those described by Hoy (1979) and Bassett and Hoy (1980) which were prepared by copolymerizing methyl methacrylate, butyl acrylate, and ethyl acrylate with an unsaturated acid such as itaconic, acrylic, or methylacrylic. The particles obtained appeared to consist of a spherical core particle sur-

I

t~

I

íf

43

1. The Stability and Instability of Polymer latices

I I / POLYION /

LAYER /

-

/'

Fig. 28. Schematic ilIustration of a latex partic1e with an adsorbed or grafted layer of polyelectrolyte. represents the extension in space of the adsorbed layer, and --- represents the extension in space of the electrical double layer. 'Reproduced with permission. of American Chemical Society.

rounded by an acid-bearing polyion shell. The latter expanded at high pH values as the acid groups were neutralized to give a structure similar to that shown schematically in Fig. 28. These authors have carried out extensive ultracentrifugation studies on this type of system as a function of pH in order to determine values for the expanded shel1thickness (see c5in Fig. 17). Latices prepared by grafting polyacrylate chains onto a polystyrene core have also been described recently by Buscall and Comer (1980). These authors also examined the stability behavior of their latices as a function of the degree of neutralization of the polyacrylate and as a function of temperature at different electrolyte concentrations. As can be seen from the results given in Fig. 29 the systems were stable over a certain range of temperature and degree of neutralization but flocculated both on heating and on cooling. The behavior at the upper temperature appears to be similar to that observed with nonionic surfactants as stabilizing molecules at or near the c10ud point of the surfactant and basically arises as a

R. H. Ottewill

44

20

Fig. 29. Data iIIustrating the temperature behavior of a latex with a polystyrene core stabilized by grafted poly(acrylic acid) in 1.10 mol dm - 3 sodium chloride solution; IX= degree of neutralization of the latex.

consequence of desolvation of the chains; in the polymer terms the cloud point would be close to the () temperature. The authors explain the lower temperature behavior as a consequence of the dissimilarity between the free volume of the polymer and the solvent. The behavior of this type of polyelectrolyte system is of fundamental interest and it is hoped that in the near future more detailed stability studies will be reported. An interesting feature from the point of view of polymer morphology is whether all the chains are on the surface or some are buried in the particle. The behavior of aqueous latex dispersions, in which the polymer core particles were stabilized by block copolymers of poly(ethylene oxide) and a vinyl or acrylic monomer, has been investigated in some detail by Napper (1969). The particles in these latices were shown by electrophoresis to be noncharged. Flocculation occurred when the solvency of the dispersion medium for the polymer chains was decreased as, for example, by raising the temperature when a critical flocculation temperature was observed that was found to be insensitive, over a limited range, to the molecular weight of the poly(ethylene glycol) chains. The influence of a number of other factors on the stability of the latices was also investigated (Napper, t970a) including the nature of the anchoring groups of the stabilizing polymers, the

r

I

~

I

1. The Stability and Instability of Polymer Latices

45

nature of the disperse phase, the particle size, the surface coverage and the molecular weight of the stabilizing polymer. It was found that to obtain colloid stability, it was necessary to use a dispersion medium that was better than a () solvent for the stabilizing chains. The implication of this observation was that the second virial coefficient of the stabilizer needed to be po sitive so that the segmental excluded volume was also positive. Under these conditions once overlap of the stabilizing polymer layers occurred, the configurational entropy of the molecules in the overlap region would become less than that of the molecules in the dispersion medium and an excess osmotic pressure would occur in the overlap volume. As a consequence of this, molecules of the dispersion medium would difIuse into the overlap region, forcing the stabilizing layers apart. This is essentially the basis upon which Eq. (24) was formulated. It was found that with this type of system that peptization could be achieved spontaneously after centrifugation or flocculation,' as would be anticipated from Fig. 27d. Napper (1970b) also investigated the flocculation of poly(vinyl acetate) particles stabilized by poly(ethylene glycol) chains with a series of electrolytes. The order for the cations was

The controlling factor appeared to be the capability of the ion to convert water into a ()solvent for the stabilizing chains.

XIX. Particle Stability in Emulsion Polymerization Any consideration of the stability of polymer latices would be incomplete without some discussion of the stability of the colloidal polymer particles formed during the course of an emulsion polymerization. As pointed out by Dunn and Chong (1970) the adsorption of the emulsifier plays a major role in determining the surface charge density of the particle and hence in determining the final particle size. The case in which there is an absence of added emulsifier has been considered by Goodwin et al. (1978) on the basis that" the particulate units initially formed contain only a small number of chains; they therefore have a low surface charge and are colloidally unstable. Hence, coagulation occurs until the particles formed reach values of surface charge density and radius sufIciently large to render them stable colloidal particles. The arguments can be developed in terms of the stability ratio starting with Eq. (13), which gives the rate of coagulation, and recapitulating that as W becomes greater than unity the rate of coagulation is reduced. Some

R. H. Ottewill

46

fundamental questions therefore arise, namely, At what size does the particle become a colloidally stable entity and at this, point what is its surface charge density and how many polymer chains does it contain? A further point of importance is to understand how this size varies with the ionic strength of the aqueous phase. In order to obtain a qualitative understanding of these points we can proceed by making some simple assumptions. These are (i) the particles formed are spherical, (ii) each polymer chain has the same molecular weight Me, (iii) each chain has two end groups, and (iv) all the end groups are anchored on the surface of the particle. Hence, if the latex particle has a molecular weight M L and a density of PL, then the number of polymer chains per particle is given by Ne

= 4na3PLNAi3Me = MdMe

(31)

The number of charged end groups per particle is therefore given by Ne

= 2Ne = 8na3 PLN Ai3Me

(32)

and the surface charge density by (33) Thus, as is directly proportional to a. For spherical particles the surface potential1/ls is given by 1/Is

= 4naa./[B(1 +

Ka)]

(34)

an equation which holds reasonably well up to 1/Isvalues of 50 mV. From this we find that, taking Me = 150,000: for a = 5 nm, 1/Is= 8 mV; for . a = 10 nm, 1/1.~ 20 mV; and for a = 22 nm, 1/1.~ 50 mV. Using a combination of Eqs. (4) and (6) to calculate VT as a function of h it is then possible to calculate W by numerical integration of Eq. (14). Since 1/Isis known as a function of particle size, then W can also be obtained as a function of r at an appropriate ionic strength. The results obtained are shown in Fig. 30 in the form of curves of log W against a. It is clear from these curves that the size of the first stable colloidal particle formed is controlled to a large extent by the ionic strength of the dispersion medium, i.e., at 4 x 10-4 mol dm - 3 log W = 2 is achieved with a = 3.7 nm, where to achieve the same Wat 4 x 10-3 mol dm-3 a has to grow to 11.3 nm. Since the size of the initial stable particles controls the nu~ber concentration of the latex during the diffusional growth period, then for the same initial monomer concentration and for the same percentage conversion of monomer, the final particle diameter in the medium of higher ionic strength will be the larger. This conforms to the clear trend found in the preparation

-

47

1. The Stability and Instability of Polymer Latices 12

10

8

:;: O>

:

+ 2

.' 5

10 Radius

15

20

(nm)

Fig. 30. Log Wagainst latex partic1eradius as a function of the concentration of 1: 1 electrolytein the system; ,4 x 10-4 mol dm-3; -, 4 x 10-3 mol dm-3.

of polystyrene latices in the absence of added emulsifier (Goodwin et al., 1976). XX.

Summary

In this chapter I have attempted to show in a broad sense how the application of the basic principIes of colloid science can be applied to develop our understanding of the various mechanisms involved in the stabilization of polymer latices. In the space available, it was not possible to go into very specific details of the many systems that have been investigated nor to deal with nonaqueous polymer latices. The latter, however, have been discussed in the recent comprehensive book by Barrett (1975). The literature on polymer colloids appears to be growing exponentially and to the authors of the many excellent papei-s which I have not quoted, I offer my sincere apologies. References Barrett, K. E. J. (1975). "Dispersion Polymerization in Organic Media." Wiley, New York. Bassett, D. R., and Hoy, K. L. (1980). In "Polymer Colloids 11" (R. M. Flitch, ed.), pp. 1-25. Plenum Press, New York.

R. H. Ottewill

48

Bee, H. (1978). B.Sc. thesis, Univ. of Bristol. Bibeau, A. A., and Matijevié, E. (1973). J. Colloid Interface Sci. 43,330. Bolt, P. (1978). B.Sc. thesis, Univ. of Bristol. Buscall, R., and Comer, T. (1980). Org. Coat. Plast. Chem. 43, 203. Cebula, D. J., Thomas, R. K., Harris, N. M., Tabony, J., and White, J. W. (1978). Faraday Discuss. Chem. Soco 65, 76. Cheung, W. K. (1979). Ph.D. thesis, Univ. of Bristol. Clint, G. E., Clint, J. H., Corkill, J. M., and Walker, T. (1973). J. Colloid Interface Sci. 44,121. Connor, P. (1968). Ph.D. thesis, Univ. of Bristol. Connor, P., and Ottewill, R. H. (1971). J. Colloid Interface Sci. 37, 642. Comell, R. M., Goodwin, J. W., and Ottewill, R. H. (1979). J. Co/loid Interface Sci. 71, 254. Derjaguin, B. V., and Landau, L. (1941). Acta Physicochim. URSS 14, 633. Derjaguin, B. V., and Muller, V. M. (1967). Dokl. Phys. Chem. 176, 738. Doroszkowski, A., and Lamboume, R. (1971). J. Polym. Sci. Part C 34,253. Dunn, A. S., and Chong, L. C-H. (1970). Br. Polym. J.2. 49. Dzyaloshinskii, 1. E., Lifshitz, E. M., and Pitaevskii, L. P. (1961). Adv. Phys. 10, 165. Evans, R., and Napper, D. H. (1973). J. Co/loid Interface Sci. 45, 138. Fitch, R. M. (1980). "Polymer Colloids 11." Plenum Press, New York. Flory, P. J., and Krigbaum, W. R. (1950). J. Chem. Phys. 18, 1086. Force, C. G., and Matijevié, E. (1968a). Kolloid Z. Z. Polym. 224, 51. Force, C. G., and Matijevié, E. (l968b). Ko/loid Z. Z. Polym. 225,33. Fowkes, F. M. (1964). Ind. Eng. Chem. 56, 40. Fowkes, F. M. (1967). In "Surfaces and Interfaces" (J. J. Burke, ed.), Vol. 1, p. 199. Syracuse Univ. Press, New York. Frens, G. (1968). Doctoral thesis, Univ. of Utrecht. Frens, G., and Overbeek, J. Th. G. (1971). J. Co/loid Interface Sci. 36, 286. Friends, J. P., and Hunter, R. J. (1971). J. Co/loid Interface Sci. 37, 548. Fuchs, N. (1934). Z. Phys. 89, 736. Gingell, D., and Parsegian, V. A. (1973). J. Co/loid Interface Sci. 44, 456. Goodwin, J. W., and Ottewill, R. H. (1978). Faraday Discuss. Chem. Soco 65, 338. Goodwin, J. W., Heam, J., Ho, C. c., and Ottewill, R. H. (1976). Co/loid Polym. Sci.60, 173. Goodwin, J. W., Ottewill, R. H., Pelton, R., Vianello, G., and Yates, D. E. (1978). Br. Polym. J.

10,173.

.

Goodwin, J. W., Ottewill, R. H., and Parentich, A. (1980). J. Phys. Chem. 84, 1580. Gregory, J. (1975). J. Co/loid Interface Sci. SI, 44. Hachisu, S., Kobayashi, Y., and Kose, A. (1973). J. Co/loid Interface Sci. 42, 342. Hamaker, H. C. (1937). Physica 4, 1058. Hardy, W. B. (1900). Proc. R. Soco London Ser. A 66, 110; Z. Phys. Chemie. 33.385. Heller, W., and Peters, J. (l970a). J. Co/loid Interface Sci. 32, 592. Heller, W., and Peters, J. (1970b). J. Co/loid Interface Sci. 33, 578. Heller, W. and de Lauder, W. B. (l97Ia). J. Co/loid Interface Sci. 35, 60. Heller, W. and de Lauder, W. B. (197Ib). J. Co/loid Interface Sci. 35, 308. Hiltner, P. A., and Krieger, 1. M. (1969). J. Phys. Chem. 73, 2386. Honig, E. P., Roeberson, G. J., and Wiersema, P. H. (1971). J. Colloid Interface Sci. 36, 97. Hoy, K. L. (1979). J. Coat. Technol. SI, 27. Kayes, J. B. (1976). J. Co/loid Interface Sci. 56,426. Kitchener, J. A., and Schenkel, J. H. (1960). Trans. Faraday Soco 56, 161. Kratohvil, S., and Matijevié, E. (1976). J. Co/loid Interface Sci. 57, 104. Krupp, H., Schnabel, W., and Walter, G. (1972). J. Co/loid Interface Sci. 39,421. Levich, V. G. (1962). "Physico-chemical Hydrodynamics." Prentice Hall, Englewood ClilTs, New Jersey.

,, ¡ ~ r

t

1. The Stability and Instability of Polymer Latices

49

Levine, S., and Bell, G. M. (1965). J. Colloid Sci. 20, 695. Lichtenbelt, J. W. Th., Pathmamanoharan, C., and Wiersema, P. H. (1974). J. Colloid Interface Sci. 49, 281. Long, J., Osmond, D. W. J., and Vincent, B. (1973). J. Colloid Interface Sci. 42, 545. Mardle, R. (1980). B.Sc. thesis, Univ. of Bristol. Matijevié, E. (1977). J. ColJoid Interface Sci. 58, 374. Matijevié, E., and Force, C. G. (1968). KolJoid Z. Z. Po/y. 225,33. Matijevié, E., Janauer, G. E., and Kerker, M. (1964). J. ColJoid Interface Sci. 19, 333. Napper, D. H. (1969). J. ColJoidInterface Sci. 29, 168. Napper, D. H. (1970a). J. ColJoid Interface Sci. 32, 106. Napper, D. H. (1970b). J. ColJoidInterface Sci. 33, 384. Napper, D. H. (1977). J. ColJoid Interface Sci. 58, 390. Neiman, R. E., and Lyashenko, O. A. (1962). ColJoidJ. USSR (English Trans.) 24, 433. Ninham, B. W., and Parsegian, V. A. (1970). J. Chem. Phys. 52, 4578. Ottewill, R. H. (1980). Chem. Ind. 377. Ottewill, R. H., and Rance, D. G. (1977). Croatica Chem. Acta SO,65. Ottewill, R. H., and Rance, D. G. (1979). Cr,oatica Chem. Acta 52, 1. Ottewill, R. H., and Shaw, J. N. (1966). Discuss. Faraday Soco 42, 154. Ottewill, R. H., and Shaw, J. N. (1967). KolJoid Z. Z. Po/y. 218, 34. Ottewill, R. H., and Walker, T. (1968). KolJoid Z. Z. Po/y. 227, 108. Ottewill, R. H., and Walker, T. (1974). J. Chem. Soco Faraday 170, 917. OttewilI, R. H., Rastogi, M. C., and Watanabe, A. (1960). Trans. Faraday Soco 56, 854. Overbeek, J. Th. G. (1952). In "ColIoid Science" (H. Kruyt, ed.), Vol. 1. Elsevier, Amsterdam. Overbeek, J. Th. G. (1977). J. ColJoid Interface Sci. 58, 408. Parsegian, V. A., and Ninham, B. W. (1971). J. Col/oid Interface Sci. 37, 332. Pelton, R. (1976). Ph.D. thesis, Univ. of Bristol. Rance, D. G. (1976). Ph.D. thesis, Univ. of Bristol. Reerink, H., and Overbeek, J. Th. G. (1954). Discuss. Faraday Soco 18, 74. Richardson, R. (1979). B.Sc. thesis, Univ. of Bristol. Richmond, P. (1975). In "ColIoid Science" (D. H. Everett, ed.), Vol. 2, p. 130. Chemical Society, London. Schulze, H. (1882). J. Prakt. Chem. 25, 431. Schulze, H. (1883). J. Prakt. Chem. 27, 320. Smitham, J. B., Gibson, D. V., and Napper, D. H. (1973). J. ColJoid Interface Sei. 45, 211. Storer, C. S. (1968). Ph.D. thesis, Univ. of Bristol. Tamaki, K. (1960). KolJoid Z. 170, 113. van den HuI, H. J., and Vanderhoff, J. W. (1970). Br. Po/y. J. 2, 121. Verwey, E. J. W., and Overbeek, J. Th. G. (1948). "Theory of the Stability of Lyophobic ColIoids." Elsevier, Amsterdam. Visser, J. (1972). Adv. ColJoid Interface Sei. 3, 331. von Smoluchowski, M. (1917). Z. Phys. Chem. 92,129. Wiese, G., and Healy, T. W. (1970). Trans. Faraday Soco 66, 490.

í

I

~¡ \

-

2 Particle Formation Mechanisms F. K. Hansen and John Ugelstad

1. 11. 111. IV. V. VI. VII.

Introduction. Micellar Nucleation: The<Smith-Ewart Theory . Radical Absorption Mechanisms. ". Micellar Nucleation: Newer Models . Homogeneous Nucleation. Partide Coagulation during the Formation Period. Nucleation in Monomer Droplets References .

51 54 56 63, 73 82 86 91

l. Introduction The nuc1eation stage constitutes the so-called Interval I in an emulsion polymerization, the initial period in which the partic1e number is changing. In Intervals n and nI the paftic1e number is believed to be essentially constant. Nuc1eation of new partic1es may in some cases also take place during Intervals n and In. This phenomenon is often referred to as secondary huc1eation and may be encountered in systems with poor stability (coagulation) or with changing composition (continuous and semicontinuous polymerizations). The present chapter will attempt to treat all mechanisms that may lead to formation of polymer partic1es, in whatever stage of the polymerization they take place. All discussions of partic1e nuc1eation start with the Smith-Ewart theory in which Smith and Ewart (1948) in a quantitative tre!ltment of Harkins' micellar theory (Harkins, 1947, 1950) managed to obtain an equation for the partic1e number as a function of emulsifier concentration and initiation and polymerization rates. This equation was developed mainly for systems of monomers with low water solubility (e.g., styrene), partly solubilized in micelles of an emulsifier with low critical micelle concentration (CMC) and p'?"Ts{lIlíedto work well for such systems (Gerrens, 1963). Other authors have, ,

however, argued against the Smith-Ewart theory on the grounds that (i) . partic1es are formed even if no micelles are present, (ii) the equation for the 51 EMULSION POLYMERIZATlON

Copyright«; 1982 by Academic Press, Inc. Al! rights of reproduction in any form reservcd.ISBN 0-12-556420-1

52

F. K. Hansen and J. Ugelstad

partic1e number gives an estimate that is a factor of 2 higher than that found experimentally even for styrene, (iii) more water-soluble monomers do not fit the theory, and (iv) a maximum in polymerization rate at the end of the nuc1eation period is predicted, but has rarely been observed. On this basis other theories for the nuc1eation have been put forward, based on the idea of self-nuc1eation of oligomer radicals produced in the aqueous phase. These mechanisms of partic1e formation were first treated quantitatively by Fitch and Tsai (1971). These ideas, which have been further elaborated by other authors, seem to have solved the first problem of the theory of partic1e nuc1eation but leave open the question of whether the micelles, when present, play any role at all in partic1e formation. More recent work seems not only to confirm the importance of the micelles but to stress the necessity of inc1uding more detailed features of absorption and reaction in micelles . and particles in order to explain the other discrepancies of the Smith-Ewart theory. Usually, monomer droplets are believed not to play any role in'emulsion polymerization other than as a source of monomer. Ugelstad and associates ha ve shown, however, that in cases with very small monomer droplets, these may become an important, or even the sole, loci for partic1e nuc1eation. The system may then be regarded as a microsuspension polymerization with water-soluble initiators. It has therefore been pointed out (Hansen and Ugelstad, 1979c) that partic1e nuc1eation models should inc1ude a~l three initiation mechanisms-micellar, homogenous, and droplet-since all these mechani~ms may compete and coexist in the same system, even if one of theni usually dominates. Figure 1 illustrates the main components and phases in an emulsion polymerization system. The arrows indicate the possible distribution ofthe components between the phases. Monomer (i) will usually exist as monomer droplets; (ii) some monomer, depending on water solubility, will be dissolved in the continuous phase; and (iii) some monomer will be solubilized in micelles. Emulsifier (i) will be partly dissolved in the continuous phase; (ii) if concentration is above the CMC, the excess will form emulsifier inicelles; and (iii) some emulsifier will be adsorbed on monomer droplets, and may even be dissolved into the droplets. lnitiator (i) will mostly be dissolved in the continuous phase as watersoluble initiators are usually applied. For special applications partly or completely oil soluble initiators may be used. These will be distributed similarily to the monomer. A so-called ordinary emulsion polymerization, Le., similar to the case treated by Smith and Ewart, is characterized by large monomer droplets

'2. Particle Formation Mechanisms

53

Honamer H

Hice//es

Drap/ets

¡nitiatar

~

Emu/sifier

, Aqueaus sa/utian Fig. 1.

Schematic iIIustrationsof the components and phases usually present in an

emulsion polymerization among phases.

system. The arrows indicate

the possible distribution

of components

with consequently smaIl total surface, miceIle-forming emulsifiers with reIatively low CMC, and ionic initiators that decompose either by a thermal andjor by a redox mechanism. In the Smith-Ewart theory as weIl as in other treatments, the primary radicals formed by decomposition of the initiator in the continuous phase were assumed to enter emulsifier miceIles and polymer partieles. It has been pointed out by several authors (Alexander and Napper; 1971, Nomura et al., 1975; Barrett, 1975) that these usuaIly ionic, very water-soluble radicals are rarelyabsorbed directIy into a miceIle or partiele but must add some monomer units in the aqueous phase to become sufficientIy oil soluble to be absorbed. It now seems to be generaIly accepted that formation of these oligomers in the aqueous phase is the first step in the nUeleation (and polymerization) process. The presen'ce and molecular weight of oligomers in some systems have been analyzed by means of GPC and spectrophotometric methods (Fitch and Tsai, 1971; GoodaIl et al., 1975; Chen and Piirma, 1980). Degrees of polymerization have been found to lie between 1 and 6070. The water-soluble oligomers may be destroyed or may nUeleate partieles. The different possibilities are listed in Table I. It should be added that oligomers that are surface active may also be adsorbed onto the surface of partielesjdropletsjmiceIles rather than being absorbed into the interior. Being on the end of the oil-soluble chain, the active site of the radical wilI no doubt' be able to propagate into the interior.

11

54

F. K. Hansen and J. Ugelstad TABLEI Reaction Possibilities of an Oligomer Radical in the Aqueous Phase and the Probable Result Process

A B e D E F G

Result Micellar initiation

Absorption into a micelle Absorption into a monomer droplet Absorption into an earlier formed particle Propagation in the aquous phase Termination in the aquous phase

Droplet initiation. Radical disappearance

(particle growth)

Higher oligomers "Dead" oligomers (may or may not lead to nucleation) HoIt1ogeneous nucleation

Precipitation in the aquous phase (self-nucleation) Mixed-micelle formation (with or without emulsifiers)

Nucleation

(homogeneous

or micellar)

This may lead to a core and shell morphology. Process E in Table 1, termination in the aqueous phase, may also lead to homogeneous nucleation if the "dead" molecules are sufficiently water insoluble. When partic1es have been formed, transfer reactions to monomer (or chain transfer agent if present) willlead to monomer radicals, which may be desorbed into the aqueous phase. This is also indicated in Fig. 1. The monomer radicals may act in a way similar to the initiator radicals. ~elow is given a detailed description of the different processes listed in Table 1 and of the theories that have been advanced for these processes.

n.

MiceUar Nucleation: The Smitb-Ewart Theory

Smith and Ewart (1948) proceed as follows: radicals are absorbed into monomer-swollen emulsifier micelles which then are transformed into polymer partic1es; the rate of radical absorption Is equal to the rate of initiator decomposition p¡, which means that dN/dt

= p¡

(1)

where N is the partic1e number. The rate of growth of a partic1e is assumed to be constant and is expressed as dv/dt

=

J1.

(2)

where v is the volume of the partic1e. The number of micelles will decrease as the partic1es grow, giving an increasing surface which will adsorb

-

2.

55

Particle Formation Mechanisms

emulsifier. Particle formation stops when a11emulsifier is adsorbed on the particles, which means that the particle surface area Ap is equal to the total surface area of emulsifier asS where as is the specific surface area per unit of emulsifier and S the amount of emulsifier. To be precise S should be the amount in excess of the CMC. The difference will not be of any importance if the CMC is much lower than S. As dv/dt is assumed to be constant, Ap may be expressed by J1.and t by integration over a11formation times from O to t (3) where (4) From Eq. (3) one obtains the time tl when Ap = asS,which inserted in Eq. = Pitl' The result is

(1)givesthe particle number N

(5)

This is the so-ca11edupper limit equation. The lower limit is derived in a similar way. In this case it is also assumed that polymer particles may absorb radicals leading to a decrease in the rate of nucleation. For computational purposes the rate of radical absorption is set as proportional to the particle surface area, Ap =

¿ api'

where

apj is the surface

area of one

particle. Equation (1) is then transforme9 to (6) The total area Ap is expressed by an integral equation in a way similar to that for the upper limito Inserting for Ap in Eq. (6) gives by a somewhat complicated integration (7) which is identical to Eq. (5) except for the constant. From diffusion laws (Fick's first law) it is expected that the radical absorption rate is proportional to the particle surface area divided by the radius (i.e., 4nrN) so that sma11 particl~s absorb more radicals per unit area than do large particles. This fact was realized by Smith and Ewart who stated that the true value of k should be between 0.37 and 0.53 and accordingly the particle number somewhere between those values predicted by Eqs. (5) and (7). It may be shown (Section IV) that this procedure is not quite correct, the constants and ~will also be slightly altered (the former decreased and the latter increased). In addition the constancy of dv/dt = is somewhat doubtfuI, both because the average radical number per particle in the lower case wilI

t

J1.

-

56

-

I

I

.

F. K. Hansen and J. Ugelstad

not be constant (willdecrease)and because the monomer concentration in the particles is expected to increase with increasing size. These two factors will counteract each other and are not expected to have a great influence on the value of N. The Smith-Ewart theory has been successful in describing the experimental results with some systems,especiallythe predicted orders of N with respect to initiator and emulsifier as obtained from double logarithmic plots of N against the two variables (Gerrens, 1963). Other authors have, however,found a wide range of exponents(citedby Fitch, 1973).AIso,other discrepancies exist, as mentioned in the introduction. The Smith-Ewart theory has been modified and recalculated by several worker~ (Parts et al., 1965; Gardon, 1968a-f, 1971; Harada et al., 1972). Parts et al. applied a numericaf integration of the nucleation equations and reached the same conclusions as Smith and Ewart. They found ihat the average number of radicals per particle ñ is approximately 1 through the entire nucleation period because the rate of radical adsorption in micelles is so much higher than that in new particles (very large number of micelles). They propose that the absorption efficiency of micelleshas to be lower in order to explain experimental findings. Gardo~ has recalculated the lower limit of the Smith-Ewart theory by a seminumerical method and finds that ñ decreases from 1 to 0.67 during the formation periodo This does not, however, significantly influence the exponents 0.4 and 0.6. The particle numbers (or more correctly, particle sizes) calculated by Gardon were found to describe some experimental results for styrene and methyl metacrylate fairly well, whereas other data on particle numbers were 2-3 times lower than predicted. Another feature of the Smith-Ewart theory is that the reaction rate at the end of the nucleation period is expected to be higher than in the steady state because ñ is higher than the steady-state value of 0.5 (Smith-Ewart Case 2 kinetics). There is little experimental evidence for such a maximum in rate (Ugelstad and Hansen, 1976), and this discrepancy may be explained by more details about the radical absorption rates in micelles and particles. Before any further discussion of particle-formation mechanisms, it therefore seems logical-to review the mechanisms responsible for radical absorption,

111. Radical Absorption Mechanisms

Gardon based his calculations on a geometric derivation of the radical absorption rate which gave the result that the rate should be proportional to the particle surface area. This derivation, which also was adopted by Fitch and Tsai (1971), has been criticized for not taking the concentration

-

2. Partiele Formation Meehanisms

57

gradient of radicals around a particle into account (as in Fick's laws) (Barrett, 1975; Ugelstad and Hansen, 1976; Hansen and Ugelstad, 1978). This was also realized by Fitch and Shih (1975) who, using seeded experiments, found the expected proportionality of the absorption rate to Nr. Similar conclusions may also be drawn from the seed experiments of Gatta et al. (1969), the kinetic results of Ugelstad et al. (1969) with vinyl chloride, and the recent findings of the authors (Hansen and Ugelstad, 1979a) using a polystyrene seed. Fick's first law for the diffusion of a component A in a stationary medium B around a spherical particle may be written (Byron Bird et al.; 1960) (8) where 1)ABis the diffusion constant for A in B, e is the total molar, concentration of A and B, and XA is the molar fraction of A. For the geometric configuration see Fig. 2. '\

\ \ \

\ \

\ 1 I

R

/

Halar fraction, Concentration

r.e

.

XAW'CW

/

---

------

o

r

R Distance

Fig. 2. Geometric and concentration conditions around an absorbing spherical partic1e. e is concentration and X is molar fraction of dilfusing species.

F. K. Hansen and J. Ugelstad

58

Assuming a stationary diffusion layer outside the particle of thickness b one has for dXAldR. dXAldR

= (r/R2)[(r + b)/b](l -

Inserting into Eq. (8) for R

XA) In[(1

-

XAa)j(l - XAw)]

(9)

=r (10)

Usually the molar fraction of the diffusing species (A) is much smaller than that of the' stationary component (B) so that 1 - XA ~ 1 and In(l - XA) ~ XA, which gives (11) where (12a) and (12b) Ir b ~ r, Eq. (11) simplifies to. ñA = -Dw4nr(Cw- Ca)

(13)

Ir one has an electrostatic repulsion between charged oligomers and equally charged particles the diffusion rate is given by (Hansen and Ugelstad, 1978) (14) ñA = -4nDwr(Cw - CaeZ)/W' where z = et/lo/kTand W' is given by ,

W=r-

.

r+~

r+ b b f.r

et/lodR

exp--

( kT ) R2

(15)

where R is the distance from the oligomer to the center of the particle, e is the electronic charge, and t/lois the (effective)surface potential. The value of W' has been calculated by numerical integr~tion for the case where b ~ r and has been given as a function of et/lo/kT and Kr where K. is the inverse double-Iayer thickness (Hansen and Ugelstad, 1978). The term eZin Eq. (14) may be considered as an activation energy (Boltzmann factor) and accounts for the fact that in order to become absorbed the charged radical s have tq surmount an energy barrier at the surface. The'term W' is the' integral of this activation energy factor giving the retardation of a unit charge upon diffusion to the surface. In the case where Cw~ Caez Eq. (14) reduces to' (16)

\.

..

2.

59

Particle Formation Mechanisms

Equation (16)describesthe situation with irreversiblediffusionand seems to apply to cases of relatively large particles. The folIowing simplifications are inherent in Eq. (16):

o

1. The concentration of the diffusing component has to be low (lower than 1 M). This condition is always fulfilIed for free radicals. 2. The stationary layer "thickness" ( 0.6, the particle number, as well as the value of Z,

2. Particle Formation Mechanisms

71

will decrease with increasing values of x and decreasing values of b. The effect of x will be relatively low if Z ~ 1, because then ñ ~ 1, which means that the particIes will grow more slowly. It may be estimated that the value of rpjrm is approximately 10 and 3 for styrene and vinyl acetatejvinyl chloride, respectively. The experimental results indicate that with a value of x = 3 the value of b is approximately unity for styrene. For vinyl chloride nnd vinyl acetate the value of b is 1.0 x 10- 3 and 1.6 x 10-4 for x = 3 and with q equal to ñ in Eq. (40), which seems to be the more probable situation for vinyl chloride and vinyl acetate. Ir q in Eq. (40) is set to unity (irreversible absorption) the value of b increases, most significantly for vinyl acetate. In any case, it appears that the apparent value of b is considerably lower for vinyl chloride and vinyl acetate than for styrene and the acrylates. This may be taken to indicate that the values of b for the radicals formed by chain transfer are considerably lower than for the polymer radicals, which in tum may be explained if the radicals formed by chain-transfer reactions are less reactive than the polymer radical s (Hansen and Ugelstad, 1979d). For styrene, a nonsteady-state calculation has been carried out without the simplifications used above (Hansen and Ugelstad, 1979a). Thus, instead of Eq. (35), the following equation was used for the rate of particIe formation (50) where PImand PMmare absorption probabilities for radicals stemming from initiator and monomer radical s, respectively. AIso, a distinction is made between Po and P1. In the calculation of the absorption probabilities, the complete expressions for nobtained from Eqs. (21) and (22) were used. In addition the monomer volume fraction in particIes (
~D =

{[f

D2n(D, t) dD

I f n(D,

t) dD

]-

[f Dn(D,

t) dD

If n(D, t) dD J}

1/2

It may appear at first sight that 15 and ~D in fact are the only measures of the PSD necessary (in a unimodal system), because of the following argument. If one supposed that each latex particle in an emulsion polymerization were to grow at constant rate in terms of the volume added per unit time, then the mean square volume deviation of the PSD would be constant after Interval I (the nucleation period). The PSD expressed in terms of volume would thus translate along the volume axis (i.e., the distributions would be superimposable) as the experiment progressed in time. On the other hand, if each particle were to grow ata constant rate in terms of radius added per second, then the radius distribution would translate along the radius axis after Interval 1. In fact, a real latex system cannot obey either type of behavior exactly because not all particles actually contain a growing free radical at any given instant. The foregoing argument involving the translation of the PSD on the size axis remains valid provided only that the fluctuations in the average polymerization times from particle to particle be negligible. This permits the real particle tha~ polymerizes for half the time to be replaced by an "equivalent" particle that grows continuously at half the rate of a particle containing one free radical. Such considerations are clearly of only limited value, and more complete measures of the PSD are required. Another measure of the PSD, one used by Gerrens (1959), is the mean square volume deviation, which is directly proportional to the mean square deviation of the diameter cubed. Gerrens was aware that such a description

Gottfried Lichti et al.

96

is only meaningful when all the PSDs to be compared have the same Gaussian or normal distributions. His experimental evaluation of several PSDs of polystyrene latexes showed that the distributions were almost Gaussian when expressed in terms of the particle volume. An additional measure of the PSD is the skewness parameter s, defined by Gardon (1968b) as the exponent of the radius such that the PSD expressed in the size parameter (radius)' is most nearly Gaussian. The preceding approaches to characterizing the PSD are adequate if the PSDs always closely approximate a particular type of distribution. If, however, the shape of the PSD changed markedly during polymerization, or if the PSD do es not conform to a recognizable functional form, greater mathematical sophistication is required.

B.

Population Balance Models of PSD

A comprehensive theory of the PSD should account for the time evolution of the PSD throughout the emulsion polymerization. The basic concepts widely accepted as describing the growth of polymer latex particles were originally advanced by Harkins (1945, 1946, 1947, 1950) and Smith and Ewart (1948). Each latex particle contains an integral number i (O,1,2, 3, ...) of polymerizing free radicals. Henceforth, i will be referred 'to as the "state" of the latex particle, The fraction oflatex particles that are in st'ate i at any time t is denoted by N¡(t). These Ni values relate directiy to the rate of polymerization through the average number of free radical s per particle, ñ = L¡ iNi (with the normalization condition L¡ Ni = 1). The Smith-Ewart equations describing the time evolution of N¡ are dN;/dt

= p(Ni-l - N¡}+ k([i + 1JNi+l - iNi) + e([i + 2J[i + lJN¡+2 - i[i -1JNi)

(2)

for all i ~ O. Here p is the first-order rate coefficient for the entry of free radicals into latex particles, k the first-order rate coefficient for exit (desorption) of free radicals from the particles and 2e pseudo-first-order rate coefficient for the annihilation of pairs of free radical s by bimolecular termination. Note that in Eq. (2), the term with a negative state coefficient (for i

= O) is

ignored and that bimolecular

termination

only contributes

if

i ~ 2. Note, too, that p and kñ give, respectively, the average number of free radical s that enter and exit from each particle in unit time and that the average lifetime of an isolated pair of free radicals in a latex particle is about !e.

¡,;

-

...

3. The Particle Size and Molecular Weight Distributions

97

The quantity N¡(t) alone contains insufficient information to specify the PSD of the growing latex because it refers to the fraction of particIes in state i at time t, irrespective ofparticle size. To circumvent this problem, it is necessary to define the number density distribution of particles of size (1in state i at time t by nM, t). The actual fraction of particIes with sizes in the range (a, b) can be evaluated from the integral of n¡«(1, t) between these limits. Thus, n¡«(1, t) is related to the previously defined N¡(t)through

N¡(t)

=

(3) 100

n¡«(1, t) d(1

from which it can be seen that the dimensions of n¡«(1, t) are (size)-l. The observable PSD n«(1,t) at any time during the experiment may be calculated from n¡«(1, t) by a summation over all states: n«(1,t) =

L¡ n¡«(1, t)

(4)

The n¡«(1, t) term is a function of two independent variables, (1and t. The equations describing them are a family of partial differential equations. The basic forro of these equations, as derived from population balance considerations, has been recognized by many authors (O'Toole, 1969; Sunberg and Eliassen, 1971; Pis'men and Kuchanov, 1971; Min and Ray, 1974; Lichti et al., 1977) and may be written in the compact but general form . onlot

= en - o(Kn)lo(1+ e

(5)

Here n«(1,t) i~ a vector whose (i + l)th component represents the distribution n¡«(1, t) and e is a square matrix formed from the kinetic parameters p, k, and c. The nature of e may be recognized by expressing Eq. (2) in matrix form and identifying the resulting square matrix with e. e is called here the Smith-Ewart coupling matrix because its elements in Eq. (5) describe how latex particIes change state as a consequence of the SmithEwart mechanisms (entry, exit, and bimolecular termination). The matrix K in Eq. (5) is a diagonal matrix ~hose (i + 1, i + l)th element specifies the rate of growth of a latex particIe in state i. The term -o(Kn)lo(1 in Eq. (5) accounts for the particIe growth process whereby particles of size (1 are lost to the population n¡«(1, t) when they grow to another size (1+ d(1,whereas particles are gained in the population n¡«(1, t) when particIes of size (1 d(1 in state i grow to size (1(Lichti et al., 1977). One common assumption regarding K is that

-

K¡+l,¡+l = iK

(6)

98

Gottfried Lichti el al.

where K = rate of growth of a partiele in state 1. This assumption, which asserts that a partiele containing i free radicals grows i times more rapidly than a partiele containing one free radical, is valid provided that partiele growth is not limited by, for example, monomer diffusion. Note that K = kpCrJNA where CMequals monomer concentration in the latex partieles and NA is Avogadro's constant. The vector c in Eq. (5) describes the creation in, andjor remo val oflatex partieles from, the system. The creation component may arise from in situ partiele formation (e.g., in Interval 1) or from the flow behavior in a continuous stirred-tank reactor system (CSTR) with an arbitrary number of reaction vessels. Partiele remo val terms may be required if coagulation occurs or in the context of CSTR operation. All the parameters in Eq. (5) may be functions of (J and t. Equation (5) may also be coupled to other mass balance equations, for example, that for the total amount of monomer present in the system. Certain parameters, e.g., the entry rate coeffcient p, may themselves be functions oe n¡«(J,t). Such would be the case if a radical that exists from one particle may enter another (Ugelstad and Hansen\ 1976). This results in highly nonlinear behavior. Equation (5) reduces to the Smith-Ewart equation [Eq. (2)] if c is sef equal to zero and if both sides of Eq. (5) are integrated between (J = O and (J = oo. This reduction further requires the assumption that all rate coef-

ficients forming the elements of a are independent of (J. It is evident that the population balance Eq. (5) are considerably more general in scope than the Smith-Ewart equation because the inelusion of the size parameter enables the formalism to model the partiele formation process, as well as both the kinetics and the evolution of the PSD. The Range of Applicability of the Populat~onBalance Equations The population balance equations are very general and may be applied to batch, semicontinuous, and continuous emulsion polymerizations. Furthermore, both seeded and ab initio polymerizations are comprehended by Eq. (5) in all (or part) of the three commonly considered polymerization intervals. The following sections show how the different possibilities are reflected in different functional forms of the elements of the matrices a and K and of the vector c. It should be remembered, however, that certain conceivable situations are not comprehended by Eq. (5); for example, if the monomer molecules are not freely exchanged between the latex partieles so that the monomer concentration inside each latex partiele is determined by its growth history.

.

3. The Partiele Size and Moleeular Weight Distributions C.

99

Batch Polymerizations 1. lnterval II

Interval 11 of an emulsion polymerization is characterized by polymerization in a constant number of latex particles in the presence of monomer droplets (Le., nucleation is absent). This situation usually exists in an ab initio polymerization immediately on completion of Interval 1; however, for a seeded system, it may exist from the commencement of polymerization. Clearly, if no new particles are formed and coagulation does not occur, the vector e in Eq. (5) is zero. Moreover, in the presence of monomer droplets, the concentration of monomer at each polymerizing site in the particles is approximately constant. The growth factor K is thus known. The remaining parameters p, k, and e in Eq. (5), or rather size-averaged values for them, can be evaluated from kinetic studies (Hawkett et al., 1980, 1981; Lansdowne et al., 1980). All the parameters in Eq. (5) are then specified and it is possible to predict uniquely the time evolution of the PSD. Of course, the initial distribution n~q, t = O)must also be specified (Lichti et al., 1977). Provided that a relatively monodisperse seed latex is used, the size-averaged kinetic parameters can be employed without introducing significant error. In this way, the predictions of Eq. (5) can be tested against experiment and, e.g., the accuracy of the kinetic parameters checked by PSD evolution data. Additional experiments with polydisperse seed latexes may permit the size dependence of the parameters to be specified. a. The Solution of the Population Bala'lce Equations. The solution of Eq. (5), given the initial conditions and all the requisite parameters, can be achieved in at least three different ways: (i) by analytic solutions, if they exist; (ii) by finite difference numerical solutions which may always be generated (Carnahan et al., 1969), although they may require considerable computing time and are prone to inaccuracy; or (iii) by the method of moments (Bamford and Tompa, 1954) which provides an efficient numerical procedure for certain systems. Analytic solutions for Eq. (5) provide the most direct path of the prediction of PSD evolution. For batch polymerizations in Interval 11, however, analytic solutions have only been achieved forothe so-called zeroone system (Lichti et al., 1981). These are systems wherein negligibly few particles contain two or more free radicals because of the rapidity of the bimolecular termination reaction (e.g., in styrene emulsion polymerizations with smalllatex particles). In this case, Eq. (5) may be written as follows: ano/at = - pno + (p + k)n¡ (7) andat = pno (p + k)nl - aKndaq (8)

-

Gottfried lichti et al.

100

The appearance of the term pnl in Eq. (7) reflects the fact that entry into a . state 1 latex particIe causes a state zero latex particIe to be formed as a consequence of rapid bimolecular termination. Ir all the paramaters p, k, and K in Eqs. (7) and (8) can be taken to be independent of time, while being permitted to be arbitrary functions of size, analytic solutions may be generated for arbitrary initial conditions (Lichti et al., 1981). These solutions encompass the more restricted cIass of analytic solutions provided by O'Toole (1969) and Watterson and Parts (1971). The use of the term analytic is retained Qere even though, for arbitrary initial conditions, one numerical integration is required; such integrations are very rapidly performed numericalIy. The importance of this analytic solution is that it enables one to make a thorough comparison between experimental PSDs and theoretical predictions, with a wide range of assumptions as to the values and functional forms of the various contributing rate coefficients. The analytic solution derived by Lichti et al. is applicable to any zeroone system in lnterval n, with no restrictions on the rate coefficients except that they be independent of time (but may depend on size). Although a zero-one description is applicable to some important systems, it is not universally

valid. For systems where ñ exceeds

t, general

analytic solutions

have yet to be developed, and we now examine methods for solving Eq. (5) for systems of arbitrary ñ. The second method for solving the PSD evolution equations is bruteforce numerical solution using first-order finite difference. Whereas a solution can always be obtained by this technique, it suffers from numerical instability, from the lack of any automatic check on accuracy, and from requiring large amounts of computer time. The third method for solving Eq. (5), the method of moments, was exploited by Katz et al. (1969) in connection with MWDs (see Section n,E,2) and by Sundberg and Eliassen (1971) in connection with PSDs. This method is numerically efficient. The kth moment mf of the population n¡(a,t) is defined by mNt) = 1-

t-

¡¡¡ Z t.s e

lb)

a: t.s ID ~ => z

VOLUME

Fig. 2.

Schematic comparison of the PSDs produced by (a) batch and (b) CSTR

emulsion polymerizations.

(1971) showed that it was possible to calculate the PSD from an expression for the distribution of residence times (and hence growth times), as well as a growth rate parameter that depends upon the particle size [see Eq. (21)]. The crux of their argument will now be presented, The lifetime distribution of any species in the CSTR g(t) is given by g(t)

=

Co exp(

-

tlr:)

(19)

if the effiux rate is uniformo Here Cois a normalizing factor. Suppose that it is possible to assign a unique value of particle size O'(t)to a particle that leaves the reactor after a residence time t. This assumption actually disregards stochastic broadening by ignoring the fact that particles that leave the reactor after residence time t will in reality possess a size distribution. The lifetime distribution expressed by Eq. (19) can then be transformedinto the steady-statePSD as follows[cf. Eq."(11)]: g(O')

=

Co exp[

- t(O')Ir:](dtldO')

(20)

where t(O') is obtained by inverting O'(t). In order to calculate t(O'),or equivalently O'(t),an equation describing the average growth rate of the particle is used: dO'ldt

= Kñ"

(21)

Gottfried Lichti et al.

W8

where ñ",is the size-dependent average number of free radicals per particle calculated from a knowledge of the size-dependent parameters p, k, and e [see Eq. (2)]. The foregoing argument is equivalent to the approach of Stevens and Funderburk (1972), who used a more abstract line of reasoning. Suppose that a summation of Eq. (5) is performed over all states i (Le., all the rows are summed). Then Eq. (5) becomes .0

L ni/ot = -oKL inJoa- L n;/r:

(22)

Equation (22) follqws from Eq. (5) if the vector e is given the form appropriate to CST,s [see Eq. (18)] for removal of latex particles in the effluent stream and if the particle creation terms are considered as boundary conditions operating at a = o. The term involvingthe Smith-Ewart coupling matrix vanishes on summation because of the conservation of particle numbers. The value of ñ",is given by (23)

ñ",= L ini/L ni so that Eq. (22) may be rewritten,as on/ot

(24)

= - oKñ", n/Ba - n/-r:

where n represents the overall PSD (Le., n = Li n¡). For steady-state conditions, Eq. (24) becomes (25)

oKñ", n/oa = -n/-r: This equation

has the general solution

n where

= coex{

Co is the number

-(f:da/Kn",-r:)JIKñ

of newly formed

particles

(26) present

at a

= O.

Equation (26) is identical to Eq. (20),ifviewed in conjunction with Eq. (21). The utility of the steady-state CSTR methods lies in the fact that the problem of the steady-state PSD may be couched in terms of uncoupled equations that are considerably easier to solve than the full problem. Some caution must be exercised in omitting the stochastic termo The results of the comprehensive calculations of Thompson and Stevens (1977) show that in certain instances, the steady-state PSD is sensitive to stochastic broadening. Finally, we note that the PSD in a CSTR is strongly sensitive to the residence time distribution, which may be varied over a wide range. Consequently, the production of a latex with a desired PSD is usually more readily achieved with a CSTR process than a batch or semicontinuous process, for the latter depend in a complex manner upon many mechanisms. The production of monodisperse latexes is an exception to this rule: these

3. The Particle Size and Molecular Weight Distributions

109

are more readily prepared by a batch or semicontinuous process since the CSTR product has a much broader distribution.

F. Experimentallnvestigations of the PSDs 1.

Batch Polymerizations

Min and Ray (1974) in their review article detailed many experimental studies on the PSDs of emulsion polymers. In this section, some more recent results that compare the predictions of the population balance equations [Eq. (5)] with experimental data will be discussed. The actual acquisition of reliable experimental PSD data is a task of considerable complexity. Such techniques as soap titration, turbidity, ultracentrifugation, and light scattering (Morton et al., 1954; Kerker, 1969; Shaw, 1970) usualIy pro vide only average values for the particle diameter, although some of these methods can provide PSD data if some functional form for the distribution function is assumed. The Coulter Counter can usualIy be exploited only for particles oflarger size (> 0.5 ,um), although it is not an absolute method (Eckhoff, 1967). Fractional creaming of latexes has also been proposed as a method for determining PSD (Schmidt and Kelsey, 1951; Schmidt and Biddison, 1960). This method is based on a relationship between the concentration of creaming agent and the size of the creamed particles, which is claimed to be independent of the nature of the dispersed particles. It has the advantage of simplicity, but again calibration is mandatory. Hydrodynamic and liquid exclusion chromatography (Nagy et al., 1980; Singh and Hamielec, 1978) have also been proposed as methods for determining the PSD, although difficulties still exist in relating the detector signal to the PSD. By far the most direct method for the measurement of PSDs is the use of transmission electro n microscopy. The measured PSD is a distribution expressed in terms of the unswolIen radius (or diameter) of the particles. AlI sizes are treated with the same statistical weight (provided they are able to be resolved); this is not the case with some of the sizing methods listed above (e.g., light scattering), which weight the larger particles more heavily. One major difficulty with electro n microscopy is that so me latex particles are adversely affected by the electro n beam. Shadowing techniques and hardening procedures (Corio et al., 1979) can sometimes be used to overcome this problem. A further difficulty inherent in any PSD measurement by electro n microscopy resides in the need to measure a large number of images of latex particles on a photographic print. Gerrens (1959) has stated that at least 3000 particles must be measured in order to obtain a statisticalIy meaningful distribution. This involves considerable human

Gottfried Lichti et al.

110

effort because current attempts to automate the procedure have thus far foundered on the difficulty of resolving the extensively overlapping particles thát inevitably occur in samples of sufficient number density to give meaningful statistics. It seems likely that sophisticated image analysis software andjor software that can resol ve overlapping particles will become commercially available in the near future. An additional complication arises with relatively mono disperse samples in that slight changes in the focus of the electro n microscope introduce random errors in the exact magnification of the final print. These problems of focus adjustment can be corrected for by the use of an internal standard (frequently another polymer latex with a significantly larger average particle size). It is usually mandatory for the PSD of the internal standard to be completely separate from the PSD of the sample. Sundberg and Eliassen (1971) ha ve compared the predictions of their population balance formalism with the experimental PSD data obtained by Gerrens (1959) for polystyrene latexes. Their model is a zero-one system with no exit (desorption) from the particles. Although the neglect of radical desorption for particles of this size is questionable (Hawkett et al., 1980; Lansdowne et al., 1980), the theoretical curves are in qualitative agreement with the experimental PSDs (see Figs. 3 and 4). It must be stressed, however, that the experimental PSDs were taken at the conclusion of the 6

32

1 2 3 4 5 6

28

"> o H "

24 20

lO' [1]0 (g cm-3) 3.61 .5.41 10.8 18.0 28.9 54.1

16 12 8 4 O

500

1000

1500

2000

1018 VOLUME (cm3)

Fig. 3.

Experimental PSDs for polystyrene latexes obtained by Gerrens (1959) for

various initiator concentrations [1]0. Gerrens' d('r.%)/dV, the derivative of the cumulative PSD, is proportional to n(O',t) with O'= volume (V). (After Gerrens, 1959; adapted with permission of Springer- Verlag.)

3. The Partide Size and Molecular Weight Distributions

111

4

z 0.6 52 1Ü Z ~

I.J... Z O

¡: 0.4 ~ m a: 1IJ)

o

w

~ 0.2 ...J O >

o Fig. 4.

0.5

1.0 1.5 VOLUME

2.0

Theoretical PSDs for polystyrenelatexes predicted by Sundberg and Eliassen

(1971). Initiator concentration curves (mol dm-3): 1, 1.1 x 10-3; 2, 2.2 x 10-3; 3, 5.5 x 10-3; 4, 1.1 x 10-2. Volume is dimensionless. (Reproduced with permission of Plenum Publishing Corp.)

polymerization reaction and SO incorporate information from the entire range of kinetic processes (e.g., nuc1eation, monomer-depleted growth, etc.). It is therefore difficult to assess whether the qualitative agreement between theory and experiment is significant or whether it results from a fortuitous cancellation of errors in the theory, especially as the data in Figs. 3 and 4 do not correspond to the same recipe and temperature conditions. Min and Ray (1978) have compared the predictions of the population balance model with the experimental results obtained by Gerrens (1959) on poly(methyl methacrylate) latexes. The correspondence between theory and experiment is qualitatively acceptable, and the same kinetic parameters model the kinetic behavior ofthe polymerization process (see Fig. 5). Again, the PSD was measured at the conc1usion of the polymerization reaction, raising once more the problem of cancelling errors in the theory. Min and Ray (1978) used literature values for all but two of the rate coefficients

112

Gottfried Lichti et al. 0.3

N I

.

o ~

0.2

>-

1Vi Z UJ e ...J

~ 0.1

Ir o Z

80 PARTlCLE

Fig. 5.

DIAMETER

160 Inm)

.

Comparison of experimer,ttaland theoretical PSDs for poly(methylmethac-

rylate) latexes prepared with various initial initiator concentrations: [1]0 = 1.8 x 10-3 g cm-3, ... [1]0 = 3.6 x 10-4 g cm-3, predictions of Min and Ray (1978); O and 1::" corresponding experimental data of Gerrens (1959). Ordinate is "normal density," proportiomi.l to n(u, t) with u = D (diameter). (After Min and Ray, 1978; adapted with permission of Journal 01 Applied Polymer Science. Also after Gerrens, 1959; adapted with permission of Springer-Verlag.) .

required in the polymerization. As noted above, such literature values frequently display large variations from one source to another. Lichti et al. (1981) examined the evolution of the PSD during a seeded styrene emulsion polymerization. The polymerization was sampled only during Interval 11, thus removing any uncertainties associated with monomer-depleted growth. The use of well-characterized seed particles obviates the difficulties associated with nucleation in Interval I and provides the initial condition for Eq. (5). Exhaustive kinetic studies on this system showed that the zero-one approximation was obeyed so that Eqs. (7) and (8) were applicable. The same kinetic studies yielded independent, precise estimates for the values of the parameters p and K, the effects of desorption being negligible; values of p and K were then chosen to optimize the agreement between the theoretical and experimental PSDs at the later time. Figure 6 shows the good agreement achieved between the calculated and measured PSDs at the later time (t = 45 min). A comparison between the values of p and K obtained by the kinetic and PSD experiments was also presented and excellent agreement found. These experiments demonstrate that consistent rate parameters may be determined using the kinetic and

113

3. The Particle Size and Molecular Weight Distributions RADIUS (om) 40 5 102 ('._1 1025 K

4

7e ~

60

50

PSD KINETICS 2.2 2.3 2.3

2.4

m3 $-1

3

2: e

Ñ

I o

2

~\

\ .\

,\

o

2

4

6

8

10

Fig. 6. Comparison of theory and experiment for a seeded emulsion polymerization. Continuous lines: smoothed partic1e size distribution obtained experimentally; broken line: theoretical fit using values of p and K shown in inset (values of p and K from kinetic study also shown) at t = 45 mino (After Lichti et al., 1981; reproduced with permission of Journal of Polymer Science.)

PSD approaches; alternatively, they demonstrate that the PSDs may be reliably calculated from o , kinetic data alone.

2. Continuous Emulsion Polymerizations De Graaf and Poehlein (1971) and Stevens and Funderburk (1972) have compared the predictions of the simple residence-time theory for the CSTR with experiment. However, the results of Stevens and Funderburk must be treated with caution in view of their use of only 30ü-400 particles to establish the PSD. Figure 7 shows the comparison of theory with experiment for the cumulative PSD obtained by De Graaf and Poehlein for a styrene CSTR with a mean residence time of 59 mino The agreement obtained was good provided that the Stockmayer solution for ñ in terms of p and e was used (k = O), rather than the Smith-Ewart Case 2 (i.e., ñ was greater than t). Note, however, that De Graaf and Poehlein assumed that free-radical desorption need not be taken into account; moreover, they assumed that the initiator capture efficiency was 100%. Both assumptions

Gottfried lichti et al.

114 1.0

z 0.8 O i= => ID

Q? 1- 0.6 VI i5 ILI >

~ 0.4 -1 =>

~

=>

00.2

O'

/

/

/

o

100.

200

300

PARTICLE DIAMETER(n m) Fig. 7. A comparison of theory with experiment for the normalized cumulative PSD, proportional to Son(O',t) dO',where O'= diameter, obtained by De Graaf and Poehlein (1971) for a polystyrene CSTR: --- ñ = 0.5; ñ > 0.5; O experimento (After De Graaf and Poehlein, 1971; adapted with permission of Journal of Polymer Science.)

are of dubious validity for the small particle sizes studied (Hawkett et al., 1980). G.

.

Conc/usions ofthe PSD

Because of the large number of mechanistic processes opera tive in emulsion polymerizations, complete theories for the PSD are necessarily complex. Nevertheless, they can be formulated by a population balance approach. Much remains to be done, however, to clarify the basic colloid science that underpins the nucleation process in Intervall. The experimental challenge in evaluating the predictions of the theory for PSDs resides not only in the attainment of agreement with experiment but also in showing that such agreement is not merely fortuitous but arises from the . correct mechanistic scheme. Considerable experimental work will be required to establish the validity of mechanistic assumptions for any particular monomer.

3. The Particle Size and Molecular Weight Distributions

115

11. Molecular Weight Distributions

A. lntroduction The molecular weight distribution (MWD) of a polymer generated by emulsion polymerizations can be fundamentally different from that generated in solution or bulk. For example, in styrene emulsion polymerization the MWD of formed polymer has a much higher average molecular weight than may be obtained using other methods. The basic reason for this was postulated by Smith and Ewart (1948) to be the compartmentalization of the polymerization reaction inside the latex particles which leads to the isolation of free radicals. This isolation reduces the probability of bimolecular terminations and hence increases the degree of polymerization. The fundamental difficulty in constructing a theory for the MWD in emulsion polymers is to account for the compartmentalized nature of the system. In the commonly occurring situation where particles contain only a few free radical s at any given time, it is obviously incorrect to consider that each latex particle behaves like a "mini-bulk" reaction vessel, and so the conventional methods used for bulk polymerizations are inapplicable. Nevertheless, some assumptions which introduce only minor errors may often be made. The most important such assumptions is that the evaluation of the MWD may be separated from that of the PSD. In other words, provided that the MWD being produced at any given moment is the same as would be formed in an equivalent set of monodispersed latex particle systems [as expressed in Eq. (27) below], then the MWD evolved in a system that is polydispersed in size may be computed trivially. Formally, this is expressed as follows. Let S(M, a, t) be the MWD formed in a monodisperse' system 'Of size a at time t; here M is the molecular weight variable. In a polydisperse system with PSD n(a, t) the overall MWD at any experimental time te, defined as S(M, te),is postulated as "" re (27) S(M, te)= O dt O da n(a, t)S(M, a, t)

ii

A detailed critique of the validity of Eq. (27) is given in Section III. In brief, two criteria are required for this validity, both of which are well satisfied in ordinary emulsion polymerization systems. These are (i) that the time required for formation of a single polymer chain be much less than that over which significant changes occur in the rate coefficients governing the MWD and (ii) that the average number of free radicals per particle of size a is close to its steady-state value.

116

Gottfried Lichti el al.

Equation (27) may be expanded to incorporate the contribution to the MWD from newly formed latex particles by assuming that these enter the system at some small volume and calculating the MWD in the new particles separately. Since latex particles grow to many times their original volume during the course of polymerization, the component of the MWD produced by nucleation is often negligible (at least, when measured as a weight average). Any aqueous-phase polymerization may also be included in the same way. We now show how to evaluate the MWD in a monodisperse compartmentalized system. It will be seen that the problem may be solved with complete generality if chain-branching reactions do not occur; moreover, analytical solutions can be obtained for the steady-state regime. B.

Elementary Concepts

In this chapter, the term MWD refers to the number density MWD, S(M), which gives the relative number of polymer molecules of molecular weight M. Another commonly used measure is the weight density MWD, W(M), giving the relative weight of polymer molecules. The two distributions are related by .

W(M)

= MS(M)

(28)

Thus, the relative number of chains in the sample with molecular weights in the range MI::;;;M ::;;;M2 is given by N[MI,M2J = rM2S(M)dM

.

(29)

JMI

and the relative number in a small range M to M + 11M by S(M) 11M. The normalization (and thus the dimensions) of S(M) will be specified later. In the ensuing development, S(M) will refer to the "instantaneous" MWD, Le., the contribution to the MWD created over a comparatively short time periodo Ir all polymerization occurs under steady-state conditions, S(M) then gives the final MWD of formed polymer directly via Eq. (27); otherwise, the final MWD is given by a simple time integration over S(M), as given in Eq. (27) and amplified in Section II,C,2. 1. Review of Bulk MWD Theory In bulk or solution polymerizations, the following free-radical mechanisms are usually considered: initiation, propagation, chain transfer (to monomer, polymer or transfer agent), and bimolecular termination (by

3. The Particle Size and Molecular Weight Distributions

117

combination or disproportionation). Except for initiation, all these reactions are bimolecular. Initiation and transfer are chain-starting reactions, whereas transfer and bimolecular termination are chain-stopping. A particularly simple theory for the MWD in bulk or solution polymeri. zations

is obtained if long-chain branching mechanismsare ignored and if

the growth of chains is envisaged as a continuous rather than a discrete process. The latter approximation is valid for almost all systems of interest, since the average degree of polymerization is usually large. The molecular weight of a chain is proportional to the growth time t' of this chain: M = kpCMMot' ==al'

(30)

where kp is the propagation rate coefficient, CMthe monomer concentration, and Mo the molecular weight of monomer. We note at this point that a may be a function of time, as it is during Interval III when CMdecreases; this point will be pursued in Section II,C,2. G(t),the distribution of growing chains of growth time t', is then G(t') = Goexp(-Jet')

(31)

where Go is a constant given by the rate of formation of new, growing chains (involving initiation and transfer rates) and Jerelates to the stopping of growing chains (involving bimolecular termination and transfer rates). Equation (31) expresses the fact that, once formed, a growing chain continues growing unless it undergoes a chain-stopping reaction. As Flory (1953) has pointed out, these kinetic probabilities imply a chain distribution that is of the most probable type, which to a good approximation is exponential. The required MWD is then given by the distribution of dead (nongrowing) chains; this is readily found as follows from G(t'). For those chainstopping reactions ih which the identity of the growing chain-stopping remains intact (i.e., transfer to monomer or to chain-transfer agent and disproportionation), the distribution of dead polymers produced which have (previously) polymerized for t' is simply proportional to G(t'). Denoting this distribution by Str.iM), we thus ha ve

.

Str.iM) oc exp( - Jet')

(32)

where we have used Eq. (31) together with Eq. (30) to relate M and t'. However, in the case of chain stoppage by bimolecular combination, pairs of free radicals join together to form a chain whose "equivalent" growth time t~q is the sum of the growth times of the two contributing chains. Now, for bulk or solution polymerization, the product G(l'¡)G(tí) specifies the distribution of pairs of growing chains of growth times t'¡ and tí, since in a bulk system any growing chain is equally likely to undergo

118

Gottfried Lichti et al.

bimolecular combination with any other chain (it will be seen that this does not hold in a compartmentalized system). The distribution of dead chains created by bimolecular combination with equivalent growth time t~q, denoted Sbc(M) (where M = at~q) is thus proportional to the sum of alI possible pairs of growth times giving this equivalent growth time:

i

t~q

Sbc(M

= at~q)oc. o

(33)

G(t' - t~)G(t~)dt~

Thus, from Eq. (31) we have Sbc(M)oc t~qexp( -

(34)

At~q)

The overalI MWD is then the sum of Sbcand Str,d' In order to facilitate comparison between MWDs, it is customary to define a dimensionless quantity P, the polydispersity of the MWD, which is the ratio ofthe weight average to the number average tnolecular weight: P = [1'X) M2S(M) dM LX)S(M) dM

J/[LooMS(M)

dM

J

(35)

If alI chains have the same molecular weight, P = 1; otherwise P > 1. For monomodal MWDs, the larger the value of P, the broader the distribution. For the MWD of Eqs. (32) and (33), P = 2 and 1.5, respectively. 2. Emulsion Polymer MWDs: Concepts and Limiting Cases The concepts introduced above require extension and revision when we turn to an emulsion polymerization. In view of the above discussion, we may take the system to be monodisperse in volume. First, we briefly consider the kinetics of such a system. As described in Section I,B, this is given by the distributions N¡(t), which are in turn given by the first-order rate coefficients for entry (p), desorption (k) and bimolecular termination (c). We note here that c

= Cbc + Cd

(36)

where Cbcand Cdare the (pseudo-first-order) rate coefficients for bimolecular combination and disproportionation, respectively. It is convenient to introduce at this point a further pseudo-first-order rate coefficient, that for transfer, given the symbolf We write f

= ktr.MCM + ktr.A[A]

(37)

where ktr.Mand ktr.Aare the rate coefficients for transfer to monomer and to chain transfer agent A; we assume that if transfer of free radical activity to A occurs, the species so produced polymerizes further. It is postulated that

3. The Particle Size and Molecular Weight Distributions

119

desorption, if it occurs, proceeds by transfer of free-radical activity to a small molecular weight species that subsequently leaves the latex particle. Thus, f as given in Eq. (37) includes desorption as well as true propagative transfer. To resolve this ambiguity, we redefine f as the rate coefficient for transfer events that do not result in desorption, i.e., we set

f = ktr.MCM + ktr.A[A]- k

(38)

We will consider the MWD in two simple cases. The first is when chain transfer is sufficiently rapid to ensure that all other chain-stopping events can be ignored. In such a situation, whereas the compartmentalized nature of the reaction may affect the rate of initiation of new chains, it will not affect the lifetime distributions of the chains once they are formed. The MWD may then be found from the bulk formulas, provided only that ñ, the average number of free radicals per particle, is known. Such an approach has been used by Friis et al. (1974) to calculate the MWD evolved in a vinyl acetate emulsion polymerization. These authors included in addition the mechanisms of terminal bond polymerization and of transfer to polymer (both of which cause broadening). The formulas required for the incorporation of these mechanisms could be taken from bulk theory. It is important to note that, even in this present limiting case of a transfer-dominated system, the chain-stoppage mechanism can be changed by compartmentalization. Thus, the MWD formed in the polymerization of styrene appears to be transfer-dominated in some emulsion systems (Piirma et al., 1975) but to be combination dominated in bulk or solution (George, 1967). This difference occurs because, in ~tyrene emulsion systems, the rate of radical entry into a particle is slow, and most particles usually contain either zero or one free radical. In the "state one" particles (Section I,B), the growing free,radical ras time to undergo several transfer reactions before a further entry causes radical annihilation. The second simple case we consider in this section arises in a zero-one system (Section I,C,l,a), where the average time between successive entries of free radicals into a latex particle is short compared with the time taken for a growing chain to undergo a transfer reaction. In this case, most chains cease growth by bimolecular termination. Ir this happens by combination with a low molecular weight entrant species, the following approach is possible. When entry occurs in a state zero particle, a free radical is formed and grows uninterrupted until the next entry event. Since the rate coefficient for entry is essentially constant, Eq. (31) gives the distribution of growing chains. Since the terminating free radical is of negligible molecular weight, the MWD will then be given by Eq. (32). The average chain growth time A.-1 is the average time between successive entries into the same particle (viz., p -1; i.e., p = },).

120

Gottfried Lichti el al.

In concluding this section, we note that except for the limiting cases given above, the phenomenon of long-chain branching has not thus far been incorporated into the models for emulsion polymer MWDs. In the general MWD formulation given below, it is assumed that these branching events occur with negligible frequency. C.

General Theory 01 Emulsion Polymer MWDs 1. "Singly Distinguished" Particles

The development here follows that of Lichti et al. (1980). By analogy with the MWD calculation for bulk and solution polymerizations presented earlier, the MWD formalism for monodisperse emulsion systems requires the evaluation of certain types of free-radical growth time distributions. Because of the variable nature of the reaction loci (depending on the state i), a separate growth time distribution is required for the population of particles in each state i. It is therefore convenient to define the distribution of singly distinguished latex pa¡;ticles in state i, denoted N;(t, t'), as the relative number of latex particles inside which a certain free radical began growth at time t, and continued growing in an uninterrupted manner for a further growth time t'. The state of the latex particle at time t + t' equals i. The free radical with growth time t' is called the distinguishing free radical. When the distinguishing free radical ceases growth, the latex particle is no longer said to be distinguished. These N; are analogous to (but quantitatively different from) the G for bulk systems (Section U.B.1). Note that for a distribution of singly distinguished free radicals N;(t, t') evaluated at experimental time te, we have

te = t + t'

(39)

from the above definition. It will be seen that this feature of the formalism has the important consequence that the evolution equations governing N;(t, t') are simply differential equations containing terms in only one of the independent variables, viz., t'. For N;, the minimum value of i is unity, not zero, since the definition requires the presence of at least one growing chain. If N;(t, t') is integrated over all values of the growth time t', the resulting integral counts each of the ¡free radicals inside the particle. Thus, from Eq. (36), we have the following normalization: rte

iNi(te)

= lo

N;(te - t, t') dt'

Since Ni is dimensionless (by the normalization condition dimensions of reciprocal time.

(40)

L Ni = 1),N; has

3.

The Particle Size and Molecular Weight Distributions

121

Because N¡(t, t') is a function of two independent variables, it would at first appear that the concomitant family of evolution equations would be, for example, coupled partial differential equations, and thus difficult to solve. Indeed, some formalisms for emulsion polymer MWDs (Katz et al., 1969) are based on distribution functions that suffer from this difficulty. However, N¡(t, t') is defined such that this problem is avoided, since the evolution equation separates into equations in t' alone with t being merely parametric. In fact, we have oN¡/ot' = pN;-l - [p + ik + f + i(i - l)cJN¡ + ikN¡+l + i(i + l)cN¡+2

(41)

This has been derived in detail by Lichti et al. (1980). In brief, this equation states that singly distinguished latex particles in state i form (i) by entry into an N¡-l type particle, (ii) by desorption of nondistinguishing radicals from an N¡+l particle, (iii) by termination among any of the (i + 1) nondistinguishing radicals in an N¡+2 particle. Similar statements hold for the loss term in Eq. (41). Since Eq. (41) involves variation in t' only, it may be solved given the initial conditions, viz., N¡(t, t' = O). These last quantities are the distributions of ordinary latex particles in state i inside which a new chain begins growth at time t: N¡(t, t' = O)= pN¡-l(t) + ifN¡{t)

(42)

Equation (42) asserts that entry and transfer (involving any of the i chains present) are the only chain starting reactions. Note that since N¡(t, t') usually decay monotonically in t', no steady-state approximatincan ~e made to simplify Eq. (41). Nevertheless, there are only a set of differential equations in one variable (as noted above) which can readily be solved (e.g., as in Section II.D.2). Having specified N¡, we now show how to compute the required MWD, or more specifically, the component of the MWD arising from the mechanisms considered in this section. This is the distribution of dead chains formed as a result of transfer, exit (desorption), and termination by disproportionation. We denote the components ofthe instantaneous MWD arising from each of these mechanisms by Str(t., M), S.it., M) and Sd(t., M), where t. is the experimental time. We have

L fN¡(t., t') S.x(t., M) = L kN;(t., t') i~l Stit., M) =

Sd(t., M) =

(43)

i;?;l

L 2(i -

i?t2

I)CdN¡(t., t')

(44) (45)

122

Gottfried lichti et al.

Here t' and M are related by Eq. (30). The right-hand sides of Eqs. (43-45) are the rates at which the distinguishing growing chain can cease growth by each of the events considered. It can be seen that Stn Sex, and Sd each has dimensions of(time)-z. The meaning of these distribution functions is as follows: the number of chains formed by transfer whose growth times are between t' and t' + dt', over the range of experimental time from te to te + dte, is given by SIrdt' dte (a dimensionless quantity), and similarly for Sex and Sd' Note that the expression for Sd is simpler than that given by Lichti et al. (1980), although the more complex expression for Sd given by these authors reduces to Eq. (45) after suitable manipulation. 2. " Doubly Distinguished" Particles The N; values are insufficient to account for the contribution to the MWD from bimolecular termination by combination: since tfiis requires a knowledge of the growth times of each of a pair of growing chains. In the bulk system considered in Section II,A,l, the distribution of free-

radical pairs with growth times '1/1 and t~ was shownto be givenby the product G(t~)G(t~).However, the equivalent relation in an emulsion system does not hold. A simple counter example will illustrate this. We consider a system containing only two particles in state 2. Let particle 1 contain growing chains with growth times t~ and tB' and particle 2 contain growing chains with growth times t~ and tD' The fraction of particles in state 2 which contain a chain of growth time t~, which we denote N~(t~), is 0.5, and similarly for N~(tB)'N~(tC),and N~(tD)' Note that, as in Eq. (40) N~(t~) + N~(tB)+ N~(t~) + N~(tD) = 2 since each particle is counted twice in this expression. Now, the fraction of particles that contain a pair of chains of growth times t~ and t~ is evidently zero, 110t N2,(t~)N~(tC)= 0.25, as would be given by the product formula. The reason for this difference between bulk and compartmentalized systems is that whereas all free radicals in a bulk system are mutually accessible, in an emulsion system only radicals inside the same latex particle are mutually accessible for bimolecular termination. We therefore define a new distribution N?(t, t', t") as being that of "doubly distinguished" latex particles in state i, Le., particles in which one free radical began growth at time t and continued growing for a time t', at which time (t + t') another free radical began growth in the same particle, and both are still growing at time t" later, the particle being in state i. From this definition, the experimental time te is given by te

= t + t' + t"

(46)

3. The Particle Size and Molecular Weight Distributions

123

The two free radical s which started at t and t + t' are termed "distinguishing," and the latex particle ceases to be doubly distinguished when either or both of these distinguishing free radicals cease growth. Obviously, the minimum value of i for which N7 is nonzero is 2. We define N;' to be normalized such that the integral of N7 over all possible values of t' and t" counts all possible pairs of growing free radicals in particles in state i. Since the number of pairs in a single particle in state i is ti(i

-

1), we have fte ftte

Jo Jo N;'(t. - t' - t", t', t") dt' dt" = ti(i - I)N¡(t.) (47) Note that this definition corrects a minor error in the work of Lichti et al. (1980). From Eq. (47),we see that N7 has dimensions of(time)-2. Although N;'(t, t', t'') is a function of three independent variables, the evolution equation governing variation with t" can be shown to contain t and t' only parametrically, analogous to that for N;(t, t'). In fact, it may be shown (Lichti et al., 1980) that ON;'/ot" = pN;'-l - [p + 2f + ik + i(i - l)c]N;' + (i - l)kN;'+l + i(i - l)cN;'+2 i ~ 2 (48) The usual population balance methods are used to derive this equation: thus entry into an N;'-l particle creates an N;' particle, and so on; note in particular that transfer occurs with arate coefficient of 2f, since f is that for transfer of a single free radical, and we here have two distinguishing free radicals. Since Eq. (48) involves variation with respect to t" only, it may be solved if the initial conditions N;'(t, t', t" = O) are known. These initial conditions are the distributions of singly distinguished particles in state i inside which a new chain starts growing at time t + t'. Thus N7(t, t', t" = O)= pN;-l(t, t') + (i - l)fN;(t, t') (49) because only entry and transfer are chain-starting reactions. The coefficient (i - 1) arises because the chain-starting transfer must not involve the first distinguishing radical. We see therefore from Eqs. (49), (48), (42), and (41) that the multiplevariable distribution functions represented by N7(t, t', n are in fact obtained as the solutions of sets of coupled hierarchical ordinary or simple partial differential equations: the solution of one set [for N¡(t)] pro vides the initial conditions for another set of partial differential equations [for N;(t, t')] which in turn pro vides the initial conditions for N7(t, t', t"). Obviously, this could be extended to n-tuply distinguished distributions, but this is unnecessary since the order of polymer-producing reactions never exceeds two.

Gottfried Lichti et al.

124

We now show how to use N¡' to obtain the contribution to the MWD arising from bimolecular termination by combination. When two distinguishing chains, one of which has been growing for time t" (this being the second chain formed)

and the other for t'

+ t"

(this being the first) undergo

combination, the resulting single dead chain has an equivalent growth time of t~q= t' + 2t". The rate coefficient for this event is 2c/(the factor of 2 arising from the usual definition of c as being per pair of free radicals). Thus, the relative number of chains of equivalent growth time t~qpresent at experimental time te formed by bimolecular combination is

f

t,~q

Sbc(te, M) =

O

dt" L: 2ccN¡'(te i

-

t~q

+ t", t~q-

2t", t")

(50)

where at~q = M as usual. Equation (50) is obtained as follows. First, the upper limit of the integral is the maximum value of t" such that a chain of growth time t~q can be produced by combination. Second, the integral is performed over all pairs of chains (after summation over all doubly distinguished particles in all states i) one of whose growth time is t" and the other whose growth time is such as to give a chain of growth time t~q, after combination has occurred. The te - t~q + t" is the starting time for such chains so that the time at which combination occurs is te; note here that since the lifetime of chains is normally much less than experimental times,

one has te - t~q + t"

~ te

to an excellent approximation.

Equation (50) defines a distribution as do Stn Sex, and Sd, so that the number of chains formed of growth time in the range t~qto t~q+ dt~q over the experimental time te to te + dte, is Sbc dt~q dte. These four contributions to the instantaneous MWD may thus be directiy summed to give the overall instantaneous MWD from all sources, S(te, M): S(te, M) = SIr + Sex + Sd + Sbc te,

(51)

Finally, we recall that the instantaneous MWD, S(te, M), may vary with The accumulatedoverall MWD of formedpolymer at time t* is simply Ct'

S(M, t*) ~ Jo S(te, M) dte

(52)

If S is independent of te, one simply has S = teS; such is the case, for example, during the Interval n steady state. In other cases (e.g., Interval In) the variation of quantities such as CM (and hence a) with time will be known from the kinetics; the evaluation of Eq. (52) is then a simple quadrature.

D.

Samp/e Eva/uation 01 MWDs The formalism given in Section n,e involves a large number

of equations

whích, although posing no difficulties so far as developing solutions, can best be understood by examining their behavior in limiting cases.

3. The Particle Size and Molecular Weight Distributions

125

1. Zero-One-Two Systems We first consider a system in which the maximum number of free radicals per latex particle is 2. This is termed a "zero-one-two" system. An analytic solution to the MWD equations for such a system in the Interval II steady state has been developedby Lichti et al. (1980). The zero-one-two model accurately describes systems wherein ñ (the average number of free radical s per particle) does not exceed 0.7; it is thus applicable to small-particle styrene (Hawkett et al., 1980),vinyl acetate (Ugelstad and Hansen, 1976), and vinyl chloride (Friis and Hamielec, 1975) emulsion polymerizations. In the Interval II steady state, the analytic solutions to the various functions involved in the MWD obtained by Lichti et al. (1980)are as follows; in these expressions,the dependence on te is suppressed since a steady state is assumed (53) N'l(t') = B1 exp( - A+t') + El exp( - Aj) N~(t') = B2 exp( where

-t{all

},+ t')

+

E2 exp(

-

Aj)

(54)

= N;(O) - E¡, E¡ = [(a¡l + },+)N'I(O) + a¡2N~(0)]/(},+- A_), A:t = + a22:t [(all + a22)2 - 4(alla22 aI2a21)]1/2}, N'¡(O) = pNo + fNI,

B¡

-

N~(O)= pNI + 2fN2, whereNo = 1- NI - N2,NI = p(p + 2k + 2c)/rx,N2= p2/rx,with rx= p(2p + 3k + 4c) + 2k(k + c), and the aij are the elements of the matrix

-p-f - k

[

p

k + p/2 -p-2k-f-2c

]

We further have N;(t',I t") = [pN'I(t') + fN~(t')] exp( - Qt")

(55)

where Q = p + 2(f + k + c). These results give the singly and doubly distinguished particle contributions for i = 1 and 2 (no higher value of i being required for our zero-one-two system) in terms of the various rate coefficients for the microscopic processes involved. The various components of the instantaneous MWD are then found from Eqs. (43-45) and (50). Using the above results, Eq. (50) is found to reduce to

Sbc(M)= 2cc{(fB2 + pBI)[exp(-A+t~q) - exp(-tQt~q)]/(Q- 2A+) + (fE2 + pE¡}[exp(-A_t~q)- exp(-tQt~q)]/(Q - 2A_)}

(56)

Finally, there is an additional contribution to the MWD arising from the artificial truncation used in a zero-one-two system, Le., from the assumption that entry into a particle in state 2 causes instantaneous termination. We denote this additional term Sit(M), and find (57)

126

Gottfried Lichti el al.

This term is in fact a numericaIly inconsequential component of the overaIl MWD (if ñ < 0.7). Eqs. (53-56), (43-45), and (30) may then be used to compute MWDs for some cases of interest; because we are considering an lnterval II steady-state system, the overall MWD is simply proportional to the instantaneous MWD: s(t., M) = t.S(t., M) = t.(Str + S.x + Sd + Sbc+ Sit). Figure 8 shows the MWD produced in a zero-one-two system where values of the various rate coefficients have been chosen to give a MWD dominated by transfer and exit: p = 0.1 sec-l, k = 0.3 sec-l,f= 0.9 sec-1, e = Osec-1. This gives a steady state ñ of 0.34. Before discussing the form of the curves, we pause to consider the axes employed in this figure. Because we are considering a steady state system, we may use as abscissa either M (molecular weight, in a.m.u.) or t' (growth time, in s). We use both

0.5

o

0.4

4

2

10-5M 6

2

( b)

(o)

0.4

N I U Q) ~ 0.3 z O ¡:: ::> ~ 0.2 a:: ti> 5

0.3

0.2

0.1

0.1

O O

O 2

341 GROWTH TIME (sec)

2

3

4

Fig. 8. Plots of instantaneous molecular weight distribution S arising from instantaneous termination, stoppage, and transfer, as a function of growth time (Iower abcissa) and molecular weight (for a typical styrene system; upper abcissa), for a steady-state transferdominated zero-one-two emulsion polymerization, with e = O, p = 0.1 sec-I, k = 0.3 sec-I, f = 0.9 sec-I. (a) Curves I and 2 give conlribulions lo S from N'I and N2, respeclively. (b) Overall MWD, S. (Mter Lichli et al., 1980; reproduced wilh permission of Journal of Polymer Science.)

127

3. The Particle Size and Molecular Weight Distributions

for illustrative purposes, with the value of a computed for styrene emulsion polymerization at 50°C, using kp = 258 dm3 mol-1 sec-1, CM = 5.-8mol dm - 3 (Hawkett et al., 1980). The ordinate of the instantaneous MWD plot has dimensions (time)-2, the dimensions of S; the conversion from S to the relative number of chains of a given molecular weight was given in Section n.c.l. We consider the curves given in Fig. 8. In Fig. 8A, curves 1 and 2 give the instantaneous MWD arising from distinguished partic1es in states 1 and 2, respectively. Fig. 8B gives the overall instantaneous MWD, which is a monotonically decreasing function approximating a single exponential. This behavior, which for a transfer dominated system is identical with that in the bulk, will be further discussed in Section n,D,3. Figure 9 shows the instantaneous MWD computed for a zero-one-two system where the rate coefficients have been chosen to give domination by combination: p = 0.1 sec-1, Cc = 1 sec-l, k = O, f= O, Cd = O. This gives a

steady state ñ of 0.55. Figure 9A showsthe contributions to the MWD from Sbc [Eq. (56)] and the artifactual Sil term [Eq. (57)]. It can be seen that Sil indeed contributes but little to the MWD, showing that the artificial truncation method used to reduce the problem to a zero-one-two case (so 0.004 (b)

(o)

.-

...

I CII

'"

5.... 0.002 ::> ID

ir \ñ e TT

o o

1.0

2.0

1.0

2.0

EQUIVALENT GROWTH TIME (sec) Fig. 9.

(a) Plots

of components

of instantaneous

molecular

weight

distribution

S

arising from bimolecular combination (Sbe' labeled BC) and instantaneous termination (Sil' labeled TI) for a steady-state zero-one-two system with termination by combination only, assuming Ce= I sec - 1, P = 0.1 sec - 1, k = f = Cd= O. (b) Plot of S, the overall instantaneous MWD, for the same system. Note ñ = 0.55. (After Lichti et al., 1980; reproduced with permission of Journal of Polymer Science.)

128

Gottfried Lichti et al.

as to obtain analytic solutions) does not affect physical content. It can be seen from Fig. 9B that the overall MWD shows a distinct maximum. This resembles the behavior in bulk or solution, but as will be seen the similarity is only qualitative. Figure 10 shows the MWD for a zero-one-two system with rate coefficients chosen to give a disproportionation-dominated process: p = 0.1 sec-l, Cd= 0.3 sec-l, Cc= O= k = f. This gives ñ = 0.64. Here the MWD is seen to be monotonically decreasing, again a resemblance to solution and bulk behavior which is only qualitative. 2. Systems of Arbitrary ñ We next consider the evaluation of the instantaneous MWD for systems where ñ may be arbitrarily large. For simplicity we confine our discussion to a steady-state system in Interval II; the minor adjustments required to extend the computation to Interval III have been mentioned in Section n,e,2. For systems of arbitrarily large ñ, analytic solutions as given in Section n,D,l are no longer possible. Nevertheless, the numerical evaluation of S for such systems will be seen to require negligible computational effort. The procedure is as follows: first, from a given set of values of p, k, and c, one numerically solves the steady-state Smith-Ewart equations for Ni, i.e., Eq.

0.03 .... I

o

(\)

'" ~

0.02

z o ~

:J CD

g:0.01 IJ)

e

o

O

10 GROWTHTIME (sec)

20

Fig. 10. Plot of instantaneous MWD S as function of growtb time in a zero-one-two Interval 11 system witb termination by disproportionation only, assuming p = 0.1 sec-1, Cd = 0.3 sec-1, c m ir 1(J)

e w >

~

~ =>

~

=> (,)

o

2

4

GROWTH TIME

Fig. 15.

Plot

combination-dominated

curve 2 ñ = oo.

of instantaneous

MWD

system, as functions

(S) and

cumulative

of growth time (arbitrary

distribution

(F) for

units). Curve 1 ñ

= 0.5;

3. The Particle Size and Molecular Weight Distributions

137

shows plots of Sbcand F [from Eq. (64)] for these two limits, with a = 2. It can be seen that plots of the MWD, Sbc' show a pronounced qualitative ditTerence between the compartmentalized and bulk limits. However, the equivalent plots of F, the cumulative distribution of the tail, are qualitatively similar, and only show smalI quantitative ditTerences.This is because the greatest ditTerencesbetween the Sbcfor the bulk and compartmentalized limits occurs at smalI values of M (or t'), to which F(M) is insensitive.

3.

Formulation of Min and Ray

Min and Ray (1974) have presented a theory in which the MWD and PSD evolutions are considered together. The proposition that these quantities may be evaluated separately is considered in Section III of ~his chapter. The treatment of Min and Ray is applicable to a system which is mohodisperse in volume if one omits both particle nucleation terms and terms involving O/OV,and if the volume dependences of any rate coefficients are ignored. For such a system, Min and Ray define a function fm(i, te), the distribution of latex particles in state i at experimental time te that contain a growing radical with m monomer units. This is directIy related to the singly distinguished distribution function Ni(te, t') by

Ni(te

-

t', t') = fm(i, te)

(67)

where t' = mMo/a, where Mo is the molecular weight of monomer. The evolution of fm(i, te) is given by simple coupled ditTerential equations in te, one for each m. There would typicalIy be thousands of such equations; this is to be compared wi'th the sma:lI number of simple coupled ditTerential equations required to evaluate Ni. The formulation of Min and Ray alIows for exit, in the form of permitting desorption of growing chains of any length, rather than the exit of smalI oligomers alone permitted in the general formulation given here. This assumption of Min and Ray does not relate desorption to any chaintransfer evento If growing chains of any length could d~sorb, it would seem as likely that dead chains would also desorb; this possibility is however not encompassed within Min and Ray's formalismo Their formalism in addition has no analog of the doubly distinguished distribution functions Ni. Instead they approximate the contribution to the MWD by combination by

a convolution expression involving products of the fm and indeed acknowledge that this assumption is open to question. It was shown in Section II,C,2 by use of a particular example that such a product approximation is

138

Gottfried Lichti et al.

invalid. We now give a more specific illustration. Consider the correct form for Sbcin a zero-one-two system:

i

it'

Sbe(te' t' ) =

o

" N"2e' (t t' - 2t" , t" ) dt

(68)

with the equivalent convolution expression of Min and Ray, which we denote Sbc:

sbe(t e' tI) =

tO

pio

NI 2e' (t

tI

-

t" ) N' 2e' (t

t") dt"

Here the normalization constant pis. chosen so that Figure 16 shows Sbc and Sbc for p

=

~~

JSbcdt' = J

SbC

dt'.

1 sec-1, Cc = 10 sec-1, k = f = Cd = O

(giving ñ = 0.55). It is clear that the product assumption is both quantitativeiy and qualitatively inaccurate. Indeed, the limiting , .value of the

2

NI o ~ ~

~

o 31 ~

2

3

4

EQUIVALENT GROWTH TIME (sed Fig. 16. Comparison of predictions of the formalism of (1) Min and Ray (1974) with that of (2) Lichti et al. (1980) for the MWD of a combination-dominated system assuming p = 1 sec -1, C, = 10 sec -1 and k = f = Cd = O. Ordinate cissa is growth time (sec).

is instantaneous

MWD

(sec - 2), ab-

3. The Particle Size and Molecular Weight Distributions

polydispersi"ty ratio P as ñ approaches 0.5 is P correct value of P = 2. F.

= 2.5 for Sbc, rather

139 than the

Experimental Determination 01 the MWD

We now briefly consider experimental methods of determining the MWD, in order both to test and to apply the theory. In relating theory and experiment, it is clearly insufficient to consider a single measure of the MWD alone, e.g., the number average, since interpretation of a single datum is prone to ambiguities. Hence, experimental techniques that only give such a datum (e.g., light scattering) are inadequate for the present purpose. Even the polydispersity P, being the weight-to-number-average ratio, is ambiguous unless it can be established unequivocally that the MWD is monomodal, and even then different-shaped MWD curves can give similar values of P. It is clear that a measure of the full molecular weight variation of the MWD is necessary to pro vide useful information. The best procedure at present for obtaining this appears to be gel permeation chromatography (GPC). As this is relatively new technique, full MWD data are sparse. Friis and Hamielec (1975) have used GPC to study the MWD development in vinyl acetate and vinyl chloride emulsion polymerizations. For these monomers, the main chain-stopping mechanism is thought to be transfer, and so the compartmentalized nature of the system is relatively unimportant. These workers found that the MWDs produced at early times, where branching reactions are unimportant, have a P value close to 2, as expected for transfer-dominated reactions. A careful experimental study of the MWD produced in a highly compartmentalized system has been carried out by James and Piirma (1976) and Piirma et al. (1975). Ah ab initio emulsion polymerization was used which produced particles small enough ('" 75 nm unswollen radius) to ensure low ñ values in the Interval 11 part of the reaction. P was found to be constant during Interval 11, with a value somewhat in excess of 2. The fact that P (which as measured experimentally is of course from the cumulative rather than the instantaneous MWD) remains constant implies that the P of the instantaneous MWD is also close to 2 throughout this interval. This value of P may be rationalized on the assumption that either transfer or combination (in the ñ = 0.5, i.e., compartmentalized limit) are the main chainstopping mechanisms. Piirma et al. showed that in fact transfer to monomer was dominant, since the number-average molecular weight observed corresponded closely to that predicted from transfer domination. This result is interesting, in view of the fact that the main chain-stopping mechanism for styrene solution or bulk polymerization is combination. This result is thus a

140

Gottfried Lichti et al.

clear example of how compartmentalization can change the dominant chain-stopping mechanism. This is because free radicals inside the growing latex particles polymerize in isolation for periods of time sufficiently long to make the MWD transfer limited. The work of James and Piirma and of Piirma et al. is important because, inter alia, it highlights the role of several commonly used surfactants (e.g., Triton X) as transfer agents. This discovery complicates the interpretation of many experimental results reported in the literature. Included in this category is the rise in molecular weight with conversion in Interval 11,used by Grancio and Williams (1970) as evidence for the core-shell model of latex particle morphology. Lin and Chiu (1979) have reported measurements of P as a function of time for an ab initio styrene emulsion polymerization. They also found that P was slightly in excess of 2 for a considerable part of the reaction, but their actual molecular weight averages were much lower than those of Piirma et al. (1975). This suggests that their surfactant may have been acting as a chain-transfer agent. De Graaf and Poehlein (1971)'have measured the MWD of styrene in a continuous stirred-tank reactor. A wide range of particle sizes were present, and thus interpretation of the data is hindered by an inadequate knowledge of the size dependence of the various rate parameters. However, the values reported for the average molecular weights suggest again a transferdominated process; this is also consistent with the value obtained for P which was close to 2, for a variety of conditions. The influence of particle size on MWD has been investigated by Morton et al. (1954). They showed that the average molecular weight of the polymer produced was insensitive to the latex particle size. This is consistent with the molecular weight being dominated by chain transfer to monomer, a conclusion that holds irrespective of any differential swelling of particles of different sizes. The foregoing experimental review includes the major studies in which polydispersity ratios (as distinct from average molecular weights) were reported. The experiments are somewhat piecemeal in nature, dealing with systems for which reliable estimates of the rate coefficients which govern the MWD (p, etc.) were not available. It would clearly be advantageous, in terms of mechanistic understanding, to carry out MWD measurements on systems that are well characterized kinetically. The simplest experiment to interpret would be on a seeded system (thereby obviating problems as to the MWD formed during particle nucleation, as will be shown later), employing a volume monodisperse latex with polymerization being carried out in lnterval 11 under steady-state conditions and without significant particle growth during the course of polymerization. Methods have been established

3. The Particle Size and Molecular Weight Distributions

141

for obtaining the various rate coefficients klP p, k, kp, and e from kinetic studies (e.g., Hawkett et al., 1980, 1981). The theoretical evaluation of the MWD is then trivial. For systems with a significant polydispersity in volume, the MWD can readily be calculated using Eq. (27), given the volume dependence of the various rate coefficients; however, it was pointed out above that uncertainties in these quantities make an unambiguous relation between theory and experiment impossible in such an experiment. Experiments on a volume-monodisperse Interval III system could again be readily compared with theory, using known values of the time variation of CM, etc. (obtained from the kinetics), and integrating the instantaneous MWD over time. The ideal seeded experiment would be where seed Iatex particles are prepared in which the MWD of the seed is significantly different from that formed during subsequent seeded polymerization. The instantaneous MWD formed at early times could then be estimated directly. Seed latexes of different sizes could be used to probe the effect of compartmentalization (we stress again that if the system is transfer-dominated, the MWD of formed polymer should be independent of particle size). The experimental strategy outlined above would give a unified set of rate parameters to describe the kinetics, MWD, and PSD. If no such consistent set could be found, new mechanisms may pro ve to be important. For example, aqueous-phase termination, which is not usually incorporated in MWD theories, would give rise to extensive low molecular weight fragments, which would significantly influence the MWD if these fragments are subsequently incorporated into the latex p~rticles.

111. Separa~iIity of 1YIWDand PSD Most of the theory surveyed in this chapter is based on the premise that, although the MWD and PSD of a system are governed by the same fundamental set of rate coefficients (p, klP kp, etc.), it is nevertheless possible to decouple the computation of these quantities. This is required, for example, for the validity of Eq. (27). The basis of this premise '¡s the separation of time scales over which the various proc~sses governing these quantities occur. This concept was first introduced by O'Toole (1969). We now consider its validity. It is frequently the case that the time required for a Iatex particle to increase its volume by a measurable amount (say by 10%) is large compared with (i) the time taken for the growth of a polymer chain and (ii) the time taken for the distribution of free radicals inside the latex particles to reach a steady-state value (for a given particle volume). This separation of time

142

Gottfried Lichti el al.

scales is plainly invalid at very early reaction times, when most of the volume of the partic1e may be that of a single chain. However, since latex partic1es usually contain many individual chains, the time scale separability assumption for volume versus chain growth [i.e., (i) above] is c1early relevant to mO$tof the growth process. We now consider the second time scale separability, (ii) above. Again, this will obviously break down at early times, but again will be valid for most of the polymeriz.ation. This occurs for three reasons. 1. As partic1es increase in volume, a fixed percentage increment of growth takes longer to achieve. 2. With increasing time, the system will attain and maintain its steady state in the distribution of free radical s in all partic1es, with the possible exception of growth during a particularly rapid Trommsdorff gel effect. 3. The PSD becomes smoother with the passage of time (e.g., Fig. 1),so that partic1e growth [which disturbs steady state through the o/ay term in Eq. (8)] becomes less significant. . Since the separate time scale aondition is c1early valid for most of the polymerization process, one may say that each polymer chain is formed inside a partic1e of unchanging size, wherein all rate coefficients are constant and the distribution of free radicals has its steady-state value, for each volume V. Any residual effect of the PSD on the MWD would reside presumably in the effects of the PSD on the kinetic parameters (e.g., p, e, and to a lesser extent k). Conversely, the MWD would possibly influence the PSD through its effects on the swelling of the partic1es by the monomer; the effect, if it exists, is likely to be small.

IV. Conclusions The theoretical tools required to describe the evolution of the MWD and PSD of an emulsion polymerization, in terms of a consistent set of rate coefficients for the various microscopic processes involved, are fairly well developed. Whereas chain branching (transfer to polymer) has yet to be inc1uded, it is likely that bulk theories for this process (Bamford and Tompa, 1954) can be readily modified to the emulsion polymer case. Although the general theory is well developed, a real understanding of the details of the processes involved is limited by the paucity of unambiguous experimental results to enable theory to be tested, modified, and applied.

143

3. The Particle Size and Molecular Weight Distributions Acknowledgments

The support of the Australian Research Grants Committee, and of an Australian Institute for Nuclear Science and Engineering postdoctoral fellowship for GL, are gratefully acknowledged.

References Ballard, M. J., Gilbert, R. G., and Napper, D. H. (1981). J. Polymer Sci. PoI. Phys. Edn., (in press). . Bamford,C. H., and Tompa, H. (1954).Trans. Faraday Soco SO, 1097-1115. Camahan, B., Luther, H. A., and Wilkes, J. O. (1969). "Applied Numerical Methods." Wiley, New York. Corio, l., Mara, L., and Salvatore, O. (1979). Makromol. Chem. 180,2251-2252. De Graaf, A. W., and Poehlein, G. W. (1971). J. Polym. Sci. Parl A-2 9, 1955-1976. Eckhoff,R. K. (1967).J. Appl. Polym. Sci. 11, 1855-1861. Ewart, R. H., and Carr, C. 1. (1954). J. Phys. Chem.58, 640-644. Flory, P. J. (1953). "Principies ofPolymer Chemistry." Comell Univ. Press, 1thaca, New York. Friis, N., and Hamielec,A. E. (1975).J. Appl. Polym. Sci. 19,97-113. Friis, N., Goosney, D., Wright, J. D., and Hamielec, A. E. (1974). J. Appl. Polym. Sci. 18, . 1247-1259. Gardon, J. L. (l968a). J. Polym. Sci. Parl A-I 6, 665-685. Gardon, J. L. (1968b). J. Polym. Sci. Parl A-I 6, 687-710. Gerrens, H. (1959).Forlschr. Hochpolym.Forsch. 1, 234-328. George, M. H. (1967). /n "Vinyl Polymerizations" (G. E. Ham, ed.), Vol. 1, part 1, p. 165. Dekker, New York. Grancio, M. R., and Williams, D. J. (1970). J. Polym. Sci. Parl A-I 8,2733-2745. Harkins, W. D. (1945).J. Chem. Phys. 13, 381-382. Harkins, W. D. (1946). J. Chem. Phys. 14,47-48. , Harkins, W. D. (1947). J. Am.Chem.Soco69, 1428-1444. Harkins, W. D. (1950). J. Polym. Sci. 5, 217-251. Hawkett, B. S., Gilbert, R. G., and Napper, D. H. (1980).J. Chem.SocoFaraday Trans. /76,

1323-1343. .

.

Hawkett, B. S., Napper, D. H., and Gilbert, R. G. (1981). J. Chem. Soco Faraday Trans. / (in press). James, H. L., and Piirma, l. (1976)./n eds.), pp. 197-209. American

"Emulsion

Polymerization"

(1. Piirma and J. L. Gardon,

Chemical

Society, Washington, D.C. Katz, S., Shinnar, R., and Saidel, G. M. (\969). Adv. Chem. Ser. 91, 145-157. Kerker, M. (1969). "The Scattering of Light." Academic Press, New York.

Lansdowne, S. W., Gilbert, R. G., Napper, D. H., and Sangster, D. F. (1980). J. Chem. Soco Faraday Trans./76, 1344-1355. Lichti, G., Gilbert, 1957-1971.

R. G., and Napper,

D. H. (1977). J. Polym. Sci. Polym. Chem. Ed. 15,

Lichti, G., Gilbert, 1297-1323.

R. G., and Napper,

D. H. (\980).

Lichti, G., Hawkett,

B. S., Gilbert,

R. G., Napper,

Sci. Polym. Chem. Ed. 19, 925-938.

J. Polym. Sci. Polym. Chem. Ed. 18,

D. H., and Sangster,

D. F. (1981). J. Polym.

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Lin, C. c., and Chiu, W. y. (1979). J. Appl. Polym. Sci. 23, 2049-2063. Min, K. W., and Ray, W. H. (1974). J. Macromol. Sci. Rev. Macromol. Chem. C 11, 177-255. Min, K. W., and Ray, W. H. (1978). J. Appl. Polym. Sci. 22, 89-112. Morton, M., Kaizerman, S., and Altier, M. (1954). J. Colloid Sci. 9, 300-312. Nagy, D. J., Silebi, C. A., and McHugh, A. J. (1980). In "Polymer Colloids II" (R. M. Fitch, ed.), pp. 121-137. Plenum Press, New York. O'Toole, J. T. (1969). J. Polym. Sci. Part C 27, 171-182. Piirma, J. Kamath, V. R., and Morton, M. (1975). J. Polym. Sci. Polym. Ed. 13,2087-2102. Pis'men, L. M., and Kuclianov, S. I. (1971). Vysokomol. Soedin. A13, 1055-1065. Schmidt, E., and Biddison, P. H. (1960). Rubber Age 88, 484-490. Schmidt, E., and Kelsey, R. H. (1951). Ind. Eng. Chem. 43, 406-413. Shaw, D. J. (1970). "Introduction to Colloid and Surface Chemistry." Butterworths, London. Singh, S., and Hamielec, A. E. (1978). J. Appl. Polym. Sci. 22, 577-584. Smith, W. V., and Ewart, R. H. (1948). J. Chem. Phys. 16,592-599. Stevens, J. D., and Funderburk, J. O. (1972). Ind. Eng. Chem. Process Res. Dev. 11, 360-369. Sundberg, D. C., and Eliassen, J. D. (1971). In "Polymer Colloids" (R. M. Fitch, ed.), pp. 153-161. Plenum Press, New York. Thompson, R. W., and Stevens, J. D. (1977). Chem. Eng. Sci. 32: 311-322: Ugelstad, J., and Hansen, F. J. (1976). Rubber Chem. Tech. 49, 53Cr609. Watterson, J. G., and Parts, A. G. (1971). Die Makromol Chem. 146, 11-20. Wood, D., Lichti, G., Napper, D. H., ~nd Gilbert, R. G. (1981), to be published.

4 Theory 01 Kinetics 01 Compartmentalized Free-Radical Polymerization Reactions D. C. Blackley

1. Introduction. A. Definitionsand Introductory Concepts B. Practical Significance of the Theory C. Scope of Chapter . 11. Reaction Model Assumed A. Description. B. Constancy and Uniformity of Monomer

111.

IV. V.

VI. VII. VIII.

Concentration within Reaction Loci C. Mechanism of Fundamental Processes; Influence of Locus Size , The Time-Dependent Smith-Ewart Differential Difference Equations; Methods Available for Their Solution A. Derivation . B. Methods Available for Their Solution . Solution for the Steady State Solutions for the Nonsteady State . A. Case in which Radical Loss 15 Predominantly by First-Order Processes B. Case in which Generation of New Radicals Ceases . C. Approximate .. Poissonian" Solution to the General Case. D. Other Approximations Predictions for Molecular-Weight Distribution and Locus-Size Distribution . Theory for Generation of Radicals in Pairs within Loci . List of Symbols. References .

146 146 147 149 149 149 151 153

156 156 160 164 167 167 172 176 177 183 185 187 189

145 EMULSJON POL YMERIZA TION Copyright ~ 1982 by Academic Press, Inc. AII rights 01 reproduction in any lorm reserved. ISBN 0-12-556420-1

146

D. C. Blackley

l. Introduction A.

Definitions and Introductory Concepts

By the term "compartmentalized free-radical polymerization reaction" is meant a free-radical polymerization reaction that is taking place in a large number of separa te reaction loci. These loci are dispersed in a contiguous external phase. They are "separate" from each other in the sense that material contained within one particular locus is presumed to be capable of transferring to another locus only insofar as it is capable of being lost from the first locus to the external phase, and then of being subsequentIy absorbed by the second locus from the external phase. The number of separate reaction loci is presumed to be "large" in the sense that it is at least of the same order of magnitude as the number of propagating free radicals present within the reaction system as' a whole. It therefore follows that the average number of radicals per single reaction locus is small; it is convenient to regard 10 as the absolute upper limit, 5 as the usual upper limit, and 0.01-2 'as the range that covers many reaction systems of practical significance. The term "compartmentalization" as applied to free-radical polymerization reactions of the type considered here seems to have first been introduced by Haward (1949). The mechanism of the polymerization reaction is presumed to be essentially that of a homogeneous bulk or solution free-radical polymerization. The concern is exdusively with the polymerization by double-bond opening of carbon compounds that contain at least one carbon-carbon double bond. The reactive species that propagates to produce the polymer chain is a free radical formed by opening of the n-bond of the carbon-carbon double bond. The basic steps of the polymerization reaction are initiation, propagation, termination (by various means), and various transfer reactions. The structure of the polymer produced is determined by the balance of the propagation, termination, and transfer reactions. Whereas the polymerization mechanism is essentially that of homogeneous bulk or solution polym~rization, there are significant differences attributable to compartmentalization as the term has been defined above. The most important of these is that, unlike free-radical polymerizations taking place in a homogeneous medium, there are physical barriers that prevent interaction between the various propagating radicals present in the reaction system at a given time. To a large extent, the propagating radicals are physically isolated from each other, each very small group of radicals being provided with its own reaction vessel. This is, in fact, the significance of the adjective "compartmentalized." An important consequence of the compartmentalization of the propagating radicals is that opportunities for

4.

Kinetics of Compartmentalized

Free-Radical Polymerization Reactions

147

the mutual termination of radicals are reduced relative to the case of a similar polymerization occurring in a homogeneous medium at the same overall concentration of propagating radicals within the reaction system as a whole. The model for the reaction system will be considered in detail in Section 11.However, it is convenient to note here that, in principie, the free radical s that initiate the polymerization may be generated either within the external phase (external initiation) or within the reaction loci themselves (internal initiation). Whereas very brief reference will be made at the conclusion of this chapter to reaction systems of the latter type, the concern here will be almost exclusively with reaction systems of the former type. Insofar as the initiating radicals are generated exclusively within the external phase (and therefore ha ve to be by some means acquired by the loci by absorption from the external phase), we have a further important distinction between homogeneous and compartmentalized reactions. In the latter case, the processes that lead to the generation of the initiating radical s are physically isolated from the propagation, termination, and transfer reactions. One minor consequence of this is that transfer-to-initiator reactions may be virtually eliminated in the latter case. The primary objective of the theory of compartmentalized free-radical polymerization reactions is to predict from the physicochemical parameters of the reaction system the nature of the "Iocus population distribution." By this latter term is meant collectively the proportions of the total population of reaction loci which at any in~tant contain O,1,2,. ", i,... propagating radicals. The theory is concerned with the prediction of these actual populations and also with such characteristics of the locus population distribution as the average number of propagating radical s per reaction locus and th~ varian~e of the distribution of locus populations. .,

B.

Practical Significance of the Theory

Apart from intrinsic interest, the theory of compartmentalized freeradical polymerization reactions is of importance primarily because it is believed that ITIost of the polymer which is formed in the course of an emulsion polymerization reaction is formed via reactions of this type. The general shape of the conversion-time curve for many emulsion polymerization reactions suggests (see Fig. 1) that the reaction occurs in three more-orless distinct stages or "intervals." The first of these, the so-called Interval 1, is interpreted as the stage of polymerization in which the discrete reaction loci are formed. In the second and third stages - Intervals 11 and 111- the polymerization is believed to occur essentially by compartmentalized freeradical polymerization within the loci which were formed during Interval l.

148

D. C. Blackley

c. o

.~ Q)

> e o

INTERVAL m

time

Fig.1.

Conversion-time

curve for typical unseeded emulsion

polymerization

reaction.

The feature that distinguishes IntervallI from Interval 111is that monomer droplets are present as a separate phase during the former only. The theory also has relevance to the so-called "seeded "emulsion polymerization reactions. In these reactions, polymerization is initiated in the presence of a "seed" latex under conditions such that new particles are unlikely to formo The loci for the compartmentalized free-radical polymerization that occurs are therefore provided principally by the particles of the initial seed latex. Such reactions are of interest for the preparation of latices whose particles have, for instance, a "core-shell" structure. They are also of great interest for investigating the fundamentals of compartmentalized freeradical polymerization processes. In this latter connection it is important to note that, in principIe, measurements of conversion as a function of time during nonsteady-state polymerizations in seeded systems offer the possibility of access to certain fundamental properties of reaction systems not otherwise available. As in the case of free-radical polymerization reactions that occur in homogeneous media, investigation of the reaction during the nonsteady state can provide information of a fundamental nature not available through measurements made on the same reaction system in the steady state.

4. Kineticsof Compartmentalized Free-Radical PolymerizationReactions 149. C.

Scope 01 Chapter

Detailed consideration will first be given to the nature of the reaction model assumed. Such consideration is important in order to gain an appreciation of the limitations of the theory that is subsequently developed. This will be followed by derivation of the equations that govern the behavior of compartmentalized free-radical polymerization reactions and a discussion ofmethods that are available for their solution. This will be followed by brief consideration of the solution of these equations for reaction systems that have attained a steady state. This part of the subject is not recent and will not be dealt with in detail. The main part of the chapter will not be concerned with the solution of the equations for reaction systems in the steady state but with those that are approaching it. This aspect of the subject is of relatively recent development. Brief reference will then be made of the evolution of the particle-size and molecular weight distribution during a compartmentalized free-radical polymerization; although this subject is of recent development, it will be dealt with in detail elsewhere in this book. The chapter will conclude with brief reference to reaction systems in which the initiating radicals are generated in pairs within the reaction loci.

11. Reaction Model Assumed A.

Description

The number of reaction loci is assumed not to vary with time. No nucleation of new reaction loci occurs as polymerization proceeds, and the number of loci is not reduced by processes such as particle agglomeration. The monomer is assumed to be only sparingly soluble in the external phase (a typical example is styrene as monomer and water as the external phase), and thus polymerization is assumed to occur exclusively withip the reaction loci and not within the external phase. The monomer is assumed to be present in sufficient quantity throughout the reaction to ensure that monomer droplets are present as a separa te phase, and the rate of transfer of monomer to the reaction loci from the droplets is .assumed to be rapid relative to the rate of consumption of monomer in the loci by polymerization. The monomer concentration within the rea

(11)

~=l

and the variance of the distribution of locus populations at any instant can be found from 00

V(t)

=

-

-

L (i -

¡=o

~ { o~

(

Y)2y¡(t)

~0'P

= L i2V¡(t) -

o~ )} ~= 1

-

0'1'

/2

2

(o~ )~=

(12) 1

162

D. C. Blackley

The function 'P(~, t) c1early has the additional property that CJ)

'P(I, t) = L: v¡(t)= 1 ¡=o

(13)

for all values of t. Thus, the time dependency of 'P(~, t) should always disappear when the substitution ~= 1 is made-this is a consequence of the assumption that the number of reaction loci in the system remains constant. The function 'P(~, t) has another interesting property which as yet hasnever been exploited as fai:as is known. Ir the substitution ~ = - 1 is made; then we obtain 'P(-I, t) = vo(t) - v¡(t) + V2(t)- v3(t) + ... (14) Thus, 'P(-I, t) gives at any time the difference between the fraction of the total number of loci that contain an even number of propagating radicals and those that contain an odd number, the fraction containing an even number inc1uding within it those containing no propagating radicals at all. Ir it is desired to know the fraction of the total number of loci that do not contain any propagating radicals, then this can be obtained from the relationship vo(t) = 'P(O,t) -(15) Thus, it is c1ear that the single function 'P(~, t) is capable of yielding a great deal of information concerning the distribution of locus populations in the reaction system. There are at least four important advantages to using locus-population generating functions as compared with the matrix approach: 1. It is quite unnecessary to truncate the set of equations contained in Eq. (2); in fact, the logic of the method requires that the set be maintained infinite in extent. There is therefore no possibility that the set of equations actually being solved is a distorted representation of the behavior of the reaction system being investigated. To the extent that the requisite mathematical operations ha ve been correctly performed, the resultant expressions for v¡(t),etc. are free from approximation errors. 2. It is possible in several cases to obtain expressions for v¡(t), etc. in c10sed analytic form rather than as somewhat unwieldy algebraic expressions. 3. The method using the locus-population generating function is more readily applicable to systems in the steady state. 4. It is in principIe possible to make inferences concerning the nature of the distribution of locus populations if the generating function can be recognized as that of the frequencies of some known distribution. It should perhaps be pointed out that in several previous publications (e.g., Stockmayer, 1957; Birtwistle and Blackley, 1977, 1978: Birtwistle et al.,

4. Kineticsof Compartmentalized Free-Radical Polymerization Reactions 163 1979) the function 'P(~, t) has been detined using the actual locus 'populations n¡(t) and not the fractionallocus populations v¡(t).The difference is trivial. The function as detined by these previous writers is merely N times the function as detined here, N being the total number of reaction loci in unit volume of the reaction system. The advantages of the present detinition over the alternative detinition are that (i) the analytic expressions for 'P(~, t) are simplitied to the extent that they do not contain N as an arbitrary multiplier and (ii) the expressions for i(t) and v(t) in terms of 'P(~, t) are somewhat simpler. It should be noted, however, that if 'P(~, t) is detined using n¡(t)in place of v¡(t),then the right-hand side of Eq. (10) gives n¡(t),and not v¡(t). In order to convert the set of differential difference equations in Eq. (2) into a single differential equation with 'P as the dependent variable, each equation for dn;/dt is multiplied by ~¡/N [the factor N-1 changes n¡(t)into the corresponding v¡(t)], and then all the equations so obtained are added together. It is then noted that

(16)

where in each case the summations cover all possible values of i. The resultant single differential equation then readily transforms to .8'P -

8t

t = O\~ - 1)'P + k(l - ~)-8'P + X(l - ~2)-82'P

8~

8~2

(17)

This is the partial differential equation for 'P(~, t) which has to be solved for each particular case. The solution has to be subject to the initial (boundary) conditions appropriate to that particular case. For the special case of reaction systems that are in a steady state, n¡(t) are invariant with time, and therefore 'P(~, t) is not a function of t. It is therefore denoted by 'P(~) in this case. Thus, in this case 8'P/8t = O,and the righthand side of Eq. (17) must be zero. What were hitherto partial differential coefficients of 'P(~, t) with respect to ~ become ordinary differential coefticients. Dividing through by the factor 1 - ~ then gives d'P d2'P a'P - k- X(l + J:) -

d~

-

..

de

=O

(18)

164

D. C. Blackley

as the ordinary differential equation to be solved for 'P in this case. This is identical to the differential equation first given by Stockmayer (1957) for the locus-population generating function that characterizes the behavior of reaction systems in the steady state. Equation (18) can also be obtained from the infinite set of equations for the steady-state as given by O'Toole (1965). It will be recalled that this is the set of which Eq. (3) with the lefthand side set equal to zero is the typical member. Again the procedure is to multiply each eq\lation by ~i/N, add together all the equations so obtained, and then note the relationships summarized in Eq. (16) together with the additional relationship . ¿¡2'P ¿(i + 1)ivi+l~1 = ~ ¿¡~2

(19)

The resultant single differential equation can then readily be transformed into Eq. (18). Thus, it becomes clear that the O'Toole (1965) formulation of the steady-state problem is exactIy equivalent to that of Smith and Ewart (1948), notwithstanding that the approach to the problem is somewhat different. 3.

Approximation Methods

As has already been explained, the matrix approach involves some degree of approximation because of the necessity of truncating the set of equations in Eq. (2). There have, however, been other attempts to solve the Smith-Ewart differential difference equations by approximation methods that have not used the matrix approach. These methods have been used for steady-state systems as well as for nonsteady-state systems. Again the procedure is to set a limit on the maximum number of propagating radicals which a single locus can contain. This maximum number is sometimes qujte low, e.g., either one or two. The resulting set of equations is then solved by conventional methods for simultaneous linear algebraic equations (in the case of equations representing the steady state) or simultaneous linear differential equations (in the case of equations representing the nonsteady state).

IV. Solution for the Steady State The general solution to the Smith-Ewart differential difference equations for reaction systems in the steady state is most readily obtained using the locus-population generating function approach. This was first demonstrated by Stockmayer (1957) and subsequentIy by O'Toole (1965). It is convenient to introduce two new parameters ¡:;and m defined as ¡:;= av/kt = a/x,

m = kv/kt = k/X

(20)

4. Kineticsof Compartmentalized Free-Radical Polymerization Reactions 165 It is subsequently convenient to introduce a third parameter h defined by the equation (21) The parameter e is a measure of the rate of radical entry relative to the rate of bimolecular termination within loci. Similarly, the parameter m is a measure of the rate of radical exit relative to the rate of bimolecular termination within loci. The ratio e/m = CT/kis a measure of the rate of radical entry relative to radical exit. The so-called Case 1, Case 2, and Case 3 of Smith and Ewart (1948) correspond to the following circumstances: Case 1: m large relative to e; Case 2: k = O,kt = 00 (Le., e = m = O); Case 3: e large relative to m. Note that both e and m increase linearly with locus volume v if CT,k, and kt remain constant, but the ratio e/m is independent of both v and kt. The first step is to solve Eq. (18) for 'P(~). Introducing the parameters e and m, this equation becomes (1 + ~) d2'P/d~2 + m d'P/d~ - e'P = O

(22)

Two new variables x and y are introduced. These are defined by the equations x

y = 'P(~)/xl-m

= 2)e(1 + ~),

(23)

Equation (22) then becomes X2 d2y/dx2 + x dy/dx

-

{(1 - , m)2

+

X2}

y

=O

(24)

This is a modified Bessel equation of order 1 - m. Except for the physically improbable case wliere 1 - m is exactly an integer or zero, the general solution to Eq. (24) is y = AIl-m(X) + Blm-l(X) (25) I

where Il-m(x) and Im-l(X) are modified Bessel functions of orders 1 - m and m - 1 respectively, and A and B are constants. Physical considerations (see O'Toole, 1965) indicate that the A must always be zero. The value of B is determined by the requirement that 'P(1) = 1. The final result for 'P(~) is

(\/+fl+1)

2(m-1j/2

'P(~) = Im-l(h) (1 +

~)(1-m)/2Im-l

(26)

Application of Eqs. (10) and (11) gives h

hi2m-1-3i and

Vi

= i! Im-l(h)

Im-l+i

(J2)

(27)

(28)

166

D. C. Blackley

Factors such as Im(h)/Im-l(h) have been convenientIy referred to by van der Hoff (1958, 1962) as "subdivision factors." In effect, they quantify the extent to which the overall concentration of propagating radical s in the reaction system as a whole is enhanced by compartmentalization of the propagation steps into a large number of small reaction loci. An important further contribution to the analysis of steady-state reaction systems has been made by Ugelstad et al. (1967). They have shown how account can be taken of the likely possibility that radicals that exit from the reaction loci contribute to the stationary concentration of free radicals in the external phase which is available for entry into a reaction loci. For this purpose, it is necessary to distinguish bimolecular mutual termination between radical s that occurs in the reaction loci (i.e., within polymer/ monomer partic1es)from that which occurs in the external phase. The rate at which the former reaction occurs is characterized by the rate coefficient ktP, the rate of the latter reaction by ktE. The total rate of entry of radicals into all loci within unit volume of reaction system is th~n expressed as the sum of three contributions. The first derives from the rate of formation of new "acquirable" radicals within the external phase; the second derives from the rate at which acquirable radicals become present in the external phase by the process of exit from the loci; the third (which is negative) allows for the fact that radicals can be lost from the external phase by bimolecular mutual termination within the external phase. The resultant equation is 00

p = p' + k

I

i= 1

ini - 2ktE e2

(29)

where p is the total rate of entry of radicals into all the N loci in unit volume of reaction system, p' is the rate of generation of new radicals within unit volume of reaction system, and e is the stationary concentration of radicals within the external phase. The form of this equation implies that all the radical s that become present in the external phase (whether by generation or by exit from the loci) are potentially available for reentry, except insofar as they are destroyed by mutual termination. Dividing through by N, setting a = p/N, a' = p'/N, and y = pie, and introducing parameters

e = av/ktP'

e' = a'v/ktp,

m = kv/ktP,

Z = 2NktpktE/yZv

(30)

Eq. (29) readily transforms to e = e' + mf - Ze2

(31)

Ugelstad et al. have used Eq. (31) in conjunction with Eq. (28) to calculate y as a function of e' for various values of m and Z. It is valid to regard the e of

4.

Kinetics of Compartmentalized

Free-Radical Polymerization Reactions

167

Eq. (31) as equivalent to the h of Eq. (28) because the reaction system is presumed to have reached a steady state in which, inter alia, ¡¡,¡¡/,and 1 have attained stationary values.

V. Solutionsfor the NonsteadyState A.

Case in which Radical Loss is Predominantly by First-Order Processes

Considerable progress has been made in recent years in obtaining solutions to the time-dependent Smith-Ewart differential difference equations for various special types of reaction system in the nonsteady state. Although it has so far not pro ved possible to give an entirely general solution to these equations, it has pro ved possible to obtain a general solution to a modified set of equations which, under certain circumstances, approximate to the exact set of equations. The simplest case for which an exact solution has been obtained is that of a reaction system in which radicals are lost from reaction loci almost exdusively by first-order processes. Initially, the reaction system is devoid of free radicals, so that no polymerization is occurring. Then, at a time taken as the zero of subsequent time, radicals begin to be generated in the external phase at a constant rateo This case has been discussed in detail by BirtwistIe and BIackley (1977, 1979) using the locus-population generating function approach, and by Gilbert and Napper (1974) using the matrix approach. The expressions for the locus-population generating function, for v¡(t), and for l(t) haye also been given by Weiss and Dishon (1976), but without discussion of the significance of the result. Setting X = O,the partial differential equation to be solved for this case is

..

o\f/ot

= c¡(~-

1)\f + k(1 - ~)o\f/o~

(32)

The boundary conditions are vo(O)= 1 v¡(O) = V2(0)

= ... = O

(33)

and, furthermore, for all t 00

L

;=0

v¡(t)

=

vo(O)

=

1

(34)

Equation (32) with these boundary conditions is amenable to solution by the method of separation of variables, in which the function \f(~, t) is assumed to be of the form 2(~)'11:t),where 2(~) is a function of ~ only, and

168

D. C. Blackley

T{t) is a function of t only. The details of the process whereby the solution can be obtained have been given by Birtwistle and Blackley (1977). The result for 'P(~, t) is 'P(~, t) = exp{(o-¡k)(~ - 1)(1 - e-k/)}

(35)

It follows immediately from this that at all times the distribution of locus populations with respect to radical occupancy is Poissonian, and that the parameter of the distribution at any instant is (O"/k)(1- e-k/). These conclusions follow because the expression for 'P(~, t) is recognizable as the frequency-generating function for a Poisson distribution (see Kendal and Stuart, 1965). It then follows that

{

v.(t) = .!. ~(1 , i! k

- e-k/)

} { ¡ eXP

-~(1

k

- e-k/)

}

(36)

and

(37)

T~t) = (O"/k)(1- e-k/)

Alternatively, the results embodied in Eqs. (36) and (37) can be derived from Eq. (35)using Eqs. (10)and (11).The general result for v¡(t) then shows that the distribution of locus populations must always be Poissonian with timedependent

parameter

(O"/k)(1- e-k/). Setting t

= 00 in

these results givcs the

following predictions for the reaction system when it has attained the steady state: 'P(~,oo)

= e(O'/k)(~-l)

1 O" i v¡(oo)=i! k e-O'/k

()

Y(oo) = O"/k

(38)

Approximate solutions for vo(t),Vl(t),and V2(t)have been given for this case by Gilbert and Napper (1974). These solutions were obtained using the matrix method. The expressions obtained are algebraically cumbersome compared to the general result embodied in Eq. (36), but can, of course, be readily handled using modern computers. Provided that O"/k is small compared to unity, the solutions given by the two methods predict almost identical numerical results for Vi(t);in fact the ratio of the values given by the two methods is e-O'/k/(1- 0"/2k)2,which is almost unity for O"/k~ 1. An example of the numerical predictions that this theory gives is shown in Figs. 5 and 6. Figure 5 shows the fractionallocus populations vo(t),Vl(t), and

V2(t)

as functions of t for a reaction system for which

= 1 x 10- 5 sec-l, k = 5 X 10-4 sec-l, and X is zero. It is seen that the steady state is reached after approximately 104 sec, and that at all times most of the reaction loci are devoid of propagating radicals. Of those loci that contain radicals, most contain only one. Figure 6 shows the prediction for T(t) as a function of t for this reaction system. Again this shows the O"

4. Kinetics of Compartmentalized Free-Radical Polymerization Reactions 169 1.0 í=O i=I

;,;-0.5

i=2

o

5

10

15

20

time (sec) x 10-3

Fig. 5.

Fraetional locus populations, v¡(t),as functions of time t for i = 0,1,2 for

reaction system for whieh radiealloss from reaction loei is exclusively by first-order processes. Values taken ror (] and k are 1 x lO-s sec-¡ and 5 x 10-4 see-¡ respeetively. The ordinates for i = Oare vo(t); those for i = 1 are 40 v,(t); those for j = 2 are 2 x 103 v2(t).(Reproduced with permission of J. Chem. SocoFaraday l.)

2

.. Q

,x

O

5

10

15

20

time (sec) x 10-3

Fig. 6. Average number of radieals per loeus I(t) as a funetion of time t for reaetion system to which Fig. 5 refers. (Reprodueed with permission of J. Chem. SocoFaraday l.)

170

D. C. Blackley 10

-

5

i

O

10

5

15

20

time (se e) x 10-3 Fig. 7. Conversion of monomer to polyrner M(t) as a function of time t for reaction system to which Figs. 5 and 6 refer. (Reproduced with permission of J. Chem. Soco Faraday l.)

steady state being reached after about 104 seco The average number of radical s per locus in the steady state is small (0.02) for this case. The amount of monomer M(t) converted in a unit volume of the reaction system after the eIapse of time t can be obtained by substituting for T(t) in Eq. (1) from Eq. (37), and then integrating

over the range t

= O to t = t.

The

result is (39) The prediction for M(t) as a function of t for the reaction system to which Figs. 5 and 6 refer is shown in Fig. 7. Marked deviation from linearity in the predicted curve for conversion as a function of time occurs over the range t < 5 X 103 seco An interesting generalization to the case X = O has been given by Birtwistle and Blackley (1979). In this generalization, the theory is extended to include cases where a and k are time dependent. In order to emphasize the time dependence of a and k they are written as a(t) and k(t). What Birtwistle and Blackley (1979) have shown is that the solution to Eq. (32) always has the form 'I'(~, t) = e 2, so that the only locus populations that have to be considered are

'°1

:+

e

y.

D

F

o

2

3

4

kt Fig. 10. Predictions for average number of radicals per locus I(t) as a function of time t obtained by numerical solution of time-dependent Smith-Ewart differential difference equations, showing effect of increasing /. keeping (1 and k constant. Values taken for (1and k are both 1 x 10-3 sec-'. Values taken for /. are as follows: A, O; B, 1 X 10-5 sec-'; e, 1 x 10-4 sec-I; D, 1 X 10-3 sec-t; E, 1 X 10-2 sec-'; F, I X 10-' sec-'. (Reproduced with per'mission of J. Chem. Sal'. Faraday /.)

182

D. C. Blackley

those for which i = O, 1, and 2. No assumptions are made concerning the variation of n2(t) with t. Brooks gives explicit expressions for vo(t),Vl(t), and V2(t)from which T(t)can be readily obtained as (78) Brooks regards his approximation as valid for reaction systems in which T((0) is smalI (up to 0.4). The matter of the decay behavior of a seeded emulsion polymerization reaction following the cessation of the generation of new radicals in the external phase has recently been treated by Lansdowne et al. (1980) using the matrix approach. Numerical procedures have been devised for solving the set of equations in Eq. (2) in particular instances. Birtwistle and Blackley (1981b) have described one such procedure. Examples of the results of their calculations are shown in Figs. 10 and 11. These calculations refer to reaction systems 0.6

0.4

-

,-

0.2

A time (sec) Fig. 11 . Predictions for average number of radicals per locus, ¡(t), as a function of time t obtained by numerical solution of time-dependent Smith-Ewart differential difference equations, showing effect of decreasing k and increasing X, keeping u constant. Value taken for u is 1 X 10-5 sec-I. The values taken for k and X are as follows: A: k = 1 X 10-3 sec-I, X = 1 X 10-8 sec-I; B: k = 1 x 1O-4sec-1, X = 1 X 10-6 sec-I; C: k = 1 X 10-5 sec-I, X=lxl0-4sec-l; D: k=lxl0-6sec-l, X=lxl0-2sec-l; E: k=lxl0-7sec-., X = 1 x 10° sec-I. (Reproduced with permission of J. Chem. SocoFaraday l.)

4.

Kinetics of Compartmentalized

Free-Radical Polymerization Reactions

183

that are initially devoid of radical s and in which, at a certain time (taken as t = O),radicals suddenly begin to be generated in the external phase at a constant rateo Figure 10 illustrates the effect of increasing Xon the variation of l(t) with t, (1,and k being held constant. As expected, an increase in the value of X leads to a reduction in the value of "t at any instant. Figure 11 shows the effect of increasing X and decreasing k on l(t) as a function of t, with (1being held constant. As expected, for large values of X/k, the value of i(t) at long times approaches the Smith-Ewart Case 2 value of 0.5. Numerical solutions have also been obtained by Brooks (1980) using his three-state model (see above). The relevant simultaneous differential equations were solved by Euler's method. Brooks has included in his examples the case where (1 decays exponentially with time. In all the cases investigated, he finds that allowance must be made for bimolecular mutual termination of radicals and for the re-entry of desorbed radicals into reaction loci; he concludes that failure to take account of these possibilities can lead to serious errors.

VI.

Predictions for Molecular-Weight Distribution and Locus-Size Distribution

This chapter has been concerned exclusively with predictions for the distribution of locus populations within a compartmentalized free-radical polymerization reaction system. Other matters of considerable interest are the distribution of polymer molecular weights which the polymerization reaction produces, and the distribution of sizes of the reaction loci at the end of the reaction. A significant literature concerning both these aspects is beginning to develop, but beca use of the complexity of the subject and limited space, oniy a brief summary of the various contributions can be given here. The distribution of molecular sizes produced by a given addition polymerization reaction is determined by the balance of processes such as propagation, termination, transfer; branching, and cross-linking. In fact, the final polymer molecule produced is the "corpse" of the kinetic chain by which it was produced, as modified by transfer, branching, and crosslinking reactions. Early attempts to consider the distribution of molecular weights produced by a compartmentalized free-radical polymerization reaction were made by Katz et al. (1969) and by Saidel and Katz (1969). These workers derived a set of coupled partial differential equations, from the solution of which the molecular-weight distribution can, in principIe, be predicted. However, except for a few simple cases, solutions were restricted to predicting the lower moments of the molecular-weight distribution, and

184

D. C. Blackley

even these predictions required moderately extensive numerical calculations. Furthermore, the treatment did not take into account several important molecular events, such as the possibility of mutual termination with disproportionation, and the various transfer reactions. An alternative treatment, given by Min and Ray (1968), is also restricted to the prediction of the lower moments of the molecular-weight distribution. More recently, Sundberg and Eliassen (1971) have attempted a prediction of molecular-weight distribution for a reaction taking place isothermally in a well-mixed batch reactor. Micelles are assumed to be the only source of reaction loci. Both micelles and loci receive radicals at arate proportional to surface area. All the radicals produced in the external phase are absorbed by micelles and loci. Termination takes place immediately when two radical s become present in a locus and combination is the outcome. Allowance is made for the reaction of transfer to monomer. The predicted molecular-weight distribution is of the Flory "most probable" type with a polydispersity ratio of 2.0. As might be expected, the value of the rate coefficient for transfer to monomer relative to that for propagation has an important effect on both the average molecular weight and the distribution of molecular weights. An extensive treatment of this subject has been given very recently by Lichti et al. (1980),and a brief summary was given in an earlier paper (Lichti et al., 1978).The model assumed for this treatment is a three-state model in which i is O,1,or 2. An earlier paper (Lichti et al., 1977)applied a similar treatment to a two-state model in which i is O or 1. The treatment allows for the possibility that mutual termination may result in either combination or disproportionation. It also allows for the possibility of transfer to monomer. It has not, however, been possible to make allowance for branching and cross-linking. Prediction of the full distribution of molecular sizes, and not merely of particular moments of the distribution, has been achieved. The conclusion has been reached that compartmentalization of the reaction leads to a broadening of the molecular-weight distribution. An early contribution to the cognate matter of the prediction of the eventual distribution of locus sizes was made by Ewart and Carr (1954). The two principal factors that affect the nature of this distribution were considered to be (i) the distribution of locus sizes present when locus nucleation ceases, and (ii) the manner in which the loci grow subsequently as a consequence of polymerization. A third factor that may influence the growth pattern is the tendency for large loci to imbibe more monomer than do smallloci. O'Toole (1969) has discussed the extent of stochastic (as contrasted with deterministic) con tribu tion s to the polydispersity of locus sizes. He con-

4. Kineticsof Compartmentalized Free-Radical Polymerization Reactions 185 eludes that for moderately large loci, stochastic contributions are significant if the ratio of propagation to termination rate coefficients is greater than about 0.1. Sundberg and Eliassen (1971) have also made predictions of locus-size distribution using their modeI. The subject has also been discussed by Watterson and Parts (1971a,b). BirtwistIe and Blackley (in press) ha ve recently applied the locuspopulation generating function approach to the problem of the evolution of the locus-size distribution in compartmentalized free-radical polymerization reactions. They have introduced a generalized locus-population generating function 'P(~, t, v), defined as 00

'P(~, t, v) =

L

v¡(t, v)~¡

¡=o

(79)

where the function v¡(t,v) is such that v¡(t,V)DVis the fraction of the total number of reaction loci which at time t contain i propagating radical s and which also ha ve a volume that lies between v and v + DV.The relationship between this locus-population generating function and the generating function 'P(~, t) which has been used previously is elearly 'P(~, t) =

Vll.

(80) Loo

'P(~, t, v) dv

Theory for Generation of Radicals in Pairs within Loci

This problem was first treated in detail by Haward (1949). He considered the case of a bulk' polymerization that has been compartmentalized by subdividing the reaction system into a large number of separate droplets, each of volume v. Radicals are generated exelusively within the droplets and always in pairs. An example would be the polymerization of styrene in emulsified droplets dispersed in water initiated by the thermal decomposition of an oil-soluble initiator which partitions almost exelusively within the monomer droplets, In the model considered by Haward, radicals are unable to exit from the droplets into the external phase. The only radical-Ioss process is in fact bimolecular mutual termination. It therefore follows that all the droplets must always contain an even number (ineluding zero) of propagating radicals, and that the state of radical occupancy will change in increments of :t2. The conelusion reached by Haward is that in this case the effect of compartmentalization is to reduce the overall rate of polymerization per unit volume of disperse 'phase. The physical reason for this is that, as the volume of the droplets is reduced, so are the opportunities for a radical to escape from the others-and hence to avoid mutual

186

D. C. Blackley

termination. Compartmentalization therefore enhances the rate of termination relative to propagation, and therefore reduces the rate of polymerization; indeed, according to Haward, polymerization can be effectively suppressed altogether if the degree of compartmentalization is sufficiently great. The average degree of polymerization of the polymer produced is also reduced by compartmentalization. The expression derived by Haward for the overall rate of polymerization per unit volume of disperse phase in the steady state is as foUows: d[M]/dt

=

-kp[MJX/2k\)1/2

tanh{(X/2k\)1/2vNA}

(81)

where X is the rate of formation of new radicals containing one monomer unit, in unit volume of monomer, k\ is the rate coefficient for bimolecular mutual termination, and NAis the Avogadro number. Comparison with the corresponding equation for bulk polymerization without subdivision shows that the effect of compartmentalization is expressed by the factor tanh(vNAi.jX/2k,), and this is, of course, always less than unity. A more detailed treatment of, this problem has been given by O'Toole (1965). In this treatment, allowance is made for the possibility that radical s can exit from the loci into the external phase. However, no allowance is made for the possibility that radicals which do so exit may reenter the reaction loci and reinitiate polymerization there. The transitions which affect the numbers of loci which cross a notional boundary between states of radical occupancy i and i + 1 are illustrated in Fig. 12. The condition for the steady state is readily found to be k(i + l)n¡+1 + x(i + l)in¡+1 + x(i + 2)(i + l)n¡+2 = !CT(n¡ + n¡-I) {j+2)(i+l)n¡+2X

(i+l)ni+lk

(i+ I)lnl+lx

(82)

1+2

i+1

---- -- - - -- -- - 1--- - --n¡cr/2 i-I nl-lcr/2 Fig. 12. Transitions between states of radical occupancy required for derivation of OToole equation for rate of transition of loci across notional barrier between states i and i + 1 in the case where radicals are generated in pairs within reaction loci.

4. Kineticsof Compartmentalized Free-Radical Polymerization Reactions 187 where (j is now the rate of formation of single radicals within a single locus, and therefore (j/2 is the rate of formation of pairs of radicals within a single locus. The remaining symbols have their previous significances. Introducirig the parameters 8 and m as defined previously, this equation becomes

(m + i)(i + l)n¡+1 + (i + 2)(i + l)n¡+2= t8(n¡ + n¡-l)

(83)

An equation of this type must be satisfied at notional boundaries between each pair of neighboring states of radical occupancy, if the reaction system as a whole is to be in a steady state. By introducing the locus-population generating function 'P(~) as defined previously, this infinite set of equations can be combined together to give the single equation d2'P/d~2 = Cm/O

+ ~)J d'P/d~ - (8/2)'1' = O

(84)

The solution of this equation consistent with the physical constraints of the reaction system is l-m

'P(~) = A(l

h

+ ~)-r-Im;1 { 4(1 + ~)}

(85)

where A is a normalizing constant and h is as defined previously. The result obtained for Viis

where the combinatorial symbols have their usual meanings, and ¿j is zero for even values of i.and unity for odd values of i. The average number. of propagating radical s per locus is found to be given by

-1 =-h I(m+ 1)/2(h/2) 4I(m-1)/2(h/2)

(87)

For the case where loss of radical s by first-order processes is of negligible significance, the expression for 't'assumes the simple form 1=

ih tanh(h/4)

List of Symbols A [AJ,(AzJ a B Bp

constant column eigenvectors (4and the TABLEI Energy Characteristics of SDDS Adsorption on WaterAlkyl Acrylate Interface Depending on Alkyl Acrylate Polarity YI2

E.d'

Acrylate

(mJ/m2)

(kJ/mol)

t:.G (kJ/mol)

Methyl Ethyl Butyl

8 13.7 23.2

20.0 22.1 25.6

1.69 1.85 2.10

253

7. Polymerizationof Polar Monomers

free energy of adsorption LlG per methyl group of the alkyl (assuming that the entire alkyl portion of the emulsifier is immersed in the monomer phase). Data presented in Table I indicate a decrease in the energy of interaction between the hydrocarbon portion of the emulsifier and the organic phase (alkyl acrylate) with an increase in the polarity of the latter. Accordingly, the overall adsorption energy also decreases. Several important practical conc1usions may be drawn from these data. On the one hand, dispersions of polar monomers are stabilized to a lesser extent than nonpolar monomers by emulsifiers containing an aliphatic hydrocarbon in the oleophilic part of the molecule. Although the overall adsorption energy increases only slightly with increasing length of the hydrocarbon portion, as follows from data presented in Table 11,the solubility of the emulsifier in water is significantly reduced, which may hinder its practical use. Therefore, on the other hand, in order to increase the energy of interaction of the oleophilic part of the emulsifier molecule with the polar organic phase, polar groups should be introduced into its structure. This has been confirmed in practice. A consideration of the adsorption kinetics is very important in an estimation of the effectiveness of surfactants under the dynamic conditions of emulsion polymerization. In a stalagmometric study of dynamic and static adsorption of emulsifiers of various structure at the air-water interface, it was established that adsorption values of micelle-forming surfactants differ significantly in the period of drop formation (Nikitina et al., 1961). This was explained by the consider~ble period needed for establishment of adsorption equilibrium connected with the kinetics of adsorption layer formation. The authors conc1uded that for usual concentrations of surfactant solutions the period of establishment of adsorption equilibrium can be taken as equal to 2 mino Figure 2 shows the adsorption isotMrms of TABLE 11

Energy Characteristics of Adsorption Alkylsulfates on Ethyl Acrylate-Water Interface8

Alkyl su,lfate C12H2SS04Na C14H29S04Na CI6H31S04Na a The

E.d. (kJjmol) 22.1 23.3 24.6

of

I:1G

(kJjmol) 1.85 1.69 1.55

absence of proportionality between I:1Gand

increase of the hydrocarbon radicallength indicates that for polar organic liquids Traube's rule is not valid.

254

V. 1. Yeliseyeva

-10

rmxa =3.ifx!O

moZ/cm2

1

.

2;5 5.0 Concfntration

Z5

10.0

of lIerosol (tOsmoZ/ dm J)

Fig. 2. Adsorption isotherms of Aerosol AT plotted according to data of (1) static (2 min) and (2) dynamic (2 sec) values of surface tension.

.

Aerosol AT solutions (sodium dioctylsulfosuccinate) .plotted according to both dynamic (drop formation period 2 sec) and static (drop formation period 2 min) values. The diff~rence in the shape of the isotherms clearIy indicates the significance of the time of surface formation for effective emulsifier adsorption and stabilization at the interface. Apart from the emulsifier structure, the process may depend on the difference in polarity between the contiguous phases. Stalagmometric determination of the SDDS adsorption at the aqueous solution-ethyl acrylate interface dependence on the rate of drop formation (volume 0.03 cm3) were carried out in our laboratory by Vasilenko. The measurements showed that establishment of adsorption equilibrium at the CMC occurs at drop formation periods of 15-20 sec, i.e., at surface formation rates not exceeding 10-6 m2/sec. Adsorption kinetics acquires considerable importance in analysis of the mechanism of particle formation during emulsion polymerization, when the rate of organic phase formation and the rate of adsorption layer formation may be commensurate. B.

Aqueous Phase-Polymer and Aqueous Phase-Latex Particle lnterfaces

The general ideas developed for adsorption of surfactants on liquidliquid and liquid-air interfaces obviously cannot be completely transferred to adsorption on interfaces with a solid phase. Wolfram (1966) caIculated the packing density of surfactant molecules on the polymer surface from the adsorption value at the CMC and showed that it varies with polymer polarity. Thus, the adsorption area of a SDDS molecule on various surfaces takes the following values (nm2): paraffin 0.41,

7. Polymerization of Polar Monomers

255

polyethylene 1.21, poly(methyl methacrylate) 1.32, poly(ethylene terephthalate) 1.42, and polycarbonate 1.48. On the basis of the results obtained the author introduced the concept of "required area" of a surfactant molecule and conc1uded that this area increases with increasing polarity of the polymer. The established relationship he explains by the existence of other interactions such as orientation, induction, and hydrogen bonds between contiguous phases, in addition to dispersion forces. Because of these interactions the molecules adsorbed on the solid phase-liquid interface may acquire a nonvertical orientation: they may be arranged with greater planarity the more intensively the field (depending on polymer polarity) acts upon the polar part of the surfactant molecule. Several investigations have determined the absorption behavior of surfactant adsorption on partic1es of aqueous polymer dispersions by adsorption titration. The results have been similar to those observed by Wolfram for adsorption on aplanar polymer surface determined from the wetting angle. Thus, Paxton (1969) established that the area occupied by a sodium dodecylbenzylsulfonate molecule in a saturated adsorption layer (AS1im)on the surface of PMMA latex partic1es is 1.31 nm2, whereas on the surface of polystyrene latex partic1es it is only 0.53 nm2. The author considers that previous studies of adsorption of this emulsifier, which gave adsorption area (ASlim)of 0.50 nm2, were carried out on interfaces with similar adsorption characteristics. The intermolecular forces involved in adsorption between water and air or water and hydrocarbon solvent or nonpolar polymer are nearly the same. In his opinion the differences observed in surfactant adsorption on polymers of different nature allows one to use the molecular adsorption area value as a polarity characteristic of the polymer phase. Dunn and Chong (1970) determined the value of ASlim for sodium dodecylsulfate on poly(vinyl acetate) latex partic1es; it was fountl to be 1.24 nm2, i.e., 2.4 times higher than the minimal adsorption area of this emulsifier. Zuikov and Vasilenko (1975) showed that the ASlim value for sodium dodecylsulfate increases with polarity of the polymer and is inversely related to the interfacial tension at the interface of the corresponding monomer with water. A correlation was established (Yeliseyeva and Zuikov, 1976) between ASlimof a given emulsifier and its adsorption energy at the corresponding monomer-water interface. The same correlation is observed between ASlimof emulsifier at the given polymer-water interface and its adsorption energy at the interface between the monomer and water for the homologous series of alkyl sulfates (Table III). It follows from the results obtained that the factors determining the adsorption value of a given surfactant at the water interface with the monomer and with the corresponding polymer are the same. Since with the increase in the adsorption energy on the emulsifier-monomer interface, the

256

V. 1. Yeliseyeva TABLE

111

Adsorption Characteristics of Surfactants on Water-Monomer and Aqueous Phase-Polymer Particle Interfaces

Monomer

Emulsifier

MA EA EA EA BA HA Sty

C12H2SS04Na C12H2SS04Na C14H29S04Na C16H31S04Na C12H2SS04Na C12H2SS04Na C12H2SS04Na

.

aAt polymer-water

1'12 monomerwater (mJjm2)

-8 13.7 13.7 13.7 23.2 27.2 33.8

a

E.d' (kJjmol)

ASHm (nm2)

20.0

1.51 0.92 0.82 0.74 0.67 0.52 0.48

22.1 23.3 24.6 25.6 -

interface.

value of the surfactant adsorption increases at the interface with the polymer (Le., ASlimdecreases) which means that the packing density of its molecule in the adsorption layer increases. In a study of the adsorption characteristics of alkyl acrylate and alkyl methacrylate latexes Sütterlin et al. (1976) obtained values for adsorption areas of sodium dodecylsulfate which support the values listed in Table III. The authors, however, attributed the established regularity of ASlimvariation not to polarity but to hydrophilicity of the polymer, the increase of which leads to reduction of the equilibrium emulsifier concentration on the latex particle surface. Since in the homologous series of alkyl acrylates and alkyl methacrylates the increase of hydrophilicity of the monomer occurs parallel to the polarity of the interface, the established relationship may also be attributed to the latter factor. In an attempt to apply Wu's (1974) ratios to calculation of the polar component of interfacial tension at the interface with aqueous phase for polymers of difIerent polarity, Vijayendran (1979) estimated ASlimin latexes of the same polymers on the basis of the obtained results. The values are in good agreement with reported experimental data (Yeliseyeva et al., 1978; Sütterlin et al., 1976). The coincidence of adsorption areas for molecules of a given emulsifier on latex particles of polymers of difIerent polarity obtained by difIerent authors points to the determining role of this factor in adsorl'tion in the case of usual latexes prepared by emulsion polymerization. However, in general when determining the adsorption areas of surfactant molecules other factors, such as electrolyte concentration, temperature, and particle size, should also be considered .(Piirma 'ind Chen, 1980).

7.

IV.

257

Polymerization of Polar Monomers

Mechanism of Particle Generation

Numerous experimental data are available which suggest that both the solubility of the monomer in water and its polarity affect the mechanism of particle generation. Whereas during a conventional emulsion polymerization of styrene (or other hydrophobic monomer) examined by Harkins (1947) and Yurzhenko and Mints (1945) the most important sites of particle generation are monomer-swollen emulsifier micelles; with the increase of monomer solubility in water particle generation in similar systems may be shifted to the aqueous phase. Thus, the possibility of preparing concentrated and stable latexes by polymerization of methyl acrylate and ethyl acrylate in the absence of an emulsifier was demonstrated by Yeliseyeva and Zaides (1965) and by Yeliseye~a and Petrova (1970). It is also well known that styrene latexes may be obtained by emulsifier-free polymerization; such latexes are characterized, however, by larger particles and much lower concentrations. In all such cases the process initiated by persulfate begins in the aqueous phase with the formation of water-soluble, surface-active polymeric radicals which, after growing to a certain critical size, precipitate to form particles; subsequent polymerization proceeds mainly within these particles. The higher the solubility of the monomer in water, the more surface-aciive radical s and therefore primary particles are formed and the higher the stability and the concentration of the latex. A kinetic curve of the emulsifier-free polymerization of ethyl acrylate confirming this scheme is shown by Curve 1 in Fig. 3 (Yeliseyeva anq Petrova, 1970). The process

.

5

15 t(mini

25

Fig. 3. Conversion kinetics (g/100 g of aqueous phase) of ethyl acrylate (1) in the absence of emulsifier and (2) in the presence of 0.5 mol % of emulsifier E-30. Ammonium persulfate, 0.33%; tert-dodecyl mercaptan, 0.33% in aqueous phase; T, 60°C; phase ratio, 1: 3.

258

V. 1. Yeliseyeva

beginning in the aqueous solution is characterized by a low initial rate which rapidly increases, evidently due to the formation of partic1es and the transfer of the polymerization mainly into these partic1es. At the initial stages the surface tension of the system decreases and then increases with conversion. Estimation of the total activation energy of emulsifier-free polymerization of ethyl acrylate showed it to be 100 kJjmol initially, but it markedly decreases in the course of the process in accordance with the increase in the termination energy occurring in the partic1es typical of emulsion polymerization. In the presence of emulsifier (Fig. 3, Curve 2), the process proceeds from the beginning at a high rate and a low activation energy (53.5 kJjmoI), suggesting formation of partic1es from the very beginning of the process and participation of the emulsifier in the mechanism of partic1e generation. The slight difference in the vaIues of the integral molecular mas s of the poIy(et~1 acrylate) oDtained in the absence and in the presence of an emulsifier supports the assumption that in both cases the process mainly follows the mechanism of emulsion polymerization; the viscosity average molecular masses of this polymer obtained under identicaI conditions, both without emulsifier and in the presence of 2% (based on monomer) SDDS, are approximately 4 x 106 and 7 x 106, respectively (Mamadaliev, 1978). The slightly lower integral molecular mas s in the former case may be explained by a larger contribution of polymerization in aqueous solution at the initial stages of the process. Hence, the difference in the mechanisms of partic1e formation of ethyl acrylate (solubility in water 2.5%) in the absence and presence of emulsifier at concentrations above CMC consists in the first case of poIymerization in the aqueous solution with the formation of" own" surfactants which precede the generation of partic1es, whereas in the second case partic1es are formed from the very beginning. Whether own surfactants are formed in the latter case and the role they play is yet to be determined. After generation of partic1es, when their concentration becomes quite high, polymerization in the aqueous phase probably results in the formation of low molecular weight oligomer radical s which are captured by the partic1es before they grow to critical size. Their role in the subsequent process consists of initiation and termination of polymerization in the formed partic1es according to the usual behavior of water-soluble, active radicals. In the case of their termination in the aqueous phase they may also act as own surfactants contributing to the stabilization of partic1es. Zuikov and Soloviev (1979) studied the effect of monomer polarity on both the size and the size distribution of partic1es in the forming latexes. As a criterio n of polarity they took the solubility of the monomer in water, which in this case has an independent significance as well. Polymerizations of styrene, methyl methacrylate, butyl methacrylate, and methyl acrylate

i

Polymerization of Polar Monomers

259

initiated by ammonium persulfate(0.05%in aqueous phase) in the presence of SDDS at concentrations below the CMC were investigated. As can be seen from the electro n microphotographs of the latexes (Fig. 4), with increasing monomer solubility in water the size of particles decreases and their distribution broadens. This is also illustrated by plots shown in Fig. 5. The results obtained may be associated with the effects of both monomer solubility in water and polarity of the forming interface on the mechanism of particle formation. Since in all experiments the emulsifier concentration was below the CMC, particle generation in the case of the water-soluble initiator began in the aqueous phase. Because of the lower solubility of styrene, the concentration of polymer radicals initially formed is lower than for MMA and MA. Accordingly, a smaller number of large particles is formed. The critical size of polymer radicals increases with the solubility of the monomer in water (according to literature data it is 8 monomer units for styrene and 65 for MMA). The period of particle formation also increases, leading to higher polydispersity. Moreover, as previously discussed, the emulsifier adsorption decreases with increasing polarity of the interface; and therefore for the formation of a given surface less emulsifier is consumed, which also increases the period of particle generation and

...

'. 2

,

1.1"~

Fig. 4. Electron micrographs of latex particles obtained by polymerization of (1) styrene; (2) methyl methacrylate; and (3) methyl acrylate. SDDS, 0.066%; ammonium persulfate, 0.05% in aqueous phase; T, 80°C; phase ratio, 1: 10.

260

V. 1. Yeliseyeva

250 200

~t:: 150f4 4'

1J

0.7

100

1 2 J '+ 5 MonomerSolu.bilitgin water (%) Monodispersity coefficient (K) and diameter(D)of latexparticlesformedby

Fig. 5.

polymerization of monomers differing in solubility in water: (1) styrene; (2) butyl methacrylate; (3) methyl methacrylate; (4) methyl acrylate. For conditions see legend to Fig. 4.

decreases the coefficient of monodispersity. This facilitates the increase of the overall surface area, because of a decrease in particle size. As can be seen from Fig. 5, the broadest particle size distribution is observed for the most polar monomer, MA. Due to the increased critical size of radical s and the smallest consumption of emulsifier by the forming interface, this corresponds to the longest period of particle formation.

""

I

0.8

"-

.

.

I

1

0.0'+

0.05

I 0.05

/3

I

0.7 I

.

I

0.01

o

.

0.02

0.03

[s1)1)8] Xc the situation is more complex, with both k.p and kde fa11ingas polymer concentration increases. For vinyl acetate, the separate monomer phase already disappears at 20% conversion. For X > Xc, Vpis almost constant; however [Mp], k.p, and kdea11decrease with conversion. These effects will be discussed in more detaillater. 2. Case 2 Kinetics Consider a monodispersed latex, where water-phase termination and radical desorption are negligible and termination is instantaneous when a radical enters a polymer particle containing one radical. By definition, N2 = N3 = ... = O and the total radical entry rate per liter of latex equals R" the rate of radical generation per liter of latex via initiator decomposition. The rate of radical termination per liter of latex is equal to twice the rate at which radicals enter polymer particles containing one radical (two radicals are consumed per radical entry). Application of the stationarystate hypothesis gives R, = 2R,NdNp

(15)

= No = tNp

(16)

In other words NI

and ñ-1. -2

(17)

324

A. E. Hamielec and J. F. MacGregor

One normally assumes that systems such as styrene and methyl methacrylate, where transfer to monomer is not prominent, follow Case 2 kinetics when latex particles are small and termination in polymer particles is instantaneous. It has recently been shown that at low initiation rates radical desorption can be significant relative to radical absorption, and as a consequence ñ values appreciably smaller than 0.5 were found (Gilbert et al., 1980).At higher initiation rates ñ = 0.5 was approached. The use of chain-transfer agents would of course increase the desorption rate and lower ñ. For Case 2 kinetics the rate of polymerization is given by Rp

= kp[Mp]Np/2NA

(18)

with the rate independent of initiator concentration for fixed Np. Ir the initiation rate is reduced to a great extent, ñ will fall below 0.5 as radical desorption becomessignificant. 3. Stage III PolymerizatiolJs In Stage III, the polymer concentration in the polymer particles increases with ever increasing chain entanglements, and if the polymerization tem-: perature is below the glass-transition temperature of the polymer being syntht;sized, a glass-state transition will occur with the diffusion coefficients of small molecules falling by several orders of magnitude. The pro paga tion reaction becomes diffusion-controlled and a limiting conversion of less than 100% is reached where the rate of polymerization is effectively zero (Friis and Hamielec, 1976). Marten and Hamielec (1978) pro po sed the following models for the effect of polymer molecular weight and free volume on the termination and propagation constants. k ~=

M:

2a

1 ---

1

VF

VFcrl

( ) [ ( ~ = exp[- B(~ - ~ )] k,o

kpo

a exp(b/RT)

~C Xccan be ca\culated using (23) To ca\culate m for X < Xc set X

= Xc in Eq. (23).

C. Polymer Quality 1. Molecular Weight Polymer can be produced via the followjng reactions: transfer to monomer to chain transfer agent and to polymer, termination reactions, and terminal double bona reactions. Transfer to polymer and terminal double bond reactions produce long-chain branches. Transfer to small molecules and termination reactions produce linear polymer chains. A major simplifiGation in the modeling of molecular weight and branching development results when it can be assumed that a negligible amount of polymer is produced via termination reactions. Fortunately, in most emulsion polymerizations, transfer reactions are relatively more important than termination reactions in the production of polymer. This is a direct consequence of the compartmentalization of radical s in polymer partic1es which permits commercial polymerization rates at relatively low radical initiation rates. When the amount of polymer produced by termination reactions is negligible, molecular weight and branching development is independent of initiator and emulsifier levels (or number and size of the polymer partic1es). The appropriate equations that describe molecular weight and branching development, which have been proved valid for the emulsion polymerization

326

, A. E. Hamielec and J. F. MacGregor

of vinyl chloride and vinyl acetate in a batch reactor, follow (Friis and Hamielec, 1975). For X < Xc

-

MR KXc 1+CM 1 - Xc

(

) 2MR 1 + KXc ( ) CM+-1+1 - Xc (1 - Xc)( 1 - XC) EN= ~ ( - )( + K)

MN=-

(25)

2

M=

l-Xc'

w

1

Xc

CM = krm/kp,

KXc

2CpXc

CpXc

CpMn MR

Cp = kr~/kp,

K = k;/kp

(26)

(27) (28)

For X > Xc dQo = CM- KQo dX 1- X

(29)

dQl = 1 dX

(30) 1

dQ2 =2 dX

(

1

KX + 1- X

)(

d(QoEN) - CpX + KQo dX 1- X IN = 1000MREN/MN MN

= MRQdQo

Mw = MRQ2/Ql

CpQ2

KX

+1=X+1=X C

CpX M+I-X

(31)

) (32) (33) (34) (35)

Equations (29-32) can be solved using standard numerical techniques. The initial conditions may be found using Eqs. (25-27). When polymer produced by termination reactions is appreciable, molecular weight and branching development depend on the initiation rate and emulsifier concentration (number of particles and their size distribution). This more general problem has never been solved and is perhaps intractable because of the long-chain branching reactions. Solutions have been obtained for the case where long-chain branching reactions are neglected;

..

9. Latex Reactor Principies: Design. Operation. and Control

327

these involved the use of population balance equations and sophisticated numerical solution techniques. The most general model that predicts particle size and molecular weight distributions was developed by Min and Ray (1974). Work of a more fundamental nature on the details of the microscopic events that control particle size and molecular weight development is being done by Lichti et al. (1977; Lichti, 1980). 2.

Particle Number and Size Distribution

The prediction of particle number and size distribution has been far less successful than the prediction of conversion time histories and molecular weight development, given these parameters as initial conditions. The nucleation and early growth of polymer particles is even today, after the comprehensive investigations of Ugelstad and Hansen (1976), not. well understood. There are several reasons for this. First, the measurement of number

and size of polymer

particles

smaller than

100

A presents

rather

difficult experimental problems. Many complex processes occur simultaneously including radical capture by micelles and polymer particles, precipitation of growing radicals from the aqueous phase, and finally particle flocculation and coagulation. To illustrate how certain variables such as initiation rate and emulsifier concentra tion might affect particle number, we will refer to a highly idealized model for particle number after Smith and Ewart (1948). To simplify the analysis, let us assume that the area occupied by an emulsifier molecule is the same on the micelle as on the polymer particle. In other words (36) For typical emulsifiers, as has values of 30-100 A2jmolecule or 1.8-6 x 107 dm2jGmol. The detailed derivation may be found in the original reference and the final result has the form (37) where (X= 0.53 when radical capture by polymer particles is neglectedand = 0.37 when micelles and polymer particles both capture radicals and assuming surface area on both kinds of particles are equally effective in the capture process. The diffusion theory of radical capture would predict that the micelles, being smaller, would provide more effective surface area for radical capture and thus give an exvalue greater than 0.37. According to Ugelstad and Hansen (1976), the capture process is more complex with the possibility of both collision and diffusion theories of radical capture applying under different limiting conditions. To develop effective control strategies for polymer particle concentration, it is of considerable importance to (X

328

A. E, Hamielec and J. F. MacGregor

establish the time scale for micellar nucleation of polymer particles. The effectiveness of any control scheme would of course depend on the nucleation time. Np is given by Np

= NAR.tr

(38)

where tr is the nucleation time (time to end of Stage 1). A value of 2fkd

=

10-6 sec-1 (K2S208

=6

at 50°C) and [/]

X 10-3

Gmol/liter

gives

an initiation rate' of 6 x 10- 9 Gmol/sec liter latex. The time required to nucleate 1018 particles per liter of latex is thus only abou't 5 mino It is clear that manipulation of initiator and emulsifier concentrations to control polymer particle nucleation would not be an easy task. Equation (37) has been shown to be valid for the emulsion polymerization of styrene (Gerrens, 1959). It does not apply to Case 1 systems such as vinyl chloride and vinyl acetate where in fact Np is independent of initiator concentration. This disagreement is not unexpected as ¡.J.is a function of initiation rate and the total radical entry rate into micelles and polymer particles is much greater than the initiation rateo Parti~le nucleation has been successfully modeled for vinyl chloride polymerization by Min and Gostin (1979). The particle size distribution (PSD) of a fully converted latex depends on the conditions in both Stages 1 and 11. It depends on the original PSD at time tr (end of Stage 1) and the subsequent growth rates of various sized particles in Stage 11.Gerrens (1959) has shown that the cumulative PSD at time tr is, for Case 2 kinetics, given by

v cum F(v) =

¡.J.tr

v = O.53¡.J.2/5(as[S]/R.NA)3/5

(39)

It is clear that the PSD at time tr narrows with increasing initiation rate and broadens with increasing emulsifier concentration. Ir the original PSD at tr is very narrow, then all particles will grow at the same rate in Stage 11 and the fully converted latex will have a narrow PSD. On the other hand, if the original PSD is broad, the final PSD will be broad. Data on the final PSD after Gerrens (1959) showing the effect of initiator concentration on the breadth of the PSD are shown in Fig. 2. As anticipated by Eq. (39), the distribution broadens with decrease in initiator concentration. To predict the PSD to the end of Stage 11 for a broad distribution requires a knowledge of the radical capture mechanism. According to the collision theory, surface area on a small particle is equally effective in capturing radicals to that on a large particle. On the other hand, according to the diffusion theory surface area on a small particle is more effective.

.

9.

Latex Reactor Principies: Design, Operation, and Control

329

1.0

0.8

A

~ 0.6 IL

E ::J () 0.4

0.2

1.0

0.5

1.5

2.0

V (LlTERS X 1016)

Fig. 2. Cumulative partic1e size distributions for styrene emulsion polymerization at different initiator concentrations [1]0 (in g/liter of water) (adapted from Gerrens, 1959): A, 5.41; B; 2.88; C, 1.80; D, 1.08; E, 0.541; F, 0.361.

D. lnitiation Systems Hydrophilic initiators such as the persulfates and hydrogen peroxide and initiators with varying solubilities in the polymer particles such as the hydroperoxides with 'or without reducing agents are employed in emulsion polymerization. A recent review of redox systems in emulsion polymerization was published by Warson (1976). The most efficient manner of adding the components of a redox system is gradually in stages or by adding the peroxide at the start and then the reducing agent gradually (Warson, 1976). A typical redox system for cold SBR production employs sodium formaldehyde sulfoxylate (reducing agent), a hydroperoxide (oxidant), and ferrous sulfate, plus' a chelating agent or a chelated irol) salt. A simplified kinetic mechanism for this redox couple follows (Wright and Tucker, 1977). ROOH + FeH Fe3+ + XH

--

RO'+M-

ROM'

330

A. E. Hamielec and J. F. MacGregor

The function of the chelator is to complexthe ferrousion and thus limit the concentration of free iron. Redox systems appear very versatile, permitting polymerization at ambient temperatures and the possibility of control of the rate of radical initiation versus polymerization time. This would thus permit control of heat generation and the minimization of reaction time. The use of the redox system ammonium persulfate (2 mmol) together with sodium pyrosulfite (Na2S20S 2.5 mmol) together with copper sulfate (0.002 mmol) buffered with sodium bicarbonate in 1 liter of water form an effective redox system for vinyl acetate emulsion polymerization. The reaction was started at 25°C and run nonisothermally to 70°C. The time to almost complete conversion was 30 min (Warson, 1976; and Edelhauser, 1975). The rational for the use of finishing initiators that have appreciable solubility in the polymer particJe follows. At high conversions, the concentration of monomer in the aqueous phase is very low and water-phase termination of hydophilic radical s becomes excessive. The rate of radical entry into polymer particJes is thus greatly reduced an,d the polymerization rate falls to a very low level prematurely.

E. Heat Removal

1. Techniquesfor Heat Removal An energy balance for a batch reactor may be written as V( - LlHp)Rp = UAw(T - Te)+ QE (40) As reactors are increased in size it is usually necessary to add supplemental heat removal capacity since the heat transfer area of the jacket only increases with reactor volume to the 0.67 power whereas heat generation rate increase is proportional to volume. Internal cooling coils and baffies are often used to increase heat transfer and in this way heat transfer area may be increased 40-50%. In some processes external heat exchangers are used where the reaction mixture is continuously pumped through the heat exchanger cooled and ret.urned. Reflux cooling is very effective in increasing heat removal capacity and is extensively used with large latex reactors. It is of interest to investigate the effect of reactor size on the magnitude of QE' The results of sample calculations indicate that for reactor volumes greater than 5000 gal additional cooling capacity would likely be required to achieve commercial production rates. 2.

Techniquesfor the Reduction of Reaction Time

To minimize the polymerization time, it is necessary that full utilization of the heat removal capacity of the reactor be made at all times during the

331

9. Latex Reactor Principies: Design. Operation. and Control

batch. In other words, the polymerization rate and associated heat generation rate should equal the heat removal capacity at all times during the batch. For an isothermal polymerization this can best be done by controlling the initiation rate using a redox system. Of course, the sinallest reaction time occurs with adiabatic polymerization. The temperature rise can be reduced by controlled heat removal to moderate the spread in molecular weights. Typical conversion and heat generation rate profiles for an isothermal polymerization with constant radical initiation are shown in Fig. 3. The heat generation rate over the batch is far from optimal as the maximum heat remo val capacity of the reactor is about 220 arbitrary units. The polymerization time could be significantly reduced by employing a higher initiation rate early in the polymerization and by letting it fall with time so as not to exceed a heat generation rate of 220 arbitrary units. A similar reduction in polymerization time could be achieved with semibatch monomer feed using monomer starvation to give a polymerization rate equal to Rpmax'A cold monomer feed would also assist in heat removal. The shortest batch time would be obtained using semibatch monomer feed giving Rpmaxand permitting the temperature to rise. In semibatch emulsion polymerizations the polymer particles are kept monomer-starved to obtain higher rates of polymerization and to permit easier control of the rate and particle size distribution. There are two aspects to the control of PSD. The controlled addition of emulsifier during particle growth stabilizes the particles without further particle nucleation. The second aspect is related to the particle sticky stage which often occurs 1.0

2.5

0.8

1.5 ~

0.6

"el. :J: C U Q) lit I

.. Q) ::

3

......

o E

2

Q. Q:

o

1.0

. 2.0

3.0

4.0

ff4] mol/ 1Iter 100

50 Conversion

Fig. 4. semibatch

J o

(%)

Polymerizationrate curve showingstable and unstable operating points for a

emulsion polymerization.

at intermediate conversion levels and certain polymerjmonomer concentrations. During the sticky stage wall fouling and formation of coagulum often occur. The use of semibatch operation under monomer-starved conditions effectively eliminates the sticky stage. To illustrate that it is easier to control a semibatch emulsion polymerization under monomerstarved conditions, consider the rate curve shown in Fig. 4. Point P1 at low monomer concentration is a stable operating point for semibatch feed of monomer, whereas P2 is unstable. A consideration of perturbations in monomer feed rate clearly illustrates this fact. Semibatch operations are commonly employed in copolymerizations to maintain a more uniform copolymer composition. The monomers are fed at rates equal to their consumption rates in the reactor.

9. Latex Reactor PrincipIes: Design, Operation, and Control

333

111. Continuous Stirred-Tank Reactors: Steady-State Operation

A. lntroduction In this section, emulsion polymerization in continous stirred-tank reactor(s) (CSTRs) operating at steady state will be treated, with emphasis on polymer quality, Le., molecular weight, long-chain branching, and polymer particle number and size distribution. The commercially important problems associated with oscillations in conversion (Le., particle number and size, weight average molecular weight, and long-chain branching) and their control will be considered in following sections. A single CSTR or a train of CSTRs may offer several advantages over batch reactors both with regard to production rate and polymer quality. The copolymer composition distribution obtained in a CSTR is generally narrower than that for a batch reactor. To obtain fairly narrow copolymer composition distributions without appreciable loss in productivity, one has the flexibility to increase the number of CSTRs in series to approach plug flow. Because of diffusion-controlled termination and propagation with concomitant Rpmaxat a particular conversion, it is possible to operate a CSTR at considerably higher production rates. Because of the additional beneficial effects of cold monomer and water feeds on heat removal, much higher production rates are possible than with a batch reactor of the same volume. It should be remembered that polymer production rates are usually limited by heat removal capacity. Quality variations in the polymer produced in batch reactors are often caused by slight vaI:iations in the reactor start up procedure. Furthermore, the polymerization rate may change considerably during the batch and this may give temperature variations that are difficult to reproduce causing batch-to-batch variations in quality. These problems would be minimized with CSTRs if the continuous reactor system could be operated for at least several weeks before wall fouling and coagulum build up become critical and require reactor shutdown for cIeanup. Ir an effective start-up procedure for a continuous reactor train is not available, the costs associated with offspec material' could make continuous operation uneconomical. In addition, with a continuous reactor system one loses the flexibility of batch reactors when a multiproduct operation, with its short productions runs, is involved. During start-up and operation of CSTRs there is a potential safety hazard created by the possibility of more than one stable steady-state

334

A. E. Hamielec and J. F. MacGregor

operating point (Ley and Gerrens, 1974). The start-up procedure should ensure that the desired steady state is reached.

B. Po/ymer Qua/ity 1. Particle Number and Size Distribution The frequency distribution of partic1e radius F(R) is. related to the exit age distribution E(t) with the identity F(R) dR

= E(t) dt

(41a)

which states that the number fraction of polymer partic1es in the exit stream of a CSTR train with radius in the range R to R + dR equals the fraction of fluid in the exit stream with age in the range t to t + dt. For a single CSTR F(R) dR

= rr1 exp( -

tlO) dt

(41b)

Equation (41b) is valid when there are no polymer partic1es flowing into the reactor with aIl the partic1es nudeated within the reactor. lt is assumed thar density changes can be neglected and that partic1es foIlow the streamlines. These are reasonable assumptions in view of the smaIl size of partic1es and the smaIl density difference between particIe and water. When two or more CSTRs are employed in series, however, one must remember that the total residence time of a polymer partic1e is made up of different times in each reactor in the train. The relative amounts of time spent in each reactor wilI not matter when the volumetric growth rate of a partic1e is the same in each. This would require that the temperature, monomer concentration, and average number of radicals per partic1e be the same for each reactor, an unlikely possibility. This idealization is useful, however, when ilIustrating the effect of increasing the number of CSTRs in series on the breadth of the partic1e size distribution. The exit age distribution function for N tanks in series is given by E(t) =

(~)((N ~ 1)!)(~r-1 exp(-tIO)

(42)

The change in partic1e radius with time is obtained as dRldt

= KñlR2

(43a)

where K = kp[Mp]MR/4nNAdp

(43b)

To solve Eq. (43a), a relationship between ñ and time or partic1e radius is required. The simplest situation is to treat Case 2 kinetics where ñ is

I

9.

335

Latex Reactor Principies: Design, Operation, and Control

independent of t and R and equals 1- Using ñ = 1, integrating Eq. (43a) and then substituting for t in Eq. (41b) for a single CSTR, one obtains the differential frequency distribution. F(R)

= (2R2/K()

(44a)

exp[ -(2/3K()R3J

and the cumulative frequency distribution cum F(R)

= 1-

(44b)

exp[ -(2/3K()R3J

In a similar manner the following PSD may be derived for Case 1 kinetics for a single CSTR (Friis and Hamie1ec, 1976). (45a) F(R) = K~() exp[ -(2~1() )R2 J cum F(R)

=

K 1

(45b)

= 1 - exp[ -C~I()

[

(4/3n)2/3 R,

kp[MpJMR

dp

)R2 J 1/2

(45c)

J[ 16n2NANpkáe J

The cumulative distribution for Case 2 kinetics [Eq. (44b)] is shown in Fig. 5 plotted for different values of the parameter K(). It is c1ear that the

0.8

~0.6

E

=' 0.4 u

4

8

R ( Al( Fig.5.

Cumulative

16

12

partic1e size distributions

20

24

10-3) for a single CSTR. Case 2 kinetics.

.

336

A. E. Hamielec and J. F. MacGregor

PSD broadens with increasing K(). For situations where a separate monomer phase exists, [Mp] is a constant independent of (/ and perhaps varying slightly with temperature. K will therefore depend on temperature alone and increase with an increase in kp. Thus, increase in temperature and () broaden the PSD. The PSD from a single CSTR is excessively broad for most applications and it is thus desirable to add additional CSTRs in the train to narrow the PSD. Eq. (42) shows the effect of increasing the number of CSTRs on the .exit age distribution which also narrows and approaches that of a plug flow reactor (PFR) (or a batch reactor). The narrowest PSD would be obtained in a PFR. The use of 5-6 CSTRs in series should give a PSD which is insignificantly narrower than that for a PFR. The total number of polymer particles per liter of emulsion (Np) in a single CSTR may be calculated by [S]o

-

[S]

Np

(4nNp/as) Loo R2F(R)

= ()R¡NA

(

as[S]

Ap

dR

(46)

=O

)

[S]

+ aseS] = ()R¡NA [S]o

(47)

The last term in Eq. (46) gives the amount of emulsifier covering the polymer particles in the exit stream. Equation (47) is obtained by equating the rate of particle outflow to the rate of particle nucleation within thé

reactor employingthe collisiontheory. For Case 2 kinetics N p is givenby ()R¡NA Np

(48)

= 1 + NAR¡(4n/as[S]0)(3K/2)2/3()S/3r(5/3)

This equation is identical to the one derived by Gershberg and Longfield (1961). Equation (48) is shown graphically in Fig. 6. It appears that for a

Q. Z

10 81hrl

Fig. 6.

Number of polymer particles as a function of emulsifier and initiator levels for a

single CSTR. Case 2 kinetics.

9.

Latex Reactor Principies: Design, Operation, and Control

337

given set of operating conditions there is a O which gives a maximum Np. Npm.xhas been observed experimentally by Veda et al. (1971) and Nomura et al. (1971) for continuous styrene emulsion polymerization. Gerrens and Kuchner (1970) have furthermore confirmed the zero-order dependence on initiator, the first-order dependence on emulsifier at long residence times, and that Np oc 0-2.3 at large O. The mechanisms of particle nucleation for commercial monomer systems such as vinyl acetate, vinyl chloride, methyl acrylate, and chloroprene have not been as extensive1y studied experimentally as has styrene. It is likely, however, that an Npmaxwould occur with these more complex systems and this fact should be considered when designing a reactor train with high volumetric efficiency (Nomura, 1980). 2.

Molecular Weight and Long-Chain Branching

For the calculation of molecular weights in CSTR emulsion reactors, a useful classification comes to mind. This includes those monomer systems whose molecular weight and branching development depends on particle size and those that do no1. Styrene falls into the former class and vinyl chloride and vinyl acetate into the latter class. Thus, in vinyl chloride emulsion polymerization where LCB is neglected, the instantaneous molecular weight distribution is given by W(M)

= (MjM~)

exp(-MjMN)

(49) -1

=

M N

1.fJ: 2

W

=M

(

krm + krx[xJ

R kp.

kp[M]

)

It should be noted that at high conversions some branched PVC may form and it may therefore be necessary to consider the transfer to polymer reaction (Hamielec et al., 1980b). When transfer reactions control molecular weight development, all polymer particles in the first CSTR in the train will have the same molecular weight distribution (MWD) (neglecting inflow of polymer particles) given by Eq. (49). It there is no CTA and each reactor in the train is at the same temperature then the MWD will be the same in all the reactors in the train. Ir a CT A is employed for an isothermal train, the contribution from the CT A will increase in downstream reactors as the monomer concentration falls, and therefore molecular weight averages will fall. Of course, the MWD in each reactor will be of the same form and given by Eq. (49), but the final whole polymer will be a composite of the polymer produced in each reactor. This can be easily calculated if the conversion in each reactor is known. It should be remembered that less than one CTA molecule is consumed per polymer chain and therefore the CT A consumption by reaction can often be neglected.

338

A. E. Hamielec and J. F. MacGregor

With vinyl acetate the situation is more complex since transfer to polymer and terminal double bond reactions are significant at all conversion levels. If one calculates the MN, Mw, and BNfor a nucleated polymer particle with different lifetimes in a single CSTR, it is found that these quantities are independent of the particle age and are given by

M = MR N.

Mw

CM

where

~

1

KWp

(50)

p

2

KW 2MR 1 + 1 -

-

( = ~ (c

BN

( + 1- W ) (

1

~)

M+1-W

Wp

)(

p

1-W

CpMn

+K

MR

(51)

p

CpWp- 2CpWp 1 p

)(

K~

+l-W

p

)

)

(52)

is the weight fraction of polymer in the polymer particle. Friis et

al. (1974) have measured the following parameters for vinyl acetate using batch emulsion polymerization. kp = 1.89 x 107 exp(- 5650/R T) krm= 3.55 x 106exp(-9950/RT) krp = 1.430 x 106exp(-9020/RT) (k:)o = 1.07 x 107 exp(-5650/RT)

liter/mol sec liter/mol sec liter/mol sec liter/mol sec

(53)

For Wp> 0.2, a decrease in k: was observed and this decrease is given by

k: = (k:)o - (l69.6~ + 479.9W; + 1014.0W;)

(54)

At 60°C with a conversion of 60% in a single CSTR, BN= 4.86 long branches per polymer molecule. Increasing the number of equal-sized CSTRs in series so as to approach plug ftow (or batch reactor) behavior, one would obtain a BNof about 1.5 long branches per polymer molecule. In other words the long-chain branching frequency can be greatly reduced by increasing the number of CSTRs in a train. This is clearly the strategy that is used for the production of SBR and polychloroprene. Even with a CSTR train it is necessary to limit the conversion to keep branching at a low level. C.

Optima/ Reactor Type and Operation lor Continuous Emu/sion Po/ymerization

This section is a review of a recent paper by Nomura (1980) who has made some interesting suggestions about the design and operation of

9. Latex Reactor Principies: Design, Operation, and Control

339

continuous latex reactors with stable operation at steady-state conditions. The conventional CSTR train is operated with a11the recipe ingredients being fed into the first reactor and the product latex removed from the last reactor in the train. For a given production rate, the size and cost of the CSTR train can be reduced by maximizing the nuc1eation of polymer partic1es in reactor one. This can be done by increasing the emulsifier and initiator concentrations in the feed stream and by lowering the temperature of the first reactor. The use of higher emulsifier levels may have deleterious effects on the polymer product, however, Other techniques of maximizing the number of polymer partic1es in the first reactor are given. The optimal choice of mean residence time will give Npm.xas already discussed earlier in this chapter. Another approach to increasing the number of polymer partic1es early in the train is to use a plug flow type reactor (PFR) as reactor one. A larger number of polymer partic1es can be produced in a PFR than in a CSTR at the optimal value of (J.The volume of the PFR could then be reduced substantia11y by reducing monomer in feed and feeding rest of the monomer into reactor two. It is also c1aimed that a PFR as first reactor in the train would substantia11y reduce oscillations and thus increase the stability of operation of a CSTR train. These design and operation suggestions by Nomura are based on calculations performed with a steady-state model. More realistica11ythe design, operation, and control of a CSTR train of latex reactors should be based on calculations with a transient CSTR model. This subject will be considered in the fo11owingsections.

IV.

Continuous S~rred-Tank Reactors: Dynamic Behavior

A. lntroduction Continuous emulsion polymerization is one of the few chemical processes in which major design considerations require the use of dynamic or unsteady-state models of the process. This need arises because of important problems associated with sustained oscillations or limit cyc1es in conversion, partic1e'nlJmber and size, and molecular weight. These oscillations can occur in almost a11commercial continuous emulsion polymerization processes such as styrene (Brooks et al., 1978), styrene-butadiene and vinyl acetate (Greene et al., 1976; Kiparissides et al., 1980a), methyl methacrylate, and chloropene. In addition to the undesirable variations in the polymer and partic1e properties that will occur, these oscillations can lead to emulsifier concentrations too low to cover adequately the polymer partic1es, with the result that excessive agglomeration and fouling can occur. Furthermore, excursions to high conversions in polymer like vinyl acetate

340

A. E. Hamielec and J. F. MacGregor

and styrene-butadiene where transfer to polymer reactions are important can lead to excessive long-chain branching and thereby result in poor processability of the rubber. Although these oscillations can be avoided by operating at sufficiently high emulsifier concentrations, the concentrations needed are often high for most commercial processes. Even if, under the conditions used, a nonoscillatory steady-state exists, it is not unusual that during start-up or other disturbance transients an oscillation is induced which qamps out only very slowly (20 or more residence times). Furthermore, oné is never certain whether steady-state models actually apply in any given situation unless one either has experimental confirmation or verifies it by first solving a dynamic model.

B.

A Dynamic Model Based on Age Distribution Functions

In this section we consider a model for the continuous emulsion polymerization of vinyl acetate based on one presented by Kiparissides et al. (1978). Latex properties such as average particle size, number of particles, total particle surface area and volume, molecular weight averages, and average number of long-chain branches per molecule will be considered. The approach is similar to that used by Dickinson (1976) and focuses on the residence time or age distribution of particles in the reactor rather than on the size distribution. The latter approach (Min and Ray, 1974) has recently been used successfully by Min and Gostin (1979) to model the semibatch emulsion polymerization of poly(vinyl chloride)~ However this approach could lead to difficulties in modeling reactors where the growth rate of a particle is not only dependent on its current size and . the current conditions in the reactor but also on the conditions prevailing during its previous history. Using an age distribution approach if we define the function n(t, -r)d-r to be the number of particles in the reactor at time t that were born in the time interval (-r,-r+ d-r),and p(t, -r)tobe a property of the latex associated with this cIass of particles, then the total property P(t) = J~ p(t, -r)n(t,-r)d-rfor all particles in a CSTR reactor will be given by dP(t) -=--+ dt

P(t) ()

1 op(t, -r) ) ( t) + f( tpt -nt ( -r) d-r , ' 1o ot

(55)

where () is the reactor residence time and f(t) is the net rate of particle generation in the reactor. Using this equation, property balances for the total number of particles, the total diameter, the area and volume of

9.

341

Latex Reactor Principies: Design, Operation, and Control

partic1es,and the conversion can be written as follows (Kiparissides et al., 1978; Chiang and Thompson, 1979): dN(t) = - N(t) + f(t) dt ()

(56)

dD (t) --f¡-

(57)

D (t)

= ---t-

+ 2~(t)N(t) + dp(t,t)f(t)

dAp(t) + 4nW)Dp(t) + tIp(t,t)f(t) dt = - A~(t) (7 dVit) dt

=-

dX(t) -¡¡¡-

= (1 - e-I/6){

V~t) (7

(58)

(59)

+ ~(t)Ap(t) + vp(t,t)f(t)

1

- X(t) ()

+ Rp(t)

(60)

[MFJ }

The exponential term in the conversion equation arises from the assumed start-up condition of a reactor filled only with degased water and then fed at time zero with a stream having monomer concentration [MFJ. Assuming both micellar and homogeneous nuc1eation the partic1e generation ratef(t) can be written as (61) and following Fitch and Tsai (1971), the homogeneous rate constant is assumed to decrease with partic1e area as k.h= kho(1- ApL/4). Applying the stationary-state hypothesis to the radical balance gives the radical concentration in the wáter phase as (62) where the rate of appearance p(t) of free radicals in the water phase is given by

(63) Assuming that radical termination takes place exc1usively in the polymer partic1e,the average number of radicals in a partic1eof volume vp(t,-r)can be obtained as (Ugelstad and Hansen, 1976) q(t, -r) =

[

R

1 2kde(t, -r)n(t,-r)d-rJ

1/2

a (t -r) n(t -r) d-r

[

p"

Ap(t)

J

1/2

(64)

342

A. E. Hamielec and J. F. MacGregor

Using Eq. (14) the desorption rate constant can be expressed as kde(t, ,) = kge/ap(t, ,) which when substituted

into Eq. (64) yields

-

q(t,,)

-

-

R,

1/2

[

2kodeJ

1/2

a;(t,')

[

A p(t)

J

(65)

Substituting these i~to Eq. (63) and integrating gives p(t) = R,(t) + (!kgeR,)1/2(N(t)/A~/2(t))

(66)

The partic1e generation rate f(t) can now be evaluated using Eqs. (61), (62), and (66),in conjunction with the property balance equations (56-60). The rate of polymerization Rp [see Eq. (2)] can be expressed here as Rp(t) = (kp[Mp]/NA) LtJ(t,,)n(t,,) d, = (kp[Mp]/NA)(RJ2kge)1/2 A~/2(Ó

(67) (68)

The time function ~(t) in the property balance equations (57-59) is given by kpdm R, ~(t) = NAdp 2kge

1/2

( )

1

cI>m(t)

A~/2(t) 1 - cI>m(t)

(69)

where cI>m(t) is the monomer volume fraction in the polymer partic1es given earlier in Eq. (23). The property balances must be coupled with the balances for the initiator and emulsifier given, respectively, by d[1]/dt

= e-1([1F]- [1]) - R,

(70)

where R, = 2fkd[1], and

d[S]/dt = e-1([SF] - [S])

(71)

The micellar area Am(t) needed in Eqs. (61) and (62) can then be obtained as Am(t) = ([S]

-

[Scmc])as- Ap(t)

(72)

where as is the coverage area of 1 mole of emulsifier.

The set of simultaneous nonlinear differentialequations defined by Eqs. (56-60) and Eqs. (70)and (71)therefore provides a dynamic model for the total latex properties of PV Ac being produced in a single eSTRo This model has been shown to provide an excellent representation of the conversion-time experimental data of Greene et al. (1976) under conditions of sustained oscillations (see Fig. 7) and of the extensive data of Kiparissides et al. (1980a,b) under both steady-state and sustained-oscillation conditions

343

9. Latex Reactor Principies: Design, Operation, and Control

30

t520 ¡¡;

Q: ILI

~ 15 o



10 1]= 0.005

2

4

6 8 10 12 14 DIMENSIONLESS TIME (t / e)

16

18

20

Fig. 7. CSTR conversion transients. Comparison between experimental results of Greene et al. (1976) and model predictions. Model (---): Greene's experimental results (-0-0-).

using sodium lauryl sulfate as an emulsifier and potassium persulfate as initiator. Figure 8 illustrates the behavior of the number of particles, the particle area, and the free soap or micellar area predicted by the model for one of Greene's runs. The reasons for the s4stained oscillation phenomenon are quite apparent from these figures. In periods where kmAm or khO(1 - ApL/4) are greater than zero, rapid generation of particles occurs leading to a large surface area and the subsequent depletion of free emulsifier. Micellar particle nucleation therefore stops and, with the explosive increase in particle are a (Ap), so does homogeneous nucleation. There follows a long period, the duration of which depends on the emulsifier feedrate and residence time of the reactor, in which particles are not generated. However, as the washout of existing particles continues, the emulsifier concentration builds up again to exceed the CMC, and a new .generation of partÍcles is formed. This periodic nucleation leads to the formation of discrete particle populations with concomitant oscillations in polymerization rate, conversion, and latex properties. In other situations where the emulsifier feedrate to the reactor is sufficient to produce a steady-state concentration in the CSTR that is above the CMC, one observes damped oscillations upon start-up followed by an eventual attainment of a steady-state, nonoscillatory condition (Kiparissides

et al., 1980a,b).

344

A. E. Hamielec and J. F. MacGregor

2.0 re TQ .. ..... .. ...

= 1.5 ...... C/) UI

.J o ~

~

1.0

el. la. O a: UI m :!! 0.5 :) z .J

~ O

1-

O O

2

4

6 8 10 12 14 DIMENSIONLESSTIME (tIa)

16

18

Fig. 8. CSTR transients.Model predictions for number of particles, free emulsifier concentration and total particle surface area. 6 = 30min, 1 = 0.01molJliter,S = 0.01molJliter, MjW = 0.43.

C.

Cons;derat;on 01 Dynam;cs ;n Reactor Des;gn

These unsteady-state experiments and the dynamic models developed to explain the observed behavior ha ve a number of important implications for the design of continuous emulsion polymerization reactor trains. In order to avoid this oscillation phenomenon and the varying latex quality that results from it, one should not design continuous flow stirred-tank reactor. trains along previous lines, namely with a number of equal or nearly equalsized CSTRs in series with all the feed streams of the receipe entering the first reactor. Most continuous industrial systems designed in this manner exhibit oscillations in the early reactors of a magnitude comparable to those shown in Fig. 7. Although the oscillations may be largely damped out by the later reactors in the train certain inhomogeneities will exist in the latex. A common industrial practice used to avoid oscillations is to seed the first reactor with small-diameter seed particles produced earlier in batch reactors. By then keeping the emulsifier concentration below the CMC one avoids further micellar generation and simply grows the seed particles. Although in some cases this may introduce additional flexibility in that a

9.

Latex Reactor Principies: Design, Operation, and Control

345

cheaper or otherwise more desirable polymer may be used for the seed partic1es, it usually results in costlier operations resulting from the need for batch reactors to produce the seed partic1es. This oscillation phenomenon, however, can be avoided quite easily without the need for seeding. By employing as the first reactor in the train a small, optimally sized CSTR reactor and splitting the emulsifier, initiator, water, and monomer feeds to enter at various points along the train it is quite possible to ensure that all partic1e generation occurs in the first small reactor under nonoscillatory or steady-state conditions, followed only by growth of these partic1es in the subsequent reactors (Pollock, 1981; Nomura, 1980). As a simple illustration one could feed most of the emulsifier to the small first CSTR "seeding" reactor along with part of the initiator and part of the water and monomer, thereby ensuring continuous partic1e generation in condition of excessemulsifier. By feeding much of the remaining water and monomer to the second reactor, the emulsifier is diluted to a value below the CMC at which partic1e generation is avoided and partic1e growth only is promoted. A good dynamic model is the key to sizing the reactors determining an optimal split of the feed streams. As will be discussed later, the dynamic model is also the key to designing an on-line monitoring and control system that will ensure that the desired conditions of latex quality and quantity are maintained. It is also possible, and in some cases more desirable,to use a continuous

tubular reactor as the first "seeding" reactor (Nomura, 1980). Oscillations should not be a problem in such a reactor, but reactor fouling might be a more important consideration.

V. On-Line Control of Continuous Latex Reactors

A. Introduct;on As indicated above a number of fundamental control problems with cOQtinuous emulsion polymerization reactors are associated with the initial .

reactor systemdesign,and such problems should, therefore,be solved at the design stage. However, even with an efficient design some on-line monitoring and control are needed to ensure that the design conditions are indeed being maintained in the face of numerous disturbances which can affectthe system.In this section we first consider some of the on-line instrumentation and techniquesfor monitoring reactor condition, and then we look at some of the techniques that are being used fOr"the control of polymer latex properties in these reactors.

346 B.

A. E. Hamielec and J. F. MacGregor

On-Line Instrumentation

The most useful on-line instruments would be those capable of measUfing the latex properties given in Eqs. (56-60), the free soap concentration (Am), and some molecular weight characteristics. In this section we will not attempt on exhaustive survey, but will only concentrate on a few of the more rugged and more promising on-line measurements that have been used.

1. Reactor Heat Balances One of the earliest attempts at monitoring the progress of emulsion polymerization reactors was to perform an on-line enthalphy balance around the reactor or the reactor cooling jacket in order to monitor the heat release by reaction. Using temperature measurements in the reactor, in the cooling jacket entrance and exit, and at any ~ther essential points, together with relevant cooling water flows one can perform steady-state or unsteady-state enthalpy balances using an on-line minicomputer. From the computed rate of heat release, the rate of reaction can be evaluated, and the conversion in the reactor can be followed using Eq. (60). A major problem in successfully implementing this scheme is to overcome the effects of the numerous measurement errors that propagate into the ca1culated reaction rate and conversion. For any given situation a simple propagation of error analysis will reveal the precision possible. Usually, significant filtering of the measurements is necessary in order to obtain worthwhile results. Two approaches are possible, both requiring an on-line mini- or microcomputer. In the first approach instantaneous steadystate balances can be performed over short periods using averaged or filtered values of many measurements. Alternatively, less-frequent measurements can be utilized with the unsteady-state balance equations in the form of a Kalman Filter (Astrom, 1970; Jazwinski, 1970). 2. Density A number of fairly rugged on-line instruments are available to follow the emulsion density variations. Examples include nuclear instruments and instruments based on mechanicaloscillator techniques(Kratky et al., 1973). By utilizing the density difference between the unreacted monomer and the polymer (providing a reasonable difference exists) the reactor conversion ca,n then be ca1culated via

x = (Pe- p~)/(p:oo- p~)

,

9.

347

Latex Reactor Principies: Design, Operation, and Control

where the emulsion densities at O and 100% conversion (p~ and be approximated

P:OO)

may

by the weighted averages

o_

Pe

{' - 1.mPm + JwPw

100 _ Pe - 1.mPp

{'

+ JwPw

Here fm and fw are the feed stream weight fractions of monomer and water, and Pm, Pw, and Pp are the densities of, respectively, monomer, water, and polymer. Using the new on-line densitometers which rely on vibrational frequencies, a precision of approximately :t 0.5% on conversion can be reasonably attained in the polymerization of methyl methacrylate (Schork and Ray, 1980) or vinyl acetate (Pollock, 1981).

3. Turbidity Spectra Light transmission has been a standard method for the measurement of the size of spherical particles for many years. Mie and Rayleigh scattering theories show how the measured turbidity can be related to particle diameter and number. In the Mie regime (1 < nD/Am < 10) information on the number of particles and the size distribution could be obtained, particularly if used in conjunction with other instruments such as a densitometer (Maxim et al., 1969). In the Rayleigh regime (nD/Am< 1) only a turbidity-average particle diameter is easily obtained. From the point of view of on-line process measurements, turbidity instruments are rugged and have been demonstrated to follow easily the oscillations in average particle diameter in continuous emulsion polymerizations (Kiparissides et al., 1980a). For example, Fig. 9 shows a plot ofthe absorbance at 350 nm and the 0.4 o~ o o

10 0.31 al

/

0.8

,~ ~,,/

~ 0.6 (J)

15

eI '

ILI

~ 0.4

~ ~

=> :::!:

e'I I

a 0.2

I

/

,,

e,' ,

,I

" ,/ .,,,,,¿;€I

O

O

200

400

600

PARTlCLE

800

1000

DIAMETER

(Al

1200

1400

Fig. 1. Particle size distribution for a styrene latex (Poehlein and De Graff, 1971). T = 70oC: 0= 15min; (S)= 2.79g/lOOg H20; (/) = 0.35g/IOOg H20. ---, Dn ($-E Case 2) = 690 Á; -, Dn(De Graaf) = 700Á: O, Dn(Exper.) = 720Á.

Combining Eqs. (10), (8b), and (2) yields the following relationship for N (R¡fJNAfN) = 1 + (C

O

!2

-0.5

- 1.0

-4

-3

-2

-1 log 01

-

O

2

3

Fig. 4. Stockmayer's equation plotted as logn as a function of logO( (Ugelstad and Hansen, 1976). m = O; O(= Pav/Nk,.

10.

369

Emulsion Polymerization in Continuous Reactors 1.0

08

z 0.6 o ¡:: :o ID ¡¡: ....

'" o 0.4 "' > ¡:: O

(22)

This causes particle size distributions in the latex from a CSTR to be broader. Rate of polymerization is directly proportional to (ñN). Since ñ can decrease as N increases the rate of polymerization may not depend strongly on N. C.

Other Particle Formation Mechanisms

Smith and Ewart (1948) assumed that particles are formed when free radical s diffuse into monomer-swollen micelles. Roe (1968) demonstrated later that the S-E Model, in a mathematical sense, did not depend on the concept of a micelle. Roe referred simply to the stabilizing capability of the

10.

371

Emulsion Polymerization in Continuous Reactors

HYDROPHILlC

-S04'

!

+M

---

-S 04M

SURFACE ACTIVE

--

I I +M

t

,-

-S04Mn

HYDROPHOBIC I HOMOGENOUS I+M NUCLEATION

ADSORPTION ONEXISTING R\RTICLES OR MONOMERDROPS CAPTUREBY MICELLES

so; FLOCCULA

FLOCCULATION OF PRIMARY R\RTICLES

TION

ONTO MATURE R\RTICLES

+MaS POLYMERIZATION GROWTH

CONTINUED GROWTH BY

POLYMERIZATION I FLOCCULATION, OR

Fig.7.

BOTH

Paths for free radicals initiated

in the aqueous

phase.

system or the free emulsifier concentration. We know that particles can be formed by a number of mechanisms. Figurs: 7 shows some potential ways in which a sulfate ion-radical might become associated with a latex particle. The rate at which 'various mechanisms proceed depends on a number of factors, including those listed below: 1. Monomer solubility in water. 2. Propagation rate of the polymerization reaction in the aqueous and organic phases. 3. Size and concentration ofmonomer drops. 4. Size and concentration of micelles if any are present. 5. Size and 'concentration of polymer particles. 6. Solubility of the growing polymer oligomers in the water phase. 7. Concentration of free emulsifiers in the water phase. 8. Concentration of electrolyte in the water phase. The monomer solubility in the water phase, discounting that located in any swollen micelles, will influence the rate at which the oligomer chains grow. Monomer solubility will, of course, also be related to oligomer solubility, or more precisely to how long the oligomer chains might grow

372

Gary W. Poehlein TABLE

11

Solubility of Monomer in Water and Polymer Concentration in (M):

Monomer

Water

Styren"e Butadiene Vinylidene chloride Vinyl chloride Methyl methacrylate Vinyl acetate Ethylene Acrylonitrile Acrolein

0.005 0.015 0.066 0.11 0.15 0.3 0.3-0.6 1.75 3.1

Polymer partic1es

5.4 6.5 . 1.1 6.0 7.0 7.6 5.0

before they precipitate. Table 11gives monomer solubilities in water and in the polymer particles for several common monomers. It should be noted that in copolymer systems the oligomers formed in the water phase would be expected to have a higher concentration of the more soluble monomer if the reaction rates are of the same order. The size and number of monomer droplets are usually ignored in emulsion polymerization kinetic stud~es. The basis for this comes from several observations. First, in CIassical recipes the monomer is located in a relatively few, large monomer droplets that represent a small total surface area when compared to the micelles or free emulsifier. Second, large particles are not usually seen in electronmicrographs of latex particles. Ir one or two are found they can be conveniently forgotten. Ugelstad et al. (1973) cIearly demonstrated that if the monomer droplets can be made small enough, they can efTectivelycompete for free radicals and form particles. Considerable work has been published (see Ugelstad and Hansen, 1976, for other references) on the preparation of finely dispersed monomer droplets for the formation of latex particles by direct polymerization. Recently,Durbin et al. (1979)have shown that a few large particles are formed by monomer droplet polymerization in cIassical styrene recipes. Their work showed that the number of oversized particles increased with increased intensity of preemulsification. Thus, in some circumstances the installation of preemulsification systems may be counterproductive. The size and number of micelles and particles will influence the path taken by free radicals in the water phase. As sizes andjor numbers increase the probability for absorption increases.

10. Emulsion Polymerization in Continuous Reactors

373

PRIMARY PARTICLE Fig. 8.

Schematic representation of the growth and precipitation of an oligomeric

radical (Fitch and Tsai, 1971).

The vertical path shown in Fig. 7 iIIustrates a rnechanisrn of particle formation which Fitch and Tsai (1971) have called "hornogeneous nucleation." Figure 8 shows this process of diffusion and addition of rnonorner units schernatically. As the oligorneric radical grows it becornes strongly hydrophobic. In this state, it rnay precipitate to forrn a prirnary particle or deposit on an existing particle, micelle, or rnonorner drop. Estirnates of the degree of polymerization necessary to cause precipitation have been given as 54 for vinyl acetate (Priest, 1952),65 for methyl rnethacrylate (Fitch and Tsai, 1971), and 30 for styrene (Peppard, 1974). Another rnechanisrn for particle forrnation involves the scheme shown in Fig.7, but it oCGurslater in sorne batch polyrnerizations. This secondary nucleation is caused by free ernulsifier which is libenited from the particle surface. Figure 9 shows sorne data of Gerrens for a batch ernulsion polyrnerization of MMA. These data were successfully modeled by Ray and Min (1976). The population of srnaller particles was forrned late in the reaction because ernulsifier was desorped beca use of the crowding of radical end groups on the surface and because of particle shrinkage caused by . rnonorner conversion to polyrner.

374

Gary W. Poehlein .30

'"

Q

"

.20

>1¡¡j Z LrJ o .J ~ ~ a: o z

.10.

o

O

40

80

DIAMETER

120 (nm)

Fig. 9. Experimental and predictedparticle sizedistribution for a MMA latex (Ray and Min, 1976).0,Gerren'sdata: -, Model prediction (97.8%conversion).

D.

Experimental Results

The model based on S-E Case 2 kinetics has been quite successful in handling steady-state data for styrene emulsion polymerizations in a CSTR. One or more of the mechanisms described above, however, generally cause other monomer systems to deviate from this simple model. The nature of these deviations varies among the different monomers. Ir published literature data are fitted to equations of the type listed below one can obtain values for the exponents a, b, and c. Rp = kRRf[S]bOc

(23)

= kNRf[S]bOC

(24)

N

Table III gives the results of this fitting for styrene, methyl acrylate, methyl methacrylate, vinyl acetate, vinyl chloride, and ethylene. Only the styrene data agree with the simple theory. The exponents on R¡ are usually greater than 0.0 (the theory prediction). This can be explained by radical desorption from the particles, by slow t.erminatiop in the particles, or by a combination of these factors. Data on the other variables ([S] and O)are incomplete but scattered results show significant deviations from the S-E Case 2 Model. More complete models, such as those presented in this book by Hamielec, have been successful with nonstyrene monomers.

10.

375

Emulsion Polymerization in Continuous Reactors TABLE

Steady-State

111

Behavior of CSTRs Approximate exponents

a Theory (Rp and N) Styrene (Rp and N) MA(N) MA(Rp) MMA (Rp) VA (Rp) VC(Rp) Ethylene (Rp)

IV. A.

0.0 0.0 0.0 0.65 0.8 0.8 0.5 or less 0.5

b

e

1.0 1.0 0.85 0.0 0.9 SmaIl ? 0-0.3

-0.67 -0.67 -0.67 +0.43 ? 0.0 SmaIl Small

Transient Behavior of CSTR Systems Experimental Observations

An understanding of the transient behavior. of continuous reactors is important for start-up and reactor control considerations. Continuous oscillations have been observed by a number of workers. Figures 10 and 11 show data for styrene and methyl methacrylate. Gerrens and Ley (1974) reported continuous, undamped oscillations in surface tension during a styrene emulsion polymerization run which lasted for more than 50 mean residence times. Nearly five complete cycles were observed during this run. Berens (1974) conducted experiments with vinyl chloride in which the measured particIe size changed with time. No steady state was achieved with the data shown in Fig. 12.

B.

Physicochemical Mechanisms

The reason for oscillations in conversion and surface tension become clear when one considers particle formation and growth phenomena. If a single CSTR is started empty or by adding initiator to a full vessel of inactive emulsion, a conversion overshoot occurs. The first free radicals generated are almost entirely utilized to form new particles. Since these particles do not grow rapidly to the steady-state size distribution, radical

376

Gary W. Poehlein 70 60

STYRENE 40 z O Cñ a: 30 UJ

> Z O u

-

-~ - -

-

-~

80

120 RUN

Fig. 10.

Conversion transients

160 TIME

at

start-up

200

7O'C

_STAGE2 - ~ 7O'C

¿.: STAGEI .100'C

20

10 O

StAGE 3

240

280

{min} for

styrene

emulsion

polymerization

(Gershfield and Longfield, 1961).

50

40

30

z 020 Cñ a: UJ >

~

U

10 METHYL

METHACRYLATE

O O

2

4

Fig. 11.

TIME

10

12

14

(t le)

MMA conversion transíent (Greene, Gonzalez, and Poehlein, 1976). T

40°C: IJ= 20 min: (NaHC03)

8

6

DIMENSIONLESS

(S) = 0.01molj\iter;

= O.OL (NaCI)

= 0.02.

(1) = 0.01molfliter; Monomer:H20

=

= 0.43;

377

10. Emulsion Polymerization in Continuous Reactors 30 i 10

/

-,

í',

,1 \

"'

3

"

\

"

-,

¡

,

'1

/

I

/

\ \ \

\,

,

z

\

I

\,

" i, I

::>

\

i I

\'

11 11

Q: 1.0 w ID

\

I I I

"-

/

1Z w rg. The initial driving force for the diffusion of Zt from b to a droplets expressed as (l:1GlbO - l:1G1ao)is then given by l:1G tbO

-

l:1G taO

= 2yV1M(l/rg-

l/raO)

(7)

As it is assumed that Z2 is almost completely insoluble in the continuous phase, the transport of Zl from the smaller to the larger droplets leads to an increase in the volume fraction CP2bin the b droplets and a corresponding decrease in CP2a'A concentration potential working in the direction opposite the interfacial free energy difference in Eq. (7) will thus build up. At equilibrium, the activity of component i is the same in the a and b droplets, and furthermore equal to the activity in the continuous phase. The semiequilibrium distribution of component i is therefore determined by l:1Gra

+ 2YV¡Mlra = l:1Grb + 2YV¡Mlrb= RTlnaiw

(8)

where the values of the interfacial tension and radii are those at equilibrium (see note on p. 413). The index w is used to denote the continuous phase, most commonly water. It should however be noted that it mayas well be a mixture of water and nonaqueous compounds or pure nonaqueous solvents. The partial molar free energy differences may be obtained from

11. The Formation of Monomer Emulsions and Polymer Dispersions

Flory-Huggins expression. Substituting semiequilibrium state discussed above In(4)la/4>lb)+ (1 - J¡jJ2)(4)2a

-

387

in Eq. (8) one obtains for the

4>2b)

+ 2V1My(1/ra - 1/rb)/RT = O

+ (4)1a -

4>1b)x12

(9)

where X12 = J1X~2is the interaction parameter per mole of Zl' Calculations based on Eq. (9) show that even minor quantities (-1%) of Z2 may prevent degradation by diffusion to the extent that the size of the resulting emulsion droplets deviate only slightly from their "instantaneous" values. As Eq. (9) describes a semiequilibrium situation, the emulsion is subject to further degradation by diffusion. The rate of this "secondary" degradation process is however determined by the rate of transport of Z2 from b to a droplets. For many practical purposes, this rate may be almost negligible. In systems that are stabilized toward coalescence by means of conventional emulsifiers, the capability of small amounts of Z2 to prevent degradation by diffusion has been demonstrated by several authors (Higuchi and Misra, 1962; Hallworth and Carless, 1974; Davies and Smith, 1974). It has also been demonstrated that the stabilizing effect of different compounds is independent of the chemical structure and determined only by those compound's solubility in the continuous phase (U gelstad et al., 1980a). Thus, the effect of straight- and branched-chain alkanes was found to be the same for compounds with equal solubility. Long-chain alcohols also stabilize aqueous emulsions toward degradation by diffusion to an extent determined mainly by their water sólubility. This observation seems to contradict the suggestion by Hallworth and Carless (1974) that the formation of a condensed layer of emulsifier and fatty alcohol at the droplet surface plays a significant stabilizing role. The effect of relatively water-insoluble additives has been utilized by Ugelstad (1978) and Ugelstad et al. (1978a,b, 1979a, 1980a,b) for the preparation of emulsions by diffusion. The main principIe of these methods is that particles that consist wholly or partly of low molecular weight, insoluble Z2 are .capable of absorbing much larger quantities of Zl than are pure polymer partitles of comparable size. In practice, tbe preparation of an emulsion by diffusional swelling is carried out by first producing an aqueous dispersion of the water-insoluble compounds. This dispersed phase may consist either of pure Z2 or of polymer particles into which Z2 has been introduced in a first step. During the subsequent swelling of these preformed dispersions with a slightly water-soluble Zl, conditions are such that the only transport process possible is the diffusion of Zl through the aqueous phase to become absorbed into the preformed droplets or particles.

388

J. Ugelstad el al.

Any diffusion of Z2 from the particles to the original phase of Z1 is hindered by the low solubility of Z2' The swelling capacity of such particles may be calculated from Eq. (8). Since in this case Zl is present as apure phase, the appropriate equation describing the semiequilibrium state is In cPla+ (1 - J¡jJ2)cP2a+ (1 - J¡jJ3)cP3a+ cPiaX12+ cPiaX13 + (X12 + X13 - X23J¡jJ2)cP2acP3a + 2V1My(1/ra

-

l/rb)/RT

=O

(10)

where raand rb are the radii of the swollen particles and the droplets of Zl at equilibrium. In case no polymer is present, the terms '1Vithindex 3 should be omitted. Figure 1 shows the volume (V1)of Z1 which, according to Eq. (10), may be absorbed per unit volume of polymer (V3)+ Z2 (V2)for different values of J2fJ1 as a function of y/ro. Here, y is the interfacial tension at equilibrium and ro is the radius of the polymer-oligomer particles prior to swelling with Zl' The values of the interaction parameters are arbitrarily chosen to be X12 = X13 = 0.5 and X23 = O. Furthermore, l/rb is considered negligible when compared to l/ra.

o 3

5

4 Lag I ~/ro)

6

(nm-2)

Fig. 1. Swelling capacity of polyrner-oligomer partic1es as a function of 'Ilrofor various values of J2fJ. as ca1culated fram Eq. (10): ro = radius of polymer-oligomer partic1e prior to swelling with ZI, V2 = V3= 0.5, X12= XI3= 0.5, X23= O, V.M= 10-4 m3/mol, T = 323 K.

11. The Formation of Monomer Emulsions and Polymer Dispersions

389

Figure 1 reveals that the swelling capacity of polymer particles containing 50% of a relatively low molecular weight, water-insoluble compound is drastically increased compared to the case with pure polymer particles of the same size (J2fJI = 00). It also appears that the swelling capacity of

polymer-Z2 particles is far more dependent on the value of y/re than is the case for pure polymer particles. This has been verified experimentally by Ugelstad et al. (1978b).Calculations of the swellingof pure droplets of Z2 with ZI give similar curves (Ugelstad et al., 1980a). Figure 2 illustrates the swelling capacity dependence on V2 at a constant value of J2fJI = 5. It appears that even low amounts of Z2 should give a substantial increasein swellingcapacity at low values of y/re. This effecthas been verified experimentally (Ugeistad et al., 1979a). In the course of the swelling process, LlGlasteadily increases. This means that

the driving

force, LlGlb

-

LlGla

--

LlGla (in case rb =

00), steadily

decreases. Figure 3 gives LlGla/RT as a function of the swelling ratio V¡/(V2+ V3)for different values of y/re. It is seen that LlGla/RT is close to zero even at values of VI that are far from the equilibrium value. This means 6 5

l

V¡:1.0 0.67 0.50

0.20

f'r 3

0.10 0.05 0.02

o ..J

21

0.01 O

o 3

t.

5

6

Log(~/ro) Fig. 2. Swellingcapacity of polymer-oligomerparticles versus y/ro for different values of v2: ro = radius of polymer-oligomer particles prior to swelling with Zt> V2+ V3 = 1, J2/J, = 5, X12= X13= 0.5, X23= O, V1M= 10-4 m3/mol, T = 323 K.

390

J. Ugelstad et al.

0.002 0.000

- 0.002 - O.OO~

1~:;:-0.006 -0.008

I

¡

-0.010 -0.012 -0.014 10

1000

10,000

VI/IV2>V3)

Fig. 3. Partialmolar free energy of ZI in swelling partic1es versus swelling ratio, for various values of "t/ro: XI2 = XI3 = 0.5, X23= O, J2/J1= 5, V2= 2/3, V3= 1/3, V1M= 10-4 m3/mol,T= 323K.

that the maximum obtainable swelling is very sensitive to factors that would cause a deviation in the entropy of mixing from that given by the Flory-Huggins expressions. At equilibrium the activity of Zl in the continuous phase is equal to its activity in the a and b droplets. Ir an excess of pure Zl is present and if the radius of the b droplets is considered to be infinite, as in the case discussed above, this activity is equal to unity. The swelling equilibrium is in this case the same as the one that would be obtained for swollen particles in equilibrium with a saturated solution of Zl in the continuous phase. Ir Zl is present .as droplets of finite size, its activity at equilibrium (equal in all phases) is larger than unity because of the interfacial free energy of the b droplets. This in turn may lead to an increase in the degree of swelling. Ir the b droplets are smaller than the a droplets, or become smaller during the swelling, one may in principie obtain "infinite" swelling of the a droplets. Finally, it may be desirable to emphasize some points regarding the general application of Eq. (8) for calculations of swelling and equilibrium distribution of Zl between the various phases. 1. Ir one operates with an excess of Zl, the swelling of each phase may be treated independently of the other phases. Ir there is a shortage of Zl,

11. The Formation of Monomer Emulsions and Polymer Dispersions

391

several cases of competitive swelling may be encountered in practice. They may easily be dealt with by applying the correct expression for dGl to the various phases. A typical example is the swelling of polydisperse droplets of Z2' The final distribution is determined by the initial droplet sizes only. The swelling of a mixture of monodisperse droplets of various Z2 cO'nstitutes another example. In this case the final size distribution will be determined by the value of J2 for the various compounds. A third possibility is the swelling of a system consisting of two types of monodisperse polymer particIes containing different amounts of the same Z2' The appropriate equilibrium equation in this case is In

13/4>13 (V2= O)(Le.,the ratio of the volume fractions of monomer in the partic1es with and without addition of Z2 to the monomer phase) for various values of vNV30,the initial.ratio of monomer to polymer. In order to illustrate the etfects one may expect and the problems in interpretation of experimental results, some ca1culationsshowing the etfect of addition of Z2 to the monomer phase have been carried out using Eq. (19). In Figures 8 and 9, 4>13/4>13 (V2= O)is given as a function of VIO/V30for various values of 4>fb = V2/(V2+ VIO)and V2, respectively.In these ca1culations, the radius of the polymer partic1es is assumed to be r~ = 5 x 10- 8 m

400

J. Ugelstad el al.

contains a given amount of polymer. The minimum droplet size corresponds to the sweIling capacity of a particIe consisting of this amount of polymer. Therefore, the minimum droplet size wiII increase with increasing amount of solvent (21), If a smaIl amount of 22 is added before homogenization, the droplets formed contain a smaIl amount of 22 in addition to 23' The sweIling capacity of a particIe in which even a smaIl amount of 23 is replaced by 22 is considerably higher than that of pure 23' Therefore, in the presence of 22 the minimum diameter that may be obtained with a given amount of solvent (21) may be drasticaIly reduced. Calculation of this minimum diameter may be carried out by applying Eg. (lO). An example of such a caIculation is given in Table 1. It appears from the table 'that the minimum diameter increases with increasing amounts of 21, It also appears that even with smaIl amounts of 22 it is possible to obtain smaIl particIes. This is also the case where without 22 one gets phase separation even at infinite diameters of the particIes. During homogenization of a solution of polymer (23), solvent (21), and water, droplets of different sizes are formed. Because of the interfacial energy difference, a transport of 21 from the smaIler to the larger droplets wiII take place. If droplets smaIler than the critical size are formed, these droplets, especiaIly, wiII rapidly lose 21 to the surrounding larger droplets. The viscosity within these smaIl droplets wiII increase and a further degradation may be mechanicaIly hindered. In the presence of 22, two effects that wiIIfacilitate subdivision of the emulsion may be encountered. 1. The degree of diffusion between smaIl and large droplets wiII be reduced because the presence of 22 means that a counteracting concentration potential is more rapidly established. TABLEI The Minimum Diameter That May Be Obtained by Homogenization of a Polymer Dissolved in a Solvent (Z1) as a function of Volume V1 of the Solvent and Volume V2 of a Z2a.b X = 0.3

x = 0.5

x = 0.6

>\'

2

5

10

2

5

10

2

O 0.01 0.02 0.10

21 20 19 16

110 103 96 66

415 370 334 190

48 44 40 24

454 338 269 105

2980 1500 1005 280

147 108 85 32

a

From Ugelstad

b X12

5

10

2545 3 x 105 146 356

(1978) with permission.

= O, XI3= X23= X, V2+ V3= 1, J2/11= 5, VIM= 10-4 m3/mol, y = SX10-3 N/m,

T= 300 K.

11. The Formation of Monomer Emulsions and Polymer Dispersions

401

2. The presence of Z2 tends to decrease the critical minimum size. In practice this means that the formation of droplets with a higher viscosity, due to transport of ZI from small to large droplets, is reduced. The presence of Z2 makes possible the formation of small droplets with relatively high amounts of ZI, and therefore low viscosity, without getting below the critical value of the droplet size.

v.

Effect of Addition ofWater-Insoluble Compoundsto the MonomerPhase

If in an ordinary emulsion polymerization, a water-insoluble Z2 is added to the monomer phase, the effect will obviously be to decrease the activity of the monomer in this phase and accordingly to decrease the concentration of the monomer in the partic1es (Azad et al., 1980; Ugelstad et al., 1980b,c). The appropriate form of the equilibrium equation for the case in which one has ZI equilibrated between Z3 partic1es and monomer droplets containing Z2 will be In I/Jla

+ 1/J3a +

I/Jfax13+ 2VIMy/raRT

= In I/Jlb+ (1 - J¡jJ2)1/J2b+ l/JibX12+ 2VIMy/rbRT

(19)

where index a and b refer to polymer partic1es and monomer droplets, respectively, and J3 is set equal to infinity. If the monomer droplet radius (rb) is very large the last term may be neglected. It appears that if y/ra is assumed to be constant, any equilibrium value-of 1/J2bwill correspond to a value of I/Jla that is independent of the initial monomer-polymer ratio. Such a situation was considered by Azad et al. (1980) in their discussion of the effect of additives to the monomer phase on the degree of swelling and on the kinetics of polymerization. A more relevant approach to this problem might be to consider the effect of a given amount of Z2, V2, or a given initial volume fraction of Z2, I/J~b'in the monomer phase on I/JIJI/JIa (V2 = O) (Le., the ratio of the volume fractions of monomer in the partic1es with and without addition of Z2 to the monomer phase) for various values of vNv30, the initiaf ratio ofmonomer to polymer. In order to ilIustrate the effects one may expect and the problems in interpretation of experimental results, some ca1culations showing the effect of addition of Z2 to the monomer phase ha ve been carried out using Eq. (19). In Figures 8 and 9, I/JlaNla (V2= O) is given as a function of VI°IV30for various values of I/Jfb = V2/(V2 + VIO)and V2, respectively. In these ca1culations, the radius of the polymer partic1es is assumed to be r~ = 5 x 10-8 m

402

J. Ugelstad et al.

1.0 0.9 0.8

-o

0.7

N >

0.6

"

o 02b

" 61 ..... !! 61

=

0.2 o 02b =0.3

0.5

0.1,

0.3

0.2

0.1

1.0

2.0

5.0

3.0 VI/V3

Fig. 8.

Ratio of volume fraction of ZI in a partic1es to the same volume fraction when

=

V2 O versus initial ratio of ZI to Z3 for various values of the initial volume fraction of Z2 in the b phase: r~ = 5 x 10-8 m, X12 = X13 = 0.5, V1M= 10-4 m3/mol, V2M= 3 X 10-4 m3jmol,

y = 5 mNjm,J2jJ¡ = 3, T = 323K.

and the monomer droplet radius is set equal to infinity. The values of X12 and X13are arbitrarily chosen to be 0.5 and the molar volumes of 21 and 22 are set equal to 10-4 and 3 x 10-4 m3/mol, respectively. The interfacial tension at equilibrium is assumed to be 5 mN/m and T = 323 K. It would

appear

from Fig. 8 that

o. el

100

100 Fíg. 12. chloride

200 TIME (min)

300

Conversions versus time curves for seeded emulsion polymerization of vinyl

with hexadecane

present

in the monomer

H20. Seed: 3.6x 1016 particles/dm3 H20, hexadecane.

phase.

[K2S20SJ

= 2.7

x 10-3 moljdm3

T= 323 K. A: without hexadecane: B: 20%

reduced by a factor that decreases as the particle radius increases. In the present example (r~ = 0.05 Jlm) this factor is initially 0.27. The relevant equation for calculating the curves in Fig. 13 is obtained from Eq. (19) by inserting the following relationships r.

= raO[(V3o + V1s)fV30] 1/3

Vs = V30(3r;c5

- 3rac52 + c53)/(r~)3

= V1sIVs

c: o Ü

50

eQ) (,)

...

,f

25

Fig. 1. Typical styrene conversion curve at 30°C (dilatometer trace). Dose rate 0.02 Mradjhr, 25%solids, 6.7%sodium lauryl sulfateon total volume.

Recently, Garreau et al. (1979) reported a careful and rather detailed study of the kinetics of the radiation-induced polymerization of styrene in emulsion with sodium lauryl sulfate under conditions found earlier to lead to c10se agreement with simple Case 2 Smith-Ewart kinetics (Smith, 1948). Most of the normal reaction variables were studied, and the rates of polymerization were found to be independent of the monomer-to-water TABLE I Chemical and Radiation Polymerization

of Styrene in Emulsion Preparation y-Radiation

Property pH Surface tension (dyne/cm) Particle size (J¡m) Molecular weight (viscosity)

K2S20S at 600C. 3.8 61.1 0.13 2,700,000

9.2 68.9 0.10 1,664,000

7.9 69.0 0.07 413,000

.25% solids and 6.7% sodium lauryl sulfate based on water content

(Araki el al., 1969).

422

Vivian T. Stannett B

24.... 22 20 18L

A

16 14 e

:o

12

10 o Cñ 8 6,

:1

O

250

-Diameter

2000

(,.\.)

Fig. 2 Differential particle size distribution of polystyrene lattices initiated by: A, radiation O°C; B, radiation 600C; C, potassium persulfate 60°C (Araki et al., 1967).

ratio and to the 0.4 power of the dose rate (Fig. 3). The activation energy (Fig.4) associated both with the rate and the molecular weight was 7.9 :t 0.6, in good agreement with Van der Hoff et al. (1958) and with the literature values for Ekp' This is reasonable since little or no activation energy is involved in the initiation and termination steps. The dependence of the rate and the molecular weight on the emulsifier concentration was about 0.7, in reasonable agreement with the Smith-Ewart value of 0.6. The number of particles, however, was of the correct order of 0.6 with respect to the emulsifier concentration. The termination rate constants were caIculated using the method of Van der Hoff (1958). The values found were in excellent agreement with those of Van der Hoff for persulfate initiation and Bradford et al. (1961) for y initiation. The propagation rate constant kp was caIculated from Case 2 Smith-Ewart kinetics and found to vary between 32 and

68 M- 1 sec- 1, well within the range of the reported literature values at 30°C. The caIculated efficiencies of initiation were low, about 40%, but similar to those discussed by Bradford et al. with y irradiation and determined by Van der Hoff (1958) with persulfate initiation. No seeded polymerizations were conducted but the above results indicate clearly that the

423

12. Radiation-Induced Emulsion Polymerization ~ .s::.

~

90 80 70 60