Structure and Properties of Polymers

1 Model Polymers for Materials Science Lewis J. Fetters Exxon Research and Engineering Co., Corporate Research Science L...

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1 Model Polymers for Materials Science Lewis J. Fetters Exxon Research and Engineering Co., Corporate Research Science Laboratory, Clinton Township, Annandale, NJ, U.S.A. Edwin L. Thomas Massachusetts Institute of Technology, Department of Materials Science and Engineering, Cambridge, MA, U.S.A.

List of Symbols and Abbreviations 1.1 Introduction 1.2 Polymer Synthesis 1.3 Living Polymers 1.4 Model Polymer Requirements 1.5 Functionalization of Model Polymers 1.6 Model Branched Polymers 1.7 Processing 1.8 Model Polymers in Polymer Physics 1.9 References

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. All rights reserved.

2 3 4 5 7 16 18 19 22 29

2

1 Model Polymers for Materials Science

List of Symbols and Abbreviations DPn c dt / GN Mt Mn Mw Mz Mv Nt R Tg wt

number-average degree of polymerization concentration of polymer, g/cm3 tube diameter contour length plateau modulus molecular weight of species i number average molecular weight weight average molecular weight Z average molecular weight viscosity average molecular weight number of moles of species i end-to-end distance of model polymer glass transition temperature weight fraction of species i

[r\\ Q

intrinsic viscosity, cm3/g density of polymer melt, g/cm3

FTIR GPC LC LCP NMR NSE RIS ROMP SANS SAXS SEC TEM

Fourier transform infrared spectroscopy gel permeation chromatography liquid crystal liquid crystal polymer nuclear magnetic resonance neutron spin echo rotational isomeric state ring-opening metathesis polymerization small angle neutron scattering small angle X-ray scattering size exclusion chromatography transmission electron microscopy

1.1 Introduction

1.1 Introduction Materials scientists like to make their measurements on large, pure, single crystal specimens. Such samples afford the opportunity to explore the full anisotropy of physical properties with crystallographic direction and thus more completely characterize structure-property relations. Indeed, often it is the spectacularly different measures of a given property with direction that make a material interesting. Man's first recognition that polymers are useful materials likely derived from the naturally occurring polymers in his environment - the cellulose in wood for housing and plant and animal fibers for his clothing. From the earliest days of polymer science, the novel and diverse behavior of the newly synthesized materials was immediately recognized - for example, after the discovery of the polymerization of nylon, researchers realized that a wide variety of properties were available from a single polymer by careful manipulation of processing conditions. Thus, nylon could be spun into a family of fibers with properties conveniently controlled by extent of draw, draw temperature, post heat setting treatment, etc. The wide range in polymeric behavior stems from the enormous spectrum of structural states that polymers can assume: solutions, melts, glasses, crystals, liquid crystals, gels and rubbers. This in turn stems from the chemical diversity of the macromolecules. Moreover, because of the tremendous conformational diversity of a polymer, a given polymer can yield a huge variety of structures! It is hard to imagine a material replacing rubber's unique physical properties, which essentially derive from conformational diversity (see Chaps. 2, 8, and 9).

In addition to technically relevant properties such as reversible deformation of several hundred per cent (rubber elasticity) and Young's moduli 1.5 times that of steel (rigid rod fibers), polymer molecules offer the scientist a fertile area for basic study. From flexible chains to rigid rods, scientists have been seeking to understand polymeric behavior. This task is very challenging since polymeric materials are highly complex. Polymers are complex because synthesis often does not result in simple, single component species, frequently their organization in the solid state takes place on several length scales, and the relatively long time scales required to reach equilibrium are frequently much longer than experimentally feasible. While a good deal of the complexity of polymers is unavoidable, the difficulties arising from their synthetic complexities has been reasonably addressed via the production of model materials by the modern synthetic polymer chemist. By model material, we mean a species for which the parameters of molecular characterization are very well-represented by the ensemble average. That is, every molecule is essentially identical to every other (monodisperse ideal population) or the collection of molecules corresponds to a valid statistical assembly of species arising from a given stochastic process (ideal random population). Experiments conducted on model materials benefit from the specificity of response to the molecular variables and allow much more precise interpretation and testing of structure-property relationships. Over the past 50 years, polymer chemists have made important advances in the control of macromolecular chain structure. Initially, most polymers were produced by polymerizations in which the chemical linking events were more or less randomly


selected from a large number of possibilities, hence making the product macromolecules highly complex species. The chemist's prime goals were high yields and high molecular weight. This fact of dispersity in composition, in stereochemistry, in sequence distribution and in molecular length meant that the resultant material could only be characterized by average molecular properties. And as became evident, different physical probes were sensitive to different averages or different aspects of the molecular distribution. For example, polydispersity causes the progression of molecular weight averages M z > M w > Mn > M n where Mz is the Z-average molecular weight (size exclusion chromatography), M w is weight average molecular weight (light scattering), Mn is the viscosity average molecular weight (viscosity) and M n is the number average molecular weight (osmometry) (for characterization methods, see Vol. 2 A, Chap. 1). Given that the polymer materials had fairly broad distributions, early workers perforce satisfied themselves with more qualitative measurements than were possible with pure materials composed of a collection of identical macromolecules. Quantitative materials science (akin to that currently practiced in, say, the field of semiconductors) is just beginning to take place for polymers. Careful, precise characterization of highly specific behavior is replacing the former measurements, which were almost always significantly broadened by the spread in the molecular response stemming from the collection of diverse species in a given "pure" sample. The availability of these model materials has facilitated studies of, for example, polymer dynamics, adsorption, block copolymer morphology, diffusion (both tracer and self-diffusion), rheology, unperturbed chain dimensions and their temperature coefficients via small

angle neutron scattering (SANS), which permits the evaluation of those parameters in the realistic environment of the polymer melt. The SANS studies have been made possible by the use of deuterium labelling, either in the form of a perdeutero monomer or by saturation of a hydrogenous polydiene by deuterium. As we approach the 21st century, three activities are especially important in the changeover from such statistically complex materials of the past to the new, precisely defined, materials of the future. Activity in novel synthetic methods, including recombinant gene technology, promises synthesis of uniquely specific macromolecules. Secondly, advances in instrumentation to accurately measure molecular properties of these substances are continually being made. Thirdly, the significant growth of computational power to model, ever more realistically, such molecules provides good prospects for future understanding of structure-property relationships (see Chap. 2). All three of these endeavors, plus the merger of ideas and techniques of the polymer scientist with the molecular biologist and the closer coupling of the polymer chemist with the materials scientist, will contribute in major ways to the success of the "materials science of macromolecules".

1.2 Polymer Synthesis The preparation of homopolymers, block copolymers and functionalized polymers is of both academic and commercial value. In this regard, anionic polymerization (living polymerization) is uniquely positioned since it alone, to date, has the capacity to prepare the above families of model macromolecules in quantitative yield. This is due to the absence, in many cases, of a spontaneous termination event,

1.3 Living Polymers

the consequence of which also allows the preparation of chains of various architecture: for example, linear, graft, H-shaped, rings and stars (see Fig. 1-1). The latter class of materials contains regular armed homo- and block copolymer stars, asymmetric stars, and most recently, mikta (mixed)-arm stars, which are either A2B or ABC type branched stars where the various homopolymer arms radiate from a common junction point (Iatraou and Hadjichristidis, 1992).

1.3 Living Polymers The so-called living anionic polymerization systems are obtainable, in the strict sense, by the use of organolithium initiators and hydrocarbon solvents under conditions that exclude oxygen and protic impurities. Under the proper conditions, initiation can be rapid and near-mono-

Table 1-1. SEC heterogeneity evaluations for low molecular weight polybutadienes. Mwxl0~3

(gmor1) 1.00 1,05 1.05 1.15 2.00 3.60 4.30 5.90 6.50 13.0 18.0 36.0

1L08 0 1L09 8 1L07 4 1L07 4 1L07 4 t.03 5 1L02 9 L02 4 L02 3 1.02 3 L02 4 1.018

1.072 1.078 1.066 1.069 1.066 1.03 3 1.028 1.024 1.023 1.022 1.023 1.018

1.062 1.076 1.059 1.069 1.066 1.03 3 1.028 1.024 1.023 1.022 1.023 1-018

Mz

disperse molecular weight distributions can be obtained. This is shown in Table 1-1, where the size exclusion chromatography (SEC) characterization results are given for 1,4-polybutadienes with molecular weights as low as 103. As expected, the heterogeneity indices decrease as molecular weight increases. tert-Butyl lithium was the initiator with cyclohexane as the polymerization solvent at 20°C. The molecular weights of these materials are controlled by the simple expedient of manipulating the monomer [M] /initiator [/] ratio since each molecule of the latter will generate a growing chain. Thus, from the standpoint of the number-average degree of polymerization, DPn, we have DP-

Figure 1-1. Schematic of macromolecules with various architectures: (a) linear, flexible chain; (b) graft; (c) H-shaped; (d) ring; (e) star.

Mz Mw

Mz+1

Mn

[M]o

(1-1)

where the subscript denotes the initial reactant concentrations. Under normal conditions the reactive chain-ends in hydrocarbon solvents, the so-called "active centers", can retain their reactivity for periods of time that far exceed the time required for complete con-


version. Thus, the various molecular weight moments can be, within experimental error, virtually identical. These are defined as follows: Number average molecular weight: (1-2) Weight-average molecular weight: (1-3) Z-Average molecular weight: Mz =

—l- =

—y

(1-4)

Z + 1-Average molecular weight: (1-5) where wt = cjc and denotes the weight fraction of species i in the polymer and Nt is the number of moles of species i with the molecular weight Mt. The viscosity average molecular weight for a near-monodisperse polymer is virtually identical to M w . As shown in Table 1-1, the ratios of R H R' I I I

H R H R' I I I I C - C = C-C-LI I H H

R"Li

H R I I C-C I I H C = Cv I H

R"Li modifier

H R H R' I I I I

c-c=c-c

I H

I H

H R I I C-C I I / H C = CV I H

these various moments serve as the required pedigree insofar as the length heterogeneity characteristics of a polymer are concerned. Another characteristic of the organolithium-based polymerization systems in hydrocarbon solvents is the control possible in the microstructure of the polydienes (see Scheme 1). The modifiers that can be used to bring about the 1,2 mode of addition are ethers (ROR'), and amines (NR3), or diamines (R 2 N-CH 2 CH 2 -NR 2 ). The saturation of these polydienes leads to polyolefin structures which retain the narrow molecular weight distribution of the parent material. Deuterium can be used in place of hydrogen, if labelling for neutron scattering studies is needed. Some of the polyolefins prepared in this fashion are shown in Scheme 2. The retention of chain-end reactivity has allowed the preparation of model starshaped materials. The most reliable and controllable route involves chlorosilanes, i.e. 4PLi + SiCl4 -> P4Si + 4LiCl, where P denotes the linear polymer. This approach has allowed stars with up to 128 arms to be prepared (Roovers, 1992). Through the se-

1,4 structure

1,2 structure

1,4 and 1,2 structures

Scheme 1

1.4 Model Polymer Requirements

H H I I

H \

H /

c=c-c=c / \

H CH, H H H I I C=C-C=C-CH, / H

Polyethylene Random (ethylene/ butene-1) copolymer

H

H \

H CH3 H I I ' /

c=c-c=c / \ H H

\

H H C9H, H

i r /

c=c-c=c

H

PLi + CH 3 OH -> PH + CH 3 OLi Atactic polypropylene

Alternating (ethylene/ propylene) copolymer

Alternating (ethylene/ butene-1) copolymer

H

Scheme 2 quential polymerization of different monomers, di- or triblock co- or terpolymer stars can be prepared. However, the conditions under which these polymerizations are conducted require attention. Generally an acceptable temperature range is 0° to 50°C. Higher temperatures will lead to active-center deactivation via, for example, lithium hydride elimination or metallation events (Antkoviak, 1971; Kern et al, 1972; Anderson etal, 1972; Roovers and By water, 1973; Neutwig and Sinn, 1980; Broske etal., 1987; Pennessi and Fetters, 1988). Solvents which can cause chain-transfer are also to be avoided. Toluene is often cited as the polymerization solvent. This is an unsuitable choice, since the following occurs (Gatzke, 1969): PLi

nol, will lead to chain coupling (Fetters and Firer, 1977). Ideally, deactivation of the active centers occurs as follows:

LiCH 2 -

Such reactions clearly will distort the molecular weight distribution. Care must also be exercised in the termination of these active centers, since trace amounts of oxygen in, for example, metha-

The presence of oxygen can cause the following side reaction:

The transition state of the chain-ends involves free-radicals which readily couple to yield "doublet" material, with the subsequent distortion of the molecular weight distribution. This dimerization event is also promoted by the fact that the polymers themselves are associated as dimers in hydrocarbon solvents as a consequence of electron deficient bonding present in organolithiums.

1.4 Model Polymer Requirements The high degree of control possible with anionic polymerization can be replicated only partially by other addition polymerization mechanisms. These include cationic (limiting in the main to isobutylene), radical Ziegler-Natta, ring-opening metathesis polymerizations (ROMP) and grouptransfer polymerization. Many of these latter systems have been described as pseudoliving, immortal, quasi-living, livingness enhancement and truly living. As Quirk and Lee (1992) have pointed out, the term "living polymerization" is to be used only for these systems which proceed with absence of the termination or chain transfer steps. The following describes these processes and presents the procedures by which the resultant macromolecules should be characterized in order that they be classified as model materials. The lack of a spontaneous termination event in anionic polymerization systems

8


Table 1-2 a. Effect of polymerization temperature on polydiene chain structure (Young et al., 1984). (1) Polyisoprenea Temp. (°C)

[s-BuLi] 103

46 20 0 25 40 25

Microstructure b

Solvent

0.01 0.01 0.01 0.01 1 1

None None None None Hexane Hexane

[s-BuLi] 103

Solvent

trans-1,4

cis-1,4

3,4

1 1 2 3 18 18

95 95 95 93 76 78

4 4 3 4 6 4

1,2

(2) Polybutadiene3 Temp.

35 20 0

0.007 0.007 0.007

Microstructure b trans-1,4

cis-1,4

9 9 7

85 86 86

None None None

3,4

1,2 6 5 5

Monomer = 2.5 M; b via 300 MHz ^ - N M R .

was recognized in the mid-1930s by Ziegler and his co-workers (Ziegler, 1936; Ziegler and Jacob, 1934; Ziegler et al., 1934, 1936) and by Abkin and Medvedeev (1936). Flory predicted in 1940 that such systems would possess the Poisson molecular weight distribution (Flory, 1940). Approximately twenty years later such polymers were prepared in the case of poly(amethyl-styrene) (Sirianni et al., 1959) and polystyrene (Wenger, 1960). Both polymerization systems utilized tetrahydrofuran as the polymerization solvent at dry-ice temperature, and respectively sodium naphthalene and sodium biphenyl as the initiators. Ethereal solvents and the sodium counter-ion are, though, of limited utility in the preparation of model polymers. The polymerization of styrene is extremely fast requiring attention to heat transfer and

mixing problems - and the use of sodium prohibits the attainment of the high 1,4 enrichment for polydienes. A more subtle problem is the short-term (hours) instability of initiators like sodium naphthalene (Schulz et al., 1970), a facet of this organometallic that went unrecognized by its advocates. Thus, the initiators of choice are the hydrocarbon soluble organolithiums, one of which, n-butyllithium, was used by Ziegler in his work. Organolithium-based hydrocarbon polymerizations readily yield nearmonodisperse molecular weight distributions (Fig. 1-2), along with polydienes of various microstructures (Tables 1-2 a and b) and active center stabilities that permit, in the absence of side-reactions, near quantitative chain-end functionalizations. The advent of SEC (also called gel permeation chromatography, GPC, see


Table 1-2 b. Microstructure of polydienes prepared in solvating media (Young et al., 1984). (1) Polyisoprene Solvent THF THF THF Ether Ether Ether Ether Dioxan TMEDAb TMEDAC DME Radical polymer

Cation

Temp. (°C)

Li free anion free anion Li Na K Cs Li Li Li Li

30 30 -70 20 20 20 20 15

cis-l,4(%)

trans-l,4(%)

3,4 (%)

1,2 (%)

12 22 10 35 a

59 47 45 52 61 43 32 68 55 45

29 31 31 13 22 19 16 18 15 30

1 ?

a

38a 52a 11 30 25

15 63

25

(2) Polybutadiene Solvent

Cation

THF THF THF Dioxan

Li Li Na Na K Cs

Temp.(°C)

trans-lA{%)

80 15 0 15 15 15

Radical polymer

25

25 13 14 15 45 59 50

1,2 (%) 70 87 80 85 55 41 25

a

Cis- and trans-1,4 not separated but predominance of trans expected; b principal solvent was benzene; base:Li ratio was 60:1; c principal solvent was hexane; base:Li ratio was 1:1.

X Mni. Thus, for a mixture of two monodisperse polymers where the MnJMni ratio equals 2, the maximum value of Mv,/Mn is 1.125 (Table 1-5) when equal weight fractions of the two components are present. For the case of n equal to three the Mw/Mn ratio is 1.33 for wx of 0.5.


11

Table 1-4. Sample calculations: equal weights of monodisperse polymers. Sample 1: Calculated:

Sample 2:

Calculated:

0.5 wt. fraction M n = 50 x 103 g/mol; 0.5 wt. fraction Mn = 100 x 103 g/mol; Mn = 67 x 103 g/mol; Mw= 75 x 103 g/mol; M w /M n = 1.125 0.33 wt. fraction M n = 50 x 103 g/mol; 0.33 wt. fraction Mn = 100 x 103 g/mol; 0.33 wt. fraction M n = 150 x 103 g/mol Mn = 81.8 xlO 3 g/mol; Mw = 100xl0 3 g/mol; M w /M n = 1.22

Table 1-5. Polydispersity indices for bimodal polymer mixtures. Singlet wt.% 100 95 90 80 70 60 50 40 30 20 10 5 0

Mn -

Doublet wt.% 0 5 10 20 30 40 50 60 70 80 90 95 100

MJMW

Mw/Mn

1.000 1.043 1.074 1.111 1.124 1.122 1.111 1.094 1.072 1.049 1.025 1.012 1.000

1.000 1.000 1.045 1.080 1.105 1.120 1.125 1.120 1.105 1.080 1.045 1.000 1.000

" *; M w = S (w; M^; Mz = X(wt- M?)/

The limitations inherent in the use of the Mw/Mn ratio as a barometer of "monodispersity" have been demonstrated in another study by Quirk and Lee (1992). They prepared synthetic mixtures by combining equimolar amounts of two and three nearmonodisperse polystyrenes as described in Table 1-6. The mixtures were then analyzed by SEC with the resulting chromatographs shown in Fig. 1-3. It is obvious that samples exhibiting the relatively narrow distributions of 1.11 and 1.15 cannot be

I 35

I 40 Elution volume (ml)

Figure 1-4 SEC chromatographs of the polymer mixtures of Table 1-4: dotted curve, sample 1; solid curve, sample 2.

described as monodisperse or even as nearmonodisperse. For these reasons, it is important to rely only on SEC data obtained from a high resolution column set. Examples of SEC data must be presented in view of the potentially deceptive nature of the Mw/Mn ratios as shown in Table 1-6. The term narrow molecular weight distribution must be limited to polymers with M z + 1 / M z , MJMW and Mw/Mn ratios of 50°C) in the anionic polymerizations of dienes and styrenes in hydrocarbon solvents. The use of such temperatures will lead to lithium

75°C/3 h

hydride elimination, metallation, chain branching and active center isomerization (Antoviak, 1971; Kern et al., 1972; Anderson et al., 1972; Roovers and By water, 1973; Neutwig and Sinn, 1980; Broske etal., 1987; Pennisi and Fetters, 1988). Figure 1-8 serves as an example. Therein, SEC chromatographs of a polybutadiene (Pennisi and Fetters, 1988), polymerized at 20 °C, are shown as a function of time after exposure to 75 °C. The onset of branched material is obvious. Other side reactions that can be encountered have been reviewed by Young et al. (1984). Clearly, the mere comment, which is often encountered, that anionic polymerization was used to prepare the polymer in question is an inadequate pedigree. Reaction conditions must be described and their limitations understood. The companion information needed, as has been noted, is proper characterization. An example of what that exercise will reveals is as follows.

10wt.% Linked polymer

0 •a

1 33 wt.% Linked polymer

i

75°C/24 h

a

Figure 1-8. SEC chromatographs of poly(butadienyllithium) as a function of time at 75 °C (Pennisi and Fetters, 1988). 40 SEC counts

15


44

48

52 36

40

44

46

Figure 1-9. SEC chromatographs of two samples of "star" polystyrene (Mays et al., 1988). The large peaks at high elution volume correspond to unlinked arms.

52

SEC counts

The SEC traces shown in Fig. 1-9 are those of polystyrene "star" polymers which were purported to contain less than 5 wt.% unlinked arm (the material exhibiting the longest retention time in the colums). The elementary SEC characterization exercise revealed a markedly different situation: approximately 70% of the total material was unlinked arm! (Mays et al., 1988.) In consonance with proper SEC protocol is the capacity to evaluate molecular weight in an absolute sense via osmometry and light scattering. Of these two methods, the latter is generally more useful - particularly if coupled with SEC evaluations. The importance of absolute molecular weight measurements is highlighted by the following elementary example. Figure 1-10 presents data for the intrinsic viscosities of polystyrene in toluene (measurement temperature not specified). The molecular weights for those samples were taken as

principle, the procedure used would work if the previously listed criteria for model polymer preparation held for the system in question. The following rudimentary evaluation reveals that this is not the case. The solid line denotes the intrinsic viscositymolecular weight behavior of properly characterized polystyrene (Fetters et al., 1992) (where M denotes Mw) of nearmonodisperse molecular weight distributions given as L

(1-8)

y 1.0

o.y

/

.•6

•

oo§-

/

°/

x 104, where M is the

monomer concentration in mol/L and / is the concentration of initiator in mol/L; in other words, the stoichiometric molecular weights were used to describe these polystyrenes prepared in tetrahydrofuran with sodium naphthalene as the initiator. In

0.1

i

10 4

10 5 M

106

Figure 1-10. Intrinsic viscosities of polystyrene as a function of molecular weight. The filled symbol denotes a data point eliminated, without explanation. (From Fig. 1 of Waack et al., 1957).

16


This equation, for a given [rj\9 yields molecular weights larger by more than a factor of two than the one based on the data (Waack etal., 1957) of Fig. 1-10: M =1.6 x 1 0 - 4 M 0 7 4 d L g - 1

(1-9)

Obviously, the erroneous nature of Eq. (1-9) is the result of the failure of the polymerization system to perform as expected, i.e. considerable termination existed, and the failure to characterize the sample in absolute terms. In short, the announcement that anionic polymerization was the synthetic tool does not supersede or replace the necessity for thorough polymer characterization. Unfortunately, encounters of this nature are still possible (e.g. Tsitsilianes and Staikos, 1992).

1.5 Functionalization of Model Polymers In recent years there has been a growing interest in the preparation of well-defined polymers with terminal functional groups.

The primary motivation for this interest in functional groups is their potential for the following applications: - chain extension, branching or crosslinking reactions; - reactions with fundamental groups on other polymer chains; - co-polymerizations with other monomers to form graft copolymers (the macromer concept); - the initiation of polymerization of other monomers; - reversible ionic-association; - endow polymers with surfactant-like properties. Some of the feasible structures are schematically given in Table 1 -7 where X indicates the location of the functional group. As is the case with the model polymers the most well-defined and exploited route to these materials to date has been lithium based anionic polymerizations. Some examples for the functional groups are given in Table 1-8. The foregoing also serves to summarize some of the non-functionalized structures that have been prepared via anionic polymerizations.

Table 1-7. Molecular architecture for functionalized model polymers (Quirk, 1992).

Y Multifunctional

Chain-end

XX X X Random copolymer

Block copolymer (Chain-end)

X X

X X Star polymer (Chain-end)

Heteroarmstar polymer (Chain-end)

X Block copolymer (Block interface)

ex Macrocyclic polymer

1.5 Functionalization of Model Polymers

Table 1-8. Polymer functional groups via anionic reactions (Quirk, 1992). Reactant

Group

CH 9 -CH,

-OH

CH,-CH2 \ / S

-SH

\ / O

Side reactions

O

CO,

so,

-COH

X

0 II -SOH

X

0 0 II

S-OH

x

X

ii

II

0 I

-COOH I -Br

Br2 CH3 N

NCH2CH2CH2Li

/

CH 3 CH3

CH3 C2H5O I CH 3 -C-O(CH 2 ) 5 CH 2 Li

-OH

H

Unfortunately, many of the functionalization reactions which have been described are not well-characterized, as discussed in a recent detailed review by Quirk (1992). Precedent from the reactions of model small molecule analogs is often misleading. Thus, adequate and necessary characterization probes embrace: - chain molecular weight; - functional group titration;

17

- NMR, UV-visible and FTIR spectroscopy; - thin layer and column chromatography; - chemical reactions. The primary need in chain-end characterization is an accurate evaluation of functionalization efficiencies. To this end, knowledge of functional group titration, spectroscopic probes (Vol. 2B, Chap. 11) and chemical reactions can all contribute to determining the efficiency of a given reaction. Chromatography has proven useful (Quirk and Chen, 1992; Quirk et al., 1989, 1992; Quirk and Yin, 1992) for analyzing unfunctionalized polymer mixed with the functionalized product as a consequence of the different migration rates that the two species exhibit. Chain molecular weight, when combined with items b), c) and e), allows a quantitative assessment of the functionalization reaction in question. One of the most useful and widely used functionalization reactions has been the carboxylation of polymeric anions using carbon dioxide. Although intrinsically a simple reaction, a complex mixture of products involving carboxylated polymer, the dimeric ketone and trimeric alcohol are, as noted previously, often obtained. Initially, it was found that the coupling and linking side reactions could be virtually eliminated in systems where the polystyryllithium was complexed with a base such as tetrahydrofuran or tetramethylethylendiamine (Quirk and Chen, 1992; Quirk et al., 1989). An alternative route involved the solid state (achieved by freeze drying the polystyryllithium) exposure of the active centers to CO 2 at atmospheric pressure. In both cases the yield of carboxylated polystyrene was >99% (Quirk etal, 1989, 1992; Quirk and Yin, 1992). The use of 13 CO 2 revealed that approximately 15% of the -COOH groups were present on the aromatic ring. In other

18


PS-CH 2 -CHLi + -^

•

PS-CH2-CHLi+ -
PISiCl2+CCH3SiCl3 (I) Cl I b) I+PSLi-*PISi(CH 3 )PS(II)+LiCl I CH3 (II) PBd I c) II + PBdLi -» PISiPS + LiCl

CH3

Scheme 4

A cornerstone of materials science is processing of materials. Indeed, Vol. 18 of this Series is entirely devoted to polymer processing. It has become increasingly realized that simultaneously combining the steps of synthesis and processing can sometimes lead to better control of microstructure. This combined approach can provide new materials of outstanding performance. Recent nonpolymer examples are chemical vapor deposition and molecular beam epitaxy, wherein precise chemical and structural control of the material is accomplished at the atomic level. In simple "sphere-pack" materials, crystallization proceeds rapidly and organizes the assembly of members at the level of the crystal grain (~one micron). Metallurgists, long ago, developed extensive methods to manipulate sample microstructure via timetemperature-transformation studies with and without the presence of applied fields (mechanical, electrical, etc.). Rational control of microstructure in polymers is a challenge, since one needs to control packing at several length scales simultaneously and because, of their inherent conformational diversity, each molecule has an enormous number of structural states of similar energy to choose from. As indicated, stereochemical and monomer sequence control during species synthesis is the sine qua non for molecular level control of structure and function.

20


The realization of desirable superstructures arising from the self assembly of individual species far more complex than simple spheres requires intermolecular assembly to proceed in a uniform manner in order to allow the carefully built-in specificity of chain structure and the applied environmental conditions to produce the prescribed, desired structural state. With polymers, one discovers even with the most simple macromolecules a wide array of structural states and length scales, e.g. for semicrystalline linear polyethylene, the orthorhombic crystalline unit cell has edge lengths about 1 nm and the all trans chains pack into lamellae typically 20 nm thick. The lamellae are separated by noncrystalline regions wherein a combination of trans and gauche states results in a random chain structure. Moreover, crystallization proceeds from nucleation centers to develop spherulites of average diameter of several microns. Inside a spherulite, the lamellae are radially arranged (see Chap. 3). Such a complex microstructural geometry involving both crystalline and noncrystalline phases makes establishment of structure-property relations difficult; furthermore, the important and practical evolution of the properties with applied force or imposed strain is challenging both to characterize and to model. For this reason, simplified geometries and uniform structures have been produced to enable exploration of fundamental structureproperty relations. Materials scientists have discovered various post-synthesis ways to influence and control the microstructure - from the earliest days of forging, rolling and annealing heat treatments to the use of thermal gradients and single crystal seeds. Kinetic control of microstructure is readily exploited in polymeric systems as in other materials. Spinodal decomposition and nucleation

and growth of phase domains is a major field of investigation (see Chaps. 3 and 7 in Vol. 5 and Chap. 6 in this Vol.). Polymer chemists have had good success in systematically manipulating polymers for liquid crystal (LC) applications. The simple joining-together of mesogens by short flexible spacer units into a long chain results in both nematic and smectic LC materials (see Chap. 5). Liquid crystalline materials can be readily oriented into monodomains via mechanical, electric or magnetic fields. Precise control of the transition temperature from the crystalline into the LC state and from the LC state into the isotropic can be made by variation of the type, length and degree of flexibility of the spacer unit, as well as the attachment of pendant groups from the chain backbone. Side chain LCPs (liquid crystalline polymers) are readily made by the attachment of mesogens to the main chain again by spacer units. These side chain LC materials have the advantage of greatly decreased response times of the mesogens to applied fields and are of increasing interest for optical applications, wherein the polymeric nature of the LC material is exploited for processing and mechanical properties of the final product. It is very easy to produce a polymer glass (see Chap. 2). The atactic distribution of chemical substituents along long, entangled chains in a high viscosity melt, provides even under relatively slow cooling conditions, access to the glassy solid state. Moreover, highly anisotropic glasses are possible. The polymer can be solidified during processing such that the stretch oriented conformations of the molecules are frozen-in by rapid cooling. It is also possible to produce room temperature glassy liquid crystalline materials by quenching a mesogenic polymer with a high glass transition from its high temperature LC state.

1.7 Processing

Self-assembly of block copolymers under an applied bias field can result in materials with single-crystal-like texture. Block copolymers segregate into microdomains composed of the respective blocks. Typical domain sizes are several hundred angstroms and these domains in turn assemble into grains with typical diameters of several microns. The grains of the cylindrical or lamellar microdomain-forming block copolymer have a preferred orientation resulting in local anisotropy of the material properties. However, since the grains are relatively small in comparison to the whole sample and since they are normally in random mutual orientations, the resultant macroscopic sample exhibits isotropic properties. When such materials are examined by scattering or subjected to mechanical testing, the net response is the superposition of the different responses to each of the various grain orientations. Fundamental understanding of structureproperty relationship is thus difficult to achieve. To overcome such limitations, one can employ special processing to convert the isotropic multigranular sample into a single crystal. Such an approach was first taken by Keller and coworkers (Keller etal, 1970; Folkes et al, 1973), utilizing extrusion to produce highly oriented single crystal-like material of a triblock copolymer. This process is akin to the texturing of polycrystalline metals by die drawing. While providing a highly oriented sample, this process suffers from the presence of grain boundary structures and other morphological defects due to the alignment of an already microphase separated material. The influence of such defects and boundaries is unknown at present, but in analogy to work with other materials, one can expect a strong influence on properties. Recently, Albalak and Thomas have devised a roll-casting process which elimi-

21

nates grain boundaries by causing the self assembly of the block copolymer molecules into the microdomains to take place in an orienting flow field (Albalak and Thomas, 1993). The roll-casting process is based on the rolling of an initially homogeneous polymer solution between two counter-rotating cylinders while removing solvent at a controlled rate. The experimental system is schematically pictured in Fig. 1-12 and consists of two adjacent rolls and is similar to conventional set-up of calendars or rollmills. The polymer solution flows between the rolls through a minimum gap, called the "nip" and coats either one or both of the rolls. The detailed velocity profile in such a geometry has been analyzed (Gaskell, 1950) and contains both shear and elongational flow components. The key to this process is that the onset of microphase separation occurs in the applied flow field so that the microdomains form in a globally oriented structure. Figure 1-13 a is a TEM micrograph of a [0001] axial view showing the well-ordered microdomain structure of a PS/PB/PS triblock copolymer containing PS cylinders in the PB matrix. The long-range hexagonal registry of solution

Figure 1-12. A schematic representation of the rollcasting apparatus. A concentrated solution of block copolymer is processed between two adjacent counter-rotating cylinders (Albalak and Thomas, 1993).

22


phology and phase transformation theories and to further the basic understanding of chain dimensions in block copolymer systems (see next section).

1.8 Model Polymers in Polymer Physics

(b)

Figure 1-13. (a) Bright field TEM micrograph of an OsO 4 stained, microtomed section of a roll-cast PS/ PB/PS film. View is along flow direction and shows the two-dimensional hexagonal lattice packing of the PS cylinders, (b) Two-dimensional small angle X-ray scattering pattern of roll-cast PS/PB/PS film with incident beam along the flow direction. Bragg reflections of a well-ordered two-dimensional hexagonal structure are evident (Albalak and Thomas, 1993).

the PS cylinders is quite evident in this projection. Low magnification TEM images show that the single crystal structure is unperturbed for at least many tens of microns. The degree of order over larger distances can also be assessed by small angle X-ray scattering (SAXS). Figure 1-13 b is a two-dimensional SAXS pattern taken with the incident beam along the [0001] direction. The pattern originates from a sample area on the order of 1 mm 2 demonstrating the near single crystal nature of the macroscopic film. Such specimens offer significant advantages in testing phase mor-

There are many examples of the use of model polymers to develop novel materials structure, properties and to test theoretical concepts in polymer physics. We first detail the employment of model polymers in testing the elegant reptation hypothesis of de Gennes, Doi and Edwards and the tube concept in reptation. According to reptation theory, individual chains are assumed to be constrained by their neighbors and are thus compelled to migrate primarily along their own contours in a snake-like manner (see Chap. 9). Thus, a polymer melt is made up of an assembly of chains all simultaneously involved in disengagement from their current cage of strands. In this disentanglement process, each chain creates a new cage and configuration as it undergoes its movement through the mesh. The cage can be visualized as a tube of diameter d and a contour length /, which encloses each chain in the mesh. The chains are assumed Gaussian, with v chains per unit volume. The chain dimensions are expressible as (R2}0/M where 0

M

n •]["

NJ

(1-10)

23


where G£ is the plateau modulus (the short time modulus of an entangled polymer liquid). The above equation can be reduced to df = 0.8

roi [Me] L M \

(1-11)

where M e denotes the entanglement molecular weight. Thus, a combination of the static [0/M] and the dynamical based [Me] macroscopic parameters allows an evaluation of dt. Until recently the direct measurement of dt had not been accomplished. For that purpose near-monodisperse, amorphous poly(ethylene-propylene) (PEP) was used (Richter et al., 1991,1992; Butera et al., 1991). The route to that polymer was via the hydrogenation saturation of anionically prepared polyisoprene. The analytical tool of choice was the IN-11 neutron spin echo (NSE) spectrometer at the Institut Laue-Langevin in Grenoble. The unique feature of the NSE technique is its ability to evaluate in a direct way the energy changes induced in neutrons during the scattering process. In essence, the dynamical data are gathered by examining the decay in an NSE as a function of an imposed precessional frequency. Therein, a magnetic field of varying strength is applied to the incident polarized neutron beam. The small energy change of each scattered neutron is analyzed as a shift in the processional phase. In view of the limited time resolution (ca. 4 x 10" 8 s) and distance resolution (5 to 100 A) of the IN-11 instrument, only the most dynamically flexible chains with rapid local motion are candidates. PEP and polyethylene (from the saturation of 1,4-polybutadiene) fulfill these criteria. These results of such an NSE investigation on PEP are summarized in Fig. 1-14, which also contains rheological and SANS findings. The tube diameter (NSE) was

300

400

500

600

T[K]

Figure 1-14. Entanglement distance dt, radius of gyration R2G, ratio of G^ax and d? G'J{Q TCJ for alternating poly(ethylene-propylene) as a function of temperature.

found to increase with temperature, while the radius of gyration (SANS) and plateau modulus (rheology) were observed to decrease. The latter in turn signaled an increase in the entanglement molecular weight, which, when combined with the change in chain dimensions, led to an increase in the tube diameter with increasing temperature. It is important to note that dt (NSE) and dt (SANS, G£) were in essential agreement, a picture also found for polyethylene (Richter et al., 1992). These results fortify the elegant Ansatz of de Gennes' reptation model.

24


Flory (1953) predicted that chain dimensions in a 9 solvent where the effect of excluded volume is just balanced by the polymer segment-solvent interaction are identical to those in a polymer melt and that the chain behaves like an ideal random walk. The use of model polymers has also been of great value in the detailed evaluation of rotational isomeric state (RIS) predictions pertaining to the polymer chain dimensions and their respective temperature coefficients (see Chap. 2). The advent of SANS has expanded the range of these evaluations, since both parameters can be measured in the melt state. Some recent findings are presented in Table 1-9. The combination of theta solvent, RIS and SANS results shows excellent agreement with regard to the parameters in question for PE and PEP. However, such agreement is absent for a-PP and a-PEE. It can be seen that neither theory (RIS) nor theta solvent measurements yield chain dimension temperature coefficients that may be related to what is observed in the melt state. From the RIS standpoint there seemingly is a problem in correctly visualizing the role of non-bonded interactions as a function of temperature. The failure of the

RIS approach is duplicated in the failure of the 0-condition results to mimic melt behavior. Thus, the assumption that modeling a polymer in a vacuum (RIS) universally yields its behavior under 6 conditions, which in turn serves as a mimic of melt behavior, is erroneous. Solvent influences on "unperturbed" chain dimensions have been previously predicted (Lifson and Oppenheim, 1960). Our current understanding lacks predictive capacities regarding which solvents can exert so-called specific solvent effects, or, conversely, which polymers are susceptible to such influences. Nonetheless, a partial rationalization as to why solution theta condition is not equivalent to melt behavior is possible. Rotational isomeric states are identified with the minima of the conformational energy maps, minima which can be influenced by local-polymer solvent interactions - the so called "specific solvent effect". This effect can be viewed as one that does not create new rotational isomeric states, but modifies the potential energy of those already existing in the melt. Thus, the location of the various minima remain unchanged while their magnitudes can be altered by the prevailing solvent-

Table 1-9. Polyolefin chain parameters at 298 K. PE = polyethylene; PEB-2 = poly(ethylene-butene) with ca. 2 ethyl branches per 100 backbone carbons. PEP = alternating poly(ethylene-propylene); a-PP = atactic polypropylene with Bernoullian chain statistics; a-PEE = atactic poly (ethyl ethylene) with Bernoullian chain statistics; q = solution intrinsic viscosity or light scattering; SANS = small angle neutron scattering of melt; RIS = rotational isomeric state (theory). Polymer

RG/M

(A mol

PE (PEB-2) PEP

a-PP a-PEE a

0.470 0.381 0.340 0.279

1/2

1/2

g-

aa

C 1/2

(A)

d(ln 0 )/dT (lO^C- 1 )

)

SANS

*

SANS

0.485 0.397 0.337 0.283

7.9 6.5 6.6 5.5

8.4 7.1 6.1 5.7

Statistical segment length: a2 = 6 Rl/Nw.

SANS 6.1 5.6 5.6 5.1

6.3 5.8 5.3 5.2

-1.1 -1.1 -2.9 -2.3

SANS

RIS

-1.15 -1.16 -0.1

-1.2 -1.0 0 to -1.8 -0.1

0.4


polymer interaction for any particular state. This scenario requires that the probabilities of occurrence of the most stable conformations - those isomeric states required by the short range intramolecular interactions - can be modified by solvent interaction. Under theta conditions such shortrange interactions can be neglected if solvent discrimination between conformations is absent. This criteria is apparently met for polyethylene and alternatingpoly (ethylene-propylene) copolymer (Table 1-9). Conversely, the difference between the solution and melt behavior for a-PP and a-PEE seemingly demonstrates that their local conformational characteristics are considerably influenced by theta-solvent (n-alkyl alcohols) interactions, interactions which lead to enhanced gauche populations with increasing temperature (and hence a negative temperature coefficient for chain dimensions). It appears that the configuration of the skeletal bonds will lead in some cases to solvent discrimination regarding the attached side groups with gauche sequences being more susceptible to such interactions than their trans counterparts. Given the presence of such favorable interactions, the net effect is a decrease in the trans /gauche ratio with increasing temperature. Similar behavior is evidenced for a-PP in either 2,4-dimethyl heptane or 2,2,4-trimethyl-pentane which gave a d(ln (R2}0)/ dT of -2.4 x 1(T 3 °C x over the temperature range of 10 to 70 °C. The use of those alkanes was based on their assumed capacity to act as athermal solvents, an assumption which the SANS work has shown to be incorrect. Interestingly, like the rc-alkyl alcohol theta solvents, these branched alkanes alter the a-PP trans/gauche ratio with increasing temperature. In a parallel fashion, IR work of Casal et al. (1986) on

25

n-tridecane 7,7-d2 shows that the average gauche population increases from 35% to 60% when the solvent is changed from 7i-heptane to hexadecane. The solution induced behavior of a-PP and a-PEE is strikingly different than that observed in the melt where the former is found to exhibit a virtually invariant population of conformer types while the latter shows that increasing temperature favors the trans sequences. Attempts have been made to account for solvent effects within the framework of the RIS model (Bahar et al., 1986; Brant and Flory, 1965; Yoon et al., 1975 a,b). This involved the adoption of a truncation procedure involving an adjustable parameter, s, which Brant and Flory (1965) took to be the sum of the van der Waals radii of the interacting atoms. Variations of this approach have subsequently led to a procedure (Bahar et al., 1986) which does not utilize adjustable parameters. However, although a-PP and a-PEE apparently acquiesce to the influence of their solvent shells, the nature and identity of these local interactions is not generally understood (Janik etal., 1987). The foregoing SANS studies were facilitated by small heterogeneity indices of the subject polymers, e.g. see Fig. 1-1. Thus, the measured RG (z-average dependent dimension) is equal to the weight-average value. In other words, polydispersity corrections are unnecessary. The types of polyolefins that have been thus far prepared via the polydiene route are shown in Table 1-10 along with the parent material. Table 1-11 lists some of their properties. Model diblock copolymers produced by anionic polymerization and processed into single crystal specimens are quite useful to probe how the microstructure influences chain dimensions. SAXS and SANS measurements, as well as theoretical predic-

26


Table 1-10. Polydiene and polyolefin structure and nomenclature. Common or IUPAC name

Structure

Nomenclature

Polypropylene

CH3 I [-CH 2 CH-]

1,4-Poly(dimethylbutadiene)

CH 3 CH 3 CH3 II' I [_CH2C = CCH 2 ] 9 7 [-CH 2 C-] 3

PP

1,4-PDMBD

C = CH2 I CH3 Head-to-head polypropylene

CH3 CH3 II"

CH3 I "

HHPP

CH2CH - CHCH 2 -] 97 [- CH2C -] 3 HCCH3 1,4-Poly(ethylbutadiene)

CH3 CH3

CH2CH3

1,4-PEB

[-CH 2 CH = CHCH 2 -] 90 [-CH 2 CH-] 10 CCH2CH3 II CH? Poly(ethylene-butene)

CH2CH3 I [-CH 2 CH 2 CHCH 2 -] 90 [-CH 2 CH-] 10

PEB-26 3

HCCH2CH3 I CH3 1,4-1,2-Poly(dimethylbutadiene)

CH3 CH3 CH3 I ' I * I [_CH2C = CCH 2 ] 55 [CH 2 C-] 45

55-PDMBD

CCH3 II CH9 Poly (1,2-dimethyl-1 -butylene-co-1 (1 -methyl-1 -isopropyl)ethylene

CH3 CH3 II"

CH3 I

- CH2CH - CHCH 2 -] 55 [- CH2C - ] 4 5 HCCH3 1,4-Polymyrcene

CH3 - C = CHCH 2 CH 2 -] 90 [- CHCH 2 -] 10 CH? I CH2 I CH II C(CH3)2

C = CH2 CH2 CH II C(CH3)2

1,4-PMYRC

27


Table 1-10. (continued) Common or IUPAC name Poly( 1 -(4-methy lpenty 1)-1 -butylene-co-1 (1,5-dimethylhexyl)ethylene)

Structure [-CHCH 2 CH 2 CH 2 -] 90 [-CHCH 2 -] 10 CH2 I CH2

CHCH3 I CH2

CH?

CH2 I CH2 I CH(CH3)2

CH(CH3)2

Poly (1 -(4-methyl-3-pentenyl-1 -buteny lene-co-1 (l-methylene-5-methyl-4-hexenyl)ethylene)

Poly (1 -(4-methylpentyl)-1 -butylene-co-1 (1,5-dimethylhexyl)ethylene

Nomenclature

[- C = CHCH 2 CH 2 -] 64 [- CHCH 2 -] 36 ' ' Ln 2 Cri? I I CH2 C = CH2 I CH CH2 II C(CH3)2 CH II C(CH3)2 [-CHCH 2 CH 2 CH 2 -] 64 [-CHCH 2 -] 36 I I CH2 CHCH3 I I CH2 CH2 I I CH2 CH2 I CH(CH3)2 CH2

H21,4-PMRYC

64-PMYRC

H.64-PMRYC

CH(CH3)2 Poly(ethylene-cc>-butene)

PEB-14a

[-CH 2 CH 2 -] 36 [-CH 2 CH-] 14 CH2 CH3

Poly(ethylene-co-butene)

PEB-40a

[-CH 2 CH 2 -] 10 [-CH 2 CH 2 -] 40 I CH2 CH3

Poly(vinylethylene)

[-CH 2 CH = CHCH 2 -],_ 2 [-CH 2 CH-] 98 _ 99

PVE

CH II CH2 Poly(ethylethylene)

[-CH 2 CH 2 -],[-CH 2 CH-] 4 9 I CH2 I CH 3

The number in the PEB series denotes the number of ethyl branches per 100 backbone carbons.

PEE

28


Table 1-11. Glass transition temperatures and density parameters for polydienes and polyolefins. Polymer

ppa

HHPP b PEP 1,4-PEB PEB-26 55-PDMBD H255-PDMBD 1,4-PMYRC H21,4-PMYRC 64-PMYRC H264-PMYRC PVE PEE

Glass transition temp. (°C) 4

1

Peak onset

Peak midpoint

Q at 20°C(gcm~ 3 )

- ci(lnrf/dT(10 K- )

-5 -28 -65 -65 -61 14 5 -71 -57 -6 -46 0 -30

-3 -26 -62 -64 -59 16 10 -68 -54 -65 -45 2 -28

0.857 0.878 0.586 0.891 0.861 0.918 0.892 0.895 0.856 0.897 0.853 0.889 0.870

7.9 7.2 7.0

5.8 6.3 6.7 9.2 6.0 7.5 6.0 6.0

a Tg values of — 6°C (Cowie, 1973) and — 5°C (Burfield and Doi, 1983) have been reported for amorphous polypropylene. b A Tg of — 27°C has been reported for HHPP. That value is via extrapolation to the 0 K m i n ' 1 heating rate. Arichi et al. (1979) gave Tg's of -10°C (HHPP-1) and -17°C (HHPP-2).

tions of the domain spacing in A/B amorphous/amorphous block copolymers, indicate that the spacing varies approximately as M 2 / 3 in the strong segregation regime where the individual phases are essentially pure components and the A/B junction is strongly localized in the narrow interfacial region between domains. The individual component block dimensions should therefore be distorted from the Gaussian behavior of a chain in a single phase melt, where RG scales as M 1/2 . Measurement of the chain dimensions in a phase separated block copolymer is, however, complicated by the strong scattering from the periodic domain microstructure, which occurs in the same angular range as the single chain scattering. To circumvent the domain scattering problem, Hadziioannou and coworkers (Hadziioannou et al., 1979,1982) employed a special shearing process to obtain near-single crystallike specimens of deuterated-PS/PI lamel-

lar diblock copolymers. Because the oriented domain scattering is highly anisotropic, the low angle scattering is dominated by the diffraction from the lamellar lattice only in the direction normal to the lamellae, while only the single chain scattering essentially dominates for the directions parallel to the A/B interface (see Fig. 1-15 a). Using Guinier analysis, Hadziioannou et al. (1982) found the block chain to be contracted to between 60% and 70% of the unperturbed component of the radius of gyration along the parallel direction (Kgll). Hasegawa et al. (1985) attempted to directly determine the chain stretching normal to the A/B interface (via Rgl) by employing both SAXS and SANS to separate out the contributions from single chain scattering and domain scattering. Unfortunately, the subtraction technique is not very precise, making the Rg ± value suspect. Subsequently, Hasegawa and coworkers (Hasegawa et al., 1987) used con-

1.9 References single chain scattering

incident beam

domain

29

scattering

incident beam

(a)

(b)

Figure 1-15. Schematic of oriented lamellar diblock copolymers and SANS patterns, (a) The beam is normal to the lamellae and the scattering results from the deuterium labeled block, enabling Guinier analysis to determine RG)I. (b) The incident neutron beam is parallel to the A/B interface and the small angle scattering is dominated by the periodic lattice structure (adapted from Hasegawa et al., 1987).

trast matching wherein the level of deuterium labelled diblock was adjusted so as to eliminate the domain interference. However, the Rg ± was still indeterminate due to incomplete contrast matching. This problem was believed to arise from nonuniform deuterium concentration within the PS domains from insufficient overlap of the deuterated blocks. The Rg^ value determined was not affected and the result confirmed the chain contraction in the parallel direction previously measured. Most recently, Matsushita et al. (1990) have used anionic synthesis to carefully deuterium label parts of a block. Their focus was on examining the dimension of a selected portion of the chain. Two types of triblock copolymers composed of deuterated PS, PS and poly-2-vinylpyridine (P2VP) were prepared: "end-labelled" (consisting of d-PS, PS and P2VP) and "centered labelled" (consisting of PS, d-PS

and P2VP). Lamellar orientation resulted from the solvent casting process rather than by shearing. The portion of the chain adjacent to the A/B diblock junction was found to be contracted along the direction parallel to the interface to the same extent as previous researchers had found for the entire block (i.e. approximately 70%). The portion of the chain at the free end of the block was localized in the middle of the PS domain and its dimension was unperturbed by the microdomain morphology. Thus, the gradient in chain stretching can be explored in microstructural systems.

1.9 References Abkin, A., Medvedev, S. (1936), Trans. Faraday Soc. 32, 286. Albalak, R., Thomas, E. (1993), J. Polym. ScL, Polym. Phys. Ed. 31, 37. Ambler, M. R., Fetters, L. I, Kesten, Y. (1977), /. Appl. Polym. Sci. 21, 2439.

30


Anderson, J. N., Kern, W. X, Bethea, T. W, Adams, H. E. (1972), /. Appl. Polym. Sci. 16, 3133. Antoviak, T. A. (1971), Polym. Prepr. (Am. Chem. Soc, Div. Polym. Chem.) 12 (2), 393. Arichi, R, Pedram, N. Y, Cowie, J. M. G. (1979), Eur. Polym. J. 15, 107, 113. Bahar, I., Baysal, B. M., Erman, B. (1986), Macromolecules 19, 1703. Brant, D. A., Flory, P. J. (1965), /. Am. Chem. Soc. 87, 2791. Broske, A. D., Huang, T. L., Allen, R. D., Hoover, J. M., McGrath, J. E. (1987), in: Recent Advances in Anionic Polymerization: Hogen-Esch, T. E., Smid, X (Eds.). New York: Elsevier, p. 363. Burfield, D. R., Doi, Y (1983), Macromolecules 16, 702. Butera, R., Fetters, L. X, Huang, X S., Richter, D., Pyckhout-Hiutzen, W, Zirekl, A., Farago, B., Ewen, B. (1991), Phys. Rev. Lett. 66, 2088. Casal, H. L., Yang, P. W, Mautsch, H. H. (1986), Can. J Chem. 64, 1544. Cowie, J. M. G. (1973), Eur. Polym. J. 9, 1041. Fetters, L. X, Firer, E. M. (1977), Polymer 18, 305. Fetters, L. X, Hadjichristidis, N., Lindner, X S., Mays, X W, Wilson, W. W. (1992), unpublished. Flory, P. X (1940), /. Am. Chem. Soc. 62, 1521. Flory, P. X (1953), Principles of Polymer Chemistry. Ithaca, NY: Cornell University Press. Folkes, M. X, Keller, A., Scalisi, F. P. (1973), Colloid and Polym. Sci. 251,1. Gaskell, R. E. (1950), /. Appl. Mech. 17, 334. Gatzke, A. X (1969), /. Polym. Sci. 7, 2281. Hadziioannou, G., Mathis, A., Skoulios, A. (1979), Colloid Polym. Sci. 257, 136. Hadziioannou, G., Picot, C , Skoulios, A., Ionescu, M.-L., Mathis, A., Duplessix, R., Gallot, Y, Lingeler, X-P. (1982), Macromolecules 15, 263. Hasegawa, H., Hashimoto, T., Kawai, H., Lodge, T. P., Amis, E. X, Glinka, C. X, Han, C. C. (1985), Macromolecules 18, 67. Hasegawa, H., Tanaka, H., Hashimoto, T., Han, C. C. (1987), Macromolecules 20, 2120. Iatraou, H., Hadjichristidis, N. (1992), Macromolecules 25, 4649. Janik, B., Samulski E. T., Toriumi, H. (1987), J. Phys. Chem. 91, 1842. Keller, A., Pedemonte, E., Willmouth, F. M. (1970), Nature 225, 538. Kern, W. X, Anderson, X A., Adams, H. E., Bouton, T. C , Bethea, T. W. (1972), /. Appl. Polym. Sci. 16, 3123. Lifson, S., Oppenheim, I. (1960), /. Chem. Phys. 33, 109. Matsushita, Y, Mori, K., Saguchi, R., Noda, I., Nagasawa, M., Chang, T., Glinka, C. X, Han, C. C. (1990), Macromolecules 23, 4387. Mays, X W. (1992), Polym. Bull. 23, 247. Mays, X W., Hadjichristidis, N., Fetters, L. J. (1988), Polymer 29, 680.

Morton, M., Helminiak, T. E., Gadkary, S. D., Bueche, F. (1962), J. Polym. Sci. 57, 471. Neutwig, N., Sinn, H. (1980), Makromol. Chem., Rapid Commun. 1, 59. Olvera de la Cruz, M., Sanchez, I. C. (1986), Macromolecules 19, 2501. Park, I., Barlow, X W, Paul, D. R. (1991), /. Polym. Sci., Polym. Chem. Ed. 29, 1329. Pennisi, R. E., Fetters, L. X (1988), Macromolecules 21, 1094. Puskas, X E., Kaszas, G., Kennedy, X P., Hager, W. G. (1992), /. Polym. Sci., Polym. Chem. Ed. 30, 41. Quirk, R. P. (1992), in: Comprehensive Polymer Science, Supplemental Volume: Eastmond, G. C , Ledwith, A., Russo, S., Sigwalt, P. (Eds.). New York: Elmsford, in press. Quirk, R. P., Chen, W.-C. (1992), Makromol. Chem. 183, 2071. Quirk, R. P., Lee, B. (1992), Polym. Int. 27, 359. Quirk, R. P., Yin, X (1992), J. Polym. Sci., Polym. Chem. Ed., in press. Quirk, R. P., Yin, X, Fetters, L. X (1989), Macromolecules 22, 85. Quirk, R. P., Yin, X, Fetters, L. X, Kastrup, R. V. (1992), Macromolecules 25, 2262. Richter, D., Butera, R., Fetters, L. X, Huang, J. S., Farago, B., Ewen, B. (1992), Macromolecules 25, 6148. Richter, D., Fetters, L. X, Huang, X S., Farago, B., Ewen, B. (1991), J. Non-Crystalline Solids 131-133, 604. Roovers, X (1992), private communication. Roovers, X E. L., Bywater, S. (1973), Polymer 14, 594. Schulz, G. V., Bohm, L. L., Chmelir, M., Lohr, G., Schmitt, B. X (1970), in: IUPAC International Symposium on Macromolecular Chemistry. Budapest: Butterworths, p. 223. Sirianni, A. F , Worsfold, P. X, Bywater, S. (1959), Trans. Faraday Soc. 55, 2124. Suter, U. W, Flory, P. X (1975), Macromolecules 8, 765. Tsitsilianes, C , Staikos, G. (1992), Macromolecules 25, 910. Waak, R., Rembaum, A., Coombe, X D., Szwarc, M. (1957), /. Am. Chem. Soc. 79, 2026. Wenger, R (1960), Macromol Chem. 36, 200. Yoon, D. Y, Sundarajan, P. R., Flory, P. J. (1975 a), Macromolecules 8, 766. Yoon, D. Y, Sundarajan, P. R., Flory, P.J. (1975 b), Macromolecules 8, 784. Young, R. N., Quirk, R. P., Fetters, L. X (1984), Adv. Polym. Sci. 56, 1 (see Tables 17 and 22 for relevant references). Ziegler, K. (1936), Angew. Chem. 49, 499. Ziegler, K., Dersch, R, Willthan, H. (1934), Justus Liebigs Ann. Chem. 511, 13. Ziegler, K., Jacob, L. (1934), Justus Liebigs Ann. Chem. 511, 45. Ziegler, K., Jacob, L., Willthan, H., Wenz, A. (1936), Justus Liebigs Ann. Chem. 511, 64.

1.9 References

31

General Reading Bovey, F. A., Winslow, F. H. (Eds.) (1979), Macromolecules — An Introduction to Polymer Science. New York: Academic. de Gennes, P. G. (1979), Scaling Concepts in Polymer Physics. Ithaca, NY: Cornell University Press. Flory, P. I (1953), Principles of Polymer Chemistry. Ithaca, NY: Cornell University Press.

Mandelkern, L. (1983), An Introduction to Macromolecules, 2nd ed. New York: Springer. Treloar, L. R. G. (1970), Introduction to Polymer Science. London: Wykeham. Wunderlich, B. (1973), Macromolecular Physics, Vols. 1-3. New York: Academic. Young, R. X, Lovell, P. A. (1991), Introduction to Polymers, 2nd ed. London: Chapman and Hall.

2 Amorphous Polymer Microstructure Frank T. Gentile Cytotherapeutics, Inc., Providence, RI, U.S.A. Ulrich W. Suter Institut fur Polymere, ETH Zurich, Switzerland

List of 2.1 2.1.1 2.1.2 2.1.3 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.4.1 2.2.4.2 2.3 2.3.1 2.3.1.1 2.3.1.2 2.3.1.3 2.3.1.4 2.3.1.5 2.3.2 2.3.2.1 2.3.2.2 2.3.3 2.3.3.1 2.3.3.2 2.3.3.3 2.4 2.4.1 2.4.2 2.5

Symbols and Abbreviations Introduction What Is Amorphous? What Is a Polymeric Glass? Why Is There a Need for the Detailed Atomistic Polymer Microstructure? History of Atomistic Modeling (Metropolis) Monte Carlo Methods Static and Quasi-Static Molecular Mechanics Methods Molecular Dynamics Methods What Kinds of Properties Should Be Predictable? Polymer Structure Properties Derived from the Structure Atomistically More Detailed Methods - Structure Prediction A Static Model Representation of Energy Initial Guess Generation Potential Energy Minimizations Example - Atactic Poly(propylene) Extension to Polar Polymers - Atactic Poly(vinylchloride) and Polycarbonates Prediction of Structure Cohesive Energy Density and Solubility Parameter Polymer Microstructure Prediction of Properties in the Quasi-Static Approximation Local Molecular Processes in Polycarbonate Diffusion of Small Molecules in Glassy Polymers Mechanical Properties Atomistically Detailed Methods - Dynamic Behavior Constrained Molecular Dynamics in Cartesian Coordinates Constrained Molecular Dynamics in Generalized Coordinates References


34 37 37 39 40 40 41 41 42 45 45 46 46 46 47 50 51 51 53 55 55 57 61 62 63 65 69 70 72 74

34

2 Amorphous Polymer Microstructure

List of Symbols and Abbreviations a a(Tg) a0 D D (r) E Eij9 Et ft ^conf F inter G q gtj (r) I (Q) hj ( 0 k k^ L Nt iVat Nej p^t P- V- T Q qt qt, qt q^;i r R R1 R2 R* #sd rf 2 + < / > 2 S ( 0

(2-13)

2

where and denote the averages of the scattering power, ft{Q\ for all scatters i in the sample and are independent of the detailed structure. Expressing the intensity and structure factor as a function of Q only implies that the scattering is completely elastic. The structure factor S(Q) contains all structural information of the material and must incorporate the effect of thermal motion. To account for shortcomings of static models to properly account for dynamic terms, Ludovice and Suter (1990) employed a spherically symmetric Gaussian distribution of thermal motion, so that (Wagner, 1978):

i= 1

rNi

J=l

-i

(2-14)

2.3 Atomistically More Detailed Methods - Structure Prediction

where Nt is the number of atomic species i9 A is the root mean square displacement of the atomic motion taking approximate account of thermal motion (Debye-Waller factor) (Azaroff et al., 1974) and Iis(Q) is the partial interference function: 0

\l

(2-15) where Q0 is the number density of atoms and gij(r) is the pair distribution function which now includes contributions that are independent of conformation. The value of A is usually between 0.3 and 0.4 A for organic molecules (Vainshtein, 1966); 0.3 was used in this case. The overall interference function may be numerically evaluated to yield S(Q) and I(Q). Figure 2-10 shows an application of this method to neutron scattering performed on PC. The experimental curve from Cervinka et al. (1987) is compared to simulations from an ensemble of 13 PC structures in periodic cubes of edge length equal to 18.5 A (Hutnik et al., 1991 a). Intensity is in arbitrary units. The comparison is quite good here, but there are often considerable discrepancies between these simple estimates and experiment. The source of these differences is currently under investigation.

61

Other Structural Features Visual examinations of space filling molecular models yield qualitative information on the degree of homogeneity of the structures. Such structural visualizations showed that there are large heterogeneities in PC in the form of "empty space" in the bulk polymer with characteristics dimensions on the order of 3-5 A in diameter. A more rigorous analysis of the empty space analysis was performed by Arizzi et al. (1990). These researchers investigated the interstitial space in the structures as a prelude to the study of diffusion in glassy polymers. This analysis confirmed the existence of a great deal of volume not occupied by atoms in PP and PC, there being, however, considerably different distributions of "empty volume". 2.3.3 Prediction of Properties in the Quasi-Static Approximation

If a molecular process is slow compared to the local vibrational time scale (i.e., r < c a . 10" 10 s), simulations that rely on the faithful reproduction of molecular motion on an atomistic level are not appropriate. An effective approach can instead be employed which departs from the static,

Experimental

Figure 2-10. Comparison between experimental and simulated PC neutron scattering behavior.

Predicted (a = 18.5 k) 0.0

1.0

2.0

Q in A"

3.0

62


fully relaxed structures described on the previous pages ("zero-temperature" structures at some reference density). These "fully relaxed" systems are then subjected to changes in one or a few of their degrees of freedom in small increments with all other degrees of freedom being continuously adjusted to retain a structure of minimal energy. This process "drives" the system along the bottom of a "valley" of potential energy towards a saddle point; the process is quasi-static and reversible in that a reversal of the process lets the system retrace its path in configuration space exactly. The change in potential energy upon these changes is a rough approximation to the mean effect a dynamic system would display for the same changes in driven degree^) of freedom. If the procedure is carried in one direction sufficiently far that a saddle-point is reached, it typically results in an uncontrolled (dissipative) change in structure and a precipitous drop in potential energy; the saddle-point is then viewed as the "activated state" which is still in "equilibrium" with the starting state and the difference in potential energy is a rough approximation to the activation energy of the process being investigated. Rates, finally, can be estimated from this activation energy and estimates of the classical partition functions of the starting and saddlepoint structures can be determined. This method to derive information on the response of detailed structural models to stimuli of various nature is termed quasistatic. In the following, some applications are discussed. 2.3.3.1 Local Molecular Processes in Polycarbonate Hutnik et al. (1989) used PC microstructures to investigate local dynamics of phenylene rings at glass densities. Experi-

ments indicate (Jones et al., 1983; Schaefer et al., 1984; Schmidt et al., 1985; Roy et al, 1986; Bayer AG, 1989) that the phenylene rings in PC undergo oscillation about their mean positions and "ring flip" processes in which the phenylene moieties rotate by about 180 degrees in a relatively short time. These processes are the focus of the computations reviewed here. The energy barrier of a "ring flip" is due to a combination of intramolecular and intermolecular forces. Molecular modeling techniques such as quantum mechanical calculations may be used to determine the intramolecular contribution to this activation energy, but at glass densities, intermolecular contributions are high and could potentially dominate the ring-flip barrier (Hutnik et al, 1989). The frequency range of this dynamic process in PC below the glass transition temperature is on the order of 108 s ~ \ indicating that a quasistatic approach would seem appropriate. To calculate the barrier, one begins with the relaxed static structure. Then, a phenyl ring is incrementally rotated and one of its torsion angles fixed in a specified position while the entire rest of the static structure is allowed to relax. Differences in energy between the "totally relaxed" structure and the newly relaxed structure (with one "fixed" phenyl ring) are recorded. The process is repeated until a flip is actually performed in a laboratory reference frame. Every ring of the microstructure is so treated, yielding a sufficient statistical sample. Microstructures of degree of polymerization of 35 (17 repeat units, cube edge length = 18.5 A) were used in this study. One can make the following observations for the polycarbonate ring flip with respect to the structure: in general, in the "ascent" to the energy "peak", the conformational changes which occur are completely reversible; after, they are not; this


behavior corresponds to the unrestrained relaxation from a saddle-point configuration in transition state theory (Hutnik et al., 1989). Furthermore, when viewing the entire structure, one sees that the conformational changes in the chain are far reaching and involve all parts of the packed sample to the borders of the periodic microstructure, indicating that a ring flip actually involves matter in a volume larger than the one modeled. The mean value of the phenyl ring flip activation energy calculated for 34 phenyl ring flips is equal to 10.4 + 6.0 kcal/mol. This is in good agreement with values determined by solid state NMR experiments: 12 kcal/mol determined by proton spin lattice relaxation (Jones et al., 1983); 11 kcal/ mol and 12 kcal/mol determined by 13 C NMR dipolar rotational spin echo experiments (Schaefer et al., 1984; Roy et al., 1986 respectively); and 9.1 kcal/mol by deuterium NMR (Schmidt et al., 1985). [It is also interesting to note that the apparent activation energy of the "^-process" at approximately — 100°C is 12 kcal/mol by dynamic-mechanical spectroscopy (Bayer AG, 1989).] A number of further experimental facts also found their matching counterpart in these simulations, e.g., frequencies of ring oscillations or variation in phenylene-flip axis (Hutnik et al., 1991 b). A similar simulation can be done to determine the changes in the carbonate groups. Hutnik et al. (1991 b) have "driven" 24 carbonate groups by the same method and determined the carbonate group flip activation energy to be 9.9 + 5.5 kcal/mol. This is in good agreement with measurements by dielectric relaxation, 9 kcal/mol (Bayer AG, 1989).

63

2.3.3.2 Diffusion of Small Molecules in Glassy Polymers Unoccupied Space Analysis In an attempt to clarify the phenomena of diffusion and sorbtion of light gases in glassy polymers, Arizzi et al. (1990) started with an analysis of the atomistic interstitial space. This analysis is based on the amount of space available to a hard-sphere "guest" in an atomistic hard-sphere matrix consisting of static amorphous packings of polymer chains, where the spatial coordinates of every atom are known. Given a spherical diffusant with a fixed radius, a quantitative determination of the space available around each polymer atom is required as well as a description of how these local portions of empty space are distributed across the system. First, a Delaunay tessellation (Dirichlet, 1850) is carried out. This is a deterministic filling of space which is dual to the Voronoi tessellation and is defined (in 3-D) as the network of tetrahedra formed by the lines joining nearest neighbors in a Voronoi sense and has the important feature of guaranteeing a unique subdivision of the sample volume into tetrahedral bits that are "empty". A very efficient algorithm of Tanemura et al. (1983) for Voronoi tessellation can be used, a side-product of which is the desired Delaunay tessellation. The space effectively available to sorbtion and diffusion in each tetrahedra is then calculated, by subtracting from the volume of each tetrahedron the pyramidal sectors occupied by the hard-sphere polymer atoms at its vertices. The diffusant (the "guest"), depending on its radius, will be able to move between the tetrahedra in some regions of the static glass (where the faces of the tetrahedra contain unoccupied area sufficient to let the guest pass), whereas its movement beyond these domains will be

64


made impossible by denser packings of matrix atoms. This defines a cluster: a part of the polymeric structure where a diffusant of a given size can move from one tetrahedron to a neighboring one. It is obvious that, for a particular structure, the clusters for the diffusants of smaller radii will be larger or more numerous or both. The cluster-analysis allows for a simple characterization of space available to guest molecules and provides opportunity for visualization of that space as well as for simple considerations in relation to diffusion (the motion of a diffusant internal to a cluster can be considered to mimic a process with an activation energy much lower than the one required when moving outside a cluster). Frequency distributions of cluster volume for He, O 2 and N 2 averaged over 10 PC structures of cube edge length 18.5 A are shown in Fig. 2-11 (Arizzi et al., 1992).

diffusion in amorphous metals, one can approach modeling of the diffusion coefficient of small molecules in a glassy polymer based on activated state theory with the static polymer microstructures. The Delaunay tetrahedra are viewed as sites of preferential residence ("ground states") from which the diffusant "jumps" to a neighboring tetrahedron via an activated process. The energy barriers that a diffusant has to overcome in order to move from a position of local energy minimum to a neighboring one are estimated from the simple force fields discussed earlier. A Monte Carlo simulation in which both the random walk direction and the residence time in each local energy minimum are stochastic variables and functions of the energy barriers is then performed. This model can be applied to different gas/polymer systems. The theory of stochastic processes (Chandresekar, 1943) states that in three dimensions

Simulation of Diffusion

D= lim/(6 0

Following Langonet al. (1985), who have developed an interesting method to model

where r2 is the squared displacement of a diffusant, achieved in time t. D is the diffu-

6-i

6

10BPA-PC structures Diffusant :He

5-

(2-16)

5-

10BPA-PC structures

10BPA-PC structures

Diffusant: 0 2

Diffusant: N2

5-

h-

4-

3-

CO

empty

o

o 2 -I

i

1

o.

Jil lii

400 800 Volume in A3

1200

400 800 Volume in A3

1200

400 800 1200 Volume in A3

Figure 2-11. Distributions of empty space for helium, oxygen and nitrogen in 10 microstructures of PC {X = 35).


sion coefficient. Given a diffusant in tetrahedron i at time t, the probability of it jumping to the neighbor tetrahedron j at time t + dt is co^dt, where co^ is the jump frequency from site i to site j , and can be written in an activated state theory form as: ij = v 0 exp < -

kT

(2-17)

where v0 is the frequency corresponding to a jump without energy barrier and can be deduced from first principles, while Etj and Et represent the energies associated with the diffusant on the boundary surface (gate) between tetrahedra i and j and internally to tetrahedron i, respectively. These energies are calculated following the approach discussed previously in modeling non-bonded interatomic interactions. The residence time in tetrahedron i is a random variable with an exponential distribution (i.e., T = Tjlnx, where x is uniform in [0,1] and zt = 1/S Wij is the mean residence time j

in tetrahedron i). The jump frequency and the associated probability of a step into any of the neighboring tetrahedra are independent of time, while the residence time distribution is independent of the next tetrahedron to be visited, i.e. the process consists of two random walks, independent of each other, in space and time. These techniques have been used so far to accurately predict the permeability selectivity of oxygen and nitrogen in PC as well as the diffusion constants (Arizzi, 1990). 2.3.3.3 Mechanical Properties

To model the response (e.g., the conformational changes in the polymer matrix) of macromolecular packings to mechanical loading, one can perform simulated "deformation" and "relaxation" experiments on

65

these static microstructures. Strain appears in this treatment as a change in the rules governing translation for spatially periodic continuation conditions. Basically, one can attempt infinitesimal or finite plastic deformation, where the latter will lead to "plastic" events, while the former can be compared to elastic deformation of real materials. The history of predicting mechanical properties of polymeric materials from first principles is not extensive and what little has been done has dealt almost exclusively with the crystalline state, since the existence of a well defined periodic structure allows for drastic simplifications. An early statistical mechanics treatment of crystalline poly(methylene) by Pastine (1968) gave aP-V-T equation of state that compared well with experimental evidence. Later, Tashiro et al. (1977 b, 1978) derived elastic constants for a variety of crystalline polymers to within 35% of observed quantities. Theoretical approaches to determining mechanical properties of amorphous polymers, until recently, were phenomenological or "group-contributional" in character. Haward and MacCallum (1971) proposed that the adiabatic compressibility of a polymeric glass were determined mainly by intermolecular forces. Using a LennardJones relationship for potential energy they showed that the adiabatic compressibility was a function of V/Vo, where V is the deformed state molar volume and Vo is the molar volume in a state where the potential energy with respect to volume at constant temperature is minimal (Theodorou, 1985). This correlation compared well with experimental evidence, indicating that intermolecular forces are relevant variables for "elastic" mechanical properties. Van Krevelen and Hoftyzer (1976) proposed the use of the Schuyler correlation to

66


estimate the compressibility for amorphous polymers from the molar voume, V9 and wRao, the Rao function or molar sound velocity. Tables (Van Krevelen and Hoftyzer, 1976) exist for group contribution estimates of V and uRao. From these, the compressibility is obtained which can then be used to determine the Young's modulus, £, and shear modulus, G, once a suitable estimate for the Poisson's ratio has been made. These correlations however fail in many cases. A more sophisticated approach was introduced by Yannas (1975) and Yannas and Luise (1982, 1983, 1991) which considers molecular level displacements brought on by mechanical deformation of the glassy polymer. Assuming that deformation of a macromolecular segment occurs primarily by rotation around skeletal bonds, rather than by bond angle or bond length distortion, Yannas coined the term "strophon" to characterize a three-virtual-bond-long segment as the smallest possible chain section that can change its end-to-end distance during mechanical loading. Forces required to overcome intramolecular and intermolecular barriers to bond rotation in the bulk were estimated based on a simple strophon model. The estimated contributions of both types of forces to the smallstrain modulus was used as a basis to distinguish which was more important during deformation. It was found that both intraand intermolecular forces were of similar significance in both vinyl polymers and non-vinyl polymers (Luise and Yannas, 1990). The calculations of Yannas and his coworkers were performed on many amorphous polymers using only the polymeric repeat unit as a basis. They pinpointed a dimensionless parameter, called the "locking factor" which is equal to the value of the radius of a circle describing the rotation of a characteristic virtual bond around

the chain backbone bond (the moment arm of the strophon), r, divided by the van der Waals distance between two strophons in neighboring chains which interact according to a Lennard-Jones potential, a0. Calculations of this locking factor can point to different mechanisms of deformation; large values of r/a0, it was found, encourage stress transfer to occur during intermolecular pathways during loading. These researchers also proposed that yielding and the glass transition are related through a critical ratio of the inter-strophon distance at Tg, a(Tg), and its value near absolute zero, a0. Both yielding and the glass transition occur at an "equivalent dilatational threshold" which is crossed either by heating a glass above its Tg, or by subjecting it to a yield stress greater than a critical value, Tm (in order to reduce the number of "intrinsic" nuclei, which are essentially heterogeneous nuclei) is taken to Tc and, as soon as it reaches thermal equilibrium, seeded with the crystals grown in a separate experiment. This procedure makes it possible to grow large crystals or mixed crystals made of successive rings of polymers differing in molecular weight or structure. Yet a further variation on this theme has led to the isochronous decoration technique, which is based on abrupt modifications of crystallization conditions of the solution: crystals grown at a given Tc are used as seeds for growth at a different Tc within the same solution. As shown by Dosiere et al. (1986), the crystallization temperature can be changed by 5 °C within 20 s in a cylindrical vessel of 1.5 mm radius and results in a sharp increase or decrease in the lamellar thickness of the single crystals which can easily be visualized by conventional shadowing in electron microscopy. Recently, an original method to produce polymer nuclei has been proposed (Ding etal., 1989). It rests on the fast quench-

94

3 Structure of Polymer Single Crystals

ing of a polymer solution [in this case polyethylene oxide)], which produces a liquid-liquid phase separation and crystallization of small polymer globs. The authors claim that, under these conditions, the globs are ideal for further seeding of polymer solutions. The complexity of the resulting morphologies and the general validity and potential improvements over the more conventional self-nucleation technique remain however to be assessed. 3.3.1.3 Other Crystallization Procedures

Crystallization procedures based on thermal treatments as described in the previous sections enable the best control of crystallization conditions and resultant morphologies. A wider palette of different crystallization procedures can however be used and may yield quite perfect crystals: - The method developed by Patel and Patel (1970) is based on the slow extraction of a glass slide from a solution of polymer. Crystallization takes place at constant temperature, in relatively thin layers, while the solvent is evaporating. Although growth conditions are ill defined, the method is suited for a number of polymers, including polyolefins considered above. - Slow change in chemical potential via progressive diffusion of a nonsolvent into the polymer solution [e.g., water or alcohol in a polymer solution in acidic solvent, for example trifluoroacetic acid (TFAA)], by progressive evaporation of the volatile solvent (e.g., TFAA in a TFAA/trifluoroethanol mixture) or by dialysis of an ionic solution (e.g., LiBr solution). These methods are well adapted for polar, strongly hydrogen-bonded polymers, notably polypeptides and polyamides with short methylene segments. To summarize, growth conditions required for the production of polymer single

crystals are essentially similar to those used for low molecular weight materials. They are based on the same overall basic approach, i.e., on a small change in the chemical potential of the solution via drift or change of concentration, temperature, solvent power, etc. The major originalities lie in the possible control of self-nuclei by thermal treatment and in the fact that only the lateral dimensions of the lamellae can be monitored to some extent (usually between ~ 1 Jim and 100 jim, i.e., no more than two orders of magnitude) whereas the third dimension - the lamellar thickness remains in the 10-30 nm range. 3.3.2 Growth of Lamellar Crystals from the Melt

Polymer single crystals can be obtained under specific crystallization conditions that imitate the melt, or directly from the melt. The first approach is due to Keith (1964), who crystallized polyethylene in a low molecular weight paraffinic diluent (typically C 32 H 66 ), which has the same chemical structure as polyethylene, but a much lower melting temperature and viscosity. Crystallization conditions therefore mimic the conditions experienced by the high molecular weight fraction of the polymer in the presence of the still molten lower molecular weight part acting as solvent. The resulting polyethylene single crystals display significant features characteristic of bulk crystallization, e.g., lenticular shapes as opposed to lozenge type and chain tilt in the lamella. However, similar morphologies have been obtained using poor solvents (e.g., n-dodecanol) which also result in rather high Tc (Khoury, 1979; Organ and Keller, 1985, 1987 b). Crystallization of polymers from the melt in the form of single crystals is very uncommon. Indeed, under usual condi-

3.3 Growth and Occurrence of Polymer Single Crystals

tions the supercooling needed to observe nucleation is ten or several tens of degrees, i.e., a temperature range where high growth rates are incompatible with single crystal growth. Furthermore, several disordering processes exist in the bulk crystallization of polymers (e.g., lamellar twisting) which result in complicated, three-dimensional morphologies. Therefore, a combination of slow growth rate and thin film growth are often sought. The first single crystals produced by bulk or thin film polymer growth were for polymers with "inherent" low growth rates because of their complex molecular architecture or size of side chains. Isotactic polystyrene single crystals (Keith et al., 1970) could be extracted from the surrounding melt after quenching and dissolution of the, by then, vitrified melt. Similarly, random block copolymers of poly(butyleneterephthalate) and poly(tetramethylene ether glycol) can form single crystals when grown from the melt in spite of their molecular complexity; fractionation by sequence length of the crystallizable poly(butyleneterephthalate) was demonstrated (Briber and Thomas, 1985). Structures with polygonal shapes reminiscent of single crystals have been reported, e.g., for polytetramethylsiloxane (Magill, 1964). A more systematic investigation of single crystals grown from the bulk had to await the development of the self-nucleation procedure. Controlled production of significant concentrations of nuclei makes it possible to overcome the low nucleation rate at very low supercoolings, i.e., in a temperature range where growth conditions are compatible with single crystal development. The two examples examined so far - polyethylene and poly (ethylene oxide) - do indeed indicate that, however complicated spherulitic textures are produced at

95

low Tc, lamellar single crystals grow at the very high end of the crystallization temperature range. In a similar manner, isotactic polypropylene produces at 160°C very extensive lamellae virtually devoid of the lamellar branching (Bassett and Olley, 1984) that occurs more profusely at lower Tc (Olley and Bassett, 1989). Examination of the resulting single crystals is difficult as they are embedded in a surrounding matrix of the same polymer, usually solidified by rapid crystallization on quenching. Staining or etching techniques may be used (Bassett et al., 1988). Conversely, rapid quenching induces crystallization of spherulites and birefringent contours on the crystal edges, which can be observed by optical microscopy in polarized light (Kovacs and Gonthier, 1972; Labaig, 1978) or selectively dissolved (DiCorleto and Bassett, 1990). 3.3.3 Single Crystals of Biopolymers

Chain folded polymer single crystals illustrate a most efficient means of packing long chain molecules. It is not surprising therefore that a number of biopolymers adopt spontaneously a lamellar morphology, or have been obtained in lamellar form. Most characteristic among them are polysaccharides, cellulose and its derivatives, various polypeptides (Padden and Keith, 1965; Keith et al., 1969) including structural models of fibrous proteins (Lotz and Brack, 1979) and antibiotics (Lotz etal., 1977) and fibrous proteins themselves, such as silk of Bombyx mori (Lotz et al., 1982). All these single crystals were obtained using experimental procedures based on melting or dissolution in appropriate (often polar) solvents followed by cooling or dialysis. The most striking illustration of chain folding among biopolymers is provided by

96


the DNA of bacterial viruses. The head of the E. coll bacteriophage T2 is a hexagonal casing containing a single molecule of DNA - 60 jam long. The DNA can be visualized in its integrity after being released via an osmotic shock (Kleinschmidt et al., 1962). From X-ray diffraction and birefringence studies combined with the shape of the head and the DNA molecular length, the picture of a head made of a chain folded "single molecule single crystal" of DNA was proposed early on (Bendet et al., 1960). Production of hexagonal single crystals of DNA has been repeated in vitro by using fragments of moderate molecular weight, but of sufficient length to establish chain folding in the platelets ~ 1 |im across and ~ 15 nm thick (Giannoni et al., 1969). The key feature of the experimental procedure appears to be heating in a temperature range where the DNA starts to "melt", i.e, where close connectedness of its two strands is partly disrupted by breaking of hydrogen bonds without however reaching the denaturation stage with its associated separation into individual strands. 3.3.4 Polymers Obtained in the Form of Single Crystals

Most crystalline polymers have been obtained in the form of single crystals by one or many of the experimental procedures described so far. A compilation made by Wunderlich (1973, 1976, 1980) describes the experimental conditions, the resulting morphologies and some structural analysis for nearly 50 polymers of various chemical structure: polyenes, polyvinyl derivatives (alcohol, chloride), polyacrylonitrile, polyacrylic acid, polyethers including polyoxymethylene and polyoxyethylene, polyesters, both aliphatic and aromatic [e.g., polyethylene terephthalate)], various polyamides and polypeptides, and cellulose and

its derivatives (e.g., cellulose triacetate). Single crystals of polysiloxanes (e.g., polydipropylsiloxane [Si(C 3 H 7 ) 2 -O] n are also described, as an example of a polymer with a non-carbon backbone. The compilation made by Wunderlich covers roughly the first 15 years after the emergence of polymer single crystals as a research domain in its own right. In the early days, the main purpose was to demonstrate the feasibility and generality of chain folding for polymers of various types. In more recent years however, the scientific motivation when producing single crystals appears to have shifted: whereas the possibility of chain folding is no longer questioned, single crystals are mainly produced as a convenient investigation tool to analyze a number of structural, kinetic, and thermodynamic aspects of polymers. Although similar motivations clearly existed also in earlier studies, this significant shift in emphasis indicates the maturity of the field. Several such applications will be developed in the following sections. The next two sections deal with the structural aspects of the crystalline core and the lamellar surface, and the interactions between these two regions of single crystals. 3.3.5 Nonfolded or Chain Extended Polymer Single Crystals

Besides the conventional chain folded single crystals described above, other types of polymer single crystal morphologies have been produced, using different routes. These morphologies have in common the fact that the chains are either fully or nearly extended, resulting in lamellar thicknesses in the micrometer range or in the absence of lamellar periodicity. Experimental routes to produce such crystals are quite specific and cannot be generalized to all polymers. We will consider in succes-

3.3 Growth and Occurrence of Polymer Single Crystals

sion: the topochemical polymerization of a monomer into extended chain polymers, crystallization under high pressure (notably of polyethylene), and the formation of crystal solvates. 3.3.5.1 Solid State Polymerization

"True" polymer single crystal fibers can be obtained by polymerization of a crystalline monomer when polymerization proceeds along a well-defined crystallographic axis of the monomer crystal and the polymer lattice has parameters close to that of the monomer: this topochemical polymerization can therefore result in polymer single crystals that have the dimensions of the monomer single crystal (which can be macroscopic) and with polymer chain lengths determined by those dimensions. An early example is the polymerization of trioxane into polyoxymethylene (CH 2 -O) n ; it is not ideal however due to differences in lattice spacings of the monomer and polymer, which results in a fiber habit with poor mechanical characteristics. The best known and most studied system is the solid state polymerization of substituted diacetylene monomer crystals to produce the crystal made of extended polydiacetylene chains. Considerable work has been devoted to these systems; the reader is referred to reviews by Wegner (1979), Bloor (1982) and Young (1987). 3.3.5.2 Crystallization Under High Pressure

Crystallization under high pressure of polyethylene produces lamellar crystals with thickness in the micrometer range, i.e, with a much reduced impact of chain folding. This morphology will be examined in more detail in Sec. 3.5.6, in Chap. 4 of this Volume, and in Sees. 2.1.4.1 and 2.1.5,

97

where the role of mobile phases involved in lamellar thickening is developed. 3.3.5.3 Inclusion Compounds and Molecular Complexes

Polymers can form inclusion compounds or molecular complexes with low M. w. organic molecules. Typical examples include PEO and paraffins or aliphatic polyesters with urea or perhydrotriphenylene, PEO with mercuric chloride (Tadokoro, 1979; Farina and Di Silvestro, 1980), and syndiotactic poly(methylmethacrylate) with, e.g., chloroacetone (Kusuyama et al., 1983). The PEO-urea complex can be formed by soaking an oriented PEO sample in a methanolic urea solution. Formation of the complex yields a macroscopic ( « 1 mm) single crystal; on desoaking the complex in a methanolic solution the original PEO sample is restored. The PEO chain conformation in the complex is essentially that of the crystalline PEO (72-helix). In these systems, polymer chains are encaged in a channel created by the crystalline solvent. The systems are ideal to investigate properties, conformational characteristics and mobility of individual chains, in the absence of their traditional surroundings. 3.3.6 Thermotropic and Lyotropic Polymers

Polymers with rigid moieties in their backbone are a class of technologically important systems. They exhibit liquidcrystal behavior either in the presence of solvents or upon heating (lyotropic and thermotropic polymers respectively). These systems are not normally expected to form lamellar, chain folded crystals. However, a lamellar organization develops in a solution of a typical rigid-rod lyotropic polymer, poly(p-phenylene benzobisthiazole)

98


(Cohen et al., 1982). The polymer slowly coagulates by gradual absorption of moisture in the solution. The crystals have thicknesses comparable to the molecular length and are presumably made of extended chains; furthermore they are most probably crystal solvates, i.e., incorporate solvent molecules bound by hydrogen bonds to the polymer: similar crystal solvates structures are known for a number of rigid rod polymers (Iovleva and Papkov, 1982; Gardner et al., 1983). Thermotropic liquid-crystal polymers have more flexibility in their backbone than lyotropic polymers, often through inclusion of a flexible spacer. When this spacer (for example, a paraffinic segment) is of sufficient length, chain folding can take place and lamellar structures may develop, for example on annealing. This lamellar decoration

technique

of liquid-

crystal polymer films (Wood and Thomas, 1986) can be used to determine director textures in the liquid-crystal structures (Hudson et al., 1990). More recently, single crystals of a related thermotropic liquidcrystal polyester with a main chain C 7 spacer (i.e., based on azeloate) have been reported by Kent and Geil (1991). Lamellar thickness is ^ 1 0 n m whereas chain

Dominant fold orientation

length is « 80 nm, and chain folding is believed to be predominantly reentrant.

3.4 The Crystalline Core of Polymer Single Crystals 3.4.1 General Features of Polymer Single Crystals 3.4.1.1 Overview of the Model

It is most convenient to define the general characteristics of lamellar crystals at the onset. The main features of a polyethylene single crystal grown from solution are represented schematically in Fig. 3-5. The crystal shown here has a lozenge shape, as it is bounded by four lateral {110} growth faces of an orthorhombic unit cell. The crystal is slightly pyramidal, with an overall tentlike morphology. The polymer chains are parallel to the axis of symmetry of the crystal (c-axis of the unit cell) and fold back and forth on the upper and lower fold surfaces (surfaces tangential to the folds), which are thus less ordered than the crystalline core. The schematic representation of the fold structure, conformation and orientation, is more speculative, as only indirect means exist to assess

Growth sector boundaries Unit-cell

Fold surface

Cilia Loops Tight fold

Fold length L (~10nm) Helical or zig-zag chain conformation

{110} Growth faces

Figure 3-5. Schematic representation of a four-sectored polyethylene single crystal with main features of molecular conformation.

3.4 The Crystalline Core of Polymer Single Crystals

99

Figure 3-6. Selected area electron diffraction pattern (center) of PE single crystal seen in bright field (a) and three dark fields obtained by selecting diffraction spots indicated. Note sectorization as revealed, e.g., by differences in (c) and (d) when diffracting planes are parallel or not to growth planes. Scale bar: 3 \im. (c)

these features. For reasons developed in some detail later (cf. Sec. 3.5.4.4), folds are represented as being oriented more or less parallel to the growth faces. They may link adjacent stems in the growth face (sharp folds or, in short, folds) or stems further apart either in the same or in nearby growth planes (loops). Chain ends are represented either impinging at or emerging from the lamellar surface in the form of cilia. Whereas some features may vary with crystallization conditions (three-dimensional shape of the crystal, overall geometry defined by growth conditions), several invariants characterize the polymer single crystal morphology and structure: exis-

(d)

tence of a lamellar thickness /, coexistence of crystalline core and less ordered surface layers resulting in a composite structure at a scale of / (~ 1 to 2 nm), preferred orientation of folds within the various growth sectors. Although the structure is more complicated than low molecular weight material single crystals, it provides a very efficient means to reconcile crystalline order with the structural diversity of polymers, as for example their molecular weight distribution. As an illustration, the crystal shown in Fig. 3-6 has lateral dimensions of 8 j^m x 5 jim for a thickness of about 10 nm. Its mass is 4 x 10~ 12 g; it is made of 2 x 108 stems (38 |im2/0.18 nm2, cross-sec-

100


tional area of a PE chain seen along its c-axis). As a stem of length 10 nm corresponds to a mass of ~ 1000 daltons, the crystal is made, e.g., of ~ 2 x l O 7 chains of molecular weight 104. It therefore comprises ~ 4 x 107 chain ends and ~ 1.6 x 108 folds or loops, divided between upper and lower lamellar surfaces. The above picture and figures indicate that analysis of the structure of polymer single crystals must be made at several levels: whereas the crystalline core structure can be analyzed rather rigorously, at best an average view of the fold surface structure can be achieved. Further, an important part of the structural analysis deals with interplay and mutual interference between crystalline core and fold structure and conformation. 3.4.1.2 The Fundamental Experimental Evidence

Analysis of polymer single crystals and establishment of chain folding rests on a by now familiar approach which supposes determination of chain orientation and comparison of molecular length with stem length in the crystal. Although this experimental evidence can be gained by X-ray diffraction (at wide and low angle) on single crystal mats, the most vivid and illustrative method remains electron microscopy in bright and dark field, and electron diffraction on individual single crystals. The crystal thickness (10-20 nm) is indeed ideal for TEM (transmission electron microscopy) observations. Figure 3-6 is a most classical illustration of electron microscopy on a single crystal of polyethylene. It displays in (a) a bright field micrograph of the crystal. This picture is characterized by dark striations roughly parallel to (130) in the growth sector bounded by the (110) growth face. The stri-

ations correspond to a Bragg contrast, i.e., to areas of the crystals which diffract electrons which are stopped by the objective aperture. Further, the lamellar thickness of the crystal can be determined by the shadow length if the crystal is shadowed with a heavy metal (e.g., Pt-Pd alloy). The single crystal diffraction pattern is represented in the center of Fig. 3-6. It corresponds to the ab reciprocal plane of the unit cell, i.e., it is only composed of reflections which are located on the equator of a PE fiber pattern (a-axis vertical). This pattern unambiguously establishes that the electrons are traveling down the c-axis of the unit cell, i.e., that the chains are normal to the broad faces of the crystal. Combination of this orientation information, of the limited lamellar thickness and of the molecular length as deduced from the molecular weight, which can be orders of magnitude larger, leads to the concept of chain folding. The dark field pictures correspond to enlargements of the diffraction spots formed by the electrons diffracted by (200), (110) and (1T0) planes (Fig. 3-6 b, c, and d respectively). They complement exactly the bright field picture, as they are formed with the electrons "missing" in (a). Information conveyed by the dark field pictures strengthens the case of sectorization (compare c and d); the origin of the diffraction contrast is analyzed below in relation to the overall shape of the single crystal. 3.4.2 The Lamellar Core and Crystal Defects

Single crystals of polymers provide an easy and quite rare opportunity to observe the crystal lattice down the chain axis, i.e., at right angles to the fiber orientation usually produced by mechanical means (stretching, etc.). They provide therefore a


suitable material to investigate most aspects of crystal structure, defects, impact of deformation, etc. The crystallography of polymers at the unit cell level is considered in Sec. 3.7.1. Here we examine: - The various defects that may affect the crystal structure. These are basically of two types: (1) Crystallographic defects such as twins, edge or screw dislocations. These have counterparts in the structure of nonpolymeric materials; original features linked with the polymer nature are underlined when appropriate. (2) Deformation of the crystal lattice induced by the presence of folds on the lamellar surface, or by the specific growth conditions, etc. - The overall conformation of polymer molecules within individual lamellae (e.g., separation of stems belonging to the same molecule in the crystal core).

101

3.4.2.1 Edge Dislocations Edge dislocations correspond to the creation or annihilation of one or several crystallographic planes. A convenient means to assess the existence and concentration of edge dislocations is via moire patterns produced by two overlapping but slightly rotated crystals (Fig. 3-7). Terminating moire fringes indicate the presence of edge dislocations. Concentration of edge dislocations varies of course significantly with crystallization conditions and increases on annealing, but it is found to be remarkably low in some single crystals. Typical figures are 1011 m~ 2 for polyethylene with a majority of dislocations resulting from creation or termination of fold planes. Further, a 103 fold increase takes place on annealing to 95 °C (Holland, 1964). Poly(4-methyl-l-

Figure 3-7. Moire pattern of superposed PEO-PS block copolymer single crystals. Note the virtual absence, in this crystal, of edge dislocations. Terminating fringes can be seen at the crystal center (from Lotzetal., 1966).

102


pentene) single crystals have comparable concentrations; in polyoxymethylene, fewer dislocations are located preferably in the vicinity of sector boundaries (Bassett, 1964). Moire patterns have been used to investigate structural changes (generation of edge dislocations) in PE single crystals during annealing (Abe et al., 1970). 3.4.2.2 Twins

Twinning is, as in low molecular weight crystals, a most common defect in polymer crystal. Twins are induced during growth or as a result of deformation. It is customary to define reflection twins, in which the twin plane introduces an extra element of symmetry in the twinned crystal, and rotation twins, in which the two parts of the crystal are rotated about an axis which may or may not be an element of symmetry of the crystal lattice (Barrett and Massalski, 1966). Molecular connectedness through successive unit cells along the chain axis seriously limits the number and frequency of possible twinning modes of polymers. Bevis (1978) considered the possibility of twin planes which either avoid or result in a change of molecular orientation (socalled chain-axis invariant and chain-axis rotation twins). This classification has been extended recently by Martin and Thomas (1991) who describe possible grain boundaries in extended-chain polymers and include the notion of axial and lateral planes between grains. It is of course directly relevant to twins in polymers. In this section, growth twins are considered, as illustrated by reflection and rotation twins, and their specificities are analyzed. Twinning induced by deformation is considered in Sec. 3.7.3.2. Reflection twins are the most frequent twins in polymer crystals and indeed the only ones considered so far. They are char-

acterized by a disturbance of the regular packing of molecular stems along the contact surface of the twin components. This disturbance is rather small as the misorientation merely involves a rotation around its axis of a chain that often has a nearly cylindrical shape. As a result, reflection twins in polymers have their composition planes parallel to the c- or helix-axis, i.e., are hkO or chain invariant twins. Single crystals are ideally suited for investigation of these twins, since observation is performed parallel to the twin plane. Investigation is further facilitated when the twin growth habit differs from that of the untwinned crystals (growth conditions which enhance such differences may be purposedly selected). Studies of twinning rely on selected area electron diffraction; twin sectors and even the twin planes may be visualized by dark field imaging. The proportion of twinned crystals can be significantly increased over its natural occurrence by using a variant of the self-nucleation method (Blundell et al., 1966 a). When highly disordered crystals (produced by initial fast quench of the solution) are used as starting material for the self-nucleation procedure, the proportion of twins and multiple twins in the resultant population is significantly increased: up to ~ 50% for polyethylene oxide (PEO, [-CH 2 -CH 2 -O-] M ) as compared to a natural occurrence of less than 5% (Kovacs et al., 1969). Twinned habits have been reported for polyethylene (Blundell and Keller, 1968 a; Wittmann and Kovacs, 1970), for polyethylene oxide (Kovacs et al., 1969), polyvinylalcohol (Tsuboi, 1968), isotactic polypropylene (iPP) (Kojima, 1967), etc. The latter investigation however deals with thin film growth and describes a number of twins which do not correspond to low index twin planes, and are not observed with any frequency in normal growth. The crys-


talline morphology of iPP is actually dominated by another twin mode, of the rotation twin type. The most extensive study deals with twins and multiple twins of PEO. PEO crystallizes in a monoclinic unit cell with, in chain axis projection, 2 a sin p = b. Two twin modes were recognized: (120) twins involving the most densely packed crystal plane, which is also the major growth face, and (100) planes, in which twinned components differ by the mirror symmetry dip of the a-axis. These two twin planes are at 45° to each other and happen to be planes of symmetry for the PEO single crystals which are square or truncated squares. They lend themselves therefore to a systematic classification. It was found that 14 possible twins and multiple twins can exist, comprising up to eight sectors in mirror relationship with their neighbors (Fig. 3-8). Using the modified self-nucleation technique referred to above, 13 out of the 14 possibilities were observed and characterized by diffraction and dark and bright field electron microscopy. A systematic study of twins and multiple twins of polyethylene (Wittmann and Kovacs, 1970) confirms that two twin modes exist for PE: (110) and (310). These twin modes can be rationalized on the basis of FriedeFs rules of twinning by noting the close analogy of the orthorhombic PE subcell and a hexagonal lattice: the two twin planes thus correspond to the two densely packed planes of the pseudo-hexagonal packing (Fig. 3-9). However, the slight departure from hexagonal symmetry of the PE lattice does not allow a systematic classification of the multiple twins as in the case of PEO. Multiple twins comprising up to six different sectors have been observed. (110) twinning leads to highly elongated crystal laths, because the twin plane acts as a continuous nucleation site for the

103

3b

4b

Figure 3-8. The rationale of PEO multiple reflection twins. Untwinned crystal (top, growth sector boundaries represented by dotted lines) and the 14 simple and multiple twins resulting from all conceivable associations of (120) and (100) twin planes (represented by full and broken lines, respectively). The arrow represents the dip of the a-axis in the monoclinic unit cell (from Kovacs et al., 1969).

104


67.4°

126.9°

**# G II0 , GIII0 h AH i / /0 k L /, ll912 /min /* §/

crystal axes coefficients defined in the rate equations for backward and forward steps in creating a nucleus free energy change on crystallization rate at which a nucleus once formed spreads across the surface crystal growth rate crystal growth rate in regime I, II, III constant governing the absolute crystal growth rate constants governing the absolute crystal growth rate in regime I, II, III Planck constant enthalpy change on crystallization surface nucleation rate overall nucleation rate constant governing the absolute nucleation rate Boltzmann constant substrate length crystal thicknesses minimum stable crystal thickness initial lamellar thickness mean crystal thickness =/;-/mio

JV0 R S(l) T Tn T^ T^ [/*

n u m b e r of molecules gas constant flux absolute temperature nucleation temperature thermodynamic constant equilibrium melting point thermodynamic constant

p

crs a os Aa A(/>* \jj

transport term representing the rate at which molecules arrive at the surface constant surface free energy surface energy at the ends of a box enclosing a molecule, where the chain emerges or folds interfacial energy between the crystal a n d the foreign surface interfacial energy between the foreign surface a n d the melt = a + 0 for all three mixtures, they thus have a UCST type phase behaviour (Fig. 6-6 a). For a given DPB (e.g., DPB-63-268), #eff is observed to increase, and hence the miscibility is suppressed with decreasing the 1,2-butadiene unit of HPB from 68 to 7 wt.% (cf. the results indicated by the straight lines numbered 1 and 3). For a given HPB (e.g., HPB-7-10 or HPB-11-52 having the nearly equal fraction of 1,2-linkages), #eff tends to increase and hence the miscibility is suppressed with increasing the 1,2-unit of DPB from 11 to 63 wt.% (cf. the results numbered 1

263

6.2 Unique Features in Polymers

Table 6-1. Characteristics of polymers. (a) Protonated polyisoprene Sample code

M

HPI-7-101 HPI-12-192 HPI-15-136

a

Microstructure b in %

xl0~

1.0, 1.03 1.05

100 192 136

3,4

Cis-1,4

Trans-1,4

7 12 15

69 66 63

24 22 22

1,2

(b) Deuterated polybutadiene Sample code

DPB-12-52 DPB-16-61 DPB-20-103 DPB-28-71

M

a

Microstructure b in %

xl0~

49 59 103 71

1.07 I.O3

1.05 1.02

1,2

Cis-1,4

Trans-1,4

12 16 20 28

36 38 36 27

52 46 44 45

a

Number average molecular weight M n and weight average molecular weight M w , determined by GPC equipped with low-angle light scattering apparatus; b microstructure determined by 13 C-NMR.

Table 6-2. Sample characteristics for HPB/DPB blends. Microstructure (%) ( 13C-NMR)

Mnaxl0~3

MJMn>

(GPC)

(GPC)

1,2

Cis-1,4

Trans-1,4

HPB-7-10 HPB-11-52C HPB-68-243

6.9 b 4.9 135

1.5 1.05 1.8

7 11 68

40 36 11

53 53 21

DPB-11-276c DPB-63-268

252 134

1.1 2.0

11 63

36

53 *

Sample code

a

M n : Number average molecular weight, M w : weight average molecular weight; b determined by vapor pressure osmometry; c from Bates, Dieker, and Wignall (1986); * undetectable due to spectral broadening.

and 2). Interestingly, the effect of the microstructure is opposite for DPB and HPB, the origin of which may well be understood in terms of the copolymer effect, as described below (e.g., using Eqs. (6-14) and (6-15), as well as the results shown in Fig. 6-6 b).

DPB and HPB are considered to be random copolymers of 1,2- and 1,4-deuterated and protonated butadiene units, respectively (designated (Dl,2) x -r-(Dl,4) 1 _ J C and (H1,2)y - r - (H1,4) x _ y, respectively, x and y being the weight fractions). Thus in the context of the mean-field theory, %eff

264

6 Structure of Polymer Blends

2,5

2.0

Temperature (°C) 100 50

150 1

30

1

i ^

—

may be expressed in terms of "fundamental" segmental x values between monomers with different microstructures (ten Brinke et al., 1983),

^ ^

1

Xeff

=

^ - - ^

1.5 o

5 i.o

(6-14)

AC, the critical composition. 6.5.1 Scaling Analyses of Im(t; T) and qm(t;T) Here we discuss the validity of the scaling postulate proposed by Chou and Goldburg (1979) for mixtures of simple liquids near the critical point. It postulates that the time changes of / m and qm obtained at various temperatures T can be scaled by the temperature dependent correlation length £(T) and mutual diffusivity D app (T). Thus the reduced wavenumber Qm(x) is defined by GmW = «m(T)5(T1)

(6-43)

The data obtained for qm {t; T) at various t and T should fall onto a single master curve when they are plotted as a function of the reduced time i, as defined by Eqs. (6-6) and (6-42). Similarly the reduced scattering intensity Tm{z) is defined by

I{q9t,T)q2dq (6-44)

272


Data Tm(r) obtained at various t and T from I(q;t,T) should fall onto a master curve when they are plotted as a function oft. In binary mixtures of simple liquids, t;(T) and D app (T) were determined by static and dynamic light scattering (Chu et al., 1969; Gulari et al., 1972) in the single-phase state as a function of T. Then the values at a given phase separation temperature T were obtained by extrapolating them using the following scaling laws, = fole T r v

critical mixture of SBR1/PB19 with a composition of 50/50 wt./wt. at deep quenches of eT ~ 1 (Izumitani et al., 1990). Figure 6-9 shows the time-evolution of light scattering profiles at 60 °C where the scattered intensity (in arbitrary units) was plotted as a function of the scattering vector g, and the time elapses in the order of (a) to (c). 180000

14-0000 -

(6-45) 100000 -

and (6-46) Equations (6-2) and (6-7) correspond to the mean-field approximation, v = l/2 and y = l. Note that the 3d kinetic Ising model for the critical regime of sT 438.85 h 500.61

3 20 o

/

in

6 0f-

No. /t 1mln ) „ o »

+

(c)

+

o

4-0-

x

_

• *

• +

64.86 73.39 81.92 109.43 137.15 192.45 220.91

1

0

1

2

x = q/qm(t)

Figure 6-12. Scaled structure factors F(x) obtained at 40°C for the SBR1/PB19 58 wt.%/42 wt.% mixture where the time elapses in the order of parts (c) to (a). In part (c), F(x) monotonically increases with an elapse of time but in parts (a) and (b) is essentially independent of time. The reduced time T is obtained by dividing the real time by tc = 2000 s. From Takenaka etal. (1990).

t. The characteristic time tc at 40 °C is 2000 s for this blend. Parts (b) and (a) cover the time scale t from 220.9 to 500 min or T from 6.6 to 15.0 and t from 500.6 to 1693.7 min or T from 15.0 to 50.8, respectively, and hence correspond, respectively, to the stage extending from the final part of the intermediate stage to the beginning of the late stage, and to the late stage. In this

276


as discussed in Sec. 6.2.1. Truly unique features of polymers beside these effects will be discussed briefly in Sec. 6.5.3. The detailed functional forms of fm(T) and Qm(x) reflect the ordering mechanisms, information on which is still limited, and should be fully explored in future for many polymeric blends. F(x) uniquely describes the morphology of the growing pattern (domain structure). F(x)'s are also only available for a limited number of systems, and hence they should be determined for various systems in future work. Figure 6-13 shows the time change of the structure factor in the intermediate stage. The intensity increase of F(x) due to the increase of (t](t)2s) is clearly seen in part (a), and the sharpening of F(x) with t is partic-

time scale F(x) becomes essentially universal with t and fits closely with one of the Furukawa's scaling functions (Furukawa, 1989), F (x) = x4/(3/2 + xx °)

(6-49)

A number of similar experimental results were also reported in the literature described in the beginning of this section. Thus the polymer blends obey the universal features found for low molecular weight mixtures (Komura et al., 1984, 1985): (i) Tm{r) and Qm(?) are universal with T, and (ii) in the late stage F(x) is universal with t. In this sense the self-assembly of polymer blends is unique only in that it has a long temporal scale (tc being long) and a large spatial scale [£ ~ l/qm(0) being large],

CD O *X 1

e

2. 8

£0°f 6 xXxxx~ R

7 X*

2. 4 2. 0

Jfc

v

xx

x grtt^

Nt. M a l i . ) 1 » 64.86 2a 73.39 34 81.92 4 • 109.43

5 , 137.15 6 * 250.77

t 1.9 2.2 2.5 3.3 4.1 7.5

xX x ^ ^

""

******* 3

7

/

/

x

x

/

O"

(F(

cr

o o

1. 6

_ •

•

•

•

—

1. 2 i

^

.

(a)

I

i

Nt. H.I..)

2.8

-

1• 2. 3.

J M k x

t

64.86 1.9 73.39 2.2 81.92 2.5

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-0.5

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.

-0.3

I

-0.1

.

, +0.1

L0G(q/qm(t,T) )

, (b) +0.3

Figure 6-13. Scaled structure factors F(x) obtained at the intermediate stage SD at 40 °C for the same mixture as in Fig. 6-12 and plotted in double logarithmic scales (a). The profiles in part (b) are shifted vertically and matched at x = l and at T = 7.5. OT is the vertical shift factor. From Takenakaetal. (1990).

6.6 Fine Structure in the Scaled Structure Factor F(x)

ularly obvious in part (b), where F(x)'s numbered 1 to 5 are superposed on number 6 by vertical shifting. 6.5.3 Unique Features of the Scaled Quantities of Polymers

In small molecular systems, qm (t; T) data obtained for various mixtures, with critical compositions, for example, become universal, independent of the particular material when they are plotted using the reduced variables Qm and 7 m . In the case of polymer blends with near critical compositions, Qm(r) showed non-universality when (i) the DP's of constituent polymers were changed (known as "N-branching") (Hashimoto et al., 1986 b; Hashimoto, 1988 a,b), (ii) a small amount of solvent was added to the mixture (known as "C-branching") (Hashimoto, 1988 a), and (iii) a small amount of the block copolymer, corresponding to the constituent homopolymers, is added (denoted as "B-branching") (e.g., A-B type di-block copolymer + A homopolymer -f B homopolymer) (Izumitani and Hashimoto, 1992). In each case, 2) tends to increase with increasing molecular weight of PB in the 50/50 wt./wt. mixtures of SBR1/PB, while Fig. 6-14b shows an example of C-branching where Qm(z) at a given T (> 2) tends to increase with increasing total polymer concentration from 50 to 100 wt.%. The early stage SD for the systems whose later stage behaviour is shown in Fig. 6-14 a were discussed in detail by

277

Takenaka et al. (1987). In the experiment shown in Fig. 6-14 b, the polymers used are SBR2 and PB273, and the polymer composition is fixed at 50/50 wt./wt. to which 0 to 50 wt.% dioctylphthalate (DOP) was added as a common solvent. The exponent oc ~1/2 for SBR1/PB273 in Fig. 6-14a and for SBR2/PB273 (in melts) in Fig. 6-14 b may be explained by the theories proposed by Kawasaki and Sekimoto (1987) and Furukawa (1988). Like the Nand C-branchings, the B-branch shows that 2 tends to increase with increasing amounts of the block copolymers (Izumitani and Hashimoto, 1992). It was suggested that the N- and C-branches arose from the entanglement effects of polymer molecules in a condensed phase (Onuki, 1986; Hashimoto et al., 1986b; Hashimoto, 1988 a, 1988 b), while the Bbranch is expected to be associated with the reduction of the interfacial tension due to the localization of the block copolymer in the interface between growing domains. The B-branch may be seen even in small molecule systems by adding a third molecule which can act as a surfactant for the constituent components. Another unique feature of polymers is observed in the fine structure in the scaled structure factor F(x) (Hashimoto et al., 1989 b, 1991b; Bates and Wiltzius, 1989; Nose, 1989; Takenaka et al., 1990) or in the spontaneous pinning (Hashimoto, 1988 b; Takenaka et al., 1989) of the domain growth for off-critical mixtures of high polymers, which will be discussed in Sees. 6.6 and 6.7, respectively.

6.6 Fine Structure in the Scaled Structure Factor F(x) So far the scaled structure factor F(x) has been investigated over a narrow range

278


-0.4 -0.8

Temp.:60.0(°C)

-1/2

O:5BR1/PB19 D:SBR1/PB55 A.SBR1/PB122 OISBR1/PB273

(a) -1.2

I

SBR2/PB273=50/50(wt/wt) 70.0°C

O -0.4

81

-0.8

-1/2

-1.2

Figure 6-14. Plots of the reduced wavenumber Qm vs. the reduced time T showing (a) N-branching for mixtures of SBR1 with PB having different N at 60 °C, and (b) Cbranching for SBR2/PB273 in DOP with a composition of 50/50 wt./wt. at 70 °C. The total polymer concentration was changed from 50 to 100wt.% in part (b). From Hashimoto (1988 b).

logr

of x of x ~ 1 both in polymer and smallmolecular systems (Komura et al., 1984, 1985; Hashimoto et al., 1986a; Takahashi etal, 1986; Tomlins and Higgins, 1989; Kyu and Saldanha, 1990; Kyu and Lim, 1990 b; Lee and Kyu, 1990). Taking advantage of studying the pattern growths in polymer systems, with their spatial and temporal scales being much larger and longer than those of small-molecular sys-

tems, it should be possible to study F(x) extremely precisely up to higher x values. Such studies enable us to explore a new aspect of the pattern growth, viz., the time change in the local features of the patterns such as the interface thickness and waviness. This is a subject which has been left essentially unexplored up to now and which has quite recently attracted some experimental and theoretical investigation

6.6 Fine Structure in the Scaled Structure Factor F(x)

(Hashimoto et al., 1989 b, 1991b; Takenaka et al., 1990; Takenaka and Hashimoto, 1992; Bates and Wiltzius, 1989; Nose, 1989; Oono and Puri, 1988 a, 1988 b; Puri and Oono, 1988; Shinozaki and Oono, 1991; Ohta and Nozaki, 1989; Chakrabarti et al., 1989 a, 1989 b; Hayakawa and Koga, 1990; Koga and Kawasaki, 1991; Koga et al., 1992; Kawasaki et al., 1992). Figure 6-15 shows some typical examples of F(x) showing a fine structure at large x (Takenaka et al., 1990). The data were obtained for SBR1/PB19 with 58/ 42wt./wt. or 57.2/42.3 vol./vol. at 40 to 60 °C at which eT ~ 1, i.e., a strong segregation, and in the late stage SD where the value of T was from ca. 10 to 50 at each T. In the small x range of 0.5<xt = 110 min (see the vertical arrow marked tc for the 50/50 mixture, tc being the crossover time between the intermediate stage and the late stage). In contrast to the near critical mixture, the off-critical mixtures show the pinning of the growth such that at t > tp both qm and Im hardly change with t. For example the time change of qm for the off-critical mixtures is given by qm(t) = qmp(t/tp)~cc

(6-56)

where a ^ 0 for t/tp < 1 and a = 0 for t/tp > 1 and the value of qm after the pinning (qmp) is given by tp)

(6-57)

The peak scattered intensity Im is found to be essentially pinned down at the level of

Impatt>tp. It is observed that the more the composition is biased from the critical composition, the earlier the stage at which the pinning occurs, yielding the higher value of qmp, the lower intensity level of Imp, and the shorter time tp. It was also found that the higher the temperature, the earlier the stage of the pinning (Hashimoto, 1988 b). The pinning was found to occur in the intermediate stage of ttc (e.g., the 30/70 mixture at 60°C). The pinning phenomenon, which is unique to the ordering process for an offcritical mixture with high molecular weights is proposed (Takenaka et al., 1989) to originate from the "dynamical percolation-to-cluster transition" occurring during the coarsening process of SD (Hasegawa et al., 1988). The phase separation via SD develops a bicontinuous periodic domain structure. The domain structure can ini-

tially maintain the macroscopic percolation, i.e., both types of the domains being continuous through the whole sample space. However, as the domains grow, one type of domain, composed of the component with a minor volume fraction, cannot maintain volumetrically the macroscopic percolation, resulting in a break-up into fragments with local percolations only. The fragments eventually degenerate into spherical droplets due to interfacial tension, generating a cluster of spheres. Although we do not have a direct realspace pattern to support this idea of the percolation-to-cluster transition for our particular mixtures SBR1/PI55, we have it for mixtures of X-7G/PET where X-7G and PET stand for a thermotropic liquid crystalline polymer and poly(ethylene terephthalate), respectively. Figure 6-20 shows the time-evolution of an unmixing structure through SD for the 50/50 wt./wt. blend at 270 °C (Hasegawa et al., 1988). At 270 °C, X-7G and PET by themselves are, respectively, anisotropic and isotropic liquids. The mixture is thermodynamically unstable and undergoes phase separation between the anisotropic liquid domains of X-7G and isotropic liquid domains of PET through the SD mechanism. The anisotropic and isotropic domains appear to be bright and dark under the crossed polarizers, respectively. The percolated domain structure self-similarly grows with time in the time domain I where the structure evolution relevant to the late stage SD is observed. As the domain grows as a consequence of the network instability, the anisotropic liquid domains can no longer maintain the macroscopic percolation, resulting in the formation of fragments of the network with local percolation only, as seen in the beginning of the unmixing structure in the time domain II. The fragmented networks then degenerate

6.7 Spontaneous Pinning for Off-Critical Mixtures

285

Figure 6-20. Time-evolution of the unmixing structure showing the dynamical percolation-to-cluster transition. In regime I the percolated structure grows with dynamical self-similarity. The transition takes place in regime II and clusters are formed in regime III. From Hasegawa et al. (1988).

into spherical droplets to minimize interfacial free energy (in the time domain II) and result in clusters of droplets (in the time domain III). The growth of the droplets occurs extremely slowly for this particular mixture of X-7G/PET. The external potential exerted by the unwetting phenomenon of X-7G gives a driving force of the diffusion-coalescence of the spheres (Nakai etal, 1992). Once the cluster structure is formed in our off-critical mixtures of SBR1/PI55, ac-

cording to the mechanism similar to that shown in Fig. 6-20, the growth of the droplets is again expected to be very slow or essentially pinned. The growth occurs as a consequence of the diffusion-coalescence of the droplets. The diffusion of the droplets, in turn, involves, the dissolution and condensation (Lifschitz and Slyzov, 1961) of the molecules forming the droplets, i.e., as shown schematically in Fig. 6-21, the molecules A in domain A dissolve into Bmatrix being rich in B molecules, translate

286

6 Structure of Polymer Blends State I

State II

B-matrix

© o

I

II

State

Figure 6-21. Schematic illustration of the kinetic energy barrier for the diffusion-coalescence process. Since state I has a higher free energy than state II, a transformation from state I to II is thermodynamically favoured. However it must overcome the barrier associated with the heat of mixing of unlike chains in the diffusion-coalescence process. From Hashimoto et al. (1992 a).

over a distance, and condense in a different position in domain A. Such fundamental molecular motions cause Brownian diffusion of the droplets, and when they coalesce, they grow into larger droplets. However this transport process involving dissolution, diffusion and condensation has to overcome a kinetic barrier associated with the enthalpy of mixing of unlike constituent polymers AiJmix. AHmix per chain is proportional to % N kB T where N is the DP, NA = NB = N being assumed here for the sake of simplified arguments. Thus when N is very large, as in the case of SBR1/PI55, xN is very large, and hence the kinetic barrier is too large to be overcome (i.e., very low solubility of A molecules in the B matrix). Thus the molecular diffusion process is kinetically frozenin, causing pinning of the growth of the droplets. Thus pinning is a unique feature associated with the ordering process of polymer mixtures consisting of high molecular weights. A similar pinning effect was

observed (Shibayama et al., 1983) for the growth of the spherical microdomains of block copolymers induced by increasing the segregation power. In the mixtures with low molecular weights for which Aifmix is relatively small (AHmix < kB T), growth of the droplets due to the diffusioncoalescence can always take place, and hence pinning will not be observed. Even for mixtures with very large molecular weights, coarsening and hence the decrease of qm with t can occur as long as the mixtures maintain the percolated structure, i.e., a continuous network structure. In this case the coarsening takes place by partial disruption of the networks and degeneration of the disrupted part of the networks into the existing networks (Nakai et al., 1986, 1992). Thus the pinning results from two effects, i.e., (i) the dynamical percolation-to-cluster transition and (ii) the high molecular weight effect which causes the kinetically freezing-in of the molecular diffusion. A detailed discussion of this effect will be given elsewhere (Hashimoto et al., 1992 a).

6.8 Interplay with Two Kinds of Phase Transition 6.8.1 Interplay with Crystallization

When the spinodal decomposition is coupled with other kinds of phase transition, such as crystallization (liquid-solid phase transition) and microphase transition of block copolymers, the resulting morphology is expected to be very rich. An interesting morphology generated by coupling SD with crystallization was reported for binary mixtures of polypropylene (PP) and poly(ethylene-r-propylene) (ethylene-propylene random copolymer, EPR) (Inaba et al., 1986, 1988). It was

287

6.8 Interplay with Two Kinds of Phase Transition

found that the pattern developed by SD in the molten liquid state at 7\ > Tm was conserved during a rapid crystallization process of PP at T2 < Tm, and that the crystallization effectively pinned down further growth of the pattern due to SD at T2, Tm being the melting temperature of PP. The coupling generated patterns composed of dual morphological entities of spherulites and periodic bicontinuous domain structures, as shown in Figs. 6-22 and 6-23. The two pictures in Fig. 6-22, which were obtained on the same field for the same sample, show the morphology of the volume-filling spherulites (a) which contain, as their internal structure, the periodic and bicontinuous structure of PP-rich and EPR-rich domains (b). The bicontinuous percolated structure was first developed by SD at Tx for a time period t1, and the spherulitic structure was developed by a subsequent lowering of the temperature from Tt to T2. At T2 rapid spherulitic crystallization occurred in the interconnected PP-rich domains, without significantly disrupting the pattern originally developed by SD. The polarized light micrograph in Fig. 6-23 shows that the pattern typically developed by the SD at Tx, i.e., a bicontinuous percolated structure of PP-rich and EPRrich domains with a characteristic periodicity of A ~ 60 jam. In this case the time spent for SD is very long, and the domains are large enough so that the PP-rich domains contain the volume filling spherulites and hence have high optical anisotropy. The EPR-rich domains have less crystallinity and hence smaller optical anisotropy. 6.8.2 Interplay with the Microphase Transition

In the case where blends contain a block copolymer at least as a component, they undergo in principle both "macrophase

Figure 6-22. (a) Polarized light micrograph (POM) and (b) phase-contrast light micrograph (PCM) for PP/EPR mixture unmixed at 200 °C for 5 min and subsequently crystallized at 125 °C for 5 min. The POM which was obtained by setting polarizer and analyzer in vertical and horizontal directions clearly shows the volume-filling spherulites, while the PCM clearly shows the fine structure developed by SD. The two pictures were obtained on the same field for the same sample. From Inaba et al. (1988).

POM

100 Figure 6-23. Polarized light micrograph showing the bicontinuous percolated structure developed by SD for a PP/EPR mixture unmixed at 200 °C for 20 min and subsequently crystallized by quenching the mixture in an ice-water bath (polarizer and analyzer being set in vertical and horizontal directions). Note that the percolated PP-rich domains contain the volume-filling spherulites and have high optical anisotropy. From Inaba et al. (1988).

288


and microphase transitions" (Tanaka and Hashimoto, 1988; Hashimoto et al., 1988 a; Ijichi and Hashimoto, 1988), interplays of which also generate a rich variety in their patterns. For example, the blends such as A-B/A, A-B/B, A-B/C, (A-B)J(A-B)2, A-B/B-C etc., as described in Sec. 6.1, belong to this class of blends. Figure 6-24 shows schematically the "microphase transition" of a pure A-B type block copolymer (a) and the "macrophase transition" of A/B blends between the two homopolymers (b), as well as equilibrium structures both in the single-phase state (upper half) and the two-phase state (lower half) (Hashimoto et al., 1988 a). In the two-

Polymer Blend

Block Polymer

...r t •U ,

Disordered '

State

no connectivity between A 8

molecular connectivity between A 8 B

B

( X A B N )C

State

Q

Or*^

£•••'

Dx/Rg Dn/Rg

Dx/Rg -*oo

D I I / Rg - » oo

(a) (b) Figure 6-24. Comparison of concentration fluctuations of (a) A-B block copolymers and (b) A/B polymer blends in the single-phase (disordered) state and two-phase (ordered) state at thermal equilibrium, and the phase transitions between the two states, i.e., order-disorder transition ("microphase transition") for the block copolymers vs. liquid-liquid phase transition ("macrophase transition") for the polymer blends. From Hashimoto et al. (1988 a).

phase state (or ordered state in the sense that it has a lower entropy) the A/B blends separate into two macroscopic phases where the domain sizes parallel and normal to the interface, D\\ and D ± respectively, are much larger than the radius of gyration Rg of individual polymer coils. On the other hand an A-B block copolymer cannot separate into the macroscopic phases owing to a covalent bond between A and B polymers; instead it separates into submicroscopic domains with D± nearly equal to Rg (Inoue et al., 1969; Meier, 1969, 1987; Helfand and Wasserman, 1976,1978, 1980; Hashimoto et al., 1980, 1983; Noolandi and Hong, 1980; Hong and Noolandi, 1981; Leibler, 1980; Hadziioannou and Skoulios, 1982; Ohta and Kawasaki, 1986; Kawasaki et al., 1988,1990). At the segregation limit the domains have to be arranged periodically in space with a long range order characteristic of "crystals" with a super-large lattice spacing D±. The variety in the domain morphology depends on the composition / of one of the components A or B (see for example, Hashimoto et al., 1983; Hashimoto, 1987 a; Thomas et al., 1986; Hasegawa et al., 1987; Hashimoto et al., 1992 b). The sketch in the bottom left hand corner of Fig. 6-24 shows only one type of domain morphology (i.e., lamella). In contrast to the ordinary liquid-liquid phase transition of polymer blends, we denote this order-disorder transition of block copolymers (Hashimoto, 1987 a) as a "microphase transition", and in contrast to this microphase transition, we denote the ordinary phase transition as a "macrophase transition". The thermodynamic stability limits for macrophase and microphase transitions, Zs, macro and zs, micro (i-e., x-parameters at the spinodal points) were discussed for blends containing block copolymer(s), based upon the mean-field random phase approxima-

6.8 Interplay with Two Kinds of Phase Transition

tion (RPA). Figure 6-25 shows an example of the phase diagram for the stability limits of A-B/A, in which A-B has N = N1 and / = NJ(NA + NB) = NJN = 0.5, and A has N = N2 = NJ2 (Hashimoto etal., 1988 a). The stability limits faiVi)s,maCro (solid line) and fa Nx)St micro (broken line) are evaluated as a function of the block copolymer volume fraction biock. (f>c is the critical value of block below which only the stability limit for the macrophase transition exists. It is needless to say that this phase diagram cannot predict the equilibrium structures; it can predict only the stability limits. The equilibrium structures may be predicted by the approach developed by Hong and Noolandi (1983) and Whitmore and Noolandi (1985). First consider the case of block > c where stability limits exist for both the microphase and macrophase transitions. In the region where XNX l M m )

Microphase Transition

I Cylinder

I/S

S

I/S

I

Lamella Cylinder Lamella Cylinder H h Amicro = O(IOOA)

transition, and if the two polymers ab and /?b are mixed at molecular level to form a microdomain, as in the case shown in Fig. 6-26, the microphase transition may develop a spatially varying morphology of microdomains, as shown in Fig. 6-27 b, with a large scale periodicity of AmacTO within which a small scale periodicity of the microdomain structure of AmiCTO of the order of 100 A is superposed. Here we assume that ab is the PS-rich block copolymer, hence possibly giving rise to the microdomains of Pi-cylinders in the matrix of the PS in the ab-rich domains. Similarly, ph is the Pi-rich block copolymer so that the microdomain of PS-cylinders in the matrix of the PI may be formed in the /?b-rich domains. In the domains where ab and /Jb are present at nearly the same concentration, the alternating lamellar microdomains may be formed. An example of such modulated patterns is shown in Fig. 6-28 for the 80/20 mixture of an SI block copolymer (HY-12 in Table 6-3) and poly(phenyleneoxide) (PPO) prepared by slow evaporation of its dilute toluene solution (Hashimoto et al., 1991 a). Pure SI has a lamellar microdomain. During the solvent evaporation process, SD

(b)

291

Figure 6-27. Possible modulated structure (superlattice structure) for a mixture of two different SI block polymers ab and /?b with a large-scale characteristic spacing Amacio of the order of 0.1 to 1 (im resulting from spinodal decomposition (a), and a small-scale characteristic spacing ^micro °f t n e order of 100 A resulting from the microphase separation (b) which could follow the spinodal decomposition. Note that the microdomain morphology itself could spatially vary with the spacing AmacTO. From Hashimoto et al. (1988 a).

occurs, which generates the spatial concentration fluctuations of SI and PPO, as shown in Fig. 6-27 a, where ab and /Jb can be regarded as SI and PPO, respectively, in this case. This macrophase transition is followed by the microphase transition, which then forms the microdomains. Since PPO is immiscible with PI but miscible with PS, PPO is selectively solubilized into PS microdomains, increasing the volume fraction of the phase comprising PS and PPO relative to that of the PI phase. Thus the Si-rich domains form the lamellae (region A), PPO-rich domains form the PI spheres in the matrix of PPO and PS (region D), and the domains intermediate between the two form PI cylinders in the matrix of PS and PPO (regions B and C), as shown in Fig. 6-28. The modulated pattern shown in Fig. 6-28 has Amacro ~ 5 jum and ^micro — 100 nm. The microdomain morphology in region B, the immediate neighbour of region A, may be regarded as a "mesh structure" (Hashimoto et al., 1992 b) in which there are catenoid channels of PS and PPO traversing through the PI lamellae, and interconnecting the lamellae comprised of PS and PPO.

292


HY12/PPO (80/20) Blend

Figure 6-28. Example of modulated structure obtained by coupling between the macrophase and microphase transitions in a solvent cast film of the 80/20 wt./wt. mixture of an SI block copolymer (HY-12) and PPO. Toluene is used as a solvent. From Hashimoto et al. (1991 a).

6.9 Morphology Control Information obtained on the various ordering processes, as described in the preceding sections, serves as a key to controlling the morphology of polymer blends. Here we summarize the processes leading to morphology control, the basis of which was discussed in the preceding sections in this chapter and which involves freezing-in of the growing patterns by means such as (i) spontaneous pinning, (ii) physical pinning, and (iii) chemical pinning. Spontaneous pinning was discussed in Sec. 6.7 and is applicable to off-critical mixtures. The morphology is controlled by the composition and the unmixing tempera-

ture for a given pair. Physical pinning involves (a) crystallization when at least one component is crystallizable and (b) vitrification. This type of pinning, as well as chemical pinning, involves pinning the domain growth by freezing-in the translational diffusion of the center-of-mass of the molecules. Physical pinning due to crystallization was discussed in Sec. 6.8.1 and the morphology control is achieved by the time and temperature for SD and crystallization. An example of a pattern controlled by vitrification is shown in Fig. 6-29 (Hashimoto et al., 1984) for the solvent cast-film of a PS/PB5.9 mixture with 20/80 wt./wt. composition, in which parts (a) and (b)

6.9 Morphology Control

show the light scattering pattern and the light micrograph of the solvent-cast film. During the solvent evaporation process, the concentration of the polymers increases, which induces the phase separation due to the SD mechanism (Sasaki and Hashimoto, 1984). The pattern developed in the solution will be frozen-in at some stage of SD at which the domains rich in PS were vitrified, causing freezing-in of further growth of the pattern. This frozen-in pattern in the solution is essentially con-

Solvent Cast Film

Light Scattering Pattern

(a)

••>mxmr: » #**

:••:••••

:

t'^;\

photomicrograph

10H (b)

Figure 6-29. Light scattering pattern (a) and photomicrograph at the solvent-cast film of PS/PB5.9 with a composition of 80/20 wt./wt. stained by O S O 4 . Toluene was used as a solvent. From Hashimoto et al. (1984).

293

served in the solvent-cast film. The scattering maximum was found to correspond to the average spacing of the PI spheres but not to the average size of the spheres themselves. Presumably the percolation-to-cluster transition, as discussed in Fig. 6-20 of Sec. 6.7, may have happened before the vitrification, generating the clusters of the spheres. Hence the reciprocal-space and real-space patterns in Figs. 6-29 a and b reflect the memory of the periodic pattern developed by the SD process (Hashimoto, 1992). Chemical pinning involves the chemical crosslinking of polymer chains of one component or both components in binary blends, for example. It was found (Hashimoto et al., 1989d) that peroxide crosslinking is quite effective at pinning down the pattern growth for the binary polymer blends of SBR2/PI55 (in Table 6-3) and DPB/HPI (DPB-20-103/HPI-15-136 in Table 6-1) both in the long and short length scales, as studied by time-resolved light and small-angle neutron scattering methods, respectively. Figure 6-30 sketches a controlled pattern for elastomers composed of two component rubber-like polymers such as SBR2/PI55. The elastomer contains bicontinuous and periodic domains with a given average spacing A, and level of unmixing A<j> (see Fig. 6-7), and with a given crosslinking density. When an applied field, such as an electric (or magnetic) field, deformation, shear (or extensional) flow, is imposed on the systems, their self-assembly and morphology control become extremely intriguing both conceptually and for practical applications. There are a large number of reports on a class of multicomponent polymer systems called "interpenetrating network" (see for example, Manson and Sparling, 1976). Their patterns are controlled by a subtle and complex balance of chemical-reaction

294


Figure 6-30. Schematic illustration of bicontinuous and percolated domains of rubber-like polymer blends formed by SD and subsequent crosslinking reaction. Note that both domains are chemically crosslinked.

induced phase separation, chemical pinning, vitrification, etc. Real-time and insitu analyses of the self-assembly during the chemical reaction badly need to be carried out as a fundamental study.

6.10 Effects of Applied Fields on the Self-Assembly There are a number of interesting and fundamental observations for binary polymer mixtures of A and B without a solvent (A/B) or with a solvent (A/B/S), and for a solution of A (A/S) under shear flow, where S stands for solvent. Homogeneous solutions (A/S) undergo phase separation under shear. The "shear-induced phase separation" or the "shear-induced concentration fluctuation" has not been reported so far for small molecular systems and has been attributed to the elastic deformation of polymer coils (Ver Strate and Phillippoff, 1974; Schmidt and Wolf, 1979; Rangel-

Nafaile et al, 1984; Kramer and Wolf, 1985; Tirrell, 1986; Vrahopolou and McHugh, 1987; Pistoor and Binder, 1988; Helfand and Fredrickson, 1989; Onuki, 1989,1990; Milner, 1991; Hashimoto and Fujioka, 1991; Wu et al., 1991; Yanase et al., 1991). In contrast to the above phenomenon, phase separated A/B/S and A/B systems become homogeneous under shear (Silverberg and Kuhn, 1954; Hashimoto et al., 1988b, 1990a; Cheikh Larbi et al., 1988; Takebe and Hashimoto, 1988; Takebe etal., 1989, 1990; Nakatani et al., 1990). This "shear-induced homogenization" was also found for a binary critical mixture of simple liquids (Beysens et al., 1979; Beysens and Gbadamassi, 1979). However, the size of the drop in the critical temperature or cloud-point temperature due to shear ATc(y) was found to be three-to-four order of magnitudes bigger in the polymeric systems than in the small molecular systems (Hashimoto etal., 1988b, 1990a; Takebe et al., 1989, 1990). The drop ATc(y) for the small molecular systems was theoretically treated on the basis of critical dynamics and explained in terms of a suppression of the critical fluctuations by shear (Onuki and Kawasaki, 1979 a, 1979 b; Onuki et al., 1981; Onuki, 1989). On the basis of this theory, the strikingly large effect of shear on ATc(y) in polymers is attributed to an enormously long molecular relaxation time for the polymer system compared with that for the simple liquid system (Takebe etal., 1989; Hashimoto etal., 1990 a). Such a shear effect on the critical dynamics of simple liquids may also be important for the polymer systems. In addition to this effect, however, one cannot overlook a shear effect on molecular deformation, the contribution of which is insignificant on the concentration fluctuations in the small molecule systems but may become significant in polymers.

6.10 Effects of Applied Fields on the Self-Assembly

295

Figure 6-31. Light scattering pattern for the as-cast film (a) and the change of the pattern with a number of repeats (n) of the 1/2 compression. Patterns (b)-(f) correspond to n = 1, 2, 4, 6 and 12, respectively. The angle mark shows 6 = 10° in air. SBR1/PB19. From Hashimoto et al. (1989 a).

In recent years an increasing number of reports have become available on quantitative small-angle light scattering (Hashimoto et al., 1986c, 1988 a, b; Takebe et al., 1989,1990; Hashimoto and Fujioka, 1991; Wu et al., 1991) and small-angle neutron scattering (Nakatani et al., 1990) for polymer systems under shear flow, which are believed to provide key information to understanding rheology and phase transition through shear-rate dependent structure factors. Deformation also has a strong influence on the pattern of the concentration fluctuations developed by SD. Phase-separated, immiscible, rubber-like polymer blends with higher molecular weights are reported to be homogenized by applying a repetitive uniaxial compression, as shown in Fig.

6-31 (Hashimoto et al., 1989 a). The unmixing rate of the high molecular weight blends is much slower than the applied deformation rate, although the latter rate itself is very slow, so that the compression did not involve significant molecular orientation. In this case, the deformation caused by repetitive uniaxial compression is cumulative, i.e., the pattern hardly grows and changes during the time period between the successive deformations. After a number of compression the wavelength of the dominant mode of the concentration fluctuations parallel to the compression axis A{1 becomes shorter than the critical value Ac = 2n/qc. Here qc is the crossover wavenumber predicted in the linearized theory of SD by Cahn (1965), the modes of the fluctuation with the wavelength A greater

296


than Ac being able to grow but the modes with A (j)a, (j)c \j/

b o n d angle; degree of crystallinity shear stress tensile strain in the a m o r p h o u s phase, crystal lattice tensile strain component parameter in Tsai-Halpin equations diffraction angle critical angle of incidence for longitudinal, transverse ultrasonic wave X-ray wavelength; width of the crystalline region (Takayanagi model) wavelength of chain LAM frequency of chain LAM Poisson's ratios Poisson's ratios in the case of fiber symmetry geometric parameter in Tsai-Halpin equations density density of the molecular chain tensile stress in the amorphous, crystal phase tensile stress components sample tensile stress shear strain inclination of object from the fiber axis <j) referring to chains in the amorphous, crystalline phases length of the crystalline region (Takayanagi model)

ASTM CNDO DMTA iPP ISO LAM LCAO LVDT MNDO NMR PBO PBZT

American Society for Testing and Materials complete neglect of differential overlap dynamic mechanical thermal analysis isotactic polypropylene International Organization for Standardization longitudinal acoustic mode linear combination of atomic orbitals linear variable differential transformer modified neglect of differential overlap nuclear magnetic resonance poly(p-phenylene benzobisoxazole) poly(p-phenylene benzobisthiazole)

v

tj

Q

303

304

7 Elastic Properties of Crystalline Polymers

PE PET PEO POM PTFE SAXS

polyethylene poly(ethylene terephthalate) poly(ethylene oxide) polyoxymethylene polytetrafluoroethylene small angle X-ray scattering

7.1 Introduction

7,1 Introduction 7.1.1 General, Scope of Chapter

Most uses of solid polymers depend to some extent on the mechanical properties of the material, and the basic mechanical properties of a solid are its stiffness, strength and toughness. Stiffness represents the resistance of a body to elastic deformation, the ratio of load over deflection. This depends on the material and on the shape of the body. The material properties themselves are expressed as a modulus, a ratio of stress over strain, or as a compliance, a ratio of strain over stress. A deformation is elastic if the sample recovers its orginal shape as soon as the load is removed. The strength of a material is the stress at which the material fails by fracture or by irreversible deformation. Toughness is a measure of the work required to fracture a material. The complex topics of plastic deformation and failure are dealt with in Chaps. 10 and 15 of this Volume. In traditional materials science, where metals are always taken as typical, the elastic properties of a solid are not very interesting. The modulus is seen as structure insensitive, while strength and toughness are the mechanical properties affected by microstructure. Modulus is very basic; it relates to the bond strength and thus to the melting point, and there are few surprises. The elastic constants can be obtained by direct measurement on a single crystal sample. They can also be calculated from bond force functions, using the known crystal structures. The elastic properties are often nearly isotropic and not sensitive to temperature, so a single number called "the modulus" is used to describe the elastic properties. In the world of polymers, the elastic properties of a solid sample are equally

305

basic, but they may be strongly structure sensitive, anisotropic, and rate or temperature dependent. Thus different samples of the same polymer tested in different ways may show moduli that are different by orders of magnitude. Most solid polymers are viscoelastic. The formal description of viscoelasticity is given in Chap. 9 of this Volume. It is sufficient here to say that viscoelastic solids show some time dependent deformation even at low stresses. Measurement of elastic properties must take this into account; in particular, the modulus will be a function of time. The slower the test, the more deformation, so the larger the strain and the softer the material appears to be. Generally the time dependent deformation is thermally activated, faster at high temperature, so that the modulus will be strongly temperature dependent as well as time dependent. Real samples of crystalline polymers are not simply polycrystalline, they contain many disordered interfaces between the small polymer crystals so that they are partially ordered at best. Crystalline polymers are usually considered to consist of two phases, being a mixture of perfect crystals and fully disordered material. They are often called "semi-crystalline" for that reason. Generally the crystals are denser and stiffer than the amorphous phase, so the crystal modulus is an upper limit for the modulus of a real specimen. Comparison of sample modulus with crystal modulus then allows a judgement of how much remains to be achieved, if the aim is a stiff material. Polymer crystals are ordered assemblies of parallel molecular chains, with bonds between chains that are much weaker than the covalent main chain bonds. The crystals are therefore intrinsically anisotropic. It is not normally possible to obtain macroscopic samples of a perfect polymer crystal, so its mechanical properties cannot

306


be obtained by direct mechanical testing. Indirect measurements using X-ray diffraction or vibrational spectroscopy need careful interpretation and often do not agree. Theoretical estimates of the elastic constants of the crystals are now quite easily obtained but may not be accurate. If the chain conformation in the crystal is such that bending or stretching of covalent bonds is required to extend the chain, it will be very stiff and the crystal anisotropy will be very high. If the conformation is a helix that can be stretched by uncoiling, which only involves rotation of the bond angles, the chain will be much softer. Modeling of the molecular stiffness will give a good first approximation to the stiffness of the crystal in the chain axis direction. This type of modeling has been done for some time. It is now possible to model the full crystal structure using semiempirical bond properties, giving a better estimate of the modulus in the chain axis direction, and a full set of elastic constants for the crystal. Computational requirements limit quantum mechanical calculations to a single chain. Amorphous polymers by definition contain a wide range of polymer chain conformations; most will be able to deform by bond rotation. The stiffness of glassy polymers then depends on the constraints and barriers to bond rotation imposed by neighboring chains. These constraints are difficult to model in a random system. The properties of amorphous polymer solids are dealt with in Chap. 2 of this Volume. In some cases, bulk samples of amorphous material can be obtained and tested, but there is no guarantee that the disordered material in the semi-crystalline sample has the same properties. In particular, the disordered material in the semi-crystalline state is constrained by the surrounding crystals and is often limited to

very small regions. This extra constraint will affect the molecular mobility and thus the stiffness. When the sample is drawn and the crystalline phase becomes oriented, there will be orientation of the amorphous phase too. Thus the word "phase" is used loosely here and in later discussion of mechanical models, and does not imply a thermodynamic equilibrium phase with well defined properties. A semi-crystalline polymer sample approximated by a two-phase model is thus a composite material where the mechanical properties of the two components are not well known. For modeling mechanical properties there is the further serious difficulty that the internal arrangement of the two phases may be unknown. Even when the arrangement is known, constructing a self-consistent mechanical model for a complex structure is not straightforward. An important case of this type is the spherulitic lamellar structure described in Chap. 4 of this Volume. Whatever model is used, the elastic properties of a real semi-crystalline polymer cannot be predicted from the crystalline and amorphous elastic properties alone. Structural parameters are required, and the most important of these are orientation and crystallinity. Clearly, if the molecular or crystal properties are highly anisotropic and the chains or crystals in a sample are aligned, then the elastic properties of the sample will also be anisotropic. As the chain direction is generally the stiffest, alignment of chains by flow or colddrawing will increase the sample stiffness in the draw direction. Generally the modulus of semi-crystalline samples will also increase with crystalline content. These are not the only structural parameters that affect modulus, as samples of the same degree of orientation and crystallinity may still have very different moduli. In high per-

307

7.1 Introduction

formance polymer fibers the fiber modulus approaches that of the perfect crystal. In this case measurement of elastic properties as a function of processing conditions and comparison with those of the perfect crystal may be used to investigate the mechanical defects that relate to imperfections in the structure of the sample. 7.1.2 Definitions and Terms in Elasticity

The oriented chains in crystalline polymers make the crystals anisotropic. To describe the elastic properties of anisotropic materials Hooke's law is generalized to three-dimensional Cartesian axes x, y, and z. The 3-D set of equations is conventionally given in a contracted form: q=6 —

X^ C q=l

q—6 P

—

P '

""S"* '

(7-1)

planes. Thus <J4 may also be written as T 2 3 , and g4 as y23. Care is required to derive the exact form of these components, as factors of 2 arise in some of them (Nye, 1957; Arridge, 1975; Ward, 1979). External stability requires that cpq == cqp and this reduces the number of independent components from 36 to 21 (Hearmon, 1961). If the material has orthotropic symmetry (sometimes called orthorhombic, as in crystallography) the number is reduced to 9, with a compliance matrix of the form n

S

12

s13

12

S22

•S23

13

S23

«33

0 0 0

0 0 0

0 0 0

0 0 0 S44

0 0

0 0 0 0 S55

0

0 0 0 0 0

(7-2)

S £L

66

q=l

C is the matrix of elastic constants, components cpq. Each cpq is the ratio opjzq when all other strain components are zero. S is the compliance matrix, components spq. Each spq is the ratio &p/oq when all other stress components are zero. Indices 1, 2, and 3 refer to normal stress and extensional strain along x, y and z respectively. 4, 5, and 6 refer to shear in the yz, xz, and xy-

Figure 7-1. Axes and compliance component directions for a polymer sheet with orthotropic symmetry.

A common case of anisotropic elasticity in polymers where orthotropic symmetry exists is a sheet biaxially drawn or drawn at constant width. Conventionally, in this case the z-axis is taken as the draw direction of the sheet, x is transverse to the draw direction and in the plane of the sheet. The y-axis is then normal to the sheet, as in Fig. 7-1. The polymer molecules tend to lie in the plane of the sheet, but more along z than along x. The extensional compliances in the machine and transverse directions in the plane of the sheet are s 33 and s11 respectively. They will be different and both will be less than that of an isotropic material, while the compliance perpendicular to the sheet, s 22 , is higher. The out-of-plane compliance is difficult to measure in real thin sheets, and is also of little practical importance in many cases. If the drawing is uniaxial and the sample dimension along x and y not too different, then the x- and y-axes become equivalent. This is called transverse isotropy or fiber symmetry, and such samples then require


308

shear experiment in the yz-plane of the orthotropic material will give the shear modulus G 2 3 = I/544. The compliance terms along the diagonal of the compliance matrix [Eq. (7-2)] are the inverses of the three Young's moduli and three shear moduli. The off-axis terms relate to the Poisson's ratios. In full for the orthotropic material

only 5 independent elastic constants. An expert in mechanics would naturally choose the x- or 1-axis as the unique (fiber draw) axis, but in polymer fibers it is the z or 3-axis. This is because the molecular chain axis becomes oriented along the fiber axis by drawing and the chain axis is conventionally labeled the c, z- or 3-axis in poly-

1/E, — v12/— v13/-

-

0 0 0

0 0 0

mer crystals* so that the compliance matrix becomes Eq. (7-3). Unless a detailed study of anisotropic elastic behavior is being made, the results of mechanical experiments are expressed not as elastic constants spq or cpq but as engineering constants. These are Young's moduli, shear moduli and Poisson's ratios. (7-3) Ll 9

s=

13

0 0 0

5

12

5

13

0

0

0

51t

S^

0

0

0

S

13

S33

0

0 0 0

0 0 0

0 0

0 0 0

S

44

0 0

S

44

0

:2(s,,

-

>12.)

Consider a material with orthotropic symmetry that has a tensile stress as applied along z. ol = os, all other stresses are zero and there will be strains e l5 e2 and e 3 . The tensile or Young's modulus in the 3-direction, E 3 , is the ratio aje3, so s 33 = 1/E3. The Poisson's ratios for this experiment are defined as v31 = — s1/s3 and v32 = — s2/s3. Thus s 31 , which by definition is ej(rs when other stresses are zero, is equal to — v 3 1 /£ 3 . This relation may also be expressed as v31 = — s 31 /s 33 . A

V32AE

0 0 0

0 0 0 VG23 0

0 0 0 0

0 0 0 0 0

(7-4)

0

since spq = spq, vtj = v^EJEj). Isotropic materials have single values of E, G and v. There are only two independent elastic constants, so for example the shear modulus may be calculated as G = E/[2(1 +v)]. The bulk modulus K = [pressure/^! + e2 + £3)L the tensile compliance D = l/£, the shear compliance J = 1/G or the bulk compressibility B = l/K may also be used to describe the material. Returning to anisotropic materials, it may be difficult to obtain all the compliance components experimentally; theoretically derived components are often used for perfect crystals, and estimates for amorphous or semi-crystalline materials. If all the compliance components are known, the elastic constants cpq can be obtained by inverting the matrix, C = S~1. For orthotropic materials the shear constants are simple, but the extensional constants are not (McCullough, 1977). For example, C44 = IA44 = G 2 3 but

1 Except in some monoclinic crystal structures, such as poly(vinyl alcohol) where crystallographic conventions confusingly require that the chain axis is b.

7.1 Introduction

^33

=

- 2(E3/Ex)v12 v23 v13 -

"?2

For fiber symmetry with unique axis 3, as in Eq. (7-3), EX=E2 = E±, E3 = £,,, = G 2 3 = G|,,

and

=

V23=V,,

v

i2 — V2i = v i - This simplifies the expressions for the elastic constants considerably. Using the same example as above,

"

l-2(£||/£1)v2

(7-6)

In a fiber used as a primary mechanical component such as a tire cord, the most important elastic constant is the tensile modulus in the fiber direction, £j ( . The tensile modulus in the transverse direction, E l9 may also be important, but all transverse properties are difficult to measure. The torsional modulus about the fiber axis G|| = l/s 4 4 is the only other property likely to be determined. Mechanical properties may be measured by the determination of longitudinal ultrasonic wave velocity, as the speed of sound depends on the density and modulus of the material. In this case the transverse strains are zero, so the experiment does determine a component of C, the elastic constant or modulus matrix, directly. In the case of fiber symmetry with z the fiber axis [Eq. (7-3)], the longitudinal ultrasonic modulus along the fiber direction, L 3 or L|( = c 33 and Lx or L± = c11. Note that Ly =(= l/s 3 3 , the usual E j|. 7.1.3 Units The SI unit of modulus and stress is the pascal, which is 1 newton • meter" 2 . Materials are likely to have moduli over 109 Nm~ 2 , or 1 GPa. Other units that may be used are dyne • cm" 2 , and in more practical work psi (lb-in" 2 ), kg mm" 2 , or

309 (7-5)

kg • cm 2. The relation between them is: l G P a = 10 1 0 dyne-cm" 2 = = 101.9 kg • mm" 2 - 145000 psi = = 145 kpsi The units of compliance are the inverse of these, m 2 /N and so on. The "practical" units use units of mass confusingly, using them to express the force "lb wt." or "kg f" that results from gravity acting on the mass. When the mass of the product is important to the application, the modulus of the material is often expressed as a specific modulus, (modulus/density). This measure makes polymers look better than metals or ceramics. The unit for this is J - k g " 1 , which may be reduced to (m/s)2. The specific modulus is unfortunately often quoted in terms of inches. This is due to writing it as modulus in units of lb • in" 2 divided by density in units of lb • in" 3 , and falsely cancelling a force in pounds weight with a mass in pounds. Specific modulus is also used in fiber science because of the difficulty in determining the cross sectional area of fine fibers. The specific modulus units are N • tex" 1 and g • denier" *. Denier and tex are units of mass per unit length, the product of density and cross sectional area and they are obtained simply by weighing a known length of fiber. The tex is 1 0 " 6 k g m or l g - k m " 1 , while 1 g • denier" 1 is 0.9 N • tex" 1 . Thus: 1 M J - k g " 1 = 1 km 2 s~ 2 = = 106m2s"2 = 1 N-tex"1 7.1.4 Non-Ideal Elasticity The definition of the modulus given above assumes a well defined ratio between stress and strain that does not depend on

310


load, that is, the elasticity is linear. The derivations use engineering stress and strain, which are related to the original cross-sectional area and original length of the specimen. All assume that the strains are small. More fundamentally, any discussion of elasticity assumes that elastic and plastic strain can be distinguished clearly. The mechanical deformation of real polymers may not meet any of these assumptions of ideal elasticity. Most important of these non-ideal behaviors is the viscoelastic nature of the semicrystalline materials, due to the disordered components in the material. It is assumed that the polymer crystals themselves are simple elastic bodies with a well defined plastic yield point. 7.1.4.1 Viscoelasticity Viscoelastic properties are of practical importance, and give insights into the molecular mechanisms of deformation (see Chap. 9 of this Volume). Here the main point is that elastic measurements and definitions must take into account the unavoidable plastic flow component of displacement. All data for sample stiffness will depend on the time scale of the experiment. Compliances measured at fixed load during creep, at fixed extension during stress relaxation and during oscillatory loading may all be different. Strictly, all are better defined than the compliance determined by the usual simple tensile test at constant cross-head speed. It is not generally possible to make useful comparisons unless the exact conditions of testing are known. For example, materials suppliers who quote a modulus for their product without giving the strain rate used or the ASTM standard test number are not being helpful. If stiffness is a selling point, they can be assumed to have used a high strain rate or ultraso-

nic measurement. This gives a high value of modulus that may or may not be appropriate to the end use being considered. Elastic deformation is immediately recovered on unloading, and plastic deformation is permanent and unrecoverable. What then of a polymeric material that deforms under load, and when the load is removed, slowly recovers the deformation so that say 24 hours later is in its original form? This behavior is quite common, as you may see when you move furniture on a carpeted floor. The weight crushes the fibers, but they should recover over a period of hours or days so that the marks disappear. This is called a delayed elastic or anelastic response. In general terms it is the long range interactions and the long relaxation times of polymer chains that cause this behavior. If a metal wire is kept under constant load and slowly extends then it is creeping. This is a plastic deformation, and the elastic modulus is a constant. A polymer slowly extending under constant load may be creeping or it may be showing how its compliance increases from the immediate elastic value Jo to the larger long term value J^. The only way to distinguish these two cases is to allow the sample to relax for long times and see how much of the deformation is recoverable (Turner, 1966). 7.1.4.2 Nonlinear Elasticity Simple materials are often linear elastic; at small strains, stress is a linear function of strain, so that the modulus or compliance is constant as the load increases. In many polymer systems this is not true, and the compliance generally increases with load. Thus measurements of modulus should also quote the maximum strain used in determining the slope of the stressstrain curve. A high modulus is obtained

7.2 Experimental Determination of Elastic Properties

by using precise and well aligned testing machines, where grip motion and joint alignment do not spoil the very first part of the test. For viscoelastic systems where the compliance will already be a function of time and temperature, the situation rapidly becomes very complex. The stiffness may change during a mechanical test because of structural changes in the material, or because the time that the material has been under stress is increasing. In most cases tests are made under standard conditions, and the nonlinear behavior is ignored. Studies that take nonlinearity into account do so largely on an empirical or curve fitting basis (Arridge, 1975) as the equations for anisotropic materials contain many unknowns. The main aim of such studies is usually to predict elastic properties under some conditions, such as creep or cyclic loading, from tests made under other conditions. For some high modulus fibers, nonlinear elastic behavior may be predicted or explained by changes in the orientation or geometry of chains under stress (Northolt and Van der Hout, 1985; Wierschke et al., 1992). Modulus has been found to increase during creep of aramid fibers (Rogozinsky and Bazhenov, 1992). 7.1.4.3 Large Strains When strains are not small, many factors in Hookean elasticity have to be altered, and so do the testing methods. Consider a tensile test in a testing machine with a constant rate of motion of cross-head. At large strains the gauge length and the cross-sectional area of the specimen do not remain constant. This means that the strain rate falls and the load variations do not follow changes in stress. To have a test at a constant rate of strain, the cross-head must

311

accelerate. As metals and most other materials fail at small elastic strains, this is not a practically important problem for them. The theory of linear elasticity at large strains has been developed (Green and Adtkins, 1970; Arridge 1975) and in polymer science is mostly used to describe rubber elasticity (Chap. 8 of this Volume). In that case extensions of several times the original length must be considered and the unstrained material is isotropic. When semi-crystalline polymers are made into very strong fibers, they can undergo elastic deformations of 1-2%. While this is far from the extensions possible in elastomers, it can be enough to cause a change in the molecular chain orientation. For highly oriented systems with very stiff molecular chains this can increase the apparent tensile modulus (Northolt, 1980).

7.2 Experimental Determination of Elastic Properties 7.2.1 Quasi-Static Measurements Quasi-static refers to all tests where the measurement is made over a comparatively long time scale, commonly in the range 1 to 100 s. There are two classes of test used. In one class the sample dimensions and all other conditions of testing are defined by some standards organization, such as ASTM or ISO. Most of these tests use standard mechanical tension/compression testers, where a cross-head is moved by screw or hydraulic control at a fixed speed. Since the sample gauge length, the crosshead speed, and the strain at which the modulus is to be determined are all defined, the time scale of testing is also standardized. ASTM D 2990-77, D 638-77 a, D 882-75 b are examples of standard tests for solid polymer samples.

312


The strain rates in such tests are generally in the range of 10~ 4 to 10" 1 s" 1 . In this range the modulus of many practical semi-crystalline polymers does not vary strongly with strain rate, but only increases slowly as the testing rate increases. Thus these tests give reliable and reproducible measurements of the elastic properties. Most of them are concerned only with tension or torsional shear and with largely isotropic samples such as compression molded sheet or injection molded bars. If the materials being tested are in short supply, the tests may be modified to use smaller samples at some cost of precision. In the other class of test, a more fundamental measure of elastic properties is sought. A constant load is applied for a given time, and the sample strain is then measured. If the time was 10 s, the result would be the 10 s creep compliance. Applying different loads allows one to build up an isochronous stress-strain curve, Fig. 7-2. This may be used for the study of nonlinear elasticity, or more simply to ensure that the loads and extensions measured are small enough to remain in the linear elastic region. In principle a static modulus or the long term compliance J^ can be obtained by testing over a range of time scales and extrapolating to infinite time; this is not often done. Testing machine design for tensile tests of this type is not very demanding. The load is applied by attaching a weight to the sample; polymer samples are not usually very stiff, so that machine compliance is not a problem. Displacement measurement by linear variable differential transformer (LVDT) is sufficiently accurate for most purposes. More accuracy may be obtained from direct extensometer measurement of sample length, but extensometers with knife edges may deform the sample locally unless the polymer is stiff. Optical

0.5

7

0.4

r

Increasing time under load

O 0.3

.2 'E. o 0.2 o PH

Sii > 533- The anisotropies are high, with s22/s33 = 15 and the maximum ratio of compliances s66/s13 almost 1000. The accuracies of measurement also range widely, from 2 - 3 % for the "easy" s339s1± and s 44 to 17% for the difficult s 22 . When the sample is a fiber, any measurement except s 33 and the torsional compliance s 44 can be a problem. The transverse stiffness is measured by compressing a fiber between two plates. The analysis requires that the fibers are circular in cross-section, and the surface constraint gives plane strain conditions. With glass plates and comparatively large monofilaments optical microscopy gives the width of the contact zone and thus the strain under a constant load, Fig. 7-4 (Pinnock et al., 1966). PoisFigure 7-3. Exploded view of the apparatus used to measure the in-plane compliance s22 of oriented sheet, from Akindayini et al. (1986). (A) piston, (B) sample, (C) clamps, (F) Hall effect magnetic field sensor, (G) permanent magnets with like poles adjacent, (H) to (K) locating and alignment devices.

uct of thickness and refractive index, so the stress-optical coefficient must also be determined. Otherwise, sensitive extensometers can be used (Darlington and Saunders, 1979; Richardson and Ward, 1978). These use capacitance or Hall effect transducers to measure displacements of less than

Microscope objective

Monofi lament

Glass blocks

Figure 7-4. Measurement of transverse fiber modulus E±. A fiber is compressed between two glass plates and the contact area is measured in an optical microscope (Pinnock et al., 1966).

0.36 ±0.01 -0.38 ±0.04 -0.018 + 0.001 0 0 0 -0.38 ±0.044 0.9 ±0.16 -0.037 + 0.005 0 0 0 -0.018 ±0.001 -0.037 ±0.005 0.066 + 0.001 0 0 0 0 0 0 9.6 ±0.3 0 0 0 0 0 0 0.56 ±0.02 0 0 0 0 0 0 14 + 0.

(1)

315


Driver

Strain Gauge

Load Cell I

Sample I

=H3

D "i

Chamber ' I

son's ratio can also be obtained from the increase in lateral fiber diameter in this case. Using optically flat steel plates to compress fibers laterally in an Instron testing machine, transverse modulus and failure strain could be measured for fibers as small as 8 jim in diameter (Phoenix and Skelton, 1974). Naturally, alignment in this compression test must be very good indeed. 7.2.2 Dynamic Measurements

Dynamic measurements of elastic properties involve oscillatory motion of the specimen. There are many methods and many commercial devices for the purpose. The commonest is where the specimen is put into forced oscillations by a driver similar to a loudspeaker coil at a frequency usually in the range 0.01 -100 Hz. The time scale of the deformation that is being measured is now controlled by this frequency. Figure 7-5 is a schematic of a dynamic mechanical analyzer where the driver and transducers are mounted on an optical bench. In this case the specimen is always tested in tension, but other instruments have a range of sample mounts for tensile, bending (a) or shear deformation (b), Fig. 7-6. The applied displacements and the resulting load are measured in both amplitude and relative phase, giving both an elastic storage modulus and a viscous loss modulus. The procedure is called dynamic mechanical analysis, or since the tests are

Adjust Tension

Figure 7-5. A schematic diagram of the Rheovibron instrument for dynamic mechanical testing in tension.

often run over a range of temperatures, dynamic mechanical thermal analysis, DMTA. In a modern instrument, automatic measurements at several frequencies will be made during the continuous heating or cooling of the sample. When a film or tape is tested in tension, there must be a static tensile load as well as an oscillatory one, to keep the sample in tension. End effects must again be corrected for, and this is done by using several samples of different length. There are many possible sources of error in obtaining a reliable absolute measurement of modulus, including machine compliance, sample slippage and misalignment that also affect static tests. In a dynamic tensile test the static load may change the sample structure, and inertia and resonance in the testing machine have to be considered. Massa

(a)

Ib)

Figure 7-6. Sample mounting arrangements for a more modern DMTA (Polymer Laboratories PLC). These are for tests in dual cantilever bending mode (a), and tension (b). The supports and measurement devices are all on one side, making it easier to design a good temperature controlled environment.

316


(1973) and Wedgewood and Seferis (1981) discuss these errors, and methods of dealing with them particularly for the models of the Rheovibron viscoelastometer available at that time. The shear attachments for DMTA instruments are generally designed for the testing of soft elastomers or foams. A comparatively stiff film of crystalline polymer will have insufficient displacement. The dynamic shear of such samples is measured with a free vibration torsion pendulum. One end of a long sample is clamped, while the other end is attached to an inertia bar. The sample is held in light tension, by the weight of the bar or other counterweight and the bar is set into oscillation, Fig. 7-7. The frequency gives the elastic shear modulus and the decay of amplitude gives the loss modulus. The frequency is commonly set to be about 1 Hz by choice of the inertial weights, but can be from 0.01 to 50 Hz. At higher frequencies the wavelength of the stress wave may become comparable to the sample dimension, producing resonant effects.

Counter balance

Inertia rod

Specimen

rflinin

Figure 7-7. Torsion pendulum apparatus for the measurement of G,,. The inverted arrangement makes it simple to place the sample in a temperature controlled bath.

7.2.3 Ultrasonic Measurements Ultrasonic modulus measurements are simple in principle and have many advantages. For unoriented materials and for oriented materials when the acoustic waves propagate down symmetry axes of the sample, the relevant modulus component is simply the density times the wave velocity squared. In an unoriented material longitudinal waves give £, transverse waves give G and so any of the other elastic properties can be calculated. Table 7-1 shows some typical results for semi-crystalline polymers obtained at room temperature and 2 MHz by Hartmann and Jarzynski (1974). Continuous wave propagation methods have been used for the measurement of wave velocity (Kolsky, 1958; Ward, 1983, Sec. 6.5). There the phase of a stress wave of frequency 1-10 kHz is measured as a function of distance along a sample. Alternatively, scanning frequency to find resonance peaks in rod shaped samples relates frequency to wavelength (Davidse et al., 1962). Today pulse propagation at 0 . 1 10 MHz is used, and the modulus may be derived from the group velocity determined from the pulse transit time. Dispersion is low, and phase and group velocity are treated in the same way; difficulties may arise in highly anisotropic materials if the pulse direction is not well defined. Higher frequencies allow faster pulse rise times and so more accurate pulse timing, but attenuation increases with frequency. The sample thickness must be kept low, negating the improvement. In the case of oriented materials, for longitudinal waves propagating down z, c33 = Q v2. If the sample has fiber symmetry and thus transverse isotropy, the velocity of transverse waves along z gives c 44 ; waves traveling along x give c11 and


317

Table 7-1. Elastic properties of unoriented semicrystalline polymers, measured ultrasonically (Hartmann and Jarzynski, 1974). Polymer

Nylon 66 Nylon 6 Polypropylene Polyoxymethylene High density polyethylene Poly(vinylidene fluoride)

Bulk modulus (GPa)

Shear modulus (GPa)

Young's modulus (GPa)

Poisson's ratio

6.53 6.45 4.37 6.59 4.54 5.18

1.43 1.43 1.54 1.43 0.91 1.07

3.99 4.00 4.13 4.01 2.55 3.00

0.40 0.40 0.34 0.40 0.41 0.40

[through c 66 , Eq. (7-3)] c 12 . The fifth elastic constant, c 13 , can be obtained from a velocity measurement at a known angle to z. Disks a few millimeters thick may be cut from an oriented sample rod and bonded between pairs of transducers (Rider and Watkinson, 1978; Leung et al., 1987), Fig. 7-8. This requires a large initial specimen, and in a general anisotropic material defines the direction of wave propagation rather than the direction of pulse propagation. Alternatively, a rectangular sample block of known dimensions is immersed in a liquid such as water or silicone oil. Then the transducers are also immersed and do not need to be bonded directly to the sample, so many arrangements are possible (Rawson and Rider, 1974; Read and Dean, 1978; Leung et al., 1980). In one setup, the transmitter and receiver are pointed directly at one another, and the sample is mounted between them on a turntable. The extra pulse delay when the sample is inserted in the path gives the velocity. Rotating the sample on the turntable allows continuously varying propagation directions, and thus curve fitting with the elastic constants as variables rather than their derivation from a few results. Refraction at the sample surface has to be taken into account in the calculation. The arrangement can be further simplified

with a reflecting surface behind the sample, so that one transducer acts as transmitter and receiver. A large number of measurements may be required, but automated systems may be used. An automated ultrasonic system was used by Habeger (1990) to measure all in-plane moduli as a test for orthotropic symmetry. If the velocity of sound in the immersion liquid is known, the velocities in the sample can be determined without any timing measurements (Treloar, 1970). The transducers are set to measure reflectivity. For this the transmitter and receiver move so

Figure 7-8. Schematic diagram showing samples cut from a polymer rod or tube with transverse isotropy for the ultrasonic measurement of moduli. A is used for c 33 , B for c 11? c 44 and c 66 , and C for c 13 (Leung etal., 1987).

318


that the angle of incidence and the detected angle of reflection from a sample surface remain equal as they vary (as in an X-ray diffractometer). At the critical angle of incidence the reflected wave reaches a maximum constant amplitude. This would be 100% of the incident amplitude, if there were no losses in the system. There are two such angular positions 6L and 9T corresponding to longitudinal and transverse wave propagation in the specimen. At each position, v/v0 = 1/sin 9, where v0 is the velocity of sound propagation in the liquid. In optical terms, which may be more familiar, this is equivalent to the determination of refractive index by measuring the critical angle for total internal reflection. More recently methods have been developed which do not require sectioning or immersion of the sample; pulsed lasers are used as point sources of ultrasonic waves. Lasers can also be used as interferometric detectors of the displacements due to ultrasonic pulses (Hutchins, 1988; Sachse and Hsu, 1989). These optical noncontact techniques remove the need for immersion of the specimen. In anisotropic sheet specimens the pulse propagation direction is defined by the vector from the point source to a small transducer. The laser is scanned over the sheet to give a range of directions and as both longitudinal and shear waves are detected, the result is a full set of elastic constants (Castagnede et al., 1991), Fig. 7-9. These techniques have been applied to polymer matrix composites (Sachse et al., 1990, 1991), but not yet to oriented crystalline polymers. Measurement of pulse propagation through the thickness of fine fibers or films less than 1 mm thick requires too much accuracy in timing, but the longitudinal properties of fibers can be obtained. Again a pulsed laser beam is focused onto the specimen and the thermal pulse creates a

Composite Specimen Receiver Scanned Ultrasonic Point-source

Pulsed .' X Laser Source

(on rear surface)

fx 2

Figure 7-9. Schematic of the use of a laser beam as a moving point source of ultrasonic pulses. The detector is of small area, and at a fixed position on the rear surface of the sample (from Sachse et al., 1991).

shock wave that moves along the fiber (Smith et al., 1987). Timing the shock wave arrival at the end of the long fiber gives the fiber modulus. The sample may be held at high temperature or in a controlled atmosphere during the test. 7.2.4 High Speed Testing

High speed testing machines use hydraulics or compressed air to move the cross head and electronic data storage to record the data. They can be used for tensile tests on specimens at strain rates of 10-100 s" 1 . In ballistic impact testing a projectile is fired at the sample, and its deformation is recorded by high speed photography or other means. If the geometry is simple, the modulus can be calculated from the record of the deformation of the sample. One example of simple geometry is a semicylindrical projectile incident perpendicular to a fiber under tension (Smith etal., 1960; Prevorsek et al., 1991). Strain rates of over 1000 s" 1 are achieved, roughly equivalent to strains of 0.01% at a frequency of 10 MHz in ultrasonic testing. High speed testing and ballistic testing are both a lot more trouble than ultrasonic

7.3 Experimental Determination of Crystal Modulus

testing, so they are used primarily to obtain data relating to failure, strength and toughness, and not elastic properties.

7.3 Experimental Determination of Crystal Modulus 7.3.1 X-Ray Diffraction Wide angle X-ray diffraction spots appear at diffraction angles 2 0 given by Bragg's Law, 2 d sin 9 = A, where X is the X-ray wavelength and d is the spacing of the crystal lattice planes. When the crystal is elastically strained during diffraction, d-+d + §d and differentiating Bragg's Law gives

^ = e

= - c o t 0-50

(7-7)

Thus measurement of the change of the diffraction angle of reflections under load gives £c, the crystal lattice strain, directly. If possible a reflecting plane that has its normal along the chain axis, such as (002) in polyethylene (PE), is chosen to give the strain in the chain direction. Other reflections such as (200) in PE can be used to investigate strains perpendicular to the chains. In principle these measurements can be made on either isotropic or oriented material with any stress state, but simple tension on a highly oriented material is almost always used. Parallel alignment of chains makes interpretation of the results feasible, and tension along the chain axis has the important practical advantage that the sample is stronger. It can then withstand larger stresses without fracture or plastic flow, and the shift of the X-ray reflection is larger and easier to measure. To determine the crystal modulus in the chain direction E% (simply called Ec from

319

now on) we need the crystal stress ac as well as sc. Experimentally the macroscopic sample stress as is measured and in principle oc is not known. Some structural model for the sample structure or some assumption must be used to calculate oc from the sample stress as. The simplest possible assumption is that these stresses are equal. In that case the ratio oj&c is the crystal modulus. The structure required to give this result of uniform or homogeneous stress is one where the crystals and any disordered material are mechanically in series. The literature is strongly divided on the validity of this assumption. On one hand some authors use the uniform stress assumption and say that X-ray diffraction determines the crystal modulus. On the other are those who do not accept the assumption and take (jjsc as an apparent modulus E app , which gives information about the structure of the material. They may use values of Ec determined by other methods, or combine X-ray diffraction with other data to determine Ec from Eapp. The basic argument in favor of the uniform stress assumption is that it gives consistent values for Ec even when samples with very different macroscopic moduli are used. There is further discussion of this point in Sec. 7.5. To measure crystal modulus by X-ray diffraction a drawn sample or bundle of fibers is mounted in a straining frame horizontally on an X-ray diffractometer, Fig. 7-10. Measurements of this type were first made by Dulmage and Contois (1958). Many materials have since been studied by Sakurada and his co-workers (Sakurada et al., 1962,1964,1966; Sakurada and Kaji, 1970) and more recently by Clements et al. (1978, 1979), Nakamae et al. (1987 a, 1990, 1991) and Matsuo and Sawatari (1986, 1988 a, 1988 b, 1990). The polymer sample is stretched under dead loading, using a pulley to transfer the load to the horizontal

320


(a)

— to Sealer

(b)

Figure 7-10. (a) Top view of a horizontal diffractometer with stretching frame mounted on it, for the measurement of X-ray modulus. The load is applied from a weight W through a pulley and cable to the sample C. (b) Simple stretching frame used in (a). Clamps (1) hold fiber sample (2); extension indicated on length scale (3) (Sakurada et al., 1962).

sample. The more recent papers listed measure the crystal modulus as a function of temperature. The accuracy of strain measurements is high. The error in determining 2 9 was first given as one minute of arc, ±0.0166° (Sakurada et al., 1962, 1966). In a more recent paper an error of + 0.005° is quoted for the case of PE (Nakamae et al., 1991). The PE (002) reflection, used to determine the strain parallel to the molecular chain,

has a lattice spacing d = 0.127 nm. With an X-ray wavelength of the usual value, X = 0.154 nm, 2 9 = 74°, 2 tan 9 = 1.5 and a 0.5% lattice strain gives a change of diffracted angle of 7.5 mrad or 0.4°. A 0.005° error in 2 9 then corresponds to a strain error of ± 5 x 10~5. The line width of the (002) reflection is normally from 0.1 to 1°, depending on the size and perfection of the crystals in the sample and the collimation and wavelength spread of the X-ray beam. Thus the position of the center of the reflection is measured to an accuracy 2 0 200 times better than the width of the reflection. The total shift is only of the same order as the linewidth. Using a double focused monochromatic X-ray beam from a powerful synchrotron source, the same type of experiment has been conducted with a single high modulus fiber as the sample. Poly(p-phenylene terephthalamide) [PPTA or Kevlar (DuPont)] and high modulus polyethylene have been used, with sample diameter as low as 12 jum (Prasad and Grubb, 1990; Li etal, 1991). Figure 7-11 shows the (002) reflection from a Spectra 1000 (Allied Signal) polyethylene sample at low and high stress. In this case the deconvoluted linewidth at low stress was less than 0.1°, and curve fitting defined the centroid of the peaks to better than ±0.001°. However, this corresponds to a positional accuracy for the reflection of ± 3 jim, and if the sample were to move, so would the reflection. Thus the true accuracy of strain measurement depends on the stability of the sample and its holder under load. This also holds for the experiments carried out on larger samples using a normal diffractometer. Matsuo and Sawatari show in Fig. 1 of Matsuo and Sawatari (1988 a) that the observed crystal strain is proportional to the applied stress, even when there is some misalignment or instability of sample posi-

7.3 Experimental Determination of Crystal Modulus

321

2.05 GPa 1.62 GPa 1.14 GPa

0.56 GPa

Figure 7-11. Polyethylene (002) reflection from a single Spectra 1000 fiber (28 (im in diameter) showing the shift to lower 2 0 and broadening as stress increases up to 1 GPa, then no change (Prasad and Grubb, 1989).

0.00 GPa V9 73.0

73.5

74.0

74.5 2©B, degrees

75.0

75.5

tion under load. Unexpected changes in diffracted intensity signaled this misalignment, and allowed the authors to discard the poor data. Without this care, values for the crystal modulus ranging from 86 to 374 GPa could have been derived, an error range of ± 60%. Not all materials are as convenient as PE. PET, for example, has no strong reflection parallel to the chain direction in the crystals. Instead the (105) reflection must be used, which is 11 ° away from the chain axis (Sakurada et al, 1966; Thistlethwaite etal., 1988). The situation is similar in f-polypropylene (iPP), where the reflection nearest to the meridian is the (T13). The modulus is measured along the (113) plane normals, and a complicated mathematical treatment is used to recover the chain modulus (Matsuo and Saw atari, 1986; Sawatari and Matsuo, 1986). The ratio ojsc varies as a function of temperature in some specimens and this may be interpreted directly as a real change in Ec. Significant temperature variations in £ c , more than that merely caused by thermal expansion of the lattice, may be expected in high modulus polymers. These

76.0

materials, PPTA, for example, contain chains extended to their maximum stable end-to-end distance, so that any bond rotation reduces the end-to-end distance of the chain. On heating, thermal excitation of torsional vibrations makes the chain become more of an open helix, on average. The thermal expansion coefficient in the chain direction of such an extended chain is thus negative. Some low energy bond rotations can extend these thermally contracted chains, so their extensional modulus is less than that of the original extended chains and Ec falls as temperature increases. This interpretation has been made for several materials, for example PE (Nakamae etal., 1987b; Nishino etal., 1992), PPTA (Nakamae etal., 1986a; Tadaoki et al., 1987), and polyoxymethylene (POM) (Nakamae et al, 1990); results are summarized in Nakamae and Nishino (1991). In polytetrafluoroethylene (PTFE) changes of Ec with temperature are more specifically associated with phase transitions of the chain. As the temperature increases the PTFE chain conformation in the crystals changes from a 13/6 helix to a 15/7 and then to an irregular helix. Fig-

322


200

150 -

100 -

- 1.300

ttf)

'o

-100

0

100

Temperature,

200

1.295

c

Figure 7-12. Modulus in the chain direction for polytetrafluoroethylene as determined by X-ray diffraction. The changes as a function of temperature are associated with changes of chain conformation, indicated by the meridional d-spacing changes shown in the lower plot (Nakamae et al., 1986 c).

ure 7-12 shows the modulus changes and the changes in meridional X-ray spacing for PTFE (Nakamae et al., 1986 c). For para-linked aromatic polymer chains where every main chain single bond is collinear, bond rotations have no effect on chain length. In principle, thermal excitation of chain bending vibrations can contract the chain and reduce the extensional modulus. As bond angle changes require a lot more energy than bond rotations, these effects will be small, and Ec will be affected by heating primarily through transverse thermal expansion of the interchain distance. If the uniform stress assumption is not made, temperature variation of ajgc may be due to changes of crystallinity or to changes in £ a , the stiffness of the disordered component. Increase of £ a p p at low temperature is then associated with an increase of Ec and Ea in PE (Clements et al.,

1978,1979) and in POM (Brew et al., 1979; Jungnitz et al., 1986; Wu et al., 1989). The ratio . The change in bond inclination, 5c/>, is Sa/2 and the effect of 80 on the chain length, 8La is Fl2 8La = 8(j) I sin 4> = —— sin2 <j> = 4/c (7-10) The total chain extension 8L is just the sum of the two expressions above and Young's modulus of the chain is given by E=

F/A 5L/L 4/Ci/CpL

A (4 fcpcos2 4> + k1 sin 2

(7-11)

where A and L are the chain area and unit length, obtained from X-ray diffraction. With A = 0.182 nm2, L-0.127 nm (per bond), fc1 = 436N/m and fcp = 35 N/m (from Rasmussen, 1948), Treloar obtained Ec = 182 GPa (Treloar, 1960 a). The simple two-constant method was also applied to Nylon 66, PET (Treloar, 1960 b), cellulose (Treloar, 1960 c) and poly(phenylene sulfide) (Unwin and Ward, 1988). Having to sum over rings and include different atoms in the main chain and different bonds rapidly increases the complexity of even the simplest calculations for planar chains. The limitations of this method were realized, and soon more complex force fields were used. Urey-Bradley force fields including effects of bond twisting and interatomic repulsions were used for polyethylene and

a number of nonplanar chains such as i-polypropylene and polyoxymethylene (Shimanouchi et al., 1962; Asahina and Enomoto, 1962). Holliday and White (1971) contains a summary of this work. The result obtained for polyethylene was 340 GPa, now recognized as high, but at the time agreeing with Raman LAM data. A major advance was the inclusion of intermolecular interactions, initially for PE with the methylene groups treated as single units (Enomoto and Asahina, 1964; Miyazawa and Kitagawa, 1964; Anand, 1967) and then with full H - H and C - H interactions for bonded and nonbonded atoms. Odajima and Maeda (1966) used Lennard-Jones and other potentials for the intermolecular interactions to derive the complete elastic constant matrix C for polyethylene. Ec was found to be 257 GPa, and intermolecular effects on this parameter were only 0.2%. Recent advances have been in techniques for dealing with crystals containing large numbers of atoms. Keeping track of co-ordinate system transformations between the bond motions, the unit cell and Cartesian axes is a problem in complex systems. As in normal mode analysis of vibrational spectra, the use of symmetry relations is important in simplifying the treatment. Tashiro et al. (1978 a) use the B matrix method (Shiro and Miyazawa, 1971) but enter the space group symmetry of the crystal at the beginning of the calculations. An orthorhombic system with an elastic constant matrix as in Eq. (7-2) is factorized into four blocks, one 3 x 3 and three l x l blocks for the remaining diagonal terms. The result is that it is not necessary to consider all the atoms in a unit cell, as one asymmetric unit implies all the others. Applied to polyethylene the result for the compliance matrix is given by Tashiro etal. (1978 b) as

7.4 Theoretical Modeling of Elastic Constants

0.145 0.0478 0.00019 0 0 0

-0.0478 0.117 -0.00062 0 0 0

-0.00019 -0.00062 0.00317 0 0 0

The a form crystal structure of nylon 6 contains 76 atoms in one asymmetric unit, and using this method would require matrices of order 230. Tashiro and Tadokoro (1981) found it necessary to approximate the crystal structure, in a manner chosen to leave nearest neighbors relatively unaffected. For a qualitative view of the transverse moduli, a graphical representation as in Fig. 7-21 is more effective than a matrix of elastic constants or compliances. The large transverse modulus associated with hydrogen bonding can be seen in nylon and in poly(vinyl alcohol), while polyethylene is closer to transverse isotropy. As computer power increases, more complex systems can be dealt with, and recently full elastic constant matrices for cellulose I and cellulose II have been reported (Tashiro and Kobayashi, 1991). If the unit cell is too large for a full crystal calculation or if the polymer lacks full three-dimensional order a molecular dynamics calculation of single chain stiffness

0 0 0 0 0 0 0.313 0 0 0.618 0 0

0 0 0 0 0 0.2

331

(7-12)

is very useful. Crystal symmetry or the helical symmetry of the chain can be used to simplify the calculation, even when intermolecular effects are neglected (Manley and Martin, 1973; Tashiro et al., 1977 b). Calculations of single chain properties often refer to high modulus materials such as aromatic polyamides (Tashiro et al., 1977 a) or poly(p-phenylene benzobisoxazole) (PBO) (Tashiro and Kobayashi, 1991). It is technologically interesting to know how close practical high modulus materials are to the theoretical limit, and for these materials intermolecular effects on the modulus along the chain direction are likely to be small. The methods outlined in the references above, which once involved a considerable effort in computation are becoming available in commercial packages running on inexpensive workstations. With a given molecular configuration, crystal parameters are rapidly derived by energy minimization and a touch of a button will gen-

PVA

nylon 6

P

'

0

IOGPQ

Figure 7-21. Graphical representation of the transverse modulus of crystals of polyethylene, poly(vinyl alcohol) and a and y forms of Nylon 6, to the same scale, superimposed on drawings of the projected molecular chains of the crystals (Tashiro and Tadokoro, 1981).

332


erate an elastic constant matrix (Yang and Hsu, 1991). The atomic force constants used are derived from a wide range of spectroscopic studies. However, there is a choice of constants and intermolecular potentials to be made by the programmer or the user. If these empirical choices were made on one set of compounds and the calculations are made on a different type, there may be doubts about the accuracy of the result. The uncertainty may be hidden to the casual user by the precision and computer presentation of the results. Interatomic potentials can be used to calculate many properties of the material, so if experimental data is available it is possible to check the calculations. The observed chain conformation and crystal structure should correspond to an energy minimum for the system if the correct potential function is being used. The vibrational spectrum of the polymer should be accurately predicted by the same potentials as are used for the modulus. The heat capacity at low temperature can also be derived from the long wavelength phonon spectrum. 7.4.3 Molecular Orbitals Quantum Mechanical Calculations

The accuracy required of the quantum mechanical calculations of modulus is very great. The energy of a covalent bond is very small compared to the total energy binding the electrons in an atom. Mechanical displacement corresponds to phonon excitation, a small perturbation of the covalent bonds. Molecular orbital calculations are more often concerned with bond stability and with electronic excitations, than with mechanical properties. A basis set of atomic orbitals is chosen, and these are combined to form the total electronic configuration (LCAO - linear

combination of atomic orbitals). For a given geometry of the chain, the combination that gives the wave function with the lowest energy is found. The geometry is altered and the energy recalculated. This gives the geometry of minimum energy. If a crystalline chain conformation is known, it acts as a check on the procedure so far. The minimum energy conformation had better be very close to that found experimentally. Next the length of the chain can be increased by some amount - by a small amount if Young's modulus is sought, by a large amount if it is the breaking strength that is to be calculated. All other geometrical parameters of the chain are again allowed to vary, and a new minimum energy found. The increase in energy on extension is the work done, and so the restoring force is the first differential of energy with respect to length, and the modulus is the second differential. If a number of length steps are made, on either side of the equilibrium position, nonlinear elasticity and differences between tension and compression can be investigated. There are many different ways to perform these calculations. All use the Hartree-Fock self-consistent field oneejectron approximation. Some use semiempirical procedures to approximate and simplify the calculations, setting some unknowns to values that fit experimental data. The methods of this type that have been used for modulus calculation are known by the acronyms CNDO and MNDO for complete or modified neglect of differential overlap. Methods without these approximations are generally called "ab initio" methods. An important check on these calculations, as for the molecular dynamics method, is to derive as many properties as possible from the same set of input data.

7.4 Theoretical Modeling of Elastic Constants

Boudreux used the more approximate CNDO method to determine Ec for polyethylene as 297 GPa (Boudreux, 1973). This is an unusually low value for molecular orbital calculations. Dewar et al. (1979) used MNDO on n-alkanes and polyethylene to obtain chain geometry, vibrational spectra, electronic band structure and £ c , which was found to be 490 GPa. According to Hong and Kertesz (1990), who obtained a value of 369 GPa by a similar method, incomplete relaxation of carbon atom position after deformation was the reason for this exceptionally high value. Klei and Stewart (1986) obtained 360 GPa for Ec in PE with MNDO. Dewar et al. (1979) regarded the MNDO method as a "useful stop-gap until such time as effective ab initio methods become feasible, which unfortunately, seem very remote at present". The first ab initio calculation of polyethylene modulus came later the same year (Crist etal., 1979). While the previous semi-empirical calculations had used cyclic boundary conditions to calculate the properties of an infinite chain, Crist et al. (1979) calculated the energy as a function of length for three linear n-alkanes, C 3 H 8 , C 5 H 12 and C 7 H 1 6 . Using these very small clusters allowed them to do geometrically optimizing ab initio calculations at that early date. The effects of the end groups were removed by subtracting the energies of adjacent n-alkanes, to give the energy as a function of extension for the -(CH 2 ) 2 unit. The result was the same for both alkane pairs, and using X-ray data for the chain cross-sectional area gave a chain modulus of 405 GPa. The calculations were later extended to C 9 H 2 0 with about the same result (Brower et al., 1980).

333

As in empirical calculations, crystal symmetry can be used to limit the number of independent atoms in the system considered. Karpfen (1981), as part of a series of ab initio calculations on polymers, used the S2 screw axis along the PE chain to reduce the atomic basic of the chain set to a single CH 2 group. Karpfen (1981) used two different basis sets of orbitals. Although the effect of basis choice on chain geometry was very small, the C-C bond stretch force constant changed. The chain modulus was 405 GPa with the minimal (STO-3G) basis and 345 GPa with a larger set. Suhai (1983) used similar methods and basis sets but added the refinement of correcting for electron correlation effects using perturbation theory. The uncorrected results for the chain modulus of polyethylene were very close to those of Karpfen (1981) and the corrections reduce the moduli by about 10%, to 303 GPa for an extended basis set. Karpfen remarked that his methods typically gave force constants about 10% too high, compared to studies including electron correlation and accurately predicted a "best value" of 300 GPa from his higher result (Karpfen, 1981). All the quantum mechanical calculations so far referred to have been on polyethylene. This is taken as the typical polymer, and it is also a simple hydrocarbon with few atoms in its basic repeat unit. Combining the theoretical results mentioned here and listed in Table 7-2 with the experimental results from that table and Sec. 7.3, and plotting as a function of year of publication gives Fig. 7-22. There are no clear trends with time, but excluding values below 200 and above 400 GPa, the rolling average has remained in the range 298 ± 7 GPa since 1963. This is close to the best ab initio value and not far from the

334


450 O A • 400 - +

X-ray Raman Neutron Scattering Mol Dynamics

x MNDO 0 ab initio

cd

\ vAvA

I !

•. .

. .

350

§ 300 250

ex, 200

150

1950

1960

1970

1980

1990

Year Figure 7-22. Theoretical and experimental values for the crystal modulus Ec of polyethylene (the single chain modulus, for ab initio and MNDO calculations), plotted as a function of the year of publication.

current Raman data. The uncorrected MNDO results are too high, and the X-ray results at room temperature are low; at 77 K values are higher. The ab initio calculations should give the elastic constants at 0 K. The empirical calculations are based on X-ray diffraction and spectroscopy performed at room temperature. Thermal vibrations have then been taken into account in some way, but small molecule compounds may not model the polymer behavior well. Theoretical modeling of thermal expansion for polymer chains remains a problem, even at the level of quasiharmomic force fields (Baughman, 1973; Kobayashi, 1979; Barron and Rogers, 1989). There are a few quantum mechanical calculations on other polymers. MNDO moduli for PBO and PBZT were obtained

by Klei and Stewart (1986), but the minimum energy geometry pedicted is not the experimental structure. Wierschke et al. (1992) have made semi-empirical calculations using a different method (AMI) that predicts the correct structure. They predict the chain modulus of ds-PBO to be 690 GPa and that of trans-PBZT to be 620 GPa, both about twice the experimental values. The calculations predict that chain stiffness drops rapidly on compression, and the chains undergo a buckling failure at 2 - 3 % compressive strain. Hong and Kertesz (1990) have made MNDO and hybrid calculations of the modulus of polyacetylene in different chain conformations. The hybrid calculations used chain geometries obtained from energy minimization with MNDO and force fields obtained from spectroscopy for PE and polyacetylene (Schachtschneider and Snyder, 1963; Takeuchi et al., 1988). The advantage over using the empirical force fields to optimize chain geometry was not made clear. As MNDO calculated force constants are too high (Fig. 7-22) Hong and Kertesz (1990) say that the alternative to empirical force fields is to "scale" the forces, that is, reduce them by some arbitrary factor to agree with experiment (Pulay et al., 1983). This process removes most of the attraction of a fundamental calculation of materials properties.

7.5 Interpretation of Results for Crystal Modulus 7.5.1 Experimental Data The modulus in the molecular chain direction is the only crystal elastic constant where there is sufficient experimental data to be concerned with interpretation. In that case, however, there is considerable

7.5 Interpretation of Results for Crystal Modulus

controversy. The reader may choose a review concluding that spectroscopy is correct and X-ray diffraction wrong (Fanconi and Rabolt, 1985) or one with the opposite conclusion (Nakamae et al., 1991). The methods do give different results, and the first question to ask is whether they measure the same thing. Song and Krimm (1990) point out that they do not. In the quasi-static X-ray measurement, each stretched chain is surrounded by other stretched chains and interchain distances and positions are altered to new equilibrium values. Raman spectroscopy measures the stiffness of a vibrationally excited molecule which moves in a crystal that remains in the zero stress form. Raman spectroscopy relates to a single chain, but it is not an isolated chain; the interchain forces will be larger in Raman than in the X-ray experiment as the neighboring chains have no time to relax. According to Song and Krimm (1990) this means that the X-ray experiment is much closer than the Raman experiment to stretching an isolated chain. Although continuum concepts do not really belong at this molecular level, the lack of lateral adjustments in Raman makes the measurement more like one of c 33 , while the "X-ray modulus" is l/s 3 3 . This analogy shows that the results need not be identical, and if there is a difference due to this effect, the Raman modulus will be the larger. Initial Raman experiments gave values much larger than those obtained from Xray diffraction. Later corrections to the simple theory have reduced the Raman values, to 20-30% greater than the X-ray values in the cases of PE and PEO described above. In some polymers disagreement is still large, as shown in Table 7-2, but in these cases less detailed consideration of corrections to the Raman results has been made. The table is far from com-

335

300 1 2 3 4 5 6 7 8 9 10 11 12 13 14

O 250

200

0 cd

-

Silk PPS i-PP PEK PEEK Technora PET Vectran Ekonol Cellulose I / Kevlar 1 4 9 / Nylon 6 /© PE / PVA1 /

in

g 100

a

(ft)

o

0

50

100

150

200

250

300

Crystal Modulus, GPa Figure 7-23. Comparison of the maximum specimen modulus that has been reported and the crystal modulus in the chain direction for a number of polymers. (After Nakamae et al., 1991.)

plete, and understates the complexities of interpretation of the X-ray data by putting the results into a single column. Only for the single crystal samples of polydiacetylenes is the direct mechanical testing result comparable with determination of crystal modulus. However, measured values for highly oriented, highly crystalline samples are a lower limit for the possible values of crystal modulus. In a number of cases the largest observed specimen modulus does approach the X-ray or other value for the crystal modulus Ec, Fig. 7-23. Experimentally the difficulty with the Raman LAM is that monodisperse oligomers or samples with a well defined lamellar structure are needed. Small angle X-ray scattering (SAXS) gives a long period in most cases, which is some measure

336


of lamellar thickness. Samples with a high fiber modulus have no long period, or an extremely weak SAXS peak that may not be representative (Adams et al., 1985). To get chain length from the long period it must be corrected for the crystallinity (the thickness of the amorphous layers) and for tilt of the molecular chains in the crystal. These factors will not be accurately known in many cases. The difficulty in interpretation of the Raman results is seen to be that various corrections can have a very large effect, at least on helical chain polymers. Confidence in the result demands confidence in the molecular dynamics used for the correction. The same molecular dynamics can be used to predict the modulus directly, and this should at least be consistent with the Raman result. The difficulty with the X-ray results remains that of determining ac, the stress on the crystals. The uniform stress model has been used for a long time. Sakurada et al. (1962,1964,1966) and then Nakamae et al. (1987 a, 1991) have found for polyethylene and other materials that the ratio ojsc remains the same, even though the structure and the macroscopic modulus vary widely. There seemed to be only two possibilities: (1) All the samples have a series structure, with uniform stress on crystal and amorphous components. They have different sample moduli because of different amounts of the components. (2) All the samples have different more complex structures which for an unknown reason happen to give the same ratio of sample stress to lattice strain. The second possibility seemed extremely unlikely to Sakurada and Nakamae, so they have taken the first to be correct, and describe ajsc as the lattice modulus Ec in every case.

Several factors have now caused the uniform stress assumption to be questioned. In some cases fibers have been prepared with a modulus E{1 higher than the original X-ray value of Ec (see Chap. 13 of this Volume; also Ward, 1985). SAXS and electron microscopy show that most well oriented samples contain parallel lamellar sheets of crystals separated by disordered material. The crystal/amorphous sandwiches are stacked along the draw direction, so the samples are largely series structures and the series model will be generally correct. Samples with modulus Ey comparable to Ec show little lamellar structure, and appear fibrous. SAXS shows a declining intensity in the meridional two-spot pattern that indicates lamellar structures as the stiffness of fibers increases (Ohta, 1983; Adams etal, 1985). Dark field and high resolution transmission electron microscopy shows that the fibrils in this type of material are not long perfect single crystals, but contain many short (< 100 nm) crystalline regions (Grubb and Hill, 1980; Sherman et al., 1982). Mechanical models for such structures are discussed in Sec. 7.6.5, but clearly a series model is no longer likely to be correct. There are also independent indications that the stress is not homogeneous at all times in all samples. As was mentioned in Sec. 7.3.1, the (002) X-ray reflection in polyethylene becomes broader as it shifts under stress, and has been observed to split into two peaks. Similarly, Raman bands (regular vibrational bands, not LAM) shift in position when the crystal is strained. The shift is proportional to molecular strain, so when the ratio (shift/. The extended chains generally have high chain moduli, so s 33 is small and can be neglected in the second term and third terms of Eq. (7-16). Often s 1 3 •(2s 13

— 2s33)sin4

(7-16)

Northolt and Van Aartsen (1978) measured the modulus of a range of aramid fibers ultrasonically, and measured the orientation distribution by X-ray diffraction. They find a good fit to Eq. (7-16) when using accepted values for the transverse compliance components. In particular, when sin2 <j> is less than 0.05 ( < 12°) the third term in Eq. (7-16) can be neglected, giving 1

1 = -^ + A sin2

(7-17)

Figure 7-25 shows that for well oriented fibers, the fiber compliance does follow Eq. (7-17). Thus the modulus of different samples can be entirely explained by the

0.00 0.01

0.02

0.03

0.04 0.05

Figure 7-25. Plot of experimental axial modulus E ^ for a range of poly(p-phenylene terephthalamide) fibers, as a function of the crystal misorientation. The aggregate model prediction is a straight line, with Ec as the intercept at sin2 4> -• 0. The dashed lines are the limits of predictions using reasonable values for the transverse elastic constants (Northolt and Van Aartsen, 1978).

7.6 Interpretation of Macroscopic Modulus

Eq. (7-12) gives the ratios s 4 4 :s 3 3 :s 1 3 as 313:3.2:0.5. For well oriented materials where the third term in Eq. (7-16) is negligible, the expression then becomes: 1

sin2

1

(7-18) £ E G This has been used to model highly oriented PE by Powell etal. (1990). G,, was measured independently, so the excellent fit of the data to Eq. (7-19) allowed an accurate estimation of sin2 for the aggregates. This was found to be consistently some 30% lower than sin2 (j) for the crystals as measured by X-ray diffraction. The implication of this is that the mechanical units are made up of a number of crystals, imperfectly ordered within the unit. Then the extrapolated value of E ~ 250 GPa should correspond to the modulus of the unit, and this should be a little less than that of the PE crystals. A recent study of iPP shows that the one-phase aggregate model fits the data for £|l measured at a temperature of — 95 °C for unoriented samples, and samples drawn up to 18 x (Unwin et al., 1990). In this case the crystal compliances obtained by X-ray diffraction were used as the properties of the anisotropic unit. These authors use a two-phase aggregate model for the E^ data at 45 °C, where the sample extension compliance / 3 3 is given by __

I

y33 = asr3% + (1 - a)s'i3

(7-19)

Here a is the degree of crystallinity, and the compliances of the two phases s'33 and s'33 are obtained by averaging over an orientation distribution as in Eq. (7-13). An obvious problem with this approach is that the elastic properties of the highly oriented but amorphous material, s*pq, are generally not known. The modulus will increase with chain orientation in an unknown manner, and the increase may be substantial. In se-

341

ries Takayanagi models (Sec. 7.6.2) values of up to 80 times the unoriented amorphous modulus are used. It is also difficult to determine the amorphous orientation function, to obtain terms such as cos4 0 a . If the orientation of the sample is not too high, and the chain is stiff so that s33 and s13 are small, the terms s33 cos 4 <j> and s13 sin2 (j) cos2 / in Eq. (7-14) will be small and so / 3 3 = sxl sin 4 4> + s44sin2 0 cos 2 4> (7-20) then

If

(sin4 (/> + sin2 (/> cos2 (j)) = s11 sin

£33

In this approximation the sample modulus in the orientation direction is simply the lateral modulus of the aggregating unit divided by the second moment of the orientation distribution of the units. This approximation is reasonable but not very accurate (Ward, 1983, p. 291). It can be seen as the basis for a much earlier twophase aggregate model for iPP (Samuels, 1974). In that model a

-r-^—

(1 — oc) 2 a sin (/> E\

(7-22)

Here In 13 k k0 kB L L S£(y) m M Mo Mc Mn Mw n n Hi n2 N NA Nv

Helmholtz free energy constants depending on molecular weight, concentration, and scattering contrast factors exponents in the Ogden equation constant in the generalized Rivlin equation cross sectional area of undeformed sample change in elastic free energy change in free energy on mixing Nx solvent molecules with N2 polymer molecules change in free energy of swelling parameters of experimental front factors length of a statistical chain segment constants in a birefringence deformation equation specific heat crack length stress optical coefficient constant in the strain energy function constants in the Mooney equation network functionality magnitude of force Gibbs free energy enthalpy scattering intensity symmetric invariants of deformation unit vector in the direction of the incident neutron beam unit vector in the direction of the scattered neutron beam Boltzmann's constant length interlamellar distance Langevin function parameter in Tschoegl-Gurer equation molecular weight monomer molecular weight molecular weight between crosslinks number average molecular weight weight average molecular weight number of chain segments in a polymer chain mean refractive index, n = {n1-\-2n2)/Z refractive index parallel to stretching direction refractive index perpendicular to stretching direction number of chains in a specimen Avogadro's number, iV A = 6.023 x 10 2 3 number of chains per unit volume

List of Symbols and Abbreviations

N1

359

i?ll R± S Ss(q) ST(q) T To u U V vx W W w(X) z

number of solvent molecules (Flory-Huggins theory of polymer solutions) number of polymer molecules (Flory-Huggins theory of polymer solutions) vapor pressure of solvent in polymer vapor pressure of pure solvent vapor pressure of solvent in polymer solution vapor pressure of solvent above crosslinked polymer probability function pressure 2nd Legendre polynomial P2 (cos 0) = (3 cos 2 9 -l)/2 wave vector magnitude of q value of q at a maximum in a plot of q2 S(q) versus q vector connecting chain ends mean value of r value of r for an undeformed chain vector connecting monomers ij on same or different polymer chains vector connecting monomers i,j on the same polymer chain ensemble average of r2 fluctuation in r ideal gas constant radius of gyration apparent radius of gyration measured at an azimuthal angle
= 90° entropy single chain scattering function total scattering function temperature threshold tearing strength mean energy of a chemical bond internal energy volume molar volume of solvent work strain energy function function in the Valanis-Landel equation number of monomer units in a statistical segment

a1 a2 P y

parallel polarizability of a statistical chain segment perpendicular polarizability of a statistical chain segment quantity in the theory of highly stretched chains, p = ^'x (r/(n b)) shear gradient

N2 Pi p? pu pc P(r,r + dr, n) P P 2 (cos 9) q q qpeak r r r0 rtj r'ij

>

c

c

0.08

u

/

t_

L_

A/

A /

tic

0.12

o

sion

,o

/

(0

JQ

E 0.04

o

(0 TJ 10 6 ) deuteropolystyrene crosslinked by y-radiation in cyclopentane solution with lower molecular weight protonated polystyrene. The Kratky plot of the deswollen network with a molecular weight between crosslinks of 35000 with a labeled chain of 2.6 x 106 molecular weight exhibited a broad peak at q&0.15 nm'1 (see Fig. 8-14). The peaks calculated for the crosslinked phantom and fixed junction networks were greater than that found in the experiment. The authors suggest that this peak could be accounted for in part by interchanges in spatial positions of geometric near neighbors with chemical near neighbors upon deswelling. In the Tsay and Ullman experiments, Kratky plots on stretched samples showed a broad peak at q « 0.5 nm" 1 for scattering in the perpendicular direction. Scattering in the parallel direction showed a slight maximum in the same q range.

i

I

383

I

I I I i i 0.4 0.6 1 q in nrrf Figure 8-14. Kratky plot comparing data obtained from a dry gel (A) and an isotropic melt (•) of deuterium labeled polystyrene to calculated form factors for the same deformation ratio: isotropic Gaussian (plain line below); junction affine model, Mmesh = 35000 (plain line above); phantom network model, Mmesh = 35OOO and Mmesh = 50000 (below), dashed lines. (Bastide et al., 1985.)

There have been many other SANS studies on rubbery networks which have not been discussed here. It has not yet been possible to determine from these studies which of the various models of rubber elasticity is most realistic. The Flory-Erman model and model of Edwards and associates have been applied to the SANS problem, but it is not possible at this stage to tell which view of rubber elasticity is most nearly in agreement with SANS measurements.

8.7 Rubber-Glass and Rubber Crystal Transitions When a rubber is cooled, it becomes a hard solid. If the rubber has an irregular chemical structure such as polystyrenebutadiene), the solid is a glass. If the elas-

384

8 Rubber Elasticity

tomer has a regular chemical structure, it may partially crystallize upon cooling. Lightly crosslinked natural rubber is in the latter category. The crystalline structure which is formed on cooling can also be induced by stretching the rubber above the crystalline melting point. Strain induced crystallization of rubber was first shown by Katz using X-ray diffraction (Katz, 1925) on natural rubber. One of the consequences of strain-induced crystallization is an increase in modulus and also in ultimate strength. Crystallinity in an oriented polymer is accompanied by a crystalline X-ray pattern at wide angles and, often, a two-point pattern in the small angle domain. When a polymer crystallizes, its density usually increases substantially since the chains pack more efficiently in the crystal than in the liquid or glassy states. In addition, a sharp endotherm is evidenced during heating since the polymer crystals melt. If a rubber is crystallized in part by stretching, the crystalline fraction can be melted by heating, in which case the modulus decreases and the density drops. The assumption that the volume of a rubber is largely unchanged by stretching, used widely here and in other studies of rubber elasticity, is only valid under circumstances where no crystallization takes place. Rubbers which are normally crystallizable can be prevented from crystallizing by increasing the extent of crosslinking. An increased density of crosslinks reduces the regularity of the chemical structure of the rubber and places severe constraints on the chains, resulting in a serious impediment to the formation of a crystalline structure. Crystallization of rubber without strain has been investigated by many research groups. In one interesting case, crystalline morphology of fractionated, uncrosslinked

natural rubber was studied by Phillips and Vatansever (1987).

8.8 Thermoplastic Elastomers Thermoplastic elastomers (TPEs) are multiblock copolymers, comprising hard segments and soft segments, which microphase separate into hard domains in a soft segment matrix (Legge et al., 1987). The crosslinks in TPE materials are physical in nature arising from aggregation of the hard segments into domains, which physically constrain the rubbery soft segments between them. The hard segment domains can be both glassy or crystalline, depending on the chemical structure of the hard segment. TPE materials such as polystyrene - polybutadiene - polystyrene triblock copolymers, now in large volume commercial use, depend for their elastomeric properties in the anchoring of the rubber chains by the hard segment domains. Polyurethanes, polyesteramides, polyetheramides, and oc-olefin TPE materials all exhibit interesting stress-strain behavior. The anionically prepared PS-P.BPS triblock materials are a useful model system since domain size, shape, and spacing are very well defined. Keller and coworkers (Folkes et al., 1973; Keller and Odell, 1985) have examined the deformation behavior at both low and high strain. For strains below about 50%, affine deformation takes place, but at higher strain levels, complications arise from hard segment domain reorientation and domain breakup. Stevenson and Cooper (1988) studied the changes which occur with deformation of the elastomeric block copolymer of poly(tetramethylene oxide) (PTMO) with poly(tetramethylene isophthalate) (PTMI) or with mixtures of PTMI and poly (tetra-

8.9 Ultimate Strength

methylene terephthalate) (PTMT) as the hard segment. The PTMO block had a number average molecular weight of 1000. The material prepared by transesterification of poly(tetramethylene ether glycol), dimethyl isophthalate, dimethyl terephthalate, and 1,4-butanediol had a molecular weight of 25000 to 30000. The ester block forms crystalline hard segment domains, while the ether block imparts flexibility to the polymer. The weight fraction of the hard segment (PTMI + PTMT) was approximately 60%. Samples were films prepared in a press by heating 30 °C above the melting point and then cooling. The crosslinks in this system are physical in nature, are localized in the hard segment aggregates, and change from glassy to partially crystalline as the system is annealed. Stress-strain measurements during crystallization at room temperature exhibit stresses which decrease markedly with time at a given elongation.

8.9 Ultimate Strength Elastomers tear when subjected to high stresses. The stress required to bring about this mechanical failure depends on a number of factors. The result may depend on viscoelastic response of the rubber, molecular weight between crosslinks, the temperature at which the stress was applied, the chemical makeup of the material, and the existence of flaws in the specimen. Rubbers which crystallize tend to be stronger than materials which remain amorphous upon stretching. In 1967, Lake and Thomas (1967) provided an analysis of strength of elastomers based on stored energy, and how this energy is released when the elastomer is torn. This work was extended by Gent and Tobias (1982) who performed many experi-

385

ments and incorporated results of earlier researchers in their studies. They derived a formula for the threshold tearing strength, a quantity which represents the smallest value of the tensile failure stress that would be measured in a real experiment. In many circumstances, the energy stored in a rubber on the verge of tearing is composed in part of stored energy which would relax viscoelastically if given sufficient time. As a result, the measured strength is greater than that expected in a threshold model. The Gent and Lake-Thomas equations take the form T0 = KM^/2c

(8-54 a)

To is the threshold tearing strength, M c , the molecular weight between crosslinks, and c is the length of a flaw perpendicular to the tearing direction. K is given by K=

3V /2 QU ~

71/2

M1'2

(8-54 b)

Here Q is density, u is the mean energy of a main chain bond, z is the number of main chain monomers per statistical segment, M is the molecular weight of a polymer chain, M o is the monomer molecular weight. K is predicted to be about 0.3 J/m2, measured values tend to be a factor of 3 higher. Threshold tearing strengths for polybutadiene and polyisoprene rubbers lie between 40 and 100 J/m 3 for a series of materials where Mc is bounded by 3 x 103 and 104. Polydimethylsiloxane networks have slightly lower threshold tear strengths. The data on siloxanes include both end-linked and randomly crosslinked materials. A number of measurements were performed on swollen specimens of the siloxanes, and, as expected, X2 To for the swollen specimens equalled To for the bulk network.

386

8 Rubber Elasticity

8.10 References Anthony, R. L., Caston, R. H., Guth, E. (1942), /. Phys. Chem. 46, 826. Ball, R. C , Doi, M., Edwards, S. E, Warner, M. (1981), Polymer 22, 1110. Bastide, X, Duplessix, R., Picot, C , Candau, S. J. (1984), Macromolecules 17, 83. Bastide, X, Herz, X, Boue, F. (1985), /. Phys. 46, 1967. Beltzung, M., Picot, C , Rempp, P., Herz, X (1982), Macromolecules 15, 1594. Beltzung, M., Herz, X, Picot, C. (1983), Macromolecules 16, 580. Beltzung, M., Picot, C , Herz, X (1984), Macromolecules 17, 663. Benoit, H., Decker, D., Duplessix, R., Picot, C , Rempp, P., Cotton, X P., Farnoux, B., Jannink, G., Ober, R. (1976), J. Polymer Sci., Polymer Phys. Ed. 14, 2119. Brotzmann, R. W. Jr., Eichinger, B. E. (1983), Macromolecules 16, 1131. Carpenter, R. L., Kramer, O., Ferry, X D. (1977), Macromolecules 10, 111. de Gennes, P. G. (1971), J. Chem. Phys. 55, 572. De la Condamine, C. M. (1751), Mem. Acad. Roy. Sci. 17, 319. DeLoche, B., Samulski, E. T. (1981), Macromolecules 14, 575. Dossin, L. M., Graessley, W. W. (1979), Macromolecules 12, 123. Duplessix, R. (1975), Thesis, Doctor d'Etat, L. Pasteur University, Strasbourg. Edwards, S. R, Vilgis, T. A. (1987), Reports on Progress in Physics 51, 243. 'Erman, B., Flory, P. X (1982), Macromolecules 15, 806. Faraday, M. (1826), Quart. J. Sci. 21, 19. Ferry, X D., Kan, H. C. (1978), Rubber Chem. Tech. 51, 731. Flory, P. X (1942), J. Chem. Phys. 10, 51. Flory, P. X (1977), /. Chem. Phys. 66, 5720. Flory, P. X, Erman, B. (1982), Macromolecules 15, 800. Folkes, M. X, Keller, A., Scalisi, F. P. (1973), Colloid and Polymer Sci. 251, 1. Fukuda, M., Wilkes, G. L., Stein, R. S. (1971), /. Polymer Sci. A2 9, 1417'. Gee, G. (1946), Trans. Faraday Soc. 42B, 33. Gent, A. (1969), Macromolecules 2, 262. Gent, A. N., Tobias, R. H. (1982), /. Polymer Sci., Polymer Phys. Ed. 20, 2051. Goodyear, C. (1844), U.S. Patent 3633. Gough, X (1805), Mem. Lit. Phil. Soc. Manchester I, 288. Granick, S., Ferry, X D. (1983), Macromolecules 16, 39. Gumbrell, S. M., Mullins, L., Rivlin, R. S. (1953), Trans. Faraday Soc. 49, 1495.

Gurer, C, Tschoegl, N. W. (1985), Macromolecules 18, 687. Guth, E., Mark, H. (1934), Monatsh. Chem. 65, 93. Hancock, T. (1844), British Patent. Huggins, M. L. (1942), /. Amer. Chem. Soc. 64,1712. Huggins, M. L. (1943), /. Phys. Chem. 46, 151. Ishikawa, T, Nagai, K. (1969), /. Polymer Sci. A2 7, 1123. James, H. M. (1947), /. Chem. Phys. 17, 651. James, H. M., Guth, E. (1943), J. Chem. Phys. 11, 455. James, H. M., Guth, E. (1947), /. Chem. Phys. 17, 669. Jarry, X P., Monnerie, L. (1978), /. Polymer Sci., Polymer Phys. Ed. 16, 443. Jarry, X P., Monnerie, L. (1979), Macromolecules 12, 316. Jarry, X P., Monnerie, L. (1980), J. Polymer Sci., Polymer Phys. Ed. 18, 1879. Joule, X P. (1859), Phil. Trans. Roy. Soc. 149, 91. Karrer, E. (1933), Protoplasma 18, 475. Katz, X R. (1925), Chem. Z. 49, 353. Keller, A., Odell, X A. (1985), in: Processing, Structure and Properties of Block Copolymers: Folkes, M. (Ed.). New York: Elsevier. Kramer, O., Carpenter, R. L., Ty, V., Ferry, X D. (1974), Macromolecules 7, 79. Kuhn, W., Grim, F. (1942), Kolloid Z. 101, 248. Lake, G. X, Thomas, A. G. (1967), Proc. Roy. Soc. A 300, 108. Legge, N. R., Holden, G., Schroeder, H. E. (1987), Thermoplastic Elastomers — A Comprehensive Review. Munich: Hanser Publishers. Llorente, M. A., Mark, X E. (1980), Macromolecules 13, 681. Meyer, K. H., von Susich, G., Valko, E. (1932), Kolloid Z. 59, 208. Mooney, M. (1940), /. Appl. Phys. 11, 582. Ogden, R. W. (1972), Proc. Roy. Soc. A 326, 565. Pearson, D. S. (1977), Macromolecules 10, 696. Phillips, P. X, Vatansever, N. (1987), Macromolecules 20, 2138. Rivlin, R. S. (1948), Phil. Trans. Roy. Soc. A 241, 379. Roe, R. X, Krigbaum, W. R. (1964), /. Appl. Phys. 35, 2215. Roth, F. L., Wood, L. A. (1944), J. Appl. Phys. 15, 749. Saunders, D. M. (1956), Trans. Faraday Soc. 52, 1414, 1425. Saunders, D. M. (1957), Trans. Faraday Soc. 53, 860. Stevenson, X C , Cooper, S. L. (1988), Macromolecules 21, 1309. Thirion, P., Weil, T. (1983), Polymer 25, 609. Thomson, W. (1857), Quart. J. Appl. Math. 1, 57. Treloar, L. R. G. (1954), Trans. Faraday Soc. 50, 881. Tsay, H. M., Ullman, R. (1988), Macromolecules 21, 2963. Tschoegl, N. W, Gurer, C. (1985), Macromolecules 18, 680.

8.10 References

unman, R. (1979), J. chem. phys. 7i, 436. Ullman, R. (1982), Macromolecules 15, 1395. Valanis, K. C , Landel, R. F. (1967), /. Appl. Phys. 38, FF

2197

Warner, M., Edwards, S. F. (1978), /. Phys. A. 11, 1649. Wood, L. A., Roth, F. L. (1944), J. Appl. Phys. 15, 781.

387

General Reading fe

Treloar, L. R. G. (1975), The Physics of Rubber Elasticit y- O x f o r d : Clarendon Press.

9 Viscoelastic and Rheological Properties Masao Doi Department of Applied Physics, Faculty of Engineering, Nagoya University, Nagoya, Japan

List of 9.1 9.2 9.2.1 9.2.2 9.2.3 9.2.4 9.2.5 9.2.5.1 9.2.5.2 9.2.5.3 9.3 9.3.1 9.3.2 9.3.3 9.3.4 9.4 9.4.1 9.4.2 9.4.3 9.4.3.1 9.4.3.2 9.4.4 9.4.4.1 9.4.4.2 9.4.4.3 9.4.4.4 9.4.4.5 9.5 9.5.1 9.5.2 9.5.3 9.5.4 9.5.5 9.5.6

Symbols and Abbreviations Introduction Linear Viscoelasticity Significance of Linear Viscoelasticity Shear Flow Stress Relaxation Boltzmann's Superposition Principle Parameters and Functions Characterizing Linear Viscoelasticity Steady Shear Flow Oscillatory Experiments Creep Experiments Characteristics of the Viscoelasticity of Polymeric Liquids Time-Temperature Superposition Rule Behavior of Relaxation Modulus Linear Polymers with Narrow Molecular Weight Distribution Effect of Molecular Weight Distribution and Branching Non-Linear Viscoelasticity Stress Tensor Basic Principles in Continuum Mechanics Non-Linear Viscoelasticity in Rheometrical Flows Shear Flow Uniaxial Elongational Flow Constitutive Equation Second Order Fluid Convective Maxwell Model Integral Form of the Convective Maxwell Model Non-Linear Convective Maxwell Model Single-Integral Constitutive Equation Molecular Theory I: The Rouse Model Polymer Motion in Concentrated Systems Basic Equation of the Rouse Model Normal Coordinates Self-Diffusion and Segmental Motion in Equilibrium Molecular Expression for the Stress Tensor Constitutive Equation for the Rouse Model

Materials Science and Technology Copyright © WILEY-VCH Verlag GmbH & Co KGaA. Allrightsreserved.

391 393 393 393 394 395 396 397 397 397 398 399 399 399 401 403 404 404 405 406 406 408 408 409 409 409 410 410 410 410 411 413 413 414 414

390

9.6 9.6.1 9.6.2 9.6.3 9.6.4 9.6.5 9.6.6 9.6.7 9.6.8 9.7

9 Viscoelastic and Rheological Properties

Molecular Theory II: The Tube Model Characteristics of the Polymer Motion in Strongly Entangled Systems . . . . Reptation and Fluctuations Reptation Model Self-Diffusion Constant Stress Relaxation Fluctuations Constitutive Equation Effect of Branching and Molecular Weight Distribution References

415 415 416 418 418 419 421 422 423 424


391

List of Symbols and Abbreviations B (t, t') E (£, f) f{P)

Finger-strain tensor deformation gradient tensor force exerted on the material below the plane j8 by the material above the plane /? G (t) relaxation shear modulus G' (co), G" (co) storage a n d loss modulus GN plateau modulus Go instantaneous relaxation modulus H (T) relaxation spectrum J(t) creep compliance Je steady state compliance kB Boltzmann constant M molecular weight Mc characteristic molecular weight Me molecular weight between the entanglement ML, M s molecular weight of long and short polymers Mx molecular weight of a polymer segment between the cross link Nx (y) first normal stress difference Qf^l tensor relating the stress and the formation tensor E R gas constant Rg center of mass of the molecule Sap orientational order parameter of a tube segment T torque T temperature Tg glass transition temperature t time U (t) orthogonal tensor representing the rotation of a frame u(s, t) unit vector tangent to the tube axis v velocity W elastic potential Z number of entanglement points per chain M/Me y (t) y (t) e & C rj rj (y) rj0 %(&) Y\ + (t, y) rj* (co), rj' (co)

shear strain shear rate elongation strain (Hencky strain) elongational rate friction constant of a Rouse bedd viscosity steady state viscosity zero shear rate viscosity elongational viscosity shear stress growth function complex viscosity

392

9 x xittP) k Q, QS a aE oa(i T rd Te TR Q co


gap angle velocity gradient tensor velocity gradient tensor evaluated at time t, at a material point P dimensionless material constant density at temperature T and Ts stress tensor elongational stress a,/? component of the stress tensor relaxation time reptation time relaxation time Rouse relaxation time angular velocity of the cone angular frequency

9.2 Linear Viscoelasticity

9.1 Introduction This chapter deals with mechanical properties of polymers in the liquid state. When a plastic is heated or when it is dissolved in a solvent, it becomes a liquid. The liquid state of polymers, generally called polymeric liquids, have unusual mechanical properties. Firstly, polymeric liquids have elasticity. This is demonstrated by chewing gum. Chewing gum is a liquid since it does not have any macroscopic equilibrium shape and can be molded into any shape. For slow deformation, chewing gum flows like usual liquids and shows viscosity. On the other hand, for quick deformation, it responds like an elastic material: for example, if we stretch the gum and release quickly the gum shrinks as if it remember the original shape. Thus chewing gum has both viscosity and elasticity. Such a material is called viscoelastic. Polymeric liquids are a typical example of viscoelastic material. Secondly, polymeric liquids have nonlinear mechanical properties. For example, the viscosity of polymeric liquids is not constant but decreases with the shear rate. This phenomena, called shear thinning, is quite dramatic: the viscosity changes over several orders of magnitude. The nonlinearity also appears as various curious flow phenomena: for example when a polymeric liquid is sheared between concentric cylinders, the liquid crawls up the inner cylinder. This effect is called the normal stress effect. The complex flow properties of polymeric liquids are important in various practical situations. Industrial processes of polymer technology involves flow of polymers: injection molding, fiber spinning, film blowing, surface coating are a few examples. In such processes, understanding

393

the flow properties is essential in designing and controlling the processes. The unique flow properties of polymers are also utilized in the industries of paints, glues, surface coating etc. The science of studying the flow and deformation of materials is called rheology. Although these problems have been discussed in the two well established sciences of fluid mechanics and elasticity theory, there are many materials which cannot be dealt with by classical continuum mechanics. Polymeric material is a typical example. Since it has both elasticity and viscosity, it can be dealt with neither by fluid mechanics nor elasticity theory. Rheology is concerned with such materials. Because of their importance in industry, polymeric liquids have been most intensively studied in rheology. Many examples of their unusual flow behaviors are given in the textbooks on rheology (Bird et al., 1987; Barnes et al., 1989). In this chapter, an introductory description of the rheology of polymeric liquids is given: in particular, it will be discussed (i) how the complex flow properties can be characterized, and (ii) how they are related to molecular structures.

9.2 Linear Viscoelasticity 9.2.1 Significance of Linear Viscoelasticity

The rheological properties of a material are characterized by the relationship between stress and strain. The stress represents the force applied to the material and the strain represents the deformation caused by it. The full description of such relations becomes rather complex for polymeric liquids, and will be postponed to a later section. Here we shall start with a simple case, called linear viscoelasticity.

394


When the stress level is low, a linear superposition principle, the precise meaning of which will be given later, holds in the relationship between the stress and strain. Such regime is called linear. In this regime, the viscoelasticity can be characterized by a single material function. In many practical applications, the stress level is not so small for linear viscoelasticity to be directly applied. However, linear viscoelasticity is important because of the following reasons: (i) Much experimental work has been done in the linear regime since material characterization is easy in this regime. Indeed most information on the relation between molecular structure and viscoelastic properties has been obtained through the study of linear viscoelasticity. (ii) Linear viscoelasticity is useful in predicting the behavior in the nonlinear regime. There are many empirical relations which predict the nonlinear behavior from linear viscoelasticity.

In shear flow, a material is sandwiched between two parallel plates, and the top plate is moved parallel to the bottom (see Fig. 9-1). If the material is homogeneous, and if the end effect is negligible, the velocity v (r, t) of the material at position r and time t is given by

9.2.2 Shear Flow

y (t, t') = J dt" y (t'f)

vv = vz = 0

(9-1)

The constant (9-2) is called the shear rate. To induce such a flow, a force has to be applied to the top plate. The shear stress a is the x component of this force per unit area. Actually, a is the x-y component of the stress tensor, and should be written as oxy. However, as we discuss only the shear stress in the next first two sections, we shall for now abbreviate it as o. The integral of y from time t' to t t

(9-3)

t'

The material function characterizing linear viscoelasticity can be obtained by various types of flow, but most commonly, shear flow is used. Therefore we shall describe linear viscoelasticity taking shear flow as an example.

Figure 9-1. Shear flow between parallel plates.

is called the shear strain applied between time t' and t. In general, if the velocity field is described by Eq. (9-1) in a certain reference frame, which may be moving with the fluid element, it is called a shear flow. Thus the flows shown in Fig. 9-2 are also shear flows: (i) Flow between concentric cylinders (see Fig. 9-2 a). Here two concentric cylinders with the radii rl9 and r2 (r1 < r2) are rotated relative to each other. The flow can be regarded as a shear flow if the gap distance (r2 ~ rt) is much less than rx. If the inner cylinder is rotated with the angular velocity Q relative to the outer cylinder, the shear rate is given by y = '1

r, Q ^ —

' 1

(9-4)


Figure 9-2. Examples of shear flow. Side and top views of (a) concentric cylinders and (b) cone and plate.

(a)

The shear stress is related to the torque T on the inner cylinder: T G = (9-5)

2nr\L

where 2nr1L is the surface area of the inner cylinder dipped in the liquid. (ii) Flow between the cone and plate (see Fig. 9-2 b). Here the liquid is sandwiched between a fixed plate and a rotating cone. The shear rate is constant throughout the fluid, and is given by y = Q/9

2nr3

the shear stress G (t). Alternatively, we can control the shear stress G(t) and measure the shear rate y (t). A particularly simple experiment is the stress relaxation, in which we apply a shear strain y0 in a very short time, and measure the stress while keeping the shear strain constant (see Fig. 9-3).

y(t-)

(9-6)

where Q is the angular velocity of the cone, and 6 is the gap angle. The shear stress is related to the torque T acting on the plate by G =

395

(a)

(9-7)

time, /•

a(t)

where r is the radius of the plate. 9.2.3 Stress Relaxation To characterize the viscoelasticity of a material, we apply a certain shear flow controlling the shear rate y {t\ and measure

(b)

time, t

Figure 9-3. The shear strain (a) and the shear stress (b) in stress relaxation.

396


If the material is an elastic material, the deformation creates a shear stress proportional to the applied shear strain y0, and the ratio G = a/y0 is called the shear modulus. If the material is viscoelastic, the flow creates a shear stress which decays with time. The stress eventually becomes zero if the material does not have an intrinsic shape. Such a material is called a viscoelastic liquid. Polymer melts and solutions belong to this class. On the other hand, for gels or rubbers, the stress approaches a constant non-zero value. Such a material is called a viscoelastic solid. In this chapter, we shall deal only with viscoelastic liquids. Chapter 7 deals extensively with the viscoelastic solids. If the shear strain y0 is small, the shear stress is proportional to y0, and is written as a(t) = G(t)y0

(9-8)

called Boltzmann's superposition principle. The principle states that provided the stress level is low, the effect of strain on stress can be added. Let us consider a shear flow characterized by the shear strain: (9-12) Suppose that when a shear strain y1(t) is applied to a material, the stress cr1(t) appears and that for a shear strain y2(0> a stress a2 (t) appears. Then the Boltzmann's principle states that when the shear strain is yi(t) + y2(i), the shear stress will be

MO + MOIf this principle holds, the stress for any strain history y (t) can be expressed by the relaxation modulus G (t). To see this, notice that any time-dependent shear strain y(i) can be regarded as a sum of step strains (see Fig. 9-4). The step strain applied be-

G (t) is called the relaxation shear modulus. For polymer melts with narrow molecular weight distribution, G (t) can be approximately written as G(0«Goexp(-t/T)

(9-9)

A/ (t')

Go is called the instantaneous relaxation modulus, and T the relaxation time. In the general case, G(t) has many relaxation times, and can be written as t/zp)

> t

(9-10)

or G(t)=

j dlnTH(T)exp(-r/T)

(9-11)

— oo

The function H (T) is called the relaxation spectrum. 9.2.4 Boltzmann's Superposition Principle We shall now explain the meaning of the linear superposition principle, which is also

r

t

Figure 9-4. The principle of calculating the stress for arbitrary strain.


397

tween t' and t' + At' is

9.2.5.2 Oscillatory Experiments

Ay (t 1, G*(co) = Go, which means that the polymeric material behaves as an (9-31) elastic material for quick deformation. On the other hand, for co T oo

Figure 9-5. The shear stress (a) and the shear strain (b) in the creep experiments.

(9-35)

399

9.3 Characteristics of the Viscoelasticity of Polymeric Liquids

9.3 Characteristics of the Viscoelasticity of Polymeric Liquids 9.3.1 Time-Temperature Superposition Rule

The mechanical properties of a polymeric material change dramatically with temperature. For example, polystyrene appears as a glassy solid at low temperature, but as the temperature is raised, it becomes soft and finally becomes a fluid. The effect of temperature on the rheological behavior can be well described by a useful empirical relation called the time temperature superposition rule. The time-temperature-superposition rule states that well above the glass transition temperature Tg, changing the temperature is approximately equivalent to changing the time scale. For example, let G (t, Ts) and G (t, T) be the relaxation moduli measured at two different temperatures Ts and T. The time-temperature superposition rule states that they can be related to each other as =

bTG(t/aT,Ts)

(9-36)

where aT and bT are constants independent of t. From Eqs. (9-18), (9-33) and (9-36), it follows

Empirical equations are available for aT and bT\ aT =exp[

-

(9-38) l

s

"T" ^ 2

and (9-39) where Q and QS are the densities of the material at temperature T and Ts respectively, and Cx and C 2 are constants. If Ts is taken as the glass transition temperature Tg, Q

and C 2 become independent of the material (Ct « 17 and C 2 ~ 52). Examples of C1,C2 and Ts can be found in the literature (Ferry, 1980). For the other viscoelastic functions, the time temperature superposition rule gives the following relations: = bTG' (coaT,Ts); (9-40) and =

l/bTJ(t/aT9TJ

(9-41)

A similar superposition rule exists for the effect of pressure = fcPG(t/flF,Ps)

(9-42)

These superposition rules indicate that there is only a single fundamental time scale which governs the dynamics of polymers, and that changing the temperature or the pressure only affects this fundamental time scale. 93.2 Behavior of Relaxation Modulus The relaxation modulus G(t) shown in the literature covers an extremely wide range of time scale; typically of the order of 1010. Data for such a wide time scale is obtained by doing the experiments at different temperatures and then synthesizing a master curve by using the time temperature superposition rule. Figure 9-6 shows a sketch of the relaxation modulus of polymer melts. The time dependence of the relaxation modulus can be broadly divided into four regions, the glassy region, transition region, rubbery region and flow region. The glassy region corresponds to the behavior at low temperature, where polymeric materials have a rather high modulus of the order of 1010 Pa. The relaxation modulus decreases in the transition region, and then becomes flat

400


flow

Figure 9-6. Sketch of the relaxation modulus of a polymeric liquid. 10-8

10-6

10-4

10-2

1

in the rubbery region, where the polymeric liquids behave like a soft rubber (G « 105 - 106 Pa for polymer melts). In the flow region, which corresponds to the time scale larger than the longest relaxation time, the polymeric materials behave as a viscous fluid. Generally speaking, the relaxation behavior at longer time scale reflects the molecular motion on larger length scale.

The behavior in the glassy and transition region is determined by the local structure of polymers, such as the chemical structure of monomeric units or the short side branches etc. On the other hand, the behavior in the rubbery and flow regions is determined by the global characteristics of polymers, such as molecular weight, molecular weight distribution, and the structure of any long branches. As the molecular weight is increased, the rubbery region extends towards longer times, but the glassy and the transition regions remain unchanged. Figure 9-7 shows a sketch of the storage and loss moduli. The above characteristics can also be seen in this graph. The flow region corresponds to the region of low frequency where G' (co) and G" (co) have a linear portion on the log-log plot with respective slopes of 2 and 1 (see Eqs. (9-24) and (9-34)) Gf(co) = rjlJeco2

Figure 9-7. Sketch of the storage modulus G' (co) and the loss modulus G" (co) of a polymeric liquid.

and Gff (co) = rj0co (9-43)

The rubbery region corresponds to the plateau of G' (co), and the transition region corresponds to the increase of G' (co) and G" (co) at higher frequency.


9.3.3 Linear Polymers with Narrow Molecular Weight Distribution

For linear polymer melts with narrow molecular weight distribution, the relaxation modulus in the rubbery and flow regions can be approximated to a single exponential curve. This is shown in Figs. 9-8

-6

-5

-4

-3

-2

401

and 9-9 in the curves of G'(co) and G"(co). According to Eq. (9-29), G' (co) becomes constant for COT ^ 1 and G"(co) shows a peak at COT = 1. Such a plateau and peak are indeed seen in experimental data for polystyrene in Figs. 9-8 and 9-9. The flat region of G' (co) is called the rubbery plateau, and its height GN, called the plateau

Figure 9-8. Storage modulus of polystyrene with narrow molecular weight distribution. (From Onogi, S., Masuda, T, Kitagawa, K. (1970), Macromolecules 3, 109; reproduced by permission of American Chemical Society.)

-1

PS 160 °C

x

6

1w yr-

J

7/.

V

-6

-5

-k

-3

Y>

-2

-1 0 log UQ T ,sec"

y

F

Figure 9-9. Loss modulus of polystyrene with narrow molecular weight distribution. (From Onogi, S., Masuda, X, Kitagawa, K. (1970), Macromolecules 3, 109; reproduced by permission of Americal Chemical Society.)

402


modulus, is independent of the molecular weight, GN oc M°

(9-44)

while the relaxation time is strongly dependent on the molecular weight; T oc M 3 - 4 (9-45) The plateau modulus GN is used to estimate M e the molecular weight between the entanglement junctions. As it is described in Chap. 8, the rigidity modulus G of a rubber can be related to the molecular weight Mx of a polymer segment between the crosslinks: M =

QRT

(9-46)

where Q is the density of the polymer, R the gas constant and Tthe absolute temperature. Likewise, from the height of the rubbery plateau, one can define a molecular weight, M =

QRT

M 3 4

M -

(M < MJ (M > Mc)

(9-48)

The characteristic molecular weight M c is about two or three times larger than M e . Table 9-1. Characteristic molecular weights for selected polymers. Polymer species Polystyrene Poly(a-methyl styrene) 1,4-Polybutadiene Polyvinyl acetate Poly(dimethyl siloxane) Polyethylene 1,4-Polyisoprene Polyisobutylene

Me 19 500 13 500 1900 12000 10000 1300 6 300 8 900

Mc

M'c

35 000 130000 28000 104000 5000 13 800 24 500 86000 24400 61000 3 800 12000 10000 35 000 15 200 —

101*-

1012-

(9-47) 1010-

which is supposed to characterize the spacing between the entanglement junctions. In the early theory, the physical entity of the entanglement junction was not clarified. However M e turns out to be an important quantity in the modern theory of entanglement (see Sec. 9.7). Values of M e for some typical polymers are listed in Table 9-1. In the region where G(co) shows a plateau, the steady state viscosity depends on M as rj0 oc M 3 4 . As the molecular weight is decreased, the plateau disappears, and the molecular weight dependence of rj0 changes (see Fig. 9-10). When the correction for the chain end effect on the monomeric friction constant is accounted for, the viscosity is proportional to M. Thus the molecular weight dependence of the viscosity can be approximately expressed by

a 10* -

106 -

10* -

102 -

100 -

107

Figure 9-10. Molecular weight dependence of the viscosity. The filled circles indicate data corrected for the chain end effects on the monomeric friction constant. The open circles indicate unadjusted result. (From Colby, R. H., Fetters, L. J., Graessley, W. W. (1987), Macromolecules, 20, 2226; reproduced by permission of American Chemical Society.)


/ DO

°V /

-o

9.3.4 Effect of Molecular Weight Distribution and Branching

oo o

?•

8-7-

/ o

ft-

LOG

5 Mw

Figure 9-11. Molecular weight dependence of the steady state compliance of polyisoprene with narrow molecular weight distribution. (From Odani, H., Nemoto, N., Kurata, M. (1972), Bull Inst. Chem. Res. 50, 117.)

Figure 9-11 shows the molecular weight dependence of the steady state compliance. The change in the molecular weight dependence is also seen in J c . For M < M'c, Je is proportional to M, while for M > M'c, Je becomes independent of M. Values of M c and M' are shown in Table 9-1.

PS 160

7

^ fit\^&T

5-

A?

o L 15 •

/

r

/ /

G

2-

1 -

-5

-4

-3

If the polymer has broad molecular weight distribution, or long side chains, the relaxation spectrum becomes broad, and the clear plateau region disappears. An example for polystyrenes is shown in Fig. 9-12. As a result of the broad relaxation spectra, the steady state compliance becomes much larger than that of narrow distribution polymers. On the other hand, the viscosity is known to be described by the empirical Eq. (9-48) even for a system with broad molecular weight distribution provided the weight averaged molecular weight M w is used for M. The effect of branching has been studied in some detail for star shaped polymers which consists of / arms of equal length connected at the central node. Figure 9-13 compares the viscosity of such polymers with that of linear polymers. With fixed molecular weight, the viscosity decreases with the increase in the arm number /,

°C

6-

3-

403

-1

-2

log

sec- 1

PS 7

s>

Figure 9-12. G'(co) and G" (co) of polystyrene of narrow molecular weight distribution (unfilled circles) and broad molecular weight distribution (filled circles). (From Masuda, T., Kitagawa, K., Inoue, T, Onogi, S. (1970), Macromolecules 3, 116; reproduced by permission of the American Chemical Society.)

404


tion. To do that, it is essential to introduce the stress tensor. The stress tensor describes the force acting within the material. The a-j? component of the stress tensor at a point P in the material is defined as follows (see Fig. 9-14). Consider a small region of area AS on a plane which includes the point P and is normal to the jff-axis. Let / (/?) be the force which the material above the plane exerts on the material below the plane through the region AS. For small AS, the force is proportional to AS. The stress tensor aap is defined by the ratio (9-49) Figure 9-13. Viscosity of linear and star branched polyisoprene in normal tetradecane: circle: linear, square: 4 arm star and hexagone: 6 arm star. (From Masuda, X, Onogi, S. (1973), Annual Rep. Res. Inst. Chem. Fibers. 33 9.)

since the molecule is becoming more compact. However with the increase of the molecular weight at fixed arm number, the viscosity of star polymers increases more rapidly than linear polymers, and eventually exceeds it at high molecular weight. This tendency is seen for other types of branched polymers. This is because at high molecular weight molecular disentanglement in branched polymers takes place much more slowly than in linear polymers.

If there is no electric or magnetic field, the stress tensor is symmetric. Gap = Gpa

(9-50)

Also since the polymeric liquids may be regarded as incompressible, the stress tensor is written as °ap = = 2t;kBTId (i -j) 5 (t - f) The differential equation (9-104) is supplemented by the boundary conditions which are derived from Eqs. (9-101 and 9-102): — = 0 at i = 0 and i = N

(9-106)

9.5 Molecular Theory I: The Rouse Model

9.5.3 Normal Coordinates The Rouse model gives a linear equation for R (i, t). A standard way of treating such a system is to use normal coordinates each capable of independent motion. In the present problem, the normal coordinates are given by

9.5.4 Self-Diffusion and Segmentai Motion in Equilibrium At equilibrium, a polymer molecule moves around by thermal motion and its speed is characterized by the self-diffusion constant defined by - RG(0))2}

= lim

(p = 0,1,2,...)

(9-107)

It then follows from Eq. (9-104), 8 dt

P

Cp

P

P

(9-114)

iv o

P

(for p = l,2,...)

(9-108)

(9-113)

where RG(t) is the center of mass of the molecule. For the Rouse chain, RG (t) is given by

1 p

413

F r o m

where

(9-115) and CP =

for p = l,2,...

and 6n2kBT Nb2

J

= 0,1,2,...)

(9-109)

Substituting this into Eq. (9-96) and using Eq. (9-110), we have (t) - RG(0))2} =

(9-116)

and/ p 's are the random variables satisfying = 0 p = 0,1,2,...;

(9-110)

Comparing this with Eq. (9-113), we have (9-117)

According to Eq. (9-108), the characteristic relaxation time of Xp is Cp/kp = TR/p2, where 3 n2 kn T

(9-111)

TR corresponds to the longest relaxation time of the Rouse model, and is called the Rouse relaxation time. Since N is proportional to the molecular weight M, Eq. (9-111) indicates that TROCM2

(9-112)

Thus the self-diffusion constant is proportional to the inverse of the molecular weight: Z)Goc Af"1

(9-118)

Although the motion of the center of mass obeys the simple diffusion law, the motion of a Rouse bead does not follow this law. It can be shown that the mean square displacement of a Rouse bead is given by (de Gennes, 1979) 6DGt

for

tp

IOr

t ^ T

TR

1/2

(9-119)

414


Notice that in a short time scale the mean square displacement increases at t1/2. 9.5.5 Molecular Expression for the Stress Tensor Whe shall now discuss the viscoelastic properties of the Rouse model. To do this, first we have to know the molecular expression for the stress tensor. This is obtained as follows: As it is explained in Sec. 9.4.1 the stress tensor aafi represents the a component of the force acting through the plane normal to the /? axis. In polymeric materials, the force has two origins: one is the molecular potential (such as the van der Waals potential) which acts between the atomic groups in proximity, and the other is the chemical bond which connects the backbone atoms. The stress arising from the molecular potential is essentially the same as for usual liquids, and will not be important for viscoelastic properties. Thus we shall consider the stress arising from the forces acting through the backbone atoms. Now if a chain is passing through the plane at a polymer segment i, the upper plane exerts a force ,Q . ^ ™..x

3fc

B^

„

,

The isotropic term represents the contribution from the molecular potential. In terms of the normal coordinates, the stress tensor is written as (9-122)

*.,(') = v £

kp<Xpa(t)Xpfi(t)y-P8afi

9.5.6 Constitutive Equation for the Rouse Model Given the molecular expression for the stress tensor, we can obtain the constitutive equation for the Rouse model. First from Eq. (9-108), we can show (see Doi and Edwards, 1986, p. 112) (f

(t)X

(t)y = k TS

(9-123)

Using the relation and Eq. (9-108), we have

±<x,MXpf(t» =

Kll(t)Xpfl(t)

Xpx(t)[

\Xpf(t))-

_ J

3kBTdR(Ut)

on the lower plane (see Fig. 9-21). If there are v chains per unit volume, the probability that the segment i of any polymer passes through the plane is

xXfl(Xpfl(t)Xpll(t)) (9-124) where

where AS is the area of the plane. Thus the stress tensor is given by

= v-

(9-125)

2K

Notice that Eq. (9-124) is similar to the convective Maxwell model. Indeed if we define (9_12g) = vkp(^Xpa(t)Xpfi(t)>

-

9.6 Molecular Theory II: The Tube Model

415

Since the Rouse relaxation time TR is proportional to M 2 , Eq. (9-132) indicates that rj0 oc M;

(9-133)

Je oc M

This is in agreement with the experimental relation Eq. (9-48) behavior for M < Mc or M<M'C (see Figs. 9-10 and 9-11). For t < TR? the sum in Eq. (9-131) may be replaced by an integral for p. Thus

AS,

Figure 9-21. Molecular origin of the stress tensor for the Rouse model.

G{t)&vkBT J dpexp

/

2n2 —t

1/2

(9-134) we can rewrite Eq. (9-124) in the form of the convective Maxwell model

B(afi

fi

(9-127)

The complex modulus is obtained from the Fourier transform of G{t). Especially for CDTRP>

1, G*(co) is given by T

\l/2

- )

<JP + *P^t°P

=

vk

(9-128)

BTTpy/

The total stress is written as oo

(9-129)

* = Z Op

As an example, let us consider the linear viscoelasticity in the shear flow. From the above constitutive equation, we obtain the relaxation modulus:

expf-

(9-130)

p=i

The factor vkBT can be rewritten as Q R T/M (Q being the weight of polymers per unit volume). Hence the relaxation modulus is written as (W31) The viscosity and the steady state compliance are then calculated by Eqs. (9-18) and (9-33). The result is K2QRT

1

"

2M J

(

tj

ll2

KvNkBT(toTs) (l

+0

(9-135)

Thus G' (to) = G" (to) at high frequency and increases as co1/2. These features are seen in Figs. 9-8 and 9-9 at high frequency period.

9.6 Molecular Theory II: The Tube Model 9.6.1 Characteristics of the Polymer Motion in Strongly Entangled Systems

The basic idea of the tube model is as follows (de Gennes, 1979; Doi and Edwards, 1986). Consider the motion of a certain test polymer in a strongly entangled system (see Fig. 9-22 a). If the test polymer attempts to move perpendicularly to its own contour, it will create a large scale motion of many surrounding polymers and encounter large resistance. On the other hand, if the polymer tends to move along its own contour, it will feel much less resistance. Thus dynamically the polymer

416


Figure 9-22. Entangled polymer system (left) and the tube model (right).

is considered to be confined in a tube (Fig. 9-22 b). The tube represents a mean field potential created by the other polymers, and gives a constraint to the polymer in it. In a simple version of the model, the tube is assumed to be fixed in the material. The motion of the polymer perpendicular to the tube is restricted by the tube wall, but its motion along the tube is free. This situation is represented by the Rouse chain confined in a tube (Fig. 9-23). At equilibrium, the tube takes a random configuration. The persistence length of the tube would be of the order of the tube diameter. To specify the model, it is usually assumed that the central axis of the tube consists of straight segments (called the tube segments) of equal length a, connected in random directions. In this model, only one new parameter is introduced. This is the length a, which characterizes the entanglement effect of the surrounding polymers. The rest of the parameters are the same as for the Rouse

model, i.e., N, b and £. All quantities in the tube model can be expressed by these parameters. For example consider the contour length L of the tube axis. At equilibrium, the tube axis is a randomly connected tube segment of length a. Since there are Z = L/a tube segments, the mean square end-to-end distance of the tube axis is Za2. This must be equal to the mean square end-to-end distance of the polymer Nb2. Thus b2 N Z = N-^ = —

and

L = Za

(9-136)

where we have introduced N =

(9-137)

As it will be shown later, JVe corresponds to the entanglement molecular weight M e . The above model is consistent with the time temperature superposition rule. Among the parameters a, N9 b and £, only C is considered to depend on the temperature or pressure sensitively. If the other parameters are assumed to be independent of temperature, the time-temperature rule is derived from the model by dimensional analysis. 9.6.2 Reptation and Fluctuations

Figure 9-23. Rouse chain in a tube.

From the model shown in Fig. 9-23, one would realize that there are three characteristic types of motion.

9.6 Molecular Theory II: The Tube Model

417

1) The lateral fluctuation

(3) Reptation

In a very short time scale, each Rouse bead can move around freely without feeling the constraint of the tube wall. This picture would be valid provided the mean square displacement of a Rouse bead is less than a, or from Eq. (9-119)

In the time scale longer than TR, only the mode of p = 0 is effective. Thus the Rouse chain may be regarded as moving along the tube with a diffusion constant

~

1/2

— I

b

1, the time scales of these modes are quite different from each other.

moves out of the tube, and can go in a random direction, whence:

r{L,t + At) = r(L9t) + v{t)

(9-148)

where v (t) is a random vector, whose mean and variance are given by 0, the segment at s = L,

(9-152) 1 r G (t + At) - vG (t) = - (r (L, t) - r (0, t)) As where we have neglected the terms of order v/L. Taking the average of the square of Eq. (9-152) and using the fact that As is independent of r (s, t\ we have = = - ^ « v { L , t) - v(0, t))2} (9-153) The first average ((r(L,t) - r(0,t))2} is equal to Nb2, and the second average is given by Eq. (9-146). Hence (9-154) Nb2 2 = C At Thus the self-diffusion constant of the polymer is (see Eq. (9-113)), Afh2

Af b J1

3L2

7

(9-155)

9.6 Molecular Theory II: The Tube Model -6-

According to Eq. (9-155), DG is proportional to M~ 2 . Figure 9-25 shows the self-diffusion constant of monodispersive polystyrene in melts. The results can be described as

DGx

2

M~

for for

M<MC M>MC

419

monodisperse PS Tn +125 °C

-8-

(9-156) ^ -10-

This is in agreement with Eqs. (9-118) and (9-155).

© Bueche 6 Bachus+Kimmich

en o

-o Fleischer

-12-

1.25 in Fig. 9-27 is shifted vertically by an amount — log [h (y)] so that it superposes on the top curve in the long time region. TJ indicates the longest relaxation time, and xk the characteristic time below which the superposition is not possible. (From Osaki, K., Nishizawa, K., Kurata, M. (1982), Macromolecules 15, 1068; reproduced by permission of American Chemical Society.)

Figure 9-30. The damping function h (y) obtained by the procedure explained in Fig. 9-29. Filled circles represent polystyrene of molecular weight 8.42 x 106 and the unfilled circles of 4.48 x 106. Directions of pips indicate concentrations which range from 0.02 gem" 3 to 0.08 gem" 3 . The solid line represents the theoretical value. (From Osaki, K., Nishizawa, K., Kurata, M. (1982), Macromolecules 15, 1068; reproduced by permission of American Chemical Society.)

useful approximation, called the independent alignment approximation which enables us to derive a simple constitutive equation. In the presence of a flow, the equation of motion for u (s, t) becomes

culate the stress tensor for general flows. The result is (9-188)

u (s,t + At) = u{s + As, t) + Awflow (9-186) The first term represents the reptation motion, and the second term the effect of the macroscopic flow. The independent alignment approximation assumes that Anflow is given by the change of a unit vector embedded in the material. Atffiow = (x(t) • u - (*:uu)u)At

(9-187)

This is not consistent with the deformation model shown in Fig. 9-26, but the error of the approximation has been shown be small as long as the direction of the flow is not changed. If the independent alignment approximation is used, it is possible to cal-

°*,(t) = G'o \ — 00

dtd(P{t~tf)QW(E(t,t>)) or

where G'o = 5/4 Go, &(t) is given by Eq. (9-171), and Q^(E) is given by

Equation (9-188) is the BKZ type constitutive equation explained in Sec. 9.4.4.5. It reproduces major characteristic aspects of the non-linear viscoelasticity. Detailed comparison of the theory with experiments has been done by Osaki and Doi (1984). 9.6.8 Effect of Branching and Molecular Weight Distribution

The crucial assumption of the model described above is that the tube can be re-

424


garded as fixed in the material. For linear polymers with narrow molecular weight distribution, this simple picture appears to hold: at least many experimental results can be accounted for qualitatively or sometimes quantitatively by this simple model. However, the simple picture breaks down for polymers with broad molecular weight distribution. The failure of the fixed tube model can be demonstrated by a simple example. Consider a mixture of short (S) and long (L) polymers. If the molecular weight of the short polymers M s , is close to that of the long polymers M L , the fixed tube assumption will be valid. However as M s decreases, the constraints imposed by the short polymers become weaker. In the extreme case of M L > M s , the constraint imposed by the short polymers will be negligible for the long polymers. Thus in a system with broad molecular weight distribution, the assumption of the fixed tube will be invalid. A mechanism for the motion of the tube is illustrated in Fig. 9-31. The topological constraints imposed on the polymer A is

Figure 9-31. Constraint release process.

released and recreated if the polymer C moves as shown in Fig. 9-31. Such a process, called the constraint release, causes the motion of the tube. The constraint release is considered to be important also for branched polymers. If the polymer has long side branches, the reptation motion is severely suppressed. In such a case, the contour length fluctuation, and the constraint release are considered to be the dominant mechanism of stress relaxation. However, it is difficult to account for the coupling between reptation, constraint release and contour-length-fluctuation, and quantitative theory is not yet available. The constraint release is discussed in detail by Graessley (1982) and Marrucci (1985).

9.7 References Barnes, H. A., Hutton, X E, Walters, K. (1989), An Introduction to Rheology. Amsterdam: Elsevier. Bird, R. B., Armstrong, R. C. Hassager, O. (1987), Dynamics of Polymeric Liquids, 2nd ed. volume 1. New York: John Wiley. Crochet, M. I, Davies, A. R., Walters, K. (1984), Numerical Simulation of Non-Newtonian Flow. Amsterdam: Elsevier. de Gennes, P. G. (1979), Scaling Concepts in Polymer Physics. New York: Cornell Univ. Press. Doi, M., Edwards, S. F. (1986), The Theory of Polymer Dynamics. Oxford: Oxford University Press. Ferry, I D. (1980), Viscoelastic Properties of Polymers, 3rd ed. New York: Wiley. Graessley, W. W (1982), Adv. Polym. Sci. 47, 67. Larson, R. G. (1988), Constitutive Equations for Polymer Melts and Solutions. Stoneham: Butterworths Publishers. Marrucci, G. (1985), Adv. Transport Processes 5: Mujumdar, A. S., Mashelkar, R. A. (Eds.), Wiley Eastern Ltd., New Delhi. Osaki, K., Doi, M. (1984), Polym. Eng. Rev. 4, 35-72. Tanner, R. I. (1985) Engineering Rheology. Oxford: Clarendon Press. Tucker III, C. L., (Ed.) (1989), Computational Modeling for Polymer Processing. Munich: Hanser Publishers. Walters, K. (1975), Rheometry. London: Chapman & Hall.

9.7 References

General Reading Astarita, G., Marrucci, G. (1974), Principles of NonNewtonian Fluid Mechanics. London: McGraw Hill. de Gennes, P. G. (1975), The Physics of Liquid Crystals. Oxford: Oxford Univ. Press. Edwards, S. F. (1976), The Configurations and Dynamics of Polymer Chains in Molecular Fluids: Balian, R., Weill, G. (Eds.). London: Gordon and Breach. Flory, P. F. (1953), Principles of Polymer Chemistry. Ithaca, N.Y.: Cornell Univ. Press.

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Janeschitz-Kriegl, H. (1964), "Polymer Melt Rheology and Flow Birefringence", Elastic Liquids: Springer, A. S., Lodge, T. P. (Eds.). London: Academic Press. Lodge, T. P., Rotstein, N. A., Prager, S. (1990), "Dynamics of Entangled Polymer Liquids: Do Linear Chains Reptate?", Adv. Chem. Phys. 79, 1-132. Osaki, K., Doi, M. (1984), "Nonlinear Viscoelasticity of Concentrated Polymer Systems", Polymer Engineering Reviews 4, 35-72. Tirrell, M. (1984), Rubber Chem. Tech. 57, 523. Tobolsky, A. V. (1960), Properties and Structure of Polymers. New York: John Wiley.

10 Plastic Deformation of Polymers Buckley Crist Department of Materials Science and Engineering, Northwestern University, Evanston, IL, U.S.A.

List of 10.1 10.1.1 10.2 10.3 10.4 10.4.1 10.4.2 10.4.3 10.5 10.5.1 10.5.2 10.5.3 10.6 10.6.1 10.6.2 10.6.3 10.7 10.8 10.9

Symbols and Abbreviations Introduction Basic Concepts of Plastic Deformation Mechanical Testing and Definitions Criteria for Yielding and Crazing Yielding and Deformation Behavior Glassy Polymers Semicrystalline Polymers Blends and Block Copolymers Fundamental Nature of Polymer Yielding Glassy Polymers Semicrystalline Polymers Crystalline and Liquid Crystalline Polymers Post-Yield Deformation and Modeling Viscoelastic Models Molecular Models Continuum Mechanics of Necking Summary Acknowledgements References


428 430 430 432 436 437 438 440 444 445 446 451 457 460 460 461 464 465 467 467

428

10 Plastic Deformation of Polymers

List of Symbols and Abbreviations a A, Ao b C E (t) Eo F G H AH k /, l0 Za lc L Lk m n P r0 R* t t' T Td 7^ Tg Tm TR w* A I/* v* w z*

radius of molecule or bundle in a double kink instantaneous loaded area, initial loaded area modulus of the Burger's vector of a dislocation exponential factor in nonlinear viscoelastic model tensile stress relaxation modulus elastic (unrelaxed) tensile modulus force elastic shear modulus hardness activation energy in absence of stress Boltzmann's constant instantaneous length, initial length thickness of amorphous layer thickness of lamellar crystal long period of semicrystalline polymer length of a kink band number of cooperatively rearranging conformers width parameter of distribution of relaxation times pressure radius of dislocation core critical radius of dislocation loop time relaxation time or average relaxation time absolute temperature temperature of tensile drawing temperature at which yield stress vanishes glass-liquid transition temperature melting temperature reference temperature for stress-strain behavior critical size of region sheared by screw dislocations activation barrier for forming nucleus of critical size activation volume height of kink band in loading direction critical size of kink pair

a 7, y 7y f a, g n , £t e, £ n , £t £p £y

fractional crystallinity shear strain, shear strain rate shear yield strain preexponential frequency factor in Arrhenius equation uniaxial strain, nominal strain, true or logarithmic strain uniaxial-strain rate, nominal-strain rate, true- or logarithmic-strain rate plastic strain uniaxial yield-strain


e* critical strain for onset of plastic flow r\ viscosity 9 fictive temperature A uniaxial stretch ratio fi pressure coefficient of shear yield-stress v Poisson's ratio

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Electrical Properties of Polymers

Electrical Properties of Polymers

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Quasicrystals: structure and physical properties

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Structure and Properties of Nonferrous Alloys

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Fundamentals of Amorphous Solids: Structure and Properties

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Water-Soluble Polymers : Solution Properties and Applications

Structure and Properties of Polymers

Structure and Properties of Crosslinked Polymers

Electrical Properties of Polymers

Electrical Properties of Polymers

Electrical Properties of Polymers

Properties of Polymers

Physical Properties of Polymers

Electrical Properties of Polymers

Electrical Properties of Polymers

Viscoelastic Properties of Polymers

Structure and Properties of Composites

Structure and Properties of Ceramics

Mechanical Properties of Polymers and Composites

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Clathrochelates.. Synthesis, Structure and Properties

Quasicrystals: structure and physical properties

Clathrochelates: Synthesis, Structure and Properties

Foldamers: Structure, Properties, and Applications

Perovskites: Structure, Properties and Uses

Structure and Properties of Liquid Crystals (Bookmarked)

Structure and properties of engineering alloys

Structure and Properties of Nonferrous Alloys

Structure and Properties of Liquid Crystals

Structure and Properties of Atomic Nanoclusters

The Structure and Properties of Ferromagnetic Materials

The Structure and Properties of Water

Fundamentals of Amorphous Solids: Structure and Properties

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