Polymer Reactivity

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POLYMER REACTIVITY: ASPECTS OF ORDER AND DISORDER No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

POLYMER REACTIVITY: ASPECTS OF ORDER AND DISORDER

G.E. ZAIKOV AND

B.A. HOWELL EDITORS

Nova Science Publishers, Inc. New York

Copyright © 2006 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Available upon request.

ISBN 978-1-61668-127-2 (E-Book)

Published by Nova Science Publishers, Inc. New York

CONTENTS Preface

vii

Chapter 1

Polymeric Media with the Gradient of the Optical Properties N. G. Lekishvili, L. I. Nadareishvili and G. E. Zaikov

Chapter 2

Fundamental Regularities of Thermal Oxidation of Heat-Resistant Heterochain Polymers E. V. Kalugina, K. Z. Gumargalieva and G. E. Zaikov

Chapter 3

Practical Stabilization of Heat Resistant Polymers E. V. Kalugina, K. Z. Gumargalieva and G. E. Zaikov

Chapter 4

Heterophase Supramolecular Model of Photochemical Transformation of Naphthalene in Cellulose Triacetate Yu. A. Mikheev and V. G. Zaikov

Index

1

103 117

173 185

PREFACE “He knew little, but did much.” The opinion of Americans on Ronald Reagan

These are well-liked and respectful words of American citizens about their president. Clearly, there are several alternatives of this expression: 1. Knew much and did much, 2. Knew much, but did little, 3. Knew little and did little. Generally, the second or third expression describes people. Let it be so that criticism and self-criticism are shown. Anton Chekhov − the famous Russian writer, once said that “a man is a fraction, where the denominator shows his personal estimation and the numerator shows his estimation by other people”. Of course, if your personal estimation is low, the fraction will not be very small, even if a person has done little. However, let us find out the readers’ opinion about the authors of this collection and then deduce the fractions for different scientists. Some part of this collection is devoted to an important topic: the order and disorder in polymers. This is a very important factor because it defines diffusion properties (molecule motion speed in polymeric matrix) and solubility of low molecular substances in polymers. Recall that the diffusion coefficient in a glass of water equals 10−4 − 10−5 cm2/s, in melted polymer − 10−10, in a solid amorphous polymer − 10−15, and in crystalline polymer − 10-21-1027 cm2/s. Thus the molecule’s motion and speed at transition from liquid low molecular compound to solid crystalline polymer may vary by 20 orders of magnitude or higher. Hence, a transition from kinetic area (when the reaction rate is defined by reactivities of the substances) to diffusion area (when the situation is completely defined by reagent delivery to the interaction site and the Frank-Rabinovich cage effect) may occur.

viii

B. A. Howell and G. E. Zaikov

The rate of chemical processes in polymers strictly depends on the order degree in the polymer matrix. In its turn, this will affect the operation properties of polymeric materials and their storage life and reliable operation. The authors would be grateful for positive comments on the materials of the current collection, which will be taken into consideration in our future work.

Prof. Bob A. Howell Central Michigan University Mount Pleasant, Michigan, USA Prof. Gennady E. Zaikov N. M. Emanuel Institute of Biochemical Physics Moscow, Russia

In: Polymer Reactivity Editors: G. E. Zaikov and B. A. Howell, pp. 1-101

ISBN 1-60021-263-8 © 2006 Nova Science Publishers, Inc.

Chapter 1

POLYMERIC MEDIA WITH THE GRADIENT OF THE OPTICAL PROPERTIES N. G. Lekishvili, L. I. Nadareishvili and G. E. Zaikov1 Javakhishvili Tbilisi State University, 3, Chavchavadze Ave.,Tbilisi, 0128, Georgia 1 N.M.Emanuel Institute of Biochemical Physics, Russian Academy of Sciences, 4, Kosygin St., Moscow, 119991, Russia

Formation of a refractive surface is not the only way of obtaining an optical image. Media with inhomogeneous distribution of the refractive index can be used for the same purpose. Such media with assigned gradient of the refractive index are named briefly as selfocs or GRIN-elements (GRIN – gradient refractive index). Some information about the possibility of image formation with the help of media possessing inhomogeneous, and in particular, radial-symmetric distribution of the refractive index, was first obtained at end of the 19th century. However, intensive development of gradient optics began three decades ago and, by now, it has become an independent direction of research in optics [1 - 7]. Achievements in gradient optics have proved quite valuable. Selfocs are widely applied in many spheres of science and technology, such as fiber-optical connection lines, fax technology, medical devices, small-scale copying machines, focusing elements of laser systems for video recording, etc. In contrast with traditional optics, in which a change of direction of a light beam occurs by refraction at the surface of the optical element, in a gradient medium the trajectory of the beam along the free path length is continuously curvilinear: this produces beam deviation and, under conditions of appropriate profile of refractive index, distribution focusing of beams, as well. GRIN-optics offers strong possibilities from the point of view of modernization of existing optical devices and of creation of fundamentally new apparatuses. Development of the gradient optics has made polymeric materials competitive with traditional optical materials (inorganic glasses) and, in some cases, even exceed them in some characteristics. Polymeric materials in gradient optics and in the fiber optics have the potential for much wider application than inorganic glasses: primarily, it is the marked high

2

N. G. Lekishvili, L. I. Nadareishvili and G. E. Zaikov

variation of the refractive index observed in polymeric selfocs (∆n may reach 0.1, which significantly exceeds ∆n for quartz glasses) and the possibility of producing elements with different diameters. The following properties are also to be noted: low density, flexibility, ease of contacting and production, and low power consumption in the synthesis of the polymers (polymeric selfocs are prepared under softer conditions and over a shorter time). Owing to the variety of basic raw materials (monomers, oligomers), the application of polymers as materials for gradient media significantly widens the functional potential of gradient optics. The following principles should be noted for development of polymeric GRIN-optics: there is a need for improvement of optical characteristics of polymeric materials by modernization of preparation methods and selection of the most valuable components of the gradient composition; development of new methods for preparation of polymeric GRINelements for special devices; and application of other materials that so far have proved recalcitrant. This requires close cooperation of specialists in calculations of optical systems and polymer chemists developing processes of creation and methods for control of the properties of GRIN-elements [7].

1. CLASSIFICATION OF THE REFRACTIVE INDEX GRADIENT It is common practice to consider three main types of light behavior with respect to trajectory in media with a refractive index gradient: a. Spherical gradient; b. Axial gradient; c. Radial gradient. In selfocs with spherical gradient, the refractive index changes with radius in such a way that surfaces with identical values of n represent concentric spheres. Classic examples of systems with spherical gradient are: so-called Maxwell’s ‘fish eye’ [8] possessing the following distribution of the refractive index (RID):

n( r ) =

n0 1 + (ra ) 2

,

(1)

i.e. spherical gradient allows creation of an optical system with extremely wide viewing angle. The Luneburg lens [9, 10], in which the refractive index distribution is described by the formula:

n(r ) = n0 2 − (r / r0 ) .

(2)

Polymeric Media with the Gradient of the Optical Properties

3

However, practical aspects of realization of these ideas in the optical diapason are not solved yet. In an optical material with axial gradient, the refractive index changes along the optical axis. A spherical lens with refractive index gradient is equivalent to a spherical lens with aberrations – a specific gradient of the refractive index can eliminate some geometrical aberrations [11, 12]. The same effect is achieved when a plate with axial gradient of the refractive index is used as a separate element [11]. In optical materials with radial gradient, the refractive index changes along the radius of the optical element. When the gradient changes in proportion to the square of the radius (∆n ~ r2), the light beam in such an element is spread sinusoidally, and the optical material as a plane-parallel plate or a cylinder without any additional fitting possesses the property of a focusing lens. This type of the gradient is the best studied, and the gradient elements discussed in this book generally relate to this type (radial RID). If the period of the sinusoid is a multiple of the length of the optical element (selfoc), then the image is transmitted from one end to the other at a one-fold enlargement; otherwise, the selfoc possesses the properties of a positive or negative lens. At the same time, the focal length may be changed within any range. A selfoc displays ideal focusing properties when the refractive index distribution follows a hyperbolic secant law [13, 14]: n(r) = n0sech(αr) or

n 2 (r ) = n02 sec h 2 (αr ) = n02 (1 − α 2 r 2 ) ,

(3)

where n(r) is the refractive index at distance r from the axis; n0 is the refractive index on the axis in the center of the selfoc distribution; α is a constant (the distribution constant); ∆n = n0 – n(r) is the gradient of the refractive index from periphery to center, and the distribution constant α in expression (3) is expressed by the formula

α=

1 ∆n 2 , R n0

where R is the selfoc radius, rmax = R.

(4)

4


2. MEASUREMENTS OF THE MAIN PARAMETERS OF SELFOCS (METROLOGY OF SELFOCS). MEASUREMENTS OF DISTRIBUTION OF THE REFRACTIVE INDEX PROFILE AND THE ABSOLUTE VALUE OF THE REFRACTIVE INDEX BY SELFOC RADIUS One of the main optical characteristics of real selfocs is distribution of the refractive index profile (RID) [15 - 26]. Described in the literature are measurement methods of both relative and absolute distribution of the refractive index profile [18, 27, 28]. These methods are subdivided into two main groups: 1. Interference; 2. Non-interference. Among these methods, the most applicable is the method 1, because of its accuracy, universality, obviousness, ease of mathematical processing of experimental data obtained, and ability for reproduction. Most often applied to measurements is the method of a fine slice using the Mach–Zender mirror (double-beam) interferometer (Figure 1) [14, 27].

3

6

5

3

2

7 2 1 2

Figure 1. Scheme of experimental set-up for studying distribution of the refractive index profile by the interference method. 1 – He-Ne laser; 2 – reflecting mirrors; 3 – semitransparent mirrors; 4 – collimating system; 5 – objective; 6 – photo camera; 7 – cuvettes with the sample studied.

On this device, measurements are performed in two regimes: a.

In the regime of formation of the refractive index distribution by radius of the prepared gradient element during regulated diffusion (kinetics of diffusion process); b. In the regime of settled RID in the sample under investigation. Samples of interference measurements represent plane-parallel cross-section slices from polymeric light focusing elements with varying profile of the refractive index from the sample center to periphery. The thickness of these slices is limited by the maximum number


5

of rings that can be resolved by photo film at appropriate enlargement of the control system and is 100–300 µm depending upon the system studied. The refractive index is measured with an accuracy of 10–5 [14]. A typical interferogram is shown in Figure 2.

Figure 2. Interferogram of selfoc based on DAIF–4FMA.

According to the data from an interference picture, determination of the refractive index profile by cross-section of the gradient element consists of calculation of the function ∆n = f(r) by the formula:

∆n =

∆a λ ⋅ , a d

(5)

where ∆a/a is the relative value of displacement of interference bands; d is the thickness of the sample measured. Accuracy of measurement is ∆a/a = 0.05 of the band, which corresponds to the relative error ∂n/∆n = (6–10)%. In practice, the formula

∆n =

N ⋅λ , d

is most often used, where N is the number of circle on the interferogram [29]. Distance r from the center of the interference image to the circle Nr is calculated by the expression [26]:

r=

d ⋅L, D

(6)

where d is the diameter of the cross-section slice of the gradient element; D is the diameter of the interference image of the slice on film; L is the distance from the interference image center to a circle.

6


After exposition of the interference image observed on the slice of light focusing polymeric element, a functional dependence ∆n = f(r) or ∆n = f(r2) appropriate to the RID of the selfoc studied is derived from the experimental points. Investigation of RID formation and performance of some practical calculations require data on absolute values of nD with respect to the selfoc radius [30]. In the literature, particular methods of measurement of these values are described in detail [24, 27]. Among them, let us describe a modified variant of the method using a fine slice [30 - 32]. According to this method, a plane-parallel glass plate with known value of nD with accuracy up to 10–5 (standard) is placed simultaneously with the sample studied in a cuvette filled with an immersion liquid. The standard and the sample are turned to each other by finely polished edges. The path of interference bands on the interferometer output is recorded, and the difference between the refractive index of a selected point of the sample and the standard is determined by displacement of these bands on the standard–immersion liquid and the sample–standard borders. Then the value of the refractive index at any point of the sample is determined [30]. Other methods of determination of absolute nD values by the selfoc radius have also been described (for example, the method of ‘saturated’ beam, the method of interferometry of the sample in the direction perpendicular to its optical axis [30], etc.). In the view of the authors of ref. [30], the methods described above (first of all, the modified variant of the fine slice method) are quite suitable and provide accuracy of the measurement (at a definite slice thickness) of 1–2⋅10–4. It should be noted that accuracy of measurement of the refractive index distribution in a selfoc is influenced by such factors as focusing of beams permeating though the selfoc slice, and the sphericity and non-plane-parallelism of the sample. Analysis performed by the authors of ref. [30] indicates that the relative error of measurements produced by focusing of the beams is below 0.5%, and deviation from the plane-parallel shape of the slice by 10 µm at its thickness of 500 µm indicates the relative error of about 4% at ∆n = 0.04 with accurate selection of immersion. It has been mentioned [30] that accuracy of RID change by the method of interferometry of fine slices may be increased by photoelectric registration of interferograms of polished plane-parallel slices and computerization of the calculation.

3. METHODS OF PREPARING GRIN-ELEMENTS There are various methods of obtaining light focusing polymeric gradient elements (LFPGE) or selfocs [33 - 60]. They may be conditionally subdivided into two main groups. The first group comprise traditional methods: the two-stage method of diffusion exchange; copolymerization (photocopolymerization) of double or triple composites of monomers (or oligomers) differing by relative activity and refractive index. The second group comprise: separation in a gravitational field [52]; gradient modification of polymeric transparent material from the surface to a definite depth by physical and chemical methods (swelling or dissolving in an organic solvent, halogenation, etc.) [40, 51, 61, 62], dipolephoresis [63, 64].


7

Obtaining LFPGE with some specific gradient of the refractive index is based on proper selection of initial monomers and polymers and their transformation under such conditions that provide the required distribution of the refractive index gradient in the final material. Factors that promote creation of a gradient distribution nD of copolymeric compositions include temperature, concentration, gravitation and electric field, the field of initiation, etc. Of great importance also is the search for technological regimes of formation and fixing (if necessary) of the profile of refractive index distribution obtained, which would provide materials with close to the ideal focusing distribution.

3.1. Method of Diffusion Exchange In practice, among the methods mentioned above, the most applicable one is the method of diffusion exchange [33] or its modified variants. The process of creation of a gradient distribution of the refractive index by the diffusion method consists of the following stages [33, 65]: obtaining of a for-polymeric matrix with high value of the refractive index (such as nD ≥ 1.5) and molecular exchange of residual monomer of the matrix and monomer-diffusate possessing a refractive index lower than the matrix component of LFPGE. As a result of the exchange, a definite distribution of monomer concentration and, consequently, appropriate distribution of the refractive index (RID) is formed in LFPGE after a definite time. Using this method, it is necessary to fix the RID formed in the sample by thermal and photo-copolymerization. To obtain high-quality selfocs by the method of diffusion exchange, it is necessary to select monomers that fulfill the following requirements [29]: -

-

-

To prepare the matrix, the initial monomer must be, at least, bifunctional and give a transparent polymer network on polymerization, which at relatively low degrees of conversions (≤ 30%) is able to keep the assigned shape, for example, cylinder, planar or spherical shape; To provide high (maximum) gradient of the refractive index ∆n over the selfoc crosssection, the monomer–diffusate and the matrix must possess possibly great difference in refractive indices; To reach high transparency of the selfoc, the monomer–diffusate and the matrix must be miscible, able to form (after exchange diffusion) a homophase random copolymer.

Because any dielectric medium (including selfocs) possesses dispersion of the refractive index, it is necessary to strive for gradient elements that produce low chromatic aberration. It is found that chromatic aberration depends on the relative optical characteristics of polymers of the matrix and diffusate separately and, consequently, proper selection of a pair of copolymers is the main factor determining low chromatic aberration [26]. Selecting a pair of monomers, M1 and M2, for preparation of selfocs possessing low chromatic aberration, it is necessary to take into account the following conditions [14, 29, 59]:

8

N. G. Lekishvili, L. I. Nadareishvili and G. E. Zaikov a.

Difference between [(nF – nC)/nD]1 (matrix) and [(nF – nC)/nD]2 (diffusate) must be low; b. Correlation [(nF – nC)/nD] of both monomers must be low; c. Difference between (nD)1 and (nD)2 must be high. Problems of decrease of the chromatic aberration, increase of ∆n, length and diameter of LFPGE have been topics demanding the constant attention of investigators. They solved the problems complexly, by searching for the most opportune pairs of monomers for obtaining selfocs with assigned optical, physicomechanical and other characteristics with simultaneous modernization of technology for obtaining gradient media [29, 66]. Initial information on obtaining polymeric selfocs appeared in the early 1970s [33]. In spite of a number of valuable properties (primarily, high input aperture angle and, secondly, low chromatic aberration), classic polymeric gradient (GRIN) elements based on accessible 1 2 monomers, such as DAIP -MMA, DEGBAC (CR-39)-4FMA [67], possessed limited lengths (L ≤ 150 nm) and diameters (d ≈ 1–3 mm) which, in turn, limited the sphere of their application. To reach high values of the gradient of the refractive index (∆n > 0.05) and increase of LFPTE diameter (d ≥ 10 mm) while preserving good opto-mechanical indices, investigations were generally performed in two directions: 1. Obtaining new copolymeric matrices from comonomers with high values of nD; 2. Selection of monomer–diffusates with minimal values of the refractive index and good optical miscibility with widely used matrix components (monomers for matrices – DAIP, DEGBAC, DMEG3, etc.). As an example of the first approach may be considered creation of a selfoc based on copolymeric matrix from DAIP-a and maleic anhydride (MA) [67, 68] with conversion of monomers of 30–60%. The polymeric matrix obtained is more plastic than the matrix from homopolymer DAIP with analogous conversion of the monomer, which allowed production of selfoc samples with relatively large diameter (d ≈ 6–8 mm) with the refractive index gradient ∆n = 0.035. Therewith, definite difficulties are encountered at fixing the created RID, associated with the reverse diffusion and partial evaporation of the monomer–diffusate that penetrated into the matrix [67, 68]. Increase of ∆n with simultaneous reduction of chromatic aberration and improvement of a series of physicochemical, mechanical and exploitation properties of selfocs were achieved by application of chlorine-containing (meth)acrylates as monomers (diffusate) with low refractive index [69]. Y. Ohtsuka et al. [59] have indicated the possibility of obtaining gradient elements with low chromatic aberration by appropriate selection of monomeric pairs – DEGBAC (CR-39) and 1,1,3-trihydrofluoropropyl methacrylate or DEGBAC and 1,1,5-trihydroperfluoropentyl methacrylate. Selfocs possessing chromatic aberration of about 8⋅10–3 were obtained.

1

DAIP – diallyl ether of isophthalic acid DEGBAC – diethyleneglycol bis(allyl carbonate) 3 DMEG – ethyleneglycol dimethacrylate 2


9

Based on perfluoroalkyl methacrylates and various matrices, selfocs with ∆n ≥ 0.06–0.08 and high thermal and humidity resistance have been created, which also display excellent optical and mechanical properties that they retain for a long time [46]. In the search for ways of broadening the functional abilities and to increase radiation and thermal resistance of selfocs, methods of obtaining them from organoboron and organosilicon dimethacrylate monomers [70] and their copolymers [61] have been developed. Investigations have indicated [26, 71] that not only relatively expensive bifunctional network-forming monomers, but also quite accessible monomers (styrene, for example) which give a linear polymer with high value of nD may be used as a matrix, if a network structure is formed in it by the use of cross-linking agents [26, 72]. A method of preparation of selfocs based on styrene and cross-linking agent, dimethacrylate ethyleneglycol (DMEG, Table 1) is suggested [72]. Among the advantages of gradient elements produced by this method are such characteristics as high light transmission, refractive index distribution close to the ideally focusing one, etc. However, the value of refractive index gradient ∆n of these elements does not exceed 0.04 [72]. Oligocarbonate dimethacrylates (OCM-2, Table 2) and organosilicon dimethacrylates [55, 61] have also been used as cross-linking agents for polystyrene. They simultaneously played the role of modifiers of the mechanical properties of rigid-chain polystyrene. From this viewpoint, optimal results are obtained, when these compounds are used in copolymerization with styrene in amount not more than 20% [55]. There is no point in increasing the proportion of dimethacrylates, because, first of all, the refractive index of the appropriate polymeric matrix decreases and, secondly, the rate of copolymerization autoacceleration at comparatively low conversion of the monomer increases, which makes production of a shapestable gel-matrix with required fraction composition less regulated. Table 1. [72] Influence of change of DMEG concentration in the initial matrix on optical properties of gradient elements, Tpolym = 75°C, [PB] = 0.75 mass. % DMEG, mass. % 10 30 50 70

Conversion, mass. % 14.8 15.0 11.5 11.5

Gel-fraction, mass. % 90 92 98 99

Tc, °C 85 75 65 65

NA 0.317 0.358 0.3105 0.286

N 0.0324 0.0405 0.0314 0.0267

F (NA = 0.1) 15.0 91.4 97.5 403

To solve some technical problems, it is necessary to use flexible rod-like selfocs. Described in the literature are two types of such flexible selfocs: 1 – flexible at bending [73, 74]; 2 – selfocs with variable focal length [75]. In this case, on the molecular level, GRINmaterials represent a cross-linked macromolecular three-dimensional network that differs from others in that, in order to be able to regulate the focal length of LFPGE by an external mechanical influence, macromolecules between network cross-linked points contain flexible fragments from different units in a definite molar ratio that exist in a partially entangled state. This enables intercross-linked point chains of macromolecules to be partially compressed or more elongated on subjecting LFPGE to a compressive or elongating force and, therewith, the

10


value of the focal length may be changed accordingly. This property is achieved without distortion of the overall LFPGE shape or changing curvature of its surface [75]. Table 2. Compositions of mixtures and conditions of OCM-2 copolymerization with styrene

No. 1. 2. 3. 4. 5. 6.

nD of monomers mixture 1.5403 1.5336 1.5260 1.5120 1.5120 1.4960

Matrix copolymers Styrene, OCM-2, mol % mol % 98.7 1.3 95.9 4.1 89.0 11.0 80.0 20.0 80.0 20.0 63.8 36.2

T, K 353 353 353 353 333 343

Gel-fraction concentration in copolymer, % 11.0 16.8 18.0 23.9 22.1 24.1

τ, min 50 35 20 15 40 26

Generally, selfocs flexible at bending are produced by two methods: selection of appropriate initial components [73, 74] which provide for formation of a flexible copolymeric network material with gradient nD or development of a specific technological regime [33] which allows regulation of the process of matrix monomer copolymerization and afterdiffusion fixing of the RID obtained in such a way that network structures with low crosslinking degree would be formed. For example, a copolymeric rod with refractive index gradient of half-period length 63 mm, possessing a bending modulus E ≤ 5.7⋅103 kg/cm2, was successfully obtained by thermal copolymerization of DAIP-MMA, performed under relatively mild conditions (60°C, 18–20 hours), after the stage of diffusion exchange (prepolymerization). Investigations on creation of flexible rod-like selfocs performed so far have indicated [73, 74] that, among existing monomers for matrix formation, those with greatest potential are bifunctional monomers or oligomers of polyethyleneglycol dimethacrylates of the following general structure:

CH2

C

C(O)O

(CH2

CH3

CH2O)l

(O)C C

CH2

CH3

where l = 1, 2, 3, …, 13. The value of l determines the rigidity of the polymeric matrix obtained. In this case, good results were also obtained at application of triallylcyanurate and 13-ethyleneglycol dimethacrylate (l = 13) as matrix comonomers. These compounds cause formation of a flexible random copolymer network with reduced frequency of cross-linking. Another way of obtaining a flexible selfoc concludes in application as a diffusate of a vinyl monomer (butyl methacrylate, for example), the polymer of which possesses low glass 0 transition temperature Tg < Troom [59].

(

)

There are data in the literature on synthesis of relatively flexible gradient mutually penetrating network polymers (MPNP) based on the pair DAIP and diallylsebacinate [14].


11

MPNP is usually synthesized by polymerization of a polymer–monomer system obtained by swelling of a polymer network in the mixture of another monomer with a cross-linking agent. The essence of this work is in the fact that the network of three-dimensional bonds is formed in a single polymer-matrix, and the second monomer (with lower nD) without a cross-linking agent is injected into the network polymer by exchange diffusion with unreacted matrix monomer. In this manner, a gradient of the refractive index may be created in the polymer. Mutually penetrating networks with gradient based on oligoetheracrylates [76], etc. are also described. Discussed above were focusing gradient elements, but many gradient polymeric elements are defocusing selfocs. The method of preparation of samples is the same as for focusing polymeric rods. The effect of defocusing is achieved when the value of the refractive index of the matrix monomer (DEGBAC, for example) is lower than that for the diffusate (vinylbenzoate or styrene, for example) [77]. However, new applications of defocusing selfocs are still being explored. The need for a variety of gradient compositions is dictated by the range of technological tasks of modern GRIN-optics, in particular, by the necessity of broadening functional and technical abilities of light focusing gradient elements. However, it should be noted that despite the existence of polymeric selfocs with a refractive index gradient ∆n ≥ 0.1 and d ≈ 20 mm, they are still less important than selfocs based on inorganic glasses with respect to resolution (~ 100 and 300 mm-1 per 1 mm, respectively) [29]. Reproduction of optical characteristics, as well as creation of GRINelements of unlimited length and possibly larger radii (for example, in the case of plate thin GRIN-elements), etc. may also be explored. Further developments are required in diffusion technology to obtain gradient elements with the required performance, and better systematic studies of synthesis of the matrix, formation and fixing of gradient medium RID, and determining the precise interconnection between RID formed and the initial gradient composition, experimental conditions, physicochemical and optical characteristics of gradient elements obtained. One of the main stages of obtaining selfocs according to the diffusion technology is preparation of a fore-polymeric gel-matrix. Generally, bifunctional vinyl monomers – dimethacrylate and diallyl ethers – are used as the matrix monomers. Among them, ethyleneglycol dimethacrylates (DMEG) [72], diallylisophthalate (DAIP) [14,28, 33, 55, 59, 67], diallylitaconate (DIT) and diallylsebacinate (DAS) [14], diallylmaleinate (DAM), diethyleneglycol bis-allylcarbonate (DEGBAC) [14, 28, 55, 67], and their copolymers find greatest application [2, 14]. Besides the requirements mentioned above, various properties of materials obtained, in particular, thermal and heat resistance, resistance to radiation, resistance to humidity, flexibility, mechanical strength, and technological effectiveness of their formation must also be taken into account in selection of the matrix monomers. It should be noted that, to obtain a shape-forming material that is able to keep the assigned form, very specific concentration of the gel-fraction of the polymer network must be achieved in the synthesis of the matrix. On the other hand, the higher is concentration of unreacted (residual) monomer in the matrix which may be exchanged by diffusate molecules, the greater is the gradient ∆n between the axis and the selfoc surface that may be obtained [29].

12


In accordance with the task set, block radical initiated polymerization (copolymerization) would be appropriate for use as a method for the matrix synthesis. A spatially cross-linked structure, swelled in its own monomer, is typical of the gel-matrix, which is the intermediate product for obtaining a selfoc. The presence of a three-dimensional structure is necessary not only to confer and preserve the given shape in the target optical element, but also for creation of definite conditions for the diffusion exchange of penetrating agent by a unreacted monomer of the matrix and formation of a specific RID. The monomers mentioned above are able, at rather low degrees of conversion (20–30%), to form shape-stable spatially crosslinked structures containing residual functional groups capable of entering the copolymerization reaction with the diffusate monomer after formation of the given distribution of the refractive index by the exchange diffusion with the residual matrix monomer. Investigations have shown that conditions of homopolymerization of bifunctional monomers generally depend on the type of monomer, i.e. on the class of compounds, dimethacrylate or diallyl, to which the given monomer relates. In practice, this factor may determine selection of an initiator and its amount. Kinetics and mechanism of polymerization of allyl monomers are extremely complicated. Besides reactions general for radical polymerization of vinyl monomers – initiation, propagation and termination of the chain, and chain transmission to monomer, polymer and admixtures – the following specific reactions are typical of allyl monomers, in particular, di(poly)functional allyl ethers: degenerate chain transmission [78], rupture of ester bond [79], cyclopolymerization [80]. A proposal has been made that the above-mentioned reactions are complicated by cross-linking reactions of macromolecules and diffusion control of the process at greater degrees of transformation. However, unfortunately, accurate methods of quantitative description of the given transformation of these complex systems have not yet been reported in the literature. This definitely restricts statement of a clear interconnection between the optimal technological regime of gel-matrix production and assigned characteristics and fractional composition, and kinetic parameters of homopolymerization of diallyl monomers. Numerous works [80 - 85] are devoted to the study of kinetics and mechanism of radical polymerization of divinyl monomers. That is why in this book, we have limited ourselves to consideration of the main regularities only, associated with formation of a gel-polymeric matrix up to the required degree of conversion (10–50%). When these regularities are considered, we base practical methods on the ideas discussed in refs. [72, 81 - 84], supporting them by data of kinetic investigations [84 - 86]. Study of homopolymerization (of DAIP, for example) by DSC4 and SPR-spectroscopy methods has shown [81] that the process is separated into two stages: quasi-linear polymerization and structuring. Monomer consumption proceeds with formation of primary polymeric chains, whereas structuring is characterized by formation of secondary polymeric chains by double bond. An important feature of divinyl monomers of the methacrylate ether (OEA) type is that, during formation of a spatially cross-linked structure, the microgel is already formed at low transformation degrees (< 1%) of the monomer, i.e. since the very beginning, the process of radical polymerization of these monomers proceeds under the conditions of 4

DSC – differential scanning calorimetry


13

microheterogeneity [72, 82, 87, 88]. In connection with specificity of the process itself, the reaction can proceed at high rates, which frequently makes the process of obtaining a gelmatrix with the required properties poorly regulated. The ratio of outputs of gel-fractions and β-polymer (zole-fraction) at the initial stages is determined by the conditions that affect the degree of aggregation of macrochains that are formed up to that time [88]. Therewith, it has been proved that processes of chemical and physical structure formation (engagement of propagating intermediate macrochains) are simultaneous (identical), and polymerization is performed by permolecular particles [72, 89]. It is stated that (in the case of OEA) [72, 87], already at low transformation depth, accumulation of new ‘isolated’ particles and further polymerization proceeds without a significant change in the number of particles due to their expansion. Therewith, the number of primary polymeric chains is limited rapidly due to competitive reactions of addition of these newly formed chains to macrochains that have accumulated in the reaction mixture up to that point [72, 90]. At the same time, formation of soluble β-polymer (zole-fraction) is highly complicated due to its low stationary concentration [88]. It has been shown [91, 92] that possible intramolecular cross-linking reactions in early stages of polymerization causes a displacement of the so-called gel formation point (the moment of gel-fraction formation) to the side of higher transformation degrees and leads to increase of the amount of β-polymer (zole-fraction) [72]. Based on the differences in mechanisms of homopolymerization of dimethacrylate and diallyl ethers mentioned above, differences in the processes of gel-matrix formation based on them are quite understandable. It is also noted [72, 84, 85] that, depending on the nature of applied allyl monomers (for example, DAIP, DAP5, DAC6, etc.), a microgel at higher degrees of transformation may be formed due to selection of conditions of homopolymerization performance (the presence of inhibitors, chain transmitters, etc.). Therewith, the proportion of soluble linear and branched products (zole-fraction) increases significantly. It was also found that the zole-fraction output up to high conversion degrees of these monomers is lower than of insoluble polymer. According to the authors of the work [72], in the pre-gel period, the reaction system represents a colloid dispersion (size of gels are quite low) in which the role of the dispersion phase is played by microgels with high transformation degrees, and the role of dispersion medium by unreacted initial oligomer (monomer). In the view of these authors, diffusion transparency of such a system will be high, and strength is very low, because the system properties in this state are generally determined by the property of the continuous phase. Proceeding of the process without diffusion limitations is typical of the gel-polymerization on this stage. Therewith, probability of defect formation during this period is minimal due to localization of homopolymerization in the liquid phase, in which the possibility to form defects of the ‘freezing’ type of internal stresses is minimal [89]. Appearance of this type of defect as, for example, defects of macromolecules formed of free ends and cyclic structure type [93], may affect some physicochemical characteristics of the network formed. It should be noted that the most significant contribution to formation of spatially cross-linked structure with definite physicomechanical properties is made by so-called morphological (permolecular) defects [72], the formation and degree of which are determined by natural and technological factors of performance of the polymerization itself. To put it differently, 5 6

DAP – diallylphthalate DAC – Diallyl ether of µ1-carboranedicarboxylic acid [55]

14


formation of defects in the structure may be minimized by a selection of appropriate favorable conditions of polymerization. During transformation of bi(poly)functional monomers, one more feature of threedimensional polymerization is displayed [87] – appearance of auto-acceleration (gel-effect) during accumulation of a polymer network at rather low transformation degrees. This may be the consequence of deterioration of chain termination conditions limited by diffusion which, in its turn, will change the ratio Kp/Kter during polymerization [72]. In contrast with usual linear polymerization (for example, monofunctional vinyl(monomethacrylate) monomers), in which the main reason for auto-acceleration is the occurrence of the so-called engagement network, in the case of bifunctional monomers, the probability of auto-acceleration increases with increase of concentration of network crosslinked points [86]. This concept may be confirmed by occurrence of auto-acceleration (geleffect) even in the case of copolymerization with participation of monomers (styrene, for example), in which the gel-effect is not typical [72, 94, 95]. This suggests that (at good optical miscibility) the gel-effect, with styrene applied as a comonomer in the mixture with vinyl monomers (to obtain the matrix), may be displaced to the side of higher transformation degree [72]. In the literature, there are various methods for analysis and control of homopolymerization processes (and hence, of the composition of the polymer obtained): refractometric [86], kinetic [85, 86], and spectroscopic [81]. Data on NMR-spectra recorded during homopolymerization of bifunctional (diallyl) monomers (for example, DIAP, DEGBAC) have been published. It is found [26] that the 1HNMR-spectrum of the system (in the case of DAIP homopolymerization, initiator [BP] = 3.81%, Treact = 80°C) is bimodal, in which the presence of a wide component in the spectrum is confirmed by formation of a network fraction, and a narrow component is confirmed by a quasi-linear structure. Intensity of the narrow component decreases during polymerization, but that of the wide one increases. Comparison of the correlation of wide and narrow components at different stages of polymerization allows determination of the ratio of fractions in samples. Based on these studies, it has been stated that copolymerization of this type of monomers proceeds up to a definite conversion degree (for example, for DAIP, S ≈ 13.5%, and for DEGBAC – S ≈ 9.5%) with formation of a soluble β-polymer as the result of splitting of a double bond in one of two allyl groups of the monomer. Above these conversion degrees, a gel-fraction (spatially cross-linked structure) is formed, which is intensified due to involvement of the second double bond of the allyl group into polymerization. Such a mechanism has also been confirmed by 13C-NMR-spectroscopy (on the example of DEGBAC polymerization) [55]. It is found that, in the second stage of the reaction (S>16–17%), comparatively fast decrease of relaxation times of the system is observed which, apparently, is due to a decrease of mobility of intercross-linked fragments which results from appearance of a spatially crosslinked structure. At the same time, new signals (33.0 m.p.) appear, typical of so-called ‘ruptured’ allyl groups [55]. Therewith, there arises the possibility of controlling the processes by obtaining a shape-stable gel with specified degree of conversion (specific concentration of the gel-fraction).


15

Refractometric control of the gel-forming process (in the case of DEGBAC polymerization) is based on determination of dependencies of conversion and nD of the reaction product on the reaction duration, as well as dependence of transformation degree on the difference in refractive indices of polymerizate and monomer [86]. Kinetic curves obtained by the refractometric method (at homopolymerization of DEGBAC) are characterized by the presence of a bending in the range of the transformation degree of ~10%, corresponded to the onset of the gel formation [86]. The above-suggested mechanism of the homopolymerization process of DMEG, DAIP, DEGBAC, etc. type monomers is also confirmed by the study of kinetics of homopolymerization at initial and high degrees of transformation with simultaneous measurement of the parameter η⋅d [84] in the polymerization product as a function of the stage of the process (Figure 3). It is indicated that at conversion of 9.2%, the character of increase of the system viscosity (η) and polymer density (d) changes. This fact may be explained only by a supposition that formation of polymer at the stage before 9.2% conversion proceeds preferentially with participation of a single double bond. Experimental determination of the residual unsaturation of polymerizate as the function on time indicates that, indeed, when 9.2% conversion is reached, consumption of double bonds increases sharply.

η d  lg ⋅ − 1  η0 d 0 

1.0 0.6 0.2 4

8

12

16

20

S, %

-0.2 Figure 3. Dependence of system viscosity multiplied by polymer density on transformation degree S at DEGBAC polymerization, [BP] = 1.5 mass%; T = 338 K

Branched and linear structures may be differentiated by value of the branching factor [96], as well as by spectroscopic studies [81]. Change in structure of polymers based on divinyl (diallyl) monomers during polymerization is also confirmed using the thermomechanical method [81]. From thermomechanical curves indicated in refs. [81], it is obvious that Tg of a product of quasilinear structure is sharply different from Tg of polymeric cross-linked structures. Therewith, softened (Tg ≈ 27°C) and non-softened spatially cross-linked polymers with significantly

16


different parts of network cross-linked points are formed [81]. Temperature increase promotes an increase in the proportion of cross-links (the proportion of chains with two open double bonds) of polymerizate with appropriate decrease of the proportion of chains possessing only one double bond (pendant double bonds), which affects the shape of the thermomechanical curves. Regulation of the temperature regime of homopolymerization allows control of crosslink rate and degree [81]. It should be noted that complexity of the investigation methods indicated in ref. [81] (DSC, ESR, thermomechanics) allows determination of not only the monomer consumption and conversion of double bonds during the process of polymerization, but also structuralmorphological characteristics which, obviously, must affect the resulting characteristics of optical materials. Besides the temperature regime [55], one of the ways of regulating gel formation and cross-linking frequency of the gel-matrix obtained is appropriate selection of the type and amount of initiator. Increase of the initiator amount over a definite value (which is, for example, 2.80–2.90% for DEGBAC, 3.80–3.85% for DAIP, 3.0–3.5% for DAC, 1.00–1.50% for DMEG, etc.), often presents difficulties in regulation of the gel-forming process, which complicates obtaining a matrix with assigned concentration of the gel fraction, required frequency of cross-links in the spatial structure and mechanical properties of optical materials based on them [55]; and vice versa, a significant decrease of the initiator amount practically in all cases of polymerization of applied diallyl monomers abruptly decreases the rate of gel formation (the fore-polymeric matrix is formed as a result of durable heating up of the reaction mass), which often causes a negative effect on optical properties of selfocs obtained (yellowing of the samples takes place, etc.). Beside the process of fixing RID, of importance for the given RID formation is the stage of molecular exchange of matrix monomers and diffusate. Experimental study of RID formation at molecular exchange of the residual monomer of a gel-polymeric matrix with a monomer-penetrant [24] has shown that, in the first approximation, mutual diffusion of monomers (in gel) can be described by the known Fick equations [24]. In the case of rod-like cylinder samples (if it is assumed that the molecular exchange proceeds in an infinite vat, and the sample volume remains unchanged), an expression describing functional dependence of concentration (C) of one of the components (diffusate) as the function of coordinates and time is presented [24]:

C ( r / R, Dt / R 2 ) = f (r / R, Dt / R 2 ) , C0

(7)

where C0 is the concentration of one of the components (for example, the penetrating agent) on the sample surface. To move from distribution of the component composition to RID, the known Gladstone–Dale equation should be used. Then: n(r/R, Dt/R2) = nM – f(r/R, Dt/R2)(nM – ndif),

(8)

where nM is the refractive index of the initial matrix; ndif is the refractive index of the matrix with completely substituted matrix monomer by the monomer–penetrating agent [24].


17

1

30 ∆P/P0⋅100%

2 3 4 5

10

5 4 3 2 1, 6

6 100

200 Time, min

24 Time, h

Figure 4. Kinetic curves of swelling of gel-polymer DAIP samples. The mixture composition (vol.%): MMA:DAIP = 100:0 (1); 87.5:12.5 (2); 75:25 (3); 62.5:37.5 (4); 37.5:62.5 (5); 0:100 (6); T = 20°C.

100

DAIP, mol%

90

1

80 70 60 50

4 2

40

3

5

30 20 10

6 10

20 30 40 50 60 70 80 90 100 t, min

Figure 5. Kinetic curves representing diffusion of fluoroalkyl methacrylales into the DAIP matrix at 50°C: 1 7 – 4FMA; 2 – 8FA; 3 – MAMC [394] ; 4 – 8FMA; 5 – 12FA; 6 – 12FMA.

With regard to these theoretical dependencies, as well as to the fact that the time of formation of assigned RID is proportional to the square of the radius of the cylinder (at constancy of all other factors – temperature, T, pressure, P, volume, V) for the DAIP–MMA system, the authors have obtained experimentally diffusion anomalies that result in acceleration of diffusion compared with the Fick one at an increase of diameter of cylinder samples. The phenomena observed are explained by the ideas of a concentration jump of the diffusion coefficient (plasticization of the matrix by a monomer – penetrating agent), which is explained in the framework of the modified model [24]. These very authors have also found [97] anomalous sorption at molecular diffusion exchange of DAIP and MMA monomers in partially polymerized DAIP matrix, which represents a gel polymer (fore-polymer network impregnated by the non-reacted self monomer). The anomaly results in the extreme character of the kinetic sorption curves (Figure 4), which approach the normal shape at swelling in the mixture of diallylphthalate and MMA as concentration of the latter increases. Gel samples impregnated by MMA up to the equilibrium state, when submersed in DAIP, first, reduce their weight and then restore it up to the initial value. Other authors [98] explain the sorption 7

CH2=C(CH3)COOCH2Si(CH3)3

18


DEGBAC, mol%

anomaly by collapse of network polymers (abrupt decrease of their volume), which is interpreted as appearance of the coil–globule transition in subchains composing the network.

Figure 6. Kinetic curves of fluoroalkyl methacrylate diffusion into DEGBAC matrix at different temperatures: 1 – 60°C; 2 – 40°C; 3 – 20°C.

Figure 7. Distribution of the refractive index in the model of DAIP-4FMA gradient element of 5 mm diameter for different temperatures at different moments of time [14, 99]: a) Tdif = 30°C: 1 – 7 min, 2 – 20 min, 3 – 30 min, 4 – 40 min; 5 – 50 min, 6 – 70 min, 7 – 110 min; b) Tdif = 60°C: 1 – 3 min, 2 – 8 min, 3 – 12 min, 4 – 17 min; 5 – 25 min, 6 – 60 min, 7 – 90 min.


19

Numerous investigations [14, 25, 26, 29, 33, 55 – 57, 61, 65, 66, 96 - 99] of kinetics of the diffusion exchange (Figures 5 and 6, Table 3) and regularities of the diffusion formation of the RID profile in model systems have been performed. Based on the analysis of curves n = f(r) (Figure 6 and 7) and n = f(r2) recorded under different conditions for various systems [24, 25, 26, 33, 37, 55, 99, 100], the general character of RID formation in a selfoc, which is practically identical to that in unitypical monomers, has been stated. Therewith, quantitative indices of the process significantly depend on the molar volume of monomer-diffusates, structure, the boiling point and molecular mobility of the diffusate and the matrix monomer, their relative activity at copolymerization, cross-linking degree of the polymeric matrix, time and temperature of diffusion, etc. [29, 33, 57, 65, 99 - 101]. The role of the ‘reverse’ diffusion at fixing of the RID formed is estimated [29, 56]. It is also indicated that the resulting RID profile is also significantly affected by the initiator concentration and experimental conditions of fixing the distribution of the refractive index obtained [29, 55, 57]. Special attention of investigators dealing with gradient elements is paid to the study of interconnection of RID with their optical and other properties. It is shown that optical characteristics of selfocs are also definitely influenced by factors such as fractional composition of the reaction mixture after obtaining the gel-polymeric matrix, conversion degree, nature of diffusate, the ratio of monomer diffusion rate and the rate of its copolymerization in partially polymerized matrix, etc. [2, 14, 29, 33, 56, 57]. Table 3. Values of activation energy and interdiffusion coefficients for DAIP – MMA, DEGBAC – FMA, DEGBAC – FMA, DAIP – FA, DEGBAC – MAMS systems System studied Diffusate Tetrafluoropropyl methacrylate (4FMA) Octafluoropentyl methacrylate (8FMA) Dodecafluorohexyl methacrylate (12FMA) Octafluoropentyl acrylate (8FA) Dodecafluorohexyl acrylate (12FA) Tetrafluoropropyl methacrylate (4FMA) Octafluoropentyl methacrylate (8FMA) Dodecafluorohexyl methacrylate (12FMA) Octafluoropentyl acrylate (8FA) Dodecafluorohexyl acrylate (12FA) Methacryloyloxymethyltrimethylsilane (MAMC)

E, kJ/mol

Diallylisophthalate (DIAP)

D⋅10–6, cm/s2 at 50°C 10.00


4.18

21.36


1.50

34.90

Diallylisophthalate (DIAP) Diallylisophthalate (DIAP)

3.20 1.60

20.01 30.20

Diethyleneglycol-bisallylcarbonate (DEGBAC) Diethyleneglycol-bisallylcarbonate (DEGBAC) Diethyleneglycol-bisallylcarbonate (DEGBAC) Diethyleneglycol-bisallylcarbonate (DEGBAC) Diethyleneglycol-bisallylcarbonate (DEGBAC) Diethyleneglycol-bisallylcarbonate (DEGBAC)

12.00

13.20

6.50

19.26

2.80

23.90

5.90



3.60



4.80



Matrix

15.10

20


Physical and mathematical models of RID formation, combined with investigation of optical characteristics of real gradient elements have been suggested [29, 66]. The physical model for studying formation of the refractive index distribution of selfocs may be described in the following manner. In the initial stage, in order to obtain a gel-matrix, polymerization is performed not in a tube, but between two glass plates with a thin (100 – 200 µm) polyethylene layering between them. A round hole is cut off in the layer, in which the component A is installed. When the initial stage is finished, the polyethylene layer is removed, and the plate together with clamped gel-polymeric tablet is placed in a cuvette with the component B. The whole system is thermostatically controlled and placed into one of the shoulders of the Mach – Zender laser interferometer [27]. The process of interdiffusion of A and B components, as well as polymerization during diffusion leads to a constant change of the refractive index distribution in the cross-section of the tablet and, further on, to a change of the interference image. Processing of interferograms recorded by a photocamera is performed by formula (5). Analysis of the results obtained according to this method enables formation and fixation of the refractive index to be studied directly during preparation of polymeric selfocs. The mathematical model of this process is presented as a totality of the following equations [57]:

1.

 f1  df1 = ∇[D( f1 + f 3 )]∇   − K1 f1 ; dt  f1 + f 3 

(9)

2.

df 2 = K1 f1 ; dt

(10)

3.

df 4 = K2 f2 ; dt

(11)

4.

∑ fi = 1 ;

4

(12)

i =1

5. n =

∑ n1 f1 ,

(13)

i =1

where {fi = fi(r, I); I = 1 – 4} are volumetric parts of components A and B and their contributions to polymer, k1 and k2 are rate constants of the polymerization reaction; {ni} are values of partial refractive indices; D is the diffusion coefficient. Each equation has some definite physical meaning. The first equation is deduced from the equation of the Fick diffusion with regard to the fact that diffusion proceeds in an inhomogeneous medium, when the diffusion flow is determined by the relative concentration of diffusing components and their mean free path. Moreover, decrease of the concentration of diffusing particles due to polymerization is also taken into account.


21

The second and third equations describe the accumulation of immovable polymer at the sacrifice of polymerization of the components A and B, respectively. The fourth equation expresses the assumption that concentration does not influence the diffusion. The fifth equation of the model forms the basis for transition from the concentration distribution to the refractive index distribution. The model equations are solved numerically by a computer. The work indicates the detailed description of functioning of the model, but it does require the participation of an expert operator [57]. Comparing the above-described experimental data and results of theoretical calculations, values of parameters of the model may be determined and the physical image of the process of RID formation may be refined [57]. The method suggested also attracts the attention of investigators, because it enables the possibility of appearance of various RID to be estimated during a relatively short time and the optimal regime to be selected [57]. Based on mathematical modeling, ref. [66] studies the technology of preparation of ‘ideal’ non-aberration spherical lenses with axial gradient of the refractive index. Firstly, the performance is modeled of the exchange diffusion of monomers with various molecular refractions through a plane surface into a semi-infinite formed medium with further polymerization. If a spherical lens is cut off from the sample obtained, then refractive index will change on the segment surface. The problem is raised to clarify such RID profile, when all beams parallel to the optical axis falling on the lens at different heights of the plane surface are focused to a single point. Application of mathematical calculations (RID demanded for preparation of non-aberrational spherical lens and a sequence of Fick’s RID profiles, i.e. the ones occurring due to diffusion, for various moments of time) to a particular DAIP-MMA polymeric system indicated that by cutting off a spherical lens with radius R from a semi-infinite sample in such a manner that the sphere center was located at a distance of 0.95R (R is the sphere radius) from the plane surface border, the optical characteristics of the gradient lens can be improved compared with a homogeneous lens with the same geometrical dimensions. Modeling the ray path in such an inhomogeneous medium shows that longitudinal spherical aberration in the gradient lens will be by some 7.6 times lower than that of an analogous homogeneous lens from PMMA [87]. The method of refractometric control of the diffusate gradient in a cylinder selfoc has been worked out [102]. For the matrix (M), organosilicon dimethacrylate or DEGBAC is taken, and 4FMA is used as diffusate (D) (Table 3). Calculation of n and spatial distribution of the diffusate concentration with respect to the selfoc radius is based on the supposition that the selfoc consists of a copolymer network of substances M and D only. Such a network consisting of monomeric units M and D can be represented (in the calculated model system) by a random alternation of dimers M-M and M-D as repeated units of copolymers. For such polymer network:

(1 − mM-D )A + mM-D RM-D 3 R n = 1 + ⋅ M -M , 2 (1 − mM-D )VM-M + mM-DVM-D

(14a)

22


where R and V are refraction and van der Waals volumes of dimers; mM-M and mM-D are their molar proportions, with mM-M = 1 – mM-D. From (14a), it follows that:

mM - D =

3( RM-M

3RM-M − 2(n − 1)VM-M , − RM-D ) − 2(n − 1)(VM-M − VM-D )

(14b)

and the diffusate concentration MD(ini) (mol%) can be calculated by the formula: MD = 100mM-D/2 – 50mM-D at any point of radius (r) of the selfoc, if the value of n(r) at this point is known (in the case of a cylinder selfoc, n(r) is determined by the interference method: see Section 2). It is shown that dependencies of mD and nD can be represented graphically and are of the symbath type (Figure 8) [102].

1.51

70 2

1.50 1

1.49 nD

60 50

1.48

40

1.47

30

1.46

20

1.45

10

1.44

0

0.4

0.8

r

1.2

1.6

mD

0 2.0 mm 2.4

Figure 8. Dependence of the refractive index nD (1) and the molar proportion of diffusate 4FMA (2) on radius of selfoc based on the matrix from DEGBAC.

Creation of estimation criteria of optical quality of selfocs is quite important for production of polymeric gradient elements. The fullest information on optical quality of gradient elements can be obtained if certain of their characteristics are known, such as numerical aperture, NA, the focal length, f ( −

1 = n0α sin αL , where n0 is the refractive f

index on the axis; α is the distribution constant; L is the selfoc length), losses for aberration, focusing focal spot, etc. However, for preliminary estimation of optical properties of selfocs in the first approximation, determination of the RID type with respect to the sample radius is quite sufficient. Several approaches to this problem have been considered. One of them [56] suggests selecting a suitable target function for optimization of the technological process.


23

This function can be, for example, the mean square deviation of the real or theoretical diffusion curve of distribution from the perfectly focusing function: n −1

σ=

∑ [n0Sech (αr1 ) − n(ri )]

2

i =0

N

,

(15)

where n(ri) are values of the refractive index at the radius point ri obtained from measurements or by solving the Fick diffusion equation for a definite moment of time; N is the number of partition points on the distribution curve (the number of experimental points). It is the authors’ point of view [29, 56] that this approach allows a successful study of the influence of various factors, including conditions of copolymerization diffusion, change of limiting conditions, etc., on the degree to which the perfectly focusing distribution of the refractive index can be approached. It is proved that this method enables optimization of conditions to obtain selfocs designed for any particular application. However, it may be that focusing properties of real gradient elements are different at the same values of σ. That is why the authors assume, as the most correct criterion of estimation of their optical quality, the method of numerical calculation of the ray path in the selfoc according to the Euler equation at the given distribution of the refractive index:

  1 n = n0 1 − α 2 r 2    2

(16)

In n = f(r2) coordinates, the dependence is linear. Distortion of the linear dependence in various points of the selfoc radius corresponds to a deviation nD of distribution from the perfectly focusing one, which is observed in real selfocs (Figure 9). This method of RID estimation [72] does not differ significantly from the aboveconsidered method of determination of the mean square deviation (σ) of the refractive index distribution from the parabolic law and is characterized by almost the same drawbacks. In practice, with regard to the above-said, the focal length, focal spot of focusing, numerical aperture, efficiency of radiation input into the guide with the help of selfoc, etc. are assumed [29, 72] as more accurate criteria for estimating optical properties of selfocs. These criteria are also the target functions in calculations of σ. There are other approaches to estimation of optical properties of selfocs, in particular, determination of size of the focusing focal spot and the value of longitudinal spherical aberration [72]. It is proved experimentally and by calculations that minimal values of the focusing focal spot and longitudinal spherical aberration correspond to parabolic (ideally focusing) distribution of the refractive index [72]. The reason for RID deviation from the ideally focusing one in real selfocs (in prepared samples) may be the occurrence of a series of technological difficulties in the production process, for example [29]: decreased diffusion rate of the penetrating agent into gel-polymeric matrix compared with the rate of homopolymerization, with respect to which this process achieves diffusion, and the glass transition of the sample appears prematurely without reaching the parabolic RID; bad miscibility of monomers; improper selection or conduction

24


of fixing during diffusion exchange of RID; distortion of the technological process of exchange diffusion; deterioration of the geometric shape of the samples during diffusion or at fixing of the RID formed (for example, bending of the sample or formation of spiral-like ‘cracks’ on their surfaces; etc [29]. 1.57

nD

1.55

1.53 3

1.51 1

1.49

2

1.47 1.45 1.43

0

1

2

3

4

2

5

r , mm

2

6

Figure 9. Dependence of the refractive index on the square of the radius of a selfoc: 1 – the ideal profile; 2, 3 – real profiles.

Based on the analysis of the known works by G.O. Karapetyan and V.I. Kosyakov, valuable conclusions [29, 30] on modernization of processes of preparation of high quality selfocs were made: 1. Processes of diffusion formation of the refractive index profile and its fixing by consequent copolymerization, with continuation of diffusion during fixing of the RID, but at already changed limiting conditions need not always be separated. This, with possible diffusate evaporation from the selfoc surface, will inevitably cause a transformation of the RID formed during diffusion. 2. The process of RID fixing should be performed during a shorter time than is required for a noticeable change of the distribution reached (for example, decreasing the fixing temperature of RID and, consequently, decelerating diffusion, or performing prepolymerization by radiation initiation); 3. The RID profile can be controlled by conduction of simultaneous diffusion of several monomers copolymerized with the matrix, which possess different atomic refractions and molecular masses; 4. To increase stability of optical characteristics of selfocs, it is necessary to select conditions that would provide possibly high conversion of comonomers of the stage of fixing of the RID formed in order to eliminate its change with time due to elimination of a residual (unreacted) monomer – diffusate from the matrix.


25

Elaboration of theoretical grounds of selection of the gradient composition and criteria of estimation of optical properties of selfocs, as well as the inclusion of results of physicochemical investigations of the diffusion formation of RID performed, allowed transition from relatively empirical approaches to creation of GRIN-elements to more scientifically established ones. This provided refinement of the diffusion technology and techniques of selfoc preparation with the required excellent properties [103 - 107]. This provided the possibility for creation of optical systems of the modern generation, in particular, those used in computing and copying devices, as well as fast-operating hardware for information processing and transmission, etc. Examples of technologies used to obtain GRIN-elements can be found in refs. [41, 103 – 108]. Among these works, special mention should be made of refs. [41, 105, 108]. Reference [105] describes the closed extrusive technology (see Section 6) for obtaining polymeric gradient materials possessing square distribution of the refractive index. It is indicated that geometry of the master form (matrix) and length of the diffusion zone affect significantly the type of refractive index distribution in the gradient material (fiber). The key parameters controlling the RID formed during this process are composition of the reaction mixture, length and temperature of the diffusion zone, etc. Mutual diffusion in a closed system of monomers with different refractive indices (for example, MMA and benzyl methacrylate (BzMA)) is strengthened due to elongation of the diffusion zone that leads to a growth of homogeneous distribution of monomers in PGs. Obtaining PGs (gradient elements) with high values of ∆n allows their application to modern facsimile (fax) equipment [105]. A method has been described to obtain polymeric rod lenses with a gradient of refractive index (Figure 10) [41] without bubbles and voids using copolymerization (swollen-gel polymerization) of methyl methacrylate (MMA, nD = 1.490) with benzyl methacrylate (BzMA, nD = 1.1570) in a polymethacrylate pipe, where diffusion of the monomer with higher refractive index into swollen PMMA-MMA gel takes place.

∆n

0.01

0.005

0.000

0.0

3.5 r, mm 7.0

Figure 10. Profile of refractive index for polymeric rod lens based on MMA-BzMA (4:1)

26


The diameter of the polymeric rod obtained is from 14.5 to 15 mm (diameter of PMMA pipe) after copolymerization. Concentration of MMA increases from the pipe wall to its center, and a gradient of refraction index is formed. In this case, defocusing selfocs are obtained. By the same methods, focusing selfocs can also be obtained [41]. Very large radial graded-index polymer materials were first prepared by two methods [106]: 1. The so-called ‘curved mold method’; 2. The method of diffusional copolymerization. These methods allowed production of GRIN-materials with a diameter of 70 mm and a difference of refractive index ∆ ≥ 0.02 [106]. The first method enables us to prepare large GRIN-materials with various values of the difference in refractive index and RID profile. Larger GRIN-materials may possibly be obtained by the same diffusional method, by modernization of the existing method of preparing polymeric materials with a gradient of refractive index. The authors suggest that GRIN-materials produced according to their method can be successfully applied to production of thin ophthalmic lenses without specific aberrations or multifocusing characteristics [106]. Y. Koike et al. [107, 108] obtained commercial high-bandwidth graded-index polymer optical fiber with quadratic RID profile and capable of transmitting optical signals at a high rate in short-wave multimedia (multimodal, multichannel) translation networks by the method of interphase gel-polymerization [109], which could not be performed using polymer optical fibers possessing a stepwise RID profile [108]. It is common knowledge that the refractive index (RI) in cylinder-shaped gradient elements does not change in the axial direction. At the same time, of special interest are gradient lenses possessing a RI gradient, in both the radial and axial directions [60]. Such lenses can be obtained by substance diffusion into the plate covered by a mask impermeable for diffusate [65]. The method seems to have potential for successful development of the planar technology in the microelectronics industry [107]. An analytical expression describing the ideally focusing RID in such lenses has not yet been derived. That is why deduction of regularities of RID profile formation control is possible on the basis of experimental data only [110]. Reference [60] describes the influence of various factors on RID formation in polymeric planar structures (forepolymeric matrix from DAIP – maneic anhydride (MA) – penetrant – MMA; MA is injected into the matrix composition for decreasing cross-link frequency, which reduces mechanical stresses occurring at diffusion). It is found by studying thin slices on the Mach-Zender interferometer by the method from ref. [30] that providing for high jump of RI and elimination of RID profile deformation demand combination of profile diffusional formation with its fixing. At short diffusion duration and low temperature, at which polymerization rate is low, deep permeation of the penetrating agent into the matrix and decrease of RI at the mask surface and the area below it are observed. The authors associate this fact with an increase of the diffusion coefficient with temperature up to the fixing point. As a result, the penetrating agent does not actively diffuse deep into the matrix until the sample is polymerized completely.


27

Reduction of the diffusion temperature from 80°C to 60°C decreases RI difference at the same level of penetrating agent permeation. Further on, as expected, it is shown that the increase of the diffusion tank volume leads to a decrease of the absolute RI value in the area bordering the diffusion tank. The relation of the initiator concentration (benzoyl peroxide – BP) in the matrix and penetrating agent defines the moment of diffusion stage transition into the stage of diffusion profile fixing and, finally, causes a significant influence on the RID type as a result. At low concentration of initiator in the matrix, the absolute RI value of the matrix – diffusion tank interface decreases in the border area due to penetrating of greater amount of MMA monomer into it, whereas in the presence of the initiator in the diffusion tank polymerization of the whole volume of the tank is accelerated, the amount of MMA molecules capable of diffusing deep into the matrix decreases, and the absolute RI value increases in the whole gradient layer [60]. Planar lenses obtained by the method described are of long focal length. Despite the large jump of RI (∆n ≈ 0.03), the numerical aperture NA ≤ 0.01 due to a short ray path in the gradient layer, and diameter of the focal spot (df ≈ 80 µm) are close to the theoretical diffraction limit for the given numerical aperture [60]. Micro-lenses are widely used in electro-optical and information transmitting systems [111 - 117]. To create lenses of this type, diffusion of a monomer is performed to a forepolymeric plane support through a mask. Therewith, the refractive index gradient is formed in both radial and axial directions. The mask shape is determined by the difference of the support refractive index – a sequence of disks is used as mask, around which diffusion of the monomer proceeds. By this method, a system of plane microlenses was prepared, in which DAIP monomer was used as the support, with MMA as the diffusate. (The focal point of the microlenses was 27 µm, the focal length was 25 mm) [113]. Attention is drawn to planar polymer lenses with diameter d ≥ 3 – 6 mm, obtained by diffusion exchange. The expediency of combining diffusion and depolymerization stages for the purpose of approximation of the RID obtained to the ideally focusing one is shown [117].

3.2. Copolymerization Method One of the traditional methods of obtaining GRIN-elements is the method of copolymerization of monomers [11, 42, 43, 118 - 123]. During usual radical-initiated copolymerization of a mixture of two or three monomers, distribution of the composition of the copolymers obtained is dependent upon the temperature gradient and different reactivity of monomers with different refractive indices. Polymeric material with a gradient of refractive index along the axis (for preparing lenses with a spherical entering surface and planar exit surface) is prepared [11] by the copolymerization method, based first on the difference in reactivity of monomers. The method of gel-copolymerization on the interface for obtaining gradient elements (polymeric media) of various geometrical forms has been described [43]: radical-symmetric, axial-symmetric, spherical gradient polymeric media, as well as gradient wavebeam guide (PG).

28


Samples obtained by copolymerization of monomer mixtures photoinitiates by UVradiation and, to fulfill certain requirements (for example, n1 ≠ n2), are also characterized by gradient distribution of the component composition. Gradient-forming factors are different activities of monomers and gradient of the initiation field [11]. References [119, 120, 122, 123] give details of a method of preparing light- focusing polymeric elements by photopolymerization of monomers possessing different relative activities and different refractive indices. The mixture of monomers existing in a soldered glassy pipe is radiated by UV-light and rotated simultaneously. A couple of monomers M1 and M2, selected for creation of selfocs by copolymerization, must fit the following conditions: in the refractive index n1 of the homopolymer M1 is smaller than the refractive index n2 of the homopolymer M2, then relative activity r1 of the monomer M1 must be greater than 1, and relative activity r2 of the monomer M2 must be smaller than 1. In this case, the copolymer formed initially at the pipe surface is more saturated by the component with higher relative activity. Concentration of molecules with lower relative activity grows in moving towards the rod axis. As a result, the rod obtained possesses a radial distribution of the polymer composition and, consequently, of the refractive index. Preparations can be obtained by copolymerization of two monomers, such as methyl methacrylate – vinylbenzoate, for which r1 > r2 and nD1 < nD2. Unfortunately, such a binary system does not produce the desired distribution of the refractive index, especially near the preparation center [121]. To obtain a selfoc possessing a radial jump of the refractive index, copolymerization is performed in a quartz ampoule with d = 112 ± 3 mm rotating around its axis in the presence 20

of a photoinitiator. Selfocs based on N-vinylphthalimide (VPI, nD = 1.62) and methacrylic 20

acid (MAC) ethers ( nD = 1.48 – 1.49) copolymers have been obtained [121]. It has been found that the refractive index jump increases with exposure time and molar proportion of Nvinylphthalimide. Dependence of ∆n on r with increase of the initiator concentration and VPI content in the initial mixture of monomers was also found [43, 121]. In the case of three monomers, they must satisfy the following conditions: {r12(M1/M2)m + 1}/{(M1/M2)m + r21} > 1.1; {r13(M1/M3)m + 1}/{(M1/M3)m + r31} > 1.1; {r23(M2/M3)m + 1}/{(M2/M3)m + r32} > 1.1,

(17)

where (M1/M2)m, (M1/M3)m, and (M2/M3)m are molar concentrations, and r12, r13, r21, r23, r31, r32 are constants of copolymerization characterizing their relative activities. Due to difference in reactivities of the monomers in the three-component system, for example, methyl methacrylate – acrylonitrile – vinylbenzoate [11, 19], at the initial stage of polymerization, the polymerizate contains a greater amount of PMMA chains (nD = 1.49), then PAN (polyacrylonitrile) concentration increases (nD = 1.52), and at the end of copolymerization vinylbenzoate homopolymer (PVB) is mostly formed (nD = 1.58). Polymerization begins on the walls of a cylinder vessel rotated around its axis and UV-radiated, and propagates to the center of the reactor. A similar result is obtained for the triple methyl methacrylate – vinyl acetate – N-vinylcarbazole system [11, 43].


29

Selfocs of various shapes – cylinder, branched, spherical – can be obtained by varying the form of the polymeric support. If several holes are made in the polymeric form, a series of samples can be prepared simultaneously [11, 19]. The advantage of the copolymerization method is that it provides the possibility of drawing gradient fibers directly from the rod, because substances forming linear thermoplastic copolymer can be used as initial monomers; so RID in PG differs insignificantly from RID in the initial preparation. The drawback of this method is the difficulty of controlling thedistribution of the refractive index in the optical element. However, in most cases, distribution of the refractive index in samples obtained by photocopolymerization of monomers is rather far from the ideally focusing one, and optical characteristics of such selfocs only poorly fit the demands imposed. The method of photocopolymerization for preparation of GRIN-elements does not produce stable characteristics due to a significant amount of residual monomer present in these elements as the result of copolymerization by UV-radiation.

3.3. Method of Gravitational Separation An original method of obtaining a gradient guide with the help of centrifugation has been suggested [52]. Monomers are selected such that the monomer M1, initially at polymerization, gives a homopolymer with low refractive index. Subsequently, the second monomer M2 gives a homopolymer with high refractive index. Placed into the centrifuge together with the initiator, the monomer and the polymer formed distribute under the influence of the centrifugal force as follows: most of the polymer moves to the periphery, whereas the monomer locates in the center of the volume. This gives the distribution of the future composite. At the centrifuge output nozzle, the temperature is kept at the level of 80°C, and then the monomer is polymerized in a special reactor. In another case, a mixture of the monomer and the polymer is centrifuged and cured up completely at the output of the centrifuge. Unfortunately, this method has not yet been widely applied.

3.4. Method of Dipolephoresis The process of natural diffusion of the cover monomer into the core polymer may be intensified by dipolephoresis (Figure 11), which is the induced diffusion of polymer molecules possessing a dipole moment under the effect of a significantly inhomogeneous electrostatic field of required configuration (dipoles in electrostatic field do not only orient, but also move in the direction opposite to the field gradient). If the direction of motion of polar molecules of the monomer under influence of the inducing field coincides with the diffusion direction proceeding under the effect of concentration forces, then velocities of motion of polar molecules are additive, diffusion is intensified, and the diffusion coefficient in this case depends not only on temperature and viscosity of the material, in which diffusion is performed, but also on the dipole momentum value of the diffusate molecules.

30


References [63, 64] indicate the principle of estimation of diffusion of polar particles under the influence of an inducing force of inhomogeneous electric field using a mathematical estimation of the type for diffusion waves of dipole particle flow in an axially symmetric field. Distribution of the molecules of diffusers in the polymeric cylinder may be described as follows [63]: C(t, r) ≈ const⋅r2.

(18)

Using the formula by Ber–Gladstone–Dale for the refractive index

n = 1 + const ⋅ N ,

(19)

where N is the density of particles of the substance studied. For the refractive index of a polymeric cylinder (with radius R) the following expressions are obtained: before diffusion:

n0 = 1 + (const)0N,

and during diffusion:

n1 = 1 + (const)1C(r1t).

(20)

Using the Landolf principle of refraction additivity, the equation for refractive index of a cylinder rod after dipolephoresis, reaching the core, was obtained [63]: n(r, t) ≈ n0 + (n1 – n0)ξ(t1)r2, where

ξ (t1 ) =

(21)

C0 ϕ (t , r ) . NR 2

Figure 11. Interferogram of fine cross-sections of a cylinder sample of polymer after introduction of a diffusate under the effect of inhomogeneous electrostatic field [63]


31

This equation indicates that the refractive index, obtained after dipolephoresis of the cylinder, grows or decreases towards the center depending on relative values of the refractive index of matrix and diffusate as the square of the radius. It should also be noted that dipolephoresis may be used as the basis for elaboration of a method of analysis and separation of a substance from mixtures (if the substances comprising the mixture components differ by values of dipole moments like the difference in diffusion velocity of molecules possessing different dipolar moments in a medium of the gel type in a significantly inhomogeneous electrostatic field).

3.5. Method of Gradient Modification of Polymeric Materials A light conducting polymeric material with ∆nD = 11% may be obtained by impregnation of the polymer with ‘microcavities’ by a resin (the refractive index of which is much higher than that of the matrix polymer) [44], and with further thermal curing. The initial polymer with ‘microcavities’ is obtained by UV-radiation of a mixture of the polymer and the monomer (for example, PMMA-MMA, i.e. components identical in chemical structure) through a hollow gage. It has been shown that repeated distillation heating of a gel-polymer based on vinyl monomers (for example, vinyl acetate–vinylidene cyanide) at 160–180°C alternating with rapid cooling down leads to an increase of the refractive index by 0.33–0.70% [45]. A gradient polymeric material may also be obtained by a specific treatment of a polymeric beam waveguide surface. Thus, for example, processing of an optical fiber consisting of the core and the cover (PMMA), by a low-molecular alcohol leads to a decrease of refractive index by diffusion of the low-molecular compound and breach of the cover structure [2]. Fluorination of surfaces of various polymeric materials by gaseous fluorine was shown to produce a significant increase of their permeability, for example, in the case of polyvinylchloride, polypropylene, polyethylene, and polyacrylonitrile in the range of 0.3 – 2.6 µm by 4% [51]. Variations of the present method are discussed below in Sect. 4.

4. CONTROLLED GRADIENT FORMATION 4.1. Main Principles of Formation of Macrosurface GRIN-Elements [124] As mentioned above, the most widespread methods of evaluation of the refractive index gradient – the two-stage (diffusion, gradual) copolymerization [33, 34, 55] and photopolymerization [119, 120], during which the gradient is formed due to the diffusion laws – produce material with parabolic change of the refractive index profile only. Moreover, diffusion, being a limiting process, limits the diameter of the objects produced up to ~20 mm. Methods are known, however, in which these restrictions are eliminated, for example, methods of obtaining of a planar lens [106, 125] and a long process for forming optical fibers [124, 126], in which the refractive index over the object radius (not limited by diffusion, in

32


this case) may be changed according to the previously given parabolic law, but not only limited to this case. The general significant drawback of all the above methods is their limiting of the selection of applied polymers, because they cannot produce selfocs from polycondensed polymers, or polymerizational monomers, synthesized from gaseous ones. Let us note some further restrictions of the traditional preparation methods. Figure 12 shows a block-scheme to obtain a gradient polymeric lens by the method of vapor/liquidphase diffusion [11]. According to this method, in order to decrease spherical aberrations, the whole spherical surface of the gel-polymeric object with a planar output surface (Figure 12) is processed simultaneously by a monomer–diffusate. As a result, a homogeneous diffusion layer is formed (Figure 12). Hence, the approach to solution of correction of such optical distortions is restricted from the point of view of the need to regulate the thickness of the gradient layer with respect to the lens radius.

Figure 12. Block-scheme of the diffusion technique method.

The technique of creation of a diffusion surface, i.e. surfaces with the given spherical gradient of the diffusion layer, includes additional operations. Such an approach was created for optical inorganic glasses [12]. First, the whole surface of the object (Figure 13, a) is processed simultaneously by a diffusate (by melting of a mixture of metals). Alkaline metals from the glass are exchanged with metal ions from the melt forming a homogeneous diffusion layer with gradual growth of n on the surface of the glassy body (Figure 13, b, c). Further on in the process, a layer of required thickness is partially removed (polished off) from the diffusion surface, and an optical element with non-homogeneous refracting surface is obtained (Figure 13, d). This widens significantly the ability to correct optical distortions and provides the possibility of producing planar lenses as well (Figure 13, e).


33

In this connection, it has been suggested that non-traditional solution of creation of gradient optical media – the method of controlled gradient formation – may provide the possibility of obtaining thin-layer materials based on polymerization and polycondensation polymers with great surface area and preliminarily given radial (axial) distribution of the refractive index, with no dependence on the aggregate state and chemical nature of appropriate monomers that determine the specificity of the polymer synthesis [61, 127 - 133].

b

a

c

d

e

Figure 13. Obtaining gradient optical elements by the exchange diffusion method: initial gelpolymeric/glassy lens (a), gel-polymeric (b) and glassy (c) lenses with homogeneous gradient layer; glassy lens with inhomogeneous gradient layer (d); planar glassy lens of positive spherical type (e).

The controlled gradient formation, realized in various technological solutions, is based on the controlled gradient chemical modification, in particular, the controlled gradient heterogeneous (solid-phase) polymer-analogous transformation and the already mentioned two-stage (gradual) copolymerization [33, 38, 55]. The main principles of the controlled gradient formation concludes in an optically transparent polymeric or gel-polymeric film/plate that serves as a matrix–carrier of the refractive index gradient. Selection of a gradient carrier is not limited by the conditions of its synthesis which, in turn, are determined by the aggregate state and chemical nature of the appropriate monomers. That is why the gradient carrier may be a polymerization or polycondensation polymer, produced by any method. The gradient carrier must possess the property that it can be subjected to a polymeranalogous transformation or two-stage (gradual) copolymerization, which results in formation of an optically transparent polymer with another (greater or lower) value of the refractive index. Therewith, the initial and newly formed polymers must not dissolve under conditions of polymer-analogous transformation. Moreover, the gradient carrier must be stable in retaining the shape of the initial sample (films, plates) and physicomechanical properties, as well as providing stable new properties. Chemical modification of the gradient carrier by polymer-analogous transformation/twostage copolymerization is performed by its interaction with the gradient former, the chemically active medium/monomer–diffusant, which may be an individual liquid substance or liquid solution of a gas, liquid or solid. In some cases, the gradient former may exist in the

34


gaseous state [134]. A liquid, inert in relation to the chemically active medium, as well as to the initial and newly formed polymer is applied in solving some specific problems. To resolve the onset problem, the inert liquid must possess higher or lower density than that of the gradient former. A constructive scheme for performing controlled gradient formation is known (see below), the application of which makes unnecessary the selection of a definite ratio of the gradient former density and inert liquid [135]. A magnetic liquid may be used as the inert liquid [129, 130]. In this case, its density is of no concern. Of special importance for realization of the controlled gradient formation is an understanding of the reaction mechanism (polymer-analogous transformation /diffusion copolymerization) so that the duration of the process can be determined in order to control the reaction product. Specifically, the refractive index change with time is investigated, i.e. the function n = f(τ) is determined, where n is the refractive index, τ is the duration of chemical reaction/diffusion. Moreover, reaction parameters must be selected that will produce more or less complete transformations within the technologically available time, i.e. when the maximal value of τ in the function n = f(τ), necessary for finishing the process, does not exceed a limit, set by technological, economic or any other reasons. At the same time, the minimal value of τ, at which transformation of the refractive index may be detected (i.e. the induction period of the transformation), is several minutes long. The starting quantitative characteristics for determining the regime of polymeranalogous/diffusion copolymerization are: -

-

Values of refractive indices n of the initial polymer (in polymer-analogous transformation) or homopolymer (the product of prepolymerization of a gelpolymer); Value of the refractive index of the process product (polymer-analogous transformation/diffusion copolymerization); Previously given profile of the radial (axial) distribution of the refractive index n, i.e the function n = f(R), where R is the radius (length) of the sample; Dependence of change of the refractive index n on the process duration, determined experimentally, i.e. the function n = f(τ) with the help of which the given dependence n = f(R) transforms to the form of n = f(R).

From the algorithm formulated, polymeric systems of given size and with given profile of distribution of the refractive index may be created. Actually, different contact duration of the gradient former (chemically active liquid/monomer-diffusate) and gradient carrier in the points of some line, for example, the straight one on the surface of the latter, will provide for an adequate change of the refractive index due to structural changes, the depth of which depends on the process duration. Interpolation of values of the refractive index, stated for discrete periods of time, produces a continuous sequence of these changes in the continuum scale. Hence, in the method of controlled gradient formation, formation of the given distribution of the refractive index is produced by gradient of the process duration on the surface of the gradient carrier. That is why, accurate regulation of this parameter in the given directions on the sample surface represents an important technical requirement in the method discussed.


35

Creation of the radial distribution of the refractive index is achieved by controlled change of the diaphragm mask size in the chemical reaction/diffusion zone on the surface of polymeric/gel-polymeric film (plate). It should be specially noted that when radial distribution of the refractive index is formed from one and the same process (polymer-analogous transformation /diffusion copolymerization), two principally different results may be achieved – obtaining of a polymeric medium with the properties of a convex or concave plate lens. Actually, if n1 (the initial polymer/homopolymer – the product of prepolymerization of gel-polymer) is greater than n2 (the product of polymer-analogous transformation/diffusion copolymerization), then decrease of the process duration with the radius of the polymeric/gelpolymeric film/plate from its periphery to center gives a medium with properties of a convex lens and, vice versa, at the increase of the process duration with sample radius from periphery to center, a medium possessing properties of a concave lens may be obtained. The opposite results are obtained, when polymer-analogous transformation /diffusion copolymerization is accompanied by an increase of refractive index, i.e. when n1 > n2. In this case, decrease of the process duration by the sample radius from the periphery to the center produces a medium with properties of a concave lens, and increase gives a medium with properties of a convex lens. In this process, the results mentioned are achieved under conditions of injection of an active medium/diffusate and an inert liquid into the reactor under different regimens.

4.2. Equipment To set the required duration of the process (polymer-analogous transformation/ diffusion copolymerization) in given directions on the surface of the gradient carrier a device is used that allows use of a diaphragm mask in the contact zone of the gradient former with the gradient carrier. Several technical solutions of application of such devices have been suggested. a) Use of the diaphragm mask in the centrifugal field [127, 128, 131] The contact zone between the gradient former and the gradient carrier may be variably masked using a diaphragm in the centrifugal field, created in the cylinder reactor rotating vertically around its axis. The main features of such a device (diametrical cross-section) are shown in Figure 14, a. The device consists of the cylinder reactor (4) possessing a round groove (11). A polymeric sample (18) (film/plate) is located in the reactor (4) between clips (3, 20). Round clips (3, 20) possess holes (2, 10, 14, 21). The face of the reactor (4) and the lid (19) possess juts–rolls (11, 16), in which pipes (6, 15) for input/output of liquid (a gradient former or an inert liquid) are located. Ends of the pipes (6, 15) in the reactor (4) are curved perpendicularly, and the external circles are connected to the pipe (12), which is connected, in its turn, with the gradient former/inter liquid tank. Introduced into the pipes (6, 15) are pipes (5, 17) for input/output of the gradient former/inert liquid. External ends of the pipes (5, 17) are connected with the pipe (1) which, in turn, is connected with the gradient former/inert liquid tank. If required, the pipes (5, 17) may serve as air eliminators. The reactor (4) possesses a hole with a plug (13) for liquid output. The rod (7) is connected with an electric motor (9) by a V-belt (8).

36

N. G. Lekishvili, L. I. Nadareishvili and G. E. Zaikov 2

21 20

1

3 4 5 6

4

19 18 17

7 8

22 27 23 26 24 25

16 15

9 13 14

10 11 12

b

a

Figure 14. The device for controlled radial-gradient polymer-analogous transfor-mation and exchange diffusion in the centrifugal field (diametrical cross-section). See text for details.

In some cases, especially at the application of gel-polymeric matrix as the gradient carrier, tight fixing of the latter in the rotating reactor must be provided. For this purpose, the gel-polymeric sample (23) (Figure 14, b) is placed in a circle framework (26) which, in turn, is fixed to immovable details (22, 24, 25, 27) in the reactor (4).

1 15

2

14 13 12

3 4 5 6 7

S

N

11 10 9 N

S

8 ~U Figure 15. The device for controlled radial-gradient polymer-analogous transfor-mation and exchange diffusion in magnetic field (diametrical cross-section). See text for details.

b) Use of the diaphragm mask in a magnetic field [129, 130] The contact zone between the gradient former and the gradient carrier may be gradually increased by use of a diaphragm mask in a device, the main features of which (the diametrical cross-section) are shown in Figure 15. The operation principle of the device is based on simultaneous application of magnetic field and ferromagnetic liquid as the inert one. The


37

device consists of a cylinder reactor (12), prepared from non-magnetic material. A round polymeric/gel-polymeric sample (11) (film, plate) is located between round clips (7, 9). The reactor (12) possesses a round groove (14) and a cover (4). Round clips possess holes (3, 6, 10, 13). Pipes (1, 2, 8) for input/output of the gradient former/ ferromagnetic liquid are connected to the reactor. If required, the same pipes may serve as air supplies. The reactor (12) is located in the circle electromagnet (5) or between poles of a horseshoe magnet (5). c) Use of the diaphragm mask between horizontal surfaces [135] In the solution suggested, the possibility of regulating the contact zone between the gradient former and the gradient carrier is based on the effect of the surface tension, under the effect of which the “inert medium/gradient former concluded in the inert medium” system not wetting the gradient carrier obtains the shape of a flattened globe – a circle, thus providing for a vertical contact between the gradient former and the inert medium. The main features of the device (diametrical cross-section) are shown in Figure 16. It consists of two parallel plates (2, 5), between which, in parallel, a polymeric/gel-polymeric sample (film/plate) (3) is placed, fixed by the circle clip (4). The plates (2, 5) and the clip (4) are connected with the control clip (6), by which they are set parallel. The distance between the sample (3) and the plates (2, 5) is selected so that if the gradient former and the inert liquid locate between them simultaneously, they contact each other in the vertical plane only. A pipe (8), connected with the curved pipe (15), is introduced to the center of the plate (5). In its turn, the pipe (15) is connected to the pipe (17) through a rubber pipe (16). Another end of the pipe (17) is introduced into the center of the plate (2). The pipe (8), through the pipe (14) introduced to the glass (7) bottom and the rubber pipe (13) is connected with the T-joint (12) connected, in turn, with tanks for gradient former/inert liquid (10, 11). The device possesses the cover (1) and branch/inlet pipes (9, 18). Plates (2, 5) and the sample (3) are disposed horizontally.

1

18

2 17

3 4 5 6 7 8

16 15 14 13

9

12 10

11

Figure 16. The device for controlled radial-gradient polymer-analogous transfor-mation and horizontal exchange diffusion (diametrical cross-section). See text for details.

38


4.3. Theoretical Requirements for Controlled Gradient Formation [127 – 131, 133 - 135] The theoretical principles underlying the controlled gradient formation are clear and independent of the diaphragm mask used in the contact zone between the gradient former and the gradient carrier. First, these reasons will be discussed in relation to use of a diaphragm mask in the centrifugal field (Figure 14, a, b). In the reactor, diaphragm masking the transformation zone may be performed by using a gradient former only or a gradient former and an inert liquid simultaneously. In the second case, the sequence of their introduction into the apparatus is determined by the nature of the problem, namely, which polymeric medium, possessing properties of a convex or concave lens, is required. When a particular problem is to be solved, the relation of densities of the gradient former and the inert liquid, the relation of values of refractive indices of the initial polymer/prepolymerized gel-polymer and the product of polymer-analogous/diffusion copolymerization should be taken into account, and the character of variation (increase or reduction) of the gradient formation duration with respect to sample radius from its periphery to center should be determined. The reactor (4) (Figure 14, a) may be filled first both by the gradient former and the inert liquid. The first liquid, with no regard to its relative density, is always injected at the periphery of the rotating reactor (4) (via pipes (12, 6,15)). That is why, an air supply is always situated in the center of the reactor (pipes (5, 17, 1)). The second liquid, if it is heavier than the first, is injected into the periphery pipes (12, 6, 15)). In this case, the first liquid is pressed off through the center (pipes (5, 17, 1)). If the second liquid is lighter that the first one, it is injected through the center, and the first liquid is ejected from the reactor periphery. The character of the duration of the change of the gradient formation over the sample radius from periphery to center is determined from the ratio of densities of the gradient former and the inert liquid. When the density of the gradient former is higher (lower) than that of the inert liquid, despite the sequence of their inlet, duration of the gradient formation decreases (increases) from periphery to center. Analogous general rules may also be formulated for processes of the gradient formation in a magnetic field. The reactor (12) (Figure 15) is first filled by a gradient former or a ferromagnetic liquid. Whatever the inlet order of liquids to the reactor when a horseshoeshaped magnet is used, the duration of the gradient formation decreases with sample radius from periphery to center; in the case of a circular magnet, it increases. When the reactor is located in the circular magnet, the ferromagnetic liquid is injected into and ejected from the peripheral (pipe (2)). If the reactor is placed between the poles of the horseshoe-shaped magnet, the ferromagnetic liquid is injected into and ejected from the center (pipe (1)). In this connection, in the case of the circular magnet, the gradient former is ejected from the center (pipe (1)), and in the case of the horseshoe-shaped magnet, it is from the peripheral pipe (pipe (2)). The reactor is always preliminarily filled by the gradient former from the periphery (pipe 8). When the ferromagnetic liquid must be pressed off by the gradient former, in the case of a horseshoe-shaped magnet, the latter is injected into the periphery (pipe 8), and in the case of a circular one, into the center (pipe 1). Table 4 shows all the above-discussed variants of controlled gradient formation, conditions of their performance, and optical properties of plane-parallel optical systems obtained.

Table 4. Creation of polymeric media with given profile of radial distribution of the refractive index, by methods of controlled heterogeneous polymer-analogous transformation and controlled two-stage exchange diffusion, in centrifugal and magnetic fields Density, ρ

Change of the process duration (polymeranalogous transformation/exc hange diffusion by radius of polymeric/ gel-polymeric sample from periphery to center

Decreases

ρ (GF) > ρ(IL)

No.

1 2 3 4

5 Increases

ρ (GF) < ρ(IL)

6

7 8 9

Field

Centrifugal (sample rotates) Centrifugal (sample is immovable) Centrifugal (sample is immovable) Magnetic (horseshoeshape magnet)

Preliminary filling

IL

Sequence of inlet of gradient former (GF)/inert liquid (IL) Regime of dosed Change of GF/IL inlet refractive index with sample radius from periphery to center On periphery To center Increases Decreases GF n(initial)> n(initial)< VGF(τ) = πhx(τ)[2R – x(τ)] GF n(product) n(product) Dozed inlet

GF

IL

GF

IL (ferromagn.)

Magnetic (horseshoeshape magnet) Centrifugal (sample rotates/immovable)

IL(ferromagn.) GF

Centrifugal (sample rotates/immovable) Magnetic (circular magnet) Magnetic (circular magnet)

IL GF IL(ferromagn.)

VIL(τ) = πhx[R – x(τ)]2

GF IL

n(initial)< n(product)

n(initial)> n(product)

VGF(τ) = πhx(τ)[2R – x(τ)] VIL(τ) = πhx[R2 – x2(τ)]

GF

VGF(τ) = πhx2(τ)

GF

VIL(τ) = πhx[R2 – x2(τ)] VGF(τ) = πhx2(τ)

IL(ferro magnetic)

40


In technical solutions of the controlled gradient formation considered, the possibility of using gaseous substances as gradient former/inert medium is somewhat restricted. These restrictions are eliminated when the process is performed in the horizontal plane between two surfaces (Figure 16). In this case, there is no need to take account of values of densities of the gradient former and the inert medium. The identity of results obtained by the latter and former methods (Table 4) will be indicated below in the discussion. a) Gradient formation with respect to sample radius from periphery to center at decrease of the process duration [127, 129]. To simplify calculations, radius of the sample (polymeric/gel-polymeric film/plate) is made equal to the internal radius of the reactor (4) (Figure 14, a). The internal volume of the reactor V0 = πR2h, where h is the internal height of the cylinder reactor. A gradient former (chemically active liquid/monomer-diffusate), injected via pipes (12, 6, 15) into the reactor rotating around the axis (Table 4, variant N1) under the effect of a centrifugal field is pressed off to the reactor wall and obtains the shape of a hollow cylinder with wall thickness x. Hence, the part of the sample radius R contacting the gradient former is also equal to x. As the gradient former is injected into the reactor, the value of x increases from 0 to R. Dependence of volume of the gradient former on x is represented by the following relations: V(x) = π(R – x)2h,

(22)

V(x) = πx(2R – x)h.

(23)

To write down the dependence of the gradient former volume, injected into the reactor, on the transformation duration τ at the given τ(x), determined by the function τ = f(R), the reverse function x(τ) is derived and then substituted into equation: V(τ) = πhx(τ)[2R – x(τ)].

(24)

This equation expresses the dynamics of the gradient former introduction into the reactor, which provides the regime of formation of the transformation product with the given radial distribution of the refractive index under conditions of spreading of the gradient formation front with decrease of its duration with respect to the radius from periphery to center. As mentioned above, the algorithm of the controlled gradient formation suggested includes two principally different (from the point of view of the process mechanism) approaches to the problem of creation of gradient systems – controlled polymeranalogous transformation and controlled exchange diffusion with consequent copolymerization. However, in the latter case, some preventive measures may become necessary. A labile gel-polymeric sample is deformed in the centrifugal field – the sample material is shifted from its center to the periphery, and at high frequencies of reactor rotation, which is necessary for providing strictly round front of the exchange diffusion, the overall shape of the sample may be distorted. This may be easily eliminated if the gelpolymeric sample is encased in a round holder, fastened to immovable parts inside the rotating reactor (4) (Figure 14, c). Moreover, before the onset of the dosed inlet of the monomer-diffusate according to equation (24), the reactor must be filled by an inert liquid, the density of which is lower than that of the monomer-diffusate (Table 4, variant N2). This will prevent the possibility of uncontrolled impact of monomer-diffusate drops


41

on the surface of the immovable gel-polymeric sample, and thereby excludes the possibility of distortion of the given profile of the radial distribution of refractive index. Further injected monomer-diffusate possessing higher density than the inert liquid is moved to the reactor wall, pressing out the inert liquid trough pipes (5, 17, 1) (Figure 14, a). When the exchange diffusion is ended, the composition gradient is fixed as usual, by completing the gel-polymeric matrix polymerization. It is desirable that the completion of the polymerization, as well as obtaining the gelpolymeric matrix, should be performed in a device in which one of two plane-parallel surfaces is moved as the gel-polymeric sample shrinks during copolymerization, providing equal decrease of the volume and plane-parallelism of the copolymerized sample [133]. Let us consider setting of conditions of the gradient formation front change with decrease of its duration by radius from periphery to center of the sample under the influence of a magnetic field, created by a horseshoe-shaped magnet, and ferromagnetic liquid as the inert medium. The reactor (12) with the gradient former (11) is placed between the poles of the horseshoe-shaped magnet (5) in the plane transverse to spreading of magnetic force lines. The reactor is first filled by a ferromagnetic liquid (Table 4, variant N5), and then the gradient former is injected, which gradually presses out the ferromagnetic liquid. The ferromagnetic liquid that remains in the reactor occupies the space where magnetic force lines are most closely packed, i.e. between the poles of the magnet, and, consequently, possesses the form of a cylinder, and the gradient former occupying the rest of the space possesses the form of a hollow cylinder. As the gradient former is injected, the thickness of the hollow cylinder increases from 0 to R, i.e. the gradient formation zone increases, and the radius of the cylinder of ferromagnetic liquid decreases from R to 0. In this variant, preliminary calculations are the same as at application of equation (24). Let us consider one more variant of performance of the gradient formation with decrease of its duration by the sample radius from its periphery to center. In some cases, introduction of the ferromagnetic liquid and the gradient former into the reactor according to the sequence mentioned may cause pollution of the gradient former surface by the ferromagnetic liquid that will lead to distortion of the given duration of the gradient former contact by radius of the gradient carrier and, consequently, to distortion of the given radial distribution of the refractive index. This also concerns the above-described case of obtaining selfocs (gradient elements) (Table 4, variant N2), when the reactor with fixed gel-polymeric matrix, placed in the centrifugal field, is filled by an inert liquid, and then the monomer-diffusate possessing higher density is injected. In both cases, the possible complications mentioned may be easily eliminated (Table 4, variants N3 and N4), if the reactor is first completely filled by the gradient former (chemically active liquid/monomer-diffusate), and then an inert (ferromagnetic) liquid with lower density is introduced, which gradually presses out the gradient former. The inert liquid is first of a cylinder shape (x = R), then it attains the shape of hollow cylinder (x < R) and, finally, is pressed out completely (x = 0). Consequently, the inlet inert ferromagnetic liquid will occupy the space in the center of the reactor and will possess the cylindrical shape with radius y = (R – x), which will be gradually increased up to R. The volume of the gradient former is given by V(x) = πR2h – π(R – x)2h = πh(2Rx – x2).

(25)

42

N. G. Lekishvili, L. I. Nadareishvili and G. E. Zaikov The volume of the inert/ferromagnetic liquid is

V(y) = πR2h – πh(2Rx – x2) = π(R – x)2h.

(26)

To record dependence of the volume of inert/ferromagnetic liquid injected into the reactor on the transformation duration τ at the given τ(x), determined by the function τ = f(R), the reverse function x(τ) is derived and substituted into equation (26): V(τ) = πh[R – x(τ)]2.

(27)

Equation (27) expresses the dynamics of inert/ferromagnetic liquid introduction into the reactor (which is preliminarily filled by the gradient former; the gradient former is fixed in the centrifugal field or the reactor is placed between poles of the horseshoeshaped magnet) providing formation of the transformation product with the given radial distribution of the refractive index under conditions of the gradient formation front spreading as its duration decreases by radius from periphery to center of the sample. b) Gradient formation at the increase of process duration by the sample radius from periphery to center [61, 128, 130] Here several variants should be also considered. Let us begin with the variant N6 (Table 4). The reactor (4) (Figure 14) is first completely and rapidly filled by the gradient former, and then dosed introduction of the inert liquid, the density of which is higher than that of the gradient former, is started. Under the influence of the centrifugal field, the inert liquid is reflected to the reactor wall and attains a hollow cylinder shape with the wall thickness of y. As the inert liquid is injected, y increases from 0 to R. The gradient former present in the central part of the reactor assumes the form of a cylinder with radius x = R – y (analogous to the variants considered above and, in this case, x is equal to the part of radius R in contact with the gradient former), gradually decreasing from R to 0. Dependence of the inert liquid volume on the radius x of the cylinder gradient former is expressed by the following equation: V(y) = πh(R2 – x2).

(28)

To record dependence of the inert liquid volume, injected into the reactor, on the transformation duration τ at given τ(x), determined by the function τ = f(R), the reverse function x(τ) is derived and substituted into equation (28): V(τ) = πh[R2 – x2(τ)].

(29)

The equation expresses the dynamics of the inert liquid introduction into the reactor, which provides the regime of transformation product formation under conditions of spreading of the gradient formation front with increase of its duration from periphery to center of the sample. The algorithm suggested is also applicable to the variant N8 (Table 3), when the magnetic field created by the circular electromagnet is used instead of the centrifugal one (Figure 15), and a magnetic liquid is used as the inert medium. The reactor (12) is placed in the circular electromagnet (15) in the plane transverse to the plane of spreading of magnetic force lines. First, the reactor is rapidly filled with the gradient former and dosed delivery of a ferromagnetic liquid is begun immediately, which occupies the volume in


43

the reactor, where magnetic force lines are most closely packed, i.e. near the internal wall of the circular magnet and, consequently, forms a cylinder-shaped ring, inside which the gradient former having the form of a cylindrical body is located. As ferromagnetic liquid is poured in, the thickness of the hollow cylinder wall increases from 0 to R, and the radius of the cylinder from the gradient former, which is gradually pressed off from the reactor, decreases, respectively. As the process ends, the electromagnet is switched off and the valve (8) is opened. Hence, the zone of gradient formation in this variant decreases gradually, while the process duration along the sample radius from periphery to the center increases. That is why preliminary calculations necessary for creation of the given radial distribution of the refractive index are the same as at application of equation (30). Let us discuss two possible variants of increase of the gradient formation duration along the sample radius from its periphery to the center that can be performed both in centrifugal (the sample rotates or is fixed) and in magnetic (circular electromagnet) fields (Table 4, variants N1 and N9). Initially, the reactor is filled with an inert/ferromagnetic liquid, and then gradient former is injected possessing lower density, which occupies the space in the reactor center and assumes a cylindrical shape. As the gradient former is injected, the cylinder radius increases from 0 to R and gradually presses off the inert/ferromagnetic liquid. The volume of the gradient former is V(x) = πx2h.

(30)

To write down the dependence of the gradient former volume injected into the reactor on duration of transformation, τ, at given τ(x) determined by function τ = f(R), the reverse function x(τ) is derived and substituted into equation (30): V(τ) = πx2(τ)h.

(31)

Equation (31) indicates the dynamics of the gradient former delivery to the reactor, which provides the regime of transformation product formation with the given radial distribution of the refractive index under conditions of the gradient formation spreading with growth of its duration by sample radius from the periphery to the center.

4.4. Particular Examples of Realization of Controlled Heterogeneous Gradient Formation The method of controlled gradient formation is illustrated below using the example of 20

the following initial polymers: poly(vinyl acetate) (PVA, nD = 1.4655, film thickness is 20

0.015 mm), poly(vinyl alcohol) (PVAl, nD

= 1.530, film thickness is 0.015 mm),

20

isotactic polypropylene (PP, nD = 1.495, film thickness is 0.015 mm), gel-polymeric matrix based on diallylisophthalate (DAIP,

nD20 = 1.5254), diethylene-glycol-

20

bisallylcarbonate (DEGBAC, nD = 1.4570). Refractive indices of initial polymers were measured according to ref. [136]. Variation of the refractive index was measured on the Mach-Zender interferometer with

44


He-Ne laser (LG-201, Russian trademark) as the light source with the active wavelength λ = 0.6328 µm. The value of ∆n was calculated by formula (5). Chlorine content was determined gravimetrically (by weighing films before and after reaction), and by chemical analysis as well [137]. a) Creation of gradient systems by polymer-analogous PVA → PVAl transformation [61, 127, 128, 133] This transformation can be performed by alcoholysis of PVA films by sodium hydroxide solution in ethanol with [NaOH] = 10 wt% at 60°C. Hence, in the present process, PVA is the gradient carrier, and sodium hydroxide solution is the gradient former:

(

CH2

)n + nNaOH O COCH3 CH

C2H5OH

(

O CH2

CH OH

)n + nCH3

C ONa

Experiments are performed in order to study function n = f(τ) – determination of influence of PVA → PVAl transformation duration on variation of the refractive index of the gradient-carrier. These data are shown in Figure 17 indicating that maximal growth of the refractive index in the gradient-carrier, reached after 4 hours, is ∆ = 0.025.

n 1.4950

1.4850 1.4750 1.4650 1

2

3

4 τ, h

Figure 17. Dependence of the refractive index of the gradient-carrier (n) on duration (τ) of polymeranalogous PVA → PVAl transformation

The function n = f(R), the radial distribution profile of the refractive index, is given. Figure 18a expresses the given profile of radial distribution of the refractive index in the PVA → PVAl transformation product of 40 mm in radius. Hence, a system possessing properties of a defocusing lens (the refractive index decreases from periphery to center) must be obtained. Based on Figures 17 and 18 (curve “a”) the function τ = f(R) is graphally depicted, representing the dependence of PVA → PVAl polymer-analogous transformation duration (τ) on radius (R) of the gradient former. This graph is shown in Figure 19 (curve “a”), according to which the transformation duration must decrease from gradient-carrier periphery to the center. Consequently, five variants of technical solution are applicable to realization of the transformation regime required (Table 4, variants N1, 2, 3, 4, 5) using equations (23) or (26). In variants N4, 5, 8, 9, a colloid solution of


45

Fe(OH)3 in isooctane (2,2,3-trimethylpentane) is used as the ferromagnetic liquid. Illustrated below is the example using variant N1.

n 1.4950

a

1.4850 b

1.4750

1.4650 10

20

30

40 R, mm

Figure 18. Given dependence of the refractive index (n) on radius (R) of the gradient-carrier at PVA → PVAl transformation: “a” and “b” are products with properties of biconcave and biconvex lenses, respectively.

τ, h 4

b

a

3 2 1 10

20

30 40 R, mm

Figure 19. Dependence of duration (τ) of polymer-analogous PVA → PVAl transformation on radius (R) of the gradient-carrier: “a” and “b” are regimes of obtaining products with properties of defocusing and focusing lenses, respectively

Table 5. V (0.5 hour) = 3.14⋅120 mm⋅1 mm (2⋅40 mm – 1 mm) = 29767.22 mm3; V (1 hour) = 3.14⋅120⋅3 (2⋅40 – 3) = 87040.8 mm3; V (1.5 hours) = 3.14⋅120⋅10 (2⋅40 – 10) = 263760 mm3; V (2 hours) = 3.14⋅120⋅17.5 (2⋅40 – 17.5) = 412125 mm3; V (2.5 hours) = 3.14⋅120⋅25 (2⋅40 – 25) = 518100 mm3; V (3 hours) = 3.14⋅120⋅30 (2⋅40 – 30) = 565200 mm3; V (3.5 hours) = 3.14⋅120⋅35 (2⋅40 – 35) = 593460 mm3; V (4 hours) = 3.14⋅120⋅40 (2⋅40 – 40) = 602880 mm3.

For reactor (4) (Figure 14), the following sizes are selected: height h = 120 mm, internal radius R = 40 mm. Further on, based on Figure 19 and equation (23), the volumes of sodium hydroxide solution, which will be delivered to reactor (4) rotating around the axis at time moments, are given in Table 5.

46


The required volumes of sodium hydroxide (Table 5) are delivered with the help of a batcher attached to pipe (12) (Figure 14). Figure 20 shows the scheme of one of the types of liquid batcher with a cam. For completeness in presentation, all calculations associated with usage of such a batcher are given below.

A 1 1

2

2

3

3

A Figure 20. Batcher with a cam for controlled delivery of gradient former/inert (ferromagnetic) liquid. See text for details.

A cylinder vessel (3) (Figure 20) is used, with internal diameter of 40 mm and height of 120 mm. Plunger (2) transposition in vessel (3) up to time τ is calculated by the equation:

H (τ ) =

V (τ ) . 3.14 ⋅ 40 2

(32)

Substituting values of V(τ) from Table 5 into equation (32), we get: Table 6. H(0.5 hour) = 5.96 mm; H(1 hour) = 17.325 mm; H(1.5 hours) = 52.5 mm; H(2 hours) = 82 mm; H(2.5 hours) = 103.1 mm; H(3 hours) = 112.5 mm; H(3.5 hours) = 118.1 mm; H(4 hours) = 120 mm.

The cam (1) profile is calculated in the usual way [138]: a circle of an arbitrary diameter is selected, a half of the circle is divided into eight parts, and radii are depicted. The following distances are laid on the radius extensions:


47

on the extension of the Ist radius – 20 mm; on the extension of the IInd radius – (20 + 6) mm; on the extension of the IIIrd radius – (20 + 17) mm; on the extension of the IVth radius – (20 + 52.5) mm; on the extension of the Vth radius – (20 + 82) mm; on the extension of the VIth radius – (20 + 103) mm; on the extension of the VIIth radius – (20 + 112.5) mm; on the extension of the VIIth radius – (20 + 112.5) mm; on the extension of the IXth radius – (20 + 120) mm. Connecting the ends of rays, we obtain the cam (1) profile (Figure 20). The cam is made from a hard metal. Further on, gradient-carrier (18) (Figure 14), i.e. the PVA film, is dipped into reactor (4). Vessel (3) (Figure 20) is filled with the sodium hydroxide solution. The plunger (2) contacts the cam (1) at the point on the extension of the radius I. The batcher (Figure 20) is attached to the pipe (12) (Figure 14). As the demanded temperature is reached (60°C), the electric engine (9) is switched on (Figure 14), the valve (4) is opened, and the electric engine (6) (Figure 20) is switched on. The cam (1), rotating by the arrow (see the Figure), presses on the plunger (2) pressing off the gradient former delivered by pipes (6, 12, 15) to the periphery of the reactor (4). After full rotation of cam (1), electric engines (9) (Figure 14) and (6) (Figure 20) are switched off, plug (13) is removed, and the product of polymer-analogous PVA → PVAl transformation is removed, washed and dried. The same polymer-analogous PVA → PVAl transformation under the conditions when duration of the chemical reaction by the gradient-carrier radius increases from periphery to the center gives the possibility of creating a medium with properties of a focusing lens (the refractive index increases from periphery to the center). Curve “b” in Figure 18 expresses the given profile of refractive index distribution in the product of gradient polymer-analogous PVA → PVAl transformation 40 mm in radius and with properties of the focusing lens. Based on Figures 17 and 18 (curve “b”), a graph of function τ = f(R) – the dependence of PVA → PVAl transformation duration (τ) on radius (R) of the gradient-carrier – is plotted. This graph is shown in Figure 19 (curve “b”). For practical realization of the demanded regime of gradient transformation, four variants of technical solution are basically applicable (Table 4, variants N6, 7, 8, 9) using appropriate equations shown in Table 6. Calculations of gradient-carrier volumes delivered up to the moment τ and cam (1) profile (Figure 20) are performed in accordance with the directions above. Mercury can be used as the inert liquid fitting the condition ρ(gradient-carrier) < ρ(inert liquid). b) Creation of gradient systems by polymer-analogous PVA → Poly(vinylbutyrate) transformation [61] In the above-considered examples, the refractive index of the gradient-carrier was lower than that of the product of polymer-analogous transformation. It is clear that with the reverse relation of refractive indices, as for example, with PVA (n = 1.53) → Poly(binylbutyrale) (n = 1.485) transformation:

48


reaction products possessing properties of defocusing and focusing lenses are formed under conditions of the opposite (comparing with the above-considered PVA → PVAl transformation) spreading of the chemical reaction front (see Table 5). The gradient former (n-butyric aldehyde) is used as a water solution [n-C4H8O] = 5 wt%. The reaction proceeds at 40°C in the presence of ~0.5% of concentrated sulfuric acid as the catalyst. During the 5 hours of the reaction process, the lowest reduction of the refractive index in the gradient former is ∆n = 0.02, which indicated incomplete acetylation of the initial PVA. This agrees with the literature data [139]. Moreover, it should be taken into account that the heterogeneous nature of the process almost excludes spreading of the polymeranalogous transformation deep into the sample, which affects the transformation of the refractive index. This judgement is true for all heterogneous polymer-analogous transformations described in the present monograph. That is why determination of the type of profile of radial distribution to be set must be based on practically realized limiting values of refractive indices. N-Hexane (ρ20 = 0.6594) and mercury, respectively, can be used as inert liquids possessing lower and higher density comparing with the gradient former. Here ferromagnetic liquid is the same (Fe(OH)3 solution in isooctane). c) Creation of gradient systems by diffusion polymerization [61, 127, 133] Gel-polymeric matrices (~2 mm thick) from diallylisophthalate (DAIP) can be used as gradient formers:

CH2

CH CH2

O C O

20

C O CH2

CH CH2

O 20

(For DAIP, nD = 1.5254; for poly(DAIP), nD = 1.569; conversion is ~35%), or diethyleneglycolbisallylcarbonate (DEGBAC):

CH2

CH CH2

O C O

(OCH2

CH2)2

C O CH2 O

CH CH2

Polymeric Media with the Gradient of the Optical Properties 20

49

20

(For DEGBAC, nD = 1.4503; for poly(DEGBAC), nD = 1.50; conversion is 20

~40%). The monomer-diffusate can be methyl methacrylate (for MMA, ρ20 = 0.943, nD 20

= 1.4146; for PMMA, nD = 1.492), or fluoroalkyl methacrylates, for example: 1,1,3-trihydrotetrafluoropropyl methacrylate (F4MA)

CH2

C C OCH2 H3C

20

(CF2)2 H

O 20

(For F4MA, ρ20 = 1.239, nD = 1.375; for poly(F4MA), nD = 1.420). These gradient formers reduce refractive index in the gradient-carrier. Poly(methylsiloxane) (PMS-100, ρ20 = 0.97) is the inert liquid for both gel-polymers and F4MA, and Fe(OH)3 solution in isooctane is the ferromagnetic liquid. d) Study of solid-phase chlorination of isotactic polypropylene(PP) and creation of poly(chloropropylene) GRIN-elements [134] It seems desirable to consider this approach of creation of gradient elements in more detail, because the chemical processes on which it is based are rather ordinary and quite well studied, especially for homogeneous polymer-analogous transformations, and this is not usually the case in producing materials for gradient optics. The principles of solid chlorination of PP-films that should be used for creation of thin-layer selfocs with the given profile of refractive index distribution are discussed below. PP films were chlorinated by UV-radiation at 90°C. Figure 21 shows curves of the dependence of the wt% of chlorine that participates in the reaction on the process duration: [Cl] = f(τ). Theoretically, the proportion of chlorine in completely monochlorinated PP is 46.4 wt% Cl, and in completely dichlorinated product is [Cl] = 64 wt%. If chlorinated in solution or suspension, up to 70 wt% Cl can be introduced into PP [140], which corresponds to the presence N (Cl) = 2.6 in every structural unit. The type of change of molar parts of structural units of various chemical structure is clearly observed from comparison of ω(Cl) ordinates and A, B, and C in Figure 21 with increase in the molar proportion of chlorine. Hence, if ω(Cl) ρ(inert)

+ r 2 − [ R − x(τ )]2

K = (H − r) + Increases

+ r 2 − x 2 (τ )

ρ(act) < ρ(inert)

Preliminary filling of the reactor 

On periphery Act.

In center 

Act.



Inert

Inert



Act.

Act.

Inert



Change of the refractive index in modified polymer layer from periphery to center of the sample Increases n(init.) > n(product)

Decreases n(init.) < n(product)

n(init.) < n(product)

n(init.) > n(product)

Regime of active/inert liquids delivery (sign “minus” corresponds to convex lens samples, sign “plus” – to concave ones) Vact(τ) = 1/6π{6hx(τ)[2R – x(τ)] ± K(τ)[6hx(τ) – 3x2(τ) + K2(τ)]} Vinert(τ) = 1/6π{3[R – x(τ)]2[2h ± K(τ) ± H] ± [H –K(τ)]3} Vinert(τ) = 1/6π{[6h ± 3K’(τ)] [R2 – x2(τ)] ± K’3(τ)} Vact(τ) = 1/6π{3x2(τ)[2h ± K’(τ) ± H] ± [H – K’(τ)]3}

58


4.5. Method of Obtaining Cylindrical (Disk-Like) Polymeric Objects with a Given Radial Gradient of the Refractive Index [124] The methods of creating gradient systems considered in the previous Section are based on the solid-phase chemical modification of a polymer or gel-polymer, which, on the one hand, gives the possibility of obtaining macrosurface GRIN-elements, but on the other hand, limits their thickness, especially at polymer-analogous gradient formation. The present Section describes devices and methods for obtaining cylindrical polymeric objects with given radial composition gradient providing a given profile of refractive index distribution. The principal features of the device are shown in Figure 26. The device contains two vessels (1) and (2), intended for liquid substances with different values of the refractive index. Rotating in a definite direction, cams (3) and (4) move plungers (5) and (6). Liquids flowing from vessels (1) and (2) are mixed by warm mixer (7). The mixture is delivered to vessel (8) rotating around its axis. The volumetric proportion of every component in the mixture is regulated and determined by the profile of cams (3) and (4). The profile of cam (3) provides for a gradual increase of the rate of plunger (5) transposition, while the profile of cam (4) provides a gradual decrease of plunger (6) transposition rate. In this way, the mixture of components, in which relative volumetric content of one of them is reduced continuously, and that of the second one is increased, is delivered to the mixer (7) and then to the vessel (8). B

A A

B

43

3

4

6 2

6 2

5

5

1

1 B

A 7 8

Figure 26. Principal components of the device for obtaining objects with radial composition (refractive index) gradient. See text for details.

As a drop of the mixture enters rotating vessel (8), it spreads as a thin layer over its vertical wall under the effect of centrifugal forces. Additional portions form further layers, and so on. The filled vessel is placed into a thermostat for polymerization. In the material obtained, due to azeotropic copolymerization, the radial gradient of structural units is preserved and, consequently, the appropriate refractive index gradient.


59

This approach is analogous in many points to that given in ref. [143]. However, as description of the calculations necessary to form the given refractive index gradient is absent in ref. [143], an adequate algorithm is discussed below. Shrinkage of the material due to polymerization can be neglected in the first approximation, and volumes of the vessel (8) and the resulting cylindrical object can both be assumed equal to W = πR1 h , where R1 is the radius of vessel (1), and h is its height. 2

Assume that such polymers are used that separate polymerization would give homopolymers with refractive indices n1 and n2, and that n1 < n2. Let us denote these monomers as Mmin and Mmax. The maximal difference in refraction that can be realized over the radius of the cylindrical object is ∆nmax = n2 – n1. Assume that an object needed with the refractive index gradient profile shown in Figure 66, where the abscissa axis represents the distance from the cylinder side to its center, and the ordinate axis is the difference in the refractive index. Obviously, the increase of n from the periphery to the center of the cylinder must be caused by an increase in the volumetric proportion of the component with higher refractive index, ϕ(Mmax), in the mixture with the component possessing lower refractive index, Mmin, from 0 up to 100%. That is why the curve shown in Figure 27 does not change if the volumetric proportion ϕ(Mmax) is displayed on the ordinate axis instead of n, and the nmax point is combined with the point of 100% content ϕ(Mmax), and the point nmin with the point of zero content (Figure 27). Filling of the vessel (8) (Figure 26) of radius R1 by the mixture of Mmax and Mmin proceeds during time τ. That is why the curve run in Figure 66 will not change, if R is substituted by τ on the abscissa axis (Figure 27).

V(Mmax) ϕ(Mmax) %

n

100

max

0

min

0

R1

τ1

R

τ

Figure 27. Dependence of the refractive index (n), volumetric proportion of component with higher refractive index ϕ(Mmax) and rate of plunger motion V(Mmax) in vessel (1) (Figure 26) on distance R from the edge to the center of vessel (8) for copolymerization and duration of its filling (τ)

60


To increase the volumetric part ϕ(Mmax) from periphery to center of the object (or the same in a stream of mixture delivered to the vessel (8)) in accordance with the curve in Figure 27, the rate of plunger (5) motion in vessel (1) (Figure 26) must change according to the same relationship. That is why V(Mmax) can be represented on the ordinate axis instead of ϕ(Mmax), and hence the maximal value of V(Mmax) must be combined with the point of 100% content of Mmax. Thus, the nature of the dependence of n on R1 is equivalent to that of V(Mmax) on τ. A graph of dependence of the volumetric proportion of the component with lower refractive index ϕ(Mmin) on R (Figure 28) is plotted from the curve given in Figure 66. Using analogous reasoning to that above, this dependence can be reduced to the one of V(Mmax) on τ, where V(Mmax) is the rate of plunger (6) motion in vessel (2) filled with Mmin. These curves are integrated graphically according to the method given in ref. [144, 145]. Results are shown in Figure 29. Based on the dependence obtained, x = f(τ), the profile of cams (3) and (4) may be calculated.

R1 R

τ1 τ

Figure 28. Dependence of volumetric proportion of the component with lower refractive index ϕ(Mmin) and the rate of plunger motion V(Mmin) in vessel (2) (Figure 26) on distance R from the edge to the center of vessel (8) (Figure 26) for copolymerization and duration of its filling (τ)

Volume of the procurement, W, is: W = W1 + W2; W1 = πr2l1, W2 = πr2l2, where W1 and W2 are the volumes of components Mmax and Mmin that are extruded from vessels (1) and (2), respectively; r is the radius of vessels (1) and (2); l1 and l2 are maximal displacements of plungers (5) and (6) in vessels (1) and (2), respectively. It is clear that R12h = r2(l1 + l2).

(37)


61

X X1 a

X2

b

τ1 τ Figure 29. Dependence of relative displacement of plungers (Figure 26) on time (τ): 1 – in vessel (1); 2 – in vessel (2)

The following relation can be derived from Figure 1:

x1 =K. x2 Values of l1 and l2 depend on r. However, in any case:

x1 l1 = =K. x2 l 2

(38)

Comparing equations (37) and (38), we get:

l1 =k•l2

l2= R12h/r2(1+ K)

(39)

In Figure 29, maximal values of x1 and x2 equal the calculated values of l1 and l2, respectively, and are based on the dependence obtained by the method already used in Section 4 [138], where profiles of cams (3) and (4) are considered. This method allows us to obtain materials with a given profile of radial distribution and other properties that depend on the composition.

4.6. Method of Obtaining Cylindrical (Disk-Like) Objects with a Given Radial Composition (Refractive Index) Gradient Based on Powder-Like Materials [146] As powder-like substances are used for obtaining cylindrical objects by the method discussed in the previous Section, several specific problems must be solved. In particular,

62


when powder-like substances are injected according to a definite program into a cylinder vessel rotating around its axis, then the density of material within the vessel will decrease from the periphery to the center with centrifugal force. That is why the real shrinkage at melting of a powder-like material will increase with radius from periphery to center which, at leveling of the melt surface, leads to a shift of melt layers and distortion of the given radial distribution of the composition. In this Section, we present solutions for several major problems encountered in preparing laminar objects with a given radial distribution of refractive index, starting from powder-like materials. Figure 30 shows the main features of the apparatus required. The appliance consists of bunker (1) with partition (2) separating areas filled by powder-like substances I and II (view A-A) possessing different refractive indices. Bunker (1) is equipped with a sliding valve (3). The appliance also contains bunker-mixer (4) with mixer (5) and electric engine (6). Further on in the processing, the appliance possesses a cylinder molding volume (7) with a cone-like lid (8) and a hole (9). Volume (7) is fixed on the table of a centrifuge (10) connected with an electric engine (11). The appliance operates as follows. Powder-like components I and II are placed in bunker (1), to areas I and II, respectively. As slide valve (3) is displaced in the direction marked by the arrow, components I and II from bunker (1) are gradually delivered to bunker-mixer (4) and, mixed by mixer (5), are delivered through whole (9) to molding volume (7) rotating around its axis. When the latter is filled, rotation is stopped, and thermal processing is performed.

A

A

A-A 1 2

1 3

4

6

3 I

II

5 1

9 8 7 10

II

I

2 3

11 Figure 30. Principal features of the appliance used to obtain a cylindrical object from powder-like substances with the given radial gradient of refractive index. See text for details

The following notations are used in this Section:


63

n is the refractive index; n1 and n2 are refractive indices of components, n2 > n1; R is the radius of the cylindrical product; H is the height of the cylinder procurement; [mon.n1] is the concentration of the component of the cylindrical object in the monolith state with refractive index n1 and density ρ1; [mon.n2] is the concentration of the component of the cylindrical object in the monolith state with refractive index n2 and density ρ2; [pow.n1] is the concentration of the powder-like component in the cylindrical vessel with refractive index n1 and density ρ1(R), i.e. dependent on radius R; [pow.n2] is the powder-like component in the cylinder vessel with refractive index n2 and density ρ2(R), i.e. also dependent on radius R; ϕ is the volumetric part of the component; ∆r is the interval of constant length on the procurement radius R; j is the number of interval ∆r, j = 0, 1, 2, 3, …, k – 1; Vj is the component volume in the j-th layer, placed between two cylinder surfaces, radii of which are j∆r and (j + 1)∆r; mj is the component mass in the j-th layer; hj is the component height in the j-th layer. Assume that a cylinder of height H and radius R is prepared, in which radial distribution of the refractive index, i.e. dependence n = f(r) is of the shape shown in Figure 31, curve ‘a’. The central cross-section of such a cylinder is shown in Figure 32.

n n2

a

n1 R

r

Figure 31. The given dependence of refractive (n) on radius (R) of cylindrical product

To obtain such a product, components I and II with refractive indices n1 and n2 (n1 < n2) and density ρ1 and ρ2 are used. The refractive index from the cylinder center to periphery decreases with the ϕ-volumetric proportion of the component with n2 and ρ2 in the monolith state [mon.n2] mixed with the component with n1 and ρ1 in the monolith state [mon.n1]. As ϕ[mon.n2] decreases, ϕ[mon.n1] increases so that the following equation is true for every value of R:

ϕ[mon.n1] + ϕ[mon.n2] = 1.

(40)

64


n n2 H

a n1

r

R

R

0

r

Figure 32. Central cross-section of a cylinder-shaped product H high and R in radius, in which dependence of the refractive index (n) on radius (R) is expressed by curve ‘a’, i.e. it possesses the shape shown in Figure 31.

ϕ[mon. n2]

That is why the shape of the curve ‘a’ in Figure 31 will not change if ϕ[mon.n2] is substituted on the ordinate axis instead of n, and the n2 point is combined with the point of 100% content of this component, and the n1 point is replaced by the zero content point (Figure 33).

1

0

R

r

Figure 33. Dependence of the volumetric part (ϕ) of component in the monolith state with n2 on radius (R) of cylindrical product

ϕ[mon. n2]

Figure 34. Central cross-section of the cylindrical product H high and R in radius, in which dependence of the volumetric part (ϕ) of the component in the monolith state with refractive index n2 on radius of the cylinder, R, is expressed by curve ‘b’, i.e. it possesses the shape shown in Figure 33

Let us suppose that the average value of refractive index n in the product is given by n = n2ϕ[mon.n2] + n1{1 – ϕ[mon.n2]}.

(41)


65

Relation (41) implies that ϕ[mon.n2] depends on r, with 0 ≤ r ≤ R. Then relation (41) is reduced to the form: n(r) = n2ϕ[mon.n2](r) + n1{1 – ϕ[mon.n2](r)}.

(42)

Solving equation (42) in relation to ϕ[mon.n2](r), we get:

ϕ[mon.n2](r) =

n(r ) − n1 . n2 − n1

(43)

If a graph of ϕ[mon.n2](r) is plotted point by point with the help of Figure 31 and expression (43), the curve shown in Figure 33 will be obtained. By the same arguments, the contour of the central cross-section of the cylindrical product in Figure 32 will not change if it is presented in the coordinates of Figure 34. This Figure can be interpreted as follows: if [mon.n1] and [mon.n2] are mentally separated from each other for every value of R and disposed according to height H above each other, then the sought-for cylindrical product can be conditionally considered somewhat to consist of two parts, where the shaded area corresponds to [mon.n1] and the clear one to [mon.n2]. Let us divide the interval [0, R] in Figure 34 into intervals of constant length ∆r, the number of which is k = R/∆r (Figure 35).

1 H 0

R

∆r 2∆r

j∆r (j+1)∆r

R

Figure 35. See text for details.

Numbers of intervals are j = 0, 1, 2, 3,…, (k – 1), and cross-link points are 0, ∆r, 2∆r, 3∆r, …, (k – 1)∆r = R. Let us consider an interval [j∆r, (j + 1)∆r]. It represents a layer formed by two cylinder surfaces, radii of which are j∆r and (j + 1)∆r. Let us denote this layer as j. In the present case, height of this layer equals the height of the cylinder. Heights of [mon.n2] and [mon.n1] in this layer are equal to:

 

hj[mon.n2] = f  j∆r +

 

∆r  ; 2 

hj[mon.n1] = H – f  j∆r +

∆r  . 2 

Let us calculate volumes of [mon.n1] and [mon.n2] in the j-th layer:

(44)

(45)

66


  ∆r   = π(2j + 1)∆r2⋅f  j∆r + . 2  

Vj[mon.n2] = π[(j + 1)∆r]2⋅f  j∆r +

∆r  ∆r   2  – π(j∆r )⋅f  j∆r +  2  2   (46)

With regard to equation (45), Vj[mon.n1] is calculated in an analogous manner:

  ∆r   = π(2j + 1)∆r2⋅[H – f  j∆r +  ]. 2  

Vj[mon.n1] = π[(j + 1)∆r]2⋅[H – f  j∆r +

∆r  ∆r   2  ] – π(j∆r )⋅[H – f  j∆r + ] 2  2   (47)

Masses of the components in the j-th layer are: mj[mon.n1] = ρ1Vj[mon.n1];

(48)

mj[mon.n2] = ρ2Vj[mon.n2].

(49)

Operating with the density notion suggests that densities of each pure component and a mixture with another component are equal. Such a suggestion applies to components in both powder and monolith states. To obtain a preform H high, the height of the molding vessel must be (H + ∆H), because material shrinkage occurs during melting of the powder-like components and monolith formation. Therewith, the real shrinkage increases with respect to radius from periphery to center proportionally to the filling density decrease stipulated by the centrifugal force decrease. This means that obtaining of a preform H high and exclusion of horizontal displacement of the material after melting of powder-like components demand ∆H changing with radius, i.e. it depends upon j. Determined experimentally are ρ1(R) and ρ2(R) – dependencies of the filling density by powder-like components with n1 and n2 on R at the given rotation frequency of centrifuge (11) electric motor (Figure 30). Let us denote powder-like substances loaded into the volume (7) (Figure 30) as [pow.n1] and [pow.n2]. Components possessing mass mj and present in the j-th layer of the monolith in the powder-like state with regard to filling density dependence on R occupy the following volumes:

V j [pow.n1 ] = V j [pow.n2 ] =

m j [pow.n1 ]

ρ1 ( R)

,

m j [pow.n2 ]

ρ 2 ( R)

(50)

.

(51)


67

As the mass of components is independent of their state, we get:

V j [mon.n1 ] = V j [mon.n2 ] =

m j [mon.n1 ]

ρ1 ( R)

,

m j [mon.n2 ]

ρ 2 ( R)

(52)

.

(53)

Volume increase, when we have [pow.n1] instead of [mon.n1], is: ∆Vj(n1) = Vj[pow.n1] – Vj[mon.n1]. With the regard to equation (52), we get:

∆V j (n1 ) =

ρ1

ρ1 ( R)

⋅V j [mon.n1 ] − V j [mon.n1 ] ,

 ρ  ∆V j (n1 ) =  1 − 1 ⋅ V j [mon.n1 ] .  ρ1 ( R) 

(54)

In an analogous way, the volume increase when [pow.n2] is taken instead of [mon.n2], is calculated as

 ρ  ∆V j (n2 ) =  2 − 1 ⋅V j [mon.n2 ] .  ρ 2 ( R) 

(55)

Let us calculate height growth of the j-th layer – ∆Hj, [pow.n1] and [pow.n2] are assumed instead of [mon.n1] and [mon.n2]: ∆Hj = ∆h’j + ∆h”j,

(56)

where ∆h’j and ∆h”j are height increments due to increase of ∆Vj(n1) and ∆Vj(n2) volumes, respectively:

∆h′j =

∆h′j′ =

4

π (∆r )2 4

π (∆r )2

⋅ ∆V j (n1 ) ,

(57a)

⋅ ∆V j ( n 2 ) .

(57b)

With the regard to equations (54) and (55), we get:

68

∆h′j =

∆h′j′ =


4

 ρ  ⋅  1 − 1V j [mon.n1 ] ,  ρ1 ( R) 

(58)

4

 ρ  ⋅  2 − 1V j [mon.n2 ] .  ρ 2 ( R) 

(59)

π (∆r )2

π (∆r )2

Transforming these equations with regard to (46) and (47), and introducing into equation (56), finally, we obtain  ρ   ρ   ∆r   ∆r  ,  ∆H j = 4( 2 j + 1)  1 − 1 h − f  j∆r +  + 4(2 j + 1)  2 − 1 ⋅ f  j∆r +  2  2    ρ1 ( R )    ρ 2 ( R)  

or

 ρ  ∆H j = 4(2 j + 1) 1 − 1  h −   ρ1 ( R )  

 ∆r    ρ 2  f  j∆r + − 1 ⋅  +  2   ρ 2 ( R)  

∆r   .  f  j ∆r +  2  

(60)

Profile of a cone-like lid (8) (Figure 30), at which horizontal displacement of the melt and distortion of the given radial distribution of the composition are excluded, is calculated by equation (60). The present Section also suggests a variant of dosed delivery of powder-like components I and II to the tank (7) (Figure 30), demanded by some other definite shape to the bunker (1). For this purpose, the length of partition (2) (Figure 30) is set equal to R – the radius of the cylinder preform. Partition (2), as in Figure 74, is divided into intervals of constant length ∆r, the number of which k =

R , and numbers of intervals j = 0, 1, 2, 3, …, (k – 1), and cross ∆r

points 0, ∆r, 2∆r, 3∆r, …, j∆r, (j + 1)∆r, …, (k – 1)∆r = R (Figure 36). The values of ρ1(H) and ρ2(H), the apparent densities of powder-like components with refractive indices n1 and n2, respectively, are experimentally determined. If the component mass in the j-th layer of the preform is known, the volume of this component in the bunker can be determined:

V j [pow.n1 , ρ1 ( H )] = V j [pow.n2 , ρ 2 ( H )] =

m j [mon.n1 ]

ρ1 ( H )

,

m j [mon.n2 ]

ρ2 (H )

(61)

.

(62)

If the bunker is filled with powder-like components up to the height l, volumes of these components can be expressed in terms of the bunker parameters (Figure 36):


69

V j [pow.n1 , ρ1 ( H )] = l∆rx j (n1 ) ,

(63)

V j [pow.n2 , ρ 2 ( H )] = l∆rx j (n2 ) .

(64)

Substituting volumes from equations (61) and (62) to (63) and (64) and determining xj(n1) and xj(n2), we get expressions for calculating the profile of the bunker walls:

x j (n1 ) =

x j ( n2 ) =

m j [mon.n1 ]

ρ1 ⋅ l ⋅ ∆r

,

m j [mon.n2 ]

ρ 2 ⋅ l ⋅ ∆r

(65)

.

(66)

Clearly the length of the partition (2) in the bunker (1) (Figure 30) should optionally equal the preform radius. Moreover, to raise accuracy of the controlled delivery of powderlike components, thickness of the powder layer (l) and the front of components delivery x(n1) and x(n2) must be reduced, and the length of partition (2) must be increased.

3

0 ∆r

Component filling zone

2∆r

[pow. n1, ρ1(H)]

Component filling zone [pow. n2, ρ2(H)]

j∆r 2

(j+1)∆r 1

x(n1)

R

x(n2)

Figure 36. Scheme for calculating profile of bunker (1) walls (Figure 30): 1 – front side of the bunker; 2 – direction of slider (3) motion (Figure 30); 3 – bunker partition.

The method suggested can be applied to obtain materials with the radial gradient not only of the refractive index, but of other physicochemical properties also.

70


4.7. Method of Obtaining Long Polymeric Cylindrical Preforms with Gradual Radial Gradient of Refractive Index [126] As mentioned above, a monofiber with a core and a cover is able to transmit light energy only, but not images. To transmit images, a fiber must possess a gradient of refractive index. Such a fiber can be molded from a selfoc, prepared by the photopolymerization method [119, 120], for which either linear or three-dimensional polymers can be used. At the same time, the method of photopolymerization enables us not to have to form a previously given radial gradient of refractive index in a selfoc. Moreover (which is no less significant), making stricter demands on the monomers (in difference in copolymerization constants), this method narrows the class of polymers suitable for fiber molding. The method of a cylinder preform production of an arbitrary diameter with the given radial gradual profile of refractive index distribution based on polymers, from which optical fibers can be molded capable of image transmission, is discussed below. The initial substances are two fluid monomers (for example, methyl methacrylate or ethyl methacrylate and styrene), which at separate polymerization form transparent homopolymers with different values of the refractive indices, n1 and n2, in which we take n2 > n1. The same monomers at any volumetric relation form transparent copolymers. It is also necessary for shrinkage coefficients ρmon/ρpow to be equal or to be of close values, because this situation affects the degree to which the theoretical calculations can be realized.

R1

Figure 37. Cylinder bar from homopolymer with high refractive index n2 and radius R1.

First, a cylindrical bar of radius R1 from homopolymer with n2 (Figure 37) and a copolymeric belt with the axial composition gradient L long, S wide, and H high (Figure 38) are prepared. On one end of the belt (marked by a cross) structural units with n1 prevail, and on the other one (marked by a circle) the units with n2 predominate. The direction of increase of the refractive index over the belt length L from n1 to n2 is indicated by an arrow. The refractive index does not change along the belt width S (dotted line), as the concentration of the structural units with indices n1 and n2 is the same.


L

71

H

S

Figure 38. Untwisted copolymeric belt with given sizes (L, S, H) and structural units with high refractive index n2 and low refractive index n1 prevailing on opposite ends of it (marked by a circle and a cross, respectively).

Such a copolymeric belt is tightly twisted on the bar from the homopolymer with n2 starting with the end where the prevailing structural units are those with index n2 (Figure 39), after which the preform is caked. Therewith, the preliminarily given distribution of the refractive index over radius R2 of the cylinder preform obtained (Figure 40) is provided.

Figure 39. Copolymeric belt partially twisted on cylinder bar from homopolymer starting with the end possessing structural units with high refractive index.

The copolymeric belt with the given profile of refractive index axial distribution is produced on a device (Figure 41), generally analogous to the one described above (Figure 36). Vessel (1) is designed for a monomer with n1, and vessel (2) for a monomer with n2. During rotation, cams (3) and (4) move plungers (5) and (6). Monomers flowing out of vessels (1) and (2) in the given permanently changing ratio are mixed in a worm mixer (7) and delivered through input sleeves (8) to casting mold (9) (L long and S wide) evenly moving in the direction indicated by an arrow. Limiting plate (10) makes for uniform

72


redistribution of the monomer mixture in the filling zone in the casting mold (9). The duration of the delivery of the monomer mixture to casting mold (9), i.e. duration of motion of the plungers (5) and (6), is the time it takes for the casting mold (9) to travel along the whole length L. As the casting mold (9) is filled, photo- and thermal polymerization of the mixture is performed.

R2

Figure 40. Long cylinder preform with given size and distribution of the refractive index over radius R2.

Calculations required for preparing a cylinder preform S high and R2 in radius representing a copolymeric belt twisted on a cylinder bar from homopolymer with radius R1 (Figure 40) are based on the following ideas. Recall that dependence n = f(R2) of refractive index on radius of the cylinder preform is, for example, shaped as in Figure 42.

2

1

3 5

4

4

3

Figure 41. Principal features of the device for producing a copolymeric belt with given sizes (L, S, H) and axial distribution of refractive index.

Obviously, decrease of n along R1 towards the preform periphery (Figure 40) is the consequence of a decrease of volumetric proportion ϕ of the polymerized component with index n2 in the copolymeric belt from 1 to 0 and of appropriate increase of the proportion by volume (1 – ϕ) of the polymerized component with index n1. That is why the curve illustrated


73

in Figure 42 will not change if n is substituted byϕ on the ordinate axis, and the value of n2 is combined with the point of 100% content of the component with index n2, and n1 with the point where it is of zero concentration (Figure 43). The justification for such substitution was discussed above (see Section 4.6).

n n2

n1 0

R1

R2 r

Figure 42. Given dependence of refractive index (n) on radius of a long cylinder preform (R2).

ϕ 1

0

R1

R2 r

Figure 43. Dependence of volumetric proportion (ϕ) of the component with high refractive index (n2) on radius of long cylinder preform (R2).

According to a change of copolymer composition along the belt length, difference in shrinkage coefficients ρ(mon)/ρ(pol) of the components with high and low values of refractive index must lead to an appropriate change in the belt thickness. However, as this change is low due to low thickness of the belt, the belt thickness h along its length is assumed to be uniform. The twisted belt represents a spiral. In the first approximation, a series of circles with alternating radii can be considered instead of the spiral. Let us designate length of the k-th loop as lk: lk = 2π(R1 + kH). Here k is the loop number; k = 1, 2, 3, …, i.

(67)

74

N. G. Lekishvili, L. I. Nadareishvili and G. E. Zaikov Total length of untwisted belt is

m 1+ m   L = ∑ 2π (R1 + kH ) = 2π  mR1 + H m , 2   k =1

(68)

where m is the number of loops; m = 1, 2, 3, …, i. In untwisted belt (Figure 38), points dislocated along the radius of the cylinder preform (Figure 40) will be disposed at the distance 2π(R1 + kH) from each other. Based on the graph (Figure 43) and with regard to equation (68), graphs [1 – ϕ](l) and ϕ(l) are plotted point by point, where 0 ≤ l ≤ L. Further on, these curves are graphically integrated. Graphs x1(l) and x2(l) are obtained in which x1 and x2 are maximum relative displacements of plungers (5) and (6) (Figure 41). Let us denote the relation: x2/x1 = z.

(69)

and introduce the following designations:

ρ1(mon) and V1(mon) are density and volume of the monomer with n1 spent, respectively; ρ1(pol) and V1(pol) are density and volume of copolymeric belt component formed from the monomer with n1; ρ2(mon) and V2(mon) are density and volume of the monomer with n2 spent, respectively; ρ2(pol) and V2(pol) are density and volume of copolymeric belt component formed from the monomer with n2. It is clear that

ρ1(mon)⋅V1(mon) = ρ1(pol)⋅V1(pol);

(70)

ρ2(mon)⋅V2(mon) = ρ2(pol)⋅V2(pol).

(71)

and V1(mon) = πR3 x1 ; 2

(72)

V2(mon) = πR3 x2 = πR3 x1Z , 2

2

where R3 is the radius of vessels (1) and (2) (Figure 41). The volume of the copolymeric belt equals: LSh = V1(pol) + V2(pol). With regard to equations (70), (71), (72) and (73), we get:

(73)

Polymeric Media with the Gradient of the Optical Properties LSH = πR3 x1 ⋅ 2

ρ1 (mon ) ρ (mon ) + πR32 x1 z 2 , ρ1 (pol ) ρ 2 (pol )

75

(74)

which gives

x1 =

LSH . ρ 2 (mon ) 2  ρ1 (mon ) +z πR3  ( ) pol ρ ρ 2 (pol )   1

(75)

Knowing x1, x2 can be calculated from equation (63). If shrinkage during copolymerization is neglected, expression (75) takes the following form:

x1 =

LSH . πR32 (1 + z )

(76)

At mentioned above, the duration of motion of the plungers (5) and (6) in vessels (1) and (2) (Figure 41), i.e. the time during which the mixture of monomers with indices n1 and n2 is delivered into the casting mold (9) equals the duration of motion of the mold (9) at a constant rate over the distance L. Designate this duration as τa. Let us select a value of R3 for vessels (1) and (2) and calculate numerical values of x1 and x2 . Plot graphs of functions x1(τ) and x2(τ), for which purpose the value of L on graphs x1(l) and x2(l) is equal to τa. Further on, profiles of cams (5) and (6) are calculated according to the methods described in ref. [138]. An example of the cam calculation is shown in Section 4.1. The method described allows us to produce polymeric cylinders, from which optical rods and fibers with the given radial gradual distribution of the refractive index can be molded, and which are suitable for image transmission.

4.8. Method of Diffusion Copolymerization of Monomers [33, 37, 61, 147 149] (See Section 3) The maximum diameter of selfocs obtained by the known methods is limited by the restrictive conditions – by the low degree of polymerization of gel-polymer, which is required to ensure the possibility of monomer diffusion with low refractive index deep into the selfoc. A flexible rod of large diameter taken out of the mold is deformed under the effect of its own weight, and a preform with distorted circular cross-section is obtained, which deteriorates the optical properties of the gradient element. However, at relatively high conversion, which is required to ensure enough rigidity in a rod of large diameter to avoid deformation under its own weight after it is taken out of the mold, diffusion of monomer with lower refractive index is hindered. Consequently, the width of the product displaying refractive index gradient, narrows.

76


A method of obtaining a gradient element, in which the stage of gel-matrix preparation and limitations associated with it are absent, is discussed below. At the same time, the method discussed includes aspects of both the method of exchange diffusion and lit-par-lit copolymerization [11, 42, 43, 119, 120], which is indicated below.

7

8

6 5

1

4 3

h

2

h1 R

r

2d Figure 44. Principal features of device for obtaining a selfoc. See text for details.

A device, the main features of which (diametrical cross-section) are shown in Figure 44, is used to realize the method. The device consists of a cylinder tumbler (2). Pipe (6) is tightly mounted at the middle of the polished base (3) of the tumbler (2). Tumbler (2) position is fixed by logs (1) and (4). The upper end of pipe (6) passes out of pipe (5) and is connected with electric motor (8) by a worm mechanism (7). Monomer-I (or a mixture of monomers) with high refractive index is poured into pipe (6) at height h. Monomer-II (or a mixture of monomers) with low refractive index and possessing density the same as the monomer-I in pipe (6) is poured into the space between the wall of tumbler (2) and pipe (6) at the same height h. Electric motor (8) is started up, and pipe (6) is carefully removed from tumbler (2) to exclude any possibility of flow formation leading to inhomogeneous mixing of monomers I and II. As the result of exchange diffusion of monomers I and II, the concentration gradient is obtained, which is fixed by carrying out the polymerization reaction. To illustrate the essence of the method suggested, assume that after pipe (6) is removed from tumbler (2), monomers I and II do not mix and just change their spatial forms – height h decreases, and R and r increase (see Figure 44). As density and level of monomers I and II are equal in the filling zones, pressure caused by these monomers on the internal and external walls of pipe (6) is also equal. It can be assumed that reduction of height h in the zones of monomers I and II is equal under the condition that R in the monomer-I zone and r in the monomer-II zone also increase equally (i.e. by d), which is achieved at definite scales of pipe (6) and tumbler (2).


77

Let us introduce the following designations: V(I) – the monomer I volume; V(II) – the monomer II volume; V(III) – the volume of pipe (6) dipped in the monomer; V*(I) – the volume of the monomer I after removal of pipe (6); V*(II) – the volume of the monomer II after removal of pipe (6). V(I) = πR2h; V(I) + V(III) = πh(R + 2d)2; V(I) + V(II) + V(III) = πh(R + 2d + r)2; V(II) = πh(R + 2d + r)2 – [V(I) + V(III)]; V(II) = πhr(2R + 4d + r).

(77)

As pipe (6) is completely removed from tumbler (2), height of the monomer-I column decreases down to h1, and radius increases up to (R + d). Analogously, height of the monomer-II ring decreases down to h1, and thickness of the ring increases up to (r + d). Volumes of monomers are equal to V*(I) = πh1(R + d)2; V*(I) + V*(II) = πh1(R + 2d + r)2; V*(II) = πh1(R + 2d + r)2 – V*(I) = πh1(R + 2d + r)2 – πh1(R + d)2; V*(II) = πh1(2R + 3d + r)(d + r).

(78a)

It is clear that V(I) = V*(I); πR2h = πh1(R + d)2; R2h = h1(R + d)2; V(II) = V*(II).

(78b)

With regard to equations (76) and (77), we get: πhr(2R + 4d + r) = πh1(2R + 3d + r)(d + r); hr(2R + 4d + r) = h1(2R + 3d + r)(d + r).

(79)

Let us divide equation (78) by (79):

( R2 R + d )2 = . r (2 R + 4d + r ) (2 R + 3d + r )(d + r )

(80)

Equation (80) expresses the condition when, at the initial equal level of monomers after removal of pipe (6) due to equality of densities, the monomers I and II fuse in the manner that only vertical planes contact with each other, i.e. a column from the monomer I (h1 high and (R + d) in radius) exists in the center of tumbler (2) surrounded by the circle from the monomer

78


II (h1 high and (r + d) in radius). This provides for the condition for radial exchange diffusion leading to formation of radial concentration gradient, fixed then by photo- and thermal polymerization. For this purpose, tumbler (2) is covered by a pipe furnace. In this case, mutual disposition of the components resembles that given in ref. [33] in appearance, in which one of the components is a cylinder gel-polymeric rod surrounded by a medium of a liquid monomer (Table 9). The method discussed also includes signs of lit-parlit copolymerization [11, 42, 43, 119, 120], when both components participating in the gradient element formation are liquids. The generality of these external signs is mentioned above. The method can be realized with the help of the following monomers: Table 9. Monomers

ρ 420

nD20

Styrene Diallylisophthalate Methyl methacrylate Ethyl methacrylate n-Propyl methacrylate iso-Propyl methacrylate n-Butyl methacrylate iso-Butyl methacrylate tert-Butyl methacrylate 1,1,3-Trihydrotetrafluoropropyl methacrylate

0.9060 1.12 0.9440 0.9135 0.9022 0.8847 0.8936 0.8858 0.8775 1.232

1.5468 1.5254 1.4142 1.4147 1.4183 1.4334 1.4240 1.4199 1.4143 1.375

The method described allows us to generate preforms with comparatively large diameter but with strictly rounded cross-section and smooth surface. If a component with lower refractive index is poured into pipe (6), and the space between the pipe and the tumbler is filled by a component with high refractive index, a selfoc is obtained equivalent to a defocusing lens.

4.9. Preparing a Selfoc with Elliptical Cross-Section [150] A selfoc able to transform radiation with asymmetric diagram of direction from band semiconductor radiation to a point irradiator with axis-symmetric radiation, as well as suitable as a preform for molding unimode wave guides capable of retaining the polarization plane of radiation passing though it, must be of elliptical cross-section. Such a selfoc is obtained in an analogous device, in which the pipe and the tumbler possess an elliptical cross-section (Figure 44). Pipe (1) with elliptical cross-section is filled by the monomer I with high refractive index, and the space between pipe (1) and tumbler (2) is filled by the monomer II with low refractive index. Monomers I and II are of the same density.


79

2d c 1 2b 2 2a Figure 45. Projection of pipe and tumbler of the device, for producing a selfoc with elliptical cross-section.

Later, as described in Section 4.8, the pipe is carefully taken out and, after exposure, photo- and thermal copolymerization are performed. Using argument analogous to that given above, it can be shown that to provide conditions such that monomers I and II will join in the vertical plane after the pipe is removed from the tumbler, the overall size of the tumbler must be defined by the following equation:

ab a+b+d = , c(a + b + 3d + c) a + b + 3d

(81)

where a and b are the major and minor semiaxes of the pipe with elliptical cross-section, respectively; c is the distance between the external pipe wall and the internal tumbler wall, also with elliptical cross-sections; and d is the semi-thickness of the pipe wall, again of elliptical cross-section.

4.10. Preparing a Multi-Channel Light Focusing Matrix [151, 152] In Sections 4.8 and 4.9, equations (80) and (81) are deduced from the condition when parameters of the device for preparing a selfoc (tumbler (2) and pipe (1) diameters, and wall thickness of pipe (1)) are selected such that after pipe withdrawal after equal shrinkage of monomers, the radius of the central cylinder monomer column (3) and peripheral monomeric circle (4) wall thickness increase by the semi-thickness of the pipe (1) (Figure 46a). In Figure 46b, the volumes of the central and peripheral monomers are designated by V1 and V4, which fill volumes V2 and V3 that are vacated by taking out pipe (1). Hence, V1 = V2 and V3 = V4. General formulation of the regularity of the overall size change of the monomers at their equal shrinkage, independent of the cross-section profile of tumbler (2) and pipe (1), will broaden significantly the potential applications of the approach described above. Let us introduce the following designations: h1 is the initial monomer height in the tumbler and the pipe; h2 is the monomer column height after pipe withdrawal; S1 and S4 are the initial areas of the central and peripheral monomers; S2 is the base area part of pipe (1), by which the area of the central monomer increases after withdrawal of the pipe; S3 is the area of the remainder of the pipe (1), by which the peripheral monomer area increases after pipe

80


withdrawal. Therewith, values of S2 and S3, and V2 and V3, respectively, are not regulated by the semi-thickness of pipe (1) in either Section 4.8 or 4.9. S2+S3

S2+S3 1 2

S1

S4

3

S1

S4

V1

V4

4

V2

h1

V3

h2 a

b

Figure 46. See text for details.

It is clear that at even shrinkage of monomers, volumes V2 = S2h2 and V3 = S3h2 that become free after pipe (1) withdrawal are occupied by the central monomer volume V1 = S1(h1 – h2) and the peripheral monomer volume V4 = S4(h1 – h2), respectively. Hence S1(h1 – h2) = S2h2, S4(h1 – h2) = S3h2, which gives S1/S4 = S2/S3,

(82)

H2(S2 + S3) = (h1 – h2)(S1 + S4).

(83)

These equations express the change in the overall size of the central and peripheral monomers, possessing equal density, after the pipe withdrawal and are true for any crosssection profile (circle, ellipse, etc.) of the tumbler and the pipe. Based on equation (83), multichannel optical devices can be prepared, a single matrix of which will contain several selfocs simultaneously, which may be similar or different in area and cross-section profile, and chemical composition. For this purpose, the required number of pipes with chosen (similar or different) profiles and values of the cross-sectional area are placed into the tumbler. Pipes are filled with the same or different unsaturated fluid monomers (a mixture of monomers) with equal (equalized) density up to the same height, and the space between the pipes and the tumbler is filled by a monomer (a mixture of monomers) possessing the same density but lower refractive index up to the same height. Then all the pipes are withdrawn simultaneously. The system is now unrestrained, so that radial gradient of the composition, which is determined by the performance of the copolymerization, is achieved by diffusion exchange. Parameters of


81

pipes and the tumbler used in this case are related by the following equation in the reduced form of expression (83):

S (1) + S 4(1) S1( 2) + S 4( 2) S1(3) + S 4(3) S1( n ) + S 4( n ) h2 , = 1(1) + + + + ... h1 − h2 S 2 + S 3(1) S 2( 2) + S 3( 2) S 2(3) + S 3(3) S 2( n ) + S 3( n )

(84)

where h1 is the initial height of monomers in the tumbler and pipes;

S1(1) , S1( 2) , S1(3) , …, S1( n ) are areas of the apertures of the first, second, third, . . . up to the n-th pipe, respectively;

S 2(1) , S 2( 2) , S 2(3) , …, S 2( n ) are the parts of the tumbler base area occupied by the first, second, third, . . . up to the n-th pipe, respectively, in contact with the central monomer;

S 3(1) , S 3( 2) , S 3(3) , …, S 3( n ) are the remaining parts of the second, third, . . . up to the n-th pipe wall areas, respectively;

S 4(1) is the area of the peripheral monomer surface, when a single pipe is introduced into the tumbler;

S 4( 2) , S 4(3) , …, S 4( n ) are increments of the peripheral monomer surface when the second, third, . . . up to the n-th pipe are introduced into the tumbler. The total area of the apertures in the tumbler is i =n

i =n

i =n

i =n

i =1

i =2

i =3

i = n −1

S = ∑ S1(i ) + ∑ S 2(i ) + ∑ S 3(i ) + ... +

∑ S n(i−)1 + S n(i ) .

(85)

Table 10. Comparison of production methods for light focusing elements

Production method Two-stage copolymerization (diffusion copolymerization) [43, 44, 46] Lit-par-lit copolymerization (photocopolymeri-zation) [104 – 106, 112 – 114] Diffusion copolymerization of monomers

Polymers used Linear Spatial

Optical fiber molding



+



Multifunctional light focusing matrix obtaining +

+

+

+



+

+

+

+

Describing conditions of the equal shrinkage of monomers, equations (84) and (85) give the possibility for simultaneous regulation of changes of monomer overall sizes (h2, S2 and S3) and, consequently, overall size of light focusing elements. The method of diffusion copolymerization of monomers includes positive aspects of the above-mentioned methods of

82


two-stage polymerization [33] and photocopolymerization [123], both from the point of view of technological realization and characteristics of light focusing matrices obtained (Table 10).

4.11. Study of the Effects of γ-Irradiation on Optical Properties of Isotactic Polypropylene [153] All the above-mentioned methods for preparing GRIN-elements, in which the GRINforming process decelerates the diffusion/polymer-analogous transformation deep into the material, are characterized by limiting of the diameter/thickness of the element, as well as the difficulty of RID variation. However, ionizing radiation produces serious structural changes in polymers – degradation, branching and network structure formation, which produce significant change of some physical properties, including optical ones [154, 155]. That is why it is of special interest is study the possibility of using ionizing radiation as a GRIN-forming influence, γ-radiation, in particular, possessing an ability to penetrate deep into materials. This ability is discussed below using the example of isotactic PP (15 µm thickness) by studying the changes that are produced by radiation in the values of the most relevant parameters the characteristics of absorption in the UV-spectrum and refractive index in the visible region of the spectrum (radiation source, 137Cs isotope; radiation temperature, 25 ± 3°C, in air; radiation power, 2.5 Gr/min; quantum energy, E = 0.662 MeV; radiation dose, R ≤ 60 kGr; the range of low doses is selected because it is essential to preserve the optical properties in the visible spectrum). Note that for PP, degradation and cross-linking effects on γ-irradiation are equally probable [154].

µ ν~ , cm -1 700 600 500 400 300 200 100 0

54

50

45

40

35 3

ν ⋅10 , cm -1

Figure 47. Absorption spectra for PP films. The continuous line marks the initial sample; dotted line – after radiation by R = 28.8 kGr dose.

Figure 47 shows spectral dependencies of general losses index in the UV-spectrum, æ ν~ , for the initial PP films and those which have been γ-irradiated with a dose R = 28.8 kGr.


83

Clearly, PP possesses an initial bright absorption band near frequency ν~ = 50⋅103 cm-1 (λ ≈ 200 nm), in which the tail spreads to ν~ = 27⋅103 cm-1 (λ ≈ 370 nm). In the visible spectrum, the samples remain transparent even after irradiation for given doses, thicknesses and spectrophotometer sensitivity. After γ-radiation in the range of low doses R ≤ 60 kGr, an increase of the refractive index in the main band near λ ≈ 200 nm is observed. Therewith, wavelengths of additional absorption maximums in spectra are almost coincident with wavelengths of extremes of the initial absorption in the main absorption band. Figure 48 shows fixed wavelengths for which ∆æ ν~ = f(R) graphs are plotted. They are: (1) and (3) – real maximums in the spectral absorption curve; (2) – the hidden maximum coincident with the bending point of the spectral absorption curve.

∆µ ν~ , cm -1 160

1

120 80

2 3

40 0

2

4

6

8

R, 3.6⋅103 Gr

Figure 48. Dependencies of additional absorption ∆æ in PP films on γ-irradiation dose R for three fixed wavelengths of absorption maximums on the dependence ∆æ = f(λ); 1 – 201 nm; 2 – 228 nm; 3 – 278 nm.

High loss intensity of æ ν~ = f(ν~ ) at frequency ν~ = 49.8⋅103 cm-1 (λmax ≈ 201 nm) in the main absorption band (Figure 47) is produced not only by absorption, but also by the contribution of scattering in crystalline PP film; and the complicated shape of the band with secondary absorption bands at λmax ≈ 228 nm and λmax ≈ 278 nm are, apparently, the result of the presence of additive chromophore groups [156, 157]. The main absorption band at λmax ≈ 201 nm consists of the superposition of an electron singlet-singlet absorption band long-wave rim and additive chromophore absorption band in spectral areas of ~60 – 200 nm and ~180 – 210 nm [158], respectively. This suggestion is confirmed by almost identical run of ∆æ ν~ = f(R, λ) dependencies for all three wavelengths of additional absorption maximums of PP (1), PP (2) and PP (3) in Figure 48. At amorphous and crystalline PP radiation, the latter possesses several-fold higher concentration of radicals at 20°C [159]. This is the reason for such high additional absorption,

84


∆æ ν~ = 110 – 190 cm–1, in the main absorption band with λmax ≈ 201 nm. On the other hand, the amount of chain breaks in PP at doses R = 25 – 50 kGr is 1.5 – 1.8 times higher in air, than in vacuum [160], i.e. the degrading influence of air oxygen is obvious, especially in the present case of after-radiation study of PP thin films, which also provides for ∆æ ν~ increase. The rate of accumulation of vinylidene groups during degradation of polymeric chains in isotactic PP is highest at low doses of γ-radiation [161]. However, variations in competitive processes involving their accumulation and degradation were apparently first observed in the present case, which is clearly seen in Figure 48. Let us discuss the problem of realization of the given radial profile of the refractive index in polymeric medium using the example of thin-film PP (Figure 49).

∆n 0.02

0.015 0.01

0.005

0

1

2

3

4

5

8 7 6 3 R, 3.6⋅10 Gr

Figure 49. Dependence of refractive index variation ∆n on γ-radiation dose R absorbed in PP film.

Occurrence of additional absorption in the far UV-spectrum in irradiated samples at the sacrifice of C-C bonds and production of vinylidene groups points out a possibility of a change in refractive index in the visible spectrum, where they are transparent, and with loss of the anomalous dispersion tail [162]. The dependence curve of ∆n = ϕ(R) at the wavelength λ = 632.8 nm is shown in Figure 50, where it may be observed that the after-radiation change of the refractive index rises with the γ-radiation dose, and at R > 14.4 kGr yields to the saturation branch, ∆nmax = 0.017. Shift of interference bands in a Mach–Zender interferometer at ∆n change of chlorinated PP films [134] and radiated PP films moves to the same side relative to the system of interference bands of the initial PP film. Hence, ∆n > 0, as may be expected due to the anomalous dispersion curve run, because ∆n > 0 at ∆æ > 0. It is common knowledge that, according to the empirical Gladstone–Dale formula and the theoretically proven Lorentz–Lorenz formula, the refractive index of optical material is proportional to its density. However, despite density decrease in the range of doses R < 6.0


85

MGr (for polyethylene, for example) a slight increase of dielectric permeation at frequencies from 10 kHz to 60 MHz is observed: ε = 2.27 for non-radiated and ε = 2.29 for polymer radiated by 640 kGr dose [163]. This indicates the predominant influence of the anomalous dispersion curve run on the refractive index change n = f(R) and the possibility of at least one more part of increment and saturation ∆n at high radiation doses of polymeric materials, isotactic PP, in particular, compared with doses indicated in this paragraph. The above-discussed experimental results provide for creating a refractive index profile of any shape in PP film by varying the exposure dose of γ-radiation on the surface sample [164]. This problem is discussed below.

4.12. Obtaining a Macrosurface GRIN-Medium by γ-Irradiation of Isotactic Polypropylene [164, 165] One of the variants for creating a given axial/radial gradient of the refractive index involves simultaneous treatment of the polymeric sample surface by γ-radiation, while a mask from a material absorbing radiation, for example, lead, is placed between the γ-radiation source and the sample. Therewith, the axial/radial distribution of the mask thickness is selected so that the real distribution of radiation by length/radius of the sample corresponds to the experimentally found doses required to achieve the given axial/radial change of the refractive index. The regime of γ-radiation to obtain the GRIN-medium is based on three graphs: 1. ∆n = f(R) – the experimentally found dependence of refractive index change, ∆n, on the dose absorbed of γ-radiation, R. This graph is shown in Figure 50. 2. K = ϕ(d) – the multiplicity dependence of γ-radiation intensity reduction, K, on lead mask thickness, d. Therewith, K = R0/R∆n, where R0 is the dose of γ-radiation affecting the lead mask; R∆n is the dose of γ-radiation passing through the lead mask. Such a graph for the quantum energy of 137Cs radiation of E = 0.662 MeV, obtained by averaging values E = 0.6 MeV and E = MeV from [164], is shown in Figure 51. 3. n = f(r) – the given RID, where r is the length/radius of the sample. Figure 52 shows the given (approximate) distribution of the refractive index difference, which must be realized in PP film of 30 mm radius. According to Figure 50, for the maximal change of the refractive index (∆n = 0.017), the PP film must be irradiated by the dose R = 14.4⋅103 Gr. For such part of the film, the lead mask thickness will be minimal, for example, d = 1 mm. As lead 1 mm thick reduces γradiation by K = 1.1 times (Figure 50), the maximal change of refractive index can be reached by increasing the γ-radiation dose affecting the lead mask up to: R0 = R(∆n = 0.017)⋅K(d = 1 mm) = 14.4⋅103⋅1.1 = 15.84⋅103 Gr. Hence, on the entire surface of the lead mask, the dose of γ-radiation is R0 = 15.84⋅103 Gr. On the part of the mask, where d = 1 mm, a part of the γ-radiation is absorbed, and R(∆n =

86


= 14.4⋅103 Gr passes through the mask and causes structural changes in the PP film, which will lead to increase of the refractive index by ∆n = 0.017. At the distance r = 3 mm from this part, ∆n = 0.0167 (Figure 51). To reach this value of ∆n, the radiation dose R(∆n = 3 0.0167) = 13.932⋅10 Gr is necessary according to Figure 50. That is why the γ-radiation dose affecting the mask surface must be decreased according to 0.017)

K=

R0 R( ∆n=0.0167 )

=

15.84 ⋅10 3 = 1.137 . 13.932 ⋅10 3

As shown in Figure 50, as K = 1.137, the lead layer thickness is d = 3 mm. Thickness of the lead mask for other points is calculated analogously. These results are summarized in Table 11. K 80

70 60 50 40 30 20 10 10

20

30 40 dPb, mm

Figure 50. Dependence of multiplicity of γ-radiation attenuation K on lead mask thickness d at radiation quantum energy E = 0.662 meV

∆n, ×10

-3

15

10

5

0

5 10 15 20 25 30 r, mm

Figure 51. Given distribution of different increments of refractive index ∆n by length/radius r of PP-film


87

The lead mask, for which dependence d = ψ(r) corresponds to the given Table 11 (instead of ∞, a thickness is conditionally chosen, for example, 40 mm), is placed between the source of γ-radiation and PP film (Figure 52). As the γ-radiation power is 2.5 Gr/min, to obtain R0 = 15.84⋅103 Gr radiation dose, the exposure time is 105.6 hours [165].

a

b

1

1

2

2

3

3

30 mm

60 mm

Figure 52. GRIN-element production with given axial (a) and radial (b) distribution of refractive index (principal features): a – front view; b – diametrical section; 1 – γ-radiation source; 2 – lead mask; 3 – PP-film

ϕ, ° 360

τ, h 100

337.5

90

300

80

262.5

70

225

60

187.5

50

150

40

112.5

30

75

20

37.5

10

0

0

40 r, mm

10

20 30

10

20 30 40 ρ, mm

Figure 53. Dependence of γ-radiation duration τ on radius r of PP-film (additional coordinates are explained in text)

88


In the technical solution described, the duration of γ-irradiation of the entire PP film surface is the same, and formation of a refractive index gradient is brought about by the specific gradient of the radiation dose over the length or radius of the sample. For completeness sake of the presentation, let us discuss another solution, when the radiation dose gradient is formed by the radiation duration gradient over the sample radius at equal specific radiation dose. Using data from Table 1, we determine that

τ=

R∆n (Gr) , 2.5 (Gr/min)

(86)

where τ is the duration of γ-radiation, during which the radial gradient of the radiation dose is provided along the sample radius, which is necessary for creating the given radial gradient of the refractive index change (Figure 51). The appropriate graph of dependence τ = f(r) is shown in Figure 53. In the general case, when the given RID, n = f(r), is described analytically, and the experimental dependence ∆n = f(R) = ∆nmax⋅(R/Rmax) is used in its linear part (∆nmax is the maximal value of ∆n on the linear part ∆n = f(R); Rmax is the maximal dose of absorbed γradiation corresponding to ∆nmax), the profile of the lead mask d = f(r) can be calculated using the expression n(R) = n0 + ∆nmax⋅eæd, where n0 is the initial (before radiation) value of the refractive index; æ = 0.087 mm–1 is the absorption coefficient for 137Cs γ-radiation in lead calculated according to table values [166] (e.g. Figure 50) with the help of expression for γ-radiation reduction: K = f(d) = eæd. Suggesting that ∆n ≈ 0, for example, at K = eæd = Rmax/R ≈ 100, we obtain æd ≈ ln100 and d ≈ 53 mm.

ρ

ϕ

(30 mm, 360°)

Figure 54. Lead mask window contour (polar coordinates: ρ - radius-vector; ϕ - angle between the radiusvector and the polar axis)


89

For practical realization of the dependence shown in Figure 54 a lead mask with a window, the contour of which can be represented by the function F(ρ, ϕ), where ρ is the radius-vector, and ϕ is the angle between the radius-vector and the polar axis. The value of ϕ corresponds to the determined values of radiation duration, and ρ = r (additional coordinates in Figure 53). Figure 54 presents function F(ρ, ϕ) in polar coordinates. A lead mask with diameter of, for example, 80 mm and with such window in the center (for the remainder with thickness of, for example, d ≥ 40 mm, at which the influence of penetrated radiation dose on properties of PP film is neglected under present conditions) is placed between the γ-ray source and PP film. The centers of the sample and the mask are superposed, around which the sample or the mask rotate for 96 hours (τmax = 96 h). The rotation frequency is chosen so that at incomplete rotations the lack of γ-radiation on some parts of the PP film can be neglected [165]. Great depth of γ-radiation penetration and continuous structural changes in the polymer thickness accompanied by GRIN-forming influence allow us to create RID profiles of any shape in optical polymeric materials by distributing the intensity or γ-radiation exposure on the surface, independent of the size and shape of the material. Table 11. Distance from one of the ends of processed part/length of the sample radius, r (mm) 0 3 6 9 12 15 18 21 24 27 30

γ-Radiation dose ∆n (see Figure 50) 0.017 0.0167 0.016 0.015 0.0139 0.0123 0.0107 0.0088 0.0065 0.0042 0

required for transfor-mation, R∆n (Gr) (see Figure 50) 14.4⋅103 13.932⋅103 10.8⋅103 9.252⋅103 7.92⋅103 5.4⋅103 4.068⋅103 2.88⋅103 1.543⋅103 0.936⋅103 0

Required multiplicity of γ-radiation subsidence, K = R0/R∆n 1.1 1.137 1.257 1.7 2.0 2.93 3.894 5.5 10.23 16.928 ∞

Lead mask thickness, d (mm) 1 2.5 3.5 5.5 8 11 15 18 23 28 ∞

The present method illustrated using the example of thin-film isotactic polypropylene can be easily extended to other types of polymers with regard to their sensitivity to γ-radiation and wider applicability to optical instrument making.

4.13. Method of Producing a Gradient Birefringent Element The general practice, world-wide, in gradient optics has been to focus attention only on the refractive index gradient. However, materials with gradients of other optical properties are of interest also, and in particular, with a gradient of birefringence (GB). Optical polymeric films with linear GB can be used in polarimeters used for various applications, including

90


equalizers for measuring birefringence value and studying character of birefringence distribution, etc. The preparation and description of a device for creating a given axial GB in polymeric films is discussed in this Section [167, 168]. It is suggested that such films should be called GB-elements or GB-materials [167, 168]. Supposedly, successful creation of GB-elements, investigation of properties and determination of fields of their application require broadening of the notion of ‘gradient optics’, subdividing it into two groups – GRIN-optics (optics of materials with refraction gradient) and GB-optics (optics of materials with birefringence gradient). It is common knowledge that deformation of polymers above the glass transition point is accompanied by uncoiling of flexible chains of macromolecules and orientation of their segments in the stretch direction. The sample thus obtains the symmetry of a one-axis crystal, the optical axis of which coincides with the stretching direction. Orientation of optically anisotropic molecular chains is the reason for the occurrence of birefringence in the polymer, which is a function of the relative stretch deformation of the sample: ∆n = n1 – n2 = γλ,

(87)

where ∆n is birefringence; n1 and n2 are refractive indices of singular and ordinary rays, respectively; γ is the optical deformation coefficient; λ is the relation of elongation to the initial sample length. At one-axis stretching of a rectangular-shaped polymer film, elongation along the width of the film is practically1 the same. That is why the orientation degree and, consequently, the value of birefringence in the direction perpendicular to the stretching are the same. If a trapezium-shaped (equilateral/non-equilateral) sample is stretched, i.e. when clips are disposed on not parallel lines, but at some chosen angle (Figure 55), then the relative elongation over height h will be different. It will increase from the longer side of the trapezium to the smaller one. This, according to equation (87), provides for GB growing in the same direction. Clearly, if at the long (free) edge of sample (2) (at the longer side of the trapezium) elongation is zero, and then the range of GB will be maximal. Described below is a device for obtaining a GB-element. It allows formation of an inhomogeneous mechanical field with the required degree of inhomogeneity, application of which to the sample forms the given axial gradient of elongation (the given gradient of orientation degree of molecular/permolecular structure) and, consequently, the given axial GB.

1

Strictly speaking, elongation along the sample surface is different even under conditions of homogeneous stretching, i.e. if a ‘neck’ does not occur. Elongation increases on both sides from the central part of the sample in the direction of their free (unfixed) edges, which shows as narrowing of the sample, which, in its turn, increases from fixed ends of the sample to its center in the stretch direction. Relative values of narrowing and their distribution in the stretch direction depend on relation of the ratio of rectangle side lengths and the deformation mode (rate, relative elongation, temperature, etc.). Nevertheless, this difference in values of elongation can be neglected, because it is insignificant for forming a practical GB.


∆l

4

h

91

1

3 ∆l

2

Figure 55. Scheme of sample stretching without elongation gradient.

The principal features of the device are shown in Figure 56. Sample (8) (a film, a plate) trapezium-shaped, x high, is fixed by two clips (3) and (6) with non-parallel chamfered rims. To demonstrate the character of elongation distribution, a grid can be applied to sample (8). Clips (3) and (6) are rotated in opposite directions around parallel axes (4) and (5) by an electric motor with reducing gear (1) and gearwheels (2) and (7). As clips (3) and (6) rotate, their rims, in fact, are moving along the surfaces of round cones, generatrices of which are disposed at angles of ϕ1 and ϕ2 to the heights of the cones, coincident with rotation axes (4) and (5) in space. Clips (3) and (6) (clip rims) can be separately set at different angles to rotation axes (4) and (5). Therewith, different variants are available: ϕ1 ≥ ϕ2, or ϕ1 > 0 and ϕ2 = 0. Points of intersection of the rims of clips (3) and (6) with rotation axes (4) and (5) are static (immovable). The distance between them equals the length of the greater side of the trapezium – the maximal length of the initial sample (8) and remains constant during elongation: lmax = l(x = 0; 0° ≤ α ≤ 90°) = const, where α is the rotary angle of clips (3) and (6), whereas moving from the longer side towards the smaller one of the trapezium, i.e. from x = 0 to x = h, the absolute elongation increases. 1 2

7 lmax 8

x 1 2 3

6

3 ϕ1

h 4

ϕ2 5

Figure 56. Scheme of device for sample stretching with elongation gradient (see text for details)

92


As shown below, change of the sample appearance on elongation is governed by complicated rules. If the process of deformation is idealized and we assume conditionally that change of distances between rims of the clips (3) and (6) (Figure 57) accurately correspond to the appearance and overall size of the sample, the following suggestions can be made: • •

As clips (3) and (6) are rotated in the range 0° ≤ α ≤ 90°, both longitudinal and transverse elongation occur, the latter reducing the degree of anisotropy; It can be shown that for the case ϕ1 = ϕ2, length and width increments are ∆l =

(

2x⋅tgϕ(1 – cosα) and ∆h = h 1 + sin •

2

)

α ⋅ tg 2ϕ − 1 , respectively;

Conditions under which the inequality

∆l/2 > ∆h,

(88)

is fulfilled must be estimated, as the criterion for providing dominant orientation and, consequently, advisability of application of the method. It can be easily shown that inequality (88) is fulfilled, when inequality cosα < h/R is true (where R is the base radius of the imaginary cones, along the surface of which clips (3) and (6) are moved). Therefrom, it follows that if h > R, then inequality (88) is true at any values in the range [0°, 90°]. If h < R, inequality (88) is true at any values of α in the range [arccos (h/R), 90°]. Clearly, if α > 90°, ∆h decreases, and the sample must be narrowed, whereas ∆l increases continuously; •

Distribution of sample widths in the general form (ϕ1 ≥ ϕ2, or ϕ1 > 0 and ϕ2 = 0) is described by the relation:

l ( x;α ) =

[x sin α (tgϕ1 − tgϕ 2 )]2 + [l ( x;0) + x(1 − cosα )( tgϕ1 + tgϕ 2 )]2 ,

(89)

where l(x; α) is the sample length on the selected part (at distance x from the longer trapezium side) at clips rotation by angle α; l(x; 0) is the initial sample length on the same part. Relation (89) is deduced from the condition that the line connecting static points on rotation axes (4) and (5) (the longer side of the trapezium in Figure 57) is parallel to the direction in which the sample is stretched. To preserve this condition, clip (6) is designed with the possibility of being displaced by the trapezium height, as well as along the stretch direction; •

If ϕ1 = ϕ2, then as clips are rotated by 90° the length of sample (8) remains the same over the whole width x and equals the maximal initial length:

l(0 ≤ x ≤ h; α = 90°) = l(x = 0; 0° ≤ α ≤ 90°); •

The sample attains a rectangle shape; its width equals the length of the generatrix of the imaginary cones, on the surface of which clips (3) and (6) are moving.

Clip rims can be not only linear, but may possess a more complicated, almost arbitrary shape, when ϕ1 and ϕ2 are different for separate parts or are changing continuously. For such


93

cases, equation (89) can be reduced to another form taking into account that r = xtgϕ, where r is the clip rim distance from rotation axis in the stretch direction on the value of x chosen. In the device described, the maximal rotation of clips (3) and (6) is αmax = 90°. As α > 90°, sample (8) will be applied to the rims of clips (3) and (6). Therewith, constructions are suggested in which αmax = 180° [267]. Obtaining of a GB-element is illustrated below using the example of an isotropic amorphous PVS film, deformed in water at 20°C, at an angular rotation rate of 0.0785 s–1.

lmax = 72 mm x 1 2 3

ϕ1 = ϕ2 ≈ 46° ϕ2

ϕ1 45 mm

h = 33 mm h

h 10 mm

a

18 mm 1

0

1

h = 24 mm ∆h = 9 mm 3 mm

b Figure 57. Topographic image of deformation distribution in PVS film at stretching with elongation gradient (see text for details)

Figure 57 shows a typical scheme of the sample with applied grid before and after elongation (rotation angle of clips α = 90°). A complex connection between features of the superposed inhomogeneous mechanical field and topographical picture of deformation distribution is observed. The sample is elongated inhomogeneously. The area of dominant deformation, localized in the sample center in its upper part, gradually spreads down at the whole width of the sample. At maximal (theoretical) elongation of the lower rim of the sample, equal to 620%, longitudinal elongation is ∆h = 8.5 – 9 mm (~75% from theoretical

94


one). Gradient deformation provides for clearly expressed dominant narrowing of the sample in the upper part. The initial horizontal rim of the sample corresponding to x = 0 is elongated by ~10%. The shortest width of the sample is 24 mm. The sample material shifts downwards, and the non-curved horizontal line represents somewhat a ‘center of gravity’ of narrowing (dotted line in Figure 57, b), which in the case of parallel clips applied is disposed in the central zone of the sample and in the present case is strongly shifted to the side of the most elongated part. Figure 58 shows quantitative characteristics of a GB-element produced by the abovementioned method. ∆n d, µm ∆l, % 140

140

700

120

120

600

100

100

500

80

80

400

60

60

300

40

40

200

20

20

100

2 3

5

10

1

15 20

25 h, mm

Figure 58. Distribution of elongation (curve 1), thickness (curve 2) and birefringence (curve 3) by width (h) of GB-element from PVS.

I 1.0 0.8 0.6 0.4 0.2

5

10 15 20

25 h, mm

Figure 59. Dependence of relative light intensity passing through a PVS film with GB by width (h) of GBelement.

Curve 1 (Figure 58) displays distribution of longitudinal elongation by PVS width deformed in an inhomogeneous mechanical field. Countdowns are made for the most deformed part of the film, marked by digits 1-1 in Figure 57, b. It is clear from the Figure that


95

a jump of elongation exists in the sample, changed by a decimal degree. Under special conditions, the elongation jump can be increased by 2 – 3 decimal degrees [167, 168]. Curve 2 (Figure 58) shows dependence of the sample thickness on its width. Countdowns are taken by the zero line located in the middle of the sample (Figure 58, b). As would be expected, the sample thickness is reduced as longitudinal elongation increases. Finally, curve 3 (Figure 58) shows the distribution of birefringence over the width (on the zero line) of the same sample. Measurements were performed with the help of a Berek capacitor. The Figure shows that the sample possesses clearly displayed axial GB in complete concordance with the character of elongation (curve 1) and thickness (curve 2). Figure 59 shows dependence of the relative light intensity passing through the same sample from PVS with GB (wavelength λ = 0.63 µm, junction Nicols). Countdowns were made by the same zero line (Figure 58b). The Figure shows that light intensity passing through the PVS film is modulated depending on values of h coordinate with quite good correspondence to the known expression [169]: I = I0sin2[(π/λ)⋅∆n⋅d],

(90)

where I0 and I are light intensities after passing polarizer and analyzer, respectively. Deviation of the experimental curve by both Imax and Imin values, and the form of expression (90) should be, apparently, related to local inhomogeneities in the film occurring during its formation as a one-axis GB-element [170]. In the authors’ point of view, these very initial results already point out unambiguously that GB-elements, polymeric films with the given axial GB, can be prepared according to the method suggested. Let us just note that recently a device has been suggested that essentially allows creation of radial GB in a polymeric film [171].

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In: Polymer Reactivity ISBN 1-60021-263-8 Editors: G. E. Zaikov and B. A. Howell, pp. 103-115 © 2006 Nova Science Publishers, Inc.

Chapter 2

FUNDAMENTAL REGULARITIES OF THERMAL OXIDATION OF HEAT-RESISTANT HETEROCHAIN POLYMERS E. V. Kalugina, K. Z. Gumargalieva1* and G. E. Zaikov2 Polyplastic Co., 14A, General Dorokhov st., Moscow 119530, Russia 1 N.N.Semenov Institute of Chemical Physics, 4, Kosygin st., Moscow 119991, Russia 2 N.M.Emanuel Institute of Biochemical Physics, 4, Kosygin st., Moscow 119991, Russia

The papers [1-4] considered phenomenological features of degradation process developing in polysulfones (PSF) and polyesterimide (PEI), poly(alkaneimide) (PAI) and polyphthalamides (PPA) in the presence of oxygen at processing temperature (300 - 400°C) and solid-phase oxidation (150 - 250°C). These processes define loss of operation properties by polymers and materials derived from them. Before passing to description of attempts to decelerate degradation processes, let us generalize the above-mentioned experimental material and present the mechanism of thermal oxidation of the studied polymers. The primary important similarity in the degradation behavior of all studied TP is their absorption of oxygen already at relatively low temperatures (150°C). At temperatures, when oxygen absorption kinetics (200°C or higher) becomes possible to trace, polymers display identical kinetic type of oxidation with no respect to the test temperature (refer to [1 - 4]), the process proceeds in two stages, subsequently obeying order one and zero laws. The existing data from literature [5] show that kinetics of the solid-phase oxidation includes one more stage, associated with self-exited acceleration of the process. Such complex kinetic type of oxidation was not previously observed. Of special interest is the initial stage of aging, during which oxygen absorption and CO2 release follow the order one kinetic law. At this very stage physicomechanical characteristics of heat-resistant polymers decrease significantly. Quick development and completion of this stage nudges possible anomalies in the structure of polymers or additives. If the first stage of aging of PEI and PSF, PAI and PPA, liquid-crystal copolyesters (LCP) etc. is the artifact, and

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it may be removed by some technical steps, Kinetics of the process leads to quite widespread type of thermal oxidation, when primarily, slowly developing thermal oxidation with the induction period transits to the self-excited acceleration mode. This indicates the radical-chain branched oxidation process. As discussed in [1 - 5], the induction period of oxidation may be observed at PSF thermal oxidation in the system with oxygen deficiency. However, correspondence of the first oxidation stage to inadequacy of the polymer structure is impossible, because it argues the experimental data. Let us discuss these contradictions in more detail. In the first stage, up to 10% of carbon is removed with carbon oxides. Relation of this percent to the content of anomalous structures in the polymer indicates that different structure degree is too high, much higher than the sensitivity of spectral methods. Meanwhile, preliminary analyses of studied thermoresistant polymeric materials (TP) showed their correspondence to postulated formulae, i.e. PSF corresponds to bisphenol A-derived polysulfone structure, PAI corresponds to fatty-aromatic polyimide derived from PPA and dodecamethylene diamine, etc. Further on, we will discuss in detail the effect of organic and inorganic additives on thermal stability of TP, which will show their negative impact on thermal transformation rate. Anyway, this effect is drastically lower than displays of the first TP aging stage. Moreover, differences in kinetics of different stages of aging are not accompanied by the change in composition of products released at every stage. No typical products of thermal transformations of solvents, used in syntheses of corresponded polymers, possess oligomers, which release should be expected under acceptance of regular structure deterioration by anomalous, thermally labile groups. Meanwhile, at high-temperature thermal oxidation a broad selection of heavy oligomeric products: all homologues in PAI and a selection of structure fragments in PSF, PEI, PPA, LCP, etc. According to [6], defect zones are the first which degrade and initiate degradation of the main polymer structure. The most practical statement is that the first stage of TP aging is associated with the features of their chemical structure. This kinetic type of oxidation is displayed by aromatic polyimide (PI), polyphenylene quinoxaline (PPQ) and copolyimidophenyl quinoxalines [5]. The second similarity in TP degradation behavior is formation of the similar type of the degradation product (with respect to structure of the elementary unit). More precisely, it is pyromellite diimide (PDI) in PAI and PI:

and it is terephthalic acid (TPA) amide for PPA

Fundamental Regularities of Thermal Oxidation …

105

It is also 2,2’-1,4-phenylene-bis-(3-phenylpyrazine) (PPP) for PPQ

Formation of these products as a result of thermal oxidation of appropriate polymers was not previously known in the literature, but gave new information on understanding of the TP aging mechanism. To separate such fragments, one should make an “impact” and “tear out” these structures from the macromolecule. The relation of PDI and CO2 outputs (in moles) at PI thermal oxidation (300°C) is close to 1/24, and for PAI (200°C) – 1/10, i.e. diamine components must be oxidized completely to CO2. If the primary oxygen attack is random, as oxidation develops a definite tendency to activation is observed. The random O2 attack on the next amine residue would cause formation of several oligomeric structures with appropriate end groups. However, besides the mentioned fragments, no other oligomeric products were detected in the “cold ring” or at PPA and PAI aging directly on the surface of mold samples. Thus, TPA amid formation in PPA, PDI in PAI and PI, PPP in PPQ and copolyimidophenyl quinoxaline (CPIPQ), and the absence of such oligomeric compounds among these products testifies that oxidation of a single amine residue activates oxidation of the nearest amine residues, remaining the oxidizing component unchanged and releasing it under the same conditions, in the form of individual compound (PDI, PPP, TPA-amide). Invulnerability of the oxidation component to O2 attack and, under these conditions, stability of the imide cycle, preserved even in degradation products, eliminates the thermohydrolytic mechanism, discussed in the literature. Thus, during TP aging oxidation is aimed at amine residues of macromolecules; oxygen is added to these structures and oxidizes them: PDI, PPP and amide-TPA are exclusive products of thermal oxidation. Therefore, the role of oxygen may not be reduced to oxidation of thermal degradation products. According to the data obtained [7], the absence of nitrogen-containing products and nitrogen content (%) increase in oxidized residue are typical of PI oxidation. This was explained [8, 9] by easiness of nitrogen introduction into combination reactions; therefore, it remains in the polymer residue until deep degradation stages. It has been found [5] that PDI

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release at PI oxidation in the solid phase under relatively “soft” conditions”, when the film transformation degree is below 5%. Intermolecular crosslinking is general for all TP which leads to full loss of polymer solubility in the solvents, where they dissolved before thermal oxidation. In the absence of oxygen crosslinking at the solid-phase oxidation, i.e. at relatively low temperatures, either proceeds or proceeds at much lower rate. During high-temperature oxidation oxygen also activates crosslinking and branching processes. Apparently, this is one more proof for the activating effect of oxygen at some place of macromolecule and then along the backbone. This is the only way to explain formation of products (TPA amide, PDI, PPP) in appropriate polymers. In this case, common hypothesis about molecular complex formation between a macromolecule and oxygen is actual. Hydrogen release was also associated with the crosslinking process. Hydrogen is the main product of purely thermal degradation of PI and other thermoresistant engineering plastics (TIP) above 500°C. In the presence of oxygen, H2 release at high temperature is abruptly decreased. Hydrogen is not released at oxidation of aliphatic hydrocarbon polymers, below 300°C, i.e. in the well-studied process reliably described by the radical-chain scheme. Hydrogen was identified among thermal oxidation products, such TP, as PEI, PAI, PI and PPQ under soft conditions, and O2 initiates its formation. Apparently, O2 interaction with the aromatic structure activates C-H-bond break, which then enter intermolecular interaction. Analysis of external manifestations of PI, PPQ, PAI, PPA, PEI, PSF, etc. thermal aging induces a conclusion about general regularities: -

kinetic features of the oxidation process are similar; with respect to structure of the elementary unit, the selection of thermal oxidation products is also similar; tendencies to branching and crosslinking of macrochains are the same.

Thus, thermal oxidation mechanisms of studied polymers are similar. Differences in thermal behavior are of the quantitative type. Let us dwell on consideration of thermal oxidation at high processing temperature. Let us start with O2 molecule. The strength of O-O bond in this molecule is 493 kJ/mol. TP degradation form products containing a half of O2 molecule: CO, H2O, quinones, i.e. O2 molecules dissociate during oxidation. In the theory of oxidation processes, including combustion theory, oxygen dissociation is possible due to energy reasons only via peroxide or peroxy radical formation. Scientists from Institute of Chemical Physics, Russian Academy of Sciences discovered the effect of isotropic enrichment by oxygen in hydrocarbon and hydrocarbon polymer oxidation [10, 11]. The theory explains this phenomenon by spin effects at the second degree recombination of peroxy radicals and macroradicals. To some extent, the phenomenon may be considered as an indirect proof of peroxide radical participation in thermal oxidation. Another reaction may compete with usual oxidation initiation in rigid structures: • RH + O2 = R• + HO 2


107

which requires energy consumption equal the bond strength (ER-H = 196 kJ/mol, for aromatic about 251 – 263 kJ/mol). Figure 79 shows the fluorescence spectrum for PI film at light excitation, λ = 320 nm. The long-wave boundary of the spectrum characterizes energy required for electron transfer from external, occupied orbital to lower, loosening orbital. For PI film, the long-wave boundary of fluorescence falls at 520 – 530 nm, corresponded to energy of 217 – 226 kJ/mol. Thus, macromolecule transition to the active state via electron excited one is by 40 kJ/mol more energetically profitable than by usual (for aliphatic polymers) reaction between P-H-bond and molecular oxygen. Electron excitation contributions were discussed in works by Belyakov et al., and light flashes were observed at thermal oxidation [12].

Figure 1. Fluorescence spectrum of PI films (a) without increments and (b) with 2 wt.% BT-5

Oxygen is added as biradical to the aromatic system in the triplet state. The internal peroxide or endoperoxide is formed. The possibility to initiate this act was assessed using quantum-chemical calculations, and reactivity of imide, quinoxaline and phthalamide structure were compared [5, 13]. Quantum-chemical calculation techniques were found efficient for studying reactivity of organic compounds in various reactions [14 - 16]. Molecular diagrams for many models of heat resistant polymers were calculated [14]. Quantum parameters of the system are low dependent on doubling or tripling of the molecule which imitates the elementary unit. Therefore these parameters may also be related to the macromolecule. A correlation between energy of higher occupied molecular orbital (EHOMO) in the system and its thermal oxidation stability was found. The authors used the Waters idea in which oxidation as the electron migration process is considered [17]. We based upon a suggestion on endoperoxide formation in thermal oxidation of heat resistant polymers. The possibility of cyclic peroxide formation is also shown [18 - 20]. Moreover, endoperoxide transformation to quinoid structures may also be imagined. Occurrence of these structures during PI, PEI, PSF etc. oxidation were proved experimentally. As forming on aromatic ring, endoperoxides change the primary π-system of the compound which corresponds to localization of two π-electrons of the ring. The appropriate change in πelectron energy may be considered as the localization energy (απ) [15]. The value of απ was

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selected as the measure of endoperoxide formation probability. The higher απ (its absolute value) is, the lower is the gain of O2 and the higher is polymer resistance to oxidation. Molecules of model compounds under consideration, PI and PPQ [5], contain 28 atoms each. Calculation of so large molecules requires taking a tremendous number of integrals. The reactivity study of a single multi-atom molecule using non-empirical calculation of the electron structure demands many hours of computer time. Usually, these tasks are solved using semi-empirical methods, in which some groups of electrons are neglected. In this case, some integrals of electron energy become zero or are reduced to different integrals; some Hamiltonian terms are neglected or expressed through some empirical parameters.

Figure 2. Charges on atoms and bond degrees in model compounds of polyimides


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In the current case, the Parizer-Parr-Pople (PPP method) semi-empiric, quantumchemical method was used [21]. This method considers only π-electrons in the calculation. It allows quick obtaining of reliable π-electron energy values, required for απ determination. Besides απ, both molecular orbital energies (EHOMO values are shown in Table 1), changes on atoms and bond orders (Figure 2) are calculated. The standard series of parameters selected for reactivity calculation is used [22]. The values of απ were determined as the difference 0 between π-electron energies of the initial compound ( Eπ ) and endoperoxide (Eπ). The value of Eπ represents a sum of π-electron energies of endoperoxide fragments and energies of two π-electrons excluded from the conjugation system due to O2 addition. The bond length between phenyl cycles equals 1.5 Å. It is suggested that all compounds have flat structure. The variation of geometrical parameters indicated independence of calculation results on bond length changes below 0.05 Å and valence angles in the range of ±5°. Therefore, average bond lengths and average valence angles were used in the calculation (Figure 3).

Figure 3. Bond lengths and valence angles in 2-phenylquinoxaline (1) and N-phenylphthalimide (2)

To compare resistance to oxidation of different macrochain fragments in polymers, απ values were calculated for endoperoxide formation on aromatic rings of acid and amine components of PI model compounds and ketone and amine fragments of PPQ and PPA model compounds. Table 2 shows that for all compounds απ values are lower for the amine component that testifies about its higher attackable by oxygen compared with ketone and acid components. These results correlate with already discussed experimental data on PI and PPQ thermal oxidation [5].

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The value of may be suggested as a criterion of both polymers and macromolecule fragments heat resistance. Table 1. Higher occupied molecular orbital energy and π-electron localization energy in model compounds

Acid component

EHOMO, eV

απ, eV Amine component

-12.04

-

6.95

-10.99

6.33

6.96

-12.68

-

7.48

N

-11.20

6.18

7.33

O

O

C C

C C N

-11.16

6.32

7.34

O

O

-10.71

5.75

-

-10.47

5.76

6.42

-10.31

5.76

6.68

Structure

O C C

N

H

O O C C

N

O

H N

H

N

O

O

C C

C C

O

O

O

O

C C

C C

O

O

N

N

N

H

N N N

N

N

N

N


111

Table 2. The amount of absorbed oxygen and release carbon oxides as a result of PI and PPQ model compound oxidation (T = 300°C, t = 100 h, P(O2) = 26.7 kPa)

Compound N-phenylphthalimide N,N’-diphenylpyromellitimide 6,6’-oxy-bis-(2,3diphenylquinoxaline) 2,2’-(1,4-phenylene)-bis-(3phenylquinoxaline) 2,3’-diphenylquinoxaline

Absorbed oxygen, mol/base-mol 0.034 0.045 1.71

Oxidation products, mol/base-mol CO2 CO 0.02 0.002 0.03 0.003 0.91 0.09

0.28

0.16

0.014

0.27

0.19

0.021

As previously shown by Kosobutsky [14], there is the opposite dependence of polyheteroarylene thermal stability in the presence of oxygen and their EHOMO. The correlation is also confirmed by calculation results obtained by the authors of the current monograph (Table 1) [5] and assessment of PI and PPQ models of thermal stability. However, the use of απ both indicates the formal substance disposition in the thermal stability sequence among other substances and binds thermal stability with particular elementary act which is O2 linking with endoperoxide formation. In the quantum-chemical study the probability of molecule ionization at TP decomposition, for example, by proton detachment, was also taken into account. In the frames of semi-empirical FNDO/2 method (Full Neglecting of Differential Overlapping) the energy of H+ detachment (Table 3) was calculated by the difference between total electron energy of the molecule and energy of its negative ion (H+ detachment). These observations do not correlate in any way with the experimental data on thermal stability. This, apparently, removes the problem of ionic state participation in the degradation process. Quantum-chemical calculations based on the endoperoxide model confirm experimental conclusions: -

oxidation is developed in the amine component of imide and quinoxaline structure; oxygen addition causes changes in electron densities on structure skeleton atoms e.g. activation or conjugation; electron effects conform to higher reactivity of the imide (or amide) or quinoxaline structure in oxidation processes. Thus, there is no direct proof (positive analysis) of endoperoxide formation in PI, PPQ, PSF, PEI and PPA, but positive calculation prerequisite and indirect experiments proofs are observed.

Endoperoxide decomposition causes occurrence of oxygen-containing structures in the phenylene structure. Deeper oxidation causes the ring break, CO2 and CO release, where carbon from the aromatic ring “combusts” (e.g. oxidizes). It is suggested that the acts of endoperoxide electron excitation, formation and dissociation form the initial oxidation stage. As degradation proceeds, paramagnetic properties of PI, PSF, PAI, etc. increase. This may be associated with accumulation of hexadienyl oxidized structures, radicals and biradicals in the macrostructure, and increase of

112


the system conjugation degree. The excitation level is reduced, and the system becomes selfcatalytic. This is the third stage of the process, described for PI and PPQ [5]. For oxygen absorption and gas product release, γ values are almost the same. This indicates that oxygen absorption and structure degradation elementary acts become maximum close in time. Apparently, the stage of endoperoxide formation disappears, whereas O2 is added to numerous elementary centers in the conjugated structure and C atom is immediately detached from already linear residue of the aromatic structure. This stage of the aging process is of just theoretical interest, because the loss of operation properties of the material is ended at the second thermal oxidation stage. At this stage, for the gross, one aromatic ring in the elementary unit decomposes (as deduced from carbon oxide yield). Table 3. The energy of H+ detachment from aromatic rings in model compounds Energy of H+ detachment, eV ∆E1 ∆E2 25.773 -

Structure H2N

N H2

H2N

N H2

O N

25.836

24.805

-

24.391

-

24.969

25.447

-

25.564

-

25.698

-

C C O

H3 C

N

O

O

C

C

C

C

O

O

O C C

N

N

CH3

CH3

O H3 C

CH3 N

N

H3 C

CH3

H3 C N H3 C N (CH 3 )2 N (CH 3 )2

Concerning the above experimental material (refer to [1 - 5]), Schemes of PI, PAI, PPQ, PEI and PSF thermal oxidation schemes may be shown as follows:


113

The problem is: what is the way of TP thermal stabilization in the context of their degradation? The classical inhibition of oxidation processes is based on kinetic chain break and deactivation of branching, intermediate products. In the case of TP, chain type of the process is not obvious. The chain break in chemical increments suggests the inertness of residual inhibitor radical. These radicals are active above 200°C. Therefore, classical antioxidants are ineffective at high temperatures. It is accepted that TP must be thermally stabilized towards the structure transition into electron-excited state (quenching these states), prevention of charge transfer complex formation (interaction or complex formation between O2 and amine residues in macromolecules for PI, PPQ, PEI, etc.) to reduce their reactivity, for example, down to acid or ketone residues and, finally, free radical deactivation.

REFERENCES [1]

Neiman M.B., Aging and Stabilization of Polymers, Moscow, Nauka, 1964, 332 p. (Rus)

114 [2] [3] [4] [5]

[6]

[7] [8] [9] [10]

[11]

[12]

[13] [14]

[15] [16] [17] [18]

[19] [20] [21]

E. V. Kalugina, K. Z. Gumargalieva and G. E. Zaikov Kuz’minsky A.S., Oxidation of Caoutchoucs and Rubbers, Moscow, Goschimizdat, 1957, 319 p. (Rus) Grassi N., Chemistry of Polymer Degradation Processes, Moscow, Inostrannaya Literatura, 1959, 252 p. (Rus) Zaikov G.E. and Moiseev V.V., Chemical Resistance of Polymers in Aggressive Media, Moscow, Khimia, 1979, 216 p. (Rus) Vdovina A.L., ‘Thermal Transformations and Stabilization of Polypyromellitimide, Polyphenylquinoxaline and Copolyimidophenyl-quinoxalines’, Candidate Dissertation Thesis, Moscow, 1988. (Rus) Pravednikov A.N., ‘Thermal degradation of polymers with rings in the cycle’, A Report at the 5th Polymeric School Devoted to Methods of Synthesis and Study of HeatResistant Polymers, Gomel’, 1972, 12 p. (Rus) Annenkova N.G., Kovarskaya B.N., and Gur’yanova V.V., ‘High-temperature oxidation of polyimides’, Vysokomol. Soed., 1975, vol. A17(1), pp. 134 - 142. (Rus) Gaudiana R.A. and Conley R.T., ‘Weak-link versus active carbon degradation of aromatic heterocyclic systems’, J. Polym. Sci., 1969, vol. 7(11), pp. 793 – 801. Gaudiana R.A. and Conley R.T., ‘Weak-link versus active carbon degradation of aromatic heterocyclic systems’, J. Macromol. Sci., 1970, vol. 4(2), pp. 441 – 480. Buchachenko A.L., Galimov E.M., Ershov V.V., Nikiforov G.A., and Pershin A.D., ‘Isotope enrichment, induced by magnetic interactionsin chemical reactions’, Doklady AN SSSR, 1976, vol. 228(2), pp. 379 – 381. (Rus) Buchachenko A.L., Yasina L.L., Makhov S.V., and Galimov E.M., ‘Magnetic isotope effect and 17O enrichment at polypropylene oxidation’, Doklady AN SSSR, 1981, vol. 260(5), pp. 1143 – 1145. (Rus) Belikov V.K., Belyakova I.V., Kozlova M.V., Okunev P.A., and Tarakanov O.G., ‘The effect of chemical structure of polyheteroarylenes on resistance to thermal oxidation’, Vysokomol. Soed., 1973, vol. A15(12), pp. 2635 – 2642. (Rus) Kalugina E.V., ‘Thermal transformations and stabilization of some heat-resistant heterochain polymers’, Doctoral Dissertation Thesis, Moscow, 2003. (Rus) Kosobutsky V.A., ‘Electron composition and some physical and chemical properties of aromatic polyamides and polyheteroarylenes’, Candidate Dissertation Thesis, Rostovna-Donu, 1973. (Rus) Bazilevsky M.B., The Method of Molecular Orbitals and Reactivity of Organic Molecules, Moscow, Khimia, 1969, 303 p. (Rus) Straightwiser E., The Theory of Molecular Orbitals, Moscow, Mir, 1965, 435 p. (Rus) Waters W., Oxidation Mechanism of Organic Compounds, Moscow, Mir, 1966, 175 p. (Rus) White E.N. and Harding M.J.C., ‘Chemiluminescence in liquid sols. – Chemiluminescence of laphine and its derives’, J. Am. Chem. Soc., 1964, vol. 86, pp. 5686 – 5698. Kurtz D.W. and Shetcher H., ‘Photooxidation of tryphenyl oxazole’, J. Chem. Soc. Ser. D. Chem. Commun., 1966, No. 7, pp. 689 – 698. Matsuuta T. and Saito J., ‘Photosensitized oxidation of hydroxylated purines’, Tetrahedron Letters, 1968, vol. 29, pp. 3273 – 3281. Parr R.G., Quantum Theory of Molecular Electronic Structure A. Lecture – note and reprint volume, N.Y., W.A. Benjamin Inc. Publ., 1964, p. 510.


115

[22] Kuthan J., Danihel J., and Skala V., ‘Quantum chemical study of dissociation of metaand para-substituted benzoic acids in π-electron approximation’, Collect Czechosl. Chem. Commun., 1978, vol. 43(2), pp. 447 – 462.

In: Polymer Reactivity ISBN 1-60021-263-8 Editors: G. E. Zaikov and B. A. Howell, pp. 117-171 © 2006 Nova Science Publishers, Inc.

Chapter 3

PRACTICAL STABILIZATION OF HEAT RESISTANT POLYMERS E. V. Kalugina, K. Z. Gumargalieva1 and G. E. Zaikov2 Polyplastic Co., 14A, General Dorokhov st., Moscow 119530, Russia 1 N.N.Semenov Institute of Chemical Physics, 4, Kosygin st., Moscow 119991, Russia 2 N.M.Emanuel Institute of Biochemical Physics, 4, Kosygin st., Moscow 119991, Russia

The papers [1-11] discuss the features of thermal and thermal oxidation transformations in highly heat resistant polymers. Polymers selected for the experiment were had “ideal” structure which contained no increments. Many experiments were performed on “model” samples, which were compounds with lower molecular mass, or oligomers. Such searching approach is required for understanding the degradation mechanism. However, under real conditions, there is no opportunity to obtain a polymer of either “ideal” structure or the above-mentioned purity. Therefore, to solve the applied tasks, the most important of which is stabilization, the investigators must base upon studies of degradation mechanisms, developed on appropriate models, and take into account “negative” contributions characterizing real polymeric structures. These are all reasons why this Chapter thoroughly discusses the effect of structure defects, end groups, molecular-mass distribution (MMD), organic and inorganic additives on thermal stability of polymers.

LABILE STRUCTURES AND ADDITIVES Polysulfones (PSF) To make the regularities not only characteristics of the studied sample, but the regularities of a class of substances, one should understand how strong are effects of labile

118


groups in the polymer structure, processing and other additives on the degradation process. To put it differently, selection of the polymer sample is of the principal meaning. Similar to other compounds, polymer may be “real” or “ideal” product. In the first case, structure and composition of the polymer, usually produced by an industrial or semi-industrial way, represents a compromise of production costs and competitive ability of the product by its quality. Most frequently, “ideal” or similar product is obtained in laboratory in vitro, using pure monomers and optimal technique. The latter should provide the absence of labile structures. However, quantity of “ideal” samples is limited, so it is not enough for performing pilot studies of aging and stabilization, in which the effect of these factors on macroscopic properties of polymers are considered. Therefore, a standard must be selected among industrial PSF. It should approach the “ideal” polysulfone in purity and the presence of labile structures. We mean not full absence of the mentioned defect, but their minimal content that causes no significant effect on thermal behavior. Data from the literature [1] and industrial laboratory tests [2 - 4] indicate specific danger from the side of hydroxyl end groups in the PSF molecule for thermal oxidation stability. Let us make an assessment, how high is the effect of end OH-group on thermal oxidation stability of rather long macromolecule, for example, the one containing 30 – 50 units, each containing two side methyl groups the fragments of isopropylidene bridge from bisphenol A residues) tender to oxidation at high temperatures. The model is the following. Oxygen molecule attacks the macromolecule according to common autooxidation initiation mechanism [5 - 8]: RH + O → R + HO2. Aromatic rings are eliminated from the consideration. Since experimental data [9] show relative stability of the aromatic structure, as compared with aliphatic one, even at thermal oxidation of alkyl-aromatic polymers, at the initial stage aromatic rings are not involved in the process. Thermal oxidation by chlorophenylene end groups (end Cl-groups) may not be initiated by this reaction. Initiation by dehydrochlorination is also improbable due to high strengths of Carom-Cl and Carom-H bonds - 400 and 450 kJ/mol, respectively [10]. The activation energy, Ea, of the above-mentioned elementary act of initiation is approximately equal to its heat [5], because −1 1 Q = E a + Ea,

1 −1 where E a and E a correspond to direct and reverse radical recombination, respectively.

According to [5], they approach zero. Generally speaking, one of the features of the solidphase oxidation is high Ea value of macroradical recombination. For instance, Ea of the second degree recombination of peroxy radicals reaches 50 – 100 kJ/mol [11, pp. 64, 69]. However, it is also known that PO recombination mechanism is superimposed with migration of free valences. Intensification of molecular movements due to plasticization or transition to amorphous systems causes an abrupt decrease of Ea to 10 – 20 kJ/mol [11, p. 64]. These values are typical of transfers in liquids [6, p. 81]. In the present case, oxidation proceeds in the polymer melt, i.e. molecular movements are unfrozen. Moreover, macroradical

Practical Stabilization of Heat Resistant Polymers

119

recombination with a small HO2 radical but not recombination of two peroxy macroradicals [7, p. 85] is considered. Therefore, the assumption that −1 E a ~Q

seems to be true, i.e. Ea ~ Q. Similar approach was used in other investigations [11, p. 126]. Initiation is endothermic reaction. The heat of it equals the difference between breaking and forming bond strengths. This means that Q = DR-H⋅221 kJ/mol [11, p. 125]. As shown [10], CH2-H bond strength in linear alkanes equals 400 – 410 kJ/mol. The same bond strength in C6H5CH3 is decreased to 356 kJ/mol (nearly by 50 kJ/mol) [10] due to delocalization energy in benzene radical. Methylene group in isopropylidene bridge of PSF is located in β-position in relation to aromatic rings. The π-system impact is weakly transferred by σ-bond chain and attenuates on β- and γ-atoms [13]. This is illustrated well by the change of chemical shifts of methyl protons in ESR spectra of alkylbenzenes, from toluene to ndecanebenzene: 2.32, 1.25, 0.89, and then the same (0.89) ppm. The latter result is similar to chemical shift of methyl protons in hexane, which is 0.99 ppm [14]. As another example we may accept the rate constant of H atom detachment from methyl group in toluene by phenyl radical. It is by an order of magnitude higher than for tertbutylbenzene [15]. Therefore, let us assume that delocalization energy of alkyl radical with free valence on β-C atom is 4 times lower than for benzene radical. Moreover, additional bond strength decrease by 4 – 8 kJ/mol in alkyl chain branchings [10] should also be taken into account. Finally, the strength of CH2-H bond in the PSF isopropylidene bridge may equal 380 – 390 kJ/mol. The bond O-H is phenol is as strong as 351.5 kJ/mol, whereas D(O-H) strength in 2,6-ditert butylphenols is about 320 – 340 kJ/mol [15]. The strength of O-H-bond in unshielded and shielded para-substituted phenols is defined by electron effects of substitutes. This is shown up by the empirical D(O-H) dependence on the Hammett constant: for example, for shielded phenols D(O-H) = 343.5 – 23. As accepted for the end hydroxyl group in PSF, σpara = 0.197 [83]. Then applying the mentioned dependence, we get: D(O-H) = 351.5 + 23×(-0.197) = 347 kJ/mol. Since we consider competing, unitypical reactions of the same macromolecule, preexponential terms of the rate constants may be equal. As a consequence, the competition between elementary initiation acts at the end hydroxyl groups or side CH3-groups is ruled by the relation of activation energies of competing reactions, which according to

120


Ea = Q and Q = D(P-H) – 221 equal 126 and 159 – 169 kJ/mol, respectively. As already shown empirically [2], end hydroxyl groups negatively affect the quality of PSF. Therefore, an excess chlorine-containing monomer is used in the synthesis. This allows a decrease of end OH-group amount in PSF to 0.02 – 0.15% at the mean molecular mass of 8,000 – 15,000. Let us estimate the ratio of side CH3- and end OH-group concentrations under the most “unfavorable” conditions for OH-groups: Mn = 15,000 and OH-group concentration 0.02 wt.%. It may be simply calculated that

[CH 3 ] ~ 400. [OH] At 350°C the ratio of competing initiation act rates equals:

vCH3 v OH

=

 OH

CH 3

[CH 3 ] exp E a − E a = RT K 0OH [OH ] CH 3

K0

 

⋅

[CH 3 ] = 0.1 ÷ 0.7. [OH]

Figure 1. TGA (1, 2) and DTA (1’, 2’) curves for blocked (1, 1’) and unblocked (2, 2’) PSF in air, the heating rate of 6°/min

Therefore, the calculation performed with assumptions of maximum favorable conditions for the main CH3-group main structure in initiation of high-temperature oxidation of PSF already indicates much higher activity of end hydroxyl groups in initiation of PSF thermal oxidation. Hydroxyl end groups are labile elements of the PSF structure. Therefore, they should be blocked, for example, by treating PSF with methyl chloride which substitutes OHgroups by OCH3-groups. Other monofunctional agents (PhCH2Cl, C2H5I, alkylarylchlorosylanes) are also used for blocking PSF. Unfortunately, blocking may not fully

Practical Stabilization of Heat Resistant Polymers

121

deactivate OH-groups. In marketable PSF their concentration reduced by an order of magnitude, but still remains at a level of 0.005 - 0.007 wt.%. Nevertheless, according to TGA the degradation initiation temperature of blocked PSF in air is by 20 - 25°C higher than for unblocked one (Figure 1). The run of two-stage TGA curves of the above-mentioned samples is similar. Effective activation energies of degradation (dynamic TGA data) equal 90 and (96 ± 20) kJ/mol. DTA curves are of similar type, except for the area corresponded to the initial degradation stage. For blocked and unblocked PSF, the one-stage degradation type in the absence of O2 with the degradation start at 300 - 400°C, coke residue formation up to 30 wt.% at 700°C with equal Ea = (192 ± 20) kJ/mol is also similar, whereas in this case blocked PSF is higher heat resistant, though not so high as in thermal oxidation. As improving the quality of Russian marketable PSF, thermal behavior of blocked and unblocked PSF was compared by analysis of both TGA data and other, higher informative degradation signs, for example, changes in molecular mass characteristics. In all cases, it was observed that at similar type of degradation, thermal oxidation damage kinetics effective kinetic constants of unblocked PSF samples degradation are greater (at similar values of effective activation energies). It is desirable to suggest that PSF thermal oxidation mechanism is independent of the end group origin. Labile hydroxyl end groups define higher degradation initiation rate, and the main process is then developed in the macrochain according to the same scheme. Similar situation is observed for PVC dehydrochlorination, when chain process initiation is defined by the type and content of allyl, ketoallyl and other labile end groups [16]. The change of the degradation mechanism proceeding via depolymerization with respect to the end group origin [17] is known for polyacetals only. PSF is not the depolymerizing polymer. Therefore, preservation of thermal oxidative degradation mechanism in samples with end groups of different heat resistance is quite clear. Though the above calculation indicates 0.02 wt.% content of labile hydroxyl groups as the boundary of the initiation mechanism change (equality of initiation rates at end OH- and side CH3-groups), one should take into account the approximate type of calculations , i.e. interlocking of hydroxyl groups up to their content of 0.005 – 0.007 wt.%, apparently, provides for preferable initiation of random thermal oxidation. As a matter of principle, substitution of OH-groups by CH3Ogroups is not unambiguous because of reduced C-H-bond strength in methoxy group as a result end ArOCH2-radical delocalization. This radical is formed after the bond break at thermal oxidation initiation. However, any empirical assessments do not conform to experimentally determined independence of thermal oxidation of PSF with end OCH3-groups on molecular mass of the polymer in the range Mw = 17,000 – 70,000 [94], i.e. thermal oxidation of interlocked PSF is initiated statistically on side CH3-groups. Instability of hydroxyl end groups is typical of all polysulfone types. Polysulfone thermal stability regularly decreases with OH-group content increase in samples with similar molecular masses. For instance, the initial degradation temperatures for PES derived from 4,4’-dioxydiphenyl with reduced viscosity equal 0.65 at hydroxyl end group content of 0.018 and 0.9 wt.% equal 425 and 330, respectively. Another example is shown by copolymer polysulfone (PSB-230 trademark, derived from bisphenol A and a mixture of 4,4’dichlorophenylsulfone and bis-(4-chlorophenylsulfonyl)bisulfone), in which end OH-group content decrease from 0.24 – 0.28 to 0.15 wt.% at close values of reduced viscosity (0.4 – 0.5) three times reduces the gross degradation rate at 400°C. Similar to PSF, the effective activation energy of mass loss in PES at dynamic heating is low-dependent on the

122


concentration of hydroxyl end groups. The determining factor for this parameter is the presence of the absence of isopropylidene e.g. aliphatic group in the aromatic structure of polysulfones. Polysulfones derived from bisphenol A (PSF and PSB-230) degrade both in the absence and in the presence of O2 with Ea equal 90 - 100 and 190 – 205 ± 20 kJ/mol, respectively. Degradation of fully aromatic PES under the same conditions is characterized by higher Ea values: by 20 and 40 kJ/mol, respectively. Experiments with injection of additives [18] show that inorganic and organic increments cause some negative effect on PSF thermal stability. The analysis of various PSF samples performed by chemical techniques, GLC, ion chromatography, atom-absorption and emission spectroscopy indicated the presence of various admixtures in PSF (Table 1). Admixtures of bisphenol A (DPP), 4,4’-dichlorodiphenylsulfone (DCDPS), dimethylsulfoxide (DMSO) and chlorobenzene (CB) represent residues of monomers and reaction mixtures in PSF polycondensation and extraction technology. Oxychlorodiphenylsulfone (OCDPS) and, apparently, dioxydiphenylsulfone (DODPS) traces represent DCDPS hydrolysis products at the polycondensation stage as a result of deviations from stoichiometric relation between 2alkaline and DPP and at water presence in the reaction mixture [18]. Anions Cl- and SO 4 form secondary products of both DCDPS hydrolysis and decomposition. Among inorganic additives, Na+ and K+ additives only may be associated with participation of alkaline agents during PSF production. The rest metals (cations) occur in the polymer from raw materials and the equipment. With respect to end OH-groups and additive content, PSF thermal stability represents a multifactor reflex surface with direct and interrelated contributions of every admixture. Analysis of data from Table 1 shows that the sample No. 1 possessing optimal thermal oxidative stability is similar to other PSF samples by concentration of organic admixtures (metals and anions). A significant difference (an order of magnitude or higher) is observed for hydroxyl end group content (blocked and unblocked samples) and organic admixtures, as well. From positions of quality, the maximum permissible content of admixtures, determined in experiments with polymer filling and further regression analysis [18], equals: DPP – 0.005 wt.%, OCDPS – 0.1 wt. %, DCDPS – 3 wt.%, NaCl – 0.005 wt.%, CB is practically inert. Experiments with DMSO additives show that concentration MFI and extrudate light transmittance dependence (320°C) is characterized by a plateau up to DMSO content of 0.05 wt.%. Further increase of DMSO content leads to MFI increase and yellowing. Degradation changes become catastrophic at DMSO content above 0.5 wt.%. As DPP content is that mentioned in Table 1, concentration of its active hydroxyl groups is, approximately, two orders of magnitude lower than at macromolecule ends. Extension of the above reactivity assessment at oxidation initiation on end OH- and side CH3-groups in PSF macromolecule leads to the border concentration of end hydroxyl groups defined by the equality of rates on the mentioned structural elements: 0.002 – 0.01 wt.%. Tests show full identity in thermal behavior of interlocked PSF samples (No. 1 and 6) with equal content of additives (Table 1). They differ by concentration of hydroxyl end groups only: 0.006 - 0.009 wt.%. These samples possess the elemental composition (C – 73.7%; H – 5.4%; S – 7.5%; O – 13.4%) approaching

Table 1. PSF sample characteristics Sample 1 2 3 4 5 6

Organic admixture content, wt.% DPP DHDPS OHDPS DMSO

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