Dimensions of labeled chains in randomly ... - ACS Publications

2576

Macromolecules 1986,19, 2576-2588

Dimensions of Labeled Chains in Randomly Cross-Linked Polystyrene Networks a t Swelling Equilibrium by Small-Angle Neutron Scattering Neil S. Davidsonf and Randal W. Richards* Department of Pure and Applied Chemistry, University of Strathclyde, Glasgow GI l X L , United Kingdom. Received November 7, 1985 ABSTRACT: The use of small-angleneutron scattering (SANS) in measuring deformation at a molecular level in polymer networks through the isotopic labeling technique is discussed. Some aspects of molecular models of rubber elasticity and swollen gels in relation to SANS experiments on polymer networks reported in the literature are reviewed. SANS measurements have been made of a constant labeled molecular path in random polystyrene networks covering a range of cross-link density at swelling equilibrium in toluene at 23 "C and in cyclohexane at temperatures in the range 35-65 O C . Results have been compared with calculations by Ullman for multilinked network chains. While some results were shown to be consistent with the phantom network model, remarkable independence of the radius of gyration of the labeled path on cross-link density in a given swelling environment was noted. Together with an apparent maximum of the molecular deformation for gels swollen in cyclohexane at ca. 48 "C these data suggest a predominant role for the local polymer-solvent interaction in determining the conformation of network chains in gels at swelling equilibrium. Measurements of the scattering law in the intermediate regime support the notion of the important role of the polymer-solvent interaction in describing the conformation in these semiconcentrated systems.

Introduction The study of rubberlike elasticity has been pivotal to the development of polymer science over the past 50 years, and much effort has been expended on the establishment of a fully comprehensive quantitative theory of the structure-property relationships of elastomers.'-' Developments that have greatly increased the understanding of polymer network properties over the past 15 years include new theoretical approaches, synthesis of networks with a carefully controlled and well-defined structure, and the development of new experimental probes enabling a more rigorous examination of theories. However, a complete understanding of network behavior in terms of elementary universal features has yet to be established. Until recently, examinations of molecular theories of polymer networks were usually made by investigations of bulk properties such as stress-strain isotherms and swelling equilibria. While such measurements are straightforward, their interpretation and relation to molecular theories can be somewhat tenuous. During the 1970s it became possible to measure unambiguously the molecular dimensions and configuration in semidilute, concentrated and bulk polymer systems with the advent of small-angle neutron scattering.&l5 Consequently it is now possible, in principle, to test the theories of rubber elasticity directly at a molecular level. We present here an overview of molecular theories of rubber elasticity and the scaling law description of gels, i.e., networks swollen by solvents. The relevance of the network chain dimensions to these theories is underlined by a review of the equations developed thus far together with a summary of the principles of SANS and its application to networks. Results obtained by other workers using SANS are presented as a backgrounbd to our experimental results on polystyrene gels which are discussed in the context of the theories presented. Molecular Theories of Elastomers The classical theory of elasticity was developed by Wd16 and Flory2 but has also been presented in a similar form

* To whom correspondence should be addressed.

Present address: Exxon Research and Engineering Co., Corporate Research Laboratories, Annandale, NJ 08801.

0024-9297/86/2219-2576$01.50/0

by others.'g3 Essentially the polymer network is modeled as an assembly of Gaussian chains embedded in an elastic continuum that deforms affinely. In early versions it was considered that the components of all length scales of the chain deformed affinely; this assumption is now regarded as being applicable only to the mean positions of the junctions between the chains (cross-links) and the endto-end vectors of the network chains (a junction affine or end-to-end pulling model). A much discussed model for elastomers is the phantom chain model, originally developed by James and Guth."-I9 The model comprises a hypothetical network of Gaussian chains devoid of material properties other than the ability to transmit forces between junctions, thereby imparting the fundamental connective properties of networks. Although the junctions represent fixed points of the structure, they are not immobile except below the glass transition temperature or when restricted by crystallization. They undergo Brownian diffusive motion over a limited region of space about their mean positions due to rearrangements in the configuration of structurally associated chains. These junction fluctuations are modeled by a Gaussian distribution about the mean junction position, and their mean value is independent of the strain imposed on the network. The mean square displacement of the junction fluctuations (( AP))is large and related to network chain unperturbed dimension by ( M 2=) ((f - O / V - 2)f)(R,2)

(1)

where (Ro2)is the mean square end-to-end distance of the unperturbed chain and f is the junction functionality, i.e., the number of chains emanating from it. Although the magnitude of the fluctuations is strain invariant, the mean positions of the junctions and of the domains of the fluctuations deform affinely; this results in the end-to-end distance of the deformed network chain being nonaffine. Since the development of these theories there has been much discussion surrounding their respective merits and demerits; neither can provide a complete description of all the experimental results. Attempts have been made to bridge the two theories by Ronca and Allegra,20F l ~ r y , ~ l - ~ ~ and Flory and Erman.25-28In these newer theories the essential feature is the imposition of packing forces to alter the spatial range of junction fluctuations in a real network relative to those in a phantom network. These models 0 1986 American Chemical Society

Macromolecules, Vol. 19, No. 10, 1986

Randomly Cross-Linked Polystyrene Networks 2577

predict that if junction fluctuations in a real network are suppressed the junction affine model results, and correspondingly if this suppression is relaxed the phantom network model prevails. One case of network deformation that has received some attention is swelling by incorporation of a low molecular weight diluent to form a gel. Using Flory-Huggins solution theory together with the classical theories of elasticity, one can obtain the Flory-Rehner e q u a t i ~ nfor ~ ~network ~~ swelling. Recently, an alternative approach to the description of gel properties has been proposed by de Gennes30 using the analogy between swollen networks and semidilute polymer solutions, i.e., the c* theorem. In a swollen gel the cross-links maintain the network chains in a relatively close-packed condition and the gel automatically maintains an equilibrium-swollen volume fraction, #e, proportional to the overlap volume fraction, #*, whereas the semidilute solution region begins: #e

= 474*

(2)

where z ( f ) is a constant dependent on the junction functionality. The dimensions of the network chain are the same as the unperturbed dimensions of the equivalent free chain. When swollen by thermodynamically good solvents to a semidilute regime the chains are predicted30to scale with concentration as R,2 c4.%, and experimental results on solutions have confirmed this.31

-

Network Chain Dimensions From the brief summary of the molecular theory of elastomers, a knowledge of the dimensions of network chains and the changes consequent on deformation is particularly germane to the examination of the theories. Radiation scattering (light, X-ray, and neutron) by a polymer chain arises from the spatial correlations of the monomers;32where the macromolecule is a network chain the scattering can be used to monitor the molecular response induced by macroscopic deformation of the network. A number of attempts have been made to calculate the single-chain scattering for a deformed network within the framework of rubber elasticity theory. In the main, these calculations are for elementary network chains, i.e., between two network junctions; however, there has been a recent extension to include multilinked network chains. Among the first attempts to calculate a scattering law for a network chain was that of Benoit et al.,33which assumed affine deformation at all levels. For an isotropic deformation such as swelling then the ratio of the radius of gyration in the strained network, R,, to that in the undeformed network, Rg0is given by

(Rg/R:I2 = A2

(3)

where A is the macroscopic deformation ratio of the network. P e a r ~ o nand ~ ~Warner and Edwards35have independently obtained the scattering law in a strained phantom network using different approaches. The calculation of Warner and Edwards is for a chain randomly cross-linked into a network in the limits of weakly and densely crosslinked networks. For the weakly cross-linked case, the result agrees with Pearson's calculation for end-linked phantom chains. For a network with junctions of functionality f and isotropic deformation these calculations yield

( R g / R g o ) 2= 1/[1

+ A2 - 2/f(A2 - l)]

(4)

Allowing f to become infinite produces the result pertaining

Figure 1. Schematic diagram of a multilinked network chain used in the Ullman model: ( 0 )network function; (0) chain end. Table I Cases Contributing to the Single-Chain Scattering Law for Multilinked Chains Encountering k Cross-Links in a Random Network case multiplicity 1. i and j on same end 2 2. i and j on different ends 2 3. one unit on end, other on internal submolecule 4(k - 1) 4. i and j on same internal submolecule k-1 5. i and j on different internal submolecules (k - l ) ( k - 2)

to affine displacement of junctions, that is, the end-to-end pulling model

( R g / R g o )=2 K(1 + A2)

(5)

In a series of publications, Ullman36-39has calculated the scattering laws for a variety of models for elastomeric networks, the equations being developed with a view to their verification by SANS. Two contemporary ideas in rubber elasticity theory have been introduced to the phantom chain model on a phenomenological basis.37 The expression for the radius of gyration of the elementary network chain is a more general form of 3

( R g / R g o )=2 1/2[1 + A*2(1- (2/f(1

K)))

+ l/f(l +

(6)

where A*2 = (1 - a)A2

+a

(7)

A* is the deformation experienced at the level of the elementary network chains rather than the macroscopic deformation and the parameter a allows for network unfolding without deformation of elementary chains, which has been proposed to explain some experimental observ a t i o n ~ . ~The , ~ ~parameter ,~~ K is a measure of the severity of restrictions on junction fluctiations with respect to those allowed in the pure phantom chain model. A major drawback to these equations is that they have been developed for end-linked network chains and are not applicable to randomly cross-linked network chains. Such chains encounter several cross-links in the path through the network and moreover are generally not cross-linked at their ends. Ullman has addressed this problem39 wherein the various submolecules between cross-links on the same precursor polymer molecule are arranged coherently as shown schematically in Figure 1. Contributions from pairs of scattering centers on different submolecules have to be included in the single-chain scattering law, which is then the weighted sum of the five cases in Table I. Expressions for the scattering law have been calculated by Ullman for such multilinked chains in the limits of phantom and end-to-end pulling models for the deformation of the internal subunits while terminal sub-


2578 Davidson and Richards

i

7 -

/

i

r

/

/

I

I

where Cht is a constant term containing instrumental and sample parameters. The scattering vector, Q, the magnitude of which for elastic scattering is given by IQI = (4a/X) sin 8 (12) where X is the neutron wavelength and 28 the angle of scattering. The term (dx(Q)/dQ),, is the macroscopic total differential scattering cross section and is the probability of neutrons being scattered into a solid angle dQ. This term can be factorized into two contributions, a coherent term and an incoherent term: (dx(Q)/dQ)tot = (dx(Q)/dQ)coh + (dZ/dQ)inc (13)

3

2 ,\1,Q"3)

i

of the technique to the whole field of polymer s c i e n ~ e . ~ ~ ~ ? ~ ~ Only a Precis of the salient points are presented here as a framework underpinning the results discussed later. The intensity of neutrons scattered by matter is given by Z(Q) = cin,t(dx(Q)/dQ)tot (11)

Figure 2. Molecular deformation of elementary end-linked network chains in a network with a tetrafunctional cross-links: (A) chain affine; (B) junction affine; (C) pure phantom; (D) good

solvent swelling.

units remained undeformed. The expressions obtained are as follows: Phantom Model ( R , / R , o ) ~= 1 ( ~ -2 i/(k + 1)3)k3- k + [(f- 2)/2fl(4k + l)(k - 1) - 6(T1 + 2T2 + T3) (8) where TI = rf(1 - (f - l)'-')

+

Tz = rr((k- 2) T3

=

+ [ l - (f-

l)'-']/(f-

2))

rf(l/z(k - 2)(k - 3) + [(k - 3) x

(f- 1) - (k - 2) + (f- 1)3-k]/(f- 2)2) Y f = 2(f - l ) / f (-f 2)

End-to-End Pulling Model

(Rg/Rg0)2=

1 + [(A2 - 1)/2(k

+ l)3][(k - 1)(2k2 + 6k + l)]

(9)

where k is the number of cross-links encountered by each chain. Lastly, Geissler et al.4M have produced an intuitive scaling model for chain dimensions of gels swollen in good solvents using the postulates of the c* theorem mentioned above. Use was made of a proposal by Bastide et a1.5*40341 that differentiated between the spatial and topological neighboring junctions of any junction in the network chosen as reference. The final equation was R; = (R,o)~(I ~ 0 . 2 5 ) / 2 (10) where Rg0 is the radius of gyration of the undeformed network chain in the bulk state and Q is the volumetric swelling ratio. The molecular deformations predicted by some of these models for elementary network chains in tetrafunctionally end-linked networks subjected to isotropic deformations are shown in Figure 2. We note that the linear deformation ratio for swollen networks is given by A = Q113,where Q (not to be confused with the scattering vector Q) is the volumetric swelling of the network. There are clear differences between the models which should, in principle, be observable from the data obtained by SANS experiments. Small-Angle Neutron Scattering from Polymers Detailed fundamental theory for SANS from polymers is available elsewhere as well as reviews of the application

+

It is only the coherent term that retains the structural information of the scattering system; the incoherent term merely gives rise to a flat background scattering. For a multicomponent system the differential coherent scattering cross section may be written as N

(dB(&)/dQ),,h = CB$j exp(iQ.(r, - r,))

(14)

ir

where N is the number of scattering centers in the sample and rr and r, are the position vectors of the scattering centers. The parameter B is the scattering length density of the scattering center i or j . The phase term of eq 14 is often called the coherent scattering law, S(Q), and it is the Fourier transform of the pair correlation function of the scattering centers and thus is a direct measure of the macromolecular configuration. I t has been shown that when deuterated and hydrogenous polymers are used which are identical in molecular weight and distribution the maximum scattered intensity is obtained from a 50150 mixture of these, and the single-chain scattering law, P(Q), can be extracted unambiguously from these data.46 Conventionally, however, the concentrations of deuterated polymer used are much lower and the labeled molecules are dilute with respect to each other, although the total polymer concentration (hydrogenous plus labeled) may be high. Hence the labeled polymer is subjected to all the thermodynamic and mechanical influences arising from the high concentration and connectivity (chemical bond and/or entanglements), but we can observe the singlechain properties by undertaking neutron scattering measurements. In the situation where there is no low molecular weight diluent then for low concentration of labeled species and in the limit of incompressibility (dx(Q)/dQIc0h (Bd - Bh)2fid2Ndp(&) (15) where nd is the degree of polymerization of the labeled polymer and Nd the number of labeled polymer molecules in the system. Where solvent is present the scattering length density of the solvent may be adjusted to that of the hydrogenous polymer by mixing the appropriate ratio of deuterated and hydrogenous solvent isomers, and Bh in eq 15 can be replaced by B,, the mean scattering length density of the solvent. For situations where solvent is present eq 15 is only correct for 9 conditions; all other cases must include a nonideality term, i.e., virial coefficients, in eq 15. The single-chain scattering law, P(Q), is given by the Debye expression47 P(Q) = 2/(Q2R,2)2(exp(-Q2R,2)- 1 + Q2Rg2) (16)


Macromolecules, Vol. 19, No. 10, 1986 and for values of QR, < 1 this reduces to

p(Q) = 1 - (Q2R,2)/3

(17)

hence since R for polymer molecules is ca. 100 A this entails Q 510-8A-1; i.e., scattering measurements must be made in the small-angle region. The majority of SANS measurements of R, have been interpreted by using this approach or the Zimm modification; examples for both are given later for our data.

Applications of SANS to Networks From the discussion on the principles of SANS it is evident that to study the molecular deformation in polymer networks they must be isotopically labeled (deuterated) somewhere. Thus far networks have been labeled by three methods: (i) labeling of the junctions, (ii) labeling of elementary chains between the junctions of model networks, and (iii) labeling of multilinked network chains of randomly cross-linked networks. Benoit et al.&reported the results of SANS experiments on anionically prepared polystyrene networks wherein the junction points were labeled. The SANS spectrum exhibited a single broad maximum removed from zero scattering vector, the existence of which is incompatible with a Gaussian distribution of chains assumed by most models. SANS experiments on labeled elementary chains in end-linked polystyreneMand poly(dimethylsiloxane)49(PDMS) networks have established that the ideal Gaussian conformation is adopted in the bulk state. Furthermore, the mean dimensions are comparable for cross-linked and un-crosslinked chains in the bulk state. Results obtained for the deformation of elementary chains in uniaxially stretched polybutadiene networkss0were shown to be in reasonable agreement with the phantom network model. Recently a comprehensive study of uniaxially stretched PDMS networks was reported by Beltzung et al.51 A range of behavior was observed, but in no case was the deformation of the elementary network chains affine with the macroscopic deformation. The effect of increasing the junction functionality was to increase the observed molecular deformation. They were able to correlate their results in terms of an estimate of the ratio of spatial to topological neighbors (degree of interpenetration) and found a continuous decrease of molecular deformation with increasing values of this ratio, in accord with the concept of an unfolding mechanism. A number of SANS experiments on swollen networks have also indicated smaller deformation of elementary network chains than would be expected from the classical statistical theories of rubber elasticity. Bastide et aL5v41found very little change in the radius of gyration of the elementary network chain until the network was nearly dry, in spite of a large charge in volume of the swollen gel during osmotic deswelling of polystyrene gels in benzene. The radius of gyration of labeled chains in anisotropically swollen polystyrene model networks was measured by Geissler et a1.44 and found to be in good agreement with their intuitive model for the swelling of gels in a good solvent. Beltzung et al.52have also reported results obtained for PDMS networks at swelling equilibrium in good solvent conditions. Although some of the results were roughly consistent with the phantom and end-to-end pulling models, comparison with dilute and semidilute solutions of the equivalent chains showed a remarkable agreement between the conformation of the elementary network chain and that of the free chain in dilute solution. They concluded that the conformation of the elementary chain in a swollen network was governed by local polymer-solvent interactions. To date the only SANS experiment on random networks reported in the

literature is that of Clough et al.53on y-irradiated polystyrene networks, containing -5 cross-links per labeled chain, uniaxially stretched at a temperature above the glass transition temperature. The results were initially compared with the model predictions for end-linked chains, and the phantom model was found to be the most appropriate description. A comparison with Ullman’s model for multilinked network chains showed the experimentally measured deformation to be less than the phantom model p r e d i c t i ~ nin~this ~ case. The above review demonstrates that although there have now been a number of SANS experiments on polymer networks reported, no current molecular model is unambiguously favored. The general trend, however, appears to indicate considerably less than affine deformation of the elementary chains and the role of some unfolding mechanism at a scale larger than the mesh size involving a reorganization of the relative positions of spatial and topological neighboring junctions. In swollen networks the local polymer-solvent interaction may be fundamental in defining the equilibrium conformation of elementary network chains while the macroscopic swelling is determined by junction functionality, network defects, and the degree of interpenetration of the chains. In an attempt to substantiate and clarify some of these ideas we have measured SANS from a monodisperse constant molecular weight labeled path randomly cross-linked in polystyrene networks as a function of the cross-link density and the degree of equilibrium swelling in solvents of varying thermodynamic quality. The potentially strong excluded volume environment and high degree of swelling was obtained with toluene as the diluent. Lower degrees of swelling were investigated with cyclohexane at different temperatures as a diluent. Results from these random networks should be complementary to those for model end-linked networks. Labeled chains in random networks pass through several cross-links and correspond to molecular paths larger than the network mesh size. The scale of observation relative to the mesh size can be varied either by doping the network with longer chains at fixed crosslink density or by increasing the cross-link density on a constant length of labeled chain; such data should help reveal the role of an unfolding mechanism at large length scales. Model elastomers prepared by end-linking of monodisperse linear chains should be regular in structure and be highly characterized with respect to molecular weight between junctions and junction functionality, but the elastic properties are strongly modified by incomplete reactions, making the interpretation of experiments somewhat vulnerable. Randomly cross-linked networks are generally more easily prepared, and although the structure tends to be irregular with variable chain lengths between junctions and many dangling terminal chains, such irregularities can be more easily accounted for than in model networks and become less important as the degree of cross-linking increases. Cross-linking is frequently accomplished by chemical reaction of specific groups on the primary chains directly with those on other chains or with suitable cross-linking agents acting as nodes. It has been observed that such groups and cross-linking agenb can give rise to specific structural artifacts, and in an attempt to avoid this complication y-irradiation has been used here to induce cross-linking. Furthermore, to avoid any complications due to solvent or concentration effects cross-linkingwas carried out in the bulk state. Polystyrene was chosen for this study because near-monodisperse polymer can be easily prepared anionically; deuterated monomer for SANS labeling is commercially available and


2580 Davidson a n d Richards

it is fairly stable to high-energy radiation but cross-linkable. Polystyrene, however, has a high glass transition temperature, so at ambient conditions one is restricted t o swelling to investigate elastomeric properties. The use of equilibrium swelling conditions should nonetheless be of interest to demonstrate over what scales in the network the polymer-solvent interaction is p r e d o m i n a n t in describing the labeled chain. Correlations within the labeled chain have also been measured to characterize the conformation more fully. Experimental Section Network Preparation. Hydrogenous linear atactic polystyrenes (PSH)with weight-average molar masses of ca. lo5 daltons, as determined by gel permeation chromatography, were synthesized in this laboratory by anionic polymerization in tetrahydrofuran solution. Anionically prepared deuterated polystyrene (PSD), M , = 1.04 X lo6 daltons, Mw/M,, = 1.10, was purchased from Polymer Laboratories Ltd., Church Stretton, Shropshire, U.K. Mixtures of hydrogenous and deuterated polymers (2,3,4, and 5% (w/w) PSD) were prepared by dissolution of the solid polymer mixture in methyl ethyl ketone (ca. 5% solution) and precipitation into methanol. The precipitated polymer mixtures were collected and thoroughly dried under vacuum for several days. Disks, 13 mm in diameter and approximately 1 mm thick, were formed by compression molding the linear polymer in an evacuable die, followed by heating to 450 K and allowing to cool below the glass transition temperature while in the press. High-concentration labeling as advocated by Akcasu et al.46 was not used because of the increased probability of two or more deuterated molecules being cross-linked to each other. Such large aggregations would be the source of a major contribution to the scattered signal and would not be characteristic of a single labeled path through the network. The polystyrene was randomly cross-linked by exposure, under vacuum, to a ' Y o y-ray source at ambient temperature, using a dose rate of 5 kGy h-* and total doses between 2 MGy, nominally the gel point dose for polystyrene of molecular weight lo6, and 10 MGy. After irradiation, sol fraction was removed from the cross-linked polystyrene networks by exhaustive extraction, first in cyclohexane and then in toluene. The gels were then carefully deswollen and thoroughly dried under vacuum so that the weight of polymer network could be established. Neither gel permeation chromatography nor infrared spectroscopy of the extracted material indicated any significant preferential extraction of PSD." In the experiments reported here networks prepared from different precursor P S H samples were used, but only one PSD was used to label all the networks. Swollen Gels. The dry extracted networks were swollen to equilibrium in cyclohexane at temperatures in the range 298-343 K and in toluene at ambient temperature, 293 K. The equilibrium swelling was determined by periodic weighing of the gels until an essentially constant weight was obtained; volume additivity was assumed for the calculation of volumetric swelling. By comparison of the equilibrium swelling for several networks prepared from the same P S H that had received the same dose, the reproducibility of swelling was evaluated. For low doses (approximately the gel point dose), the standard deviation from the mean swelling was large a t ca. 25%; this decreased rapidly with increasing dose to give an average value of ca. 5 % No specific effects were noted for samples labeled with a low concentration of PSD. Comparison of experimentally measured R with values calculated from Ullman's model for multilinked network chains necessitates a knowledge of the average number of cross-links per chain. The effective number of cross-links per chain was determined by substitution of the polymer volume fraction a t swelling equilibrium into the Flory-Rehner equation. For SANS experiments 13-mm-diameter specimens were cut from the swollen networks. These specimens were dried and then reswollen in the mixture of hydrogeneous and deuterated solvents used for the scattering experiments. The ratio of hydrogenous to deuterated solvent was chosen to give a mean scattering length density equal to that of the hydrogenous polymer component in

.

the network. This required mixtures of 3:l (v/v) hydrogenous to deuterated (Aldrich 99.5 atom %D) cyclohexane and 9 1 (v/v) hydrogenous to deuterated (Aldrich 99+ atom % D) toluene. Deuterated solvents were used as received, and it was assumed that there was no influence on the degree of swelling due to the presence of deuterated solvent. Small-Angle Neutron Scattering. SANS measurements were made at the Institut Laue-Langevin, Grenoble, France, using the D11 and D17 diffractometers. Use of the two diffractometers enabled a wide range of scattering vector to be explored, 0.005 5 Q/k' 5 0.1. The swollen gels and an excess of solvent were placed between quartz windows separated by a %-"-thick PTFE spacer, and the whole assembly contained in a cylindrical brass cell. A thermocouple placed close to the cell permitted temperature measurement; temperatures were maintained to & 2 "C of the set temperature. Gels were swollen for a t least 24 h in the appropriate solvent before SANS measurements were begun. However, for the measurements made in cyclohexane as a function of temperature, restricted time allocation for measurements meant that a minimum time of only 4 h could be allowed for the swelling of each gel from one temperature to the next highest temperature. Separate experiments had established that this time was sufficient for equilibrium swelling to be attained. Calibration of each diffractometer was made by measuring the scattering from a 2-mm path length water sample. Coherent scattering from the networks was generally relatively weak due to the low concentration of labeled species. Consequently times of the order of ca. 0.75-4 h were used to collect data of sufficient statistical accuracy. T o account for the incoherent scattering present, measurements were made of control samples that were the appropriate swollen hydrogenous network, Le., no labeled chains present. Data Reduction and Analysis. Data reduction and analysis were performed with a suite of programs a t the ILL55and programs developed on mainframe computers a t the University of Strathclyde. The scattering from the empty cell and that due to the incoherent scattering in the specimens were subtracted (normalizing for neutron transmission) from that of the swollen network specimens. For the subtraction of incoherent scattering, corrections were also made for the differences in thickness of labeled and unlabeled gels. The resultant excess scattering is (d2(Q)/dQ),ohfor the labeled chains multiply cross-linked into the network. In the region of low scattering vector the single-chain differential coherent scattering cross section was interpreted via the classical Zimm56s57extrapolation to yield R,:

(18) where A2 is the second virial coefficient and R, is the z-average radius of gyration. The radius of gyration of a single chain is obtained from the slope a t zero concentration of labeled species, the second virial coefficient from the slope at zero scattering vector, and the number of scattering centers in the particle (i.e., the molecular weight) from the intercept at zero scattering vector and zero concentration. The data were fitted by linear regression over various ranges of scattering vector to obtain the "best fit" values of the slope and intercept, from which the radius of gyration and molecular weight were extracted. A useful check on the validity of the fit is the comparison of the molecular weight obtained to that obtained with an independent method. An explicit expression for extracting the molecular weight in a SANS experiment has been given by Richards and ThomasonS8assuming the approximation that the scattering of the calibrant was totally incoherent.

where Z&O)/Z,(O) is the normalized intensity a t zero scattering vector, K2 = (Bd - B,)2is the contrast factor per unit volume, cd is the concentration of labeled chains, u is the specific volume of polymer, T , is the sample transmission, T, is the transmission of calibrant (water), D, is the sample thickness, M , is the weight-average molecular weight, and N Ais the Avogadro Number.



Table I1 Radius of Gyration and Molecular Weight of Marked Chain (PL/PSD) in Un-Cross-Linked Hydrogenous Polystyrene Matrices Obtained by Zimm Plots of SANS Data (A Assumed Zero) ~~

M , x 10-3 PSH matrix NDPSH8 NDPSH16 NDPSH16 a

treatment

matrix 164 154 145

compression molded, T > Tg compression molded, T > Tg compression molded, T > Tg

labeled fraction 0.05 0.037 0.037

Q range, A-'

A

@JZ,

a 0.0065-0.0212 a

a

99 f 3 a

M , x 10-3 a 140 a

Anomalously high scattering.

Table I11 Radius of Gyration and Molecular Weight of Marked Chain (PL/PSD) in Cross-Linked Dry Extracted Networks Obtained from Zimm Plots of SANS Data (A Assumed Zero) approx no. of cross-links per M, X code dose, lo6 Gy chain matrix labeled fraction Q range, A-' @Jz, A M , x 10-3 0.04 0.0100-0.0211 93 f 2 111 N4.04D400 4 3 115 a a 98 f 12 126 Bl.D400 4 3 164 117 a a 93 f 2 B2.Dl000 10 4 162 0.05 B1.05D400 4 3 164 b b b 0.04 B1.04D600 6 5 164 0.05 B1.05D800 8 6 164

See Table IV. *Anomalously high scattering. Notwithstanding the use of near-monodisperse labeled polymer, a correction59to the value of R, obtained should be made to produce weight-average values of R,: (QW/(RJ2 = [ ( h+ U / ( h

+ 2)1'/'

0.3-

(20)

where h = [(M,/M")

- 11-1

(21) 0.2-

Results Un-Cross-LinkedBulk Polymer and Dry Extracted Networks. It was important to establish the unperturbed Gaussian configuration of the labeled chains in the uncross-linked polymer matrix and make a comparison with the configuration of the same chains randomly cross-linked in bulk extracted networks. The results of three attempts to measure the radius of gyration and molecular weight for the labeled polymer in an un-cross-linked hydrogenous polymer matrix are summarized in Table 11. The data were analyzed by a simple plot of reciprocal intensity against the square of the scattering vector at a single low concentration of labeled species and the second virial coefficient was assumed to be zero as reported for bulk amorphous polystyrene in the literature.@ In one case the molecular weight obtained by SANS was in modest agreement with GPC analysis, but in the other two samples extremely large values were obtained from the Zimm extrapolations over any selected fitting range in the appropriate region of low scattering vedor. The unperturbed weight-averageradius of gyration of bulk amorphous polystyrene is reported in the literature to scale with molecular weight according toso (R,O),/A

0 0

4

Figure 3. Zimm plots for dry extracted B2.Dl000 network. PSD content as marked. Table IV

Apparent Dependence of (R& and M , of Marked Chain (PL/PSD) on Concentration of Labeled Species in Dry Extracted Networks Obtained from Zimm Plots of SANS Data code BLD400

= 0.275(M,)0.5

Using the GPC analysis for the weight-average molecular weight and the polydispersity of the deuterated polymer and making the correction of eq 20, one predicts the unperturbed z-average radius of gyration to be 93 A. Table I1 again shows only one sample yielded a value close to that expected. Possible reasons for the observed anomalous scattering are discussed below. The results of Zimm extrapolations for the labeled species in bulk extracted networks covering a range of cross-link density and y-radiation dose are presented in Table 111. For three samples the measured M , and (RJZ

2

Q2,10-4~;-2

B2.Dl000

labeled fraction 0.05 0.04 0.03 0.02

Q range, A-' 0.0106-0.0201 0.0106-0.0201 0.0100-0.0206 0.0100.0206

(R&, A 194 f 9 165 f 8 139 f 6 124 f 6

0.05 0.04

0.0065-0.0189 0.0094-0.0189

92 f 1 95 f 1

0.03

0.0077-0.0201

0.02

0.0094-0.0195

90 f 1 94 f 2

M,

X

281 180 172 155 130 119 115 104

were in moderate agreement with the GPC value of M , and the predicted unperturbed dimensions, but for the other three samples an anomalously high level of scattering was again measured. For two of the networks in Table I11 the influence of the concentration of labeled chains, up to 5% of the total polymer chains, incorporated into the network was in-


Figure 4. Zimm plots for dry extracted Bl.D400 network. PSD


Figure 5. Scattered intensity as a function of Q for hydrogenous network Bl.H400: (A) dry extracted network; (B) at swelling equilibrium in 9:l H:D toluene mixture.

content as marked.

vestigated. The correlations of reciprocal intensity against the square of the scattering vector are shown in Figures 3 and 4, and the results of the linear regression to obtain M , and ( R ) z are given in Table IV. The behavior of the two networks was very different. In the B2.Dl000 samples the influence of the concentration of labeled chains was small. At each concentration the value of (Rg)zwas in agreement with the unperturbed dimensions, and the values obtained for M, were in agreement with GPC. The trend was a slight decrease in M, with decreasing concentrations of labeled species, indicating a small negative second virial coefficient,but this is probably not significant and may be regarded as zero within experimental error. The Bl.D400 samples, however, showed a remarkably strong dependence of both (R& and M , on the concentration of labeled species. This concentration dependence would correspond to a significant negative value of the second virial coefficient on a full Zimm plot, but an empirical extrapolation yielded (Rg)zand M, at zero concentration of labeled species, in agreement with unperturbed dimensions and GPC molecular weight. The criterion used to assess the validity of the fit to the measured scattering was whether a reasonable value of the molecular weight could be obtained from the Zimm extrapolation, but as seen above some samples exhibited an upturn in intensity a t low scattering vector, resulting in large values of (Rg)zand M,. This effect was strongest in the bulk state and diminished with increasing degree of swelling. No trend of this phenomenon with cross-link density or radiation dose was distinguishable and indeed it was observed in un-cross-linked material so it could not be solely attributed to either the radiation treatment or any artifact of network junctions. Furthermore, since the effect was not totally reproducible it suggests the source was randomly introduced during the sample preparation. Figure 5 compares the scattering of purely hydrogenous network in the dry and swollen states. In the bulk network there is a low-angle coherent signal that falls off rapidly and becomes insigificant for Q 1 0.01 A-l. This upturn is not exhibited a t all in the swollen gel (the contrast between hydrogenous polymer and solvent has of course been matched out) and the signal is independent of Q. Figure 6 compares the scattering of a bulk sample labeled with a low concentration of deuterated chains to that of an equivalent unlabeled sample. The upturn in scattering of the unlabeled sample is only a small fraction of that arising

05

i

NDPSH 16

01

Figure 6. Water-normalized scattered intensity for un-crosslinked polymer showing relative scattering of sample and background.

from the coherent contrast between labeled and unlabeled chains, but the signal from a single-componentamorphous sample should be incoherent and Q independent. Increased scattering at very small Q has been discussed in terms of the scattering from microvoids. Initially this was only considered to be important for samples containing a large concentration of deuterated chains,6O but it has more recently been further discussed by Bou6. et al.61 They concluded that the effect of voids was sample dependent and could be suppressed to some extent by compression molding at elevated temperature. The effect was present in both hydrogenous and deuterated samples but more observable in the latter due to the larger coherent scattering length of deuterium. The effect of voids an the spectrum from a labeled polymer sample could largely be accounted for by subtraction of the spectrum from an equivalent unlabeled sample. Many of the bulk samples exhibited an increase in scattering at low Q, which can be reconciled with the notion of scattering from microvoids. As this is a sample-dependent feature it could not rigorously be accounted for by a background substration, but discarding the data at lowest angles generally rendered the spectra useful. The above reasoning, however, was not satisfactory to account for all the samples exhibiting anomalous scattering



Table V Radius of Gyration and Molecular Weight of Marked Chain (PL/PSD) in Cross-Linked Gels at Swelling Equilibrium in Toluene at 293 K Obtained from Zimm Plots of SANS Data and Comparison with Phantom and Affine Network Model Predictions of Eq 8 and 9 approx no. of (Rg/Rs")' cross-links per code swelling ratio chain ( R g L8, M, x 10-3 measd phantom affine Bl.D800 8.5 6 155 f 16 117 2.8 2.5 3.5 110 2.7 1.6 4.0 11.3 5 152 f 14 Bl.D600 11.6 4 158 f 1 2 112 2.9 2.3 3.8 B2.Dl000 156 f 19 108 2.8 2.3 2.9 Bl.D400 14.9 3 23.6 2 151 f 12 100 2.6 2.4 3.8 N2.D400

i'= 0

4t

l 2

4

6

8

10~02+2000~~

Figure 7. Z i " plot for N2.D400 network at swelling equilibrium in toluene at 293 K.

at low angles. An un-cross-linked sample molded at ambient temperature, which looked inhomogeneous even on a macroscopic scale, gave a high level of scattering over the entire low-angle range, but of two samples molded above the glass transition temperature (both of which appear homogeneous to the eye), one also displayed an anomalously high signal while one did not. A few other samples molded above the glass transition temperature also displayed an increased level of scattering over the entire range of low scattering vector. Moreover, the sample Bl.D400 in the dry state showed a dependence of this effect on the concentration of labeled chains which was too pronounced to be explained satisfactorily by an increased contrast for void scattering. This sort of behavior seems better explained by some form of clustering of the labeled chains. Schelten and c ~ - w o r k e r s have ~ ~ J ~pointed out that relatively few labeled intermolecular contacts above those expected statistically are required to produce a substantially higher molecular weight than would be expected for the individual labeled chains. This was not a reproducible feature in these samples and, although the exact explanation is uncertain, we speculate that it may be related to inefficient dispersion of deuterated and hydrogenous chains in some batches. To recapitulate, samples exhibiting anomalous low-angle scattering could be divided into two groups: (i) those whose spectra could be rationalized by neglecting the data collected at lowest angles, the anomalous scattering being attributed to microvoids, and (ii) samples from which no sensible interpretation in terms of the single-chain scattering could be made over the low scattering vector regime, but for which the single-chain parameters could apparently be extracted by an empirical extrapolation to zero concentration of labeled species. Swollen Gels. A typical Zimm plot for a network swollen in toluene is shown in Figure 7. The maximum concentration of deuterated chains in the bulk state was 5% and since the networks were highly swollen the concentration of labeled chains was very low. This meant the coherent scattered intensity was reduced and data had to

0' 0

2

4

6

0

10

104Q2+ 4 0 0 c d

Figure 8. Zimm plot for B2.Dl000 network at swelling equilibrium in cyclohexane at 308 K.

be collected for long time to obtain data of adequate statistical quality. Table V summarizes the results obtained from full Zimm plots for five gels, covering a range of cross-link density, at swelling equilibrium in toluene. In each case the second virial coefficient was zero within experimental error, and the molecular weight obtained from the intercept of the Zimm extrapolation was in excellent agreement with the GPC value. The consistency between the SANS and GPC values of the molecular weight of the deuterated polymer provides some support for the validity of the fit used to obtain the radius of gyration and further confirms that there was no preferential extraction of labeled chains and no radiation damage leading to degradation. The value of ( R J Zquoted for each network is the mean of the values at four concentrations of labeled chains (i.e., assuming A, = 0) and the estimated error is the standard deviation in the mean. The value of ( R J 2was increased from that obtained for the bulk state, indicating the polymer coil had been expanded, and (Rg)? was constant within the limits of experimental uncertainty over the range of cross-link density investigated. The average value of ( R J 2of the labeled chain of ca. 155 A is larger than the interpolated value for an equivalent free chain in a good solvent extracted from the literature. Light scattering studies of polystyrene in toluene62indicate a value of 116 A for a free chain of this molecular weight while SANS of polystyrene in carbon disulfide31yields a value of 132 A. Furthermore, the concentration dependence R,2 c4,156recently found by SANS for polystyrene solutions in Toluene63is not indicated by these results either. A stronger dependence was earlier reported for polystyrene in carbon d i ~ u l f i d e . ~ ~ The same procedure described above was applied to gels at swelling equilibrium in cyclohexane at 308 K, approximately the Flory 8 temperature for dilute solutions of linear polystyrene in cyclohexane. Figure 8 illustrates a typical Zimm plot. The statistical quality of the data was generally better than obtained for the gels in toluene since the degree of swelling was less, the concentration of labeled

-



Table VI Radius of Gyration and Molecular Weight of Marked Chain (PL/PSD) in Cross-Linked Gels at Swelling Equilibrium in Cyclohexane at 308 K Obtained from Zimm Plots of SANS Data and Comparison with Phantom and Affine Network Model Predictions of Ea 8 and 9 code B1.04D800 B2.Dl000 N4.04D600 Bl.Df300 Bl.D400 N4.D400 N2.05D400

swelling ratio 2.9 3.5 3.8 4.3 5.0 4.9 6.0

approx no. of cross-links per chain 6 5 4 4 3 3 2

( R AA 95 f 10 116 f 4 95 i 9 112 f 8 112 f 8 92 f 5 113 f 3

chains was consequently higher, and data of adequate statistical quality could be accumulated in a shorter time. The problem of anomalously high scattering was a complicating feature for some samples as discussed above for the bulk state. The results of Zimm extrapolations for a range of cross-link densities are summarized in Table VI. Some of the gels were investigated with a full Zimm plot, and where no anomalous scattering was observed the second virial coefficient was essentially zero; the remainder of the gels were studied at a single concentration of labeled species. The molecular weight determination, although prone to errors from the measurement of sample thickness and swelling, again generally gave values in satisfactory agreement with GPC analysis of the linear deuterated polymer. As expected since the degree of swelling in cyclohexane at 308 K was much reduced from that in toluene, the radius of gyration was also reduced. Some of the values were in close agreement with the unperturbed dimensions while others were more consistent with a slight expansion. However, within the experimental uncertainty and considering there was no immediately obvious trend with cross-linking density it appears legitimate to average all the results. This yields a mean value of 105 f 12 A, which is indistinguishable from the mean bulk value of 96 f 7 A although the gels have been swollen by a factor of 3-6 times, depending on the cross-link density. The behavior of the gels at intermediate degrees of swelling between the excluded volume limit, corresponding to toluene as the solvent, and the 8 condition, Le., cyclohexane as the diluent at 308 K, was studied by following the radius of gyration of two networks of different crosslink density in cyclohexane as a function of temperature over the range 308-338 K. Following the results discussed above the second virial coefficient was assumed to be zero and the radius of gyration was obtained from plots of reciprocal scattered intensity at a single concentration of labeled chains (4% of the polymer chains deuterated). The results are collected in Table VI1 and the radius of gyration is correlated as a function of temperature in Figure 9. The molecular weight obtained from the extrapolation was again generally in moderate agreement with GPC analysis. The equilibrium degree of swelling increased monotonically for both networks over the range explored but the radius of gyration displayed an apparently more complex behavior. Table VI1 shows that the chains expanded at first as the networks increased in swelling but above 320 K the value of (RJZdecreased although the macroscopic swelling of the gels continued to increse. The value decreased catastrophically and fell below the unperturbed dimensions at ca. 335 K. The results obtained for gels in toluene and in cyclohexane at 308 K did not show any influence of the cross-linking density on the measured ( R ) z and the deviation was ca. 10% so in Figure 9 error gars showing a 10% relative standard deviation have been drawn through

M , x 10-3 146 105 105 110

(R,/R,0)* phantom 1.5 1.5 1.5 1.5 1.4 1.4 1.4

measd 1.0 1.6 1.0 1.4

1.4 1.0 1.5

110 166

affine 1.8 2.0 2.0 2.1 2.1 2.1 1.9

Table VI1 Radius of Gyration and Molecular Weight of Labeled Chain in Cross-Linked Gels at Swelling Equilibrium in Cyclohexane over the Range 308-333 K code N4.04D400 ( k = 3)

temp, K 308 313 318 325 333 338

swelling ratio 4.9 5.6 6.2 6.9 7.7 8.0

N4.04D600 ( k = 4)

308 313 318 325 333 338

3.8 4.1 4.5 4.9 5.3 5.7

(R&, 92 110 125 123 113 66 95 123 131 127 114

76

M, X 79 92 68 107 101 114 105 97 100 103 98 92

0

0

l L

80 I-

1

1 --

oN4.04D400

.N4.04D600

308

328

318

33 8

/K Figure 9. Radius of gyration of labeled chains in cross-linked gels at swelling equilibrium in cyclohexane as a function of temT

perature.

the mean value of the two networks at each temperature. Scattering in the Intermediate Regime. The single-chain scattering function was also measured in the intermediate regime of scattering vector, over the range 0.034 = Q/A-' I0.123, to characterize more fully the conformation of the network chains. The measurements were made for one concentration of labeled species only. At these intermediate values of Q the monomer-monomer correlations correspond to shorter range effects than probed in the Guinier regime and the behavior inside the coil is explored. In general the curvature of P(Q)-l gives



1 r

B

A

--

- e;;d2linked

202.01.0-

4-

1.0-

0.5

0.5-

0.03 0.05

0.1

0.03 Q

0.05

0.1

I

la-'

c'1-

. cn

3-

'0

LY

Lz

Figure 10. Reciprocal scattered intensity in the intermediate Q range: (A) un-cross-linked containing 3.7% PSD in NSD PSH16; (B) dry extracted network containing 5% PSD in

2.

B1.05D800.

100

A (=Q

Figure 12. Comparison of molecular deformation of labeled network chains as measured by SANS with phantom model predictions for k cross-links (f = 4) per chain. Solid lines calculated from eq 8 with number of cross-links per chain as marked: (0) k = 6: (a)k = 5: ( 8 )k = 4: ( 0 )k = 3: lo) k = 2.

J

0.03

0.05

0.1

0.03 0.05

0.1

1

Q 18.' Figure 11. Reciprocal scattered intensity in the intermediate Q range for N2.05D400 gel: (A) in toluene at 293 K (4e= 0.044); (B) in cyclohexane at 308 K = 0.166).

the higher moments of the distribution of effective scattering centers and provides information on the particle shape. In particular, for random coils where the segment distribution is of the ideal Gaussian form the scattered intensity follows the Debye form31

-

P(Q) Q-'

(23) and when the coil is expanded by relatively long-range excluded volume effects the scattering law follows excluded volume statistics and scales with Q according to31

-

P ( Q ) Q-5/3 (24) Typical correlations of scattered intensity in the intermediate range for an un-cross-linked and a bulk network are shown in Figure 10. In both cases Gaussian behavior was observed and the reciprocal scattered intensity scaled as the square of the scattering vector. No influence of cross-linking density was observed. Typical correltions of intermediate regime scattering behavior for gels swollen in (a) toluene and (b) cyclohexane at 308 K are shown in Figure 11. In the intermediate region the coherent signal approached the incoherent background level and therefore data of high statistical quality and careful background subtractions were necessary. The highest concentration of labeled species in the network (5% of chains labeled) was used but in spite of relatively long data collection times the high degree of swelling meant that data of poorer quality than desired were obtained. The gels in toluene were all in agreement with the excluded volume scaling of the reciprocal scattered intensity of eq 24. The gels in cyclohexane exhibited a scaling in agreement with Gaussian statistics in the lower range of scattering vector but then displayed a more rapid falloff in intensity. The reason

for this behavior is unknown but may merely reflect inaccuracies in accounting for the incoherent background level. Once again no trend with cross-link density was observed. Finally no crossover behavior between Gaussian and excluded volume statistics was seen for any of the gels in toluene or cyclohexane over the Q range explored.

Discussion The molecular deformation of the labeled chains in the gels swollen in toluene and in cyclohexane at the 8 temmultilinked chains in the limiting cases of the pure phantom and end-to-end pulling (or affine deformation of junctions) models. The results are reported in Tables V and VI. In the calculations it was assumed that the junctions were tetrafunctional and that the deformation was isotropic. The measurement of the degree of equilibrium swelling and the calculation of the approximate number of cross-links per chain using the Flory-Rehner equation introduces errors into the calculations. The number of cross-links per chain and the isotropic deformation ratio are inextricably linked for these equilibrium-swollen networks (i.e., the cross-link density determines the degree of swelling at equilibrium in a given solvent) and the models do not predict much variation in the molecular deformation with increase in cross-link density across the range covered. For the most part the experimental results lie between the predictions of the two models. The ability of the phantom model, albeit better than the affine model, to predict the deformation of the multilinked network chain in a gel at swelling equilibrium is far from convincing even allowing for the large experimental error. These results are further compared with the phantom model deformation in Figure 12, and this serves to highlight the difficulty in differentiating between the various forms when the experimental uncertainty is high. It is of interest to note that Figure 12 shows the predicted deformation of multilinked chains with only 2 or 3 crosslinks per chain falls below that for the end-linked chain due to the influence of the end subchains, which are considered not to deform. Ullman remarked in his discussion



Table VI11 Comparison of Labeled Path Deformation by SANS for Gels at Swelling Equilibrium in Toluene at Ambient Temperature with Predictions for End-Linked Model Networks

(RS/RgO)* ref

N2.400 B1.400 B2.1000 B1.600 B1.800

SANS 2.64 2.81 2.89 2.67 2.78

affine 8.24 6.05 5.11 5.02 4.16

end-to-end Dulline 4.62 3.52 3.06 3.01 2.58

Dhantom 2.81 2.26 2.02 2.00 1.79

affine 0.32 0.46 0.56 0.53 0.67

(RgIRgO)iANSI (R,I~,O);Oddel end-to-end Dullina Dhantom 0.32 0.94 0.80 1.24 0.94 1.43 1.34 0.88 1.08 1.55

of the modeP8 that the assumptions employed become more valid as the number of cross-links per chain increases and in particular are best if It is at least 10. There are, 0 N4.04D400 k= 3 however, some practical difficulties in trying to achieve b N 4 . 0 4 D 6 0 0 k.4 /Ak= 4 3 these conditions for a SANS experiment. Restrictions on / the experimental accessibility of very low scattering vector / /Ak=3 limit the size of labeled chain that can be measured easily in the Guinier regime and hence the absolute size of the labeled pathway through the network. On the other hand, imposing a high degree of cross-linking on a short chain will severely restrict its elastomeric properties. Moreover, the difference between the predicted molecular deformation experienced by chains with different numbers of cross-links is greatest at low values of k, i.e., less than 10 cross-links per chain. b It is believed that the molecular deformation should 0 approach affine behavior as the scale of observation is increased with respect to the network mesh size. This was not observed within the context of the model for multilL inked chains. One possible explanation may be that the labeled topological route investigated was not large enough h (=01'3) to cross some threshold distance for affine behavior. Figure 13. Deformation of R, of labeled chains in cross-linked Bearing in mind, however, Ullman's qualification on the gels at swelling equilibrium in c clohexane as a function of isotropic deformation ratio (A = Q13, and comparison with junction validity of his model for multilinked chains, it is worth making a brief comparison of the SANS results with the affine (A) and phantom models (P)for multilinked chains with k cross-links per chain. (O,.) Experimental data for specimens predictions of models for end-linked network chains. A noted swollen in cyclohexane at different temperatures. comparison of the molecular deformation in gels swollen in toluene with models assuming (i) affine deformation of lecular behavior which cannot be easily explained by these the entire chains, (ii) affine deformation of the mean theories include essentially unperturbed dimensions at 308 junction positions with no fluctuations (i.e., end-to-end K in spite of significant macroscopic swelling and an appulling model), and (iii) affine deformation of the mean parent maximum in the molecular deformation curve at junction positions with fluctuations of junction positions ca. 320 K which is not reflected in the macroscopic about the mean (Le., pure phantom model) is presented swelling. in Table VIII. In no case did the molecular deformation In an attempt to rationalize at least some aspects of this reach that of the bulk network; for the more highly behavior one might care to invoke the recently advanced cross-linked networks agreement of the experimental reconcept of some unfolding mechanism without deformation sults with the predictions for the affine displacement of of the elementary network chains. This implies the exjunctions was quite reasonable and for the networks of low istence of some reference state beyond which the elecross-link density the phantom behavior was a more satmentary network chains must also experience deformation isfactory description. This trend is at least qualitatively swells. One conceivable choice for such a consistent with Flory's constrained junction m ~ d e l : ~ as ~ the ~ ~network ~ affine behavior is expected at low strains, phantom bereference state is the degree of swelling at the 0 temperhavior is expected at higher strains, and equally affine ature; this selection also satisfies the criterion that the effect of unfolding decreases with increased cross-link behavior, through the Flory-Erman parameter, K F , is exdensity. Table IX shows the data obtained for the gels pected to be approached with increased cross-link density. It would be dangerous to regard this treatment too optiswollen in toluene normalized to this reference state and mistically as the corresponding analysis for the gels in a comparison with the models. In all cases the measured molecular deformation exceeded the model predictions. cyclohexane is much less favorable. Such an analysis relies heavily on the choice of reference Figure 13 shows the molecular deformation in the two state, and until it becomes clearer how this should be networks swollen in cyclohexane at different temperatures defined the treatment is necessarily rather tentative. In correlated against the bulk deformation and a comparison any event the postulation of such a reference state does with the model predictions for multilinked chains. Clearly none of the statistical theories of rubber elasticity for either not appear to be able to explain the maximum in the molecular deformation of the multilinked chains around multilinked or end-linked chains, assuming a purely geometrical deformation of the chains, is capable of accounting 320 K. for the behavior of the labeled path in gels of fixed According to de Gennes' c* theorem the size of the elcross-link density at swelling equilibrium in cyclohexane ementary chain between junctions in swollen gels is comat various temperatures. Prominent features of the moparable to the size of an isolated coil of the same molecular .V

7


Macromolecules, Vol. 19,No.IO, 1986

Table IX Comparison of Labeled Path Deformation by SANS for Gels at Swelling Equilibrium in Toluene at Ambient Temperature with Predictions for End-Linked and Multilinked Network Models Assuming a Reference State with Respect to an Unfolding Mechanism at the Equilibrium Degree of Swelling at 8 Conditions

(R./R.0)2 vol swelling ref N2.400 B1.400 B2.1000 B1.600 B1.800

Qfni

QA

23.6 14.9 11.6 11.3 8.5

5.96 5.03 3.48 4.30 2.90

QdQA 3.96 2.96 3.33 2.62 2.93

k 2 3 4 5 6

SANS 2.64 2.81 2.89 2.67 2.78

weight in dilute solution. Recent SANS results of Beltzung et al.42on end-linked poly(dimethylsi1oxane)networks at swelling equilibrium in a good solvent have shown excellent agreement between the size of the elementary network chain and the equivalent free coil in dilute solution. These workers concluded that the local conformation of the network chains in gels at swelling equilibrium was governed by the polymer-solvent interaction and the macroscopic swelling by cross-link density, junction functionality, etc., presumably via some unfolding mechanism. In experiments on multilinked chains it is conceivable that rather than the free coil dimensions a reduced size might be observed consistent with the behavior in semidilute solutions. In any event as has already been pointed out above, the measured radius of gyration of the multilinked chains in these random polystyrene gels proved larger than the free coil estimation except possibly at the 8 temperature where it could be argued that unperturbed dimensions were observed. A possible explanation could be postulated analogous to Ullman's argument for the effect of statistical theories of rubber elasticity on a multilinked chain that correlations between monomers on different submolecules must be taken into account. Geissler et aL4 have already formulated a simple model for the swelling of end-linked network chains in a good solvent, invoking an unfolding mechanism and semidilute solution scaling behavior that presumably could be extended to the multilinked chain case. The above comparison between the behavior of gels and solutions still does not account for the maximum in the molecular deformation of the chains in the gels swollen in cyclohexane. The dramatic collapse is difficult to rationalize from the limited data available. Certainly the polystyrene-cyclohexane system is known to have a lower critical solution temperature (LCST) so that the solvent quality will not improve indefinitely with increasing temperature, but at ca. 486 K the LCST is far removed from the conditions studied here. A maximum in the radius of gyration for polystyrene in cyclohexane solution has been observed by SANS at ca. 413 K,@and a maximum in the swelling of cross-linked polystyrene spheres has been measured at 423 K.65It is unlikely that the observed maximum in radius of gyration of these network chains could be attributed to a LCST unless the permanent cross-links were to drastically alter the conformational entropy of the system compared to a normal solution. On the other hand, the 8 temperature (defined as the condition when the second virial coefficient, A2, becomes zero) in the region of cross-links and in the middle of the chain would not be expected to be the same due to the different spatial distribution of chain segments in the two cases.66 Data in the literature indicate that the 8 temperature for star polymers in dilute solution is reduced compared to that for their linear co~nterparts.~'-~~ The effect increases with increasing functionality and decreasing arm length, but nothing indicates as dramatic an effect as observed

end-linked multilinked affine end-to-end pulling phantom end-to-end pulling phantom 2.50 1.75 1.38 1.58 1.29 2.06 1.53 1.26 1.61 1.24 2.22 1.61 1.30 1.84 1.40 1.90 1.45 1.42 1.68 1.37 2.04 1.52 1.26 1.83 1.43

here for the networks. Furthermore, the severity of the constraints defining the 8 condition is reduced for the entangled compared to the dilute regime. Any difference between deuterated compared to hydrogenous chains is not strongly indicated either since their 8 temperatures in cyclohexane have been shown to be within a few deg r e e ~ . ' ~We have no evidence of phase separation of labeled and unlabeled species in these samples. The presence of moderate amounts of pendant chains has been reported to strongly affect mechanical properties while having only slight influence on chain conformation and equilibrium degree of swelling.72

Conclusions Neglecting samples that gave anomalously high scattering at low angles, the dimensions of the multilinked chain in the bulk network were the same as for an equivalent chain in the un-cross-linked material. This result was independent of cross-link density over the range studied. The value of the radius of gyration was in agreement with the prediction for an unperturbed Gaussian chain. The Q2 dependence of the scattered intensity of both cross-linked and un-cross-linked chains over the intermediate range of scattering vector further confirmed their Gaussian nature. The salient feature of the measurements on gels swollen to equilibrium in solvents of varying thermodynamic quality was the remarkable constancy of the radius of gyration of the labeled molecular pathway in networks of different cross-link density under given conditions in spite of differences in the degree of swelling. It was shown to some extent that this was consistent with a model suggested by Ullman for multilinked network chains. The bulk deformation and the number of cross-links per chain are inseparable in these random networks under conditions of equilibrium swelling and these parameters have opposing effects on the chain deformation in the model. The molecular deformation was always less than the affine prediction and the phantom model generally gave a better description. In the light of recent theory and SANS experiments, the results were also compared with the behavior of polymer solutions. Although a strictly quantitative comparison was not found, the predominant influence of the polymer-solvent interaction certainly remains very attractive in explaining the weak influence of crosslink density on the radius of gyration. The behavior of the scattering law in the intermediate regime provides a further indication that the local polymer-solvent interaction is fundamental in determining the chain conformation. There is also some indicating that an unfolding mechanism may be important in the swelling process. None of the models discussed could explain an apparent maximum at ca. 320 K in the radius of gyration of the labeled pathway in gels swollen to equilibrium in cyclohexane at different temperatures. For the present, in the absence of further data, the cause of this feature remains



unresolved. The discussion illustrates the difficulty in arguing the case for one model above the others as being the most appropriate to describe the behavior of individual network chains in swollen gels. It can be misleading to rely on one experiment, and behavior over a range of conditions must be viewed to obtain a complete understanding.

(33) Benoit, H.; Duplessix, R.; Ober, R.; Daoud, M.; Cotton, J.-P.; Farnoux, B.; Jannink, G. Macromolecules 1975,8, 451. (34) Pearson, D. S. Macromolecules 1977, 10, 696. (35) Warner, M.; Edwards, S. F. J. Phys. A 1978, A l l , 1649. (36) Ullman, R. J . Chem. Phys. 1979, 71, 436. (37) Ullman, R. Macromolecules 1982, 15, 582. (38) Ullman, R. Macromolecules 1982, 15, 1395. (39) Ullman, R. In Elastomers and Rubber Elasticity; Mark, J. E.;

Acknowledgment. N.S.D. thanks the Science and Engineering Research Council for the provision of a research studentship. We thank Dr. Ann Maconnachie, Imperial College, London, for the radiation cross-linking of the polystyrene samples, and the assistance of Dr. S. Bantle at the ILL, Grenoble, was much appreciated throughout the work reported here.

Lal, J., Eds.; American Chemical Society, Washington, DC, 1982; ACS Symp. Ser. No. 193. (40) Bastide, J.; Picot, C.; Candau, S. J . Macromol. Sci., Phys. 1981,

Registry No. PSH, 9003-53-6; PSD, 27732-42-9; neutron, 12586-31-1.

References and Notes Treloar. L. R. G. The Physics of Rubber Elasticity. - . 3rd ed.: Clarendon: Oxford, 1975: Flory, P. J. Principles of Polymer Chemistry; Cornel1 University: Ithaca, NY, 1953. Dusek, K.; Prins, W. Adu. Polym. Sci. 1969, 6 , 1. Smith, T. L. In Treatise Mater. Sci. Technol. 1977, IOA. Candau, S.; Bastide, J.;Delsanti, M. Adu. Polym. Sci. 1982,44, 30.

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