Structural, elastic, and dynamic properties of swollen ... - Springer Link

Structural, Elastic, and Dynamic Properties of Swollen Polymer Networks We dedicate this review to Prof. Manfred Gordon on the occasion of his 65th birthday.

Sauveur Candau 1, Jacques Bastide2 and Michel Delsanti3 1 Laboratoire d'Acoustique Moldculaire, Universit6 Louis Pasteur, Institut de Physique, 4, rue Blaise Pascal, F-67070 Strasbourg Cedex 2 Centre de Recherches sur les Macromol6cules, 6, rue Boussingault, F-67083 Strasbourg Cedex 3 C.E.N. Saclay S. DN, 91191, F-Gif-sur-Yvette Cedex

This article reviews some recent developments in the physics of gels, due to both new methods of synthesis and modern techniques for the study of microscopic properties of gels. The review consists of four major sections. In the first section, some of the recent methods of synthesis allowing to prepare labelled networks are described. The second section is concerned with the structural properties of networks. A critical discussion of both classical and scaling theories in the light of small-angle neutron scattering data is presented. In the third section, scaling relations for the elastic moduti of networks swollen in good solvents are discussed. The fourth section deals with the dynamic properties of swollen networks with special emphasis on inelastic light-scattering experiments. The conclusion of this review stresses the important progress made in the understanding of the static and dynamic properties of swollen networks and describes possible future developments.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.

Synthesis of Polymer Networks

B.

Microscopic Structure of Networks . . . . . . . . . . . . . . . . . . . . . . B.I. Properties of Polymer Solutions in G o o d Solvents . . . . . . . . . . B.I. 1. Swelling of a Single Coil . . . . . . . . . . . . . . . . . . B.1.2. T h e r m o d y n a m i c Properties of P o l y m e r Solutions . . . . . . B.II. Conformations of the E l e m e n t a r y Elastic Chains of Networks . . . . B.II. 1. Radius of Gyration of Network Chains . . . . . . . . . . . B.II.2. Swelling Equilibrium D e g r e e of Polymeric Networks "c* T h e o r e m " . ...................... B.III. Pair Correlations in Gels . . . . . . . . . . . . . . . . . . . . . . . B.III. 1. Dry Networks . . . . . . . . . . . . . . . . . . . . . . . B.111.2. Networks Swollen at Equilibrium in a G o o d Solvent . . . .

. . . . . . . . . . . . . . . . . . . . . . . .

30 30 32 32 32 33 35 36 44 46 46 47

Ad-~'ancesin Polymer Science 44 © Springer-Verlag Berlin Heidelberg 1982

28 C.

D.

S. Candau et al. Elastic Properties of Networks Swollen at Equilibrium in a Good Solvent . . . C.I. Scaling L a w for Elastic Moduli of N e t w o r k s Swollen at E q u i l i b r i u m in G o o d Solvents . . . . . . . . . . . . . . . . . . . . . . . . . . . C.II. E x p e r i m e n t a l M e t h o d s for t h e D e t e r m i n a t i o n of t h e Elastic Moduli C.II.1. Shear Modulus . . . . . . . . . . . . . . . . . . . . . . . C.II.2. Compressional Modulus .................. C.II.3. Longitudinal Modulus ................... C.III. E x p e r i m e n t a l Results . . . . . . . . . . . . . . . . . . . . . . . . . C.III.1. S h e a r M o d u l u s . . . . . . . . . . . . . . . . . . . . . . . C.III.2. C o m p r e s s i o n a l M o d u l u s . . . . . . . . . . . . . . . . . . C.III.3. L o n g i t u d i n a l M o d u l u s . . . . . . . . . . . . . . . . . . .

49 51 51 52 52 53 53 55 56

Dynamic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D,I. Frictional P r o p e r t i e s . . . . . . . . . . . . . . . . . . . . . . . . . D.II. C o o p e r a t i v e Diffusion Coefficient . . . . . . . . . . . . . . . . . . D . I I I . Kinetics of Swelling or Deswelling of N e t w o r k s . . . . . . . . . . . .

56 58 60 64

Conclusions

49

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

List of Symbols and Abbreviations a3 al b c c~

c* f

_f _fro g g(r) h i~ kB

size of the site; volume of the monomet; volume of the chain link diluent activity radius of a spherical gel sample polymer concentration expressed in g cm -3 polymer concentration in a network at swelling equilibrium, expressed in g cm -3 cross-over concentration between dilute and semi-dilute regimes functionality of the crosstinks effective frictional coefficient per monomer driving force per unit volume force exerted on a monomer number of links in a btob spatial pair-correlation function between monomers volume fraction of polymer in the reference state photocurrent due to the polarized scattered light Boltzmann constant exponent in the scaling form of the elastic free energy

m*

apparent mass of the monomer immersed in a solvent n total number of neighbours of a given crosslink n~ number of statistical units in a Gaussian subchain p exponent of the scaling law ~ (d#) q scattering wave vector (transfer momentum) r modulus of the vector position s shear deformation sd sedimentation coefficient t time tp number of topological neighbours of a given crosslink u 1/2 - Z u' interaction parameter between blobs u* effective interaction parameter in a semi-dilute regime u (L, t) displacement fluctuation of the polymer network from its equilibrium position at point r and time t _u(L, t) displacement velocity tim drift velocity of a monomer uq(t) longitudinal component of u(L, t) with wave vector q

Structural, Elastic, and Dynamic Properties of Swollen Polymer Networks fl~ v1 xs, Xk, XD

velocity of the solvent molecular volume of the solvent exponents of the scaling laws as a function of the polymer concentration for sedimentation, permeability, and diffusion coefficients, respectively A background of c (t) A0 numerical constant in the expression of the photocurrent due to the polarized light scattered B prefactor in the scaling form of the elastic free energy c (t) time autocorrelation function of the photocurrent D diffusion coefficient E Young modulus E0 (t) scattered electric field at an angle 0 F'~ elastic free energy density (per unit volume of the gel) F't elastic free energy density (per site) G total Gibbs energy G' Gibbs energy density (per unit volume of the solution) G" Gibbs energy density (per site) G~iI Gibbs energy density of mixing (per site) G~0 Gibbs energy density (per unit volume) of a gel under shear deformation Io incident light intensity K compressional modulus Ket inverse of the elastic contribution to the compressional modulus arising from permanent linking of the chains Kgel compressional modulus of a gel Kp permeability coefficient K~o~ compressionai modulus of a solution (at zero frequency) M longitudinal modulus Mn number average molecular weight Mw weight average molecular weight N polymerization index N' N/g P pressure R end-to-end distance of a chain Ro unperturbed end-to-end distance of a chain Rg radius of gyration of a chain Rgo radius of gyration of a network chain in the reference state Rgll radius of gyration of a network chain parallel to the stretching Rg~ radius of gyration of a network chain perpendicular to the stretching

R~ (R2d) (P~)

(R~) RE S (q) T TR 6 e r/~ 0 x

2 2i

/xi #~o

zc a as r ~o ~*

D0 (a c d~ Z Z0 CGD PDMS PS QELS SANS

29

radius of gyration of an elastic chain for unstretched network mean-square end-to-end distance of a network chain in bulk mean-square end-to-end distance of a network chain in the reference state mean-square end-to-end distance of a network chain in the swollen state end-to-end distance of a single chain in a good solvent scattering function temperature lifetime of physical entanglements numerical constant characteristic of chain packing in the swollen network dielectric constant viscosity of solvent scattering angle correction term in the expression of S (q) for semi-dilute solutions in a good solvent deformation ratio wavelength of the light in the scattering medium shear modulus chemical potential of the solvent in the solution or in the gel chemical potential of the pure sop vent screening length osmotic pressure number density of monomers compressional stress shear stress characteristic swelling time of the network circular frequency volume fraction of polymer cross-over volume fraction of polymer between dilute and semi-dilute regimes volume fraction of polymer in the reference state volume fraction of polymer in the solution prior to crosslinking equilibrium volume fraction of polymer in a swollen network Flory-Huggins interaction parameter osmotic compressibility classical gradient diffusion polydimethyisiloxane polystyrene quasi-elastic light scattering small-angle neutron scattering

30

S. Candau et al.

Introduction Despite the very large number of studies carried out on gels during the three last decades, the properties of these materials are not yet well understood and none of the current theoretical models enables a satisfactory interpretation of the experimental data. The more controversial point concerns the dependence of the macroscopic properties of gels on the molecular network parameters. In this regard, an important progress has been recently achieved, owing to both new synthesis methods and modern techniques for the study of microscopic properties of gets. The first purpose of this paper is to review the information on the local conformation of the network chains as a function of the state of strain or swelling of the networks, as inferred from small-angle neutron scattering results. The mechanism of macroscopic deformation of gels wilt be discussed and compared with the current theoretical models. The second goal of this paper is to describe the static and dynamic properties of the gels as seen from the viewpoint of the scaling approaches. The scaling concepts recently introduced into the theory of polymer solutions were subsequently extended to the description of swollen networks. We will summarize the current status concerning the scaling laws in swollen networks and semi-dilute solutions of linear polymers, with special emphasis on the concentration dependences of elastic moduli and diffusion coefficients. Only systems consisting of neutral flexible polymers and very good solvents have been considered. The physical gels in which the chains are crosslinked via the formation of helical structures or microcrystals are not considered either. This paper consists of four major sections. In the first section are described some of the recent synthesis methods allowing the preparation of calibrated networks with labelled chains. The second section deals with the microscopic structure of networks. The recent small-angle neutron scattering results are reviewed and the deformation mechanisms in gets discussed. In the third section, scaling relations derived for the elastic moduli of networks swollen at equilibrium are compared with the existing experimental data. Finally, in the fourth section a comparison between the dynamic properties of networks swollen at equilibrium and semi-dilute solution is made within the framework of a scaling approach.

A. Synthesis of Polymer Networks Any attempt to test a theoretical model relating the thermodynamic and dynamic properties of polymeric gets to their internal structure requires the use of well-defined networks. An "ideal" network for such investigations should consist of a three-dimensional random collection of identical coils with their ends connected to crosslinks. This implies that the network chain possess a narrow molecular weight distribution. Moreover, the functionality of the crosslinks, that is the number of elastic network chains tied to one given crosslink, should be constant throughout the sample. The microscopic defects most commonly encountered in gels are dangling chains (with only one end attached to the network), loops (i.e. chains linked at both ends to the same crosslink) or entanglements trapped in between two adjacent crosslinks.

Structural, Elastic, and Dynamic Properties of Swollen Polymer Networks

31

Microscopic heterogeneities associated with non-random crosslinking are also frequently observed in gels. The most spectacular evidence of such heterogeneities is provided by the micro-syneresis1' 2), that is the formation of a dispersion of droplets of liquid inside the get. This effect generally results in a phase separation induced by an increase of the degree of crosslinking in the course of the reaction. Syneresis does not generally occur, if the network is prepared in the absence or at a low content of diluent. Nevertheless, it seems difficult to synthesize macroscopically homogeneous gels, where both segment density and crosslink density remain constant throughout the sample. The methods generally used to synthesize polymer networks cannot provide any control of the microscopic structure of the network. For instance, radical copolymerization or polycondensation 3' 4) are basically random processes. As a consequence, the molecular weight distribution of the network chains is necessarily broad. Moreover, various chain transfer processes may lead to the formation of an appreciable number of dangling chains. The vulcanization processes5' 6), carried out on linear "primary" macromolecules, lead also to a very broad molecular weight distribution of the elastic chains. The number of pendant chains which is roughly equal to twice the number of "primary" chains can be reduced by using very long chains. However, loops cannot be avoided, as well as trapped entanglements. Recently, new" methods based on an "end-linking" procedure have been developed in various laboratories in order to synthesize model networks fulfilling as much as possible the requirements of ideality. These methods have been described in detail elsewhere 7-1°) (cf. also review paper of J. Mark in the same issue and references therein) and we shall merely recall their principle and discuss the main features of the resulting networks. In the first step, a precursor polymer with reactive functions (such as organometallic sites, alcohol functions or hydrosilane groups) at both ends is synthesized, generally by anionic polymerization 11). The polymer obtained exhibits a narrow molecular weight distribution; the average value can be determined by conventional methods (gel permeation chromatography, light scattering, osmometry ...). In the second step, the precursor polymer molecules are reacted with a multifunctional reagent added in stoichiometric amounts. The functionality of the network is that of the crosslinking reagent. Alternatively, the second step is achieved by anionic block copolymerization: the carbonionic living sites at both ends of the precursor chains initiate the polymerization of a small amount of a bifunctional monomer (generally divinylbenzene). This polymerization yields small, tightly crosslinked nodules, each of which being connected with f chain ends. This method does not yield networks of known average functionality f. Another end-linking method involves end-linking of star-shaped macromolecules with functional groups at the end of all branches through a bifunctional reagent 12' 13). Most of the end-linking methods yield networks with precisely known average molecular weight of the network chains joining two neighbouring crosslinks and sometimes the average functionality of the crosslinks. In addition, the molecular weight distribution is rather narrow. However, the possibility to prepare labelled networks is the main advantage of these methods. For instance, networks containing a small amount of entirely deuterated elastic chains have been synthesized TM 15) Small-angle neutron scattering experiments performed on these networks allow a direct measurement of the mean-square radius of gyration of an individual elastic chain in the network. Alterna-

32

S. Candau et al.

tively, the crosslinks can be labelled 16'17) but in this case the interpretation of the SANS data is more complex. The end-linking methods provide some control of the structure defects on the molecular scale. Under the usual experimental conditions, the number of dangling chains is expected to be rather low provided the stoichiometry is accurate. Moreover, it is possible to prepare gels containing controlled amounts of dangling chains by using a mixture of monofunctional and bifunctional chains 18'19) as a precursor polymer. The formation of loops in the end-linking reaction must be taken into account, especially when the crosslinking reaction is carried out at low polymer concentration or when the molecular weight of the precursor chain is low 2°'21). As a matter of fact, the polymer concentration at which crosslinking is achieved is a fundamental parameter as far as the internal structure of the network is concerned. At very low polymer concentrations, syneresis occurs, yielding strongly inhomogeneous gels; even if syneresis is avoided, a large proportion of dangling chains and loops is produced. Higher concentrations of precursor polymer increase the probability for permanent trapped entanglements to occur. Moreover, a very high viscosity of the medium may be favorable for the formation of inhomogeneities within the gel. Apparently, the optimal concentration should be of the order of the overlap volume fraction ~* at which the interpenetration of chains begins. This volume fraction depends on the molecular weight of the polymer precursor. It should be noticed that the so-called "model" networks exhibit generally very poor mechanical properties 9) and break readily under extension. This can be due mainly to the presence of a rather large amount of submicroscopic inhomogeneities. It is obvious that, when non-randomly crosslinked networks are mechanically loaded, some portions of the elastic chain would undergo excessive strain so that poor mechanical performance may result as a whole.

B. Microscopic Structure of Networks In this section, we discuss the relation between macroscopic deformations of gels and local conformation of the network chains, as inferred from recent small-angle neutron scattering (SANS) experiments. More specifically, we consider the isotropic deformation associated with the swelling of a network in a good solvent for the polymer chains. It will be seen that crosstinked gels exhibit some features analogous to those of semi-dilute solutions of linear high-molecular-weight polymers. Therefore, we first recall briefly some fundamental properties of the solutions of linear chains in good solvents. For a more detailed analysis, the reader is referred to the recent book of de Gennes 22).

B.I. Properties of Polymer Solutions in Good Solvents

B. L 1. Swelling of a Single Coil This problem has been treated long ago by Flory within the framework of a mean field theory 3).


33

Let us consider a chain in a good solvent, of end-to-end distance R, polymerization index N and internal average volume fraction q5 = Na3/R3, where a 3 is the volume of the monomer. Following Flory, the total Gibbs energy can be written as the sum of two terms: G/kBT =

(1

-Z

~

+ ~

(B-l)

where Z is the Flory-Huggins interaction parameter. The first term of Eq. (B-l) represents the repulsive interaction between monomers (mean field approximation). The last term is the elastic energy of an ideal chain. The minimization of Eq. (B-l) with respect to R leads to the Flory relationship between the end-to-end distance RF and the polymerization index N of a single chain in a good solvent

(S-2) This result, well verified experimentally, has also been obtained from numerical methods 23-251 and also from the theoretical approaches based on the analogy between polymer solutions and magnetic systems22'26-32)'. However, it is important to remark that the above derivation gives the correct result only because of the cancellation of the effects caused by the two following strong approximations: (a) the correlations between segments are neglected; and (b) the elastic energy of distortion is overestimated as a consequence of the Gaussian chain approximation.

B.L2. Thermodynamic Properties of Polymer Solutions The behaviour of a polymer solution depends strongly on its concentration. At high dilution, the coils are separated and behave as a gas of hard spheres. In good solvents, the relevant parameter describing the interactions is the excluded volume. At high concentrations, the chains overlap each other. The threshold concentration, at which the chains begin to interpenetrate, corresponds to a close packing of the coils, that is (:I)* = NaB/R3F

(B-3)

Although the transition between the dilute and entangled regimes is not sharp, the concentration ~* defined as the cross-over between the two asymptotic regimes qb ~ alp* and qb ~> ~* presents interesting scaling properties. More especially, Eqs. (B-2) and (B-3) yield the following dependence of ~* on N for solutions in very good solvents ~* ~ N -4/5

1 The more recent results give the value 0.588 for the exponent of the power law RF(N)33)

(B-4)

34

S. Candau et al.

We are mainly interested in the semi-dilute regime that is the case where • exceeds do*, but is sufficiently low so that the system can still be described by one interaction parameter. In this regime, the thermodynamic properties are no longer dependent on the molecular weight of the polymer. The relevant parameter is the screening length introduced by Edwards 34). The theoretical studies by de Gennes and des Cloiseaux, based on the analogy between ferromagnets and polymer solutions established that the correlation length ~ in the semi-dilute solutions obeys the following scaling l a w 22' 35) e ~ do-3/4

(B-5)

and is independent of N. Roughly speaking, the correlation length measures the average distance between nearest chain contacts. Thus the semi-dilute solution can be simply visualized as a polymeric net with the mesh size ~ which decreases rapidly as the concentration do increases. At the cross-over concentration do*, the correlation length ~ can be identified with RF. A scaling argument can be given to predict the thermodynamic properties of semidilute solutions and more especially the concentration dependence of the osmotic pressure. The osmotic pressure of a semi-dilute solution was originally derived from the Gibbs energy of mixing given by the Flory-Huggins lattice model 1' 3). The expansion to the second order of the density per unit volume of the Gibbs energy of mixing G' is: 1 G'/kRT = •/N lnQ + ~ ua3• e

(B-6)

where u = 7i - X, Q is the number of chain links per unit volume and is related to dO through do = oa 3. o/N lnQ corresponds to the free energy of translation of the chains into the solvent. The chemical potential Pl of the solvent can be derived from the following relation 22'36)

/t 1 =

a3

OG' G' - O 3Q ] = 6 " -

do--

where G" is the density per site of size pressure is given by :r = -/~1 vi-l

G"

5do a3

(B-7)

of the Gibbs energy of mixing. The osmotic

(B-8)

where vl is the partial molecular volume of the solvent. Equation (B-6) and Eq. (B-8) lead to the classical er ~ do2 dependence in the semi-dilute regime (in which the translational term and the three-body contribution proportional to Q3 can be neglected). De Gennes 22) has shown that this mean field model is not adapted to the semi-dilute regime because of the large fluctuations of the local polymer concentration. From the analogy between polymeric and magnetic systems, des Cloiseaux derived the following concentration dependence of the osmotic pressure35): 37 0c (I)9/4

(B-9)


35

An alternative derivation of Eq. (B-9) is based on the decimation procedure. Rigourously, a detailed analysis in terms of renormalization group trajectories (cf. Ref. 22, Chap. 11 and Ref. 37) is required. This type of procedure is the theoretical basis behind the so-called blob model. The excluded volume effects are important at short-range scale, within a "blob" containing g monomers. At larger scale excluded volume interactions are screened. The mean-field approach will, therefore, be valid if the blob of size and volume ~3 is taken as the site. Then, in order to describe the thermodynamics, the following transformations must be carried out in Eq. (B-6) 0

~ 0' = 0/g

N

) N' = N/g

a

)~

U

) U~

(B-IO)

u' is the interaction parameter between blobs and ~ oc g3/5. This relation yields the following expression of the Gibbs energy per site a 3 ap G"/kBT = ~ - lnq~ + u*

(I)9/4

(B-11)

where u* is the effective interaction parameter. All terms linear in ap have been omitted in Eq. (B-11) as they give a constant contribution to the total Gibbs energy. From Eqs. (B-7) and (B-8), the relation (B-9) is easily obtained.

B.II. Conformations of the Elementary Elastic Chains of Networks The essential question underlying any microscopic statistical theory of rubber elasticity is the description of the local conformation of the chains in relation to the deformation of the network. This subject has given rise to much controversy regarding mainly the role of the fluctuations of the junction points 1' 3s). In this respect, from the earliest works on the elasticity of networks two opposite points of view have been developed, both based on the Gaussian statistics approximations. The first model, proposed by James and Guth 39-41), is the phantom network model in which the sole constraints considered are the forces exerted by the chains on the junctions to which they are attached. On the other hand, Hermans 42), Flory and W a l l 43' 44) assumed that the individual end-to-end vectors of the elastic chains are affine in the strain, the junction points being spatially fixed for a given swelling degree. In more recent formulations, the network is assumed to behave in a fashion intermediate to those two opposite extremes38'45, 46) m detailed analysis of the thermodynamic behaviour of both phantom and real networks has been developed by Staverman and is given in the same issue. Another still debated point concerns the effect of crosslinking on the conformation of the elementary chains between the two first neighbouring crosslinks. This effect is generally taken into account through the "memory term", first introduced as "front factor" by Tobolsky47) and defined, according to Dugek and Prins, as 1)

36 hZ3

S. Candau et al.

is the mean square end-to-end distance of the network chains in the dry" network, (R~) the mean square end-to-end distance of the corresponding free chains (after severage of crosslinks) and characterizes the reference state. Therefore, h represents the volume fraction ~0 of the polymer in the reference state if, as usually assumed, the endto-end distances are deformed affinely in macroscopic volume changes. Generally, h is identified with the volume fraction qbc of the "nascent" network, that is, under the conditions under which it is formed 4). A fundamental criticism, already raised by Dugek and Prins 1), concerns the Gaussian approximation. Actually, it is now well established that in the usual equilibrium swelling concentration range, the elementary chains exhibit excluded volume statistics. This also applies to chains in the reference state, when the network is crosslinked in the presence of a large amount of diluent. This section is concerned with a critical discussion of theoretical predictions concerning the radius of gyration of elementary network chains in the light of the neutron scattering experimental results. We consider networks both in dry state and swollen at equilibrium. The swelling equilibrium degree of a network can be expressed as a function of molecular parameters by using a lattice model. Several slightly different relationships have been obtained, depending on the choice of the memory term and the assumption made on the influence of the fluctuations of the junction points 1). Recently, de Gennes postulated that the swelling of the elementary chains of the network in a very good diluent is the same as that of an equivalent chain at infinite dilution in the same solvent 22). As a consequence, the equilibrium volume fraction of polymer q~e can be identified with the cross-over volume fraction q~* between the dilute and semi-dilute regions of a solution of macromolecules of the same molecular weight as the strands of the network ¢ * ~ Na3/R~

(B-13)

The relation (B-13) which has been called the "c* theorem" has also been derived from the classical mean-field theory with an additional assumption on the reference state which is described as an assembly of closely packed Gaussian coils 48). As we shall see later, although the conclusions of the c* theorem are verified in the first approximation, the swelling mechanism cannot be described by an affine deformation model. Therefore, we present a recently proposed derivation of the c* theorem based on a phenomenotogical scaling approach which does not require the detailed knowledge of the variation of microscopic parameters during the swelling process 36). In order to simplify the equations, we shall only consider highly swollen calibrated gels where the number N of monomers between adjacent crosslinks is well defined.

B.II.1. Radius of Gyration of Network Chains Small-Angle Neutron Scattering (SANS) is a very efficient technique for characterizing the conformation of polymer chains in a bulk system or of elastic chains in a crosslinked polymer. The technique of perdeuteration of polymer chains allows the visualization of a deuterated chain in an undeuterated polymer matrix. The measurement of the angular


37

dependence of the neutron scattering intensity from networks containing small amounts of perdeuterated elastic chains enables the determination of the average radius of gyration of a network chain in the dry state as well as in the swollen state. The knowledge of the variation of the radius of gyration of the elastic chains as a function of the macroscopic strain is of particular interest because it provides direct information on the deformation mechanism. Moreover, this variation can be compared with the theoretical predictions of the current models, which are described below.

B . I I . 1.1 D e f o r m a t i o n M o d e l s

Affine deformation This rather crude model is based on the assumption that the elementary chains of the network move as if embedded in an affinely deforming continuum. For uniaxially stretched networks, the molecular deformations are characterized by the radii of gyration Rgip and Rg± parallel and perpendicular to the stretching direction, respectively, and given by Rgll/Rgi = 2

(B-14)

Rgi]Rgi

(B-15)

= /~-1/2

where Rgi is the radius of gyration for unstretched networks. 2 the extension ratio, and the network is assumed to be incompressible. For isotropic swelling, the affine deformation assumption yields Rg oc ~b-v3

(B-16)

End-to-End Pulling Deformation 14'49~ This model assumes that the positions of the junctions are fixed and transformed affinely when the network is strained. The chains, only compelled to join the junctions, are free to rearrange between themselves. This model can be considered as a limiting case of the phantom network behaviour, which is discussed below. Phantom Network 1' 39-41, 50, 51)

A phantom network is described as an ensemble of volumeless chains interacting only at the crosslinks. The distribution of the end-to-end vectors of the elastic chains is Gaussian in the undeformed state. The average positions of the crosslinks are transformed affinety when the network is strained. By contrast, the fluctuations of a junction about its average position are independent of the strain. These assumptions result in a non-affine transformation of the instantaneous end-to-end distances and of the average radius of gyration of the labelled elastic chains. For tetrafunctional networks under uniaxial stretching, Pearson derived the following expression of RgII/Rgi52): Rgll/Rgi =

(B-17)

38

S. Candau et al.

The above equation applies also to isotropic swelling, if 2 is replaced by (~el@)t/3:

Rg/Rg0 =

"(qb°/qb)z/3 + 3

) 1/2

(B-18)

4 where Rgo is the radius of gyration of the elementary chains in the reference state. However, this generalization is not rigorous, taking into account the following considerations: (a) Equation (B-18) assumes unrestricted fluctuations of the junction. It seems unrealistic to assume that the fluctuations of the junction points are strictly independent of the concentration of the network, i.e. the state of swelling. Therefore the predictions of the phantom network model for isotropic swelling can be considered only as an extreme lower limit of the molecular deformation. As the magnitude of the fluctuations of the junction point positions is reduced, the behaviour of Rg/Rg0 tends to approach the theoretical prediction of the end-to-end pulling model. (b) The swelling is considered as an "external field" which induces a purely geometrical change of the chain dimensions. In other words, the role of the excluded volume in the swelling mechanism is neglected and consequently, no valuable predictions can be made on the local chain statistics of swollen networks. (c) A further difficulty arises from the choice of the reference state in which the chains are in their relaxed Gaussian configuration. It is important that the variation of the radius of gyration calculated from Eq. (B-18) as a function of the degree of swelling depends critically on the value of qb0. Finally, the three preceding models predict that the elastic chains of a gel prepared in a good solvent undergo a supercoiling under deswelling. As a consequence, the end-to-end distance of the elastic chains of a dry network should be smaller than the unperturbed dimensions.

B.II. 1.2. Experimental Results Here, we summarize the results of recent SANS experiments performed on one polystyrene (PS) network 53) and a series of polydimethylsiloxane (PDMS) networks 5~). The PS network was synthetized by the end-linking procedure using the anionic block copolymerization of a mixture of linear PS (H) chains, containing 5% of perdeuterated chains of the same molecular weight, with divinylbenzene (three molecules per living end). The crosslinking reaction was performed in the presence of an aprotic solvent (equal volume of tetrahydrofuran and toluene), the overall polymer volume fraction being equal to 0.1. The PDMS networks were prepared by end-linking of a-t~ functional precursor chains containing 20% of perdeuterated chains, either in the dry state or in toluene at PDMS volume fractions ranging from 0.6 to 1. Both initial mixtures of linear chains and the corresponding crosstinks were investigated by SANS. The study dealt with the following points:

(a) Effect of Crosslinking on the Chain Dimensions of Networks in the Dry State The comparison between the radius of gyration of the elastic chain and that of the corresponding free chain in a PDMS (D) - PDMS (H) mixture is given in Table 1. For


39

Table 1. Comparison of the radius of gyration of the elastic chain in a dry network and of the corresponding free chain for (a) a series of PDMS networks and the corresponding chains in the mixture PDMS(D)-PDMS(H) (Beltzung et alJ4>; (b) a PS network and the corresponding free chain in bulk (Duplessix14)) Type of network ,~

(a) PDMS

MJM, - 1.5

M,. (kg tool- I>) Rg (-~) mixtures

Rg (A) networks 0~= 1

0~=0,71

O~=0,6 (I)c=0.1

9.9 19 34

31 41 61

32 41 61

31 39 60

-

27

44

-

-

-

39 60

(b) Polystyrene Mw/Mn : 1.2

44

PDMS networks synthetized in the dry state (q)c = 1), no appreciable change of RG between the crosslinked and uncrosslinked state has been observed. This means that crosslinking performed under these conditions has no significant influence on the chains dimensions. Therefore, the memory term as defined in Eq. (B-12) must be equal to 1 since polymer chains in bulk exhibit Gaussian conformation 16). Also, it can be observed in Table 1 that for PDMS networks prepared at relatively high qbc as well as for the PS network synthetized at qbe = 0.1, the polymer molecules are Gaussian and unperturbed after all solvent is removed• This very important result shows that, under deswelling, the chain deformation is not affine in the macroscopic volume and consequently, no supercoiling occurs. Therefore, the memory term must be taken unity• This result contradicts the conclusion h ----0.1, currently inferred from conventional analysis of thermodynamic and stress-strain data 55-57).

(b) Molecular Deformation for Uniaxially Stretched Networks The influence of uniaxial stretching on the dimensions of elastic chains has been investigated for the same series of PDMS. In Fig. 1 the ratio Rgll/Rgi (Rgll is the radius of

~.,//

1.2

/

// "

"Rgi/

J*/

,/

...... ;" °o~'° •

1.1

Fig. 1. Molecular deformation (Rgll/Rgi) for uniaxially stretched PDMS networks, • M~ = 6100, • Mn = 10500, • Mn = 23 000, ~c = 0.7. The full, broken and dotted lines refer to affine, end-to-end pulling, and phantom network models, respectively• From Beltzung et al. ~)

/

i

I

.,f

Az

,,-A

....

..--"

.../u)

1.25 '

,

1~5 0

+

+

1.75 '

40

S. Candau et al.

gyration of the chain parallel to the stretching direction and Rg the radius of gyration of the unstretched chain) is plotted vs. the macroscopic deformation ratio 3~of the sample. In the same figure are plotted the theoretical curves corresponding to the three models described previously. It can be observed that all the experimental points are located far below the theoretical line corresponding to the affine behaviour. For very short chains, the molecular deformation could be roughly described by the end-to-end pulling model. Since the molecular weight increases, the chain deformation at a given ). is reduced so far that values even lower than the theoretical predictions of the phantom network model are reached.

(c) Chain Dimensions of Swollen Networks The SANS technique allows to investigate the deformation process of network chains under swelling of the network. The radius of gyration of the elastic chains of the PS network has been measured as a function of the swelling degree in benzene TM. The following experimental procedure was used: First, the network was swollen at equilibrium in pure benzene. Then, the gel was osmotically deswotlen by the addition of large linear polymer chains (Mw = 760 000) to the surrounding solution. Figure 2 describes the

60

50

40

1

1.5

2,5

Fig, 2. Dependence of Rg on q~for a PS network in an osmotic deswelling experiment. The full and dotted lines refer to the affine behaviour and the phantom network model, respectively. From Bastide et al. 53)


41

dependence of the radius of gyration on the volume fraction of the polymer in the gel. The radius of gyration obtained for the network immersed in pure benzene is equal, within experimental accuracy, to that of the corresponding free chains in benzene at infinite dilution sS) Rg (network)

= 58 + 3 A

Rg linear PS

= 58 + 3 fi,

Mw = 26000 This result is in good agreement with the conclusion of the c* theorem. In Fig. 2 are also reported the theoretical variations Rg = f(~) calculated from the affine deformation and the phantom network models, respectively. For the latter model, the memory term has been taken equal to unity, since the dimensions of the elastic chains are unperturbed in the dry state. It is seen from Fig. 2 that Rg does not show any appreciable dependence on the volume fraction in the range 0.05 < • < 0.2 and its variation remains significantly smaller than any theoretical prediction. It must be emphasized that a slightly better agreement between experimental data and phantom network behaviour may be attained if q~0 is taken as an adjustable parameter. Under this condition, the best fit of the experimental data to Eq. (B-18) would lead to d~0 < 1. This would imply a slight supercoiling of the chains in the dry state, which is of the same order of magnitude as the experimental accuracy. However, this procedure is both arbitrary and unrealistic, since it assumes that the fluctuations of the junction point positions are independent of the swelling of the network. Besides the osmotic deswelling experiments reported above, other SANS studies have demonstrated the capability of a network to undergo large macroscopic deformation without change of chain dimensions. Measurements from Duplessix on a PS network swollen in cyclohexane at room temperature have shown that the radius of gyration of the elastic chains is the same as in bulk even though the network was swollen by a factor of three14, a6). One must also mention the recent surprising results of Richards et al. 59) which evidence a decrease of the radius of gyration of PS network chains in cyclohexane, if the swelling degree is increased by raising the temperature. Such a behaviour has to be connected undoubtedly with topological reorganization of the network chains.

(d) Discussion of the Deformation Mechanism From the preceding results, three main conclusions can be drawn: (a) In dry networks, the elementary chains have the unperturbed dimensions. (b) In the investigated PS network swollen at equilibrium in a good solvent, the radius of gyration of the elastic chain is equal to that of the corresponding free chain in the same solvent6°). (c) The molecular deformation induced by a strain or a swelling is much less pronounced then that calculated from the affine deformation model. The phantom network model, although less unrealistic, also cannot account for the ensemble of the experimental data. In view of these conclusions, it appears that new mechanisms should be forwarded to explain the experimental results. To this end, it has been proposed that large macroscopic deformations could be produced through topological rearrangements at a scale

42

S. Candau et al.

Fig. 3. Schematic representation of a dry network. The sphere has a radius equal to the end-to-end distance of a chain. The stars and the circles represent the topological and the spatial neighbours of the central junction, respectively. The striped line represents the shortest path between two spatial neighbours. From Bastide et al. 60

larger than the mesh size involving only small molecular deformations. The starting point of this line of argument is the remark originally made by Flory that in a dry network the elementary chains are widely interspersed 38). As a consequence, a given crosslink has two kinds of first neighbours (cf. Fig. 3). The topological first neighbours which are directly connected by one elementary chain to the reference crosslink. Each junction is surrounded by f topological first neighbours. The spatial neighbours. Although their mutual distance is shorter or equal to the distance between topological neighbours, they are connected by a longer path through the network. The ratio n/f of the total number of neighbours over the number of topological neighbours of a given crosslink is easily shown to be 61) n _f ~ R 3N-1 f-z ~ _ .1.5 ~ _ N 1/2 QcN1/z

(B-19)

In Table 2 are listed some values of n/f calculated from Eq. (B-19) for tetrafunctional networks and by means of the following Rg(Mw) dependence obtained by SANS experiments on linear PS molecules 15).


43

Table 2. Calculation of the ratio n/f of the total number of first neighbours over the number of topological first neighbours for dry tetrafunctional PS networks with different mesh sizes. The experimental formula of Rg(Mw) is taken from Ref. 15 Mw (kg mo1-1)

10

15

20

25

30

35

n/f

10.5

13

15

16.5

18

19.5

Rg = 0.275M~ 5 Over the range of mesh size considered, n/f considerably exceeds unity. This implies that, on an average, the shortest path connecting two neighbouring junctions is much longer than the elementary chain and is therefore embedded in the network through several crosslinks (cf. Fig. 3). From this observation, it was conjectured that in a network subjected to macroscopic strain, the topological paths constrained by a large number of crosslinks are more deformed than the elementary chains 61). Consequently, under stretching or swelling, the distance between spatial pairs is drastically increased with respect to that of topological pairs, resulting in a rearrangement of the positions of junction pairs. This three-dimensionai accordion-like folding can be illustrated by the following limiting case: let us consider a network formed by rigid rods freely jointed at the crosslinks. Although the end-to-end distance of the elementary unit is fixed, the network can be deformed uniaxially or even tridimensionally through rearrangement of the spatial neighbours 61). In real networks, the deformation mechanism is undoubtedly more complicated. Nevertheless, this kind of crude argument enables a qualitative interpretation of the experimental results. In the tridimensional case, the osmotic deswelling appears as a progressive interspersion of the chains whose dimensions are not significantly modified owing to the topological rearrangement. The rearrangement of the junctions in the deformation process is expected to be strongly dependent on the mesh size of the network. For highly crosslinked bulk networks, n/f is of the order of unity and few rearrangements of the neighbours are allowed. Consequently, the elementary chain deformation is expected to be affine with the strain or at least to follow the prediction of the end-to-end pulling model. This is consistent with the behaviour of the PDMS sample 2 (cf. Fig. 1). Since the molecular weight of the chains and hence the parameter n/f are increased, the spatial rearrangement of the junctions for a given 2 will become more and more favoured. As a consequence, the molecular deformation is progressively reduced. This description is well supported by the experimental behaviour of PDMS sample 4 which exhibits a molecular deformation smaller than that predicted by the phantom network model. These complicated deformation processes make difficult the application of the hitherto proposed statistical mechanics models. Nevertheless, some conclusions about thermodynamics and elastic properties can be reached without reference to a particular chain deformation mechanism.

44

s. Candau et al.

B.H.2. Swelling Equilibrium Degree of Polymeric Networks "c* Theorem" The equilibrium swelling of a network immersed in a diluent results from a balance between the osmotic pressure and a restoring elastic pressure. This balance can be expressed by the equality between the chemical potentials Pl and #10 of the solvent inside and outside the get ~1 - - P l 0 = 0

(B-20)

The chemical potential ~l is obtained from the Gibbs energy per site G"ite according to the relation (B-7). In the current thermodynamic theories of gel swelling, G's'ite is written as the sum of the Gibbs energy density of dilution and an elastic contribution G's'ite= G~il + F'l

(B-21)

The dilution term G~i I is the same as that for semi-dilute solutions (cf. Eq. (B-11) except for the translational term which vanishes when a network is formed G~il = u* 4p9/4 ksT

(B-22)

where u* is an effective interaction parameter which can be determined, for instance, from osmotic pressure studies (cf. Sect. B.t.1.). It has been proposed to take for Fel the chain deformation 48) free energy F~,/kBT = ( ~ / N ) ( ( R 2 ) / ( R 2 ) )

(B-23)

where (R 2) is given by the following packing condition, which expresses the affine character of the deformation

= 6Na3/(R2) 3/2

(B-24)

where 8 is a constant which depends on the functionality of the crosslinks. However, as pointed out in the previous section, the swelling mechanism is much more complicated than that described by the packing condition (B-24). We have discussed (Eq. (B-19)) how the macroscopic swelling of the gel can be interpreted in terms of chain desinterspersion. The upper limit of the swelling involving only a neighbouring rearrangement is reached if n/f is the order of unity. As n/f scales like N re, the corresponding volume fraction is given by: (O(n/f-- 1))o ~ N -in which should represent as a first approximation the volume fraction of a network swollen at equilibrium in a theta solvent. In a good solvent the elementary chains are expanded and ~(n/f --- 1) has to be corrected as follows:

Structural, Elastic, and Dynamic Properties of Swollen Polymer Networks ( ~ ( n / f - 1))gs = ( ~ ( r # f - 1))o" (R~'R~)

cc N -4f5 oc ~ *

45 (B-25)

where the indices gs and 0 refer to the good solvent and the theta conditions, respectively. From a thermodynamical point of view (~(n/f = l ) ) g s - alp* characterizes the reference state of swollen networks with respect to the topological rearrangement mechanism. According to de Gennes, the chain deformation process refers also to ~*. Therefore, qb* can be taken as the volume fraction of the polymer in the reference state whatever the relative contributions of either one mechanism to the overall swelling. These considerations are the basis of the phenomenological expression for the elastic part of the free energy 36), ¢b F~,/kBT = ~ - F (qb/qb*)

(B-26)

where F is a dimensionless function which may depend on the gel functionality. In Eq. (B-26), it is assumed that Fel is proportional to the chain concentration as in the case of the affine deformation model. However, no packing condition is postulated so that the network is allowed to deswell (or swell) through topological rearrangements of the crosslinks as well as through chain deformation, Thus, ~/q~* represents the degree of interpenetration and deformation of chains with respect to the reference state ~*. Taking only the dominant term, F~I may be written as: F~I/kBT = B ~ - ( ~ / ~ , ) - x

0 < x < 1

and then " F~/kBT -- B ~e ~p,9/4-e - -

(B-27)

with ( = 1 - x, where B depends on the functionality of the network. N is related to ~* through Eq. (B-4). The equilibrium volume fraction (I) e in a pure solvent is obtained from Eqs. (B-25), (B-7), (B-11), (B-21) and (B-27) = r (1 - g)B]l*(9/4-e)q~,

(B-28)

This result expresses the c* theorem. It i m l ~ s that in a good solvent and at the swelling equilibrium, the end-to-end distance V(R~) of a network chain of the gel scales with N like the end-to-end distance R F of a free macromolecule of the same molecular weight in the same solvent. ~ Rv ~ N a/5

(B-29)

40

S. Candau et al.

At this stage, two remarks should be made. (a) Some derivations of the free energy of swollen networks take into account an entropy of crosslinking 3'42-46). This introduces an additional logarithmic term into the expression of the Gibbs energy which would affect significantly the overall thermodynamic properties only in the case of a strictly affine deformation of the network chains 45'46). As shown before, such behaviour has not been detected. Therefore, in the derivation leading to Eq. (B-28) and (B-30) the logarithmic term has been neglected. (b) The c* theorem was derived under the assumption that the volume fraction ~0 of the polymer in the reference state was equal to ~*. In previous approaches, q50was assumed to be independent of molecular weight and was generally identified by the volume fraction of the polymer prior to crosslinking. Under this condition, the James and Guth model, based on the Gaussian behaviour assumption and the meanfield approximation, leads to the following form of the equilibrium volume fraction:

(B-30)

B.III. Pair Correlations in Gels The information obtained from SANS experiments reported in the preceding section enables the prediction of the spatial pair correlation function for both dry networks and networks swollen at equilibrium in a good solvent.

B.III.1. Dry Networks We consider here networks containing a small proportion of labelled elastic chains. As the network chains in bulk exhibit unperturbed dimensions, the pair correlation between monomers of an elastic chain is expected to be that of an ideal chain, 22) g(r) ac ~

3

(r ~ Ro)

(B-31)

The scattering function S(q) is given by the Fourier transform of g(r) 12 S(q) oc _-7z7 qa

(qR0 "> 1)

(B-32)

where q is the scattering vector. The dependence of S-l(q) on qZ has been well verified for the PS network previously described (cf. Sect. B.II) as illustrated in Fig. 4 in which qZS(q) is plotted as a function of q214). It can be seen that for high q values (qR0 "> 1) the experimental points tie on an horizontal line as predicted from Eq. (B-44). This result confirms the Gaussian character of the chain conformation in dry networks.


47

f

f Fig. 4. Plot of q2S(q) vs. q2 for a partially labelled PS network in the dry state. From Duplessix 14/

q2

B.III.2. Networks Swollen at Equilibrium in a Good Solvent In a good solvent, the network chains are no longer Gaussian because of the excluded volume interactions, the range of which is equal to RF as shown before. This situation is comparable to that of a semi-dilute solution of high molecular weight linear polymers in which the excluded volume correlation length ~ can be identified with the average distance between two successive entanglements. In the latter case, the following expression of the correlation function for all pairs of monomers has been derived by de Gennes 22)

g(r) = o 7 exp(- r/i) g(r) = ~

1

for r > ~

(B-33)

for r < ~

(B-34)

The Fourier transform is S(q) oc ~ S(q) oc ~

1

(q~ < 1)

(B-35)

(q~ > 1)

(B-36)

These laws were derived under the assumption of very long chains at high dilution (but still in the semi-dilute regime). For finite values of ~, Farnoux et al. 62' 63) have shown that the scattering function S(q) obeys the following law 1

S(q) ~ q_se + x5/3

(B-37)

where x is a correction term taking into account finite dilution and chain length. In the regime q~ < 1, S(q) is still given by Eq. (B-35) 62'63)

48

S. Candau et aL

Recently, the scattering function has been measured for polystyrene networks swollen in CSz64) and polyacrylamide networks swollen in D2O65). The polystyrene networks were prepared by 7-irridiation of a concentrated solution of totally deuterated polystyrene chains in cyclohexane. After crosslinking, cyclohexane was removed and the networks were swollen to equilibrium with CSz. The PSD-CS2 pair provides the best signal over the noise ratio for polystyrene systems, which is necessary to get reliable information in the regime q~ > 1, where the signal is generally weak. The results indicate that both gels at equilibrium and semi-dilute solutions follow the law (B-37) in the q~ > 1 range. This result is illustrated for one sample in Fig. 5 which describes the dependence of S-I(q) on q5/3. The inset of Fig. 5 shows the variations of S-l(q) as a function of q2 in the regime q~ < 1. The semi-dilute solution exhibits the q2 dependence predicted by Eq. (B-35). On the other hand, a large excess of scattering in observed in gels at low angles. This effect has been attributed to the presence of static inhomogeneities in gels, which have also been evidenced by light scattering experiments 66'67) Because of this stray signal, the correlation length ~, which in principle should be obtainable from the q~ < 1 data, cannot be determined. A similar study was performed by Geissler et at. 65) on unlabelled polyacrylamide networks prepared by radical polymerization and then swollen in DzO. In the range q~ > 1, the scattered signal was not strong enough to allow a precise characterization of the scattering function. On the other hand, the situation in the q~ < 1 regime appears more favorable than for polystyrene gels. Indeed, the excess of scattering associated with inhomogeneities does not extend over the whole range corres-

025

s'l(q) (arbitrary units)

,L A • A,LA

0.05

A * A A A ' ~ A '~

AA

°

•

o

"

* °

q2

t

0 I

0

0.002

I

I

0101

0.001 I

I

I

I

t

o%

Fig. 5. Plot of S(q) -1 vs. q5/3for a totally deuterated PS network swollen at equilibrium in CS2 and the corresponding semi-dilute solution at the same concentration. In the inset, plot of S(q) -1 vs. q2 in the q~ < 1 regime. From Bastide and Picot64)


49

ponding to q~ < 1 so that the validity of Eq. (B-35) has been verified in a q2 range extending approximately from 10 -3 to 3 x 10 -3 ~ - 2 The measurements have been performed on a series of gels prepared at various concentrations and various crosslinking densities. The correlation length ~ was found to vary with (I)3/4 for gels, both in the state after preparation and at the swelling equilibrium. This behaviour is similar to that observed in semi-dilute solutions and consistent with the predictions of the scaling approach (cf. Eq. (B-5)).

C. Elastic Properties of Networks Swollen at Equilibrium in a Good Solvent The conventional analysis used for the studies of the elastic properties of swollen networks are based on models developed for dry networks and generalized to include polymer-diluent interactions 0. When the diluent is a very good solvent for the polymer chains, two basic assumptions, namely Gaussian statistics and mean-field approximation, are incorrect because of the excluded volume effect. The same criticism has been raised against the description of the thermodynamic behaviour of semi-dilute solutions 2z' 35). As a matter of fact, the classical theories were unable to interpret some fundamental properties of solutions in the semi-dilute regime like, for instance, the concentration dependence of the osmotic pressure. In this respect, the introduction of the scaling concept into the physics of polymers has led to a substantial progress. With regard to the osmotic pressure, the scaling form of its concentration dependence :~ oc ~p9/4 has been verified with a very good accuracy by both recent data 6s-7°) and reanalyzed old data 71). In gels swollen at equilibrium, the elastic moduli are closely connected with the osmotic pressure and an identical scaling form has been proposed for the concentration dependence of both parameters 22'36,48). In this section, the more recent measurements of elastic moduli in series of equilibrium swollen networks are reviewed and the data are compared with the scaling prediction.

C.I. Scaling Law for Elastic Moduli of Networks Swollen at Equilibrium in Good Solvents The experimentally attainable elastic moduli are the bulk compressional modulus K, the shear modulus ,u, and the longitudinal modulus M given by the combination 4 M = K + ~-/z

(C-1)

K is defined as the inverse of the osmotic compressibility and given by

K=~

--~

T

50

S. Candau et al.

Taking the scaling form of the free energy Eqs. (B,26), (B-27), the following expression for the osmotic pressure of a swollen gel is found 7t'gel= ((5/4)u* (I)9/4 - - (1 - () Bt~ .9/4-~ qbe) kBT

(C-3)

Vl

Under the condition of equilibrium swelling, i.e.

(1

(5/4)u* ~9,4 _

- e) B ~,,9,~-,~ ¢/e = 0

~gel =

0, we obtain: (C-4)

This equation is equivalent to Eq. (B-28). The compressional modulus Kgel of a network swollen at equilibrium is obtained from Eqs. (C-2), (C-3), and (C-4): kBT Kgel = (9/4 - ~) 5/4 u* (I)9/4 ..........

(C-5)

Vt

For a semi-dilute solution, the isothermal bulk compressional modulus K~ol at zero frequency as derived from Eq. (B-tl) (in which the logarithmic term is neglected) is given by 9

5 u* qb9j4 kBT

K~o, = ~- x ~

(C-6)

Vx

The comparison of Eqs. (C-5) and (C-6) shows that for a given polymer volume fraction, the modulus of the crosslinked gel is reduced with respect to the solution

K~ol-

Kgel =

(C-7)

Kel

where Kel characterizes the compressibility associated with the permanent linking of the chains Kel = g x 5/4 u* (I)9/4

kBT

~9> 0

(C-8)

VI

Formally, the shear modulus is given by 72) /2

-

1 OG'(s) s as

(C-9)

where G'(s) is the total free energy density (per unit volume) of the gel under a shear deformation s = (2 - 1/2) (2 denotes the deformation ratio). Since the deformation is performed at constant volume, we may write: aG'(s)

aF;~(s)

~s

~s

(C-10)


51

Assuming that, as in the case of the affine deformation, F~i(s) can be factorized as follows 36) Fel(S) = B(S) q/d~ .9/4-t vi< kBT

(c-11)

the modulus can be expressed as # ( ~ ) _ 1 a B ( s ) B(o)_ 1 5/4 u* kB___T_T09/4 s cqs 1 - ( Vl

(C-12)

The free energy is generally written as a simple linear function of the first invariant of the deformation tension. Under this assumption and taking into account the normation condition

S(o)

B(s) = --5-- (s2 + 3)

(C-13)

we obtain 2 5/4 U* kBT 09/4 P ( ~ ) - 3 1 - t' V 1

(C-14)

In this expression both functionality and crosslinking density have been included into the equilibrium volume fraction ~ . The important feature of the scaling approach is that it predicts the same scaling law • 9/4for shear and compressional moduli of a crosslinked gel and for the osmotic compressional modulus of a semi-dilute solution. The above derivation has been performed assuming that O* corresponds to the reference state. It should be emphasized that the conclusions are more general since the scaling behaviour is independent of the exponent so that any scaled form of Fel would lead to - We ~9t4 scaling for the moduli, whatever the reference state. As an illustration, a model based on the assumption of affine deformation also leads to the same scaling behaviour provided G~jl is taken to be proportional to i;i)9/4 48)

C . I I . E x p e r i m e n t a l M e t h o d s for t h e D e t e r m i n a t i o n o f t h e E l a s t i c M o d u l i

C.II. 1. Shear Modulus The more direct determination of the shear modulus is provided by the measurement of the sample deformation resulting from the application of a simple shear. The corresponding stress-strain relationship is73) as = ~(2s - 1/22)

(C-15)

where as is the shear stress and ~ the deformation ratio in a direction perpendicular to the applied shear.

52

S. Candau et al.

As a matter of fact, stress-strain experiments on swollen networks are more frequently performed by means of the uniaxial compression technique 57). The compressional stress a per unit undeformed area of swollen gel may be generally written as 73) E a = -~- (2 - 1/22)

(C-16)

where 2 is the deformation ratio in the direction of the applied stress and E the Young modulus. Under the current experimental conditions, the size of the investigated samples is rather large (>i 1 cm 3) and consequently the deswelling caused by the compression requires a long time (typically a few hours) 74). Therefore, it can be assumed that the volume of the gel sample remains constant under compression provided the measurements are performed rapidly enough (within about 15 min following the application of the compressional force). However, this time must be sufficient for stress relaxation (on creep) in order to obtain the stress value very close to equilibrium. In this case, E = 3~ TM and Eq. (C-16) becomes: o = it(2 - 1/22)

(C-17)

The validity of the constant volume assumption in rapid compression experiments has been checked experimentally 75). Recently, a new method for the determination of the shear modulus has been proposed 76). It is based on the measurement of the phase velocity of an axially symmetrical dilatational mode in a gel cylinder. This method has been applied to the study of the shear modulus of polyacrylamide gels. The results obtained in the frequency range 200-2000 Hz are in very good agreement with the equilibrium values measured by conventional stress-strain methods.

C.II.2. Compressional Modulus The compressional modulus of networks being in equilibrium with pure diluent can be determined by means of a method based on a decrease in the equilibrium swelling using different swelling agents 77' 78). The compressional modulus is obtained from the variation of the polymer volume fraction with the diluent activity a, through the relation ( ~3:r ) t

kBT/alnal~ ]

(C-18)

In recent experiments 78), solutions of low molecular weight polymers of known activity were used to decrease the activity of the diluent. The swollen gel was not directly contacted with the solutions but only through a semipermeable membrane.

C.H.3. Longitudinal Modulus The longitudinal modulus of a swollen network can be obtained by measurement of the polarized scattered light arising from collective excitations of the network 67' 79). The photocurrent is due to the polarized scattered light is given by

Structural, Elastic, and Dynamic Properties of Swollen Polymer Networks . Iok.T q~2(Se~ 2 is = , ' % ~ \5¢]

53 (C-19)

where Ao is a constant which depends on the wavelength of the incident light, the geometry used, and the quantum efficiency of the photomultiplier; e is the dielectric constant. As a matter of fact, light scattering from gels arises from concentration fluctuations of the network as well as quasistatic macroscopic inhomogeneities. The latter contribution to the total scattering should be eliminated in the determination of M from scattered intensity measurements. Some devices, using frequency filtering techniques have been described 66'80). The determination of the absolute value of the longitudinal modulus from light scattering experiments is not straightforward because it requires a normalization procedure. However, the relative variation of the longitudinal modulus as a function of the concentration can be quite well described. It must also be mentioned that light-scattering probes network fluctuations in a frequency domain ranging typically from 102 to 105 Hz. For permanently crosslinked networks swollen at equilibrium, the corresponding logitudinal modulis should not be different from those measured under equilibrium conditions. This is not the case for semi-dilute solutions of linear chains which do not exhibit low frequency elasticity.

C.III. Experimental Results A check of the scaling predictions of the elastic moduli requires the investigation of a series of networks with different mesh sizes, swollen at equilibrium with the same solvent. However, it must be kept in mind that the scaling laws have been derived under the assumption of calibrated networks of given functionality. The prefactor of the power law concentration dependence of elastic moduli presumably depends on the functionality of the crosslinks. Moreover, network defects at the molecular scale may cause deviations from the scaling behaviour. Several sets of data concerning the three moduli (~, K and M) are available in the literature.

C.III.1. Shear Modulus The first attempt to interpret shear modulus results in terms of scaling behaviour was performed by Munch et al. sl) who reanalyzed earlier data of Belkebir-Mrani et al. on polystyrene networks swollen with benzene 82). The investigated networks were prepared by end-linking polymer precursors of different molecular weights, using as crosslinking agents either a tri-functional reagent or DVB in variable proportions. The shear modulus data could be satisfactorily fitted to a q~9/4power law, independently of the functionality as a first approximation. This result is confirmed by the analysis of the data obtained for several other neutral polymer networks swollen with very good diluents: natural rubber-n-decane 83), polyisoprene-n-decane 84), polydimethyl-siloxane-cyclohexanesS), polystyrene (networks prepared in a radical manner)-benzene s6), poly(vinyl acetate)-toluene 871,poly(vinyl acetate)acetone 88), polydimethylsiloxane-heptane 75).

54

S. Candau et al.

ln~ 12

11

10

9

8 slope 2.38

7

6

5

......

i

1.s

i

~

2.s

,

- In C e

Fig, 6. Log-log representation of the shear modulus as a function of the equilibrium volume fraction of polymer for PVAC gels swoUen in toluene. The dotted line describes the dependence of the osmotic pressure on concentration for semi-dilute solutions of linear poty(vinyl acetate) in toluene, calculated from an experimental fit. From Zrinyi et al. sty. M, n are expressed in N m-2, c in g cm-3. From Vink 9°~ For all these systems, the shear modulus data fit well a power law as a function of the equilibrium volume fraction, with an exponent ranging from 2.1 to 2.689~. An example is given in Fig. 6 which shows a log-log representation of the shear modulus as a function of the equilibrium volume fraction for poly(vinyl acetate) networks swollen with toluene 87~. The straight line with the best fit of the experimental data has a slope of 2.38 rather close to the predicted value 9/4. In the same figure, the osmotic pressure is plotted against the concentration for semi-dilute solutions of linear poly(vinyl acetate) 9°). It can be seen that the concentration dependence of the osmotic pressure of semi-dilute solutions and the equilibrium concentration dependence of the shear modulus of crosslinked networks exhibit a very similar behaviour, as expected from the scaling theories. Generally, the ~9/4 power law fits more accurately the osmotic pressure data than the shear modulus results. This can be related to the difficulty for preparing homologous series of networks containing a negligible amount of structural defects. If crosslinking is performed in bulk or concentrated solutions, the formation of trapped entanglements is favoured in a way which presumably depends on the molecular weight of the polymer precursor. On the other hand, networks prepared at high dilution may contain non-negligible amounts of dangling chains and elastically inactive rings. The role played by the trapped entanglements has been recently investigated on polydimethylsiloxane networks swollen with heptane 75). The samples were prepared by end-linking at volume fractions ~c ranging from 0.5 to 1. The results of shear modulus


55

and equilibrium swelling degree measurements demonstrate the presence of an increasing number of trapped entanglements when ~Pc increases. Furthermore, the modulus exhibits the same polymer volume fraction dependence for both the series prepared at variable q~c and another one prepared at given q~c in which the mesh size was changed, using different molecular weights of the precursor polymers. The results strongly support the model in which the trapped entanglements act as supplementary crosslinks in the swollen network. Consequently, trapped entanglements should not considerably modify the scaling behaviour of elastic moduli in networks swollen with a good diluent. The thermodynamic properties of polystyrene networks containing controlled amounts of pendent chains have also been investigated is' 19). It has been shown experimentally that such gels in which some of the elementary chains are connected only by one end to the network have practically the same swelling equilibrium degree than the gel in which all the elementary chains of the same length are connected by both ends. This behaviour can be understood qualitatively owing to the C* theorem: This situation is equivalent to that of an ideal gel in which some chains are randomly cut near crosslink points. When increasing number of cuts, the elementary mesh of the network is progressively changed to a star or a branched molecule. The equilibrium volume fraction of the gel is then expected to be equal to the cross-over volume fraction ~*esh of the new branched mesh. In the case of a star molecule in a good solvent, it can be easily shown that its cross-over volume fraction is approximatively equal to that of an arm of the star 18). Since this arm is, in the case of gels, the elementary chain of the equivalent ideal network (i.e. without any cut), the swelling equilibrium degree of a gel will approximatively remain constant when some chains are cut. On the other hand, the elastic shear modulus will be decreased proportionally to the number of effectively elastic chains present in the system. In terms of equilibrium between the osmotic pressure and the elasticity of the network, this behaviour may look surprising: the decrease of the elastic shear modulus is intuitively associated with a swelling of the gel. Actually, when chains are cut, the osmotic contribution in the swelling equilibrium equation (B-21) can no more be identified as that of linear chains, but more likely as that of branched systems. It is a well-known experimental fact that the osmotic pressure of star molecules is smaller than that of linear chains, even in the semidilute regime 69'91), and does not follow the des Cloizeaux law/~ oc ff~9/4. This result arises from different screening mechanisms in the two systems. Therefore, in the case of a get with pendant chains both the osmotic and the elastic term are reduced as compared with an ideal gel and this may qualitatively explain the result. However, the presence of pendant chains markedly affects the validity of the scaling laws which have particularly been established for cases in which the ratio of the pendant chains is not constant for all the studied samples. It should be pointed out that for networks formed by end-linking as well as by free-radical polimerization, the fraction of pendant chains may be higher when the mesh size is large. Consequently, the apparent exponent in the /~ ~ q)~ law may be expected to be larger than 9/4 when no particular care is taken in order to avoid pendant chains.

C.III.2. Compressional Modulus Osmotic deswelling experiments, performed on a series of poly(vinyl acetate,) networks swollen either with toluene or acetone have revealed that the compressional modulus

56

S. Candau et al.

/../

10-4K(Nm-2) 5

4

3

2

1

i

i

I

1

2

3

10-4/•.

4

Fig. 7. Dependence of K on/~ for poly(vinyl acetate) networks swollen in acetone. From Zrinyi et al. 921

obeys the same power law dependence on the equilibrium volume fraction of polymer as the shear modulus 92~. This fact is illustrated in Fig. 7, where the variation of K as a function of p is shown for the poly(vinyl acetate) networks swollen with acetone.

C.II1.3. Longitudinal Modulus The longitudinal modulus M = K + 4/3/~ of networks swollen at equilibrium is also expected to scale as ~09/4since both compressional and shear modulus obey this law. As a consequence, the photocurrent associated with the polarized scattered light should vary, for a given network-diluent system, according to (cf. Eq. (C-19)) is oc ~beTM

(C-20)

Figure 8 shows the log-log plot of is versus q~e for a series of polystyrene networks swollen with benzene. Within experimental accuracy, the data can be fitted to a straight line with the negative slope 1/4. Light scattering experiments were also performed in semi-dilute solutions of linear polystyrenes in benzene 93~. The osmotic compressibility Z0 of these systems was determined from the extrapolation of the scattered intensity to q = 0, in Fig. 9, which shows the plot of CZ0 against concentration it can be seen that the data are also in good agreement with the theoretical prediction c -1/4.

D. Dynamic Properties In the preceding chap. B and C, we saw that a similarity exists between static properties of gels and semi-dilute solutions. In this section, we show that the similarity can be extended to dynamical properties. The comparison between gets and polymer solutions is


57

40

20

10

£

5

l

2

I

5

I

i,

10

20

102Ce(g cm"3) Fig. 8. Log-log representation of is as a function of the equilibrium concentration for a series of polystyrene gels swollen in benzene, The straight line has a negative slope 1/4. From Candau et al 66)

cxo 5

0.25 _+0.04

"~"~ -

2 Fig. 9. Log-log dependence of the osmotic compressibility (multiplied by C) on concentration. The data are taken from Ref, 93. The representation is taken from Farnoux62)

I

10"3

I

2

i

8(gcm_3 )

of long date. The pioneering quasi-elastic light scattering experiment was carried out by Mc A d a m et at. 94), but unfortunately no clear conclusion could be drawn due to experimental difficulties. A set of experiments by Tanaka et al. 79), Munch et al. 81' 95) and A d a m et al. 96'97) provided a series of confirmations on the analogy between the dynamical behaviours of gels and semi-dilute solutions. Our aim is not to review all experimental work but to focus our attention only on experiments which allow to compare the behavior of gels and semi-dilute solutions in good solvents.

58

S. Candau et al.

D.I. Frictional Properties To determine the friction coefficient between the polymer network (or the linear polymer) and the solvent, let us suppose that each monomer is subjected to an external force _fro. Each monomer acquires a drift velocity, "_urn,which is proportional to -fro l~lm = ~ra/f

(D-l)

is the effective frictional coefficient per monomer. Assuming that hydrodynamic interactions between monomers due to backflow dominate, de Gennes 98~has found the following relation between the dynamic quantity f and the static correlation function g(r) ~-1 =

f d3r ~

1

g(r),

(D-2)

where ~h is the viscosity of the solvent. This relation is valid if the monomer-monomer friction can be neglected, i.e. if the volume fraction of the solvent is large (~> 90%). In chapter B it was experimentally shown through analysis of the structure factor S(q) that for semi-dilute solutions and chemically crosslinked gels a correlation length exists such that g(r) = Qh ( ~ ) ,

(D-3)

where h is a dimensionless function; ~ is the screening length for excluded volume effects equal to the effective mesh size in a chemically crosslinked gel. Inserting Eq. (D-3) into Eq. (D-2), we obtain oc ~]s ~/~) ~3 .

(D-4)

is the screening length for excluded volume effects and hydrodynamic interactions. Experimentally, sedimentation or permeation allow the effective frictional coefficient per monomer to be measured. In a sedimentation experiment 99) measuring the drift velocity of monomers subjected to a centrifugal acceleration 7, the sedimentation coefficient, defined as So = (JmlT, is determined. Each monomer experiences a force f m = m*Z, where m* is the apparent mass of the monomer immersed in the solution; using Eq. (D-l), we have m ~

sD = -T-"

(D4)

The effective frictional coefficient per monomer may be obtained by sedimentation experiments. In a permeation experiment 1°°' t01), a constant pressure gradient VP is maintained in a tube containing the polymer. A measurement of the velocity of the solvent fi~ flowing through the polymer permits the determination of the permeability coefficient. This coefficient is defined by


59

Kp -- t~s~]s VP

(D-6)

The gradient pressure applied counterbalances the frictional forces per unit volume exerted on the solvent. The frictional coefficient per unit volume is 0f and we have VP oc 0f t~.

(D-7)

Following expression (D-7), (D-6), and (D-4) we get j°2)

I% o~ Q~/,~ ~ ~ .

(D-S)

Permeation experiments measure the frictional coefficient per unit volume 0f. For semidilute solutions and gels in an athermal solvent (X = 0), we have ~ ~: 0v/O-3v); with v = 0.6, this leads to the following scaling laws sv ~ 0 -xs,

xs = 0.5

and KpOCO-xk,

(D-9) x k = 1.5 .

Sedimentation experiments on semi-dilute solutions are appropriate and many experiments have been performed on neutral polymers like polystyrene and poly(a-methylstyrene) in good solvents 1°1'103-a06) It has been found that the effective exponent xs increases from 0.59 up to 0.8 as the concentration rises from 0.1 to 10%. Good solvents used in these experiments (benzene, bromobenzene and toluene) are far from athermal conditions (Z = 0.45). Two monomers, belonging to a subchain of size ~ and separated by n monomers, experience excluded volume effects when n > no, where nc o~ (1 2Z) -2. As the concentration decreases, the number of monomer per subchain ~, g, increases and excluded volume effects become more and more important. The effective exponent xs, which is a combination of effective dynamic and static exponents l°s' 109), tends monotically to the asymptotic value 0.5 (g -> no). Inversely, if the concentration increases, g decreases; when g < no, the subchain exhibits purely Gaussian behaviour, and v = 0.5 which leads to ~ cc 0 -1 and SD oc 0 -1. This cross-over between excluded volume and Gaussian behaviour qualitatively explains the increase of x~, if 0 increases. Details on the dependence of x~ on the concentration can be found in Ref. 110. Whatever the exact value of the exponent, these experiments show that the frictional properties of semidilute solutions depend only on the concentration; they are independent of the molecular weight of the polymer used a°l' 103-106, 110) The frictional properties of the gel are appropriately determined by permeation experiments. However, only few studies on the dependence of the permeation on concentration have been made. Moreover, it appears that the permeation is very sensitive to the preparation procedure and thus to the structural inhomogeneities of the gel m). One experiment 112) was performed on polyacrylamide-water gel at concentrations between 5% and 35% (g/g). This study shows that the permeability factor is only controlled by the concentration, it is independent of the degree of crosslinking, and Kp decreases as 0 -

60

S. Candau et al.

increases. However, on the basis of these data it is not possible to extract the exponent Xk. To our knowledge there is no direct experimental evidence of the analogy between frictional properties of gels and semi-dilute solutions; they only depend on the concentration in both systems. In order to verify directly the analogy concerning frictional properties, it would be interesting to perform systematic sedimentation experiments on gels and on the same polymer-solvent system in the same range of concentration. In the section, D.II. we will see that quasi-elastic light scattering experiments show indirectly the validity of this analogy.

D.II. Cooperative Diffusion Coefficient In the preceding section, we have examined the frictional properties of semi-dilute solutions and gels and the analogy between the two systems. In this section, our interest is focused on the dynamics of concentration fluctuations of the polymer. Let us consider first the case of a gel. The polymer network fluctuating around its equilibrium position is subjected to two driving forces: The osmotic force tends to equalize the concentration and the elastic force tends to keep the network in its position. The fluctuations are damped by the frictional force between the polymer network and the solvent. At a macroscopic distance larger than the distance ~ between two crosslinks, the gel can be considered as a continuous medium. Under these conditions, the equation of motion of the polymer network can be derived using the viscoelasticity theory of an isotropic medium 72). Let u(r, t) be the displacement fluctuations of the polymer network from its equilibrium position at point r and time t. The driving force per unit volume is related to u as follows f = (M - #) gra d d i v u + p V2u

(D-10)

where M and p are the longitudinal modulus and shear modulus of the gel, respectively. The dissipation force per unit volume is proportional to the displacement velocity f- = ~ f f l .

(D-11)

At low frequency, which is of interest here, the inertial force is negligible and displacement fluctuations obey the following equation offl_ = (M - p ) grad d i v u +/~ V 2 u .

(D-12)

Solving this equation for longitudinal modes of wave vector q we get M

Uq(t) = e --~- qzt

(D-13)

The expression (D-13) shows that displacement fluctuations diffuse with a cooperative diffusion coefficient M D = '-v. Qr

(D-14)


61

This equation of motion was derived and tested experimentally for the first time by Tanaka et al. 79) on polyacrylamide-water gels. From quasi-elastic light scattering and permeation and uniaxial compression experiments, they determined the order of magnitude of the three parameters D, Qf, and M. They found that the ratio M/Qf is close to the measeured D values. Now, let us discuss the case of semi-dilute solutions and point out the properties which are comparable for gels and semi-dilute solutions. Following the de Gennes model,22,113) polymer solutions in the semi-dilute region have physical entanglements with a finite life time Tr. The mean distance between two successive entanglements is ~. At frequencies w ~> T;-1, the entanglements do not relax and the polymer solution behaves like a chemical gel with permanent crosslinks (this regime is sometimes called pseudogel). In this regime, displacement fluctuations of the polymer diffuse with a diffusion coefficient similar to that given in expression (D-14). In Sect. D.I., we have seen that at the same concentration the effective frictional coefficient per monomer f can be considered as identical in gels and semi-dilute solutions. As we will see below, the elastic modulus depends on concentration in the same way in both systems, but it can differ by a numerical factor. Following the classical theory of elasticity n) we have 4 M = K +~-/~.

(D-15)

where K and ,u are the bulk and shear modulus, respectively. Using the expressions (C-5), (C-6), (C-7), and (C-14), we can write for a gel

M=a--U-

(D-16)

where k0, ke and ~e are numerical factors. Pseudogels and gels have the same form for the energy (see Sect. B) and qbe must be replaced by the monomer volume fraction q~ of the semi-dilute solution. The distance between chemical crosslinks R is replaced by ~ ~ •-°'75 which is supposed to be the distance between two physical entanglements. We derive for the elastic modulus M an expression similar to relation (D-16), obtained for a chemical gel. The first term in the modulus expression comes from the energy mixing, it is related to the osmotic force which tends to equalize the concentration. This term is identical whatever the situation considered, a semi-dilute solution or a gel. The second term is due to the elastic free energy. It is connected with the elastic force which tends to keep the network in its position. It is not evident that the numerical factors k e and kte are the same for gels and pseudogels. Thus, it appears that the diffusion coefficient, which controls the decay time of the polymer fluctuations, are similar in gels or in semi-dilute solutions (pseudogel regime). The diffusion coefficients measured in both systems, at the same monomer concentration 6), must be identical or at least their concentration dependence must be the same. Under athermal conditions, combining Eq. (D-4), (D-14), and (D-16) we obtain the following power law

62

S. Candau et al.

D oc

kBT

oc ~)xD

XD = 0.75 .

(D-17)

Experimentally, the diffusion coefficient is measured by quasi-eleastic light scattering experiments (QELS). We recall the principle of the experiment. In semi-dilute solutions or gels, the polarized quasi-elastic light scattering is due to the local concentration fluctuations of the polymer. At a scattering angle 0, the autocorrelation function of the scattered electric field is 79)

(Eo(t) E~-(0)) = !Uq(t) u~-(0)) = e_Dq~t

where

(D-18)

4~

q = ~-- sin 0/2

~.i is the wavelength of incident light in the polymer solution. The expression (D-t8) is only valid if q-1 > ~, and for this condition we probe the macroscopic properties of the polymer. The autocorrelation function of the scattered electric field is experimentally observed by light beating spectroscopy 114). Gels and often also semi-dilute solutions contain some inhomogeneities, which scatter the light and the evolution of their local concentration gives rise to a spectrum with a characteristic time much longer than the characteristic time (DqZ) -1. Experimentally, it was shown that this stray light, which can be considered as a local oscillator, heterodynes with the light scattered by the concentration fluctuations of the polymer 96' 115). Then, the experimental autocorrelation is Ce~p(t) ~ (e -Dq:t + A)

(D-19)

From an experimental point of view, in the time range where Cexp(t ) is analyzed, A is a constant coincidental background. In the fitting procedure, it must be considered as an ajustable parameter in order to avoid errors in the determination of the decay time (Dq2) -1. To check the predicted power law of the diffusion coefficient (Eq, (D-17)), experiments on polyacrylamide-water gets were carried out. It has been found that the exponent XD lies between 0.65 and 0.75, and that these values are not too far from the predicted value 0.75116' 117). However, the system studied is polar and a polyetectrolyte behaviour due to partial hydrolysis of the amide group cannot be excluded 1t7). The most suitable way to test the analogy between pseudogels and gels is to perform systematically QELS. experiments on both systems with the same polymer-solvent pair. Munch et al. sl' 95,115,118,119) carried out this study on different non-polar systems: PSbenzene, PS-ethyl acetate, and PDMS-toluene. From this set of experiments, two remarkable properties can be inferred (see Fig. 10):


63

!

"l

I

t

!

I

lo7D (cm2s-1) 20

1(]

•

C(g era-3)

•

lo-3

lo -a

...........

1o-+

Fig. 10. Diffusion coefficient D as a function of concentration C(g/cm~) for the PS-benzene system in dilute and semi-dilute solutions and swollen networks. • Solutions, Mw = 7 x 105. I swollen networks of different functionnalities. From Munch m)

First, in each system it is observed that the diffusion coefficient of the gel is larger than that of the pseudogel. This difference may be due to the fact that the numerical factor 4/3 Pe - ke in expression (D-16), is larger for a gel than for a pseudogel. Secondly, the dependences of the diffusion coefficient on the equilibrium concentration, lie on the same master curve for a given polymer-solvent pair regardless of the way of gel preparation and the functionnality of the crosslinks. Thirdly, the values of the effective exponent xD found for semi-dilute solutions and gels are identical for a given polymer-solvent pair. They are as follows: xo = 0.68 + 0.01

for PS-benzene, m)

xD = 0.66

for PS-ethyl acetate, s])

xD = 0.77 _+0.03

for PDMS-toluene. 1~5)

The difference in xo for various solvent-polymer pairs and the slight differences between the predicted and found values may have a different origin. The fact that the velocity of the solvent is neglected in the theory given above may be one of the reasons. Taking into account the solvent displacement, we can show that lt7~ D ~c (1 - ago) ,~o.75, where a is an unknown positive parameter. This neglect can explain the low value of the effective exponent XD. Another reason for the difference may be due to the fact that the athermal condition is not fulfilled and the experiments are performed in the cross-over region between excluded volume and Gaussian behaviour (see Sect. D.I.). The effective exponent xo varies non-monotonically vg' ]20)between the asymptotic values 0.75 and 1, if

S. Candau et al.

64

2: (or concentration) increases. This could explain the variation of the observed xD values for the three systems studied which have different Z values.

D.III. Kinetics of Swelling and Deswelling of Networks In Chap, B., the static properties and phase equilibria of gels in an excess of solvent were described. In this section we describe the parameters which control the kinetics of swelling of gets and the corresponding counterparts in semi-dilute solutions. Let us consider a gel which is almost at swelling equilibrium in a good solvent and place it at time t = 0 in the same good solvent bath. At t < 0, the chemical potential of the solvent inside the polymer network is higher than that of the pure solvent. At t I> 0, the gel absorbs solvent molecules to equalize the chemical potential of the solvent inside and outside the gel phase. The macroscopic linear dimensions and the mesh size of the polymer network increase. Following the fluctuation dissipation theorem, m) it is expected that macroscopic displacement (under swelling) or microscopic displacement fluctuations of the polymer network are controlled by the same parameters; they obey the equation (D-12). This fact was realized for the first time by Tanaka et al. 74), who found that under swelling, the radius b(t) of a small spherical gel sample increases with time according to b(t) = b ( 0 ) + [ b - b(0)] [ 1 -

~ e-t£/r],

ift > r/4

(D-20)

where b is the radius of the gel at swelling equilibrium, and the characteristic swelling time, r, is: r = b2/D.

(D-21)

To obtain the expression (D-20) 74), they assumed that (1) before the transfer into the solvent bath, the network is under uniform stress; (2) after the transfer, the normal stress on the gel surface is zero; (3) the shear modulus is small compared to the bulk modulus. Tanaka et al. 74) examined a set of spherical polyacrylamide-water gets with different radii (between 1 and 3 mm), having initial degrees of swelling smaller than the equilibrium value by a factor of 2/3. Measuring the time dependence of the diameter of the swelling spheres, they determined the characteristic swelling time, r. A plot of r vs. b z (see Fig. 11) yields a straight line in agreement with the expression (D-21). On the same samples at swelling equilibrium, the authors measured the diffusion coefficient using the QELS experiments. The diffusion coefficients measured by the two methods agree within 7%. These experiments show that the swelling kinetics is controlled by the diffusion coefficient of the polymer network. The same conclusions were also inferred by Geissler et al. 123) from deswelling experiments on polyacrylamide water gels. A comparable situation exists in semi-dilute solutions. Let us consider two semi-dilute solutions with the monomer volume fractions cp+ and q~_. At time t = 0, the two solutions are brought into contact. At t < 0, chemical potential of the solvent in the solution "~p+" is higher than that in the solution "q~_". At t 1> 0, solvent molecules move toward

Structural, Elastic, and Dynamic Properties of Swollen Polymer Networks I

I

I

65 I

I

I

I

I

3 -

.I"

I-'

.i"

1

pO

Fig. l 1, Characteristic time r of swelling of

spherical polyacrylamide-water gets as a function of the square of the final radius b. From Tanaka et al. TM

~

0

I

0

0 1

0.02

!

I

0.04

,

I

0.06

I

I

I

0.08

0 J0

b2(¢m 2)

the region rich in polymer to equalize the concentration. The mean distance between two successive physical entanglements increases in the region rich in polymer (swelling) and • decreases in the region poor in polymer (deswelling). Practically, a macroscopic step-like concentration gradient can be established in a double-sector capillary-type cell using an ultracentrifuge and the evolution of the concentration gradient is followed by means of the Schlieren optical system. Usually, this kind of experiment is called a classical gradient diffusion (CGD) experiment. The diffusion coefficient is obtained from photographs of the Schlieren pattern at different times 124). The time span, in which the diffusion is measured, is typically 5 x 103 s for a semi-dilute solution of PS in benzene (1%). The situation considered here for semi-dilute solutions is comparable with the case of a chemical gel under swelling (or deswelling), but it is not equivalent. Indeed, whatever the time scale of the observation for a gel the situation remains identical, because crosslinks are permanent. For semi-dilute solutions, this is not the case; they are pure viscous liquids at the observation time scale (e.g. 5 x 103 s) of CGD experiment. The elastic part of the longitudinal modulus (Eq. (D-16)) does not play any role because M(a~Tr ~ 1) = a-TkBTk0 ~)2,25 .

(D-22)

In contrast, in the QELS experiment at o)Tr ~ 1, the elastic part must play a role and M must have the form proposed in Eq. (D-16). These two types of moduli are sometimes called isothermal and adiabaticl°2L Adam et al. 97' 125)carried out systematic measurements of the diffusion coefficient of semi-dilute solutions of polystyrene in benzene using QELS and CGD methods. They found (cf. Fig. 12) that the diffusion coefficients obtained by both methods are equal within experimental error. The diffusion coefficient scales with the concentration as C °67-+°ce and depends only on concentration. These results are identical with those obtained by Munch et al. 81) from QELS experiments performed on the same system PSbenzene (cf. Section D-II) and the results of J. Roots et alJ z6), who found for PS-toluene by means of CGD experiments that D ~ c°7°-*°°1. These studies show that for PS systems in good solvents the isothermal and adiabatic longitudinal moduli are identical in the first approximation and scale with the monomer

66

S. Candau et al. O ~ 10 s (cm2s j )

0.5

C x 102(gig) I

I

l

1

L

~

I

t

5

t

I

I

I

J

.

10

Fig. 12. Diffusion coefficient D of semi-dilute PS-benzene solutions as a function of concentration C(g/g). Results obtained from QELS experiments: ~ Mw = 1.27 x 106, ~ Mw = 3.8 x 106, ~, Mw = 8.4 × 106; results obtained from G C D experiments: + Mw = 3.2 × 105,41~Mw= 5.8 × 106,-t-Mw = 3.2 x 105; + best fit obtained from QELS results, From Adam et alJ TM

fraction in the same way. However, this experimental fact does not lead to the conclusion that the elasticity of a pseudogel is negligible. Indeed, one possibility is that, like in PSbenzene gels, the elastic part of the bulk modulus approximately counterbalances the shear modulus. From an experimental point of view, it appears that the kinetics of swelling of gels and that of semi-dilute solutions by a solvent are similar.

Conclusions The aim of this paper was to review the information recently gained on both microscopic and macroscopic aspects of the properties of polymeric networks. The microscopic aspect mainly concerned the local conformation of the network chain. In this respect, the analysis of the recent results of small-angle neutron scattering experiments leads to some important conclusions. More specifically, the conformation of network chains in bulk is found to be Gaussian whatever the concentration at which the crosslinking took place. No supercoiling of the elastic chain occurs if a network is prepared in solution and subsequently deswollen. If swollen at equilibrium in a good diluent, the network chains exhibit a characteristic excluded volume conformation. However, it was shown that the large macroscopic deformation resulting from swelling as well as from


67

an external strain only produces a small deformation of network chains. This behaviour cannot be interpreted within the framework of the classical phantom network or affine deformation theories. Although the main features of the deformation process can be qualitatively explained on the basis of topological considerations, a profound understanding of the microscopic properties of networks requires the development of a new statistical theory and offers a challenge to the theoreticians. Also, additional experiments are necessary to give a more precise description of the topological rearrangements in a network subjected to a strain. Up to now, only the size of the strand in the network was considered. Experiments on samples containing labelled paths of variable length are currently being performed in Strasbourg 127) and Glasgow12s). They should enable a characterization of the deformation at different length scales and thus provide an experimental basis for a theoretical model. However, even without a detailed microscopic model of swelling, the understanding of many properties of gels has been improved due to the analogy with semi-dilute solutions. The advance in the solution of the excluded volume problem has been transposed to gels, in correlation with topological considerations. This has led to a formal separation of the contributions of the macroscopic strain and of the local polymer-solvent interactions to the dimensions and statistics of the elementary chains. In other words, in the classical theories the nature of the polymer-solvent interactions was believed to determine macroscopic swelling and the microscopic configuration through the deformation model. In the new approach, the local configuration is mainly determined by the polymer-solvent interaction, and the macroscopic swelling may differ significantly, depending on the boundary conditions. This can be illustrated as follows: Let us imagine two identical parts (a) and (b) of the same dry network. They can be swollen at the same concentration in two different solvents. For example, sample (a) is placed in a moderately good solvent and (b) in a mixture of a very good solvent and very large chains of the same chemical nature as the network (osmotic deswelling). Although the swelling degrees of (a) and (b) are the same, the dimensions and the statistics of the elementary chains of (a) and (b) will be different. On the other hand, the swelling degree of (b) can be modified very strongly by changing the concentration of the linear chains in the mixture, but with an almost negligible change in the configuration of the elementary chains. An important contribution to the swelling is therefore due to the tridimensional unfolding related to the complicated topology of polymer networks. The influence of the conditions of preparation of the network has also been reconsidered. Their effect on the amount of trapped entanglements and therefore on the length of the effective elastic chain has been recognized. On the other hand, within these restrictions, the reference state is no longer believed to depend on the concentration at which crosslinking was performed. Because of this fact, and assuming only a very general form for the free energy of deformation, thermodynamic properties such as elastic moduti, swelling equilibrium concentration, sedimentation coefficient, and diffusion constant can be described by simple scaling laws. More specifically, the experimental results concern the elastic moduli, and the cooperative diffusion constants are well compatible with the predictions. The fit of the experimental data to the scaling laws is considerably better for semi-dilute solutions than for gets. This finding must be associated with the presence of pendant chains in the networks the content of which varies. It must also be pointed out that scaling predictions have been verified experimentally only in cases where the diluent was a very good solvent for the polymer; this holds for semi-dilute solutions as well as for

68

S. Candau et al.

crosslinked gels. In this respect, the verification of the variation of the osmotic pressure in semi-dilute solutions is suitable test for the applicability of the scaling approach. To conclude, it must be stressed that many other important studies on the physical properties of gels have not been discussed in this review. However, the problem of the critical fluctuations occurring close to microphase separation t29-t39) and the very attractive field of ionic gels, in which spectacular experiments have been recently performed, should be mentioned134k Also, the present knowledge about the structure and the thermodynamic properties of physical gels is rather limited and important developments on this topic can be expected in the near future.

Acknowledgements. The authors are grateful to Dr. L. Leibler for his comments and critical survey of the manuscript. We would also like to thank Dr. M. Adam for fruitful discussions and Dr. M. Beltzung, J. Herz and C. Picot for making their unpublished experimental data available. We are also indebted to Prof. Du~ek, who carefully reviewed this article, for his helpful suggestions.

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Received October 26, 1981 K. Dugek (editor)