A simple model for the swelling of polymer networksPaul C. Painter and Suresh L. Shenoy Citation: The Journal of Chemical Physics 99, 1409 (1993); doi: 10.1063/1.465385 View online: http://dx.doi.org/10.1063/1.465385 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/99/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Rubber elasticity: A contactprobability model with harmonic entanglement constraints J. Chem. Phys. 105, 8352 (1996); 10.1063/1.472690 Optical switching in polymer gels J. Appl. Phys. 80, 131 (1996); 10.1063/1.362768 Efficient, fast response lightemitting electrochemical cells: Electroluminescent and solid electrolytepolymers with interpenetrating network morphology Appl. Phys. Lett. 68, 3218 (1996); 10.1063/1.116442 Equilibrium swelling properties of polyampholytic hydrogels J. Chem. Phys. 104, 8713 (1996); 10.1063/1.471560 Nanotribology on a polymer network film J. Vac. Sci. Technol. A 14, 1864 (1996); 10.1116/1.580351

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A simple model for the swelling of polymer networks Paul C. Painter and Suresh l. Shenoy Department 0/ Materials Science and Engineering, Penn State University, University Park, Pennsylvania 16802

(Received 23 November 1992; accepted 9 April 1993)

The Flory-Rehner approach to the solvent swelling of polymer networks is modified so as to abandon the affine deformation assumption. Instead, it is assumed that swelling occurs by a process of disinterspersion of cross-link junctions, and relationships between chain expansion and segment concentration are established using the ideas of de Gennes. Using this approach, we also examine osmotic deswelling, aspects of phase equilibria, and the maximum observed in differential swelling measurements.

INTRODUCTION

The initial aim of the work we will report here was to be an exploration of the role of hydrogen bonding in the swelling and collapse of gels. Recent work in this labora­tory has focused on the effect of such strong, specific in­teractions on the miscibility of polymer mixtures and we have developed a lattice model (1) that apparently pro­vides a good description of the phase behavior of these materials (much of this work is collected in Ref. 2). An extension of this work to gels should therefore follow in a straightforward manner. We can make the usual Flory­Rehner assumption of the separability of the elastic and mixing components of the free energy (3) and modify the mixing term to account for hydrogen bonding interactions. This is the type of approach that has generally been taken in studies of phase transitions in polymer gels. The elastic free energy has been described using classical rubber elas­ticity theory, while some form of the Flory-Huggins equa­tion, modified to account for interactions between ionic species or free volume, e.g., has usually been used to de­scribe the free energy of mixing (see Refs. 4-10 and cita­tions therein).

However, as a result of various studies reported over the last ten years or so (e.g., Refs. 11-15), it is now clear that there are major problems with this type of model. Although some data have been interpreted so as to bring into question the fundamental assumption of the separabil­ity of the elastic and mixing free energies,16,17 the most significant deficiency of the Flory-Rehner theory appears to involve the assumption that the deformation of the ele­mentary chains of the network is in some fashion affine with the macroscopic deformation (swelling) of the sam­ple. The neutron scattering work of Bastide et af. 13 has demonstrated that the network elementary chains have ap­proximately the same dimensions as equivalent non-cross­linked chains in solutions of the same concentration, which is much less than would be expected on the basis of an affine deformation model. These results are in good agree­ment with the c* theorem of de Gennes,18 who proposed that in a good solvent, the swollen coils of the network largely exclude one another from a volume that is (more or less) defined by their radius of gyration, but because the chains are forced into contact at their cross-link points, the gel is analogous to the situation at the overlap threshold in

a semidilute solution. Accordingly, Bastide et af.12 pro­posed that the swelling of a gel proceeds by a process of topological rearrangement or disinterspersion of the cross­link points or nodules and demonstrated that an analysis based on a scaling approach is in good agreement with experimental observations.

In this communication, we will argue that the classical Flory-Rehner approach can be modified in a simple man­ner by abandoning the affine assumption. We will use the packing conditions that are a consequence of the c* theo­rem and a scaling law to define a relationship between the degree of swelling and elementary chain extension. The model appears to provide a good description of various swelling and -deswelfing experiments and provides a simple foundation on which we can (in future work) explore the role of hydrogen bonding in swelling and gel collapse.

EQUILIBRIUM SWELLING IN A GOOD SOLVENT

The approach we will use is similar to that described by Brochard19 more than ten years ago, but we will con­sider "packing conditions" in somewhat more detail. The expression we will obtain for equilibrium or maximum swelling is similar in form to the affine model described by Flory (3), differing only in the exponent of a single term. Because the alternative classical treatment, involving the assumption of so-called phantom chains, provides a better description of the neutron scattering data,13 it might at first seem that we are getting off on the wrong foot. The prin­ciple problem with the Flory treatment is the assumption that the chains deform affinely with the degree of swelling, however, and once disinterspersion of the cross-link points is introduced into the Flory model, far more reasonable results are obtained.

The treatment for eqUilibrium swelling in a good sol­vent, where the swollen chains are at the threshold between the dilute and semidilute regimes, is straightforward and based on three basic and familiar assumptions

( 1) The free energy of the gel can be written as the sum of two separate components, describing the elastic free energy and mixing free energy, respectively.

(2) These components of the free energy can be ex­pressed in terms of the classic elastic free energy and the Elory-Huggins theory (modified later to account for hy­drogen bonding).

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1410 P. C. Painter and S. L. Shenoy: Swelling of polymer networks

"UNOCCUPIED" VOLUME

DANGLING END

FIG. 1. A pictorial representation of the c* model of a gel.

(3) Following de Gennes, we assume that at equilib­rium in a good solvent, the chains expand to the extent that they would in a dilute solution of the same solvent. The cross-link points disintersperse to the extent that the gel is a collection of spheres of individual network chains that as far as possible exclude segments from other chains from their volume, but are forced into contact at their cross:..link points, as illustrated in Fig. 1.

Using the first two assumptions, the equations for the free energy can be written down immediately, while the final assumption provides the essential connection between the volume fraction of polymer segments (Le., the degree of swelling) and the chain expansion factor. We are going to take a somewhat longer path in the derivation, however, in order to address a particular problem. This problem is an old one and involves a logarithmic term that occurs in the affine model, but is absent in the phantom network model. There are other problems, of course, notably the validity of the assumption of the separability of the elastic and mixing free energy terms and various other points that have arisen as a result of the development of the scaling law approach. We will address these later and in this sec­tion confine our concerns to the logarithmic term and the derivation of an expression for the free energy.

The necessity for including the logarithmic term would seem to follow intuitively from the c* theorem. If the net­work swells largely by a process of "unfolding" or topo­logical rearrangement of the cross-link points, then this is surely accompanied by some sort of entropy change in the system. The inclusion of such a term in the treatment of swelling follows formally from the use of a hypothetical scheme introduced by Flory2o in order to rederive the re­sults of the James and Guth phantom network treatment of rubber elasticity. This was applied by Flory to the mechan­ical deformation of a network and it is useful to reproduce the arguments here, as they illustrate (within the confines of this model) that the logarithmic term vanishes for such mechanical deformations, but must be included in order to describe swelling.

The scheme utilized by Flory is shown in Fig. 2 and describes alternative routes to the formation of a deformed network from an undeformed acyclic tree. The acyclic tree

UNDEFORMED ACYCLIC TREE

affine deformation

~

3 I join S ,. Junctions

UNDEFORMED NETWORK, CYCLE

R~

deformation of phantom network

affine deformation

• 4

DEFORMED TREE

21 join S .. junctions

DEFORMED NETWORK,AFFINE CYCLE RANKS

~ 51 relax labeled

.. points

STRAINED PHANTOM NETWORK

FIG. 2. Flory scheme illustrating the formation of a deformed networJc.

has junction points illustrated as dots, but these are not linked sufficiently to form any of the closed paths charac­teristic of a network. We will start with the free energy change associated with step (3), which described the join­ing of S junctions to form the closed paths characteristic of the underformed network (of cycle rank s). Following Flory,20 this is simply given by

1lF3 kT = -sln(<5V/VO), (1)

where <5 V is the volume in which two junction points must occur to become joined and va is the volume of the unde­formed network. If there are v chains (hence v+ l:::::v ini­tial junction points), the number of labeled points is re­duced to v-S+l:::::v-s by this process. Paths (4) and (6) then lead to the deformed affine network and deformed phantom network, respectively. The free energy changes involved in these steps are obtained by difference, however, so we must next consider the free energy change associated with deformation of the acyclic tree, identified as step 1 in Fig. 2. Flory points out that it is essential to first deform the acyclic tree in an affine manner, so that the network formed by then joining S junctions in the deformed net­work, step (2) is equivalent in structure to that formed by joining S junctions in the undeformed network, described above. Again following Flory, the free energy changes for steps (1) and (2) can be written

(2)

and

(3)

where V is the volume of the deformed network. The elas­tic free energy change in step (1) is for a deformation of ail

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P. C. Painter and S. L. Shenoy: Swelling of polymer networks 1411

chains of the network by the same factor a (because our interest is in swelling) and includes a In a3 term as a result of the combinatorial method Flory used to obtain the free energy change for the deformation of a network.

The free energy associated with an affine deformation of the network !:Ji' 4 [step (4)] is given by

(4)

Hence

!:Ji'4 [3 2 3] kT=V "2 (a -I)-Ina +sln(VIVo)· (5)

Flory then obtains the phantom network results by allowing the remaining v-s junction points to relax to their equilibrium position [step (5) in Fig. 2]

!:Ji's [3 2 3] kT=-(v-s) "2 (a -I)-Ina. (6)

The free energy change associated with the deforma­tion of a phantom network [step (6) in Fig. 2] is equal to !:Ji' 4 + !:Ji' 5' hence

Il.F6 3 2 kT=s"2(a -1). (7)

The logarithmic terms are eliminated because in the affine assumption

V 3 1 -=a =-. Vo CPp

(8)

This is a simple treatment of rubber elasticity, and in subsequent work, Flory introduced the constrained junc­tion model, where the free energy is described as the sum of this phantom network behavior and the effect of con­straints on junction fluctuations.21 Erman and Flory22 then determined that the phantom network limit provides a bet­ter description of swelling. Partly on this basis, and partly because we wish to see how far the simplest possible treat­ment can take us, we will neglect the effect of constraints and proceed by using the simple Flory scheme described above.

We can account for two situations, one where the net­work has been formed by cross linking in the dry (unswol­len) state and the other where the network has been formed by cross linking in solution. We will only discuss the former in detail, but the treatment is easily extended to the latter situation and it can be shown that it is not nec­essary to include a "front factor" or "memory term." For cross linking in the dry state, we can use a Flory scheme similar to that shown in Fig. 2 and first connect S junction points to form the undeformed network [step (3d), where d stands for "dry"]

(9)

The steps taken to form the swollen network start with an affine expansion of the chains by an amount a', appro-

priate to the volume of the swollen network (V1Vo=a/}, _ followed by mixing with solvent [equivalent to step (1) in Fig. 2]

AF1d [3 . ]!:Ji' m --=V - (a /2 -I)-lna,3 +--kT 2 kT '

(10)

where !:Ji'm is the free energy of mixing solvent molecules with the acyclic tree. We now link S junction points to form a network that has an equivalent structure to that formed by cross linking in the dry state

Il.F2d kT =~slnoVIV. (11 )

Finally the chains are allowed to relax to the expansion (a) found in dilute solutions of the same (good) solvent, at which time we assume that the junction points adjust their positions relative to one another to form a disinter­spersed arrangement

~;d =v{~ (a2-I) -In a 3

] -v[~ (a /2-I) -In ad].

(12)

The free energy change associated with swelling of a network formed in the dry state is then given by

Il.F kT =!:Ji' Id+ Il.F 2d+!:Ji' 4d-!:Ji' 3d

(13)

where the Flory-Huggins equation has been used for the free energy of mixing (subscript sand p prefer to solvent and polymer, respectively).

. There are several points concerning this treatment that are important. First, to reiterate, CPp (= V oIV) is not equal to l!a3

• It still remains for us to establish a relationship between CPp and a. Second, the expression for the mixing free energy will depend on the assumptions we make con­cerning the distribution of segments within each "blob" or expanded coil. It is convenient for us to leave the mixing . term in the familiar form used in Eq. (13) for now, how­ever, and consider this point in more detail later. Third, when we considered isotropic expansion (Fig. 2), the phantom network result was obtained by allowing the v-s . junction points remaining after cross linking to relax, so that fluctuations of the junction points are independent of the strain. If we assume that the c* theorem is correct, however, there is now a considerable rearrangement of these junction points and the constraints on fluctuations in the presence of solvents are very different to those found in the dry network. Collective and perhaps cooperative fluc­tuations of the junctions linking spheres are known to be significant,23 but the extent to which segments from one network chain fluctuating within a "blob" can interpene­trate adjacent blobs is affected by excluded volume consid­erations and is presumably very limited. This would pre­clude any relaxation step equivalent to step (5) in Fig. 2.

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1412 P. C. Painter and S. L. Shenoy: Swelling of polymer networks

Equation (13) is therefore our final result, describing the free energy change associated with the swelling of a net­work formed by cross linking in the dry state. The loga­rithmic terms do not cancel and are related to the entropy of disinterspersion of the cross-link junctions.

Some additional points that should be mentioned con­cern factors characteristic of real networks, such as the effect of pendant chains ("dangling ends") and entangle­ments. In the deformation of dry networks, pendant chains (those chains connected at only one end to a junction point) do not contribute to the elastic properties of the network (unless they are significantly entangled with other chains), and it is necessary to define an "effective" number of chains. In the c* model, however, these chains will still swell and exclude other chains from their sphere of influ­ence. Accordingly, the effect of a limited number of such chains should be small, as the experimental work of Bas­tide et aL 12 indeed suggests. A large fraction of such dan­gling ends would affect what we will refer to as the packing conditions, however, and we will come back to this point later.

The role of entanglements is more complicated and presumably determines the degree to which the junction points can disintersperse. We are neglecting entanglements here, but their effect is presumeably diminished in net­works formed in solution. A similar Flory scheme to the one shown in Fig. 2 can be devised to describe this situa­tion and it is a relatively simple matter to show that we obtain the same result as before [Eq. (13).]

The chemical potential

The equation for the chemical potential follows in the usual way by differentiating Eq. (13) with respect to ns the number of solvent molecules to obtain

Af.Ls [( I) ¢;] aa S¢p -=- 3y a-~ - --+-·-+In(1-¢) kT a Ny a¢p Ny p

+¢p+¢;X, (14)

where N is the number of statistical segments in the poly­mer chain and we assume that all the network chains have the same length.

In models that assume an affine deformation, an ex­pression for aala¢p is obtained from the condition ¢p= lIa3

• In using the c* model, we can obtain a similar relationship from the packing condition, but first we must consider the appropriate definition of concentration for use in this equation. If the chains act as swollen coils, they effectively exclude neighbors from their volume (apart from some minor overlap where they are forced into con­tact at their junction points). It would then seem necessary to use two concentration terms ¢p and ¢g. The first of these reflects the overall or average concentration of polymer segments within the spheres occupied by each of the swol­len coils and is simply given by

(1T16)NP 1 (1T16)(aNI12[) 3 a3N1I2' (15)

where we have used the end-to-end distance of the chains as a measure of coil diameter and the chains are assume to consist of spherical beads of diameter l. The second con­centration variable ¢g reflects the average concentration of segments in the gel as a whole and will depend on how the swollen coils pack. For example, if the cross-link density or cross-link functionality f is such that the swollen coils are arranged in a manner similar to random close packing of spheres, then we would expect that ¢g would be less than ¢p by a factor of about 0.637, the "filling factor" or fraction of space occupied by such a collection of spheres. For net­works where f is only 3 or 4, however, we would expect that in a good solvent, the coils would attempt to get as far away from one another as possible and the volume of the gel would be considerably greater (for the same N). For a perfect network of cross-link functionality f equal to 4, e.g., we could envisage the coils being arranged tetrahe­drally relative to each cross-link point. The volume frac­tion of polymer segments in the gel would then, to a first approximation, be determined from a simple calculation of the number of segments in a sphere of radius aN1I2[ sur­rounding a given cross-link point or nodule

(1T16)PfN fl8 ¢g (4/3)1T(aN1!21) 3 d.3N 1I2 •

(16)

More often, the packing conditions will be affected by entanglements, pendant chains, etc., so that

(17)

where P is a geometric or packing factor that should de­pend on f and the extent of network "defects," but should be largely independent of a.

Accordingly, the use of the c* model to describe swell­ing would seem to require that we describe thermodynamic

. interactions within each sphere (or later, blob) defined by the excluded volume of a network chain (or part of a chain) and then sum the free energy over all such spheres. With one exception, the terms in the expressions for the free energy and chemical potentials [Eqs. (13) and (14)] remajn the same, but we have to keep in mind that ¢p is not in most cases equal to ¢g. The exception is the term de­scribing the entropy of disinterspersion of the junction points, which depends on the volume of the gel as a whole, and therefore should be written as -Sin ¢g. The chemical potential can now be obtained using

aa a4N 1I2 a a¢p = -3-= 3¢p (18)

and by assuming that the packing factor P is independent of ¢p at concentrations close to maximum swelling in a good solvent, so that

a¢g a¢p =P. (19)

We therefore obtain for the chemical potential

Af.Ls 2 ¢p S ¢p 2 kT=(a -1) N+v N+ ln (1-¢p) +¢p+¢pX, (20)

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P. C. Painter and S. L. Shenoy: Swelling of polymer networks 1413

TABLE 1. (A) Poly(styrene}/benzene [functionality(f} =3 packing factor(P} =f/8]. (B) Poly (dimethyl siloxanelheptane [functionality(f) =4 packing factor(P) =f/8].

Degree of swelling N (A) Samplea Me (Q) ( calculated) C'"

GPST3 7000 11.00 10.82 6.22 GPST2 10600 14.25 17.08 5.97 GPST4 13000 15.10 18.88 6.62 GPST9 15 400 17.63 24.54 6.03 GPST5 19900 20.25 30.90 6.19 GPST6 30 300 25.60 45.23 6.44

Degree of swelling N (B) Meb (Q) (calculated) C'"

3 775 5.50 10.86 4.7 4 500 5.90 12.37 4.92 8 700 9.60 27.24 4.32

13 250 11.20 34.10 5.25 17 100 13.20 43.03 5.37

"Reference 25. bReference 26.

(21 )

This is very similar in form to the Flory result, differ­ing only in the first term, where the assumption of an affine deformation results in a tP~/3 IN term.

As one would expect, the use of the c* assumption results in the expected dependence of a and tPp (at maxi­mum swelling) upon N. Substituting for tPp in Eq. (18),

a 5 _ NJ·5 ( 1/2 - X) + other terms (22)

so that

a_NJ·l,

and using Eq. (16)

A. N-4/5 'I'p- •

Some comments on the mixing term

(23)

(24)

In the treatment described above we have used the Flory-Huggins equation to describe mixing within each coil, essentially following the "simplest possible treatment" used by Di Marzio in treating the polymer collapse tran­sition,24 where it is assumed that the polymer segment con­centration is uniform throughout the spheres defined by the end-to-end distances of the network chains. We could also have used the Flory treatment, where it is assumed that the polymer segment concentration is uniform over small volume elements, but overall has a Gaussian distri­bution within each sphere. Alternatively, we could follow the arguments of de Gennes and used an interaction term of the form U*tP;·25, so as to take into account the effect of monomer correlations. 18 In order to be simple, we will maintain the Flory-Huggins description, but note here that the effective interaction coefficient u* in the de Gennes

treatment can be obtained from measurements such as os­motic pressure. Notwithstanding the compelling theoreti­cal arguments for using an interaction term of this form, the same data can be fit to a simple Flory-Huggins model, provided that X is allowed to be composition dependent; it is these experimentally determined values of X that we will use in our calculations.

Comparison to experiment

For-equilibrium or maximum swelling in a good sol­vent we have

(25)

For a theoretically perfect model network of known functionality I, the factor s/v is given by

~=1-~ v I

(26)

and tPp is related to the measured degree of swelling Q through

1 1 Q=-=--tPg PtPp '

(27)

while N is related to the degree of polymerization (DP) by

DP N=-·-.

Coo (28)

We can illustrate how Eq. (25) works in two ways. First, we will assume that various model networks that have been synthesized are indeed perfect, so that we can use Eq. (16) to obtain the "packing factor" (P-118). We can then calculate N and hence Coo and see if this gives us reasonable numbers. Of course, if there are pendant chains and entanglements, then this could affect the packing fac­tor to an unknown extent. Accordingly, we will also use values of Coo reported in the literature and calculate the packing factor P.

We will first examine pata obtained on the swelling of trifunctional polystyrene (PS) network in benzene and tet­rafunctional poly(dimethyl siloxane) (PDMS) in heptane reported by Belkebir-Mrani et al. 25,26 Assuming P= 118, values of Coo were calculated from the experimental de­grees of swelling and the results are reported in Table 1. It can be seen that very consistent results are obtained over a range of primary chain molecular weights with Coo for polystyrene having a calculated values near 6, while the Coo values for POMS were of the order of 5. The agreement within each set is very encouraging, but the values are somewhat lower than those determined using rotational isomeric states calculations (PS-lO, PDMS-6). Obvi­ously, even model networks are not perfect and we would anticipate that factors such as incompletely reacted cross­link sites would result in a larger packing factor than that calculated on the basis of P= 118. Accordingly, if we as~ sume values of Coo 7= 10 and Coo = 6.3 for atactic PS and PDMS, respectively, the packing values listed in Table II

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1414 P. C. Painter and S. L. Shenoy: Swelling of polymer networks

TABLE II. Po1y(styrene)lbenzene [functionality(f) =3, C", = 10]. (B) Po1y(dimethy1 siloxane/heptane [functionality(f) =4, C", =6.3].

Degree of swelling Packing factor (A) Samp1ea Me (Q) N (P)

GPST3 7 ()()() _11.00 6.73 0.289 GPST2 10600 14.25 10.19 0.279 GPST4 13000 15.10 12.50 0.296 GPST9 15400 17.63 14.81 0.279 GPST5 19900 20.25 19.13 0.2818 GPST6 30300 25.60 29.13 0.286

Degree of swelling Packing factor (B) Meb (Q) N (P)

3775 5.50 8.10 0.4316 4500 5.90 9.65 0.4385 8700 9.60 18.66 0.3911

13 250 11.20 28.42 0.4410 17100 13.20 36.68 0.4465

aReference 25. bReference 26.

are calculated. As expected, the calculated values of Pare smaller than those calculated for perfect networks, but are about what we would intuitively expect.

We would anticipate that as the functionality of the network increases, the value of P should also increase. Un­fortunately, the experimental results cannot be compared easily with our calculations because of the nature of most of the PS networks that have been studied. These consist of divinylbenzene (DVB) nodules whose functionality f is unknown and described in terms of the ratio of DVB mol­ecules to living ends (DVB/LE). Because f and Pare related variables, we have to make assumptions about f in order to calculate P. One crude way is to assume that P is equal to some constant multiplied by f /8, then calculate this constant from the results obtained from the known trifunctional network. This gives the estimated valued listed in Table III. The values of f are smaller than those originally estimated by Belkebir-Mrani et al.,25 but are of the same order as those estimated in a later review by Rempp et al. 27 (Note that these authors pointed out that for polystyrene networks, a limiting functionality is ob­tained for DVB/LE ratios of the order of 6, so the esti­mated value of f = 6 for DVB/LE ratio's equal to 10 is about right).

Also shown in Table III are the results of various swelling studies reported by Belkebir-Mrani et al.,25 Benoit et al., 28 and Bastide et aZY·29 The calculated packing fac­tors are largely in good agreement for samples with differ­ent primary chain molecular weights but the same f, and also vary systematically with f. The exceptions are those networks polymerized so as to have pendant chains and also apparently a set of samples with DVB/LE ratios equal to 5.

OSMOTIC DESWELLING

Upon osmotic deswelling of a gel initially in equilib­rium with excess of a good solvent, the swollen coils over-

TABLE III .. Poly (styrene) !benzene (C", = 10).

Sample DVB/LE f (estimated) Me Q N P

GPS Fe 9a 3 4 13 500 12.33 12.98 0.3347 GPS Fe 13a 3 4 16500 14.06 15.87 0.3290 GPS Kia 3 4 22 ()()() 15.20 21.15 0.3596 GPS Fe 12' 3 4 24000 015.53 23.08 0.3707 GPS Fe 10" 3 4 28000 19.04 26.92 0.3315 GPS Fe 11" 3 4 39500 24.40 37.98 0.3194 Ab 3 4 26 ()()() 17.40 25.00 0.3470 64b 3 4 12500 9.80 12.00 0.4030 65b 3 4 12500 8.50 12.00 0.4646 Blc 3 4 26 ()()() 17.40 25.00 0.3470

GPS Fe 6" 5 4.5 14 ()()() 7.94 13.46 0.5136 GPS Fe 3" 5 4.5 22500 12.14 21.63 0.4417 GPS Fe 4" 5 4.S 36 ()()() 14.44 34.62 0.4934 GPS Fe Sa 5 4.5 58000 22.41 55.77 0.4293 54b 5 4.5 34500 20.5 33.17 0.3385 56b,d 5 4.5 35 ()()() 20.4 33.65 0.3432 58b,d 5 4.5 30 ()()() 23.8 28.85 0.2677

- --

GPS Fe 16" 6 5 16500 11.03 15.87 0.3959 GPS Fe IS" 6 5 22000 13.20 21.15 0.3908 GPS Fe 14" 6 5 33500 17.20 32.21 0.3863 B2c 6 5 26000 14.90 25.00 0.3824

.GPS Fe 8" 10 6 28 ()()() 12.94 26.92 0.4435 Alc 10 6 21 ()()() 9.80 20.19 0.4935 A2c 10 6 22 ()()() 10.14 21.15 0.4902 A3c - 10 6 30000 12.60 28.85 0.4749 A4C 10 6 44 ()()() 17.50 42.30 0.4333 ASc 10 6 sO 000 19.50 48.08 0.4217 B3a 10 6 26 ()()() 12.50 25.00 0.4391

"Reference 25. bReference 13. cReference 28. dGels with pendant chains.

lap and the simple relationship between a and ¢p given by the c* model no longer holds. Here again, however, we can use the ideas of de Gennes to obtain a relationship that can be incorporated into a Flory-type model. We will assume that the overlapping coils essentially act as a series of blobs of size (diameter) b and utilize the scaling law (18)

(29)

where R I is the root mean square end-to-end distance of the elem.entary chail1s of the network in a dilute solution of a good solvent and we now let ¢: represent the volume fraction of polymer segments found in the individual coils ih the equilibrium (fully swollen) state. The exponent ms is equal to 3/4 on the basis that powers of N from R I( ~a*ptJ·5 ~ptJ.6) cancel with those from ¢: ( ~ 1/(a*)3ptJ.5 ~ l/ptJ.8).

The size of each blob b is given by

(30)

where N b is the number of monomers per blob. If the monomers within each blob do not interact with those from other chains or in other blobs,

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P. C. Painter and S. L. Shenoy: Swelling of polymer networks 1415

(1T'/6)N J3 1 epp (1T'/6)(aNf21) 3 a3NY2' P_- (31)

Substituting for Nb from Eq. 30 and beep) from Eq. (29), we obtain

(32)

Substituting for the chain expansion factor a* found at equilibrium sw6'lling in the pure solvent

= 1 (ep:) 1/3 (epp) 3/4 epp ~ N ep: (33)

and, as before

aa a aepp = - 3epp . (34)

We again assume that mixing and chain extension de­pend on the local segment concentration inside each blob epp rather than the average segment concentration within the gel as a whole epg. In addition, for any given value of the concentration of polymer solution in equilibnuni- with the gel, we again assume that epg varies with epp as

aepg epg=Pepb and aepb =P,

so that the chemical potential of the solvent is given by;

!.. (ep:) 1/3(epp) 3/4 ( s) epp 2 N N ep: - 1-:;; N+ ln(1-epp) +epp+eppXsp

(35)

where epe is the volume fraction of the polymer in the so­lution in contact with the gel. We have assumed that by the design of the experiment, this polymer does not penetrate the network. This equation can be modified in the obvious manner to account for situations where, in principle, some mixing within the gel could occur, but in the work of Bas­tide et al., 12,13 which we will consider below, no such mix­ing of polymer and network was detected. Note that if epe is equal to zero, epp = ep: and Eq. (30) reduces to Eq. (20), as it should.

We have applied Eq. (35) to a calculation of the pack­ing factor P for the osmotic deswelling of three samples labeled A, 54, and 58 by Bastide et al. 13,29 All three net­works were prepared at the same polystyrene concentra­tions and with the same ratio DVB/LE (=5), but the chains have somewhat different molecular weights (26 000, 34500, and 30000) and sample 58 was prepared from polystyren'e with a fraction of nonfunctional ends (0.6). Nevertheless, the values of P calculated as a function of swelling Q are consistent for all three samples and vary systematically, approaching the expected value of 1 for values of Q just less than 5, as illustrated in Fig. 3.

PHASE BEHAVIOR

When using the affine model, the calculation of phase behavior follows directly from the equations for the chem-

1.0 r-----;.;;-----------------,

0.8 g sample 54 , • sample 58

m • sample A

~ ..:g 0.6 •

Ill) ·.m ~ '" • ~ .m

0.4 • • Ii

• 0.2

0 10 20 30

degree of swelling

FIG. ~. Variation of the packing factor with the degree of deswelling for osmotically deswollen gels. (Deswelling data are taken from Refs. 11 and 13.)

ical potential. The deformation of the chains is simply re­lated to the swelling through a=ep-1I3. In applying the disinterspersion model, however, w~ have a much more complex situation. We can still write a general equation for the chemical potential of the solvent as

because all the models we have considered give

aa a aepp = - 3epp , (37)

but the relationship between a and ep will now vary with X. At high temperatures, where the s~lvent can be consid­ered to be "good," a relationship can be found from con­siderations of simple geometry, as described above. When the solvent becomes poor CX:> 1/2), individual chains in dilute solution would collapse to a compact configuration, but at the concentration of segments found in deswollen gels, we can assume that the chains have their unperturbed dimensions (a = 1). Indeed, Beltzung et aI., 30 e.g., found a= 1 for poly(dimethylsiloxane) networks swollen in tol­uene (X> 0.5 over most of the composition range). Ac­cordingly, for a network in a poor solvent, the only "elas­tic" contribution to the free energy comes from the Flory logarithmic term describing the interspersion of the cross­link points over the volume of the gel relative to the dry network.

The situation that is not well described is the crossover from a good to a poor solvent where there is no simple way to obtain a relationship between a and epp. In order to model phase behavior over the entire range of temperature, we will therefore make a crude assumption that the net­work behaves as a collection of excluded volume spheres with

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1416 P. C. Painter and S. L. Shenoy: Swelling of polymer networks

TABLE IV. Natural rubber/solvent X=Xl +X2tPP+X3tP;.

Solvent Xl X2 X3

Benzene 0.41 0.0 0.0 MEK 1.42 -2.29 2.28 Ethyl acetate 1.61 -2.52 2.16 Acetone 7.27 -15.81 10.45

1 CPp= a3N1!2 (38)

and that this relationship holds up to concentrations cor­responding to cpp_N-1I2 (i.e., a=l). At higher concen­trations it is assumed that a is always equal to 1 and there­fore there is no contribution to the free energy from the elastic deformation of the chains. Clearly, calculations of this region of the phase diagram would be improved by using some sort of blob model to account for chain overlap, as in the treatment of osmotic deswelling, but at this point, we wish to maintain maximum simplicity, so for now we will use this assumption and proceed to examine calculated phase behavior.

We will illustrate the application of this simple model by considering a reinterpretation of experimental results obtained by McKenna et af. 17 rather than a presentation of phase diagrams, as this points to the heart of the difference between affine and disinterspersion type models. These au­thors examined the swelling of dicumyl peroxide cross­linked natural rubber. Four different solvents were used to swell four samples that differed from one another in cross­link density. Of the four solvents, benzene, methyl ethyl ketone, ethyl acetate, and acetone, only benzene can be considered a "good" (or nearly good) solvent with a X value (for non-cross-linked chains) of less than 0.5. The values of the X parameter for these polymer solvent sys­tems are listed in Table IV. McKenna et af. used the affine assumption, but calculated the elastic free energy from pa-

rameters determined from mechanical measurements. They found that the elastic and mixing terms could only be equated if the X term was made cross-link dependent. In contrast, if we use the disinterspersion model, we would require a value of a= 1 for swelling in poor solvents. Un­fortunately, we do not know the average cross-link func­tionality or the value of P for these networks, cross linked in tlie solid state: We therefore assume that P= 1 and use the results obtained from the swelling in benzene to calcu­late N, the average number of segments between cross-link points for each sample. In effect, we assume that the effect of entanglements (hence packing), etc. can be included in this parameter. Using these values of N, the volume frac­tion of the rubber segments present in the networks swol­len in the "poor" solvents can be calculated and are com­pared to experimental values in Table V. It can be seen that in spite of the simplicity of the model, there is good agree­ment. This suggests that the degree of swelling observed in poor solvents is predominantly a reflection of a phase be­havior that is determined largely by the mixing terms.

DIFFERENTIAL SWELLING MEASUREMENTS AND THE SWELLING ACTIVITY PARAMETER

Finally, although our principal aim is the description of a simple model that can be used to explore phase behav­ior, for completeness we wish to consider the observations of Gee et af. 31 and later Eichinger and co-workers. 32,33

These authors measured the solvent activity of samples in the cross-linked (ae ) and uncross-linked (au) states, pa­rameters that can be combined in the form of what has subsequently been called a swelling activity parameter S by McKenna et af. 37

(39)

where As is the swelling deformation. It has been experi­mentally observed that S has a maximum when plotted against As and also a nonzero intercept at As= 1. There

TABLE V. (A) Natural rubberlbenzene (P= 1). (B) Natural rubber/solvent (P= 1, a= 1).

(A)

(B) Sample

APHRI (n=27.04)

APHR~

(n= 11.77)

APHR5 (n=7.16)

APHRIO (n=4.94)

Sample

APHRI APHR3 APHR5

APHRlO

MEK

tPg(calc.) tPg(obs.)

0.6231 0.575

0.6657 0.648

0.7058 0.709

0.742 0.759

Me tPg N

21800 0.158 27.04 _ 5450 0.256 11.77

3115 0.335 7.160 1500 0.405 4.940

Ethyl acetate Acetone

tPg(calc.) tPg(obs.) tPg(calc.) tPgCobs.)

0.6175 0.527 0.8302 0.854

0.6476 .- 0.602 0.8423 0.882

0.678 0.665 0.855 0.907

0.7085 0.725 0.868 0.943

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P. C. Painter and S. L. Shenoy: Swelling of polymer networks 1417

have been a number of interpretations of these results; McKenna et al,34 e.g., argue that at least in part this phe­nomena is due to a cross-link dependence of X. We wish to suggest an alternative interpretation that is based on the model presented here, but is also inspired by an observa­tion made by Bastide et af. 13 that there is a threshold dis­tance for affineness in the macroscopic strain. It was pro­posed that at scales larger than the threshold distance, the deformation is affine, but at smaller scales, it is not. What we propose is something different, but along the same lines, that at low degrees of swelling, entanglements are such that the swelling mechanism is largely affine,- with little disin­terspersion of the cross-link points, but as the concentra­tion of solvent increases, disinterspersion becomes more significant and eventually dominates the swelling progress.

To see how this idea works, we first write the equation for the solvent activity in the most general form

(41)

(recall that for both the affine and disinterspersion model aa/a¢p= -a/3¢p; it is the relationship between a and ¢p that is different).

There are two points concerning these equations that should be noted. First, in the affine model ¢p=¢g' but this is not necessarily so in the disinterspersion model we are using. However, we have seen that the factor P approaches values of 1 when Q is about 4 or 5, so at fairly low degrees of swelling, where the maximum in the swelling activity parameter is observed, we can assume ¢p=¢g. Second, and crucially, even if the network is formed by end linking the uncross-linked chains, the number of segments N defined for the purposes of describing the elastic free energy is not the same as the number of segments N m defined for the mixing free energy term. The former is defined in terms of the number of chemical repeat units that would give be­havior equivalent to that of a freely"jointed chain, while the latter is equal to the molar volume of the chain divided by the molar volume of the solvent.

We can now write the swelling activity parameter as

S=~ [a2 ¢p +! ¢p - ¢p (l-~)], ¢p N vN N N m

(42)

where the substitution As= 11¢;13 has been made, but it must be kept in mind that As is only equal to the chain expansion factor a in the affine model. We have also as­sumed that there is no cross-link dependence of X. We can now substitute for a 2 to obtain for the affine model

SaIf=~+ ¢;13 [!_ (1-~) ] N N v N m

(43)

and for the disinterspersion approach, using the value of a 2

determined from the blob model [Eq. (32)]

0.05

0.04 ~

$ e S 0.03 a.

~ ;; 0.02 'ii .. .. ,g O,o! ;; I: ..

0.00 1.0

0.03

:0 ;; E 1! .. a.

~ :E (;

O,o! .. co ,g ;; I: ..

0.00 1.0

.......... ........... w ... • .. •••• ..... ;:.·=-------,

• btob modal

.. afffne model A Oala from ref 31

A

A

-----------..;.-:"" .. -...................... ; ................ :: .T. :

1.2 1.4 1.6

............... _..................... .

blob model affine model

6, Oat. from ref 32

A

'-..- '

............ --".--.• - ...... ~ ...................... '" ....... A ••

1.2 1.4 1.6 1.8

".

FIG. 4. Variation of the swelling activity parameter with As for the affine and disinterspersion models. Data points have been taken from from Refs. 31 and 32.

1 {I (¢:)1/3 (¢p)3/4 ¢p[s ( N)]} SdiS=¢1/3 N N + ¢* +N ~- I-Nm .

p p (44)

If we first examine the affine model, it can be seen that Saff will increase with As (Le., decreasing ¢p) because, in general, N / N m is of the order of 11 Ceo, as illustrated on the two plots shown as Fig. 4 (we assumed /=4 and cal­culated N from ¢: in making these calculations). If we had made the erroneous assumption of N =N m' then we would have obtained the opposite result Saff decreasing with in­creasing As' In contrast, Sdis generally decreases with in­creasing As, as also shown in Fig. 4. These calculated re­sults are compared to the experimental results of Gee et af. 31 and Brotzman and Eichinger32 and it can be seen that a crossover from affine to disinterspersion type behav­ior would give the observed maximum. Furthermore, there is an intercept at As= I that should be roughly of the order of (s/v)/N. The crossover does not appear to proceed to completion for the peroxide cross-linked sample used by Gee et af., but this sampl~ is relatively highly cross linked (maximum swelling ~2.2) and entanglements are presum­ably significant. For the end-linked samples used by Brotz­man and Eichinger, where entanglements would be much less of a factor, there is a good agreement between exper­iment and the model proposed here.

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1418 P. C. Painter and S. L. Shenoy: Swelling of polymer networks

SUMMARY AND CONCLUSIONS

( 1) Weare proposing a model that is based on the general approach described by Flory, but which does not include the affine deformation assumption.

(2) Swelling is assumed to proceed by a process of disinterspersion and the c* and blob models of de Gennes are used to obtain a relationship between chain expansion and swelling.

(3) A key feature of the model is that in a good sol­vent, the network chains act as excluded volume spheres, which means that in hypothetically perfect networks where the cross-link functionality is only 3 or 4, there is "unoc­cupied volume" within the gel. By this, we mean that be­cause of packing and excluded volume considerations, the volume of the gel is greater than the sum of the volumes occupied by the excluded volume spheres.

(4) The amount ·of "unoccupied volume" within the gel will be decreased by entanglements, but increased by the presence of pendant chains.

(5) It is proposed that the maximum observed in dif­ferential swelling measurements may reflect a crossover from affine to disinterspersion behavior at low solvent con­centrations.

ACKNOWLEDGMENT

We gratefully acknowledge the support of the office of Basic Energy Sciences, Department of Energy, under Grant No. DE-FG02-86ER13537.

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