Parameters of the Swelling Equation and Network Structure BY K. DUSEK Institute of Macromolecular Chemistry, Czechoslovak Academy of Sciences, 162 06 Prague 6, Czechoslovakia Received 30th November, 1973 The functionality of a crosslink which appears in the swelling equation differs from the chemical functionality and represents the number average of elastically-active chains issuing from an elastically-active cross-link, independently of the type of the crosslinking process. Relationships have been derived for the polycondensation of an $functional monomer, for alternating copoly- condensation of two monomers, and for the crosslinking of primary chains which allow the determina- tion of the effective functionalityf, either from the composition of the gel phase or its calculation by means of crosslinking statistics based on the theory of cascade processes (Gordon-Good).In close proximity of the gel point,f, is always three and its increase is the faster the more positive the substitu- tion effect in the monomer unit. Up to comparatively low values of the gel fraction (10-20 %), fe in most cases does not depart too much from three and increases steeply to the final value only at a virtually negligible sol fraction. The change in the free enthalpy AG for mixing of a solvent with polymer systems arising by crosslinking reactions may be written 1. as a sum of two contributions, AG = AGmi,+AGe1, (1) where AGmjx relates to the mixing of a solvent with a crosslinked polymer having un- stretched chains, while AGel is a contribution due to a change in the dimensions of the elastically-active network chains (EANC).The term AGmi, has been derived on the basis of a lattice model and the quasi-chemical equilibrium approach, when primary structural units by which the network is formed (monomer units or primary chains) bearing specific contact points (surfaces) are placed on a lattice. Among these contact points there are some (crosslinkable) which are capable of quantitative coupling (at the expense of a entropy change), either with contact points of the same kind, or with similar contact points of a different kind. This simulates the actual process of network formation by various types of reactions. The other contact points, the number of which also varies as the reaction proceeds, interact physically with each other and with the contact points of the solvent.The magnitude of the equilibrium sorption of the solvent is determined by the equality of the chemical potentials of the solvent in both phases. We have demon- strated l * that the contribution of mixing to the chemical potential of the solvent may be obtained in a closed form as a function of surface and volume frac- tions of the solvent and polymer, of the fractions of the contact surfaces, and of the interaction free energies for the formation of the respective contacts. If one assumes for AGel the Gaussian behaviour of EANC (Wall-Flory's approach 3, and carries out a partial expansion of (AP&~ into a power series of the volume fraction of the polymer, &, one obtains a relationship for the total change in the chemical potential of the solvent Aps, co APS/RT = In (1-4p)+4: C Q i ~ ~ + [ ~ - ( 1 / ~ ) ( ~ ~ / ~ ~ ) ( ~ ~ / 2 - 1 ) ] ~ ~ + i = O ve(VsIQ[Ch$(+,")' - 4 g I 7 1011 02 PARAMETERS OF THE SWELLING EQUATION where f is the number average number of segments per primary unit (monomer unit, primary chain), G, vp and 6, respectively, are the molar volume of the solvent and the average molar volume of the polymer segment in the system and in the gel; +g is the volume fraction of the gel component in the system, and 4: is its value at net- work formation (assuming that the network chains are in the relaxed state during network formation), NJ2 is the number of bonds formed per primary unit, and 17, is the number of EANC per segment.The sum Qi& has the meaning of a concectra- tion-dependent interaction parameter x ; the coefficients Q, comprise the effect of thermodynamic interactions and are dependent on the size of the primary units and on the number of the bonds formed.They may be expressed analytically if the respective interaction energies and the magnitudes of the contact surfaces are knowii. For the case of crosslinking of simple primary chains bearing only one type of non- crosslinkable contact points it has been demonstrated that the parameter 3~ increases with increasing & and the degree of crosslinking, if the interaction free energy is positive, but that it may decrease with concentration for a negative value. The situation observed for polycondensation networks is more complex, since here one has to consider more types of non-crosslinkable contact points and major changes during the reaction (e.g., formation of an ester bond due to the reaction between the carboxyl and the hydroxyl group).Eqn (2) may be used directly for measuring sorption equilibria between a cross- linked system (containing both sol and gel) and, e.g., solvent vapours, as long as the crosslinked system does not separate into two condensed phases. N, is known from the course of the reaction, v,, e.g., from mechanical measurements, and +g may be determined from parallel extraction of soluble fractions. Thus for the determination of x the treatment of a closslinked system may be sometimes more useful than the treatment of the swelling equilibrium of an extracted gel. If we now pass to the sorption of solvent by an extracted gel (e.g., swelling in pure solvent), all quantities must be related to the gel phase (index g).A simple trans- formation of eqn (2) gives a relationship analogous to Flory’s swelling equation (cf. ref. (5)) : 00 i = O ~,-m- = in (1 -4g) + ~,+X42,+V,(~1~)~~~(4g0)8-~4~i, (3) where x is a variable corresponding to the sum in eqn (2), and B replaces 2/fin the original version (f is the chemical functionality of the crosslink), so that (4) B = 1 - (NJ2 - l ) / ~ , f ~ . It is evident that B varies with the extent of reaction (crosslinking) ; as a consequence, f cannot be constant either. Ng in eqn (4) has then the following meaning for various types of crosslinking reaction * * : No =fag Ng = nlga,,f, + n 2 p 2 , f 2 for homopolycondensation, or for alternating polycondensation of monomer 1 and 2, or Ng = v,Y, for the crosslinking of the primary chains In the above expressions, a, is the con- version of the functionalities in the gel, f is the chemical functionality of the cross- link, nip is the molar fraction of the monomer units i in the gel, and vp is the fraction of the crosslinked segments in the gel.K .DUSEK 103 In principle, N , may be determined experimentally or calculated by means of cross- linking statistics. In the following section we shall apply crosslinking statistics based on the theory of cascade processes 6-8 (cf. a.lso ref. (2), (4)) and derive relation- ships for the effective functionality of the crosslink-f, in the swclling equation. MEANING AND DERIVATION OF THE FUNCTIONALITY f e I N THE SWELLING EQUATION By analogy with Flory's swelling equation we write .f, s 2/B = 2veFg/(v&-Ng/2+ l), (8) and express the parameters on the right-hand side by using the crosslinking statistics with the first shell substitution effect (fsse), iizmely, using the probability generating function (pgf) for the ties T(8).The pgf for the number of offsprings of a primary unit in the root of probability trees for a reaction of units of one type is given by Fo(8) = C piei, (9) i where p i is the probability that i bonds lead from one unit in the root to the units in the first generation. The pgf for the number of units borne by the unit in the first and fol1owir.g generations is and the extinction probability u is defined by Pgf for ties (or active bonds, active reacted functionalities) i.e., those that are part of an infinite sequence of bonds may be written as For a reaction of several types of primary units, it is possible to perform generalization in the form of vectorial generating functions (cf.ref. (7)) which for alternating poly- condensation of two monomers of type 1 and 2 degenerates to if the coefficients p l i or p 2 j denote the probability that from a monomer unit of type 1 or 2, i o r j bonds lead to monomer units of type 2 or 1, respectively. These relation- ships may also involve cyclization, if the coefficients pr are related to units giving rise to i intermolecular bonds. In the appendix,f, is expressed in terms of the moments of T(8) and the proof is given that fe is the average number of EANC issuing from elastically-active cross- links (i.e., from those issuing at least three EANC); consequently,I 0 4 PARAMETERS OF THE SWELLING EQUATION if zt is the fraction of crosslinks with i EANC.(A10) and (A15) of the Appendix we obtain for homopolycondensation, f e = CWG(1)- T W ) - G ( O ) ) + n2(7X)-- G(0)- T’;(0))3/ for alternating polycondensation, and for crosslinking of the primary chains. The symbol T’(k) or T”(k) designates the first or second derivatives of T(8) with respect to 8 at 8 = k. By using relationships (9)-( 16) one may transform eqn (18)-(20) into (1 8a)-(20a) : By rearrangement of eqn (A4), fe = (T’( 1) - T’(0) - T”(O))/( 1 - T(0) - T’(0) - T”(0)/2) (18) [ 1 - n,(T,(O) + TXO) + T;(O)) - n,(T,(O) + W O ) + 7-;(0)/2)1 (19) fe = 4(1 +T’(l)-T(O)-2T’(0))/(2+T’(l)-2T(O)-3T’(O)) (20) fe = [( 1 - vI2fa( 1 - F;(v))lI[ 1 - Fo(u) --fa( 1 - fe = C[2(1 - U 1 ) ( 1 - % ) - 4 - 2)2)2F;1(v2)-n2(l -vd2Fi2(v1)11 + (+)(I - u)Fi(u))l; (184 [l - ~ l ~ o l ( ~ 2 ) - ~ 2 ~ 0 2 ( ~ 1 ~ + ~ ( ( ~ -v2)u1 +(I -u2)”l;;1(u2)/2+ (1 - U , > U , + (1 - vd21;.i2(u1)/2)1 ; fc = 4[1 -Fo(~)+y(l -~)(1-2~)]/[2-2~0(~)+~(1 -~)(1-3~)], (194 (204 c n l a l f l = n2a2f2> where y = v,,v is the crosslinking index.DISCUSSION The finding that the effective functionality dfe) in the swelling equation is a number- average number of EANC originating from an elastically-active crosslink in the gel is in accord with the conception of crosslinking entropy derived by F l ~ r y , ~ if applied to real systems. However, the quasi-chemical equilibrium approach involves surface fractions of the contact points ; consequently, crosslinking entropy contains also terms with higher powers of the volume fraction of the polymer which affect x, but not f e .In real systems the fractions and effective functionalities of elastically-active crosslinks are not obvious, but the analytis offered here demonstrates that fe may be calculated either from experimental data (eqn (5)-(8)), or from crosslinking statistics At the gel point,f, = 3 independently of the chemical functionality of crosslinks The increase infe with proceeding crosslinking after the gel point is illustrated in fig. 1 (a-c) and 2 by several examples of ring-free equilibrium-controlled homopoly- condensation and alternating polycondensation of two monomers involving both a linear positive (N > 1) or negative (N < 1) fsse (as well as random crosslinking of the primary chains with different distributions).(The definition of the respective pgf of the substitution parameter N, as well as further treatment, have been explained else- where 2-4* 7-9.) Thef, is plotted against reduced conversion where a, is the conversion at the gel point. The examples make it clear that : (a) the increase inf, from 3 to the final value is the steeper the more positive the substitution effect ; (b) for alternating copolycondensation and stoichiometrically non-equivalent ratio of the functional groups (fig. I(c)), f e will never attain a value corresponding to the chemical functionality of the crosslink, f; (c) f e does not depart greatly from 3 (eqn (1 8)-(20))* %d = (a - (21)K. DUSEK 105 within a comparatively broad range of the experimentally-available weight fraction of the gel, w,, if the substitution effect is not strongly positive, or the distribution very wide.For instance, for chemically tetrafunctional crosslinks, fe increases in most cases to attain 3.1-3.2 at wg = 0.85, which may be used for estimatingf, of sys- tems with a measurable sol content. The procedure described above may also be used in cases when cyclization becomes operative. Then, quantities in eqn (5)-(8) of the coefficients of pgf must be related to intermolecular connections. In crosslinking statistics, cyclization up to the gel 4.0 3.5 3.0 4.0 q 3.5 3.0 4.0 3.5 i 2 A I "'"0 0 .5 I .o 0 . 5 I %ed, w g FIG. l(a-c).-Dependence of the effective functionality fe in the swelling equation on the reduced conversion of functional groups Crred and on the weight fraction of the gel, wg, for networks obtained by homopolycondensation or by alternating ring-free copolycondensation. (a) homopolycondensa- tion of a tetrafunctional monomer with a linear fsse ; numbers near curves designate the value of the substitution parameter N ; (b) copolycondensation of a tetrafunctional (1) and bifunctional (2) monomer with a linear fsse, 1 iVl = 1.5, N2 = 1.5,2 Ni = 1, N2 = 1, 3 Nl = 0.5, N2 = 0.5 ; (c) co- polycondensation of a tetrafunctional and bifunctional monomer in a stoichiometrically non- equivalent ratio of the functional groups without fsse ; the initial ratio of the number of functional groups: 1 1 : 1, 2 1 : 1.1, 3 1 : 1.5, 4 1 : 2.106 PARAMETERS OF THE SWELLING EQUATION point may be treated satisfactorily lo ; the simplest assumption after the gel point is that all gel-gel reactions are intermoIecuIar and that the cyclization proceeds only within the s01.'~ This assumption was used for the treatment of the kinetically- controlled chain polymerization of a bisunsaturated monomer with a strong cycliza- tion.12 Fig. 3 shows that the dependences offe on the reduced conversion of the double bonds differ from each other, but the dependences on the gel fraction are practically identical.The experimental data that may be employed for the determination of& are scarce. It would be possible to use a procedure suggested by Mark et al.l3* l4 according to which the coefficient B (eqn (3)) may be determined from the equilibrium elastic moduli of samples having different network density and dilution at crosslinking, but i ---l----- Yhfc, "'g FIG.2.-Dependence of the effective functionality fe in the swelling equation on the relative cross- linking index y/yc and the weight fraction of the gel wg for networks obtained by ring-free random crosslinking of the primary chains. Number average degree of polymerization of the primary chains r, = 1 0 0 ; distribution type: 1, homodisperse chains; 2, most probable distribution; 3, Schulz distribution with the polydispersity index rw/rn = 5 . 4. w c, 0 . 5 0.5 ared, wg FIG. 3.-The effect of cyclization on the effective functionality f, during radical crosslinking polymer- ization of a bisunsaturated monomer in dependence on the reduced conversion of double bonds a,,d and on the weight fraction of the gel wg.Monodisperse primary chains with m = 50 ; 1, without cyclization ; 2, with cyclization using the spanning-tree approximation and Gaussian statistics for cyclization probability l 2 (distance between the double bonds 9 links, number of links in a statistic segment 2).K. DUBEK 107 swollen in equilibrium to the same degree. While using this method for loosely crosslinked diethyleneglycol methacrylate networks (the Mooney-Rivlin constant C2 which is practically zero) we found that in most casesf, varies between 3 and 4. The polydimethylsiloxane networks investigated by Johnson and Mark l 4 are not very suitable because of the high value of the constant C,.If one assumes v, to be pro- portional to the elasticity modulus (cf. ref. (14)), f, varies from 4 up to physically unwarranted high values. On the other hand, for ve cc C1 (Mooney-Rivlin),f, de- creases for strongly diluted networks below the physically reasonable value of 3. One must not forget, however, that the method is based on the assumption that the para- meter 31 is independent of cross-linking density, which may be a source of error. APPENDIX EXPRESSION OF THE EFFECTIVE FUNCTIONALITY .f, I N TERMS OF CROSSLINKING STATISTICS The effective functionality fe is defined by eqn (S), and the quantities N,, ve and F, for the individual cases (eqn ( 3 4 7 ) ) are expressed in terms of the moments of probability generating functions (9)-(16).The summing indices are i or j . (a) HOMOPOLYCONDENSATION OF AN f-FUNCTIONAL MONOMER (4 =fag, OdiGf) The number of reacted functionalities in the gel per monomer unit in the system is the difference between the number of all reacted functionalities and reacted functionalities in the sol. This difference is divided by the number of monomer units in the gel (cf. eqn (10)-(12)) : By using pgf (12) and the definition equation (11) for u, the numerator may be rearranged to become Only monomer units with more than two active reacted functionalities contribute to the number of EANC; each such bond contributes to the number of EANC by 3, so that We obtain from eqn (A2) and (A3) after substitution into eqn (8) and rearrangements, which is just the average number of EANC issuing from one elasticaly-active monomer unit, i.e., those which gives rise to three and more EANCs.(6) ALTERNATING COPOLYCONDENSATION OF MONOMER 1 A N D 2 (Ng = f1a1,n1,+f2~2gn20, 0 < i < fl, 0 d j < f2). The fraction of monomer units 1 in the gel is given by the relationship, because the gel contains units having at least one active reacted functionality. The number of reacted functionalities of type 1 per monomer unit in the gel is obtained similarly to the case of homopolycondensation :108 so that PARAMETERS OF THE SWELLING EQUATION and This again is the average number of EANC issuing from one elastically-active monomer unit. (c) RANDOM CROSSLINKING OF PRIMARY CHAINS (N, = VeFg, 0 < i < 00) In this case the pgf& and Timplicitly include the distribution of the degrees of polymer- ization of the primary chains (cf.ref. (8)). The number average degree of polymerization of the primary chains in the gel is 7;0 = wgrn/(C t i ) (A111 1 (wg is the weight fraction of the gel and r,, is the number average degree of polymerization of the primary chains), since the numerator is the number of segments in the gel and the de- nominator is the number of particles in the gel per primary chain in the system. The fraction of crosslinked segments in the gel is obtained by simiIar reasonings as for po 1 ycondensa t ion vg = (C i t i + tl)/rnwg* (A 12) VeFg = Ne/(C ti> 6413) 1 The product vei, is the number of EANC per primary chain in the gel, so that if N, is the number of EANC per primary chain in the system. The calculation of Ne com- pared to polycondensation is somewhat different, since the EANC are considered to be also segment sequences within the primary chain between active crosslinks.The procedure has been explained in ref. (8) or ref. (4) (N, was designated there by pe) : On substitution into eqn (8) and rearrangement we obtain fe = 4(C i t i + C ti)/(C i4+2 C t i ) . 2 2 2 2 It is possible to prove thatf, has again the same meaning as for polycondensation. We consider a primary chain having i active crosslinked segments (ACU) (from which a sequence of segments leads to infinity), of which i-2 are inner and 2 are end segments. If an inner ACU is connected with an end ACU, the crosslink is effectively trifunctional, similarly to the case when an end ACU is connected with an inner ACU. Connection between two inner ACU gives an effectively tetrafunctional crosslink. Their fractionK. DUSEK 109 and the fraction of effectively trifunctional crosslinks is It is evident that is identical with eqn (Al5). K. DuSek, J. Polymer Sci. (Polymer Symp.), 1972, 39, 83. K. DuSek, J. Polymer Sci. (Polymer Phys.), 1974, 12, 1089. F. T. Wall and P. J. Flory, J. Chem. Phys., 1951, 19, 1435. K. DuSek, J. Polymer Sci. (Polymer Symp.), 1973, 42, 701. P. J. Flory, J. Chem. Phys., 1950, 18, 108. M. Gordon, Proc. Roy. SOC. A, 1962,268,240. G. R. Dobson and M. Gordon, J. Chem. Phys., 1965, 43, 705. M. Gordon and G. R. Scantlebury, Trans. Faraday Soc., 1964, 60, 604. M. Gordon, T. C . Ward, and R. S . Whitney, Polymer Networks. Structure and Mechanical Properties, ed. A. J. Chompff and S. Newman (Plenum Press, New York, 1971), vol. 1. ’ M. Gordon and G. N. Malcolm, Proc. Roy. Soc. A, 1966,295,29. lo M. Gordon and G. R. Scantlebury, J. Polymer Sci. C, 1968, 16, 3933. l 2 K. DuSek and M. Ilavsky, J. Polymer Sci. (Polymer Symp.), in press. l3 J. E. Mark, J. Amer. Chem. Soc., 1970,92,7252. l4 R. M. Johnson and J. E. Mark, Macromolecules, 1972, 5, 41.