Model Polystyrene Networks BY G. ALLEN, P. A. HOLMES AND D. J. WALSH Chemistry Dept., University of Manchester, Manchester MI 3 9PL Received 28th December, 1973 Model networks have been prepared, at different concentrations in an inert solvent, in such a way that the number of crosslinks is accurately known, and the topology of the network can be varied in a controlled way. The contribution of various network defects to the modulus of the net- work has been examined. A theoretical estimate of their effect has been made and compared to the experimental result. Two tentative conclusions have been made ; first, that the front factor A in the equation for the free energy of deformation of the network has a value of 3 and not the alternative value 1 ; and secondly that the contribution of entanglements to the modulus of the network varies as the concentra- tion squared rather than as the concentration to the first power.The various theories of rubber elasticity lead to a general expression for the free energy change on deformation of a network of the form : where v = the number of elastically effective network chains, k = the Boltzmann constant, T = the absolute temperature, Ax, Ay, ,Iz = the deformation ratios in the directions specified by the x, y and z axes and A and B are constants. Flory and Wall predict that A = 1 and B = +,1-4 whereas James and Guth 5-8 give a value of A which is dependent on the crosslinking process and for the random crosslinking of existing polymer chains is 3, and a value of B = 0. Edwards also obtains a result which is consistent with that of James and Guth, with A = 3, without assuming any crosslinking process.The values of the constants A and B differ for the various theories. The theory results in a value of G, the shear modulus, of G = AvkT = 2AnkT (2) where rz = the effective number of crosslinks per unit volume. It should also be pointed out that these equations would have to be corrected for the change in dimensions of the network chains if the network was prepared in the presence of a solvent and the concentration of this solvent changed before the modulus was determined. But as long as the network is prepared in solution at a high enough concentration for the chains to exhibit their unperturbed dimensions, and the modulus is measured at the same concentration, this can be ignored.Thus, by measuring the modulus of a network containing a known number of elastically effective chains it is, in principle, possible to determine the value of A. Previous attempts to measure the number of elastically effective chains have met two main problems, the difficulty of knowing precisely the number of chemical cross- links, and the difficulty of assessing the effect of network defects on the modulus. 1920 MODEL POLYSTYRENE NETWORKS The three commonly considered network defects are lo : (i) unreacted functionalities and free chain ends, (ii) closed loops, and (iii) entanglements. In a previous paper l1 we described a procedure for preparing networks of poly- styrene containing a known number of crosslinks, at various concentrations in an inert solvent.The moduli of the network were measured and the results were inter- preted in terms of the above network theory and the effect of network defects. The results were consistent with a value of A = 3. In this paper we report the preparation of similar networks to those prepared in the previous paper under conditions in which we have been able to alter the number, and hence effect, of network defects in a controlled manner. The results have again been compared to a theoretical prediction. EXPERIMENTAL 1. THE PREPARATION AND CHARACTERIZATlON OF THE PARENT SUBSTITUTED POLYSTYRENE Polystyrene containing a small, accurately known number of amine groups randomly placed along the polymer chain was obtained in the same way as in previous papers.11* l2 Polystyrene (of the desired molecular weight and distribution) in solution in carbon tetra- chloride was reacted with chloromethyl methyl ether in the presence of anhydrous stannic chloride, thus introducing chloromethyl groups into the para positions. Any required degree of substitution can be obtained by stopping the reaction after a given time.The pro- duct was extracted and purified by repeated precipitations from chloroform solution into methanol and then dried. The product was reacted with a primary amine (n-butylamine or isopropylamine) thus converting the chloromethyl sites into secondary amine groups. A small part of this aminated polystyrene was reacted with 1 fluoro, 2,4 dinitrobenzene and the degree of substitution determined from the ultra-violet spectra of the product. 2.SIMPLE NETWORK FORMATION Simple polystyrene networks were prepared in the same way as described in a previous paper.l1 The amine substituted polymer was dissolved in tetralin at a concentration to give a final concentration in the range 5-25 %. To the solution was added a known quantity of hexamethylene diisocyanate (calculated to be about 5 % less than the stoichiometrically required quantity) dissolved in a little tetralin, the mixture was stirred and poured into the required mould where gelation took place. Networks were prepared from polystyrene with greatly differing degrees of substitution, thus not only producing networks with different crosslink densities, but also with different percentages of the crosslinks wasted in closed-loop network defects.3. PREPARATION OF ISOCYANATE-SUBSTITUTED POLYSTYRENE Polystyrene containing a small number of isocyanate groups along the chain was prepared from the arnine substituted polymer. " Aminated " polystyrene (20 g) was dissolved in benzene (200cm3, dried over calcium hydride) and stoppered in a flask. Hexamethylene diisocyanate (50 to 100 times the required stoichiometric quantity) was dissolved in dry benzene (200 an3) in a stoppered flask. The two flasks and all the other requirements of the preparation were placed in a large dry-bag, which was then flushed with dry air. Inside the dry-bag the solution of polymer was slowly poured with stirring into the solution of diisocyanate and the mixture was allowed to stand for 1 h. The polymer was then pmipi- tated into isooctane (41., dried over calcium hydride), filtered, redissolved in benzene, re- precipitated, filtered and washed with dry isooctane.The polymer was broken into as fine lumps as possible and placed in a flask which was then transferred to a drying line where it was exhaustively dried under vacuum at a pressure of Torr of mercury.G. ALLEN, P. A. HOLMES AND D. J . WALSH 21 4. PREPARATION OF NETWORKS CONTAiNING NO LINKS FROM ONE PRIMARY CHAIN TO ITSELF By reaction of polymer containing amine groups with polymer containing isocyanate groups networks were formed which contained no simple closed loops. The isocyanate group containing polymer was dissolved at a suitable concentration (between 5 % and 25 %) in tetralin (dried over calcium hydride).Aminated polystyrene, of the same batch from which the isocyanate containing polymer had been made, was dissolved at the same con- centration in the same volume of the same solvent. The two solutions were quickly mixed and poured into a mould. Typically the mould was a 3 i in. crystallizing dish, into which was poured a total of 150 cm3 of solution. A most suitable polymer, optimized to keep the viscosity of the solution prior to cross- linking, and the chain end corrrtion of the resultant network, both to a minimum was prepared from polystyrene of M , = 100 0o0 and M,/Mn = 1.06. The use of isopropyl- amine in the amination reaction caused the gelation to proceed slower and allowed gels of higher concentration to be prepared. 5. MEASUREMENT OF EFFICIENCY OF REACTION The number of unreacted isocyanate groups remaining in the gel after completion of the reaction was determined by reacting them with carbon-14 labelled methanol.After each sample of gel was prepared and tested, an approximately 10 g part of it was broken into small lumps and to it were added tetralin (10 cm3) and C-14 labelled methanol (2 cm3 of a suitable activity). The mixture was then left to stand for a week to ensure com- pletion of reaction. The gel was separated from the excess liquid by decanting, and then extracted by putting it into a large excess of first methanol and then tetrahydrofuran, on alternate days for ten days ; finally it was dried, swollen in benzene, and freeze dried. Samples of the gel swollen in toluene and mixed with a " scintillation cocktail " of 0.3 % diphenyl oxazole and 0.03 % bis(5 phenyl oxazol-2-yl) benzene in xylene, were counted on a Packard Model 3320 Liquid Scintillation Spectrometer, and compared with a dilute solution of the original C-14 methanol and a blank.The efficiency of counting, found from a comparison of the counts of the two channels of the spectrometer was found not to be reduced by more than 1 % by the inhomogeneity of the gel sample. From the results, the number of free isocyanate groups remaining in the gel could be calculated, and a correction to the number of crosslinks made. This method of extracting excess C-14 methanol was found to be the best of several methods tried. Measurements were taken after successivechanges of solvent until a constant value was obtained, and 10 days extraction was generally found to be sufficient. 6.MEASUREMENT OF MODULI The moduli of the networks were determined in the same way as described in a previous paper. The method is based on the identation of the gel by a rigid sphere. A modified dial gauge measures the indentation of the gel by a ball bearing produced by a series of loads. The modulus G of a gel into which a sphere of radius R is caused to indent a distance d by a load P is found to be l3 CALCULATION AND RESULTS In order to compare our results with the various theories we must calculate the number of chemical crosslinks per unit volume, which for an ideal network is equal to n. This factor is given by22 MODEL POLYSTYRENE NETWORKS where N is Avogadro's number, c is the concentration of polymer in kg dm-3, N is the mean number of monomer units between crosslinks, K is a correction for the number of crosshks required to join the chains into an infinite network, given by 1.5 h b 3 1.0- 0.5 2 x i 0 4 X ~ an K = 1- where M,, is the number average molecular weight of the original polymer.- '-, - sample 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 polymer A A A A C B B D C D D E D E* E* E* E' E* E* TABLE 1 nkTl Ng concn./kg dm-3 N Nm-2 X lo4 G/Nm-2 X lo4 65 0.05 76 0.695 0.265 65 0.075 76 1.05 0.56 65 0.05 71 0.765 0.345 62 0.075 71 1.14 0.64 60 0.125 68 1.64 1.18 66 0.175 75 2.28 2.03 66 0.20 75 2.59 2.68 164 0.106 191 0.245 0.225 149 0.125 180 0.40 0.45 164 0.159 185 0.42 0.56 164 0.212 189 0.54 0.86 1 50 0.250 177 0.825 1.15 164 0.265 193 0.64 0.99 150 0.05 175 0.165 0.165 150 0.095 167 0.355 0.41 140 0.150 154 0.665 0.98 140 0.170 156 0.715 1.02 150 0.200 166 0.765 1.155 140 0.250 156 1.05 1.67 A Mw = 240 O00 M w / g n 1.6 (n-butylamine) €3 a, = 70 OOO Hw/Mn 1.06 (n-butylamine) C a, = 110 000 A Z w / g n 1.06 (isopropylamine) D Ew = 85 OOO @,/M, 1.20 (isopropylamine) E M , = 90 OOO H w / E f n 1.20 (isopropylamine) * polymers produced containing no simple closed loops.2.0} 1 - 1 , I 0 0.05 0.10 0.15 Ci.2? 0.25 0.30 concentration/kg d m 3 Glnkr 0.38 0.53 0.45 0.56 0.72 0.89 1.03 0.92 1.13 1.32 1.59 1.40 1.56 1 .00 1.17 1.45 1.42 1 S O 1.58 FIG. 1.-Experimental results ; a plot of GpkTagainst concentration. A-N, = 63 ; 0--samples with Ng = 157 ; 0- samples with Ng = 145 containing no closed loops.G.ALLEN, P . A . HOLMES AND D . J . WALSH 23 N is calculated from the amount of diisocyanate added and is thus 1.05 x N,, where No is the average number of monomer units between reactive groups on the polymer, in the ideal case. A correction to N is also made for incomplete reaction of isocyanate and incomplete transfer of crosslinking agent where applicable. In table 1 the results are presented for three series of gels ; the first has values of N, = 63 +3 ; the second has values of N, = 157+8 ; the third is a series prepared by the method used to ensure that no simple closed loops are formed. The poly- styrene from which they were prepared and the amine used to inti-oduce the amine groups are indicated. For each sample the measured modulus G is given and a value of nkT calculated.A value for G/nkT is calculated, and in fig. 1 we have plotted G/nkT against the concentration of the gel for each of the three series of gels. INTERPRETATION OF RESULTS The results obtained for the three series of gels are explained in terms of network defects. The two main types of network defects are closed loops, which will result in a reduced modulus, and entanglements which will result in an increased modulus. These may be represented as simple closed loop closed loop involving two chains entanglement The probability of formation of simple closed loops can be estimated from chain statistics in the manner of Kuhn,14 and Jacobson and Stockmayer,ls and the fraction of chemical crosslinks which will take the form of a simple closed loop can be shown to be l 1 K K + c where c is the concentration at which crosslinking occurs and K is a constant for the polynier system.P This expression can be generalized for any value of No to give P+cNZ' (7) The contribution of closed loops involving two chains is calculated in a similar way but the average loop size will be twice as big and hence is P P+c(2Ng)+' Using the arguments presented in a previous paper l1 we can show P to have a value of approximately 30. Larger loops involving three or more chains can also be considered, but their effect will become progressively smaller. The larger loops could also be partially elastically effective as indeed may be the smaller ones. Just considering the first two must therefore be considered an approximation.24 MODEL POLYSTYRENE NETWORKS The calculation of the effect of entanglements can be done in two alternative ways : (a) Assuming a constant molecular weight between entanglements (Me), which for polystyrene has been calculated as 35 O0O,l6 the contribution of entanglements to the modulus of the gels will be proportional to the concentration.This gives a final expression for the modulus : G = 2AnkT 1- 1- ( P + l N t ) ( or G P nkT - = 2A[( l- P+J(' - P + c(*N,)J+ 4 (9) where Q can be shown to be 3.3 x lo4. In fig. 2 this theoretical equation has been plotted for our three types of network assuming a value A = 3. For the network containing no simple closed loops the appropriate term is eliminated. 2 0 . 0 5 0.10 0.15 0 . 2 0 0.25 (n3C I 1 I I I concentration/kg dm-3 FIG.2.-A theoretical plot of G/nkT against concentration, based on eqn (10). - is a plot for N g = 63; - - - is a plot for N, = 157 ; - - - is a plot for Ng = 145 containing no closed loops. (b) Edwards1' has suggested that entanglements should be proportional to c and experiments on the dynamic modulus of swollen uncrosslinked rubber give a similar dependence.18 This would give an expression : or G P nkT R has been given a value of lo5 in order to produce a best fit with the results. Measurements of the modulus of swollen uncrosslinked natural rubber would predict a value of R between 2 x lo5 and 3 x lo5 but polystyrene chains would not be expected to give the same value. In fig. 3 this theoretical equation has again been plotted for our three types of network.The simple closed loop correction is again eliminated from the appropriate plot.G. ALLEN, P. A. HOLMES AND D. J . WALSH 25 In each series of gels, an average value of c/nkT has been used to calculate the en- tanglement contribution. This factor which should be constant for each value of Ng shows a slight variation due to scatter in N, and differences in chain end correction I i I I I I 9 C.05 0.10 0.15 3.20 0.25 0.30 concentration/kg dm-j N, = 157 ; - - - is a plot for N, = 145 containing no closed loops. FIG. 3.-A theoretical plot of G/nkT based on eqn (12). - is a plot for Ng = 63 ; - - - is a plot for arising from the use of different molecular weight polymers. This effect is, however, small, and only at the lower degrees of substitution where the relative contribution of chain entanglements is greater, could they account for a significant amount of scatter in the results.1. K = P/N? except that K is an approximation including both types of closed loop. 2, K' = QclnkT 3. K" = Rc'InkT. Q and R are also affected by using different units. The constants P, Q, R are related to the constants K, K' and K" of a previous paper by : CONCLUSION Previously we reported that two series of simple polystyrene gels having N, of 63 and 75 respectively gave results which implied that the factor A in eqn (1) tended towards 3 rather than 1. There was also tentative evidence that the contributions of physical entanglements to the observed modulus were more consistent with a c2 dependence rather than with a constant molecular weight between entanglements.In the present work we find additional support for both of these conclusions. A series of simple gels having N, = 145 and a series in which the probability of the formation of simple loops is very small also produce results which are consistent with GlnkT + 3. We have also made a more confident analysis of the effect of physical entangle- ments on the moduli of the gels. We find clearer evidence than previously reported that the c2 concentration dependence of chain entanglements gives an equation of the right general form though it must be stressed that eqn (12) is not an exact representa- tion. Apart from uncertainties introduced by experimental errors, we have made two major assumptions in the analysis of our results : (i) that physical entanglements are independent of the degree of crosslinking, i.e., of N,, and (ii) that closed loops do not contribute at all to the observed modulus.Neither of these assumptions may be strictly true and we have not been able to assess their physical significance.26 MODEL POLYSTYRENE NETWORKS It is also worthwhile noting that eqn (12) predicts a value of G/nkT greater than 2 for networks formed at 100 % concentration, which would explain why a value of A = 1 might appear to fit experimental results better for networks formed in this way, when network defects are ignored. We wish to thank Prof. S. F. Edwards and G. Gee for many helpful discussions. P. J Flory, Princfples of Polymer Chemistry (Cornell University Press, 1953). P. J. Flory, J. Chem. Phys., 1950, 18, 108, 112. F. T. Wall and P. J. Flory, J. Chem. Phys., 1951, 19, 1435. F. T. 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