Scattered intensity by a cross-linked polymer blend

zy zyxwvut zyxwvuts zyxwvu zyxwvu zyxwvu Macromol. Theory Simul. 4, 67- 76 (1995) 67 Scattered intensity by a cross-linked polymer blend Amina Bettachy, Abdelali Derouiche, Mabrouk Benhamou Universite de Casablanca, Laboratoire de Physique des Liquides et Polymeres, B. P. 6621, Ben Msick, Casablanca, Maroc zyxwvu Mustapha Benmouna, Thomas A . Vilgis Max Planck Institut fur Polymerforschung, Postfach 3448, D-55021 Mainz, Germany Mohamed Daoud * Laboratoire Leon Brillouin (C. E. A.-C. N. R. S.), C. E. Saclay, 91 191 Gif/Yvette Cedex, France (Received: April 19, 1994; revised manuscript of June 20, 1994) SUMMARY: Cross-linked mixtures of polystyrene and poly(viny1methyl ether) exhibit a non-vanishing zeroangle intensity in small-angle neutron scattering experiments. A possible explanation is that fluctuations in composition in the mixture may be frozen by the presence of cross-links. Assuming this, we introduce a screening length K by the condition that the scattered intensity should not be changed by cross-linking. We find K’ C/k-xi), where C is an elastic constant, and xi, respectively, the inverse temperature and that where cross-linking is performed. When the temperature is varied, we find three regimes. In the first one, the scattered intensity is monotonously decreasing. In the second one, it has a finite maximum. In the last one, the maximum eventually diverges. - zy x Introduction A model system for interprenetrating networks I * * ) was proposed by de Gennes3) some years ago. He considered a mixture of two chemically different polymers that is cross-linked in the one-phase region of the phase diagram, and then brought to the immiscibility region by a change in [email protected] the system is crosslinked, there is then a competition between the monomer-monomer repulsion and the elastic forces that resist the phase separation. There results a microphase separation, where finite domains rich in each of the components are present. Renewed interest in this model arose recently when Briber and Bauer7) found an actual mixture of polymers, namely polystyrene and poly(viny1 methyl ether) (PWPVME), that is miscible at room temperature and may be crosslinked in a controlled way by irradiation. It was then possible to test de Gennes’ predictions about microphase separation by small-angle neutron scattering experiments. The general conclusion was that there is an acceptable agreement between the experimental results and theory. There was however one notable exception concerning the zero-angle scattering. The latter was assumed to be vanishing in the calculations, whereas it was definitely non-vanishing and rather large in the experiments. The purpose of the present paper is to discuss this 0 1995, Huthig & Wepf Verlag, Zug CCC 1022-1344/95/$05.00 68 zyxwvutsrq A. Bettachy, A. Derouiche, M. Benhamou, M. Benmouna, T. Vilgis, M. Daoud discrepancy, to suggest an alternative form for the scattered intensity and to study its consequences. The basic difference between actual systems and the de Gennes model resides, in our opinion, in the fact that he assumed that no fluctuations are present in the initial mixture, and that there is perfect mixing of both components. In an actual blend such as the PS/PVME system that was studied, however, fluctuations are present because one is not very far from phase separation. When the system is cross-linked, these fluctuations are frozen-in. Therefore for very small angles, one should still observe them. Thus we are led to assume that two temperatures are present in the system we are considering, and not only one as in de Gennes’ approach. These are the initial one where the cross-linking process is taking place, Ti, and the final one, at which the system is quenched after cross-linking. In this respect, de Gennes’ approach implicitly assumes that the initial temperature Tiis infinite. In the following, we will first recall de Gennes’ approach and its comparison with Briber’s and Bauer’s results. We will then turn to the more general case when cross-linking is made in a solution where concentration fluctuations are quenched by the cross-linking that is performed. We will argue then that cross-linking should not affect the scattering if no other parameter is changed. More precisely, we will argue that the scattered intensities by the solution and by the cross-linked gel are equal if the system is kept at a constant temperature. This assumption implies that no phase separation is induced by the procedure. zyxwv zyxwvu zyxwv zyxw zyxwvutsrq zyxw zyxwvu De Gennes’ limit Consider a mixture of two polymers of different chemical nature. For simplicity, we will assume that they are of equal length N a n d that there is a symmetric composition where the compositions @a and @b in both components are equal. The temperatureconcentration diagram of such a mixture usually has a large two-phase region, where demixing occurs between the two components. In the following, we assume that such demixing occurs at low temperatures. For high temperatures, both polymers mix in a one-phase region. For very large temperatures, one may assume, following de Gennes, that mixing is perfect and that no fluctuations are present in the system. Cross-linking is performed on this mixture at random, and we assume that the resulting gel is rather tightly bound: Let n be the average chemical distance (i. e., along the contour of the chain) between two consecutive cross-links. We assume it to be much smaller than the length N of the polymers. The free energy of the system is F 1 = --(@In@ N + (1 - @)ln(l - @)I + x @ . ( l - @) + C P z (1) where the first two terms are the ideal gas contribution, the third one the interaction between monomer units and the last one an elastic contribution to be discussed below. @isthe monomer concentration in one of the species, x the Flory interaction parameter and is inversely proportional to temperature. C is an elastic constant that was shown to be inversely proportional to nz by de Gennes, and P the displacement of the center of masses due to the incipient phase separation zy zyxw zyxwvut zyxwvutsrq zyxwvuts zyxwvu zyxwvu zy 69 Scattered intensity by a cross-linked polymer blend where the sum is extended to all monomers i a n d j belonging, respectively, to polymers A and B. In the one-phase region, the centers of masses of A’s and B’s are identical. When phase separation starts, they tend to separate from each other, and the elastic term appears because cross-linking resists this. De Gennes made an equivalence between this and an electrostatic problem, where charges of opposite sign would be on the polymers. In this analogy, polarization is equivalent to P, and it is related to the density fluctuation by div P (3) = -p where p is the density fluctuation, p ( r ) = @ ( r ) - 4 where 6 is the average composition. Inserting relation (3) into Eq. (l), Fourier-transforming, and keeping only the fluctuating part gives (4) where xc - 2 / N is the critical value of the interaction parameter for the uncrosslinked mixture and pp is the Fourier transform of p ( r ) . Note that we also included a gradient term in relation (4) in order to take into account the spatial variation of @. Relation (4) has several implications. The first one is that the scattered intensity vanishes for q = 0, and decreases as q-’ for large values of q. It exhibits a maximum, as shown in Fig. 1 a, for a value q* such that This maximum diverges for a value x s such that Note that the scattered intensity in a radiation scattering experiment vanishes as q goes to zero as S(q-0) q2 -C (7) Finally, a second length may be found in this problem, related to the width of the scattering curve. Expanding S ( q ) in the vicinity of q*, one finds ( S ( q ) = S ( q * ) [l (q - q*)2(’1 with - * This length is identical to q*-’ for C f 01, - x ) , and crosses over to 01, - x)-’/’ in the opposite limit, far from the critical point of the binary mixture. As we are 70 zyxwvutsrq A. Bettachy, A. Derouiche, M. Benhamou, M. Benmouna, T. Vilgis, M. Daoud zyxwv zyxw interested in the vicinity of the spinodal however, only the first of these two cases, namely the existence of a single length is valid, as predicted by de Gennes’). Recently, Bauer and Briber’) found an actual system, namely a mixture of polystyrene (PS) and poly(viny1methyl ether) (PVME) that mixes at room temperature and demixes at higher temperatures, around 60 “C. Furthermore this mixture may be crosslinked by irradiation. Thus they could check de Gennes’ ideas. Assuming that the density of cross-links is proportional to the radiation dose, they found a good agreement between their experimental results and the above predictions. There was however one notable deviation concerning relation (7). Whereas the zero-angle intensity was predicted to vanish, it was found to have a non-zero value. We believe that the reason for this lies in the fact that it was implicitly assumed that no fluctuactions in composition were present when cross-linking is performed: so far, the only fluctuations present in the system are those that appear when it is quenched to its final temperature. In the next section, we will assume that at the initial inverse temperature, xi, fluctuations are already present. Initially fluctuating mixture z The previous section considered the case when no fluctuations are present in the initial mixture. This implies that the temperature where cross-linking is made is very far from any critical temperature, or that it is very high. We will now assume that crosslinking is performed at a finite temperature, on a blend where fluctuations are present. These may be related to the fact that such a system usually has a coexistence region in its phase diagram, and that the critical temperature is not too far from room temperature, or at least that critical effects are still present in the uncross-linked mixture. We assume that cross-linking is made at a first temperature Ti x ;’, and that the system is subsequently brought to the final temperature T x - I . We argue that because of the cross-linking, long-range fluctuations corresponding to the initial temperature are trapped. As a result, the zero-angle scattering deviates from zero. We expect therefore this deviation to become more and more important as xi becomes closer to the critical temperature T, x;‘. In other words whereas de Gennes’ argument includes only one temperature, namely the one where the system is quenched, we assume that two temperatures are present because of the presence at the final temperature of frozen fluctuations from the initial state. In this section, we estimate the scattered intensity by assuming the existence of an extra inverse length K that was already discussed previously*). More precisely, we assume that the scattered intensity S ( q ) of the cross-linked blend has the following form - zy zy - - where K was interpreted as an inverse screening length. The form of Eq. (9) is necessitated by the experimental observation that the zero-angle intensity does not vanish. In order to determine this length, we assume that the scattering at zero angle is not changed by the cross-linking procedure. Moreover, because the existing longrange fluctuations are frozen by the cross-links, we will assume that the scattered zyxwv zy zyx zyxwvuts zyxwvu zyxwvutsrqp Scattered intensity by a cross-linked polymer blend 71 intensity at zero angle remains unchanged for all temperatures after cross-linking has taken place. We are thus led to identify the scattered intensities at q = 0 by the uncrosslinked mixture at the temperature Ti and by the cross-linked system at a lower temperature T. The former is given by a relation similar to Eq. (9), with C = 0. This leads to: K2 C - (10) I x - xi We notee that l / f l and hence the corresponding length K-' is inversely proportional to the average distance between crosslinks which, for loosely cross-linked mixtures, corresponds to the radius of the initial chains, as already pointed out in our previous work *). We expect this relation to hold only when the corresponding length is smaller than the correlation length ti of the initial mixture zy In the opposite case, we expect to recover de Gennes' results that the zero-angle scattered intensity becomes negligible. A similar result may be obtained if we apply a self-consistent approach similar to the Debye-Hiickel theory for electrolytes 3*g-11). This may be done by considering the monomer-monomer pair correlation function T(r - J ) for the actual system and calculating it using the correlation function g ( r - r') for the uncrosslinked one. In this approximation we have T(r - J ) = g(r - J) zyxwv zyxwv + 5 g ( r - r") V(J' - J") T ( f " - J ) dJ' dJ" (12) where V ( r ) is a potential corresponding to cross-linking, leading - in Fourier space - to wheref(q), S(q) and So@) are, respectively, the Fourier transforms of V ( r ) ,T ( r ) and and is the correlation funtion for the uncross-linked mixture. The self-consistent equation for S ( q ) in terms of S o ( q ) has the following form where the minus sign is due to the fact that the integration is repulsive. This may also be written in the following equivalent form 72 zyxwvuts A. Bettachy, A. Derouiche, M. Benhamou, M. Benmouna, T. Vilgis, M. Daoud zyxw zyxwvutsr where the potentialf(q) may be obtained by applying a constraint to the system. In de Gennes’ case for instance, this is the condition that the zero-angle scattering vanishes because of the electroneutrality condition. The latter is related to the electrostatic equivalence. Assuming that for small wavevectors we have S ( q ) = q”c (17) we findf(q) = C/qz,and de Gennes’ result for the scattered intensity: zyxw z zyxwv A shortcoming of the previous relation is that it leads to a divergent component in the free energy for zero wavevector. Moreover, this is equivalent to an incompressibility condition for the binary mixture. Whereas this is justified when one considers a system that is far from its critical point, it becomes less and less true as the blend approaches its critical point. Because of these reasons, one is led to introduce a cut-off length, and to consider the following form for the scattered intensity S-’(q) = xc - x + 42 C +q2 + Kz where the inverse cut-off length K is to be determined. Comparing relations (14), (15) and (19), we obtain We estimate K by setting the condition that the scattered intensity by the cross-linked blend for q = 0 at any temperature is equal to the one scattered by the uncross-linked mixture at the cross-linking temperature. As discussed above, the reason for this is that we assume that the existing long wavelength fluctuations are quenched by the crosslinks, and should be observable at any temperature below the initial one. Using relations (14) and (19), we find relation (10). The shape of the scattered intensity curve depends stronlgy on the relative values of x and xi. Fig. 1b shows the various cases as the temperature is varied. Three different cases are found, as x is varied. For large values of x, S ( q ) has a maximum and eventually diverges, whereas for small values it is a continuously decreasing function. For intermediate values of x, as the location of the maximum vanishes, we find that S ( q ) goes through a maximum without diverging. For large values of x, the scattering curve exhibits a maximum for a finite value q* of the wavevector. The latter is smaller than the one evaluated by de Gennes. Whereas the latter is qd fC, we find in the present case - zy zyxwv zyxwvutsrq zyxw zy zyxw zyx 73 Scattered intensity by a cross-linked polymer blend q*2 - $- K2 (21) with S-'(q*) - Xc - X + 2 $ (22) - K2 zyxw When temperature is varied, the maximum of the scattered intensity diverges for a special temperature xs obtained by solving S - ' ( q * ) =O. This corresponds to the spinodal and to the appearance of an instability. - w Q- 0.50 - _-- ' f=l, (b) K2=1 : f=3/2.K2=2 : f:-0,5.KZ:0.L ........... 0.25 0 zyxwvu zyxwvu I 0 I 1 I I I I 8 12 0' 0 , I I 1 I I 2 L 6 8 10 12 q2 9 Fig. 1. (a): Comparison of de Gennes results (- - -) for the scattered intensity - as a function of wavevector transfer q - and the present ones (-). The former goes to zero for vanishing q while the latter remains finite. (b): The scattered intensity S ( q ) as a function of wavevector transfer q. Various cases are shown, with or without a maximum, depending on the interaction parameter x. f is the reduced temperaturex c - x . S ( q ) is dimensionlessand q is expressed in u-' units, where u is the segment length The location q* of the maximum vanishes for C f interaction such that = K ~ that , is for a value xxof the For smaller values of x , the scattering curve does not show a maximum any longer, and because of our condition that the scattered intensity a t zero wavevector is fixed, there is n o more any divergence of the scattered intensity. Therefore for lower values of x, we d o not expect any spinodal to appear, except for xi xc, when the uncrosslinked system is at criticality. Conversely, for a fixed value of xi, the cross-linked mixture has a maximum in the scattered intensity only if C is smaller than a special value C, 01 - xi)2.For values of C larger than C,, n o maximum is present. As C is related to the chemical distance n between successive cross-links o n a chain, this implies that the mixture has to be rather weakly cross-linked in order to have a maximum. In other words, if the system is too tightly cross-linked, no maximum is present, and this implies that the mixture has become miscible. - - zyxwvuts zy zyxwvuts zyxwvuts zyx zyxwv zyxwvu A. Bettachy, A. Derouiche, M. Benhamou, M. Benmouna, T. Vilgis, M. Daoud 14 When a maximum is present, as in the de Gennes case above, one may expand S ( q ) in the vicinity of its maximum. We find with t2 - and 4 *2 IrC.Ol, - x + IrC + P2) < where the width corresponds, as in de Gennes' case, to a coherence length of the system. Note that the third-order term is present so that the curve is not symmetric in the vicinity of the maximum. The relative values of q* and < - I may be determined. It may be shown easily that we have < - I fl 4 q*2. It is interesting to look at the behavior of the maximum scattered intensity as q* vanishes. The reason is that we found that there may be a divergence as a function of temperature when q* is non zero, whereas we imposed a constraint that S ( 0 ) does not diverge. Expanding q*, and S - ' (q*), relatioils (24) and (25), in the vicinity of the inverse temperature xxwhere q* vanishes, we find + and zyx where E = x - xi - C f was assumed to be small. This implies that q* has to be larger than a special value q$ in order to make S - ' ( q * ) vanish. A very rough approximation is obtained by using relations (27). We obtain G2 - IrC.[xi - XI and zyxwvut for the value of the interaction parameter when this occurs. In the interval between xx and xo where a maximum exists without the existence of a divergence in S ( q ) , the inverse scattered intensity at the maximum varies as zy zyxwv zyxwvutsrq zyxwvuts Scattered intensity by a cross-linked polymer blend 15 These results are summarized in the diagram of Fig. 2. Fig. 2. The various regimes in a &, @) plane. In region I, for low values of ,y, no maximum is present. In region 11, a maximum is present but the scattered intensity does not diverge. Regime 111 covers the case of the presence of a maximum and a divergence corresponding to an instability Conclusions In order to take into account the eventual fluctuations that are present in the initial mixture and are trapped by the cross-links, we introduced an inverse screening length K by imposing to the system the constraint that the scattered intensity at zero angle is a constant, independent of the cross-linking state of the blend. This implies that two temperatures are present, instead of one as was formerly assumed. One possible way of interpreting this length is as follows: The free energy F of the cross-linked mixture may be written as the sum of that of the uncross-linked blend at the initial temperature Fi = (ti + q2)and an extra term, corresponding both to the change in tempera9 zy zyxwvutsrqpo zyxwvu zyxwvuts ture and to the elastic effects due to cross-linking: F = Fi + [x - xi + C/(q2 + K2)] 9 When q is much smaller than K , it may be neglected in the sum, and because of the value of K , the difference in free energies between both systems may be neglected. Therefore K may be considered as the inverse distance where the fluctuations in the cross-linked system are those of the initial uncross-linked one. In other words, this corresponds to the case where the elastic contribution is dominated by the frozen fluctuations of the initial blend for large distances. As a result, we find several possible regimes, depending on temperature T, or the interaction parameter x, which is assumed to be inversely proportional to 7: For x smaller than xx,the scattering curve exhibits a maximum for a value q* that we find smaller than in de Gennes’ case. This maximum eventually diverges at the spinodal. Between xx and xo, the curve has a maximum, but no divergence may be found. Finally, for x larger than xo, no maximum is present. These results are in rough qualitative agreement with recent neutron scattering results on interpenetrating networks j2). zyxwvu 76 zyxwvutsrq zyxwvu zyxwvut zyxwvuts zyxwvu zyxwv zyxwv zy A. Bettachy, A. Derouiche, M. Benhamou, M. Benmouna, T. Vilgis, M. Daoud M. Daoud and M. Benhamou thank the University of Casablancaand L. L. B. for their warm hospitality during their visits. )’ 3, 4, 6, ’) *) 9, lo) ‘I) 12) L. H. Sperling,“Znterpenetrating polymer networks and related materials’: Plenum Press, New York 1981 H. L. Frisch, D. 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