Symmetric blends

Figure 36. The scaled distributions of mean, P(H/Y ) (a), and Gaussian, P(K/Y]2) (b), curvatures scaled with the inteface area density, computed at several time intervals of the spindal decomposition of a symmetric blend. There is no scaling at the late times because the amplitude of the thermal undulations does not depend on the average growth of the domains, and therefore the scaled curvature distributions functions broaden with rescaled time.

In Fig. 41 we plot the minority phase volume fraction, fm, versus the Euler characteristic density for a large number of simulation runs performed at different quench conditions. For the symmetric blends (< )0 = 0.5), fm = 0.5 and is independent of time and XEuier/ is always negative. For the asymmetric blends, fm decreases with time and xEu cr/F may change the sign. We have not observed the bicontinuous morphology for fm < 0.29, nor have we observed the droplet morphology for fm > 0.31. This observation suggests that the percolation occurs at fm = 0.3 0.01 and that the percolation threshold is not very sensitive to the quench conditions (noise intensity). 

Rigby et al. (1985) showed using a Flory-Huggins model that for symmetric blends, the spinodal and critical temperatures decrease linearly with increasing content of a symmetric diblock for blends with equal volume fractions of homopolymers (with the same molecular weight). The condition for a linear decrease of the binodal was less restrictive, not requiring equal concentrations of homopolymer in the blend. 

As an example, consider a uniform parent distribution p(°) (a) = const, which can be written as p (o) = p /2, using the fact that pj, = do p (a). Then pf 1 — pff / 3 and p =p /5 and so a tricritical point occurs if the overall density (i.e., the copolymer volume fraction) is pf = 3/(2r + 3). Figure 13 shows the coexistence curve calculated for this parent (with r = 1), which clearly shows the tricritical point at the predicted value X1 = l/(2/4° ) = r + 3/2 = 2.5. Our numerical implementation manages to locate the tricritical point and follow the three coexisting phases without problems we take that as a signature of its robustness [58]. Note that the tricritical point that we found is closely analogous to that studied by Leibler [57] for a symmetric blend of two homopolymers and a symmetric random copolymer that is, nonetheless, chemically monodisperse (in the sense that o = 0 for all copolymers present). In fact, in our notation, the scenario of Ref. 57 simply corresponds to a parent density of the form p (o) S(o — 1) + S(o +1), with the copolymer (o = 0) now playing the role of the neutral solvent. 

The following problems have been studied experimentally using different SFM techniques (i) the surface topography and its dependence on the preparation conditions, (ii) the chemical composition of the surface, and (iii) the role of the substrate structure in polymer demixing. Here, one has to distinguish between asymmetric blends, where one of the components concentrates at the surface and the other at the substrate interface, and symmetric blends where one of the components segregate at both interfaces. 

Figure 14 exemplifies two computational methods to determine the probability distribution of composition for binary polymer blends described by the bond fluctuation model [67]. Phase coexistence can be extracted from these data via the equal-weight rule. For the specific example of a symmetric blend, the coexistence value of the exchange chemical potential, A/u, is dictated by the S3munetry. One can simply simulate at A oex = 0 and monitor the composition. Nevertheless, the probability distribution contains additional information, as discussed in Sect. 3.5. 

Within mean field theory, for a symmetric blend the excess free energy of mixing per volume is given by the Flory-Huggins expression ... 

In the insets of Fig. 15 we show binodal curves for the symmetric blend. Again, we And deviations in the immediate vicinity of the critical point but for larger incompatibilities, xN 2, the mean held predictions provide an adequate description of the phase boundary utilizing the Flory-Huggins parameter extracted from the composition fluctuations in the one-phase region, xN < 2. 

The above equation can be solved for the interaction parameter corresponding to the phase boundary—the binodal (solid line in the bottom part of Fig. 4.8) of a symmetric blend ... 

The overall free energy of Eq. (49) could be rewritten [60] for a symmetric blend (with NA=NB=N) ... 

Fig. 5.6 F/C intensity ratio in function of depth in symmetrical blend of fluoro a,(D-end-functional protonated polystyrene (a,(D-/ PS(Rf)2) and deuterated PS, measured by angle dependent XPS. Reproduced with authorization from [57]...

FIGURE 7 Morphologies in symmetric blends of PET (a) and PHBV (b) with ENR, the arrows mark ENR domains (Chan et al. 2011 permission by Whey). 

Fig. 2.18 A schematic of the phase diagram for liquid mixtures with the upper and lower critical solution temperature, UCST and LCST, respectively. The placement of the critical compositions at about 4>cr = 0.5 denotes that this is a symmetric blend (Ni = A2). In the general case, the phase diagram is qualitatively the same, but much less symmetric with respect to

Yun, S. I. Melnichenko, Y. B. Wignall, G. D., Small-Angle Scattering from Symmetric Blends of Poly(dimethylsiloxane) and Poly(eth)dmethylsiloxane). Polymer 2004,45, 7969-7977. 

In the vicinity of the critical point of a binary mixture one observes universal behavior, which mirrors the divergence of the correlation length of composition fluctuations. The universal behavior does not depend on the details of the system but only on the dimensionahty of space and the type of order parameter. Therefore, binary polymer blends fall into the same imiversality class as mixtures of small molecules, metalHc alloys, or the three-dimensional Ising model, hi the vicinity of the critical point, Xc = 2 for a symmetric blend [ 14], the difference of the composition of the two coexisting phases—the order parameter m—vanishes like m - XcN), where the critical exponent... 

To illustrate the application of the method to study the dynamics of collective composition fluctuations, we now consider the spontaneous phase separation (i.e., spinodal decomposition) that ensues after a quench into the miscibility gap at the critical composition, a = b = 1/2, of the symmetric blend. Different time regimes can be distinguished During the early stages, composition fluctuations are amphfied. Fourier modes with a wavevector q smaller... 

Curro and Schweizer have carried out numerical [60,61] and analytical [23] studies of the symmetric blend using the Mean Spherical Approximation (MSA) closure successfully employed for atomic, colloidal, and small molecule fluids [5,6]. This closure corresponds to the approximation ... 

Fig. 12. Numerical predictions of PRISM/R-MPY for the variation with composition of the symmetric blend incompressible chi-parameter (normalized by its values for 0.5) 176]...

Fig. 13. Intermolecular structure of the < > = 0.5 symmetric blend close to the critical temperature [70]. The curve labeled g (r) is the corresponding homopolymer melt (infinite temperature) result The inset depicts a measure of the non-random packing expressed as Ag(r) = SaaW 8abW- The behavior of the latter ftinction is qualitatively different wten using the atomic MSA closure of the PRISM equations [23,61,68]. The radius-of-gyration is 2.91 for N = 32...

Fig. 14, Selected predictions of PRISM/R-MPY theory for the dependence of the symmetric blend g (r> on ccmiposition and total padcing fraction at fixed temperature [70]...

For the symmetric blend the reference system is the homopolymer melt and hence the reference correlation functions are independent of species label (M, M ). It is important to emphasize that Eq. (8.11) is not valid for polymers of nonzero hard core thickness. Thus, the thread idealization is a very special limit characterized by a unique simplification of the integral equation theory. These equations have the analytical structure of a high temperature and/or... 

Figures 6.2 and 6.3 show theoretical calculation of the phase diagram for a symmetric blend where both chains have the same length [1]. The solid lines show the binodal, the broken lines MST, and dotted lines the SP. MST and SP meet at the two symmetric points (indicated by LP) at which the two conditions (6.17) and (6.21) reduce to a single one.

For a symmetric blend, the conditions when the solution to the quadratic equation is one root is derived. This may denote the miscible blends. At this point the entropy of mixing can be seen to be zero. For a symmetric blend when jc ... 

The change in entropy of mixing at the glass transition temperature of the blend can be used to account for the observations. A quadratic expression for the mixed glass transition temperature is developed from the analysis. For a symmetric blend the conditions when the solution is one root denoting the miscible blends is derived. For a symmetric... 

The development of new molecular closure schemes was guided by analysis of the nature of the failure of the MSA closure. In particular, the analytic predictions derived by Schweizer and Curro for the renormalized chi parameter and critical temperature of a binary symmetric blend of linear polymeric fractals of mass fractal dimension embedded in a spatial dimension D are especially revealing. The key aspect of the mass fractal model is the scaling relation or growth law between polymer size and degree of polymerization Ny cr. The non-mean-field scaling, or chi-parameter renormalization, was shown to be directly correlated with the average number of close contacts between a pair of polymer fractals in D space dimensions N /R if the polymer and/or... 

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