Symmetrical mixture

Specializing to symmetrical mixtures (Na = Nb = N) for simplicity, it is easy to also find the order parameter i ocx of the unmixing transition... 

Recently efficient techniques were developed to simulate and analyze polymer mixtures with Nb/Na = k, k > I being an integer. Going beyond meanfield theory, an essential point of asymmetric systems is the coupling between fluctuations of the volume fraction (j) and the energy density u. This coupling may obscure the analysis of critical behavior in terms of the power laws, Eq. (7). However, it turns out that one can construct suitable linear combinations of ( ) and u that play the role of the order parameter i and energy density in the symmetrical mixture, ... 

Figure 17. Different stages of the spinodal decomposition in a symmetric mixture (4>0 = 0.5) r is the dimensionless time. The Euler characteristic is negative, which indicates that the surfaces are bicontinuous. The Euler characteristic increases with dimensionless time. This indicates that the surface connectivity decreases.

As already mentioned, in solutions that are not symmetric mixtures it is conventional to treat the solute B (or solutes B, C,...) differently from the treatment accorded to the solvent A. The activity coefficient is still defined as the ratio Ub/xb, but the hmit at which it equals unity is taken as the infinite dilute solution of B in A ... 

Fig. 31 Theoretical phase diagramm for a symmetric mixture of positive polyelectrolytes. s measures the effect of ionic strength and t that of the incompatibility...

Here b is the effective lattice spacing of the underlying Flory-Huggins lattice [199-206], and the coefficient k(( >) of the gradient energy term for a symmetric mixture (effective monomeric units have linear dimensions oA=oB=b, chain lengths are NA=NB=N) is [186,206,207]... 

Fig. 14 Normalized Dcoii/Bo of PDMS coated silica suspension with = 0.3 in a symmetric mixture of toluene and heptane (solid circles) along with hard sphere suspension (open squares) at similar volume fraction. The hydrodynamic interactions expressed in H(q) for the two systems (solid squares for the hard sphere suspension) are shown in the inset [101]. This system is crystallized by sedimentation as seen in the photograph...

This process is further illustrated by the plots in Fig. 4.15(b) where the moan dorusity J> of thermodynamically stable confined pha.ses is plotted as a function of z (i.c., the pore width). Three different branches arc discernible. For small z < 8, p is relatively high indicating that the pore is filled with liquid. A corresponding plot of the local densities of a representative phase for z = 5 shows that this liquid consists locally of A- (or B-)rich, high-density fluid (because the two cannot be distinguished in a symmetric mixture). Hence, for 2 < 8, we observe (local) decomposition of liquid mixtures. 

Other quantities can be obtained from suitable derivatives of the free energy or direct statistical mechanics averages e.g. in the semi-grandcanonical ensemble of a symmetrical mixture (NA = NB = N) (where the chemical potential difference Ap = pA — pB per effective monomer is a given independent thermodynamical variable) the order parameter defined as the relative excess in the number of A-chains (nA) over the number of B-chains (nB) in the system... 

For the simplest case of a symmetric mixture (NA = NB = N) this reduces to... 

As a first application for the use of the free energy functional, we will discuss the calculation of interfacial concentration profiles (x) between coexisting unmixed phases and interfacial tension [132-134]. For a symmetric mixture (Na = Nb = N) phase coexistence occurs for p = 0, and since the interfacial profile 4HX) also must be found by minimizing Eq. (47), we look for a solution of (cja = ctb = a)... 

Due to the restriction Eq. (48) which leads to q2a2 o) — 1 < 1. In this limit, the characteristic wavelength kc is much larger than the gyration radii of the polymer coils, i.e. in this regime the behavior of the polymer mixture is qualitatively the same as that of a mixture of small molecules. Using the full q-dependence of A(q) and Scon(q) in Eq. (77) this linearized theory of spinodal decomposition can be extended to deep quenches as well [78]. Here we quote the result for symmetrical mixtures (NA = NB = N, cta = crB = a) only. Then [78]... 

Similar Ginzburg criteria can also be worked out in the metastable region of the phase diagram (Figs. 1,2) The correlation length in the metastable state diverges when the spinodal curve is approached, see Eq. (39). For a symmetric mixture one obtains... 

While the study of fully symmetric mixtures with Monte Carlo methods is relatively simple due to the fact that the coexistence curve occurs for Ap = 0 by symmetry against interchange of A and B, the problem is more difficult when this symmetry is destroyed. Deutsch [93, 266] has considered asymmetry in the... 

Another very interesting problem concerns the concentration dependence of the effective Flory-Huggins parameter XeffM T) that can be extracted from small angle scattering data by fitting Eqs. (37), (46) or (55) to them. It has been found that Xrff T) for symmetric mixtures (such as mixtures of protonated and... 

Here we rather focus on effects of external surfaces (e.g. hard walls) on polymer blends. In general, one expects that the forces between the wall and monomers of type A will differ from those between the wall and monomers of type B, as it generally occurs at the surfaces of small-molecule mixtures as well [365]. For polymer mixtures that are partially compatible, the interactions in the bulk (as described by the Flory-Huggins x-parameter) must be relatively small, however, since the entropy of mixing is down by a factor of N (for simplicity, the following discussion is restricted to a symmetric mixture, Na = Nb = N). However, there is no reason that the difference of wall-A and wall-B forces is similarly small [125]. Thus one may expect rather pronounced surface enrichment effects in polymer mixtures [125], Indeed some experimental evidence for this prediction has been found [37, 38, 126, 127]. 

Fig. 55. Surface phase diagram in the plane of variables g, and <)> for three values of g. The region where the surface is non-wet (at small gj is separated from the wet region by a phase boundary which describes the wetting transition. For > (second-order wetting) this is just the straight line giril = — g(l—) The region of first order wetting is shown for symmetrical mixtures with Na = NB = N = 10 and N = 100, respectively, and the first-order transitions are shown by dash-dotted curves. In this regime metastable wet and non wet phases are possible up to the stability limits ( surface spinodals ) denoted by dashed curves. Assuming that g, and g are essentially independent of temperature T, variation of T essentially means variation of <)>, . From Schmidt and Binder [125],...

To access a pyrazine in this way one needs a 1,2-diamine and a 1,2-dicarbonyl compound, and a subsequent oxidation, but if neither component is symmetrical, mixtures are formed. The dimerisation of 2-aminocarbonyl compounds also generates symmetrically substituted dihydropyrazines - perhaps the best known examples of such dimerisations involve the natural amino acids and their esters which dimerise to give dihydropyrazine-2,5-diones - diketopiperazines . 

Fig. 12. Various regions in the temperature-composition plane near T. Since the situation for a symmetric mixture is symmetric around the line ( ) = only the regime is showm...

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