New approach to the theory of spinodal decomposition

A new approach to the study of spinodal decomposition for a scalar field is proposed. The approach is based on treating this process as a relaxation of the one-time correlation function G(q,t)=∫d r<Φ (0, t)Φ (r,t)>exp(i q·r), which plays the role of an independent dynamical object (a unique two-point order parameter). The dynamical equation for G(q,t) (the Langevin equation in correlation-function space) is solved exactly in the one-loop approximation, which is the zeroth approximation in the approach proposed. This makes it possible to trace the asymptotic behavior of G(q,t) at long and intermediate times t (from the moment of onset of the spinodal decomposition). The values obtained for the power-law growth exponents for the height and position of the peak in G(q,t) at the intermediate stage is in satisfactory agreement with the data obtained by a number of authors through numerical simulation of the corresponding stochastic equations describing the relaxation of the local order parameter. Pis’ma Zh. éksp. Teor. Fiz. 66, No. 6, 432–437 (25 September 1997)     

Macro-and microphase separation in nonaqueous solutions of binary multiblock copolymers

In terms of the random phase approximation, a global analysis of the thermodynamic stability of solutions of multiblock copolymers (A_nB_m)_k and (A_nB_m)_kA_n was performed. Phase diagrams for various parameters of the copolymer structure including the macrophase and microphase separation regions were obtained. Critical lines near which this approximation still holds and the microphase separation corresponds to the weak segregation mode were also constructed. The formation of ordered structures with a period of the order of a wavelength of visible light becomes possible under certain conditions. It was shown that the homogeneous state of solutions of multiblock copolymers in certain narrow ranges of values of the parameters breaks down through a simultaneous growth of fluctuations with substantially different wavelengths, a phenomenon which must lead to the appearance of structures ordered at two length scales. The width of such regions increases with an increase in the number of blocks k and a decrease in the degree of polymerization n + m of a block. The potential use of these multiblock-copolymer solutions as photonic crystals is discussed.     

The Influence of Ion Pair Formation on the Phase Behavior of Polyelectrolyte Solutions

In this paper we calculate the phase diagrams of polyelectrolyte solutions taking into account the possibility of ion pair formation between counterions and ions on polymer chains with their subsequent aggregation in multiplets. Multiplets play a role of reversible physical crosslinks between different polymer chains. Furthermore, we assume that the dielectric constant, and hence the binding energy of ion pairs and multiplets, depends on the volume fraction of polymer in the solution. We have shown that ion pairing and multiplet formation leads to instability of the homogeneous phase, so that at some conditions the phase separation can proceed even in good solvent. Another qualitatively new feature of the behavior of polyelectrolyte solutions is the emergence of a triple point on the phase diagram. The effect of the solvent polarity and the polarity of the dry polymer on the general form of the phase diagram is studied in detail.     

Non‐linear dynamics of spinodal decomposition

A new approach is proposed to describe the spinodal decomposition, in particular, in polymer binary blends. In the framework of this approach, the spinodal decomposition is described as a relaxation of one‐time structure factor S(q,t) treated as an independent dynamic object (a peculiar two‐point order parameter). The dynamic equation for S(q,t), including the explicit expression for the corresponding effective kinetic coefficient, is derived. In the first approximation this equation is identical to the Langer equation. We first solved it both in terms of higher transcendental functions and numerically. The asymptotic behaviour of S(q,t) at large (from the onset of spinodal decomposition) times is analytically described. The values obtained for the power‐law growth exponent for the large‐time peak value and position of S(q,t) are in good agreement with experimental data and results of numerical integration of the Cahn‐Hilliard equation.     

Spontaneous breaking of molecular identity and phase diagrams of thermally reversible association systems with alternating molecules

The global phase behavior of a mixture of molecules A _f and B _f, each containing f functional groups of, respectively, types A and B capable of forming thermally reversible chemical bonds, is considered. Contrary to the traditional approach based on the consideration of an infinite cluster of labile bonds that appears in such systems on the Bethe lattice (i.e., in the Cayley tree approximation) under certain conditions, we additionally take into account the contribution to the thermodynamics from the cluster fragments forming mesoscopic cycles. It is shown, within the framework of the suggested mesoscopic cyclization approximation, which is based on the concept of spontaneous breaking of molecular identity upon the formation of an infinite cluster, that this contribution is finite. Phase diagrams are constructed for the systems considered. The presence of a point of equal concentrations, where two liquid phases coexist and one of them contains an infinite cluster of thermally reversible bonds, is the specific feature of the phase diagrams in the approximation suggested.     

The ideal polymer chain near planar hard wall beyond the Dirichlet boundary conditions

We present a new ab initio approach to describe the statistical behavior of long ideal polymer chains near a plane hard wall. Forbidding the solid half-space to the polymer explicitly (by the use of Mayer functions) without any other requirement, we derive and solve an exact integral equation for the partition function G _D(r,r′, N) of the ideal chain consisting of N bonds with the ends fixed at the points r and r′ . The expression for G(r,r′, s) is found to be the sum of the commonly accepted Dirichlet result G _D(r,r′, N) = G ₀(r,r′, N) - G ₀(r,r”, N) , where r” is the mirror image of r′ , and a correction. Even though the correction is small for long chains, it provides a non-zero value of the monomer density at the very wall for finite chains, which is consistent with the pressure balance through the depletion layer (so-called wall or contact theorem). A significant correction to the density profile (of magnitude 1/is obtained away from the wall within one coil radius. Implications of the presented approach for other polymer-colloid problems are discussed.     

Statistics of ideal randomly branched polymers in a semi-space

We investigate the statistical properties of a randomly branched 3-functional N-link polymer chain without excluded volume, whose one point is fixed at the distance d from the impenetrable surface in a 3-dimensional space. Exactly solving the Dyson-type equation for the partition function Z(N, d )= N^-θe^γN in 3D, we find the “surface” critical exponent θ = , as well as the density profiles of 3-functional units and of dead ends. Our approach enables to compute also the pairwise correlation function of a randomly branched polymer in a 3D semi-space.     

Incorporation of a cluster into a lattice of point charges in the SCF-CI formalism (ideal crystal)

Urals Polytechnic Institute. Translated from Zhurnal Strukturnoi Khimii, Vol. 32, No. 4, pp. 17–21, July–August 1991.