Late Stage Of Spinodal Decomposition Research Articles

Digital image analysis of droplet patterns in polymer systems: Point pattern

New techniques of digital image analysis which are especially suitable for analyzing particle-distribution patterns are presented. Statistical mathematical methods are applied to the quantitative analysis of a spatial distribution of points in polymer systems. The techniques are useful for studying the spatial patterns of particles or droplets which are commonly observed in nucleation-growth-type phase separation and late-stage spinodal decomposition in polymer mixtures, crystallization process, microphase separation in block copolymers, incompatible polymer alloys, and composite materials, etc. We can divide point patterns into three typical point patterns: the Poisson pattern, the clustered pattern, and the regular pattern. The kind and strength of interaction or force between points can be determined from spatial point-distribution patterns. The point-pattern analysis has been applied to phase-separated structures of a polymer mixture, and it has been revealed that the pattern belongs to the regular pattern. The appearance of the regular pattern is probably a result of the Brownian coalescence mechanism for the droplet growth and it might also be due to a long-range interaction among droplets through diffusion field.

Exact renormalization-group results for domain-growth scaling in spinodal decomposition.

Scaling in late-stage spinodal decomposition is discussed using standard renormalization-group techniques. The equal-time correlation function has the scaling form C(r,t)=f(r/${t}^{1/z}$), with z=d+2-y, where z and (-y) are, respectively, the dynamical exponent and thermal eigenvalue for the zero-temperature fixed point and d is the spatial dimensionality. For a scalar order parameter, y=d-1 gives z=3, implying a domain growth law L(t)\ensuremath{\sim}${t}^{1/3}$. For a vector order parameter, y=d-2 gives z=4 and L(t)\ensuremath{\sim}${t}^{1/4}$.

Monte Carlo renormalization-group study of spinodal decomposition: Scaling and growth.

The kinetics of domain growth during the late stages of spinodal decomposition is studied by the Monte Carlo renormalization-group technique. A block-spin transformation is applied to the evolving configurations of the two-dimensional kinetic Ising model with conserved order parameter. This acts to renormalize the growing domains, the moving interfaces between them, and the coupled long-range diffusion fields. The growth law for the average size of domains, R(t)\ensuremath{\sim}${t}^{n}$, where t is time, is determined self-consistently by a matching condition. The result for the growth exponent, n=0.338\ifmmode\pm\else\textpm\fi{}0.008, is consistent with the classical result of Lifshitz and Slyozov for Ostwald ripening, namely, n=(1/3. A scaling form for the structure factor is obtained which is invariant under the renormalization-group transformation, to the accuracy of our study. For large wave numbers k, it is found that the scaled form of the structure factor F is in good agreement with Porod's law; i.e., F(kR)\ensuremath{\sim}1/(kR${)}^{d+1}$, in d=2 dimensions.

Late stage spinodal decomposition in an oligomeric blend of polystyrene/polybutadiene: A test of the scaling law for the structure function

The late stages of phase separation by spinodal decomposition have been studied by light scattering in an oligomeric blend of polystyrene (PS) with polybutadiene (PB) as a function of both quench depth and composition. The scattering function I(q,t) follows a dynamical scaling law of the form S(q,t)=qm(t)−3S̃[q/qm(t)] where the wave vector qm corresponds to the maximum in I(q,t). The profiles of the normalized scaled structure functions are compared with that predicted by Furukawa for critical mixtures (percolation regime) which is given by S̃(x)=4x2/(3+x8)[S̃(1)=1]. Apart from one exception, Furukawa’s function adequately describes the form of the normalized structure functions for a 60% PS mixture. However there are problems with some of the comparisons with the 50% and 70% PS blends. These disparities may in part be attributable to the fact that some of the data may lie outside the ‘‘true’’ late stages.

Late stages of spinodal decomposition in a three-dimensional model system.

We present results from a numerical study of the Cahn-Hilliard model for spinodal decomposition in a three-dimensional system. Details of the numerical integration method and the late-time field configurations are discussed. We find that the late-time behavior of the system is well described in terms of scaling with a characteristic length, R(t). The data for both the pair-correlation function and the structure function show scaling behavior at sufficiently late times. The time dependence of R(t) is analyzed extensively and found to be consistent with a modified Lifshitz-Slyozov law; i.e., R(t)=c+${\mathrm{dt}}^{1/3}$.

Observation of Micellar Formation in the Cavity of Porous Silica Glass

ABSTRACTThe small angle neutron scattering intensity distribution from the empty pores of the silica glass Vycor can be satisfactorily interpreted using a simplified late stage spinodal decomposition model, due to Cahn', if one allows for a distribution of wave numbers in the material's density function. The result confirms that the porosity of the glass is 28% and gives the mean wave number of the density wave to be k0 = 0.0285 Å-−1 with a dispersion, δk / k0, of about 0.29. Small angle neutron scattering measurements of the anionic surfactant SDS in the Vycor pores indicate that when the surfactant is introduced into the pores at 80 × CMC in bulk solution, the surfactant forms micelles with the same average micelle separation distance as in the bulk solution, but the distribution of micelle separations is much greater in the glass pores.

Numerical study of the late stages of spinodal decomposition.

A numerical simulation of a two-dimensional system with a continuous order parameter (model B) is used to study phase separation for a critical quench. Domain growth and scaling are investigated through the pair correlation function. At late stages, the characteristic domain size grows as R(t)\ensuremath{\sim}${t}^{1/3}$. It is only in this time regime that scaling of the correlation function is established. The scaling function and the asymptotic growth law are found to be independent of the strength of thermal fluctuations. Details of the finite-difference scheme used to simulate the dynamics are discussed.

Late-stage phase separation and hydrodynamic flow in a binary liquid mixture.

Light-scattering measurements were made in a binary mixture in the late stage of spinodal decomposition, when gravity has become important. We observe macroscopic flow in the mixture, following which the scattering I(k,t) falls as ${k}^{\mathrm{\ensuremath{-}}4}$. This implies that the scattering is mainly from surfaces between growing domains. The measurements reveal an unexpectedly large rate of disappearance of the surface area per unit volume in the phase-separating system. The effect is probably related to a gravity-driven instability which gives rise to the flow and which was also investigated.

Late stage spinodal decomposition of a binary polymer mixture. I. Critical test of dynamical scaling on scattering function

Unmixing kinetics of quenched critical polymer mixture of polystyrene and poly(vinyl methyl ether) was investigated by time-resolved light scattering technique. In the late stage of spinodal decomposition τ>60, τ being a reduced time, a simple dynamic scaling law was found on the scattering function I(q,t); I(q,t)∼ξ(t)3S̃[qξ(t)] with ξ(t)∼ta, where q is the scattering vector and ξ(t)=q−1m(t) is the correlation length of the periodic unmixed structure at time t which is measured by the position qm(t) of the scattering maximum at time t. The universal scaling functions (or the scaled structure function) F[qξ(t)] experimentally determined from the scattering functions for the lower quench depths (i.e., unmixing at 96.8, 97.3, and 98.2 °C, the spinodal temperature Ts being 95.8 °C) are in good agreement with that predicted by Furukawa, S̃(x)∼x2/(3+x8) for the critical mixtures (percolation regimes) in three dimensions [x=qξ(t)], although the scaling functions F(x) experimentally determined for the higher quench depths (i.e., unmixing at 102.5 and 106.8 °C) are much broader than the predicted function. Thus the decomposition of the polymer mixtures is proposed to obey universality found for metallic alloys, inorganic glasses, and small-molecular liquids in that the decomposed structures obtained at different time scale in the late stage has the self-similarity and unmixing dynamics is scaled with a single length parameter ξ(t).

Late stage spinodal decomposition in polymer mixtures

Late stages are examined in the spinodal decomposition process of polymer blends. The coarsening due to surface tension is much delayed in the entangled case even on the dimensionless time scale. Before its inception the hydrodynamic interaction is strongly suppressed and the coarsening should be analogous to that of the kinetic Ising model.

Hardening during the late stages of spinodal decomposition

The microstructural changes inherent in the late stages of spinodal decomposition have been taken into account in modelling the hardening process. The force balance equation for a mixed dislocation lying in the slip plane has been modified to take into account the pronouced wave squaring and coarsening commonly observed. Calculated solutions show a marked saturation in the peak hardening which can be achieved and demonstrate that this value is dependent on the wavelenght of the decomposition and the sharpness of the wave squaring.

Theory of spinodal decomposition in relaxational tricritical models

We present a theory for the late-stage spinodal decomposition for two relaxational tricritical models in which the order parameter is nonconserved and the subsidiary order parameter is either conserved or nonconserved. We find that for $d$-dimensional systems (for $d>1$) the characteristic domain size grows in proportion to ${t}^{\frac{1}{3}}$ (Lifshitz-Slyozov) and ${t}^{\frac{1}{2}}$ (Cahn-Allen-Chan), respectively, where $t$ is the time. A discussion is also given for the growth mechanism of a one-dimensional model. Our analysis in general involves linearizing the dynamical equations of motion around a stationary, but unstable state, which describes coexisting phases.

Late stages of spinodal decomposition in binary mixtures

Late stages of spinodal decomposition in binary mixtures