Cluster kinetics and dynamics during spinodal decomposition

Spinodal decomposition (barrierless phase transition) is a spontaneous phase separation caused by conditions that force the system to become thermodynamically unstable. We consider spinodal decomposition to occur under conditions of large supersaturation S and/or small ratio of interfacial to thermal energies omega, such that the computed number of monomers in a critical nucleus xi*=(omega/ln S)3 is less than unity. The small critical nucleus size is consistent with a negligible energy barrier for initiating condensation. Thus, in contrast to conventional opinion, it is suggested that the spinodal decomposition is related to the homogeneous nucleation of metastable fluids. Population balance equations show how clusters aggregate and rapidly lead to phase separation. Different mass dependences of aggregation rate coefficients are proposed to investigate the fundamental features of spinodal decomposition. When the mass dependency is an integer, the equations are solved by the moment technique to obtain analytical solutions. When the mass dependency is a noninteger, the general cases are solved numerically. All solutions predict the two time regimes observed experimentally: the average length scale of condensed-phase domains increases as a power law with an exponent of 1/3 at early times, followed by a linear increase at longer times.     

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.     

Computational analysis of spinodal decomposition dynamics in polymer solutions

In this work a model, composed of the nonlinear Cahn-Hilliard and Flory-Huggins theories, is used to numerically simulate the phase separation and pattern formation phenomena of oligomer and polymer solutions when quenched into the unstable region of their binary phase diagrams. This model takes into account the initial thermal concentration fluctuations. In addition, zero mass flux and natural nonperiodic boundary conditions are enforced to better reflect experimental conditions. The model output is used to characterize the evolution and morphology of the phase separation process. The sensitivity of the time and length scales to processing conditions (initial condition c) and properties (dimensionless diffusion coefficient D) is elucidated. The results replicate frequently reported experimental observations on the morphology of spinodal decomposition (SD) in binary solutions: (1) critical quenches yield interconnected structures, and (2) off-critical quenches yield a droplet-type morphology. As D increases, the dominant dimensionless wave number k increases as well, but the dimensionless transition time t from the early stage to the intermediate stage decreases. In addition, t is shortest when c is at the critical concentration, but increases to infinity when c is at one of the two spinodal concentrations. These results are found when the solute degree of polymerization N₂ is in the range 1 ≤ N₂ ≤ 100. When N₂ > 100, however, a problem of numerical nonconvergence due to diverging relaxation rates occurs because of the very unsymmetric nature of the phase diagram. A novel scaling procedure is introduced to explain the phase separation phenomena due to SD for any value of N₂ during the time range explored in this study.     

Small-angle polarized light scattering by spherulites

The light-scattering matrix for a three-dimensional spherulite is derived within the Ray-leigh-Gans-Debye light-scattering approximation. New expressions for the polarized, small-angle light-scattering intensities I_VV and I_VH are derived from the scattering matrix. These expressions are compared with the I_VV and I_VH expressions derived for a spherulite by Stein and Rhodes. For the case of a weakly anisotropic spherulite having an average refractive index mismatch with its surroundings, the two sets of expressions predict different I_VV and I_VH intensities. In particular, our expressions show that the I_VVand I_VH patterns usually attributed to the spherulitic anisotropy and crystallinity are also predicted for an isotropic sphere. This is in accord with recent experiments.     

The phase dynamics and wetting layer formation mechanisms in two-step surface-directed spinodal decomposition

The two-step quench process of surface-directed spinodal decomposition is numerically investigated by coupling the Flory-Huggins-de Gennes equation with the Cahn-Hilliard-Cook equation. The phase dynamics and formation mechanisms of the wetting layer in two-step surface-directed spinodal decomposition have been concerned in detail. The results demonstrate that a parallel strip structure forms near the wetting layer and propagates into the bulk, when the first quench depth is very shallow and the bulk does not undergo phase separation, and the second quench depths are various points with deeper quench depths. In this case, the wetting layer turns to be unchangeable at the intermediate and later stages of the second quench process, compared to the growth with a time exponent 1/2 during the first quench process. When the first quench depth is deeper and phase separation occurs in the bulk during the first quench process, it is found that a deeper second quench depth can stimulate a more obvious secondary domain structure, and the formation mechanism of the wetting layer changes from logarithmic growth law to Lifshitz-Slyozov growth law.     

Small-angle polarized light scattering from amorphous spheres

Small-angle light-scattering measurements have been made using micron-diameter isotropic and spherical polymer latex particles placed between crossed polarizers. Four-leaf clover patterns are obtained, reminiscent of those commonly found for spherical birefringent scatterers. The experimental results compare closely with predictions of Mie scattering theory for isotropic spheres.     

Dynamic light scattering study of the dynamics of a gelled polymeric micellar system

The dynamics of the E(92)B(18)/water system are studied by dynamic light scattering (DLS) in the liquid, soft gel, and hard gel phases. Both the liquid and the soft gel phases are micellar phases, although the structural order is higher in the soft gel phase than in the liquid phase. The hard gel phase corresponds to a face-centered cubic arrangement of micelles. DLS results show that the dilute liquid phase is characterized by a single characteristic time tau(1) associated with the diffusion of the micelles. In addition, a second characteristic time tau(2) associated with the presence of micellar clusters in the system is identified in the concentrated liquid and in the soft gel phases. According to these results, DLS suggests that the structure of the soft gel phase comprises micellar clusters coexisting with micellar fluid, in good agreement with hypotheses from our previous work. The dynamics of the system slows down as the hard gel phase is approached and a plateau is observed in the DLS correlation function. The structure of the hard gel is "softened" upon increasing temperature and/or decreasing concentration.     

Small-angle scattering of circularly polarized light from an anisotropic sphere

A theory of the small-angle scattering of circularly polarized light from an anisotropic sphere has been derived. The validity of the theory has been verified, and a relationship between the structural information thus obtained and the structural information obtained with linearly polarized light has been demonstrated.     

Nonlinear kinetics of spinodal decomposition,and dissolution of inhomogeneities formed by spinodal decomposition in polymer blends

Nonlinear kinetics of both spinodal decomposition at early stages, and the dissolution of homogeneities formed during spinodal decomposition, is studied. Variation of the scattering intensity during a complete cycle consisting of a step temperature change from T₁ in the one-phase region to T₂ in the two-phase region, a period of spinodal decomposition followed by a temperature drop from T₂ back to T₁, and the subsequent relaxation to the original equilibrium state, is investigated at various wavenumbers. Step temperature changes within one-phase region are also investigated.     

Structuring in mixed solvents: Study by polarized light scattering

The structure of mixed amide solvents used for the preparation of polyimide Langmuir-Blodgett films and the structural organization of water, on which nanofilms are formed, have been studied by polarized light scattering. Fluctuation supramolecular structures are found to exist in water and in its mixtures with amide solvents, and the dimensions of the structures are comparable with the light wavelength. When a mixture contains more than 20 vol % water, associates show microanisotropic properties. In snowmelt water, the dimensions of associates are larger than those in distilled water. In the DMAA-water mixture, the dimensions of supramolecular structures are virtually independent of composition.     

Small-angle scattering of polarized light. IV. Isotropic to birefringent transition for spheres

The Rayleigh-Gans-Debye approximation is used to investigate the small-angle, cross-polarized light scattering by a sphere of radius a, birefringence Δμ and relative index μ. If θ_max is the polar scattering angle of the intensity maxima, the quantity Û_max = 4πa/λ × sin (θ_max/2) behaves in two different ways according to the signs of Δμ and μ ? 1. When Δμ > 0, μ > 1 or Δμ < 0, μ < 1, then Û_max varies from 2.8 to 4.1 as Δμ increases from zero. If Δμ < 0, μ > 1 or Δμ > 0, μ < 1, then Û_max goes from 2.8 to about 6, thereafter decreasing to 4.1. Another interesting result is that the value of Û_max for a highly briefringent sphere is 4.1 only for large diameters. It decreases to 4.0 when the diameter decreases.     

Small-angle scattering of polarized light. II. experiments with isotropic spheres

New experimental measurements are reported of small-angle polarized and depolarized light scattering from almost monodisperse isotropic, spherical particle, polystyrene latexes. The shape and intensity distribution of the scattering patterns is shown to compare closely with the calculated patterns based on the exact Mie scattering theory from a single sphere.     

On the behaviour of small clusters near the spinodal decomposition

The canonical average of the Boltzmann factor of the interaction potential, as measured by a test particle, is shown to be equal to the inverse of the fraction of the average number of 1-particle Mayer clusters. The potential distribution theory is used to derive an analytic expression for a mean number of small clusters 1≤n<N, in anN-particle system) in the mean-field expression. Near the spinodal density, the average number of small clusters undergo a sharp change. Computation of pressure shows that only the first four clusters produce surprisingly good agreement with known pressure even beyond the spinodal density. Contribution No. 439 from the Solid State and Structural Chemistry Unit     

The simulation of the growth of colonies in the spinodal decomposition of metastable phases

The generalized continual hole gas model and Monte Carlo simulations were used to study the growth of colonies and lamellar segregations in the decomposition of metastable phases. The morphology of segregations qualitatively changed as the temperature decreased and depended substantially on the initial state (the presence of grain boundaries and critical equilibrium phase nuclei). The simulation results were compared with the kinetics of growth of eutectic colonies and morphologies of the decomposition of metastable austenite.     

Manifestation of spinodal decomposition in oscillatory shear measurements

The linear dynamic moduli of a PαMSAN/PMMA blend have been measured during spinodal decomposition and the subsequent coarsening of the co‐continuous morphology. The feasibility of probing this morphology development by rheological measurements has been investigated. During phase separation, the storage modulus shows a power law behaviour at low frequency and its value decreases as time proceeds. However, effects of the dynamic measurement on the morphology development have been observed, even for strain amplitudes as low as 0.01. This effect of oscillatory shear on the coarsening of a co‐continuous structure is consistent with the predictions of the Doi‐Ohta theory.     

Size distribution analysis of polymer latex systems by use of the extrema in the angular light scattering pattern. III. Unpolarized and horizontally polarized scattered light

The extrema in the unpolarized and the horizontally polarized angular scattering patterns are used as a basis for size-distribution analysis of polymer latex systems composed of single, spherical, optically isotropic particles. The method of analysis is similar to that recently proposed for vertically polarized scattered light and the polarization ratio, where the modal diameter and a distribution width parameter are determined from the experimental angular scattering pattern and prepared theoretical diagrams obtained by Mie theory calculations. The method of analysis, based on all four scattering functions, is illustrated by use of a Dow polystyrene latex sample.     

Kinetically stable structures in the nonlinear theory of spinodal decomposition

A nonlinear diffusion equation is used to study the spinodal decomposition of binary polymer mixtures. The structure of steady-state solutions is analyzed. The free energy of the system for the solutions is shown to be a nonincreasing time function. The solution with the minimum free energy is the only stable solution and corresponds to complete phase separation in the system. A numerical analysis shows that transition to complete equilibrium takes place through a succession of fast concentration structure changes interspersed by periods in which the process is abruptly showed down. The free energy of the system is almost constant in the kinetically stable periods. All these facts indicate a relaxation process of an absolutely new type. Slow variables are introduced to describe the spinodal decomposition in macroscopic regions. These variables govern the spatial transformation of the forms and sizes of microphases. The perturbation theory is used to derive the equations for slow variables. The final relations of the simplest type give an exponential time dependence of the average size of a microphase. The problem of a contact of two pure components that are not compatible in the entire composition range is set.     

伯硝胺酸催化分解动力学的理论研究

用SCF-MO AM1方法全优化计算了伯硝胺化合物(RNHNO~2, R=Me, Et, Pr^i,Bu^t和CH~2CO~2H)的几何构型和电子结构, 并求得15个反应的过渡态和活化能 . 分析静态结构和动力学特征, 发现取代基(R)供电性愈强, 标题物的酸催化分解速率愈大, 即相对速率为R=Bu^t＞Pr^i＞Et＞Me＞CH~2CO~2H; 溶剂化效应将降低标题物质子化过程的活化能, 有利于加速酸催化分解。     

The use of light scattering for precise characterization of polymers for DNA sequencing by capillary electrophoresis

The ability of a polymer matrix to separate DNA by capillary electrophoresis (CE) is strongly dependent upon polymer physical properties. In particular, recent results have shown that DNA sequencing performance is very sensitive to both the average molar mass and the average coil radius of the separation matrix polymers, which are affected by both polymer structure and polymer-solvent affinity. Large polymers with high average molar mass provide the best DNA sequencing separations for CE, but are also the most challenging to characterize with accuracy. The methods most commonly used for the characterization of water-soluble polymers with application in microchannel electrophoresis have been gel permeation chromatography (GPC) and intrinsic viscosity measurements, but the limitations and potential inaccuracies of these approaches, particularly for large or novel polymers and copolymers, press the need for a more universally accurate method of polymer molar mass profiling for advanced DNA separation matrices. Here, we show that multi-angle laser light scattering (MALLS) measurements, carried out either alone or in tandem with prior on-line sample fractionation by GPC, can provide accurate molar mass and coil radius information for polymer samples that are useful for DNA sequencing by CE. Wider employment of MALLS for characterization of novel polymers designed as DNA separation matrices for microchannel electrophoresis should enable more rapid optimization of matrix properties and formulation, and assist in the development of novel classes of polymer matrices.     

Universal growth exponent at the early nonlinear stage of spinodal decomposition

The nonlinear Cahn's model of spinodal decomposition is considered within the framework of the Langer's ‘mean-field approximation’. It is argued that a universal growth exponent of 1/4 characterizes the early nonlinear stage of spinodal decomposition, regardless of the space dimensionality and the spectral characteristics of the gradient energy matrix. This is a contrary to our earlier predictions made for isotropic and anisotropic decomposition (Mitlin, 1989, 1990). The temporal dependence of the magnitude of the structure factor follows a power law with the exponent of d/2 (d is the space dimensionality). The temporal dependence of the semi-width of the structure factor follows a power law with the exponent of −1/4 (with logarithmic corrections).