A model composed of a synthesis of the nonlinear Cahn−Hilliard and Flory−Huggins theories for spinodal decomposition (SD) and a second-order rate equation for the self-condensation of a trifunctional monomer is presented and used to analyze polymerization-induced phase separation (PIPS). The numerical results replicate frequently reported experimental observations on the PIPS of a binary monomer−solvent solution. These observations include a transient periodic concentration spatial profile with a wavelength that decreases with increasing rate constant. In addition, the time evolution of the maximum value of the structure factor exhibits an exponential growth during the early stage, but then slows down in the intermediate stage of SD. Computational analysis shows that, in the PIPS method, the wavelength of the phase-separated structure depends on the complex interaction between the competing polymerization and phase separation processes. The effects of these two competing processes on the characteristic time and length scales of the phase separation phenomena depend on the magnitudes of a scaled diffusion coefficient D for phase separation and a scaled rate constant K 1 for polymerization. As D increases, the dominant dimensionless wavenumber k m * also increases, but the phase separation lag time decreases. Similarly, as K 1 increases, k m * also increases, but the polymerization lag time decreases. On the basis of these two dimensionless parameters, the dominant wavelength selection mechanism in the PIPS process is identified, which enables the control of morphology during the PIPS phenomena.

SUMMARY: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 wavenumber k& increases as well, but the dimensionless transition time t: from the early stage to the intermediate stage decreases. In addition, f; is shortest when c8 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 N, is in the range I 5 N2 5 100. When N2 > 100, however, a problem of numerical nonconvergence due to diverging relaxation rates occurs because of the very unsymrnetric 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.

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A model composed of the nonlinear Cahn-Hilliard and Flory-Huggins theories for spinodal decomposition (SD) and a second-order rate equation for polymerization for the self-condensation of a trifunctional monomer is used to study the polymerization-induced phase separation (PIPS) phenomena. The numerical results are consistent with experimental observations. These observations include the formation and evolution of a droplet-type morphology. In addition, the time evolution of the maximum value of the structure factor S(km,t) exhibits an exponential growth during the early stage but saturates during the intermediate stage of SD. Moreover, the dominant dimensionless wavenumber km* decreases during the intermediate stage. The numerical results, however, also indicate that km* increases during the early stage, which has not yet been observed experimentally. Furthermore, the morphological analysis is also consistent with experimental observations. The droplet size and shape distributions indicate that the average droplet size and shape prevail during the PIPS phenomena, and statistical analysis of the Voronoi polygons indicates that the droplets are randomly positioned within the matrix. Lastly, the characteristic time τ, average dimensionless equivalent droplet diameter 〈d*〉, and droplet number density Nd depend on the magnitudes of a scaled diffusion coefficient D for phase separation and a scaled rate constant K1 for polymerization. Consistent with experimental observations, τ and 〈d*〉 decrease while Nd increases as K1 increases. Similarly, as D increases, τ and 〈d*〉 decrease while Nd increases. The parameters K1 and D have no effect on the average shape factor.

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The influence of spatial temperature gradients on the morphological development in polymer solutions undergoing thermally induced phase separation was studied using mathematical modeling and computer simulation. The one‐dimensional mathematical model describing this phenomenon incorporates the nonlinear Cahn‐Hilliard theory for spinodal decomposition (SD), the Flory‐Huggins theory for polymer solution thermodynamics, and the slow‐mode theory and Rouse law for polymer diffusion. The resulting governing equation and auxiliary conditions were solved using the Galerkin finite element method. The temporal evolution of the spatial concentration profile from the computer simulation illustrates that an anisotropic morphology (see Figure) results when a temperature gradient is maintained along the polymer solution sample. The final anisotropic morphology depends on the overall phase separation time. If phase separation is terminated at very early stages, smaller (larger) droplets are formed in the lower (higher) temperature regions due to the deep (shallow) quench effect. On the other hand, if phase separation is allowed to proceed for a long period of time, then larger droplets are formed in the low‐temperature regions, whereas smaller droplets are developed at higher temperatures. This is due to the fact that the low‐temperature regions have entered the late stage of SD, while the high temperature regions are still in the early stage of SD. The presence of a temperature gradient during thermally induced phase separation introduces spatial variations in the change of chemical potential, which is the driving force for phase separation. These numerical results provide a better understanding of the control and optimization during the fabrication of anisotropic polymeric materials using the thermally induced phase separation technique.