Polymer blends simulations

Monte Carlo simulations, which include fluctuations, then yields Simulations of a coarse-grained polymer blend by Wemer et al find = 1 [49] in the strong segregation limit, in rather good... 

Muller M 1999 Misoibility behavior and single ohain properties in polymer blends a bond fluotuation model study Macromol. Theory Simul. 8 343... 

The wall-PRISM theory has also been implemented for binary polymer blends. For blends of stiff and flexible chains the theory predicts that the stiffer chains are found preferentially in the immediate vicinity of the surface [60]. This prediction is in agreement with computer simulations for the same system [59,60]. For blends of linear and star polymers [101] the theory predicts that the linear polymers are in excess in the immediate vicinity of the surface, but the star polymers are in excess at other distances. Therefore, if one looks at the integral of the difference between the density profiles of the two components, the star polymers segregate to the surface in an integrated sense, from purely entropic effects. 

Figure 41. The percolation threshold determination for polymer blends undergoing the phase separation. Minority phase volume fraction, fm, is plotted versus the Euler characteristic density for several simulation runs at different quench conditions, /meq- = 0.225,..., 0.5. The bicontinuous morphology (%Euier < 0) has not been observed for fm < 0.29, nor has the droplet morphology (/(Euler > 0) been observed for/m > 0.31. This observation suggests that the percolation occurs at fm = 0.3 0.01.

Simulation Study of Relaxation Processes in the Dynamical Fast Component of Miscible Polymer Blends. 

The focused laser beam is scanned along an arbitrary path within the xy-plane as sketched in Fig. 10. The perspective view with the cross section through the scan path shown in Fig. 10a visualizes the color-coded concentration change due to the Soret effect according to the numerical simulation discussed later on. On the right hand side a phase contrast micrograph is shown where the word Soret has been written into the polymer blend. 

In this section it will be demonstrated how spinodal decomposition patterns in the two phase region can locally be manipulated in a controlled way by heating a polymer blend PDMS/PEMS by a focused laser beam. It will also be shown that the essential spatial and temporal phenomena, as observed in the experiments, can only be reproduced in numerical simulations if thermodiffusion (Soret effect) is taken into account in the basic equations. 

Fig. 18 Temporal evolution of a pattern in a polymer blend at T = 37.2°C < Tc which was exposed locally to laser light during the period 0 < t < 200 s. Images are taken at t = 0 (A), t = 300 s (B), and t = 700 s (C). The corresponding images (a-c) are obtained by simulations with and the images (ia — y) without taking the Soret effect into account. Figure from [119]. Copyright (2005) by The American Physical Society...

To analyze this phenomenon further, 2D numerical simulations of (49) and (50) were performed using a central finite difference approximation of the spatial derivatives and a fourth order Runge-Kutta integration of the resulting ordinary differential equations in time. Details of the simulation technique can be found in [114, 119]. The material parameters of the polymer blend PDMS/PEMS were used and the spatial scale = (K/ b )ll2 and time scale r = 2/D were established from the experimental measurements of the structure factor evolution under a homogeneous temperature quench. 

Computer Simulation of Polymer Blends in Thin Films... 

It is clear that the interpolation between the calibration lines cannot be applied to mixtures of polymers (polymer blends). If the calibration lines are different, different molar masses of the homopolymers will elute at the same volume. The universal calibration is also not capable of eliminating the errors which originate from the simultaneous elution of two polymer fractions with the same hydrodynamic volume, but different composition and molar mass. Ogawa [33] has shown by a simulation technique that the molar masses of polymers eluting at the elution volume Ve are given by the corresponding coefficients K and a in the Mark-Houwink equation. 

In some cases, one is interested in the structures of complex fluids only at the continuum level, and the detailed molecular structure is not important. For example, long polymer molecules, especially block copolymers, can form phases whose microstructure has length scales ranging from nanometers almost up to microns. Computer simulations of such structures at the level of atoms is not feasible. However, composition field equations can be written that account for the dynamics of some slow variable such as 0 (x), the concentration of one species in a binary polymer blend, or of one block of a diblock copolymer. If an expression for the free energy / of the mixture exists, then a Ginzburg-Landau type of equation can sometimes be written for the time evolution of the variable 0 with or without flow. An example of such an equation is (Ohta et al. 1990 Tanaka 1994 Kodama and Doi 1996)... 

Figure 14 exemplifies two computational methods to determine the probability distribution of composition for binary polymer blends described by the bond fluctuation model [67]. Phase coexistence can be extracted from these data via the equal-weight rule. For the specific example of a symmetric blend, the coexistence value of the exchange chemical potential, A/u, is dictated by the S3munetry. One can simply simulate at A oex = 0 and monitor the composition. Nevertheless, the probability distribution contains additional information, as discussed in Sect. 3.5. 

M. Muller (1999) Miscibility behavior and single chain properties in polymer blends a bond fluctuation model study. Macromol. Theory Simul. 8, pp. 343-374 M. Muller and K. Binder (1995) Computer-simulation of asymmetric polymer mixtures. Macrrmolecules 28, pp. 1825-1834 ibid. (1994) An algorithm for the semi-grand-canonical simulation of asymmetric polymer mixtures. Computer Phys. Comm. 84, pp. 173-185... 

M. Muller, K. Binder, and W. Oed (1995) Structural and thermodynamic properties of interfaces between coexisting phases in polymer blends - a monte-carlo simulation. J. Chem. Soc. Faraday Trans. 91, pp. 2369-2379... 

In recent work Jerry and Dutta [176] reanalyzed, with the mean field approach, conditions [8] of the second order wetting transition. They have found that critical wetting transition must be accompanied by a prerequisite phenomenon of an enrichment-depletion duality it is expected that the surface is enriched in the given component when bulk composition ( )M is below a certain value Q and is depleted in the same component for >Q. Such an effect, easily predicted by simple lattice theory [177] and observed in Monte Carlo simulations [178, 179], has been very recently determined by us for a real polymer blend [175] (see Sect. 3.1.2.4). 

The picture presented above is not complete as it neglects non-mean field behavior of polymer blends in the temperature range close to Tc [149]. The Ising model predicts phase diagrams of thin films, which are more depressed and more flattened than those yielded by mean field approach (as marked in Fig. 31d). Both effects were shown by Monte Carlo simulations performed by Rouault et al. [150]. In principle, critical regions of phase diagrams cannot be described merely by a cross-over from a three- to two-dimensional (for very thin films) situation. In addition, a cross-over from mean field to Ising behavior should also be considered [6,150]. 

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