Brownian cluster dynamics with short range patchy interactions. Its application to polymers and step-growth polymerization

We present a novel simulation technique derived from Brownian cluster dynamics used so far to study the isotropic colloidal aggregation. It now implements the classical Kern-Frenkel potential to describe patchy interactions between particles. This technique gives access to static properties, dynamics and kinetics of the system, even far from the equilibrium. Particle thermal motions are modeled using billions of independent small random translations and rotations, constrained by the excluded volume and the connectivity. This algorithm, applied to a single polymer chain leads to correct static and dynamic properties, in the framework where hydrodynamic interactions are ignored. By varying patch angles, various chain flexibilities can be obtained. We have used this new algorithm to model step-growth polymerization under various solvent qualities. The polymerization reaction is modeled by an irreversible aggregation between patches while an isotropic finite square-well potential is superimposed to mimic the solvent quality. In bad solvent conditions, a competition between a phase separation (due to the isotropic interaction) and polymerization (due to patches) occurs. Surprisingly, an arrested network with a very peculiar structure appears. It is made of strands and nodes. Strands gather few stretched chains that dip into entangled globular nodes. These nodes act as reticulation points between the strands. The system is kinetically driven and we observe a trapped arrested structure. That demonstrates one of the strengths of this new simulation technique. It can give valuable insights about mechanisms that could be involved in the formation of stranded gels.Comment: 55 pages, 32 figure

Depletion from a hard wall induced by aggregation and gelation

Diffusion-limited cluster aggregation and gelation of hard spheres is simulated using off-lattice Monte Carlo simulations. A comparison is made of the wall-particle correlation function with the particle-particle correlation function over a range of volume fractions, both for the initial system of randomly distributed spheres and for the final gel state. For randomly distributed spheres the correlation functions are compared with theoretical results using the Ornstein-Zernike equation and the Percus-Yevick closure. At high volume fractions (φ > 40%) gelation has little influence on the correlation function, but for φ < 10% it is a universal function of the distance normalized by correlation length (ξ) of the bulk. The width of the depletion layer is about 0.5ξ. The concentration increases as a power law from the wall up to r ≈ ξ, where it reaches a weak maximum before decreasing to the bulk value

Monte Carlo simulation of particle aggregation and gelation: II. Pair correlation function and structure factor

Diffusion-limited cluster aggregation and gelation are studied using lattice and off-lattice Monte Carlo simulations. The pair correlation function g(r) and the structure factor S(q) of the particle gels were investigated as a function of the volume fraction (0.5\mbox{--}49\%) and time. At volume fractions below $5\%$, the gel structure is fractal on small length scales with $d_{\rm f} = 1.8$. g(r) shows a weak minimum at the correlation length ($\xi$), before reaching the average concentration at large length scales. The cut-off function of g(r) varies during the aggregation process, but at a given $t/t_{\rm g}$, where $t_{\rm g}$ is the gel time, it is a universal function of $r/\xi$. At high volume fractions, the structure is dominated by excluded-volume interactions, while at low volume fractions, it is determined by the connectivity

Structure and size distribution of percolating clusters. Comparison with gelling systems

3d lattice Monte-Carlo simulations were done to obtain the mass distribution (N(m)) and pair correlation function (g(r)) of percolating clusters. We give analytical expressions of the external cut-off functions of N(m) at the z-average mass and of g(r) at the radius of gyration. The simulation results were compared with experimental results on gel forming systems reported in the literature. The comparison shows that the experimental results are compatible with the simulation results. However, more experiments are needed before we can be confident that the percolation model is a good model for the sol-gel transition

Influence of the Brownian step size in off-lattice Monte Carlo simulations of irreversible particle aggregation

The influence of the Brownian step size in off-lattice Monte Carlo simulations of the aggregation and gelation of spheres is studied. It is found that the kinetics are strongly influenced if the step size is larger than the mean smallest distance between the sphere surfaces. The structure of the clusters and the gels is influenced, but only over length scales smaller than the step size. Using large step sizes leads to a narrower size distribution of the clusters. Implications of the present results are discussed for simulations reported in the literature in which the Brownian step size was chosen equal to the sphere diameter

Monte Carlo simulation of particle aggregation and gelation: I. Growth, structure and size distribution of the clusters

Lattice and off-lattice Monte Carlo simulations of diffusion-limited cluster aggregation and gelation were done over a broad range of concentrations. The large-scale structure and the size distribution of the clusters are characterized by a crossover at a characteristic size ($m_{\rm c}$). For $m < m_{\rm c}$, they are the same as obtained in a dilute DLCA process and for $m \gg m_{\rm c}$ they are the same as obtained in a static percolation process. $m_{\rm c}$ is determined by the overlap of the clusters and decreases with increasing particle concentration. The growth rate of large clusters is a universal function of time reduced by the gel time. The large-scale structural and temporal properties are the same for lattice and off-lattice simulations. The average degree of connectivity per particle in the gels formed in off-lattice simulations is independent of the concentration, but its distribution depends on the concentration