Reaction-diffusion fronts in systems with concentration-dependent diffusivities

The Kinetics of Polymer Brush Growth in the Frame of the Reaction Diffusion Front Formalism

We studied the properties of a reaction front that forms in irreversible reaction-diffusion systems with concentration-dependent diffusivities during the synthesis of polymer brushes. A coarse-grained model of the polymerization process during the formation of polymer brushes was designed and investigated for this purpose. In this model, a certain amount of initiator was placed on an impenetrable surface, and the "grafted from" procedure of polymerization was carried out. The system consisted of monomer molecules and growing chains. The obtained brush consisted of linear chains embedded in nodes of a face-centered cubic lattice with excluded volume interactions only. The simulations were carried out for high rafting densities of 0.1, 0.3, and 0.6 and for reaction probabilities of 0.02, 0.002, and 0.0002. Simulations were performed by means of the Monte Carlo method while employing the Dynamic Lattice Liquid model. Some universal behavior was found, i.e., irrespective of reaction rate and grafting density, the width of the reaction front as well as the height of the front show for long times the same scaling with respect to time. During the formation of the polymer layer despite the observed difference in dispersion of chain lengths for different grafting densities and reaction rates at a given layer height, the quality of the polymer layer does not seem to depend on these parameters.

Molecular transport in systems containing binding obstacles

We studied the movement of particles in crowded environments by means of extensive Monte Carlo simulations. The dynamic lattice liquid model was employed for this purpose. It is based on the cooperative movement concept and allows the study of systems at high densities. The cooperative model of molecular transport is assumed: the motion of all moving particles is highly correlated. The model is supposed to mimic lateral motion in a membrane and therefore the system is two-dimensional with moving objects and traps placed on a triangular lattice. In our study the interaction (binding) of traps with moving particles was assumed. The conditions in which subdiffusive motion appeared in the system were analysed. The influence of the strength of binding on the dynamic percolation threshold was also shown. The differences in the dynamics compared to systems with impenetrable obstacles and with systems without correlation in motion were presented and discussed. It was shown that in the case of correlated motion the influence of deep traps is similar to that of impenetrable obstacles. If the traps are shallow a recovery to normal diffusion was observed for longer time periods.

Diffusion of small particles in polymer films

The motion of small probe molecules in a two-dimensional system containing frozen polymer chains was studied by means of Monte Carlo simulations. The model macromolecules were coarse-grained and restricted to vertices of a triangular lattice. The cooperative motion algorithm was used to generate representative configurations of macromolecular systems of different polymer concentrations. The remaining unoccupied lattice sites of the system were filled with small molecules. The structure of the polymer film, especially near the percolation threshold, was determined. The dynamic lattice liquid algorithm was then employed for studies of the dynamics of small objects in the polymer matrix. The influence of chain length and polymer concentration on the mobility and the character of motion of small molecules were studied. Short- and long-time dynamic behaviors of solvent molecules were also described. Conditions of anomalous diffusions' appearance in such systems are discussed. The influence of the structure of the matrix of obstacles on the molecular transport was discussed.

Comparison of different models of motion in a crowded environment: a Monte Carlo study

In this paper we investigate the motion of molecules in crowded environments for two dramatically different types of molecular transport. The first type is realized by the dynamic lattice liquid model, which is based on a cooperative movement concept and thus, the motion of molecules is highly correlated. The second one corresponds to a so-called motion of a single agent where the motion of molecules is considered as a random walk without any correlation with other moving elements. The crowded environments are modeled as a two-dimensional triangular lattice with fixed impenetrable obstacles. Our simulation results indicate that the type of transport has an impact on the dynamics of the system, the percolation threshold, critical exponents, and on molecules' trajectories.

Simulation of Molecular Transport in Systems Containing Mobile Obstacles

In this paper, we investigate the movement of molecules in crowded environments with obstacles undergoing Brownian motion by means of extensive Monte Carlo simulations. Our investigations were performed using the dynamic lattice liquid model, which was based on the cooperative movement concept and allowed to mimic systems at high densities where the motion of all elements (obstacles as well as moving particles) were highly correlated. The crowded environments are modeled on a two-dimensional triangular lattice containing obstacles (particles whose mobility was significantly reduced) moving by a Brownian motion. The subdiffusive motion of both elements in the system was analyzed. It was shown that the percolation transition does not exist in such systems in spite of the cooperative character of the particles' motion. The reduction of the obstacle mobility leads to the longer caging of liquid particles by mobile obstacles.

Diffusive properties of solvent molecules in the neighborhood of a polymer chain as seen by Monte-Carlo simulations

The influence of both polymer chain length and concentration on the mobility of solvent molecules in polymer solutions was studied by Monte Carlo simulations with the use of the dynamic lattice liquid (DLL) model. The poly(vinylmethylether)-water system was used as a model. Two different solvent (water) states with differing mobilities were distinguished in polymer solutions. The first one with high molecular mobility independent of polymer concentration corresponds to bulk solvent in real systems. The second state relates to so called bound solvent. In this case the solvent diffusivity decreases with polymer content. For diluted solutions the diffusion of bound solvent is affected by polymer chain length, precisely, by the ability of the polymer chain to undergo coil formation.

Computational studies of intermolecular interactions in aqueous solutions of poly(vinylmethylether)

Thermo-responsive materials, such as poly(vinylmethylether) (PVME), attract a common attention because of their unique physical properties resulted from metastable equilibrium between various types of interactions. In this work Monte Carlo (MC) and quantum-mechanical (QM) methods were used to study excluded volume and electrostatic interactions respectively. The graining procedure of PVME-water system was proposed. Its implementation to MC calculations allowed to distinguish how two water fractions differ on dynamics. The QM calculations showed that the formation of cyclic clusters leads to the lengthening of the hydrogen bonds and consequently to higher energies in comparison to linear forms, which is crucial looking at an application of QM results to MC calculation considering thermal interactions.

Simulation of diffusion in a crowded environment

We performed extensive and systematic simulation studies of two-dimensional fluid motion in a complex crowded environment. In contrast to other studies we focused on cooperative phenomena that occurred if the motion of particles takes place in a dense crowded system, which can be considered as a crude model of a cellular membrane. Our main goal was to answer the following question: how do the fluid molecules move in an environment with a complex structure, taking into account the fact that motions of fluid molecules are highly correlated. The dynamic lattice liquid (DLL) model, which can work at the highest fluid density, was employed. Within the frame of the DLL model we considered cooperative motion of fluid particles in an environment that contained static obstacles. The dynamic properties of the system as a function of the concentration of obstacles were studied. The subdiffusive motion of particles was found in the crowded system. The influence of hydrodynamics on the motion was investigated via analysis of the displacement in closed cooperative loops. The simulation and the analysis emphasize the influence of the movement correlation between moving particles and obstacles.

Higher-order large-time asymptotics for a reaction of the form nA+mB-->C

This study examines the large time asymptotic behavior for the family of reactions of the form nA+mB-->C when the reactants A and B are initially separated. Once the reactants are brought into contact they are assumed to react with a kinetic rate proportional to A;{n}B;{m} . A planar reaction front forms and usually moves away from its initial position to invade one of the reactant solutions. The position of the reaction front relative to the initial position where the reactants were put in contact x_{f} for large times t , is found theoretically to satisfy the expansion x_{f}=2sqrt[t][alpha+alpha_{2}t;{-2sigma}+alpha_{3}t;{-3sigma}+O(t;{-4sigma})] , where sigma=1(n+m+1) and alpha , alpha_{2} , and alpha_{3} are constants. This expansion is valid provided that n and m are positive constants less than or equal to 3. The implication of this is that when n+m not equal3 , x_{f} either tends to zero or infinity, while if n+m=3 then there exists the possibility of x_{f} tending to a finite nonzero constant. For fractional order kinetics, n and m are arbitrary positive constants, however, for simple reactions n and m are positive integers. Hence, the reaction A+2B-->C is the only reaction of the form nA+mB-->C with n and m being positive integers less than 4 in an infinite domain that can lead to a reaction front approaching at a finite but nonzero distance from the position at which the two liquids first met.

Analytical asymptotic solutions of nA+mB-->C reaction-diffusion equations in two-layer systems: a general study

Large time evolution of concentration profiles is studied analytically for reaction-diffusion systems where the reactants A and B are each initially separately contained in two immiscible solutions and react upon contact and transfer across the interface according to a general nA+mB-->C reaction scheme. This study generalizes to immiscible two-layer systems the large time analytical asymptotic limits of concentrations derived by Koza [J. Stat. Phys. 85, 179 (1996)] for miscible fluids and for reaction rates of the form A;{n}B;{m} with arbitrary diffusion coefficients and homogeneous initial concentrations. In addition to a dependence on the parameters already characterizing the miscible case, the asymptotic concentration profiles in immiscible systems depend now also on the partition coefficients of the chemical species between the two solution layers and on the ratio of diffusion coefficients of a given species in the two fluids. The miscible time scalings are found to remain valid for the immiscible fluids case. However, for immiscible systems, the reaction front speed is enhanced by increasing the stoichiometry of the invading species over that of the species being invaded. The direction of the front propagation is found to depend on the diffusion coefficient of the invading species in its initial fluid but not on its value in the invading fluid. Hence, a reaction front in immiscible fluids can travel in the opposite direction to the reaction front formed in miscible fluids for a range of parameter values. The value of the invading species partition coefficient affects the magnitude of the front speed but it cannot alter the direction of the front. For sufficiently large times, the total amount of product produced in time is independent of the rate of the reaction. The centre of mass of the product can move in the opposite direction to the center of mass of the reaction rate.

Exponential temporal asymptotics of the A+B-->0 reaction-diffusion process with initially separated reactants

We study theoretically and numerically the irreversible A+B-->0 reaction-diffusion process of initially separated reactants occupying the regions of lengths LA, LB comparable with the diffusion length (LA,LB approximately sqrt[Dt], here D is the diffusion coefficient of the reactants). It is shown that the process can be divided into two stages in time. For t<<L2/D the front characteristics are described by the well-known power-law dependencies on time, whereas for t>L2/D these are well-approximated by exponential laws. The reaction-diffusion process of about 0.5 of initial quantities of reactants is described by the obtained exponential laws. Our theoretical predictions show good agreement with numerical simulations.