Monte Carlo simulation curved surfaces

With the help of Monte Carlo simulations, the surface peak area can be associated with the number of molten atomic layers. Figure 8.1.13 shows the result for both surfaces. The curve expected from bulk lattice vibrations and bulk interlayer distances is denoted I in Figure 8.1.13b and lies below the data. Curve II accounts, in addition, for the enhanced surface vibration amphtudes and the relaxation of the first two interlayer distances both are manifestations of surface anharmonicity as we have seen above. This curve describes the low-T data quite well. From 500 K on, the number of visible layers is significantly enhanced compared with the expectation from a weU-ordered vibrating crystal. The only way this could be reconciled in the Monte Carlo simulations was to include molten layers on the vibrating soHd. It is seen that the surface has up to 15 such layers at. The surface... 

In Fig. 9.16 the coverages of A and B and the production rate Rco2 are shown as a function of Yqo- We assume a large desorption rate kA = 0.05 and no diffusion (D = 0). For curve 1 we have EAA = EAb = 0, which corresponds to the case of the ZGB-model incorporating A-desorption. The value of yi is not shifted by the desorption because at this point too few A particles are present. Complete coverage of the lattice by A does not occur because at every time step A particles have the chance to desorb from the surface. Both facts are in agreement with the corresponding Monte Carlo simulations [15]. 

Monte Carlo simulation for diblock copolymers confined in curved surfaces... 

FIG. 4 Comparison between MGC theory (solid curves) and Monte Carlo simulation (circles and triangles) of the diffuse-ion swarm on a planar charged surface. Distributions of cations (c+) and anions (c ) are shown for a 1 1 electrolyte solution and two surface charge densities (oq). 

Abstract Configurational-bias Monte Carlo simulations in the Gibbs ensemble have been carried out to determine the vapor-liquid coexistence curve for a pentadecanoic acid Langmuir monolayer. Two different force fields were studied (i) the original monolayer model of Karaborni and Toxvaerd including anisotropic interactions between alkyl tails, and (ii) a modified version of this model which uses an isotropic united-atom description for the methylene and methyl groups and includes dispersive interactions between the tail segments and the water surface. 

Figure 15 (a) Phase diagram of a binary polymer blend N= 32) as obtained from Monte Carlo simulations of the bond fluctuation model. The upper curve shows the binodais in the infinite system the middle one corresponds to a thin film of thickness D=2.8/ e and symmetric boundary fields [wall = 0.16, both of which prefer species A (capillary condensation). The lower curve corresponds to a thin film with antisymmetric surfaces (interface localization/delocalization). The arrow marks the location of the wetting transition. Full circles mark critical points open circles/dashed line denotes the triple point, (b) Coexistence curves in the (T, A/y)-plane. Circles mark critical points, and the diamond indicates the location of the wetting transition temperature. It is indistinguishable from the temperature of the triple point. Adapted from Muller, M. Binder, K. Phys. Rev. 2001, 63, 021602. ... 

Fig. 5 Results of Monte Carlo simulations with Grand Canonian Potential (GCMC) for the reduced solvation pressure at surface potentials 0, —40, —80 (Sihca) and —120 and —160 mV (mica). The solid lines are fit functions obtained from Eq. 1. For clarity the curves are shifted along the Y-axis. The abscissa at the bottom represents the distance normtilized with respect to the diameter of the nanoparticles (26 nm). The graph is taken from [13]...

Figure 1. Density and structure factor profiles of a planar sheet of Lennard-Jonesium fluid in equilibrium with its own vapor at 110 K. The origin is chosen to be the Gibbs equimolecular dividing surface. The curves are obtained from Monte Carlo simulation using one million configurations. Circles denote the density profile and the crosses denote the structure factor, r.

Kelvin lengths are typically twice the diameter of the molecules in a liquid (Table 5.1). It is questionable if at such length scales the liquid behaves like a continuum. Experiments with the SPA showed that the discrete molecular nature of the liquid does not seem to play a crucial role down to dimensions of 0.8 nm for hexane and 1.4nm for water, or even lower [506, 507, 534]. Molecular dynamics simulations of two silica surfaces, interacting across a water bridge agreed with predictions using Kelvin s equation [542]. Monte Carlo simulations of the interaction between a sphere and a flat surface in a vapor showed that either the adhesion force increases with humidity or the force versus humidity curve shows a maximum [543, 544]. Such simulations are, however, limited to sphere sizes of the order of at most few 10 molecular diameters. They complement continuum theory, which is applicable only for larger particle radii. 

SO far in detail. From the point of view of pure theory, or of Monte Carlo simulations, it is practical to regard temperature T, bond probability p, and monomer concentration (j> as three independent variables and to study phase transition surfaces in this T - p - space. (The special plane p = 1 corresponds to Fig. 5 above, the limit T = < to Fig. 6.) At a fixed temperature T above the critical consolute temperature Tc, i.e. in the one-phase region one has curves similar to the T = > limit of Fig. 6 only the end point at p = 1 is shifted slightly to lower concentrations

quantitative results for these percolation line in the simple cubic lattice on the basis of Monte Carlo simulations. (At temperatures appreciably below the phase separation temperature T the system is separated into one phase with very few monomers where even for p = 1 no gelation is possible, and another phase with very few solvent molecules where the system is approximated well by random-bond percolation, 0 = 1.)... 

In this review, almost all of the simulations we have described use only classical mechanics to describe the nuclear motion of the reaction system. However, a more accurate analysis of many reactions, including some of the ones that have already been simulated via purely classical mechanics, will ultimately require some infusion of quantum mechanical methods. This infusion has already taken place in several different types of reaction dynamics electron transfer in solution, > i> 2 HI photodissociation in rare gas clusters and solids,i i 22 >2 ° I2 photodissociation in Ar fluid,and the dynamics of electron solvation.22-24 Since calculation of the quantum dynamics of a full solvent is at present too time-consuming, all of these calculations involve a quantum solute in a classical solvent. (For a system where the solvent is treated quantum mechanically, see the quantum Monte Carlo treatment of an electron transfer reaction in water by Bader et al. O) As more complex reaaions are investigated, the techniques used in these studies will need to be extended to take into account effects involving electron dynamics such as curve crossing, the interaction of multiple electronic surfaces and other breakdowns of the Born-Oppenheimer approximation, the effect of solvent and solute polarization, and ultimately the actual detailed dynamics of the time evolution of the electronic degrees of freedom. 

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