Thermodynamics and Molecular Simulation

In the previous chapter we have learned that analytic calculations on the basis of microscopic interactions can become very dififlcult or even impossible. In such cases computer simulations are helpful. And even though thermodynamics is not the theory of many particle systems based on microscopic interactions. Statistical Mechanics is this theory, it possesses noteworthy ties to computer simulation. In the following we want to discuss some of them. 

There are two main methods in this field. One is Molecular Dynamics and the other is Monte Carlo. Additional simulation methods are either closely related to one or the other aforementioned methods or they apply on spatial scales far beyond the molecular scale. Molecular Dynamics techniques model a small amount of material (system sizes usually are on the nm-scale) based on the actual equations of motion of the atoms or molecules in this system. Usually this is done on the basis of mechanical inter- and intra-particle potential functions. In certain cases however quantum mechanics in needed. Monte Carlo differs from Molecular Dynamics in that its systems do not follow their physical dynamics. Monte Carlo estimates thermodynamic quantities via intelligent statistical sampling of (micro)states. Capabilities and applications of both methods overlap widely. But they both also have distinct advantages depending on the problem at hand. Here we concentrate on Monte Carlo—which is the more thermodynamic method of the two. 

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Hideo Nishiumi, Molecular Thermodynamics and Molecular Simulation, MTMS 97. Based on the Second International Symposium held 12-15 January 1997, Hosei University, Tokyo, Japan, in Fluid Phase Equilib., 144 (1-2), Elsevier, Amsterdam, The Netherlands, 1998. 

Michael E. Paulaitis is Professor of Chemical and Biomolecular Engineering and Ohio Eminent Scholar at Ohio State University. He is also Director of the Institute of Multiscale Modeling of Biological Interactions at Johns Hopkins University. His research focuses on molecular thermodynamics of hydration, protein solution thermodynamics, and molecular simulations of biological macromolecules. 

To establish the molecular thermodynamic model for uniform systems based on concepts from statistical mechanics, an effective method by combining statistical mechanics and molecular simulation has been recommended (Hu and Liu, 2006). Here, the role of molecular simulation is not limited to be a standard to test the reliability of models. More directly, a few simulation results are used to determine the analytical form and the corresponding coefficients of the models. It retains the rigor of statistical mechanics, while mathematical difficulties are avoided by using simulation results. The method is characterized by two steps (1) based on a statistical-mechanical derivation, an analytical expression is obtained first. The expression may contain unknown functions or coefficients because of mathematical difficulty or sometimes because of the introduced simplifications. (2) The form of the unknown functions or unknown coefficients is then determined by simulation results. For the adsorption of polymers at interfaces, simulation was used to test the validity of the weighting function of the WDA in DFT. For the meso-structure of a diblock copolymer melt confined in curved surfaces, we found from MC simulation that some more complex structures exist. From the information provided by simulation, these complex structures were approximated as a combination of simple structures. Then, the Helmholtz energy of these complex structures can be calculated by summing those of the different simple structures. 

As expected, the total interaction energies depend strongly on the van der Waals radii (of both sorbate and sorbent atoms) and the surface densities. This is true for both HK type models (Saito and Foley, 1991 Cheng and Yang, 1994) and more detailed statistical thermodynamics (or molecular simulation) approaches (such as Monte Carlo and density functional theory). Knowing the interaction potential, molecular simulation techniques enable the calculation of adsorption isotherms (see, for example, Razmus and Hall, (1991) and Cracknell etal. (1995)). 

The calculations reported in this paper and a related series of publications indicate that it is now quite feasible to obtain reasonably accurate results for phase equilibria in simple fluid mixtures directly from molecular simulation. What is the possible value of such results Clearly, because of the lack of accurate intermolecular potentials optimized for phase equilibrium calculations for most systems of practical interest, the immediate application of molecular simulation techniques as a replacement of the established modelling methods is not possible (or even desirable). For obtaining accurate results, the intermolecular potential parameters must be fitted to experimental results, in much the same way as parameters for equation-of-state or activity coefficient models. This conclusion is supported by other molecular-simulation based predictions of phase equilibria in similar systems (6). However, there is an important difference between the potential parameters in molecular simulation methods and fitted parameters of thermodynamic models. Molecular simulation calculations, such as the ones reported here, involve no approximations beyond those inherent in the potential models. The calculated behavior of a system with assumed intermolecular potentials is exact for any conditions of pressure, temperature or composition. Thus, if a good potential model for a component can be developed, it can be reliably used for predictions in the absence of experimental information. 

The Kirkwood-Buff theory of solutions was originally formulated to obtain thermodynamic quantities from molecular distribution functions. This formulation is useful whenever distribution functions are available either from analytical calculations or from computer simulations. The inversion procedure of the same theory reverses the role of the thermodynamic and molecular quantities, i.e., it allows the evaluation of integrals over the pair correlation functions from thermodynamic quantities. These integrals Gy, referred to as the Kirkwood-Buff integrals (KBIs), were found useful in the study of mixtures on... 

Figure 0.2 By combining molecular theory, thermodynamics, experimental data, and molecular simulation, thermodynamic modeling simplifies and extends descriptions of physical and chemical properties. This contributes to the reliable and accurate design, optimization, and operation of engineering processes and equipment. Note the distinction between molecular models used in molecular simulation and macroscopic models used in thermodynamics.

Many additional criteria have been defined. For examjde, in tte context of a temperature-jump simulation one can study tte rate of apprc ich of various thermodynamic and molecular properties to their final equilibrium values. Such properties are the configurational internal energy and the density of the system (for NPT simulations), as well as the average end-to-end distance and radius of gyration of the chains. Such a study was not undertaken in this work. 

Atomic and molecular simulation methods can generally be categorized as either equilibrated or dynamic. Static simulations attempt to determine the structural and thermodynamic properties such as crystal structure, sorption isotherms, and sorbate binding. Structural simulations are often carried out using energy minimization schemes that are similar to molecular mechanics. Elquihbrium prop>erties, on the other hand, are based on thermodynamics and thus rely on statistical mechanics and simulating the system state function. Monte Carlo methods are then used to simulate these systems stochastically. 

Maginn [87] and Margulis [24, 88] presented their studies on the transport properties of ionic liquids and explained why these media have to be regarded as a different class of solvents. Several other groups associated spectroscopic techniques and molecular simulation calculations to assess the interfacial behaviour of the solutions containing ionic liquids. The use of AA force fields allows an accurate balance between the specific intermolecular interactions and the structural/conformational effects responsible for the properties determined experimentally, whether they are thermodynamic or spectroscopic. 

Recent advances in statistical thermodynamics and better understanding of intra- and intermo-lecular interactions, thanks to accurate experimental measurements and molecular simulations using realistic force fields, have contributed significantly to this end. Many of the recent thermodynamic models based on statistical mechanics are rooted in the pioneering work of Guggenheim [1] and Flory [2] on lattice models for complex fluids, including polymers. The lattice fluid (LF) theory of Sanchez and Lacombe [3,4] is probably one of the most widely used and successful lattice models. 

Many experiments and molecular simulations of the freezing of fluids confined in nanoporous solids have been reported [1]. This effort is devoted to the understanding of the effect of confinement, surface forces, and reduced dimensionality on the thermodynamics of fluids. These works are also of practical interest for applications involving confined systems (lubrication in nanotechnologies, synthesis of nano-structured materials, phase separation, etc). Beside the abundant literature for pure fluids in nanopores, few studies [2-7] have focused on the freezing of confined mixtures. As in the case of pure substances, the pore width H and the ratio of the wall/fluid to the fluid/fluid interaetions (parameter a [8]), play an important role in the phase behavior of the mixture. The ratio of the wall/fluid interaction for the two species is also a key parameter in describing freezing of these systems. 

Statistical thermodynamics and computer simulations showed that the density profiles of hard-sphere and Lennard-J ones fluids normal to a planar interface oscillate about the bulk density with a periodicity of roughly one molecular diameter [1079-1086], The oscillations decay exponentially and extend over a few molecular diameters. In this range, the molecules are ordered in layers. The amplitude and range of density fluctuations depend on the specific boundary condition at the wall and on the size and interaction between the molecules. A steep repulsive wall-fluid... 

Here, Aq is the maximal amplitude of the oscillation. Such an exponentially decaying, oscillating density is justified by statistical thermodynamics and computer simulations of simple fiuids [1079-1086]. The exponential factor with the characteristic decay length X ensures that the force decreases with distance and vanishes for large distance. A cosine (or sine) is the first-order approximation for any periodic function. Here, do is the layer thickness, which in the case of simple liquids is close to the molecular size. We assumed that the density is maximal (or minimal for negative Aq) directly at the surface at = 0. 

Two simulation methods—Monte Carlo and molecular dynamics—allow calculation of the density profile and pressure difference of Eq. III-44 across the vapor-liquid interface [64, 65]. In the former method, the initial system consists of N molecules in assumed positions. An intermolecule potential function is chosen, such as the Lennard-Jones potential, and the positions are randomly varied until the energy of the system is at a minimum. The resulting configuration is taken to be the equilibrium one. In the molecular dynamics approach, the N molecules are given initial positions and velocities and the equations of motion are solved to follow the ensuing collisions until the set shows constant time-average thermodynamic properties. Both methods are computer intensive yet widely used. 

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