Continuum representation

In this section, we discuss applications of the FEP formalism to two systems and examine the validity of the second-order perturbation approximation in these cases. Although both systems are very simple, they are prototypes for many other systems encountered in chemical and biological applications. Furthermore, the results obtained in these examples provide a connection between molecular-level simulations and approximate theories, especially those based on a dielectric continuum representation of the solvent. 

Sampling the conformational space of solute(s) by MC or MD algorithms requires many intramolecular and solvation calculations and accordingly simplicity in the solute Hamiltonian and computer efficiency in the continuum method used to compute solvation are key requirements. This implies that, with some exceptions [1], MD/MC algorithms are always coupled to purely classical descriptions of solvation, which in order to gain computer efficiency adopt severe approximations, such as the neglect of explicit electronic polarization contributions to solvation (for a discussion see ref. [1]). In the following we will summarize the major approaches used to couple MD/MC with continuum representations of solvation. 

We shall present in this contribution to the Annals a survey of some recent advancements in the description of the properties of molecules in solution obtained by using a continuum representation of the solvent [1], The exposition will be kept at a relatively low formal level, our intention being to give a contribution not addressed to specialists in the field. In this introduction, we will report some comments to make the understanding of the approach easier. 

Pore networks in 2-D and 3-D are still being developed as computer-aided representations of real porous materials since the idea was first proposed some 40 years ago (2). Subsequently, 2-D networks were applied to porosimetry (5) and low-temperature gas adsorption (4), and 2-D and 3-D models have been compared (5). More recent work has applied 3-D networks to porosimetry (< ), to flow and transport behaviour (7), as well as to diffusion and reaction in catalysis (S). The equivalence of pore networks to a continuum representation for porosity has lately been established (9) and a review of recent developments and applications is available 10). 

Two-way embedded interfacing methods. These involve embedding an atomistic or CG model within a continuum representation. Implicit solvent models fall into this category. New multiscale methods, which capture hydrodynamic and mechanical effects have now also been developed. 

Finally, on time scales that include many thermal impulses, the overdamped dynamics can be cast in a continuum representation where the density of states p x, t) at location x and time t obeys Smoluchowski transport, ... 

Since the dielectric continuum representation of the solvent has significant limitations, the molecular dynamics simulation of PCET with explicit solvent molecules is also an important direction. One approach is to utilize a multistate VB model with explicit solvent interactions [34-36] and to incorporate transitions among the adiabatic mixed electronic/proton vibrational states with the Molecular Dynamics with Quantum Transitions (MDQT) surface hopping method [39, 40]. The MDQT method has already been applied to a one-dimensional model PCET system [39]. The advantage of this approach for PCET reactions is that it is valid in the adiabatic and non-adiatic limits as well as in the intermediate regime. Furthermore, this approach is applicable to PCET in proteins as well as in solution. 

Bulk solvent alone is not able to describe these effects which are to be ascribed to the quantum behaviour of a single water molecule. Bulk solvent, in its continuum representation, indeed modifies the picture of the reaction mechanism we have resumed here. Its effect on the reaction energy profile... 

The basic assumptions in fluid mechanics are thus that for lengths and time scales much larger than the characteristic molecular lengths and times, the continuum representation provides a quantitative correct description of the fluid dynamic behavior of the system. In general, the differential description is useful for processes where there is a wide separation of scales between the smallest macroscopic scales of interest and the microscopic scales associated with the internal structure of the fluid. The mean free path which is of the order of for gases is commonly used as a suitable characteristic... 

The formulations of the population balance equation based on the continuum mechanical approach can be split into two categories, the macroscopic- and the microscopic population balance equation formulations. The macroscopic approach consists in describing the evolution in time and space of several groups or classes of the dispersed phase properties. The microscopic approach considers a continuum representation of a particle density function. 

To avoid complications resulting from a detailed description of the atomic structure of the heterogeneous substrate surfaces, we employ a continuum representation of the fluid substrate potential. Details of its derivation can be found in Appendix E.2. Combining the expressions in Eqs. (E.21), (E.24), (E.26), (E.27), and (E.29), we obtain for the potential energy of a film molecule in the continuum representation of the substrate k (= 1,2)... 

As a first illustration we consider the model discussed in Section 1.3.3, namely a fluid of simple molecules confined between chemically striped solid surfaces (see Fig. 5.2). As before in Section 5.4 we treat the confined fluid as a thermodynamically open system. Hence, equilibrium states correspond to minima of the grand potential 11 introduced in Eqs. (1.66) and (1.67). The fluid fluid interaction is described by the intermolecular potential ug (r) introduced in Eq. (5.38) where the associated shifted-force potential is introduced in Eq. (5.39). The fluid substrate interaction is described by 1 1 (x, z) in the continuum representation [see Eq. (5.68)], where x replaces x because of the misaligmncnt of the sul)stratcs relative to each other [see Eq. (5.103)]. 

Because we are concerned in this tutorial with the effects of chemical heterogeneity at the nanoscale on the behavior of the confined film, we expect the details of the atomic structure not to matter greatly for our purpose. Therefore, we adopt a continuum representation of the interaction of a film molecule with the substrate, which we obtain by averaging the film substrate interaction potential over positions of substrate atoms in the x-y plane. The resulting continuum potential can be expressed as... 

The reactor shown in Figure 7.3 has enough catalyst so that a continuum representation is adequate. 

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