Toward efficient chemical potential calculations by expanded ensemble simulations; to make the free energy pathway fairly level

A scheme is suggested of how to construct good bias potentials ("balancing factors") to be used in expanded ensemble (EE) calculations of chemical potentials of solutions. A combination of two strategies are used: (i) to use a pathway for particle insertions that avoids large variations in free energy and (ii) to use calculated free energy derivatives to construct a bias potential that makes the pathway fairly level. Only a few very short simulations are needed to accomplish the latter, and then, a full EE simulation is done to obtain the chemical potential. By practical calculations of the chemical potential of benzene, cyclohexane, and benzylamine in water, it is shown that this method is at least equally efficient to the recent adaptive EE (AEE) method by Aberg et al. (J. Chem. Phys. 2004, 120, 3370). Furthermore, the new method provides an alternative strategy that complements existing EE methods.     

A stepwise simulation approach to the chemical potential of fluids

A new approach to the computation of the chemical potential of fluids is presented. In this method the particle-insertion operation in the conventional test particle method is replaced by the growth of a specific particle. Application of the new technique to hard sphere and Lennard-Jones fluids shows that it is capable of providing reliable estimates of the chemical potential, even at high density where the conventional test particle methods are difficult to apply.     

A topological stochastic approach to the study of multidimensional potential energy surfaces of chemical reactions

In this article we propose a new approach for investigating the properties of multidimensional potential energy surfaces in chemical reactions, based on relations of each multidimensional surface to its one-dimensional image which is the chemical reaction tree. This approach makes it possible to find a common number of independent channels in chemical reactions for complex systems and to construct the probable models.     

Mapping patterned potential energy landscapes with diffusing colloidal probes

We report a new method for mapping patterned surfaces based on monitoring the interactions of freely diffusing colloidal probes with pattern features to generate measured potential energy landscapes. Evanescent wave scattering and video microscopy are used to track 3D center positions of nominal 2 microm silica colloids as they diffuse over 5-20-nm-thick patterned gold films. An analysis of ensemble-averaged particle height histograms on different pattern features using Boltzmann's equation produces local electrostatic and van der Waals potentials in good agreement with independent measurements and predictions. Absolute separation is obtained from theoretical fits to measured potential-energy profiles and direct measurement by depositing silica colloids onto gold surfaces via electrophoretic deposition. As colloidal probe and pattern feature dimensions become comparable, potential energy profiles suffer some distortion due to the increased probability of probes sampling pattern feature edges. An analysis of interfacial colloidal probe diffusion in conjunction with potential energy measurements demonstrates a consistent interpretation of dissipative and conservative forces in these measurements. Future extensions of this work should produce similar approaches for interrogating physical, chemical, and biomolecular heterogeneous/patterned surfaces and structures with diffusing colloidal probes.     

Monte‐Carlo program for simulating segregation and diffusion utilizing chemical potential calculations

The surface segregation phenomenon was simulated by means of a Chemical Potential Monte‐Carlo (CPMC) model, which is based on the Darken model. The chemical potential equations are rewritten to include the segregation energy associated with the surface layer. Simulations involving the change in chemical potential are performed on a two‐dimensional matrix containing two elements: the solute and the solvent elements. A random selection of an atom inside the matrix initiates the model. The change in chemical potential due to an atomic jump of a randomly selected atom to an adjacent layer is calculated. The largest change in chemical potential directs the atomic motion, complying with the conditions associated with lowering of the Gibbs free energy. Inclusion of the segregation energy (for jumps involving the surface layer) limits the number of atomic jumps from the surface layer to the bulk. Simulated segregation profiles generated by the CPMC model were compared with profiles calculated with both the modified Darken and Fick models. These profiles show that the CPMC model successfully describes both the kinetic and equilibrium conditions associated with surface segregation. Copyright © 2004 John Wiley & Sons, Ltd.     

Quantum mechanical guides to the potential energy curves of some simple molecules

The molecular orbital expression for the bond energy of a chemical bond is used to obtain some insight into the factors which produce the potential energy curves of a number of simple bonds. The resulting picture of bond formation and the potential energy curve is an electrostatic one and it depicts the potential energy curve as the sum of a long range attractive curve and a short range repulsive one. Broadly speaking, that part of the curve to the long bond length side of the minimum is determined essentially by the two electrons which form the bond and, in particular, by the `binding energy' of these two electrons. The position of the minimum and the shape of the short bond length side of the curve do depend in general on the other valence electrons of the two atoms. The long range attractive curve is easily calculated but it is difficult to get the short range repulsive curve accurately. The results may prove useful in the construction of potential energy surfaces where the long bond length side of the potential energy curve is the important part.     

Fermionic chemical potential

This paper gives a critical review of the physical meaning of the chemical potential, perhaps the most abstract of all thermodynamic quantities. To show its basic behavior, thereby to illustrate its physical significance, we have derived the chemical potential of a system of free electrons as a function of the density and temperature in different spatial dimensions. We have shown how to obtain the isothermal compressibility given the chemical potential. To emphasize the usefulness of the knowledge of dimensional dependence, both the compressibility and average kinetic energy are expressed as simple dimensional relationships of the density and, hence, the chemical potential. Finally, there is a certain temperature at which the chemical potential should identically vanish. Physical implications of zero chemical potential are discussed.     

Adsorption of heterogeneously charged nanoparticles on a variably charged surface by the extended surface complexation approach: charge regulation, chemical heterogeneity, and surface complexation

Adsorption of randomly branched polyelectrolytes, "hairy" particles and internally structured macromolecules, collectively denoted as heterogeneously charged nanoparticles, on charged surfaces is important in many technological and natural processes. In this paper, we will focus on (1) the charge regulation of both the nanoparticle and the surface and (2) the surface complexation between the particle functional groups and the surface sites and will theoretically study the adsorption using the extended surface complexation approach. The model explicitly considers the electrochemical potential of a nanoparticle with an average (smeared-out) structure and charge both in bulk solution and on the surface to obtain the equilibrium adsorption. The chemical heterogeneity of the particle is described by a distribution of the protonation constant. Detailed analysis of the chemical potential of the adsorbed nanoparticle reveals that the pH and salt dependence of the adsorption can be largely explained by the balance between an energy gain resulting from the particle and surface charge regulation and the surface complexation and an energy loss from the unfavorable interparticle electrostatic repulsion close to the surface. This conclusion is also supported by the strong impacts that the chemical heterogeneity of the particle functional groups, the magnitude of the surface complexation, the number of the functional groups, and the size of the particle have on the adsorption.     

Direct first-principles chemical potential calculations of liquids

We propose a scheme that drastically improves the efficiency of Widom's particle insertion method by efficiently sampling cavities while calculating the integrals providing the chemical potentials of a physical system. This idea enables us to calculate chemical potentials of liquids directly from first-principles without the help of any reference system, which is necessary in the commonly used thermodynamic integration method. As an example, we apply our scheme, combined with the density functional formalism, to the calculation of the chemical potential of liquid copper. The calculated chemical potential is further used to locate the melting temperature. The calculated results closely agree with experiments.     

The chemical potential for dilute hard-sphere mixtures

Accurate Monte Carlo evaluation of the probability of inserting an additional particle of arbitrary size into a hard-sphere fluid at various densities allows a quantitative check on the scaled particle interpolation formula for this probability, which is rigorously known when the added particle is either very small or very large. The simple scaled particle formula is remarkably accurate due to a favorable choice of the functional dependence of the surface tension on curvature. The biggest deviation occurs at liquid-like densities where the insertion probability is about 20% larger for larger particles, indicating a larger probability of occurrence of larger density fluctuations, and resulting in a smaller (3%) excess chemical potential than the simple theory predicts. On the other hand, at lower densities the insertion probability for large particles is slightly smaller than the theory predicts.     

On the importance of the "density per particle" (shape function) in the density functional theory

The central role of the shape function sigma(r) from the density functional theory (DFT), the ratio of the electron density rho(r) and the number of electrons N of the system (density per particle), is investigated. Moreover, its relationship with DFT based reactivity indices is established. In the first part, it is shown that an estimate for the chemical hardness can be obtained from the long range behavior of the shape function and its derivative with respect to the number of electrons at a fixed external potential. Next, the energy of the system is minimized with the constraint that the shape function should integrate to unity; the associated Lagrange multiplier is shown to be related to the electronic chemical potential micro of the system. Finally, the importance of the shape function for both molecular structure, reactivity, and similarity is outlined.     

The potential energy landscape contribution to the dynamic heat capacity

The dynamic heat capacity of a simple polymeric, model glassformer was computed using molecular dynamics simulations by sinusoidally driving the temperature and recording the resultant energy. The underlying potential energy landscape of the system was probed by taking a time series of particle positions and quenching them. The resulting dynamic heat capacity demonstrates that the long time relaxation is the direct result of dynamics resulting from the potential energy landscape. Moreover, the equilibrium (low frequency) portion of the potential energy landscape contribution to the heat capacity is found to increase rapidly at low temperatures and at high packing fractions. This increase in the heat capacity is explained by a statistical mechanical model based on the distribution of minima in the potential energy landscape.     

Reactions over multiple, interconnected potential wells: unimolecular and bimolecular reactions on a C3H5 potential

In this article we analyze quantitatively and discuss in detail a number of reactions that take place on a C3H5 potential. These reactions include the reaction of hydrogen atoms with allene and propyne, the reaction of methyl with acetylene, the isomerization of cyclopropyl to allyl, and the dissociation of allyl, 1-propenyl, and 2-propenyl. The theory employs high-level electronic-structure methods to characterize the potential energy surface, RRKM theory to calculate microcanonical, J-resolved rate coefficients, and master-equation methods to determine phenomenological rate coefficients, k(T,p). The agreement between our theory and the experimental results available is very good. The final theoretical results are cast in a form that is convenient for use in chemical kinetics modeling.     

Efficient solvent boundary potential for hybrid potential simulations

A common challenge in computational biophysics is to obtain statistical properties similar to those of an infinite bulk system from simulations of a system of finite size. In this work we describe a computationally efficient algorithm for performing hybrid quantum chemical/molecular mechanical (QC/MM) calculations with a solvent boundary potential. The system is partitioned into a QC region within which catalytic reactions occur, a spherical region with explicit solvent that envelops the quantum region and is treated with a MM model, and the surrounding bulk solvent that is treated implicitly by the boundary potential. The latter is constructed to reproduce the solvation free energy of a finite number of atoms embedded inside a low-dielectric sphere with variable radius, and takes into account electrostatic and van der Waals interactions between the implicit solvent and the QC and MM atoms in the central region. The method was implemented in the simulation program pDynamo and tested by examining elementary steps in the reaction mechanisms of two enzymes, citrate synthase and lactate dehydrogenase. Good agreement is found for the energies and geometries of the species along the reaction profiles calculated with the method and those obtained by previous experimental and computational studies. Directions in which the utility of the method can be further improved are discussed.     

Optimal performance of a generalized irreversible four-reservoir isothermal chemical potential transformer

A new cyclic model of a four-reservoir isothermal chemical potential transformer with irreversible mass transfer, mass leakage and internal dissipation is put forward in this paper. The optimal relation be-tween the coefficient of performance (COP) and the rate of energy pumping of the generalized irre-versible four-reservoir isothermal chemical potential transformer has been derived by using finite-time thermodynamics or thermodynamic optimization. The maximum COP and the corresponding rate of energy pumping, as well as the maximum rate of energy pumping and the corresponding COP, have been obtained. Moreover, the influences of the irreversibility on the optimal performance of the iso-thermal chemical potential transformer have been revealed. It was found that the mass leakage affects the optimal performance both qualitatively and quantitatively, while the internal dissipation affects the optimal performance quantitatively. The results obtained herein can provide some new theoretical guidelines for the optimal design and development of a class of isothermal chemical potential trans-formers, such as mass exchangers, electrochemical, photochemical and solid state devices, fuel pumps, etc.     

Sedimentation velocity and potential in concentrated suspensions of charged porous spheres

The body-force-driven migration in a homogeneous suspension of polyelectrolyte molecules or charged flocs in an electrolyte solution is analyzed. The model used for the particle is a porous sphere in which the density of the hydrodynamic frictional segments, and therefore also that of the fixed charges, is constant. The effects of particle interactions are taken into account by employing a unit cell model. The overlap of the electric double layers of adjacent particles is allowed and the relaxation effect in the double layer surrounding each particle is considered. The electrokinetic equations which govern the electrostatic potential profile, the ionic concentration (or electrochemical potential energy) distributions, and the fluid velocity field inside and outside the porous particle in a unit cell are linearized by assuming that the system is only slightly distorted from equilibrium. Using a regular perturbation method, these linearized equations are solved for a symmetrically charged electrolyte with the density of the fixed charges as the small perturbation parameter. An analytical expression for the settling velocity of the charged porous sphere is obtained from a balance among its gravitational, electrostatic, and hydrodynamic forces. A closed-form formula for the sedimentation potential in a suspension of identical charged porous spheres is also derived by using the requirement of zero net electric current. The dependence of the sedimentation velocity and potential of the suspension on the particle volume fraction and other properties of the particle-solution system is found to be quite complicated.     

Molecular dynamics simulation in the grand canonical ensemble

An extended system Hamiltonian is proposed to perform molecular dynamics (MD) simulation in the grand canonical ensemble. The Hamiltonian is similar to the one proposed by Lynch and Pettitt (Lynch and Pettitt, J Chem Phys 1997, 107, 8594), which consists of the kinetic and potential energies for real and fractional particles as well as the kinetic and potential energy terms for material and heat reservoirs interacting with the system. We perform a nonlinear scaling of the potential energy parameters of the fractional particle, as well as its mass to vary the number of particles dynamically. On the basis of the equations of motion derived from this Hamiltonian, an algorithm has been proposed for MD simulation at constant chemical potential. The algorithm has been tested for the ideal gas, for the Lennard-Jones fluid over a wide range of temperatures and densities, and for water. The results for the low-density Lennard-Jones fluid are compared with the predictions from a truncated virial equation of state. In the case of the dense Lennard-Jones fluid and water our predicted results are compared with the results reported using other available methods for the calculation of the chemical potential. The method is also applied to the case of vapor-liquid coexistence point predictions.     

Reaction path potential for complex systems derived from combined ab initio quantum mechanical and molecular mechanical calculations

Combined ab initio quantum mechanical and molecular mechanical calculations have been widely used for modeling chemical reactions in complex systems such as enzymes, with most applications being based on the determination of a minimum energy path connecting the reactant through the transition state to the product in the enzyme environment. However, statistical mechanics sampling and reaction dynamics calculations with a combined ab initio quantum mechanical (QM) and molecular mechanical (MM) potential are still not feasible because of the computational costs associated mainly with the ab initio quantum mechanical calculations for the QM subsystem. To address this issue, a reaction path potential energy surface is developed here for statistical mechanics and dynamics simulation of chemical reactions in enzymes and other complex systems. The reaction path potential follows the ideas from the reaction path Hamiltonian of Miller, Handy and Adams for gas phase chemical reactions but is designed specifically for large systems that are described with combined ab initio quantum mechanical and molecular mechanical methods. The reaction path potential is an analytical energy expression of the combined quantum mechanical and molecular mechanical potential energy along the minimum energy path. An expansion around the minimum energy path is made in both the nuclear and the electronic degrees of freedom for the QM subsystem internal energy, while the energy of the subsystem described with MM remains unchanged from that in the combined quantum mechanical and molecular mechanical expression and the electrostatic interaction between the QM and MM subsystems is described as the interaction of the MM charges with the QM charges. The QM charges are polarizable in response to the changes in both the MM and the QM degrees of freedom through a new response kernel developed in the present work. The input data for constructing the reaction path potential are energies, vibrational frequencies, and electron density response properties of the QM subsystem along the minimum energy path, all of which can be obtained from the combined quantum mechanical and molecular mechanical calculations. Once constructed, it costs much less for its evaluation. Thus, the reaction path potential provides a potential energy surface for rigorous statistical mechanics and reaction dynamics calculations of complex systems. As an example, the method is applied to the statistical mechanical calculations for the potential of mean force of the chemical reaction in triosephosphate isomerase.     

Influence of roughness and capillary size on the zeta potential values obtained by streaming potential measurements

Streaming potential measurements are performed to determine the zeta potential of flat surfaces, particles, or fibers. Although the zeta potential is a well-defined property of solid surfaces in a liquid, there are indications that the absolute values of the zeta potential calculated using the Helmholtz-Smoluchowski equation are affected by surface roughness and—in case of particle or fiber assemblies—their packing density. The study at hand investigates these influences using flat polymer surfaces with different roughness and topography and assemblies of basalt spheres. It was found that increasing roughness of the flat surface and larger size or smaller number of particles in particle assemblies result in flatter slopes of the streaming potential versus pressure and thus lower apparent absolute values of the zeta potential. The interpretation of streaming potential measurements should therefore not focus on absolute zeta potential values but on trends in pH- and concentration-dependent measurements.