Rate theory of gas-liquid nucleation: Quest for the elusive quantitative accuracy.

The task of a first principles theoretical calculation of the rate of gas-liquid nucleation has remained largely incomplete despite the existence of reliable results from unbiased simulation studies at large supersaturation. Although the classical nucleation theory formulated by Becker-Doring-Zeldovich about a century ago provides an elegant, widely used picture of nucleation in a first-order phase transition, the theory finds difficulties in predicting the rate accurately, especially in the case of gas-to-liquid nucleation. Here, we use a multiple-order parameter description to construct the nucleation free energy surface needed to calculate the nucleation rate. A multidimensional non-Markovian (MDNM) rate theory formulation that generalizes Langer's well-known nucleation theory by using the Grote-Hynes MDNM treatment is used to obtain the rate of barrier crossing. We find good agreement of the theory with the rate obtained by direct unbiased molecular dynamics simulations-the latter is feasible at large supersaturation, S. The theory gives an experimentally strong dependence of the rate of nucleation on supersaturation, S. Interestingly, we find a strong influence of the frequency-dependent friction coefficient at the barrier top. This arises from multiple recrossings of the barrier surface. We find that a Markovian theory, such as Langer's formulation, fails to capture the rate quantitatively. In addition, the multidimensional transition state theory expression performs poorly, revealing the underlying role of the friction coefficient.

Exploration of Stokes hydrodynamic law at molecular length scales.

The celebrated generalized Stokes law predicts that the velocity of a particle pulled through a liquid by an external force, Fex, is directly proportional to the force and inversely proportional to the friction ζ acted by the medium on the particle. We investigate the range of validity of the generalized Stokes law at molecular length scales by employing computer simulations to calculate friction by pulling a tagged particle with a constant force. We thus calculate friction for two model interaction potentials, Lennard-Jones and soft sphere, for several particle sizes, ranging from radius (a) smaller than the solvent particles to three times larger. We next obtain friction from diffusion (D) by using Einstein's relation between diffusion and friction ζ in an unperturbed liquid. We find a quantitative agreement between the two at a small-to-intermediate pulling force regime for all the sizes studied. The law does break down at a large pulling force beyond a threshold value. Importantly, the range of validity of Stokes' scheme to obtain friction increases substantially if we turn off the attractive part of the interaction potential. Additionally, we calculate the viscosity (η) of the unperturbed liquid and find a good agreement with the Stokes-Einstein relation ζ = Cηa for the viscosity dependence with a value of C close to 5 π, which is intermediate between the slip and stick boundary condition.

Sensitivity of nonequilibrium relaxation to interaction potentials: Timescales of response from Boltzmann's H function.

We investigate, by simulations and analytic theory, the sensitivity of nonequilibrium relaxation to interaction potential and dimensionality by using Boltzmann's H function H(t). We evaluate H(t) for three different intermolecular potentials in all three dimensions and find that the well-known H theorem is valid and that the H function exhibits rather strong sensitivity to all these factors. The relaxation of H(t) is long in one dimension, but short in three dimensions, longer for the Lennard-Jones potential than for the hard spheres. The origin of the ultraslow approach to the equilibrium of H(t) in one-dimensional systems is discussed. Importantly, we obtain a closed-form analytic expression for H(t) using the solution of the Fokker-Planck equation for velocity space probability distribution and compare its predictions with the simulation results. Interestingly, H(t) is found to exhibit a linear response when vastly different initial nonequilibrium conditions are employed. The microscopic origin of this linear response is discussed. The oft-quoted relation of H function with Clausius's entropy theorem is discussed.

Diffusion in a two-dimensional energy landscape in the presence of dynamical correlations and validity of random walk model.

Diffusion in a multidimensional energy surface with minima and barriers is a problem of importance in statistical mechanics and it also has wide applications, such as protein folding. To understand it in such a system, we carry out theory and simulations of a tagged particle moving on a two-dimensional periodic potential energy surface, both in the presence and absence of noise. Langevin dynamics simulations at multiple temperatures are carried out to obtain the diffusion coefficient of a solute particle. Friction is varied from zero to large values. Diffusive motion emerges in the limit of a long time, even in the absence of noise. Noise destroys the correlations and increases diffusion at small friction. Diffusion thus exhibits a nonmonotonic friction dependence at the intermediate value of the damping, ultimately converging to our theoretically predicted value. The latter is obtained using the well-established relationship between diffusion and random walk. An excellent agreement is obtained between theory and simulations in the high-friction limit but not so in the intermediate regime. We explain the deviation in the low- to intermediate-friction regime using the modified random walk theory. The rate of escape from one cell to another is obtained from the multidimensional rate theory of Langer. We find that enhanced dimensionality plays an important role. To quantify the effects of noise on the potential-imposed coherence on the trajectories, we calculate the Lyapunov exponent. At small friction values, the Lyapunov exponent mimics the friction dependence of the rate.

Ethanol exchange between two graphene surfaces in nanoconfined aqueous solution: Rate and mechanism.

We observe, by computer simulations, a remarkable long-distance, rare, but repetitive, exchange of ethanol molecules between two parallel graphene surfaces in nanoconfined, aqueous, ethanol solutions. We compute the rate of exchange as a function of the separation (d) between the two surfaces. We discover that the initiating (or, the launching) step in this exchange is the attainment of an instantaneous orientation of the carbon-oxygen bond vector relative to the graphene surface. This observation led us to construct a two-dimensional free energy surface for this exchange, with respect to two order parameters, namely, (i) the perpendicular distance of ethanol molecule from the graphene surfaces, z, and (ii) the orientation of the O-C bond vector, θ, of the tagged ethanol molecule. For d = 3 nm, the rate of exchange is found to be 0.44 ns-1 for the force field used. We also vary the force field and determine the sensitivity of the rate. From the free energy landscape, one could determine the minimum energy pathway. We use both, the transition state theory and Kramers' theory, to calculate the rate. The calculated rate agrees well with the simulated value as mentioned above. We find that the rate of exchange phenomenon is sensitive to the interaction strength of graphene and the hydrophobic group of ethanol. The free energy landscape exchange shows dependence on the distance separation of the two hydrophobic surfaces and reveals interesting features.

Non-Markovian rate theory on a multidimensional reaction surface: Complex interplay between enhanced configuration space and memory.

A theory of barrier crossing rate on a multidimensional reaction energy surface is presented. The theory is a generalization of the earlier theoretical schemes to higher dimensions, with the inclusion of non-Markovian friction along both the reactive and the nonreactive coordinates. The theory additionally includes the bilinear coupling between the reactive and the nonreactive modes at the Hamiltonian level. Under suitable conditions, we recover the rate expressions of Langer and Hynes and establish a connection with the rate treatment of Pollak. Within the phenomenology of generalized Langevin equation description, our formulation provides an improvement over the existing ones because we explicitly include both the non-Markovian effects along the reaction coordinate and the bilinear coupling at the Hamiltonian level. At intermediate-to-large friction, an increase in dimensionality by itself tends to reduce the rate, while the inclusion of the memory effects increases the rate. The theory predicts an increase in rate when off-diagonal friction terms are included. We present a model calculation to study isomerization of a stilbene-like molecule using the prescription of Hochstrasser and co-workers on a two-dimensional reaction energy surface, employing Zwanzig-Bixon hydrodynamic theory of frequency-dependent friction. The calculated rate shows a departure from the predictions of Langer's theory and also from the two-dimensional transition state theory.

Tug-of-War between Internal and External Frictions and Viscosity Dependence of Rate in Biological Reactions.

The role of water in biological processes is studied in three reactions, namely, the Fe-CO bond rupture in myoglobin, GB1 unfolding, and insulin dimer dissociation. We compute both internal and external components of friction on relevant reaction coordinates. In all of the three cases, the cross-correlation between forces from protein and water is found to be large and negative that serves to reduce the total friction significantly, increase the calculated reaction rate, and weaken solvent viscosity dependence. The computed force spectrum reveals bimodal 1/f noise, suggesting the use of a non-Markovian rate theory.

Structural Stability of Insulin Oligomers and Protein Association-Dissociation Processes: Free Energy Landscape and Universal Role of Water.

Association and dissociation of proteins are important biochemical events. In this Feature Article, we analyze the available studies of these processes for insulin oligomers in aqueous solution. We focus on the solvation of the insulin monomer in water, stability and dissociation of its dimer, and structural integrity of the hexamer. The intricate role of water in solvation of the dimer- and hexamer-forming surfaces, in long-range interactions between the monomers and the stability of the oligomers, is discussed. Ten water molecules inside the central cavity stabilize the structure of the insulin hexamer. We discuss how different order parameters can be used to understand the dissociation of the insulin dimer. The calculation of the rate using a recently computed multidimensional free energy provides considerable insight into the interplay between protein and water dynamics.

Rate of Insulin Dimer Dissociation: Interplay between Memory Effects and Higher Dimensionality.

We calculate the rate of dissociation of an insulin dimer into two monomers in water. The rate of this complex reaction is determined by multiple factors that are elucidated. By employing advanced sampling techniques, we first obtain the reaction free energy surface for the dimer dissociation as a function of two order parameters, namely, the distance between the center-of-mass of two monomers (R) and the number of cross-contacts (Q) among the backbone Cα atoms of two monomers. We then construct an orthogonal 2D reaction energy surface by introducing the reaction coordinate X to denote the minimum energy pathway and a conjugate coordinate Y that spans the orthogonal direction. The free energy landscape is rugged with multiple maxima and minima. We calculate the rate by employing not only the non-Markovian multidimensional rate theory but also several other theoretical approaches. The necessary reaction frequencies and the frictions are calculated from the time correlation function formalism. Our best estimate of the rate is 0.4 µs-1. Our study reveals interesting opposite influences of dimensionality and memory in determining the rate constant of the reaction. We gain interesting insights into the dimer dissociation process by looking directly at the trajectories obtained from molecular dynamics simulation.

Unfolding of Dynamical Events in the Early Stage of Insulin Dimer Dissociation.

The dissociation of an insulin dimer is an important biochemical event that could also serve as a prototype of dissociations in similar biomolecular assemblies. We use a recently developed multidimensional free energy landscape for insulin dimer dissociation to unearth the microscopic and mechanistic aspects of the initial stages of the process that could hold the key to understanding the stability and the rate. The following sequence of events occurs in the initial stages: (i) The backbone hydrogen bonds break partially at the antiparallel ß-sheet junction, (ii) the two α-helices (chain B) move away from each other while several residues (chain A) move closer, and (iii) a flow of adjacent water molecules occurs into the junction region. Interestingly, the intermonomeric center-to-center distance does not increase, but the number of native contacts exhibits a sharp decrease. Subsequent steps involve further disengagement of hydrophobic groups. This process is slow because of an entropic bottleneck created by the existence of the large configuration space available in the native state (NS), which is inhabited by low-frequency conformational fluctuations. We carry out a density-of-states analyses in the dimer NS to unearth distinctive features not present in the monomers. These low-frequency modes are also responsible for a large entropic stabilization of the NS. Hydrophobic disengagement in the early stage leads to the formation of a twisted intermediate state which itself is a metastable minimum (IS-1). The subsequent progress leads to another dimeric complex (IS-2), which is on the dissociative pathway and characterized by a further decrease in the native contacts. The dissociation process provides insights into the workings of a biomolecular assembly.

Study of entropy-diffusion relation in deterministic Hamiltonian systems through microscopic analysis.

Although an intimate relation between entropy and diffusion has been advocated for many years and even seems to have been verified in theory and experiments, a quantitatively reliable study and any derivation of an algebraic relation between the two do not seem to exist. Here, we explore the nature of this entropy-diffusion relation in three deterministic systems where an accurate estimate of both can be carried out. We study three deterministic model systems: (a) the motion of a single point particle with constant energy in a two-dimensional periodic potential energy landscape, (b) the same in the regular Lorentz gas where a point particle with constant energy moves between collisions with hard disk scatterers, and (c) the motion of a point particle among the boxes with small apertures. These models exhibit diffusive motion in the limit where ergodicity is shown to exist. We estimate the self-diffusion coefficient of the particle by employing computer simulations and entropy by quadrature methods using Boltzmann's formula. We observe an interesting crossover in the diffusion-entropy relation in some specific regions, which is attributed to the emergence of correlated returns. The crossover could herald a breakdown of the Rosenfeld-like exponential scaling between the two, as observed at low temperatures. Later, we modify the exponential relation to account for the correlated motions and present a detailed analysis of the dynamical entropy obtained via the Lyapunov exponent, which is rather an important quantity in the study of deterministic systems.

Rotation of small diatomics in water-ethanol mixture: Multiple breakdowns of hydrodynamic predictions.

We study the rotational and translational dynamics of three small important linear molecules, namely, carbon monoxide (CO), nitric oxide (NO), and cyanide ion (CN-) in water-ethanol mixtures, at different compositions. Here, we report a detailed study of the dynamics of these diatomics in water-ethanol binary mixtures for the first time. We find multiple anomalous results, namely, (i) faster rotational motion of CO and NO than CN-, (ii) larger translational diffusion of CO and NO in pure ethanol than in water but the reverse for CN-, (iii) a pronounced anomaly in the composition dependence of translational-rotational dynamics at low ethanol composition, and (iv) a re-entrant type behavior in the viscosity dependence of orientational relaxation. We compare our simulation result-based observations with the existing experimental results wherever available and find that the simulation results are in reasonably good agreement with the experiments. We implement, for the first time, a calculation of the rotational binary friction following the sophisticated scheme of Evans and co-workers. We also calculate the hydrodynamic predictions for the solute molecules. On comparison with the simulation and experimental results, we find that neither the rotational binary friction nor the hydrodynamic expressions (both stick and slip) can reproduce the results. To rectify the situation, we develop a detailed mode-coupling theory and suggest that such an approach if completely implemented can provide a more reliable description than the hydrodynamic approach. Many of our results could be tested in experimental studies because these linear molecules are amenable to spectroscopic studies, such as 2D-IR.

DNA Solvation Dynamics.

Experiments have revealed that DNA solvation dynamics is characterized by multiple time scales ranging from a few picoseconds to a few hundred nanoseconds and in some cases even up to several microseconds. The last part of decay is not only slow but can also be described by a power law (PL). The microscopic origin of this PL is yet to be clearly established. Here we present a theoretical study employing multiple approaches from time dependent statistical mechanics and computer simulations. The present study shows that water dynamics may not account for the slow PL decay because the longest time scales describing water dynamics could be at most of the order of 100 ps. We find that the DNA solvation dynamics is complex, due to multiple different contributions to solvation energy. Our investigations also show that the primary candidates for this exotic nature of solvation dynamics are the response of the counterions and ions of the buffer solution. We first employ the well-known Oosawa model of polyelectrolyte solution that includes effects of counterion fluctuations to construct a frequency dependent dielectric function. We use it in the continuum model of Bagchi, Fleming, and Oxtoby (BOF). We find that it fails to explain the slow PL decay of DNA solvation dynamics. We then extend the Oosawa model by employing the continuous time random walk technique developed by Scher, Montroll and Lax. We find that this approach could explain the long time PL decay, in terms of the collective response of the counterions. To check the nature of random walk of counterions along the phosphate backbone, we carry out atomistic molecular dynamics (MD) simulations with a long (38 base pair) DNA. We indeed find frequent occurrence of random walk of tagged counterions along the phosphate backbone. We next propose a generalized random walk model for counterion hopping on phosphate backbone (observed in our MD simulations) and carry out kinetic Monte Carlo simulations to show that the nonexponential contribution to solvation dynamics can indeed come from dynamics of such ions. We also employ a mode coupling theory (MCT) analysis to understand the slow relaxation that can originate from ions in solution due to the use of the buffer. Explicit evaluation suggests that buffer ion contribution could explain a logarithmic time dependence in the nanosecond time scale but not a power law. To further understand the nonexponentiality of solvation dynamics at relatively shorter times (less than 100 ps) we carry out atomistic MD simulations with explicit water molecules. Log-normal distributions of relaxation times of water dynamics inside the grooves may be responsible for the initial multiexponential decay of solvation dynamics. We find that the observed faster solvation of groove bound probe than that of the intercalated probe could arise from the self-motion of the probe.

Enhancement of reaction rate in small-sized droplets: A combined analytical and simulation study.

Several recent mass spectrometry experiments reveal a marked enhancement of the reaction rate of organic reactions in microdroplets. This enhancement has been tentatively attributed to the accumulation of excess charge on a surface, which in turn can give rise to a lowering of activation energy of the reaction. Here we model the reactions in droplets as a three-step process: (i) diffusion of a reactant from the core of the droplet to the surface, (ii) search by diffusion of the reactant on the surface to find a reactive partner, and finally (iii) the intrinsic reaction leading to bond breaking and product formation. We obtain analytic expressions for the mean search time (MST) to find a target located on the surface by a reactant in both two- and three-dimensional droplets. Analytical results show quantitative agreement with Brownian dynamics simulations. We find, as also reported earlier, that the MST varies as R2/D, where R is the radius of the droplet and D is the diffusion constant of the molecules in the droplet medium. We also find that a hydronium ion in the vicinity can substantially weaken the bond and hence lowers the activation barrier. We observe a similar facilitation of bond breaking in the presence of a static dipolar electric field along any of the three Cartesian axes. If the intrinsic reaction is faster compared to the mean search time involved, it becomes primarily a diffusion-controlled process; otherwise the reaction cannot be accelerated in the droplet medium. The air-droplet interface provides a different environment compared to the interior of the droplet. Hence, we might also expect a completely different mechanism and products in the case of droplet reactions.