Statistical Mechanical Theory Research Articles

Influence of gravitational potential on the thermodynamic stability of pure and mixed clathrate hydrates

Clathrate hydrates are an ice-like material consisting of gas molecules confined within cavities in a crystalline water lattice. Phase equilibria of clathrate hydrates systems was described using the statistical mechanical theory of van der Waals and Platteeuw. This theory makes use of the fractional occupancy of cavities within the clathrate hydrate lattice in the determination of chemical equilibria. Classical density functional theory with intermolecular interactions restricted to the first hydration shell was employed to determine the fractional occupancy. In addition to the external field describing the gas-water interactions, the effect of a gravitational field was introduced. The results of the calculations show that although the gravitational potential term may be orders of magnitude smaller than the thermal kinetic energy of the gas species or the hydrogen-bond energy holding the clathrate lattice together, it can nevertheless influence the phase equilibrium of the clathrate hydrate system to some degree. The effect of the magnitudes of both the gravitational potential and the local gravitational field are considered too.

Ethylene Dehydrogenation on Pt4,7,8 Clusters on Al2O3: Strong Cluster Size Dependence Linked to Preferred Catalyst Morphologies

Catalytic dehydrogenation of ethylene on size-selected Ptn (n = 4, 7, 8) clusters deposited on the surface of Al2O3 was studied experimentally and theoretically. Clusters were mass-selected, deposited on the alumina support, and probed by a combination of low energy ion scattering, temperature-programmed desorption and reaction of C2D4 and D2, X-ray photoelectron spectroscopy, density functional theory, and statistical mechanical theory. Pt7 is identified as the most catalytically active cluster, while Pt4 and Pt8 exhibit comparable activities. The higher activity can be related to the cluster structure and particularly to the distribution of cluster morphologies accessible at the temperatures and coverage with ethylene in catalytic conditions. Specifically, while Pt7 and Pt8 on alumina have very similar prismatic global minimum geometries, Pt7 at higher temperatures also has access to single-layer isomers, which become more and more predominant in the cluster catalyst ensemble upon increasing ethylene co...

Relationship between Pore-size Distribution and Flexibility of Adsorbent Materials: Statistical Mechanics and Future Material Characterization Techniques.

Measurement of the pore-size distribution (PSD) via gas adsorption and the so-called "kernel method" is a widely used characterization technique for rigid adsorbents. Yet, standard techniques and analytical equipment are not appropriate to characterize the emerging class of flexible adsorbents that deform in response to the stress imparted by an adsorbate gas, as the PSD is a characteristic of the material that varies with the gas pressure and any other external stresses. Here, we derive the PSD for a flexible adsorbent using statistical mechanics in the osmotic ensemble to draw analogy to the kernel method for rigid materials. The resultant PSD is a function of the ensemble constraints including all imposed stresses and, most importantly, the deformation free energy of the adsorbent material. Consequently, a pressure-dependent PSD is a descriptor of the deformation characteristics of an adsorbent and may be the basis of future material characterization techniques. We discuss how, given a technique for resolving pressure-dependent PSDs, the present statistical mechanical theory could enable a new generation of analytical tools that measure and characterize certain intrinsic material properties of flexible adsorbents via otherwise simple adsorption experiments.

Impact characterization and modelling of cement and asphalt mortar based on SHPB experiments

The dynamic behaviour of cement and asphalt mortar (CA mortar, including types CAI with the asphalt/water-to-cement ratios of 0.85/0.78 for the Shinkansen track and CAII with the asphalt/water-to-cement ratios of 0.30/0.50 for the Bögl track) was experimentally investigated under impact loading using a Split Hopkinson Pressure Bar (SHPB). The strain rate effects on the compressive strength, elastic modulus, peak strain and specific energy absorption were obtained. The results showed that the compressive strength and specific energy absorption increased with increasing strain rate, whose strain rate sensitivity for CAI was stronger than that for CAII. The strain rate sensitivity of the elastic modulus and peak strain scattered. Under similarly high strain rates, the compressive strength, elastic modulus, peak strain and specific energy absorption of CAII were larger than those of CAI. A statistical continuous damage constitutive model involving the strain rate for CA mortar was proposed based on the theory of thermodynamics and statistical damage mechanics. The theoretical results showed a consistent comparison with the experimental data.

Dynamic cross-correlations between entangled biofilaments as they diffuse

Entanglement in polymer and biological physics involves a state in which linear interthreaded macromolecules in isotropic liquids diffuse in a spatially anisotropic manner beyond a characteristic mesoscopic time and length scale (tube diameter). The physical reason is that linear macromolecules become transiently localized in directions transverse to their backbone but diffuse with relative ease parallel to it. Within the resulting broad spectrum of relaxation times there is an extended period before the longest relaxation time when filaments occupy a time-averaged cylindrical space of near-constant density. Here we show its implication with experiments based on fluorescence tracking of dilutely labeled macromolecules. The entangled pairs of aqueous F-actin biofilaments diffuse with separation-dependent dynamic cross-correlations that exceed those expected from continuum hydrodynamics up to strikingly large spatial distances of ≈15 µm, which is more than 104 times the size of the solvent water molecules in which they are dissolved, and is more than 50 times the dynamic tube diameter, but is almost equal to the filament length. Modeling this entangled system as a collection of rigid rods, we present a statistical mechanical theory that predicts these long-range dynamic correlations as an emergent consequence of an effective long-range interpolymer repulsion due to the de Gennes correlation hole, which is a combined consequence of chain connectivity and uncrossability. The key physical assumption needed to make theory and experiment agree is that solutions of entangled biofilaments localized in tubes that are effectively dynamically incompressible over the relevant intermediate time and length scales.

Strontium and Barium in Aqueous Solution and an Ion Channel Blocking Site

Ion hydration structure and free energy establish criteria for understanding selective ion binding in potassium (K+) ion chan- nels, and may be significant to understanding blocking mechanisms as well. Recently, we investigated the hydration properties of Ba2+, the most potent blocker of K+ channels among the simple metal ions. Here, we use a similar method of combining ab initio molecular dynamics simulations, statistical mechanical theory, and electronic structure calculations to probe the fun- damental hydration properties of Sr2+, which does not block bacterial K+ channels. The radial distribution of water around Sr2+ suggests a stable 8-fold geometry in the local hydration environment, similar to Ba2+. While the predicted hydration free energy of −331.8 kcal/mol is comparable with the experimental result of −334 kcal/mol, the value is significantly more favorable than the −305 kcal/mol hydration free energy of Ba2+. When placed in the innermost K+ channel blocking site, the solvation free energies and lowest energy structures of both Sr2+ and Ba2+ are nearly unchanged compared with their respective hydration properties. That result suggests that differences in blocking behavior may arise due to kinetic properties associated with exchange of water ligands for channel ligands instead of equilibrium thermodynamic properties.

How to estimate the signs' configuration in the directed signed social networks?

Inspired by the ensemble theory in statistical mechanics, we introduce a reshuffling approach to empirical analyze signs' configuration in the directed signed social networks of Epinions and Slashdots. In our reshuffling approach, each negative link has the reshuffling probability prs to exchange its sign with another positive link chosen randomly. Many reshuffled networks with different signs' configuration are built under different prss. For each reshuffled network, the entropies of the self social status are calculated and the opinion formation of the majority-rule model is analyzed. We find that Souts reach their own minimum values and the order parameter |m⁎| reaches its maximum value in the networks of Epinions and Slashdots without the reshuffling operation. Namely, individuals share the homogeneous properties of self social status and dynamic status in the real directed signed social networks. Our present work provides some interesting tools and perspective to understand the signs' configuration in signed social networks, especially in the online affiliation networks.

A new paradigm for atomically detailed simulations of kinetics in biophysical systems.

The kinetics of biochemical and biophysical events determined the course of life processes and attracted considerable interest and research. For example, modeling of biological networks and cellular responses relies on the availability of information on rate coefficients. Atomically detailed simulations hold the promise of supplementing experimental data to obtain a more complete kinetic picture. However, simulations at biological time scales are challenging. Typical computer resources are insufficient to provide the ensemble of trajectories at the correct length that is required for straightforward calculations of time scales. In the last years, new technologies emerged that make atomically detailed simulations of rate coefficients possible. Instead of computing complete trajectories from reactants to products, these approaches launch a large number of short trajectories at different positions. Since the trajectories are short, they are computed trivially in parallel on modern computer architecture. The starting and termination positions of the short trajectories are chosen, following statistical mechanics theory, to enhance efficiency. These trajectories are analyzed. The analysis produces accurate estimates of time scales as long as hours. The theory of Milestoning that exploits the use of short trajectories is discussed, and several applications are described.

Similarity examination of truncated virial equation of state correspondence to linear isothermal regularity (LIR) by applying square-well (SW) potential

Linear isotherm regularity works very well for fluids at high densities, and it has been shown that it is compatible with the EOSs based on statistical–mechanical theory. On the other hand, at low densities, the first few terms of virial EOS have the most contribution to express the deviations from ideal behavior. For finding similarities between dense and dilute states, experimental p–v–T data of 14 fluids (He, Ne, Ar, Kr, H2, O2, N2, CO, NH3, CH3OH, CH4, C2H4, C2H6 and C3H8) are examined. Comparing the thermal dependencies of the attraction and repulsion terms (A and B) of the LIR with the second and third virial coefficients (B 2 and B 3) in liquid and supercritical regions (0.7 < T r < 3.0) shows a remarkable similarity. Square-well potential is applied to examination and comparison of theoretical results with experimental results. It is shown that in liquid and supercritical regions, (1) the short-range potential governs among particles in dense fluids, and the long-range interactions become important in the less dense fluid, (2) similar to Boyle temperature, T B, in dilute state, there is a temperature as T′B (in dense fluids) that the attractive forces and the repulsive forces acting on the dense-fluid particles balance out; thus, probably there is a maximum σ (molecular diameter) at nearly 2T c (T′B), and (3) in the liquid and supercritical regions (0.7 < T r < 3.0), in the first-order approximation, there are no significant interactions higher than triple interactions in dense-fluid particles.

Network Formation and Mechanical Properties of Telechelic Associating Polymers with Fixed Junction Multiplicity

A statistical–mechanical theory of thermoreversible gelation which is formed by monodisperse telechelic associating polymers with junction multiplicity of three is developed. In the present theory, the effect of loop formation is considered. Using the theory, the properties of the system are obtained, such as the state of association of polymers, the sol–gel transition line, and the number concentration of elastically effective chains. In addition, a Monte Carlo simulation of a bead‐spring model of monodisperse telechelic associating polymers is performed. The simulation results are in good agreement with the present theoretical results. Furthermore, the shear modulus is calculated by an application of phantom network theory and compared with the experimental data. The theoretical results agree well with the experimental results. It is shown that the loop formation occurs especially in dilute regime and causes the decrease of the modulus in the regime. image

Equivalence between particles and fields: A general statistical mechanics theory for short and long range many‐body forces

While in vacuum fundamental and dispersion inter‐molecular forces can be in principle of infinite range, in the bulk, however, they are of rather short‐range, typically between a few angstroms to a few hundreds of angstroms. Yet colloidal particles, nano‐particles, and supramolecules, like enzymes, thousand or hundreds of thousands of angstroms apart, seem to recognize each other and self‐assembly, according with their shapes and sizes, in a wide variety of complex structures. In this paper we discuss this long‐range molecular recognition mechanism through a general liquid theory, based on the recognition of the equivalence between particles and fields, and molecular simulations. In particular we show results of a counter‐intuitive attraction between like‐charged colloidal particles and compare them with experimental results.

Computational Models for Nanoscale Fluid Dynamics and Transport Inspired by Nonequilibrium Thermodynamics.

Traditionally, the numerical computation of particle motion in a fluid is resolved through computational fluid dynamics (CFD). However, resolving the motion of nanoparticles poses additional challenges due to the coupling between the Brownian and hydrodynamic forces. Here, we focus on the Brownian motion of a nanoparticle coupled to adhesive interactions and confining-wall-mediated hydrodynamic interactions. We discuss several techniques that are founded on the basis of combining CFD methods with the theory of nonequilibrium statistical mechanics in order to simultaneously conserve thermal equipartition and to show correct hydrodynamic correlations. These include the fluctuating hydrodynamics (FHD) method, the generalized Langevin method, the hybrid method, and the deterministic method. Through the examples discussed, we also show a top-down multiscale progression of temporal dynamics from the colloidal scales to the molecular scales, and the associated fluctuations, hydrodynamic correlations. While the motivation and the examples discussed here pertain to nanoscale fluid dynamics and mass transport, the methodologies presented are rather general and can be easily adopted to applications in convective heat transfer.

Finding equilibrium in the spatiotemporal chaos of the complex Ginzburg-Landau equation.

Equilibrium statistical mechanics allows the prediction of collective behaviors of large numbers of interacting objects from just a few system-wide properties; however, a similar theory does not exist for far-from-equilibrium systems exhibiting complex spatial and temporal behavior. We propose a method for predicting behaviors in a broad class of such systems and apply these ideas to an archetypal example, the spatiotemporal chaotic 1D complex Ginzburg-Landau equation in the defect chaos regime. Building on the ideas of Ruelle and of Cross and Hohenberg that a spatiotemporal chaotic system can be considered a collection of weakly interacting dynamical units of a characteristic size, the chaotic length scale, we identify underlying, mesoscale, chaotic units and effective interaction potentials between them. We find that the resulting equilibrium Takahashi model accurately predicts distributions of particle numbers. These results suggest the intriguing possibility that a class of far-from-equilibrium systems may be well described at coarse-grained scales by the well-established theory of equilibrium statistical mechanics.

Understanding decomposition and encapsulation energies of structure I and II clathrate hydrates.

When compressed with water or ice under high pressure and low temperature conditions, some gases form solid gas hydrate inclusion compounds which have higher melting points than ice under those pressures. In this work, we study the balance of the guest-water and water-water interaction energies that lead to the formation of the clathrate hydrate phases. In particular, molecular dynamics simulations with accurate water potentials are used to study the energetics of the formation of structure I (sI) and II (sII) clathrate hydrates of methane, ethane, and propane. The dissociation enthalpy of the clathrate hydrate phases, the encapsulation enthalpy of methane, ethane, and propane guests in the corresponding phases, and the average bonding enthalpy of water molecules are calculated and compared with accurate calorimetric measurements and previous classical and quantum mechanical calculations, when available. The encapsulation energies of methane, ethane, and propane guests stabilize the small and large sI and sII hydrate cages, with the larger molecules giving larger encapsulation energies. The average water-water interactions are weakened in the sI and sII phases compared to ice. The relative magnitudes of the van der Waals potential energy in ice and the hydrate phases are similar, but in the ice phase, the electrostatic interactions are stronger. The stabilizing guest-water "hydrophobic" interactions compensate for the weaker water-water interactions and stabilize the hydrate phases. A number of common assumptions regarding the guest-cage water interactions are used in the van der Waals-Platteeuw statistical mechanical theory to predict the clathrate hydrate phase stability under different pressure-temperature conditions. The present calculations show that some of these assumptions may not accurately reflect the physical nature of the interactions between guest molecules and the lattice waters.

From Tools in Symplectic and Poisson Geometry to J.-M. Souriau’s Theories of Statistical Mechanics and Thermodynamics

I present in this paper some tools in symplectic and Poisson geometry in view of their applications in geometric mechanics and mathematical physics. After a short discussion of the Lagrangian an Hamiltonian formalisms, including the use of symmetry groups, and a presentation of the Tulczyjew’s isomorphisms (which explain some aspects of the relations between these formalisms), I explain the concept of manifold of motions of a mechanical system and its use, due to J.-M. Souriau, in statistical mechanics and thermodynamics. The generalization of the notion of thermodynamic equilibrium in which the one-dimensional group of time translations is replaced by a multi-dimensional, maybe non-commutative Lie group, is fully discussed and examples of applications in physics are given.

Exploring the aggregation free energy landscape of the amyloid-β protein (1–40)

A predictive coarse-grained protein force field [associative memory, water-mediated, structure, and energy model for molecular dynamics (AWSEM)-MD] is used to study the energy landscapes and relative stabilities of amyloid-β protein (1-40) in the monomer and all of its oligomeric forms up to an octamer. We find that an isolated monomer is mainly disordered with a short α-helix formed at the central hydrophobic core region (L17-D23). A less stable hairpin structure, however, becomes increasingly more stable in oligomers, where hydrogen bonds can form between neighboring monomers. We explore the structure and stability of both prefibrillar oligomers that consist of mainly antiparallel β-sheets and fibrillar oligomers with only parallel β-sheets. Prefibrillar oligomers are polymorphic but typically take on a cylindrin-like shape composed of mostly antiparallel β-strands. At the concentration of the simulation, the aggregation free energy landscape is nearly downhill. We use umbrella sampling along a structural progress coordinate for interconversion between prefibrillar and fibrillar forms to identify a conversion pathway between these forms. The fibrillar oligomer only becomes favored over its prefibrillar counterpart in the pentamer where an interconversion bottleneck appears. The structural characterization of the pathway along with statistical mechanical perturbation theory allow us to evaluate the effects of concentration on the free energy landscape of aggregation as well as the effects of the Dutch and Arctic mutations associated with early onset of Alzheimer's disease.

SAMPL5: 3D-RISM partition coefficient calculations with partial molar volume corrections and solute conformational sampling.

Implicit solvent methods for classical molecular modeling are frequently used to provide fast, physics-based hydration free energies of macromolecules. Less commonly considered is the transferability of these methods to other solvents. The Statistical Assessment of Modeling of Proteins and Ligands 5 (SAMPL5) distribution coefficient dataset and the accompanying explicit solvent partition coefficient reference calculations provide a direct test of solvent model transferability. Here we use the 3D reference interaction site model (3D-RISM) statistical-mechanical solvation theory, with a well tested water model and a new united atom cyclohexane model, to calculate partition coefficients for the SAMPL5 dataset. The cyclohexane model performed well in training and testing ([Formula: see text] for amino acid neutral side chain analogues) but only if a parameterized solvation free energy correction was used. In contrast, the same protocol, using single solute conformations, performed poorly on the SAMPL5 dataset, obtaining [Formula: see text] compared to the reference partition coefficients, likely due to the much larger solute sizes. Including solute conformational sampling through molecular dynamics coupled with 3D-RISM (MD/3D-RISM) improved agreement with the reference calculation to [Formula: see text]. Since our initial calculations only considered partition coefficients and not distribution coefficients, solute sampling provided little benefit comparing against experiment, where ionized and tautomer states are more important. Applying a simple [Formula: see text] correction improved agreement with experiment from [Formula: see text] to [Formula: see text], despite a small number of outliers. Better agreement is possible by accounting for tautomers and improving the ionization correction.

Formation of a Three-Electron Sulfur-Sulfur Bond as a Probe for Interaction between Side Chains of Methionine Residues.

The mechanism of oxidation processes of l-Met-(Pro)n-l-Met peptides that contain two Met residues located on the N- and C-terminal and separated by a defined number (n = 0-4) of proline residues was investigated in aqueous solutions using pulse radiolysis. The use of such peptides allowed for distance control between the sulfur atoms located in the side chains of the Met residues. The formation of a contact between the side chains of the Met residues was probed by the observation of transients with σ*-type 2c-3e S∴S and S∴O bonds as well as of α-(alkylthio)alkyl radicals (αS). This approach enabled the monitoring, in real time, of the efficiency and kinetics of interactions between methionine side chains. Such knowledge is important, inter alia, for long-distance electron transfer processes because methionine side chains can serve as relay stations and also for many aspects of protein folding when the formation of a contact between two amino acid residues in an unfolded polypeptide chain plays a central role in protein-folding mechanisms. The yields of these transients (measured as G-values) were found to be dependent on the number of Pro residues; however, they were not dependent in a simple way on the average distance ⟨rS-S⟩ between the sulfur atoms in Met residues. A decrease in the yield of the (S∴S)(+) species with an increase in the number of Pro residues in the bridge occurred at the expense of an increase in the yields of the intramolecular three-electron-bonded (S∴O)(+) radical cations and αS radicals. A detailed understanding of these trends in the chemical yields was developed by modeling the underlying chemical kinetics with Langevin dynamical simulations of the various oligoproline peptide chains and combining them with a simple statistical mechanical theory on the end-to-end contact rates for polymer chains. This analysis showed that the formation of a contact between terminal Met residues in the peptides with 0-2 Pro residues was controlled by the activated formation of (S∴S)(+) whereas in the peptides with 3 and 4 Pro residues, by the relative diffusion of the sulfur radical cation and unoxidized sulfur atom. In this picture, the dynamics of the other radical products can be seen to be only indirectly dependent on the length of the proline bridges because their formation is in competition with (S∴S)(+) formation.

The initial magnetic susceptibility of polydisperse ferrofluids: A comparison between experiment and theory over a wide range of concentration.

Temperature dependencies of the static initial magnetic susceptibility for ferrofluids at various concentrations are studied using experiment and statistical-mechanical theories. Magnetic susceptibility measurements are carried out for twelve samples of magnetite-based fluids stabilized with oleic acid over a wide range of temperatures (210 K ≲T ≲ 390 K); all samples have the same granulometric composition but different volume ferroparticle concentrations (0.2 ≲ φ ≲ 0.5). Experimental results are analyzed using three theories: the second-order modified mean-field theory (MMF2) [A. O. Ivanov and O. B. Kuznetsova, Phys. Rev. E 64, 41405 (2001)]; its correction for polydisperse ferrofluids arising from Mayer-type cluster expansion and taking into account the first terms of the polydisperse second virial coefficient [A. O. Ivanov and E. A. Elfimova, J. Magn. Magn. Mater 374, 327 (2015)]; and a new theory based on MMF2 combined with the first terms of the polydisperse second and third virial contributions to susceptibility. It turns out that the applicability of each theory depends on the experimental sample density. If twelve ferrofluid samples are split into three groups of strong, moderate, and low concentrated fluids, the temperature dependences of the initial magnetic susceptibility in each group are very precisely described by one of the three theories mentioned above. The determination of a universal formula predicting a ferrofluid susceptibility over a broad range of concentrations and temperatures remains as a challenge.

Understanding the viscosity of supercooled liquids and the glass transition through molecular simulations

We discuss a potential energy landscape (PEL) approach to calculate the characteristically sluggish viscous behaviour of supercooled liquids. This phenomenon is central to the theoretical understanding of the liquid-to-glass transition, a long-standing challenge in non-equilibrium statistical mechanics of amorphous states of matter. Experimentally, the shear viscosity of supercooled glass-forming liquids shows strong variations with the quench temperature. In particular, two types of behaviour are observed, Arrhenius and highly super-Arrhenius. Conceptually, the molecular description of the structural and dynamical processes underpinning the viscosity behaviour has become a topic of interest because of general implications for the kinetics of slow relaxation in the glassy state. We regard the supercooled liquid as an assembly of interacting particles undergoing thermal fluctuations such that the system evolves in space and time by crossing a series of potential energy barriers. To find these barriers and their corresponding atomic configurations, we apply an energy landscape sampling algorithm that maps out the evolution trajectories in the form of alternating sequences of local energy minima and saddle points. The energy sequences and the atomic coordinates constitute a body of atomic-scale data which we then process using two distinct methods. One is based on an effective activation barrier extracted from the transition-state pathway data, and the other is based on the linear response theory of statistical mechanics. The former is heuristic by being more physically transparent, while the latter is theoretically more rigorous. Through these two complementary calculations, an understanding of the temperature variations of shear viscosities of supercooled liquids, as well as the nature of fragile and strong behaviour of the glass transition, emerges. Our calculation provides a molecular-level account of the viscosities of supercooled liquids in a unified and consistent manner without invoking ad hoc assumptions. Relative to the nature of the glass transition, the usefulness of the PEL perspective is demonstrated, along with the concept of crossover between strong and fragile behaviour. In terms of advancing atomistic simulation capability, we believe the time-scale limitations of traditional molecular dynamics may be significantly extended through the use of metadynamics algorithms for sampling transition-state pathways.