Nonlinear Solvation Research Articles

Upscaling poromechanical models of coalbed methane reservoir incorporating the interplay between non-linear cleat deformation and solvation forces

We construct herein a three-scale coupled mechanical model for naturally fractured coalbed methane reservoir with the ability of describing the stress balance between the solvation force, arising from the gas adsorption in nanopores, and the restoration stress stemming from the elastic response of the cleats. To determine the cleat porosity, the non-linear hyperbolic Barton–Bandis (BB) law, which captures increase in joint stiffness induced by the cleat closure due to matrix swelling, is postulated for the fracture mechanical response. At the microscale, the theory incorporates the coupling between the effects of the solvation force and the elastic response of the matrix. Such system of governing equations is coupled with the fluid pressure in the discrete cleat system with dependency of aperture with the normal stress dictated by the aforementioned BB-model. A reiterated homogenized procedure is pursued and capable of providing the constitutive response of the homogenized poromechanical parameters on gas pressure. Numerical simulations illustrate the performance of the proposed model.

Marcus’ electron transfer rate revisited via a Rice-Ramsperger-Kassel-Marcus analogue: A unified formalism for linear and nonlinear solvation scenarios

In the pioneering work by R. A. Marcus, the solvation effect on electron transfer (ET) processes was investigated, giving rise to the celebrated nonadiabatic ET rate formula. In this work, on the basis of the thermodynamic solvation potentials analysis, we reexamine Marcus’ formula with respect to the Rice-Ramsperger-Kassel-Marcus (RRKM) theory. Interestingly, the obtained RRKM analogue, which recovers the original Marcus’ rate that is in a linear solvation scenario, is also applicable to the nonlinear solvation scenarios, where the multiple curve-crossing of solvation potentials exists. Parallelly, we revisit the corresponding Fermi’s golden rule results, with some critical comments against the RRKM analogue proposed in this work. For illustration, we consider the quadratic solvation scenarios, on the basis of physically well-supported descriptors.

Mobility of large ions in water

Mobility of ions in polar liquids is diminished when the ionic charge is increased. This phenomenon, known as dielectric friction, is caused by the retarded response of the liquid's dipoles to the charge movement. Linear response theories predict linear scaling of the inverse diffusion coefficient with the squared ionic charge. This prediction is analyzed here by molecular dynamics simulations of model ions with fractional charge q in the simple point charge water and by microscopic theory formulated in terms of the dynamic electric-field susceptibility of the solvent. The results of the analytical theory, and of its dielectric continuum limit, are in excellent agreement with simulations at sufficiently small charges q < 0.5 when linear response holds. At higher ionic charges, the hydration shell contracts, resulting in deviations from linear response in both static and dynamic properties of the electric field produced by water at the ion. Nevertheless, dielectric friction continues to rise in the nonlinear regime, resulting in an overall factor of 3.7 slower diffusion upon placing a single charge q = 1 on the solute. An approximately linear scaling of the inverse diffusion coefficient with the squared ionic charge comes from a mutual compensation between nonlinear solvation and correlations between non-electrostatic and electrostatic forces. Mobility of common electrolyte ions in water is predicted to occur in the regime of nonlinear dielectric friction.

Decompositions of Solvent Response Functions in Ionic Liquids: A Direct Comparison of Equilibrium and Nonequilibrium Methodologies.

Time-dependent Stokes shift (TDSS) measurements provide crucial insights into the dynamics of liquids. The interpretation of TDSS measurements is often aided by molecular dynamics simulations, where solvent response functions are computed either with an equilibrium or nonequilibrium approach. In the nonequilibrium approach, the solvent is at equilibrium with the ground electronic state of the solute and its charge distribution is instantaneously changed to that of the first excited state. The solvation response function is then calculated as a nonequilibrium average of the subsequent evolution of the solvent influence on the electronic energy gap. In the equilibrium approach, the normalized time correlation function of the fluctuations of the solvent-perturbed electronic energy gap is calculated. If the linear response approximation is valid, then the nonequilibrium solvation response function is identical to the equilibrium time correlation function. The nonequilibrium methodology conceptually mimics the experiment, but it is significantly more computationally expensive than the equilibrium approach. In multicomponent systems such as ionic liquids, it is natural to inquire how the various components affect the observed relaxation dynamics. When utilizing the nonequilibrium methodology, the solvation response naturally decomposes into a sum of responses for each component present in the system. However, the equilibrium time correlation function does not decompose unambiguously. Here, we have evaluated a decomposition strategy that is consistent with the linear response approximation for the study of solvation dynamics of coumarin 153 (C153) in the 1-ethyl-3-methyl imidazolium tetrafluoroborate, [emim][BF4], ionic liquid. The agreement of the equilibrium and nonequilibrium solvation response functions demonstrates the validity of the linear response approximation for the C153/[emim][BF4] system. Moreover, decompositions of the equilibrium time correlation function into contributions of the translational and rovibrational motions of the anions and cations are essentially identical to the same decompositions of the nonlinear solvation response.

Hydration thermodynamics beyond the linear response approximation

The solvation energetics associated with the transformation of a solute molecule at infinite dilution in water from an initial state A to a final state B is reconsidered. The two solute states have different potentials energies of interaction, and , with the solvent environment. Throughout the A B transformation of the solute, the solvation system is described by a Hamiltonian that changes linearly with the coupling parameter ξ. By focusing on the characterization of the probability density that the dimensionless perturbational solute–solvent interaction energy has numerical value y when the coupling parameter is ξ, we derive a hierarchy of differential equation relations between the ξ-dependent cumulant functions of various orders in the expansion of the appropriate cumulant generating function. On the basis of this theoretical framework we then introduce an inherently nonlinear solvation model for which we are able to find analytical results for both and for the solvation thermodynamic functions. The solvation model is based on the premise that there is an upper or a lower bound (depending on the nature of the interactions considered) to the amplitude of the fluctuations of Y in the solution system at equilibrium. The results reveal essential differences in behavior for the model when compared with the linear response approximation to solvation, particularly with regards to the probability density . The analytical expressions for the solvation properties show, however, that the linear response behavior is recovered from the new model when the room for the thermal fluctuations in Y is not restricted by the existence of a nearby bound. We compare the predictions of the model with the results from molecular dynamics computer simulations for aqueous solvation, in which either (1) the solute–solvent electrostatic interactions, or (2) the shorter-range attractive interactions are switched-on in the A B process.

Intrinsic advantages of packed capillaries over narrow-bore columns in very high-pressure gradient liquid chromatography

250μm×100mm fused silica glass capillaries were packed with 1.8μm high-strength silica (HSS) fully porous particles. They were prepared without bulky stainless steel endfittings and metal frits, which both generate significant sample dispersion. The isocratic efficiencies and gradient peak capacities of these prototype capillary columns were measured for small molecules (n-alkanophenones) using a home-made ultra-low dispersive micro-HPLC instrument. Their resolution power was compared to that of standard 2.1mm×100mm very high-pressure liquid chromatography (vHPLC) narrow-bore columns packed with the same particles. The results show that, for the same column efficiency (25000 plates) and gradient steepness (0.04min−1), the peak capacity of the 250μm i.d. capillary columns is systematically 15–20% higher than that of the 2.1mm i.d. narrow-bore columns. A validated model of gradient chromatography enabled one to predict accurately the observed peak capacities of the capillary columns for non-linear solvation strength retention behavior and under isothermal conditions. Thermodynamics applied to the eluent quantified the temperature difference for the thermal gradients in both capillary and narrow-bore columns. Experimental data revealed that the gradient peak capacity is more affected by viscous heating than the column efficiency. Unlike across 2.1mm i.d. columns, the changes in eluent composition across the 250μm i.d. columns during the gradient is rapidly relaxed by transverse dispersion. The combination of (1) the absence of viscous heating and (2) the high uniformity of the eluent composition across the diameter of capillary columns explains the intrinsic advantage of capillary over narrow-bore columns in gradient vHPLC.

Anatase TiO2 nanowires functionalized by organic sensitizers for solar cells: A screened Coulomb hybrid density functional study

The adsorption of two different organic molecules cyanidin glucoside (C21O11H20) and TA-St-CA on anatase (101) and (001) nanowires has been investigated using the standard and the range separated hybrid density functional theory calculations. The electronic structures and optical spectra of resulting dye–nanowire combined systems show distinct features for these types of photochromophores. The lowest unoccupied molecular orbital of the natural dye cyanidin glucoside is located below the conduction band of the semiconductor while, in the case of TA-St-CA, it resonates with the states inside the conduction band. The wide-bandgap anatase nanowires can be functionalized for solar cells through electron-hole generation and subsequent charge injection by these dye sensitizers. The intermolecular charge transfer character of Donor-π-Acceptor type dye TA-St-CA is substantially modified by its adsorption on TiO2 surfaces. Cyanidin glucoside exhibits relatively stronger anchoring on the nanowires through its hydroxyl groups. The atomic structures of dye–nanowire systems re-optimized with the inclusion of nonlinear solvation effects showed that the binding strengths of both dyes remain moderate even in ionic solutions.

Hybrid functional calculated optical and electronic structures of thin anatase TiO2 nanowires with organic dye adsorbates

The electronic and optical properties of thin anatase TiO2 (101) and (001) nanowires have been investigated using the screened Coulomb hybrid density functional calculations. For the bare nanowires with sub-nanometer diameters, the calculated band gaps are larger relative to the bulk values due to size effects. The role of organic light harvesting sensitizers on the absorption characteristics of the anatase nanowires has been examined using the hybrid density functional method incorporating partial exact exchange with range separation. For the lowest lying excitations, directional charge redistribution of tetrahydroquinoline (C2-1) dye shows a remarkably different profile in comparison to a simple molecule which is chosen as the coumarin skeleton. The binding modes and the adsorption energies of C2-1 dye and coumarin core on the anatase nanowires have been studied including non-linear solvation effetcs. The calculated optical and electronic properties of the nanowires with these two different types of sensitizers have been interpreted in terms of their electron–hole generation, charge carrier injection and recombination characteristics.

Range-Separated Hybrid Density Functional Study of Organic Dye Sensitizers on Anatase TiO2 Nanowires

The adsorption of organic molecules coumarin and the donor-$\pi$-acceptor type tetrahydroquinoline (C2-1) on anatase (101) and (001) nanowires have been investigated using screened Coulomb hybrid density functional theory calculations. While coumarin forms single bond with the nanowire surface, C2-1 additionally exhibits bidentate mode giving rise to much stronger adsorption energies. Nonlinear solvation effects on the binding characteristics of the dye chromophores on the nanowire facets have also been examined. These two dye sensitizers show different electronic charge distributions for the highest occupied and the lowest unoccupied molecular states. We studied the electronic structures in terms of the positions of the band edges and adsorbate related band gap states and their effect on the absorption spectra of the dye-nanowire combined systems. These findings were interpreted and discussed from the view point of better light harvesting and charge separation as well as in relation to more efficient charge carrier injection into the semiconductor nanowire.

Calculated and experimental chromatograms for distorted gradients and non-linear solvation strength retention models

Computer calculations of gradient chromatograms were performed by taking into account the adsorption behavior of the strong eluent in RPLC and the true Henry constant of the analytes. This improves the accuracy of classical gradient calculations, which all assume no affinity of the eluent modifier for the stationary phase and that the linear solvation strength model (LSSM) applies. The excess adsorption isotherm of acetonitrile with respect to water was measured by the minor disturbance method onto a Symmetry-C18 RPLC adsorbent. The variations of the Henry constants of a nine compound mixture with the volume fraction of acetonitrile in the aqueous mobile phase were measured. The equilibrium dispersive model of chromatography combined with orthogonal collocation on finite elements was used to calculate chromatograms of the sample mixture for four gradient times decreasing from 25 to 1min. The results predict a loss of resolution for the less retained analytes when the gradient times becomes smaller than 4min. They also predict that this behavior can be eliminated when applying a quadratic gradient profile rather than a classical linear gradient. The predictions were validated by the agreement between the calculated and experimental chromatograms.

A study of the density functional methods on the photoabsorption of Bodipy dyes

• Pure exchange-correlation functionals fail to describe the optical features of Bodipy molecules. • The long-range part of exact exchange has a significant effect on the HOMO–LUMO gaps. • The calculated absorption thresholds blue-shift by ∼200 meV in chloroform solution. • HSE absorption spectra of Bodipy dyes compare well with the experimental results. Tunability of the photoabsorption and directional charge injection characteristics of Bodipy-based dye molecules with different carbonyl groups make them promising candidates for photovoltaic applications. In order to study the effect of screening in the Coulomb interaction on the electronic and optical properties of two Bodipy derivatives, we have used linear response time-dependent and exact exchange hybrid density functional approaches. The effect of linear and non-linear solvation models on the electrochemical properties of the dyes has also been discussed.

Operator splitting ADI schemes for pseudo-time coupled nonlinear solvation simulations

This work introduces novel operator splitting alternating direction implicit (ADI) schemes to overcome numerical difficulties in solving pseudo-time coupled nonlinear partial differential equations (PDEs) for biomolecular solvation analysis. Based on the variational formulation, a pseudo-transient continuation model has been previously formulated to couple a nonlinear Poisson–Boltzmann (NPB) equation for the electrostatic potential with a generalized Laplace–Beltrami equation defining the biomolecular surface. However, the standard numerical integration of the time dependent NPB equation is known to be very inefficient. Moreover, it encounters instability issues for smoothly varying solute–solvent interfaces so that a filtering process has to be conducted. In the present work, we propose to solve the unsteady NPB equation in an operator splitting framework so that the nonlinear instability can be completely avoided through an analytical integration. Central finite differences are employed to discretize the nonhomogeneous diffusion term of the generalized NPB equation to form tridiagonal matrices in the Douglas and Douglas–Rachford type ADI schemes. The proposed time splitting ADI schemes are found to be unconditionally stable for solving the NPB equation in benchmark examples with analytical solutions. For the solvation analysis involving two pseudo-time coupled nonlinear PDEs, the time stability of the NPB equation can be maintained by using very large time increments, so that without sacrificing the accuracy, the present biomolecular simulation becomes over ten times faster.

Surface Polarity and Nanoscale Solvation.

Linear solvation theories are well established to describe electrostatic hydration of small solutes when the hydration free energy is dominated by the electrostatic free energy of the solute multipole. In contrast, hydration of nanometer solutes is driven by surface hydration. We address the question of whether the linear-response thermodynamics established for small multipolar solutes applies to surface hydration. To this end, molecular dynamics simulations are carried out on a model C180 solute that carries no global multipole, but the surface of which is decorated with radially pointing dipoles. Linear response is dramatically violated in this case. Further, two crossovers in the solvation thermodynamics are discovered as the surface polarity is increased. Both transformations produce strongly nonlinear solvation response. The second, more collective, crossover leads to a dramatic slowing down of the interfacial dynamics, reaching the time-scales of nanoseconds. Our picture offers the possibility of flipping water domains at interfaces of nanoparticles and biomolecules.

Giving physical insight into the Butler–Volmer model of electrode kinetics: Part 2 – Nonlinear solvation effects on the voltammetry of heterogeneous electron transfer processes

Electrochemical and spectroscopic experiments have shown the weaknesses of the symmetric Marcus–Hush model to model electrode processes, which is likely related to “asymmetric” oxidation and reduction processes. In a previous paper (M.C. Henstridge, E. Laborda, Y. Wang, D. Suwatchara, N. Rees, A. Molina, F. Martínez-Ortiz, R.G. Compton, J. Electroanal. Chem. 672 (2012) 45) the effects arising from different vibrational force constants have been considered, and in the present work we focus on those associated with solvent reorganization.The effects of different solvent frequencies in the oxidized and reduced forms are analyzed for the Gibbs energy surface, the oxidation/reduction rate constants and the voltammetry of solution-phase and surface-bound redox systems. Criteria to assess these effects by electrochemical experiments are given based on the analysis of the value of the Butler–Volmer transfer coefficient.

Standard electrode potential, Tafel equation, and the solvation thermodynamics

Equilibrium in the electronic subsystem across the solution-metal interface is considered to connect the standard electrode potential to the statistics of localized electronic states in solution. We argue that a correct derivation of the Nernst equation for the electrode potential requires a careful separation of the relevant time scales. An equation for the standard metal potential is derived linking it to the thermodynamics of solvation. The Anderson-Newns model for electronic delocalization between the solution and the electrode is combined with a bilinear model of solute-solvent coupling introducing nonlinear solvation into the theory of heterogeneous electron transfer. We therefore are capable of addressing the question of how nonlinear solvation affects electrochemical observables. The transfer coefficient of electrode kinetics is shown to be equal to the derivative of the free energy, or generalized force, required to shift the unoccupied electronic level in the bulk. The transfer coefficient thus directly quantifies the extent of nonlinear solvation of the redox couple. The current model allows the transfer coefficient to deviate from the value of 0.5 of the linear solvation models at zero electrode overpotential. The electrode current curves become asymmetric in respect to the change in the sign of the electrode overpotential.

On the validity of dielectric continuum models in application to solvation in molecular solvents

We report Monte Carlo simulations of solvation of a point dipole in dipolar–quadrupolar solvents of varying dipole moment and axial quadrupole. The simulations are carried out to test the prediction of dielectric solvation models of a monotonic increase of the absolute value of the solvation chemical potential |μp| with the solvent dielectric constant ε. Dielectric constants are obtained from pure liquid simulations carried out for each solvent used in solvation simulations. A raising dependence of |μp| on ε, in qualitative agreement with dielectric solvation models, is seen when the solvent dipole moment is varied at constant solvent quadrupole. An increase in the axial quadrupole at constant solvent dipole reduces the dielectric constant at the same time leading to higher |μp| values. The simulations and dielectric models thus give the opposite dependence on the solvent quadrupole for any solvent dipole. We also show that for solvation in dipolar–quadrupolar solvents the saturation limit |μp|→const at ε≫1 predicted by linear response dielectric continuum models actually occurs in the range of nonlinear solvation.

Calculated Hydration Free Energies of Small Organic Molecules Using a Nonlinear Dielectric Continuum Model

Prediction of solvation free energies is an important subject in fundamental natural science but also important to the pharmaceutical and food industry. A popular modeling approach is to treat the solution by an implicit solvent model. The solute molecule is rigid with a fixed effective charge distribution localized at the atomic nuclei positions. The hydration free energy is described by the van der Waals energy, the solute cavity formation energy in the water phase, and the change in electrostatic solute−solvent interaction energy. The dielectric continuum is generally assumed to be a simple medium, that is, linear, homogeneous, and isotropic. However, this approximation is quite severe and will give too hydrophilic solvation free energies. We show here that the simple medium approximation must be relaxed and nonlinearity must be taken into consideration. In strong electric fields, the solvent polarization becomes saturated and the dielectric no longer responds linearly in the applied field. This effect is well-described by the modified Langevin−Debye model. This nonlinear solvation model is used to study the hydration of 181 small organic molecules. Atomic charges and radii of the solute molecule are described by a standard classical force field. We apply the optimized potentials for liquid simulation all atom (OPLS-AA) force field, which is parametrized to reproduce both structural and thermodynamical data. This leads to a mean unsigned error of 0.6 kcal/mol, which is a 25% improvement compared to a simple medium approach. The nonlinear solvation model is further improved by introducing a few charge-scaling parameters for some functional groups that show a systematic deviation from their experimental data. This yields a mean unsigned error of 0.4 kcal/mol, which is only twice the experimental uncertainty. Hence, we conclude that nonlinear dielectric effects are indeed important to incorporate in implicit solvent models, even for neutral polar molecules.

Dipole Solvation: Nonlinear Effects, Density Reorganization, and the Breakdown of the Onsager Saturation Limit

Linear cavity solvation models predict saturation of the solvation chemical potential, μp → constant, at high solvent polarity. This qualitative prediction is tested on computer simulations of dipole solvation in dipolar hard-sphere solvents in the liquid and solid phase states. We find that solvation saturation does exist for solid dipolar solvents, but does not exist for liquid dipolar solvents when the linear solvent response holds. Solvation saturation occurs due to nonlinear solvation in liquid solvents when solvent−solvent attractions exceed solute−solvent attractions. Nonlinear solvation is caused by electrostriction resulting in dewetting of the solute surface of a nonpolar or weakly polar dipolar solute. Solvation thermodynamics is affected by a combination of orientational and density solvent reorganization. The relative contribution of each component is strongly dependent on solvent polarity. In highly polar solvents, the orientational and density reorganization approximately equally contribute...

Modeling the free energy surfaces of electron transfer in condensed phases

We develop a three-parameter model of electron transfer (ET) in condensed phases based on the Hamiltonian of a two-state solute linearly coupled to a harmonic, classical solvent mode with different force constants in the initial and final states (a classical limit of the quantum Kubo–Toyozawa model). The exact analytical solution for the ET free energy surfaces demonstrates the following features: (i) the range of ET reaction coordinates is limited by a one-sided fluctuation band, (ii) the ET free energies are infinite outside the band, and (iii) the free energy surfaces are parabolic close to their minima and linear far from the minima positions. The model provides an analytical framework to map physical phenomena conflicting with the Marcus–Hush two-parameter model of ET. Nonlinear solvation, ET in polarizable charge-transfer complexes, and configurational flexibility of donor-acceptor complexes are successfully mapped onto the model. The present theory leads to a significant modification of the energy gap law for ET reactions.

A perturbation theory for solvation thermodynamics: Dipolar–quadrupolar liquids

The thermodynamics of solvation of a dipole in hard sphere solvents with dipoles and quadrupoles is studied by using the Padé approximation for the perturbation expansion of the solvation chemical potential and compared to Monte Carlo simulations. Solvation chemical potentials, energies, and entropies of solvation are obtained at different dipolar and quadrupolar solvent strengths. The effect of nonlinear solvation is analyzed and found not to exceed 10% in the parameter range studied. An agreement between the simulations and the analytical theory is obtained by an empirical rescaling of the triple perturbation integrals of the perturbation expansion. This rescaling does not, however, provide a quantitatively correct partitioning of the solvation free energy into the energy and entropy of solvation.