Vikram Jadhao

Iterative Monte Carlo for quantum dynamics

We present a fully quantum mechanical methodology for calculating complex-time correlation functions by evaluating the discretized path integral expression iteratively on a grid selected by a Monte Carlo procedure. Both the grid points and the summations performed in each iteration utilize importance sampling, leading to favorable scaling with the number of particles, while the stepwise evaluation of the integrals circumvents the exponential growth of statistical error with time.

A variational formulation of electrostatics in a medium with spatially varying dielectric permittivity

In biological and synthetic materials, many important processes involve charges that are present in a medium with spatially varying dielectric permittivity. To accurately understand the role of electrostatic interactions in such systems, it is important to take into account the spatial dependence of the permittivity of the medium. However, due to the ensuing theoretical and computational challenges, this inhomogeneous dielectric response of the medium is often ignored or excessively simplified. We develop a variational formulation of electrostatics to accurately investigate systems that exhibit this inhomogeneous dielectric response. Our formulation is based on a true energy functional of the polarization charge density. The defining characteristic of a true energy functional is that at its minimum it evaluates to the actual value of the energy; this is a feature not found in many commonly used electrostatic functionals. We explore in detail the charged systems that exhibit sharp discontinuous change in dielectric permittivity, and we show that for this case our functional reduces to a functional of only the surface polarization charge density. We apply this reduced functional to study model problems for which analytical solutions are well known. We demonstrate, in addition, that the functional has many properties that make it ideal for use in molecular dynamics simulations.

Ionic structure in liquids confined by dielectric interfaces

The behavior of ions in liquids confined between macromolecules determines the outcome of many nanoscale assembly processes in synthetic and biological materials such as colloidal dispersions, emulsions, hydrogels, DNA, cell membranes, and proteins. Theoretically, the macromolecule-liquid boundary is often modeled as a dielectric interface and an important quantity of interest is the ionic structure in a liquid confined between two such interfaces. The knowledge gleaned from the study of ionic structure in such models can be useful in several industrial applications, such as in the design of double-layer supercapacitors for energy storage and in the extraction of metal ions from wastewater. In this article, we compute the ionic structure in a model system of electrolyte confined by two planar dielectric interfaces using molecular dynamics simulations and liquid state theory. We explore the effects of high electrolyte concentrations, multivalent ions, dielectric contrasts, and external electric field on the ionic distributions. We observe the presence of non-monotonic ionic density profiles leading to a layered structure in the fluid which is attributed to the competition between electrostatic and steric (entropic) interactions. We find that thermal forces that arise from symmetry breaking at the interfaces can have a profound effect on the ionic structure and can oftentimes overwhelm the influence of the dielectric discontinuity. The combined effect of ionic correlations and inhomogeneous dielectric permittivity significantly changes the character of the effective interaction between the two interfaces.

Electrostatics-driven shape transitions in soft shells

Significance Shape is a fundamental property of an object that influences its interaction with the environment and often determines the object’s functional capabilities. Understanding how to generate and control shape by modifying the environmental conditions is of primary importance in designing systems that respond to external cues. We show here that electrostatic interactions can be used to change the equilibrium shape of soft, nanometer-sized shells. We find that a uniformly charged, spherical shell undergoes shape changes, transforming into ellipsoids, discs, and bowls, as the electrolyte concentration in the environment is decreased. This electrostatics-based shape design mechanism, regulated by varying properties external to the shell, can be used to build efficient nanocontainers for various medical and technological applications. Manipulating the shape of nanoscale objects in a controllable fashion is at the heart of designing materials that act as building blocks for self-assembly or serve as targeted drug delivery carriers. Inducing shape deformations by controlling external parameters is also an important way of designing biomimetic membranes. In this paper, we demonstrate that electrostatics can be used as a tool to manipulate the shape of soft, closed membranes by tuning environmental conditions such as the electrolyte concentration in the medium. Using a molecular dynamics-based simulated annealing procedure, we investigate charged elastic shells that do not exchange material with their environment, such as elastic membranes formed in emulsions or synthetic nanocontainers. We find that by decreasing the salt concentration or increasing the total charge on the shell’s surface, the spherical symmetry is broken, leading to the formation of ellipsoids, discs, and bowls. Shape changes are accompanied by a significant lowering of the electrostatic energy and a rise in the surface area of the shell. To substantiate our simulation findings, we show analytically that a uniformly charged disc has a lower Coulomb energy than a sphere of the same volume. Further, we test the robustness of our results by including the effects of charge renormalization in the analysis of the shape transitions and find the latter to be feasible for a wide range of shell volume fractions.

Iterative Monte Carlo path integral with optimal grids from whole-necklace sampling.

The efficiency of the iterative Monte Carlo (IMC) path integral methodology for complex time correlation functions is increased through the use of optimal grids, which are sampled from paths that span the entire path integral necklace. The two-bead marginal distributions required in each step of the IMC iteration are obtained from a recursive procedure. Applications to one-dimensional and multi-dimensional model systems illustrate the enhancement in stability effected by the use of grids based on whole-necklace sampling.

Probing large viscosities in glass-formers with nonequilibrium simulations

Significance As a liquid cools, molecules move more slowly and the viscosity rises. A fundamental question is whether this trend continues smoothly down to zero temperature, or if flow stops at a finite temperature where the material undergoes a transition to a glass phase. Direct measurements of growing viscosities become difficult as the time for motion exceeds years or centuries. We describe and test an approach for obtaining large viscosities using nonequilibrium molecular dynamics simulations. Results agree with existing experiments on the model glass-former squalane and allow viscosities over 10 orders of magnitude larger to be predicted. The temperature dependence at fixed pressure or density is consistent with a gradual slowing of dynamics, rather than a finite-temperature divergence in viscosity. For decades, scientists have debated whether supercooled liquids stop flowing below a glass transition temperature Tg0 or whether motion continues to slow gradually down to zero temperature. Answering this question is challenging because human time scales set a limit on the largest measurable viscosity, and available data are equally well fit to models with opposite conclusions. Here, we use short simulations to determine the nonequilibrium shear response of a typical glass-former, squalane. Fits of the data to an Eyring model allow us to extrapolate predictions for the equilibrium Newtonian viscosity ηN over a range of pressures and temperatures that change ηN by 25 orders of magnitude. The results agree with the unusually large set of equilibrium and nonequilibrium experiments on squalane and extend them to higher ηN. Studies at different pressures and temperatures are inconsistent with a diverging viscosity at finite temperature. At all pressures, the predicted viscosity becomes Arrhenius with a single temperature-independent activation barrier at low temperatures and high viscosities (ηN>103 Pa⋅s). Possible experimental tests of our results are outlined.

Generating true minima in constrained variational formulations via modified Lagrange multipliers

Variational principles are important in the investigation of large classes of physical systems. They can be used both as analytical methods as well as starting points for the formulation of powerful computational techniques such as dynamical optimization methods. Systems with charged objects in dielectric media and systems with magnetically active particles are important examples. In these examples and other important cases, the variational principles describing the system are required to obey a number of constraints. These constraints are implemented within the variational formulation by means of Lagrange multipliers. Such constrained variational formulations are in general not unique. For the application of efficient simulation methods, one must find specific formulations that satisfy a number of important conditions. An often required condition is that the functional be positive-definite, in other words, its extrema be actual minima. In this article, we present a general approach to attack the problem of finding, among equivalent variational functionals, those that generate true minima. The method is based on the modification of the Lagrange multiplier which allows us to generate large families of effective variational formulations associated with a single original constrained variational principle. We demonstrate its application to different examples and, in particular, to the important cases of Poisson and Poisson-Boltzmann equations. We show how to obtain variational formulations for these systems with extrema that are always minima.

Free-energy functionals of the electrostatic potential for Poisson-Boltzmann theory

In simulating charged systems, it is often useful to treat some ionic components of the system at the mean-field level and solve the Poisson-Boltzmann (PB) equation to get their respective density profiles. The numerically intensive task of solving the PB equation at each step of the simulation can be bypassed using variational methods that treat the electrostatic potential as a dynamic variable. But such approaches require the access to a true free-energy functional: a functional that not only provides the correct solution of the PB equation upon extremization, but also evaluates to the true free energy of the system at its minimum. Moreover, the numerical efficiency of such procedures is further enhanced if the free-energy functional is local and is expressed in terms of the electrostatic potential. Existing PB functionals of the electrostatic potential, while possessing the local structure, are not free-energy functionals. We present a variational formulation with a local free-energy functional of the potential. In addition, we also construct a nonlocal free-energy functional of the electrostatic potential. These functionals are suited for employment in simulation schemes based on the ideas of dynamical optimization.

Reply to Bair: Crossover to Arrhenius behavior at high viscosities in squalane

As noted in our paper (1), values of the Newtonian viscosity, η N, of squalane from Deegan et al. (2) are well fit by the Vogel–Fulcher–Tammann (VFT) equation over the measured range of temperature, T . This is entirely consistent with the fairly straight line shown for their data in the Stickel function plot by Bair (3). As shown in figure 1 of ref. 1, our calculated values of η N are consistent with Deegan et al.’s data (2). Thus, any derived quantity such as the Stickel function will be consistent within statistical uncertainties. The line attributed to our data by Bair (3) is not consistent with our plotted data. It appears to reflect a misunderstanding of the crossover we describe, which we are happy to address below. The difficulty … [↵][1]1To whom correspondence should be addressed. Email: mr{at}jhu.edu. [1]: #xref-corresp-1-1

Coulomb energy of uniformly charged spheroidal shell systems

We provide exact expressions for the electrostatic energy of uniformly charged prolate and oblate spheroidal shells. We find that uniformly charged prolate spheroids of eccentricity greater than 0.9 have lower Coulomb energy than a sphere of the same area. For the volume-constrained case, we find that a sphere has the highest Coulomb energy among all spheroidal shells. Further, we derive the change in the Coulomb energy of a uniformly charged shell due to small, area-conserving perturbations on the spherical shape. Our perturbation calculations show that buckling-type deformations on a sphere can lower the Coulomb energy. Finally, we consider the possibility of counterion condensation on the spheroidal shell surface. We employ a Manning-Oosawa two-state model approximation to evaluate the renormalized charge and analyze the behavior of the equilibrium free energy as a function of the shells aspect ratio for both area-constrained and volume-constrained cases. Counterion condensation is seen to favor the formation of spheroidal structures over a sphere of equal area for high values of shell volume fractions.