Self-consistent Integral Equation Research Articles

Breakup a droplet passing through an obstacle in an orthogonal cross-section microchannel

In this work, the breakup of a droplet passing through an obstacle in an orthogonal cross section is numerically investigated. The relevant boundary data of the velocity field is numerically computed by solving the depth-averaged Brinkman equation via a self-consistent integral equation using the boundary element method. To study the dependence of the droplet breakup on the obstacle shape, two different shapes of obstacle, circular and elliptical, are considered in the present work. We investigate the effect of obstacle size, obstacle position, and capillary number on the breakup treatment of the droplet. Numerical results indicate that the critical capillary number depends on the obstacle shape, obstacle position and droplet size. In the elliptical obstacle, in addition, the results also show that the area ratio of daughter droplets depends on the capillary number. Results show that the area ratio of daughter droplets depends on the capillary number, obstacle shape, and obstacle position. Our results is in a good agreement with the previous studies.

A numerical study of droplet deformation and droplet breakup in a non-orthogonal cross-section

In this work, we numerically investigate the deformation and breakup of a droplet flowing along the centerline of a microfluidic non-orthogonal intersection junction. The relevant boundary data of the velocity field is numerically computed by solving the depth-averaged Brinkman equation via a self-consistent integral equation using the boundary element method. The effect of the capillary number, droplet size, intersection angle, and ratio of outlet channel width to inlet channel width on maximum droplet deformation are studied. We study droplet deformation for the capillary numbers in the range of 0.08-0.3 and find that the maximum droplet deformation scales with the capillary number with power law with an exponent 1.10. We also investigate the effect of droplet size and intersection angle on the maximum droplet deformation and observe that the droplet deformation is proportional to droplet volume and square root of intersection angle, respectively. In continue, we study the droplet breakup phenomenon in an orthogonal intersection junction. By increasing the capillary number, the deformation of a droplet traveling in the cross-junction region becomes larger, until the droplet shape is no longer observed and droplet breakup takes place at a critical value of capillary number. We present a phase diagram for droplet breakup as a function of undeformed droplet radius.

Large effects of tiny structural changes on the cluster formation process in model colloidal fluids: an integral equation study

The formation of aggregates is commonly observed in soft matter such as globular protein solutions and colloidal suspensions. A lively debated issue concerns the possibility to discriminate between a generic intermediate-range order taking place in the fluid, as contrasted with the more specific presence of a clustered state. Recently, we have predicted by Monte Carlo simulations of a standard colloidal model — spherical particles interacting via a short-range attraction followed by a screened electrostatic repulsion at larger distances — the existence of a tiny structural change occurring in the pair structure. This change consists in a reversal of trend affecting a portion of the local density as the attractive strength increases, that is shown to take place precisely at the clustering threshold. Here, we address the same issue by refined thermodynamically self-consistent integral equation theories of the liquid state. We document how such theoretical schemes positively account for the observed phenomenology, highlighting their accuracy to finely describe the aggregation processes in model fluids with microscopic competing interactions.

On the structure, property, and phase behaviour of the symmetric Yukawa mixtures: testing of the consistent integral equation theories

ABSTRACTYukawa mixtures are important microscopic models of liquids. In order to obtain their properties, we must have accurate and consistent theories, without the issues arising from the multi-valued pressures and chemical potentials due to inconsistencies in the theory. This work tests two self-consistent integral equation theories (IETs): the modified hypernetted-chain (MHNC) theory and the zero-separation (ZSEP) based theory. These equations, by construction, automatically satisfy the pressure consistency the Gibbs-Duhem relation, as well as the zero-separation theorem. To verify, new Monte-Carlo (MC) simulations are carried out for temperatures 0.7 < T* < 1.1 and densities 0.4 < ρ* < 0.5. The Yukawa interactions uij are symmetric (u11 = u22). The unlike u12 is made weaker than the like-type by a factor α < 1. The structures, exemplified by the radial distribution functions, are accurately reproduced by both MHNC and ZSEP. The internal energies are predicted to within 1% of the MC data, the pressures to within 4%; and the chemical potentials to less than 3%. The vapour-liquid phase envelope is determined for a typical mixture. In this case the performance of both IETs is also shown to be satisfactory.

Modeling droplet deformation through converging–diverging microchannels at low Reynolds number

In this work, the deformation of a droplet flowing through planar diverging and converging channels connected by a narrow straight microchannel is studied numerically. The solution of the depth-averaged Brinkman equation is obtained numerically via a self-consistent integral equation using the boundary element method. The droplet experiences a converging/diverging radial flow field as it flows in/out of the straight channel. Therefore, droplet deformation strongly depends on channel diverging angle. Our numerical results indicate that the maximum deformation of the droplet depends on the droplet size and the modified capillary number, and the channel diverging angle can be scaled with power laws with exponents 3.00, 0.70, and 0.50, respectively.

Theory of Plasmons for Two-Dimensional Materials in the Random Phase Approximation

A theory is derived for plasmons in two-dimensional (2D) materials by using three-dimensional (3D) plasmon theory, which was reported previously in the random phase approximation under high frequency conditions. When the 3D local electron density is expressed by the 2D local electron density n 2 D multiplied by the delta function in the thickness direction, a self-consistent integral equation for the scalar potential is derived using only n 2 D and the 2D Coulomb potential. The integral equation consists of the edge and planar plasmon terms which give their resonant frequencies. These frequencies are analytically calculated for uniform 2D atomic layers and nanodisks with step function-like electron densities at their edges. The light emission intensities from the nanodisks are also calculated. These frequencies are compared with those for the 2D and 3D Weyl fermions, i.e., massless Dirac fermions.

Equilibrium theory of the hard sphere fluid and glasses in the metastable regime up to jamming. I. Thermodynamics

We formulate and apply a non-replica equilibrium theory for the fluid-glass transition, glass thermodynamic properties, and jamming of hard spheres in three and all higher spatial dimensions. Numerical predictions for the zero complexity glass transition and jamming packing fractions, and a "densest" equilibrium glass, are made. The equilibrium glass equation of state is regarded as the practical continuation of its fluid analog up to jamming. The analysis provides a possible resolution to the inability of any fluid virial series re-summation based equation of state to capture jamming at a reasonable volume fraction. The numerical results are quantitatively compared with various simulation data for equilibrium hard sphere glasses in 3 to 12 dimensions. Although there are uncertainties in this comparison, the predicted zero complexity or configurational entropy and corresponding jamming packing fractions do agree well with two characteristic packing fractions deduced from the dynamic simulation data. The similarities and differences of our approach compared to the replica approach are discussed. The high dimensional scaling of the equilibrium glass transition and jamming volume fractions are also derived. The developments in this paper serve as input to Paper II [R. Jadrich and K. S. Schweizer, J. Chem. Phys. 139, 054502 (2013)] that constructs a self-consistent integral equation theory of the 3-dimensional hard sphere pair structure, in real and Fourier space, in the metastable regime up to jamming. The latter is employed as input to a microscopic dynamical theory of single particle activated barrier hopping.

Extension of the KLI approximation toward the exact optimized effective potential

The integral equation for the optimized effective potential (OEP) is utilized in a compact form from which an accurate OEP solution for the spin-unrestricted exchange-correlation potential, Vxcσ, is obtained for any assumed orbital-dependent exchange-correlation energy functional. The method extends beyond the Krieger-Li-Iafrate (KLI) approximation toward the exact OEP result. The compact nature of the OEP equation arises by replacing the integrals involving the Green's function terms in the traditional OEP equation by an equivalent first-order perturbation theory wavefunction often referred to as the "orbital shift" function. Significant progress is then obtained by solving the equation for the first order perturbation theory wavefunction by use of Dalgarno functions which are determined from well known methods of partial differential equations. The use of Dalgarno functions circumvents the need to explicitly address the Green's functions and the associated problems with "sum over states" numerics; as well, the Dalgarno functions provide ease in dealing with inherent singularities arising from the origin and the zeros of the occupied orbital wavefunctions. The Dalgarno approach for finding a solution to the OEP equation is described herein, and a detailed illustrative example is presented for the special case of a spherically symmetric exchange-correlation potential. For the case of spherical symmetry, the relevant Dalgarno function is derived by direct integration of the appropriate radial equation while utilizing a user friendly method which explicitly treats the singular behavior at the origin and at the nodal singularities arising from the zeros of the occupied states. The derived Dalgarno function is shown to be an explicit integral functional of the exact OEP Vxcσ, thus allowing for the reduction of the OEP equation to a self-consistent integral equation for the exact exchange-correlation potential; the exact solution to this integral equation can be determined by iteration with the natural zeroth order correction given by the KLI exchange-correlation potential. Explicit analytic results are provided to illustrate the first order iterative correction beyond the KLI approximation. The derived correction term to the KLI potential explicitly involves spatially weighted products of occupied orbital densities in any assumed orbital-dependent exchange-correlation energy functional; as well, the correction term is obtained with no adjustable parameters. Moreover, if the equation for the exact optimized effective potential is further iterated, one can obtain the OEP as accurately as desired.

A theoretical study of structure and thermodynamics of fluids with long-range competing interactions exhibiting pattern formation

We study the structure and phase behavior of a model fluid with competing short-range attraction and long-range repulsion, constituted by hard spheres interacting by means of two opposite Kac potentials. We use, to this purpose, a thermodynamically self-consistent integral equation approach developed by one of the authors [J.-M. Bomont and J.-L. Bretonnet, J. Chem. Phys. 119, 2188 (2003)], which proven accurate in predicting the properties of other competing fluids. We choose the potential parameters in such a way that, upon appropriate thermodynamic conditions, the fluid displays microphase separation terminating, at sufficiently low temperatures, with a phase transition into an ordered-pattern fluid. The propensity toward the pattern formation is indicated by long-wavelength, slowly decaying oscillations in the pair correlation function, and by the presence of a sharp peak in the structure factor S(q) at a small but finite wavevector q(c). The limits of stability of the micro-separated phase are identified by a drastic, diverging-like, increase of S(q(c)) as the temperature drops. The behavior of S(q) in the disordered-pattern phase suggests that different morphologies of the ordered patterns should be expected, depending on the ratio between the strengths of competing interactions. The structural predictions are confirmed, at the thermodynamic level, by the change of sign observed in the "residual multi-particle entropy," according to the one-phase ordering criterion developed by Giaquinta and Giunta [Physica A 187, 145 (1992)], and by the trend shown by the chemical potential. Our self-consistent approach succeeds in describing the thermodynamic regime where the phase transition occurs, whereas, as reported in the literature, other sophisticated schemes within the same theoretical framework generally fail; reasons of this outcome and putative remedies are discussed.

Plasmons and optical excitations in graphene rings

A simple semiclassical drude-like conductivity of graphene is employed to describe plasmon excitations of graphene in the ring structures. A quasi-static self-consistent integral equation approach is performed, allowing the calculation of all the plasmon modes with different angular momentum l. Among them only the dipole modes (l = 1) will couple out to the radiation modes, which in turn can be excited optically by the plane waves, and the excitation energies as a function of the ratio of the radius of the inner hole to that of the outer ring have also been investigated. It is demonstrated that the energy of symmetric modes will monotonically decrease as the ratio rises, and the energy of antisymmetric modes does not exhibit a monotonically increasing behavior as in a three-dimensional metallic ring, but first reduces and then increases. These predictions are tested by full-wave simulations using the optical conductivity of graphene that was obtained from the random phase approximation (RPA).

Theoretical description of cluster formation in two-Yukawa competing fluids

We investigate the temperature threshold whereupon two-Yukawa systems with microscopic competing interactions transform from normal fluids to ‘cluster fluids’. Specifically, we apply two refined, thermodynamically self-consistent integral equation theories of the liquid state to study a family of two-Yukawa models having the same interaction strength, independent of the choice of potential parameters. We find, over the range of potential parameters and fluid densities investigated, an almost linear scaling between the temperature threshold for the cluster formation and the height of repulsive interactions, with higher barriers stabilizing the cluster fluid at higher temperatures. Cluster–cluster correlation lengths seem instead to be influenced by the long-range features of the interactions.

A macroion correlation effect on the structure of charged stabilized colloidal suspensions: a self-consistent integral equation study

Using the mean field Poisson–Boltzmann (PB) and the Ornstein–Zernike (OZ) integral equation theories, we have determined the macroions effective interactions and the structure of charged stabilized colloidal suspension for a large charge range of macroion and screening parameter values. The renormalized parameters are calculated by solving the PB equation written in the framework of the modified Jellium model. The structures have been determined by solving the OZ equation coupled with a self-consistent integral equation, which is related to the Verlet’s modified closure. Our results of the effective parameters are in good agreement with the experimental data, also the structure presents acceptable improvement compared to the Monte Carlo simulation data, against the HNC structure results, PACS: 61.20 Gy, 82.70 Dd, and 82.70 Kj.

Temperature study of cluster formation in two-Yukawa fluids

An accurate thermodynamically self-consistent integral equation theory of the liquid state is used to investigate model fluids with competing attractive interaction at short distances and long-range repulsion, focusing on thermodynamic states where the formation of clusters is expected to occur. We find a remarkable accuracy of theoretical predictions, through a detailed assessment against results of Monte Carlo simulations. The behavior of theoretical radial distribution functions and structure factors faithfully follows the onset and growth of cluster aggregates in the homogeneous dense-vapor phase. The thermodynamic properties of the system sensitively depend on the ratio between the repulsive barrier and the attraction strength. We elucidate the role of accurate theoretical tools to investigate the properties of fluids with complex phase behaviors.

Structural and dynamical properties of a core-softened fluid in a supercritical region

We present a theoretical study of the structural, thermodynamic, and transport properties of a supercritical fluid comprising particles interacting via isotropic attractive core-softened potential. The shear viscosity and self-diffusion coefficient are computed on the basis of the mode-coupling theory, with required structural input obtained from the thermodynamically self-consistent integral equation theory. We also consider dilute solutes in a core-softened fluid and use the anisotropic integral equation theory to obtain the solute-solute potential of mean force, which yields the second virial coefficient. We analyze its dependence on the solvent density and solute-solvent interaction strength.

Waterlike dynamic anomalies in a liquid described by a core-softened potential

We present a theoretical study of transport properties of a liquid comprised of particles interacting via isotropic core-softened potential. Shear viscosity and self-diffusion coefficient are computed on the basis of the mode-coupling theory, with required structural input obtained from thermodynamically self-consistent integral equation theory. Both self-diffusion coefficient and viscosity display waterlike anomalous density dependence, with diffusivity increasing and viscosity decreasing with density within a particular density range along several isotherms below a certain temperature. Our theoretical results for both transport coefficients are in good agreement with the simulation data.

Fifth-order thermodynamic perturbation theory of uniform and nonuniform fluids

A recently proposed numerical third-order thermodynamic perturbation theory (TPT) is extended to its fifth-order counterpart. Extensive performance evaluation based on several model potentials indicates that the fifth-order version is generally superior to both the third-order version and a well-known second-order macroscopic compressibility approximation TPT, and is at least comparable to an accurate self-consistent Ornstein-Zernike integral equation approximation (SCOZA) with regard to predictions of varying bulk thermodynamic properties. A bulk second-order direct correlation function (DCF) free of numerical solution of a bulk OZ integral equation, is proposed, which, in combination with the framework of classical density functional theory (DFT) and a thermodynamic consistency condition, extends the uniform fifth-order TPT to nonuniform case. Grand canonical ensemble Monte Carlo simulation is carried out to produce density profile of a core-softened fluid confined in a hard spherical cavity, the resultant density profile is employed to evaluate the performance of the present nonuniform fifth-order TPT and a recently proposed third +second-order perturbation DFT. It is found that the present nonuniform fifth-order TPT is generally more accurate than the third+second-order perturbation DFT. Additional advantages of the present nonuniform fifth-order TPT over the third +second-order perturbation DFT is discussed.

Allovalency revisited: An analysis of multisite phosphorylation and substrate rebinding

The utilization of multiple phosphorylation sites in regulating a biological response is ubiquitous in cell signaling. If each site contributes an additional, equivalent binding site, then one consequence of an increase in the number of phosphorylations may be to increase the probability that, upon dissociation, a ligand immediately rebinds to its receptor. How such effects may influence cell signaling systems is not well understood. Here, a self-consistent integral equation formalism for ligand rebinding, in conjunction with Monte Carlo simulations, is employed to further investigate the effects of multiple, equivalent binding sites on shaping biological responses. Multiple regimes that characterize qualitatively different physics due to the differential prevalence of rebinding effects are predicted. Calculations suggest that when ligand rebinding contributes significantly to the dose response, a purely allovalent model can influence the binding curves nonlinearly. The model also predicts that ligand rebinding in itself appears insufficient to generate a highly cooperative biological response.

Thermodynamics of a long-range triangle-well fluid

The long-range triangle-well fluid has been studied using three different approaches: firstly, by an analytical equation of state obtained by a perturbation theory, secondly via a self-consistent integral equation theory, the so-called self-consistent Ornstein–Zernike approach (SCOZA) which is presently one of the most accurate liquid-state theories, and finally by Monte Carlo simulations. We present vapour–liquid phase diagrams and thermodynamic properties such as the internal energy and the pressure as a function of the density at different temperatures and for several values of the potential range. We assess the accuracy of the theoretical approaches by comparison with Monte Carlo simulations: the SCOZA method accurately predicts the thermodynamics of these systems and the first-order perturbation theory reproduces the overall thermodynamic behaviour for ranges greater than two molecular diameters except that it overestimates the critical point. The simplicity of the equation of state and the fact that it is analytical in the potential range makes it a good candidate to be used for calculating other thermodynamic properties and as an ingredient in more complex theoretical approaches.

Theory of the spatial structure of nonlinear lasing modes

A self-consistent integral equation is formulated and solved iteratively which determines the steady-state lasing modes of open multi-mode lasers. These modes are naturally decomposed in terms of frequency dependent biorthogonal modes of a linear wave equation and not in terms of resonances of the cold cavity. A one-dimensional cavity laser is analyzed and the lasing mode is found to have non-trivial spatial structure even in the single-mode limit. In the multi-mode regime spatial hole-burning and mode competition is treated exactly. The formalism generalizes to complex, chaotic and random laser media.

Phase behavior of attractive and repulsive ramp fluids: Integral equation and computer simulation studies

Using computer simulations and a thermodynamically self-consistent integral equation we investigate the phase behavior and thermodynamic anomalies of a fluid composed of spherical particles interacting via a two-scale ramp potential (a hard core plus a repulsive and an attractive ramp) and the corresponding purely repulsive model. Both simulation and integral equation results predict a liquid-liquid demixing when attractive forces are present, in addition to a gas-liquid transition. Furthermore, a fluid-solid transition emerges in the neighborhood of the liquid-liquid transition region, leading to a phase diagram with a somewhat complicated topology. This solidification at moderate densities is also present in the repulsive ramp fluid, but in this case inhibits the fluid-fluid separation.