Virial Theory Research Articles

Evidence of T-type structures of hard square boards in capillary confinement.

We employ Onsager's second virial density functional theory combined with the Parsons-Lee theory within the restricted orientation (Zwanzig) approximation to examine the phase structure of hard square boards of dimensions (L×D×D) uniaxially confined in narrow slabs. Depending on the wall-to-wall separation (H), we predict a number of distinctly different capillary nematic phases, including a monolayer uniaxial or biaxial planar nematic, homeotropic with a variable number of layers, and a T-type structure. We determine that the favored phase is homotropic, and we observe first-order transitions from the homeotropic structure with n layers to n+1 layers as well as from homeotropic surface anchoring to a monolayer planar or T-type structure involving both planar and homeotropic anchoring at the pore surface. By increasing the packing fraction, we further demonstrate a reentrant homeotropic-planar-homeotropic phase sequence in a particular range (i.e., H/D=1.1 and 0.25≤L/D<0.26). We find that the T-type structure is more stable when the pore is wide enough with respect to the planar phase. The enhanced stability of the mixed-anchoring T-structure is unique for square boards and becomes manifest at pore width exceeding L+D. More specifically, the biaxial T-type structure emerges directly from the homeotropic state without intervention of a planar layer structure as observed for other convex particle shapes.

The role of the second virial coefficient in the vapor-liquid phase coexistence of anisotropic square-well particles

We examine the role of the second virial coefficient in the vapor-liquid (VL) phase coexistence of anisotropic hard bodies with square-well attractions of variable range. According to the extended law of corresponding states, the parameters of hard body interactions and attractions can be built into the reduced density and the reduced second virial coefficients, respectively, which gives rise to the collapse of all VL binodals in the reduced second virial coefficient vs. reduced density plane. The second virial perturbation theory shows that the shape dependence appears as an extra parameter in the phase behavior of anisotropic particles, which does not make possible the perfect collapse of the VL binodals for varying shapes. Interestingly, the binodal curves go closely together and even cross each other in the liquid side allowing to define a quasi-master curve. The existence of an almost perfect master curve is confirmed by replica-exchange Monte Carlo simulations for oblate square-well ellipsoids with several shape anisotropies.

Thermodynamic perturbation theory coefficients for hard spherocylinders and cylinders

Thermodynamic perturbation theories are indispensable in the development of equations of state based on Statistical Mechanics. Recently, a new statistical thermodynamic model has been proposed in which molecules are represented as cylindrically symmetric hard bodies (ellipsoids of revolution, spherocylinders, and cylinders) interacting with identical neighboring molecules through a spherically symmetric square-well potential. This novel equation of state embeds at least two approximations to the Onsager’s second virial theory: one restrains the equation of state to systems with an isotropic (disordered) orientation distribution, while the other decouples the translational and orientational degrees of freedom, being only valid at low densities. Using canonical Monte Carlo simulations, we investigated the feasibility of such approximations by determining the uniaxial nematic order parameter of the systems and calculating the first- and second-order perturbation coefficients of the high-temperature series expansion (HTSE) of the Helmholtz free energy. Our results indicate that both approximations are reasonable for a certain range of packing fractions and molecular anisotropies. We have also calculated the full Helmholtz free energy of the perturbed system, which can be used for determining high-order perturbation terms to improve the predictive capability of the equation of state.

Reexamining equationsof state of oblate hard ellipsoids of revolution: Numerical simulation utilizing a cluster Monte Carlo algorithm and comparison to virial theory.

We provide highly accurate equation-of-state data determined by means of cluster Monte Carlo simulations for the isotropic phase of oblate hard ellipsoids of revolution. Both equation-of-state data and phase boundaries of the isotropic phase are obtained from relatively large ensembles with typically 1000 particles. The comparison of simulation data with a virial approach gives evidence for the importance of high-order so-far-unknown virial coefficients and therewith many-particle interactions in dense, isotropic systems of anisotropic particles. While a virial approach with a rescaled Carnahan-Starling correction for the unknown, higher-order virial coefficients reproduces the simulation data of moderately anisotropic particles with high accuracy, we suggest for highly anisotropic shapes a simple, heuristic equationof state as a suitable approach.

Extended law of corresponding states: square-well oblates

The vapour–liquid coexistence collapse in the reduced temperature, T r = T/T c , reduced density, ρ r = ρ/ρ c , plane is known as a principle of corresponding states, and Noro and Frenkel have extended it for pair potentials of variable range. Here, we provide a theoretical basis supporting this extension, and show that it can also be applied to short-range pair potentials where both repulsive and attractive parts can be anisotropic. We observe that the binodals of oblate hard ellipsoids for a given aspect ratio (κ = 1/3) with varying short-range square-well interactions collapse into a single master curve in the –ρ r plane, where , B 2 is the second virial coefficient, and v 0 is the volume of the hard body. This finding is confirmed by both REMC simulation and second virial perturbation theory for varying square-well shells, mimicking uniform, equator, and pole attractions. Our simulation results reveal that the extended law of corresponding states is not related to the local structure of the fluid.

Failure of standard density functional theory to describe the phase behavior of a fluid of hard right isosceles triangles.

A fluid of hard right isosceles triangles was studied using an extension of scaled-particle density-functional theory which includes the exact third virial coefficient. We show that the only orientationally ordered stable liquid-crystal phase predicted by the theory is the uniaxial nematic phase, in agreement with the second-order virial theory. By contrast, Monte Carlo simulations predict exotic liquid-crystal phases exhibiting tetratic and octatic correlations, with orientational distribution functions having four and eight equivalent peaks, respectively. This demonstrates the failure of the standard density-functional theory based on two- and three-body correlations to describe high-symmetry orientational phases in two-dimensional hard right-triangle fluids, and it points to the necessity to reformulate the theory to take into account high-order body correlations and ultimately particle self-assembling and clustering effects. This avenue may represent a great challenge for future research, and we discuss some fundamental ideas to construct a modified version of density-functional theory to account for these clustering effects.

Enhanced two-dimensional nematic order in slit-like pores

The effect of out-of-plane positional freedom is examined on the stability of two-dimensional (2D) nematic order of hard non-spherical particles using the second virial density-functional theory. The particles are allowed to move and rotate freely in the plane of confining walls and can move between the two parallel walls. The wall-to-wall distance (H) is varied between the strictly 2D and the two-layer forming cases, i.e. σ < H < 2σ, where σ is the particle’s shortest length. As expected, we observe that more and more particles are required for the formation of 2D nematics with increasing H when the rod-like particles are hard ellipsoids. Surprisingly, we found that the opposite tendency is observed in the case of hard cylinders, i.e. fewer and fewer particles are needed to stabilize the nematic order with increasing H. This paradox can be understood by projecting the three-dimensional system into a 2D mixture of particles having position-dependent aspect ratios and molecular areas. However, the complex phase behaviour found for plate-like cylindrical particles with increasing H cannot be explained in terms of the same simple geometrical arguments.

Nanofracture of stretched hexagonal boron nitride strip with an edge crack

The tensile fracture behavior in hexagonal boron nitride (h-BN) strip with an introduced zigzag edge crack is investigated by molecular dynamics (MD) using a modified Tersoff potential. The far field remote stress σremote at crack initiation for various original crack lengths is examined, which shows that Griffith criterion overestimates σremote for short cracks < 2.5 nm while underestimates that for longer cracks. Thus we propose a modified exponent factor 0.28 for Griffith criterion rather than 0.5. The asymptotic stress field around nano-crack is calculated by MD atomic virial stress theory, comparing with continuum theory of stress intensity factor K and stress concentration factor, which indicates that elastic continuum overestimates local stresses around the crack front as the bond breaking stress remains ~ 54.4 GPa at the tip. The critical energy release rate Gc at initiation for small cracks < 1.5 nm is also much higher than theoretical toughness GIc. Our results provide a clear understanding of the nano-crack fracture in a h-BN strip for the continuum limitation in discrete hexagonal lattices.

Second-Virial Onsager Theory and Its Limitations in the Prediction of the Ordering Transitions of Confined Hard Rods between Two Parallel Hard Walls

We studied the phase behavior of hard rectangular rods with edge lengths L, D, and D. All the calculations have been carried out through Onsager second virial theory within the three-state Zwanzig ...

Insights into excluded volume and percolation of soft interphase and conductivity of carbon fibrous composites with core-shell networks

Interphase configurations affect conductive properties of carbon fibrous composites (CFCs), specifically the percolation of interphase triggers the dramatic change of conductive properties. It has been a key but unresolved issue how to capture the percolation of soft interphase interacted by anisotropic-shaped carbon fibers and its quantitative effect on the conductivity of CFCs. This work proposes a theoretical framework for the accurate predictions of the excluded volume and percolation threshold of soft interphase and the effective conductivity of CFCs. This framework contains three theoretical models: (1) a coupling scheme between the second virial coefficient theory and Monte Carlo simulations for determining the excluded volumes; (2) an excluded-volume-based percolation model for obtaining the percolation threshold of soft interphase; (3) a continuum percolation-based micromechanical model (CPMM) for predicting the effective conductivity of CFCs with spherocylindrical carbon fibers of high volume fraction and their surrounding interphase percolation. Comparison against numerical and experimental data validates that the theoretical framework can well predict these properties of CFCs. This framework can be regarded as a general scheme that is readily applicable to other core-shell networks. Furthermore, we use the framework to probe various factors effects on the percolation threshold of interphase and electrical conductivity of CFCs.

Elliptical Gaussian beam propagation in nonlinear fibres with focusing and defocusing refractive profiles

The paper shows systematic analysis of elliptical Gaussian beam (GB) propagating in self-focusing/self-defocusing media of Kerr type with focusing/defocusing linear refractive profiles. Instead of solving commonly accepted Nonlinear Schrödinger Equation (NLS) we propose equations of geometrical optics: complex eikonal equation and complex transport one. Eikonal equation let us derive immediately the ordinary differential equations for principle widths omitting this way complicated variational process and virial theory used in nonlinear optics. From the transport equation we obtain first order ordinary differential equation for complex amplitude evolution and conservation principle for energy flux in the elliptical GB cross-section. For combined effects of diffraction, nonlinear refraction and focusing/defocusing graded-index factor we observe three types of the solutions for the elliptical radius of principle beam widths: self-focusing or self-defocusing together with diffraction widening, stationary and oscillatory ones under and without collapse effect. We prove here that principle curvatures of the wave front can introduce into our description the additional ellipticity of the beam, which can effectively limit collapse effect. Moreover, we discuss the evolution of elliptical Gaussian beam in nonlinear self-defocusing fibre with focusing refractive profile presenting conditions for existence of elliptical spatial solitons. We discuss here also the influence of initial wave front curvatures on elliptical Gaussian beam evolution in nonlinear inhomogeneous fibres of Kerr-type.

Hydroelastic solitary waves with constant vorticity

In this work, two-dimensional hydroelastic solitary waves in the presence of constant vorticity are studied. Time-dependent conformal mapping techniques first developed for irrotational waves are applied subject to appropriate modification. An illustrative high-order Nonlinear Schrödinger Equation is presented to investigate whether a given envelope collapses into a singular point in finite time by using the virial theory. Travelling solitary waves on water of infinite depth are computed for different values of vorticity and new generalised solitary waves are discovered. The stabilities of these waves are examined numerically by using fully nonlinear time-dependent computations which confirm the virial theory analysis.

Ordering transitions of weakly anisotropic hard rods in narrow slitlike pores.

The effect of strong confinement on the positional and orientational ordering is examined in a system of hard rectangular rods with length L and diameter D (L>D) using the Parsons-Lee modification of the second virial density-functional theory. The rods are nonmesogenic (L/D<3) and confined between two parallel hard walls, where the width of the pore (H) is chosen in such a way that both planar (particle's long axis parallel to the walls) and homeotropic (particle's long axis perpendicular to the walls) orderings are possible and a maximum of two layers is allowed to form in the pore. In the extreme confinement limit of H≤2D, where only one-layer structures appear, we observe a structural transition from a planar to a homeotropic fluid layer with increasing density, which becomes sharper as L→H. In wider pores (2D<H<3D) planar order with two layers, homeotropic order, and even combined bilayer structures (one layer is homeotropic, while the other is planar) can be stabilized at high densities. Moreover, first-order phase transitions can be seen between different structures. One of them emerges between a monolayer and a bilayer with planar orders at relatively low packing fractions.

Complex geometrical optics and nonlinear absorption/gain effects

The paper presents systematic analysis of the influence of nonlinear absorption/gain phenomena on Gaussian wave field evolution in Kerr type medium with additional consideration of the initial light beam convergence and divergence. To perform efficient analysis of the joint contribution of the initial curvature of the wave front and nonlinear absorption/gain effects on self-focusing/self-defocusing phenomena I propose to apply the method of complex geometrical optics (CGO) which at once reduces Gaussian beam diffraction and self-action effects (including nonlinear absorption/gain) to domain of ordinary differential equations (ODEs), which are base (output) equations. Description in output ODEs causes that CGO possess great superiority over well known analytic methods of nonlinear optics such as: variational method and method of moments which every time demand to solve Nonlinear Schrodinger Equation (NLS) by applying integral variational procedure or virial theory to obtain resulting equations describing evolution of amplitude, beam width and wave front curvature which happen to be identical with those obtained by CGO method. CGO method dealing with output ODEs is time consuming physical approach comparing to the numerical methods of nonlinear and wave optics. CGO with output evolutionary ODEs enables one to apply basic mathematical computer software such as: Matlab/Octave, Mathcad and Mathematica commonly available on every University, Colleague and secondary school nowadays. Full Text: PDF References N. N. Akhmediev and A. Ankiewicz, Solitons. Nonlinear Pulses and Beams. (Chapman and Hall, London 1997). N. N. Akhmediev and A. Ankiewicz ed. Dissipative Solitons. Lecture Notes in Physics, Vol. 661, p. 1-443, ISBN: 978-3-540-23373-2, (Springer Berlin Heidelberg, 2005). CrossRef E. Jaatinen and M. W. Jones, "Theoretical description of low divergence Gaussian fields in self-defocusing photorefractive media", Optics Communications, 281, p. 3201-3207 (2008). CrossRef P. Berczynski, and S. Marczynski. Book Chapter 1: Gaussian Beam Propagation in Inhomogeneous Nonlinear Media. Description in Ordinary Differential Equations by Complex Geometrical Optics , (Edited by: P. W. Hawkes) , Advances in Imaging and Electron Physics, 185, pages 1-111, ISBN 9780128001448, Elsevier New York, (2014). Y. Silberberg, "Solitons and two-photon absorption", Optics Letters, vol. 15, No. 18, p.1005-1007 (1990). CrossRef D. Ianetz, Yu. Kaganovskii, A. D. Wilson-Gordon, and M. Rosenbluh, "Propagation of an asymmetric Gaussian beam in a nonlinear absorbing medium", Phys. Rev. A 81, 53851, pages-1-6 (2010). CrossRef D. Mc Morrow, W. Lotshaw, J. Melinger, S. Buchner, and R. Pease, "Subbandgap laser-induced single event effects: carrier generation via two-photon absorption", IEEE Trans. Nucl. Sci., vol. 49, no. 6, pp. 3002 ? 3008 (2002). CrossRef

Orientational ordering of confined hard rods: the effect of shape anisotropy on surface ordering and capillary nematization.

We examine the ordering properties of rectangular hard rods with length L and diameter D at a single planar wall and between two parallel hard walls using the second virial density-functional theory. The theory is implemented in the three-state Zwanzig approximation, where only three mutually perpendicular directions are allowed for the orientations of hard rods. The effect of varying shape anisotropy is examined at L/D=10,15,and20. In contact with a single hard wall, the density profiles show planar ordering, damped oscillatory behavior, and a wall-induced surface ordering transition below the coexisting isotropic density of a bulk isotropic-nematic (I-N) phase transition. Upon approaching the coexisting isotropic density, the thickness of the nematic film diverges logarithmically, i.e., the nematic wetting is complete for any shape anisotropy. In the case of confinement between two parallel hard walls, it is found that the continuous surface ordering transition depends strongly on the distance between confining walls H for H<L, while it depends weakly on H for H>L. The minimal density at which a surface ordering transition can be realized is located at around H∼2D for all studied shape anisotropies due to the strong interference effect between the two hard walls. The first-order I-N phase transition of the bulk system becomes a surface ordered isotropic I_{B} to capillary nematic N_{B} phase transition in the slit pore. This first-order I_{B}-N_{B} transition weakens with decreasing pore width and terminates in a critical point for all studied shape anisotropies.

Interactions in protein mixtures. Part II: A virial approach to predict phase behavior

A virial theory was used to relate molecular interactions (in terms of second virial coefficients, B′) and molecular size ratios to liquid–liquid phase separation. Application of the virial theory to binary hard sphere mixtures (additive and non-additive) confirmed the applicability of this simple approach towards predicting phase behavior based on two-particle interactions.Experimentally, second cross virial coefficients were obtained for dextran/gelatin, whey protein isolate (WPI)/gelatin mixtures and whey protein aggregate (WPA)/gelatin mixtures using membrane osmometry at varying ionic strength. From this, solvent conditions where interactions between proteins are dominated by electrostatics and solvent conditions where interactions are dominated by hard body interactions could be determined.Using experimentally obtained second virial coefficients, the liquid–liquid phase separation for gelatin/dextran mixtures was successfully predicted. Second cross virial coefficients for gelatin/whey protein isolate and for gelatin/whey protein aggregate could be related to the absence of phase separation in these mixtures. This could be related to a similar size of the proteins and their non-additive behavior at conditions where they mainly interact via hard body interactions.

The isotropic-nematic and nematic-nematic phase transition of binary mixtures of tangent hard-sphere chain fluids: an analytical equation of state.

An analytical equation of state (EoS) is derived to describe the isotropic (I) and nematic (N) phase of linear- and partially flexible tangent hard-sphere chain fluids and their mixtures. The EoS is based on an extension of Onsager's second virial theory that was developed in our previous work [T. van Westen, B. Oyarzún, T. J. H. Vlugt, and J. Gross, J. Chem. Phys. 139, 034505 (2013)]. Higher virial coefficients are calculated using a Vega-Lago rescaling procedure, which is hereby generalized to mixtures. The EoS is used to study (1) the effect of length bidispersity on the I-N and N-N phase behavior of binary linear tangent hard-sphere chain fluid mixtures, (2) the effect of partial molecular flexibility on the binary phase diagram, and (3) the solubility of hard-sphere solutes in I- and N tangent hard-sphere chain fluids. By changing the length bidispersity, two types of phase diagrams were found. The first type is characterized by an I-N region at low pressure and a N-N demixed region at higher pressure that starts from an I-N-N triphase equilibrium. The second type does not show the I-N-N equilibrium. Instead, the N-N region starts from a lower critical point at a pressure above the I-N region. The results for the I-N region are in excellent agreement with the results from molecular simulations. It is shown that the N-N demixing is driven both by orientational and configurational/excluded volume entropy. By making the chains partially flexible, it is shown that the driving force resulting from the configurational entropy is reduced (due to a less anisotropic pair-excluded volume), resulting in a shift of the N-N demixed region to higher pressure. Compared to linear chains, no topological differences in the phase diagram were found. We show that the solubility of hard-sphere solutes decreases across the I-N phase transition. Furthermore, it is shown that by using a liquid crystal mixture as the solvent, the solubility difference can by maximized by tuning the composition. Theoretical results for the Henry's law constant of the hard-sphere solute are in good agreement with the results from molecular simulation.

The isotropic-nematic phase transition of tangent hard-sphere chain fluids—Pure components

An extension of Onsager's second virial theory is developed to describe the isotropic-nematic phase transition of tangent hard-sphere chain fluids. Flexibility is introduced by the rod-coil model. The effect of chain-flexibility on the second virial coefficient is described using an accurate, analytical approximation for the orientation-dependent pair-excluded volume. The use of this approximation allows for an analytical treatment of intramolecular flexibility by using a single pure-component parameter. Two approaches to approximate the effect of the higher virial coefficients are considered, i.e., the Vega-Lago rescaling and Scaled Particle Theory (SPT). The Onsager trial function is employed to describe the orientational distribution function. Theoretical predictions for the equation of state and orientational order parameter are tested against the results from Monte Carlo (MC) simulations. For linear chains of length 9 and longer, theoretical results are in excellent agreement with MC data. For smaller chain lengths, small errors introduced by the approximation of the higher virial coefficients become apparent, leading to a small under- and overestimation of the pressure and density difference at the phase transition, respectively. For rod-coil fluids of reasonable rigidity, a quantitative comparison between theory and MC simulations is obtained. For more flexible chains, however, both the Vega-Lago rescaling and SPT lead to a small underestimation of the location of the phase transition.

Measurement and Modeling of Adsorption of Lower Hydrocarbons on Activated Carbon

This work reports adsorption isotherms for C1 to C6 hydrocarbons on activated carbon at three different temperatures (293 K, 318 K, and 358 K) over a wide range of pressure (0 bar to 100 bar). The isotherms were measured using a standard gravimetric method. The experimental data were correlated and compared using Toth, modified virial, and potential theory models. On the basis of the adsorption potential, characteristic curves were also generated for methane, ethane, propane, isobutane, n-pentane, and n-hexane on activated carbon over broad ranges of pressure and temperatures. The micropore volume of the activated carbon predicted from potential theory was in good agreement with those obtained using a N2 isotherm measured at 77 K. The enthalpy of adsorption at zero loading was found to increase linearly with the carbon number.

46. The validity of using a grouped intracellular solute when calculating chemical potentials with the multi-solute osmotic virial solution theory

46. The validity of using a grouped intracellular solute when calculating chemical potentials with the multi-solute osmotic virial solution theory