Square-gradient Term Research Articles

Liquid nucleation around charged particles in the vapor phase.

We theoretically investigate the nucleation of liquid droplets from vapor in the presence of a charged spherical particle. Due to field gradients, sufficiently close to the critical pointof the vapor-gas system, the charge destabilizes the vapor phase and initiates a phase transition. The fluid's free energy is described by the van der Waals expression augmented by electrostatic energy and a square-gradient term. We calculate the equilibrium density profile at arbitrary temperatures, particle charges, and vapor densities. In contrast to classical nucleation theory, here, both liquid and vapor phases are different from the bulk phases because they are spatially nonuniform. In addition, the theory applies to both sharp and diffuse interfaces and calculates the surface tension self-consistently. We find the composition profiles and integrate them to get the adsorption near the particle. We find that the adsorption changes discontinuously at a first-order phase transition line. This line becomes a second-order phase transition at high enough temperatures. We describe the transition pointnumerically and provide approximate analytical expressions for it. Similarly to prewetting, the adsorption diverges at the binodal phase boundary. We construct a phase diagram indicating changes in the binodal, spinodal, and critical temperature. It is shown that the field gradient enlarges the range of temperature and vapor density where liquid can nucleate.

Revisiting the Θ Point

Using the first-order perturbation theory, we compute the osmotic second and third virial coefficients, the mean-square end-to-end distance ⟨Re2⟩, and the mean-square radius of gyration ⟨Rg2⟩ of a polymer near the Θ point. Our model is based on the discrete Gaussian chain model and includes a square-gradient term accounting for the finite-range interaction (characterized by κ), in addition to the usual monomer second and third virial coefficients (characterized by v and w, respectively). The use of the discrete model avoids the divergence problems encountered in previous studies using the continuous model. Our study identifies four special temperatures in the Θ regime: the temperature ΘN where the osmotic second virial coefficient vanishes, the critical temperature ΘNcr for phase separation, and two compensation temperatures ΘN(e) and ΘN(g) at which ⟨Re2⟩ and ⟨Rg2⟩ reach their respective ideal values. In the infinite chain-length limit N → ∞, all of these four temperatures approach Θ∞, the Θ temperature for the infinitely long chain. These temperatures differ from each other by terms of order N–1/2. In general, these temperatures follow the order ΘN > ΘNcr and ΘN > ΘN(e) > ΘN(g). Furthermore, ΘN > Θ∞, in agreement with the result obtained by Khokhlov some time ago. On the other hand, depending on the ratio w/κb, Θ∞ can be higher than ΘN(e) (for w/κb < 9.45), lower than ΘN(g) (for w/κb > 11.63), or in between ΘN(e) and ΘN(g) (for 9.45 < w/κb < 11.63). ΘNcr can be either higher or lower than Θ∞ depending on whether w/b6 is larger or smaller than 0.574. From the order of these temperatures, we conclude that the chain is mostly expanded relative to the ideal chain at its ΘN. However, at Θ∞, the chain can be either expanded or contracted, depending on the relative position of Θ∞ with respect to ΘN(e) and ΘN(g) and depending on whether the chain dimension is measured by ⟨Re2⟩ or ⟨Rg2⟩.

The Cassie-Wenzel transition of fluids on nanostructured substrates: Macroscopic force balance versus microscopic density-functional theory.

Classical density functional theory is applied to investigate the validity of a phenomenological force-balance description of the stability of the Cassie state of liquids on substrates with nanoscale corrugation. A bulk free-energy functional of third order in local density is combined with a square-gradient term, describing the liquid-vapor interface. The bulk free energy is parameterized to reproduce the liquid density and the compressibility of water. The square-gradient term is adjusted to model the width of the water-vapor interface. The substrate is modeled by an external potential, based upon the Lennard-Jones interactions. The three-dimensional calculation focuses on substrates patterned with nanostripes and square-shaped nanopillars. Using both the force-balance relation and density-functional theory, we locate the Cassie-to-Wenzel transition as a function of the corrugation parameters. We demonstrate that the force-balance relation gives a qualitatively reasonable description of the transition even on the nanoscale. The force balance utilizes an effective contact angle between the fluid and the vertical wall of the corrugation to parameterize the impalement pressure. This effective angle is found to have values smaller than the Young contact angle. This observation corresponds to an impalement pressure that is smaller than the value predicted by macroscopic theory. Therefore, this effective angle embodies effects specific to nanoscopically corrugated surfaces, including the finite range of the liquid-solid potential (which has both repulsive and attractive parts), line tension, and the finite interface thickness. Consistently with this picture, both patterns (stripes and pillars) yield the same effective contact angles for large periods of corrugation.

Dynamical mean-field theory and weakly non-linear analysis for the phase separation of active Brownian particles

Recently, we have derived an effective Cahn-Hilliard equation for the phase separation dynamics of active Brownian particles by performing a weakly non-linear analysis of the effective hydrodynamic equations for density and polarization [Speck et al., Phys. Rev. Lett. 112, 218304 (2014)]. Here, we develop and explore this strategy in more detail and show explicitly how to get to such a large-scale, mean-field description starting from the microscopic dynamics. The effective free energy emerging from this approach has the form of a conventional Ginzburg-Landau function. On the coarsest scale, our results thus agree with the mapping of active phase separation onto that of passive fluids with attractive interactions through a global effective free energy (motility-induced phase transition). Particular attention is paid to the square-gradient term necessary for the phase separation kinetics. We finally discuss results from numerical simulations corroborating the analytical results.

Scalar φ4 field theory for active-particle phase separation

Recent theories predict phase separation among orientationally disordered active particles whose propulsion speed decreases rapidly enough with density. Coarse-grained models of this process show time-reversal symmetry (detailed balance) to be restored for uniform states, but broken by gradient terms; hence, detailed-balance violation is strongly coupled to interfacial phenomena. To explore the subtle generic physics resulting from such coupling, we here introduce 'Active Model B'. This is a scalar φ(4) field theory (or phase-field model) that minimally violates detailed balance via a leading-order square-gradient term. We find that this additional term has modest effects on coarsening dynamics, but alters the static phase diagram by creating a jump in (thermodynamic) pressure across flat interfaces. Both results are surprising, since interfacial phenomena are always strongly implicated in coarsening dynamics but are, in detailed-balance systems, irrelevant for phase equilibria.

The interplay between boundary conditions and flow geometries in shear banding: Hysteresis, band configurations, and surface transitions

We study shear banding flows in models of wormlike micelles or polymer solutions, and explore the effects of different boundary conditions for the viscoelastic stress. These are needed because the equations of motion are inherently non-local and include “diffusive” or square-gradient terms. Using the diffusive Johnson–Segalman model and a variant of the Rolie-Poly model for entangled micelles or polymer solutions, we study the interplay between different boundary conditions and the intrinsic stress gradient imposed by the flow geometry. We consider prescribed gradient (Neumann) or value (Dirichlet) of the viscoelastic stress tensor at the boundary, as well as mixed boundary conditions in which an anchoring strength competes with the gradient contribution to the stress dynamics. We find that hysteresis during shear rate sweeps is suppressed if the boundary conditions favor the state that is induced by the sweep. For example, if the boundaries favor the high shear rate phase then hysteresis is suppressed at the low shear rate edges of the stress plateau. If the boundaries favor the low shear rate state, then the high shear rate band can lie in the center of the flow cell, leading to a three-band configuration. Sufficiently strong stress gradients due to curved flow geometries, such as that of cylindrical Couette flow, can convert this to a two-band state by forcing the high shear rate phase against the wall of higher stress, and can suppress the hysteresis loop observed during a shear rate sweep.

Square-gradient scattering mechanism in surface-corrugated waveguides

In this work we study the surface scattering mechanisms from rough surfaces of a multimode quasi-1D waveguide (conducting quantum wires). The square-gradient scattering mechanism, which was missed in existing studies of the transport through surface-corrugated waveguides, is discovered. The main attention is paid to the interplay between the new mechanism and the known one, as well as its effect on the waveguide transport properties. For any value of the roughness height s, the square-gradient terms in the expression for the wave-scattering length (electron mean-free path) are dominant, provided the correlation length Rc of the surface disorder is small enough.

Manifestation of the roughness-square-gradient scattering in surface-corrugated waveguides

We study a new mechanism of wave/electron scattering in multi-mode surface-corrugated waveguides/wires. This mechanism is due to specific square-gradient terms in an effective Hamiltonian describing the surface scattering, that were neglected in all previous studies. With a careful analysis of the role of roughness slopes in a surface profile, we show that these terms strongly contribute to the expression for the inverse attenuation length (mean free path), provided the correlation length of corrugations is relatively small. The analytical results are illustrated by numerical data.

Gradient and amplitude scattering in surface-corrugated waveguides

We investigate the interplay between amplitude and square-gradient scattering from the rough surfaces in multi-mode waveguides (conducting quantum wires). The main result is that for any (even small in height) roughness the square-gradient terms in the expression for the wave scattering length (electron mean free path) are dominant, provided the correlation length of the surface disorder is small enough. This important effect is missed in existing studies of the surface scattering.

Anomalous corresponding-states surface tension of hydrogen fluoride and of the Onsager model

In a corresponding-states analysis of the liquid-vapor surface tension originally suggested by Guggenheim, we study the behavior of different simple (i.e., nonpolar), polar and ionic fluids. The results are compared to the corresponding ones for model fluids of each of the three types. For simple and weakly polar fluids (both real and model), the data map onto a master curve, as demonstrated by Guggenheim. For strongly dipolar, associating fluids, which also exhibit hydrogen bonding, one finds deviations from the master curve at low temperatures and, thus, observes the characteristic sigmoid behavior of the reduced surface tension as a function of temperature. The same is obtained for the model ionic fluid, the restricted primitive model. Truly exceptionally low values of the reduced surface tension are found for hydrogen fluoride and for the Onsager model of dipolar fluids, the surface tension of which we evaluate using an approximate hypernetted chain relation to obtain the square-gradient term in a modified van der Waals theory. Remarkably, in the corresponding-states plot, the surface tensions of HF and of the Onsager model agree very closely, while being well separated from the values for the other fluids. We also study the gradual transition of a model fluid from a simple fluid to a strongly dipolar one by varying the relative strength of dipolar and dispersion forces.

Lattice density-functional theory of surface melting: the effect of a square-gradientcorrection

I use the method of classical density-functional theory in the weighted-densityapproximation of Tarazona to investigate the phase diagram and the interface structure ofa two-dimensional lattice-gas model with three phases—vapour, liquid and triangular solid.While a straightforward mean-field treatment of the interparticle attraction is unable togive a stable liquid phase, the correct phase diagram is obtained when including a suitablychosen square-gradient term in the system grand potential. Taking this theory for granted,I further examine the structure of the solid–vapour interface as the triple point isapproached from low temperature. Surprisingly, a novel phase (rather than the liquid) isfound to grow at the interface, exhibiting an unusually long modulation alongthe interface normal. The conventional surface-melting behaviour is recoveredonly by artificially restricting the symmetries being available to the density field.

Interfacial properties deduced from nucleation experiments: A Cahn–Hilliard analysis

We apply a single-order-parameter Cahn–Hilliard theory to deduce properties of the fluid–crystal interface from nucleation experiments: The two Cahn–Hilliard parameters (the free energy scale and the coefficient of the square-gradient term) are chosen so that the experimentally determined interfacial free energy of nuclei is recovered. The theory is then used to predict the thickness and free energy of the equilibrium planar interface, and other quantities such as the Tolman length and characteristic thickness, which describe the curvature dependence of the interfacial free energy. The accuracy of the method is demonstrated on systems (Lennard-Jones and ice-water) for which these properties are known. Experimental data available for five stoichiometric oxide glasses are then analyzed. The reduced interfacial free energy (Turnbull’s α) and the interface thickness, we obtained, cover the α=0.28–0.51 and the d=0.8–1.6 nm ranges. For oxide glasses we find that α scales with n−1/3, where n is the number of molecules per formula unit. In agreement with computer simulations, the Tolman length is strongly size dependent, while far weaker though still perceptible temperature dependence is observed for the characteristic interface thickness used in Gránásy’s phenomenological diffuse interface theory. In some cases bulk crystal properties prevail at the center of nuclei, while in other systems the nuclei are ramified, and the local properties significantly deviate from those of the macroscopic crystal. The accuracy of these results rests on a hypothesized temperature independence of the Cahn–Hilliard parameters, an assumption whose validity remains to be seen at large undercoolings.

Fluctuations and instabilities of model amphiphile interfaces

We study the stability of planar, cylindrical and spherical interfaces with respect to shape and width fluctuations for a model amphiphile solution described by a free energy density functional with square-gradient and square-Laplacian terms. That is, we determine the stability matrix when the stationary state consists of an interface with given geometry that separates two immiscible solvent phases. From the spectrum and the related eigenfunctions of this matrix we establish where lamellar and micellar domain-structured phases occur, and contrast our results with those for a simple square-gradient fluid model for which these phases are always unstable. We also characterize some instability properties such as the buckling of lamella, the undulation of cylindrical structures and the nucleation of micelles.

The liquid-vapour interface of the restricted primitive model (RPM) of ionic fluids

The liquid-vapour interface of the restricted primitive model (RPM) of ionic fluids is investigated within a square-gradient theory. We compute density profiles and interfacial tensions for different temperatures using Debye-Hückel (DH) theory and its recent extension for ion-pair formation and interactions between the dipolar ion pairs and free ions developed by Fisher and Levin. This Fisher-Levin (FL) theory is known to give an accurate description of the coexistence curve of the RPM. To account for the inhomogeneities in the interfacial region, the local free-energy density is expanded in terms of the density gradient. For small gradients, e.g. reasonably close to the critical point, such an expansion can be truncated after the square-gradient term. The coefficient of the latter is calculated from the direct correlation function using an approximate (quadratic) hypernetted-chain (AHNC) relation and, alternatively, from an extended van der Waals approach in conjunction with different approximations to the local density. The results from the AHNC relation and various local density approximations in the thermodynamic framework of DH theory and FL theory, respectively, are compared, and it is asserted that the AHNC relation in conjunction with FL theory predicts reliably the interfacial properties of the RPM even within this simple square-gradient theory. In contrast to the situation for simple fluids, the local density approximation must be chosen carefully for ionic fluids since properties such as the interfacial thickness and the surface tension may vary by a factor of three or four depending on the applied local density approximation.

Cahn–Hilliard-type density functional calculations for homogeneous ice nucleation in undercooled water

The free energy of ice nuclei and the nucleation rate in undercooled water are calculated in the framework of a single-order parameter Cahn–Hilliard theory. The coefficients of the quartic free energy-order parameter relationship and of the square-gradient term are chosen in order to reproduce the measured Gibbs free energy difference, the ice–water interfacial free energy, and the interface thickness predicted by molecular dynamics simulations. Without adjustable parameters, the model reproduces fairly the experimental nucleation rates down to −40°C. A simple one-parameter model is adopted to describe the specific heat anomaly of liquid water at lower temperatures. The range of nucleation rates allowed by thermodynamically consistent choices of this parameter encloses extreme experimental values (∼1030m−3s−1) Bartell and Huang obtained at deep undercoolings (∼− 73°C). The temperature and size-dependencies of the interfacial properties (free energy, Tolman-length, etc.) are discussed.

Ginzburg criterion for the crossover behavior of model fluids

The Ginzburg criterion, which is based on the three coefficients in the Landau–Ginzburg (LG) expansion of the Helmholtz free energy density of a nonuniform system, is believed to give a reasonable estimate for the temperature scale on which crossover occurs. To compute the contribution of the square-gradient term in the LG expansion, we extend the van der Waals theory of surface tension and, in contrast to our earlier treatment, account for the dependence of the pair distribution function on the spatially varying density. Via this approach we calculate and compare the Ginzburg temperatures of ionic, dipolar, and simple model fluids, namely the restricted primitive model (RPM), the Onsager model, and the square-well fluid (the second and third virial coefficients, for which we also present exact results). To compute the properties of the RPM, we employ the Fisher–Levin theory and its recent extension for Debye-shielded dipole–dipole interactions and a state-dependent dielectric constant that was developed by us. In contrast to the results of our earlier work and in accordance with the calculations of Fisher and Lee, we now find that the RPM has no exceptionally small region in which mean-field theory fails.

On the Ginzburg temperature of ionic and dipolar fluids

Critical fluctuations in fluids are investigated within the framework of the generalized van der Waals theory. The square-gradient term—added to the Landau expansion of the Helmholtz free energy density—is obtained following a procedure similar to that originally proposed by van der Waals in the theory of surface tension, however replacing the Heaviside step function originally used by an approximative pair distribution function. Representative for ionic fluids we choose the restricted primitive model (RPM) and treat it within the Debye–Hückel theory, thus neglecting effects of ion pairing. The other approximative extreme—complete ion pairing resulting in a fluid of hard dipolar dumbbells—is mimicked by a fluid composed of dipolar hard spheres (DHS). For this case we use the Onsager reaction field and the second pressure virial coefficient. We calculate the amplitudes of the correlation length and the Ginzburg temperatures, and find (in reduced quantities) ξ0*=3.50 and ΔTGi*=0.0087 for the ionic system, and ξ0*=0.82 and ΔTGi*=1.63 for the dipolar fluid. For calibration we compute the same quantities for simple neutral fluids and obtain ξ0*=0.50 and ΔTGi*=2.89 for a Sutherland fluid (hard core term plus attractive r−6-potential) and ξ0*=0.43 and ΔTGi*=8.50 for a square-well fluid. The result of a smaller Ginzburg temperature for the ionic fluid than for nonionic fluids in a treatment that neglects ion pairing is clearly at variance with the results of other groups. The correlation length in the low-density limit obtained from our approach has the same functional dependencies as the Lee–Fisher expression, but differs by a numerical factor of 5.7.

Free energy expressions for a spherical interface

The free energy is analysed of a spherical interface generated in a fluid described by a local free energy density functional that features square-gradient and square-Laplacian terms. The bulk, the surface tension and the bending rigidity terms are investigated, and the position is found for the dividing surface that satisfies the generalized Laplace equation that incorporates bending terms. The results agree with those obtained previously from the general expression for the stress tensor of an interface of arbitrary shape (Romero-Rochín, V., Varea, C., and Robledo, A., 1991, Phys. Rev. A, 44, 8417; 1993, Phys. Rev. E, 46, 1600). Also, when a comparison is made between spherical and planar interfaces the expressions obtained are those derived by Gompper, G., and Zschocke, S. (1991, Europhys. Lett., 18, 731) and by Blokhuis, E. M., and Bedeaux, D. (1993, Molec, Phys., 80, 705). These expressions correspond to the length of Tolman and to similar higher-order correction terms.

Length-scale analysis of the coefficient of the square-gradient term in the density-functional expansion for nonuniform classical plasmas.

The coefficient of the square-gradient term, given by the second moment of the direct correlation function, is determined by three different length-scale parameters, each dominating a different range of the plasma coupling parameter \ensuremath{\Gamma}. It is the interplay between the relative contributions of these scales that provides the full form of the square-gradient term as a function of \ensuremath{\Gamma}.

The density-gradient expansion and surface properties of the one-component classical plasma

The surface properties of the one-component classical plasma (OCP) are re-examined by an approximate method using the density-gradient expansion truncated at second order. The coefficient of the square-gradient term is calculated via the modified hypernetted-chain (MHNC) scheme. It is found that the result is quite different from the previous ones obtained by the Baus-Hansen and mean spherical approximations. The origin of these differences and certain difficulties in the evaluation of this coefficient are discussed in some detail. The present MHNC result for the square-gradient term is used in the variational calculations for the surface density profile and the surface energy. The results of these calculations are in much better agreement than previous ones with Monte Carlo results. The authors also consider the OCP of finite size with a spherically symmetric charge background, for which Monte Carlo simulations have been performed. It is found that the surface energy appropriately defined for such systems shows appreciable size dependences especially for the OCP with small plasma parameters.