Wall Free Energy Research Articles

Chemo-elasto-electro free energy of non-uniform system in the diffuse interface context

In the present work, we propose an alternative approach for deriving the free energy formulation of a non-uniform system. Compared with the work of Cahn and Hilliard (1958 J.Chem. Phys.28258-67), our approach provides a more comprehensive explanation for the individual energy contribution in a non-uniform system, including entropy, interaction energy, and internal energy. By employing a fundamental mathematical calculus, we reformulate the local composition within the interface region. Utilizing the reformulated local composition as well as classic thermodynamic principles, we establish formal expressions for entropy, interaction energy, and the internal energy, which are functions of both composition and composition gradients. We obtain a comprehensive free energy expression for a non-uniform system by integrating these energy density formulations. The obtained free energy expression is consistent with the formula type of Cahn and Hilliard and prodives more deeper physical interpretation. Moreover, using the same approach, we derive formulations for elastic energy and electric potential energy in a non-uniform system. However, the proposed approach encounters a limitation in the special case of a non-uniform fluid contacting a solid substrate. Due to the significant difference in the length scales between the solid-fluid and fluid-fluid interfaces, the wall free energy formulation based on the aforementioned concept is unsuitable for this multi-scale system. To address this limitation, we reformulate the wall free energy as a function of the average composition over the solid-fluid interface. Additionally, the previous derivation relies on an artificial treatment of describing the composition variation across the interface by a smooth monotone function, while the true nature of this variation remains unclear. By utilizing the concept of average composition, we circumvent the open question of how the composition varies across the interface region. Our work provides a thorough understanding for the construction of free energy formulations for a non-uniform system in condensed matter physics.

Wetting Effect Induced Depletion and Adsorption Layers: Diffuse Interface Perspective.

When a multi-component fluid contacts arigid solid substrate, the van der Waals interaction between fluids and substrate induces a depletion/adsorption layer depending on the intrinsic wettability of the system. In this study, we investigate the depletion/adsorption behaviors of A-B fluid system. We derive analytical expressions for the equilibrium layer thickness and the equilibrium composition distribution near the solid wall, based on the theories of de Gennes and Cahn. Our derivation is verified through phase-field simulations, wherein the substrate wettability, A-B interfacial tension, and temperature are systematically varied. Our findings underscore two pivotal mechanisms governing the equilibrium layer thickness. With an increase in the wall free energy, the substrate wettability dominates the layer formation, aligning with de Gennes' theory. When the interfacial tension increases, or temperature rises, the layer formation is determined by the A-B interactions, obeying Cahn's theory. Additionally, we extend our study to non-equilibrium systems where the initial composition deviates from the binodal line. Notably, macroscopic depletion/adsorption layers form on the substrate, which are significantly thicker than the equilibrium microscopic layers. This macroscopic layer formation is attributed to the interplay of phase separation and Ostwald ripening. We anticipate that the present finding could deepen our knowledge on the depletion/adsorption behaviors of immiscible fluids.

Effect of wall free energy formulation on the wetting phenomenon: Conservative Allen-Cahn model.

Cahn introduced the concept of wall energy to describe the interaction between two immiscible fluids and a solid wall [J. W. Cahn, J. Chem. Phys. 66, 3667-3672 (1977)]. This quintessential concept has been successfully applied to describe various wetting phenomena of a droplet in contact with a solid surface. The usually formulated wall free energy results in the so-called surface composition that is not equal to the bulk composition. This composition difference leads to a limited range of contact angles which can be achieved by the linear/high-order polynomial wall free energy. To address this issue and to improve the adaptability of the model, we symmetrically discuss the formulation of the wall free energy on the Young's contact angle via Allen-Cahn model. In our model, we modify the calculation of the fluid-solid interfacial tensions according to the Cahn's theory by considering the excess free energy contributed by the distorted composition profile induced by the surface effect. Additionally, we propose a semi-obstacle wall free energy which enforces the surface composition to be the bulk composition within the framework of bulk obstacle potential. By this way, the accuracy of the contact angle close to 0° and 180° is significantly improved in the phase-field simulations. We further reveal that the volume preservation term in the conservative Allen-Cahn model has a more significant impact on the wetting behavior on superhydrophobic surfaces than on hydrophilic surfaces, which is attributed to the curvature effect. Our findings provide alternative insights into wetting behavior on superhydrophilic and superhydrophobic surfaces.

Phase diagram of the square 2D Ising lattice with nearest neighbor and next-nearest neighbor interactions

We have determined the temperature dependent phase diagram of the square 2D Ising lattice with anisotropic nearest neighbor () and isotropic next-nearest neighbor () interactions. The phase boundaries between the various ordered phases (ferromagnetic, antiferromagnetic and striped antiferromagnetic) and the disordered phase (paramagnetic) are obtained by considering the domain wall free energy. Although the phase boundary equations are not exact, they provide a very accurate description when the nearest neighbor interactions are stronger than the next-nearest neighbor interaction. The square 2D Ising lattice does not exhibit a phase transition when if and have the same sign or if and have opposite signs.

Phase field modeling and computation of multi-component droplet evaporation

Droplet evaporation is an interesting physical phenomenon and it has extensive applications in natural world or industrial fields. To investigate the evaporation of a single droplet and multiple droplets on a substrate, we develop novel and effective models based on phase field method. The interface of droplets can be implicitly captured by a specific level-set of order parameter (phase field function). The contact angle boundary condition derived from a wall free energy is adopted to reflect different wetting conditions. To reflect the effect of evaporation, we derive an evaporation term based on the relation between volume change and surface area. In each time step, the operator splitting based temporal discretization is adopted and all nonlinear terms are explicitly treated. Therefore, we can update each component in a step-by-step manner and the algorithm is efficient to implement. The proposed models and algorithms are validated by comparing the contact angle, spreading length, change of radius, and change of volume with analytical results. The effect of contact angle on a single droplet is investigated. Furthermore, the evaporations of multiple droplets under different contact angle conditions and pinning condition are simulated.

Dissipative spatial discretization of a phase field model of multiphase multicomponent isothermal fluid flow

This work is devoted to the development of a dissipative spatial discretization of a phase field model describing the dynamics of a multicomponent multiphase isothermal viscous compressible fluid with a resolved interface and effects associated with it (surface tension, wetting). Mass densities of mixture components are used as order parameters. The total Helmholtz free energy incorporates wall free energy providing wetting effects. The model under consideration is preliminary regularized, which allows one to increase time step in explicit time-marching methods. A finite-difference semi-discrete method is proposed and a discrete counterpart of the dissipativity theorem is proven. All the statements remain valid in the absence of regularization as well. 3D simulations of two-phase two-component and three-phase three-component fluids using explicit Euler method are performed. The simulation results confirm theoretical predictions.

Statistical mechanics of quantum error correcting codes

We study stabilizer quantum error correcting codes (QECC) generated under hybrid dynamics of local Clifford unitaries and local Pauli measurements in one dimension. Building upon 1) a general formula relating the error-susceptibility of a subregion to its entanglement properties, and 2) a previously established mapping between entanglement entropies and domain wall free energies of an underlying spin model, we propose a statistical mechanical description of the QECC in terms of "entanglement domain walls". Free energies of such domain walls generically feature a leading volume law term coming from its "surface energy", and a sub-volume law correction coming from thermodynamic entropies of its transverse fluctuations. These are most easily accounted for by capillary-wave theory of liquid-gas interfaces, which we use as an illustrative tool. We show that the information-theoretic decoupling criterion corresponds to a geometric decoupling of domain walls, which further leads to the identification of the "contiguous code distance" of the QECC as the crossover length scale at which the energy and entropy of the domain wall are comparable. The contiguous code distance thus diverges with the system size as the subleading entropic term of the free energy, protecting a finite code rate against local undetectable errors. We support these correspondences with numerical evidence, where we find capillary-wave theory describes many qualitative features of the QECC; we also discuss when and why it fails to do so.

Free energy of domain walls and order-disorder transition in a triangular lattice with anisotropic nearest-neighbor interactions.

We have derived exact expressions for the domain wall free energy along the three high-symmetry directions of a triangular lattice with anisotropic nearest-neighbor interactions. The triangular lattice undergoes an order-disorder phase transition at a temperature T_{c} given by e^{-(ε_{1}+ε_{2})/2kT_{c}}+e^{-(ε_{2}+ε_{3})/2kT_{c}}+e^{-(ε_{3}+ε_{1})/2kT_{c}}=1, where ε_{1},ε_{2},ε_{3} are the nearest-neighbor interaction energies, and ε_{1}+ε_{2}>0,ε_{2}+ε_{3}>0,ε_{3}+ε_{1}>0. Finally, we have derived expressions for the thermally induced meandering of the domain walls at temperatures below the phase transition temperature. We show how these expressions can be used to extract the interaction energies of two-dimensional systems with a triangular lattice.

A phase-field-based lattice Boltzmann method for moving contact line problems on curved stationary boundaries in two dimensions

In this work, we propose a phase-field-based lattice Boltzmann method to simulate moving contact line (MCL) problems on curved boundaries. The key point of this method is to implement the boundary conditions on curved solid boundaries. Specifically, we use our recently proposed single-node scheme for the no-slip boundary condition and a new scheme is constructed to deal with the wetting boundary conditions (WBCs). In particular, three kinds of WBCs are implemented: two wetting conditions derived from the wall free energy and a characteristic MCL model based on geometry considerations. The method is validated with several MCL problems and numerical results show that the proposed method has utility for all the three WBCs on both straight and curved boundaries.

Simulation of binary fluids exposed to selectively adsorbing walls: a method to estimate contact angles and line tensions

For an understanding of interfacial phenomena of fluids on the nanoscale a detailed knowledge of the excess free energies of fluids due to walls is required, as well as of the interfacial tension between coexisting fluid phases. A description of simulation approaches to solve this task is given for a suitable model binary (A + B) fluid. Sampling the order parameter distribution of the system without walls, the curvature dependent and flat interfacial tensions of coexisting ‘bulk’ phases is extracted. In a thin film geometry, the difference in wall free energies is found via a new thermodynamic integration method. Thus the contact angle θ of macroscopic droplets is estimated from Young's equation, for varying interactions between the fluid particles and the walls, which compares well with direct observations of inclined interfaces in ultrathin slit pores. Studying two-phase situations where a wall-attached droplet exists in the slit pore in thermal equilibrium, the excess free energy due to the droplet is found as a function of the droplet size, which is compatible with the classical theory for heterogeneous nucleation, if a line tension correction to the contact angle is made. For θ = 90○ an alternative method for extracting the line tension is also presented.

WALL FREE ENERGY BASED POLYNOMIAL BOUNDARY CONDITIONS FOR NON-IDEAL GAS LATTICE BOLTZMANN EQUATION

In the non-ideal gas lattice Boltzmann equation (LBE), the intermolecular forces between solid and fluid can be represented by the inclusion of the wall free energy in the expression of the total free energy. We derived and investigated three types of polynomial (linear, quadratic, and cubic) wall free energy based boundary conditions for the non-ideal gas LBE method. Static cases with a liquid drop sitting on solid surfaces are examined. All the proposed boundary conditions are able to predict the equilibrium states very well in the range of moderate contact angles by incorporating the potential form of the intermolecular forces and the bounce-back rule that guarantees mass conservation. Simulations with different boundary conditions are carried out and the results are compared concerning the accuracy as well as the applicability of different polynomial boundary conditions.

Heterogeneous nucleation at a wall near a wetting transition: a Monte Carlo test of theclassical theory

While for a slightly supersaturated vapor the free energy barrierΔFhom*, which needs to be overcome in a homogeneous nucleation event, may beextremely large, nucleation is typically much easier at the walls of the container inwhich the vapor is located. While no nucleation barrier exists if the walls arewet, for incomplete wetting of the walls, described via a nonzero contact angleΘ, classical theory predicts that nucleation happens through sphere-cap-shapeddroplets attracted to the wall, and their formation energy isΔFhet* = ΔFhom*f(Θ), withf(Θ) = (1−cosΘ)2(2+cosΘ)/4. This prediction is tested through simulations for the simple cubic lattice gas modelwith nearest-neighbor interactions. The attractive wall is described in terms ofa local ‘surface field’, leading to a critical wetting transition. The variation ofthe contact angle with the strength of the surface field is determined by usingthermodynamic integration methods to obtain the wall free energies which enter Young’sequation. Obtaining the chemical potential as a function of the density for a systemwith periodic boundary conditions (and no walls), the droplet free energy of aspherical droplet in the bulk is obtained for a wide range of droplet radii. Similarly,ΔFhet* is obtained for a system with two parallel walls. We find that the classical theory is fairlyaccurate if a line tension correction for the contact angle is taken into account.

Domain wall properties of FePt: From Bloch to linear walls

An investigation of the orientation and temperature dependence of domain wall properties in FePt is presented. We use a microscopic, atomic model for the magnetic interactions within an effective, classical spin Hamiltonian constructed on the basis of spin density functional theory. We find a significant dependence of the domain wall width and the domain wall energy on the orientation of the wall with respect to the crystal lattice. Investigating the temperature dependence, we demonstrate the existence of elliptical as well as linear domain walls in FePt. The calculation and further analysis of the domain wall free energy results in the evaluation of a thermodynamic exchange stiffness and anisotropy constant.

The anisotropic 3D Ising model

An expression for the free energy of an (001) oriented domain wall of the anisotropic 3D Ising model is derived. The order--disorder transition takes place when the domain wall free energy vanishes. In the anisotropic limit, where two of the three exchange energies (e.g. Jx and Jy ) are small compared to the third exchange energy (Jz ), the following asymptotically exact equation for the critical temperature is derived, sinh(2Jz /k B T c)sinh(2(Jx + Jy )/k B T c)) = 1. This expression is in perfect agreement with a mathematically rigorous result (k B T c/Jz = 2[ln(Jz /(Jx + Jy ))−ln(ln(Jz /(Jx + Jy )) + O(1)]−1) derived earlier by Weng, Griffiths and Fisher (Phys. Rev. 162, 475 (1967)) using an approach that relies on a refinement of the Peierls argument. The constant that was left undetermined in the Weng et al. result is estimated to vary from ∼0.84 at ((Hx + Hy )/Hz ) = 10−2 to ∼0.76 at ((Hx + Hy )/Hz ) = 10−20.

Modulation of Local Magnetization in Two-Dimensional Axial-Next-Nearest-Neighbor Ising Model

The axial next-nearest-neighbor Ising model is studied in two dimensions at finite temperature using the density matrix renormalization group. The model exhibits phase transition of the second-order between the antiphase in low temperature and the modulated phase in high temperature. Observing the domain wall free energy, we confirm that the modulation period in high-temperature side is well explained by the free-fermion picture.

Dual geometric worm algorithm for two-dimensional discrete classical lattice models.

We present a dual geometrical worm algorithm for two-dimensional Ising models. The existence of such dual algorithms was first pointed out by Prokof'ev and Svistunov [Phys. Rev. Lett. 87, 160601 (2001)]]. The algorithm is defined on the dual lattice and is formulated in terms of bond variables and can therefore be generalized to other two-dimensional models that can be formulated in terms of bond variables. We also discuss two related algorithms formulated on the direct lattice, applicable in any dimension. These latter algorithms turn out to be less efficient but of considerable intrinsic interest. We show how such algorithms quite generally can be "directed" by minimizing the probability for the worms to erase themselves. Explicit proofs of detailed balance are given for all the algorithms. In terms of computational efficiency the dual geometrical worm algorithm is comparable to well known cluster algorithms such as the Swendsen-Wang and Wolff algorithms, however, it is quite different in structure and allows for a very simple and efficient implementation. The dual algorithm also allows for a very elegant way of calculating the domain wall free energy.

Temperature chaos, rejuvenation, and memory in Migdal-Kadanoff spin glasses.

We use simulations within the Migdal-Kadanoff approach to probe the scales relevant for rejuvenation and memory in Ising spin glasses. First we investigate scaling laws for domain wall free energies and extract the chaos overlap length l(T,T'). Then we perform out of equilibrium simulations that follow experimental protocols. We find that (1) a rejuvenation signal arises at a length scale significantly smaller than l(T,T'), and (2) memory survives even if equilibration goes out to length scales larger than l(T,T'). Theoretical justifications of these phenomena are then considered.

Coupling-anisotropy and finite-size effects in interfacial tension of the two-dimensional Ising model

Exact expressions of the Bloch wall free energy are obtained for the Ising model on a rectangular lattice and infinitely long cylinder. The interfacial tension amplitudes are obtained for different coupling and aspect ratios. Finite-size scaling theory is used to analyse the effects of coupling anisotropy and finite size in the interfacial tension.

An Equilibrium Lattice Model of Wetting on Rough Substrates

We consider a semi-infinite 3-dimensional Ising system with a rough wall to describe the effect of the roughness r of the substrate on wetting. We show that the difference of wall free energies Δτ(r)=τAW(r)−τBW(r) of the two phases behaves like Δτ(r)∼rΔτ(1), where r=1 characterizes a purely flat surface, confirming at low enough temperature and small roughness the validity of Wenzel's law, cos θ(r)≈r cos θ(1), which relates the contact angle θ of a sessile droplet to the roughness of the substrate

The equilibrium shape of a two-dimensional crystal between parallel planes

By applying rather standard techniques for equilibrium crystal shapes (Wulff construction), we derive a construction for the equilibrium shape of a 2D crystal grown between two parallel plane substrates. The critical distance of the substrates at which this crystal splits into two parts is computed as a function of the wall free energy of the substrates. This may open new perspectives for the measurement of wall free energies.