Binary fluid mixture

A binary fluid mixture contains two molecular species whose local concentration difference can undergo mixing or phase separation.

If a nonconserved surfactant polarization has local energy $\nu |\mathbf{p}{|}^2/2+\lambda \mathbf{p}\cdot \nabla \varphi$, eliminating $\mathbf{p}$ changes the composition-gradient coefficient from $\kappa$ to $\tilde{\kappa }=\kappa -{\lambda }^2/\nu$. This expresses the surfactant's reduction of interfacial energy.

A microemulsion is a thermodynamically stable mixture of immiscible fluids and surfactant with structure on a mesoscopic length scale. In a gradient theory, a negative square-gradient coefficient stabilized by a positive fourth-gradient term favors fluctuations at nonzero wavevector.

The Cahn--Hilliard equation transports a conserved composition field down gradients of its chemical potential: ${\partial }_t\varphi =\nabla \cdot (M\nabla \mu )$ with $\mu =\delta F/\delta \varphi$. Advection adds $\mathbf{v}\cdot \nabla \varphi$ to the left-hand side.

In Ostwald ripening, the higher interfacial chemical potential of small droplets drives diffusion toward larger droplets, increasing the characteristic domain size. The Gibbs--Thomson relation supplies the curvature-dependent surface value, while conserved order-parameter dynamics controls transport through the surrounding phase.

For a three-dimensional droplet with concentration jump $\Delta \varphi$, constant mobility $M$, and quasi-static exterior chemical potential,The first formula solves the exterior Laplace equation. The second follows from conservation: the total outward flux $4\pi MR({\mu }_s-{\mu }_{\infty })$ removes excess composition at the rate $-4\pi R^2\Delta \varphi \dot{R}$. For a symmetric mixture, $\Delta \varphi =2{\varphi }_B$ and the Gibbs--Thomson relation gives ${\mu }_s=\gamma /({\varphi }_{B}R)$.

A quasi-static chemical potential outside a fixed-shape droplet solves the same Laplace equation as an electrostatic potential outside an equipotential conductor. If $\mu ={\mu }_s$ on the droplet and zero on the surrounding reservoirs, its outward diffusive current iswhere $C$ is the corresponding electrostatic capacitance in a medium of permittivity $ϵ$. The arbitrary $ϵ$ cancels against the capacitance's proportionality to $ϵ$. Thus electrostatic geometry directly determines evaporation rates.

For a spherical droplet at fixed center height $h>R$ above a flat equilibrium reservoir, the diffusion-capacitance analogy and sphere-plane capacitance giveAt large separation, $c(R/h)=1+R/(2h)+O((R/h{)}^2)$, soThe nearby reservoir increases the total flux and shortens the lifetime. The spherical-shape assumption controls the geometry; the local current density is not uniform over the surface.

Model H couples a conserved scalar composition field to an incompressible momentum density. It combines the advective Cahn--Hilliard equation with the Navier-Stokes equation and the thermodynamic force density $-\varphi \nabla \mu$.

For constant positive order-parameter mobility $M$, dynamic viscosity $\eta$ and mass density $\rho$, deterministic Model H dynamics dissipates the sum of composition free energy and kinetic energy:Assume smooth fields and periodic or closed, no-work boundaries. Using the chemical potential of a composition field, integration by parts and incompressible flow gives $\dot{F}=\int \varphi \mathbf{v}\cdot \nabla \mu -M\int |\nabla \mu {|}^2$. Dotting the momentum equation with velocity gives ${\dot{E}}_{\mathrm{kin}}=-\int \varphi \mathbf{v}\cdot \nabla \mu -\eta \int |\nabla \mathbf{v}{|}^2$. Pressure and convective work vanish under the boundary assumptions; the two capillary transfer terms cancel. This proves both the sign of the Korteweg force and its reversible exchange of energy with the composition field.

The Korteweg force is the capillary body-force density generated by composition gradients in a diffuse-interface fluid. Up to a pressure gradient it can be written $-\varphi \nabla \mu$, where $\mu$ is the chemical potential.

In bicontinuous phase separation, both fluid phases form interpenetrating connected domains. During late-stage coarsening their morphology is often statistically self-similar after lengths are divided by one characteristic domain size $L(t)$.

The dynamical-scaling hypothesis replaces all macroscopic lengths by one scale $L(t)$ and the characteristic velocity by $\dot{L}$. Then inertial, viscous, and capillary force densities scale respectively as $\rho \ddot{L}+\rho {\dot{L}}^2/L$, $\eta \dot{L}/L^2$, and $\sigma /L^2$.

Replacing domain-scale viscous stresses by a local linear drag density $-\bar{\eta }\mathbf{v}$ gives the scales $L_1=(\sigma \rho /{\bar{\eta }}^2{)}^{1/3}$ and $t_1=\rho /\bar{\eta }$. The single-scale force model isFor a regular power-law asymptotic $g\sim C{u}^y$, $g^{\prime }\sim Cy{u}^{y-1}$, $g^{\prime \prime }=Cy(y-1)u^{y-2}+o(u^{y-2})$, $y>0$, the two inertial-to-drag ratios scale as $(y-1)/u$ and $y/u$. Thus drag dominates inertia at late time, and drag-capillary balance integrates to $g^3\sim 3(c_s/c_d)u$. This proves the displayed $1/3$ exponent and its independence of mass density. The scaling assumes a homogenized stationary network and dominant advective transport. Diffusive Ostwald ripening can have the same exponent, so its omission must be justified separately; order-parameter mobility can otherwise enter the prefactor.

Under single-length dynamical scaling of binary-fluid coarsening, mass density $\rho$, dynamic viscosity $\eta$ and surface tension $\sigma$ determine the displayed unique length and time scales by dimensional analysis. Writing $L=L_{0}f(t/t_0)$ reduces the force-density estimate to $\alpha f^{\prime \prime }+\beta f^{\prime 2}/f=\gamma f^{\prime }/f^2+\delta /f^2$. Resistive and driving terms need opposite signed geometric coefficients. Viscous hydrodynamic coarsening has $f\sim u$ and inertial hydrodynamic coarsening has $f\sim u^{2/3}$. These follow respectively from density-independent and viscosity-independent powers in $L\propto {\eta }^{2-3p}{\rho }^{p-1}{\sigma }^{2p-1}{t}^p$. They require thin interfaces, statistically self-similar bicontinuous domains, dominant advection and no additional relevant macroscopic scales. A degeneracy in the signed inertial coefficients can invalidate the pure-power force balance without invalidating the dimensional calculation.

Balancing viscous and capillary forces gives $L(t)\sim (\sigma /\eta )t$. This regime is independent of the mass density and applies below the viscous-to-inertial crossover scale.

Balancing inertia and capillarity gives $L(t)\sim (\sigma /\rho {)}^{1/3}{t}^{2/3}$. This regime is independent of viscosity and applies above the viscous-to-inertial crossover scale.

If stirring imposes a time scale $\tau$, replacing time derivatives by $1/\tau$ predicts a steady domain size. Viscous-capillary balance gives $\Lambda \sim \sigma \tau /\eta$, while inertial-capillary balance gives $\Lambda \sim (\sigma {\tau }^2/\rho {)}^{1/3}$.