Reference Work Entry

Encyclopedia of Microfluidics and Nanofluidics

pp 428-433

Dynamic Density Functional Theory (DDFT)

  • Markus RauscherAffiliated withMax-Planck-Institut für MetallforschungInstitut für Theoretische und Angewandte Physik, Universität Stuttgart

Synonyms

Time-dependent Density Functional Theory (TDFT)

Definition

Dynamic Density Functional Theory (DDFT) is, on the one hand a time-dependent (dynamic) extension of the static Density Functional Theory (DFT), and on the other hand, the generalization of Fick's law to the diffusion of interacting particles. The time evolution of the ensemble-averaged density of Brownian particles is given as an integro-differential equation in terms of the equilibrium Helmholtz free energy functional (or the grand canonical functional). DDFT resolves density variations on length scales down to the particle size but only works for slow relaxing dynamics close to equilibrium.

Overview

One can prove that in thermal equilibrium, in a grand canonical ensemble (i. e., volume, chemical potential, and temperature are fixed), the grand canonical free energy \( \Omega\,(\rho({\mathbf{r}})) \) of a system can be written as a functional of the one-body density \( \rho({\mathbf{r}}) \) alone, which will depend on the position \( {\mathbf{r}} \) in inhomogeneous systems. The density distribution \( \rho _{{{\textrm{eq}}}}({\mathbf{r}}) \) which minimizes the grand potential functional is the equilibrium density distribution. This statement is the basis of the equilibrium density functional theory (DFT) for classical fluids which has been used with great success to describe a variety of inhomogeneous fluid phenomena, in particular the structure of confined liquids, wetting, anisotropic fluids, and fluid–fluid interfaces. For a historical overview and further references see [1,2].

Out of equilibrium there is no such rigorous principle. However, macroscopically one can find a large variety of phenomenological equations for the time evolution which are based on macroscopic quantities alone, e. g., the diffusion equation, the heat transport equation, and the Navier–Stokes equations for hydrodynamics. A microscopic dynamical theory for the time evolution of slow variables such as the momentum density or the particle density with molecular spatial resolution is highly desirable.

In perhaps one of the simplest microscopic cases, a system of interacting Brownian (i. e., diffusing) particles, and in a local equilibrium approximation, one can write the time evolution of the ensemble-averaged one-body density as a functional of the density [3]
$$ \frac{\partial\rho\left({{\mathbf{r}};t}\right)}{\partial t}=\nabla\cdot\left({{\textrm{D}}\rho\left({{\mathbf{r}};t}\right)\nabla\left.{\frac{\delta F\left[\rho\right]}{\delta\rho}}\right|_{{\rho({\mathbf{r}};t)}}}\right) $$
(1)
where D is the diffusion constant, and the Helmholtz free energy functional, \( F\left[{\rho({\mathbf{r}})}\right]=\Omega\left[{\rho({\mathbf{r}})}\right]+\mu\rho({\mathbf{r}}) \). The chemical potential \( \mu \) is constant throughout the system and the free energy functional has the form
$$ F\left[{\rho({\mathbf{r}})}\right]=\\ {\int\rho({{\mathbf{r}}^{{\prime}}})\left({\left({k_{{\textrm{B}}}T\ln\Lambda^{3}\rho\left({{\mathbf{r}}^{{\prime}}}\right)-1}\right)+V\left({{\mathbf{r}}^{{\prime}}}\right)}\right)\mathrm{d}^{3}{\mathbf{r}}^{{\prime}}}\\ +F_{{{\textrm{ex}}}}\left[{\rho({\mathbf{r}})}\right] $$
(2)
where \( \Lambda \) is the thermal wavelength, \( k_{{\textrm{B}}}T \) is the thermal energy, and \( V({\mathbf{r}}) \) is an external potential. The excess part of the free energy \( F_{{{\textrm{ex}}}}\left[{\rho({\mathbf{r}})}\right] \) depends on the interactions among the particles and, apart from the trivial case of noninteracting (ideal) particles for which \( F_{{{\textrm{ex}}}}\left[{\rho({\mathbf{r}})}\right]=0 \), it is only known exactly for a one-dimensional system of hard rods [3]. The key to the success of the equilibrium DFT as well as the nonequilibrium DDFT is to develop "good" functionals which can be tested against experiments and various simulations. Fundamental measurement theory has been very successfully used to construct functionals for hard sphere systems (a model for colloids in suspensions) [4]. Soft interactions, e. g., between polymer coils in a solution, can be treated in a nonlocal mean field (random phase) approximation as used by Penna et al. [5]. Their \( F_{{{\textrm{ex}}}}\left[{\rho({\mathbf{r}})}\right] \) is written in a quadratic form in \( \rho({\mathbf{r}}) \). For particles with a relatively hard core and an additional long-range interaction one can combine both concepts and write \( F_{{{\textrm{ex}}}}\left[{\rho({\mathbf{r}})}\right] \) as the sum of a functional for hard spheres and a quadratic functional containing only the soft long-range part of the interaction [6].

Various extensions to DDFT have been proposed (see below). The generalization to mixtures of different particle types is straightforward. A background flow and a position-dependent diffusion constant can also be included.

To date, there has only been one attempt to develop a dynamic density functional theory for systems in which inertia plays a role [7]. However, it has been shown that the formal proof for the existence of a quantum mechanical dynamical density functional theory by Runge and Gross can be applied to classical systems [8]by starting from the Liouville equation for Hamiltonian systems (instead of the time-dependent Schrödinger equation), which therefore includes inertia terms. However, the proof is not of practical use (see below).

Dynamical density functional theories for the spatially coarse-grained density (to be distinguished from the ensemble-averaged density \( \rho({{\mathbf{r}};t}) \) discussed here) based on Zwanzig–Mori projection techniques have been suggested by many authors, but they are only valid on length scales that are large compared to the particle size. Their functional form is similar to Eq. (1) but with a conservative multiplicative noise term added. But in general, one is interested in ensemble-averaged quantities, i. e., quantities averaged over many realizations of the noise or the numerical experiment, such that a theory for ensemble-averaged quantities is of more practical use. For a discussion of the role of noise in DDFT see [9].

Basic Methodology

We now derive the DDFT (Eq. (1)) for the simplest case, i. e., a system of \( N \) pairwise interacting indistinguishable Brownian particles, following the derivation given in [9], before giving the DDFT equations (corresponding to Eq. (1)) for mixtures and for particles in a flowing solvent.

DDFT for Interacting Brownian Particles

A starting point for developing a DDFT for \( N \) interacting Brownian particles is the Fokker–Planck (or Smoluchowsky) equation for the probability density \( W({\left\{{{\mathbf{r}}_{i}}\right\} _{{i=1\ldots N}}}) \) for finding particle \( 1 \) at position \( {\mathbf{r}}_{1} \), particle \( 2 \) at position \( {\mathbf{r}}_{2} \), … , particle \( i \) at position \( {\mathbf{r}}_{i} \), and finally particle \( N \) at position \( {\mathbf{r}}_{N} \). For pairwise interactions among the particles we have
$$ \begin{aligned}[b]\frac{\partial W(\left\{{\mathbf{r}}_{i}\right\};t)}{\partial t}=\sum\limits _{{i=1}}^{N}\nabla _{i}\cdot\Biggl(\sum\limits _{{j=1}}^{N}&\textrm{\textbf{F}}\left({\mathbf{r}}_{i}-{\mathbf{r}}_{j}\right)\\ +&\textrm{\textbf{G}}\left({\mathbf{r}}_{i}\right)-k_{{\textrm{B}}}T\nabla _{i}\Biggr)W\left(\left\{{\mathbf{r}}_{i}\right\};t\right)\end{aligned} $$
(3)
where the interaction force \( {\rm\textbf{F}}({\mathbf{r}})=-\nabla\Phi({\mathbf{r}}) \), the interaction potential is \( \Phi({\mathbf{r}}) \), and the force due to the external potential \( {\rm\textbf{G}}({\mathbf{r}})=-\nabla V({\mathbf{r}}) \). The generalization to many-particle interactions and to time-dependent external potentials is straightforward. The time evolution of the ensemble-averaged density, i. e., the density averaged over many realizations of the thermal noise and initial conditions samples from the initial \( W({\left\{{{\mathbf{r}}_{i}}\right\};t=0}) \), can be calculated by integrating out \( N-1 \) degrees of freedom in Eq. (3) to obtain
$$ \begin{aligned}[b]\frac{\partial\rho({{\mathbf{r}};t})}{\partial t}&=-\nabla\cdot\left(\vphantom{\int}\left({{\rm\textbf{G}}({\mathbf{r}})-k_{{\textrm{B}}}T\nabla}\right)\rho({{\mathbf{r}};t})\right.\\ &\hphantom{=-\nabla\cdot xx}\left.+\int{g({{\mathbf{r}},{\mathbf{r}}^{{\prime}};t}){\rm\textbf{F}}({{\mathbf{r}}-{\mathbf{r}}^{{\prime}}})\mathrm{d}^{3}{\mathbf{r}}^{{\prime}}}\right)\end{aligned} $$
(4)
with the nonequilibrium two-point correlation function \( g({{\mathbf{r}},{\mathbf{r}}^{{\prime}};t}) \). Equation (4) is only the starting point of a hierarchy of \( N \) time evolution equations for correlation functions of increasing order.
https://static-content.springer.com/image/prt%3A978-0-387-48998-8%2F4/MediaObjects/978-0-387-48998-8_4_Part_Fig77_HTML.gif
Figure 1

Illustration of the local equilibrium approximation involved in the development of the DDFT. The left-hand side illustrates the nonequilibrium evolution of the density \( \rho({{\mathbf{r}};t}) \) (thin lines) up to time \( t \) (thick line). For the time evolution the equal-time correlation function \( g({{\mathbf{r}},{\mathbf{r}};t}) \) is needed. It is approximated by the equilibrium correlations \( g_{{{\textrm{eq}}}}\left({{\mathbf{r}},{\mathbf{r}}^{{\prime}}}\right) \) in an equilibrium system (right-hand side) with the same density \( \rho _{{{\textrm{eq}}}}({\mathbf{r}}) \) (thick line) as in the nonequilibrium system. For given particle interactions and density such an equilibrium system can be obtained by applying an external potential \( U_{{\rho({{\mathbf{r}};t})}}({\mathbf{r}}) \)

The key to DDFT is to truncate this hierarchy and to express \( g({{\mathbf{r}},{\mathbf{r}}^{{\prime}};t}) \) in terms of the density. In order to arrive at Eq. (1) one assumes that \( g({{\mathbf{r}},{\mathbf{r}}^{{\prime}};t}) \) can be approximated by the two-point correlation function of an equilibrium system with the equilibrium density distribution \( \rho _{{{\textrm{eq}}}}({\mathbf{r}})=\rho({{\mathbf{r}};t}) \), as illustrated in Fig. 1. This is possible because for every given interaction potential \( V\left({\mathbf{r}}\right) \) and density \( \rho({{\mathbf{r}};t}) \) one can find an external potential \( U_{{\rho\left({{\mathbf{r}};t}\right)}}({\mathbf{r}}) \) such that the equilibrium density distribution \( \rho _{{{\textrm{eq}}}}({\mathbf{r}}) \) of the system with \( U_{{\rho\left({{\mathbf{r}};t}\right)}}({\mathbf{r}}) \) is equal to \( \rho({{\mathbf{r}};t}) \). Moreover, the excess parts of the free energy functionals of both systems, with and without \( U_{{\rho\left({{\mathbf{r}};t}\right)}}({\mathbf{r}}) \), are identical. One can therefore use a sum rule relating the integral in Eq. (4) (with \( g({{\mathbf{r}},{\mathbf{r}}^{{\prime}};t}) \) replaced by \( g_{{{\textrm{eq}}}}({{\mathbf{r}},{\mathbf{r}}^{{\prime}}}) \)) via the first direct correlation function \( c^{1}({\mathbf{r}}) \) to the functional derivative of the excess part of the free energy.
$$ \begin{aligned}[b]\int g_{{{\textrm{eq}}}}({{\mathbf{r}},{\mathbf{r}}^{{\prime}}}){\rm\textbf{F}}({{\mathbf{r}}-{\mathbf{r}}^{{\prime}}})\mathrm{d}^{3}{\mathbf{r}}^{{\prime}}&=k_{{\textrm{B}}}T\rho({{\mathbf{r}};t})\nabla c^{{(1)}}({\mathbf{r}})\\ &=-\rho({{\mathbf{r}};t})\nabla\left.{\frac{\delta F_{{{\textrm{ex}}}}\left[\rho\right]}{\delta\rho}}\right|_{{\rho({{\mathbf{r}};t})}}\end{aligned} $$
(5)
Writing the first term on the right-hand side of Eq. (4) as a variational derivative one arrives directly at Eq. (1).

For clarity we only derive Eq. (1) for the simplest case. However, Eq. (1) is also valid for arbitrary many-body interactions [6] as well as time-dependent external potentials \( V({{\mathbf{r}};t}) \). The latter simply leads to a time-dependent free energy functional \( F\left[{\rho;t}\right] \) which is of the same form as the functional in Eq. (2).

DDFT for Particle Mixtures

The generalization to mixtures of \( M \) different particle types is straightforward and for the density \( \rho _{X}({{\mathbf{r}};t}) \) of component \( 1\le X\le M \)one obtains
$$ \begin{aligned}[b]&\frac{\partial\rho _{X}({{\mathbf{r}};t})}{\partial t}\\ &=\nabla\cdot\left({D_{X}\rho _{X}({{\mathbf{r}};t})\nabla\left.{\frac{\delta F\left[{\left\{{\rho _{Y}}\right\} _{{Y=1\ldots M}}}\right]}{\delta\rho _{X}}}\right|_{{\left\{{\rho _{Y}({\mathbf{r}};t)}\right\}}}}\right)\end{aligned} $$
(6)
where the diffusivity \( D_{X} \) of component \( X \) and the joint free energy functional depend on the density distribution of all \( M \) components.

DDFT in a Flowing Solvent

Flow in the medium in which the Brownian particles are embedded (e. g., the solvent for colloids) can also be included, as long as the back-reaction of the Brownian particles on the flow field \( {\rm\textbf{u}}({{\mathbf{r}};t}) \) is not taken into account. This results in a coupling of the density field to the Navier–Stokes equations. In the case most relevant for microfluidics, i. e., for vanishing Reynolds numbers and incompressible flows, one obtains
$$ \displaystyle\begin{aligned}[b]&\frac{\partial\rho({{\mathbf{r}};t})}{\partial t}+{\rm\textbf{u}}({{\mathbf{r}};t})\cdot\nabla\rho({{\mathbf{r}};t})\\ &=\nabla\cdot D\rho({{\mathbf{r}};t})\nabla\left.{\frac{\delta F\left[\rho\right]}{\delta\rho}}\right|_{{\rho({{\mathbf{r}};t})}}\end{aligned} $$
(7)
$$ \displaystyle 0=-\nabla p+\eta\nabla^{2}{\rm\textbf{u}}({{\mathbf{r}};t}) $$
(8)
$$ \displaystyle\nabla\cdot{\rm\textbf{u}}({{\mathbf{r}};t})=0 $$
(9)
where \( p \) is the pressure and \( \eta \) is the viscosity. Here we neglect the displacement of the solvent by the Brownian particles. A more complete theory would also include the osmotic pressure of the particles in terms of a density dependent pressure tensor in Eq. (8), similar to the coupled Cahn–Hilliard/Navier–Stokes systems used in phase field modeling of multiphase fluid systems.

DDFT in Inhomogeneous Media

A spatially varying diffusivity can model not only inhomogeneities in the medium but also hydrodynamic interactions between the Brownian particles and channel walls. The diffusivity is then replaced by a diffusion tensor \( {\rm\textbf{D}}({\mathbf{r}}) \) and instead of Eq. (1) one obtains
$$ \frac{\partial\rho({{\mathbf{r}};t})}{\partial t}=\nabla\cdot{\rm\textbf{D}}({\mathbf{r}})\rho({{\mathbf{r}};t})\cdot\nabla\left.{\frac{\delta F\left[\rho\right]}{\delta\rho}}\right|_{{\rho({{\mathbf{r}};t})}} $$
(10)
Eqs. (6)–(10) only give the simplest forms of the generalization of Eq. (1) discussed in the corresponding section. All the generalizations can be combined with each other.

The local equilibrium approximation for the two-point correlation function involved in the development of the DDFT has two issues. First, it is not a priori clear when it is justifiable to approximate the nonequilibrium correlations by equilibrium correlations. It has been shown that there are cases in which this approximation breaks down, in particular in driven steady-state systems. Second, the equilibrium sum-rule in Eq. (5) gives the two-point correlations in a grand canonical ensemble. But the locally conserved dynamics underlying Eq. (1) requires a canonical ensemble. In a grand canonical ensemble each point in space is connected to a reservoir fixing the chemical potential by adding or removing particles, while in a canonical ensemble the number of particles is fixed.

Key Research Findings

As DDFT is a relatively young technique only a few results are available. The remaining part of this section is organized in terms of the particle system under consideration, i. e., in terms of the excess free energy functional \( F_{{{\textrm{ex}}}}\left[{\rho({\mathbf{r}})}\right] \) used.

One-Dimensional Systems

https://static-content.springer.com/image/prt%3A978-0-387-48998-8%2F4/MediaObjects/978-0-387-48998-8_4_Part_Fig78_HTML.gif
Figure 2

Schematic phase diagram of a system of Brownian particles with an attractive interaction. \( \Phi _{{\min}} \) is the depth of the attractive minimum of the interaction potential and \( \bar{\rho} \) is the particle density (averaged of the system volume). Only states above the full line, the bimodal, are stable. States between the full and the dashed lines, the spinodal, are metastable, and states below the spinodal are linearly unstable. The system considered in [6] is quenched from a stable state (circle) into the unstable region as indicated by the vertical arrow

The difference between the correlation functions in a canonical and a grand canonical ensemble is largest for one-dimensional systems with hard core interactions. This occurs because in such a system the particles cannot pass each other. This also leads to a significantly different diffusion behavior (single file diffusion). In a grand canonical setting, however, particles can pass each other via the reservoir which keeps the chemical potential fixed. Therefore, an ensemble of hard rods was the first test system used for DDFT, also because the free energy functional for hard rods is the only nontrivial functional for interacting systems which is known exactly. Some discrepancies between DDFT and Brownian dynamics simulations of the same system were observed, mainly for extreme heterogeneities, i. e., very far from equilibrium [3]. However, the intermediate and long-term behavior was captured correctly by DDFT, at least for large enough systems. The same system has been considered as a model for the diffusion of colloids in a narrow channel with an obstacle inside the channel [10]. The goal of this study was to investigate the interplay between the external driving forces, the interparticle interactions, and the potential barriers in nanochannels.

Soft Particles

Polymer chains in solution form a loose coil such that two coils can interpenetrate. The resulting effective interaction potential \( \Phi(|{\mathbf{r}}|) \) is repulsive and is well approximated by a Gaussian potential of strength \( \Phi _{0} \) and with a range \( r_{0} \) which is proportional to the radius of gyration \( R_{g} \). The excess free energy functional for such soft systems can be approximated by a quadratic form
$$ F_{{{\textrm{ex}}}}\left[{\rho({\mathbf{r}})}\right]=\tfrac{1}{2}\iint{\rho({\mathbf{r}})\Phi\left({|{\mathbf{r}}-{\mathbf{r}}^{{\prime}}|}\right)\rho({{\mathbf{r}}^{{\prime}}})\mathrm{d}^{3}{\mathbf{r}}\mathrm{d}^{3}{\mathbf{r}}^{{\prime}}} $$
(11)
where \( \Phi(r)=\Phi _{0}\exp(-r^{2}/r_{0}^{2}) \). This has been used in order to describe the dynamics of polymer solutions in the vicinity of colloidal particles driven through the solution [10]. The main finding is that the layering of polymer coils around the colloid can be broken up, leading to a strongly inhomogeneous polymer density in the vicinity of particles. The influence of this inhomogeneity on the depletion interaction between particles has been studied [11]. A tendency for the particles to align, a result most relevant for sedimentation processes and in line with the simulations, was found.

This type of model potential has been also used to study the phase separation dynamics of binary mixtures in cavities [12]. An excellent agreement between DDFT and Brownian dynamics simulations was found. In this particular example the fact that DDFT is a theory for the ensemble-averaged density becomes most evident, i. e., the density shows the same symmetry as the underlying system, even though the actual realizations may have a broken symmetry. In this example this means that since the probability of having one phase on the left-hand side of the cavity and the other phase on the right-hand side is equal to the probability of the mirrored situation the ensemble-averaged densities are symmetric even if every realization of the system is asymmetric.

Hard Particles

Hard particles with an additional soft attractive interaction have been used in a study of spinodal decomposition [6]. Such a system can be realized in a colloidal suspension and exhibits a fluid–fluid phase separation between a colloid-rich and a colloid-poor phase (see the phase diagram in Fig. 2). Quenched below the spinodal, the system undergoes spinodal decomposition which leads finally to the formation of dense and less dense clusters. The main progress, as compared to nonlinear Cahn–Hilliard theories, is the more accurate description of the short wavelength correlations, a property which will become most relevant in confined systems.

Future Directions for Research

Applications of DDFT

DDFT is a relatively new technique and therefore much effort will be needed toapply DDFT to microfluidic and biological systems. In particular, the transport of colloidal suspensions and polymer solutions in channels that are a few times larger than the typical particle diameter is a problem for which the DDFT method is well suited. Mixing of suspended interacting particles and polymers in such channels is also well modeled by DDFT.

The internal dynamics of complex fluids, e. g., the diffusion of colloids in polymer solutions, is another problem for which DDFT could be useful. It has been shown recently that the structure of the polymer solution in the vicinity of a diffusing colloidal particle strongly influences its diffusivity [13]. DDFT could be used to determine this structure.

Further Development of DDFT

The greatest challenges for DDFT are hydrodynamic interactions and inertia. Either one, or both of these, are needed for most systems of interest. The following paragraphs briefly outline the status of the development and the obstacles to further progress.

Hydrodynamic Interactions

Currently, hydrodynamic interactions between suspended particles cannot be included in a DDFT. However, it is well known, that, e. g., the rheology of suspensions cannot be explained without taking these into account. Hydrodynamic interactions in a simple approximation based on Oseen-tensors have been included in the Fokker–Planck equation (Eq. (3)) and the equivalent of Eq. (4) has been derived and discussed [14,15]. However, this equation contains three-point and two-point correlations in a form such that the sum rule in Eq. (5) cannot be used.

Simple Fluids

The DDFTs discussed in this article describe the time evolution of the density of Brownian particles. Although there are formal arguments for the existence of a DDFT for a system of particles which follow a Newtonian dynamics [8], in other words a system of particles for which inertia is important, to date, only one effort has been made to extend the concept of DDFT in this direction [7]. However, hydrodynamic modes were still excluded in that work.

The greatest obstacle for developing a hydrodynamic DDFT is that, the viscosity of a liquid is determined by the deviation of the two-point correlation function \( g({{\mathbf{r}},{\mathbf{r}}^{{\prime}};t}) \) from the equilibrium value \( g_{{{\textrm{eq}}}}({{\mathbf{r}};{\mathbf{r}}^{{\prime}}}) \). Approximating the first with the latter, as in the derivation of the DDFT for Brownian particles discussed above, would therefore result in a theory almost without viscosity.

Cross References

Active Mixer

Brownian Motion

Brownian Diffusion

Mixing Efficiency

Concentration Gradient Generation and Control

Demixing

Diffusion

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