Density-Functional Theory for Complex Fluids

Annu. Rev. Phys. Chem. 2007.58:85-112. Downloaded from arjournals.annualreviews.org by UNIVERSITY OF CALIFORNIA - RIVERSIDE LIBRARY on 04/19/07. For personal use only.

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Density-Functional Theory for Complex Fluids Jianzhong Wu and Zhidong Li Department of Chemical and Environmental Engineering, University of California, Riverside, California 92521; email: [email protected]

Annu. Rev. Phys. Chem. 2007. 58:85–112

Key Words

First published online as a Review in Advance on October 19, 2006

phase transitions, self-assembly, thin films, polymers, colloids, liquid crystals

The Annual Review of Physical Chemistry is online at http://physchem.annualreviews.org This article’s doi: 10.1146/annurev.physchem.58.032806.104650 c 2007 by Annual Reviews. Copyright All rights reserved 0066-426X/07/0505-0085$20.00

Abstract Density-functional theory (DFT) and its variations provide a fruitful approach to the computational modeling of the microscopic structures and phase behavior of soft-condensed matter. The methodology takes deep root in quantum mechanics but shares a mathematical similarity with a number of classical approaches in statistical mechanics. This review discusses different strategies commonly used to formulate the free-energy functional of complex fluids for either phenomena-oriented applications or as a generic description of the thermodynamic nonideality owing to various components of intermolecular forces. We emphasize the connections among different schemes of DFT approximations, their underlying assumptions, and inherent limitations. We also address extensions of equilibrium DFT to phenomenological theories for the dynamic properties of complex fluids and for the kinetics of phase transitions. In addition, we highlight the recent literature concerning applications of DFT to diverse static and time-dependent phenomena in complex fluids.

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INTRODUCTION


DFT: density-functional theory Intrinsic Helmholtz energy: a Legendre transform of the grand potential that shifts the independent variable from the one-body potential to the one-body density profile Grand potential: the free energy of an open system at fixed volume and temperature that is minimized at equilibrium

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Density-functional theory (DFT) has emerged as a unified theoretical framework for computational modeling of both quantum and classical systems. This generic methodology is built upon a mathematical theorem that states, for an equilibrium system at a given temperature and at given chemical potentials of all constituent molecules/particles, there is an invertible mapping between the external potential and the one-body density profiles (1, 2). This mathematical theorem enables a definition of an intrinsic Helmholtz energy functional that is independent of the external potential and, subsequently, a solution of the equilibrium density profiles by the minimization of the grand potential. Functional derivatives of the grand potential or Helmholtz energy lead to multibody correlation functions that allow the determination of both the structure and thermodynamic properties of the equilibrium system. DFT has become, by far, the most prominent tool in applied quantum mechanics, in particular for calculating the ground-state properties of electronic systems (3). Conversely, researchers have profitably used its classical counterpart in the study of phase transitions, interfacial phenomena, and microscopic structures of polymeric fluids, liquid crystals, crystalline solids and glasses (4–8). Invertible mappings among the time-dependent density profiles, momentum distributions, and external field have also been established for both quantum and classical systems (9, 10), thereby facilitating DFT applications to various dynamic and off-equilibrium problems. Whereas both quantum and classical DFTs are based on a formally exact mathematical framework, their practical use inevitably requires effective approximations for the exchange-correlation energy to account for fluctuations of the electronic density and for the excess Helmholtz energy functional to account for intermolecular interactions in classical systems. These approximations must capture faithfully the underlying physics of a particular problem under consideration, and the numerical performance has to be calibrated by direct comparison with experimental or simulation data. For classical systems, researchers commonly use two basic strategies to construct an approximate Helmholtz energy functional: The first relies on diagrammatic expansions of the grand partition function as used extensively in statistical mechanics, and the second seeks direct approximations of the free-energy functional by a localdensity ansatz or by a functional Taylor expansion relative to a reference density profile or an interparticle potential (5). Along with these first-principle methods is a number of phenomenological approaches that express the unknown free-energy functional in terms of a few carefully selected order parameters (11). Typical physical phenomena in a complex fluid entail multiple time- and length scales affiliated with the diverse intramolecular architectures of the constituent molecules and complicated solvent-mediated intermolecular interactions. To capture the multiple aspects of the rich phenomena in complex fluids, DFT applications often require a combination of various strategies to formulate a reliable Helmholtz energy functional. In comparison with alternative methods in molecular modeling, the advantages of DFT include its theoretical versatility, physical clarity, and a concomitant of

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computational efficiency for representing various static and dynamic phenomena in many-particle systems. Even with modern computers, direct applications of the ¨ Schrodinger equation or atomistic molecular-dynamics simulations are limited to relatively small systems. Conversely, interesting physical processes in biological systems and soft materials often involve multiple time- and length scales that cannot be addressed directly with exact methods. By contrast, for quantum and classical systems, DFT deals directly with the average system properties and thus avoids enumeration of the microscopic fluctuations of individual particles. Besides, in DFT the thermodynamic properties are embedded in the formulation of the free-energy functional, whereas in standard molecular simulations they are indirectly calculated from mechanical variables. For describing the equilibrium and dynamic properties of complex fluids, DFT provides a unified theoretical basis for various phenomenological methods originally developed in different contexts. Furthermore, DFT offers an effective theoretical means to predict the mesoscopic/macroscopic parameters used in conventional phenomenological models or in continuous equations. In this review, we discuss different strategies commonly used to formulate the Helmholtz energy functionals and related dynamic equations for representing the equilibrium and dynamic properties of complex fluids; we do not consider quantum effects. We give special attention to the merits and limitations of different DFT approximations, their underlying interconnections, and relationships to conventional phenomenological methods in equilibrium and nonequilibrium statistical mechanics. In addition, we highlight recent practical applications of DFT concerning the static and dynamic properties and phase transitions in colloids, polymeric fluids, surfactant solutions, liquid crystals, and selected biological systems.

HELMHOLTZ ENERGY FUNCTIONAL The Helmholtz energy functional plays a central role in both equilibrium DFT and in its extensions to dynamic equations for modeling time-dependent processes. In general, the formulation of an effective Helmholtz energy functional must be guided by the specific forms of the intermolecular interactions in the particular system under consideration. These interactions typically consist of contributions from short-range repulsions owing to molecular excluded volumes, van der Waals attractions, Coulomb interactions, and various forms of electron donor-acceptor interactions responsible for chemical associations and hydrogen bonding. From an atomistic viewpoint, a generic Helmholtz energy functional must account for the contributions to the thermodynamic nonideality from each element of the intermolecular forces. In addition, it must reflect faithfully the microscopic architectures of the constituent molecules (i.e., molecular shape, chain connectivity, and conformation). Complex fluids are characterized by high asymmetries in the microscopic characterizations of the constituent molecules, including molecular size, internal architecture and geometry, polarity, and charge status. Typical dynamic processes in a complex fluid span spatiotemporal scales ranging from fractions of a picosecond and a few angstroms for atomic motions, to a few microseconds and nanometers

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for the displacement or configurational rearrangements of colloidal particles and macromolecules, to seconds and micrometers for meso- or macroscopic phase transitions, and to hours and even years for nonequilibrium processes. A single theoretical model that accounts for all atomistic details is clearly improbable and often unnecessary. In modeling the hierarchical, multiscale phenomena occurring in a complex fluid, the Born-Oppenheimer-type approximations are often adequate to decouple the spatiotemporal scales affiliated with the local motions of the solvent molecules or polymer side chains from those related to the configurational changes of the polymer backbone or displacement of colloidal particles, and from those connected to the dynamics of various phase transitions. Toward that end, we often use various coarse-grained models to represent macromolecules and colloidal particles and treat the solvent as an assembly of spherical particles or as a continuous medium. The coarse-grained models may differ enormously in terms of microscopic details, ranging from nearly precise united-atom models to simplistic lattice representations. Typically, we use bead-spring chains for linear polymers and polyelectrolytes, various convex hardparticle models for colloids and liquid crystals, and rigid rods or spheres for DNA and proteins. As in atomistic modeling, we may use a range of semiempirical interand intramolecular potentials to describe the effective interactions (potential of mean force) between coarse-grained particles or molecular segments in a complex fluid. Unlike atomic models, however, the solvent-mediated interactions are much more complicated than that between two atoms in a vacuum. Indeed, the development of accurate theoretical models for the solvent-mediated interactions remains a major challenge in the molecular modeling of complex fluids. Within a typical coarse-grained model, the Helmholtz energy functional of a polymeric fluid can be decomposed into two parts: The first corresponds to that of ideal chains free of nonbonded interactions, and the second accounts for the thermodynamic nonideality owing to various inter- and intramolecular interactions. For the ideal part, the Helmholtz energy functional is known exactly: F id = k B T (1) d RρM (R)[ln ρM (R) − 1] + d RρM (R)VM (R),


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M

M

where k B stands for the Boltzmann constant, T is the absolute temperature, and R ≡ (r1 , r2 , . . . rmM ) is defined by the positions of mM segments from a polymeric molecule M. The bond potential VM (R) specifies the chain connectivity of the coarsegrained segments. The multidimensional molecular density ρM (R) entails information on both the average molecular configurations and the microscopic structure of the entire system. An exact expression for the excess Helmholtz energy functional is not attainable for most systems of practical concern. Nevertheless, we can make effective approximations using a combination of different strategies accounting for various components of the nonbonded inter- and intramolecular interactions. These strategies vary from simplistic local-density approximations that ignore any short-range correlations, to more sophisticated versions of DFT derived from different levels of mean-field approximations or from a number of functional expansion methods. Although for some

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specific problems equally satisfactory results may be attained using different versions of DFT, the connection between different theoretical methods and their inherent limitations is less obvious. In this section, we discuss four generic approaches used routinely to formulate the Helmholtz energy functional for inhomogeneous polymeric systems. We also discuss their extensions to anisotropic fluids.


Density-Expansion Method

Vertex functions: the coefficients in the functional Taylor expansion of the intrinsic Helmholtz energy with respect to the one-body density profile

The density-expansion method originated in van der Waals’ theory of capillarity; it is connected closely to a number of conventional theories of interfacial phenomena and phase transitions, including the Cahn-Hilliard theory, the Ginzburg-Landau theory, and the Flory-Huggins-de Gennes-Lifshitz theory. Because of its simplicity, the Helmholtz energy functionals derived from density-expansion methods remain attractive in many practical applications. The central idea of density-expansion methods is to represent the Helmholtz energy functional of a system by a Taylor expansion with respect to a differential density profile. Within the framework of the Flory-Huggins theory for polymeric systems, the polymer density profiles are specified by the segmental volume fractions instead of the number densities. In this case, the Helmholtz energy functional is expressed as 1 β F = d r f φi0 (r) + d rd r i j (r − r )φi (r)φ j (r ) + · · · , (2) 2 i j where φi0 (r) represents the local volume fraction of i-type segments in the reference system, φi (r) ≡ φi (r) − φ¯ i is the differential volume-fraction profile, φ¯ i stands for a local average in the reference system, and β = 1/(k B T ). In the first term on the right side of Equation 2, f (φi ) stands for the Helmholtz energy density of a uniform system with composition (volume fraction) φi . The vertex function i j (r) specifies correlations between different segments from the same chain and from different chains; it depends on both chain connectivity and inter- and intrachain correlations. If there is a difference between the total number of molecules in the real system and that in the reference (i.e., d rφ = 0), Equation 2 must also include a linear-order term affiliated with the excess chemical potentials of the segments in the reference. If the density profiles of the reference are selected identical to those of the real system and if φi0 (r) = φ¯ i , Equation 2 reduces to a local-density approximation for the Helmholtz energy functional. We can apply a similar expansion of the Helmholtz energy functional to simple fluids (12) and to off-lattice models of polymeric systems (13). In those cases, the Helmholtz energy functional truncated after the quadratic term becomes equivalent to that used in the Ramakrishnan-Yussouff theory of freezing (14). In a continuous model, we use the number density, instead of the volume fraction, as the independent variable. Meanwhile, we express the vertex functions in terms of the density-density correlation function χ¯ Mi Mj (r) or the direct correlation function c¯ Mi Mj (r): −1 Mi Mj (r) = χ¯ M (r) = δi j δMM δ(r)/ρ Mi − c¯ Mi M (r), j iM j

(3)


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where δi j and δ(r) designate the Kronecker and Dirac delta functions, respectively; superscript −1 denotes functional inversion, the subscript Mi refers to the i-th segment of molecule M, and ρ Mi stands for the segment density. Even within the framework of the Flory-Huggins theory, a number of different ways have been proposed to define the Helmholtz energy density f (φi ) and the differential volume-fraction profile φi . These methods differ in the definition of the reference system and in the selection of the segment volume fraction for f (φi ) and for i j (r). For example, Leibler’s (15) pioneering work on the microphase segregation of block-copolymer melts expands the Helmholtz energy functional relative to that of a uniform system, which defines the polymer volume fraction for both f (φi ) and i j (r). Because a quadratic expansion relative to a uniform density is insufficient to represent phase transitions, Leibler’s theory also includes higher-order terms in the density expansion of the Helmholtz energy functional with the affiliated vertex functions calculated from a Gaussian-chain model. Leibler’s theory is most useful under weak-segregation conditions in which the variation of segment composition is small. In an extension of Leibler’s theory to strong segregations, Ohta & Kawasaki (16) constructed a Helmholtz energy density that exhibits double minima in terms of the local volume fraction, whereas they derived the vertex functions from a uniform fluid at a macroscopically averaged density. Bohbot-Raviv & Wang (17) followed a similar approach in studying the microscopic morphologies of triblock-copolymer melts. Whereas the double minima embedded in the local Helmholtz energy density facilitate phase transitions, the inconsistent use of the reference density profile in f (φi ) and i j (r) is responsible for incorrect predictions concerning the effect of blockcopolymer composition on the interfacial tension. Following a procedure similar to that used by Ebner et al. (12) for inhomogeneous simple fluids, Uneyama & Doi’s (18) recently proposed DFT theory removes such inconsistency. The Uneyama-Doi theory is applicable to macroscopic phase separations of polymer blends as well as to mesoscopic phase separations in block-copolymer melts, at both strong- and weaksegregation regions. With some further approximations, we can reduce it to the Flory-Huggins-de Gennes-Lifshitz theory for polymeric systems at inhomogeneous conditions (19, 20). According to Uneyama & Doi (18), Equation 2 is truncated after the quadratic terms, and the reduced Helmholtz energy per segment is given by the Flory-Huggins theory (i.e., φi0 (r) = φi (r)): φi (r) f = ln φi (r) + χi j φi (r)φ j (r), (4) N i i i j


RPA: random-phase approximation

where Ni denotes the degree of polymerization for i-type segments, and χi j is the Flory energy parameter. The locally averaged segment volume fraction, φ¯ i = φi (r)φi (r ), is used to evaluate the second-order vertex function in Equation 2. The latter is represented by a random-phase approximation (RPA): i j (r) ≈ i0j (r) + χi j , (5) where i0j (r) is the vertex function of a Gaussian chain that has a backbone structure identical to that of the real polymer. i0j (r) = 0 if segments i and j belong to different chains; otherwise, it is determined from the single-chain density correlation function

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ωi j (r). In Fourier space, the vertex function for a Gaussian chain is given by (18) ˆ i0j (q) =

NP ωˆ i−1 j (q) Ni Nj φ P

,

(6)

where NP is the degree of polymerization for the entire polymer under consideration, −1 φ P is the polymer volume fraction, and ωˆ i−1 j is defined by the inverse matrix {ωi j (q)} . In Equation 6, the single-chain density correlation function is defined as


ωi j (r) =

Nj Ni 1 δ[r − (rα − rβ )] , Ni Nj α=1 β=1

SCFT: self-consistent-field theory

(7)

where is averaged over all configurations of an isolated polymer. For a Gaussian chain consisting of multiple blocks, we can obtain an analytical expression of the single-chain density correlation function in Fourier space:

−x i= j 2(e i − 1 + xi )/xi2 ωi j (q ) = , (8) −xi −x j −xi j − 1)e /(xi x j ) i = j (e − 1)(e where xi ≡ q 2 Ri2 , Ri is the radius of gyration for an ideal chain with Ni segments, and xi j is defined by the radius of gyration of the intermediate segments connecting i and j blocks. If Equation 8 is further expanded in a power series of q and truncated after the second-order term, the RPA approximation of the second-order vertex function leads to a Helmholtz energy identical to that obtained from the square-gradient approximation (21, 21a). Uneyama & Doi (18) derived a concise expression for i0j (q) based on its asymptotic behavior at q → 0 and q → ∞. The density-expansion method represents a popular approach in practical DFT applications for describing a wide variety of phenomena in polymeric systems. However, the density-expansion method is accurate only when a small differential density φi (r) can be justified, such as in systems with weak inhomogeneity or slowly varying density profiles. Because the vertex function obtained from RPA ignores both the intermolecular correlations and the nonideal part of the intrachain interactions, RPA is not useful for systems in which both inter- or intramolecular correlations are significant. Interestingly, for polymeric systems, the parameters used in the Uneyama-Doi theory are the same as those in the conventional self-consistent-field theory (SCFT), which consists of a path-integral formulation of the Gaussian-chain partition function and a nonlocal mean-field representation of the intersegment interactions. Whereas the scope of SCFT is probably more generic, Uneyama & Doi (18) obtained similar results concerning mesoscopic phase separations in symmetric block-copolymer melts. Because the Uneyama-Doi theory avoids numerical evaluation of the single-chain partition function, it is computationally more efficient than SCFT. A comparison of SCFT, DFT, and simulations can be found in a recent review (22).

Integral-Equation Theories We may consider integral-equation theories as a special ramification of the densityexpansion method discussed above. Early applications of integral-equation theories


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RISM: reference interaction site model


PRISM: polymer reference interaction site model Direct correlation functions: the nonideal part of the vertex functions; they can be understood as the effective potentials (in negative, dimensionless form) owing to the intermolecular interactions Density correlation functions: the second functional derivatives of the grand potential with respect to the one-body external potential that specify correlated density fluctuations in space

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are mostly limited to the liquid-state theories of simple fluids, in particular at uniform conditions (23). The value of integral-equation theories for polymeric systems was largely unrecognized until the publication of the reference interaction site model (RISM) by Chandler and coworkers (24), and its successful extension to polymeric systems (PRISM) by Schweizer & Curro (25). Cummings & Stell (26) proposed a similar theoretical framework in the context of a site-site Ornstein-Zernike equation. As for simple fluids, the polymer integral-equation theories are most useful for bulk systems. Nevertheless, the direct correlation functions derived from integral-equation theories enable applications of density-expansion methods to various off-lattice models of inhomogeneous polymeric systems. In RISM or PRISM, the density-density correlation function of a polymeric system is partitioned into an ideal-gas part (i.e., free of intermolecular interactions) and a remainder: 0 χ¯ Mi Mj (r) = χ M (r) + χ Mi M (r). j iM j

(9)

Because the ideal-gas component retains all intramolecular interactions of a real polymer, the corresponding density-density correlation function is not identical to that of a Gaussian chain; instead, it is related to a single-chain correlation function: 0 χM (r) = ρ Mi δMM ω Mi j (r). iM j

(10)

In Equation 10, ρ Mi represents the number density of the i-th segment from molecule M, and the single-chain correlation function, defined as (11) ω Mi j (r) = δ r − r Mi − r Mj , can be evaluated by using molecular simulation for a single polymer chain. The nonideal part of the density-density correlation function can be used to define the total correlation function h Mi Mj (r): χ Mi Mj (r) = ρ Mi ρ Mj h Mi Mj (r).

(12)

We can apply a similar partitioning of the intra- and intermolecular contributions to the direct correlation function: c¯ Mi Mj (r) = c 0Mi M (r) + c Mi Mj (r). j

(13)

Following Equation 3, we obtain a relationship between the direct and density-density correlation functions, in a matrix notation, given by χ¯ˆ (r)−1 = ρˆ −1 δ(r) − c¯ˆ (r),

(14)

where the elements of the density matrix ρˆ are defined as ρ Mi δi j δMM . The same relation also holds between the ideal-gas part of the direct and density-density correlation functions: χ¯ˆ 0 (r)−1 = ρˆ −1 δ(r) − c¯ˆ 0 (r). (15) A comparison of Equations 14 and 15 yields ¯ˆ −1 = χˆ 0 (r)−1 − cˆ (r). χ(r)

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(16)

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After substituting χˆ 0 (r) and χ¯ˆ (r) into Equation 16 with the intramolecular and total correlation functions defined in Equations 10 and 12, we find a final connection between the total and direct correlation functions, which, in Fourier space, is given by ˆ ). ˆ ) = ω(q ˆ )cˆ (q )ω(q ˆ ) + ρˆ ω(q ˆ )cˆ (q )h(q (17) h(q Equation 17 represents one of the key equations in RISM/PRISM. For an atomic ˆ fluid, ω(q) = I, Equation 17 reduces to the familiar Ornstein-Zernike equation. As in the Ornstein-Zernike equation for simple fluids, applications of the RISM/PRISM equations require a closure (i.e., an additional relation between the total and direct correlation functions). In general, we can derive the closure using a density expansion of the Helmholtz energy functional as discussed in the previous section, or by heuristic means. The best known closure derived from the former approach is probably the hypernetted-chain approximation (HNC): ln[g Mi Mj (r)] = −βu Mi Mj (r) + ρ Mi d r c¯i j (r − r )h Mi Mj (r ), (18)

Total correlation functions: the normalized nonideal part of the density-density correlations that provides a measure of the nonrandom spatial distributions of the molecular densities HNC: hypernetted-chain approximation

Mk

where the radial distribution function is given by g Mi Mj (r) = h Mi Mj (r) + 1. Because HNC is affiliated with a quadratic approximation for the Helmholtz energy, it is not directly applicable to phase transitions. Conversely, the Percus-Yevick approximation provides a familiar example of the heuristic approach, which, in a hard-sphere model for interacting segments, is given by ⎧ ⎨g Mi M (r) = 0 r < σ Mi Mj j , (19) ⎩ c M M (r) ∼ r > σ Mi Mj =0 i j where σ Mi Mj stands for the segment collision diameter. In the DFT of polymeric fluids developed by Chandler, McCoy, and Singer (CMS) (13), the density-expansion method, as given in Equation 2, is applied to the Helmholtz energy functional of an inhomogeneous polymeric fluid and to that of an ideal-gas reference with the same density profiles. In both expansions, the reference Helmholtz energy corresponds to that of a uniform system (real or ideal gas) with the same average polymer number densities, and the higher-order terms beyond the quadratic are ignored. The difference in the Helmholtz energy between the real system and the ideal-gas reference, both at inhomogeneous conditions, is given by 1 βF = βF0 − (20) d rd r c Mi Mj (r − r )ρ Mi ρ Mj , 2 M M i

j

where F0 stands for the difference in the Helmholtz energy in the uniform conditions, and c Mi Mj (r − r ) is, as given in Equation 16, the intermolecular site-site direct correlation function of the uniform fluid. In Equation 20 the linear term is absent because the average molecule density of the uniform system matches that of the inhomogeneous system. The direct correlation functions in the CMS theory are taken from the RISM/PRISM theories for bulk fluids. The CMS theory yields an equilibrium density profile of polymeric fluids equivalent to that of the ideal-gas polymers in an effective external field, which depends


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Cavity correlation function: the correlated density distribution of cavities (i.e., particles interacting with the system but not with themselves); it specifies the probability of solvation

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on the density profile in a self-consistent manner. As in RISM/PRISM theories for bulk polymeric systems, the ideal-gas reference retains all intramolecular interactions of the real polymer. As a result, numerical implementation of the CMS theory requires a computationally demanding iteration procedure in which each step requires a single-chain simulation. Hooper et al. (27) proposed a more efficient computational procedure using umbrella-sampling and histogram-reweighing methods for the single-chain simulations. Another way to avoid the single-chain simulation is to treat the ideal-gas polymers as Gaussian chains (28). The CMS theory has been successfully applied to the study of the structure and thermodynamic properties of homopolymers and block copolymers at inhomogeneous conditions (29–31). The theoretical predictions are often in good agreement with simulation results. Although both CMS and Uneyama-Doi theories are based on quadratic expansions of the Helmholtz energy, the correlation functions derived from RISM/PRISM theories are, in general, more accurate than those from RPA in the Flory-Huggins theory framework. By using realistic bond and intermolecular potentials, RISM/PRISM theories can reproduce the density-density correlation functions of polymeric fluids quantitatively (32). However, as in the density-expansion method discussed above, truncation of the Helmholtz energy after the quadratic terms provides a good approximation only when a small differential density profile can be justified. Furthermore, a quadratic approximation such as HNC is insufficient to capture wetting and other phase transitions that entail multiple minima in the Helmholtz energy. On the contrary, because the quadratic expansion is applied relative to a local Helmholtz energy density, the Uneyama-Doi theory is immune to such limitations.

Thermodynamic Perturbation Theory As first suggested by Chandler & Pratt (33), a polymeric fluid may be considered as a system of monomeric particles linked together by a complete chemical reaction. By following the standard cluster-diagrammatic procedures, these authors developed an exact theory for describing the chemical equilibrium and intramolecular structures of nonrigid molecules in condensed phases. In particular, they identified an exact relation between the equilibrium constant for the polymerization, K M , and the multibody-cavity correlation function of the polymerizing segments in the medium y(r1 , r2 , . . . , rmM ): ρM r1 , r2 , . . . , rmM = y r1 , r2 , . . . , rmM K M0 , (21) KM ≡ ρ M1 (r1 )ρ M2 (r2 ) . . . ρ Mm rmM where ρM and ρ Mi refer to the polymer and monomer densities, respectively, and superscript 0 stands for the ideal-gas limit (i.e., polymerization of the same segments in the ideal gas). The thermodynamic cycle proposed by Zhou & Stell (34) best illustrates the connection between chemical association and chain connectivity in a polymeric system. As shown in Figure 1, we can decompose the excess Helmholtz energy of a polymeric system (uniform or inhomogeneous) into three contributions: (a) the free energy of

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Figure 1

ΔFII = FRex

The excess Helmholtz energy of a polymeric fluid can be calculated from a three-step thermodynamic cycle: (a) dissociation into monomers at the ideal-gas state, (b) solvation of the individual particles into a dense monatomic fluid, and (c) polymerization of the particles in the dense fluid.

b c

a ΔFI

ΔFIII


F ex

Ideal gas

Dense fluid

dissociation at the ideal-gas state, (b) the excess Helmholtz energy of a monomeric fluid where the segment density profiles are identical to those in the polymeric fluid, and (c) the free energy of polymerization in the dense fluid. The free energy of polymerization in the ideal gas includes contributions from the bond formations and from nonbonded interactions among the polymerizing segments. Whereas the properties of a single chain solely determine the bond energies and the intramolecular interactions, polymerization in the dense fluid depends also on the potential of mean force to bring the polymerizing segments together (35). The additional contribution to the free energy of polymerization arises from solvation (i.e., the interactions of the polymerizing segments with the surrounding medium). The difference between the free energy of association in the ideal gas and that in the dense fluid is therefore related to the multibody-cavity correlation function: FI + FIII = −k B T ln y(r1 , r2 , . . . , rmM ).

(22)

Apparently, Equations 21 and 22 are equivalent. The thermodynamic cycle suggests that polymerization in the dense medium differs from that in the ideal gas in terms of the work to bring the segments together. We can account for this difference by subtracting the direct intramolecular energy from the potential of mean force. A combination of these two contributions leads to the cavity correlation function shown in Equation 22. Based on the thermodynamic cycle discussed above, we can express the Helmholtz energy functional of a polymeric system in terms of an ideal-gas term free of nonbonded interactions, an excess term identical to that of a monomeric fluid with the same segment density distributions, and a cavity correlation contribution accounting for the effect of the medium on the polymerization: β F = β F id [ρM (R)] + β F ex [ρ Mi (r)] − (23) d R ρM (R) ln yM (R). M

Equation 23 is formally exact; approximations are made only in the expressions for the excess Helmholtz energy functional of the monomeric reference system and for the multibody-cavity correlation functions.


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Wertheim’s (36) thermodynamic perturbation theory (TPT) provides a systematic procedure to approximate the intersegment correlations owing to chain connectivity. In its lowest-order approximation (TPT1), the multibody-cavity correlation function is represented by a superposition of the two-body correlation functions in the monomeric fluid: m M −1 ln yM (R) ≈ ln y Mi Mi+1 (ri , ri + 1 ). (24)

TPT: thermodynamic perturbation theory

i=1


Although sophisticated procedures have been proposed for the calculation of the twobody-cavity correlation functions at inhomogeneous conditions (37, 38), semiempirical expressions obtained from heuristic modifications of the corresponding expressions for bulk systems are often most convenient for practical applications (39–41). In comparison with alternative methods, TPT has a great virtue in the decoupling of the nonbonded intermolecular interactions and polymer-chain connectivity. The former defines all short-range correlations that can be effectively treated by various density-functional methods developed for monatomic systems (42). With a reliable model for the excess Helmholtz energy of monomeric fluids, TPT provides a satisfactory description of the structure and thermodynamic properties for a wide variety of polymeric systems, including block and branched copolymers and polyelectrolytes (39–47). Although the correlations owing to chain connectivity are captured only at the level of two-body correlation functions, the long-range intrachain correlations are at least partially preserved in the ideal-chain contribution. Indeed, in TPT the idealchain contribution is equivalent to that included in the density-gradient methods or in a polymer SCFT. For concentrated polymeric systems, the local segment densities primarily determine the thermodynamic properties, and, therefore, the Helmholtz energy functional is relatively insensitive to the approximations of the multibodycavity correlation function. Conversely, in a dilute polymer solution, the polymer structure is mainly determined by the intramolecular interactions that depend on both bond connectivity and on intramolecular nonbonded interactions. Whereas TPT accounts for the polymer backbone structure exactly, it predicts incorrect long-range polymer-polymer correlations. As a result, similar to a typical mean-field theory, TPT is not able to capture the correct polymer scaling behavior, such as the variation of the osmotic second virial coefficient versus the degree of polymerization. Nevertheless, even in the dilute limit, TPT accounts for the local-density and short-range correlations owing to the nonbonded intersegment interactions. Another notable advantage of TPT is that for bulk fluids, it reduces to a generic equation of state that has been used extensively for phase-equilibrium calculations (48).

Weighted-Density Approximations In the weighted-density method, we assume a priori the excess Helmholtz energy functional of an inhomogeneous system is identical to that of a uniform system, except the bulk density is replaced by a locally averaged or weighted density. As extensively used for simple fluids, the weighted-density approximation provides a simple yet effective way to extend the theory of bulk polymeric fluids to inhomogeneous systems.

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Woodward (49) proposed the earliest application of the weighted-density approximation to polymeric systems. In this theory, the Helmholtz energy of a polymeric system is expressed as a functional of the segment density ρ Mi (r), which can be decomposed into a monomeric ideal-gas term and an excess: F = F id ρ Mi (r) + F ex ρ Mi (r) . (25) Equation 25 is formally exact even though it yields a self-consistent equation for the equilibrium density profile similar to that from a mean-field theory (49). The monomeric ideal-gas system is defined such that it reproduces the exact Helmholtz energy of ideal chains as given in Equation 1. With the weighted-density approximation, the excess Helmholtz energy is expressed as F ex [ρ Mi (r)] = (26) d rρ Mi (r) f Mi (ρ¯ Mi ), Mi

where f Mi is the excess Helmholtz energy per particle in the corresponding uniform system, and the summation applies to all different monomeric species. Intuitively, one may imagine that the purpose of a weighted-density ρ¯ Mi is to smooth out the sharp variation of the local density to justify the application of a local free energy. In Woodward’s (49) original work, the weighted density is simply calculated from the average density within the volume occupied by a spherical particle of diameter σ as proposed by Nordholm et al. (50) for hard-sphere fluids: 3 ρ¯Mi (r) = (27) d r ρMi (r ) (σ − |r − r |), 4πσ 3 where (r) is the Heaviside step function. With an equation of state for hard spheres and one adjustable parameter to rescale the excess Helmholtz energy, Woodward (49) predicted, in good agreement with simulations, the density profiles of hard-sphere chains confined in slit-like pores. As in applications of the weighted-density methods to simple fluids, more sophisticated expressions of the weight function can be devised on the basis of the direct correlation functions of the corresponding bulk systems (51). However, except in the presence of strong oscillations in the density profiles, different weight functions yield similar results. Over the past few years, Woodward and Forsman (52, 53) have made several improvements to the weighted-density approach for polymers. The later versions use an accurate equation of state for hard-sphere chains instead of that for hard spheres (54). In addition, Patra & Yethiraj (55) have proposed a generalization of the hardsphere-chain model that includes van der Waals attraction at the mean-field level or adopts the direct correlation functions calculated from RISM/PRISM. As in TPT, the weighted-density methods account for the contributions of ideal chains exactly. They have been used successfully to examine the surface forces mediated by polymers of different architectures. The numerical performance of a weighted-density method depends on inputs from the bulk equation of state and on the formulation of the weight functions.


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Anisotropic Fluids In an inhomogeneous fluid of rigid molecules, we can specify the density profile ρ(R) in terms of the center of mass r and orientational angle ω; i.e., ρ(R) = ρ(r, ω). A simple expression for the Helmholtz energy functional is provided by the virial expansion truncated after the second-order term: βF = d rd ω ρi (r, ω){ln[4πρi (r, ω)] − 1} i


1 − d rd ω d r d ω ρi (r, ω)ρi (r , ω )Mi j (r − r , ω, ω ), (28) 2 i, j where Mi j (r, ω) = e −βui j (r,ω) − 1 is the Mayer function, and ui j (r, ω) is the pair potential between the rigid molecules. Onsager (56) first proposed Equation 28 for modeling the isotropic-nematic transition of lyotropic liquid crystals. By including the third-order terms, Harnau & Dietrich (57) recently demonstrated that the virialexpansion method provides an efficient means for studying the complicated phase behavior of inhomogeneous platelet and rod fluids. Despite its simplicity, the virial-expansion method retains the coupling of the orientational and translational degrees of freedom. Such coupling is indispensable for modeling the properties and phase behavior of anisotropic fluids (58). However, the second-virial equation is generally insufficient to capture various phase transitions in concentrated liquid crystals. To account for multibody interactions, Parsons (59) and Lee (60) used an effective hard-sphere model to rescale the second-order virial expansion of the Helmholtz energy functional. The Parsons-Lee method is remarkably accurate for a wide variety of lyotropic liquid crystals (61– 64). Unlike Onsager’s theory, the rescaled Helmholtz energy functional can be used to describe sophisticated phase transitions of lyotropic liquid crystals, including those with both position and orientational orderings. With the athermal systems as a reference, intermolecular attractions can be introduced using a van der Waals–like mean-field approximation. In addition to the virial equations, we can use the density-expansion method to formulate the Helmholtz energy functional of anisotropic fluids (65). With an isotropic phase of average density ρ0 as the reference, we can express the Helmholtz energy of an ordered state in terms of a functional Taylor expansion with respect to a differential density ρ(r, ω) = ρ(r, ω) − ρ0 : d rd ωρ(r, ω) F = Fiso + μex 0 kB T d rd r d ωd ω c 0 (r − r , ω, ω )ρ(r, ω)ρ(r , ω ). (29) − 2 We can calculate the direct correlation function in Equation 29 from an integralequation theory for isotropic fluids (23). If both the density profile and the direct correlation function are expressed in terms of the lowest-order spherical harmonics and with some further approximations, Equation 29 reduces to the familiar Maier-SaupeMcMillan theory of liquid crystals (66). Unlike the Maier-Saupe-McMillan theory, however, the density-expansion method accounts for both attraction and short-range steric effects explicitly. Although the quadratic theory is not able to capture multiple

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minima in the free energy, a direct comparison of the free energies of different phases enables the construction of a complete phase diagram for liquid crystals.


TIME-DEPENDENT EQUATIONS Following the Hamiltonian of classical fluids and the Liouville equation, Chan & Finken (10) established a rigorous time-dependent density-functional theory (TDDFT) that describes the evolutions of both the particle number density ρ(r, t) and the affiliated velocity field v(r, t) at arbitrary conditions. They also introduced three different implementation schemes that are potentially useful for practical applications. Although TDDFT is promising in providing a unified theoretical foundation for a wide range of phenomenological equations of time-dependent processes, a microscopically based dynamic theory has yet to be developed for practical applications. At present, conventional phenomenological equations remain most useful in describing various time-dependent phenomena in complex fluids. All these models require a thermodynamically-defined Helmholtz energy obtained from equilibrium DFT.

Liouville equation: the equation of continuity for the evolution of the phase-space distribution function in systems with a conservative Hamiltonian TDDFT: time-dependent density-functional theory

Generalized Diffusion Equation The generalized diffusion equation is most useful for describing the relaxation of Brownian particles in a medium. It applies to situations where the density profile of the particles ρ(r, t) varies slowly in time, such that the affiliated momentum distribution reaches a steady state and the thermodynamic properties of the system can be represented by a local equilibrium. Following the Langevin equation for the Brownian motion of particles, we may write the generalized diffusion equation as (67, 68) ∂ρ(r, t) = D∇ · {ρ(r, t)∇ [δ F/δρ + ϕ(r, t)] + ζ(r, t)} , (30) ∂t where D represents the single-particle diffusion coefficient, which is related to the particle mobility coefficient 0 according to the Einstein equation D/ 0 = k B T; ϕ(r, t) is a one-body external potential; and ζ(r, t) is a random noise arising from fast collisions of the solvent molecules. The random-noise term satisfies the fluctuation-dissipation theorem ζi (r, t)ζ j (r , t ) = 2ρ(r, t)δ(r − r )δ(t − t )δi j /Dβ 2 , (31) where the subscripts i, j denote three-dimensional coordinates. Unlike the conventional dynamic theories of nonconserved (model A) or conserved (model B) systems, the random-force term is proportional to the particle density. Whereas Equation 30 is given in terms of a monomeric density, we can write a similar expression for polymeric systems on the basis of segment density profiles (69, 70): ∂ρ Mi (r, t) = ∇ · d r { Mi ,Mj (r, r )∇[μ Mj (r , t) + ϕ Mj (r , t)]} + ζ Mi (r, t), (32) ∂t M j

where Mi ,Mj (r, r ) is the Onsager kinetic coefficient, which depends on the conformation distribution of polymer chains. Because of the topological constraints of


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Memory effects: the correlation of random forces in Langevin dynamics

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polymer segments or the entanglement effects, the dynamics of a polymeric system in general is much more complicated than that of spherical particles (71, 72). If short chains or the short-time dynamics is of concern, the kinetic coupling of polymer segments becomes less important. In that case, we can represent Mi ,Mj (r, r ) by a local-coupling approximation, and the diffusion equation becomes (73) ∂ρ Mi (r, t) = β DMi ∇ρ Mi (r, t) μ Mi (r, t) + ϕ Mi (r, t) + ζ Mi (r, t), ∂t

(33)


where DMi is the segmental diffusion coefficient. We can view Equation 33 as a generalization of Equation 30 for multicomponent systems. However, because the segment density profiles are thermodynamically coupled through the free-energy density functional, diffusions of polymer segments are not independent, even within the local-coupling approximation. A number of researchers have indicated that the random-force term in Equation 30 is unnecessary if the Helmholtz energy is expressed in terms of the noise-averaged one-particle density (74, 75). If we use an instantaneous density in Equation 30, conversely, the Helmholtz energy functional is different from that corresponding to the equilibrium DFT (68, 76). The two free-energy functionals agree at the longwavelength limit of density fluctuations but differ in the short-scale fluctuation limit (77). Without the random-force term, the generalized diffusion equation becomes deterministic in describing the time evolution of the density profile driven by the gradient of the chemical and external potentials: ∂ρ(r, t) = D∇ 2 ρ(r, t) + D∇ρ(r, t) · ∇ [δβ F ex /δρ + βϕ(r, t)] . ∂t

(34)

If the Helmholtz energy is formulated by a square-gradient approximation and there is no external field, Equation 34 reduces to the Cahn-Hilliard theory for spinodal decompositions. In its applications to solvation dynamics, the generalized diffusion equation is also known as the Smoluchowski-Vlasov theory (78–80). The generalized diffusion equation predicts relaxation dynamics in excellent agreement with results from Brownian-dynamics simulations for a number of colloidal systems (81, 82). It can capture the dynamic responses of soft Brownian particles to various time-dependent external potentials (e.g., optical trap) quantitatively (83, 84). Augmented with a streaming term for the transport of mass, the generalized diffusion equation has also been successfully used to predict the shear viscosity, friction constant, and self-diffusion constant from the stationary density profile around a particle in a shear flow (85, 86). The generalized diffusion equation is useful for the relaxation dynamics of interacting Brownian particles in which the momentum distribution plays no significant role. Because the evolution of the time-dependent density profile leads to a monotonic decrease of the Helmholtz energy in time, the generalized diffusion equation is not able to capture various kinetic processes with a free-energy barrier, nor is the method useful for systems at unstable conditions in which the functional derivatives of the Helmholtz energy may diverge. Besides, the steady-state assumption in the generalized diffusion equation ignores the memory effect (i.e., time correlations of the moment distributions). As a result, it is generally not applicable to dynamic processes

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occurring in a supercooled liquid and glass transitions. Indeed, the agreement between the theory and simulation deteriorates for supercooled fluids (85).


Hydrodynamic Effects In concentrated colloidal dispersions or polymer solutions, the decoupling of stress and diffusion dynamics as assumed in the diffusion equation becomes invalid. At those conditions, we must consider the time evolution of the colloidal or polymer-density profiles along with the change of the particle velocity. For a monodisperse colloidal dispersion, we can specify the variation of the density profile by the equation of continuity: ∂ρ(r, t) = −∇ · ρ(r, t) v (r, t). ∂t

(35)

If the solvent reaches quasi-equilibrium at the timescale relevant to colloidal particles and if the random forces are short-lived, we can express the momentum balance of colloidal particles using the generalized Langevin equation (87): M

∂ρv δF + M ∇ · (ρv · v) = −ρ∇ − 0−1 ρv + ζ. ∂t δρ

(36)

In Equation 36, M denotes the particle mass; the density ρ, velocity v, and random force ζ all depend on position and time. We can apply a similar expression to polymeric systems. In that case, we must replace the friction term (the second term on the right side of Equation 36) with a stress proportional to the gradient of the velocity field multiplied by the shear viscosity (88). Hydrodynamic interactions are not included in Equation 36. To account for such an effect, we may replace 0 , the mobility coefficient in a free solution, with the short-term mobility coefficient (89). In this case, the free-solution mobility coefficient appearing in the fluctuation-dissipation equation (Equation 31) must also be replaced by the short-term mobility coefficient. This simple rescaling method is most useful when hydrodynamic interactions affect only the short-term behavior of the colloidal particles (i.e., in the timescale shorter than that for one colloidal particle to diffuse the mean separation with its neighbors but longer than the relaxation time of the particle velocity). Whereas the particle mobility in a free solution is independent of the colloidal concentration, the short-term mobility coefficient may decline drastically as the colloidal concentration increases. The generalized Langevin equation provides a useful starting point for constructing the mode-coupling theory for supercooled fluids (87, 88, 90). For hard-sphere fluids near the glass transition, Equation 36 predicts the characteristic time increasing with particle density, in good agreement with the empirical Volgel-Fulcher law (91). In addition, Equation 36 captures the stretched exponential decay and two-stage relaxation of the density correlation functions. Nevertheless, the generalized Langevin equation is not applicable to situations in which the fluctuation-dissipation theorem fails or the hydrodynamic interactions and colloidal forces are strongly coupled.


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Glass Transition


Glass formation in a supercooled liquid is characterized by dynamic heterogeneity, which is coupled with short-term local relaxation and with long-term phase-transition-like kinetics (92, 93). Whereas a generalized diffusion or Langevin equation can describe various downhill relaxation processes, it is, in general, not applicable to free-energy-barrier-crossing events such as crystal nucleation or other first-order phase transitions. However, as in a thermodynamic theory of nucleation (94), DFT provides an efficient means to characterize the free-energy landscape during glass-formation processes. Unlike a typical crystallization process, the glass transition ends with thermodynamically metastable amorphous structures. Even in a simple system such as hard spheres, a large number of metastable disordered states have been identified that minimize the local free energy (95–97). The free energies of these morphological states are higher than that of the crystalline state but smaller than that of an equilibrium liquid. Although a completely rigorous microscopic theory of glass transitions has yet to be established, Wolynes and coworkers (98, 99) demonstrated that DFT provides a useful theoretical framework for analyzing the freezing of a supercooled liquid into a large number of disordered structures. Based on the universality of the Lindemann criterion for solid stability, the random first-order transition theory devised by Wolynes and coworkers (100, 101) provides a unified framework for describing a wide variety of empirical correlations, including the Adam-Gibbs theory for strong and fragile liquids and the Vogel-Fulcher law for the relaxation time.

PERSPECTIVES In this review, we discuss a number of strategies for the formulation of an effective Helmholtz energy functional to account for solvent-mediated intermolecular interactions and correlation effects in complex fluids. These free-energy functionals are useful for describing not only equilibrium properties, but also a wide variety of dynamic phenomena. From a fundamental point of view, one imperative task for future work is to develop a generic TDDFT for describing hierarchical dynamic phenomena in complex fluids. Such a theory will have broad impacts in materials science by providing a much-needed theoretical foundation of molecular self-assembly. Significant progress has been made in establishing a generic scheme from first principles and from the generalization of phenomenological equations. The recent developments allow us to comprehend a wide range of time-dependent physical processes. Another important task for future work is to incorporate ideas of the renormalization-group theory into the DFT framework. Long-range correlations have been largely ignored in essentially all existing formulations of the Helmholtz energy functional, but they are important not only for fluids near the critical point but also for applications to the kinetics of phase transitions that entail both stable and unstable regions of the phase diagram. Furthermore, we need a renormalizationgroup correction to account for the long-range inter- and intrachain bond-bond

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Table 1

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Some representative applications of density-functional theory for complex fluids


Systems

Equilibrium

Dynamics

Colloids

Force and stability (104, 105), interfacial behavior (106, 107), phase separation (108), crystallization (109), fractionation (110), elastic constants (111), crystal vacancies (112)

Shear viscosity (85), metastability (95), structure formation (84), phase separation (113), nucleation (94), ionic conductivity (114)

Surfactants

Mesostructures (115), phase diagram (116), surface tension (117), adsorption (118), stabilization (119), lipid bilayers (31)

Micellization (120)

Polymers

Interfacial properties (44, 121), steric forces (122), adsorption (123), wetting (124)

Shear viscosity (86), single chain (125)

Polyelectrolytes

Solvation forces (126), conformation (127), interface (41), phase transitions (128), ion distribution (129)

Block copolymers

Interfacial behavior (130), micelles (131), phase diagram (132), thin films (133)

Liquid crystals

Phase transitions (136), mixtures (137), columnar (138), capillary smectization (139), interfacial behavior (140), polydispersity (141), carbon nanotube (142), elastic constant (143)

Proteins

Folding/aggregation (144), polymer-protein mixtures (145), actin filaments (146)

Folding/aggregation (147)

Ion channels

Selectivity (148)

Ion flux (149)

Shear-induced phase transitions (134), thin-film stability (135)

correlations in polymeric systems beyond a mean-field approximation (102, 103). Future research on phase transitions will benefit from a generic TDDFT to capture multiple spatiotemporal scales affiliated with the solvent and with complicated solute-solvent effects, together with a good expression for the Helmholtz energy functional that accounts for long-range correlations, including those corresponding to the thermodynamically unstable regions of the phase diagram. DFT provides a new paradigm for coherent modeling of multiscale equilibrium and dynamic phenomena within a unified theoretical framework. It promises a number of important applications in materials science and biotechnology. A few prominent examples include predictions of the microscopic structures and thermodynamic stabilities of polymer composites, phase and interfacial behavior of semiflexible polymers, ion-specific interactions between biomacromolecules and their influence on conformational transitions in nucleic acids and proteins, and aggregation and transport behavior of biomacromolecules in a crowded cellular milieu. Table 1 summarizes recent literature on practical applications of DFT to equilibrium and dynamic properties of colloids, polymers, liquid crystals, and some selected biological systems. We direct the reader to the original papers (see Table 1) or a parallel review (42) for details. Future DFT applications hinge on the development of effective Helmholtz energy functionals that account for the essentials of rich physical phenomena in complex fluids and, more importantly, a user-friendly computational scheme for the numerical implementation of DFT in multiple dimensions.


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SUMMARY POINTS 1. DFT opens a new avenue for describing multiscale equilibrium and timedependent phenomena within a unified theoretical framework. The rigorous mathematical formalism is equally applicable to electronic, atomic, and coarse-grained models.


2. DFT complements traditional phenomenological theories of statistical mechanics and atomistic simulation methods, particularly for systems that entail multiple spatiotemporal scales. 3. A number of generic strategies have been established for the formulation of a nonideal Helmholtz energy functional; they are interconnected, but not all are equivalent. Practical applications often require a combination of different strategies to capture the multiple aspects of complex physical phenomena. 4. The generalized Langevin equation has been used successfully to describe various relaxation dynamics and time-dependent responses of a density profile to an external field, but, in general, it is not suitable for supercooled liquids and glass transitions. 5. Effective strategies are yet to be devised to account for long-range correlations and for describing hierarchical dynamic processes in complex fluids.

ACKNOWLEDGMENTS ¨ The authors are grateful to Siegfried Dietrich, Masao Doi, Marcus Muller, John Prausnitz, Roland Roth, and Akira Yoshimori for critical reading of this manuscript and inspiring discussions. This work was supported by NSF grants CTS-0406100 and CTS-0340948.

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12. First application of DFT and nonlocal quadratic–expansion method to classical systems.

13. First application of DFT to polymers with a concise description of the RISM theory.

18. Bridges DFT and SCFT for block copolymers.

22. Provides an instructive comparison between SCFT, DFT, and simulations.

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42. A comprehensive review of recent applications of DFT to complex fluids.

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67. A brief but clear discussion of TDDFT.

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91. Discusses the connection of TDDFT with mode-coupling theory and with the phenomenological laws for glass transition.

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108. Schmidt M, Fortini A, Dijkstra M. 2004. Capillary evaporation in colloidpolymer mixtures selectively confined to a planar slit. J. Phys. Condens. Matter 16:S4159–68 109. Groh B, Mulder B. 2000. Hard-sphere solids near close packing: testing theories for crystallization. Phys. Rev. E 61:3811–22 110. Xu H, Bellier-Castella L, Baus M. 2002. Density functional theory of soft matter. J. Phys. Condens. Matter 14:12147–58 111. Laird BB. 1992. Weighted-density-functional theory calculation of elasticconstants for face-centered-cubic and body-centered-cubic hard-sphere crystals. J. Chem. Phys. 97:2699–704 112. Groh B. 2000. Density-functional theory for vacancies in hard-sphere crystals. Phys. Rev. E 61:5218–22 113. Archer AJ. 2005. Dynamical density functional theory: binary phase-separating colloidal fluid in a cavity. J. Phys. Condens. Matter 17:1405–27 114. Chandra A, Bagchi B. 2000. Frequency dependence of ionic conductivity of electrolyte solutions. J. Chem. Phys. 112:1876–86 115. Christopher PS, Oxtoby DW. 2003. Classical density functional study of multisite amphiphile mesostructures. J. Chem. Phys. 119:10330–38 116. Napari I, Laaksonen A, Strey R. 2000. Density-functional studies of amphiphilic binary mixtures. I. Phase behavior. J. Chem. Phys. 113:4476–79 117. Stoyanov SD, Rehage H, Paunov VN. 2003. Novel surface tension isotherm for surfactants based on local density functional theory. Phys. Rev. Lett. 91:086102 118. Pizio O, Bucior K, Patrykiejew A, Sokolowski S. 2005. Density-functional theory for fluid mixtures of charged chain particles and spherical counterions in contact with charged hard wall: adsorption, double layer capacitance, and the point of zero charge. J. Chem. Phys. 123:214902 119. Patel N, Egorov SA. 2005. Dispersing nanotubes with surfactants: a microscopic statistical mechanical analysis. J. Am. Chem. Soc. 127:14124–25 120. Talanquer V, Oxtoby DW. 2000. A density-functional approach to nucleation in micellar solutions. J. Chem. Phys. 113:7013–21 121. Bryk P, Roth R. 2005. Bulk and inhomogeneous mixtures of hard rods and excluded-volume polymer: a density functional approach. Phys. Rev. E 71:011510 122. Cao DP, Wu JZ. 2006. Surface forces between telechelic brushes revisited: the origin of a weak attraction. Langmuir 22:2712–18 123. Tscheliessnig R, Billes W, Fischer J, Sokoowski S, Pizio O. 2006. The role of fluid wall association on adsorption of chain molecules at functionalized surfaces: a density functional approach. J. Chem. Phys. 124:164703 124. Forsman J, Woodward CE. 2005. Prewetting and layering in athermal polymer solutions. Phys. Rev. Lett. 94:118301 125. Takahashi T, Munakata T. 1997. Solvent effects on polymer conformation: density-functional-theory approach. Phys. Rev. E 56:4344–50 126. Jonsson B, Broukhno A, Forsman J, Akesson T. 2003. Depletion and structural forces in confined polyelectrolyte solutions. Langmuir 19:9914–22 127. Wang H, Denton AR. 2005. Effective electrostatic interactions in solutions of polyelectrolyte stars with rigid rodlike arms. J. Chem. Phys. 123:244901


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128. Kyrylyuk AV, Lohmeijer BGG, Schubert US. 2005. Predicting the morphology of metallo-supramolecular block copolymers with bulky counter ions. Macromol. Rapid Comm. 26:1948–54 129. Wang K, Yu YX, Gao GH. 2004. Density functional study on the structures and thermodynamic properties of small ions around polyanionic DNA. Phys. Rev. E 70:011912 130. Hamm M, Goldbeck-Wood G, Zvelindovsky AV, Fraaije JGEM. 2004. Nematic-amorphous polymer interfaces in the presence of a compatibilizer. J. Chem. Phys. 121:4430–40 131. Uneyama T, Doi M. 2005. Calculation of the micellar structure of polymer surfactant on the basis of the density functional theory. Macromolecules 38:5817– 25 132. Xia JF, Sun MZ, Qiu F, Zhang HD, Yang YL. 2005. Microphase ordering mechanisms in linear ABC triblock copolymers: a dynamic density functional study. Macromolecules 38:9324–32 133. Cao DP, Wu JZ. 2005. Surface-induced lamellar-lamellar phase transition of block copolymer thin films. J. Chem. Phys. 122:194703 134. Zvelindovsky AV, Sevink GJA, Lyakhova KS, Altevogt P. 2004. Dynamic density functional theory for sheared polymeric systems. Macromol. Theory Simulat. 13:140–51 135. Sun DC, Huang L, Liang HJ. 2005. Unstable inverted phases of di- and triblock copolymers on solution-casting films. Macromol. Res. 13:152–55 136. Wessels PPF, Mulder BM. 2004. Entropy-induced microphase separation in hard diblock copolymers. Phys. Rev. E 70:031503 137. Cinacchi G, Mederos L, Velasco E. 2004. Liquid-crystal phase diagrams of binary mixtures of hard spherocylinders. J. Chem. Phys. 121:3854–63 138. Coussaert T, Baus M. 2002. Density-functional theory of the columnar phase of discotic Gay-Berne molecules. J. Chem. Phys. 116:7744–51 139. de las Heras D, Velasco E, Mederos L. 2005. Capillary smectization and layering in a confined liquid crystal. Phys. Rev. Lett. 94:017801 140. Martinez-Raton Y, Velasco E, Mederos L. 2005. Effect of particle geometry on phase transitions in two-dimensional liquid crystals. J. Chem. Phys. 122:064903 141. Martinez-Raton Y, Cuesta JA. 2003. Phase diagrams of Zwanzig models: the effect of polydispersity. J. Chem. Phys. 118:10164–73 142. Somoza AM, Sagui C, Roland C. 2001. Liquid-crystal phases of capped carbon nanotubes. Phys. Rev. B 6308:081403 143. Srivastava A, Singh S. 2004. Elastic constants of nematic liquid crystals of uniaxial symmetry. J. Phys. Condens. Matter 16:7169–82 144. Kinjo AR, Takada S. 2003. Competition between protein folding and aggregation with molecular chaperones in crowded solutions: insight from mesoscopic simulations. Biophys. J. 85:3521–31 145. Denton AR, Schmidt M. 2005. Mixtures of charged colloid and neutral polymer: influence of electrostatic interactions on demixing and interfacial tension. J. Chem. Phys. 122:244911


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146. Borukhov L, Bruinsma RF, Gelbart WM, Liu AJ. 2005. Structural polymorphism of the cytoskeleton: a model of linker-assisted filament aggregation. Proc. Natl. Acad. Sci. USA 102:3673–78 147. Kinjo AR, Takada S. 2002. Effects of macromolecular crowding on protein folding and aggregation studied by density functional theory: dynamics. Phys. Rev. E 66:051902 148. Roth R, Gillespie D. 2005. Physics of size selectivity. Phys. Rev. Lett. 95:247801 149. Gillespie D, Nonner W, Eisenberg RS. 2002. Coupling Poisson-Nernst-Planck and density functional theory to calculate ion flux. J. Phys. Condens. Matter 14:12129–45


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Annual Review of Physical Chemistry


Contents

Volume 58, 2007

Frontispiece C. Bradley Moore p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p xvi A Spectroscopist’s View of Energy States, Energy Transfers, and Chemical Reactions C. Bradley Moore p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 1 Stochastic Simulation of Chemical Kinetics Daniel T. Gillespie p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 35 Protein-Folding Dynamics: Overview of Molecular Simulation Techniques Harold A. Scheraga, Mey Khalili, and Adam Liwo p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 57 Density-Functional Theory for Complex Fluids Jianzhong Wu and Zhidong Li p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p 85 Phosphorylation Energy Hypothesis: Open Chemical Systems and Their Biological Functions Hong Qian p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p113 Theoretical Studies of Photoinduced Electron Transfer in Dye-Sensitized TiO2 Walter R. Duncan and Oleg V. Prezhdo p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p143 Nanoscale Fracture Mechanics Steven L. Mielke, Ted Belytschko, and George C. Schatz p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p185 Modeling Self-Assembly and Phase Behavior in Complex Mixtures Anna C. Balazs p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p211 Theory of Structural Glasses and Supercooled Liquids Vassiliy Lubchenko and Peter G. Wolynes p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p235 Localized Surface Plasmon Resonance Spectroscopy and Sensing Katherine A. Willets and Richard P. Van Duyne p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p267 Copper and the Prion Protein: Methods, Structures, Function, and Disease Glenn L. Millhauser p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p299

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Aging of Organic Aerosol: Bridging the Gap Between Laboratory and Field Studies Yinon Rudich, Neil M. Donahue, and Thomas F. Mentel p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p321 Molecular Motion at Soft and Hard Interfaces: From Phospholipid Bilayers to Polymers and Lubricants Sung Chul Bae and Steve Granick p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p353


Molecular Architectonic on Metal Surfaces Johannes V. Barth p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p375 Highly Fluorescent Noble-Metal Quantum Dots Jie Zheng, Philip R. Nicovich, and Robert M. Dickson p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p409 State-to-State Dynamics of Elementary Bimolecular Reactions Xueming Yang p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p433 Femtosecond Stimulated Raman Spectroscopy Philipp Kukura, David W. McCamant, and Richard A. Mathies p p p p p p p p p p p p p p p p p p p p p461 Single-Molecule Probing of Adsorption and Diffusion on Silica Surfaces Mary J. Wirth and Michael A. Legg p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p489 Intermolecular Interactions in Biomolecular Systems Examined by Mass Spectrometry Thomas Wyttenbach and Michael T. Bowers p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p511 Measurement of Single-Molecule Conductance Fang Chen, Joshua Hihath, Zhifeng Huang, Xiulan Li, and N.J. Tao p p p p p p p p p p p p p p p535 Structure and Dynamics of Conjugated Polymers in Liquid Crystalline Solvents P.F. Barbara, W.-S. Chang, S. Link, G.D. Scholes, and Arun Yethiraj p p p p p p p p p p p p p p p565 Gas-Phase Spectroscopy of Biomolecular Building Blocks Mattanjah S. de Vries and Pavel Hobza p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p585 Isomerization Through Conical Intersections Benjamin G. Levine and Todd J. Martínez p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p613 Spectral and Dynamical Properties of Multiexcitons in Semiconductor Nanocrystals Victor I. Klimov p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p635 Molecular Motors: A Theorist’s Perspective Anatoly B. Kolomeisky and Michael E. Fisher p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p675 Bending Mechanics and Molecular Organization in Biological Membranes Jay T. Groves p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p697 Exciton Photophysics of Carbon Nanotubes Mildred S. Dresselhaus, Gene Dresselhaus, Riichiro Saito, and Ado Jorio p p p p p p p p p p p p719

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Contents

Density-Functional Theory for Complex Fluids

Density-Functional Theory for Complex Fluids

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