Density-Functional Theory for Complex Fluids

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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 First published online as a Review in Advance on October 19, 2006 The Annual Review of Physical Chemistry is online at http://physchem.annualreviews.org This article’s doi: 10.1146/annurev.physchem.58.032806.104650 Copyright c 2007 by Annual Reviews. All rights reserved 0066-426X/07/0505-0085$20.00 Key Words phase transitions, self-assembly, thin films, polymers, colloids, liquid crystals Abstract Density-functional theory (DFT) and its variations provide a fruitful approach to the computational modeling of the microscopic struc- tures and phase behavior of soft-condensed matter. The methodol- ogy takes deep root in quantum mechanics but shares a mathematical similarity with a number of classical approaches in statistical me- chanics. This review discusses different strategies commonly used to formulate the free-energy functional of complex fluids for ei- ther phenomena-oriented applications or as a generic description of the thermodynamic nonideality owing to various components of intermolecular forces. We emphasize the connections among differ- ent 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. 85 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.

Transcript of Density-Functional Theory for Complex Fluids

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Density-Functional Theoryfor Complex FluidsJianzhong Wu and Zhidong LiDepartment of Chemical and Environmental Engineering, University of California,Riverside, California 92521; email: [email protected]

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

First published online as a Review in Advanceon October 19, 2006

The Annual Review of Physical Chemistry isonline at http://physchem.annualreviews.org

This article’s doi:10.1146/annurev.physchem.58.032806.104650

Copyright c© 2007 by Annual Reviews.All rights reserved

0066-426X/07/0505-0085$20.00

Key Words

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

AbstractDensity-functional theory (DFT) and its variations provide a fruitfulapproach to the computational modeling of the microscopic struc-tures and phase behavior of soft-condensed matter. The methodol-ogy takes deep root in quantum mechanics but shares a mathematicalsimilarity with a number of classical approaches in statistical me-chanics. This review discusses different strategies commonly usedto formulate the free-energy functional of complex fluids for ei-ther phenomena-oriented applications or as a generic descriptionof the thermodynamic nonideality owing to various components ofintermolecular forces. We emphasize the connections among differ-ent schemes of DFT approximations, their underlying assumptions,and inherent limitations. We also address extensions of equilibriumDFT to phenomenological theories for the dynamic properties ofcomplex fluids and for the kinetics of phase transitions. In addition,we highlight the recent literature concerning applications of DFTto diverse static and time-dependent phenomena in complex fluids.

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DFT: density-functionaltheory

Intrinsic Helmholtzenergy: a Legendretransform of the grandpotential that shifts theindependent variable fromthe one-body potential tothe one-body density profile

Grand potential: the freeenergy of an open system atfixed volume andtemperature that isminimized at equilibrium

INTRODUCTION

Density-functional theory (DFT) has emerged as a unified theoretical frameworkfor computational modeling of both quantum and classical systems. This genericmethodology is built upon a mathematical theorem that states, for an equilibriumsystem at a given temperature and at given chemical potentials of all constituentmolecules/particles, there is an invertible mapping between the external potentialand the one-body density profiles (1, 2). This mathematical theorem enables a def-inition of an intrinsic Helmholtz energy functional that is independent of the ex-ternal potential and, subsequently, a solution of the equilibrium density profiles bythe minimization of the grand potential. Functional derivatives of the grand po-tential or Helmholtz energy lead to multibody correlation functions that allow thedetermination of both the structure and thermodynamic properties of the equilib-rium system. DFT has become, by far, the most prominent tool in applied quantummechanics, in particular for calculating the ground-state properties of electronic sys-tems (3). Conversely, researchers have profitably used its classical counterpart inthe study of phase transitions, interfacial phenomena, and microscopic structures ofpolymeric fluids, liquid crystals, crystalline solids and glasses (4–8). Invertible map-pings among the time-dependent density profiles, momentum distributions, and ex-ternal field have also been established for both quantum and classical systems (9,10), thereby facilitating DFT applications to various dynamic and off-equilibriumproblems.

Whereas both quantum and classical DFTs are based on a formally exact math-ematical framework, their practical use inevitably requires effective approximationsfor the exchange-correlation energy to account for fluctuations of the electronicdensity and for the excess Helmholtz energy functional to account for intermolec-ular interactions in classical systems. These approximations must capture faithfullythe underlying physics of a particular problem under consideration, and the numer-ical performance has to be calibrated by direct comparison with experimental orsimulation data.

For classical systems, researchers commonly use two basic strategies to constructan approximate Helmholtz energy functional: The first relies on diagrammatic ex-pansions of the grand partition function as used extensively in statistical mechanics,and the second seeks direct approximations of the free-energy functional by a local-density ansatz or by a functional Taylor expansion relative to a reference densityprofile or an interparticle potential (5). Along with these first-principle methods isa number of phenomenological approaches that express the unknown free-energyfunctional in terms of a few carefully selected order parameters (11). Typical physicalphenomena in a complex fluid entail multiple time- and length scales affiliated withthe diverse intramolecular architectures of the constituent molecules and complicatedsolvent-mediated intermolecular interactions. To capture the multiple aspects of therich phenomena in complex fluids, DFT applications often require a combination ofvarious strategies to formulate a reliable Helmholtz energy functional.

In comparison with alternative methods in molecular modeling, the advantagesof DFT include its theoretical versatility, physical clarity, and a concomitant of

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computational efficiency for representing various static and dynamic phenomenain many-particle systems. Even with modern computers, direct applications of theSchrodinger equation or atomistic molecular-dynamics simulations are limited torelatively small systems. Conversely, interesting physical processes in biological sys-tems and soft materials often involve multiple time- and length scales that cannotbe addressed directly with exact methods. By contrast, for quantum and classicalsystems, DFT deals directly with the average system properties and thus avoids enu-meration of the microscopic fluctuations of individual particles. Besides, in DFTthe thermodynamic properties are embedded in the formulation of the free-energyfunctional, whereas in standard molecular simulations they are indirectly calculatedfrom mechanical variables. For describing the equilibrium and dynamic properties ofcomplex fluids, DFT provides a unified theoretical basis for various phenomenolog-ical methods originally developed in different contexts. Furthermore, DFT offers aneffective theoretical means to predict the mesoscopic/macroscopic parameters usedin conventional phenomenological models or in continuous equations.

In this review, we discuss different strategies commonly used to formulate theHelmholtz energy functionals and related dynamic equations for representing theequilibrium and dynamic properties of complex fluids; we do not consider quantumeffects. We give special attention to the merits and limitations of different DFTapproximations, their underlying interconnections, and relationships to conventionalphenomenological methods in equilibrium and nonequilibrium statistical mechanics.In addition, we highlight recent practical applications of DFT concerning the staticand dynamic properties and phase transitions in colloids, polymeric fluids, surfactantsolutions, liquid crystals, and selected biological systems.

HELMHOLTZ ENERGY FUNCTIONAL

The Helmholtz energy functional plays a central role in both equilibrium DFT andin its extensions to dynamic equations for modeling time-dependent processes. Ingeneral, the formulation of an effective Helmholtz energy functional must be guidedby the specific forms of the intermolecular interactions in the particular system underconsideration. These interactions typically consist of contributions from short-rangerepulsions owing to molecular excluded volumes, van der Waals attractions, Coulombinteractions, and various forms of electron donor-acceptor interactions responsiblefor chemical associations and hydrogen bonding. From an atomistic viewpoint, ageneric Helmholtz energy functional must account for the contributions to the ther-modynamic 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 char-acterizations of the constituent molecules, including molecular size, internal archi-tecture and geometry, polarity, and charge status. Typical dynamic processes in acomplex fluid span spatiotemporal scales ranging from fractions of a picosecondand 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 andmacromolecules, to seconds and micrometers for meso- or macroscopic phase tran-sitions, and to hours and even years for nonequilibrium processes. A single theo-retical model that accounts for all atomistic details is clearly improbable and oftenunnecessary.

In modeling the hierarchical, multiscale phenomena occurring in a complex fluid,the Born-Oppenheimer-type approximations are often adequate to decouple the spa-tiotemporal scales affiliated with the local motions of the solvent molecules or polymerside chains from those related to the configurational changes of the polymer back-bone or displacement of colloidal particles, and from those connected to the dynamicsof various phase transitions. Toward that end, we often use various coarse-grainedmodels to represent macromolecules and colloidal particles and treat the solvent asan assembly of spherical particles or as a continuous medium. The coarse-grainedmodels may differ enormously in terms of microscopic details, ranging from nearlyprecise united-atom models to simplistic lattice representations. Typically, we usebead-spring chains for linear polymers and polyelectrolytes, various convex hard-particle models for colloids and liquid crystals, and rigid rods or spheres for DNAand proteins. As in atomistic modeling, we may use a range of semiempirical inter-and intramolecular potentials to describe the effective interactions (potential of meanforce) between coarse-grained particles or molecular segments in a complex fluid.Unlike atomic models, however, the solvent-mediated interactions are much morecomplicated than that between two atoms in a vacuum. Indeed, the development ofaccurate theoretical models for the solvent-mediated interactions remains a majorchallenge in the molecular modeling of complex fluids.

Within a typical coarse-grained model, the Helmholtz energy functional of a poly-meric fluid can be decomposed into two parts: The first corresponds to that of idealchains free of nonbonded interactions, and the second accounts for the thermody-namic nonideality owing to various inter- and intramolecular interactions. For theideal part, the Helmholtz energy functional is known exactly:

F id = kB T∑

M

∫dRρM (R)[ln ρM (R) − 1] +

∑M

∫dRρM (R)VM (R), (1)

where kB stands for the Boltzmann constant, T is the absolute temperature, andR ≡ (r1, r2, . . . rmM ) is defined by the positions of mM segments from a polymericmolecule M. The bond potential VM (R) specifies the chain connectivity of the coarse-grained segments. The multidimensional molecular density ρM (R) entails informa-tion on both the average molecular configurations and the microscopic structure ofthe entire system.

An exact expression for the excess Helmholtz energy functional is not attainable formost systems of practical concern. Nevertheless, we can make effective approxima-tions using a combination of different strategies accounting for various componentsof the nonbonded inter- and intramolecular interactions. These strategies vary fromsimplistic local-density approximations that ignore any short-range correlations, tomore sophisticated versions of DFT derived from different levels of mean-field ap-proximations or from a number of functional expansion methods. Although for some

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Vertex functions: thecoefficients in the functionalTaylor expansion of theintrinsic Helmholtz energywith respect to theone-body density profile

specific problems equally satisfactory results may be attained using different versionsof DFT, the connection between different theoretical methods and their inherentlimitations is less obvious. In this section, we discuss four generic approaches usedroutinely to formulate the Helmholtz energy functional for inhomogeneous poly-meric systems. We also discuss their extensions to anisotropic fluids.

Density-Expansion Method

The density-expansion method originated in van der Waals’ theory of capillarity; itis connected closely to a number of conventional theories of interfacial phenomenaand phase transitions, including the Cahn-Hilliard theory, the Ginzburg-Landautheory, and the Flory-Huggins-de Gennes-Lifshitz theory. Because of its simplicity,the Helmholtz energy functionals derived from density-expansion methods remainattractive in many practical applications.

The central idea of density-expansion methods is to represent the Helmholtzenergy functional of a system by a Taylor expansion with respect to a differentialdensity profile. Within the framework of the Flory-Huggins theory for polymericsystems, the polymer density profiles are specified by the segmental volume fractionsinstead of the number densities. In this case, the Helmholtz energy functional isexpressed as

βF =∫

dr f[φ0

i(r)

] + 12

∑i

∑j

∫ ∫drdr ′�i j (r − r ′)�φi (r)�φ j (r′) + · · · , (2)

where φ0i(r) represents the local volume fraction of i-type segments in the reference

system, �φi (r) ≡ φi (r) − φi is the differential volume-fraction profile, φi stands for alocal average in the reference system, and β = 1/(kB T ).

In the first term on the right side of Equation 2, f (φi ) stands for the Helmholtzenergy density of a uniform system with composition (volume fraction) φi . The ver-tex function �i j (r) specifies correlations between different segments from the samechain and from different chains; it depends on both chain connectivity and inter- andintrachain correlations. If there is a difference between the total number of moleculesin the real system and that in the reference (i.e.,

∫dr�φ �= 0), Equation 2 must also

include a linear-order term affiliated with the excess chemical potentials of the seg-ments in the reference. If the density profiles of the reference are selected identicalto those of the real system and if φ0

i(r) = φi, 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, theHelmholtz energy functional truncated after the quadratic term becomes equivalentto that used in the Ramakrishnan-Yussouff theory of freezing (14). In a continuousmodel, we use the number density, instead of the volume fraction, as the independentvariable. Meanwhile, we express the vertex functions in terms of the density-densitycorrelation function χMi M ′

j(r) or the direct correlation function c Mi M ′

j(r):

�Mi M ′j(r) = χ−1

Mi M ′j(r) = δi j δMM ′δ(r)/ρMi − c Mi M ′

j(r), (3)

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RPA: random-phaseapproximation

where δi j and δ(r) designate the Kronecker and Dirac delta functions, respectively; su-perscript −1 denotes functional inversion, the subscript Mi refers to the i-th segmentof molecule M, and ρMi stands for the segment density.

Even within the framework of the Flory-Huggins theory, a number of differentways have been proposed to define the Helmholtz energy density f (φi) and the dif-ferential volume-fraction profile �φi. These methods differ in the definition of thereference system and in the selection of the segment volume fraction for f (φi) andfor �i j (r). For example, Leibler’s (15) pioneering work on the microphase segrega-tion of block-copolymer melts expands the Helmholtz energy functional relative tothat 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 insufficientto represent phase transitions, Leibler’s theory also includes higher-order terms inthe density expansion of the Helmholtz energy functional with the affiliated vertexfunctions calculated from a Gaussian-chain model. Leibler’s theory is most usefulunder weak-segregation conditions in which the variation of segment composition issmall. In an extension of Leibler’s theory to strong segregations, Ohta & Kawasaki(16) constructed a Helmholtz energy density that exhibits double minima in terms ofthe local volume fraction, whereas they derived the vertex functions from a uniformfluid at a macroscopically averaged density. Bohbot-Raviv & Wang (17) followed asimilar approach in studying the microscopic morphologies of triblock-copolymermelts. Whereas the double minima embedded in the local Helmholtz energy den-sity facilitate phase transitions, the inconsistent use of the reference density profile inf (φi) and �i j (r) is responsible for incorrect predictions concerning the effect of block-copolymer composition on the interfacial tension. Following a procedure similar tothat used by Ebner et al. (12) for inhomogeneous simple fluids, Uneyama & Doi’s(18) recently proposed DFT theory removes such inconsistency. The Uneyama-Doitheory is applicable to macroscopic phase separations of polymer blends as well as tomesoscopic phase separations in block-copolymer melts, at both strong- and weak-segregation regions. With some further approximations, we can reduce it to theFlory-Huggins-de Gennes-Lifshitz theory for polymeric systems at inhomogeneousconditions (19, 20).

According to Uneyama & Doi (18), Equation 2 is truncated after the quadraticterms, and the reduced Helmholtz energy per segment is given by the Flory-Hugginstheory (i.e., φ0

i(r) = φi (r)):

f =∑

i

φi(r)Ni

ln φi(r) +∑

i

∑j

χi j φi(r)φ j (r), (4)

where Ni denotes the degree of polymerization for i-type segments, and χi j is the Floryenergy parameter. The locally averaged segment volume fraction, φi = √

φi(r)φi(r ′),is used to evaluate the second-order vertex function in Equation 2. The latter isrepresented by a random-phase approximation (RPA):

�i j (r) ≈ �0i j (r) + χi j , (5)

where �0i j (r) is the vertex function of a Gaussian chain that has a backbone structure

identical to that of the real polymer. �0i j (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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SCFT: self-consistent-fieldtheory

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

�0i j (q) = NP ω−1

i j (q)Ni Nj φP

, (6)

where NP is the degree of polymerization for the entire polymer under consideration,φP is the polymer volume fraction, and ω−1

i j is defined by the inverse matrix {ωi j (q)}−1.In Equation 6, the single-chain density correlation function is defined as

ωi j (r) = 1Ni Nj

Ni∑α=1

Nj∑β=1

⟨δ[r − (rα − rβ )]

⟩, (7)

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

ωi j (q ) ={

2(e−xi − 1 + xi)/x2i i = j

(e−xi − 1)(e−x j − 1)e−xi j /(xi x j ) i �= j, (8)

where xi ≡ q 2 R2i , 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 andj blocks. If Equation 8 is further expanded in a power series of q and truncated afterthe second-order term, the RPA approximation of the second-order vertex functionleads to a Helmholtz energy identical to that obtained from the square-gradientapproximation (21, 21a). Uneyama & Doi (18) derived a concise expression for �0

i j (q)based on its asymptotic behavior at q → 0 and q → ∞.

The density-expansion method represents a popular approach in practical DFTapplications for describing a wide variety of phenomena in polymeric systems. How-ever, 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 varyingdensity profiles. Because the vertex function obtained from RPA ignores both theintermolecular correlations and the nonideal part of the intrachain interactions, RPAis not useful for systems in which both inter- or intramolecular correlations are signif-icant. Interestingly, for polymeric systems, the parameters used in the Uneyama-Doitheory 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 functionand a nonlocal mean-field representation of the intersegment interactions. Whereasthe scope of SCFT is probably more generic, Uneyama & Doi (18) obtained similar re-sults concerning mesoscopic phase separations in symmetric block-copolymer melts.Because the Uneyama-Doi theory avoids numerical evaluation of the single-chainpartition function, it is computationally more efficient than SCFT. A comparison ofSCFT, 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 density-expansion method discussed above. Early applications of integral-equation theories

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RISM: referenceinteraction site model

PRISM: polymer referenceinteraction site model

Direct correlationfunctions: the nonidealpart of the vertex functions;they can be understood asthe effective potentials (innegative, dimensionlessform) owing to theintermolecular interactions

Density correlationfunctions: the secondfunctional derivatives of thegrand potential with respectto the one-body externalpotential that specifycorrelated densityfluctuations in space

are mostly limited to the liquid-state theories of simple fluids, in particular at uni-form conditions (23). The value of integral-equation theories for polymeric systemswas largely unrecognized until the publication of the reference interaction site model(RISM) by Chandler and coworkers (24), and its successful extension to polymeric sys-tems (PRISM) by Schweizer & Curro (25). Cummings & Stell (26) proposed a similartheoretical framework in the context of a site-site Ornstein-Zernike equation. As forsimple fluids, the polymer integral-equation theories are most useful for bulk systems.Nevertheless, the direct correlation functions derived from integral-equation theo-ries enable applications of density-expansion methods to various off-lattice modelsof inhomogeneous polymeric systems.

In RISM or PRISM, the density-density correlation function of a polymeric systemis partitioned into an ideal-gas part (i.e., free of intermolecular interactions) and aremainder:

χMi M ′j(r) = χ0

Mi M ′j(r) + χMi M ′

j(r). (9)

Because the ideal-gas component retains all intramolecular interactions of a realpolymer, the corresponding density-density correlation function is not identicalto that of a Gaussian chain; instead, it is related to a single-chain correlationfunction:

χ0Mi M ′

j(r) = ρMiδMM ′ωMi j (r). (10)

In Equation 10, ρMi represents the number density of the i-th segment from moleculeM, and the single-chain correlation function, defined as

ωMi j (r) = ⟨δ[r − (

rMi − rMj

)]⟩, (11)

can be evaluated by using molecular simulation for a single polymer chain. Thenonideal part of the density-density correlation function can be used to define thetotal correlation function hMi M ′

j(r):

χMi M ′j(r) = ρMiρM ′

jhMi M ′

j(r). (12)

We can apply a similar partitioning of the intra- and intermolecular contributions tothe direct correlation function:

c Mi M ′j(r) = c 0

Mi M ′j(r) + c Mi M ′

j(r). (13)

Following Equation 3, we obtain a relationship between the direct and density-densitycorrelation functions, in a matrix notation, given by

ˆχ (r)−1 = ρ−1δ(r) − ˆc (r), (14)

where the elements of the density matrix ρ are defined as ρMiδi j δMM ′ . The same rela-tion also holds between the ideal-gas part of the direct and density-density correlationfunctions:

ˆχ0(r)−1 = ρ−1δ(r) − ˆc 0(r). (15)

A comparison of Equations 14 and 15 yields

ˆχ (r)−1 = χ0(r)−1 − c (r). (16)

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Total correlationfunctions: the normalizednonideal part of thedensity-density correlationsthat provides a measure ofthe nonrandom spatialdistributions of themolecular densities

HNC: hypernetted-chainapproximation

After substituting χ0(r) and ˆχ (r) into Equation 16 with the intramolecular and totalcorrelation functions defined in Equations 10 and 12, we find a final connectionbetween the total and direct correlation functions, which, in Fourier space, is givenby

h(q ) = ω(q )c (q )ω(q ) + ρω(q )c (q )h(q ). (17)

Equation 17 represents one of the key equations in RISM/PRISM. For an atomicfluid, ω(q) = I, Equation 17 reduces to the familiar Ornstein-Zernike equation.

As in the Ornstein-Zernike equation for simple fluids, applications of theRISM/PRISM equations require a closure (i.e., an additional relation between thetotal and direct correlation functions). In general, we can derive the closure usinga density expansion of the Helmholtz energy functional as discussed in the previ-ous section, or by heuristic means. The best known closure derived from the formerapproach is probably the hypernetted-chain approximation (HNC):

ln[gMi M ′j(r)] = −βuMi M ′

j(r) + ρMi

∑M ′′

k

∫dr ′ c i j (r − r ′)hMi M ′

j(r ′), (18)

where the radial distribution function is given by gMi M ′j(r) = hMi M ′

j(r) + 1. Because

HNC is affiliated with a quadratic approximation for the Helmholtz energy, it is notdirectly applicable to phase transitions. Conversely, the Percus-Yevick approximationprovides a familiar example of the heuristic approach, which, in a hard-sphere modelfor interacting segments, is given by⎧⎨

⎩gMi M ′

j(r) = 0 r < σMi M ′

j

c Mi M ′j(r) ∼= 0 r > σMi M ′

j

, (19)

where σMi M ′j

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 tothe Helmholtz energy functional of an inhomogeneous polymeric fluid and to thatof an ideal-gas reference with the same density profiles. In both expansions, the ref-erence 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 beyondthe quadratic are ignored. The difference in the Helmholtz energy between the realsystem and the ideal-gas reference, both at inhomogeneous conditions, is given by

β�F = β�F0 − 12

∑Mi

∑M ′

j

∫ ∫drdr ′c Mi M ′

j(r − r ′)�ρMi�ρM ′

j, (20)

where �F0 stands for the difference in the Helmholtz energy in the uniform condi-tions, and c Mi M ′

j(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 absentbecause the average molecule density of the uniform system matches that of the in-homogeneous system. The direct correlation functions in the CMS theory are takenfrom the RISM/PRISM theories for bulk fluids.

The CMS theory yields an equilibrium density profile of polymeric fluids equiv-alent to that of the ideal-gas polymers in an effective external field, which depends

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Cavity correlationfunction: the correlateddensity distribution ofcavities (i.e., particlesinteracting with the systembut not with themselves); itspecifies the probability ofsolvation

on the density profile in a self-consistent manner. As in RISM/PRISM theories forbulk polymeric systems, the ideal-gas reference retains all intramolecular interactionsof the real polymer. As a result, numerical implementation of the CMS theory re-quires a computationally demanding iteration procedure in which each step requiresa single-chain simulation. Hooper et al. (27) proposed a more efficient computa-tional procedure using umbrella-sampling and histogram-reweighing methods forthe single-chain simulations. Another way to avoid the single-chain simulation is totreat the ideal-gas polymers as Gaussian chains (28).

The CMS theory has been successfully applied to the study of the structure andthermodynamic properties of homopolymers and block copolymers at inhomoge-neous conditions (29–31). The theoretical predictions are often in good agreementwith simulation results. Although both CMS and Uneyama-Doi theories are basedon quadratic expansions of the Helmholtz energy, the correlation functions derivedfrom RISM/PRISM theories are, in general, more accurate than those from RPA inthe Flory-Huggins theory framework. By using realistic bond and intermolecular po-tentials, RISM/PRISM theories can reproduce the density-density correlation func-tions of polymeric fluids quantitatively (32). However, as in the density-expansionmethod discussed above, truncation of the Helmholtz energy after the quadraticterms provides a good approximation only when a small differential density profilecan be justified. Furthermore, a quadratic approximation such as HNC is insufficientto capture wetting and other phase transitions that entail multiple minima in theHelmholtz energy. On the contrary, because the quadratic expansion is applied rela-tive to a local Helmholtz energy density, the Uneyama-Doi theory is immune to suchlimitations.

Thermodynamic Perturbation Theory

As first suggested by Chandler & Pratt (33), a polymeric fluid may be considered asa system of monomeric particles linked together by a complete chemical reaction.By following the standard cluster-diagrammatic procedures, these authors developedan exact theory for describing the chemical equilibrium and intramolecular struc-tures of nonrigid molecules in condensed phases. In particular, they identified anexact relation between the equilibrium constant for the polymerization, KM , and themultibody-cavity correlation function of the polymerizing segments in the mediumy(r1, r2, . . . , rmM ):

KM ≡ ρM(r1, r2, . . . , rmM

)ρM1 (r1)ρM2 (r2) . . . ρMm

(rmM

) = y(r1, r2, . . . , rmM

)K 0

M, (21)

where ρM and ρMi refer to the polymer and monomer densities, respectively, andsuperscript 0 stands for the ideal-gas limit (i.e., polymerization of the same segmentsin the ideal gas).

The thermodynamic cycle proposed by Zhou & Stell (34) best illustrates the con-nection between chemical association and chain connectivity in a polymeric system.As shown in Figure 1, we can decompose the excess Helmholtz energy of a polymericsystem (uniform or inhomogeneous) into three contributions: (a) the free energy of

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ΔFII = FRex

Fex

ΔFIIIΔFI

Ideal gas Dense fluid

b

ac

Figure 1The excess Helmholtzenergy of a polymeric fluidcan be calculated from athree-step thermodynamiccycle: (a) dissociation intomonomers at the ideal-gasstate, (b) solvation of theindividual particles into adense monatomic fluid, and(c) polymerization of theparticles in the dense fluid.

dissociation at the ideal-gas state, (b) the excess Helmholtz energy of a monomericfluid 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 ofpolymerization in the ideal gas includes contributions from the bond formationsand from nonbonded interactions among the polymerizing segments. Whereas theproperties of a single chain solely determine the bond energies and the intramolecularinteractions, polymerization in the dense fluid depends also on the potential of meanforce to bring the polymerizing segments together (35). The additional contributionto the free energy of polymerization arises from solvation (i.e., the interactions ofthe polymerizing segments with the surrounding medium). The difference betweenthe free energy of association in the ideal gas and that in the dense fluid is thereforerelated to the multibody-cavity correlation function:

�FI + �FIII = −kB T ln y(r1, r2, . . . , rmM ). (22)

Apparently, Equations 21 and 22 are equivalent. The thermodynamic cycle suggeststhat polymerization in the dense medium differs from that in the ideal gas in termsof the work to bring the segments together. We can account for this difference bysubtracting the direct intramolecular energy from the potential of mean force. Acombination of these two contributions leads to the cavity correlation function shownin Equation 22.

Based on the thermodynamic cycle discussed above, we can express the Helmholtzenergy functional of a polymeric system in terms of an ideal-gas term free of non-bonded interactions, an excess term identical to that of a monomeric fluid with thesame segment density distributions, and a cavity correlation contribution accountingfor the effect of the medium on the polymerization:

βF = βF id[ρM (R)] + βF ex[ρMi (r)] −∑

M

∫dR ρM (R) ln yM (R). (23)

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

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TPT: thermodynamicperturbation theory

Wertheim’s (36) thermodynamic perturbation theory (TPT) provides a systematicprocedure to approximate the intersegment correlations owing to chain connectivity.In its lowest-order approximation (TPT1), the multibody-cavity correlation func-tion is represented by a superposition of the two-body correlation functions in themonomeric fluid:

ln yM (R) ≈mM−1∑

i=1

ln yMi Mi+1 (ri, ri + 1). (24)

Although sophisticated procedures have been proposed for the calculation of the two-body-cavity correlation functions at inhomogeneous conditions (37, 38), semiempir-ical expressions obtained from heuristic modifications of the corresponding expres-sions for bulk systems are often most convenient for practical applications (39–41).

In comparison with alternative methods, TPT has a great virtue in the decouplingof the nonbonded intermolecular interactions and polymer-chain connectivity. Theformer defines all short-range correlations that can be effectively treated by variousdensity-functional methods developed for monatomic systems (42). With a reliablemodel for the excess Helmholtz energy of monomeric fluids, TPT provides a satis-factory description of the structure and thermodynamic properties for a wide varietyof polymeric systems, including block and branched copolymers and polyelectrolytes(39–47). Although the correlations owing to chain connectivity are captured only atthe level of two-body correlation functions, the long-range intrachain correlations areat least partially preserved in the ideal-chain contribution. Indeed, in TPT the ideal-chain contribution is equivalent to that included in the density-gradient methods orin a polymer SCFT. For concentrated polymeric systems, the local segment densitiesprimarily determine the thermodynamic properties, and, therefore, the Helmholtzenergy functional is relatively insensitive to the approximations of the multibody-cavity correlation function. Conversely, in a dilute polymer solution, the polymerstructure is mainly determined by the intramolecular interactions that depend onboth bond connectivity and on intramolecular nonbonded interactions.

Whereas TPT accounts for the polymer backbone structure exactly, it predictsincorrect long-range polymer-polymer correlations. As a result, similar to a typicalmean-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 poly-merization. Nevertheless, even in the dilute limit, TPT accounts for the local-densityand short-range correlations owing to the nonbonded intersegment interactions. An-other notable advantage of TPT is that for bulk fluids, it reduces to a generic equationof 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 energyfunctional of an inhomogeneous system is identical to that of a uniform system, exceptthe bulk density is replaced by a locally averaged or weighted density. As extensivelyused for simple fluids, the weighted-density approximation provides a simple yeteffective 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 approx-imation to polymeric systems. In this theory, the Helmholtz energy of a polymericsystem is expressed as a functional of the segment density ρMi (r), which can be de-composed into a monomeric ideal-gas term and an excess:

F = F id[ρMi (r)

]+ F ex

[ρMi (r)

]. (25)

Equation 25 is formally exact even though it yields a self-consistent equation forthe equilibrium density profile similar to that from a mean-field theory (49). Themonomeric ideal-gas system is defined such that it reproduces the exact Helmholtzenergy of ideal chains as given in Equation 1. With the weighted-density approxima-tion, the excess Helmholtz energy is expressed as

F ex[ρMi (r)] =∑Mi

∫drρMi (r) fMi (ρMi ), (26)

where fMi is the excess Helmholtz energy per particle in the corresponding uniformsystem, and the summation applies to all different monomeric species.

Intuitively, one may imagine that the purpose of a weighted-density ρMi is tosmooth out the sharp variation of the local density to justify the application of alocal free energy. In Woodward’s (49) original work, the weighted density is simplycalculated from the average density within the volume occupied by a spherical particleof diameter σ as proposed by Nordholm et al. (50) for hard-sphere fluids:

ρMi (r) = 34πσ 3

∫dr ′ρMi (r

′) (σ − |r − r ′|), (27)

where (r) is the Heaviside step function. With an equation of state for hard spheresand one adjustable parameter to rescale the excess Helmholtz energy, Woodward (49)predicted, in good agreement with simulations, the density profiles of hard-spherechains confined in slit-like pores. As in applications of the weighted-density methodsto simple fluids, more sophisticated expressions of the weight function can be devisedon 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 severalimprovements to the weighted-density approach for polymers. The later versions usean 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 hard-sphere-chain model that includes van der Waals attraction at the mean-field level oradopts 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 polymersof different architectures. The numerical performance of a weighted-density methoddepends on inputs from the bulk equation of state and on the formulation of theweight 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 virialexpansion truncated after the second-order term:

βF =∑

i

∫drdω ρi(r, ω){ln[4πρi(r, ω)] − 1}

− 12

∑i, j

∫drdω dr ′dω ′ρi(r, ω)ρi(r ′, ω ′)Mi j (r − r ′, ω, ω ′), (28)

where Mi j (r, ω) = e−βui j (r,ω) − 1 is the Mayer function, and ui j (r, ω) is the pairpotential between the rigid molecules. Onsager (56) first proposed Equation 28 formodeling the isotropic-nematic transition of lyotropic liquid crystals. By includingthe third-order terms, Harnau & Dietrich (57) recently demonstrated that the virial-expansion method provides an efficient means for studying the complicated phasebehavior of inhomogeneous platelet and rod fluids.

Despite its simplicity, the virial-expansion method retains the coupling of the ori-entational and translational degrees of freedom. Such coupling is indispensable formodeling the properties and phase behavior of anisotropic fluids (58). However, thesecond-virial equation is generally insufficient to capture various phase transitions inconcentrated liquid crystals. To account for multibody interactions, Parsons (59) andLee (60) used an effective hard-sphere model to rescale the second-order virial ex-pansion of the Helmholtz energy functional. The Parsons-Lee method is remarkablyaccurate for a wide variety of lyotropic liquid crystals (61– 64). Unlike Onsager’s the-ory, the rescaled Helmholtz energy functional can be used to describe sophisticatedphase transitions of lyotropic liquid crystals, including those with both position andorientational orderings. With the athermal systems as a reference, intermolecularattractions 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 for-mulate the Helmholtz energy functional of anisotropic fluids (65). With an isotropicphase of average density ρ0 as the reference, we can express the Helmholtz energy ofan ordered state in terms of a functional Taylor expansion with respect to a differentialdensity �ρ(r, ω) = ρ(r, ω) − ρ0:

F = Fiso + μex0

∫drdω�ρ(r, ω)

− kB T2

∫drdr ′dωdω ′c 0(r − r ′, ω, ω ′)�ρ(r, ω)�ρ(r ′, ω ′). (29)

We can calculate the direct correlation function in Equation 29 from an integral-equation theory for isotropic fluids (23). If both the density profile and the direct cor-relation function are expressed in terms of the lowest-order spherical harmonics andwith some further approximations, Equation 29 reduces to the familiar Maier-Saupe-McMillan theory of liquid crystals (66). Unlike the Maier-Saupe-McMillan theory,however, the density-expansion method accounts for both attraction and short-rangesteric effects explicitly. Although the quadratic theory is not able to capture multiple

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Liouville equation: theequation of continuity forthe evolution of thephase-space distributionfunction in systems with aconservative Hamiltonian

TDDFT: time-dependentdensity-functional theory

minima in the free energy, a direct comparison of the free energies of different phasesenables 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 introducedthree different implementation schemes that are potentially useful for practical appli-cations. Although TDDFT is promising in providing a unified theoretical foundationfor a wide range of phenomenological equations of time-dependent processes, a mi-croscopically based dynamic theory has yet to be developed for practical applications.At present, conventional phenomenological equations remain most useful in describ-ing various time-dependent phenomena in complex fluids. All these models requirea thermodynamically-defined Helmholtz energy obtained from equilibrium DFT.

Generalized Diffusion Equation

The generalized diffusion equation is most useful for describing the relaxation ofBrownian particles in a medium. It applies to situations where the density profile ofthe particles ρ(r, t) varies slowly in time, such that the affiliated momentum distri-bution reaches a steady state and the thermodynamic properties of the system can berepresented by a local equilibrium.

Following the Langevin equation for the Brownian motion of particles, we maywrite the generalized diffusion equation as (67, 68)

∂ρ(r, t)∂t

= D∇ · {ρ(r, t)∇ [δF/δρ + ϕ(r, t)] + ζ(r, t)} , (30)

where D represents the single-particle diffusion coefficient, which is related to the par-ticle mobility coefficient �0 according to the Einstein equation D/�0 = kB T; ϕ(r, t) isa one-body external potential; and ζ(r, t) is a random noise arising from fast collisionsof the solvent molecules. The random-noise term satisfies the fluctuation-dissipationtheorem ⟨

ζi(r, t)ζ j (r ′, t ′)⟩ = 2ρ(r, t)δ(r − r ′)δ(t − t ′)δi j /Dβ2, (31)

where the subscripts i, j denote three-dimensional coordinates. Unlike the conven-tional 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 sim-ilar expression for polymeric systems on the basis of segment density profiles (69, 70):

∂ρMi (r, t)∂t

=∑M ′

j

∇ ·∫

dr ′{�Mi,M ′j(r, r ′)∇[μM ′

j(r ′, t) + ϕM ′

j(r ′, t)]} + ζMi (r, t), (32)

where �Mi,M ′j(r, r ′) is the Onsager kinetic coefficient, which depends on the con-

formation distribution of polymer chains. Because of the topological constraints of

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Memory effects: thecorrelation of randomforces in Langevin dynamics

polymer segments or the entanglement effects, the dynamics of a polymeric systemin general is much more complicated than that of spherical particles (71, 72). If shortchains or the short-time dynamics is of concern, the kinetic coupling of polymersegments becomes less important. In that case, we can represent �Mi,M ′

j(r, r ′) by a

local-coupling approximation, and the diffusion equation becomes (73)

∂ρMi (r, t)∂t

= βDMi∇ρMi (r, t)[μMi (r, t) + ϕMi (r, t)

] + ζMi (r, t), (33)

where DMi is the segmental diffusion coefficient. We can view Equation 33 as ageneralization of Equation 30 for multicomponent systems. However, because thesegment density profiles are thermodynamically coupled through the free-energydensity functional, diffusions of polymer segments are not independent, even withinthe local-coupling approximation.

A number of researchers have indicated that the random-force term in Equation 30is unnecessary if the Helmholtz energy is expressed in terms of the noise-averagedone-particle density (74, 75). If we use an instantaneous density in Equation 30,conversely, the Helmholtz energy functional is different from that corresponding tothe equilibrium DFT (68, 76). The two free-energy functionals agree at the long-wavelength limit of density fluctuations but differ in the short-scale fluctuation limit(77). Without the random-force term, the generalized diffusion equation becomesdeterministic in describing the time evolution of the density profile driven by thegradient of the chemical and external potentials:

∂ρ(r, t)∂t

= D∇2ρ(r, t) + D∇ρ(r, t) · ∇ [δβF ex/δρ + βϕ(r, t)] . (34)

If the Helmholtz energy is formulated by a square-gradient approximation and thereis no external field, Equation 34 reduces to the Cahn-Hilliard theory for spinodaldecompositions. In its applications to solvation dynamics, the generalized diffusionequation is also known as the Smoluchowski-Vlasov theory (78–80).

The generalized diffusion equation predicts relaxation dynamics in excellentagreement with results from Brownian-dynamics simulations for a number of col-loidal systems (81, 82). It can capture the dynamic responses of soft Brownian par-ticles to various time-dependent external potentials (e.g., optical trap) quantitatively(83, 84). Augmented with a streaming term for the transport of mass, the generalizeddiffusion equation has also been successfully used to predict the shear viscosity, fric-tion constant, and self-diffusion constant from the stationary density profile arounda particle in a shear flow (85, 86).

The generalized diffusion equation is useful for the relaxation dynamics of inter-acting Brownian particles in which the momentum distribution plays no significantrole. Because the evolution of the time-dependent density profile leads to a mono-tonic decrease of the Helmholtz energy in time, the generalized diffusion equationis not able to capture various kinetic processes with a free-energy barrier, nor is themethod useful for systems at unstable conditions in which the functional derivatives ofthe Helmholtz energy may diverge. Besides, the steady-state assumption in the gen-eralized diffusion equation ignores the memory effect (i.e., time correlations of themoment 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 be-tween the theory and simulation deteriorates for supercooled fluids (85).

Hydrodynamic Effects

In concentrated colloidal dispersions or polymer solutions, the decoupling of stressand diffusion dynamics as assumed in the diffusion equation becomes invalid. At thoseconditions, we must consider the time evolution of the colloidal or polymer-densityprofiles along with the change of the particle velocity. For a monodisperse colloidaldispersion, we can specify the variation of the density profile by the equation ofcontinuity:

∂ρ(r, t)∂t

= −∇ · ρ(r, t) v (r, t). (35)

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

M∂ρv∂t

+ M∇ · (ρv · v) = −ρ∇ δFδρ

− �−10 ρv + ζ. (36)

In Equation 36, M denotes the particle mass; the density ρ, velocity v, and randomforce ζ all depend on position and time. We can apply a similar expression to polymericsystems. In that case, we must replace the friction term (the second term on the rightside of Equation 36) with a stress proportional to the gradient of the velocity fieldmultiplied by the shear viscosity (88).

Hydrodynamic interactions are not included in Equation 36. To account for suchan effect, we may replace �0, the mobility coefficient in a free solution, with theshort-term mobility coefficient (89). In this case, the free-solution mobility coefficientappearing in the fluctuation-dissipation equation (Equation 31) must also be replacedby the short-term mobility coefficient. This simple rescaling method is most usefulwhen hydrodynamic interactions affect only the short-term behavior of the colloidalparticles (i.e., in the timescale shorter than that for one colloidal particle to diffusethe mean separation with its neighbors but longer than the relaxation time of theparticle velocity). Whereas the particle mobility in a free solution is independent of thecolloidal concentration, the short-term mobility coefficient may decline drasticallyas the colloidal concentration increases.

The generalized Langevin equation provides a useful starting point for construct-ing the mode-coupling theory for supercooled fluids (87, 88, 90). For hard-spherefluids near the glass transition, Equation 36 predicts the characteristic time increasingwith particle density, in good agreement with the empirical Volgel-Fulcher law (91).In addition, Equation 36 captures the stretched exponential decay and two-stage re-laxation of the density correlation functions. Nevertheless, the generalized Langevinequation is not applicable to situations in which the fluctuation-dissipation theoremfails 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 hetero-geneity, which is coupled with short-term local relaxation and with long-termphase-transition-like kinetics (92, 93). Whereas a generalized diffusion or Langevinequation can describe various downhill relaxation processes, it is, in general, not ap-plicable to free-energy-barrier-crossing events such as crystal nucleation or otherfirst-order phase transitions. However, as in a thermodynamic theory of nucleation(94), DFT provides an efficient means to characterize the free-energy landscape dur-ing glass-formation processes.

Unlike a typical crystallization process, the glass transition ends with thermody-namically metastable amorphous structures. Even in a simple system such as hardspheres, a large number of metastable disordered states have been identified thatminimize the local free energy (95–97). The free energies of these morphologicalstates are higher than that of the crystalline state but smaller than that of an equilib-rium liquid. Although a completely rigorous microscopic theory of glass transitionshas yet to be established, Wolynes and coworkers (98, 99) demonstrated that DFTprovides a useful theoretical framework for analyzing the freezing of a supercooledliquid into a large number of disordered structures. Based on the universality ofthe Lindemann criterion for solid stability, the random first-order transition theorydevised by Wolynes and coworkers (100, 101) provides a unified framework for de-scribing a wide variety of empirical correlations, including the Adam-Gibbs theoryfor 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 effec-tive Helmholtz energy functional to account for solvent-mediated intermolecularinteractions and correlation effects in complex fluids. These free-energy functionalsare useful for describing not only equilibrium properties, but also a wide variety ofdynamic phenomena.

From a fundamental point of view, one imperative task for future work is to de-velop a generic TDDFT for describing hierarchical dynamic phenomena in complexfluids. Such a theory will have broad impacts in materials science by providing amuch-needed theoretical foundation of molecular self-assembly. Significant progresshas been made in establishing a generic scheme from first principles and from thegeneralization of phenomenological equations. The recent developments allow us tocomprehend a wide range of time-dependent physical processes.

Another important task for future work is to incorporate ideas of therenormalization-group theory into the DFT framework. Long-range correlationshave been largely ignored in essentially all existing formulations of the Helmholtzenergy functional, but they are important not only for fluids near the critical pointbut also for applications to the kinetics of phase transitions that entail both stableand unstable regions of the phase diagram. Furthermore, we need a renormalization-group correction to account for the long-range inter- and intrachain bond-bond

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Table 1 Some representative applications of density-functional theory for complex fluids

Systems Equilibrium DynamicsColloids 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), surfacetension (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), phasediagram (132), thin films (133)

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

Liquid crystals Phase transitions (136), mixtures (137), columnar(138), capillary smectization (139), interfacialbehavior (140), polydispersity (141), carbonnanotube (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)

correlations in polymeric systems beyond a mean-field approximation (102, 103).Future research on phase transitions will benefit from a generic TDDFT to cap-ture multiple spatiotemporal scales affiliated with the solvent and with complicatedsolute-solvent effects, together with a good expression for the Helmholtz energyfunctional that accounts for long-range correlations, including those correspondingto the thermodynamically unstable regions of the phase diagram.

DFT provides a new paradigm for coherent modeling of multiscale equilibriumand dynamic phenomena within a unified theoretical framework. It promises a num-ber of important applications in materials science and biotechnology. A few promi-nent examples include predictions of the microscopic structures and thermodynamicstabilities of polymer composites, phase and interfacial behavior of semiflexible poly-mers, ion-specific interactions between biomacromolecules and their influence onconformational transitions in nucleic acids and proteins, and aggregation and trans-port behavior of biomacromolecules in a crowded cellular milieu. Table 1 summarizesrecent literature on practical applications of DFT to equilibrium and dynamic prop-erties of colloids, polymers, liquid crystals, and some selected biological systems. Wedirect the reader to the original papers (see Table 1) or a parallel review (42) fordetails. Future DFT applications hinge on the development of effective Helmholtzenergy functionals that account for the essentials of rich physical phenomena incomplex fluids and, more importantly, a user-friendly computational scheme for thenumerical implementation of DFT in multiple dimensions.

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SUMMARY POINTS

1. DFT opens a new avenue for describing multiscale equilibrium and time-dependent phenomena within a unified theoretical framework. The rigor-ous mathematical formalism is equally applicable to electronic, atomic, andcoarse-grained models.

2. DFT complements traditional phenomenological theories of statistical me-chanics and atomistic simulation methods, particularly for systems that entailmultiple spatiotemporal scales.

3. A number of generic strategies have been established for the formulationof a nonideal Helmholtz energy functional; they are interconnected, butnot all are equivalent. Practical applications often require a combinationof different strategies to capture the multiple aspects of complex physicalphenomena.

4. The generalized Langevin equation has been used successfully to describevarious relaxation dynamics and time-dependent responses of a density pro-file to an external field, but, in general, it is not suitable for supercooledliquids and glass transitions.

5. Effective strategies are yet to be devised to account for long-range correla-tions and for describing hierarchical dynamic processes in complex fluids.

ACKNOWLEDGMENTS

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

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119. Patel N, Egorov SA. 2005. Dispersing nanotubes with surfactants: a microscopicstatistical mechanical analysis. J. Am. Chem. Soc. 127:14124–25

120. Talanquer V, Oxtoby DW. 2000. A density-functional approach to nucleationin micellar solutions. J. Chem. Phys. 113:7013–21

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122. Cao DP, Wu JZ. 2006. Surface forces between telechelic brushes revisited: theorigin of a weak attraction. Langmuir 22:2712–18

123. Tscheliessnig R, Billes W, Fischer J, Sokoowski S, Pizio O. 2006. The role offluid 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 polymersolutions. 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

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Annual Review ofPhysical Chemistry

Volume 58, 2007Contents

FrontispieceC. Bradley Moore � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �xvi

A Spectroscopist’s View of Energy States, Energy Transfers, andChemical ReactionsC. Bradley Moore � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 1

Stochastic Simulation of Chemical KineticsDaniel T. Gillespie � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 35

Protein-Folding Dynamics: Overview of Molecular SimulationTechniquesHarold A. Scheraga, Mey Khalili, and Adam Liwo � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 57

Density-Functional Theory for Complex FluidsJianzhong Wu and Zhidong Li � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 85

Phosphorylation Energy Hypothesis: Open Chemical Systems andTheir Biological FunctionsHong Qian � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �113

Theoretical Studies of Photoinduced Electron Transfer inDye-Sensitized TiO2

Walter R. Duncan and Oleg V. Prezhdo � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �143

Nanoscale Fracture MechanicsSteven L. Mielke, Ted Belytschko, and George C. Schatz � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �185

Modeling Self-Assembly and Phase Behavior inComplex MixturesAnna C. Balazs � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �211

Theory of Structural Glasses and Supercooled LiquidsVassiliy Lubchenko and Peter G. Wolynes � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �235

Localized Surface Plasmon Resonance Spectroscopy and SensingKatherine A. Willets and Richard P. Van Duyne � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �267

Copper and the Prion Protein: Methods, Structures, Function,and DiseaseGlenn L. Millhauser � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �299

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Aging of Organic Aerosol: Bridging the Gap Between Laboratory andField StudiesYinon Rudich, Neil M. Donahue, and Thomas F. Mentel � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �321

Molecular Motion at Soft and Hard Interfaces: From PhospholipidBilayers to Polymers and LubricantsSung Chul Bae and Steve Granick � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �353

Molecular Architectonic on Metal SurfacesJohannes V. Barth � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �375

Highly Fluorescent Noble-Metal Quantum DotsJie Zheng, Philip R. Nicovich, and Robert M. Dickson � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �409

State-to-State Dynamics of Elementary Bimolecular ReactionsXueming Yang � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �433

Femtosecond Stimulated Raman SpectroscopyPhilipp Kukura, David W. McCamant, and Richard A. Mathies � � � � � � � � � � � � � � � � � � � � �461

Single-Molecule Probing of Adsorption and Diffusion on SilicaSurfacesMary J. Wirth and Michael A. Legg � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �489

Intermolecular Interactions in Biomolecular Systems Examined byMass SpectrometryThomas Wyttenbach and Michael T. Bowers � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �511

Measurement of Single-Molecule ConductanceFang Chen, Joshua Hihath, Zhifeng Huang, Xiulan Li, and N.J. Tao � � � � � � � � � � � � � � �535

Structure and Dynamics of Conjugated Polymers in Liquid CrystallineSolventsP.F. Barbara, W.-S. Chang, S. Link, G.D. Scholes, and Arun Yethiraj � � � � � � � � � � � � � � �565

Gas-Phase Spectroscopy of Biomolecular Building BlocksMattanjah S. de Vries and Pavel Hobza � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �585

Isomerization Through Conical IntersectionsBenjamin G. Levine and Todd J. Martínez � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �613

Spectral and Dynamical Properties of Multiexcitons in SemiconductorNanocrystalsVictor I. Klimov � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �635

Molecular Motors: A Theorist’s PerspectiveAnatoly B. Kolomeisky and Michael E. Fisher � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �675

Bending Mechanics and Molecular Organization in BiologicalMembranesJay T. Groves � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �697

Exciton Photophysics of Carbon NanotubesMildred S. Dresselhaus, Gene Dresselhaus, Riichiro Saito, and Ado Jorio � � � � � � � � � � � �719

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