Classical density functional theory methods in soft and hard matter.

Abstract

Herein we provide a brief summary of the background, events and results/outcome of the CECAM workshop 'Classical density functional theory methods in soft and hard matter held in Lausanne between October 21 and October 23 2009, which brought together two largely separately working communities, both of whom employ classical density functional techniques: the soft-matter community and the theoretical materials science community with interests in phase transformations and evolving microstructures in engineering materials. After outlining the motivation for the workshop, we first provide a brief overview of the articles submitted by the invited speakers for this special issue of Journal of Physics: Condensed Matter, followed by a collection of outstanding problems identified and discussed during the workshop. 1. Introduction Classical density functional theory (DFT) is a theoretical framework, which has been extensively employed in the past to study inhomogeneous complex fluids (CF) [1-4] and freezing transitions for simple fluids, amongst other things. Furthermore, classical DFT has been extended to include dynamics of the density field, thereby opening a new avenue to study phase transformation kinetics in colloidal systems via dynamical DFT (DDFT) [5]. While DDFT is highly accurate, the computations are numerically rather demanding, and cannot easily access the mesoscopic temporal and spatial scales where diffusional instabilities lead to complex solidification morphologies. Adaptation of more efficient numerical methods would extend the domain of DDFT towards this regime of particular interest to materials scientists. In recent years, DFT has re-emerged in the form of the so-called 'phase-field crystal' (PFC) method for solid-state systems [6, 7], and it has been successfully employed to study a broad variety of interesting materials phenomena in both atomic and colloidal systems, including elastic and plastic deformations, grain growth, thin film growth, solid-liquid interface properties, glassy dynamics, nucleation and growth, and diffusive phase transformations at the nano- and mesoscales [8-16]. The appealing feature of DDFT (as applied to solid-state systems) is that it automatically incorporates diffusive dynamics with atomic scale spatial resolution, and it naturally incorporates multiple components, elastic strains, dislocations, free surfaces, and multiple crystalline orientations; all of these features are critical in modeling the behavior of solid-state systems. Similarities between the problems of interest to the two communities and the complementary nature of the methods they apply suggest that a direct interaction between them should be highly beneficial for both parties. Here we summarize some of the discussions during a three-day CECAM workshop in Lausanne (21-23 October 2009) which was organized in order to bring together researchers from the complex fluids and materials science communities and to foster the exchange of ideas between these two communities. During the course of the workshop, several open problems relevant to both fields (DFT and PFC) were identified, including developing better microscopically-informed density functionals, incorporating stochastic fluctuations, and accounting for hydrodynamic interactions. The goal of this special issue is to highlight recent progress in DFT and PFC approaches, and discuss key outstanding problems for future work. The rest of this introductory paper is organized as follows. In section 2, we give a brief overview of the current research topics addressed in this special issue. Then, in section 3, we present a collection of outstanding problems, which have been identified as important for further developments of the two fields and intensely debated at the CECAM workshop. Finally, we close the paper with a few concluding remarks. 2. Research topics addressed in this special issue This special issue consists of research papers that cover a broad range of interesting subjects, about a half of which are related to the theoretical materials science community and the other half came from the soft-matter community. We begin by discussing papers related to PFC. Diverse subjects related to the phase-field crystal model include exciting topics such as predicting/controlling the equilibrium phase behavior [19, 18, 17] and kinetics of epitaxial island formation on nano-membranes [20]. Moreover, phase-field crystal modeling has proved to be very successful in simulating homogeneous and heterogeneous crystal nucleation and growth, and several aspects of these phenomena are discussed in this issue [18, 21]. Finally, it is shown how to incorporate additional orientational degrees of freedom within the PFC approach to model liquid crystals [22]. On the DFT side, the other papers in this special issue deal with problems associated with advanced DFT techniques and applications. The existence of a structural instability in sub-critical crystalline fluctuations in a supercooled liquid within a square-gradient theory is discussed in [23]. Fundamental measure theory for hard-body systems is improved by discussing a correction term in detail, as discussed in [24]. A mean-field-like density functional for charges is applied to the effective interaction between charged colloids obtained within a cell model [25]. The remaining articles provide fundamental insight into how to supplement DDFT-type methods with hydrodynamics [26, 27], highlight the role of the projection operator technique in deriving dynamical density functional theories [28], and demonstrate how perturbation methods can be employed to compute the properties of solid-liquid interfaces [29]. This particular collection of papers demonstrates rather convincingly the significant potential that classical density functional techniques possess in modeling complex systems built of either soft or hard matter (or combinations thereof). While the PFC approach offers a simple and appealing means to simulate evolving microstructures in spatially extended system with atomic scale spatial resolution over diffusive time scales, DFT provides both its theoretical underpinning and (hopefully) the means to construct microscopically more quantitative density functionals for use in engineering materials. Outstanding issues within the PFC and DFT approaches, discussed next, will provide further opportunities for interactions between the PFC and DFT communities. 3. Important open issues and exciting avenues for further research In the following we summarize some of the exciting topics for future research, which were discussed during the CECAM workshop. They concern both fundamental problems and applications, all within the framework of DFT and PFC. Addressing these issues will provide a framework for future work in these two overlapping fields. (a) How to construct a reliable density functional (DF) for soft repulsions? Most of the recent developments in classical density functional theory were focussed on hard-sphere-like interactions in the framework of fundamental-measure-theory (FMT) [30-33]. While this approach can be extended to additive and nonadditive mixtures [34, 35] and to non-spherical hard objects [36, 37], it is much more difficult to include soft-core interactions, such as inverse-power-law pair-potentials. There have been attempts to include those, mainly using the Ramakrishnan-Yussouff [38] or the weighted-density [39-41] approximation, or other modifications (see e.g., [42, 43]), but the accuracy of these functionals are inferior to that of FMT for hard spheres. Clearly the FMT of Rosenfeld needs an extension for the hard-core Coulomb system. A complementary approach is to start from a density functional for hard orientable objects [36] and to integrate out the orientational degrees of freedom. This would lead to a softened effective repulsion between spherical objects. We mention finally that in the extreme limit of ultrasoft pair potentials, which are penetrable, the mean-field approximation provides a reliable functional [44]. (b) How to construct a reliable DF beyond perturbation theory? This is the key to developing accurate, predictive functionals for use in materials science problems. Typically an attractive tail in the interparticle interaction is treated within thermodynamic hard-sphere perturbation theory [45, 46], in most cases at the mean-field level. As this perturbative approach is only justified for weak attraction strengths, there is a great need to go beyond this perturbation theory. A general non-perturbative route, which could be helpful here, is to consider a functional for a mixture and reducing it to an effective one-component system. Following this idea, for example effective depletion attractions can be modeled for a one-component system by starting from the binary Asakura-Oosawa functional [34, 35]. This idea still needs to be exploited in a more general sense, i.e. for more general cross-interactions in the mixture. It could also be combined with the idea of using non-spherical hard objects and integrating out the orientational degrees of freedom. (c) How to apply the fundamental measure theory to the full phase diagram of lyotropic liquid crystals? There are already density-functional investigations of liquid-crystal phases of hard spherocylinders [47, 48], but the novel fundamental-measure-theory which was recently proposed for non-spherical objects[36] has never been applied to this problem. In fact, this new functional now needs numerical evaluation for liquid-crystal phases different from isotropic and nematic ones, such as smectic, columnar, plastic crystalline and full orientational ordered crystalline phases [49, 50]. This is mainly a pure numerical resolution problem since the density fields are sharply peaked in the solid phases and need enough grid points, which is at the moment a rather formidable challenge in three spatial dimensions. (ABSTRACT TRUNCATED)

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Classical density functional theory methods in soft and hard matter
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IOP PUBLISHING
JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 22 (2010) 360301 (8pp)doi:10.1088/0953-8984/22/36/360301
PREFACE
Classical density functional theory
methods in soft and hard matter
Guest Editors
Mikko Haataja
Department of Mechanical
and Aerospace Engineering,
Institute for the Science and
Technology of Materials
(PRISM) and Program in
Applied and Computational
Mathematics (PACM),
Princeton University,
Princeton NJ 08544, USA
L´ aszl´ o Gr´ an´ asy
Research Institute for Solid
State Physics and Optics,
H-1525 Budapest, P.O. Box
49, Hungary and BCAST,
Brunel University, Uxbridge,
Middlesex, UB8 3PH, UK
Hartmut L¨ owen
Department of Theoretical
Physics,
Heinrich-Heine-Universit¨ at
D¨ usseldorf, D-40225
D¨ usseldorf, Germany
Herein we provide a brief summary of the background, events and
results/outcome of the CECAM workshop ‘Classical density functional theory
methods in soft and hard matter’ held in Lausanne between October 21 and
October 23 2009, which brought together two largely separately working
communities, both of whom employ classical density functional techniques: the
soft-matter community and the theoretical materials science community with
interests in phase transformations and evolving microstructures in engineering
materials. After outlining the motivation for the workshop, we first provide a brief
overview of the articles submitted by the invited speakers for this special issue of
Journal of Physics: Condensed Matter, followed by a collection of outstanding
problems identified and discussed during the workshop.
1. Introduction
Classical density functional theory (DFT) is a theoretical framework, which has
been extensively employed in the past to study inhomogeneous complex fluids
(CF) [1–4] and freezing transitions for simple fluids, amongst other things.
Furthermore, classical DFT has been extended to include dynamics of the density
field, thereby opening a new avenue to study phase transformation kinetics in
colloidal systems via dynamical DFT (DDFT) [5]. While DDFT is highly
accurate, the computations are numerically rather demanding, and cannot easily
access the mesoscopic temporal and spatial scales where diffusional instabilities
lead to complex solidification morphologies. Adaptation of more efficient
numerical methods would extend the domain of DDFT towards this regime of
particular interest to materials scientists.
In recent years, DFT has re-emerged in the form of the so-called ‘phase-field
crystal’ (PFC) method for solid-state systems [6,7], and it has been successfully
employed to study a broad variety of interesting materials phenomena in both
atomic and colloidal systems, including elastic and plastic deformations, grain
growth, thin film growth, solid–liquid interface properties, glassy dynamics,
nucleation and growth, and diffusive phase transformations at the nano- and
mesoscales [8–16]. The appealing feature of DDFT (as applied to solid-state
systems) is that it automatically incorporates diffusive dynamics with atomic
scale spatial resolution, and it naturally incorporates multiple components, elastic
strains, dislocations, free surfaces, and multiple crystalline orientations; all of
these features are critical in modeling the behavior of solid-state systems.
Similarities between the problems of interest to the two communities and the
complementary nature of the methods they apply suggest that a direct interaction
between them should be highly beneficial for both parties. Here we summarize
some of the discussions during a three-day CECAM workshop in Lausanne
(21–23 October 2009) which was organized in order to bring together researchers
from the complex fluids and materials science communities and to foster the
exchange of ideas between these two communities. During the course of the
0953-8984/10/360301+08$30.00
1
© 2010 IOP Publishing Ltd Printed in the UK & the USA

Page 3

J. Phys.: Condens. Matter 22 (2010) 360301Preface
workshop, several open problems relevant to both fields (DFT and PFC) were
identified, including developing better microscopically-informed density
functionals, incorporating stochastic fluctuations, and accounting for
hydrodynamic interactions. The goal of this special issue is to highlight recent
progress in DFT and PFC approaches, and discuss key outstanding problems for
future work.
The rest of this introductory paper is organized as follows. In section 2, we
give a brief overview of the current research topics addressed in this special issue.
Then, in section 3, we present a collection of outstanding problems, which have
been identified as important for further developments of the two fields and
intensely debated at the CECAM workshop. Finally, we close the paper with a
few concluding remarks.
2. Research topics addressed in this special issue
This special issue consists of research papers that cover a broad range of
interesting subjects, about a half of which are related to the theoretical materials
science community and the other half came from the soft-matter community. We
begin by discussing papers related to PFC.
Diverse subjects related to the phase-field crystal model include exciting
topics such as predicting/controlling the equilibrium phase behavior [19,18,17]
and kinetics of epitaxial island formation on nano-membranes [20]. Moreover,
phase-field crystal modeling has proved to be very successful in simulating
homogeneous and heterogeneous crystal nucleation and growth, and several
aspects of these phenomena are discussed in this issue [18,21]. Finally, it is
shown how to incorporate additional orientational degrees of freedom within the
PFC approach to model liquid crystals [22].
On the DFT side, the other papers in this special issue deal with problems
associated with advanced DFT techniques and applications. The existence of a
structural instability in sub-critical crystalline fluctuations in a supercooled liquid
within a square-gradient theory is discussed in [23]. Fundamental measure theory
for hard-body systems is improved by discussing a correction term in detail, as
discussed in [24]. A mean-field-like density functional for charges is applied to
the effective interaction between charged colloids obtained within a cell model
[25]. The remaining articles provide fundamental insight into how to supplement
DDFT-type methods with hydrodynamics [26,27], highlight the role of the
projection operator technique in deriving dynamical density functional theories
[28], and demonstrate how perturbation methods can be employed to compute the
properties of solid–liquid interfaces [29].
This particular collection of papers demonstrates rather convincingly the
significant potential that classical density functional techniques possess in
modeling complex systems built of either soft or hard matter (or combinations
thereof). While the PFC approach offers a simple and appealing means to
simulate evolving microstructures in spatially extended system with atomic scale
spatial resolution over diffusive time scales, DFT provides both its theoretical
underpinning and (hopefully) the means to construct microscopically more
quantitative density functionals for use in engineering materials. Outstanding
issues within the PFC and DFT approaches, discussed next, will provide further
opportunities for interactions between the PFC and DFT communities.
3. Important open issues and exciting avenues for further research
In the following we summarize some of the exciting topics for future research,
which were discussed during the CECAM workshop. They concern both
fundamental problems and applications, all within the framework of DFT and
2

Page 4

J. Phys.: Condens. Matter 22 (2010) 360301 Preface
PFC. Addressing these issues will provide a framework for future work in these
two overlapping fields.
(a) How to construct a reliable density functional (DF) for soft repulsions? Most
of the recent developments in classical density functional theory were
focussed on hard-sphere-like interactions in the framework of
fundamental-measure-theory (FMT) [30–33]. While this approach can be
extended to additive and nonadditive mixtures [34,35] and to non-spherical
hard objects [36,37], it is much more difficult to include soft-core
interactions, such as inverse-power-law pair-potentials. There have been
attempts to include those, mainly using the Ramakrishnan–Yussouff [38] or
the weighted-density [39–41] approximation, or other modifications (see e.g.,
[42,43]), but the accuracy of these functionals are inferior to that of FMT for
hard spheres. Clearly the FMT of Rosenfeld needs an extension for the
hard-core Coulomb system. A complementary approach is to start from a
density functional for hard orientable objects [36] and to integrate out the
orientational degrees of freedom. This would lead to a softened effective
repulsion between spherical objects. We mention finally that in the extreme
limit of ultrasoft pair potentials, which are penetrable, the mean-field
approximation provides a reliable functional [44].
(b) How to construct a reliable DF beyond perturbation theory? This is the key
to developing accurate, predictive functionals for use in materials science
problems. Typically an attractive tail in the interparticle interaction is treated
within thermodynamic hard-sphere perturbation theory [45,46], in most cases
at the mean-field level. As this perturbative approach is only justified for
weak attraction strengths, there is a great need to go beyond this perturbation
theory. A general non-perturbative route, which could be helpful here, is to
consider a functional for a mixture and reducing it to an effective
one-component system. Following this idea, for example effective depletion
attractions can be modeled for a one-component system by starting from the
binary Asakura–Oosawa functional [34,35]. This idea still needs to be
exploited in a more general sense, i.e. for more general cross-interactions in
the mixture. It could also be combined with the idea of using non-spherical
hard objects and integrating out the orientational degrees of freedom.
(c) How to apply the fundamental measure theory to the full phase diagram of
lyotropic liquid crystals? There are already density-functional investigations
of liquid–crystal phases of hard spherocylinders [47,48], but the novel
fundamental-measure-theory which was recently proposed for non-spherical
objects [36] has never been applied to this problem. In fact, this new
functional now needs numerical evaluation for liquid–crystal phases different
from isotropic and nematic ones, such as smectic, columnar, plastic
crystalline and full orientational ordered crystalline phases [49,50]. This is
mainly a pure numerical resolution problem since the density fields are
sharply peaked in the solid phases and need enough grid points, which is at
the moment a rather formidable challenge in three spatial dimensions.
However, if only orientational degrees of freedoms are considered, the
computational effort is greatly reduced; see, e.g., [36,51,52].
(d) The role of fluctuations in DDFT and PFC. There is a continuing debate about
the role of noise in the dynamical density functional theory (see e.g. [53]) and
correspondingly also in the phase-field crystal models. Derivations of DDFT
from the Smoluchowski level [54] and also within the projection operator
technique [5] quite naturally lead to a deterministic equation without any
noise. Clearly this is an approximation, which becomes problematic in the
3

Page 5

J. Phys.: Condens. Matter 22 (2010) 360301 Preface
vicinity of a critical point or in the case of nucleation problems, where the
system has to leave a metastable minimum of the free energy; in the former
case, fluctuations are required in order to capture the correct critical behavior
(i.e., critical exponents), while in the latter case, fluctuations are needed to
establish an escape route of the system from a metastable phase. Other
approaches add noise on a more phenomenological level. However, the actual
strength of the noise, though fundamentally correlated with the thermal
energy, is not known exactly and is treated in most applications as a
phenomenological fit parameter; see, e.g., [55,56]. This problem is a very
fundamental one, and, of course, shared by the DDFT and PFC approaches.
In more general terms, the addition of noise to the equation of motion in
continuum models is not without conceptual difficulties (see [57]), even if
noise is properly discretized in the course of the numerical integration. With
the noise added, the equilibrium physical properties of the system change.
Furthermore, transformation kinetics generally depend on the spatial and
temporal steps, and in the limit of infinitely small steps an ultraviolet
‘catastrophe’ (divergence of the free energy) may occur. Evidently, an
‘ultraviolet cut-off’, i.e. filtering out the highest frequencies, is required to
regularize the unphysical singularity. In the PFC case, a straightforward
choice for the cut-off length is the interparticle distance, which is expected to
remove the unphysical, small wavelength fluctuations [58,16,59,18].
Perhaps a more elegant way to handle this problem is via renormalizing the
model parameters so that with noise one recovers the ‘bare’ physical
properties (see the application of this approach for the Swift–Hohenberg
model in [60]). However, further systematic investigations are needed in
order to settle this issue.
(e) The need to clarify the role of the adiabatic approximation. While DDFT can
be derived from more microscopic equations, such as the Smoluchowski
equation [54] or the Langevin equations [61] for the individual particles, a
major approximation is invoked in the derivation, namely the so-called
‘adiabatic approximation’. This approximation assumes that all other
observables relax much faster than the one-particle density field [5].
Therefore, the nonequilibrium correlations are replaced by equilibrium ones
corresponding to an inhomogeneous reference one-particle density [54]. This
enables one to formulate the theory in terms of the time-dependent
one-particle density field alone. What is still needed here is a more general
theory which provides the next-leading order beyond the adiabatic
approximation. This improved theory would not only provide more
fundamental insight into the DDFT itself; it would also pave the way to many
applications where the simple DDFT fails.
(f) How to apply and exploit DDFT for active matter? The collective behavior of
self-propelled particles with internal driving motors is a topic of active
research [62,63]. Given that the particle dynamics can be described in terms
of driven Brownian motion, a dynamical density functional theory can be
derived in a straightforward manner. In a first application, DDFT was
employed to describe aggregation phenomena near system boundaries for
driven rod-like colloidal particles [64]. The potential of DDFT for ‘active’
particles should be exploited more in the future, as it provides a microscopic
approach to investigate nonequilibrium effects, such as swarming and
jamming.
(g) How to construct a PFC model for inhomogeneous liquid crystals? The
traditional PFC model [6,7] describes a two-dimensional one-component
solid phase by a single inhomogeneous sinusoidal density field. The PFC
approach has been generalized to mixtures by including more than a single
4