Chapter 15
Density-Functional Theory for Nonuniform Polyatomic Fluids
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E. Kierlik, S. Phan, and M. L. Rosinberg Laboratoire de Physique Théorique des Liquides, Unité de Recherche Associée 765, Université Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris Cedex 05, France
T h e structure of molecular and polymer fluids near surfaces and in t h i n films is a topic of great fundamental and practical interest which is still not well understood. We present a density functional theory which is a generalization to inhomogeneous polyatomic flu ids of Wertheim's thermodynamic perturbation theory for associ ating fluids. In the local density approximation, this theory takes a very simple form which can be used to study the structure and the thermodynamics of long chains at the free surface. A s an ap plication, we compute the variations of the surface tension w i t h temperature and chain length and we investigate the surface seg regation effects due to side branching, segment size, or isotopic substitution.
In the case of classical simple fluids, it is known that density functional the ory i n the van der Waals approximation where attractive forces are treated i n a mean-field fashion while repulsive forces are represented by an equivalent hard sphere interaction, captures the essential physics of interfacial phenom ena, such as adsorption or wetting at solid/fluid or liquid/vapor interfaces (1). T h e simplest Local Density A p p r o x i m a t i o n ( L D A ) where the free energy of the inhomogeneous hard sphere fluid is just the spatial integral of the local free energy density of the bulk fluid already gives the gross features of the structure and phase equilibria. Somewhat better results are obtained when using one of the so-called "weighted-density approximations" which have flourished i n the past few years (1). In some instances, one can get quantitative predictions for experimental quantities like the surface tension, although the results seem to be rather sensitive to the choice of the interatomic potential or to the m i x i n g rules i n the case of mixtures. 0097-6156/96/0629-0212$15.00/0 © 1996 American Chemical Society In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
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15.
KIERLIK ET AL.
DFT & Nonuniform Polyatomic Fluids
213
T h e question that we want to investigate i n this paper is the following: can we build the same type of theory and reach the same level of accuracy i n the case of inhomogeneous liquid hydrocarbons or long chain polymeric fluids ? Let us first briefly summarize the present theoretical situation. Leaving apart the scaling approaches which only focus on the universal large scale behavior (£), we have on one hand self-consistent field ( S C F ) type of calculations based on coarse-grained lattice models, and on the other hand Landau-Ginzburg treat ments which use a C a h n - H i l l i a r d - d e Gennes free energy functional gradient expansion i n analogy with similar theories of simple liquids. T h e S C F meth ods (3) which are extensions of the F l o r y mean-field picture to inhomogeneous situations, may be quite successfull i n describing qualitatively the structure and the thermodynamics of the interfaces. B u t the use of a lattice model is a very crude representation of a real fluid, especially when packing or freevolume effects are important, as they are near a solid surface or at the free interface, respectively. In the L a n d a u - G i n z b u r g approaches (4,5), one usually assumes that the surface terms contain unknown phenomenological parame ters which are supposed to describe both entropie and enthalpic contributions. Moreover, the homogeneous part of the free energy is usually described by the simplest Flory-Huggins expression which cannot distinguish between different molecular architectures or orient at ional effects. Therefore, there is a need for a theory which could be used to interpret the experimental results i n terms of fundamental quantities such as interact ing parameters, rather than phenomenological parameters. B y "interacting" parameters, we mean parameters which have a physical meaning at the level of the structural units and are reasonably independent of temperature and concentration. W h a t are the ingredients that should contain this theory ? a) It should be a continuum (i.e. off-lattice) theory i n order to describe packing and compressibility effects (what polymerists call equation-of-state effects), b) it should be able to describe the influence of chain length and chemical architecture on the various interfacial properties, c) it should describe the conformational changes of the molecules near a sur face, d) and of course, it should contain energetic effects, at least i n a mean-field fashion, as done for simple fluids. Density functional theory seems to be a good candidate for that and sev eral theories have been proposed i n the last few years (6Ί0). T h e one that we shall discuss i n the following is called Perturbation Density Functional Theory ( P D F T ) (11-13). Indeed, we shall first explain how we can treat perturbatively the influence of connectivity i n the Helmholtz free energy by generalizing Wertheim's theory of chemical association (14) to inhogeneous fluids. It has been shown elsewhere (13,15,16) that for purely repulsive or athermal chains, this yields good predictions for the structure near a solid. Here we shall add attractive interactions i n a van der Waals fashion and use the local density approximation to study liquid/vapor interfaces. We shall see that the theory
In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
214
CHEMICAL APPLICATIONS OF DENSITY-FUNCTIONAL THEORY
is able to give the correct trends for the surface tension as a function of tem perature and chain length. F i n a l l y , we shall study the influence of segment size and side branching and consider the surface behavior of isotopic blends.
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Perturbation density functional theory If one wants to extend D F T to polyatomic molecules, the first thing to be noted is that there are different ways of expressing the m i n i m i z a t i o n principle because one can define several local densities. For instance, for a molecule composed of M monomers or units or interaction sites, one can first define the single molecule density />Μ(ΓΙ,Γ , . . , I * M ) which is the M - p o i n t joint probabil ity distribution function for finding atom 1 at Γχ, atom 2 at r , etc... B u t one can also define the individual site densities pi(r) = Jdridr ...drA/£(r — ΓΙ)/?Λ/(ΓΙΪΓ2Ϊ . . , Γ Μ ) , Ζ = Ι , . , . , Μ , the average site or monomer density p(r) = Σ ί ί ι Λ ( ) Ϊ the joint probability distribution function for 2 sites, 3 sites, etc...It is clear that these contracted distribution functions contain less information about the conformational structure than the single molecule density pM · 2
2
2
γ
It turns out that one can express the free energy either as a functional of the site densities pi(r) or as a functional of the single molecule density £ Λ / ( Γ Ι , Γ , . . , Γ Μ ) · T h e problem w i t h the first version which has been intro duced by Chandler, M c Coy, and Singer some years ago (6) is that even the computation of the ideal part of the free energy is a nontrivial problem. T h i s difficulty does not occur w i t h a free energy expressed as a functional of the single molecule density. T h e general formulation has been stated by P r a t t and Chandler i n 1976 (17,18) i n the framework of an interaction site cluster expansion. T h e y showed that one can write formally the intrinsic Helmholtz free energy of a homonuclear chain fluid as 2
- 1 + βιν {ΐΜ)]
fiF = J d\ p (l )[\np (\ ) M
M
M
M
M
Μ
+ β^ [ρ ], χ
Μ
(1)
where 1M denotes the positions r i , r , ..,ΓΛ/ of the monomers collectively and U>M(1M) is the intramolecular energy for an isolated molecule. T h e first t e r m is just the ideal part of the free energy and F [p\f] is the excess part, ex pressed i n terms of an infinite set of cluster diagrams built w i t h the contracted intramolecular distribution functions and the Mayer function associated w i t h the interaction potential between sites i n different molecules. O f course, this is a formal expression and one needs some rule to classify the diagrams i n a systematic and sensible way. Such an approximation scheme is provided to us by Wertheim's work on associating fluids. W h a t we shall use is essentially a generalization of Wertheim's thermodynamic perturbation theory ( T P T ) (14) to non-uniform fluids, i n the l i m i t of complete association where a l l molecules are fully formed. 2
ex
W h a t is the perturbation scheme ? Consider for instance a linear chain of M monomers. T h e m a i n idea consists i n building progressively the molecule from a reference hypothetical fluid i n which a l l monomers are totally dissociated at the same temperature and monomer density as the real system. For instance, i n
In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
15.
K I E R L I K ET A L
DFT & Nonuniform Polyatomic Fluids
215
the case of pearl-necklace hard-sphere chain, the reference fluid is just the hardsphere fluid at the same packing fraction. N o w the perturbation procedure consists i n computing the work which is required to build up chain segments of increasing length i n the reference system. M o r e precisely, one can order the infinite sum of graphs F as follows ex
F' [PM) X
= F%[p)
+ F?[PM}
+ .··.,
+ F?\PM)
(2)
where the first term is the excess Helmholtz free energy of the reference monomeric fluid, F{ represents the work to build a single chain i n the refer ence fluid, Fl + F\ the work to build two chains together, etc... Keeping only the first two terms corresponds to what is called the "single chain approxima t i o n " ( S C A ) : this means that the atoms on the chain see the rest of the fluid as an atomic system. Usually, one stops at this level. It can be shown that F{ has the following expression (13),
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x
x
x
x
βΡ?[ΡΜ]
= ~ J dl p (l ) M
M
M
ln(y
r e /
(l ; M ) ,
(3)
M
where y f is the M - p o i n t cavity function i n the reference system. T h e n it is clear that all phenomena involving interactions between two or more chains cannot be described. T h i s is the case for instance of the isotropic to nematic transition i n the case of semi-flexible molecules. If one wants to study this problem, one must work at least at the level of the two-chain approximation. However, even staying at the level of the S C A does not provide a tractable theory because the intramolecular potential WM still contains a l l the M - b o d y excluded volume effects. One then breaks the chain into smaller segments and considers the work needed to build these segments independently. A t the lowest order, the chain is just a succession of M — 1 dimers and this is WertheinVs T P T 1 theory. T h e n , the density functional has the following expression; re
fiFrPTilPM]
- 1+
=
j
