Chemical Applications of Density-Functional Theory - American

Chapter 16

Downloaded by UNIV MASSACHUSETTS AMHERST on October 9, 2012 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/bk-1996-0629.ch016

A Density-Functional Approach to Investigation of Solid-Fluid Interfacial Properties D. W. M . Marr and A. P. Gast 1

2

Chemical Engineering and Petroleum Refining Department, Colorado School of Mines, Golden, CO 80401-1887 Department of Chemical Engineering, Stanford University, Stanford, CA 94305-5025

1

2

We present a formulation of density-functional theory ideally suited for investigation of solid-fluid interfacial properties. W e use this ap proach to investigate the role of interactions i n determining both the energy and structure of the interface by examining a number of systems including hard-sphere, adhesive-sphere, and Lennard-Jones fluids. In addition, we study the orientational dependence of interfacial properties i n the adhesive-sphere system. Despite its ubiquity i n nature, remarkably little is known of the solid-fluid i n terfacial structural and energetic properties. T h e reason for this lies i n the difficulty i n experimentally assessing the interface; most materials of technolog ical importance (e.g. metals) are not transparent m a k i n g observation extremely difficult [1]. Efforts to experimentally study the solid-fluid interface often cen ter around removal of the solid phase from the e q u i l i b r i u m fluid but lead to a modified interface, frustrating efforts to examine the e q u i l i b r i u m structure. One can also employ a rapid temperature quench, some k i n d of sectioning, and then microscopy but this method is applicable only to multi-component sys tems. Methods of experimentally determining solid-fluid interfacial tensions can involve the measurement of grain boundary intersection angles or the study of grooves i n the crystal surface but are normally applicable only to a few systems. Results from computer simulation are also l i m i t e d . T h e complexity of the solid-fluid interface makes computational approaches extremely costly and has limited study to only a few systems, for which a nice review is available [2]. Because of the l i m i t e d amount of computational or experimental study, little is known of the influence of the interaction potential on the structure and energy of the interface. In general, we expect that the interface to become extremely sharp at low temperatures due to its low entropy. A s temperature increases, however, the interface w i l l widen and the transition from solid to fluid w i l l occur over a larger distance. T h e interface w i d t h w i l l be l i m i t e d , though, by the high 0097-6156/96/0629-0229$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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CHEMICAL APPLICATIONS OF DENSITY-FUNCTIONAL THEORY

energetic cost of producing regions having a density different from that of either coexisting e q u i l i b r i u m phase. A t a given temperature for a specific interaction these factors combine to give a m i n i m u m i n the free energy of the interface at a particular interfacial w i d t h . A number of questions immediately arise: How w i l l the interactions between particles influence the broadening of the interface? W i l l attractions have a strong influence or do repulsions alone determine i n terfacial structure? How do these influences change w i t h interface orientation? Answering these questions requires a technique that can readily determine i n terfacial thermodynamic properties as a function of interaction potential i n a computationally tractable manner. Theoretical Approach T h e thermodynamic properties of a variety of model fluid systems are well es tablished. In general, liquid-state theory allows one, w i t h knowledge of the interaction potential, to determine correlations between particles w i t h i n the ho mogeneous fluid phase. Combined w i t h a closure appropriate for the interaction potential of interest, one can obtain thermodynamic properties such as the pres sure and chemical potential of the homogeneous fluid. Determining the thermodynamic properties of the solid phase is much more difficult. One cannot apply the equations of liquid-state theory directly and the use of computer "experiments" such as Monte C a r l o or molecular-dynamics s i m ulations is both difficult and computationally demanding. Due to the success of liquid-state theory, there has been much effort i n describing the solid state w i t h liquid properties, leading to the development of density-functional theory. R e cently, density-functional theory has provided a means to describe the structure and energetics of the solid phase. In addition to facility w i t h the homogeneous solid phase v i a the principles of liquid-state theory, density-functional theory has allowed description of interfaces and other inhomogeneous systems. W h i l e there are a number of implementations of density-functional theory for studying phase transitions, all of them seek to describe the structure and proper ties of the solid phase from information about the fluid. Several excellent reviews of the different density-functional approaches have recently appeared [3-8]. B a sically, we can place the density-functional theories into two categories: i) T h e description of the solid phase through a truncated functional Taylor expansion of the η-particle direct correlation function [9,10] and ii) the description of the solid phase through appropriate choice of an effective liquid approximating its ther modynamics [11-13]. T h e latter approach, while somewhat ad hoc, is quite suc cessful i n the description of hard sphere solids [11,12,14-20]. Once the effective liquid density is chosen, the description of the solid becomes a matter of apply ing information available for the liquid state. There are a range of approaches for choice of the effective liquid from the early effective l i q u i d approximation ( E L A ) of Baus and Colot [12], where an ad hoc but physically appealing com parison of structure was invoked, to the computationally demanding weighted density approximation ( W D A ) of C u r t i n and Ashcroft [13]. M o r e recent criteria for choosing an effective liquid density bring the Baus and Colot effective liquid

In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.


16.

MARR&GAST

Solid-Fluid Interfacial Properties

231

approximation ( G E L A and S C E L A [11]) into accord w i t h the weighted-density approaches [8]. We focus on the approach of C u r t i n and Ashcroft [13] who de fine a weighting function which links the solid and l i q u i d states. T h i s approach, known as the weighted density approximation ( W D A ) , involves the determina tion of a spatially variant weighted density where fluid properties approximate those of the solid. T h i s method has been effective i n predicting solid properties and phase coexistence but its application to more complex problems has been hindered by the computational requirements i n the determination of weighted densities. To overcome these difficulties, Denton and Ashcroft have developed the modified weighted density approximation ( M W D A ) [15]. In contrast to the W D A , this approach requires only calculation of a spatially invariant weighted density and significantly lowers computation t i m e . In order to study the structural details and energetics of the solid-liquid inter face, one must determine the appropriate weighted density to model each density through the interface. T h i s can be done w i t h the W D A [16,17]; however, the calculation requires tremendous computational effort m a k i n g it i m p r a c t i c a l for complex situations. We are interested i n systems i n c l u d i n g finite interparticle i n teractions where the densities of coexisting phases w i l l depend on temperature. In this situation, the interfacial structure and energy must be determined for a variety of temperatures, significantly increasing the required amount of com putation and motivating the development of a tractable approach to describe the interface. Encouraged by the success of the M W D A i n decreasing the com putational requirements of the W D A , we developed a planar weighted density approximation ( P W D A ) to describe the interface [21]. One begins by separating the total H e l m h o l t z free energy of the solid phase into two components (1)

F\p] = F \p] + F M ia

K

representing the ideal and excess contributions to the t o t a l free energy. ideal t e r m can be calculated for any given density d i s t r i b u t i o n p(r) from FiM

= β~

ι

J < M r ) [ l n ( , ( Γ ) Λ ) - 1] 3

The

(2)

where β = 1/kT and Λ is the de Broglie wavelength, and the total excess free energy can be expressed as the sum of local contributions F [p) tx

= J άτρ(τ)φ(τ;[p])

(3)

where φ is the local excess free energy per particle. W e i g h t e d - D e n s i t y A p p r o x i m a t i o n . C u r t i n and Ashcroft [13] approximated the local excess solid free energy per particle w i t h the excess free energy per particle of a homogeneous fluid, indicated by a subscript 0, evaluated at some effective l i q u i d density F r e

A

M =

/drp(r)VoG>(r))


(4)

232


where the effective l i q u i d density is a weighted average of the local solid densities in the v i c i n i t y of r . In the W D A , the spatially variant effective l i q u i d density is defined by (5)

p(T) = JdT'p(T')w(T-v';p(r))

where the weighting function w is introduced w i t h the normalization requirement


J drw(r-r';p)

= 1

(6)

and is determined by requiring the exact 2-particle direct correlation function c to be recovered i n the homogeneous l i m i t , -β

^ ( r - r';p ) = l i m 0

e x

P->PO d/)(r)()/)(r')

.

(7)

Solving for w is most readily done i n Fourier space, leading to the following differential equation -β

c (k;p )

1

0

0

= 2^-w(k;p ) 0

+ p S~ 0

9φο

w

2

(k;p ) 0

(8)

M o d i f i e d W e i g h t e d - D e n s i t y A p p r o x i m a t i o n . T h e M W D A [15] differs from this approach i n that the excess Helmholtz free energy is calculated from a spatially invariant weighted density,

F-

WDA

W =A W )

(9)

where Ν is the number of particles and the weighted density determined from P=jfJ

drp(r) J dr'p(r')w(r - r ' ; p).

(10)

Using the normalization condition (equation 6) and imposing the exact fluid free energy i n the homogeneous l i m i t (equation 7) one obtains -β-'^ρο)

= 2Û;(*;A,) + < W o ^ f .

(H)

T h i s equation for w(k,p) is easier to solve than the W D A ; i t is proportional to the direct correlation function (for non-zero k) and does not involve the solution of a non-linear differential equation. T h e M W D A requires a great deal less com putation and is therefore the preferred approach for the study of the transition from l i q u i d to solid [4,7,8,19,22-24]. P l a n a r W e i g h t e d - D e n s i t y A p p r o x i m a t i o n . In systems such as the solidfluid interface where the bulk density varies w i t h position, one cannot apply the M W D A because of the need to retain a spatially-varying weighted density.



16.

M A R K & GAST


233

C u r t i n [17] has applied the W D A to the interfacial problem w i t h good results but after significant computational effort. One can, however, lower these require ments while still retaining the physical approach to the problem by incorporating the M W D A into the interface [21]. T h i s is done by realizing that the bulk den sity parallel to a planar interface remains constant. In the spirit of Denton and Ashcroft's reduction of the computational requirements for W D A models of bulk systems, one may approach the interface w i t h a planar-aver aged spatiallyvariant weighted density that w i l l significantly decrease calculation costs. Other authors have succeeded i n modelling systems such as a hard-sphere fluid next to a hard wall [25-28] w i t h a one-dimensional weighted density; however, efforts to describe the freezing transition w i t h such a weighted density have not led to a stable solid. One begins by expressing the excess free energy i n terms of the planaraveraged density p(z) and a planar-averaged free energy F [p] = j άτρ(ζ)φ(ζ)

(12)

p{z) = 1 J

(13)

ex

dxdyp{v).

T h i s l o c a l ' free energy is now approximated w i t h that of a homogeneous fluid evaluated at some planar weighted density p(z) i

F!r [p] A

(14)

= J άνρ(ζ)φ {p{ζ)), ΰ

determined self-consistently from / dxdypjr) P y

- r ' ; p(z))

/ d^pj^wjr Jdxdyp(r)

}

{

]

where once again the weighting function is determined from the normalization condition (equation 6) and the requirement on the l i m i t i n g behavior (equation 7). In Fourier space one obtains -β

Cv(k;p )

1

0

+

= 2^w(k;po)

S ôp ^ k

0

(16)

reducing to the W D A weighting function i n the k|| = 0 case and the M W D A weighting function when k|| is non-zero. T h i s approach incorporates many of the computational savings inherent i n the M W D A and yet is applicable to the determination of solid-fluid interfacial properties. B u l k P r o p e r t i e s . In order to solve these equations for the solid free energy one must first model the solid structure. A s proposed by Tarazona [20], the solid phase density distribution can be represented as the s u m of normalized Gaussians

*«=(j)

3/2

Ee- a(r

R)2


() 17



234

Figure 2. E x a m p l e solid-fluid interface (The peaks have been cut short to better illustrate the transition).


16.

M A R K & GAST


235

where R are the Bravais lattice vectors, or i n Fourier space as Ρ.(τ) = Ρ.+ ΣΡοέ * GÔ

( )

β

18

where G are the reciprocal lattice vectors and p = p e~ l . T h e parameter α describes the structure of the solid; the higher the value of a the more localized the structure and a value of zero corresponds to the homogeneous fluid. Figure 1 illustrates this solid-phase parameterization. To determine the solid-phase thermodynamic properties one minimizes the total free energy for a given solid density w i t h respect to a. A global m i n i m u m occurring at a non-zero α indicates a stable solid phase, determining both the stable α and excess free energy corresponding to a given solid density. T h e total free energy is found by adding the ideal and excess contributions. Phase coexistence occurs when the chemical potential μ and pressure Ρ of the solid and fluid phases are identical.


G

Ρ = ( -Ρ[p]/Ν)

G2 Aa

s

(20)

= p > ^ 1

p μ

I n t e r f a c i a l P r o p e r t i e s . C u r t i n [16,17] has developed a convenient two p a rameter model i n his application of the W D A to the solid-fluid interface. He represents the solid as the sum of Fourier components as before but now allows these components to decay as one makes the transition from solid to l i q u i d along the ζ direction across the interface: p(r) =

P l

+ (p. -

+ ΣPofa(z)e

)fo(z)

(21)

iGr

Pl

G where f (z)=\ G

[1 1(1+cos ( π ^ ) ) 10

\z\< z z ] + F£[p}. id

e

(33)

h

T h e ideal and hard sphere excess terms can be calculated as shown previously and the attractive functional approximated as a density dependent function w i t h equation 32 giving


F:?[p] ~ 2πΑβ-> J dzp(z)p(z) jT

(34)

dr'g (r'; p{z)-d)u^r')r . n

0

I n t e r f a c i a l P r o p e r t i e s . Using this approach we have computed the phase diagram, determining bulk properties, μ, Ρ , and the solid localization parameter, a , as a function of temperature. We then use the P W D A w i t h equations 33 and 34 to determine the interfacial weighted densities and their associated free energies. We summarize the resulting interfacial tensions i n Table III.

Table III: Calculated interfacial tensions for the Lennard-Jones solid-liquid (S-L) and solid-vapor (S-V) systems

T* 1.15 0.617 0.44 0.44 0.40 0.36

transition S-L S-L S-L S-V S-V S-V

interface width Δζ/δ 3-4 3-4 3-4 2 2 2 ιη

fluid density pa 0.992 0.962 0.950 1.7· 1 0 " 3.6 · 1 0 " 5.1 · Ι Ο " f

3

solid density Ρ*σ 1.104 1.063 1.046 1.046 1.060 1.073 3

5

6

7

interfacial tension >yo /kT 0.87 ± .02 0.82 ± .02 0.83 ± .02 2.38 ± .01 2.72 ± .01 3.15 ± .01 2

Ί/ΔΗρ]!

3

0.66 0.69 0.76 0.14 0.11 0.10

It is interesting to note the failure of the T u r n b u l l empiricism for this interac tion potential. In fact, as the triple point is approached from above, agreement becomes progressively worse, indicating the influence of strong long-range at tractions. Interfacial Orientation As discussed previously, the density-functional approach can be used to examine the various crystal structures and their solid-fluid interfaces. One issue not yet addressed is how and whether the various possible interfacial structures, for a given lattice, w i l l have different surface energies. How these structural differences w i l l impact the interfacial energetics is a question now accessible using densityfunctional theory.




242

There has been relatively little investigation of the orientational dependence of interfacial properties i n solid-fluid model systems because of the tremendous computational requirements of traditional approaches. Because of these l i m i t a tions only the hard-sphere system has been investigated using density-functional techniques. One may expect a priori, little orientational dependence of the interfacial tension i n this system because of its entropy dominated, high temperature nature [58]. It is clear from previous studies [ 1 7 , 5 9 - 6 1 ] however, that there is little consensus on the variation of the interfacial tension w i t h orientation as well as its absolute magnitude, properties that both strongly influence the equilib r i u m crystal structure. For comparison to these theoretical predictions the only simulation study available is the molecular dynamics investigation of Broughton and G i l m e r [62] who studied the interfacial free energy i n the Lennard-Jones system and found that the crystal-melt interface at the triple point is nearly isotropic. F r o m the work of Wulff we know that the relative values of the interfacial tension determine the crystal shape, including facet size and stability. Ideally, to determine the hard-sphere equilibrium crystal shape one would calculate η(Θ) at a l l interaction strengths r . T h i s would allow determination of both the state of faceting and the roughening transition v i a a Wulff-type construction. Unfor tunately, and despite the reduced dimensionality inherent i n our approximation, our approach to determining interfacial properties still requires a large compu tational effort m a k i n g such a calculation impractical. O n e c a n , however, look at higher index interfaces and ask the question of their stability as a function of r . For example, from a Wulff-type construction we know that the 211 inter face w i l l be stable (that is, not facet into a combination of 100 and 111 faces) i f 7211 < 0.3887ioo + 0 . 6 7 2 7 . Similarly for the 311 interface the stability condition m

is 7

3 1 1

< 0.6037

l o o

+0.5227 . m

A d h e s i v e S p h e r e s . A s discussed previously, the adhesive-sphere system is a convenient model because the attractive strength can be varied from the purely repulsive hard-sphere limit to a potential which includes a deep attractive well. T h e phase behavior i n the adhesive-sphere system is a strong function of the strength of interaction (see Figure 3 ) , resulting i n a large increase of the interfacial tension i n the f c c ( l l l ) direction w i t h increasing attraction strength. T h e question however remains: How w i l l the various crystalline orientations influence this behavior? We begin by first examining the hard-sphere l i m i t and then gradually increase the attractive strength, determining the structure and energy of the resulting equilibrium interfaces. A s seen previously for the fee (111) interface,increasing the strength of attraction i n the system causes the fee 110 and 100 interfaces to increase their interfacial free energy and decrease their interfacial thickness. Structurally, i n fact, these interface becomes sharper (Az/a = 1.95 and ν = 0.33 at τ = oo, decreasing to Az/a = 1.30 and ν — 0.13 at τ = 1.5). There appears to be little dependence on crystalline orientation i n this sys tem; the surface free energies are nearly identical as r is decreased from the


M A R K & GAST

16.


243

Table I V : Interfacial tensions 7 for the low and high-index crystal face orien tations at various adhesive-sphere strengths τ τ oo


3 2 1.7 1.5

ll™o*lkT 0.70 ±.01 0.79 ± .01 0.91 ± .01 1.01 ± .01 1.05 ± .01

0.70 0.80 0.92 1.02 1.07

± ± ± ± ±

.01 .01 .01 .01 .01

0.70 0.79 0.92 1.03 1.09

± ± ± ± ±

.01 .01 .01 .01 .01

PWDA /211 0.70 0.80 0.91 0.99 1.03

U

2

± ± ± ± ±

PWDA /311 0.70 0.80 0.91 1.01 1.04

/JfeT .01 .01 .01 .01 .01

U

2 /kT

± ± ± ± ±

.01 .01 .01 .01 .01

hard-sphere l i m i t but begin to show a small amount of anisotropy under condi tions of the strongest attractions studied. T h e origin of this anisotropy remains unclear as it does not appear to directly correlate w i t h the surface density p, where ρ = 4/y/3a , ρ = 2 / α , and ρ = y/2/a . One t h i n g to note is that the lowest tension corresponds to that interface w i t h the highest surface density suggesting the importance of interplanar interactions i n determining interfacial tension. 2

ιη

ιοο

2

ιιο

2

We list i n Table I V the values we calculate for both the low and high index interfaces where, once again, there is little anisotropy. A c c o r d i n g to the stability condition developed i n the previous section, both the higher order 211 and 311 interfaces are stable under the stengths of attraction studied here, indicating that the adhesive sphere crystal structure is nonfaceted. T h i s apparent lack of a transition from a spherical to a faceted e q u i l i b r i u m crystal shape as the attractive interactions increases may suggest that the interaction potential must have range in order to have a nonroughened, faceted e q u i l i b r i u m crystal structure. One must be careful however not to generalize since we are unable to study attractions stronger than those found at τ of 1.5. R e t u r n i n g to the work of Broughton & G i l m e r on the Lennard-Jones system, they obtain nearly isotropic values for the interfacial free energy at the triple point (T* = 0.617). E q u a t i n g v i r i a l coefficients [63] allows us to approximate an equivalent τ v i a r = 1 + 2/T*, giving r ~ 0.2, a value significantly lower than that investigated here. e q u i v

e q u i v

Summary Density-functional theory and the P W D A allow one to examine both the inter action strength and orientational dependence of solid-fluid interfaces. We have investigated the influence of interactions on interfacial properties, including hard spheres, adhesive spheres, and the Lennard-Jones system. We have also used the adhesive-sphere system to investigate both low and high index surfaces and found a small amount of anisotropy i n the interfacial tension at the highest attraction strengths. We see no direct evidence of faceting i n the adhesive-sphere system for the conditions investigated here.



244


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Chemical Applications of Density-Functional Theory - American

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