Integral Equation Analysis of Homogeneous ... - ACS Publications

Chapter 2

Integral Equation Analysis of Homogeneous Nucleation

Downloaded by NORTH CAROLINA STATE UNIV on May 6, 2015 | http://pubs.acs.org Publication Date: September 21, 1990 | doi: 10.1021/bk-1990-0438.ch002

Günther H. Peters, John Eggebrecht, and Maurice A. Larson Department of Chemical Engineering and Center for Interfacial Materials and Crystallization Technology, Iowa State University, Ames, IA 50011 The assumptions of homogeneous nucleation theory are examined for the liquid-vapor transition of a Lennard -Jones fluid. Approximate solutions of the first Yvon -Born-Green integro-differential equation in a spherically symmetric and finite volume provide the dependence of the density profile of a small droplet on temperature and supersaturation. The structure provides a mechanical approach via the pressure tensor to the interfacial properties. Classical thermodynamic and statistical mechanical expressions for the surface tension are compared. This approach allows the calculation of the free energy of formation of a droplet from a metastable vapor, avoiding most of the usual assumptions of homogeneous nucleation theory. These theoretical results, which are tested by compar ison with molecular dynamics simulations, indicate that the droplet size dependence of the interfacial free energy is sufficiently strong that, for the state points considered, the free energy barrier prior to the nucleation is absent. The atomic kinetics of condensa tion are examined visually using molecular dynamics temperature quenching experiments, providing insight into the kinetic hindrance of the nucleation process. Recent theoretical and simulation studies of cluster structure and solvation in ionic solutions are discussed. When a phase t r a n s i t i o n occurs from a pure single state and i n the absence of wettable surfaces the embryogenesis of the new phase i s referred to as homogeneous nucleation. What i s commonly referred to as c l a s s i c a l nucleation theory i s based on the following physical picture. Density fluctuations i n the p r e - t r a n s i t i o n a l state result i n l o c a l domains with c h a r a c t e r i s t i c s of the new phases. I f these fluctuations produce an embryo which exceeds a c r i t i c a l s i z e then this embryo w i l l not be dissipated but w i l l grow to macroscopic s i z e in an open system. The concept i s applied to very diverse phenomena: 0097-6156/90/0438-0016$06.00A) © 1990 American Chemical Society

In Crystallization as a Separations Process; Myerson, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.


2.

PETERS ETAL.

Integral Equation Analysis ofHomogeneous Nucleation

vapor condensation, thin f i l m growth, polymer p r e c i p i t a t i o n , l a t t i c e defect formation, and of p a r t i c u l a r interest here, to crystallization. The a p p l i c a b i l i t y of the concept to a l l of these processes r e l i e s upon several postulates: 1. Thermodynamics remains relevant as phase dimensions are reduced to a molecular scale. 2. Embryos assume the symmetry of lowest s p e c i f i c surface area. 3. Embryos are s u f f i c i e n t l y d i l u t e that interactions between them may be neglected. 4. Equilibrium thermodynamic functions can be used to represent nonequilibrium states. The implementation of the concept usually involves a d d i t i o n a l approx imations: a. The phases are i d e a l and/or incompressible. b. The phase boundary i s a density d i s c o n t i n u i t y . c. The surface tension i s independent of curvature. d. The system i s i n contact with a material reservoir which instantaneously replenishes molecules depleted by the growing embryo. It i s d i f f i c u l t to judge the l i m i t a t i o n s of the postulates while imposing these a d d i t i o n a l assumptions. Our interest l i e s i n the development of a molecular treatment of c r y s t a l l i z a t i o n . However, i t i s useful as a preliminary to that effort to consider the e f f e c t s , when these commonly used assumptions are l i f t e d , upon the basic physical picture of the process. For this purpose we examine the condensing vapor, upon which applications to other transitions are based by analogy. I f we accept the f i r s t three postulates, we can l i f t each of these approximations using s t a t i s t i c a l mechanics and the companion techniques of computer simulation. But to do so we must consider a material for which complete thermodynamic and the necessary s t r u c t u r a l information i s a v a i l a b l e . Ve, therefore, consider the Lennard-Jones f l u i d i n most of the following discussion. Ambiguities associated with rates of mass transport are removed by investigating a f i n i t e system. F i n i t e rates of mass transport w i l l result i n a depletion zone surrounding a growing embryo and i f these are not d i l u t e i n the transforming system then these embryos w i l l compete for material. At some stage i n the process, perhaps near completion, interactions of growth centers cannot be neglected. The general thermodynamic formulation of the model (1) i s not r e s t r i c t e d to a p a r t i c u l a r ensemble and conclusions reached should be transferable to other ensembles. The approximation of vapor i d e a l i t y i s e a s i l y removed. Retaining the approximations of an incompressible l i q u i d phase, a discontinuous density p r o f i l e and curvature independent surface tension the conditions are those studied by Rao, Berne and Kalos (2). The e s s e n t i a l physics was unchanged from the usual treatment i n an open system, except that a minimum i n the free energy of formation i s found which corresponds to the unique equilibrium phase separated state whose symmetry, i n the absence of an external f i e l d , i s spherical. The remaining approximations of l i q u i d phase incompressibility and a discontinuous mass d i s t r i b u t i o n can be removed through the use of the Yvon-Born-Green (YBG) equation (3), which i s simply a


17


18

CRYSTALLIZATION AS A SEPARATIONS PROCESS

molecular statement of the conservation of l i n e a r momentum. Ve have recently obtained solutions of this equation i n a f i n i t e system by appending the constraint of mass conservation (4). The method i s summarized belov. For a planar liquid-vapor interface the density p r o f i l e and surface tension have been shown to be accurately predicted by solutions of the YBG equation through comparisons with molecular dynamics computer simulations ( 5 , 6 ) . Such comparisons are e s s e n t i a l i n the development of molecular theories of the structure of matter since simulations can provide the exact r e s u l t , for a given model, within l i m i t s of precision imposed by f i n i t e sampling. Here we use recent simulation studies (7) to test our solutions of the YBG equation for the Lennard-Jones droplet. The YBG equation i s a two point boundary value problem requiring the equilibrium l i q u i d and vapor densities which i n the canonical ensemble are uniquely defined by the number of atoms, N, volume, V, and temperature, T . I f we accept the a p p l i c a b i l i t y of macroscopic thermodynamics to droplets of molecular dimensions, then these densities are dependent upon the i n t e r f a c i a l contribution to the free energy, through the condition of mechanical s t a b i l i t y , and consequently, the droplet s i z e dependence of the surface tension must be obtained. Ve must also determine the location of the surface of tension, R . Although expressions for this parameter e x i s t , they are derived by a hybrid of molecular mechanical and thermodynamic arguments which are not at present known to be consistent as droplet s i z e decreases (8). An analysis of the s i z e l i m i t a t i o n of the v a l i d i t y of these arguments has, to our knowledge, never been attempted. Here we evaluate these expressions and others which are thought to be only asymptotically c o r r e c t . We conclude, from the consistency of these apparently independent approaches, that the surface of tension, and, therefore, the surface tension, can be defined with s u f f i c i e n t certainty i n the s i z e regime of the c r i t i c a l embryo of c l a s s i c a l nucleation theory. Having described the equilibrium structure and thermodynamics of the vapor condensate we then re-examine homogeneous nucleation theory. This combination of thermodynamics and rate k i n e t i c s , i n which the free energy of formation i s treated as an a c t i v a t i o n energy in a monomer addition reaction, contains the assumption that equilibrium thermodynamic functions can be applied to a continuum of non-equilibrium s t a t e s . For the purpose of elucidating the effects of the removal of the usual approximations, we r e t a i n this assumption and calculate a r a d i a l l y dependent free energy of formation. We f i n d , that by removing the conventional assumptions, the presumed thermodynamic b a r r i e r to nucleation i s absent. F i n a l l y we consider recent investigations of e l e c t r o l y t e solutions. Larson and Garside (9) have proposed the existence of stable clusters i n the p r e t r a n s i t i o n a l solution as an explanation of the observed r a p i d i t y of c r y s t a l l i z a t i o n . Ionic clusters i n subsaturated solutions have long been thought to exist (10,11), though d i r e c t experimental observation i s d i f f i c u l t . Their existence has provided useful models for extensions of the d i l u t e solution l i m i t i n g behavior of a c t i v i t y and conductance to f i n i t e concentrations. Monte Carlo computer simulation results are presented which characterize i o n i c c l u s t e r s i z e d i s t r i b u t i o n s i n sub-saturated solutions and which examine the morphology of large i o n i c clusters i n super-saturated g


2.


PETERS ETAL.

solutions and solvation structure as the saturation l i n e approached (12).

is

The Droplet Density P r o f i l e In this section ve consider the manner i n vhich the r a d i a l l y dependent density d i s t r i b u t i o n , p(r), of the equilibrium droplet can be obtained from the Yvon-Born-Green equation (3)


\

p ( r ) = -0 p ( r ) J d r 1

2

2

P

\

(? ) 2

u(r ) g t f ^ )

(1)

1 2

vhich i s the f i r s t i n a hierarchy of equations r e l a t i n g n-par^ticle density d i s t r i b u t i o n s to n + l - p a r t i c l e d i s t r i b u t i o n s . Here r . i s the position vector of p a r t i c l e i ( i - 1 , 2 ) ; 0 i s 1/kT vhere k i s y i e ^ Boltzmann constant and T i s the absolute temperature; and g ( r ^ , r ) i s the pair c o r r e l a t i o n function. The statement of the molecular model to vhich this equation i s applied i s contained i n the intermolecular p o t e n t i a l , u ( r ^ ) . Here ve consider only the p a i r - v i s e additive Lennard-Jones potential 2

2

• P(r )) S ( 1

2

r

L

1 2

+

>

v [p(? ), p(? ) j g ( r > y

x

2

v

1 2

3

vhere the veights are simple functions of the l o c a l densities

W

?

?

L0>< 1>"< 2>)

-

JU^)

K >

+

?

2

>< 2> " >v]

4a

< >

and

W

?

v("< l>"'< 2>) - 2 T ^ ) ?

2p

[ L"

?

'< 1>

"

P

?

< 2>]

The constraint of a f i n i t e volume requires that the one-particle density d i s t r i b u t i o n s a t i s f y the condition

N = 4n

dr r* p(r)


(5)

19

20


vhere N i s the t o t a l number of atoms contained in the system vhose volume has a r a d i a l dimension R . . Details of our numerical method of solving Equations 1 through 5 are given i n reference 4. In Figure 1 ve compare our numerical solutions v i t h the molecular dynamics computer simulations of Thompson, et a l . (7). In this comparison ve use l i q u i d and vapor densities obtained from the simulation studies. In the next section ve obtain the required boundary values by approximate evaluation of v a p o r - l i q u i d equilibrium for a small system.


Droplet Thermodynamics In a f i n i t e , one component, system defined by the number of molecules, N, the volume, V, and the temperature, T, the Helmholtz free energy of formation, M , of a stable droplet i s v r i t t e n f

A

A

P

N

P

V

f " L L " L L

+

V*V " V V P

V

+

4 , 1

R

s

Y (

V

" ^

+

P

V

(

6

)

vhen the conditions of thermal, chemical and mechanical equilibrium are s a t i s f i e d : T

U

L

L

- T

=

M

(7)

y

(

V

8

)

and 2Y(R ) (9)

PL " Pv

The l a t t e r i s referred to as the Laplace-Young equation, u^, p^, and are the chemical p o t e n t i a l , pressure and volume of phase i , i-(L,V). In a purely thermodynamic formulation ve must regard these as properties of a macroscopic f l u i d at the temperature and vapor and l i q u i d d e n s i t i e s , jx. and p , of the microscopic phases, u and p are the chemical potential and pressure of the homogeneous metastable, or unstable vapor, p r i o r to condensation. The Laplace-Young equation refers to a spherical phase boundary knovn as the surface of tension vhich i s located a distance R from the center of the drop. Here the surface tension i s a minimum and a d d i t i o n a l , curvature dependent, terms vanish (15). The molecular o r i g i n of the d i f f i c u l t i e s , discussed i n the introduction, associated v i t h R can be seen i n the d e f i n i t i o n of the l o c a l pressure. The pressure tensor of a s p h e r i c a l l y symmetric inhomogeneous f l u i d may be computed through an integration of the one and tvo p a r t i c l e density distributions. L

g

g

{3 p(r) = p(r) I -

§ \ a~* 2 J 12 dr

r

r

12 12 r 1 2

8 U < r

12 3r

)

1 2


2.

Integral Equation Analysis ofHomogeneous Nucleation 21

PETERS ETAL.

da p ( r - a r ) p ( r + ( l - a ) r ) 1 2

g(r-ar ,r+(l-a)r )

12

1 2

(10)

1 2

vhere I i s the unit tit matrix matrix. The pressure tensor i n a s p h e r i c a l symmetry has tvo independent components, P ( ) and P ( r ) , r

N

T

p(r) = p ( r ) r r + p ( r ) 0 9 Downloaded by NORTH CAROLINA STATE UNIV on May 6, 2015 | http://pubs.acs.org Publication Date: September 21, 1990 | doi: 10.1021/bk-1990-0438.ch002

N

(11)

T

vhere r and 6 are unit vectors i n spherical polar coordinates. The l N

< >

T

co 2 Y(R ) = J d r (^-) [p (r) - p (r)] S

N

(13)

T

Equations 12 and 13 are tvo different expressions for the surface tension (21) vhich at the present time are not knovn to be consis tent. Although Equation 12, and Equation 9, are independent of the form of the l o c a l i z a t i o n of the pressure tensor, Equation 13 i s not (22). With the assumption of the equivalence of Equations 12 and 13 an expression for the surface of tension i s obtained

f R

( P > S

- M

CO

dr r

^1/3

00

2

[ p ( r ) - p ( r ) ] / J dr r " N

T

1

[p (r) - p ( r ) ] j N

T

(14)

The superscript (p) v i l l be used to d i s t i n g u i s h the surface of tension defined by Equation 14 from other d e f i n i t i o n s introduced belov. Equations 12 through 14 may be derived from thermodynamic and mechanical expressions for the transverse force and torque acting on a surface vhich intersects the i n t e r f a c i a l region (21). Moments of an i s o t r o p i c pressure force and a surface tension acting at R are



22

equated v i t h moments of the transverse component of the pressure tensor. However, vhen the r a d i a l dimension of a droplet i s smaller than the range of the intermolecular force the pressure near the center of the drop departs s i g n i f i c a n t l y from the thermodynamic limit. Replacement of the uniform pressure of the Gibbs model v i t h the normal component of the pressure tensor leads to the following a d d i t i o n a l expressions for the surface of tension, 00

=


R

s

0 >

0

J

00

'J

dr l p

N

( r )

P

" T

( r > I

00

1}

RJ

d r r

(

2

1

I p

N

( r )

P

" T

(

r

)

1

- J d r r [ p ( r ) - p ( r ) ] / J d r [ p ( r ) - p

R (P e

L

- P )1 V

Again this equation i s s e n s i t i v e to uncertainties i n the pressure drop. The truncated series expansion of Tolman's equation may be expected to introduce s i g n i f i c a n t error for small drops.

The potential model used in these simulations was truncated at 2.5 atomic diameters, while in our calculations the potential was truncated at 8.0 diameters. The effect of this difference in the model may be significant, particularly for small droplets. However, given the considerable difficulties in the evaluation of the surface tension by either Equation 24 or 25, the qualitative agreement with simulation reinforces the observation which is essential to an analysis of nucleation theory: the radial dependence of the surface tension is much stronger than previously thought.


(25)

2.

PETERS ETAL.


-

q=e

Primitive supersaturated states

i Monte Cario

0.5

1

•

Molarity

i

1.5

Figure 10. The compressibility factor for a charged and dipolar hard sphere mixture predicted by perturbation theory is compared with the results of Monte Carlo simulation. Tfye elementary electronic charge is denoted e.


3


32


A c e n t r a l assertion of homogeneous nucleation theory i s that i n t e r f a c i a l free energy costs induce a spherical symmetry i n the phase embryo. However, these simulation studies indicate that i n t e r molecular interactions may not permit the development of spherical symmetry when these interactions are strong and highly asymmetric. The morphologies of the large i o n i c clusters observed i n these simulations rather suggest free chain end folding to produce rudimentary l a t t i c e structure as a possible p r e - t r a n s i t i o n a l mechanism. An important consideration about which l i t t l e i s currently known i s the dehydration which must accompany c r y s t a l l i z a t i o n . In our simulations we observe that solvent species adjacent to an ion are strongly bound and o r i e n t a t i o n a l l y ordered due the to the intense l o c a l e l e c t r i c f i e l d at i o n i c concentrations both above and below the saturation l i n e . In Figure 12 the variance i n the order parameter cos8 i s close to zero near contact for the 0.4 and 2.0 molar s t a t e s . At a concentration approaching the s o l u b i l i t y l i m i t , from the subsaturation side, solvent molecules i n the primary solvation s h e l l develop an enhanced o r i e n t a t i o n a l mobility which i s greater than the free rotor l i m i t of 1/3. Ve interpret this loss of o r i e n t a t i o n a l r i g i d i t y as a cancellation of l o c a l i o n i c f i e l d contributions as i o n i c c l u s t e r i n g occurs. Solvent orientation near an ion dimer i s frustrated by the presence of two energetic minima which are separated by a b a r r i e r whose height i s defined by the orientations of other adjacent solvent species. Vhen the ion dimer shown in the figure i s joined by a third ion of the appropriate sign the height of the b a r r i e r separating l o c a l energy minima i s increased. Hence, at super-saturated states a variance reduction results on i n t e r i o r segments of a charged i o n i c chain while dehydration i s permitted at free chain ends. Conclusions Integral equation theory and computer simulation have been used to examine current assumptions through which the physical picture of homogeneous phase nucleation has been constructed. Ve conclude that the importance of i n t e r f a c i a l free energy effects have been over stated i n the c l a s s i c a l theory, that strong anisotropy i n i n t e r molecular forces may play a c e n t r a l role i n p r e - t r a n s i t i o n a l c l u s t e r structure and dynamics and that growth i s not monomeric and c l u s t e r c l u s t e r interactions must be taken into account. Basic research, i n areas suggested by these methods, i s required before a firm understanding of the molecular mechanics of phase t r a n s i t i o n s , and techniques for the control of these mechanisms, can be developed. Acknowledgments Support for that portion of this work dealing with e l e c t r o l y t e solution theory has been provided to J . E . by the National Science Foundation (CBT-8811789) and by a grant of Cray X-MP time at the National Center for Supercomputing Applications. The authors wish to express appreciation for assistance provided by Mr. John Potter with the development of computer software used i n the production of a movie of a nucleation event shown at this symposium.


2.

PETERS ETAL.



J 4

1

2

3

4

5

6

7

ion cluster size Figure 11.

The ion cluster s i z e d i s t r i b u t i o n obtained from computer simulations of the charged and dipolar hard sphere mixture at several states: half charge, 1 Holar (A); f u l l y charged, 1 Molar (B); f u l l y charged, 0.4 Molar (C)? and half charge, 0.4 Molar (D).

Figure 12.

The mean (a) and variance (b) of the o r i e n t a t i o n a l ordering of a dipolar solvent molecule as a function of distance from an ion for the indicated solution s t a t e s . Configurations vhich enhance and reduce o r i e n t a t i o n a l mobility are displayed.


33


34 Literature Cited 1. 2. 3. 4. 5.


6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

32.

Gibbs, J.W. The Scientific Papers of J.Willard Gibbs; Dover Publications, New York, 1961. Rao M.; Berne, B.J.; Kalos, M.H. J . Chem. Phys. 1978, 68, 1325. Born, M.; Green, H.S. Proc. Roy. Soc. 1946 A 188, 10. Peters, G.H.; Eggebrecht, J . Submitted to J . Phys. Chem. Eggebrecht, J.; Gubbins, K.E.; Thompson, S.M. J . Chem. Phys. 1987, 86, 2286. Eggebrecht, J.; Thompson, S.M.; Gubbins, K.E. J . Chem. Phys. 1987, 86, 2299. Thompson, S.M.; Gubbins, K.E.; Walton, J.P.R.B.; Chantry, R.A.R.; Rowlinson, J.S. J . Chem. Phys. 1984, 81, 530. Henderson, J.R. In Fluid Interfaces; Croxton ed., 1987. Larson, M.A.; Garside, J . J . Crys. Growth 1986, 76, 88. Friedman, H.L.; Larsen, B. Pure and Appl. Chem. 1979, 51, 2147. Fuoss, R.M.; Accasina, F. Electrolytic Conductance; Interscience: New York, 1959; Chap. 18. Eggebrecht, J.; Ozler, P. Accepted by J . Chem. Phys. Toxvaerd, S. Mol. Phys. 1973, 26, 91. Hansen, J.P.; MacDonald, I.R. Theory of Simple Liquids; Academic Press: New York, 1976. Tolman, R.C. J . Chem. Phys. 1948, 16, 758. Irving, J.H.; Kirkwood, J.G. J . Chem. Phys. 1950, 18, 817. Schofield, P.; Henderson, J.R. Proc. Roy. Soc. 1982, A 379, 231. Buff, F. J . Chem. Phys. 1955, 23, 419. Bakker, G. Wien-Harms' Handbuch der Experimentalphysik, Band VI; Akademische Verlagsgesellschaft, Leipzig, 1928. Ono, S.; Kondo, S. Handbuch der Physik, Band X; Springer Verlag, Berlin, 1960. Rowlinson, J.S.; Widom, B. Molecular Theory of Capillarity; Claredon Press, Oxford, 1982, Chap. 4. Hemingway, S.J.; Henderson, J.R.; Rowlinson. Faraday Sympos. Chem. Soc. 1981, 16, 33. Percus, J.K.; Yevick, G.T. Phys. Rev. 1958, 110, 1. Tolman, R.C. J . Chem. Phys. 1949, 17, 333. Falls, A.H.; Scriven, L.E.; Davis, H.T. J . Chem. Phys. 1981, 75, 3986. Rasmussen, D.H.; Sivaramakrishnan, M.; Leedom, G.L. AIChE Symposium Series, Crystallization Process Engineering, 1982, 78(215), 1. Abraham, F.F. Homogeneous Nucleation Theory; Academic Press, New York, 1974, Chap. 2. Walton, J.P.R.B. Doctoral Dissertation, University of Oxford, 1984. Rao, M.; Berne, B.J. Mol. Phys. 1979, 37, 455. Eggebrecht, J.; Ozler, P. To be submitted. Zwanzig, R. J . Chem. Phys. 1954, 22, 1420; Pople, J.A. Proc. Roy. Soc. 1954, A 221, 498; Stell, G.; Rasaiah, J.C.; Narang, H. Mol. Phys. 1972, 23, 393 (1972); Rushbrooke, G.S.; Stell, G.; Høye, J.S. Mol. Phys. 1973, 26, 1199. Stell, G.; Lebowitz, J.L. J . Chem. Phys. 1968, 49, 3706.

RECEIVED May 12, 1990 In Crystallization as a Separations Process; Myerson, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

Integral Equation Analysis of Homogeneous ... - ACS Publications

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