Download THEORY AND PRACTICE OF AEROSOL SCIENCE

TOLMAN LENGTH AND RIGIDITY COSTANTS FOR THE HARD-CORE YUKAWA FLUID
J.C. BARRETT1
1
Nuclear Department, Defence Academy, HMS Sultan, Military Road Gosport PO12 3BY, United
Kingdom.
Keywords: Nucleation, Density Functional Theory, Surface Tension, Tolman Length.
INTRODUCTION
Classical density functional theory has proved to be a very popular approach to determining
thermodynamic properties of small clusters for use in nucleation theories. The simplest approaches, the
squared gradient approximation (SGA) and mean field theory, seem to give the correct qualitative
behaviour for simple fluids, although their quantitative accuracy is unclear. Early work indicated a
significant difference between these two approaches, but more recent calculations for the cut-off LennardJones fluid showed that the SGA gave values for the planar surface tension that agreed with computer
simulations when used with an accurate equation of state (Lutsko, 2011). Here the SGA and a mean fieldtype approximation are compared to an approach based on the first order mean spherical approximation
(FMSA) for the hard-core Yukawa fluid.
METHODS
According to Tang (2005), the free energy of a non-uniform fluid can be approximated by,



F [  ]   dr1 f (  (r1 ))  12 kT  dr2 cb (r1  r2 )  (r1 )  (r2 )   (r1 ) 2 ,
(1)
where f is the free energy of a uniform fluid of density , k is Boltzmann’s constant, T is the temperature,
and cb is the direct correlation function of the bulk fluid at an appropriately chosen density. The precise
density to use in evaluating cb is unclear but, following Tang, it will be assumed here that the density is
constant, with a value to be discussed below. The Euler-Lagrange equation that determines the density
profile is then,
f (  (r1 ))   (r1 )    kT  dr2 cb (r1  r2 )  (r2 ) ,
(2)

where, µ is the chemical potential,   kT drcb (r ) and the prime denotes differentiation with respect to
ρ. Equation (2) is solved for the hard-core Yukawa fluid, with interaction potential,
  e   ( r / d 1)
u (r ) 
r  d,
(r / d )

r  d,
(3)
where d is the hard-sphere diameter and  and  are the attractive potential parameters. Tang’s (2004)
FMSA expression (together with the Carnahan-Starling form for the hard-sphere part) is used for the free
energy of the uniform fluid, and three approximations for cb were considered: (i) the FMSA correlation
function given by Tang (2005), (ii) the SGA, where kTcb is approximated by   m 2  (r1  r2 ) , with


m  16 kT  drr cb (r ) , and (iii) a Yukawa approximation (YA), where kTcb is approximated by a Yukawa
2
form for all r (not just r>d), chosen to give the same α and m as the FMSA cb, i.e.
kTcb   2 exp(   / m r ) /( 4mr) r  0 . It should be emphasized that in both the SGA and YA, the
parameters α and m are determined using the FMSA expression for cb. In the SGA, the Euler-Lagrange
equation becomes a second order differential equation for ρ:
m2   f (  (r1 ))   .
(4)
Furthermore, in the YA, applying the operator  2 to the Euler Lagrange equation gives the following
equation for  (r )  f (  (r ))   (r ) :
m 2       ( )  ,
(5)
where  ( ) is found by numerically inverting  (  ) . Thus, both the SGA and YA involve solving a
differential equation for the density profile, which is simpler than solving the integral equation in the
FMSA.
The surface tension, γ, of a spherical droplet of radius R can be expanded in powers of R-1 as:
 ( R)    
2  2k  k

 ...
R
R2
(6)
Where   is the planar surface tension, δ is the Tolman length (in the planar limit) and k and k are the
rigidity constants of bending and that associated with Gaussian curvature, respectively. The rigidity
constants depend on the precise definition of R (Barrett, 2008): here the equimolar dividing surface is used
and the associated rigidity constants are indicated by a subscript “e.” Expressions for γ , δ, and
( 2k e  k e ) are given by Barrett (2008) and by Blokhuis and van Giessen (2013). In the SGA (and hence
also the YA, by straightforward substitutions), the expressions given by Wilhelmsen et al. (2015) can be
used. All these expressions involve the planar profile, ρ0(z) and, in the case of ke, the first order (in powers
of R-1) correction to it, ρ1(z), both of which can be found from the relevant Euler-Lagrange equation (Eq.
(2), (4) or (5)): see Blokhuis and van Giessen (2013) for more details.
RESULTS
Calculations were performed for the Yukawa fluid with =1.8. Quantities were made dimensionless using
the appropriate combinations of d and . Tang (2005) suggested using the average of the equilibrium
liquid and vapour densities (ρl and ρv, respectively) to evaluate cb. However, it was found that this did not
lead to converged solutions for the Yukawa fluid. A similar problem was noted by Lutsko (2007) for the
Lennard-Jones fluid, but at higher densities. Here cb was evaluated at a density of 0.3ρl: values were
insensitive to the density used in the range 0.2ρl to 0.4ρl. Figure 1 shows the variation of the planar surface
tension with temperature. The three approaches give similar results, with the SGA giving the largest
values and the YA giving the smallest. However, all approaches overestimate γ compared to the
molecular dynamics results of González-Melchor et al. (2001). Figures 2 and 3 show the Tolman length
and rigidity constants, respectively. Once again, the SGA and YA values bracket the FMSA values.
CONCLUSIONS
Both the SGA and the YA give reasonable agreement with the FMSA for the surface tension, Tolman
length and rigidity constants of the liquid-vapour interface in the non-uniform Yukawa fluid, provided the
same equation of state is used in all approximations. However, for the surface tension, agreement with
molecular dynamics results is less good than that found by Fu and Wu (2004), using a more complicated
density functional. It would be of interest to compare the YA with alternative approaches for other fluids
such as the Lennard-Jones fluid.
Figure 1: scaled surface tension  *    d 2 /  vs scaled temperature T*=kT/ for the hard core Yukawa
fluid from the FMSA (solid line), the SGA (dotted line) and the YA (dashed line). The triangles show the
molecular dynamics results of Gonzáles-Melchor et al. (2001).
Figure 2: scaled Tolman length δ*=δ/d vs scaled temperature T*=kT/ for the hard core Yukawa fluid
from the FMSA (solid line), the SGA (dotted line) and the YA (dashed line).


Figure 3: scaled rigidity constants, 2ke*  ke*  (2ke  ke ) /  , vs scaled temperature T*=kT/ for the
hard core Yukawa fluid from the FMSA (solid line), the SGA (dotted line) and the YA (dashed line).
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