Determination of Surface Thickness Assuming a Linear-Density Determination of Surface Thickness Assuming a Linear-Density Profile Profile

Assuming a one-parameter model for the two-particle distribution function of a surface, one can choose a, value of the "width parameter" to yield a correct surface tension. In order to then identify the width of the surface layer with the value of the width parameter, as is often done, one can check the validity of the model by verifying that other properties (e. g. , surface energy) are correctly calculated, or, as proposed herein, by demonstrating that different formulas for the surface tension give identical results. A new formula is derived, and the relation between the different formulas discussed. Calculations are performed for the Ar system, For the model used, no choice of the parameter can yield identical results for both ways of calculating surface tension. This points to problems in the interpretation of the width parameter. Misleading results may be obtained if one- and two-particle distributions not related by the Born-Green-Yvon equation are used together.


I. INTRODUCTION
For most systems studied, it now appears' that the variation of density through the region of a vapor-liquid interface is monotonic. A suggestion of Fitts' for determining the width of the interfacial region is thus applicable'. Assume a monotonic form for the density variation, including a parameter representing the width, and find the value of the parameter required to give agreement of the calculated surface tension with the experimental value. Fitts applied this method to 'He and 'He. For identification of this parameter with the width of the interfacial region, however, one should at least check that, with the same value for the parameter, the modelused can give some other property correctly. Shih and Uang' used surface energy as a second property for Ar, and showed that, with a single value of the width parameter-, both surface energy and surface tension could be calculated in agreement with experiment.
In the present paper, I point out that the second property could be surface tension itself, as calculated by a different formula. The formula is derived and calculations are carried out for Ar with a "linear density profile. " It is found that it is impossibl. e to obtain identical surface tension with both. formulas. This throws doubt on any conclusions based on interpretations of the model used. The surface energy is given' ' by the expression (for a multicomponent system) rr=-, ' Q f r."(r")rpz(r")dr,.
Here i and j sum over components, the interaction potential Ptt between a particle of species i and a particle of species j i.s assumed to depend only on the interparticle distance, and the surface excess We consider the interface between a liquid and its vapor with the equimolar dividing surface" at= 0; p&'& is the two-particle distribution function for species z and j, and &'"'( .) =p", '"( .)I. l-e(,)]+p""'( .)e(,), with e (z) the unit step function, and with p, ',. ') the two-particle distribution function for bulk liquid and p$')'" that for bulk vapor. We assume p't' (z"r") = p'; " (r")f(z,)f(z,), with f(z) the one-particle density and Pts (r")=P(Ptgtt(rts), (2, r) where p& is the density of species i, and g&, is the bulk-liquid correlation function between species i and j Igtt(r)-1 for r~]. The vapor density is neglected.
These assumptions permit all calcul. ations to be made in terms of properties of the bulk liquid, which are assumed known. While more sophisticated models have been used, they require considerably more information. ' The choice of lows because P,(x,) =I', for all x,. It is (7) that usually serves as the starting point for calculations of surface tension. With the linear profile model of Eq. (5) it leads to""" The parameter d is to be identified with the halfwidth of the interfacial region. Gther one-parameter forms for f(z), to be used in Eqs. (1)-(4), have been suggested. '

II. SURFACE-TENSION FORMULAS
Gne definition for the surface tension y is ob- However, the resulting expression for y is not the expression normally used. Instead, one notes that, because of the properties of p;, "'" ) pbulk g~$ i 12 (6) p f dz f d-"pu) du"g g ew 00 12 I 2 (2) Pox» &i r»Pij 12 for the surface tension. The second equation fol-where P, and I', are components of the pressure, and are equal and independent of position in an isotropic bulk fluid. The exact two-particle distributions for the interface make the s component of the pressure (but not the x component) equal at any point in the interface to the pressure of either of the bulk phases, ' "" as required by mechanical equilibrium (this is guaranteed by the Born-Green-Yvon equation). Therefore one may use l, for the quantity on the left-hand side of (6) which appears in the expression for the surface tension, obtaining OO 2 oo , =-g f dz fdc pu~d"»" f dz) ma 00

12
The usual formula for surface energy, is obtained by inserting Eqs. (5) into (1). This gives u=, P Plu; fdr Pu(r"")du(r") which, after some lengthy algebra (see below), leads"'" to A surface-tension formula similarly derived from the equation obtained by replacing (t)"by (t);&x»/r» in (1) will, differ from (8). The derivation of (7) does not follow if an approximation is used for the p~'&it requires that the Born-Green-Yvon equation be satisfied, and none of the simple forms for p&'& does this. We now proceed to obtain the surface tension for the linear profile model, starting from the expression analogous to (1). In (9), therefore, we replace (()), & by p,& x2»/r» To. carry out the integrations, we use cylindrical coordinates, so that r' =r» --s»+ (x,z, )'. The term not involving g(x) is 0 0O 0 2' --, g p p, . dz, f ds, s ds" f dp"d,'z(r")du (r")( (z,z, r)']cps'pr, , ' t18d'r', , 8dr' , P p, p, dry, ', g"~-2d'r'+ + -) 16d, o~g t'-4d' 8d'r' 32d 'r' » 8dr' r' 't~" 4d' + dr 0'l'Apl 15 + 3 8 +2d 15 +18 I + dr 4fJglj 8 9g Combining y~w ith the following terms, we have our surface tension -8dr' 2r' r' " I -r' 2r'd ' 8d"I 4~P'P'~"g" g 15d 72d'~" g" 2 3 45)~( 12) + dr g (13) Again, the difference between (12) and (13) is due to the fact that they are obtained by the insertion of the approximate forms of (4) and (5)   should be made such that the various expressions for the surface tension give the same results. This would constitute a route to the calculation of surface tension, surface energy, and other properties, for which knowledge of a second experimental property such as surface energy would be unnecessary. The equality of the expressions is a necessary, but not sufficient, condition for p ff and P~P= f(&) to obey the Born-Green-Yvon equation, as the exact twoand one-particle distributions for a surface. must do. Therefore comparison of (8), (12), and (13)  We employ the same interatomic potential and correlation function as did Shih and Uang: P is the Barker-Fisher-Watts" potential, and g is the experimental correlation function tabulated by Yarnell et a,l. " In Table I, surface tensions calculated according to (8) and (12) are given for various values of d. According to (5), 2d is the thickness or width of the interfacial region. It is clear from the results shown that no value of d can make (8) and (12) agree. Perhaps this is already evident from the expressions themselves. Clearly, (8) can give reasonable surface tensions with the present model, if d is correctly chosen, ' while (12) cannot.
However, there is no a Priori reason for preferring one to the other.
Writing p~'~a s the bulk-liquid correlation func-