Estimation of non-equilibrium surface tension

SURFACE

SCIENCE 27 (1971) 41 l-418 0 North-Holland

ESTIMATION

OF NON-EQUILIBRIUM

Publishing Co.

SURFACE

TENSION*

CRAIG MAZE and GEORGE BURNET Institute of Atomic Reseach and Department of Chemical Engineering, Iowa State University, Ames, Iowa 50010, U.S.A. Received 16 July 1970; revised manuscript

received 1 June 1971

A mathematical description of surface tension-temperature behavior of liquid metals under non-equilibrium conditions has been developed based upon enthropy production caused by evaporation of liquid. Model parameters were obtained from experimental data using least squares. Surface tension-temperature data for lead (99.999 %) and bismuth (99.9999 %) were obtained from sessile drop tests conducted at pressures in the low 1O-g torr range. Data from White5), for zinc, were used to determine variation of the non-equilibrium parameters of the model as a function of degree of departure from equilibrium. Estimates of equilibrium surface entropy and surface tension at the melting point were also obtained.

1. Introduction Non-linear surface tension-temperature behavior lead and bismuth under non-equilibrium conditions. Whites), using zinc, that this non-linear behavior is from the liquid surface. The degree of non-linearity at the melting point both increase as the system’s brium increases. These non-equilibrium surface tension values are surface concentration caused by net evaporation surface. This results in an entropy increase relative to Assuming Gibbs’ equation for a pure fluid,

has been observed for It has been shown by a result of evaporation and the surface tension departure from equilia result of changes in losses from the liquid equilibrium conditions.

dy = - SAdT,

(1)

y = surface tension SA = surface entropy T = temperature is valid for non-equilibrium additional entropy produced surface entropy.

(erg/cm”), (erg/cm'-K)

,

(K) ,

situations, it can be modified to account for by liquid evaporation by adding a term to the

* Work was performed in the Ames Laboratory of the U.S. Atomic Energy Commission. Contribution

No. 2773. 411

412

C. MAZE AND G. BURNET

dy =-

(SA + S;) dT,

(2)

Sf = irreversible portion of the surface entropy. The next step is to determine the realationship between the evaporation rate, relative surface concentration, and the irreversible portion of the surface entropy. 2. Derivation of the model The transfer of 6n molecules gives rise to irreversible following fashion [Lewis and Randalls)] : dS, = Sn [ - d (/x/T) + Ed (l/T)]

entropy in the

= 6~ dS,

(3)

s = molal entropy for the irreversible process, For the process of evaporation dSis related to the latent heat, ,I,, by dS=d,d(l/T),

(4)

and eq. (3) becomes dSi =6ni,d(l/T).

(5) Since evaporation takes place from the surface region, the entropy, eq. (S), must pe put on a unit area basis before insertion into eq. (2). Assuming Langmuir adsorption can be used to describe evaporation and condensation, the net evaporation rate can be expressed in the following way [Adamson I)] : evaporation rate = kls, , (6) condensation rate = kzPv (s - sl) ,

(7)

k,, k, = rate constants, P, = vapor pressure, s = total number of sites on the surface, s1 = number of occupied sites.

The rate of evaporation is proportional to the number of occupied sites, and the condensation rate to the number vacant. The net rate, r, is the difference between eqs. (6) and (7). r = kls, - k,P,(s

- sl).

(8)

The rate constants are given by kinetic theory as k 1 = l/r,

(9)

kz = No”/J(27cMRT),

W)

1: = 7. exp (A,/RT) ,

(11)

NON-EQUILIBRIUM

SURFACE TENSION

413

z = average time an atom spends in the surface zone, z. = period of vibration for a surface atom,

N = Avagadro’s number, CT’= area per site, M = molecular weight, R = gas constant = 8.31 x 10’ erg/mole-K.

The evaporation rate, 2, is also given by kinetic theory as Z = PJJ(2xRMT).

(12)

Substituting eqs. (9) (lo), (1 l), and (12) into (8) gives the net rate r = sl/z - NZo’(s

- sl).

(13)

The fraction of occupied sites, 0, is defined by

e = s,ls,

(14)

and at equilibrium r =O; consequently, the equilibrium fraction of occupied sites is 8, =

ZTNCO

1 + ZzNo’

(15)

from eq. (13). Substituting eqs. (14) and (15) into (13) and rearranging gives the net rate in terms of evaporation rate and relative surface coverage r = ZNsa’

[ 8,’

To simplify matters let

e - ee B=

7,

[

where

1 1

e - 0,

e

(16)

(17)

At equilibrium p will be zero, and its magnitude in an open system will depend upon the material under study. It is not likely that p will ever approach - 1. The product (so “) appearing in eq. (16) is the total surface area, and the number of moles evaporated per unit area in t seconds can be calculated by rearranging eq. (16) to give s,,A =

5 _ NsaO Z$.

(18)

414

C. MAZE

AND

G. BURNET

Eq. (18) can be simplified further by substituting eqs. (11) and (12) for 2 and r, and using the integrated form of the Clausius-Clapeyron equation for the vapor pressure P, : P, = exp (A - &/RT), A = constant of integration. Carrying

out these substitutions

(19)

gives

(20) where the constant

K is defined by the following K

Substituting basis

=

expression

:

“_oexp (A) J(2rcMR)’

(21)

eq. (20) into eq. (5) for 6nA puts the entropy

on a unit area

(22) which when integrated of the surface entropy

holding /I constant gives the irreversible caused by evaporation loss from the liquid

portion surface. (23)

C (p) = an unknown

function

of j.

Substitution of eq. (23) into eq. (2) and integration from the melting point (T,) holding both /I and SA constant results in the final expression for the surface tension :

1, (/3) = surface tension

at the melting

point.

At equilibrium /I and C(p) are zero, and eq. (24) reduces to an equilibrium expression for surface tension and temperature with constant surface entropy. 3. Experimental Surface tension values for lead and bismuth, fig. 1, were determined from sessile drop measurements using a non-linear curve fitting procedure developed by the authorsb). These two metals were vacuum melted in tantalum crucibles, and the drops were bottom poured onto spectrographic grade graphite. All tests were conducted in an oil free, ion pumped ultrahigh vacuum system at pressures less than 5 x 10m9 torr.

NON-EQUILIBRIUM

SURFACE

415

TENSION

I I

0

MEASURED CALCULATED EQ(241

-

420 aDa

-.__LEAD 0

400 380 “I

360

c

34oL--1 500

Fig. 1.

/

2‘\

\

BISMUTH --I

600

700 TEMPERATURE

000 (K)

Surface tension of lead and bismuth as a function of temperature.

4. Discussion The parameters in eq. (24) were determined from experimental surface tension-temperature data using least squares, and the results appear in table 1. Since the parameter @A,K is linear in /I, relative values of /I can be determined from the zinc data by taking ratios of this parameter. These results also appear in table 1. It is not possible to obtain an absolute value for fi in that it never approaches - 1, even in an open system, so no definite value can be assigned to it.

RELATIVE p

Fig. 2.

Linear portion of the surface entropy appearing in eq. (24) asa function of relative departure from equilibrium.

99.999 +

99.999 +

99.9999 D.Z.R.b 99.99 +

99.9999

99.999

Zinc - open

Zinc - closed

Zinc Zinc Zinc

Bismuth - open

Lead - open

* Zinc data from Whites). b Double zone refined.

Purity (%)

System*

427

371

760 156 769

159

793

Ym(B) Cerg/cm2)

0.778

1.53

1.I9 0.205 1.21

2.35

7.55

SA + C(B) (erg/cm2-K)

-0.246

-0.480

-0.918 -0.262 -0.636

-1.16

-3.03

4B~VKi3 x 10-s

5.41

3.38

3.01 2.88 3.17

3.52

2.61

Standard deviation

Parameters appearing in eq. (24) for zinc, lead, and bismuth

TABLE1

0.303 0.0865 0.210

0.383

1.0

Relative B

601

544

693

% P CC! g r;

ti

.o 3

NON-EQUILIBRIUM

A plot of the parameter

SURFACE

417

TENSION

SA + C (/I) versus relative /I should give the equili-

brium surface entropy at /I =0 since C(0) = 0. Fig. 2, which is fortunately linear, is a plot of this type and a value of -0.58 is obtained for SA. This is in good agreement with the value of -0.64 given by White5). However, the latter value does not correspond to an equilibrium condition (ref. 5, p. 799); consequently, the extrapolated surface entropy should be somewhat different as it represents an equilibrium value. A similar plot for y,(p), fig. 3, results in y,,,(O)=752 erg/cm’, which is lower than the 761 given by Whites). Surface tension at the melting point is very sensitive to small changes in surface concentration as evidenced by the large slope, 38.7, for the line in fig. 3. It is not surprising that y,,, is so sensitive

SLOPE

= 38.7

750 TM(O) 7401

= 752

t0

0.5 RELATIVE

1.0 p

Fig. 3. Surface tension of zinc at the melting point as a function of relative departure from equilibrium. as the potential energy for surface atoms is strongly dependent upon separation distance. During evaporation the chemical potential of surface atoms is slightly lower than those in the bulk. In order to maintain constant surface area the average separation between atoms in the surface increases, and as a consequence, the surface tension increases. The results for lead and bismuth also appear in table 1. As can be seen, the parameter s/?&K appears to decreasing relative vapor pressure for open systems : Zn>Bi>Pb. 4B4K : It is tempting to generalize at this point and contend that it would be to detect non-linear y-T behavior in low vapor pressure liquid metals tin and indium. However, the maximum departure of p from zero for substance, as well as the other factors appearing in eq. (21) for

difficult such as a given K, will

418

C. MAZE

AND G. BURNET

determine the degree of non-linearity of the y-T relationship. It is possible that the open system p will never be appreciably different from zero for high vapor pressure substances. If the non-linear term in eq. (24) is too small to be of significance, the remaining two linear terms, y,,, and C(p), can still result in misleading y-T results. Departure of these terms from their equilibrium values will rotate and displace the y-T line from its equilibrium position. A glance in the literature, Whites) and Flintz), will illustrate this point. A family of parallel or nearly parallel y-T lines, decreasing linearly with temperature can be found for zinc and copper. These results were from the efforts of several investigators, each using different equipment, and each, no doubt, with a different degree of non-equilibrium (p) prevailing during the course of their experiments. 5. Conclusions The surface tension model, eq. (24), adequately describes non-equilibrium surface tension-temperature behavior for three liquid metals. Some of the variation in liquid metal surface tension data reported by other investigators can be accounted for, problems of cleanliness aside, in terms of net evaporation loss from the liquid surface and the resulting generation of entropy. Estimates of equilibrium surface entropy and y,(O) can be made by extrapolation of the parameters in eq. (24) to /I =O. Increases in y,(P) with relative p can be explained, in a qualitative way, in terms of increases in the average separation distance between surface atoms with p. The results of this model confirm the assumption that SA is constant for zinc, lead, and bismuth over a limited temperature range, but more work is needed to determine the linearity of SA + C@) near /I =O.

References 1) A. W. Adamson, Physical Chemistry of Surfaces, 2nd ed. (Interscience, New York, 1967). 2) 0. Flint, J. Nucl. Mater. 10 (1965) 233. 3) G. N. Lewis and M. Randall, Thermodynamics, Rev. by K. S. Pitzer and L. Brewer, 2nd ed. (McGraw-Hill, New York, 1961). 4) C. Maze and G. Burnet, Surface Sci. 24 (1971) 335. 5) D. W. G. White, Trans. Met. Sot. AIME 236 (1966) 796.