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Theoretical determination of surface thickness of liquid 4He and 3He
Physica
42 (1969)
205-212
0 North-Holland
THEORETICAL SURFACE
Publishing
Co., Amsterdam
DETERMINATION
THICKNESS
OF LIQUID
OF
4He AND
sHe
D. D. FITTS Department
of Chemistry, Philadelphia,
Received
University
of Pennsylvania,
Pennsylvania,
USA
17 June 1968
Synopsis The quantum statistical-mechanical theory of surface tension is applied to liquid 4He and to liquid sHe at 1.4 K. This theory relates the surface tension of the thickness of the region between the liquid and vapor phases. By a comparison of the calculated surface tension as a function of interfacial thickness with the experimental value, the surface thickness of liquid 4He is found to be 4 to 5 molecular diameters and that of liquid sHe 7 to 8 molecular diameters.
Introdzlctios. From the principles of quantum statistical mechanics, Fit&r) and Brout and Nauenbergs) have independently formulated a quantum-mechanical molecular theory of surface tension between two fluid phases for the case of a planar interfacial zone. They obtained a general expression relating the suface te.rsion to the kinetic energy density, the potential of intermolecular force, and the molecular distribution functions. In the application of this expression to a liquid phase in equilibrium with a vapor phase, it is necessary to consider the thi:kness of the interfacial region. Thus, this theory in effe-_t relates the surface tension to the surface thickness. The purpose of this article is to apply the quantum theory of surface tension to liquid 4He and liquid sHe in order to study their respective surface thicknesses. By comparing calculated and experimental values of the surface tension in each case, a theoretical estimate of the interfacial thickness can be made. Quantum theory of surface tension. The system under consideration consists of two homogeneous fluid phases a and fi separated by a planar transition zone. A Gibbss) dividing surface between the two phases is selected and a Cartesian coordinate system (x, y, Z) is defined with the dividing surface as the (x, y) plane and with the z axis normal to the dividing surface and directed from phase a to phase j3. The choice of a specific dividing surface is arbitrary and will be deferred. 205
ID. I). FITTS
206
The theory
makes
use of two molecular
distribution
functions4):
the
singlet density p(1)(R) and the pair density p(2)(R1, R12). The quantity #1)(R) dv is the average number of molecules in volume element dv at a point R in the fluid and the quantity p@)(Rl, Rlz) dv1 dv12 is the average number of molecular pairs, one member of which is located in volume element dv1 at RI and the other in dv12 at point R12 in the relative configuration space of the pair. The pair correlation function g(R, Rlz) is defined by the relation /J”)(R1, R12) = P(R1) For a planar interface, z and the pair density interface becomes P(~)(z~, R12)
=
@(Rz)
g(Rl>
R12).
(1)
the singlet density depends only on the coordinate only on z1 and R12. Therefore, eq. (1) for a planar
p(l)(a)
p(l)@1
+
212) &l,
R12).
(4
In the interior of each phase, the singlet density becomes independent of z, while the pair density and the pair correlation function depend only on the the singlet density is the macroscopic pair distance R12. Furthermore, molecular or number density and the pair correlation function is identical with the radial distribution function of the theory of liquids. Fitts1) and Brout and Nauenberg2) showed that the surface tension y for a quantum fluid is given by
-
[tT(Zl)
-
tN(Zl)]
(3)
h,
s --oo
where V(R12) is the potential of intermolecular force acting between a pair of molecules separated by a distance R12 and tT, tN are the tangential (x or y) and normal (z) components, respectively, of the average kinetic energy density. The quantities r!” are defined by r;‘)
= ~z’&A2)(z1, R12) -cc
&;]
dz1,
where p$ has the value pF)(R12), the pair density in the interior of phase a, for z1 negative and the value ~r)(R1~), the pair density in the interior of phase /?, for z1 positive. Thus, r, ~2) has a value different from zero only in the interfacial region, where the singlet and pair densities change from the uniform values in phase cc to the uniform values in phase j3. In deriving eq. (3) Fitts1) employed the equations of quantum hydrodynamicss) and the relationship derived by Buff6) between the surface
SURFACE
tension
THICKNESS
and the macroscopic
OF LIQUID
stress tensor.
Brout
*He AND
3He
and Neuenberg2),
other hand, used the surface energy for a system in a single quantum
207
on the state
to obtain eq. (3). The two deriviations are completely equivalent. The second integral on the right-hand side of eq. (3) arises from the anisotropy of the average kinetic energy density. The quantity [TV - TV] vanishes in the interior of each phase, and for an interfacial region of zero width the integral is zero. Moreover, for a classical fluid, the average kinetic energy density is isotropic and the integral vanishes identically. For the purpose of calculation, we now follow the original treatmentslps) and introduce the approximation that the average kinetic energy density is locally isotropic, so that tr(zr) = tN(Zr). Under this assumption the second integral in eq. (3) vanishes and eq. (3) becomes y=
s dV(R12) 1
(z~~T’~~’ + &fJ~‘)
R12
dur2.
dRl2
This expression for the surface tension is identical with that obtained by Kirkwood and Buff 7) for classical fluids except that here the pair density p(s)(zr, Rrs) must be determined by a quantum-mechanical theory. Evalzlation of T, (‘) . We now evaluate eq. (5) for a particular Gibbs dividing surface. Since eq. (5) is independent of the choice of dividing surface between phases cc and 8, we may use that surface which makes the superficial density of matter zero; i.e., such that
rp = y
[p(l)(z) - p$] dz = 0,
--oo
(2) This surface is called the equimolecuwhere p$) is defined analogously to pas. lar dividing surface because a hypothetical system in which the singlet density p(l)(z) changes discontinuously from its value in the interior of phase u to its value in the interior of phase /I at the dividing surface would contain the same number of molecules as the actual system. At this point we specialize our discussion to a liquid phase a in equilibrium with a vapor phase ,i3 and assume that the number density in the vapor phase pf’ is essentially zero. Kirkwood and Buff 7) have evaluated eq. (5) for a transition zone which has been shrunk to a mathematical surface coincident with the Gibbs dividing surface, so that p(r)(z) = p(r) OL’ = 0.
(2 < 0) (2 ’
0)
(7)
We now solve eq. (5) for a liquid to vapor transition zone of nonzero thickness. We assume that the boundaries of the interfacial region are
D. D. FITTS
208
distinct and that the composition of each phase remains homogeneous up to the boundary of the interfacial zone. Since the exact behavior of the singlet density p(i)(z) in the interfacial region is not known, we must assume a specific functional dependence on z. For convenience of calculation, we assume that the singlet density in the interface is continuous and is antisymmetric about the point z = 0, p(i)(z) = pf’/2. Thus, eq. (6) is obeyed. We also assume that the pair correlation function g(zi, Rls) in the interfacial zone is equal to the radial distribution phase a. In this approximation, we have p(z)
=
g(z1, RlZ)
p’l’ a
function gLy(Ris) of the liquid
(z I --b)
J
= ppl(z),
(-b
= 0;
(z > b)
=
ga(R12),
(zl I
b)
=
0,
(21 >
4
< zI
b)
(8)
where b is the half-width of the interfacial region. From eqs. (2), (4) and (8), we find that ri2’ is b-ma rZ2)
=
p:)2ga_cb!;in(zl
+
212) bl
-
j"4 w*
-b-a. (2b
-
p'lisga_~&A(zl
,[$
+
212) da
+y[:hn)
-=c
4~1
,712) +
212) dz1 -
(0 < 212 I 2b)
d.4 -b--se
=
P:‘2ga
{Jbz;A(z,)
dzl
$&@d
421
+
212) da
(-2b = fp2ga{_[
Z",&l)
s" z; da}.
dzl -
(212 I
<
-
212
I
0)
-2b)
(9)
-b
We first assume that p(i)(z) decreases linearly in the interface value in the liquid phase to zero, its value in the vapor phase, A(z) = &[I -
(z/b)].
From eqs. (9) we obtain rh2’ = pC)2ga( -Zis), =
p:)2ga[-(b/3)
(2b -C 212) -
(z12P) -
(z:,W)
+
(&Wb2)1, (0-c212
5
2b)
from its
SURFACE
THICKNESS
= &)“g&-(b/3)
-
(Z12/2)
OF LIQUID -
(zQ4b)
-
4He AND
209
(2f2/24b2)l, (-2b
= 0; I’!‘)
3He
< 212 I
(212 <
0)
-q
(-212/2) I’h2) + (b2/6) ~f’~ga.
=
(10)
Substitution of eqs. (10) into eq. (5) and integration over the surface of the unit sphere result in the final equation for the surface tension, y
4$+
_&)Rsg,(R)dR
+
0
(11) 2b
We now assume that p(i)(z) decreases in the interfacial region as a cubic, A(z) =
(23/4b3)
(32/4b) + &.
-
For this function the first derivative of p(i)(z) is also continuous at the boundaries of the inter-facial region. In this case we find from eqs. (9) rh”’ = pyga( -zrs), p&‘j2ga[-(93/35)
=
=
(2b -C 212) -
(.zr2/2)-
(3z;,/lOb) + (zt2/16b3)
(3zf2/160b4) + (zi2/2240b’J)], (93/35) -
pf’2ga[-
+ (3zf2/160b4) -
(z12/2) -
(32:,/l
(z:,/2240bQ
= 0, r$“’
=
(0 < 212 I
Ob) + (z;,/l6b3) + (-2b
<
(212 I
( -z12/2)
2b)
212 I
0)
--2b)
rb2) + (b2/10) pL1j2g,.
(12)
Substituting eqs. (12) into eq. (5) and integrating over the surface of the unit sphere as before, we obtain 2b
y,z?.-
(1s
R4
b
12800b5
R5ga(R) dR +
0
,p + --f
s =dV m
R4ga(R) dR.
2b
Eqs. (11) and (13) relate the surface tension to the interfacial thickness for two assumed functional dependencies of p(l)(z) in the interfacial region. If desired, other forms for A(z) could be assumed and used in eqs. (9). We
210
D. D. FITTS
shall, however, limit the discussion here to the linear and cubic cases. In the limit of vanishing width of the interfacial zone, cqs. (11) and (13) both reduce to the equation for surface tension derived by Kirkwood and Buff 7) under the approximation of eq. (7). Application to liquid helium. We now apply eqs. (11) and (13) to the calculation of the surface tension of liquid 4He and sHe at 1.4 K. By comparing the calculated values as a function of the width parameter b with the experimental values, we can estimate the interfacial thickness in each case. We use the Lennard-Jones potential of interatomic force with a hard core, V(R) = 4C[(U/R)i2 -
(o/R)6],
(R > 0)
= co,
(R i
C)
(14)
and the de Boer4) constants for helium, E = 14.10 x lo-16 erg and G = = 2.56 k. The densities at 1.4 K are 0.1452 g/cm3 for 4He as determined by Keesom*) and 0.08093 g/cm3 for aHe as determined by Kerr-a). Theoretical radial distribution functions have been tabulated for both 4He and sHe at 1.4 K by Mazo and Kirkwood 109ii). Because of the hard core in eq. (14), the first term in both eqs. (11) and (13) is zero whenever the interfacial width is equal to or less than one molecular diameter 0.
TABLE Surface 2b/a
tension
I
(dyne/cm)
yfromeq.
of 4He at 1.4 K y from eq. (13)
(11)
0.690
0.690 0.617
3
0.579 0.449
4
0.367
0.510 0.429
5
0.310
0.370
0 2
0.34
exp. value
TABLE Surface tension
II
(dyne/cm)
of sHe at 1.4 K
2b/o
y from eq. (11)
y from eq. (13)
5
0.173
0.206
6 7
0.150 0.132
0.180 0.161
8
0.118
0.144
exp. value
0.125; 0.132
SURFACE
THICKNESS
OF LIQUID
4He AND sHe
Values of the surface tension of 4He and 3He calculated (13) for various
values of the interfacial
211
from eqs. (11) and
width 2b are presented
in tables
I
and II. The calculated surface tension is strongly dependent on the interfacial width and is dependent to a lesser degree on the assumed shape of the singlet density distribution function in the interfacial zone. Tables I and II also list the experimental values for the surface tension of liquid 4He13) and 3Hel3114) at 1.4 K. Kirkwood and Buff 7) were able to reproduce the surface tension of liquid argon under the assumption of eq. (7), i.e., zero interfacial width. In order to reproduce the surface tensions of liquid 4He and 3He, we require interfacial widths of approximately 4 to 5 and 7 to 8 molecular diameters, respectively. The structures of both liquid 4He and liquid 3He are abnormally expanded as compared with a classical fluid such as liquid argon because of the high zero-point energy of the quantum fluids. Since quantum effects in 3He are greater than in 4He, the structure of 3He is more expanded than that of 4He. Thus, the progressively increasing values of surface thickness as estimated from the surface tension of liquid argon, liquid 4He, and liquid 3He are in qualitative agreement with expectation. According to Fresnel’s law of reflection, if the transition between the vapor and liquid phases is absolutely abrupt, light reflected from the surface of the liquid phase is plane polarized when the angle of incidence is Brewster’s angle. If the transition is gradual, the reflected light is elliptically polarized. Using this principle, Rayleighls) has determined that the interfacial width for water at room temperature is approximately one molecular diameter. Even though water molecules are polar, this experiment is consistent with Kirkwood and Buff’s7) consideration of liquid argon. Experiments along these lines on liquid 4He and liquid 3He might throw some light on the validity of our conclusions regarding their interfacial widths.
REFERENCES 1) Fitts, D. D., Thesis Yale University (1957). 2) Brout, R. and Nauenberg, M., Phys. Rev. 112 (1958) 1451. 3) Gibbs, J. W., Collected Works, Yale University Press (New Haven, 1948) Vol. I, p. 219. 4) de Boer, J., Reports Prog. Phys. 12 (1949) 305. 5) Irving, J. H. and Zwanzig, R. W., J. them. Phys. 19 (1951) 1173. 6) Buff, F. P., J. them. Phys. 23 (1955) 419. 7) Kirkwood, J. G. and Buff, F. P., J. them. Phys. 17 (1949) 338. 8) Keesom, W. H., Helium, Elsevier (Amsterdam, 1942) p. 207. 9) Kerr, E. C., Phys. Rev. 96 (1954) 55 1. 10) Mazo, R. M. and Kirkwood, J. G., Proc. Nat. Acad. Sci. (Washington) 41 (1955) 204. 11) Mazo, R. M. and Kirkwood, J. G., Phys. Rev. 100 (1955) 1787.
212 12) 13) 14) 15)
SURFACE
THICKNESS
OF LIQUID
4He AND 3He
Allen, J. F. and Misener, A. D., Proc. Camb. Phil. Sot. 34 (1938) 299. Lovejoy, D. R., Canad. J. Phys. 33 (1955) 49. Zinov’eva, K. N., Soviet Physics J.E.T.P. 1 (1955) 173; 2 (1956) 774. Rayleigh, Lord, Phil. Mag. 33 (1892) 1; see also Adam, N. K., The Physics and Chemistry of Surfaces, Oxford University Press (Oxford, 1938), second edition, p. 5.
42 (1969)
205-212
0 North-Holland
THEORETICAL SURFACE
Publishing
Co., Amsterdam
DETERMINATION
THICKNESS
OF LIQUID
OF
4He AND
sHe
D. D. FITTS Department
of Chemistry, Philadelphia,
Received
University
of Pennsylvania,
Pennsylvania,
USA
17 June 1968
Synopsis The quantum statistical-mechanical theory of surface tension is applied to liquid 4He and to liquid sHe at 1.4 K. This theory relates the surface tension of the thickness of the region between the liquid and vapor phases. By a comparison of the calculated surface tension as a function of interfacial thickness with the experimental value, the surface thickness of liquid 4He is found to be 4 to 5 molecular diameters and that of liquid sHe 7 to 8 molecular diameters.
Introdzlctios. From the principles of quantum statistical mechanics, Fit&r) and Brout and Nauenbergs) have independently formulated a quantum-mechanical molecular theory of surface tension between two fluid phases for the case of a planar interfacial zone. They obtained a general expression relating the suface te.rsion to the kinetic energy density, the potential of intermolecular force, and the molecular distribution functions. In the application of this expression to a liquid phase in equilibrium with a vapor phase, it is necessary to consider the thi:kness of the interfacial region. Thus, this theory in effe-_t relates the surface tension to the surface thickness. The purpose of this article is to apply the quantum theory of surface tension to liquid 4He and liquid sHe in order to study their respective surface thicknesses. By comparing calculated and experimental values of the surface tension in each case, a theoretical estimate of the interfacial thickness can be made. Quantum theory of surface tension. The system under consideration consists of two homogeneous fluid phases a and fi separated by a planar transition zone. A Gibbss) dividing surface between the two phases is selected and a Cartesian coordinate system (x, y, Z) is defined with the dividing surface as the (x, y) plane and with the z axis normal to the dividing surface and directed from phase a to phase j3. The choice of a specific dividing surface is arbitrary and will be deferred. 205
ID. I). FITTS
206
The theory
makes
use of two molecular
distribution
functions4):
the
singlet density p(1)(R) and the pair density p(2)(R1, R12). The quantity #1)(R) dv is the average number of molecules in volume element dv at a point R in the fluid and the quantity p@)(Rl, Rlz) dv1 dv12 is the average number of molecular pairs, one member of which is located in volume element dv1 at RI and the other in dv12 at point R12 in the relative configuration space of the pair. The pair correlation function g(R, Rlz) is defined by the relation /J”)(R1, R12) = P(R1) For a planar interface, z and the pair density interface becomes P(~)(z~, R12)
=
@(Rz)
g(Rl>
R12).
(1)
the singlet density depends only on the coordinate only on z1 and R12. Therefore, eq. (1) for a planar
p(l)(a)
p(l)@1
+
212) &l,
R12).
(4
In the interior of each phase, the singlet density becomes independent of z, while the pair density and the pair correlation function depend only on the the singlet density is the macroscopic pair distance R12. Furthermore, molecular or number density and the pair correlation function is identical with the radial distribution function of the theory of liquids. Fitts1) and Brout and Nauenberg2) showed that the surface tension y for a quantum fluid is given by
-
[tT(Zl)
-
tN(Zl)]
(3)
h,
s --oo
where V(R12) is the potential of intermolecular force acting between a pair of molecules separated by a distance R12 and tT, tN are the tangential (x or y) and normal (z) components, respectively, of the average kinetic energy density. The quantities r!” are defined by r;‘)
= ~z’&A2)(z1, R12) -cc
&;]
dz1,
where p$ has the value pF)(R12), the pair density in the interior of phase a, for z1 negative and the value ~r)(R1~), the pair density in the interior of phase /?, for z1 positive. Thus, r, ~2) has a value different from zero only in the interfacial region, where the singlet and pair densities change from the uniform values in phase cc to the uniform values in phase j3. In deriving eq. (3) Fitts1) employed the equations of quantum hydrodynamicss) and the relationship derived by Buff6) between the surface
SURFACE
tension
THICKNESS
and the macroscopic
OF LIQUID
stress tensor.
Brout
*He AND
3He
and Neuenberg2),
other hand, used the surface energy for a system in a single quantum
207
on the state
to obtain eq. (3). The two deriviations are completely equivalent. The second integral on the right-hand side of eq. (3) arises from the anisotropy of the average kinetic energy density. The quantity [TV - TV] vanishes in the interior of each phase, and for an interfacial region of zero width the integral is zero. Moreover, for a classical fluid, the average kinetic energy density is isotropic and the integral vanishes identically. For the purpose of calculation, we now follow the original treatmentslps) and introduce the approximation that the average kinetic energy density is locally isotropic, so that tr(zr) = tN(Zr). Under this assumption the second integral in eq. (3) vanishes and eq. (3) becomes y=
s dV(R12) 1
(z~~T’~~’ + &fJ~‘)
R12
dur2.
dRl2
This expression for the surface tension is identical with that obtained by Kirkwood and Buff 7) for classical fluids except that here the pair density p(s)(zr, Rrs) must be determined by a quantum-mechanical theory. Evalzlation of T, (‘) . We now evaluate eq. (5) for a particular Gibbs dividing surface. Since eq. (5) is independent of the choice of dividing surface between phases cc and 8, we may use that surface which makes the superficial density of matter zero; i.e., such that
rp = y
[p(l)(z) - p$] dz = 0,
--oo
(2) This surface is called the equimolecuwhere p$) is defined analogously to pas. lar dividing surface because a hypothetical system in which the singlet density p(l)(z) changes discontinuously from its value in the interior of phase u to its value in the interior of phase /I at the dividing surface would contain the same number of molecules as the actual system. At this point we specialize our discussion to a liquid phase a in equilibrium with a vapor phase ,i3 and assume that the number density in the vapor phase pf’ is essentially zero. Kirkwood and Buff 7) have evaluated eq. (5) for a transition zone which has been shrunk to a mathematical surface coincident with the Gibbs dividing surface, so that p(r)(z) = p(r) OL’ = 0.
(2 < 0) (2 ’
0)
(7)
We now solve eq. (5) for a liquid to vapor transition zone of nonzero thickness. We assume that the boundaries of the interfacial region are
D. D. FITTS
208
distinct and that the composition of each phase remains homogeneous up to the boundary of the interfacial zone. Since the exact behavior of the singlet density p(i)(z) in the interfacial region is not known, we must assume a specific functional dependence on z. For convenience of calculation, we assume that the singlet density in the interface is continuous and is antisymmetric about the point z = 0, p(i)(z) = pf’/2. Thus, eq. (6) is obeyed. We also assume that the pair correlation function g(zi, Rls) in the interfacial zone is equal to the radial distribution phase a. In this approximation, we have p(z)
=
g(z1, RlZ)
p’l’ a
function gLy(Ris) of the liquid
(z I --b)
J
= ppl(z),
(-b
= 0;
(z > b)
=
ga(R12),
(zl I
b)
=
0,
(21 >
4
< zI
b)
(8)
where b is the half-width of the interfacial region. From eqs. (2), (4) and (8), we find that ri2’ is b-ma rZ2)
=
p:)2ga_cb!;in(zl
+
212) bl
-
j"4 w*
-b-a. (2b
-
p'lisga_~&A(zl
,[$
+
212) da
+y[:hn)
-=c
4~1
,712) +
212) dz1 -
(0 < 212 I 2b)
d.4 -b--se
=
P:‘2ga
{Jbz;A(z,)
dzl
$&@d
421
+
212) da
(-2b = fp2ga{_[
Z",&l)
s" z; da}.
dzl -
(212 I
<
-
212
I
0)
-2b)
(9)
-b
We first assume that p(i)(z) decreases linearly in the interface value in the liquid phase to zero, its value in the vapor phase, A(z) = &[I -
(z/b)].
From eqs. (9) we obtain rh2’ = pC)2ga( -Zis), =
p:)2ga[-(b/3)
(2b -C 212) -
(z12P) -
(z:,W)
+
(&Wb2)1, (0-c212
5
2b)
from its
SURFACE
THICKNESS
= &)“g&-(b/3)
-
(Z12/2)
OF LIQUID -
(zQ4b)
-
4He AND
209
(2f2/24b2)l, (-2b
= 0; I’!‘)
3He
< 212 I
(212 <
0)
-q
(-212/2) I’h2) + (b2/6) ~f’~ga.
=
(10)
Substitution of eqs. (10) into eq. (5) and integration over the surface of the unit sphere result in the final equation for the surface tension, y
4$+
_&)Rsg,(R)dR
+
0
(11) 2b
We now assume that p(i)(z) decreases in the interfacial region as a cubic, A(z) =
(23/4b3)
(32/4b) + &.
-
For this function the first derivative of p(i)(z) is also continuous at the boundaries of the inter-facial region. In this case we find from eqs. (9) rh”’ = pyga( -zrs), p&‘j2ga[-(93/35)
=
=
(2b -C 212) -
(.zr2/2)-
(3z;,/lOb) + (zt2/16b3)
(3zf2/160b4) + (zi2/2240b’J)], (93/35) -
pf’2ga[-
+ (3zf2/160b4) -
(z12/2) -
(32:,/l
(z:,/2240bQ
= 0, r$“’
=
(0 < 212 I
Ob) + (z;,/l6b3) + (-2b
<
(212 I
( -z12/2)
2b)
212 I
0)
--2b)
rb2) + (b2/10) pL1j2g,.
(12)
Substituting eqs. (12) into eq. (5) and integrating over the surface of the unit sphere as before, we obtain 2b
y,z?.-
(1s
R4
b
12800b5
R5ga(R) dR +
0
,p + --f
s =dV m
R4ga(R) dR.
2b
Eqs. (11) and (13) relate the surface tension to the interfacial thickness for two assumed functional dependencies of p(l)(z) in the interfacial region. If desired, other forms for A(z) could be assumed and used in eqs. (9). We
210
D. D. FITTS
shall, however, limit the discussion here to the linear and cubic cases. In the limit of vanishing width of the interfacial zone, cqs. (11) and (13) both reduce to the equation for surface tension derived by Kirkwood and Buff 7) under the approximation of eq. (7). Application to liquid helium. We now apply eqs. (11) and (13) to the calculation of the surface tension of liquid 4He and sHe at 1.4 K. By comparing the calculated values as a function of the width parameter b with the experimental values, we can estimate the interfacial thickness in each case. We use the Lennard-Jones potential of interatomic force with a hard core, V(R) = 4C[(U/R)i2 -
(o/R)6],
(R > 0)
= co,
(R i
C)
(14)
and the de Boer4) constants for helium, E = 14.10 x lo-16 erg and G = = 2.56 k. The densities at 1.4 K are 0.1452 g/cm3 for 4He as determined by Keesom*) and 0.08093 g/cm3 for aHe as determined by Kerr-a). Theoretical radial distribution functions have been tabulated for both 4He and sHe at 1.4 K by Mazo and Kirkwood 109ii). Because of the hard core in eq. (14), the first term in both eqs. (11) and (13) is zero whenever the interfacial width is equal to or less than one molecular diameter 0.
TABLE Surface 2b/a
tension
I
(dyne/cm)
yfromeq.
of 4He at 1.4 K y from eq. (13)
(11)
0.690
0.690 0.617
3
0.579 0.449
4
0.367
0.510 0.429
5
0.310
0.370
0 2
0.34
exp. value
TABLE Surface tension
II
(dyne/cm)
of sHe at 1.4 K
2b/o
y from eq. (11)
y from eq. (13)
5
0.173
0.206
6 7
0.150 0.132
0.180 0.161
8
0.118
0.144
exp. value
0.125; 0.132
SURFACE
THICKNESS
OF LIQUID
4He AND sHe
Values of the surface tension of 4He and 3He calculated (13) for various
values of the interfacial
211
from eqs. (11) and
width 2b are presented
in tables
I
and II. The calculated surface tension is strongly dependent on the interfacial width and is dependent to a lesser degree on the assumed shape of the singlet density distribution function in the interfacial zone. Tables I and II also list the experimental values for the surface tension of liquid 4He13) and 3Hel3114) at 1.4 K. Kirkwood and Buff 7) were able to reproduce the surface tension of liquid argon under the assumption of eq. (7), i.e., zero interfacial width. In order to reproduce the surface tensions of liquid 4He and 3He, we require interfacial widths of approximately 4 to 5 and 7 to 8 molecular diameters, respectively. The structures of both liquid 4He and liquid 3He are abnormally expanded as compared with a classical fluid such as liquid argon because of the high zero-point energy of the quantum fluids. Since quantum effects in 3He are greater than in 4He, the structure of 3He is more expanded than that of 4He. Thus, the progressively increasing values of surface thickness as estimated from the surface tension of liquid argon, liquid 4He, and liquid 3He are in qualitative agreement with expectation. According to Fresnel’s law of reflection, if the transition between the vapor and liquid phases is absolutely abrupt, light reflected from the surface of the liquid phase is plane polarized when the angle of incidence is Brewster’s angle. If the transition is gradual, the reflected light is elliptically polarized. Using this principle, Rayleighls) has determined that the interfacial width for water at room temperature is approximately one molecular diameter. Even though water molecules are polar, this experiment is consistent with Kirkwood and Buff’s7) consideration of liquid argon. Experiments along these lines on liquid 4He and liquid 3He might throw some light on the validity of our conclusions regarding their interfacial widths.
REFERENCES 1) Fitts, D. D., Thesis Yale University (1957). 2) Brout, R. and Nauenberg, M., Phys. Rev. 112 (1958) 1451. 3) Gibbs, J. W., Collected Works, Yale University Press (New Haven, 1948) Vol. I, p. 219. 4) de Boer, J., Reports Prog. Phys. 12 (1949) 305. 5) Irving, J. H. and Zwanzig, R. W., J. them. Phys. 19 (1951) 1173. 6) Buff, F. P., J. them. Phys. 23 (1955) 419. 7) Kirkwood, J. G. and Buff, F. P., J. them. Phys. 17 (1949) 338. 8) Keesom, W. H., Helium, Elsevier (Amsterdam, 1942) p. 207. 9) Kerr, E. C., Phys. Rev. 96 (1954) 55 1. 10) Mazo, R. M. and Kirkwood, J. G., Proc. Nat. Acad. Sci. (Washington) 41 (1955) 204. 11) Mazo, R. M. and Kirkwood, J. G., Phys. Rev. 100 (1955) 1787.
212 12) 13) 14) 15)
SURFACE
THICKNESS
OF LIQUID
4He AND 3He
Allen, J. F. and Misener, A. D., Proc. Camb. Phil. Sot. 34 (1938) 299. Lovejoy, D. R., Canad. J. Phys. 33 (1955) 49. Zinov’eva, K. N., Soviet Physics J.E.T.P. 1 (1955) 173; 2 (1956) 774. Rayleigh, Lord, Phil. Mag. 33 (1892) 1; see also Adam, N. K., The Physics and Chemistry of Surfaces, Oxford University Press (Oxford, 1938), second edition, p. 5.