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Interfacial energies
JOURNAL
OF COLLOID SCIENCE
11,501-507 (1956)
INTERFACIAL ENERGIES E. A. Moelwyn-Hughes Department of Physical Chemistry, University of Cambridge, Cambridge, England Received December 20, 1955
The energy of molecules in an interface, like t h a t of molecules generally, consists of two parts: the kinetic energy depends on molecular motion, and the potential energy on the relation in which a molecule stands to the other molecules in the system. In this paper we consider only the potential energy. We shall review briefly, and in some respects extend, the theories hitherto advanced. The Energy of Adsorption on a Plane Surface
Let a molecule be placed at a point P (Fig. 1) which is at a vertical distance d above the plane surface of a condensed phase containing n , molecules per cubic centimeter. It is clear from Fig. 1 that its energy of interaction with all the molecules in the condensed phase is (~ = 2~n,~
/7/7 dz
~d
¢(a)ada,
[1]
~z
where ¢(a) is the energy of interaction of the molecule placed at P with an isolated molecule of the kind appearing in the condensed phase. This we shall take to be given by Mie's equation (1) [2]
¢(a) = Aa-'* - Ba -m, where n and m are integers (n > m), and A and B are constants. Then 4, = 2~rn,
(n-
2)(n-
3)
d ~-3
(m-
2)(m-
3) " d ~-3 "
[3]
If we ignore the energy due to repulsion, and take m = 6, we obtain ~rn. B 6d a
[41
This is the result obtained by F. London (2), and successfully applied by him to compute the energy of desorption of various gases from charcoal. His well-known dispersion theory allows of an independent evaluation of B. Among the striking experiments confirming London's theory m a y be 501
502
E . A. M O E L W ~ Y N - H U G H E S
Gas Pha~e
Adsorbent.
FIG. 1. To determine the interaction energy of a molecule of gas with a solid, when the molecule is at a vertical distance d above a plane surface. P
Ga~ Pha~e
FIG. 2. A molecule adsorbed in the plane of the surface, above a hemispherical cavity. m e n t i o n e d those of R. M. Barrer (3) (hydrogen on graphite) a n d W . J. C. 0 r r (4) (argon on potassium chloride).
The Energy of Adsorption on a Hemispherical Cavity in a Plane Surface W h e n a molecule is centrally situated on a hemispherical c a v i t y in an otherwise plane surface (Fig. 2), its energy of interaction with the condensed phase is clearly
f/
'~ = 2~rn,~
¢(a)a2da B
--2~'n,
(n-3)
d ~-3
(m -
o
3)
1]
[51
INTERFACIAL ENERGIES
503
If we again ignore repulsion and take m as 6, we find
=
27rn~B --
3d 3 ,
[6]
which is four times as great as the former result (J. It. de Boer and J. F. H. Custers (5)). The force attracting a molecule to a curved surface is thus greater than that attracting it, at the same distance, to a plane surface. If, therefore, a surface thought to be planar is in fact marred by cracks and crevices, adsorption first occurs in them. As the fissures are gradually filled up, there is progressively more adsorption on the fiat surface, and the heat of adsorption consequently gets smaller. J. K. Roberts (6), however, has shown t h a t the experimental heat of adsorption from a plane surface increases as the surface becomes more and more covered. The two effects are in opposite directions, and taken jointly can account for the fact that the heat of adsorption in certain instances passes through a minimum (7).
A Comparison of the Two Energies of Adsorption Equations [3] and [5] hold for arbitrary values of the distance d. The nearest distance to which a molecule can be drawn to the surface is given by the equation d~-~ _ (m - 2)A (n -- 2)B
[71
in the planar system, and A d~-~ -
B
[S]
in the hemispherical system. The corresponding minimum energies are
2~rn~B ¢0-
(m--2)~0
[ -s
l (n-
1 3)
(m-3)
and ~o
-
2~rn~B (n d~0 -3
3)
(m
3) --
]
1 '
[9]
[10] "
Hence
~o (curved) I (m - 2~-37 ( 1 • o(planar) = ~.(n 2im-3J ~--~/
[11]
This ratio is smaller and more exact than t h a t found using arbitrary values of d. Thus, for example, with m = 6 and n = 9, the ratio is 16/7 = 2.29.
The Frequency of Vibration of an Adsorbed Molecule B y expanding • - ¢0 in a power series of (d - do), the gain in energy when the adsorbed molecule is displaced slightly from its equilibrium posi-
50Ax
E. A. MOELWYN-ItUGHES
tion is found to v a r y in proportion to (d - d0)2. Its motion in the dimension normal to the surface is consequently a simple harmonic vibration, with a frequency in the planar model given by 1
/ w / ( m -- 3)(n -- 3)~o
[12]
where X0 = - ~ 0 , and a is the mass of the adsorbed molecule. The Energy of Cohesion between Planar Surfaces
Instead of the molecule indicated by P in Fig. 1, let us consider a number of molecules placed in a horizontal slab of thickness ~d and of extension 0 cm. 2 at the same height d from the surface of the condensed phase c~. The number of molecules in the slab is Ono~d , where no is the concentration of molecules in the upper phase. The total interaction energy between all the molecules in one phase with all the molecules in the other phase is U =
[13]
I m o 08d,
where 4~ is given b y Eq. [3]. Hence
[ U = 2~.n~O
(n -
1
2)(n -
3)(n - 4) "d ~-s
[14] -
(m -- 2) (m -- 3) (m -- 4) The nearest distance to which the two phases approach is given by the equation
d~__~ = ( m -
2)(m-
3) A
(n - 2)(n -- 3) " ~ '
[151
and the minimum potential energy of the system is Uo = -
(n -
m ) 2 n n , noBO
= - b n , noO,
[16]
(m - 2)(m - 3)(m - 4)(n -- 4)d~ -4 where b is a positive constant. This is the energy of cohesion. The reference state of zero energy is that of the isolated phases at an infinite distance apart. An expression resembling Eq. [16] has been derived by T. S. Wheeler (8), who regarded it, however, as only one of three components in the total interaction energy. We are now in a position to calculate the excess potential energy, U , , which a pure substance possesses when it exists in two phases, separated b y a plane surface of area 0, over the sum of the energies of the two isolated phases. I t follows from Eq. [16] that U~ = (n~ -
no)2bO.
An equation of this form has been derived by R. H. Fowler (9).
[17]
INTERFACIAL ENERGIES
505
The excess energy due to the creation of an interracial area of extent 0 between two phases of different substances is U~ = (b~n~ 2 + b~n~ 2 - 2b~n~n~)O.
[18]
When m = 6 and the forces of attraction are due to dispersion, the constants b can be calculated using London's theory (2). The Surface Tension of a Pure Liquid
When a liquid is in thermal equilibrium with its vapor, the excess free energy per unit area of surface is numerically equal to the surface tension. In the absence of thermal motion, the free energy equals the total energy. Hence the extrapolated value of the surface tension at T = 0 is [191
U~ 0 _ (n~ - n~)2b.
70-
When the concentration of molecules in the vapor phase can be neglected, we have U~ _ n2b. "Yo- 0
On using Eq. [16], we obtain the result (n - m ) 2 ~ n ~ B ~'o =
(m -- 2)(m
-
3)(m
-
-
4)(n
- - 4 ) a~0-4"
[201
Hildebrand, Wakeham, and Boyd (10) have shown that n = 9 and m = 6 for mercury. Hence ~o =
(~) ~
n~2B doz •
[21]
Moreover, for this liquid, B = 4.90 X 10-Ss erg-cm. G(11) and do = 1.944 A. The computed value of ~0 is thus 344 ergs/cm. 2. B y applying to molecules in the surface layer, and in the layers immediately beneath it, the radial distribution function established by Hildebrand for the liquid, G. Jura (12) calculates an excess surface potential energy of 500 ergs/cm. 2, in closer agreement with the experimental value of "/0, which is 463.6 ergs/cm? (13). Stef an ' s Ratio
J. Stefan (14) argued that the energy gained by a molecule on being transferred from the bulk of the liquid to its surface must be equal to the energy gained during a further transfer from the surface to the vapor. His argument has been corrected on theoretical (15) and experimental (16) grounds. The ratio of the excess energy for molecules in the surface to the latent heat of vaporization has been given an extremely simple interpreta-
506
E. A. MOELWYN-HUGtIES
tion in terms of the coordination numbers in the interface and in the bulk liquid (17). Another means of arriving at Stefan's ratio is provided by the general theory of intermolecular forces. If the molecules in a fluid are uniformly distributed, and fluctuations in the concentration n, are ignored, the total potential energy of N , molecules of a liquid is
t221
U = 2~rN,n~ f ¢(a)a~da. On using Mie's equation [2], the minimum value of U is found to be
2~'N"n~B I
U0 =
a ~ -3
I
(m -
3-)
1
(n - 3)
1
'
[23]
where n-~
=
a0
A _ B"
[24]
The excess energy ascribable to a surface of extension 0 cm. 2 is found from Eqs. [16] and [17] to be
27rn~(n - m)BO U~ =
(m -
2)(m
-
3)(m
-
4)(n
-
4) d~ -4'
[25]
where do is given by Eq. [15]. The ratio of the heat of vaporization (at the absolute zero of temperature) to the excess energy due to the surface is thus Go
(~vo/odo)
- - U--~ =
n.
(m - 2)(m - 4)(n - 4 ) . (do~ m-~ (n -
3)
\~/
"
[26]
The first term on the right-hand side is clearly unity. Hence, Stefan's ratio becomes ___
=
UOu~
(m - 4)(n - 4)
(
m -
~
2
(~__~)
(~3~(~) . \n - 2] ~-~
[27]
With m = 6, we thus anticipate ratios in the neighborhood of 1.91 (n = 9), 2.60 (n = 12), and 3.12 (n = 15). These agree with the experimental ratios, as given by R. Haul (18) and tt. Volkmann (19), which are 2.20 (He), 2.37 (Ne), 2.38 (A), 2.15 (Hg), 3.10 (CS2), and 3.20 (CC14). This article has been written as a tribute to Sir Eric Rideal. While writing it, I had also inevitably in mind the memory of his friend, the late Sir John Lennard-Jones. It has been one of the author's greatest privileges to enjoy over a number of years a close collaboration and friendship with both.
INTERFACIAL ENERGIES
507
I~EFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
MIE, G., Ann. Physik 11,657 (1903). LONDON, F., Z. physik. Chem. B l l , 222 (1930). BARBER, R. M., Proc. Roy. Soc. (London) A161, 476 (1937). ORB, W. J. C., Trans. Faraday Soc. 35, 1247 (1939). DE BOER, J. H., AND CUSTERS, J. F. H., Z. physik. Chem. B25,225 (1934). ROBERTS, J. K., "Some Problems in Adsorption." Cambridge, 1939. ~)OLANYI,M., AND WELKE, V., Z. physik. Chem. A132, 371 (1928). WHEELER, T. S., Trans. Natl. Inst. Sci. India 1,333 (1938). FOWLE~, R. H., Proe. Roy. Soc. (London) A159, 229 (1937). HILDEBRAND, J. H., WAKEHAM,H. R. R., AND BOYD, R. N., J. Chem. Phys. 7, 1094 (1939). MOELWYN-ttUGHES, E. A., J. Phys. & Colloid Chem. 55, 1246 (1951). JURA, G., J. Phys. & Colloid Chem. 52, 40 (1948). BInCu~sHAw, L. L., Phil. Mag. 12,596 (1931). STEFAN, J., Wied. Ann. 29,655 (1886). OSTWALD,W., Z. physik. Chem. 1, 45 (1887). HAn~INS, W. D., AND ROBEna'S, L. E., J. Am. Chem. Soc. 44, 653 (1922). WOLF, K. L., AND KLAPPI~OTH,K., Z. physik. Chem. B46, 276 (1940). HAUL, I~., Z. physik. Chem. A53, 331 (1943). VObK~ANN, H., F I A T Rev. Get. Sci. Phys. Chem. 30, 205 (1947).
OF COLLOID SCIENCE
11,501-507 (1956)
INTERFACIAL ENERGIES E. A. Moelwyn-Hughes Department of Physical Chemistry, University of Cambridge, Cambridge, England Received December 20, 1955
The energy of molecules in an interface, like t h a t of molecules generally, consists of two parts: the kinetic energy depends on molecular motion, and the potential energy on the relation in which a molecule stands to the other molecules in the system. In this paper we consider only the potential energy. We shall review briefly, and in some respects extend, the theories hitherto advanced. The Energy of Adsorption on a Plane Surface
Let a molecule be placed at a point P (Fig. 1) which is at a vertical distance d above the plane surface of a condensed phase containing n , molecules per cubic centimeter. It is clear from Fig. 1 that its energy of interaction with all the molecules in the condensed phase is (~ = 2~n,~
/7/7 dz
~d
¢(a)ada,
[1]
~z
where ¢(a) is the energy of interaction of the molecule placed at P with an isolated molecule of the kind appearing in the condensed phase. This we shall take to be given by Mie's equation (1) [2]
¢(a) = Aa-'* - Ba -m, where n and m are integers (n > m), and A and B are constants. Then 4, = 2~rn,
(n-
2)(n-
3)
d ~-3
(m-
2)(m-
3) " d ~-3 "
[3]
If we ignore the energy due to repulsion, and take m = 6, we obtain ~rn. B 6d a
[41
This is the result obtained by F. London (2), and successfully applied by him to compute the energy of desorption of various gases from charcoal. His well-known dispersion theory allows of an independent evaluation of B. Among the striking experiments confirming London's theory m a y be 501
502
E . A. M O E L W ~ Y N - H U G H E S
Gas Pha~e
Adsorbent.
FIG. 1. To determine the interaction energy of a molecule of gas with a solid, when the molecule is at a vertical distance d above a plane surface. P
Ga~ Pha~e
FIG. 2. A molecule adsorbed in the plane of the surface, above a hemispherical cavity. m e n t i o n e d those of R. M. Barrer (3) (hydrogen on graphite) a n d W . J. C. 0 r r (4) (argon on potassium chloride).
The Energy of Adsorption on a Hemispherical Cavity in a Plane Surface W h e n a molecule is centrally situated on a hemispherical c a v i t y in an otherwise plane surface (Fig. 2), its energy of interaction with the condensed phase is clearly
f/
'~ = 2~rn,~
¢(a)a2da B
--2~'n,
(n-3)
d ~-3
(m -
o
3)
1]
[51
INTERFACIAL ENERGIES
503
If we again ignore repulsion and take m as 6, we find
=
27rn~B --
3d 3 ,
[6]
which is four times as great as the former result (J. It. de Boer and J. F. H. Custers (5)). The force attracting a molecule to a curved surface is thus greater than that attracting it, at the same distance, to a plane surface. If, therefore, a surface thought to be planar is in fact marred by cracks and crevices, adsorption first occurs in them. As the fissures are gradually filled up, there is progressively more adsorption on the fiat surface, and the heat of adsorption consequently gets smaller. J. K. Roberts (6), however, has shown t h a t the experimental heat of adsorption from a plane surface increases as the surface becomes more and more covered. The two effects are in opposite directions, and taken jointly can account for the fact that the heat of adsorption in certain instances passes through a minimum (7).
A Comparison of the Two Energies of Adsorption Equations [3] and [5] hold for arbitrary values of the distance d. The nearest distance to which a molecule can be drawn to the surface is given by the equation d~-~ _ (m - 2)A (n -- 2)B
[71
in the planar system, and A d~-~ -
B
[S]
in the hemispherical system. The corresponding minimum energies are
2~rn~B ¢0-
(m--2)~0
[ -s
l (n-
1 3)
(m-3)
and ~o
-
2~rn~B (n d~0 -3
3)
(m
3) --
]
1 '
[9]
[10] "
Hence
~o (curved) I (m - 2~-37 ( 1 • o(planar) = ~.(n 2im-3J ~--~/
[11]
This ratio is smaller and more exact than t h a t found using arbitrary values of d. Thus, for example, with m = 6 and n = 9, the ratio is 16/7 = 2.29.
The Frequency of Vibration of an Adsorbed Molecule B y expanding • - ¢0 in a power series of (d - do), the gain in energy when the adsorbed molecule is displaced slightly from its equilibrium posi-
50Ax
E. A. MOELWYN-ItUGHES
tion is found to v a r y in proportion to (d - d0)2. Its motion in the dimension normal to the surface is consequently a simple harmonic vibration, with a frequency in the planar model given by 1
/ w / ( m -- 3)(n -- 3)~o
[12]
where X0 = - ~ 0 , and a is the mass of the adsorbed molecule. The Energy of Cohesion between Planar Surfaces
Instead of the molecule indicated by P in Fig. 1, let us consider a number of molecules placed in a horizontal slab of thickness ~d and of extension 0 cm. 2 at the same height d from the surface of the condensed phase c~. The number of molecules in the slab is Ono~d , where no is the concentration of molecules in the upper phase. The total interaction energy between all the molecules in one phase with all the molecules in the other phase is U =
[13]
I m o 08d,
where 4~ is given b y Eq. [3]. Hence
[ U = 2~.n~O
(n -
1
2)(n -
3)(n - 4) "d ~-s
[14] -
(m -- 2) (m -- 3) (m -- 4) The nearest distance to which the two phases approach is given by the equation
d~__~ = ( m -
2)(m-
3) A
(n - 2)(n -- 3) " ~ '
[151
and the minimum potential energy of the system is Uo = -
(n -
m ) 2 n n , noBO
= - b n , noO,
[16]
(m - 2)(m - 3)(m - 4)(n -- 4)d~ -4 where b is a positive constant. This is the energy of cohesion. The reference state of zero energy is that of the isolated phases at an infinite distance apart. An expression resembling Eq. [16] has been derived by T. S. Wheeler (8), who regarded it, however, as only one of three components in the total interaction energy. We are now in a position to calculate the excess potential energy, U , , which a pure substance possesses when it exists in two phases, separated b y a plane surface of area 0, over the sum of the energies of the two isolated phases. I t follows from Eq. [16] that U~ = (n~ -
no)2bO.
An equation of this form has been derived by R. H. Fowler (9).
[17]
INTERFACIAL ENERGIES
505
The excess energy due to the creation of an interracial area of extent 0 between two phases of different substances is U~ = (b~n~ 2 + b~n~ 2 - 2b~n~n~)O.
[18]
When m = 6 and the forces of attraction are due to dispersion, the constants b can be calculated using London's theory (2). The Surface Tension of a Pure Liquid
When a liquid is in thermal equilibrium with its vapor, the excess free energy per unit area of surface is numerically equal to the surface tension. In the absence of thermal motion, the free energy equals the total energy. Hence the extrapolated value of the surface tension at T = 0 is [191
U~ 0 _ (n~ - n~)2b.
70-
When the concentration of molecules in the vapor phase can be neglected, we have U~ _ n2b. "Yo- 0
On using Eq. [16], we obtain the result (n - m ) 2 ~ n ~ B ~'o =
(m -- 2)(m
-
3)(m
-
-
4)(n
- - 4 ) a~0-4"
[201
Hildebrand, Wakeham, and Boyd (10) have shown that n = 9 and m = 6 for mercury. Hence ~o =
(~) ~
n~2B doz •
[21]
Moreover, for this liquid, B = 4.90 X 10-Ss erg-cm. G(11) and do = 1.944 A. The computed value of ~0 is thus 344 ergs/cm. 2. B y applying to molecules in the surface layer, and in the layers immediately beneath it, the radial distribution function established by Hildebrand for the liquid, G. Jura (12) calculates an excess surface potential energy of 500 ergs/cm. 2, in closer agreement with the experimental value of "/0, which is 463.6 ergs/cm? (13). Stef an ' s Ratio
J. Stefan (14) argued that the energy gained by a molecule on being transferred from the bulk of the liquid to its surface must be equal to the energy gained during a further transfer from the surface to the vapor. His argument has been corrected on theoretical (15) and experimental (16) grounds. The ratio of the excess energy for molecules in the surface to the latent heat of vaporization has been given an extremely simple interpreta-
506
E. A. MOELWYN-HUGtIES
tion in terms of the coordination numbers in the interface and in the bulk liquid (17). Another means of arriving at Stefan's ratio is provided by the general theory of intermolecular forces. If the molecules in a fluid are uniformly distributed, and fluctuations in the concentration n, are ignored, the total potential energy of N , molecules of a liquid is
t221
U = 2~rN,n~ f ¢(a)a~da. On using Mie's equation [2], the minimum value of U is found to be
2~'N"n~B I
U0 =
a ~ -3
I
(m -
3-)
1
(n - 3)
1
'
[23]
where n-~
=
a0
A _ B"
[24]
The excess energy ascribable to a surface of extension 0 cm. 2 is found from Eqs. [16] and [17] to be
27rn~(n - m)BO U~ =
(m -
2)(m
-
3)(m
-
4)(n
-
4) d~ -4'
[25]
where do is given by Eq. [15]. The ratio of the heat of vaporization (at the absolute zero of temperature) to the excess energy due to the surface is thus Go
(~vo/odo)
- - U--~ =
n.
(m - 2)(m - 4)(n - 4 ) . (do~ m-~ (n -
3)
\~/
"
[26]
The first term on the right-hand side is clearly unity. Hence, Stefan's ratio becomes ___
=
UOu~
(m - 4)(n - 4)
(
m -
~
2
(~__~)
(~3~(~) . \n - 2] ~-~
[27]
With m = 6, we thus anticipate ratios in the neighborhood of 1.91 (n = 9), 2.60 (n = 12), and 3.12 (n = 15). These agree with the experimental ratios, as given by R. Haul (18) and tt. Volkmann (19), which are 2.20 (He), 2.37 (Ne), 2.38 (A), 2.15 (Hg), 3.10 (CS2), and 3.20 (CC14). This article has been written as a tribute to Sir Eric Rideal. While writing it, I had also inevitably in mind the memory of his friend, the late Sir John Lennard-Jones. It has been one of the author's greatest privileges to enjoy over a number of years a close collaboration and friendship with both.
INTERFACIAL ENERGIES
507
I~EFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
MIE, G., Ann. Physik 11,657 (1903). LONDON, F., Z. physik. Chem. B l l , 222 (1930). BARBER, R. M., Proc. Roy. Soc. (London) A161, 476 (1937). ORB, W. J. C., Trans. Faraday Soc. 35, 1247 (1939). DE BOER, J. H., AND CUSTERS, J. F. H., Z. physik. Chem. B25,225 (1934). ROBERTS, J. K., "Some Problems in Adsorption." Cambridge, 1939. ~)OLANYI,M., AND WELKE, V., Z. physik. Chem. A132, 371 (1928). WHEELER, T. S., Trans. Natl. Inst. Sci. India 1,333 (1938). FOWLE~, R. H., Proe. Roy. Soc. (London) A159, 229 (1937). HILDEBRAND, J. H., WAKEHAM,H. R. R., AND BOYD, R. N., J. Chem. Phys. 7, 1094 (1939). MOELWYN-ttUGHES, E. A., J. Phys. & Colloid Chem. 55, 1246 (1951). JURA, G., J. Phys. & Colloid Chem. 52, 40 (1948). BInCu~sHAw, L. L., Phil. Mag. 12,596 (1931). STEFAN, J., Wied. Ann. 29,655 (1886). OSTWALD,W., Z. physik. Chem. 1, 45 (1887). HAn~INS, W. D., AND ROBEna'S, L. E., J. Am. Chem. Soc. 44, 653 (1922). WOLF, K. L., AND KLAPPI~OTH,K., Z. physik. Chem. B46, 276 (1940). HAUL, I~., Z. physik. Chem. A53, 331 (1943). VObK~ANN, H., F I A T Rev. Get. Sci. Phys. Chem. 30, 205 (1947).