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The surface tension of ionic solutions
Bellemans, A. 1964
Physica 30 924-930
THE SURFACE TENSION OF IONIC SOLUTIONS I. DERIVATION OF THE LIMITING LAW AT INFINITE DILUTION FROM THE POISSON-BOLTZMANN EQUATION
by A. BELLEMANS Facult6 des Sciences, Universit6 libre de Bruxelles, Belgique
Synopsis The limiting law at infinite dilution for the change of surface tension of a solvent due to the addition of a strong electrolyte is derived in the most general case from the linearized Poisson-Boltzmann equation. The analytic form of the limiting law is shown to be extremely sensitive to the value of the dielectric permittivities of the solvent and the surrounding medium. The limiting law previously derived by Onsager and Samaras is recovered as a special case.
1. Introduction. The present paper is the first one of a series devoted to the surface tension of strong electrolyte solutions. Our purpose is to express the increase of the surface tension of a given solvent in terms of the concentration of the solute. As a starting point we will use the systematic cluster expansion of the surface tension derived in a preceding series of papers l). The first step of our program is however the derivation of the limiting law at infinite dilution, and this is done in the present paper by using the Poisson-Boltzmann equation. The method used is similar to the earlier treatment of Onsager and Samaras 2); these authors, however, confined themselves to the special situation where the dielectric constant of the solvent is much larger than unity so that the limiting law they derived (and which was recovered recently by Buff and Stillinger 3)) is not the most general one. Of course the use of the Poisson-Boltzmann equation makes this kind of derivation somewhat unsatisfactory, and the correctness of the so obtained limiting law needs to be confirmed by an exact statistical treatment. This will be the object of a subsequent paper. 2. The potential of average force acting on an ion near the interface. The increase of the surface tension y of a given solvent due to the addition of a strong electrolyte can be computed by integrating the Gibbs equation
dy = -
C r, djua 924 -
(2.1)
THE
SURFACE
TENSION
with respect to the chemical
OF IONIC
potentials
SOLUTIONS.
I
925
(u, of the various kinds of ions cc,
once the specific adsorptions r, are known. This method was used previously by Wagner 4) and Onsager and Samaras 2). For a plane interface coinciding with the plane z = 0,
ra = fi,_F{exp [-BW&)l - I} dz
(2.2)
0
where n, and W,(z) are, respectively, the concentration of ions o! far from the interface and the potential of average force acting on such an ion at a distance z from the interface; /I = l/kT. This reduces the problem to the determination of W,(z). As usually we approximate the actual electrolyte solution by the following scheme: the ions behave as point charges; the solvent is equivalent to a continuum (1) of dielectric constant D extending in the half space z > 0; the other half space z < 0 is filled up with a continuum (II) of dielectric constant D’ (usually equal to 1); all ions are confined to the half space 2 > 0. L
%I
y--3-
1
Fig.
For a single point charge Ea located at ra = (0, 0, za) in medium I, the electrostatic potential y(‘; rbl) is determined by the Poisson equations V’2yJr(‘; Y&!)= -
4Z&, - D 6(r -
(medium I)
rf%)
(medium II)
V2yII(r ; r,) = 0 supplemented
by the cvnditions
at the interface:
(YI)z-0
D ($)zeo
=
(YII)z=O,
Cf. fig. 1. The well-known
r,) =
where ri = (0, 0, -za)
=
D’
($)&
solution of these equations is 5)
yr(r; r,) = 2 D yII(r;
(2.3)
1 /r -
ral
+ -2.. D 1
2&x D + D’
Ir -
ral ’
is the image-point
D-D’
1
D+D’
jr-r;]’
(2.5)
(2.6)
of ra with respect to the plane
926
A. BELLEMANS
z = 0. Hence yI(r; r,) is formally equivalent to the sum of the electrical potentials due to a charge ELI at rn and a charge E~(D - D’)/(D + D’) at r: (image-charge), both embedded in an infinite medium of dielectric constant D. The image term clearly takes care of the inhomogeneity of the space. From (2.5) it follows that the charge Ed is subject to a force (2.7) derived from the potential
energy
(2.8) where we put il = (D - D’)/(D + D’). As in actual cases we have D > D’ 1: 1, this force repels ea from the interface. Now the simplest approximation to W,(z) would be to take W,(z) thereby completely lution. This gives
forgetting
r,
= %X(Z),
about
= lz,Jyexp
the presence
[-BE:
(2.9) of other ions in the so-
A/4Dz] -
(2.10)
l} dz
0
which diverges logarithmically in the integrand :
because
of the long-range
term
(2.11)
-_16~; 1/4D~. Of course we may expect that this term will be screened when of the other ions is taken into account. An approximate way for introducing this screening effect y(r ; r,) from the so-called Poisson-Boltzmann equation which the probability of finding a charge ~0 at point r in the medium ng exp [--BED yI(r; The linearized Poisson-Boltzmann PyI(r;
ra) = -
ATD ea d(r -
equation
the influence is to deduce assumes that I is
r&)1.
(2.12)
gives then :
rai) + K2pI(r; ra)
V21yrr(r ; r,) = 0
contained
(medium II)
(medium I)
(2.13a) (2.13b)
The second term of the r.h.s. of (2.13a) accounts for the presence of all ions but tc, in tlze linear a~fwoximation (compare to (2.3)); K is the charac-
THE
teristic
SURFACE
Debye reciprocal
TENSION
OF IONIC
SOLUTIONS.
927
I
length (2.14)
K2 = (47$/D) c n&f c( and it has been assumed that the condition far from the interface (i.e. 2 naEa = 0).
of electroneutrality
is satisfied
The general solution of (:. 13) subjected to the boundary conditions is easily obtained by using the Hankel transform of y; one finds: yr(r +I@,
; ra)
=
Th(p,
.z; 4
pJ~(fv~)
(2.4)
(2.15)
dp
2; &%) = =-
&u
D
e
_d_ p~+K~Iz-z.l
1/p
1
0452
+ lc2 +
+
-___ Dl/p+
K2 _
D’
K2
D’
+
e-%-G%+z~)
(2.16)
d$2 +
where p = 2/z? + ys and Jn(t) is the Bessel function average force acting on &a is again given by
K2
I
of the first kind. The
(2.17) and from (2.16-15)
on finds the following expression
for W,(z) :
m
W,(z)
=
6
s
PdP
DA@ + ~ Dz/f+ +
D’
K2-
eXp
[-2Zdp2
+
K2] .
K2
+
D’
l/p2 +
(2.18)
K2
0
One verifies
immediately
that ~&)Ilc’O
(2.19)
= Q(Z).
However, for K > 0, W,(z) becomes short-ranged. It can indeed be rigorously proved that the asymptotic value of W,(z) for KZ > 1 is (E:/~D)
(eC2”/2z) (1 -(D’/D)
d/n/K2
+
O(z-1)).
(2.20)
Hence, as expected, there is a screening effect destroying the long-range tail (2.12). Equation (2.18) can of course be integrated; however, the expression obtained is but simple in two special cases: D=D’:
W,(z) = 0%
& :;{(I + -&)2_2e2K'K+!Kz)},
(2.21 a)
D’: (2.21b)
where Ks(t) is the modified
Bessel function
of the second kind.
928
A. BELLEMANS
3. The analytical form of the limiting law. We now are able to compute r, from (2.2). However, we should keep in mind that W,(z) has been computed from the linearized Poisson-Boltzmann equation and that it is made actually of two parts: W,(z)
= V%(z) + {J%(z)
-
%X(z)}.
(3.1)
The first one is exact and takes care of the image charge of E~; the second one cares for the effects due to other charges and can only be expected to be correct to the linear appoximation. This means that the integrand of (2.2) exp [-BW&)l is correctly -
-
[--P~&)P/fi!
1= F
given up to
#IW,(z) + z [2
@a(z)]“/n!
= {e+‘*(‘)
-
1) -
/3{W,(z) -
V&(z)}
(3.2)
when expression (2.18) is used for W,(z). It turns out from the discussion given at the end of the preceding section that the second term of the r.h.s. of (3.2) is just what is needed to destroy the long range tail of its first term and insure the convergence of the integral (2.2) giving m; hence the exact limiting law for r, will be obtained from the integral
ra= nay& {exp Wh-&)l Details concerning the integration dix. The final result is r&=s{Aln(-$$)+(2C-
-
1 - ~[W,(Z)
- v~(~)I>.
(3.3)
of this expression are given in the appen-
l)A-(l
+A)[In2-+ln(l
where C = 0.5772157 (Euler constant). From the limiting value of the chemical dilution
+A)]},
potential
(3.4)
of ions CLat infinite
,uu,=&+kTlnsz,,
(3.5)
it is found, taking into account (2.1), that the change in y with respect to the surface tension yo of the pure solvent is given by (y -
yo)/kT = - /I;
a
(r&J
dn,.
_
(From (3.4) one may check that the integrand is a total differential).
(3.6) Carrying
THE
SURFACE
out the integration
TENSION
OF IONIC
SOLUTIONS.
929
I
we obtain the limiting law for y :
-(l
+A)[In2--*ln(l
If we assume D > D’ i.e. 1 + Onsager and Samaras a) Y - YO ___ = kT
+A)]).
(3.7)
1, we recover the limiting law derived by
1 &g?K ~n&9 4D ,ln - 2D + 2C - $
T
(3.9)
which is approximately valid for aqueous solutions (1 N 0.97). On the other hand if we take D = D’ = 1, we find Y -
YO
kT=?
-!I!$!? (In
2 -
4) =
-f& (In 2 -
4).
(3.9)
This shows how much the analytic form of the limiting to the dielectric constants D and D’.
law is sensitive
It is our pleasure to thank interest in this research.
for his constant
Professor
I. Prigogine
APPENDIX
The integral (3.3) is finite as a whole but its various parts diverge either at z + 0 or at z -+ 00. Reexpress first W,(z) as W,(z)
= (@/2D) +
e-2Kz/2z +
(&W(l
- 12)[P dP 0
(cf. (2.18)) by noticing
-
dp2
+ K2 -
p
d/p2+K2( 1+A)+$(1 -A)
that
e-2Kz/2Z =J-p
dp e-25’~z/2/ps
+ K2.
We then have r,
=Aa+Ba,
A, = j%z {exp [-_8&zA/402] 0
1-
[/?.$/4Dz]
(e-2KZ -
l)},
-dp2+K2
930
THE
SURFACE
TENSION
OF IONIC
SOLUTIONS.
I
Both A, and B, are finite. For A, we find: A, = (@$/4D)
{ln(&&/2D)
For B, we first integrate over z, next put this gives B, = -
Received
(~340)
(1 + A) In 2 -
+ (2C KP
=
1)).
sh q~and integrate over q;
1+1 2il
In (1 + A)
24- 1O-63
REFERENCES 1) Bellemans,
A., Physica
28 (1962) 493, 612 and 29 (1963) 548. N. N. T., J. them. Phys. 2 (1934) 528.
2)
Onsager,
3) 4)
Buff, F. P. andstillinger, F. H., J. them. Wagner, C., Phys. 2. 25 (1924) 474.
5)
See for example: Cambridge
L. and Samaras,
J. J. Thompson,
University
Phys. 25 (1956) 312.
“Elements Press (1909) p. 169-171.
of Electricity
and Magnetism”,
Fourth
Edition,
Physica 30 924-930
THE SURFACE TENSION OF IONIC SOLUTIONS I. DERIVATION OF THE LIMITING LAW AT INFINITE DILUTION FROM THE POISSON-BOLTZMANN EQUATION
by A. BELLEMANS Facult6 des Sciences, Universit6 libre de Bruxelles, Belgique
Synopsis The limiting law at infinite dilution for the change of surface tension of a solvent due to the addition of a strong electrolyte is derived in the most general case from the linearized Poisson-Boltzmann equation. The analytic form of the limiting law is shown to be extremely sensitive to the value of the dielectric permittivities of the solvent and the surrounding medium. The limiting law previously derived by Onsager and Samaras is recovered as a special case.
1. Introduction. The present paper is the first one of a series devoted to the surface tension of strong electrolyte solutions. Our purpose is to express the increase of the surface tension of a given solvent in terms of the concentration of the solute. As a starting point we will use the systematic cluster expansion of the surface tension derived in a preceding series of papers l). The first step of our program is however the derivation of the limiting law at infinite dilution, and this is done in the present paper by using the Poisson-Boltzmann equation. The method used is similar to the earlier treatment of Onsager and Samaras 2); these authors, however, confined themselves to the special situation where the dielectric constant of the solvent is much larger than unity so that the limiting law they derived (and which was recovered recently by Buff and Stillinger 3)) is not the most general one. Of course the use of the Poisson-Boltzmann equation makes this kind of derivation somewhat unsatisfactory, and the correctness of the so obtained limiting law needs to be confirmed by an exact statistical treatment. This will be the object of a subsequent paper. 2. The potential of average force acting on an ion near the interface. The increase of the surface tension y of a given solvent due to the addition of a strong electrolyte can be computed by integrating the Gibbs equation
dy = -
C r, djua 924 -
(2.1)
THE
SURFACE
TENSION
with respect to the chemical
OF IONIC
potentials
SOLUTIONS.
I
925
(u, of the various kinds of ions cc,
once the specific adsorptions r, are known. This method was used previously by Wagner 4) and Onsager and Samaras 2). For a plane interface coinciding with the plane z = 0,
ra = fi,_F{exp [-BW&)l - I} dz
(2.2)
0
where n, and W,(z) are, respectively, the concentration of ions o! far from the interface and the potential of average force acting on such an ion at a distance z from the interface; /I = l/kT. This reduces the problem to the determination of W,(z). As usually we approximate the actual electrolyte solution by the following scheme: the ions behave as point charges; the solvent is equivalent to a continuum (1) of dielectric constant D extending in the half space z > 0; the other half space z < 0 is filled up with a continuum (II) of dielectric constant D’ (usually equal to 1); all ions are confined to the half space 2 > 0. L
%I
y--3-
1
Fig.
For a single point charge Ea located at ra = (0, 0, za) in medium I, the electrostatic potential y(‘; rbl) is determined by the Poisson equations V’2yJr(‘; Y&!)= -
4Z&, - D 6(r -
(medium I)
rf%)
(medium II)
V2yII(r ; r,) = 0 supplemented
by the cvnditions
at the interface:
(YI)z-0
D ($)zeo
=
(YII)z=O,
Cf. fig. 1. The well-known
r,) =
where ri = (0, 0, -za)
=
D’
($)&
solution of these equations is 5)
yr(r; r,) = 2 D yII(r;
(2.3)
1 /r -
ral
+ -2.. D 1
2&x D + D’
Ir -
ral ’
is the image-point
D-D’
1
D+D’
jr-r;]’
(2.5)
(2.6)
of ra with respect to the plane
926
A. BELLEMANS
z = 0. Hence yI(r; r,) is formally equivalent to the sum of the electrical potentials due to a charge ELI at rn and a charge E~(D - D’)/(D + D’) at r: (image-charge), both embedded in an infinite medium of dielectric constant D. The image term clearly takes care of the inhomogeneity of the space. From (2.5) it follows that the charge Ed is subject to a force (2.7) derived from the potential
energy
(2.8) where we put il = (D - D’)/(D + D’). As in actual cases we have D > D’ 1: 1, this force repels ea from the interface. Now the simplest approximation to W,(z) would be to take W,(z) thereby completely lution. This gives
forgetting
r,
= %X(Z),
about
= lz,Jyexp
the presence
[-BE:
(2.9) of other ions in the so-
A/4Dz] -
(2.10)
l} dz
0
which diverges logarithmically in the integrand :
because
of the long-range
term
(2.11)
-_16~; 1/4D~. Of course we may expect that this term will be screened when of the other ions is taken into account. An approximate way for introducing this screening effect y(r ; r,) from the so-called Poisson-Boltzmann equation which the probability of finding a charge ~0 at point r in the medium ng exp [--BED yI(r; The linearized Poisson-Boltzmann PyI(r;
ra) = -
ATD ea d(r -
equation
the influence is to deduce assumes that I is
r&)1.
(2.12)
gives then :
rai) + K2pI(r; ra)
V21yrr(r ; r,) = 0
contained
(medium II)
(medium I)
(2.13a) (2.13b)
The second term of the r.h.s. of (2.13a) accounts for the presence of all ions but tc, in tlze linear a~fwoximation (compare to (2.3)); K is the charac-
THE
teristic
SURFACE
Debye reciprocal
TENSION
OF IONIC
SOLUTIONS.
927
I
length (2.14)
K2 = (47$/D) c n&f c( and it has been assumed that the condition far from the interface (i.e. 2 naEa = 0).
of electroneutrality
is satisfied
The general solution of (:. 13) subjected to the boundary conditions is easily obtained by using the Hankel transform of y; one finds: yr(r +I@,
; ra)
=
Th(p,
.z; 4
pJ~(fv~)
(2.4)
(2.15)
dp
2; &%) = =-
&u
D
e
_d_ p~+K~Iz-z.l
1/p
1
0452
+ lc2 +
+
-___ Dl/p+
K2 _
D’
K2
D’
+
e-%-G%+z~)
(2.16)
d$2 +
where p = 2/z? + ys and Jn(t) is the Bessel function average force acting on &a is again given by
K2
I
of the first kind. The
(2.17) and from (2.16-15)
on finds the following expression
for W,(z) :
m
W,(z)
=
6
s
PdP
DA@ + ~ Dz/f+ +
D’
K2-
eXp
[-2Zdp2
+
K2] .
K2
+
D’
l/p2 +
(2.18)
K2
0
One verifies
immediately
that ~&)Ilc’O
(2.19)
= Q(Z).
However, for K > 0, W,(z) becomes short-ranged. It can indeed be rigorously proved that the asymptotic value of W,(z) for KZ > 1 is (E:/~D)
(eC2”/2z) (1 -(D’/D)
d/n/K2
+
O(z-1)).
(2.20)
Hence, as expected, there is a screening effect destroying the long-range tail (2.12). Equation (2.18) can of course be integrated; however, the expression obtained is but simple in two special cases: D=D’:
W,(z) = 0%
& :;{(I + -&)2_2e2K'K+!Kz)},
(2.21 a)
D’: (2.21b)
where Ks(t) is the modified
Bessel function
of the second kind.
928
A. BELLEMANS
3. The analytical form of the limiting law. We now are able to compute r, from (2.2). However, we should keep in mind that W,(z) has been computed from the linearized Poisson-Boltzmann equation and that it is made actually of two parts: W,(z)
= V%(z) + {J%(z)
-
%X(z)}.
(3.1)
The first one is exact and takes care of the image charge of E~; the second one cares for the effects due to other charges and can only be expected to be correct to the linear appoximation. This means that the integrand of (2.2) exp [-BW&)l is correctly -
-
[--P~&)P/fi!
1= F
given up to
#IW,(z) + z [2
@a(z)]“/n!
= {e+‘*(‘)
-
1) -
/3{W,(z) -
V&(z)}
(3.2)
when expression (2.18) is used for W,(z). It turns out from the discussion given at the end of the preceding section that the second term of the r.h.s. of (3.2) is just what is needed to destroy the long range tail of its first term and insure the convergence of the integral (2.2) giving m; hence the exact limiting law for r, will be obtained from the integral
ra= nay& {exp Wh-&)l Details concerning the integration dix. The final result is r&=s{Aln(-$$)+(2C-
-
1 - ~[W,(Z)
- v~(~)I>.
(3.3)
of this expression are given in the appen-
l)A-(l
+A)[In2-+ln(l
where C = 0.5772157 (Euler constant). From the limiting value of the chemical dilution
+A)]},
potential
(3.4)
of ions CLat infinite
,uu,=&+kTlnsz,,
(3.5)
it is found, taking into account (2.1), that the change in y with respect to the surface tension yo of the pure solvent is given by (y -
yo)/kT = - /I;
a
(r&J
dn,.
_
(From (3.4) one may check that the integrand is a total differential).
(3.6) Carrying
THE
SURFACE
out the integration
TENSION
OF IONIC
SOLUTIONS.
929
I
we obtain the limiting law for y :
-(l
+A)[In2--*ln(l
If we assume D > D’ i.e. 1 + Onsager and Samaras a) Y - YO ___ = kT
+A)]).
(3.7)
1, we recover the limiting law derived by
1 &g?K ~n&9 4D ,ln - 2D + 2C - $
T
(3.9)
which is approximately valid for aqueous solutions (1 N 0.97). On the other hand if we take D = D’ = 1, we find Y -
YO
kT=?
-!I!$!? (In
2 -
4) =
-f& (In 2 -
4).
(3.9)
This shows how much the analytic form of the limiting to the dielectric constants D and D’.
law is sensitive
It is our pleasure to thank interest in this research.
for his constant
Professor
I. Prigogine
APPENDIX
The integral (3.3) is finite as a whole but its various parts diverge either at z + 0 or at z -+ 00. Reexpress first W,(z) as W,(z)
= (@/2D) +
e-2Kz/2z +
(&W(l
- 12)[P dP 0
(cf. (2.18)) by noticing
-
dp2
+ K2 -
p
d/p2+K2( 1+A)+$(1 -A)
that
e-2Kz/2Z =J-p
dp e-25’~z/2/ps
+ K2.
We then have r,
=Aa+Ba,
A, = j%z {exp [-_8&zA/402] 0
1-
[/?.$/4Dz]
(e-2KZ -
l)},
-dp2+K2
930
THE
SURFACE
TENSION
OF IONIC
SOLUTIONS.
I
Both A, and B, are finite. For A, we find: A, = (@$/4D)
{ln(&&/2D)
For B, we first integrate over z, next put this gives B, = -
Received
(~340)
(1 + A) In 2 -
+ (2C KP
=
1)).
sh q~and integrate over q;
1+1 2il
In (1 + A)
24- 1O-63
REFERENCES 1) Bellemans,
A., Physica
28 (1962) 493, 612 and 29 (1963) 548. N. N. T., J. them. Phys. 2 (1934) 528.
2)
Onsager,
3) 4)
Buff, F. P. andstillinger, F. H., J. them. Wagner, C., Phys. 2. 25 (1924) 474.
5)
See for example: Cambridge
L. and Samaras,
J. J. Thompson,
University
Phys. 25 (1956) 312.
“Elements Press (1909) p. 169-171.
of Electricity
and Magnetism”,
Fourth
Edition,