- Email: [email protected]
Range of the screened coulomb interaction in electrolytes and double layer problems
CHEMICAL
Volume 53, number 2
RANGE
OF THE SCREENED
IN ELECTROLYTES
D. John MITCHELL
COULOMB
PHYSICS
LETTERS
15 January 1978
INTERACTION
AND DOUBLE LAYER
PROBLEMS
and Barry W. NINHAM
Department of Applied Mathenuztics. Research School of PhysicaI Sciences, Institute of Advanced Studies, l%e Austratin National Universiry. Gznberra. A.C_T_ 2600. Australia Received 10 June 1977 Revised manuscript received 23 November 1977
1: G-&own that the range of the Coulomb interaction in unsymmetric electrolytes at large distances is determined not by the Debye screening length, but by a rather more complicated functionof the density, which is evaluatedexplicitly.
In a
not very recent calculation
[ 1,2] the authors
derived the asymptotic form of the pair distribution
function of a classical electron gas. We have extended that result to the primitive model of electrolytes_ For the one-component electron gas (jellium mode:) the pair distribution finction has the asymptotic form 121 812(4
sary steps. Consider a solution of a strong electrolyte Cii Ai: in an aqueous solvent of dielectric constant E at density p molecules/cm3 _ C and A denote cation and anion with charge ~~4, z2q respectively- We have P’, =
P1 = VIP, =
AC-+-
In3 +:)A2
B= 1 +$Ain3
+d(A3
V2P,
vl=l +y7z7 - _ = 0,
(4)
the last being a consequence of charge neutrality. The Debye screening length for the system is given by
1 - (A/x)emBX ,
where if A = (47rfl 346P)“2 is the dimensionless plasma parameter (A is denoted by e in ref. [2]); K&I = (47ipq2P)i/2 is the Debye screening length; and x = KO~; r = Irl ‘2 1, A and B have the explicit asymptotic expansions A=A+(i
trolyte. The argument very closely parallels the treatment in ref. [2] and we simply summarise the neces-
ln A),
(2)
+A2(~lnA+0.66)+G(A31nA).(3)
Here 4 is the unit charge, fl= l/kT, and P is the electron density. This result follows from the MontrollMayer cluster expansion method. The first terms of eq. (3) are already implicit in the hypemetted chain approximation which neglects socalled bridge, or nonelementary diagrams, although this is not generally recognised because most solutions for the primitive model of electrolytes are numerical, and deal with tbermodynamic properties only. We have extended eq. (1) to a two-component elec-
K;
(qz:
=(47&p/~)
+ ZJ~Z;)~>-
(3
indirect correlation functions fZij(T) =gii(r) related to the direct correlation functions cV through the Omstein-Zernike relations
1
hv
(6)
The are
= cu + $Z
PkJ
Ci,+(Z -
S)hki(S)&
)
which in Fourier space ha-/e the soIution Ell
c:l
= 1 -PIFll
- P2(Cl1CZ2 -P3&
fP1P?&$2
similar expressions for the asymptotic forms of hii the zero of the denominator calculate the CV_To do this bond as wth
[email protected]
v ’
-&) -2 --Cl2
) 90)
z12, E22_ As in ref. [2], or gii are determined by in eq. (7). It remains to we expand any Mayerfi
03) 397
Volunk
53, number
CHEMICAL
2
PHYSICS
B=l
j$ = H(r
- Ri - Rj) [eXp (+ZiZiq
2/W)- 1]
*
-’ -
Cij -
s
d3reiksr fif$
where here and hereafter SL= k/tco, x = “co, Rii = Ri -f RF This contribution is negligible for KHAN Q 1. The second contribution to FV is given by the bare
Coulomb diagram and is -l=
=ij
s
d3 eikor (-pzizj~
2/er) H(r - Rii>
= _C4nZiZj/K~n2)~q2/~)COS(KoR~~)*
(11)
The cosine term can be replaced by unity for K* Rii < I_ Next in order is the simple diagram denoted by fig_
6a of ref. [2]. It consists of two screened Coulomb bonds in parallel which connect particles i andi_ Again, fouowing rirf. [2], its contribution for KQRij Q 1 is ~~2’=
(2rrlKo~)(~i=jp42/E)’
tan-’
~~
(13)
with
A =
K&q2/E
_
(14)
Inversion of eq. (7) then yields gV(x) u e-BxIx where
,
(VI
2: + v2z;)2
(V&
+ O(A3 ln A).
(16)
+ V2z;)2
The correction term to the Debye screening length vanishes tientically for symmetric electrolytes. It _is obvious that the mean spherical model cannot give out the correct asymptotic form for the pair distribution function and therefore it is no surprise that the mean spherical model is inadequate especially for unsymmetric electrolytes. To obtain an estimate of the region of validity of eq. (16), it is sufficient to note from eq. (3) that the ratio of the third to second terms is s A = ~~~q~/~. The result also breaks down when KARL becomes significant compared with unity. At room temperature flq2/E = 7 A, so that the necessary condition (K,-,R~)Z -G 1 is always satisfied for real electrolytes whenever A < l_ For unsymmetric electrolytes we have from eq. (16) B=
l
+?&_?Ini/2 (q 2: +v2z;)2 2442
(17)
2 312 ’
(VI 21 f v23)
where c is the salt concentration in moles per litre, and we have explicitly B=
K(eff)
(18)
(12)
and substituting into the denominator of eq. (7) we require the zero of
+ ZJ&,
1978
KO
_
Collecting together eqs. (10)-(12)
K; = (4r$f72&)(rJylZ’:
++hln3
8)
Here His the step function and Ri and Rj denote radii of cation or anion. The Coulomb par&f; can be expanded further into bare Coulomb diagrams and a resummation effected as in ref. [2]. In lowest order we have two contriiutions:
15 Januig
LETTERS
W)
1O-2 M
1.06 (2 : 1)
1.12 (2 : 3)
1.32 (3 : 1)
IO-lM
1.18 (2 : i)
l-39(2
2
: 3)
(3 I 1)
At 10-l M the effective screening length formula has certainly broken down. The result eq. (17) h&sclear implications for double layer problems and questions of cohoid stability where the long range nature of the screened Coulomb interaction is important. Recently Israelachvili and Adams [3] have measured double layer forces operating between molecularly smooth mica surfaces in various aqueous electrolyte solutions. For 1 I 1 electrolytes the screening length agrees with the theoretical ~~ within experimental error up to 10-l molar in agreement with expectation. On the &her hand, for 2 : 1 electrolytes, the screening length deviates by 20% at-10-4 M and by 40% at 10m3 M from KO in the direction indicated by eq- (18). The density expansion for i (eff) is clearly
Vidume 53, r.umber 2
CHEMICAL
PHYSICS LETJXRS
of limited validity, and e.g. a variational approach to the hypemetted chain equation might be expected to do better. Nonetheless, the results of this paper do underline the inadequacy bf methods of treating double layer interactions based on the Poisson-Boltzmarm equation; which assumes K = K~, and point to a possible reconciliation within the framework of existing theory.
15 January 1978
References
111 D.J.
MitcbeU and B.W. Niiam, Phys. Letters 37A (1968) 154. VI DJ. Mitchell and B-W. Niiam, Phys. Rev. 174 (1968) 280. r31 J.N. Israelachvilli and G-E. Adams, J. Chem. Sot. Faraday II, to be published.
We thank D.Y.C. Chan and B.A. Pailthorpe for discussions.
399
Volume 53, number 2
RANGE
OF THE SCREENED
IN ELECTROLYTES
D. John MITCHELL
COULOMB
PHYSICS
LETTERS
15 January 1978
INTERACTION
AND DOUBLE LAYER
PROBLEMS
and Barry W. NINHAM
Department of Applied Mathenuztics. Research School of PhysicaI Sciences, Institute of Advanced Studies, l%e Austratin National Universiry. Gznberra. A.C_T_ 2600. Australia Received 10 June 1977 Revised manuscript received 23 November 1977
1: G-&own that the range of the Coulomb interaction in unsymmetric electrolytes at large distances is determined not by the Debye screening length, but by a rather more complicated functionof the density, which is evaluatedexplicitly.
In a
not very recent calculation
[ 1,2] the authors
derived the asymptotic form of the pair distribution
function of a classical electron gas. We have extended that result to the primitive model of electrolytes_ For the one-component electron gas (jellium mode:) the pair distribution finction has the asymptotic form 121 812(4
sary steps. Consider a solution of a strong electrolyte Cii Ai: in an aqueous solvent of dielectric constant E at density p molecules/cm3 _ C and A denote cation and anion with charge ~~4, z2q respectively- We have P’, =
P1 = VIP, =
AC-+-
In3 +:)A2
B= 1 +$Ain3
+d(A3
V2P,
vl=l +y7z7 - _ = 0,
(4)
the last being a consequence of charge neutrality. The Debye screening length for the system is given by
1 - (A/x)emBX ,
where if A = (47rfl 346P)“2 is the dimensionless plasma parameter (A is denoted by e in ref. [2]); K&I = (47ipq2P)i/2 is the Debye screening length; and x = KO~; r = Irl ‘2 1, A and B have the explicit asymptotic expansions A=A+(i
trolyte. The argument very closely parallels the treatment in ref. [2] and we simply summarise the neces-
ln A),
(2)
+A2(~lnA+0.66)+G(A31nA).(3)
Here 4 is the unit charge, fl= l/kT, and P is the electron density. This result follows from the MontrollMayer cluster expansion method. The first terms of eq. (3) are already implicit in the hypemetted chain approximation which neglects socalled bridge, or nonelementary diagrams, although this is not generally recognised because most solutions for the primitive model of electrolytes are numerical, and deal with tbermodynamic properties only. We have extended eq. (1) to a two-component elec-
K;
(qz:
=(47&p/~)
+ ZJ~Z;)~>-
(3
indirect correlation functions fZij(T) =gii(r) related to the direct correlation functions cV through the Omstein-Zernike relations
1
hv
(6)
The are
= cu + $Z
PkJ
Ci,+(Z -
S)hki(S)&
)
which in Fourier space ha-/e the soIution Ell
c:l
= 1 -PIFll
- P2(Cl1CZ2 -P3&
fP1P?&$2
similar expressions for the asymptotic forms of hii the zero of the denominator calculate the CV_To do this bond as wth
[email protected]
v ’
-&) -2 --Cl2
) 90)
z12, E22_ As in ref. [2], or gii are determined by in eq. (7). It remains to we expand any Mayerfi
03) 397
Volunk
53, number
CHEMICAL
2
PHYSICS
B=l
j$ = H(r
- Ri - Rj) [eXp (+ZiZiq
2/W)- 1]
*
-’ -
Cij -
s
d3reiksr fif$
where here and hereafter SL= k/tco, x = “co, Rii = Ri -f RF This contribution is negligible for KHAN Q 1. The second contribution to FV is given by the bare
Coulomb diagram and is -l=
=ij
s
d3 eikor (-pzizj~
2/er) H(r - Rii>
= _C4nZiZj/K~n2)~q2/~)COS(KoR~~)*
(11)
The cosine term can be replaced by unity for K* Rii < I_ Next in order is the simple diagram denoted by fig_
6a of ref. [2]. It consists of two screened Coulomb bonds in parallel which connect particles i andi_ Again, fouowing rirf. [2], its contribution for KQRij Q 1 is ~~2’=
(2rrlKo~)(~i=jp42/E)’
tan-’
~~
(13)
with
A =
K&q2/E
_
(14)
Inversion of eq. (7) then yields gV(x) u e-BxIx where
,
(VI
2: + v2z;)2
(V&
+ O(A3 ln A).
(16)
+ V2z;)2
The correction term to the Debye screening length vanishes tientically for symmetric electrolytes. It _is obvious that the mean spherical model cannot give out the correct asymptotic form for the pair distribution function and therefore it is no surprise that the mean spherical model is inadequate especially for unsymmetric electrolytes. To obtain an estimate of the region of validity of eq. (16), it is sufficient to note from eq. (3) that the ratio of the third to second terms is s A = ~~~q~/~. The result also breaks down when KARL becomes significant compared with unity. At room temperature flq2/E = 7 A, so that the necessary condition (K,-,R~)Z -G 1 is always satisfied for real electrolytes whenever A < l_ For unsymmetric electrolytes we have from eq. (16) B=
l
+?&_?Ini/2 (q 2: +v2z;)2 2442
(17)
2 312 ’
(VI 21 f v23)
where c is the salt concentration in moles per litre, and we have explicitly B=
K(eff)
(18)
(12)
and substituting into the denominator of eq. (7) we require the zero of
+ ZJ&,
1978
KO
_
Collecting together eqs. (10)-(12)
K; = (4r$f72&)(rJylZ’:
++hln3
8)
Here His the step function and Ri and Rj denote radii of cation or anion. The Coulomb par&f; can be expanded further into bare Coulomb diagrams and a resummation effected as in ref. [2]. In lowest order we have two contriiutions:
15 Januig
LETTERS
W)
1O-2 M
1.06 (2 : 1)
1.12 (2 : 3)
1.32 (3 : 1)
IO-lM
1.18 (2 : i)
l-39(2
2
: 3)
(3 I 1)
At 10-l M the effective screening length formula has certainly broken down. The result eq. (17) h&sclear implications for double layer problems and questions of cohoid stability where the long range nature of the screened Coulomb interaction is important. Recently Israelachvili and Adams [3] have measured double layer forces operating between molecularly smooth mica surfaces in various aqueous electrolyte solutions. For 1 I 1 electrolytes the screening length agrees with the theoretical ~~ within experimental error up to 10-l molar in agreement with expectation. On the &her hand, for 2 : 1 electrolytes, the screening length deviates by 20% at-10-4 M and by 40% at 10m3 M from KO in the direction indicated by eq- (18). The density expansion for i (eff) is clearly
Vidume 53, r.umber 2
CHEMICAL
PHYSICS LETJXRS
of limited validity, and e.g. a variational approach to the hypemetted chain equation might be expected to do better. Nonetheless, the results of this paper do underline the inadequacy bf methods of treating double layer interactions based on the Poisson-Boltzmarm equation; which assumes K = K~, and point to a possible reconciliation within the framework of existing theory.
15 January 1978
References
111 D.J.
MitcbeU and B.W. Niiam, Phys. Letters 37A (1968) 154. VI DJ. Mitchell and B-W. Niiam, Phys. Rev. 174 (1968) 280. r31 J.N. Israelachvilli and G-E. Adams, J. Chem. Sot. Faraday II, to be published.
We thank D.Y.C. Chan and B.A. Pailthorpe for discussions.
399