1977
J . Phys. Chem. 1986, 90, 1977-1979
7
\
0.8-
0 3
\ r\
0.6-
r
t!
E
0.4-
0 0
+-.-----0
200
_i
_ _ . _ I 1
400
800
600
PRESSURE (DYNESICM
1000
')
Figure 4. Plot of eq 27 as 1 (cm) vs. p (dyn cm-2),exhibiting the instability which initiates at p = 938 dyn cm *.
Since p I is a pressure, applied in the same direction as p , it should be added to the left side of eq 17. We obtain
or p = 'y/l.- ~ @ ~ / 8 7 1 '
(27)
Using the values of y, K , and @ listed in eq 14 and 20 we may use eq 27 to calculate p vs. I or I vs. p . Figure 4 exhibits the plot of p in dyn cm-2 vs. I in cm, obtained in this manner. This interesting curve exhibits an infinite slope at p = 938 dyn ern-,, or 0.71 torr, where I = 3.1 X cm. Furthermore, at values cm, I decreases as pressure decreases. of I less than 3.1 X This clearly indicates a mechanical instability in which, as the pressure is increased, the film collapses away from the compressive stress! Of course, the collapse cannot continue indefinitely since the elastic component of the film's viocoelastic behavior (which has not been considered in the theory thus far) will act to prevent this catastrophe.
Nevertheless, the occurrence of the instability is strongly indicated. The fact that it occurs at p = 0.71 torr, close to the pressure at which the discontinuity in slope appears in Figures 1 and 2, is, of course, a result of the magnitudes of the various parameters we have chosen. However, these magnitudes are not unreasonable and demonstrate that the theory is viable. If the film collapses to another thickness, discontinuously at the pressure of the instability, this is accompanied, according to eq 24, by an abrupt change in p I . It is well-known that a solute with a positive partial molar volume will exhibit an increase in partial vapor pressure upon an increase in external pressure. At constant partial vapor pressure this implies a reduction in solubility. This could be one reason for the discontinuous saturation appearing in the isotherms of Figures 1 and 2. Another possibility could involve the reorientation of rodlike polymer molecules from a configuration in which they are oriented with their long axes perpendicular to the gold surface to one in which the long axes are parallel to that surface. This is almost a requirement if the "spreading" of the film is to take place in the manner assumed by the above theory. Such a liquid-crystallike transition could result in a phase in which the solubility of I, was decreased. Our model does not entirely explain the approximate independence of the discontinuity in slope from temperature or I2 content. Off-hand one might expect 9 and other parameters to depend on both temperature and iodine content. Perhaps these parameters are not too sensitive (relative to the overall phenomenon) to these variables, or possibly the changes due to temperature are partly compensated by changes due to iodine content. The resolution of this problem will have to await further studies. Acknowledgment. This work was supported by N S F Grants C H E 82-07432 and DMR 84-21383. The authors are grateful to Drs. L. Warren and D. P. Anderson of the Rockwell International Science Center who prepared the polythiophene films and measured their masses. We also acknowledge the assistance of Professor F. Wudl of the University of California, Santa Barbara, who together with his co-workers prepared the polythiophene powder. Registry No. I,, 7553-56-2;polythiophene, 25233-34-5.
A Noniterative Solution of the WLMB Integral Equation for the Electrical Double Layer Pedro J. Colrnenares and Wilrner Olivares* Grupo de Qdmica TeBrica, Departamento de Quimica, Facultad de Ciencias, Uniuersidad de Los Andes, MCrida 51 01, Venezuela (Received: October 19, 1985)
The WLMB integral equation is solved for the electrical double-layer problem by using the noniterative method. The analytical series expression obtained for the contact potential gives values that fare very well with the Monte Carlo data for a 1-1 electrolyte in the restricted primitive model (RPM).
Lovett et al.' and independently Wertheim, introduced an integral equation for the one-particle distribution function of inhomogeneous systems. The characteristics of this equation were studied for the planar interacting electrical double layers by Olivares and McQuarrie3 and for one charged interphase by Blum et aL4 and Colmenare~.~For the electrical double-layer problem this equation can be written as (1) Lovett, R.;Mou, C.; Buff, F. J . Chem. Phys. 1976, 65, 570. (2) Wertheim, M . S . J . Chem. Phys. 1976.65, 2311. (3) Qlivares, W.; McQuarrie, D. A. J . Phys. Chern. 1980, 84, 863. (4) Blum,L.; Hernando, J.; Lebowitz, J. J . Phys. Chem. 1983,87,2825. ( 5 ) Colmenares, P. J. Thesis for Assistant Professor, Universidad de Los
Andes, MErida, Venezuela,
1984.
O022-3654/86/2090-1977$01.50/0
d In gi(x) dx -2aipj~x+R j=i
(r,x,xd J" gj(x2)rdCij"8x2 dx, d r -
max[O,x-R]
Ix-x~l
d W ) &ziedx (1) where pi is the particle density of ionic species i , gi(x) is the particle-wall distribution function. and C," is defined by pzizje2
Cijo(r,x,x2)= C,(r,x.x2)+ (2) tr where C,(r,x,x2)is the direct two-particle inhomogeneous cor0 1986 American Chemical Society
1978 7he Journal of Physical Chemistry, Vol. 90, No. 9, 1986
relation function. \k(x) is the mean electrostatic potential
In the absence of short-range particle-particle correlation, the second term in eq 1 vanishes and one immediately obtains the Gouy-Chapman limit. In this work we obtain an analytical expression and calculate the contact potential q ( 0 ) as a function of the surface charge density for an electrified interphase immersed in an aqueous electrolyte solution in the restricted primitive model, using the simple doniterative method of Henderson and Blum.6 Following these authors, the function g,(x) can be written as In gf(x) = H x ) - Bz,e(X(x) + W x ) )
Colmenares and Olivares function. By making a sensible approximation of g d and g,, such as the MGC, we obtain from eq 5 and 6 a first iterate for [(x) and X(x) which corresponds to the noniterative solution. We then use the analytical expression for the MGC distribution function and obtain after integration and collection of terms 1
-
2
pzeX(O) =
A
7 2 (KR)
A
h
(9)
,42n+l
(2n
S ( 2 n + 1)
+ 1)3
(IO)
where P(n) = a,
+ ~ ~ ( n +x a) , ~( n ~ )+~ a 4 ( n ~ )-S e-nx(al + a,nx
(4)
where E(x) and X(x) are short- and long-range potentials that correct the mean electrostatic potential to give the total potential of mean force. Using eq 1 and 4, one obtains for a z-z electrolyte
m
{(o) = a0 + --Lc) ~-P(2nj 1 6 ( ~ Rn=l ns
S ( n ) = b,
+ as(nx)2 + ~ ~ ( n x (1) ~1) )
+ b2nx + b , ( n ~ +) ~b , ( n ~ ) ~e n*(bl -
-
bsnx) (12)
[(K); x < R (5a)
@zeX(x) =
PzeX(R); x
< R (6a)
(6b)
where R is the ionic diameter in the restricted primitive model. The subindices s and d denote the semisum and semidifferences of the correlation function, respectively. The contact condition of Henderson and Blum is g,(O) = a b2/2 where b = PzeE/K is a dimensionless measure of the surface charge density tE/47r on the wall and a is the reduced osmotic pressure o f the electrolyte solution when g, is exact. The quantity a clearly depends on the given approximation of g, and is dominated by the term b2/2 for most practical values of E and K . When this condition and the definition of [ and X are used, the contact potential is given as
+ where q = (r/6jpR3 is the reduced density, R is the ionic diameter, and p is the density number. The constant a in eq 8 is evaluated from eq 9 in the limit b 0 as
-
I11 U
where
((0) and X(0) are obtained from eq 5 and 6 evaluated at x = 0. Since the value of a is not very critical, we assume that it is indepttident of the value of b and calculate it consistently with the contact condition as the limit of g,(O) as 6 -* 0, namely, a = exp(t(0)) evaluated with b = 0. In order to solve the set of equations (4)-(6), one needs a closure for the inhomogeneous ion-ion direct correlation function Cijo(r,x,x2). The importance of the dependence of this function on the distance of the two interacting particles to the wall was pointed out by Croxton et al. in the solution of the BGY e q ~ a t i o n . ~ Id favor of the simplicity of the analytical results obtained hereafter, we use both the most naive approximation CijD(r,x,x2) Cijo(r)and the MSA homogeneous approximation for this _______I__
--___I_.
( 6 ) Henderson, D.; Blurn, L. J . Electroanal. Chem. Interfacial Electrochem. 1980. I 1 1 , 217.
1 = -(3D,
12
+ 8D2)
(15)
The potential q ( 0 ) is then calculated from eq 7 . In Figure 1 we have plotted the contact potential \k(O) as a function of the surface charge density for several concentrations of a 1-1 electrolyte in the restricted primitive model. We compare the potential calculated through eq 7-10, labeled WLMB/MSA, with the Monte Carlo data of Torrie and Valleau' (MC) and with the noniterative solution of the H N C equation of Henderson and Blum, labeled HNC/MSA. We have used the same arameters as in the Monte Carlo calculations, namely R = 4.25 , = 78.5, and T = 25 O C . For concentrations in the range of 0.01'1 M the WLMB/MSA approximation falls within 10% or less of the Monte Carlo data and gives essentially the same results as the HNC/MSA approximation of Henderson and Blum, while the MGC approximation deviates up to 20%. For concentrations as low as 0.0001 M all of the above-mentioned approximations are within 10% or less.
8:
( 7 ) Torrie, G , Valleau, J J Chem
Phvs. 1980, 73, 5807.
The Journal of Physical Chemistry, Vol, 90, No. 9, 1986 1979
WLMB Integral Equation for the Electrical Double Layer
TABLE I: Comparison of the Predictions of the Contact Potential q ( 0 ) Given by the Different Theories for a 1 M Electrolyte Solution"
WO), v charge, pC/cm2
MC
8.83 22.18 37.69
0.028 0.055 0.079
' R , T, and
0.4
e
WLMB/ MSA 0.03 1
HNC/MSA
BGY
0.025 0.050 0.063
0.056 0.067
+ En
0.027 0.054 0.082
MPB4
MGC
0.026 0.052
0.036 0.072 0.097
are those of Figure 1.
1
TABLE 11: Test of the Contact Condition for Different Theories" ~~
~
~~
~
g m
I
- --
__+----
+1
concn. M
MC
WLMBIMSA
HNCIMSA
MGC
1.o 1.9616
0.094 0.346
0.048 0.258
0.54 1.57
0.00 0.00
"The Monte Carlo data are the osmotic coefficients of 1-1 electrolyte solution of Rasaiah, Card, and Valleau."
Figure 1. Potential at x = 0, \k(O), as a function of the charge density on the wall. Curves are for 1-1 restricted primitive model ionic solutions (R = 4.25 A, T = 298.15 K, and t = 78.5) at several concentrations. The continuous curve (-) represents the WLMB/MSA approximation solved in this work, and the curves (---) and (---) correspond to the noniterative HNC/MSA and MGC approximations, respectively. The dots ( 0 )correspond to the Monte Carlo data of Torrie and Valleau.'
elaborate theories such as the MPB48 and BGY + En9 for a 1 M solution and several surface charges. We can see that the simple analytical scheme presented in this work is quite accurate in this range of physically significant surface charges. In Table I1 we test the exactness of the contact condition of Henderson and Blum in the limit of b 0. The M C values formally correspond to the reduced pressure of the bulk electrolyte as obtained by Rasaiah et al." We see that even though this is a demanding test, the WLMB/MSA approximation gives values of the same order of magnitude as the M C expected values, while the MGC and HNC/MSA approximations give unsatisfactory values. We thus have solved the WLMB integral equation for the double-layer problem in an approximated but analytical form and have shown the adequacy of such an equation in the description of this problem. The iterative solution of this equation will be the subject of a future report.
We observe that as the concentration increases, there is a small improvement in the WLMB/MSA approximation over the H N C / M S A values when compared to the M C data. In Table I we compare the numerical values obtained for the contact potential for the WLMB/MSA approximation and more
(8) Outhwaite, C. W.; Bhuiyan, L. B.; Levine, S. J . Chem. SOC.,Faraday Trans. 2 1980, 76, 1388. (9) Croxton, T. L.; McQuarrie, D. A. Mol. Phys. 1981, 42, 141. (10) Croxton, T. L.; McQuarrie, D. A. J . Phys. Chem. 1979, 83, 1840. (1 1) Rasaiah, J. C.; Card, D. N.; Valleau, J. P. J . Chem. Phys. 1972, 56, 248.
-
0
10
c h a r g e density
20
30
(microcoul/m*)
