1573
Flow of Electric Current through a Solution Interface
Adsorption Kinetics during the Flow of a Constant Electric Current through a Nitrobenzene/Water Interface P. Joos* and M. Van Bockstaele Department of Cell Blology, Universitalre lnstelllng Antwerpen, Wilrljk, Belglum (Received November 2 1, 1975)
The interface between a water and a nitrobenzene phase is considered. In both phases some surface active electrolyte, cetyltrimethylammonium bromide, is dissolved and both phases are in equilibrium. When a constant electric current flows through that interface the interfacial tension changes. This variation in interfacial tension is explained by a difference in transport numbers of the cation in both phases. A mathematical analysis is given. For depletion experiments good agreement between experiments and theory is observed; for accumulation experiments good agreement occurs only for high concentrations or for short times.
Introduction When an electric potential difference is applied across a liquid-liquid interface, the concentration of ions at that interface may change. This change of interfacial composition, hence of interfacial tension, may be due to two entirely different mechanisms; this problem is discussed in detail by Blank and Feig1.l One of these mechanisms, known as electrocapillarity, is a pure thermodynamic phenomenon. For the other mechanism, which is considered here, the cations and anions are not confined to one bulk phase, but they can freely move across the interface, which is therefore said to be nonpolarizable. In this case changes in interfacial composition are due to a difference in transport number of the cations (or the anions) in both phases. We considered the following system: two liquid phases are brought in contact with each other, water and a polar oil such as nitrobenzene. In both phases a surface active electrolyte is dissolved, e.g., cetyltrimethylammonium bromide (CTAB), and the system is in thermodynamic equilibrium. Two electrodes one in each phase, are mounted parallel to the interface in such a way as to extend over its whole area. A potential difference is applied causing a current, I,to flow through the interface. The flux of cations migrating from the aqueous solution to the interface is Inl+/FQ (nl+ is the transport number of the cation in the aqueous phase, F the Faraday number, il, the area of the interface). At the same time the flux of cations transported from the interface to the nitrobenzene phase is Inz+/FQ (nz+ is the transport number of the cation in nitrobenzene). Hence the net change of cations at the interface is I ( n l + - nz+)/FQ.Depending on the direction of the current cations are accumulated or depleted at the interface. Similarly the amount of anions accumulated or depleted at the interface is -I(nl- - nz-)/FQ (nl- and n.2are the transport numbers of the anion in water and nitrobenzene, respectively). Since the sum of the transport numbers in each phase equals one (nl+ + nl- = 1;n2+ n2- = 1) it is seen that the same amount of anions and cations accumulate at the interface, hence the interface remains electrically neutral. At the beginning of this century concentration changes due to interfacial electrical transference were studied by Nernst and Riesenfeld2 using common salts such as KCl in a phenol/water system. These experiments were repeated by Guastalla3 and Blank4 with surface active electrolytes. These author^^-^ also studied the effect obtained with a second common salt added to the system. The situation with a supporting electrolyte will, however, not be considered here, be-
+
cause the supporting electrolyte is confined to one phase only and a quantitative treatment is more complicated.
Theory As outlined above, the passage of an electric current causes CTAB to be accumulated or depleted at the nitrobenzene/ water interface. This interface remains electrically neutral. We will assume that the current, I, through the interface is constant. The transport by electric migration is partly balanced by diffusion, to or from the interface depending on the direction of the current. The diffusion process is ruled by the ordinary diffusion equations, since the interface and the bulk phases are electrically neutral it is not the diffusion of each ion (electrodiffusion) which must be taken into account, but the diffusion of the electrolyte as a whole. This fact is particularly well discussed in Newman’s books8 The diffusion equations are: for phase 1 ( z < 0 ) aC1a2C1 _ at - D I T for phase 2 (z
> 0)
The z coordinate has its origin in the interface and points from the water phase (subscripts 1) to the nitrobenzene phase (subscript 2). The above diffusion equations must be integrated taking into account the appropriate boundary and initial conditions. These boundary conditions can be found assuming local equilibrium between the two subsurfaces and the interface, and the conservation of mass at the i n t e r f a ~ e . ~ The condition of thermodynamical equilibrium in the subsurfaces is expressed by KCi(0, t ) = Cz(0, t )
(2)
K being the distribution coefficient. The distribution coefficient of CTAB between water and nitrobenzene was experimentally determined by Blank5 and equals 1.6. The second boundary condition deals with the conservation of mass at the interface. The amount of surface active electrolyte transported by migration from or to the interface is IAnlFQ (An = nl+ - n2+). Part of this amount is used to change the adsorption d r l d t and the remaining part is transported by diffusion, whence The Journal of Physical Chemistry, Vol. 80, No. 14, 1976
1574
P. Joos and M. Van Bockstaele
I
I
I
I
60
120
180
240
1
1
I
I
300
360
420
480
t (SI 540
6(1
Figure 1. Ion accumulation experiments. Increase of interfacial pressure, K ,as a function of time t (in seconds)for constant currents flowing through the interface. Solid line (a):concentrationof CTAB in water 3 X I O - * mol C W 3 ; (la) 300 MA; (2a)200 MA; (3a) 100 @A; (4a) 50 MA. Broken line (b): concentrationof CTAB in water mol ~ m - (~l b; ) 300 PA; (2b) 200 yA; (3b) 100 MA; (4b) 50 PA. Dotted line (C): concentration of CTAB mol ~ m - ~(IC) ; 300 MA; ( 2 ~200 ) MA: (3b) 100 yA; (4c)50 MA. in water 3 X
dr (3) dt D1(%)dz 0 - Dz az o This equation without the source or sink term I A n l F Q is discussed by LevichlO who also considered interfacial diffusion and convection. These terms are omitted here because the interface remains uniform. Equation 3 without the term drldt was already used by Nernst and Riesenfeld.2 In the cases studied by Nernst and Riesenfeld neglection of the term drldt is certainly justified since they studied common inorganic electrolytes which are not surface active. The two remaining boundary conditionsstate that far away from the interface the concentrations do not change I- A = n-
FQ
(%)
+
(4)
0 r = ro;KCP = c20 z = o z
(5)
The last condition in eq 5 follows from the fact that initially both phases are in equilibrium. With the above boundary and The Journal of Physical Chemistry, Vol. 80, No. 14, 1976
An r = r o + -ItFQ
+2
[ a + K a
6
where {is an auxiliary variable. This result bears some similarity with that of Ward and Tordai,12who studied simple diffusion controlled adsorption kinetics. In fact we made the same assumption of Ward and Tordai: no adsorption barrier. We will denote Y ( t ) = clod7 - ~ t / t C 1 ( 0 t, - j-1 d
The initial conditions are at t = 0
c1=c10
initial conditions the diffusion equations are integrated. As* pointed out by Hansenll such a problem can conveniently be solved by the Laplace transform method. The result is (see Appendix)
e
(7)
The convolution integral may be evaluated graphically and in this paper the method of Lange13 will be followed. By introducing numerical values of A n and + K a ) in eq 6 it turns out that r - ro can be neglected, whence
(a
6
3 ( t ) = -2FQ
AnI
[a +K a
]
Flow of Electric Current through a Solution Interface
60
1575
120
180
240
300
Figure 2. ton depletion experiments.Decrease of interfacial pressure, a,as a function of time t(in seconds)for constant current flowing through the interface. Full line (a):concentration of CTAB in water 3 X loB8mol ~ m - (la) ~ ; 300 pA; (2a)200 pA; (3a) 100 pA; (4a)50 pA. Broken line (b): concentration of CTAB in water lo-' mol ~ 1 1 1(lb) ~ ~ 300 ; MA; (2b)200 pA; (3b) 100 p A ; (4b) 50 pA.
From this expression it follows that the plot of 3 ( t ) as a function of time should yield a straight line and from its slope A n / [ G+ K a is obtained. In eq 8 the term r - ro is neglected, hence one should expect that an equivalent expression should be obtained by assessing d r l d t = 0 in eq 3. Integration with the same boundary and initial conditions yields Sand's equation14
However we have some doubt about the application of this equation to our system because small changes in adsorption correspond with rather important changes in the subsurface concentration Cis. Therefore we prefer the rather tedious graphical integration for calculating Cis.
Experimental Section The materials used in this investigation were nitrobenzene (grade puriss) obtained from U.C.B. and a pure CTAB purchased from Baker. Clean glassware and double distilled water was used throughout. The interfacial tension between pure nitrobenzene and water was 24.4 dyn cm-l. Interfacial tensions were measured by the Wilhelmy plate method using a Cahn electrobalance. The Wilhelmy plate was siliconized. The measuring cell was similar to that described by Blank.4 The area of the interface was 42 cm2. The potential difference between the electrodes was obtained by means of a constant current source. Generally the following intensities were used: 300,200,100, and 50 MA.The output of the balance was connected to a recorder enabling us to follow the variations of interfacial tension with time. The Journal of Physical Chemistry, Vol. 80, No. 14, 1976
1570
P. Joos and M. Van Bockstaele
-J (t
40
30
2c
1c
50
100
250
500
750
b
Figure 3. The function J ( t )as defined in text for accumulation experiments. Full line (a): concentration of CTAB in water 3 X mol ~ m - (la) ~ ; 300 PA; (2a) 200 FA; (3a) 100 PA; (4a) 50 wA. Broken line (b): concentration of CTAB in water 3 X lo-’ mol ~ m - ~ ( l b; ) 300 wA; (2b) 200 wA; (3b) 100 FA; (4b) 50 FA.
Results The interfacial pressure, a, Le., the lowering of the interfacial tension as a function of the CTAB concentration in water, was measured. As argued before the concentration in nitrobenzene was 1.6 times higher. Since no indifferent electrolyte was added the Gibbs equation reads: da p=- 1 2RT d In C1 The experimental a - C curve could be fitted by the von Szyszkowski equation15
where r” is the saturation adsorption (r”= 8.7 X cm-2) and a is the Langmuir constant ( a = 3.5 X The Journal of Physical Chemistry, Vol. 80, No. 14, 1976
mol mol
cmm3).The other symbols have their usual meaning. From the von Szyszkowski equation, the Langmuir equation is obtained using the Gibbs equation
,. r = p- C1 a
+ C1
(12)
Three CTAB concentrations were investigated 3 X and 3 X mol ~ m - In ~ .experiments where the current flows from the nitrobenzene phase to the water phase the interfacial tension decreases and the adsorption increases (accumulation experiments), when the current flows in the inverse direction the interfacial tension increases (depletion experiments). Accumulation experiments are summarized in Figure 1,depletion experiments in Figure 2. No experiments were carried out when the interfacial pressure attains a value of about 15 dyn cm-l because this pressure corresponds with
1577
Flow of Electric Current through a Solution Interface
J (t)
loa
5c
Figure 4. The function J(t) as defined in text for depletion experiments for a CTAB concentration in water of lo-’ mol ~ 1 1 1 (1) ~ ~300 : PA; (2)200 PA; (3) 100 MA; (4) 50 PA.
TABLE I: Values of the Parameter A n / [ d K f- K d K ] and An Obtained from the Slopes of the J( t ) Curves a
+K
C10, mol cm-3
I , WA
3 x 10-8
+300 +200
0.57 0.33
+50 -300
0.21b 0.06 0.07 0.12 0.15 0.216
1.20c O.7Oc 0.31 0.44 0.13 0.14 0.25 0.32 0.47
O.llb
0.25
+loo
-200 -100 10-7
-50 +300
+200 +loo
+50
-300
-200 -100 3 x 10-7
-50
+300 +200
+500
An/(d&
0.X6
a )
An
0.096 0.096
0.19 0.19 0.08 0.16 0.09 0.19 0.09 0.19 0.16 0.32 0.10 0.21 0.12 0.26 0.09 0.19 Mean An = 0.255
+
a Accumulation experiments are noted by before the current and depletion experimentsby -. 6 Extrapolated from initial slope of J(t). Not considered for calculating the mean value of An.
a subsurface concentration of about mol cm-3 and at this concentration mice1 formation sets in.l6 Discussion Starting from the experimental T - t curves (Figures 1and 2) the subsurface concentration in the aqueous phase was calculated as a function of time by means of eq 11 and the adsorption using eq 12. Assuming reasonable values of Dl,D2,
and An it appeared that the contribution of Ar in eq 6 could be safely neglected, hence eq 6 may be simplified to eq 8. Knowing the subsurface concentration as a function of time the back diffusion integral of Ward and Tordai can be evaluated and also the function 3(t)as defined in eq 7 . Some typical 3 ( t )- t curves are given in Figures 3 and 4. It appears that, for both depletion and accumulation experiments at C = 3 X lo-’ mol cmm3,3 ( t )is indeed a linear function oft as expected from eq 8. From these data the value of A n / [ a K m and assuming values for D1(= cm2 s-l), D2 (= 5 X cm2s-l), and K (= 1.6) the difference in transport number of the cation in water and nitrobenzene can be estimated. Although there is some scatter, the experimental values for An seem quite reasonable. However, for accumulation experiments a t concentrations of 3 x and 10-7 mol no linear relation for 3(t)- t is obtained. At present we can offer no explanation for this deviation, but probably double layer effects become operative which are neglected in our theory. The more important the deviation becomes the higher the current and the lower the concentration. Describing his experiments with Sand’s equation, Blank4 also observed some deviations, without assessing numerical values, however. We assume our theory is correct as long as the deviations from the initial state are not too large. Therefore we calculated the parameter An/[z/i5; K G ] from the initial slope of 3 ( t ) at t 0: this procedure yields more consistent results. We have also carried out experiments with alternating current which will be published e1~ewhere.l~ In this case the oscillations around the equilibrium state were extremely small. These experiments yielded a value for An = 0.25. The mean value from our present experiments is also 0.25 (see Table I). Not too much importance should, however, be given to this agreement in view of the scatter of values for An. This scatter is mainly due to experimental and computing errors, since the determination of the initial slope is rather inaccurate. Finally we plotted in Figure 5 the variation of adsorption,
+
-
+
The Journal of Physical Chemistry, Vol. 80, No. 14, 1976
1578
P. Joos and M. Van
Bockstaele
I&
1.:
1
0.5
60
120
180
240
300
360
420
480
540 t [ s ) W
Flgure 5. Absolute value of adsorption variation as a function of time for a CTAB concentration in water of IO-’ mol ~ mulation; (2)I = 300 pA depletion;(3) I = 50 MA accumulation and depletion.
calculated from the Langmuir equation, as a function of time. It is seen that for high concentrations (C = 10-7 mol cm-2) and small deviations from the equilibrium state (small currents) the same value of Ar was found for depletion and accumulation experiments. For higher currents, however, this is not longer true.
Appendix Integration of Diffusion Eq 2 with the Boundary and Initial Conditions (Eq 1-4). Making use of the Laplace transform the diffusion equations are transformed to
(1) I = 300 PA accu-
( 1 1 ~ ~ :
(C is the Laplace transform of c ) . Solutions of ( A l ) and (A2) are
where A1 and A2 are integration constants. In view of boundary condition 3 one has KCl(0) = C2(0)
(A5)
yielding A2 = KA1
The Laplace transform applied to the boundary condition for conservation of mass at the interface yields The Journal of Physical Chemistry, Vol. 80, No. 14, 1976
CI- Exchange in AgCl Suspensions
1579
References and Notes (y being the Laplace transform of I'). Substitution of eq A6 and A7 in A3 yields for the concentration in the subsurface at
z=o
r0 - s y + ( I A n / F Q s )+-CIO 4[vz+KdDJ
s
(A8)
whence
Taking now the inverse transform of eq A9 and considering the convolution theorem
yields eq 5.
(1) (2) (3) (4)
M. Blank and S. Feigl, Science, 141, 1173 (1963).
W. Nernst and E. H. Rlesenfeld, Ann. Phys., 8, 600 (1902) J. Guastalla, Proc. int. Congr. Surf. Act., Znd, 3,Ill, 122 (1957). M. Blank, Proc. lnt. Congr. Surf. Act., 4th. B 11/10 (1964). (5) M. Blank, J. ColloidSci., 22, 51 (1966). (6) CI. Gavach and F. Henry, Electr. Chem. lnterface Electrochem., 54, 361 (1974). (7) CI. Gavach and B. d'Epinoux, Electr. Chem. lnterface Electrochem., 55, 59 (1974). (8) J. Newman, "Electrochemical Systems", PrenticeHall, EnglewocdsCliffs, N.J., 1973, Section 69. (9) J. C. Crank, "The Mathematics of Diffusion", Clarendon Press, Oxford, 1963. IO) V. Levlch, "Physicochemical Hydrodynamics", Prentice Hall, Englewood Cliffs, N.J., 1962, p 591. 11) R. S. Hansen, J. CoiloidSci., 16, 549(1961). 12) A. F. H. Ward and L. Tordai, J. Chem. Phys., 14, 453 (1946). 13) H. Lange, J. ColloidSci., 20, 50 (1965). 14) H. J. S. Sand, Phil. Mag., 1,45 (1900). 15) E. H. Lucassen-Reynders, J. Phys. Chem., 70, 1777 (1966). 16) J. Lucassen and M. van den Tempel, Chem. Eng. Sci., 27,1283 (1972). 17) P. Joos and R. van den Eogaert, J. Colloid lnterface Sci., accepted for publication.
Mechanism of the Exchange of Chloride Ions in Colloidal Suspension of Silver Chloride Tadao Sugimoto" and Goro Yamaguchi Depatfment of lndustrialChemistry, Faculty of Engineering, University of Tokyo, Hongo, Bunkyo-ku, Tokyo, Japan (Received September IO, 1975; RevisedManuscriptReceivedMarch 29, 1976) Publication costs assisted by Tokyo University
The exchange of chloride ions in a colloidal suspension of silver chloride at 25 O C was investigated by means of the radioactive chloride ion, and quantitative relationships were obtained between the degree of exchange and the change in the specific surface area or in the average particle volume. It has thus been shown that there are two steps in the halide ion exchange between a solid and a solution in a colloidal suspension of silver chloride. The first step is an instantaneous process due to the exchange in the surface layer of a solid less than 2 atomic layers deep, as derived from the linear relationship between the degree of exchange extrapolated to t = 0 and the specific surface area. The second one is a relatively slow process ascribed to the Ostwald ripening of microcrystals, in which no self-diffusion was found to be involved even at the early stage of aging, for the following three reasons: (1) the minimum amount of inhibitor needed to stop the exchange agreed with that for stopping the particle growth; (2) some inhibitors of particle growth adsorbed on the surface of silver halide particles freely allowed the surface exchange, but they did not permit any further exchange at all; and (3) the degree of exchange was a function of only the change in the average particle volume and was independent of time.
Introduction Silver halide is a well-known material for conventional photographic emulsions; it has wide industrial uses and, at the same time, very interesting properties for scientific investigation. However, despite its technological importance, the details of the mechanism of nucleation and recrystallization process during the preparation of the emulsion are still not clear enough probably because of its complexity. In the present work, radioactive halide ions have been used to study mainly the recrystallization process in colloidal suspensions of silver chloride, for it appears to be one of the most excellent techniques for this kind of investigation.
In relation to the halide ion exchange in silver halide suspension, Kolthoff and his c o ~ o r k e r s lreported -~ an extremely rapid exchange of bromide ions in freshly prepared silver bromide suspensions. They attributed it to the great thermal mobility of bromide in the extremely imperfect structure of solid, or to the Schottky defects which were supposed to cause high mobility of silver and bromide. They appear to have assumed an instantaneous diffusion for the bromide ion exchange. Pitts6 suggested, from his quantitative analyses, that the bromide exchange in a silver bromide suspension might occur by recrystallization, though he did not refer to the question whether a self-diffusion process was involved in the recrystallization. Mirnik and Vlatkovic' pointed out that fresh silver The Journal of PhysicalChemistry, Vol. 80, No. 14, 1976
