THE SURFACE TENSIOX O F CHARGED SURFACES BY OSCAR KKEFLER RICE
Introduction The surface tension at the boundary of two immiscible solvents is affected by the electrical and chemical forces concerned ryith the adsorption of the solute. If the solute is ionized the electrical forces become particularly important, and, also, they are easier t o handle theoretically. STe propose t o investigate these forces and their effect on the surface tension from the standpoints of electrostatics and thermodynamics, with particular reference to the relation between the two points of view. We can imagine a number of situations which might arise at a charged mrface. For exarrple, suppose we had two adjacent immiscible solutions, which we will call A and B,each containing ions of the same kind. Suppose ions could not of themselves cross the boundary but we had some device for taking them across, If we took more charge of one sign than the other across, and if the ions were free to m0.i-e in their respective solutions, negative ions would be attracted to the surface from the negative side of the boundary and positive ions would be repelled, and uice-iersn for the other side. These ions would form two diffuse atmospheres, one on either side of the boundary. Gradually we would build a potential difference, q5, across the boundary, which we will consider as positive if the potential of h is greater than that of B. If the solutions were so extensive that our transfers did not appreciably alter the composition of either, the change in the surface tension, y, at the boundary would be given by the differential equation dy = - Qdq5 (1) where Q is the excess of positire over negative charge per unit surface on the X side of the boundary (or, what amounts to the same thing, the excess of negative over positive charge on the B side). It is thus seen that y has a maximum where Q is zero. q5 will in general be zero when Q is zero unless there are forces acting m-hich tend to separate charges which are on the same side of the boundary and so create a double layer which does not extend across it. This has been discussed by Gouyl and also by Frumkin2. The capillary electrometer, though ihe details are somewhat different, presents a situation which is essentially similar to the above. Equation ( I ) was long ago derived for the electrometer on the assumption that the mercury meniscus was completely polarized3. IT-e can imagine another situation which may be of some importance in connection with colloid theory. Suppose the double layer t o consist of a monomolecular layer of ions adsorbed from the solution X a t the boundary of the solution and a dielectric B, and a diffuse layer of ions in the solution A. Gouy: Ann. Physique, (9) 7, 129 (191;). Frumkin: Phil. Mag. (6) 40, 363, 375 (1920). See Freundlich: “Kapillarchemie”, 394.
SURFACE TEPI’SIOh- O F CHARGED SURFACES
I349
Let the chemical potential of the ions which are adsorbed be p, that of those in the interior of A far from the boundary p‘. p will be a function of Q and hence of (which are now defined as the charge per unit surface of the monomolecular layer and the excess of its potential over that of the solution, respectively), provided we assume equilibrium to be established instantaneously in the diffuse layer. p‘ will be independent of Q provided the amount of solution is so great that the composition in its interior is not appreciably changed in the processes to be considered. (It is to be noted that we are still considering a polarized surface ; that is, equilibrium is not necessaril! established between the monomolecular layer of ions on the wrface and those in the solution, but only in different parts of the solution.) Let E be the charge per unit quantity for the adsorbed ions. K e then have
+
This n-e may easily prove. Increase the surface by unity, maintaining the charge per unit surface a t Q. During the increase of surface Q E ,inits with a charge of Q are brought up to the surface. The chsnge in free energy, clue to the surface tension and the transfer of ions to the surface is consequently y+Q++ (2 E (p-p’), Son.. holding the extent of surface constant, increase the potential by d+. =It the same time the cheniical potential will increase by clp = ( d p (14) dq5 and the charge per unit surface will increase by d Q = (dQ dq5) d+. The increase of free energy in this change n-ill be +clQ+cp-p’) Son. decrease the surface by unity with a change in free energy of -
(~+dy)- (Q+dQ) i++d+)
-
(Q ~ + t l Q e)
clQ
E
(p--p’+dp)
S o n the change in free energy in the n-hole cycle should be zero, and neglecting second order quantities we find that thiq gives 11s i z ) . practically identical derivation will hold for (I), the work due to chemical potential drorping out (if we nssiinie that there iq so much of both solutions that the chemical potential of neither is appreciahly changeJ in the processes consic1ered.l If there are ?i different kinds of ions in the monomolecular layer at the surface we get instead of ( 2 )
where the subscript 2‘ is appended t o quantities belonging to the zth ion. Since the electrical potential is the same for all ions we may n-rite (3) in the form
where Q is the total charge per unit area.
*
I 3 so
We can write equation
OSCAR KNEFLER RICE (2)
in the integrated form
where yo is the surface tension n-hen $ is zero and y1when $ is $]. 11-e can write equation ( I ) in the integrated form 71 =
YO
- J!'Qd$
(4a)
If the only forces due which act on ions in the neighborhood of the boundary are due to ions which have crossed the boundary then Q and $ are zero together, as already stated. Also yo, the surface tension when Q is zero, will not vary at all in solutions in which the ions are present in different concentrations. For Gibbs's adsorption equation, which says that the change in surface tension with the concentration is proportional to the amount of solute adsorbed, will hold under the restriction that Q is zem. But if Q is zero thera will be, by hypothesis, no forces on the ions near the boundary, hence no adsorption. Electrostatics of the Surface Let us now proceed t o investigate the meaning of the various terms in equations (4, 4a). We shall show that the term - ,;!'Qd$ gives 11s just the change in surface tension that we would expect from the electrostatic forces involved. Our general method of attack will be t o consider the pressure in a sphere whose surface is charged and which is surrounded by an atmosphere of ions whose density is a function of the distance from the center. n'e can then relate surface tension to pressure and pass to the limiting flat surface.' TTe first note that there is a negative or outward difference of pressure between the inside of the sphere and the solution immediately outside it, due t o the action of the field on the surface charge. The magnitude of this pressure, P", is zirQ2/K,where Q is the surface density of charge, and K is the dielectric constant of the medium surrounding the sphere. Besides this there will be a positive difference in pressure between a point a t a distance T from the center of the sphere and a point a t an infinite distan e from the shpere, whose mzgnitude we will call P'. The charge in the solution surrounding the sphere has a force pulling it toward the sphere. whose value per unit volume is-pE, \There p is the volume density of charge and E is the magnitude of the electric force vector. (E is to be considered as positive when a positive charge tends t o move away from the sphere.) If T is the distance from the center of the sphere, then we may write dP' dr = - p E , the minus sign indicating that the pressure decreases when T increases. If n is the radius of the sphere P'(at a) = /,"pEdr The volume of a shell of small thickness, d r , and at a distance T from the center of the sphere is 47rr2dr, and the amount of charge included in the shell For a consideration of the electrostatics of the surface from a different point of view, see Herzfeld: Physik. Z., 21, 31 (1920);v. Laue: Jahrb. Radio. Elektronik, 15, 238 (1918); RIoller: Ann. Physik, 27, 670 (1908).
SURFACE TENSIOK O F CHARGED SL-RF.ICES
1351
is A7rr2pdr. Sow Kr2Eis equal to the total amount of charge within the distance r ; or Kr2E = 47rQa2 - ,/i4nr2pdr Differentiating with respect to
T
K--d(r2E)- - 4nr2p dr or p = - -
B
d(r2E) 47rr2 dr
SO, putting into the equation for P’ (at a )
-
zE2
2nQ?
I
b) is exactly the same as the distribution back of So would be were its charge per unit surface Qb, provided, and only provided, there are no forces acting which are not clue t o the charge on the surface but belong t o the surface itself and are a function of the distance from the surface. If there are no such forces the potential $,, at b above the potential at x = w is exactly the same as the potential a t So~ v o u l dhe were its charge Qb. It is thus seen that Q,d$,= fQd$ where In the first integral we integrate by moving u p t o the surface and in the secorid we integrate by charging the wrface. This equality will hold even if the ions d o not form perfect eolutions, for the forces the ions esert on each other at any point will depend only on their concentration and not on where they are. Thus we see that the integral which contains only electrical terms gives the surface tension which we would calculate directly from the electrostatic forces, but only when there are no other surface forces involved. YO
= -
1;:1
I:&$,
SURFACE TEKSION O F CHARGED SURFACES
I353
A slight modification of this line of reasoning will enable us to deal with the case of a diffuse layer on either side of the boundary. I n this case equation (4a) nil1 hold, and we will show that the rhange in surface tension is entirely accounted for by the electrostatic effects. It is easy t o see that ,'E?ds may be expressed in terms of effective charge and potential, much as before, but that it breaks up into the sum of two integrals
where the primes refer to one side of the boundary, the double primes t o the other, and where m' and x f ' , respectively, have the value zero at the boundary and increase as we go from it. S o w we may choose a series of positions on one side of the boundary which is in one to one correspondence with a set of position%on the other side of the boundary, such that the effective charge per unit wrface Q,, for the position on one side is in equal in magnitude and opposite in sign to the effective charge per unit surface Qyn for the corresponding position on the other side. (TTe assume that the potential is continuous and always increases in the same direction in $pace.) I t is then seen, assuming the qysteni as a whole to he neutral, that the h i t s of the two integrals are respectively corresponding points in the sense indicated. K i t h the understanding that n-e are t o tabe the proper corresponding values for the double prime side of the boundary we may write x" as a function of m' and
Q,.
= -
Here. for a given (1s'. 4c$lf and d+," are corresponding dif-
Q,!.
~ 1 4 ~dq5,., ' dq, ferentials. \T7e may write Qx,simply as Qx, m' as .T. and let I = __ cis ds' ds ~
Our integral then liecomes
,";=O-
~ ~ and 4 it, is seen that
X =G
,, = x
equal to the total potential diFererce acrcss the boundary while
rlq5x is
I X =G , I=,
dq5, is
the potential difference across the boundary minus the potential difference which a surface of charge Q, would have.' 'Ibe same correspondence as liefore is now seen t o exist n i t h the thermodynamic equation.
Kinetics of the Adsorbed Ions Let us now attempt to interpret the second integral of (A). This we easily do for a n ideal case n i t h the help of the Boltzmann probability principle, dS
=
K -
?;
tl log P, where S is the entropy per unit mass of the ions
adsorbed in the monomolecular layer, 11 the gas constant, S the number of ions in unit mass, and P the thermcdynaniic probability of the system consisting of unit mass of iocs. 7k.k gives us. as is rea6.ily seen hy dividing the surface I n comparing Tvith equation ( p i >which is t h e integral of equation ( I I , notice the arbitrary manner in which th- signs of thc quantities in t h a t equation n-ere determined.
I354
OSCAR KNEFLER RICE
into surface cells in the same way that me divide the volume of a gas into volume cells, and computing the change in probability when the concentration is changed1, dS = - R d log Q, Q being analogous to the concentration of a perfect gas. Since d p = dH - TdS, where H is the heat content per unit amount of adsorbed substance (T being considered constant), and since for an ideal adsorbed substance as for a perfect gas we may assume dH to be zero, we get dp = RTd log Q = RTdQ, Q Hence
Q* E d+
d+
R T dQ
= - -d+ E d+
and where
Q
=
Q1 when
+=cPl
So for the ideal case.
This may be readily generalized if more than one kind of ion is adsorbed.
Summary An electrical charge may arise at the surface bounding two liquids in various mays. Three cases are considered. (I) Suppose we have two solvents, each of which contains various ions, all the kinds of ions being present in both solvents. If ions cannot cross the boundary, but if they can be transferred in some way from one solvent to the other, we may build up a potential difference between the solvents by transferring more positive ions than negative from one side of the boundary t o the other, as the ions will form a double layer at the surface. ( 2 ) The capillary electrometer is essentially similar to the above. (3) K e might conceive of ions adsorbed a t the boundary of two liquids and forming there a monomolecular layer, while one of the liquids contains dissolved ions (the ions a t the boundary not necessarily being in equilibrium with those in the solution), I n the first two cases the expression for the surface tension is
where yo is the surface tension of the surface when its potential 4 is 0, y1its surface tension when is equal to 41, Q being the charge per unit surface. I n the third case (if we assume that only one kind of ion is adsorbed a t the surface) the expression for the surface tension is
+
1 Such a change could be effected, for example, by the transfer of all the ions from the given surface to a n empty surface of slightly different area. There would then be a different number out of the sum total of all the cells in the tn-o surfaces filled a n d each one vould then contain a different number of ions, the actual calculation of the change in probability being readily carried out in the usual manner. See IT.C. 31.Lewis: “A System,of Physical Chemistry,” Vol. 111, p. 1 1 (1924) and Leviis a n d Randall: “Thermodynamics” p. 1 2 0
(1923).
SURFACE TEXSION O F CHARGED SURFACES
I355
where p is the chemical potential of the adsorbed ion and E ita charge per unit mass. (This expression can be generalized if several kinds of ions are adsorbed at the surface.) A proof for this esprewion is given. By considering the pressure inside a charged sphere which is surrounded I q a diffuse atmosphere of ions symmetrically distributed about the sphere, the surface tension is found in the case in which the sphere is very large. It is then possible to show that, if no forces other than the electrical forces act on the ions in the neighborhood of the boundary, that the term JT'Qd$ is
*
equal t o the surface tension thus found.
f'g E
d$ is discussed in terms of the kinetics of the ions d+ forming the monomolecular layer at the boundary. The term
Department of C h e m i s t r ~ , Cni 1 erszty o j Calijorrlza, Berkeleg, Calijor7i 1 a .
