Electric structure of the interface in an electrolyte near the critical point

Electric Structure of the Interface in an Electrolyte near the Critical Point V. M. N A B U T O V S K I I AND N. A. N E M O V

Institute of Inorganic Chemistry, Siberian Division, USSR Academy of Sciences, Novosibirsk, USSR Received August 23, 1984; accepted January 22, 1986 The existenceof the interfacearisingdue to the phase separationin an electrolyteleadsto the appearance of the electric double layer. The thickness of this layer near the critical point is of the order of magnitude of the greater of two lengths: correlation length or Debye radius. Spatial distributions of charge and dipole moment density, electrostaticpotential near the interface, as well as potential jump between two phases are obtained. The dependencies of these quantities on temperature, ion concentration, dielectric constant, and interaction between molecules are studied. The relevant experiments are proposed such as phase-boundary potential measurement, interface-electriccurrent interaction, and generation of the electromagneticwavesdue to the interface osdUation. The results are extended to the strong fluctuation case using the critical exponents in the scaling theory style. © 1986AcademicPress,Inc, INTRODUCTION It is known that in two-phase electrolytes a double electric layer arises near the interface due to the difference of interaction energies of ions in different solvent phases. We consider the spatial distributions of the charge density p(r) and the electrostatic potential ~(r) in the region adjacent to the interface of two phases of the pure solvent near its liquid-vapor or phase separation critical point. Inhomogeneous distributions of p(r) and ~(r) induced by the external potential ~0 in the region adjacent to the electrolyte boundary have been investigated in our works (1, 2). The solvent in the vicinity of the critical point m a y be characterized by the small, dimensionless, macroscopic order parameter 7/(r) ,~ 1 (3). The physical nature of the order parameter depends upon the system under consideration. Near the liquid-vapor critical point ~(r) = [N(r) - Ne]/Nc, where N(r) and Arc are the density of the solvent and its critical value, respectively. Near the phase separation critical point ~(r) = [X(r) - Xc]/X~, where X(r) and X~ are the concentration of one of the solvent component and its critical value, respectively. We consider the dilute solution of the electrolyte x ~ 1, where x = n/N~. Here n is the

effective total density of ions [1 ]. Limit ourselves at first by the more restricting condition

r3 >>r3, r2 = ~okT 41me2,

ro = n -1/3,

[l]

1 n = ~ ~e2ni. i

Here rD is the Debye radius, r0 is the mean interionic distance, ni and ei are, respectively, density and charge of ith sort of ions, e is the elementary charge, Eois the dielectric constant of the solvent, k is the Boltzmann constant, and Tis the temperature. Condition [ 1] makes it possible to consider ions as a continuous media and simplifies the derivation of the basic equations. However, this condition restricts essentially the concentration of ions under consideration. For example, for the electrolytes referred to in the Conclusion section the inequality [ 1] holds for the effective concentrations x ~< 10-5-10 -6. On the other hand condition [ 1] is sufficient rather than necessary. It m a y be shown in more sophisticated way that far less restrictive condition x ~ 1 leads practically to the same results (see Appendix). As for the order parameter ~(r), it varies from one bulk value ~ ( - ~ ) to another ~(oo)

208 0021-9797/86 $3.00 Copyright © 1986 by Academic Press, Inc. All rights of reproduction in any form reserved.

Journal of Colloid and Interface Science, VoL 114, No. 1, November 1986

209

ELECTRIC STRUCTURE OF THE INTERFACE

at the distances ~ >> a0, where ~ is the correlation length. So one may consider the solvent as a continuous media. BASIC E Q U A T I O N S

Minimization of functional [2] with respect to rl(r) and ni(r) in combination with the electrostatic Poisson equation for the inhomogeneous medium [7] leads to the set of equations

(2)

To obtain the basic equations we represent the free energy functional • as follows

--cAB + arrl(r) + bB3(r) -

8~ (V~o)2 +

[2]

(I~ = ~1 q- C]~2q- C])3.

Here ~1 is Ginzburg-Landau functional (3)

h

~gini(r) = 0,

div[ffrl)grad ~p(r)] = -4~- ~

eini(r),

hi(r) = n0iexp[ ei~p(r)+girl(r)] b 4( r ) +~rl

2hrl(r)]dr,

~r

~(°)(r)=~

1)+

I~ini(r)

f eini(r')dr'

Oe(rl)

Eolr-r'['

~= 0----~'

[4]

where #i is the chemical potential of ith sort of ions, e(rl) is the dielectric constant of the solvent. The last term in functional [41 arises due to the dependence fin). Here t0 -= e(0) and ~(°)(r) is the first approximation to the ions electrostatic potential ~(r). The interaction between ions and molecules of the solvent we consider as a short-range (4, 5)

[81

Here we introduce the equilibrium density of ions noi at the critical point. In regions far from interface and from other sources of inhomogeneity noi = N exp(-zdkT) if ~ = 0 and rl=0. Taking into account the fact that some parameters are small one can simplify the set of equations [6]-[8] for the investigation of the electric structure of the interface. The inequality g~rl(r)/kT~ 1 holds as far as both Irl(r) l 1 and gi/kT ~ 1. Since the order parameter inhomogeneity (profile of the interface) is a single source of inhomogeneity, the value of potential I~l oc I,I and ~ois small in the vicinity of the critical point, too. So one can consider the condition e~o(r)/kT ~ 1 to be true. Using these inequalities one can write the exponential expression in Eq. [8] as a power series expansion. The set of Eqs. [6] and [7] linearized by g; may be represented in the form

_crl,, q_arrl Wbrl3 ~l_ 8_~(~p,)2- ~ -G- ~

[5]

where g; are the constants of interaction of ions species i with the order parameter.

= 0,

[9] 4~rG

k(rl)~o']'- ~oV2~---~-rl = 0, ~o = 40),

[10]

h=h-Zginoi, G=~gieinoi,

[11]

i

• 3=~gifni(r)rl(r)dr,

.1

[3]

where a, b, and c are the expansion coefficients, r = (T - To)~Tc, Tc is the critical temperature, h is the external field conjugate to rl(r). The physical meaning of h depends on rl. For example, if n is the relative deviation of density or concentration from its critical values then h is the relative deviation of pressure or chemical potential from its critical values, respectively (2). The contribution of ions we represent as

Zig [f[kTni(r){ln N

[7]

i

~l = ~ f [c(Vrl)2 + arrl2(r)

~2 =

[6]

i

v=rD 1.

i

Here h is the renormalized external field. We have also taken into consideration the fact that near the plane interface (placed at z = 0) any quantity depends on one coordinate z. Journal of Colloid and Interface Science, Vol. 114, No. 1, November 1986

210

NABUTOVSKII AND NEMOV

Using the inequality 17[ ~ 1 one can write integrodifferential equation which may be e(7) as e0 + ~'n(r) and the first term in Eq. [10] solved for two limiting cases. as COCO"+ ~7~o"+ ~'7'~P'.The second term in this (1) Not too close to the critical point, where expression is smaller by factor 7 than the first [~1 is so large that the correlation length is less one (~" - 1). The ratio of the last term to the than the Debye length, ~v ~ 1, one can neglect first one is of the order of 7rD/( and may be the terms proportional to ~o and (~,)2 in Eq. large far from the critical point, if ( ~ 7rD, [9]. These terms are of the same order of where ( = ( - c / a r ) m. It may be proved how- smallness as (~v)2(g/Eo) ,~ 1 and (~v)(vao)(g/ ever that the contribution to the solution ~o(z) Eo) ~ 1 with respect to other terms of [9]. Here from every term is proportional to its value ao is the interatomic distance, Eo and g are, multiplied by the region dimension in which respectively, the solvent molecule-solvent this term is nonvanishing. The region dimen- molecule and solvent molecule-ion characsion for the last term is of the order of ( and teristic interaction energies at interatomic disfor the first one is rD. Their contribution tances. We assume that Eo ~> g. In this case ratio is (nrD/O(UrD) = 7 and one can neglect the solution of [9] describes the interface the last term with respect to the first one. It z gives us the opportunity to substitute eofor e(7) 7(z) = n0tanh x/2~" [ 14] in [10]. The electrostatic potential ~(z) is determined by the substitution [14] for n(z)into [13]

ELECTRIC STRUCTURE OF THE DOUBLE LAYER The set of Eqs. [9] and [10] represents the modification of the Ginzburg-Landau and Poisson-Boltzmann equations connected with each other by the interaction terms proportional to the interaction constants G and ~'. Far from the inhomogeneity sources (z = + ~ ) one can neglect the derivative terms in Eqs. [9] and [10] and obtain

= -~Oo7,

47rG a'~7 + b73 = f~, ~Oo= eokTv2,

c 2 ~=r+-~, a

~,2

47rG2 Ceo(kT)2v2"

[121

Thus the critical temperature decreases by the value T~g2c/a and if h = 0 and ~ < 0 the interface arises. Two different phases arising at both sides of the interface may be characterized far from the interface by the order parameters 7( + oo) representing two solutions of Eq. [ 12], 7(+or) = +no, where 7o = ( - a ~ / b ) 1/2. The solution ~o(z) of Eq. [ 10] under conditions that ~o(+oo) is finite and ~o(0) = 0 is

if;

~(z) = -~o0n0(1 - e-"lZl)sign z.

[ 15]

Charge density p(z) = Zi ein~(z) is determined by [15] and the electrostatic relation p(z) = -~o"(z)/4r: 0(z) - ~0p2 z n°4cr [~(1 _ e_,lZl)sign Z -- tanh ~-~] -

-

m

[16] The plots of 7(z), ~o(z), and p(z) versus z given by [14]-[16] are represented in Fig. la. (2) Close to the critical point where the correlation length is greater than the Debye length, ~v >> 1, Eq. [13] leads to the relation

~p(Z)=_~O0[7(2)q _1 d27(z).]

v2 dz 2 j.

[171

The term proportional to (~o')2 in [9] is of the order of smallness (g/Eo)2(ao/~) ~ 1 and may be neglected. Equation [9] takes the form ~d27+ -c~--Sz2 a~n + bn 3 = 0,

g=

c(1-~) >0. [181

~o(z) = - ~OoV

e-'l~-z'lT(z')dz'.

[ 131

ot~

Substitution of Eq. [ 13] into Eq. [9] yields the Journal of Colloid and Interface Science, VoL 114, No. 1, November 1986

The solution of Eq. [18] describes the interface and has the form given by [14] with the re-

ELECTRIC STRUCTURE OF THE INTERFACE

211

noted that the phase-boundary potential arises under thermodynamic equilibrium conditions. This phenomenon is similar to the contact potential between two metals and may be measured experimentally by the similar methods. Other experiments may be based on the existence of the electric double layer: the interface displacement due to the effect of nonuniform external electric field, interface interaction with the electric current, generation of the electromagnetic waves induced by the oscillation of the dipole moment of the interface due to the interface oscillation. consider some typical aqueous electrolytes with the separation critical point and E - 50: phenol-water (C6H5OH-H20) with Tc = 339 K and xc = 26 wt% (6), isobutyric acid-water (CaH7COOH-H20) with T~ = 299 K and xc FIG. 1. Spatialdistributionsof the interfaceprofile~(z) = 39 wt% (7), here xc is the critical concen(curve 1), electrostaticpotential~(z) (curve 2) and charge tration. The parameters of these aqueous elecdensity o(z) (curve 3) for the cases~v ~ 1 (a) and (v ,> 1 trolyte solutions are rather similar (b). g-~ (0.01-0.1) eV, Eo - (0.1-1) eV, normalized correlation length ~ = (-~/a'7) 1/2. a 0 - 10-7 cm, ~o -~ (0.01-0.1) V, Functions ~(z) and o(z) are (Fig. lb) G ~ x e g / a 3, a ~- b ~- Eo/a~, ~(z) = -f0~70[1 + (~v)-2(tanh2~2~ - 1)] 70 N_ T1/2, ,~ xl/2/ao, a)

Z

X tanh ~-~, 2

Z

--- ao~-1/2, [191

Z

[201 CONCLUSION The interface arising between two different phases in electrolyte has the electric structure, i.e., the thermodynamically equilibrium spatial distribution of the charge and electrostatic potential. The j u m p of the phase-boundary potential ~ = ~(oo) - ~(-oo) and corresponding effective surface density of the dipole moment d are determined by the expressions [15]-[19] and equal to /a \1/2 6 ~ = 4 r r d = 2~po~-~r) . [21] (Here and below we use r for ~.) It should be

~5~o,.~ 2~po.7-m.

Remember that quantity x is the total effective ion concentration including both own solution ions (let their concentrations be Xown) and some additional ions Xada, X = Xown + Xadd >~Xown- For example, for the phenol-water solution Xow. --- (xH+) + (XcrHsO) ----- 10 -6, for the isobutyric acid solution Xow. - (xn+) + (Xc.H70~) -- 2 × 10 .3 (8), and if some quantity of, say, NaC1 is dissolved in these electrolytes then Xadd = (XNa+) + (Xcl-)It is known that in the vicinity of the critical point, Irl ~ Gi, the strong fluctuation arise. Here Gi is the Ginzburg parameter (3), Gi = (ao/~o) 4, where ~0 is the correlation length far from the critical point (z ~ 1) (5). Though rigorous solution of the problem in this case is extremely complex, there exists a simple way to obtain the results with a good accuracy by using the modification of the Ginzburg-LanJournal of Colloid and InterfaceScience, Vol. 114, No. 1, November 1986

212

NABUTOVSKII AND NEMOV

dau functional (4, 5). This functional may be obtained from the free energy scaling expression and differs from [2] only by substitution (neglecting the anomalous dimensionality exponent) al~-2-~'2#, bl r2-"-4#, and ¢1T2-a'23-2v f o r at, b, and c, respectively. Here a, 3, and v are the critical exponents of the heat capacity, order parameter, and correlation length, respectively (5). All our formulas remain valid with this substitution. Taking into account the relation between the critical exponents one can obtain that with a good accuracy the substitution r --~ r ' -= r 2", and b --~ b' --- br" in all formulas can be performed to estimate the effects.

+ ~E(r). In our case 6e(r) - (&lOn)n(r) < ~o (0el07 - 1, 7(r) ~ 1). The potential of a single ion satisfies the equation [7], where p(r - r') = e6(r - r'). Using the perturbation theory one can represent the potential ~o(r - r') as ~l(r - r') + ~o2(r- r'), where e

~ol(r-r')-e(r,)lr_r,i, ~o2(r-

v"=-O/Or"

[All

1 \ " /V" 7)[ ~ ) .

[A2I

r')

1 eOe g d r3,, 4-r~-~J~(v

An ion energy is U(r) = Ul(r) + U2(r), where Ul,2(r) = ei

E1,2(r- r')d~,

APPENDIX

If condition (1) is not valid then n(r) may not be defined in a conventional way. Indeed, according to the usual definition one has to divide the space in a small volumes A V/ = Ax. Ay. Az such that (a) any volume contains many ions AN/, (b) the dimensions of any volume are so small that any macroscopical quantity varies insignificantly in the limits of this volume. Then n(ri) = lim~v--,o( A Ni/ A Vi). All dimensions of any volume usually are supposed to be of the same order of the magnitude. If rD >> ro then by choosing ro ~ Ax, Ay, Az < rD one makes both (a) and (b) true. Otherwise if ro >~ rD then choosing Ax, Ay, Az rD ~< ro in accordance to (b) one obtains ANi~_ rff3AVi~ 1 and (a) is not valid. To avoid this difficulty one can choose the volumes A V,- extended along x and y axes in such a way that Ax. Ay. Az >>ro3 and Az < rD. In this case one may lose one more effect: the inhomogeneity of the e(r) yields the dependence of an ion energy on z not only due to the everaged self-consistent field of all ions but also due to the interaction of ions with their own "images." The reason is the fact that in this case the ions of a certain volume may be more close to their own images than to their neighboring ions. To estimate this effect we consider an ion in the inhomogeneous medium e(r) = Eo Journal of Colloid and Interface Science, Vol. 114, No. 1, November 1986

Ei,2(r-- r') = --V~l,2(r- r').

[A3]

It is clear that both UI and U2 are infinite due to the infinity of~ol,2(r- r') as r--~ r'. To avoid this infinity one has to take into consideration the space dispersion of the permittivity at interatomic distances a0 and the finite size (---ao) of ion. Practically it means that all distances in [A 1]-[A3] are defined with the accuracy to a0. Then one obtains e2

e 2 Oe

Ul(z) ~- - -

0ao ora0gT(z).

[A4]

The first term in [A4] represents Born selfenergy of ion in the medium with permittivity Co. The second term is the variation of Born energy due to the variation of the permittivity. It is clear that this term is one of the contributions in the functional 03. It has to be included (and has been included) in functional 03. By simple differentiation and integration U2 may be reduced to the form U2(Z) • e 2 0e ,,oo dT(z') dz'

dz' z - z , ' Zt

d 7 ( z ' ) _ 70 cosh-2 4 ~ " dz'

[ASI

Here f - • • dz denotes the principal value of integral. This integral is equal to one taken over the cycle in the complex plane of z. The function cosh -2 w has the second order poles

ELECTRIC STRUCTURE OF THE INTERFACE in the points w = ~ri(n + ½), n = 0, + l , +2, . . . . So Uz(z) represents the s u m

Uz(z)oo

S(z) = ~

7r2 e 2 Oe no

(2n - 1)V

reaches its m a x i m u m in region ~ - rD -- Z, where F1 -~ 1 and is small everywhere else. So ~ad/~ is always small. On a n o t h e r h a n d F2 is not small in the region z >>ro, ~:

z, r D " ~ ,

z

n=l = v2+_~_(2n _ 1) 2

Z,~,rD, T h e potential energy U2(z) has to be added to the n u m e r a t o r of the expression [8]. Expanding this expression one obtains the additional t e r m (the second) in Eq. [10]

~,rD~Z,

FI-~ -rDZ ~ - ' ~ 1,

F 2 - - - - - ~ 1;

z F1 -------~ 1, rD

/72--

47rG S~ 4~reinoi = k T ~(z)+ , 7 , ~ U 2 i ( z ) .

[A7]

Here U2i is the potential energy o f / t h ion species obtained f r o m [A6] by substitution ei for e. So the full potential ~ofconsists o f two terms: the m a i n ~o(z) given by [ 13] a n d the additional 'Pad(Z) ~'ad(Z) = A

S(z' )e-~l~-~'l& ', oo

~r3(~ e 3 noi)

The additional charge density Pad(Z) m a y be obtained from ~aa(Z) Pad(Z) =

1 d2~aa 47r dz 2 "

[A9]

T o c o m p a r e ~ad a n d Pad with ~ and p we represent their relation as ~0ad = B" FI(~, rD, Z), ~0

Pad=B'F2(~,rD,z). P

[A10]

Here "a"2 1 0e B = -~- ~ ~ ew/oAO07/o-~ x I/2/e0 ~ 1

and F1 and F2 are zero-order h o m o g e n e o u s functions o f ~, ro, and z. T h e function FI

~+z rD

1;

rD Fl~----~.l, Z z3

(E0~O')' - - ~0/12~O

213

~--rD~Z,

exp

FI~-- I,

;

F2~- l.

T h e difference appears due to the fact that though ~ is large in c o m p a r i s o n with ~ad its variation at large z decreases as e x p [ - z / ( ( + tO)], whereas ~aa(Z) pc 1/z decreases not so quickly. So far as p(z) is d e t e r m i n e d by the second derivative of ~,(z) the small "tail" o f Pad(Z) exceeds that o f p(z). However, these "tails" contain only small part of double layer charge and are not essential for the physical p h e n o m e n a . So one can use both ~,(z) and o(z) as a good a p p r o x i m a t i o n to ~f(z) and pf(z) to describe different physical p h e n o m e n a . REFERENCES 1. Nabutovskii, V. M., and Nemov, N. A., Poverkhn. Fiz. Khim. Mekhanika No. 2, 68 (1983). 2. Nabutovskii, V. M., and Nemov, N. A., J. Phys. C. 17, 3849 (1984). 3. Landau, L. D., and Lifshitz, E. M., "Statisticheskaya Fizika" (Statistical Physics), Part 1, Chap. 14. Nauka, Moscow, 1976. 4. Nabutovskii, V. M., Nemov, N. A., and Peisakhovich, Yu. G., Phys. Lett..4 79, 98 (1980). 5. Nabutovskii, V. M., Nemov, N. A., and Peisakhovich, Yu. G., Soy. Phys. JETP (Engl. Transl.) 52, 1111 (1980). 6. Amirkhanov, Kh. I., Gurvich, I. G., and Matizen, E. V., Dokl..4kad. Nauk SSSR 100, 735 (1955). 7. Woermann, D., and Sarholz, W., Bet. Bunsenges. Ges. Phys. Chem. 69, 319 (1965). 8. "Stability Constants of Metal-Ion Complexes." Suppl. No. 1. Special Publication No. 25. The Chemical Society, Burlinslon House, London, 1971. 9. Nabutovskii, V. M., Nemov, N. A., and Peisakhovich, Yu. G., Mol. Phys. 54, 979 0985). Journal of Colloid and Interface Science, Vol. 114, No. 1, November 1986