: V01unie12; numb6r 3 . CHEMICAL PHYSICS LETTERS • . " " ~: . " 7 : , .

H A L L C O E F F I C I E N T IN B R O W N I A N - L I K E S O L U T I O N S

.... ~ : ...... S. HARRIS: . : : " : ' " • + : : College o.f Engineering, State UMversity o[New York,

Stony Brook, New "forl¢ i1790, USA

Received 25 October 1971

We calculate the Hall coefficient for an infinitely dilute electrolyte solution in which the ratio (solvent molecule mass/carrier particle mass) is small but finite, The first correction beyond the classical brownian motion result is found and shown to have a value identical to the brownian motion value.

The possible effect of liquid structure on second or- der dc transport coefficients can ~e assessed by stuying Hall conduction in electrolyte solutions [ 1 ] . At in- finite dilution these effects are reflected in a value of the Hall coefficient, R = eoH/O 2 , appreciably differ- e.nt from unity, where o O an.~t o H are the ohmic trod Hall conductivities and e is the charge on the carrier particle, hereafter set equal to one. When structural effects can be neglected the complex carrier particle dynamics can be replaced by a brownian motion de- scription, and bo,h o O and Oil can be expressed in terms of the frict!on coefficient, ~, which describes this motion; then, i fM is the carrier mass, o O = (M~') -1 Oil = (M~~ --2 and accordingly R = I. Prelim- inary studies [2,31 based on model calculations indi- cate th~,t when M is small compared to m (the mass of a solute molecule) appreciable structural effects can occur, however in the converse case when y = (m/M) 1/2 < 1 such effects are negligible. Thus, al- though the brownian motion description is only strictly valid in the limit 7 + 0, this last result implies that its use is also justified in treating brownian.like solt, tions, i.e., sol,ations which are not strictly brown- tan but in which r is still small, e.g., l - (aq) . The pur- pose of this letter is to describe recent results for the case of brownian like solutions which corroborate ~L . . . . . . . , L , ; . . . . " - . t , t , ~ p~mnmary li;~umgs; the fuii details wiii be pub- lished elsewhere [41.

A statistical mechan cal theory for the linear (ohm- ic) stationary response 6f an infinitely dilute brown-

imp-like electrolyte to an applied electric field has been recently developed by several authors [ 5 - 7 ] , In lowest order,in 3' they find the usual brownian motion result; a prescription for obtaining higher order results : in 7 together with an expression for the first correc- tion term has also been given [5]. From these results we find that the ohmic current is given as

i 0 = ooE =- {(~)-II(I -72v01~)}E +0(74)~, (1)

where ~ = l /kT and D = M~'/] - I . Tile diffusion coeffi, cient D is given in terms of molecular properties by a time correlation function expression, and v 0 is an- other, more complicated time correlation func t ion t . In lowest order in 7 we recover the brownian motion expression tot o, and by appropriately defining a gen- eralized friction coefficient ~'* it is clear that in the next order the brownian motion functional depen- dence of o O on the friction coefficient, now ~'*, can be maintained, i.e., we have

o O =/M~'*) -1 +0('7'4). (2)

This result is to be expected since it is known that first order transport coefficients are insensitive to the detailed system dynamics and can be described in terms of a single adjustable parameter. What is not known is whether o H is given by the corresponding

t No estimate of this term is given in refs. 15, 7]. and we have not a!tglllptgd to do ~ilis ci~,|~Ol due ~o its t.~ompli~led strut> t ure.

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Volume 12. number 3 CHEMICAL PHYSICS LETTERS 1 January 1972

browrkn motion functional of {*, so that R = 1, or

whether liquid structure effects occur. Our results in- dicate that the former is the case.

The procedure followed in refs. 15-71 was to first linearize the Liouville and reduced Liouville equations describing the system in the external field, and then treat the linearized equations. If we omit the linearization, but follow the same procedure, we obtain the following equations in place of 3.7-3.9 in ref. [7].

(3)

X exp!-t’i[Lty(l-~(Jt~‘)])POF. $+-fiv . (4) C >

The notation here is the same as that used in ref. [7]. E and H are the applied dc electric and magnetic fields, 7~ is the carrier particle velocity, and z is the

stationary renormalized carrier parL:-le distribution function. The terms appearing in the ;?crator K are F, the force on the carrier particIe due to its interac- tion with the solvent; PO, the equilibrium conditional distribution function for the solvent in the presence of the carrier partic!e; the three Liouville operators are if,, describing the motion of the solvent in the presence of a fixed carrier particle, i? describing the

motion of the carrier particle in the solvent, and X’ which describes the effects of t!le external fields on the carrier particle motion; finally, the prqjection op- erator ?- = P, 1 d-v- withy a shorthand notation for the solvent fluid phase. The y dependence of K is ex- p!icit while its full field dependence is through &‘.

An expansion in the extcmal Gelds (assumed sma11)

can now be effected by directly expanding the expo- nential operator in K and formally expanding 5:

Fj = ;I;, t G, -+ . . . )

K=Kot+l f.... (5) where the subscript indicates the order of the given term-in the applied fields. The following set of equa- tions results:

494

etc. (6)

The linear equation obtained earlier [5-71 is re- covered in first order, and a new set of equations is found for the higher orders. These new equations can be solved by using the same general method used in ref. [S] in treating the linear equation. This entails a re-expansion in 7; we re-expand the exponentia1 oper- ators in the Ri and formally expand the ~5~:

mi=(l/~~)(5i0+y5il+...),

Ei = y’(Eio +& + . ..) . (7)

For our present purpose we need only consider the first moment of the equation for s2, sincejH = Jdw&;,_ We find that

i, = o~(EAH) = (W)-2 {l +(~-Y~~,/B)+OCY~)}E AH

= ({*hi)-” E A H + 6(r4)E A H.

Thus in lowest.order in 7, uH is given by its brownian motion value. while in next order it is given by an identical functional of the friction coefficient with 5 replaced by c*. Accordingly R = 1 +O(y4), and the generalized friction coefficient effectively serves to describe both the ohmic and Hall conductivities; we can thus conclude that liquid structure effects are not significant in brownian-like solutions.

References

[ 1 J H.L. Friedman, J. Chim. Phys. SN (1969) 75. [2] H.L. Friedman and A. Ben-Naim, J. Chcm. Phys. 4E

(1958) 12c. [3] S. Harris, J. Stat. Phys. 2 (1970) 379. [4] S. Harris, tu be published. [S] J.L. LeboGtzand E. Rubin, Phys. Rev. 131 (1963) 2381 [6] P. Resibois and H. Davis, Physic3 30 (1964) 1077. (71 J.L. Lebou itz and P. Resibois, Phys. REV. 139 (1965)

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