Solid State Ionics 25 (1987) 21-25 North-Holland, Amsterdam

Vaneica YOUNG Department of Chemistry, University ofFlorida, Gainesville, FL 32611, USA

Received 1 May 1987; accepted for publication 4 June 1987

In this paper, we investigate the effect of the incorporation of silver on the elect.onic conductivity of silver sulfide-silver chloride and silver sulfide- silver bromide ion selective electrode membranes. The theoretical treatment involves an extension of the Smith-Anderson model to a three component system. Theoretical results are then compared with experimental results.

1. Introduction

In the first paper of this series, it has been dem- onstrated that the Smith-Anderson model for the conductivity of particulate systems can be applied to

selective electrode membrane [ I

to facilitate extension of the theory to general n-com- ponent system, we will introduce the following alter- nate notation. Let us denote the component as 1 and

as 6 I9 o12, and ~r22.

n

P13J i=l,2 ,..., n, r=l

w

(k,, k2 ,...) k,,) = (k, ,...

ck , ,.I?, k,) = -m!- k, !k2 L.. k,!

w

22 r/. Young/Electronic conductivity of Ag2S/AgX

and m is a positive integer such that

Using the above equations, eq. (1) can be written as

.!;“(a) =P(2,O)&J-C) +P(l,l )&a-U

+PwM(~-%*)~ (6)

where raz = 2 and ( kl, kz) is an element of the set {(2,0), (l,l), (0,2)). If we have a three component system, n = 3, then

1 Pi =P1 +P2 +P3 =l p1>/0 i=l,2,3. (7) 1 ‘= 1

It is important to realize that a contact is formed by two sp.lwa in contacr, rherer‘src in must always be equal to 2. There are now three kts and we have (k,,

k2, k3) anelementoftheset ((2,0,0), (l,l,O), (l,O,l), (0,2,0), (O,l,l), (0,0,2)). Notice that (2,0,0) cor- responds to a sphere of component 1 forming a con- tact with another sphere of component 1, (l,l,O) corresponds to a sphere of component 1 forming a contact with a sphere of component 2: etc. Then, for the three component system we have the following distribution function

+P(o,l,’ )&-323) +p(

Calculating the p( k,, k2: k3

this function becomes

now e obtai

A sixth order polynomial is obtained as follows:

c~a~+cg7~+c4~~+c3~~

+c&+C,6,+C()=O, (11)

where each coefficient is a function of the Pi’s, Q’S, and z. Using Descartes rule of signs, it can be shown that for each set of such values, only a single positive root is obtained. Therefore, the solution is unique for each set of conditions. Consistent with paper 1 in this series, the interparticle conductivities have been calcallatA llc;nn the eq.;ation: -_--VW WY-.+

Bjj = JG, i# j. (12)

Whenever pl, p2, or p3 is fixed at zero, the system becomes a two component one, therefore the three component computer codes must give results iden- tical to those of the two component computer codes in those cases. These conditions have been used in debugging the program. As indicated, the theory can be extended to an n component system. The order of the polynomial obtained after integration according to eq. (10) is given by

polynomial order = n + (13)

One sees that for a 4 compo nt system, a tenth order for a 5 component

are several cases where n= contemplate systems where (e.g. a membrane after exposure to a solution con- taining multiple inte rents). However, a great deal of effort will have to expended to treat tkese cases

V. Young/Electronic conductivity of Ag$TIAgX 23

replicate the errors in the first and second order coef- ficients of the two component polynomial into appropriate coefficients of the 6th order polynomia:. When pl, p2, or p3 is fmed at zero, C2, C,, and Co in eq. (11) become zero. This implies that the error in the first order coefficient for the two component sys- tem polynomial must be replicated in C4 of the three component system polynomial. Likewise the error in the second order coefficient must be replicated in Cs. Since the mathematical expression for Cs is much smaller than that for C4, it shall be used as an exam- ~1”. Cs without errors is given by:

+2p,p3~23 -da331 1 ( t z- 1 14* (14)

Cs for the sharp break case is given by

Fig. 1. Electronic conductivity surface for silver doped silver sui- fide -silver chloride electrode membranes.

e;=(h, +f-h2 f&3 +-a22 +023 +033)

+2P2P3423+P3fl331) (4 z-1)4. (15)

As before, the modified computer codes for the three component system has been debugged by furing pl,

p2, p3 at zero and making certain that results iden- tical to the sharp break two component system are obtained. The values used for the conductivities of silver, silver sulfide, silver chloride, and silver bro- mide can be found in the two previous papers [ 1,2].

The stoichiometry of the membrane for the three component system requires two independent vari-

/ I

/

.’

Asi 2-x*y S ,_X F:rx

Fig. 2. Electronic conductivity surface for silver doped silver sul- fide-silver bromide eizctrode membranes.

24 V. Young/Electronic conductivity of AgrSIAgX

ables for its expression - one to denote the silver excess, y, and a second to denote the relative changes between Ag,S and AgX, x. Thus, the general equa- tion for the stoichiometry is Agz_x+v S, -JCm where X=Cl or Br. Three dimensions are now required to present the results, and these conductivity surfaces are shown in figs. 1 and 2. The surfaces obtained using the sharp break polynomial are identical except at y=O. Suppose that one is unaware that elemental silver is present in the membrane, i.e. the apparent stoichiometry is Agz__v S, _-\-Xx, but that a constant silver excess equivalent to y=O.2 is present. The conductivity that one’observes is the intersection of the plane y=O.2 with the appropriate surface. Such cuts can be compared with the experimental results of Vlasov and Kocheregin [ 51. It is evident that a constant silver excess for all x will not show good

agreement with the experimental data, because an adequate range of conductivities cannot be spanned as x goes from zero to one. Surface cuts can be sub- jected to other boundary conditions; two simple ones are investigated in this paper. First, it can be assumed that only the silver sulfide component becomes con- taminated with excess silver. Then silver excess is directly proportional to the amount of silver sulfide, which in turn is directly proportional to sulfide ion. The cutting plane is given by the equation:

(I -x)ly=k,,

where k, is a constant. Such cuts are compared to experimental results in figs. 3a and 4a. Secondly, it can be assumed that silver excess is contained in both components, but more is present in the silver sulfide component. Silver excess in the silver halide com-

l Experimental 0 Calculated

k, = 3.125 k, = 3.125

‘E U

k,= 4.167

Fag. 3. comparison of experimental results to apparent clcct:o?;- . ,& ccnd,~ctix~ity for silver sulfide-silver chloride membranes: (a) silver wess dftcctl) proportional to amount of silver sulfide, Qb) silver excess in silver sulfide component > silver excess in silver chloride component.

V. YoungfHectronic conductitlity of Ag2SL4gX 25

0 Experimental

k, = 3.125

O Calculated

k, = 3.125

k, = 4.167

Fig. 4. Comparison of experimental results to apparent electronic conductivity for silver sulfide-silver bromide membanes: (a 1 silver excess directly pro*- ,,rtiona! to amount of silver sulfide, (b) silver excess in silver sulfide component > silver excess in silver bromide component.

. . IS assume. Eswever, iii i&the;: case 1 ‘s the exact

ctional form of the experimental results mat&e

undation (Grant Nu

Fellow. All s

[ B ] v. Young, Solid Sta [ 2 ] $I. Young, Solid Sta [ 3 ] D.P.H. Smith and Anderson, Phil. Msg. B -43 (I?81 1

797. [ 41 E. Pearson,ed., in: andbook of applied mathematics ( Van

Nostrand/Reinhold, New York, 1974) p. 119 1. [ 51 Y.U. Vlasov and S.G. Kocheregin, Ion Obmen Ionometriya

(Leningrad) 2 (1979) 343