Solid State Ionics 25 ( 1987) 9- 19 North-Holland, Amsterdam

Vaneica YOUNG Department of Chemistry, University of Florida, Gainesviiie, FL 3261 E, USA

Received 23 October 1986; accepted for publication 1 June 1987

In a previous paper, the Smith-Anderson particulate conductivity model has been extended to particulate 1% membranes. In this paper, theoretically calculated results are compared with experimentally measured results for silver halide-silver sulfide ISE membranes. Calculated and experimenta! ionic conductivities are in very good agreement. Calculated and experimental electronic conductivities are in poor agreement. The possible reasons for the latter are discussed.

1. Introduction

In a previous paper [ 11, we have discussed the potential usefulness of a theoretical model which would allow researchers to predict figures of merit

of the Smith-An

10 K YoutgKonductivities of silver halide-silver sulfide ISE membranes

ferent points in the medium, then we can write the effective medium conductivity, which is indepen- dent of r, as

flln= (d~dd~2)...~(rn)> > (1)

where the brackets indicate the ensemble average. In order to calculate the term in brackets, two different boundary conditions may be employed. In one case, a constant, non-zero electric field is applied across the composite and the average current density is cal- culated. This boundary condition allows an upper bound on bm to be calculated. In the second case, a constant, non-zero current density through the com- posite is assumed and the average electric field strength is calculated. This boundary condition allows a lower bound on b, to be calculated. Both E, the electric field strength, and the current density, are vector quantities, but they are not necessarily coli- near. Therefore, we are forced to treat a(r) as a ten- sor quantity. Although this can be done quite rigorously, the ease of implementation of this method is such that an analytical chemist would be unlikely to use it.

In the Finite Element Approach, the composite is divided into a countable set of convex solid poly- gons. Assume for the purpose of illustration that we have a set of n convex solid polygons. Then some

of these polygons a a conductivity of

satisfy the equation

(2)

where the brackets in icat\: average values. These average values are talc the volume element dv. complicated enough that be unlikely to use this me conductive two co

h the Variational

1.2.

where x 1 =volume fraction of component 1, x2= the volume fraction of component 2, crl = the conduc- tivity of component 1, and cr2 = the conductivity of component 2. This is a result which can be easily applied by the researcher whose skill lies predomi- nantly in fabrication and measurement. Unfortu- nately, it is identical to that obtained by earlier workers, and Smith and Anderson [ 31 have already shown that it does not give good agreement with experimental results on metal-insulator composites. (Nor is it expected to give good agreement, since for some compositions, a metal-insulator composite is not very conductive). Composites based on intrinsic semiconductor materials are also not very good con- ductors, so eq. (3) will not be useful.

The Smith-Anderson model involves the evalua- tion of a single integral co

f(a) 6m-* G+(tZ-l)0m

da=O, 0

(4)

where fl a) is the distribution function for the inter- particle boundary conductivities and z is the coor- dination number for each particle. The ease with which this integral can be evaluated depends on the choice off(a). For some choices off(a), the integral will not have an analytic solution. In these cases,

lo techniques can be used to evaluate the 3]. Smith and Anderso’n have used con-

istribution functions in their For the distribution functions which t they were able to evaluate the integral analytically. In our applications, we use a d.iscrete distribution for f(a) based on the following model. The precursor powders of systems which w

al particles. We expect rs prepared by others

necessary for the formation of good interpa contacts. Unfortunately, most researchers in this area

V. YoungKonductivities ofsilver halide-silver sulfide ISE membranes 11

of individual particles has been performed, so each particle is assumed to have a composition which reflects the composition of the solutions used to pro- duce the particles. Thus a second assumption is that the particles are heterophasic, but each particle has an identical composition. A third assumption is that a composite containing n such particles can be regarded as equivalent to a composite containing N particles of pure component 1 and n-N particles of pure component 2. In this equivalent composite, there are three types of interparticle contacts l-l, l-2, and 2-2, each having a characteristic conduc- tivity. The true composite is thus an effective medium of the equivalent composite. A fourth assumption is that the distribution function for the characteristic conductivities of the equivalent composite is a binomial distribution. Substituting a binomial dis- tribution function for&) into eq. (4) gives an inte- gral whose solution is an analytical function. That function is a polynomial in Q, so we haqe

P(a,)=O. (5)

The zeroes of the polynomial are the allowed con- ductivities of the composite. Fcx this two compo- nent system, a third order polynomial is obtained. It is shown by Descarte’s Y- nomial has only one positive root. Thus, cr, can be uniquely determined for each composition. computer code has been rewritten to modore C64 rsonal c Z-Newton. meth [14] has polynomial equation. This program’s results have been tested against those talc mainframe using the subroutine IMSL libraries [ I5 1. In every case tested, the two programs gave identical out ut. For convenience, all

culations are now being run on a Commodore C personal computer.

Table 1

ROOm temperature ionic conductivities for silver chloride.

Form QI (Q-l cm- I)

Refs.

single crystal 2.5x lO-9 WI thin films 1.0x 1o-7 to

2.0x 1o-9 1181 single crystal 2.6x lO-9 11918’ single crystal 8.1 x lo-‘O [ 201 b’ single crystal 6.3x 1O-‘o [ 201 c, single crystal 2.7 x lO-9 [ 20]*) single crystal 8.2x lo-‘O [ 201 c’ single crystal 2.4x 1O-9 [21] 0 single crystal 1.6x lO-9 1211 o single crystal 1.4x 10-9 [211 o

a) Obtained from a fit of their data. Log 0 ~0.0276 T

( “C) - 9.268; correlation coefficient = 0.996. b, Calculated from parameters of Corish and Jacobs in table V.

These are for doped AgCl. ‘) Calculated from parameters of Corish and Mulcahy in table V.

These are for doped AgCl. d1 Caiculated form parameters of Aboagye and Friauf in table V. e, Calculated from parameters of Kao and Friauf in table V. o Calculated from parameters for pure AgCl using three differ-

ent models.

ers (entries 8, 9 and II 0

nductivity CP~ 1.8~ W9

conductivitj; ~,n” a silve oride single cry resents the limiting con ctivity of silver chl the disorcler goes to ze ic conductivity increases

the concentration of defects increases, t d a

imes greater tbal t

12 V. YoungKonductivities of silver halide-silver sulfide ISE membranes

Table 2 Room temperature ionic conductivities for silver bromide.

Form 01 (Q-l cm-‘)

Refs.

single crystals 4.9x 1o-8 ~241 single crystals 5.4x 1o-9 [25] p)

1.0x lo-* [2S] b, single crytals 7.8x lo-* [ 201 c) single crystals 5.6x lo--” [ 201 d’ polycrystals c; 6.2~ lo-’ [261 polycrystals r, 7.3x lo-’ t261 single crystal 2.6x 10-8” 1271

a) Frequency not specified. vO= 3.0 x 10” Hz assumed. b, Frequency not specified. vo= 5.8 x 10” Hz assumed. c) Calculated from parameters of Aboagye and Friauf in table VI. d, Calculated from parameters of Kao and Friauf in table VI. Cl Containing 7 different cqfstal orientations. (‘) Containing 20 different crystal orientations. 8) Average of 5 values; ionic solution contacts used.

coeffkient of 0.988 has been obtained from this equation, a room temperature conductivity of 5.4 x lo- lo Sz- I cm - I has been calculated for silver chloride crystals. This represents the upper limit for the electronic conductivity of the pellet, since increasing disorder lowers electronic conductivity. We have used this value as the electronic conductiv- ity of pure, pressed silver chloride pellets.

ranging from 2.2x 10-sto 3.0x lo-‘0-l cm-‘. The single crystal data (entries 1, 3,4, 5, and 6) give an average value of 4.4 x 10 - 8 - I cm- ’ with a relative

deviation of 60%. Consistent with our

room temperaLure, these researchers obtained hole conductivities which extend from N 1 .OX lo-’ W ’ cm-* to w 1.0~10-~ Q-’ cm-‘, with a mean of 5.0x 1O-7sZ-’ cm-‘. Thus we have taken 5.0x lo-’ Q-l cm-’ as the electronic conductivity of silver bromide.

At roo;n temperature and pressures less than 2500 atmospheres, silver iodide can exist as either the y or the /3 polymorph, although the thermodynamically stable polymorph is /3-silver iode [ 32,39 1. Both sin- gle crystal /3-silver iodide, a hexagonal phase con- taining two inequivalent crystal spacings, and pellets of /I-silver iodide have been prepared. The average ionic conductivity values of single crystal /I-silver iodide (table 3) is 5.5~10-~ S&-l cm-’ with a rel- ative standard deviation of 44%. For the pellet, an assumed value of 5.5 x low6 Sz- ’ cm-’ is approxi- mately the midpoint of the range reported in the lit- erature (table 3). Essentially pure y-silver iodide can be prepared only as pellets. The average value of the literature conductivity data (table 3) is 9.0 x 1 O-’ Q-l cm-‘, and this value has been used in the cal- culation for silver sulfide/y-silver iodide. The elec- tronic conductivity of p-silver iodide has been reported as 1.48~10-~ Q-’ cm-’ by Mazumdar et al. [ 401. No electronic conductivity data has been reported for y-silver iodide, however band structure

dieted identical optical b silver iodide polymorghs [

ductivities of these two polymorphs are identical.

3.

Experimental data for the ionic conductivity, the electronic conductivity, and the silver ion transpo

K YoungKonductivities ofsilver halide-silver s&fide ISE membranes 13

Table 3 Room temperature ionic conductivities for /?-silver iodide and of y-silver iodide.

Form QI (W’ cm-‘)

Refs.

single crystals

single crystals

single crystals

single crystals

pellet (8) pellet (B)

pellet (B) pellet (v) pellet (Y)

pellet (Y)

3.5 x 10-8”’ [321 2.3 x10-8 ( la)b) 3.3 x IO-* ( Ila) h, 2.0 x lo-’ (Ila) t331 2.1 X low8 (La) 1.2 x IO-’ (Ila) 1341 6 ~lO-~(l.a) 2.3 X lOm8 (Ila) 1351 5.5 x 1O-8 ( la) 8.7 x10-’ [361 8 x10-“’ [371 1.2 x 10-6d’ 3.5 x10-s 1381 1.3 xlo-4 E361 2 x10-s 1371 4 xlo-S” 8 x~O--~” 1.9 x Io-s8’ [381 1.28x 10-4h’ 1.51 x lo-4”

_P ‘) Assuming two types of interstials present. Parameters from

bottom of table 1. b)Parametersfromtopoftablel. 1,=11,and Il,=la. ‘) Measured with ascending temperature. d, Measured with descending temperature. ‘) Pellets pressed at 1 ton/cm* to 2 4 ton/cm’. ‘) Pellets pressed at 4 tons/cm* ? lo 6 ton/cm2. $’ For purest y-AglI pellets. h’ For essentially y AgI pellets wifh small amounts of /3-AgI

pressed at 281 kg/cm*. ‘) For essentially y-AgI pellets with small amounts of /hAgI

pressed at 720 kg/cm*.

tacts (see the A~~endix). All e ures reflect the reproducibility wit determine the data from the figures o The calculated and experimental ionic conductivi-

tion of stoichiometry for silver sul-

n the range ofxe (

culated and experimental ioni

by a factor of 5, and at x= 1, of 10. The deviation is greater on t

our choice for the ionic conductivities chloride is 10 times smaller than that measured by the authors. For larger lues of x, the ionic con- ductivity of silver chto e m&es a larger mntr& bution to the calculated conductivity, the deviation starts getting larger for x> the binomial distribution, calculated an mental ionic conductivities differ by les equal to a factor of 2 for x< 03, by a factor XX (0.55, 0.9S), and by a factor of 10 at x= 1.

For silver sulfide-silver bromide (modified binomial distribution), calculated and experimental ionic conductivities differ by a factor of 2 at x= 0, this increases tc a factor of 4.5 at x= 0.90, and then falls to a factor of 1.S at x= 1. Thus on the range excluding XC (0.70, 0.90), the relative deviation is c60%, which is within the precision of the meas- ured ionic conductivities of pure silve the binomial distribution, calculate

icle contacts in a modified binom

14 t: YoungKonductivities of silver halide-silver sulfide ISE membranes

lonlc Conductivity

Ag,-, St-x Clx

0 Cakulated (Z=G?)

050 i Cm x (Ag,-, S-x Cl,)

108-w Conductwty

b h-x St-x Clx

Experbmentol

0 Calculated (Z= I2 I

h b

iV! . I I 1 I 1 I 1 . . 050 100

x (Ag,., 5-x Cl,)

Fig. 1. ionic conductivities of silver sulfide-silver chloride as a function of stoichiometry: (a) old polynomial equation and (b) corrected polynomial equation.

e l~~as~r~ of t

to do for the ionic conductivity. In spite of the fact that the data on the electronic conductivities of the pure components is quite limited, we unlikely that the values used are in error of 1000. A more likely reason for the la Mfi-r.0 rm* Lr, clorrrr.orJ 4 lllbz.b baL1l vb uabbu ~0 the experime

theoretical calculatior~s are for int fide-silver halide membranes. Experimentally, the

of the membranes have beein

yte 1 solid ekttrolyte ! 67, Iie a

usPheekct~Onicco

urn with silver is actually me

in silver sulfide an enhance the electronic conductivity [ I]. The

problem is more acute for membranes richer in the silver sulfide component. This is most evident in the comparison of calculated silver ion transport num- bers (fig. 5 j. We can propose three approaches which might be of benefit in testing this hypothesis. approach would be to calculate the electronic

V. YounglConductivities of silver halide-silver sulfide ISE membranes 15

Ionic Conductivity

a Ionic Conductwity

b

Fig. 2. Ionic conductivities of silver sulfide-silver bromide as a function ofstoichiometry: (a) old polynomial equation and (b) corrected polynomial equation.

to completely eliminate the use of metal contacts when measuring the electronic conductivities. The isothermal relaxation current measurement metho

] might be one means ofim e

Finally, we would like to comment on the results obtained for silver sulfide-y-silver iodide and silver sulfide+silver iodide. Although ion selective eleo trodes for chloride ion or bromide ion can be pre- pared from the corresponding pure silver halide, an ion selective electrode for iodide cannot be prepa from pure silver iodide, because these powders not give pellets of suffkient mechanical stability. Thus coprecipitates of silver sulfide-silver iodide are

give coprecipirates

morph is cubic wi G= 6.502 1 while the j? iodide polymorph is hexagonal with Q = 4.59 c= 7.49 A [JO]. Hence contacts along two directions of y-silver iodide are broken when a transformation

-silver iodide occurs.

16 V. Young/Condwtivities of silver halide-silver sulfide ISE membranes

a Electromc ConductMy b

0 Colculoted Kz=12;

Electromc Conductivity

0 Colculoted (2=12)

IO”

,O-‘O 1 0.50 0.00

x V&,%x CL)

IO”

lO’9

1o-‘O ! 1 . . . I . . . . r 050 I 00

x (&I 2-x St- x CL)

Fig. 3. Electronic conductivities of silver sulfide-silver chloride as a function of stoichiometry: (a) old polynomial equation and (b) corrected polynomial equation.

@ 2=...+~pz-1)(-~p2z+l)o,+ . . . .

ts calculated using hot

a

V. YoungX’onductivities ol,Fsilver halide-silver sulfide ISE membranes 17

EiPitrOiilL cOn6itlvlty E~ectrow ConductMy 0

0 Calculated (Z=l2)

E xpenmental

b 0 Calculated iZ=I21

E xperlmentul

Fig. 4. Electronic conductivites of silver sulfide-silver bromide as a function of stoichiometry: (a) old polynomial equation and (b) corrected polynomial equation.

0) 0 Colculolcd 2 = I2 0 Eaperlmental

100

. ; 050

b) 0 Colculoted 2 = I2 e Enperlmentol

0 CQlculaled L= 12

@ Enper~mentol

Fig. 5. Silver ion transport number for (a) silver sulfide-silver cl-loside, (b) silver sulfide-silver bromide hnd (c) silver sulfide-p-silver iodide as a function ofstoichiometry. Calculated based on old polynomial equation. ~~sig~~~~~~t changes smith new ~~~~~~i~~~~~ squation.

K YoungKonductivities of silwr halitie-silver sulfide ISE membranes

a

lomc CcmJctlvlty

0 B-AgI

6-W

lo-6 2 050 I 00

x W~_,%,Ix)

--i 1 I 1 1 I I I r

050 too

x (Ag,,S,.,U

Fig. 6. Calculated ionic conductivities for silver sulfide-/3-silver iodide and silver sulfide-y-silver iodide as a function of stoichiometry: f a ) 01Cr plynomial equation and (b) new polynomial equation.

b

lO-6

b

Ionic Conductivity

0 B- AgI

6-AgI

0

(6,, a) is continuous and finite on ( a) bound and continuous, the fini

approximation can be used to

n

G I=1

d=l,2,pT, j=l,2,p;t,

scontinuities, the que integrable. Th

dia6erence approximation to evaluate f for 1 L i # 0.

For our case, it can be shown by direct evaluation that 1 E I# 0. Thus there is a unique j(c) that cor- responds to the polynomial function used in pa 1. However, since most of the components of vectorfwill evaluate to zero, a

K YoungKonductivities of silver halide-silver sulfide ISE membranes 19

mainframe computers. In summary, we can prove the existence of a distribution function giving rise to the original polynomial, but we cannot evaluate it using standard techniques. The alternative is simply to evaluate P(a,) for various modified binomial distribution functions and see what we get. Since this method will allow us to see what happens when var- ious degrees of order are introduced into the system, we are conducting such a study. However, the prob- ability that we shall hit upon the distribution which gives the polynomial used in paper 1 is nil.

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