Periodic precipitation (liesegang phenomenon) in solid silver—i. experimental

PERIODIC

PRECIPITATION

(LIESEGANG PHENOMENON) EXPERIMENTAL? R. L. KLUEHI

IN

SOLID

SILVER-I.

and W. W. MULLINS

The Liesegang phenomenon (periodic precipitation) has been studied experimentally in a solid state system. Single crystals of silver were annealed in some partial pressure of oxygen to saturation and were then annealed in a partial pressure of hydrogen at 800°C. Hydrogen diffuses into the silver and reacts with the dissolved oxygen to form water vapor which precipitates in the form of bubbles. Under certain conditions these bubbles have been found to form in bands which obey Jablczynski’s relationship, that is, the spacing coefficient, which is defined as, the ratio of the distance to successive bands from the diffusion interface, is a constant. The spacing coefficients were studied in three silvers of different purities as a function of the concentration of the hydrogen and oxygen. It was found that the spacing coefficient decreased when either the concentration of hydrogen (keeping the oxygen concentration fixed) or oxygen (keeping the hydrogen concentration fixed) was increased, and it increased when the purity was decreased. PRECIPITATION

PERIODIQUE SOLIDE-I.

(PHENOMENE DE LIESEGANG) ETUDE EXPERIMENTALE

DANS

L’ARGENT

Le phenomene de Liesegang (precipitation periodique) a et& Btudie experimentalement pour un systeme a l’etat solide. Des monocristaux d’argent ont 6th recuits a 800°C d’abord sous une pression partielle d’oxygene jusqu’a saturation, puis sous une pression partielle d’hydrogene. L’hydrogene diffuse dans l’argent et reagit avec l’oxygene dissous pour former de la vapeur d’eau qui precipite sous forme de bulles. Lea auteurs ont trouve que dans certaines conditions ces bulles se rangent suivant des bandes qui obeissent a la relation de Jablczynski suivant laquelle le coefficient d’espacement, defini comme &ant le rapport de la distance des bandes successives it l’interface de diffusion, est constant. Les coefficients d’espacement ont et& Studies en fonction de la concentration en hydrogene et en oxygene, pour trois Bchantillons d’argent de differentes puretes. Les auteurs ont trouve que le coefficient d’espacement diminue quand on augmente la concentration en hydrogene (a pression d’oxygene constante) ou en oxygene (a pression d’hydrogene constante), et qu’il augmente quand la purete de l’argent diminue. PERIODISCHE

AUSSCHEIDUNG

(LIESEGANG-PHANOMEN) EXPERIMENTE

IN

FESTEM

SILBER-I.

Das Liesegang-Phlnomen (periodische Ausscheidung) wurde in einem Festkorpersystem experimentell untersucht. Silbereinkristalle wurden bei mehreren Sauerstoffpartialdrucken bis zur Sattigung und anschlie5end bei einem Wasserstoffpartialdruck bei 800°C angelassen. Wasserstoff diffundiert in das Silber, reagiert mit dem geliisten Sauerstoff und bildet Wasserdampf, der sich in Form von Blasen ausscheidet. Unter bestimmten Voraussetzungen ordnen sich diese Blasen in Bandern an, die der Jablczynski-Baziehung gehorchen; d.h. der Abstandskoeffizient, definiert als das Verhaltnis der Abstiinde der Diffusionsgrenz-f&he von aufeinanderfolgenden Bandern, ist konstant. Die Abstandskoeffizienten wurden in Silber verschiedener Reinheit als Funktion der Wasserstoff- und Sauerstoffkonzentration gemessen. Es ergab sich, da5 der Abstandskoeffizient kleiner wurde wenn entweder die Wasserstoffkonzentration (bei konstanter Sauerstoffkonzentration) oder die Sauerstoffkonzentration (bei konstanter Wasserstoffkonzentration) erhoht wurde und da5 er mit abnehmender Reinheit des Silbers zunahm.

INTRODUCTION Seventy

years

precipitation

ago

the

the precipitate

phenomenon

was discovered

of

periodic

direction

by R. E. Liesegang.(1,2)

perpendicular

Liesegang rings or bands are formed when a substance, usually

a salt,

convective upon

solution

reacting

insoluble

is allowed

to

containing

diffuse

a dissolved

with the diffusing

precipitate.

into

Under

Liesegang

a non-

electrolyte

salt which

material

forms

the proper

ACTA

METALLURGICA,

VOL.

cI

17, JANUARY

in bands

of diffusion.

are usually

studied

in gelatin-

systems by putting a dilute gelatin solution

and pouring

conditions

containing

B ions into a test tube

on top of this a concentrated

aqueous

solution containing an electrolyte with A ions. The A ions diffuse into the gelatin and react to form an insoluble precipitate

A,AB,B (e.g., an aqueous solution

of AgNO,

on a dilute

poured

gelatin

solution

of

KsCrO, to form Ag,Cr0,).3 Figure 1 shows a schematic l~Pl~“~&l”LIIV ~~==-~+%on of the experimental results. After a region of continuous precipitation, the precipitate

,

1969

bands

along the

but rather it forms

to the direction

of some electrolyte

an

t Received April 8, 1968. This paper is based on a portion of a thesis submitted by R. L. Klueh in partial fulfillment of the requirements for the degree of Doctor of Philosophy at Carnegie Institute of Technology 2 Oak Ridge National Laboratorv, Oak Ridge. Tennessee. 4 College of Engineering and Science, Carnegie-Mellon University, Pittsburgh, Pennsylvania. ”

does not form continuously

of diffusion,

59

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17, 1969

to be the same as the normal solubility that it is larger in numerical

xn

?=k

product except

value.

For the A-B system, A diffuses into the solution and

%-1

forms a supersaturated

solution

of A,*B_.

Precipi-

tation then begins a short distance behind the diffusion

I

front,

and the supersaturated

point

of nucleation

AVABVBahead of

aids the growth

This means a region immediately precipitation Liesegang

rings

A must

(diagrammatic)

FIG. 1. Schematic representation of Liesegang bands. begins to form discontinuously that the distance

between

in bands.

successive

The

It is found

bands increases

as the distance from the origin of diffusion

increases,

ahead of the point of

has been depleted of reactants, and hence,

diffuse

saturation

the

of the nuclei.

through

this

and precipitation

coagulation

region

before

super-

are again possible.

theory,‘15)

on

the

other

hand,

assumes that the reaction product is first produced a metastable

colloidal

dispersion;

the

forms because the colloid is coagulated

by an excess

such that, if the distance from the diffusion interface

of the diffusing

to the nth band

when the colloidal sol in the vicinity of the precipitate

formation

is called

x,

and the time

of its

(measured from when diffusion began) is

t,,

is

coagulated

electrolyte.

as

precipitate

and

Several difficulties Xn -= X

(Jablczynski’s(4)

k

relationship), =

/.

under

k’

(2)

3

(k is generally referred to

has

been

varying

have

only over a narrow (dissolved)

observed

outer (diffusing)

conditions(8*g)

and the

reactant

magnitude.

Generally,

with

many and

media.ol)

concluded

that

bands

in

of the

the band spacing is found to in the

concentration

of

adherents, have been advanced emphasis on the gel medium,

still have

to account for periodic

of these theories

placed

major

while others disregarded

the gel and referred to concentration

changes resulting

from the diffusion of the dissolved substances involved in the reaction. The two theories which are still of importance are the supersaturation theory and the theory.

The supersaturation theory, introduced by Wilhelm 0stwalde4) in 1897, assumed that a precipitate does is reached.

until a critical

band formation

in solids.

In 1941

alloys

which

he initially

Contrary

however,

the

decreased,

interpreted

to the

distance

as Liesegang

normal

Liesegang

bands,

between

successive

bands

and Druyvesteyne’)

later

and Meijering in the on-off

Recently, aluminum

and arsenic

observed migrated

cycle of the furnace.

in some counterdiffusion bands

of

AlAs;

with time.

in copper,

in copper, these

Kelly,(is)

observed

experiments

of

et aZ.f18)

Gerrard bands,

however,

in similar experiments

a band

on

each

side

of the

diffusion interface in the counterdiffusion

of aluminum

and antimony,

and gallium

gallium

and antimony,

and arsenic ; these bands also migrated with time and

THEORY

not form

IN SOLIDS

variation

by several orders of

Several theories,(13) only two of which

coagulation

BANDS

form

should be greater than that

an increase

Some

still use the theory to explain

showed that this banding was caused by a temperature

either the inner or outer reactant.

precipitation.

precipitate.

Most of inner

concentration

of the inner reactant-preferably decrease

in

range of concentration

reactant,

of any Liesegang

bands.

aqueousuO) and other nongelatinous investigators(12)

but some investigators LIESEGANG

as the spacing coefficient). systems(6*7)

the

by this theory,@)

Rhineso6) observed bands in internally oxidized copper

where L and k’ are constants phenomenon

by

their results.

andc5)

vtn The

adsorbed

are encountered

The literature to date contains very little evidence

X

2

(1)

n-1

The clear space forms

“supersaturation

This supersaturation

product

product” is defined

eventually

disappeared.

Chenot,(20) on the other hand, in a counterdiffusion experiment

of gallium

and arsenic in silver observed

stationary

bands

GaAs.

however, observed.

that

a

Spacing

of

constant

It

does

spacing

coefficients

not

appear,

coeficient

calculated

from

was the

data given are found to decrease with distance from the diffusion interface, indicating a decrease in distance between successive diffusion interface.

bands

EXPERIMENTAL

on

moving

from

the

SYSTEM

Although there have been several derive theoretically the conditions

attempts to for periodic

KLUEH

AND

MULLINS:

PERIODIC

PRECIPITATION

precipitation, there has never been a rigorous comparison between any of these models and experiments. In a subsequent paper,c21)Carl Wagner’s mathematical analysis,(22) which is based on the supersaturation theory, is modified to fit the boundary conditions of the present investigation and compared with the experimental results. Previously the chief obstacle to obtaining agreement between theory and experiment was the lack of values for important parameters in the gelatin-electrolyte systems studied (i.e., solubilities, diffusion coefficients, and solubility products). Furthermore, complex ions and hydrogen ions often form, thus affecting the equilibrium and supersaturation conditions of the system. In the system chosen for these experiments, it is believed that these obstacles have been largely avoided, and thus comparison between theory and experiment should be more meaningful. Liesegang band formation on the basis of the Wagner analysis requires a substantial nucleation barrier and a linear growth rate of the precipitate particles that is small compared to the diffusion-controlled advance of the reaction front. In addition to these conditions, it is believed that the growth rate of the individual particles cannot be markedly inhibited by an interface-controlled reaction as it would then be possible for supersaturation conditions to prevail in the immediate vicinity of the particles, leading to nucleation and continuous precipitation. This latter condition is believed to prevent Liesegang band formation in most solid state systems (e.g., internal oxidation systems, etc.) ; coherency relationships, strains, etc., make the transport from solution to precipitate particles difficult and precipitation continuous. It was expected that this difficulty would not be encountered in a gaseous precipitate. The system chosen was the formation of water vapor bubbles in solid silver. The bubbles were formed by annealing the silver in some partial pressure of oxygen and then annealing in some partial pressure of hydrogen.? The hydrogen diffuses into the silver, reacts with the dissolved oxygen to form water which is insoluble in silver, and precipitates in the form of water vapor bubbles. With this system it was also possible to experimentally determine the supersaturation product. EXPERIMENTAL

PROCEDURE

For the periodic precipitation studies and the supersaturation product determinations, single crystals of 7 The sequence of anneals was always, oxygen followed hydrogen. The reverse sequence gave bubbles which were not large enough to see with the light microscope. 5

IN

SOLID

SILVER-I

61

99.999+, 99.99+, and 99.9% Ag were grown in splitgraphite molds in a modified Bridgeman furnace(23) (the specimen sizes were 1R0x Ia6 x 1 in. long and I$ x & x 4 in. long for the periodic precipitation and supersaturation product determinations, respectively). To get the partial pressures of hydrogen (pu,) and oxygen (po,) desired, hydrogen and air were mixed with nitrogen which is insoluble in silver.(24) The switch from an oxygen to a hydrogen atmosphere was made by transferring the specimens from a tube furnace containing the oxygen atmosphere to one containing the desired hydrogen atmosphere. All experiments described herein were carried out at 800°C. After the bubbles had been formed the specimens were sectioned by spark cutting, silver plated for edge preservation, mounted in cold mount, mechanically ground and polished through 0.3-,u alumina powder, and finally chemically polished.(25) The bubbles were then visible without etching. To determine a spacing constant for the various specimens, the average was taken of measurements made at high magnification from the edge of the specimen to the end of the region of continuous precipitation and from the edge to the beginning of the other bands. Accurate measurements were often difficult to make because the bands were discontinuous about the periphery of the specimen and the edge of a band was not well defined. EXPERIMENTAL

RESULTS

Critical supersaturation product The results of the determination of the critical supersaturation product, Km*, for the three silvers of different purities are shown in Figs. 24. In these graphs the bubble density (the number of bubbles per unit volume), has been plotted against the maximum possible value K, of the solubility product, K,, achieved under the experimental conditions and determined from K m = 0 .256 cOocHae --

(3)

where coo and cn” are the solubilities of oxygen and hydrogen at the respective partial pressures used.: K, was determined by maximizing the normal solubility product K&G t) = co@, t)[&, _ -

$)I2

(4)

$ The solubility of hydrogen and oxygen in silver are related to the partial pressures by Sievert’s law, i.e., AS’= lcpV*. At SOO’C, k is approximately 2 x IO-” and 1.5 x 1O-6 moles of gcts/cmS/atm for hydrogen’2B’ and oxygen(z7’, respectively.

62

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with respect to x, assuming that no precipitation occuw and the silver specimen is semi-infinite, c,(x, t) and co (x, t) are the concentrations of hydrogen and oxygen,respectively. The detailed calculation of K, is given in the appendix. The critical supersaturation product, Km*, was taken as the value of K, a;t which the density decreased sharply. For all three specimens, Km* was determined twice (three times in the case of the 99.999+% Ag), using widely different partial pressure ranges of oxygen and hydrogen. To determine where the density counts would be made, each specimen cross section was examined for

/Y

4’.

d m8 I

MAXIMUM

SOLUBILITY

PRODUCT

FIG. 3. The experimentally determined bubble density aa a function of the maximum solubility product for the 99.99+y0 Ag.

of specimens (99.999+% Ag), pea was held at 1.32 x 10-S atm (1 mm Hg) and pHz was varied. Examination of Figs. 24 discloses that for each silver purity the K,* determined by vaqing the partial pressure of hydrogen, pHB,was larger than the one determined by varying the partial pressure of

MAX IMUM

SOLUBILITY

PRODUCT

FIG. 2. The experimentally determined bubble density as a function of the maximum solubility product for the 99.999+~~ Ag.

the regions of highest bubble density, and the counts were made in these regions. These regions of highest density generally consisted of one or more bands of bubbles parallel to the edge; the bands nearest the edge were chosen to make the counts. For each specimen, 80 density counts were made using a grainsize eyepiece, and the average of these 80 values was used to calculate the density. The Km* values determined for the three silvers are compiled in Table 1. The X series of specimens consisted of different specimens annealed first in air and then in varying ps,, and the XX series of specimens consisted of different specimens annealed in varying poz and then in 1 y0 H,. For the XXX series

IO9

1

,I,

10"'

,/Itt/l

1

IO-*" MAXIMUM

SOCUBILITY

PRODUCT

The experimentally determined bubble density as a fun&ion of the maximum solubility product for the 99.9% Ag. !&a.

4.

KLUEH

AND

MULLINS:

PERIODIC

PRECIPITATIOS

TABLE 1. Experimentally determined supersaturation products

Silver purity

(mo$&,

Ci!gH.)

K,*

5.40 2.68 4.27 5.40 1.6’7 5.40 1.12

x x x x x x x

1OW lo-’ IO-’ 1O-6 lo-’ 1OF 10-7

6.0 1.95 1.39 5.9 1.95 5.65 1.95

x x x x x x x

10m8 4.9 lo-’ 2.6 lo-’ 2.0 1OW 4.6 10-7 1.8 1Om8 4.4 10-1 1.1

x x x x x x x

lo-= lo-= 1O-21 1O-21 10-21 lo-= 1O-21

If the experimental difficulties are PO,. considered, however, the agreement is quite good. Table 1 shows that the difference in K,* values for the three silvers is rather small, and it has been concluded that there is just one critical supersaturation product for the three silvers and an average value for the three XX series of specimens of o=xw

K,*

gg 1.6 x 1O-21

SOLID

SILVER-I

63

was used; the solubilities of hydrogen and oxygen were in units of moles of hydrogen and oxygen per cubic centimeter of silver at S.T.P. Band formation

Series

QQ.QQQ+% 50X 50XX 50XxX 99.99+% 51X 51xx 99.9% 52X 52XX

IN

Banding was studied in the three silvers of different purity as a function of both the concentration of hydrogen (outer reactant) and oxygen (inner reactant). In Figs. 5 through 7 are shown some representative photomicrographs of the banded microstructures observed. The spacing coefficients, k, determined in these studies are plotted in Figs. 8 and 9 as functions of pn:/’ and poi/s, respectively.l_ One of the kticulties with the above-described method for studying periodic precipitation was that it was impossible to form many bands before the concentration of oxygen in the specimen fell to a value where it was impossible to form any further bands. t The solubilities of hydrogen and oxygen are, according to Sievert’s law, proportional to p~,‘/~ and po,W, respectively.

x4

FIG. 5. An example of the bands formed at 800°C in the 99.99+% Ag saturated with oxygen by annealing in air and then 2 hr in 1% H,. The method used to determine k is indicated. 576 x.

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-----x3

----x5 --x‘l -----x3 _oX1

-----x2 -x

.X2 X,i

Xc)\?1

-d

(4 Fra. 6. Examples of bands formed at 800°C in (a) 99.999+% Ag and (b) 99.99+% Ag by annealing in air until saturated with oxygen and then annealing for 2 hr in (a) pa, 2 8.9 X 10_a atm and (b) pe, z 5 x 1O-3 tatm. 676 x .

(b) FICJ. 7. Examples of bands formed at 800°C by annealing 99.9% Ag in air until sakuratsd with oxygen and then annealing for 2 hr in (e) pi% z 1.4 x 10-l atm (240x), (b) 131~E 1 x 10-l atm (288 x) and (c) paS s 1 x lo-* atm (288x).

KLUEH

MULLINS:

AND

PERIODIC

PRECIPITATION

IN

SOLID

SILVER-I

65

FIU. 9. Experiment&y determined values for & - 1 as -“’ for three silvers of different purity. a function of po,

FIG. 7 (c)

In an effort to form more bands, a method was devisedt2s) so that a cylindrical specimen could first be saturated with oxygen by annealing in air, and then it was possible to maintain one of the flat surfaces in air while the other was maintained in a hydrogen atmosphere (the cylindrical surface was insulated from both the hydrogen and oxygen atmospheres). This method, which was called the ‘Vycor tube experiment” because of the procedure used, proved to be only partially successful. LSome 16 I4

0

W--J- 99 999%L\g C-99.99% c---99

6lg 9%

I 5

results obtained by this method are shown in Figs. 10 and 11. Figures 5, 6, 10 and 11 have been marked to show how the spacing ~oe~eients were determined; it should be remembered, however, that the actual & values were determined from an average of several measurements. This average value of the spacing coefficient was then rounded off to the nearest 0.05. By examining the bands it can be seen how some amount of individual “judgement” was involved in making these measurements. In many cases it was difficult to tell what constituted the zone of continuous precipitation, for when this zone was closely examined, it was found that there were often bands within what would ordinarily be called the zone of continuous

tsg

IO &ATM--)

, 15

I

20

I

25

:

FIG. 8. Experimentally determined values for z - 1 a.s & function of r)q -I’* for the three silvers of different purity.

FIG. 10. Bands formed in the “vycor tube cxperiment.” The specimen was 99.99+% Ag annealed in air until sat,urated wit,h oxygen and then for 8 hr one surfttce w&sexposed to 1 o/oHe while the other surface remained in sir. The edge shown was the one exposed to 1 y0 Hz. 390 x

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METALL’URGICA,

X,(?l

FIG. 11. Bands formed by the “vycor tube experiment.” Specimen was saturated with oxygen by annealing in air and then for 4 hr one surface was exposed to 1% Hz while the other remained in air. The spacing od&o&mt is different because exnerimantal diffi<i& changed the conditions under which the bands were formed. The edge shown is the one exposed to 1%. Hz. 414x.

precipitation (Fig. 5). Thus, whenever possible, the measurements were carried out for bands beyond the zone of continuous precipitation. For all three silvers of different purity, it was found that for specimens first annealed in air and then in increasing r)nz (which varies as cno9) the spacing of successive bands decreased, and at some pH2, precipitation became continuous. At very low pnZ there were few bands, and it was impossible to measure the spacings. Likewise, when po, was increased to po, = 0.21 atm (highest value used), the spacing decreased. At small values of PO, the bubbles became smaller and more difficult to see, and it was impossible to measure the spacing of successive bands. It was found that a decrease in silver purity led to an increase in the average size and density of the bubbles within the bands. The effect of changing the silver purity on the spacing coefficients is shown in Figs. 8 and 9. For a given pnZ and (po,, the measured spacing coefficient increases as purity decreases; however, there was essentially no difference between the 99.99+ and 99.9% Ag with variation of po, (Fig. 9). Spacing ~oe~~ients were especially diEcult to measure in the 99.904 Ag specimens which were annealed in air and then various values of pnz because of the fissures and blisters which formed (Fig. 7). Because of experimental Iimitations, there was not much of a range of pul and po; values over which to study the banding in the 99.999% Ag. DISCIJSSION

A relationship between the supersaturation product and the nucleation of bubbles is believed to exist, and

VOL.

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it follows from Figs. 2 through 4. For K, > Km" the number of bubbles formed, and hence the nucleation rate was essentially constant, but at K, = K,* there was a sharp drop in the number of bubbles formed, indicating a change in the scaIe of nucleation. Hence, for K, > Km* the supersaturation is such that nucleation can occur on the scale observed (the exact nature of the nucleation sites is unknown) ; whereas, for K, < Km* this is no longer true. Bubbles still appear at Ki,< Km* because of heterogeneous nucleation sites which nucleate bubbles at smaller supersaturations (e.g., small voids). Band formation Although banded structu~s were previously observed in solid state systems, there was always some uncertainty as to whether they were the result of Liesegang precipitation or some other process. There appears to be little or no question that the bands observed in this investigation were in fact Liesegang bands because: (i) The distance between successive bands increased as the distance from the diffusion interface increased. (ii) The spacing coefficient (i.e., k = x~/x,__~) was reasonably constant. (iii) The effect of changes in the inner or outer reactants on the spacing coefficient was the same as that observed in the gelatin systems. In connection with the previously studied gelatin systems, two interesting conclusions can be drawn, First, the bands are not formed by the coagulation of colloids as postulated by the coagulation theory of Dhar and Chatterji.u5) Previously such a statement could not be made with certainty because of the nature of the systems studied; at least this appears to be the stand taken by adherents of the coagulation theory. It is concluded that the bands form as the result of supe~aturation as postulated by Ostwald.(r*) Secondly, one of the conditions previously believed necessary for band formation was not achieved. Most previous investigatorso2) have concluded that in order to observe banding, the concentration of the outer electrolyte should be greater than that of the inner electrolyte; it was usually observed that the outer electrolyte would have to be several orders of magnitude greater than the inner electrolyte before banding was observed. In the present investigation, just the opposite was true; that is, the concentration of the inner reactant (oxygen) was greater than that of the outer reactant (hydrogen). In fact, when the concentration of hydrogen was made equal to or larger than that of oxygen, continuous precipitation was observed. Thus, it would appear that some other criterion is necessary to determine the conditions

KLTTEH

AKD

PERIODIC

MULLINS:

under which banding will ocour; will be discussed later.@11 APPENDIX

PRECIPITATION

these conditions

a=0

for

x = 0

st at

s>O

t> 0

(Al)

t=O

(A21

for a&, tf, and b=O

for

x=0

at

t>O

(A3)

b = b,

for

x > 0

at

t= 0

fA4)

for 6(x, t). Hence

afx,5) = and

b(z, t) = b, erf

z

( ) (-.._Lf_)

a0 erfc

61

SILVER-I

for the Ag-H,O system at 800°C where (ref. 30)

D, = .Di = 2.12 X low5 cm2/see (ref. 27) _ where DE and Do are the diffusion coeficients

A

It is desired to calculate the spatial maximum value that the solubility product, as conventionally defined, can achieve under the conditions where a substance A is diffusing into a semi-incite medium saturated with a substance B to form a compound A.& The calculation will be carried out assuming that no precipitation ooc~urs. The concentrations a@, t) and b(z, tf of A and B, respectively, are the well-known error fu~~~tionsolutions to Fick’s second law with the initial conditions given by for

SOLID

L), = r>, = 8.39 X 10” cm2/sec

.MuxG&ution of 8o~~bil~~~ pro&&

a = a,

IN

---=== 22/&t

(A51

21[D,t

where Lf-4 and DB are the diffusion E~~G~ents of A and B, respectively. The solubility product is then given by

Differentiating (A?‘) and equating the result to zero gives

This transcendental equation can now be solved for the constant

Morse and Piereefs) first showed that z&&is constant and this can be seen from equation (A7), for if K,, D A¶ and D, a.re constant, then @v% must also be constant. Equation (AS) has been solved numerically

hydrogen-and found that,

oxygen in silver at 800°C.

of

It was

%3 7; = 1E,= 4.96 x IW3

and substit~ition of this value of k3into equation (AT) gives the. maximum possible solubility product as K, s

a.256 c~02coo _._ _ where es0 and coo are the equilibrium s~lub~ities of hydrogen and oxygen, respectively. ACKNOWLEDGMENTS

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