Periodic precipitation (liesegang phenomenon) in solid silver—II. Modification of wagner's mathematical analysis

PERIODIC

PRECIPITATION

(LIESEGANG

MODIFICATION

OF

PHENOMENON)

WAGNER’S

R. L. KLUEHI

IN

MATHEMATICAL

SOLID

SILVER-II.

ANALYSIS?

and W. W. MULLIN@

In the previous paper”) an experimental study of Liesegang precipitation in solid silver (Ag-H,O experiment) was described. Single crystals of silver were annealed at SOO’C in some partial pressure of oxygen to saturation and were then annealed in a partial pressure of hydrogen. Hydrogen diffused into the silver and reacted with the dissolved oxygen to form water vapor bubbles which under certain conditions formed in bands obeying Jablczynski’s relationship. Carl Wagner’s mathematical’analysis of periodic precipitation has been modified, and when the spacing coefficients calculated with this theory were compared with those found experimentally, the agreement was satisfactory with one exception. It was found that the variation of the spacing coefficient with the variation of the oxygen concentration predicted by theory was opposite to the variation observed. Some suggestions have been advanced for this discrepancy, and it has been shown how some previous results in other systems oan be explained by the modified Wagner analysis. PRECIPITATION PERIODIQUE (PHENOMENE DE LIESEGANG) DANS L’ARGENT SOLIDE-II. MODIFICATION DE L’ANALYSE MATHEMATIQUE DE WAGNER Dans le precedent article l’auteur a decrit une etude experimentale de la precipitation de Liesegang dans l’argent solide (experience Ag-H,O). Des monocristaux ont Bte recuits jusqu’a saturation it SOO’C SOUS une pression partielle d’oxygene, puis ensuite recuits sous une pression partielle d’hydrogene. L’hydrogene diffuse dans l’argent et reagit avec l’oxygene dissout pour former des bulles de vapeur d’eau qui se rangent en bandes en obeissant a la relation de Jablczynski. L’analyse mathematique de la precipitation periodique, de Carl Wagner, a 8tB modifiee, et quand les coefficients d’espacement calcules avec cette theorie sont compares avec les coefficients determines experimentalement l’accord est satisfaisant, mais il subsiste malgre tout une exception. Les auteurs trouvent en effet que la variation du coefficient d’espacement prevue par la theorie, en fonction de la concentration en oxygene, est l’oppose de la variation observee experimentalement. Quelques suggestions sont proposees pour expliquer cette divergence, et les auteurs montrent comment certains resultats obtenus precedemment pour d’autres systemes peuvent Btreexpliques a l’aide de cette analyse de Wagner modifiee. PERIODISCHE

AUSSCHEIDUNG (LIESEGANG-PHANOMEN) IN FESTEM SILBER-II. EINE MODIFIKATION DER WAGNER-ANALYSE In der vorangegangenen Arbeit wurden eine experimentelle Untersuchung der Liesegang-Ausscheidung in festem Silber (Ag-H,O-Experiment) beschrieben. Silbereinkristallewurden bei 800°C in verschiedenen Sauerstoffpartialdrucken (bis zur Sattigung) und dann in einem Wasserstoffpartialdruck angelassen. Wasserstoff diffundierte in das Silber, reagierte mit dem gel&ten Sauerstoff und bildete Wasserdampfblasen, die sich unter bestimmten Bedingungen in Bandern anordneten und der Jablczynski-Beziehung gehorchten. Die von Carl Wagner durchgeftirte mathematische Behandlung der periodischen Ausscheidung wurde modifiziert. Beim Vergleich der mit dieser Theorie berechneten mit den experimentell gefundenen Abstandskoeffizienten ergab sich bis auf eine Ausnahme eine befriedigende Ubereinstimmung. Die von der Theorie vorausgesagte und die beobachtete iinderung des Abstandskoeffizienten mit der Variation der Sauerstoffkonzentration waren entgegengesetzt. Einige Vorschlage zur Behebung dieser Diskrepanz werden gemacht und es wird gezeigt, wie einige friihere Ergebnisse an anderen Systemen mit der modifizierten Wagner-Analyse erkart werden konnen.

INTRODUCTION Several

attempts f2-5) have

theoretically most

the conditions

notably

that

supersaturation encountered

Hence, values of spacing coefficients

been

made

for periodic

of Carl Wagnerc2)

theory.@)

Because

to derive

precipitation,

interface)

based

qualitative

on the

of the difficulties

in the gelatin-electrolyte

systems

have usually

their observations

visible rings or bands.

was

VOL.

17, JANUARY

1969

bands

and only

agreement between mathematical

discussed

largely avoided.

analyses

has been achieved. paper(l)

where

an experimental

these

latter

difficulties

In this paper, Wagner’s

system were

analysis will

be modified

for the conditions

of this system and the

predictions

will be compared

with the experimental

results obtained in the bubbles in solid silver.

formation

CARL WAGNER’S Wagner’s) confined 69

(i.e., the ratio of

from the diffusion

are rarely given in the literature,

In the preceding

confined

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. $ Oak Ridge National Laboratory, Oak Ridge, Tennessee operated by the Union Carbide Corporation for the U.S. Atomic Energy Commission. 5 College of Engineering and Science, Carnegie-Mellon University, Pittsburgh, Pennsylvania. ACTA METALLURGICA,

to successive

and experiment

gen-

erally studied,c2) investigators to counting

the distance

of water

vapor

ANALYSIS

assumed two reactants, A initially (t = 0)

to x < 0 at the

concentration

as, and B

70

ACTA

METALLURGICA,

initially confined to x > 0 at the concentration 6, both obeying Fick’s law and reacting at a moving front x = X to form an insoluble precipitate AVABVB. All A and B reaching x = X(t) is assumed to react, that is

aa

vBDA -

= -vADB

ax

ab

-

ax

at

x=X,

(1)

where a, D, and 6, D, are the concentrations and diffusion coeflicients of A and B, respectively; for simplicity Wagner assumed D, = D, = D. Note that Wagner omitted the va and vn from equation (l), which, however, does not affect his final conclusions. Under these conditions, a self-consistent solution is given by X = y * 22/(Dt) (2) and a = &(a,,- b,) a=0

&a,, + 6,) erf [x/Z.$/(Dt)]for x < X for 2 > 0 (3)

b = -i(a,, b=O

- b,) + $(a0 + b,)erf [x/2d(Dt)]forx > X for x
where the dimensionless constant y is the solution of 1-

erf (y)

b,

VOL.

17, 1969

successive bands, and where y = 5,,J2(D6)1’2 with G = (4 + y2)[1 -

(note that for y to be positive, a0 > b,). To discuss the banding, Wagner observed that the isotropically growing precipitate particles will not be able to keep contact with the moving reaction front and that nucleation of new particles will cease when the supersaturation product just ahead of the front falls below the critical value (saturation conditions being assumed at the interface of any growing particle). When the nucleation of new particles ceases, the reaction front stops advancing at t = T and x = X(T) = X0 (Fig. la). Using equations (3) and (4) for a(T) and b(T) as initial conditions and assuming all B is consumed at X,, for t > T, Wagner applied Fick’s second law to calculate the distance fN ahead of the stopped reaction front at X, at which the supersaturation product a”*b”B again reaches the critical value causing a new band to form. The result is

y&/2 exp (-y2);

(7)

5, is the value of the coordinate [ at which the concentration product as a function of 5 has a maximum. MODIFICATION

OF WAGNER’S

ANALYSIS

To adapt Wagner’s analysis to our experimental case, the following modifications or generalizations of his treatment were required : (1) Diffusion coefficients were assumed to be unequal. (2) The condition of a virtually insoluble compound A,B_ was used throughout [i.e., va and vn are included in equation (I)]. (3) Instead of assuming two semi-infinite solutions t,o be in contact at x = 0 and t = 0 the concentration (of A) at x = 0 was assumed constant (depending only on the temperature and hydrogen pressure in the Ag-Hz0 experiment(l)) at all time t > 0. With these modified conditions the solutions corresponding to equations (3) and (4) are now:

(5)

1 + erf (y) = 6

erf (y)] -

[erf

[$W/(Dd)l erf (Y/v’DA)

b

=

b

0

1

x <

[erfIxlW(W)l- erfbWDB) erfc(y/dDB)

I

x, (8)

x >

x 7

(9)

where the constant X

Y=Sz/t and is determined from

exp (-y2/W exp

(-y2/Da)

erf(Y/v'DA) edc

=

(y/d&J

(11)

where E

=

VB exp(-Y~/DA)

erf

(12)

(Y/~/DA)

and is always a decreasing function of y and where (13)

yz”A,~1~2~~vA+vB~[~~p

(y2)]‘““+““&,*

= G”Ay(3”A+vB)(ao

+

b,)(‘A+“B)

l/@‘A+‘B) (6)

where 6 = x - X,,, 6 = t - T, and k is the spacing coefficient(i) defined as the ratio of the distance to __

and is always an increasing function of y. Therefore a,/b, is always an increasing function of y. Note that y is always positive, regardless of the relative magnitudes of a0 and b,.

KLUEH

AND

MULLINS:

PERIODIC

PRECIPITATION

Ql

b)

b(x)

IN

SOLID

71

SILVER-II

causing a continued growth of the precipitate at X0 (all B is assumed to be consumed at X0). In the region 2 > X, a new precipitation zone is formed only if the critical supersaturation product is reached. The position of the beginning of the new precipitation zone is obtained by calculating the position of the spatial maximum of the product of the e~~nentiated wn~entrations as a function of time and then determining the position X, + EN at which the maximum of this product equals the critical supersaturation product. Immediately after the formation of new nuclei, B is prevalent and thus the concentration of A decreases to zero as indicated in Fig. l(c). The diffusion rate of A arriving at X0 + tN then increases and conversely that of B decreases until the diffusion rates of A and B are again equivalent (Fig. Id). The details of the calculation are given in the Appendix. The result of the calculation is 1-g

k-

Cl

= [

or

UW.~-l-%) 1

0 (qQ&“BpK

*

044

Hy(3’A+“B)(Pbo)“B(Eao m+ Fbo)“A

k-l

=

dl

&*(DpnBYB)1/2

!

.

Hy(3YA+"B)(~bo)("A+YB)

where

(15)

1. concentration of A and B [a(z) and b(x), respectively] at different times as functions of the distance from the end of the zone of continuous precipitation.

FIG.

In order to derive an equation for the spacing of bands, we assume following Wagner that the growth of the reaction zone has ceased for the reasons discussed previously resulting in an initial state described by equations (8) and (9) evaluated for t = T as shown in Fig. l(a) ; a = b = 0 at x = X0 and the diffusion fluxes are equivalent and opposite according to equation (1). Since a(z) is concave upward and a(g) is concave downward, the concentration a(x) increases in the course of time while b(x) decreases with time. Furthermore, more A than B arrives at X0 for t > T leaving a surplus of unreacted A to diffuse past X0 (Fig. lb); the discontinuity in a(z) reflects the portion of A continuously consumed by the B arriving there,

and G and ye. are the same as those in equation (7) with D, replacing D. Numerical values for G, H, and ga for various values of vg, are given in Table 1. c”

1. Numerical values of constants

TABLE

G

'A

IB

YA

1 1 2 :

1 2

0.37 0.66 0.20 0.14 0.89

0.20 0.090 0.31 0.36 0.042

0.52 0.26

0.13 0.27

2 3

;1 :

DISCUSSION

OF MATHEMATICAL

H 0.928 0 306 10.80 411 12.90 0.116 12.80

ANALYSIS

TO investigate the mathematical behavior of equation (14b) as a function of the concentrations ao, b,, we first examine E(y/Dy2) and F(y/Dg2) : for values of r/DgZ< 1 we obtain by standard approximations from equation (13) F s VA [l + Zy/r( D&] whereas

12

ACTA

METALLURGICA,

for y/D!” > 1, F g v*z/(n) y/D;“. Therefore F is essentially a linear function of its argument y/Dg” with an intercept of vA changing gradually from the slope Sv&/rr to 2/(r) vA as the argument passes through unity (for the case of water vapor formation in silver to be discussed in the following section, vA = 2 and the slope changes from 2.26 to 3.35). The behavior of E, on the other hand, changes drastically as the argument y/Dp passes through unity since for y/Dj’ < 1 equation (12) gives E g v&(nDa)/2y whereas for y/Dz” > 1 it yields E E vB exp ( -y2/D,). This change in behavior of E marked by the transition value y/D?” g 1 divides the qualitative behavior of k - 1 [equation (14)] as a function of the concentrations into two corresponding regimes. To see this, suppose first y/Dy’ < 1; then equation (11) becomes

VOL.

17, 1969

the two regimes of behavior and using this condition

to rewrite equation (ll), we find the corresponding transition value of so/b, to be exp (-DA/D,)

vA erf(l) a0 b,=
erfo (DT”/D&‘“)

or

a0 - = 2.29 Y_a % 60

(18)

The expression in brackets has a minimum as a function of ( D,/D,)112 which is also a lower bound and is approximately 1.94. Therefore, regardless of the values of the diffusion coefficients we must have for the transition “_O2 4.44 vA .

(19)

vB

b0

Furthermore for (DA/DB)lt2 > 1, application of the approximation formula for F to the bracket of equation

For fixed b,, an increase in a,, will from equation (16) and the functional form of F increase both y and F. Since these quantities are both in the denominator of equation (14b), k - 1 will decrease. For fixed a,, equation (16) shows that for an increase in b,, both y and F will decrease but in such a way to leave Fb,y = constant. Substituting from this relation for Fb, in equation (14b) and combining y’s, we are left with y 2vAin the denominator showing that k - 1 must increase. The second regime of behavior of k - 1 occurs when y/D?” > 1; equation (11) then becomes

(18) gives

equation ator

(14b), the y dependent

becomes

y3*A+*Bexp [-(vA

y/Dy” > 1, the logarithmic sion shows it is dominated therefore

increases

part of the denomin+ vs)y2/D,].

derivative

by the exponential

with decreasing

For

of the expres-

y causing

of tN/Xo for

Calculation

the Ag-H,O

experiment

DA = Dg = 8.39 x 1O-s cm2/sec

(ref. 7)

D, = Do = 2.12 x 10-s cm2/sec

(ref. 8)

and

vn = 1,

where Do and DH- are the diffusion coefficients of oxygen and hydrogen in silver, respectively. Equation (14) becomes for this case

1

K *Da3Do1’2 m Hy’ FcQo(EcHo + FcoO)~ -

k-_l&+ 0

l/5 (21)

where cn” and coo are the solubilities of hydrogen and oxygen-in silver, where y is to be evaluated from equation (11) which becomes

exp (-y21&J erf(y/dDg) exp (-r21D~) erfc (yll/Do) ’

to decrease. We conclude

(22) and E and F are evaluated

using the definitions

(fixed aO) however, k - 1 decreases only if y/D2” > 1; otherwise, if y/D112 A < 1, k - 1 increases. Taking the

equations

condition y/D,II2 = 1 to mark the transition

found that reversing this procedure,

between

AND

For the experiment of the preceding paper(l) in which Liesegang bands were formed in single crystals of silver at 800°C,

1

that k - 1 always decreases (closer bands) for increasing a,, (fixed b,). For increasing b,

(20)

COMPARISON OF THEORY EXPERIMENT

and k -

4.06 < ,

a transition value independent of diffusion coefficients.

vA = 2

For fixed b,, an increase in a,, will again cause an increase in both y and F and hence from equation (14b) a decrease in k - 1. For fixed a,,, an increase in b, will cause y and F to decrease in such a way as to keep Fb, exp (y2/Da) = constant. Substituting for Fb, from this relation into

vA

a0 b, g

(12) and

always hydrogen

(13).

Note

and oxygen,

that

of

a0 and b, are

respectively.

It was

that is, annealing

KLUEH

MULLINS:

AND

PERIODIC

PRECIPITATION

IN

SOLID

SILVER-II

73

TABLE 4. Effect of hydrogen concentration on spacing coefficient for 99.99% Ag c# _ (moles/cc) 4.45 3.95 1.95 1.72 1.38

x x x x x

z (measured)

k (calculated)

1.50 1.60 1.75 1.95 2.10

2.4

10-1 10-7 10-7 lo-’ lo-’

E.3 s:9 11.4

coo _ = 5.4 x 10-e moles/cc. TABLE 5. Effect of oxygen concentration on spacing coefficient for 99.99% Ag coo (moles/cc) 5.4 3.92 3.58 2.58 1.26

x x x x x

the specimen in oxygen hydrogen,

after it had been saturated in

did not produce

bubbles.oO)

In Fig. 2 the parameter function These

y has been plotted

of cnO/coO as determined y values

spacing

have

coefficients

described

then

from equation

been

used

paper.(l)

8OO’C are given by 1.18 x 10-5dpo,

(22).

calculate

using the pHs, po, values,

in the previous

coo z

to

as a

etc.,

coo _ and cue - at

moles OS/cc Ag

The

critical

determined

moles Hz/cc Ag

supersaturation

in the previous

product

(ref. 9). Km*

was

paper(l) to be

Km* N 1.6 x 10-21.

moles/cc.

c$ (moles/cc) 8.71 7.3 6.16 4.37 1.95

x x x x x

1.95 1 85 1.68 1.38 1.07

x x x x x

k (measured)

lo-’ 10-1 lo-’ lo-’ lo-’

1.35 1.45 1.55 1.75 1.90

k (calculated) 5.2 5.5 5.8 7.2 10.9

TABLE 3. Effect of oxygen concentration on spacing coefficient for 99.999% Ag

3.16 2.86 2.23 7.91 5.27

x x x x x

K (observed)

1O-6 10-e 10-C lo-’ lo-’

CHO= 1.95 x

1.40 1.50 1.60 2.10 2.25 lo-’

moles/cc.

k (calculated)

1.30 1.60 1.75 1.90 2.00

1.6 1.7

10-1 10-1 10-1 lo-’ IO-’

;.: 512

coo = 5.4 x 1OV moles/cc. TABLE 7. Effect of oxygen concentration on spacing coefficient for 99.9% Ag

5.4 3.01 2.15 1.24 1.05

x x x x x

k (measured)

k (celculeted)

2.00 2.10 2.20 2.25 2.30

5.2 3.8

10-G 10-B 1O-6 1O-6 10-G

The

calculated

mentally

k (calculated) ::: 3.4 2.2 2.0

k values

determined

;.: 2:3

along

with

the

calculated

obtains

between

the

modified

the spacing

or better coefficients

by the modified Wagner analysis and those

observed experimentally. with

2-7

Ag, respectively.

Tables 2-7 show that order of magnitude agreement

experi-

values are given in Tables

for 99.999+, 99.99+, and 99.9%

not completely

coo _ = 5.4 x 1O-6 moles/cc.

coo _ (moles/cc)

k (measured)

cn” = 1.95 x 10-r moles/cc.

TABLE 2. Effect of hydrogen concentration on spacing coefficient for 99.999% Ag c$ (moles/cc)

5.2 4.4 4.0 3.4 2.5

TABLE 6. Effect of hydrogen concentration on spacing coefficient for 99.9% Ag

coo _ (moles/cc)

1.95 x 10-6dpn,

1.75 1.90 2.00 2.25 2.80

(ref. 8)

and ego s

k (calculeted)

IO-6 10-G 10-C 1OV 1O-6

cn” = 1.95 x lo-’ FIG. 2. Parameter y ss a function of the concentration ratio c&o for the silver-water vapor experiment. --

% (measured)

This agreement, however, is

satisfactory

oxygen

analysis is opposite

investigation

because the variation

concentration

predicted

to that observed

by

of E the

in this

(and most other investigations).

This discrepancy is clearly indicated by the mathematical discussion following equation (14b) since our experimental

values of a,/b, are usually approximately

10-l and are always less than unity which in turn is much

less than

the

8.88 for the transition;

minimum

value

therefore

4.44 x 2/l =

our concentration

74

ACTA

values k -

place

us in the regime

1 should

increase

concentration) of k -

with

METALLURGICA,

r/Dy”

< 1 in which b,

increasing

contrary to observation.

1 with a, (hydrogen

(oxygen

The variation

concentration)

is in agree-

ment with the prediction.

VOL.

17,

therefore

1969

appear unlikely that all of the discrepancy

is caused by such errors. The change

in spacing

purity can be explained

coei%cient

with change

by the mathematical

in

analysis

if CH’, ego, DR and Do are known as a function of silver

To explain the discrepancy,

two possibilities

will be

purity.

Such data are not available.

examined : (1) The discrepancy

is a result of the assumptions

made in the modified

analysis.

(2) The discrepancy values of parameters If the major

is the result of errors in the used to calculate k.

assumptions

are examined,

stands,

that the modified analysis, as it now

appears correct to a first approximation.

can be used to explain

of the modified

analysis

it appears that only the assumptions

made in the derivation

CONCLUSIONS

It is concluded

of a(t, 6) could

be used to

all experimental

without invoking any phenomena and

precipitation.

mentioned

In

that most

this

It

observations

other than diffusion

respect

it

investigators(llJ2)

should

be

have found

To

(as was found here) that an increase in b, leads to a

calculate a(t, S) it was assumed that there was a sink

decrease in k. Kant,(13) on the other hand, has found that, under certain conditions, bands of PbCr,O, that

explain the type of discrepancy

which was found.

at x = X, such that an equivalent to keep b(0, 6) = 0.

removed

may overestimate and

thus

the amount of A removed at 6 = 0,

underestimate

a([, 6)

would make the calculated acute

possibly

with

the

[ > 0.

This

Further-

of a(E, 6) would become

an increase

explain

centration

for

Ic value too large.

more, such an underestimate more

amount of A was

Such an assumption

in b,, and this could

disagreement

when

the

con-

in gelatin displayed

is in error, it is because it does

could only be explained Offhand,

for band formation

and diffusion

coeficients-indeed,

over any concentration possible

that vacancy

diffusion is apparently

step in bubble

growth.(lO)

the rate limiting

to

extremely

into

the bubbles

it will not affect the conclusions

at this position. is in error,

drawn in the preceding

section.) the parameters doubtedly

to calculate

k.

There

precipitation

Km*

determined

as a result

make

the

the

Errors contributed experimentally

appear larger than it actually improve

of

experimental

and because Km* was calculated

an infinite specimen.

the agreement

determined

is-a

between

assuming

in this manner Km*

fact which would theory

and experi-

ment. As a result of the discussion in the previous section, an error in the solubilities discrepancies An

increase

magnitude between

between

could be used to explain the

the

theory

in c~~/c~O of more is required

theory

range.

The only limitations

the

k,

only

individual the

bands

zone

of

and

at

continuous

may be observable.

to

explain

and experiment.

and than the

experiment. an order

of

discrepancy

Offhand,

ACKNOWLEDGMENTS

The financial

it would

support

poration

(fellowship

and

U.S.

the

gratefully

is un-

some error present in the experimentally

difficulties would

to assess the errors inherent in

used

to the

Thus, it is quite probable

(It should be noted that if this assumption

It is quite difficult

according

that is, at small k it may not be

resolve large

that all of the oxygen which arrives at & = 0 will not be incorporated

to place any

on the concentrations

analysis, it should be possible to form bands

that is, it assumes that the precipitate

at 6 = 0 can

in terms of the coagulation

it does not appear possible

conditions

are experimental,

It has been shown

k with an

theory.(lJ)

not take into account the growth rate ofthe precipitate, accept all of the B arriving there.

an increasing

increase in b,. Kant assumed that such an observation

modified

of oxygen was varied.

If this assumption

formed

of the Union

to R. L. Klueh

Army

Research

Carbide from

Office,

Cor-

1962-65)

Durham,

is

acknowledged. REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

R. L. KLUEH and W. W. MULLINS, Acta Met. 17,59 (1969). C. WAGNER, J. Colloid Sci. 5, 85 (1950). P. B. MATHUR, Bull. them. Sot. Japan. 34, 437 (1961). S. PRAOER, J. them. Phys. 25, 279 (1956). M. KAELWEIT, 2. phys. Chem. 32, 1 (1962). W. OSTWALD, Lehrbuch der Allgemeiner Chemie, p. 778. Engleman (1897). W. EICHENAUER, H. KUNZIG, and A. PEBLER, 2. Met&k. 49, 220 (1958). W. EICHENAUER and G. MUELLER,2. Met&k. 53, 321 (1962). K. H. LIESER and H. WITTE, 2. Elektrochem. 61, 367 (1957). R. L. KLUEH, Ph.D. Thesis, Carnegie Institute of Technology. E. S. HEDGES, Liesegang Rings and Other Periodic Structures. Chapman & Hall (1932). K. H. STERN, Chem. Rev. 54, 79 (1954). K. KANT, Kolloidzeitschrift 189, 151 (1963). R. N. DHAR and A. C. CRATTERJI, Kolloidzeitschrift 31, 15 (1922).

KLUEH

AND

MULLINS:

PERIODIC

PRECIPITATION

APPENDIX DERIVATION

Equation Wagner

OF

u/~(D&)~/~

SPACING

as variable of integration

of equations

discussed in the text. under the

(1) and (10) and that at z = X(t),

a = a0

at x = 0 and t > 0,

b = b,

x>o

of distance

to equations

(8) and (9).

and time are introduced

[=x-X0, (Al)

at

Consider now the function a( 5, 6). The initial distri-

vBDa !f = -vaD,e a5

a5

as opposed

and

at

condition

5 = 0

of which is given by the rate of component

and

6 = 0, = vADB $ 0

a([, 6)

(Note that Wagner corresponding

b( E, 6 = 0) = f(E)

is

given by equation (9). Because of the excess of species A arriving at E = 0, it is assumed that b remains zero

Fick’s

for 6 > 0. Under these conditions

boundary

for 5 > 0, an

integration

variable.

For

i . c +r I

(A7)

did not have vA and ys in his

equations.)

A solution position

is

B diffusing

6=0,

to find the concentrations

and b(E, 6) at 6 > 0. Initially, the distribution

In this case there is the

that there is a sink at 5 = 0, the intensity

(A2) and it is required

(8) ;

of b(E, d),

to the investigation

from 5 > 0, to 5 = 0. Thus,

are now t=O

W)

(y/Z/DB) ’

bution a(5, 6 = 0) at 5 < 0 is given by equation

a(5 = 0,6 > 0) is not given.

6=t-T

a=b=O

2b,y-exp(-y2/DB)

E

r. (7-rDB)l12erfc

=

however,

conditions

and substi-

tution for b gives

such that

The boundary

(A5) on time is disregarded,

t=o.

exist according

New variables

to b(t, d), the dependency

As a first approximation of equation

a--brO

distributions

a

give

(8) and (9), which are used in the derivation,

It is now assumed that the reaction front has stopped moving at 2 = X0 and t = T and concentration

where

75

SILVER-II

(14) will be derived using the procedure of

have been derived by several investigators? conditions

SOLID

COEFFICIENT

with the modifications

Equations

I?;

for a(t, 6) is now obtained

of a,(f, second

6) and a,(l, law

and

by super-

6), each of which satisfy

also

satisfy

the

following

conditions :

a,(5,6 = 0) = a([, 6 = 0),

(W

al(2J = 0, 6 > 0) = 0,

(W

a,(E,6 = 0) = 0,

WO)

(A3) f(t)

=

b(5, 6 = 0), the series expansion For a,(E, 6) the solution b([, 6) except

the initial

is similar to the solution concentration

5 < 0, which means that involving uneven derivatives is used equation equation

and

similarly

for f(a).

Differentiation

at

of

(A3) with respect to 5, introduction of (A4) and evaluation of the integrals with

i C. WAGNER, Diffusion in Solids, by W. JOST, Academic Press (1952).

is given

the sign of the terms must be changed, that is,

Liquids, Gases, edited

+2(y)Llp (~)~=,,,~, -.. W2)

ACTA

76

METALLURGICA,

For E < 0, a, vanishes because a([ > 0, 6 = 0) = 0 and equation (A9) ; thus = 0

for

6 > 0.

(Al3)

VOL.

17,

1969

concentration product a”*b”B, must equal the critical supersaturation product Km*. The coordinate 6, at which a”*b”Bhas a maximum value is determined from the condition

Equations (A5), (A9), and (A12) are now substituted into (All) and rearranged, to give

(A17) After rearranging and introducing equations (A6) and (A16) and the auxiliary value ya = ~J2(DA6)i’s, equation (A17) becomes -

a In a/at

a In alat 2 [yaKi’

=

(4 +

yA2)

exp (-yA2) - YA2erfc @A)] erfc (ya) - yA+‘” exp ( -ye’)

(A14) After substituting for a, b, and y, (A14) becomes

(A14 Equation (A15) is equivalent to a source at 5 = 0 with an intensity proportional to 61j2. Integration with respect to time of the product of the known solution for an instantaneous source and an intensity factor proportional to Bn2 as given by equation (A14) gives

YdVB

a2(Ey ‘) = (nDA)1/2T

where E and E” are given by equations (12) and (13) of the text. Since a,(t, 6) vanishes at t > 0, a,(,$, 6) represents a(E, 6) for [ > 0. For the beginning of a new precipitation zone, the

= z

’

(AW The auxiliary value ya which depends only on the ratio vB/vA is determined by this equation (note that this is exactly the same as Wagner’s equation (36) with y replaced by yA; however, Wagner’s equation contains a typographical error and the right side should be VB/VA instead of va/vn). Combining equations (A6) and (A16) for E = t,, eliminating 6 and T by using equation (10) and ya = ~,/2(DA6)1’2 one arrives at avA6YB =

x

H(D~DBY~)-~~2y3v~+YB(~bO)YB

(EaO

+

Fb,)“* 2 ( 0)

2v,+“B at t = t,,

(AN)

where CTis given by equation (7) with yA replacing y, DA replacing D and H is given by equation (15). When the spatial maximum value of the concentration product calculated in equation (A19) equals the critical supersaturation product Km*, the formation of new nuclei and thus a new precipitation zone is expected. Setting equation (A19) equal to Km*, letting 5, = tN as the distance to the new precipitation zone and solving for [JX, one obtains

5N

( D>DBVB)1'2 Km*

Hy(3~‘*A+VB)(Fbo)“B(Eao f Fb,)“A1 x,== C

l/(%,

+“B)

’

WW