Crystal growth in gels and Liesegang ring formation

Journal of Crystal Growth 75 (1986) 203—211 North-Holland, Amsterdam

203

CRYSTAL GROWTH IN GELS AND LIESEGANG RING FORMATION II. Crystallization criteria and successive precipitation H.K. HENISCH and J.M. GARCIA-RUIZ

*

Department of Physics and Materials Research Laboratory, The Pennsylvania State University, University Park, Pennsylvania 16802, USA Received 19 December 1985; manuscript received in final form 24 February 1986

This paper considers the conditions under which two diffusing reagents can form precipitates (continuous or periodic) in a gel, on the basis of numerical solutions obtained in Part I. For a precipitate to form, not only must the local concentration product exceed a certain value K~,but the two reacting entities A and B must be nearly equal to form nuclei of structure AB. With these dual conditions, the behavior of gel-diffusion systems is correctly reproduced. The periodic nature of their fulfillment (Liesegang Ring formation) is demonstrated. As a demonstration of principle, ring masses and ring spacings are calculated on the basis of a simple algorithm, suitable for microcomputers.

1. Introduction A previous paper [1] (“Part I”) has dealt with the analysis of various diffusion systems and, in particular, with the application of numerical methods to the solution of the diffusion equation. Such methods can be used, in the first instance, to ascertain the location of the first precipitation in a system of the kind shown in fig. 1, where two counter-diffusing reagents meet to generate a sparingly soluble reaction product. Of course, appropriate criteria have to be available for such an assessment, and in the past it has been widely believed that crystallization begins when the concentration product exceeds a critical value K. Indeed, the gel literature is full of papers, e.g., Morse and Pierce [2], Wagner [3], Venzl and Ross [4], and Ortoleva [5], that are based on this assumption, mostly by analogy with precipitation from solution. If this criterion were correct, then the first crystallization (or Liesegang Ring formation) would always have to take place where the concentration product has a maximum, and fig. 8 of Part I shows that place is always in the middle *

Permanent address: Departamento de Geologia, Universidad de Câdiz, Apartado 40, Puerto Real (Cãdiz), Spain.

of the system (x = L/2), no matter what the boundary concentrations of A and B may be. This is, however, in gross conflict with observations, as Garcia-Ruiz and Miguez [6] have already pointed out. The position of the first precipitate (x1) is in fact dependent on the reservoir concentrations AR and BR. This well-known fact had remained unexplained, precisely because the simple gel growth systems in wide use did not lend themselves to critical tests. By now we known that the minimum product condition is insufficient for a stochastically controlled system, like nucleation in a gel, even though it is applicable to precipitation in solutions. Two questions therefore arise: (a) how well-founded is the notion that the condition A(x1 t)B(x~ t)>_K’ has any significance for crystal growth processes in the gel, and (b) what other condition has to be satisfied for a crystal (or Liesegang Ring) to form? We shall here be concerned with both issues, and the related one of how K~’is to be interpreted. Up to a point, crystals and rings can be discussed under the same heading, since they differ from one another mainly in terms of nucleation density. One can see this very clearly in systems which exhibit ring formation, with each ring consisting

0022-0248/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

204

H. K Henisch, f.M Garcia-Ruiz

RESERVOIR CONCENTRATIONS

AR

/

Growth in gels and Liesegang Ring formation. II

A(X,t)

B(X1t)

BR

RESER~IRA

~ESERVOIR B

•1

gel column XzO

XZL

Fig. 1. Gel growth systems used as a basis of the present calculations, unless otherwise stated. Infinite liquid reagent reservoirs.

only of a few major crystals [7]. The results here described are based on a microcomputer implementation of the simple, one-dimensional algorithms described in Part I, the purpose being to demonstrate principles, rather than describe particular situations. The structure and use of more elaborate (non-determinate) algorithms which also make provision for re-solution will be discussed in a later paper.

considered the driving force of the nucleation process. The supersaturation itself is defined as AB /( AB ~ AB /K .~ =

=

“ ‘

‘

/

s’

where (A B) is the concentration product at saturation. Complete ionization has here been assumed for simplicity. Eq. (1) has the well-known form shown in fig. 2a. Though ~N must eventually saturate (since it cannot be greater than unity), it is clear that the probability of nucleus formation is negligible up to a certain value s s~.At this point, there is a virtually abrupt start, and so can therefore serve to define a quantity K~,such that s~ K~/KS. Crystallization is possible at lower values of s, but it is then highly unlikely. This constitutes the link between schematic, semi-empirical notions of K~ and modern microscopic models. For precipitation from solution this appears to be entirely sufficient, but despite the superficial successes occasionally scored by the K~hypothesis, e.g., see Morse and Pierce [2], and for the reasons given above, s ~ s~is evidently an insufficient condition for precipitation in gels. That condition suggests, for instance, that the A/B ratio is irrelevant, which is absurd for any system of limited particle mobility. In this recept, the situations which prevail in a homogeneous mixture of solutions (e.g., as encountered in many laboratory and industrial crystallization processes) and in =

2. Crystallization criteria The conventional solubility product AB K~is derived from the Law of Mass Action. For AB < K~crystals dissolve, for AB> K~they grow, always assuming that they have nucleated. We are here concerned with the conditions under which this homogeneous nucleation can take place. Modem theory [8] suggests that the probability ~N of nucleus formation is dependent on the supersaturation s in accordance with an expression of the form: =

p

=

N

exp

—

M (log ~)2

(1) ‘

where M is a constant dependent on the morphology of the building block and on the surface energy of the clusters. The term log s may be

=

H.K. Henisch, f.M. Garcia-Ruiz

/

Growth in gels and Liesegang Ring formation. II

y~exp[—M/(Iog X)2]

~0:_______________

205

+

~

10 20 30 40 50 60 70 80 90

/

.

(SUPERSATURATION)

.

(a)

12/0

~‘I

0 (b)

I

I

I

I

I

I

I

0.5 A/B RATIO

1.0

Fig. 2. Conditions of preparation: (a) Nucleation probability as a function of supersaturation; form of the relationship predicted by modern theory [6]. Here M = 1. (b) Schematic, two-dimensional (30 X 30) simulation of nucleus formation; probability of finding a BA/AB or AB/BA configuration as a function of A/B. Results of three runs.

gels are quite different. In a gel we are dealing with time-dependent macroscopic concentration gradients in which A/B takes every kind of value 0
arise is shown in fig. 3, which also reflects the fact that small departures from A B are allowed (“equality range”), but large departures are forbidden. AB K5 represents the solubility curve;

That the A/B ratio does in fact matter is clear even from the simplest and most schematic of models, relying on geometry alone; all energetic and three-dimensional pace related arguments would be additional. W~may, for instance, populate a rectangular matrix of cells randomly with A and B items, and ask about the probability of finding an AB/BA or BA/AB configuration that we could regard here as a nucleus. That probability goes through a sharp maximum at A B, and tends to zero rather rapidly as A/B departs significantly from unity. Fig. 2b shows the results of such a simulation. For practical purposes, the bell-shaped curve could, in the first instance, be approximated as a rectangle. Without taking the geometrical model too seriously, it is at once plausible that the nucleation probability should be very small, unless the reagent concentrations exceed certain minimum values. How these minima

AB K~marks the boundary of likely precipitation, and AB K~’gives the range in which precipitation is likely to arise. The two asterisks denote the minimum and maximum A/B ratio. As K~’ K~, these boundary values must tend to coincide, leading to A B A mm ~ as the second precipitation criterion, AB ~K~being the first. A three-dimensional simulation would show the equality range to be narrower than fig. lb suggests. Only half the “equality range” is actually effective. This will be clear from fig. 4, in which the full line represents the actual A and B values pairs between x 0 and x L, at the time (say) when the first precipitation point is reached. Only half the range of values which satisfy the I A/B 1 K~condition. The outcome is that when the precipitation point is approached from the side of the lower boundary

=

=

=

=

=

—*

=

=

=

=

=

—

206

H.K Henisch, J.M. Garcia-Ruiz

/

Growth in gels and Liesegang Ring formation. II

100

A8 2600

—

\AB

8min~

200 ~~BR ~

60~ 80

20-40

B

150 z A.8~K 0

//

10

2030

40

bO

z ~l00.

I

Amin A Fig. 3. Concentration relationships I. K~= solubility product. K~= precipitation product. K~>K~. Definition of the “equality range”, leading to Amia and B,mn.

Z 0 0

~ 50-

concentration A (as it is in the present computations), the simple precipitation criterion A B is still correct, despite the symmetry of fig. 2b, and despite the unsharpness of Bmin. If the concentration contours were more complicated (e.g., as they are expected to be after the first precipitation), this simplification would be impermissible. Fig. 4 also suggests that the rising concentrations do not (or, at any rate, not significantly) overshoot the criterion for first precipitation. Fig. 4 further shows that when AB K~is only just satisfied, the equality condition has to be accurately fulfilled, but when AB K 5” (where K~’> K~),then precipitation is possible over a whole range of A/B values. Within that range, precipitation is, however, most probable where A B. In the course of double diffusion, the actual concentration product increases with time everywhere, and can greatly exceed K’5 (see fig. 4 insert), but unless there is within the gel a locality which also falls into the equality range, precipitation cannot take place. When it does, its consequences depend, of course, on the concentration product contours in the vicinity of the precipitation point. In earlier work, Kirov [9] noted that the first precipitate always occurs within a certain distance range without, however, addressing the concentration ratio issue. Earlier, Salvinien and Moreau [10] and Lendvay [11] ascribed a signifi-

i I

=

I 150 Amin

C

AB =1<” $

100

CONCENTRATION A Fig. 4. Concentration relationships II. “Equality range” here assumed to be from A/B = 0.8 to A/B = 1.2. Full curve: actual relationship between solute concentrations A and B at the time (and place) when A B = 50, for AR = 100, BR = 200.

=

cant role to the ratio of diffusion coefficients (here

=

taken as equal, for simplicity).

3. First precipitation in a gel

=

On the basis of the above, and using the methods described in Part I, we can ask the computer to ascertain at what point x~in the system the concentration product first reaches a certain value K~while satisfying the condition A B, as the reagent diffusion progresses. Computer methods demand an accuracy limit for the equality condition; the results which follow were based on =

IA—BI <(A+B)/ER, with ER 10 or 20, giving a 20% or, alternatively, a 10% tolerance, corresponding to the non-zero =

H.K Henisch, f.M. GarcIa-Ruiz

/

Growth in gels and Liesegang Ring formation. II

width of the curve in fig. 2b. In this way, the computer finds the first precipitation point for various values of K~and BR/AR. The calculated x~is not in the center of the system, but on the side of the weaker boundary concentration (fig. 5). That is where the first precipitate is in fact ob-

~ ~ 50 ~ ‘— 40

Analytic calculations based on the assumption of semi-infinite concentration profiles [1] predicted a fairly sensitive dependence of x1 on BR/AR.

~ 20 ~

However, in the present system the boundary concentrations are kept constant on both sides [A(0) AR, A(L) 0, B(0) 0, B(L) BR], and this “pinning” of the concentration contours must be expected to diminish the dependence of x1 on BR/AR. In fact it does so. Fig. 5 shows that the differences in the position of the first deposit amount to only a few millimeters in a system of 100 mm length, even though BR/AR is varied by a factor of 10. Fig. 6 gives the trend of x1 with varying ratio of the diffusion constants. In terms of the algorithm described in Part I, a diffusion constant ratio other than unity can be simply introduced by transacting several iterations for the more mobile component while transacting only =

=

IOU

L

=

Bfl/AR~2\

\

0x-

\\

10 -

10

.

10

.

IO_3

.

BR/AR~10

\ 45

46

BR/ARK

BR/ARE I

BR/ARE

-

10

~

~ 30

-

-

I

I

I

I

I

I

I 2 ~ ~ ~ RATIO OF DIFFUSION CONSTANTS Fig. 6. Computed of the diffusion constant ratio the position x1 if the first precipitate.

DB/DA

on

one for the less mobile. For equal diffusion constants and BR/AR 1, the first deposit would, of course, be in the middle (here x1 50 when L 100). If in any particular experimental system the sensitivity of x~to changes of BR/AR were found to be greater than that here calculated, that would constitute reasons for believing that the two reservoirs are becoming contaminated, i.e., that progressively B(0) > 0 and A ( L) > 0. We know that this often happens in practice, leading to crystal growth within the liquid reservoirs. The methods here described, of course, can be used to calculate x1 for liquid source reservoirs of any size. The present assumptions amount to “infinite” reservoirs. =

=

.

.

.

=

.

analysis of successive precipitation phenomena is not yet available. To provide it would be a formidable task, even though the strategy is obvious enough, and a very instructive beginning can be

I

47 48 POSITION X

I

4. Successive precipitation, ring positions As far as is known, a comprehensive computer

\ \

\

I0~ .

44

=

-

207

49

50

1 Fig. 5. Computed effect on the source concentration ratio (BR/AR) on the position (x1) of the first precipitate, for different values of K~.Growth system as in fig. 1.

made by simple microcomputer techniques. After finding the position of the first precipitates, as above, the local solute concentrations must be decremented, after which the iterations can resume, until a second point is found at which the precipitation conditions are satisfied. When the first precipitate is formed, the local concentration product drops from K, to (presumably) K5 which,

208

H.K Henisch, f.M. Garcia-Ruiz

/

Growth in gels and Liesegang Ringformation. II BR •I0~

STARTING SETUP

90

~:

100

~

60

-

EQUALITY RANGE ER

DIFFUSION

10

111111

Fig. 7. Computed time-dependence of ring formation for an open system slopes. Time in computer units.

(AR

for simplicity (and in the present context only) is regarded as zero, in view of the fact that K5 << K~.

Fig. 7 gives examples of two runs, for a particu-

A precipitate, once formed, is assumed to grow (continued decrementation of the local concentrations), until the next precipitation occurs. In practice, it may continue to grow for a little longer, but this is believed to be a minor aspect, because one or other of the reagents will then be present in very low concentration, In the simple (one-dimensional) algorithm so far implemented, precipitates have positions and mass, but no explicit thickness. Where a precipitate forms, and when, are outcomes of a complex interplay of processes which depend, for instance, not only on local concentrations, but on the steepness of the supersaturation front and the speed of its advance. They also depend on the equality range (defined above) and on the value of K’. Thus, precipitates are continuous when K~is very low, and do not form at all when K~is very high. For a certain intermediate range (here typically 0.01
=

1), as defined by the insert. Note linearity and equal

lar set of start-up conditions (a closed-tube variant of fig. 1), with position plotted against the square root of time. The expected linearity is accurately fulfilled, despite the local complexities. Fig. 8 demonstrates that ring positions do, indeed, depend on whether the growth tube has one end closed, or has liquid reservoirs on both sides. The effects of reagent depletion and of mutual reagent

GEL COLUMN B11~IO STARTING SETUP

____________________

35 RING POSITIONS I I I I I 0 SYSTEM CLOSED ~

o

50 I

I I I I I I IIIII I

50

50 SYSTEM OPEN AT X0 A(O)~I,B(O)0 Fig. 8. Comparison of ring positions for a closed and an open system.

H.K Henisch, J.M. Garcia-Ruiz

~

/

Growth in gels and Liesegang Ring formation. II

209

~

C.)

“~

o

I

I- AB

I 35

I

I

I

I

I/A

I I 40 DISTANCE ‘~~q~’

I

I

~

I

45

I

________

~

0.04

-

0.03

-

~*.S..A.~~BEFORE PRECIPITATION S., OFRINGI2

z

O~I-AFTE

P~P

A1~oN-\

I,’

DISTANCE Fig. 9. Local concentration contours (a) and concentration products (b) in the vicinity of a precipitation point. Full lines: immediately before precipitation. Broken lines: soon after precipitation.

contamination of the reservoirs can be similarly documented. Under some conditions successive rings exhibit diminishing spacings, in harmony with the occasional observation of “revert” ring formation [9]. Fig. 9 presents concentration contours (a) and concentration products (b) immediately before and soon after the formation of a particular ring in position 40, the previous ring having position 46. Position 40 becomes a sink, and the A-contour to the left of it quickly stabilizes to near-linearity. The B-contour, fed by the abundant source BR, continues to increase there, without giving rise to further precipitation, because the concentration product is far below K’ and because the concentrations of A and B are very unequal (A being the lower). At a later time and somewhere to the left of position 40, the precipitation conditions will again be satisfied, and another ring will then form. When that happens, the concentration product may greatly exceed K~.In other situations, the opposite applies: there is always an equality point, but that point may not be within the range X in which AB ~ K~.Moreover, some rings are formed

in an A > B regime, and others in a B > A regime. Thus, rings which are superficially similar may have very different formative histories.

L 50 = 0.05 4

A >8 o

-

~

A>B A >8

~ 3

—

i±~

-

~

o

2

—

~ U

°

STARTING SETUP

°

~

I

CLOSED TUBE AT X~0

-

A I ________________

0

40

I

~o

.0 R

35 I

50 I

POSITION Fig. 10. Computed mean distribution over a series of rings; accumulated mass versus ring position. System as defined by the insert. B > A unless otherwise stated.

210

/

H.K. Henisch, f.M. Garcia-Ruiz

Growth in gels and Liesegang Ring formation. II

5. Successive precipitation; mass distribution and growth rates

subsequent growth of very localized precipitate. In practice, that mass can vary in complicated ways (e.g., see fig. 1.2 of ref. [5]), which has so far defied analysis by non-numerical means. The actual mass must depend, of course, on the (ever-varying) conditions of growth, as well as its duration, here the

It is a simple matter (within the framework of the present assumptions) to ascertain the total mass of material involved in the formation and

o -°

RING NO. I ~

•

6 10 II

x +

4-

I

POSITION 62

I-

iI

29 16 II

I I I I I

/ /

/ / /

L~5O K5’=O.05

-

4

3

I’

,,.#.~( LAST RING

,+

—

/

Ui

I— 4

X

x I, t /,fr

—

—j

o

4

/. .1

+

+

STARTING SETUP

+

çC +

R

RIO

A

I

/////‘/7.’I

0

F

-I-s

35

50

CLOSED END AT X~O

x

I 0

I

I

I

100 200 TIME (COMPUTER UNITS)

____I

700

Fig. 11. Computed growth rates for different ring positions. System as defined by the insert. Only the last ring is (asymptotically) completed; earlier rings are “stochastically terminated” by subsequent depositions.

H.K Henisch, f.M. Garcia-Ruiz

/

Growth in gels and Liesegang Ringformation. II

time between precipitation of the ring under review and appearance of the next ring, For a case (L 50, K~ 0.05) and configuration as shown, fig. 10 shows the accumulated mass as a function of ring position. The results are in harmony with common observations, according to which the largest crystal is usually near the bottom of a closed growth tube. That crystal tends also to be the best, because its growth rate is lowest; fig. 11 shows this. Since precipitation as such occurs everywhere under very similar conditions, the initial growth rates are also very similar, but average and final growth rates can vary a great deal. For earlier rings, the growth appears here to be terminated while the growth rate is still high, but this is merely a peculiarity of the present algorithm, which (a) neglects re-solution and (b) “abandons” each ring as soon as the next ring is formed. A more sophisticated simulation can take account of the fact that precipitates tend to re-dissolve when the actual (local concentration) product falls below K5. Secondary growths as a result =

=

of these shifting conditions, as well as the apparent movement of existing rings. Similarly, for an exploration of ring thickness, as distinct from total mass, it would be necessary to take the shape of the probability relationship of fig. 2b into account, providing for a gradual onset of ring growth. It is also recognized that the rate at which the actual solubility product and supersaturation ratio changes with time has an important effect on the morphology of the precipitate. Later papers will deal with these matters.

6. Summary It is shown that, contrary to frequently made assumptions, a concentration product minimum (K~)is not an adequate criterion for the formation of precipitates in gels. Though the K~product remains relevant and important, a second concept, the “equality range”, has to be introduced to account for observations, and the relationship of this concept to modern nucleation theory is outlined. On the basis of a numerical method described

211

in a previous paper (Part I), and a few simple assumptions about growth, it is here shown how the positions of the first and subsequent precipitates can be predicted for a great variety of conditions, boundary concentrations and system configurations. In this way, end effects can be documented, and the consequences of reagent depletion explored. Mass distributions over a ring system are computed, as well as the growth rates of stochastically terminated as well as (asymptotically) completed deposits. The present microcomputer analysis is only the beginning of a wider study, but it is quite sufficient to reveal the inter-related complexities of gel growth systems and, by the same token, more than sufficient to discourage attempts at any algebraic description of the phenomena.

Acknowledgment One of the authors (J.M. G.-R.) wishes to thank the NATO Spanish Committee for the financial support to visit the Materials Research Laboratory of the Pennsylvania State University.

References [1] H.K Henisch and J.M. Garcia-Ruiz, J. Crystal Growth 75 (1986) 195. [2] H.W. Morse and G.W. Pierce, Z. Physik. Chem. 45 (1983) 589. [31C. Wagner, J. Colloid. Sci. 5 (1950) 85. [4] G. Venzl and J. Ross, J. Chem. Phys. 77 (1982) 1302. [5] P. Ortoleva, in: Chemical instabilities, NATO—ASI Series, Series L, Vol. 120, Eds. G. Nicolis and F. Baras (Reidel, Dordrecht, 1984). [6] J.M. GarcIa-Ruiz and F. Miguez, Estudios Geol. 38 (1982) 3. [7] H.K. Henisch, Crystal Growth in Gels (Pennsylvania State University Pres, University Park, PA, 1970). [8] AC. Zettlemoyer, Ed., Nucleation (Dekker, New York, 1969). [9] G.K. Kirov, Compt. Rend. Acad. Bulg. Sci. 30 (1977) 559. [10] J. Salvinien and J.J. Moreau, J. Chim. Physique 57 (1960) [11] 518. E. Lendvay, Acta Phys. Hung. 17 (1964) 315. [12] N. Kanniah, Revert and Direct Liesegang Phenomenon, Doctoral Thesis in Chemistry, Anna University, Madras, India (April 1983).