Chapter 3

The Moving Boundary Model: A New Approach

The mechanism of rhythmic pattern formu tion in reaction-diffusion systems

is investigated theoretically by in trorlucing a new concept. The bo~~ndnry

that sepur~1te.s the rerlcting species virtiia/ly migrates as the diff~/sinn

y roceeds into the gelatina~rs medium. Based on this boundury nziy rotion

scenrrrio, crll the well-established empirical relations on Liesegong pulterns

could be proved, in a rather modified way. The ideu of formation of

irrtermediate colloidal haze prior to patterning ulnng with the moving

boundury model proved to he ef Jicient in y redicting the concentration

dependence of the width of the spatiotemporal patterns.

3.1 Introduction

Many investigators have developed several competing theories for

explaining the mechanism of Liesegang phenomenon [I-211. Most of the

theories perform well in deriving the time law and spacing law but are not

equivalent when applied to derive width law and Matalon-Packter relation.

The problem with almost all the said theories is that the mechanisms

involved in the explanation are too detailed and complicated and an

exhaustive study is cumbersome due to a large number of parameters

involved.

The present mathematical formulation aims at describing the

mechanism of band formation in a bit simpler way without employing

56 Chapter 3

unrealistic terms but at the same time keeping the true spirit of recurrent

precipitation.

3.2 The moving boundary model

In all the theoretical approaches and studies proposed [l-211 the

interface of the reagents is considered to be stationary. In this new model,

the formation of the periodic band systcms is treated as a moving boundary

problem. Accordingly i t is assumed that the boundary which separates the

outer ions and the inner electrolyte virtually migrate into the positive

direction of the advancement of the A type ions. The important features of

the Liesegang reaction diffusion mechanism will not be lost by making the

following simplifying approximations:

1. The initial concentration of the outer electrolyte CAo is assumed to

be much larger than the initial concentration Cm of the inner

electrolyte and that Ca ( x = 0, t ) is kept fixed at CAO. This assumption

is valid as in typical Liesegang experiments the patterns are

deterministic only when 0.005 5 CBO I CAO 5 0.1 [ 191.

2. Initially, i.e., at t = 0 the boundary which separates the outer and

inner electrolytes (gel-solution interface) is located at the (x = 0, y, z)

plane. Also the B type ions are assumed to be uniformly distributed

inside the gel medium and all the A type ions are on the negative side

of the boundary. In terms of the ion concentrations this means that

The Moving Boundary Model: A New Approach 57

- " C, (x, t =* ax x<o,~=o

With these initial conditions the system becomes effectively one

dimensional, since the concentrations depend only on x at all time.

Hence the experimental processes can be described by one-

dimensional reaction-diffusion equations.

3. When the first precipitation band was formed, it is assumed that the

concentration level of A type ions reaches the reservoir

concentration Cao up to the band position. This implies that the

boundary of A type ions has shifted to the region of the precipitation

front. This assumption holds good as the reservoir CAO of the A type

ions is sufficiently large compared to the initial concentration Ceo of

the B type ions. This process will continue and consequently the

boundary shifts from one band to the other. The concentration level

of A species after a bmd is formed at x, are:

where n denotes the band number. The distances xl, XZ, ... are the

positions of the first, second and subsequent bands.

4. Till the boundary advances to a new band position, it is assumed

that a steady state condition is established within the region.

Statistical fluctuations and thermal instabilities within this range are

taken to be minimum. The collective motion of the particles from

one band position to the other is more or less uniform and therefore

58 Chapter 3

it is noteworthy to assume that the boundary layer shifts from one

band position to the next with uniform speed v, (Figure 3.1).

With these initial conditions and assumptions, one can proceed further in

fixing a profile for the concentration.

Distance - Figure 3.1: Concentration distribution with moving boundary. The gel solution

interface is denoted by x = 0. The positions of the nth and the (n +I)' rings are x,(t) and x,+,(t) respectively.

To account for the quantity of isotopes diffusing into a medium

having a moving boundary, Lother Senf 122, 231 assumed a cubical

concentration profile for the isotopes. Peterlin 1241, while studying the

moving boundary problems, observed that the concentration within the first

medium declines because of the mass transfer across the boundary. Hence

according to him, the amplitude of the concentration profile of the diffusant

within the second media declines as a function of time. It seems that an

The Muving Boundary Model: A New Approach 59

exponential profile may be a generic form for the concentration profile to

describe the Liesegang phenomenon. For the present calculations an

exponential profile with const ant pre exponential factor and index

flexibility is chosen. This assumption is not much deviated from the actual

situation, since the initial concentration Cao of the outer electrolyte is much

larger than the initial concentration CRO of the inner electrolyte and Cao is

kept fixed up to the position of the band or ring according to the condition

given by equation 3.2.

When a new ring is established at x,,(t), the concentration profile of A

type ions in the gel medium is assumed to be:

where B (> 0) is regarded as a constant for a system, called the

concentration profile index and is the separation between the rth and

(n + I ) ' ~ bands or rings. The region between the nth and (n +I)" rings is th referred to as the Cn+, zone. For studying the formation of periodic

precipitation band at x,, we consider the diffusion of ions from the

immediate neighboring zones (&Lt'l and lh) on1 y.

For an infinitesimal boundary layer advancing in to the positive

x-direction, the equilibrium condition for the amount of diffusant

exchanged per unit area per unit time can be expressed as follows: the

amount of substance diffusing into the boundary layer augmented by the

amount of substance gathered by the advancement of the boundary layer is

equal to the amount of substance diffusing out. In mathematical terms:

60 Chapter 3

where DA is the coefficient of diffusion for the ionic species A and v is the

velocity with which the boundary layer shifts from one band position to

another. For A type ions, only unidirectional diffusion is considered and

hence

Thus the amount of substance diffusing in the positive x-direction follows

the concentration gradient of the system and the simplified balance

equation is:

-.

From equation 3.3

Substituting equations 3.3 and 3.7 in equation 3.6 and applying the above

boundary conditions

This is a significant relation, which connects the boundary migration

velocity v with the ring separation <. Since the effective diffusion

coefficient of A type ions DA in the gel is a constant, one easily finds

The Moving Boundary Model: A New Approach 61

which characterizes the nature of the boundary migration. If r,+l is the time

taken by the boundary to travel from the nth band position to the (n +1)lh

band position, the velocity of migration,

Making the substitution for the boundary migration velocity in equation

3.9, one sets

2

'n+1 = constant %+1

For the boundary migration in the tnth zone, the above relation can be

written as

This is the modified time law. In all the existing theories, the

distances were ~neasured from the gel-solution interface. The concentration

of the outer ions gradually builds up in the gel column and attains a

maximum value Cao up to the band position and hence it may not be proper

to measure the diffusion length from the gel-solution interface (x,), once a

band is formed. The formation of the band is enough to conclude that the

boundary of A type ions has been advanced into the gel medium up to the

band position. Also the theoretical analysis is based on Brownian motion

where we consider the 'random walk' at the molecular level and the

transitions are between 'closely neighboring states'. This implies that the

distance measurement cannot be done from the initial interface if one wants

to assume Einstein's solution to the problem. Hence 5. is a better choice of

distance than x, and hence the modified time law (equation 3.12) is more

meaningful.

From equation 3.9 it is evident that

As the boundary layer is assutncd to shift from one band position to

the next with uniform velocity, the ratio v , / V , ~ + I can be taken as a constant.

The assumption that the velocity is constant in each zone is trivial and

requires more meaningful predictions. For the moving front, the gel column

is regarded as homogeneous. However, the physical situation encountered

by the diffusing species in each zone may be different. But the constant

velocity approach in a particular zone is safer at this initial stage and which

will lead to simpli fied formulations.

Writing v, I V , ~ + ] = (1 + p3,

This is the modified spacing law and ( I+ p' ) is the new spacing

constant. For regularly spaced patterns one can write,

Substituting in equation 3.15

The Moving Boundary Model: A New Approach 63

This relation seems to be in a better position when compared to the

width law kv, - x,. It is to be noted that the At, is not exactly the width w,

of the nth bend. It is clearly evident from equations 3.15 and 3.17 that the

bands become more and more separated as it moves away from the gel

solution interface.

All the above modified relations picturize the fact that the

precipitation pattern front obeys the characteristic equation for boundary

migration.

3.3 The new model and the width law

Until now, much attention has been paid to the spacing law. On the

contrary, the width law has largely been ignored for several reasons. From

the experimental point of view, it is difficult to make precise measurements

of the width of the bands since the usually developed bands are not sharp

and well defined. From a theoretical point of view, a complete description

of the dynamics of band formation requires the knowledge of the detailed

mechanisms involved in the coarsening process. Accordingly, two different

predictions for the width law have been previously obtained [13,18]. The

purpose of this section is to focus on the width law aspect of the Liesegang

phenomenon. A theoretical argument, independent of the nature of the

detailed mechanisms involved in the band formation, is developed to

support the experimental findings.

The substance precipitated in the gel matrix is in the colloidal form

at its initial stage. The rate of production of the colloid species is supposed

to be proportional to the local concentration product of the reactants. The

mathematical analysis of simplified models based on the intermediate

compound theory has indicated that a spatially homogeneous colloid is

unstable against perturbations [25]. Large particle flux into this

homogeneous colloid, from outside generates spatial instabilities. Such

instabilities can remove the stability of the system and its kinetics. These

perturbations will initiate a phase separation process, which may eventually

lead into band formation.

The dynamics of phase separation and the formation of bands can

be explained as follows: As the outer electrolyte A reacts with the inner

electrolyte B, an intermediate compound Q of constant concentration is

formed. Antal et al. 1261 has proposed that a moving diffusion reaction

front will be developed as a result of the A+B-Q reaction-diffusion

processes. Once the unstable state is reached, phase separation takes place

on a short time scale resulting in the formation of a Liesegang band. This

band may act as a sink for the nearby particles and hence in the vicinity of

the band, the local concentration of the particles depletes. Thus the front

comes out of the unstable phase. When the front moves far enough, the

depleting effect of the band diminishes. The concentration of the particle

grows and again the system falls down into the unstable phase resulting in

the appearance of the next band. A repetition of this process should lead to

the formation of regular Liesegang pattern.

Even after the redistribution and precipitation of the initially

homogeneous sol, a low density of precipitates exists between the bands [13].

The alternating high and low density regions of the precipitate (Fig. 1.13)

appear to be frozen spatiotemporally or said to be static on the time scale.

The visual appearance of the precipitate patterns can also be taken as

The Moving Boundary Model: A New Approach 65

evidence of phase separation. According to the moving boundary model,

the formation of the nth band implies that the boundary of the A type ions

has been shifted to that position and that the concentration of A particles

reaches the reservoir concentration CAO up to that position. It is to be noted

that initially the B particles are uniformly present inside the gel medium

before the diffusion process. Thus the different processes, viz., the

diffusion of A particles, formation of colloidal dispersion, the coagulation

and formation of precipitation band, etc., in one zone are not influenced by

the neighboring zones. Hence it can be assumed that the intermediate

species Q fulfills the mass conservation law in a zone during the above

processes.

It is assumed that the phase separation mechanism redistributes the

homogeneous colloidal haze Q of uniform initial concentration co into a

band having a concentration cb and a gap having a concentration c,. As the

quantity of Q particles in the zone will be conserved, one can easily write

Thus the width of the nth band is

The concentration function or width coefficient T(c) = [(c , - c , ) / ( c , - c , )] will

be more or less constant for a steady pattern and hence

This seems to be a good approximation than the empirical relation

w,, - x:, where the concentration dependence is assigned only for the

66 Chapter 3

exponent p. The width of the precipitation bands, according to the present

theory, depends excIusively on the concentration function T(c) of the

intermediate colloidal particles. Matalon and Packter [7] established the

exclusive dependence of concentration of reacting components on spacing

coefficient. The moving boundary model in fact successfully links both the

spacing coefficient and the width coefficient with the concentration of the

respective entities.

The width of the (n + I ) ' ~ band is given by

From equations (3.2 I), (3.22) and (3.15) one can safely conclude that

Experimental studies and verifications of the reformulated laws are

discussed in the following chapter.

3.4 Conclusion

The formation of the ring is enough to conclude that the

boundary of the diffusant has advanced into the gel medium up to the

ring position. Based on the moving boundary scenario, the well-

established time law and spacing law were proved, in a rather modified

way. The less understood mechanism for the variations in the width is

also discussed in a more efficacious manner and the law related to the

width dependence is established. It is suggested that the intermediate

colloidal formation and the phase separation mechanism leads to the

formation of the bands in the gel medium.

The Moving Boundary Model: A New Approach 67

The moving boundary model is safer and simpler in many respects.

It delineates scenery with distinctive assumptions and boundary conditions.

A slight twist from the conventional approach provides safer place to drive

at Einstein's solution for diffusion equation. Thus a better understanding of

the basic facts of pattern forming processes and the geometrical positioning

of the bands is made possible with the new model.

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