Formation of periodic precipitation patterns: A moving boundary problem

Article abstract of DOI:10.1016/S0009-2614(02)01026-6

A new scenario for the formation of Liesegang patterns is proposed. The periodic precipitation pattern formation in a gel column is interpreted as a moving boundary problem. The existing time law, space law, and width law are revisited and reformulated on the basis of a moving boundary assumption and more meaningful explanations are given. All the new equations suggested were found to be in good agreement with experimental observations.

Full text of DOI:10.1016/S0009-2614(02)01026-6

13 August 2002
Chemical Physics Letters 362 (2002) 8–12
www.elsevier.com/locate/cplett
Formation of periodic precipitation patterns:
a moving boundary problem
a
Jacob George , George Varghese
b,
*
a
Crystal Physics Division, Christian College, Chengannur, Kerala 689122, India
Crystal Physics Centre, St Berchmans’ College, Chenganacherry, Kerala 686101, India
b
Received 30 January 2002
Abstract
Anew scenario for the formation of Liesegang patterns is proposed. The periodic precipitation pattern formation in
a gel column is interpreted as a moving boundary problem. The existing time law, space law, and width law are revisited
and reformulated on the basis of a moving boundary assumption and more meaningful explanations are given. All the
new equations suggested were found to be in good agreement with experimental observations. Ó 2002 Elsevier Science
B.V. All rights reserved.
1. Introduction
tures, possibly because of the variety and com-
plexity of the observed phenomena. The
mechanisms responsible for these structures are
still under discussion [2–6].
The formation of rhythmic precipitation pat-
terns in physical as well as biological systems is a
subject of extensive research ever since its re-
markable discovery by R.E. Liesegang in 1896 [1].
When anions diffuse into a gel medium impreg-
nated with cations, the possible reaction product
segregates into concentric rings or bands depend-
ing upon the geometry of the system. The phe-
nomenon has since been examined by a large
number of investigators and many theories have
been suggested to explain the rhythmic structure.
So far, no single theory of periodic precipitation
seems to be able to account for all observed fea-
In a simple ring system, as one observed in a gel
column inside a test tube, the species of Atype
ions diffuse from the outer electrolyte into the gel,
having almost uniformly distributed B type ions,
and the two co-precipitates react chemically re-
sulting in the product C inside the gel medium.
Precipitation occurs when the local product of the
ion concentrations CACB exceeds some threshold
value Ksp, if the reaction is of the form A þ B ! C
[7]. Once the precipitation front is formed, the
concentration of Atype ions reaches its maximum
value – the reservoir concentration CA0 – up to the
region of the precipitation front. This assumption
holds well provided the reservoir concentration
CA0 of the Aions is sufficiently high compared
with the concentration CB0 of the B type ions. The
* Corresponding author. Fax: +91-481-411472.
E-mail addresses: jgeo@rediffmail.com (J. George), gvar@
sify.com (G. Varghese).
0009-2614/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved.
PII: S0009-2614(02)01026-6

J. George, G. Varghese / Chemical Physics Letters 362 (2002) 8–12
9
Atype ions further penetrate into the gel medium
and a new precipitation zone is formed at a later
time, when the concentration condition is further
achieved. As the diffusion advances into the gel,
consecutive bands of precipitate are observed in
the tube. Three laws, characterizing the Liesegang
phenomenon were framed by phenomenological
investigators. The first one is the so-called time law
given by Morse and Pierce [8] which states that
x2n ꢀ tn where xn is the position of the nth band
measured from the interface of the reagents and tn
is the time of its formation. This result is analo-
gous to the well known Einstein–Smoluchowski
relation for Brownian motion interpreted in terms
of random walk in a homogeneous space [9]. Sec-
ondly, the positions of the rings follow a geometric
progression, xnþ1=xn ! ð1 þ pÞ for large enough n,
known as Jablczynski law [10] or spacing law,
where ð1 þ pÞ being known as the spacing coeffi-
cient. Later it has been found that the spacing
coefficient is a non-universal quantity depending,
among other parameters, on the experimentally
controllable initial electrolyte concentrations CA0
and CB0. This dependence was expressed further by
Matalon and Packter [11] as
formation of periodic ring systems is treated as a
moving boundary problem. Accordingly we shall
assume that the boundary which separates the
outer ions and inner electrolyte virtually migrates
into the positive direction of the advancement of
the Atype ions. Initially, the boundary which
separates Aand B ions was the gel solution in-
terface. When the first precipitation zone (ring)
was formed, Atype ions occupy up to this region
and its concentration gradually reaches the reser-
voir concentration CA0. This implies that the
boundary of Atype ions has been shifted to the
region of the precipitation front. This will repeat in
time and the boundary region moves from one ring
to the other. For a one dimensional system the
concentration levels of Aspecies at the boundary
are
CAðx; tÞ ¼ CA0
;
at t 6 tn and x0 6 x 6 xn;
ð1Þ
where n denotes the number of the ring, which is a
positive integer. The initial value x0 corresponds to
the gel solution interface and x1, x2, etc. are the
positions of the first, second, etc. rings.
Till the boundary advances to a new ring po-
sition, a steady state condition is assumed to be
established within the region. Statistical fluctua-
tions and thermal instabilities within this range are
minimum. The collective motion of the precipit-
ants from one ring to the other is more or less
uniform and therefore it is noteworthy to assume
GðCB0Þ
p ¼ F ðCB0Þ þ
;
CA0
where F and G are decreasing functions of their
argument CB0. Finally the width wn of the band has
been observed to increase with n and obey a simple
linear relation wn ꢀ xn [12]. Based on detailed ex-
perimental observations with a large number of
bands, a more accurate behavior was framed by
Chopard et al. [13] and expressed by a new law,
wn ꢀ xan, where the constant parameter a depends
only on CA0 and CB0 whose value ranges from 0.49
to 0.61. Later Droz et al. [14] combining scaling
properties of the density of precipitate in the
bands, found that a ranges from 0.90 to 0.99.
2. The moving boundary model
It is important to note that in all the theoretical
approaches proposed [2,3,5–8,10–23], the interface
of the reagents is considered to be stationary. In
this letter, we report a new scenario where the
Fig. 1. Concentration distribution with moving boundary. The
gel solution interface is denoted by x0. The positions of the nth
and (n þ 1)th rings are xn and xnþ1, respectively.

10
J. George, G. Varghese / Chemical Physics Letters 362 (2002) 8–12
that the boundary layer shifts from one ring po-
sition to the next with uniform velocity vn.
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 sub-
stance diffusing out. In mathematical terms
The concentration profile of the Atype ions
within the gel medium will establish an exponential
distribution (Fig. 1). Peterlin [24], while studying
moving boundary problems, observed that the
amplitude of the concentration profile may also
decline as a function of time. However, at least for
the present calculations, we assume a profile with
constant pre-exponential factor. This is, in fact not
a realistic picture of the problem. However, this
assumption is not much deviated from the actual
situation, since the reservoir concentration CA0 of
the Atype ions is sufficiently high compared with
the initial concentration CB0 of the B type ions. It
has been known since the earliest experiments of
Liesegang that optimum results for ring or band
formation are obtained when the concentration of
the outer electrolyte is much higher, preferably by
several orders of magnitude than that of the inner
electrolyte. In regular Liesegang experiments one
typically has 0:005 6 CB0=CA0 6 0:1 [2]. In a similar
situation, to account for the quantity of isotopes
diffusing into a medium having a moving bound-
ary, Lothar Senf [25] assumed a cubical concen-
tration profile. According to the authors an
exponential profile with index flexibility is found
to be more appropriate to describe the Liesegang
phenomenon.
DAoxCAðx ; tÞ þ vCAðx ; tÞ ¼ DAoxCAðx ; tÞ:
ð3Þ
þ
þ
À
The boundary migration velocity is denoted as v.
Here only unidirectional diffusion is considered
and hence the amount of substance diffusing in the
positive x-direction follows the concentration
gradient of the system and the simplified balance
equation is
DAoxCAðx ; tÞ þ vCAðx ; tÞ ¼ 0:
ð4Þ
Substitution of (2) in (4) and applying the above
þ
þ
boundary condition, we get
nnþ1 ¼ bDA=vnþ1
:
ð5Þ
This is a significant relation which connects the
boundary migration velocity v with the ring sepa-
ration n. Since the effective diffusion coefficient of
Atype ions DA in the gel is a constant, one easily
finds
vnþ1nnþ1 ¼ constant;
ð6Þ
which characterizes the nature of the boundary
migration. If the boundary traverses inside a zone
of length n within a time s, the velocity of migra-
tion
After a new ring is established at xnðtÞ, the
concentration profile of Atype ions at any point
between nth and (n þ 1)th ring is assumed to be
v ¼ n=s:
Making a substitution for velocity in Eq. (5) one
ð7Þ
ꢀ
ꢁ
gets
½x À xnðtފ
CAðx;tÞ ¼ CA0 exp À b
; xn 6x6xnþ1
;
n2 ꢀ sn:
ð8Þ
nnþ1
n
ð2Þ
This is in fact a better relation than the time law
x2n ꢀ tn. In all the existing theories, the distances
were measured from the gel solution interface. The
concentration of the outer ions builds up in the gel
column and attains the maximum value, CA0 up to
the ring position and hence it may not be proper to
measure the distance from the gel solution inter-
face after the formation of a ring. The formation
of a ring is enough to conclude that the boundary
of Atype ions has been advanced into the gel
medium up to the ring position. This implies that
the distance measurement cannot be done from the
initial interface, if one really wants to assume
where bð> 0Þ is regarded as a constant for a sys-
tem, called the concentration profile index and nnþ1
is the separation between the nth and (n þ 1)th
rings. The region between the nth and (n þ 1)th
rings is referred to as the nnþ1th zone. In studying
the formation of precipitation band at xn, we
consider the diffusion of ions from the immediate
neighboring zones only (nnth and nnþ1th zones).
For an infinitesimal boundary layer advancing
into the positive x-direction, the equilibrium con-
dition for the amount of diffusant exchanged per
unit area per unit time can be expressed as follows:

J. George, G. Varghese / Chemical Physics Letters 362 (2002) 8–12
11
Einstein’s solution to the problem. Hence nn is a
better choice of distance than xn and hence the
modified relation (8) is more meaningful. This
conclusion leads to two more relations.
As the boundary layer shifts from one ring
position to the next with uniform velocity, from
Eq. (6) it is also evident that
nnþ1=nn ¼ ð1 þ p0Þ;
ð9Þ
which is the modified spacing law and ð1 þ p0Þ is
the new spacing constant. Writing nnþ1 ¼ nn þ Dnn
in Eq. (9) gives
Dnn ꢀ nn:
ð10Þ
Even though Eq. (10) is not necessarily related to
the width law wn ꢀ xn, the result obtained using
this seems to be better. It is clearly evident from
Eqs. (9) and (10) that the rings become more and
more separated as it is away from the gel-solution
interface. All the above modified relations pictur-
ize the fact that the precipitation pattern front
obeys the characteristic equation for boundary
migration.
Fig. 2. Verification of the modified time law with Liesegang
rings of BaMoO4. Experimental details: silica gel of density 1.03
g cmÀ3, pH ꢀ 6, outer electrolyte BaðNO3Þ – 0.25 M, inner
2
electrolyte ðNH4Þ MoO4 – 0.1 M.
2
3. Experimental
The penetration of BaðNO3Þ into silica gel
2
containing ðNH4ÞMoO4 results in a sharp bound-
ary interface of BaMoO4 precipitate bands. In all
the experiments we have obtained more than 15
bands. The time of formation tn and distance of
each ring from the gel solution interface xn were
recorded. The data were analyzed and the results
are summarized in Figs. 2–4. The relatively large
number of bands allowed a rather accurate deter-
mination of the constants involved in the modified
relations (8)–(10).
Fig. 2 depicts the plot of n2n against sn. The
straight line shows the validity of the modified
time law. To determine the value of the new
spacing coefficient, we plotted nnþ1 against nn (Fig.
3). Note that the new spacing coefficient remains
constant throughout the entire range of the band
system as indicated by the slope and the value is
found to be 1.077. The linear relationship Dnn ꢀ nn
is also verified (Fig. 4) and the value of p0 so ob-
tained is 0.0715. Slight deviations observed were
Fig. 3. Determination of the new spacing coefficient ð1 þ p0Þ
given by the slope of the curve nnþ1 versus nn. The value of
ð1 þ p0Þ is found to be 1.0704.
due to the experimental error in the judicious
measurement of ring positions and the time of ring
formation. Within the limits of error the results
suggest that a virtual migration of boundary takes
place from one ring to the other.
4. Conclusions
To study the Liesegang pattern formation in
gelatinous media, we have developed a one di-
mensional model based on the moving boundary

12
J. George, G. Varghese / Chemical Physics Letters 362 (2002) 8–12
Acknowledgements
One of the authors (J.G.) is thankful to the
University Grants Commission, Government of
India for a fellowship.
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