A model of the formation of liesegang rings under stimulated precipitation conditions

Article abstract of DOI:10.1134/S0036024406050098

A mathematical apparatus for describing the kinetics of chemical reactions accompanied by phase transitions is suggested. The phenomenon of stimulated precipitation is singled out. Together with spontaneous precipitation-solution, it allows periodic sedimentation (Liesegang rings) to be correctly described and the concept of solubility product to be corrected and generalized. Pleiades Publishing, Inc., 2006.

Full text of DOI:10.1134/S0036024406050098

ISSN 0036-0244, Russian Journal of Physical Chemistry, 2006, Vol. 80, No. 5, pp. 714–725. © Pleiades Publishing, Inc., 2006.
Original Russian Text © G.A. Skorobogatov, A.V. Kamenskii, 2006, published in Zhurnal Fizicheskoi Khimii, 2006, Vol. 80, No. 5, pp. 826–838.
CHEMICAL KINETICS
AND CATALYSIS
A Model of the Formation of Liesegang Rings
under Stimulated Precipitation Conditions
G. A. Skorobogatov and A. V. Kamenskii
St. Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 199164 Russia
E-mail: 2Gera.Skor@pobox.spbu.ru
Received May 30, 2005
Abstract—A mathematical apparatus for describing the kinetics of chemical reactions accompanied by phase
transitions is suggested. The phenomenon of stimulated precipitation is singled out. Together with spontaneous
precipitation-solution, it allows periodic sedimentation (Liesegang rings) to be correctly described and the con-
cept of solubility product to be corrected and generalized.
DOI: 10.1134/S0036024406050098
Currently, chemical kinetics is a comprehensive sci-
ence of relaxation atom-molecular processes in the
region of large (nonlinear) deviations from thermody-
namic equilibrium [1]. It has an adequate mathematical
apparatus for describing chemical processes in both the
gas phase [2–4] and condensed media [3, 4]. Statistical
physics and quantum mechanics methods are used to
calculate absolute rate constants for elementary chemi- defined Liesegang rings are observed in the reactions
cal events [5, 6]. Against this background, the science
of chemical processes accompanied by phase transi-
tions disappointingly lags behind. For instance, a sin-
gle page is given to such processes in monograph [4],
Periodic Sedimentation
More than 100 years ago, periodic sedimentation
was discovered by Liesegang in the reaction
2
7
–
+
2
Ag + Cr O
Ag Cr O ↓.
(2)
2
2
2
7
Many dozens of reactions that exhibit spatial periodic-
ity have been discovered since [8]. For instance, well-
2
KI + Pb(NO3)2
PbI ↓ + 2KNO ,
(3)
2
3
2
NH OH + MgCl
Mg(OH) ↓ + 2NH Cl. (4)
2 4
4
2
The “supersaturation mechanism” by Ostwald [8]
where their poor reproducibility is noted and attrib- has long been considered responsible for periodic pro-
cesses of types (2)–(4). This opinion was seemingly
substantiated by successful mathematical modeling [9–
4] of periodic sedimentation by the supersaturation
mechanism. It was, however, found in [15] that works
11–14] contained mathematical errors. We can add
what was unnoticed by the author of [15]: in [12–14]
and earlier models [9, 10], the same inadmissible math-
uted to a strong but uncontrollable influence of micro-
impurities on the formation and growth of reaction
centers. We even lack a mathematical apparatus for
describing the time behavior of a component that
simultaneously experiences chemical and phase trans-
formations. It is true that the degree of precipitation
1
[
(
or solubility) of some salt M Xm in an aqueous ematical procedure was applied. Namely, these authors
n
medium at equilibrium is quantitatively characterized threw off derivatives with respect to time in the corre-
sponding kinetic equations. This transformed diffusive
parabolic-type equations, which had no oscillatory
(1) solutions, into hyperbolic-type wave equations with a
wide set of oscillatory solutions.
by the solubility product (SP) [7],
m+ n
n– m
[
M ] [X ] = SP(M X ),
n m
but this concept is internally contradictory. For
The quantum-mechanical “explanation” [8, 16, 17]
instance, how do the M and X ions “know” that of Liesegang rings as a visualization of de Broglie
m+
n–
somewhere at the bottom of a vessel there is an M X
waves cannot be considered serious. Of course, serious
attempts at solving the problem were also made [18–
n
m
precipitate ensuring solution saturation? Or how much
M X salt should lie on the bottom to ensure aqueous
2
0]. Of these, the “postnucleation” model of structuring
n
m
of colloids during their aging [21–24] was most suc-
cessful. But the postnucleation model was obviously
mechanistic and could not pass the very simple test for
quantitative calculations of periodic structure depen-
solution saturation at which the solubility product value
is determined? For instance, is it enough to have two or
three M X molecules at the bottom for (1) to be ful-
n
m
filled in the volume? We begin our consideration with dences on either component concentrations or the type
periodic sedimentation principles (Liesegang rings).
of reacting components. We are led to conclude that the
7
14

A MODEL OF THE FORMATION OF LIESEGANG RINGS
715
mechanism of formation of Liesegang rings is still the necessary condition for the appearance of concen-
unknown.
At the same time, the following direct theorem was
proved [25] in the theory of homogeneous oscillatory
tration oscillations in a spatially homogeneous cyclic
kinetic scheme is n ≥ 3.
Every stage of overall process (6) looks like an ele-
reactions (of the Belousov–Zhabotinskii type): any pro- mentary reaction of gas-phase kinetics. In the liquid
cess periodic in time can be transformed into a regime phase, even the first reaction of scheme (6) is, however,
of spatial periodicity. This result was repeatedly repro- a complex process of solvated ions, and the third and
duced in numerical experiments [26–30]. In [31], the fifth stages are heterogeneous processes of nanoparti-
hypothesis of the validity of the inverse theorem was cles comprising thousands and millions of molecules.
put forward: any spatially periodic process is based on Separate stages of overall process (6) will therefore be
a time-periodic process. It follows that, provided a called stoichiometrically elementary reactions. This is
kinetic scheme with time-oscillatory solutions is con- logical, because every stoichiometrically elementary
structed for reactions of type (2), (3), such a scheme reaction of scheme (6) corresponds to a kinetic differ-
will ensure spatially periodic sedimentation by virtue ential equation term determined by the law of mass
of the direct theorem if a spatial gradient of some com- action for the truly elementary reaction of the same sto-
ponent is created.
ichiometry.
Periodic sedimentation is observed either in media
We will restrict our consideration to reactions of
type (3), (4) described by the common stoichiometric containing thickening agents (gelatin or agar–agar) or
equation
in thin capillaries, that is, under the conditions that pre-
vent the removal of the MX ↓ precipitate from the reac-
2
KX + MY2
MX ↓ + 2KY.
(5)
2
2
tion medium. For a fairly small volume (when diffusion
limitations on component supply and consumption can
be ignored), we therefore have a focussed system (we
use the terminology of [8, 25]) with kinetic scheme (6),
which can be put in correspondence with the kinetic
differential equations
For (5), we assume the following minimum cyclic
kinetic scheme (åï is the so called petergofator):
2
2
+
–
k1
+
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
M + X
MX
k–1
k2
+
–
MX + X
MX2
k–2
–
+
–
2+
d[X ]/dt = k [MX ] – k [X ][M ] + k [MX ]
k'
–1
1
–2
↓][X–]
[MX
2
2
s
MX + (A) k –' MX ↓ + (A)
2
2
–
3
s
–
+
k3
– k
2
[X ][MX ]+ k–6[MX
] – k
6
MX + MX ↓
2MX2↓
2
2
+
3
2
–
+
k [MX ] – k [M X ][X ],
2
+
k4
k–4
+
–8
2
8
2
MX ↓ + M
2MX
2
(6)
2+
2+
–λt
+
d[M ]/dt = [M ] λe + k [MX ]
∞
–1
2
+
k5
+
MX + M
2MX
–
2+
+ 2
2+
2
k–5
k6
– k [X ][M ] + k [MX ] – k [MX ↓][M ]
1
–4
4
2
–
–
3
+ 2
][M2+],
MX ↓ + X
MX
+ k–5[MX ] – k
5
[MX
2
2
k–6
k7
+
–
2+
+
+
+
3
d[MX ]/dt = k [X ][M ] – k [MX ]
1 –1
MX + MX
M X
2
2
k–7
k8
–
+
2+
+
k [MX ] – k [X ][MX ] + 2k [MX ↓][M ]
–2 2 2 4 2
+
3
–
M X + X
2MX2.
+ 2
2+
+ 2
2
k–8
– 2k [MX ] + 2k [MX ][M ] – 2k [MX ]
–4
5
2
–5
+
+
According to the terminology suggested by
Rusanov [32], MX denotes solid phase “nuclei,” and
+
k [M X ] – k [MX ][MX ],
3
–7 2 7 2
2
–
+
d[MX ]/dt = k [X ][MX ] – k [MX ]
(7)
2
2
–2
2
MX ↓, “mature” micellated solid phase particles. An
2
+ k–s[MX
↓] – k [MX
] + k–5[MX+]2
objective criterion for such a gradation is the observa-
2
s 2
tion that MX nuclei do not settle down in the gravita-
2+
2
– k [M ][MX ] – k [MX ↓][MX ]
5
2
3
2
2
tional field and are invisible to the naked eye, whereas
+
3
+ k–7[M
X
] – k
[MX
][MX+]
2
MX ↓ particles scatter light in the visible range and
2
7
2
experience sedimentation. This allows us to treat MX2
+
3
2
–
–
2k [MX ] + 2k [M X ][X ],
–
8
2
8
2
as a dissolved component and MX ↓ as a dispersed
2
crystalline solid phase. The fictitious component A cor-
d[MX
2
↓]/dt = k
s
[MX ] – k–s[MX ↓]
2 2
responds to spontaneous crystallization centers of the
+ 2
+
k [MX ↓][MX ] + k [MX ]
3
2
2
–4
] – k
–
dissolved MX component; that is, this is either density
2
–
3
2
+
↓][X–],
fluctuations or some microimpurities. Since the con-
centration [A] does not change during the overall pro-
– k
4
[MX
2
↓][M ] + k–6[MX
[MX
6 2
–
3
–
3
d[MX ]/dt = +k [MX ↓][X ] – k [MX ],
6
2
–6
cess, we can denote the product k' [A] as a new first-
s
+
3
+
+
order rate constant k and ignore component A in what
d[M X ]/dt = k [MX ][MX ] – k [M X ]
s
2
7
2
–7
2
3
follows. Scheme (6) is cyclic, it contains positive feed-
+
3
–
]2
back (k > 0), and it has n = 4 units. According to [33],
– k
8
[M
2
X
][X ] + k–8[MX
2
3
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY Vol. 80 No. 5 2006

7
16
SKOROBOGATOV, KAMENSKII
(
‡)
(d)
1
0
0
0
0
.0
.8
.6
.4
.2
1.0
0.8
X (a), M2+(b)
–
MX (y)
2
+
MX (c)
MX2(x)
MX (z)
3
0.6
0.4
0.2
b + c + x + y + z + 24
1
2
3
4
0.2
0.4
0.6
0.8
1.0
1
0
0
0
0
.0
.8
.6
.4
.2
1.0
0.8
0.6
0.4
0.2
(
b)
(e)
0
.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
1
0
0
0
0
.0
.8
.6
.4
.2
1.0
0.8
0.6
0.4
0.2
(
c)
(f)
0
.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
–
Fig. 1. Dimensionless time (τ = λt) dependences of relative concentrations (N = [N]/[X ] ) in kinetic scheme (6) obtained by solving
0
5
4
Cauchy problem (10), (11) with the following dimensionless parameter (9) values: m = 1, K = 10 , K = 5, K = 10 , K = 1, K =
1
s
3
–4
6
5
1
00, K = 2000, K = 10 , K = K = K = K = K = K = K = K = K = 0. (a)–(f) The K constant was varied, the best
7 8 –1 –2 –s 4 5 –5 –6 –7 –8 2
oscillations were obtained at 0.01 ≤ K ≤ 0.1; k = (a) 0, (b) 0.00001, (c) 0.01, (d) 0.1, (e) 1, (f) 10. (g)–(l) The K constant was set
2
2
2
at 0.01, the rate constant K was varied, the best oscillations were obtained at K = 15000 ± 5000; k = (g) 1000, (h) 5000, (i) 10000,
3
3
3
(
j) 20000, (k) 50000, and (l) 100000.
+
–
–
with the initial conditions
c = [MX ]/[X ] , x = [MX ]/[X ] ,
0 2 0
2
+
+
–
–
3
–
[
M ]
0
= [MX ]
0
= [MX ] = [MX ↓ ]
y = [MX ↓ ]/[X ] , z = [MX ]/[X ] ,
2
0
2
0
2
0
0
(8)
(
9)
–
+
3
–
+
3
–
2+
–
=
[MX ] = [M X ] = 0, [X ]
0
> 0 at t = 0.
u = [M X ]/[X ]
0
,
m = [M ]
∞
/[X ] ,
0
3
0
2
0
2
–
1
–
In (7), the constant λ (in s units) characterizes the K = (k /λ)[X ] for i = 1, 2, 3, 4, –4, 5, –5, 6, 7, 8, –8,
i
i
0
2
+
rate at which the external component [M ] is supplied
to the system as its concentration increases from 0 to
K = k /λ for j = –1, –2, s, –s, –6, –7.
j
j
2
+
the limiting value [M ] (which is, however, only pos-
∞
Equations (7) and (8) take a more compact form in
terms of denotations (9),
–
sible if [X ] = 0). Let us introduce dimensionless time
0
τ = λt and dimensionless concentrations and rate con-
stants
da/dτ = K c – K ab + K x – K ac
–
1
1
–2
2
2
–
–
2+
–
a = [X ]/[X ] , b = [M ]/[X ] ,
+ K–6z – K
ay + K–8x – K au,
6 8
0
0
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY Vol. 80 No. 5 2006

A MODEL OF THE FORMATION OF LIESEGANG RINGS
717
1
0
0
0
0
.0
.8
.6
.4
.2
1.0
(
g)
(j)
0.8
0.6
0.4
0.2
0
.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
1
0
0
0
0
.0
.8
.6
.4
.2
1.0
0.8
0.6
0.4
0.2
(
h)
(k)
0
.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
1
0
0
0
0
.0
.8
.6
.4
.2
1.0
0.8
0.6
0.4
0.2
(
i)
(l)
0
.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
Fig. 1. (Contd.)
–
τ
dz/dτ = K ay – K z,
db/dτ = me + K c – K ab
6
–6
–
1
1
2
2
2
du/dτ = K cx – K u + K x – K au,
+
K c – K by + K c – K bx,
7
–7
–8
8
–
4
4
–5
5
where
dc/dτ = K ab – K c + K x – K ac
1
–1
–2
2
a = 1, b = c = x = y = z = u = 0 at τ = 0.
(11)
It is easy to check that differential equations (10)
have two integrals corresponding to the conservation of
the sum of both X and M,
2
2
+
2K by – 2K c + 2K bx – 2K c + K u – K cx,
4 –4 5 –5 –7 7
dx/dτ = K ac – K x + K y – K x – K xy
2
–2
–s
s
3
(10)
a + c + 2x + 2y + 3z + 3u = 1 for all τ ≥ 0,
–
τ
b + c + x + y + z + 2u = m(1 – e ) for all τ ≥ 0.
2
2
+
K c – K bx + K u – K cx + 2K au – 2K x ,
–5 5 –7 7 8 –8
Details of the numerical integration of (10) on a PC
were described in [34, 35]. It was found that differential
equations (10) with initial conditions (11) have a region
of rate constant values at which Cauchy problem (10),
(11) has oscillatory solutions (Figs. 1, 2; Table 1).
dy/dτ = K x – K y + K xy
s
–s
3
2
+
K c – K by + K z – K ay,
–4 4 –6 6
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY Vol. 80 No. 5 2006

7
18
SKOROBOGATOV, KAMENSKII
Table 1. Number of kinetic curve c(τ) and y(τ) maxima (N) as a function of dimensionless rate constant K in Cauchy prob-
i
lem (10), (11) at m = 1
K1
N
0
0
1
10
5
100
6
1000
100000 100000000
0
6
6
6
1000000
0
K–1
N
0
0.1
1
1000
6
10000
100000
6
6
6
6
1
10
1
K2
N
0
0.00001
0.01
0.1
1
0
5
6
6
4
K–2
N
0
1
100
500
1000
6
6
6
3
0
Ks
N
0
0.01
5
1
5
100
5
5
6
2
K–s
N
0
0.1
6
1
5
50
500
6
6
3
1
1
100000
3
K3
N
1000
1
5000
10000
6
20000
50000
4
9
7
K4
N
0
0.1
6
10
6
100
10000
1000000
6
10000000 100000000
6
6
7
6
6
K–4
N
0
1
10
5
50
200
6
6
2
0
K5
N
0
1
50
6
1000
100000
1000000
10000000 100000000
6
6
6
6
3
0
K–5
N
0
1
10
4
100
1000
6
5
1
1
K6
N
0
1
10
1
50
100
200
5
500
2
1000
1
1
1
5
6
K–6
N
0
0.1
6
5
50
100
500
0
6
3
2
1
K7
N
0
100
300
1
500
1000
2000
6
10000
3
100000
1
0
0
10
6
4
100000
4
5
K–7
N
0
1000
6
1000000
6
0
100000
6
K8
N
0
100
1
1000
4
10000
5
1000000
0
6
1000000
5
K–8
N
0
1
10
6
1000
6
100000
6
10000000 100000000
6
6
2
0
Note: Trial variation of every rate constant was performed with the other constants set at their optimum values, namely, K = 100000,
1
K = 0.01, K = 5, K = 10000, K = 1, K = 100, K = 2000, K = 100000 (the other constants were set at zero). The constants and
2
s
3
–4
6
7
8
their optimum values without which oscillatory solutions disappear are typeset in boldface.
Table 1 shows that, of all the reactions from scheme ates spatially periodic structures by virtue of the direct
(
6), only the reactions with the rate constants k , k , k , theorem.
1 2 s
k , k , k , and k are absolutely necessary for the
appearance of oscillatory regimes. Scheme (6) leads
to the appearance of oscillations in the focussed case.
For this reason, in the presence of a spatial gradient of extremely low solubility products [8] (10 for NiS,
one of the components, scheme (6) inevitably gener- 10 for Ag S, 10 for SnS , etc. [7]).
3
6
7
8
Note that overall scheme (6) presupposes that the
precipitation of salt MX ↓ is reversible. Indeed, peri-
2
odic sedimentation was never observed for salts with
–
28
–
49
–70
2
2
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY Vol. 80 No. 5 2006

A MODEL OF THE FORMATION OF LIESEGANG RINGS
719
0
0
0
0
.8
.6
.4
.2
(a)
I–
PbI2
PbI+
PbI2
Pb2+
PbI–
3
0
.05
0.10
0.15
0.20
0
.1 M I–
1 M Pb2+
1
.6 mm, 25 min
a
b
0
0
.5
.4
(b)
Pb2+
I–
0
0
0
.3
.2
.1
PbI+
PbI2
PbI2
0
.2
0.4
0.6
0.8
1.0
0
.014 å
0
.148 I–
2
+
Pb
2
7 mm, 96 h
‡
b
Fig. 2. Computer fitting of the topology and number of variable y oscillations of åï petergofator to the experimental PbI precip-
2
2
itate picture in process (3) in (a) water and (b) 1% aqueous agar–agar at 25°ë: (a) front of Pb2+ cations and (b) front of anions I–
transferred into the volume of (a) solution of KI and (b) solution of Pb(NO ) in aqueous agar–agar; b is the boundary between the
3
2
–
2.1 ± 0.1 –1
–
19.78
3
–4.9 ± 0.2 –1
s , [Pb2+]0 = 1018.93
solutions of KI and Pb(NO ) ; (a) λ = 10
s , [I ] = 10
particles/cm and (b) λ = 10
3
2
0
particles/cm3.
The fourth reaction in scheme (6) (stimulated pre- which obeys the detailed equilibrium principle. The
cipitation) occurs irreversibly in one direction as dis- removal of the fourth reaction from scheme (6) (that is,
tinct from the third reaction (spontaneous precipitation the assumption that k = 0) removes positive feedback
3
on microimpurities and other crystallization centers A), from (7) (or (10)), which results in the disappearance of
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY Vol. 80 No. 5 2006

7
20
SKOROBOGATOV, KAMENSKII
solutions oscillatory in time. Accordingly, spatially cally elementary reactions (6*) have rate constants of
periodic solutions also disappear.
the same order of magnitude as the other reactions in
water at 25°ë studied in [38],
Let us use the results obtained to extract informa-
tion about the rate constants for the stoichiometrically
elementary reactions of overall process (3). According
to Table 1, the following stoichiometrically elemen- Cu + é
tary reactions are then necessary for an oscillatory
regime:
+
+
0
2+
–20.3
3
Cu + Cu
Cu + Cu , k= 10
Òm /(molecules s),
–
2
+
2+
–16.1
3
Cu + O , k = 10
Òm /(molecules s).
2
Solubility Product
k1
2
+
–
PbI+,
Pb + I
The concept of stimulated precipitation is of key
importance for constructing a model of sedimentation
periodic in time and, therefore [25–30], space. If this is
a realistic concept rather than a formal construction that
can only be used to obtain oscillatory solutions to sys-
tem (7) of differential equations, it should also be useful
for solving other problems not related to dynamic phe-
nomena of periodic sedimentation. This can be illus-
trated by the static problem of describing the solubility
of poorly soluble salts.
k–1
k2
+
–
PbI + I
PbI2,
k–2
ks
PbI2
PbI ↓,
2
k–s
k3
PbI + PbI ↓
2PbI ↓,
(6*)
2
2
2
k4
To be specific, let us consider scheme (5) of the pre-
2
+
2PbI+,
PbI ↓ + Pb
2
cipitation of a poorly soluble salt MX ↓. Our analysis
2
will be restricted to the simplest case when other elec-
trolytes are absent in the solution, the total ionic
strength is low, and the replacement of concentrations
by activities is unnecessary. Currently, it is assumed
k7
+
2
3
PbI + PbI+
Pb I ,
2
+
3
k8
–
Pb I + I
2PbI2.
2
(
see [7] or [39]) that the precipitation of salt MX ↓ is
2
preceded by the upper three reactions from (6). The lat-
ter are described by differential equations (7) with
The results of computer fitting the topology and
number of oscillations of variable y to the periodic pic-
ture of PbI (that is, the åï ↓ component in scheme (6))
2
2
λ = k = k = k = k = k = k = k
–6
3
4
–4
5
–5
6
precipitation observed experimentally are shown in
Fig. 2. Of course, strictly, it would be good to augment
kinetic differential equations (7) by diffusion terms
(12)
=
k7 = k = k = k = 0,
–7 8 –8
2
+
–
[
M ] = m, [X ] = x.
(13)
2
2
0 0
D (∂ [N]/∂ x) for each component N and integrate the
N
resulting partial differential equations, that is, consider When equilibrium is attained, all the derivatives in dif-
the full-scale distributed problem. This is, however, a ferential equations (7) should be zero. The equilibrium
very complex approach, which is beyond the scope of concentrations are therefore described by the equations
the present work (our purpose is more modest and
+
–
2+
k 〈MX 〉 = k 〈X 〉〈M 〉,
(14)
(15)
(16)
(17)
(18)
–
1
1
reduces to finding a new focussed oscillatory scheme).
According to the direct theorem, time oscillations of a
focussed scheme inevitably give a spatial periodicity
picture in the distributed case. At not too large space
and time intervals, spatial periodicity should repeat
time periodicity, as is shown in Fig. 2. For instance, in
Fig. 2a, when the experiment continued for 25 min
–
+
k 〈MX 〉 = k 〈X 〉〈MX 〉,
–
2
2
2
k 〈MX ↓〉 = k 〈MX 〉,
–s
2
s
2
2
+
+
〈
M 〉 + 〈MX 〉 + 〈MX 〉 + 〈MX ↓〉 = m,
2 2
+
–
〈
MX 〉 + 2〈MX 〉 + 2〈MX ↓〉 + 〈X 〉 = x,
2 2
only, the experimental spatial periodicity [PbI ↓] is
2
where 〈N〉 is the equilibrium concentration [N]eq of
almost indistinguishable from the calculated time oscil-
lations of variable y in the focussed model. Conversely,
in Fig. 2b, when the duration of the experiment was
component N. After the elimination of the 〈MX 〉 con-
2
centration, we find from (15) and (16) that
〈
MX ↓〉
k k
2 s
9
6 h, the last bands of the PbI ↓ precipitate more and
2
2
-
---------------------------- = ----------- -- = K K ,
(19)
c, 2 c, s
more deviate from the theoretical oscillations of the
–
+
k k
–2 –s
〈
X 〉 〈 MX 〉
[åï ↓] variable of the focussed model.
2
where Kc, 2 = k /k and K = k /k are the equilibrium
2
–2
c, s
s
–s
The rate constants found for stoichiometrically
+
constants. Eliminating the 〈MX 〉 concentration from
14) and (19) and introducing the equilibrium constant
Kc, 1 = k /k yields
elementary reactions (6*) are listed in Table 2. The
last two columns of Table 2 contain the rate constants
for second-order stoichiometrically elementary reac-
tions (6*). We see that the rate constant for the same
reaction is from three to four orders of magnitude
lower in agar–agar than in pure water. Stoichiometri-
(
1
–1
〈
MX ↓〉
2
----------------------------- = Kc, 1
K
K
.
(20)
c, 2
c, s
–
2
2+
〈 X 〉 〈 M 〉
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY Vol. 80 No. 5 2006

A MODEL OF THE FORMATION OF LIESEGANG RINGS
721
Table 2. Absolute reaction (6*) rate constants for overall process (3)
logKi logλK [s ]
–
1
3
–logk [cm /(molecules s)]
i
i
i
in agar–agar
Fig. 2b)
from Fig. 2a
from Fig. 2b
from Fig. 2a
from Fig. 2b in water (Fig. 2a)
(
1
1
2
2
s
5.0 ± 0.3
3.0 ± 0.2
–2.0 ± 0.3
0 ± 1
5.0 ± 0.2
4.0 ± 0.2
–3 ± 1
2.9 ± 0.4
0.9 ± 0.3
–4.1 ± 0.4
–2.1 ± 1.0
–5 ± 3
0.1 ± 0.4
–0.9 ± 0.4
–7.9 ± 1.2
–4.9 ± 0.7
–6.9 ± 1.2
≤ –5
16.9 ± 0.4
23.9 ± 0.4
18.8 ± 0.4
–
–
26.8 ± 1.2
0 ± 0.5
–2 ± 1
–2.5 ± 2.5
≤ 3
–
s
≤0
≤1
3
4
7
8
4.3 ± 0.1
6.0 ± 0.2
3.3 ± 0.2
5 ± 1
4.3 ± 0.1
2.0 ± 0.1
3.0 ± 0.1
4.0 ± 0.3
2.2 ± 0.2
3.9 ± 0.3
1.2 ± 0.3
3 ± 1
–0.6 ± 0.3
–2.9 ± 0.3
–1.9 ± 0.3
–0.9 ± 0.5
17.6 ± 0.2
15.9 ± 0.3
18.6 ± 0.3
17 ± 1
19.5 ± 0.3
21.8 ± 0.3
20.8 ± 0.3
19.8 ± 0.5
Currently (see [7] or [39]), the concentration of the is established after the completion of the “logarithmic
solid phase 〈MX ↓〉 is identified with the activity relaxation stage” [1], when all MX ↓ nanoparticles
2
2
coalesce to form a single crystal.)
In denotations (21), the strange equation
MX ↓〉 = K K K SêMX2
a
, which is assumed to equal one. The well-known
MX2↓
(erroneous, as is shown below) equation
〈
,
(22)
2
c, 1 c, 2 c, s
a
2
+
–
2
MX2↓
〈
M 〉 〈 X 〉 = ----------------------------- = Sê
,
(21)
MX2
follows from (20). Equation (22) means that the amount
K
K
K
c, 1 c, 2 c, s
of the MX component in the solid phase does not
2
depend on the amounts of the initial reagents in reac-
tion (5). On the other hand, straightforward use of (20)
leads to the equation
then follows from (20). Here, the constant (at a given
temperature) SêMX2 value is called the solubility prod-
uct. This is a fundamental notion in the modern physi-
cal chemistry of aqueous solutions, which allows the
2+
– 2
–1
〈
M 〉〈X 〉 = (K K K ) 〈MX ↓〉,
(23)
c, 1 c, 2 c, s
2
solubility of salt MX to be calculated in the presence which is at variance with experimental data.
2
2
+
–
of arbitrary other salts containing the M cation or ï
In order to remove these discrepancies, we make a
anion. This useful notion, however, has substantial lim- single refinement of the mechanism of precipitation of
itations. In essence, the SP value is an equilibrium con- the MX
stant, and, by definition, any equilibrium constant tion to the first three reactions in scheme (6), we include
> 0.
↓ component in reaction (5). Namely, in addi-
2
relates the equilibrium concentrations of the initial and stimulated precipitation with the rate constant k
3
final components and allows the final concentrations to Equations (14), (15), (17), and (18), which describe
be calculated from initial and vice versa. In the deriva- instantaneous equilibrium, then follow from differen-
tion of SP according to (21), the concentration of the solid tial equations (7), and (16) is replaced by the more gen-
eral equation
k [A] 〈MX ↓〉 = k [A] 〈MX 〉 + k 〈MX 〉〈MX ↓〉,(16')
phase is, however, replaced by the activity aMX2↓ = 1
irrespective of how much solid phase is dispersed in the
solution (or settled down as a precipitate). Of course,
–s
0
2
s
0
2
3
2
2
where 〈MX ↓〉 is the concentration of the solid phase
2
the concentration of the solid substance MX ↓ in itself
2
MX ↓ dispersed in the whole solution volume. After
2
+
is 100%, and aMX2↓ = 1 for the MX ↓ component lying the elimination of the 〈MX 〉 concentration from (14)
2
and (15), we obtain
as a compact mass at the bottom of a vessel and not par-
ticipating in equilibria (14)–(18). We, however, suggest
〈
MX 〉
2
that the concentration [MX ↓] of the solid MX ↓ sub-
----------------------------- = Kc, 1
K
c, 2
,
(24)
2
2
–
2
2
+
2
+
〈 M 〉 〈 X 〉
stance dispersed in a solution containing the M and
–
ï ions should be considered. The current approach [7, and the elimination of 〈MX 〉 from (16') and (24) results
2
3
9] to solubility product completely removes the prob- in the replacement of (23) by the more general depen-
lem of calculating the amount of the dispersed solid dence
phase MX ↓ present in instantaneous equilibrium with
2
2
k 〈 MX ↓〉
–s 2
2
+
–
2+
–
the given amount of M and ï ions. (Instantaneous
equilibrium should be distinguished from final, which
--------------------------------------------------------------
.
〈
M 〉 〈 X 〉 =
(25)
K
K
(k + k 〈 MX ↓〉 )
c, 1 c, 2 s 3 2
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY Vol. 80 No. 5 2006

7
22
SKOROBOGATOV, KAMENSKII
Equation (25) does not lead to paradoxes and con- Substituting (31) into the left-hand side of (25) yields
tradictions. The solid phase MX ↓ should not be aged, the sought generalization
2
should not lie at the bottom, and, moreover, should not
2
2
+
1
2
be a piece of a single crystal. If the concentration of
⎛
〈 M 〉 + --x – m⎞
2
+
suspension 〈MX ↓〉 in the volume where reaction (5)
⎜
⎟
2
〈
M 〉 -------------------------------------
⎜
2+
⎟
occurs is so high that
1
– K 〈 M 〉
c, 1
(32)
⎝
⎠
k ꢀ k 〈MX ↓〉,
(26)
(27)
s
3
2
k 〈 MX ↓〉
–
s
2
=
-----------------------------------------------------------------.
(k + k 〈 MX ↓〉 )
we find from (25) that
4
K
K
c, 1 c, 2
s
3
2
2
+
–
2
k
–s
If x = 2m, (32) is simplified to
〈
M 〉 〈 X 〉 = ----------------------- -- ≡ Sê
.
MX2
K
K
k
3
c, 1 c, 2
3
2+
M 〉
k 〈 MX ↓〉
〈
–s
2
-
(
-------------------------------------- -- = -----------------------------------------------------------------.
2
+
2
4K
K
(k + k 〈 MX ↓〉 )
However, if sedimentation in the gravitational field
c, 1 c, 2 s 3 2
1 – Kc, 1 〈 M 〉 )
causes the precipitation of the MX ↓ solid phase and its
2
(
32')
removal from the sphere of reactions (6), then
Equation (32') is exact, and (29) approximate.
If inequality (26) holds, (32') can again be simplified
to obtain
k ꢁ k 〈MX ↓〉
(28)
s
3
2
and (25) therefore gives dependence (23), which does
not contradict experimental observations, because the
2+
M 〉
3
〈
k
–s
1
4
volume concentration of 〈MX ↓〉 is low to the extent
--------------------------------------- -- = ---------------------------- = --SêMX2. (32'')
2
2
+
– 2
2+
2
4k K K
c, 1 c, 2
that the concentration product 〈M 〉〈X 〉 is not bound
to reach the value given by (27).
(1 – Kc, 1 〈 M 〉 )
3
This equation is a refinement of (27).
As distinct from the equations of modern physical
It becomes clear that the solubility of poorly soluble
salt MX obeys the above law (Eq. (27)) not only under
2
chemistry [7, 39], (27) and (32") express SêMX2
the condition of solution saturation [7, 39] with salt
MX but also if two other conditions are met. First, the through the rate constants for the stoichiometrically
2
MX ↓ component should be dispersed over the whole elementary reactions from scheme (6). If a solution
2
solution volume to participate in scheme (6), because contains extraneous ions in fairly high concentrations,
the MX ↓ sediment is removed from scheme (6) and (14)–(32) remain valid if the concentrations 〈N〉 are
2
–
2+
the X and M ions do not “know” about its presence. everywhere replaced by the activities aN.
Secondly, even in the presence of partial sedimentation,
Let us use (25) and (27) to obtain information about
the amount of the MX ↓ component in solution volume
2
the rate constants for stoichiometrically elementary
reactions (6*). Table 2 only contains the upper bound-
ary of the rate constant k . Equation (27) allows us to
should be sufficiently large for condition (26) to be sat-
isfied. If these three conditions are fulfilled, (27) fol-
lows from (25); in (27), the solubility product is repre-
sented by a combination of the rate constants of stoichi-
ometrically elementary reactions.
–
s
obtain a more exact value. According to [39], we have
SP = 10 molecules /cm for PbI at 25°ë. Next, it
54.3
3
3
2
follows from Table 2 and Fig. 2a that
If a solution of MX does not contain other electro-
– 17.8 ± 0.5
2
Kc, 1 = k /k = 10
,
1
–1
lytes with common ions, we have x = 2m in (13) and
–
2+
〈
X 〉 ꢀ 2〈M 〉 and (25) takes the simpler form
– 22 ± 1
– 17.6 ± 0.2
Kc, 2 = k /k = 10
,
k3 = 10
.
2
–2
3
k 〈 MX ↓〉
2
+
–s
2
Using these data in (27) yields
k (25°C, in water)
〈
M 〉 = -----------------------------------------------------------------.
(k + k 〈 MX ↓〉 )
(29)
4
K
K
c, 1 c, 2
s
3
2
–
s
(
33)
However, if x ≠ 2m, (25) must be refined using (17)
and (18). Namely, subtracting (17) from (18) two times
yields
– 3.1 ± 2.0 –1
=
Kc, 1Kc, 2k3SêPbI2 = 10
s .
Similarly, according to the data given in Table 2 and
Fig. 2b,
+
–
2+
〈
MX 〉 = 〈X 〉 – x + 2m – 2〈M 〉.
(30)
k (25°C, in 1% agar–agar)
–
s
Dividing the left- and right-hand sides of (30) by the
left- and right-hand sides of (14), respectively, we
obtain an equation that does not contain the concentra-
(
34)
–
4.9 ± 2.2 –1
=
Kc, 1Kc, 2k3SêPbI2 = 10
s .
+
tion 〈MX 〉. It follows from this equation that
RESULTS AND DISCUSSION
It follows from (27) that the rate constant can be
written as
2
+
–
2 〈 M 〉 + x – 2m
〈
X 〉 = --------------------------------------- -- .
(31)
2
+
1
– K 〈 M 〉
c, 1
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY Vol. 80 No. 5 2006

A MODEL OF THE FORMATION OF LIESEGANG RINGS
723
ecules. The kinetics of reactions (39) can then be
(35) described by the following system of differential equa-
tions:
K K SêMX
2
3
2
k–2 = λK–2 = λKc, 1--------------------------,
–
3
0
K
–
s
[
X ]
–
where [ï ] is the internal component concentration in
d[(MX ) ]
2
n
0
-
------------------------- = k µ[(MX ) ] – k [(MX ) ],
kinetic scheme (6) for which the dimensionless rate
n 2 –n 2
n – 1 n
dt
(40)
constants K , K , and K were numerically optimized
2
3
–s
(
Fig. 2). The temperature dependence of the rate con-
n = 2, 3, 4, …, nmax.
−1
stant k (s ) can be represented in the Arrhenius form
–
2
Quasi-elementary reactions (39) occur at a much higher
rate than any of the overall process (6) stages. For this
reason, the concentration µ in differential equations
E (XM – X)
1
4 ± 1
D
⎛
⎞
k–2 = 10
exp –----------------------------- -- ,
(36)
⎝
⎠
RT
(
40) can be treated as a slowly varying µ value. In
0
where 1014 ± 1 s–1 is the preexponential factor value most addition and for the same reason, reactions (39) rapidly
typical of monomolecular reactions [2–5] and E is the reach quasi-stationary conditions, when all the deriva-
D
+
–
dissociation energy of the XM–X to the ïå and ï
tives on the left-hand sides of differential equations (40)
ions in an aqueous medium. The inversion of (36) are zero. As a result, differential equations (40) reduce
yields
to the system of algebraic equations
–
14 ± 1
[
(MX ) ] = (k /k )µ [(MX ) ] .
(41)
2
n stat
n
–n
0
2 n – 1 stat
E (XM – X) = –RT ln(10
k ).
(37)
D
–2
At small n values, the rate constants k and k can
–
2.1 ± 1
s–1 value (see Table 2 and
n
–n
Next, the k (25°ë) =10
–
2
have singularities at “magic” n values, n = 4, 12, 24, 60,
etc. At n values on the order of thousands, millions, and
billions, the k /k ratio is, however, virtually indepen-
Fig. 2) can be used to calculate the dissociation energy
ED(IPb–I) = 92 ± 11 kJ/mol for the second reaction in
scheme (6*-2) in water,
n
–n
dent of n. This simplifies (41) to
PbI (on the surface of a nucleus)
2
[
(MX ) ] = q[(MX ) ] ,
(42)
2
n stat
2 n – 1 stat
(38)
k–2
+
–
PbI + I – E .
where q = µ (k /k ). If q < 1, the sum of geometric pro-
0 n –n
gression (42) terms is
D
A similar procedure applied to the data from Table 2 and
n*
Fig. 2b yields E (IPb–I) = 108 ± 10 kJ/mol. The mean
D
q
q – 1
value is E (IPb–I) = 100 ± 11 kJ/mol = 24 ± 3 kcal/mol =
S(q, n*) =
[(MX ) ] = --------- -- [(MX ) ] ,
stat
n n* stat
2 2
D
∑
1
.0 ± 0.1 eV. This result can be compared with the gas-
n = 2
(43)
phase value E (IPb–I) = 7.2 ± 0.5 eV obtained in [40].
D
because the boundary n* between nuclei and mature
nanoparticles is on the order of millions. Mature nano-
particles (as distinct from nuclei) have not only the con-
densation but also the coagulation channel of growth,
Importantly, according to Lindemann [2–6], reac-
tion (38) is first-order only in the limit of high concen-
trations even in the gas phase. In the liquid phase, reac-
tion (38) is much more complex, because the Pb and I
ions are hydrated in aqueous media, and PbI nucleus
particles can contain thousands and millions of mole-
cules. Why can we describe such a complex process in
scheme (6) as a formally elementary first-order reac-
tion? And why is the fourth reaction in scheme (6) irre-
versible? Does not this contradict the detailed equilib-
rium principle? And, which is the main thing, is it
admissible to describe complex spontaneous and stim-
ulated precipitation processes in differential equations
7) as simple first- and second-order reactions, respec-
tively?
To answer these questions, let us consider the
+
–
k(n1, n2; n1 + n2)
2
(
MX2)n1 + (MX2)n2
(MX2)n1 + n2 . (44)
To describe the state of the whole disperse system of
particles (åï ) at time t, let us introduce the coarse-
2
n
grained distribution function f(m, t), where m is the
mass of the (åï ) nanoparticle. The time evolution of
2
n
this distribution function obeys the Boltzmann-type
kinetic equation
∂
f (m , t) ∂k(m )µ f (m , t)
1
1 0 1
(
-
-------------------- + -------------------------------------------
∂
t
∂m
1
max max
detailed kinetics of processes 6 ± s and 6 + 3. Both
nuclei åï and mature nanoparticles åï ↓ are aggre-
gates (åï ) which grow (diminish) as the next åï
molecule is added (detached),
(45)
=
[k(m , m ; m ) f (m , t) f (m , t)
2 3 1 2 3
2
2
∫ ∫
0
0
2
n
2
–
k(m , m ; m ) f (m , t) f (m , t)]dm dm ,
1 2 3 1 2 2 3
kn
åï + (åï )
(åï ) .
(39) where k(m ) is the condensation (see (39)) rate constant
2
2 n – 1
2 n
1
k–n
k for particles (åï ) with mass m and k(m , m ; m )
is the coagulation (see (44)) rate constant for nanopar-
n
2 n
1
1
2
3
Even these reactions are also not elementary, because
all the particles that participate in them are hydrated.
Reactions (39) will therefore be called quasi-elemen-
ticles (MX ) , (MX ) , and (MX ) with masses
2
n
2
n
2
n1 + n
2
1
2
tary. Let µ be the concentration of individual åï mol-
m , m , and m = m + m , respectively.
1 2 3 1 2
2
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY Vol. 80 No. 5 2006

7
24
SKOROBOGATOV, KAMENSKII
Let us divide the distribution function into two
At q > 1, we have m*f(m*, t) ꢀ [MX ], and (49) takes
2
regions: all (åï ) particles with mass smaller than m* the form
2
n
(
to which n = n* corresponds) will be treated as nuclei,
max
m*
d[MX ↓ ]
2
m*
-
---------------------- = βm*
yf (y, t)dy xf (x, t)dx
∫
∫
dt
S(q, n*) =
f (m, t)dm,
0
0
∫
0
(46)
m*
–
∫ yf (y, t)dy ∫ xf (x, t)dx
[
MX ] = mf (m, t)dm,
2
y ≤ m*
x ≤ m*
∫
0
max
m*
2
and all (åï ) particles heavier than m* will be treated
+ βm*
y f (y, t)dy f (x, t)dx
(50)
2
n
∫
∫
as mature nanoparticles,
0
0
max
2
[
MX ↓ ] =
∫ mf (m, t)dm.
(47)
2
–
y f (y, t)dy
∫ f (x, t)dx
x ≤ m*
∫
m*
y ≤ m*
The coagulation rate constants can be written in the
form taking into account the law of conservation of
mass, k(m , m ; m ) = δ(m + m – m )κ(m , m ). Let us
+ µ (k(m*) – k(0))[MX ].
0
2
Using notation (46), (47) and
2
3
1
2
3
1
2
1
multiply both sides of (45) by m and integrate the
1
k = µ (k(m*) – k(0)), k3 = βm*
(51)
(52)
s
0
result in m from 0 to m*,
1
we can rewrite (50) in the simpler form
m*
∂
d
dt
-
--- m f (m , t)dm + m*k(m*)µ f (m*, t)
----[MX ↓ ] = k [MX ]
1
1
1
0
2
s
2
∫
∂
t
0
2
+
k ([MX ] + [MX ↓ ])[MX ] – k [MX ]
3
2
2
2
3
2
m*
max
–
µ0 k(m ) f (m , t)dm
1 1 1
∫
2
2
+
k3S(q, m*)
(y – m ) f (y, t)dy .
0
∫
(48)
m*
max
m*
=
m dm
κ(m , m ) f (m , t) f (m – m , t)dm
2
The right-hand side of (52) contains the variance of the
distribution function D{f(m, t)} (the term in brackets),
which eventually gives
1
1
2
1
2
1
2
∫
∫
0
0
m*
max
d
-
---[MX ↓ ] = k [MX ]
–
m dm f (m , t) κ(m , m + m ) f (m , t)dm.
2
s
2
1
1
1
1
1
2
2
∫
∫
dt
k [MX ][MX ↓ ] + k S(q, m*)D{ f (m, t)}.
(
53)
0
0
+
3
2
2
3
We will use notation (46), (47) and the approximate rate
constants
The first term on the right-hand side of (53)
describes the condensation growth of åï ↓ nanoparti-
2
k(m) = k(0)m, κ(m , m ) = βm*m .
cles, and the second term describes coagulation growth.
It follows that we obtained a “microscopic” substantia-
tion of the formally kinetic description of stoichiomet-
rically elementary reactions 6 + s and 6 + 3 by kinetic
differential equations (7) and reduced the rate constants
for these reactions to combination (51) of more elemen-
tary reactions. Interestingly, the right-hand side of (53)
contains a term including the variance D{f(m, t)},
which is absent in kinetic differential equations (7).
49) This is an expected result. Indeed, the variance of con-
2
1
1
As a result, (48) gives
d[MX ↓ ]
d[MX ]
2
2
-
---------------------- = –-------------------
dt
dt
=
µ k(m*)m* f (m*, t) – µ k(0)[MX ]
0
0
2
max
m*
(
+
βm* m f (m , t)dm (x + m ) f (x, t)dx
2
2
2
2
centrations is absent in deterministic differential equa-
tions (7). It only appears when the chemical process is
described in more detail using the distribution function
formalism or stochastic occupation numbers [31].
Formal kinetics equation (53) is an averaged result
of an extended chain of millions and billions of reac-
∫
∫
0
0
max
m*
–
βm* m f (m , t)dm xf (x – m, t)dx.
2 2 2
∫
∫
0
0
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY Vol. 80 No. 5 2006

A MODEL OF THE FORMATION OF LIESEGANG RINGS
725
tions. It is therefore no surprise that reaction 6 + 3, 17. G. I. Kuz’menko, Zh. Fiz. Khim. 51 (10), 2607 (1977).
which formally has a simple form of a second-order 18. A. V. Shubnikov and V. F. Parvov, Nucleation and
reaction, does not obey the detailed equilibrium princi-
ple. This principle should only be satisfied by truly ele-
mentary atom-molecular processes.
Growth of Crystals (Nauka, Moscow, 1969) [in Rus-
sian].
19. V. M. Zhukovskii and A. N. Petrov, Thermodynamics
and Kinetics of Reactions in Solids: A Textbook (Izd.
Ural’skogo Univ., Sverdlovsk, 1987), Vol. 2 [in Russian].
0. I. M. Lifshits, Physics of Real Crystals and Disordered
Systems: Selected Works (Nauka, Moscow, 1987) [in
Russian].
Similarly, reaction (6-2) is a result of averaging over
all nuclei åï of a huge series of the reactions
2
2
k–2(n)
+
–
(åï2)n
(åï )
+ MX + X .
2 n – 1
k2(n)
Reaction (38) should therefore be considered the disso-
2
2
1. G. Venzl, J. Chem. Phys. 85 (4), 2006 (1986).
ciation of the PbI molecule on the surface of (PbI )
2
2 n
2. Y. Brechet and J. S. Kirkaldy, J. Chem. Phys. 90 (3),
+
–
nuclei with the transfer of the PbI and I ions into the
aqueous phase. Analytic representation (36) is admissi-
ble because it is admissible for each of the k (n) rate
1
499 (1989).
2
3. B. Chopard, P. Luthi, and M. Droz, J. Stat. Phys. 76 (1–2),
61 (1994).
–
2
6
constants, which very weakly depend on n if n is not too
2
2
4. M. Droz, J. Stat. Phys. 101 (1–2), 509 (2000).
small.
5. A. M. Zhabotinskii, Concentration Autooscillations
(Nauka, Moscow, 1974) [in Russian].
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RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY Vol. 80 No. 5 2006