Liesegang patterns : Studies on the width law

7 Pages·0.15 MB·English

by  M. Droz

Liesegang patterns: studies on the width law
M. Droz1, J. Magnin1, M. Zrinyi2
1 D´epartement de Physique Th´eorique, Universit´e de Gen`eve, CH-1211 Gen`eve 4, Switzerland.
2 Department of Physical Chemistry, Technical University of Budapest, H-1521 Budapest, Hungary.
The so-called width law for Liesegang patterns, which states that the positions xn and widths
wn of bands verify the relation xn ∼ wnα for some α > 0, is investigated both experimentally and
9 theoretically. Weprovideexperimentaldataexhibitinggoodevidenceforvaluesofαcloseto1. The
9 value α=1 is supported bytheoretical arguments based on a generic model of reaction-diffusion.
9
1
UGVA DPT 1999/01-1024

n
a I. INTRODUCTION 2. The ratio between the position of a band and the
J
square root of the time elapsed until it appears,
1
denoted t , is constant. This law is a direct conse-
1 Inrecentyears,pattern-formingchemical,physicaland n
quenceofthediffusivenatureofthedynamics(time
biological processes came to the foreground. The three
law) [5]:
] main causes of this strong interest are the exotic nature
t
f oftheseprocesses,designedpattern-formingexperiments
o x √t (2)
s turnedupinphysicalandchemicallaboratoriesand,last n ∼ n
. butnotleast,thedramaticevolutionofcomputingpower,
t
a allowing to perform extensive numerical simulations as 3. Itisalsogenerallyobservedinexperimentsthatthe
m
well as to solve numerically the reaction-diffusion differ- widthw ofthebandsincreaseswiththeirposition
n
- ential equations which underly these phenomena [1]. as a power law (width law) [6]:
d
When an electrolyte diffuses into a gel containing an-
n
o other electrolyte, the eventual formation of a rhythmic wn xnα (3)
∼
c patternofprecipitate (aweaklysolublesalt)isknownas
[ the Liesegang phenomenon [2,3]. This pattern strongly
with α>0.
dependsonthedimensionalityaswellasonthegeometry
1
of the experimental setup.
v Untilnow,alotofattentionhasbeenpaidtothespac-
9 The simplest and usual way to perform a Liesegang
inglaw(see [7],[8],[9],[10]). Onthe contrary,the width
7 experiment is to take a tube and fill it with a gel which
lawhaslargelybeenignoredforseveralreasons. Fomthe
0 contains the so-called internal electrolyte dissolved in it.
experimentalpoint of view, it is difficult to make precise
1
The other reactant, refered to as the outer electrolyte,
0 measurements of the width of the bands without hav-
is poured onto the surface of the gel at one extremity of
9 ing recourseto sophisticateddigitizing methods. From a
thetube. Usually,theconcentrationofthelatterismuch
9 theoretical point of view, a complete description of the
/ higherthanthatoftheinnerone. Themainroleofthegel dynamics of bands formation requires the knowledge of
t
a in this experiment is to prevent convection of solutions the detailed mechanisms involved in the coarsening pro-
m and sedimentation of precipitate. Thus the only mech-
cess. Accordingly, two different predictions for α have
anism allowed here for the transportation of material is
- been previously obtained [1,11].
d diffusion, i.e. the whole process is diffusion-limited.
It is our purpose,in this paper, to focus on the width-
n
Although the Liesegang phenomenon has been known
o law aspect of the Liesegang phenomenon, and to try
formorethanonehundredyears,therearestillmanydis-
c hereby to remove some ambiguities that still surround
: cussionsonthedetailedmechanismsresponsibleforthese this basic feature of this class of systems. In section II,
v
static structures. Nevertheless, most of the Liesegang
i we will expose and discuss the results of Liesegang ex-
X patterns one obtains in experiments or observes in na-
periments performed by one of us (M.Z.), for which the
ture are characterizable by the following generic laws:
r widths of the bands were measured. The analysis of the
a
data provides an accurate estimation of the exponent α.
1. The ratio between the positions x , x of two
n n+1 InsectionIII,atheoreticalargument,independentofthe
successivebands, measuredfrom the surface of the
nature of the detailed mechanisms involved in band for-
gel, tends to converge to a finite value that is usu-
mation, is developed to support the experimental find-
ally larger than one (spacing law) [4]:
ings. Finally, concluding remarks are made in section
IV.
xn+1 nlarge
1+p (1)
x −→
n
II. EXPERIMENTAL PART
with p>0.
1
A. Materials and Sample preparation containing the MnSO electrolyte. The following reac-
4
tion occurs when the spatially separated reactants come
We have experimentally studied the formation of into contact:
Mn(OH) precipitate by diffusion of NH OH into chem-
ically cro2ss-linked polyvinyl(alcohol) (P4VA) hydro-gel Mn2++2OH− ⇀↽Mn(OH)2 (4)
FIG.1. Picture of the resulting pattern obtained in one of thethree experimentsdescribed in Section 2.
The highly swollen PVA-hydrogels were prepared by linking agent, GDA was: (VA)/(GDA)= 250. In order
cross-linking of primary PVA -chains with glutaric alde- tostudythe precipitationandbandformationinswollen
hyde (GDA) in aqueous solution, containing dissolved networks, one of the reactants, MnSO , was mixed with
4
MnSO . Commercial PVA (Merck 821038) and solu- the polymer solution containing the cross-linking agent.
4
tion of 25mass% GDA (Merck) were used for prepara- The concentration of MnSO in the gel phase was 0.03
4
tion. The initial polymer concentration, as well as the mol/dm3. The gelation was induced by decreasing the
cross-linking density, were kept constant. The polymer pH of the system by H SO (Carlo Erba). The solution
2 4
content of the PVA gels was in every case 4.0 mass%. was then poured into glass tubes. The gelling process
The ratio of monomer unit of PVA (VA) to the cross- usually took 3-4 hours.
Sample label a [mol/dm3] b [mol/dm3] α
0 0
3.0 0.03 0.94
A
5.0 0.03 0.99
B
14.7 0.03 0.90
C
Table I. Concentrations used in each experiment. a is the concentration of the the outer electrolyte (NH OH)
0 4
and b , the one if the the inner electrolyte (MnSO ). The last column gives the width exponent α.
0 4
The tubes were sealed, placed vertically, and allowed a real-time video digitizer card. We also used a scanner
◦
tostandundisturbedataconstanttemperaturesof17 C. to digitize the experimental results.
After completionofthe networkformation,the gelswere We first determined the coordinate of the gel surface.
brought into contact with the outer reactant, NH OH, Then, the positions of the N bands, x , were
4 n n=1..N
{ }
thus defining the initial time t = 0 for the experiment. evaluated. Note that x stands here for the position of
n
Theconcentrationofouterelectrolytewasvariedbetween the upper limit of the precipitate zone. We finally mea-
3 and 14.7 mol/dm3. We maintained the constancy of suredthepositions x˜ ofthelowerboundariesof
n n=1..N
{ }
boundaryconditionbyrefreshingthe outer solutioncon- theprecipitatelayers. Thedifferenceofthesetwoquanti-
tinuously. Let us underline that due to the choice of the ties providesthe width of the n-thband : w x˜ x .
n n n
≡ −
initial conditions, these experiments are effectively one- Itmustbementionedthatduetothesharpboundaries
dimensional. The relevant space coordinate is the one ofthe precipitate regions,the uncertaintieson the width
that goes along the longitudinal axis of the tube. values are estimated ranging from 2% to 5%.
B. Experimental data analysis C. Experimental results
The penetration of NH OH into the gel containing Figures2-4showtheresultsofthreedifferentLiesegang
4
MnSO resultsinasharpboundaryinterfaceofMn(OH) experiments for which the concentrations of the elec-
4 2
precipitatebands. Inmanycasesweobtainedmorethan trolytes are given in Table 1. They take the form of
20 bands. Figure 1 shows a typical example. log-log plots relating the widths w to the positions x
n n
Inordertodeterminethespacecoordinateofthebands ofthebands,asmeasuredfromthedigitizedimages. The
obtained,weusedadigitalvideosystem. ACCDcamera dotted lines are least square linear fits whose slope gives
with an 1/3” video chip was connected to a PC through the width exponent α. These slope values are shown in
2
Table 1. For all the samples, the width exponent is near α=1.
6.5
6.0
w)n5.5
ln(
α = 0.99
5.0
4.5
8.5 9.0 9.5 10.0 10.5
ln(xn)
FIG. 2. Log-log plot of the widths wn of the bands versus their positions xn for the sample A. Lengths are in arbitrary
units. The dotted line corresponds to a least square linear fit.
5.0
4.5
w)n4.0
ln(
α = 0.94
3.5
3.0
6.5 7.0 7.5 8.0 8.5
ln(xn)
FIG.3. Log-logplotofthewidthswn ofthebandsversustheirpositionsxn forthesampleB. Lengthsareinarbitraryunits.
The dotted line corresponds to a least square linear fit.
8.5
8.0
w)n
ln(
7.5
α = 0.90
7.0
8.5 9.0 9.5 10.0
ln(xn)
FIG.4. Log-logplotofthewidthswn ofthebandsversustheirpositionsxn forthesampleC. Lengthsareinarbitraryunits.
The dotted line corresponds to a least square linear fit.
3
In these experiments, the two chemical species react 2. The secondcategorycontainswhatwe will callthe
andproduceanintermediatecompound(calledC)which intermediate species-type theories. We group into
also diffuses inside the gel. Its associated traveling front this set all the schemes that involve the existence
can be put in evidence by light transmission techniques. of an intermediate state, usually denoted by the
letter C, prior to the formation of the precipitate
zones. This compound is supposed to be produced
continuously, with a certain rate proportional to
III. THEORETICAL APPROACHES
the local concentration product of the reactants.
It thus gives rise to a front of C particles that
A. Two classes of competing theories for Liesegang propagates in the system. Such processes can be
patterns taggedwiththefollowingsymbolicreactionscheme
: A+B C D. It is natural to suppose that
→ →
thisintermediatecompoundalsodiffusesinsidethe
The several mechanisms one can consider for explain-
gel medium, with a diffusion coefficient D .
ing the formation of Liesegang patterns will be, in the c
context of this paper, classified in two categories [10]. We recently considered two particular schemes of
Both have in common the following set of “free” param- this type in a paper mainly devoted to a deeper
eters : understanding of the spacing law [10]. The first
one, called the nucleation and growth theory, sup-
i). The concentration of the outer reactant A, de- poses that the intermediate compound undertakes
•
noted a . It is generally kept fixed at the open a local nucleation process at the point where its
0
extremity ofthe tube during the whole durationof concentrationisgreaterorequaltoacertainnucle-
the experiment. It therefore leads to the follow- ation threshold c∗. An extension of this theory to
ing initial-and-boundary condition for the field a a case where counter-ions associated to the flow of
describing the concentration of species A : the outer reactant play a catalytic role in the cre-
ation of a new band was also examined under the
a(0<x L,t=0)=0 (5) denomination of induced sol-coagulation theory.
≤
a(x=0,t 0)=a (6)
≥ 0 As we noticed in Section II, the experiments we per-
formed show unambiguously the presence of an interme-
where L stands here for the length ofthe test-tube diate C compound. We will therefore base our theoreti-
hosting the gel. calconsiderationsonthe classofmodels including the C
species.
ii).Theinitialconcentrationoftheinnerelectrolyte,
•
b . One has:
0
B. Prediction of the intermediate-species class for
b(0 x L,t=0)=b (7) the width law
0
≤ ≤
The determination of the width law for a particular
iii). Thediffusioncoefficientsofbothspeciesinthe modelwith intermediate species canbe addressedatdif-
•
gel, D and D . ferent levels.
a b
A detailed description of the dynamics of band for-
Eachofthesetwoclassesofmodelswedistinguishhere mationcanbe attempted. However,thisrouterequiresa
try to capture a certain family of microscopical mecha- preciseknowledgeofthemicroscopicalmechanismsasag-
nisms leading to the formationof bands. In this respect, gregation,flocculation,etc. presentinthe system,which
they both can be characterized by their own set of pa- is clearly too ambitious a goal : very few informations
rameters associated to these particular mechanisms : are usually available about which of these mechanisms
are present in a given experimental setup.
1. In the first category, we put the ion-product su- Nevertheless, if one is only interested in general prop-
persaturation type theories. They have in com- erties of the bands system, as its width exponent α, a
mon the fact that the outer and inner reactants qualitative analysis basedon generalproperties like con-
directlyturnintoprecipitatewheretheirlocalcon- servationlawsshouldalreadyprovideagoodapproxima-
∗
centrationproductreachessomethresholdq . One tion for the width exponent.
can symbolize this process by the reaction scheme We are therefore in search of a general feature of all
A+B DwhereDstandsherefortheprecipitate these intermediate species models that could allow us to
→
state. The typicaland most studied scheme of this computeα. Alongthisline,wefoundageneralargument
type is Prager’s theory for periodic precipitation based on the shape and dynamics of the C front [10].
[8],firstlyproposedbyWagner[7],andinvestigated Let t be the time at which band number n starts to
n
later also by Zel’dovitch et al [9]. appear. Over the interval [t ,t ], the C profile adopts
n n+1
4
a typical shape characterized by a diffusive tail at the where k,κ and κ are reaction rates (κ ,κ 1), and
1 2 1 2
≫
right of its (growing)maximum, and a quasi-linearly de- Θ is the Heaviside step function.
creasingshapeattheleft,asshowninFigure5. Thisfact
Note that in this simple basic model, we just require
can be easily verified by solving numerically the system
thatC israpidlygoingtozerowhereversomeprecipitate
of partial differential equations modeling the simplest
alreadyexists. Thisingredientisaimedhereatincluding
intermediate-species theory producing Liesegang bands,
crudely all the microscopic effects like aggregation.
namely :
To discuss the width-law problem, we will balance the
∂ a(x,t)=D ∂2a(x,t) ka(x,t)b(x,t) (8)
t a x − quantity of C diffusing out of the C front with the total
∂tb(x,t)=Db∂x2b(x,t)−ka(x,t)b(x,t) (9) amount of matter inside a band after its completion.
∂ c(x,t)=D ∂2c(x,t)+ka(x,t)b(x,t) (10)
t c x − Let’s begin by summing up all the contributions to
∗
κ1Θ(c c )c(x,t) κ2c(x,t)d(x,t) the quantity of C entering band n, once it is formed at
− −
∂td(x,t)=κ1Θ(c c∗)c(x,t)+κ2c(x,t)d(x,t) (11) x=xn. They can be divided in two parts :
−
c*
n
o C profile
ati
ntr
e
c
n
o
c
0
x x position along
n n+1
the tube
FIG.5. Shapeof theC profile just before band number n+1 appears. x is thecoordinate along thetube.
Firstly, we can evaluate the quantity of C that flows turns out that once the next band appears at x=x ,
n+1
into band number n during the time interval (t ,t ). where the C profile has its maximum at t = t , the
n n+1 n+1
This quantity, denoted here Q , is equal to the time left side of the front is trapped between two absorbing
n,1
integral over the flux of C toward x , evaluated from t boundaries. At t=t +, the total quantity of trapped C
n n n
to t . This flux is proportional to the slope of the C can be geometrically evaluated as (see figure 5):
n+1
profile at the right of x . Numerical estimations of this
n
slope show that it scales with time as [10]: 1 ∗
δC c (x x ) (15)
n+1 n
≈ 2 −
1
∂ C(x=x +,t <t<t ) (12)
x n n n+1
∼ √t tn a finite fraction of which, say η (η <1), will diffuse into
−
bandn. Hence,usingonceagainthespacinglaw,wecan
We therefore have
evaluate Q as :
n,2
tn+1
Q D ∂ C(x=x +,t)dx D t t
n,1 ∼Ztn c x n ∼ cp n+1− n Q =ηδC 1ηpc∗x (16)
n,2 n
≈ 2
(13)
Invoking now the time and spacing laws (supposing that Finally, summing up the two contributions, we obtain :
n is large enough), leads to
Q =Q +Q x (17)
n n,1 n,2 n
∼
Q D x 2 x 2 D p2+2px D 2px
n,1 c n+1 n c n c n
∼ − ∼ ≈
p p p Note that typically, Q is smaller than Q by four
(14) n,2 n,1
order of magnitude.
We goonthenby examining the contributionQ as- The final stepofour discussiondealswith the relation
n,2
sociated to the flux into band n at times t > t . It between Q and w .
n+1 n n
5
The basic model described by eq. (8)-(11) produces as obtained fromnumericalcomputation seems to follow
onlyinfinitelythinbands. Ifonewantstogeneratebands ascalingformwithanamplitudeapproachingaconstant
with nonzero widths, it is necessaryto supplement these after some transient regime. The width exponent that
equations with new terms accounting for growth mecha- could be extracted from the set of (only) six bands pro-
nisms. Whatever the nature and details of these mech- vided by the simulation is clearly compatible with the
anisms may be, one can make the following reasonable value α=1.
assumption :
The terms one has to add to equations (8-11) in or-
der to model growth lead, for t t , to a stationary
n+1 C. Summary and conclusion.
profile of the precipitate concent≫ration D inside the nth
bands which adopts a natural scaling form:
We have reported pattern-forming experiments of
x xn Liesegang type, in which a careful measurement of the
D =A D( − ), x <x<x +w (18)
n n n n
w width of the bands leads to an exponent α close to 1.
n
By considering the most basic theoretical model of the
whereA isanamplitudewhich,ingeneral,coulddepend
n type related to the experiments we performed, we could
on n.
explain, through a simple argumentation based on con-
Byequatingthe quantityQ (the totalamountofma-
n servationlawsandmacroscopicpropertiesofthe concen-
terial that has nourished the band until its completion)
tration profiles, how such a value close to unity can be
with the total mass of precipitate inside the same band
understood, and which condition it implies on the shape
once formed, we have :
of the precipitate profile inside bands. However, only a
firstprinciple derivation,basedonthe underlyingmicro-
xn+wn x x
A dxD( − n)=γA w =Q x (19) scopical mechanisms of coarsening could give a clear cut
nZ w n n n ∼ n
xn n determination of α.
We believe therefore that the microscopical theory of
where gamma is a finite constant defined as
band formation and growth in Liesegang experiments
1/2 constitutes an extremely rich and still unexplored sub-
γ = D(z)dz (20) jectofinvestigationinitself,forwhichlotsremainstobe
Z
−1/2 done.
To reproduce a relation of the form w xα, the am-
plitude A should be of the form A n x∼β. nWe obtain
hereby n n ∼ n ACKNOWLEDGMENTS
α+β =1 (21)
We thank B. Chopard and Z. Ra´cz for useful dis-
cussions. This work has been partially supported by
Thus the experimental value α = 1 corresponds (up to
the Swiss National Science Foundation in the frame-
logarithmiccorrections)toaconstantamplitudeA. This
work of the Cooperation in Science and Research with
prediction could be tested experimentally by investigat-
CEEC/NIS, and by the Hungarian Academy of Sciences
ing quantitatevely the internal structure of the bands.
(Grant OTKA T 019451).
We can now return to the two theoretical predictions
existing in the literature [1,11] for the exponent α. In
the first one, by Chopard et al., Lattice-Boltzmann cel-
lular automata simulations including fluctuations were
performed, in which microscopical mechanisms account-
ing for aggregation were also included, in a phenomeno-
logical way. They obtained bands patterns, which ex-
[1] B.Chopard,P.LuthiandM.Droz,Phys.Rev.Lett.72,
hibited a width exponent varying between α = 0.5 and
1384 (1994); Jour. Stat. Phys. 76, 661-676 (1994).
α = 0.6. Such a value, quite far from 1, can indeed be
[2] R.E. Liesegang, Naturwiss. Wochenschr. 11, 353 (1896);
explainedby the simple fact that their modelallowedan
[3] H.K.Henisch,“Periodicprecipitation”,PergamonPress,
unbounded quantity of precipitate to form at one lattice
(1991).
site. In this model, the amplitude A increased with n
n [4] K.Jablczynski,Bull.Soc.Chim.France33,1592(1923).
andaccordingto (21),constrainedthe exponentαtobe
[5] Harry W. Morse and George W. Pierce: Diffusion and
smaller than 1. Supersaturation in Gelatine, Proceedings of the Ameri-
Inthesecondpaper,Deesolvedasystemofmean-field can Academy of Artsand Sciences, 38 625-647 (1903).
equationssimilarto (8)-(11)inwhichthe depletionterm [6] S. C. Mu¨ller, S. Kai, and J. Ross, J. Phys. Chem., 86,
for the C front was build from physical considerations 4078 (1982)
relatedtothe nucleationanddropletgrowththeories. In [7] C. Wagner, J. Colloid Sci., bf 5, 85 (1950).
this model, the distribution of precipitate inside a band [8] S. Prager, J. Chem. Phys. 25, 279 (1956).
6
[9] Ya .B. Zeldovitch, G.I. Barrenblatt and R.L. Salganik, forLiesegangpatterns”,J.Chem.Phys.109,9479-9486,
Sov.Phys. Dokl.6, 869, (1962). (1998).
[10] T. Antal, B. Chopard, M. Droz, J. Magnin, Z. R´acz [11] G.T. Dee, Phys. Rev.Lett. 57, 275 (1986).
and M. Zrinyi, “Derivation of the Matalon-Packter law
7