Guiding-fields for phase-separation: Controlling Liesegang patterns
Tibor Antal, Ioana Bena, Michel Droz, Kirsten Martens, Zoltan Racz
aa r X i v : . [ c ond - m a t . o t h e r] A ug Guiding-fields for phase-separation: Controlling Liesegang patterns
T. Antal, I. Bena, M. Droz, K. Martens, and Z. R´acz Program for Evolutionary Dynamics, Harvard University, Cambridge, MA 02138 Theoretical Physics Department, University of Geneva, CH-1211 Geneva 4, Switzerland Institute for Theoretical Physics–HAS, E¨otv¨os University, P´azm´any s´et´any 1/a, 1117 Budapest, Hungary (Dated: October 20, 2018)Liesegang patterns emerge from precipitation processes and may be used to build bulk structuresat submicron lengthscales. Thus they have significant potential for technological applications pro-vided adequate methods of control can be devised. Here we describe a simple, physically realizablepattern-control based on the notion of driven precipitation meaning that the phase-separation isgoverned by a guiding field such as, for example, a temperature or a pH field. The phase-separationis modeled through a non-autonomous Cahn-Hilliard equation whose spinodal is determined by theevolving guiding field. Control over the dynamics of the spinodal gives control over the velocityof the instability front which separates the stable and unstable regions of the system. Since thewavelength of the pattern is largely determined by this velocity, the distance between successiveprecipitation bands becomes controllable. We demonstrate the above ideas by numerical studiesof a 1 D system with diffusive guiding field. We find that the results can be accurately describedby employing a linear stability analysis (pulled-front theory) for determining the velocity – local-wavelength relationship. From the perspective of the Liesegang theory, our results indicate that theso-called revert patterns may be naturally generated by diffusive guiding fields. PACS numbers: 05.70.Ln, 64.60.My, 85.40.Hp, 82.20-w
I. INTRODUCTION
Pattern formation is an ubiquitous phenomenon inout-of-equilibrium systems, and ordered structures of-ten emerge in the wake of a moving reaction front [1].There has been recently increasing interest, both exper-imental and theoretical, in the study of various typesof chemically-generated patterns. The main reason isthat they are expected to provide new bottom-up , self-assembling technologies for engineering bulk patterns onmesoscopic and microscopic scales (for illustration seee.g. [2, 3, 4, 5, 6, 7] from a rapidly-growing bibliography),for which the traditional top-down methods (i.e., remov-ing material in order to create a structure) are reachingthe limits of their capabilities.Detailed understanding of the mechanisms responsiblefor pattern formation is a key-element in developing tech-nological applications since it helps constructing the ap-propriate tools for the control of the characteristics of theemerging patterns. In this paper we shall focus on design-ing a simple method to control the so-called
Liesegangstructures [8, 9]. In particular, this method should beuseful for a recently-proposed experimental set-up thatallows to create stamps of such structures [4].Depending on the geometry, Liesegang precipitationpatterns are bands (in an axially-symmetric configura-tion), rings or shells (in a circular, respectively spherical-symmetric configuration), clearly separated in the di-rection of motion of a chemical reaction front. Severalgeneric experimental laws characterize the patterns, seee.g. [10, 11] for reviews. In particular, it is found that thepositions of the bands usually obey simple laws; e.g. theyform, with a good approximation, a geometric series withincreasing distance between consecutive bands. This is the so-called regular banding situation, which has beenrecently explained [12] using the phase separation in thepresence of a moving front as the underlying mechanism.Briefly, the reaction front, which moves diffusively, leavesbehind a constant concentration c of the reaction prod-uct , that we shall conventionally name hereafter C parti-cles [13, 14, 15]. At a coarse-grained level, the dynamicsof the C particles (that can diffuse, and are also attract-ing each other) can be described by a Cahn-Hilliard equa-tion [16, 17, 18] with a source term corresponding to themoving reaction front. Starting with a system free of C ’s,the dynamics of the front brings locally the system acrossthe spinodal line, provided that c is inside the unstableregion of the phase diagram. A phase separation takesthen place on a short time scale and a band of precipitateis rapidly formed just behind the front. This band actsas a sink for the C particles. Then the local concentra-tion of C ’s decreases, bringing the system locally in thestable phase again. Thus Liesegang patterns are formedsince the state of the system at the front is locally andquasi-periodically driven into the unstable regime.The characteristics of these regular patterns can becontrolled to some extent through an appropriate choiceof the concentration of the reagents [19], of the natureof the gel that is filling the reaction container [20], theshape of the container [5], or through an applied electricfield [21, 22, 23, 24].The spinodal decomposition scenario has proved itspower by describing regular patterns and, furthermore,by explaining how those patterns can be influenced bythe concentration of the outer and inner electrolytes andby an external electric field. We will show that it canbe extended to describe other situations, as well. In-deed, there is experimental evidence of Liesegang-typeprecipitation patterns with decreasing distances betweensuccessive bands [25] which is termed inverse banding . Inthe borderline case between regular and inverse banding,the distances between successive bands are constant, asituation called equidistant banding [26]. In our attemptsof describing the above patterns we were lead to a mech-anism which may provide a simple, experimentally real-izable control tool of the emerging pattern.As described in detail in Sec. II, our proposal is basedon a phase separation mechanism in a space- and time-dependent guiding field , which could represent, for ex-ample, a temperature or a pH field. The pattern for-mation is thus modeled through a non-autonomous CHequation , whose spinodal line is controlled by the guid-ing field. Note that the present design of the guidingfield is different from the homogeneous (overall) cooling that was used in most of the previous studies of non-autonomous CH models, see e.g. [27, 28, 29, 30]. As weshall demonstrate, a simple guiding field is sufficient togenerate crossover between regular and inverse patterns.For example, such a guiding field can be a temperaturefield evolving diffusively due to a temperature differenceat the boundaries of the system, whose characteristics aredetailed in Sec. III. The features of the correspondingemerging patterns are analyzed in Sec. IV. As discussedin Sec. V, our numerical findings can be justified by theo-retical arguments relating the velocity of the front of theguiding field to the pulled-front velocity resulting from alinear stability analysis of the phase-separation process.Other, more flexible ways to control the patterns are alsobriefly presented in Sec. VI. Finally, conclusions and per-spectives are discussed in Sec. VII. II. THE MODEL
Let us consider a tube filled with gel, and an initiallyuniform concentration c of C particles throughout thetube. Assuming axial symmetry along the x -axis of thetube, we shall consider that the C -particle concentra-tion c ( x, t ) evolves in time according to the Cahn-Hilliard(CH) equation in one dimension. After rescaling thespace and time variables, this equation can be writtenin the following dimensionless form ∂c ( x, t ) ∂t = − ∂ ∂x (cid:20) εc ( x, t ) − c ( x, t ) + ∂ c ( x, t ) ∂x (cid:21) , (1)with 0 x L , where L is the dimensionless lengthof the tube. Note that we also performed an appropri-ate shift and scaling of the concentration that allows usto write the CH equation in a form that is more con-venient for the exposition of our problem; namely, thisform is symmetric with respect to the change in sign of c , c ↔ − c (see e.g. [10, 12] and footnote [32] for a more de-tailed discussion of this point). The shifted and rescaledconcentration can take both positive and negative val-ues, and the stable configurations are symmetric around c = 0. The parameter ε measures the deviation of the temper-ature from the critical temperature T c ; it is negative fortemperatures above T c (for which no phase-separation ispossible), while it is positive for temperatures T < T c .Below the critical temperature, a uniform concentrationprofile c inside the spinodal decomposition domain, i.e. | c | c s = p ε/
3, is linearly unstable. A small, lo-calized perturbation of the concentration can then trig-ger a phase separation throughout the system, throughthe amplification of the unstable modes of wavenumbers | k | < √ ε . A large body of work (see e.g. [31]) has beendevoted to the study of the phase separation process inthe simple case of a uniform parameter ε taking the samevalue throughout the system.Here we shall concentrate on a different situation,namely when ε is a field , which evolves according to itsown dynamics. Moreover, it is possible to control its evo-lution. For a simple realization of this control considerthe following situation. Suppose that at a time t = 0 thetemperature at one end of the tube is lowered and keptat a constant value T < T c thereafter. The other end ofthe tube is supposed to be thermally isolated [33]. Thetemperature profile then evolves in time along the tubeaccording to the usual Fourier law of heat conduction,and so does the related ε ( x, t ) field, ∂ε ( x, t ) ∂t = D ∂ ε ( x, t ) ∂x , (2)where D is the dimensionless thermal diffusion coeffi-cient. Through an appropriate scaling of the temper-ature, the value of ε can be set to − t = 0, while ε = +1 at t > x = 0)end of the tube; at the right ( x = L ) end of the tubethere is no heat flow: ε ( x > , t = 0) = − ,ε ( x = 0 , t ) = +1 ,∂ε∂x ( x = L, t ) = 0 . (3)These equations (2), (3) define completely the evolutionof ε ( x, t ) along the tube, from the onset of the coolingprocedure till reaching the asymptotic uniform profile ε =+1 throughout the tube [32].For the value ε = −
1, the uniform concentration profile c is stable, while for ε = +1 it tends to phase-separate(i.e., the initial concentration | c | < p / C -particle concen-tration becomes locally unstable with respect to phaseseparation. As a consequence, a pattern made of alter-nating low- and high-density phases of C appears simul-taneously to the propagation of the cooling front alongthe tube. Its properties and characteristics result thusfrom the CH equation (1) coupled to the evolution equa-tion (2) for ε ( x, t ). Appropriate boundary conditions (i)guarantee the conservation of C particles inside the tube(more precisely, zero-particle fluxes J c at the edges), and(ii) associated to the initial condition, they also ensurethe uniqueness of the solution. The boundary conditionswe used in our numerical discretized procedure amount,in the continuum limit, to J c ( x = 0 and L, t ) = 0 ,∂ c∂x ( x = 0 and L, t ) = 0 , (4)with J c ( x, t ) = ∂ (cid:0) εc − c + ∂ c/∂x (cid:1) /∂x . Setting ∂ c/∂x = 0 means that c at the boundaries relaxes to c = ±√ ε determined by the boundary value of ε . Moredetailed considerations, including other types of bound-ary conditions and appropriate discretization schemes,are discussed e.g. in Refs. [34].The field ε ( x, t ) related to the diffusive temperatureprofile is thus playing the role of a guiding field . Onecan think of, however, other types of fields ε ( x, t ), otherboundary and initial conditions for an experimentalsetup. As an example, one can assume that a chemi-cal agent is diffusing from one reservoir at the x = 0end of the tube, its concentration changes the local pH of the system, and thus may drive the C particles tophase separation, etc. Accordingly, we call hereafter ε the guiding field, and thus shall not restrict ourselves tothe temperature-like interpretation. III. CHARACTERISTICS OF THE DIFFUSIVEGUIDING FIELD ε ( x, t ) During its time evolution, the guiding field ε ( x, t ) willmodify locally the position of the spinodal line. At a fixedtime t , the spinodal density c s = p ε ( x, t ) / | c | at a given point x f = x f ( t ), therefore initi-ating locally a phase separation. The point x f ( t ) definesthe position of the instability front , which is thus deter-mined by the condition ε ( x = x f , t ) = 3 c . Behind thefront, which propagates to the right, the system becomeslocally unstable, and phase separates into a precipitationpattern of alternate high- and low-density regions of C .The diffusion equation (2) for ε ( x, t ) with the pre-scribed boundary and initial conditions (3) can be solvedthrough a simple Laplace transform method [35]. Oneobtains for x f an implicit equation comprising an infi-nite sum, ∞ X n =0 ( − n n + 1 exp (cid:20) − (2 n + 1) π (cid:18) tDL (cid:19)(cid:21) × cos (cid:20) (2 n + 1) π (cid:16) − x f L (cid:17)(cid:21) = π (1 + 3 c )8 . (5)The resulting trajectory of the instability front x f ( t ), aswell as its velocity v f ( t ) = dx f ( t ) /dt for a particularchoice of c are represented in Fig. 1. Note that whenthe spatial, temporal, and velocity variables are rescaled,respectively, by L , L /D , and D/L , as indicated on theaxis of these plots, then the curves for the trajectory and t D/L x f / L P . PSfrag replacements v f ( L/D ) x f /L . P PSfrag replacements v f ( L / D ) FIG. 1: Upper panel: Time evolution of the front position for c = − .
05 (solid line). The front moves diffusively for small t , x f ( t ) ∼ √ t (dotted line), while the large-time asymptoteis given by Eq. (6) (dashed line). Lower panel: The velocityof the instability front v f as a function of the front position x f (solid line). The short-time asymptote v f ( t ) ∼ / √ t (dot-ted line) and large-time behavior given by Eq. (7) (dashedline) are also displayed, with P denoting the crossover point.The scaling of the spatial, temporal, and velocity variables isdescribed in the text. velocity of the instability front are universal (for a givenvalue of c ).As can be seen in Fig. 1, the front moves diffusively atthe beginning, and it accelerates past a crossover point P where the acceleration is zero. The large-time asymptotefor the front position can be obtained by keeping onlythe leading n = 0 term in the sum (5), x f ( t ) ≈ L (cid:26) − π arccos (cid:20) π (1 + 3 c )8 exp (cid:18) π Dt L (cid:19)(cid:21)(cid:27) . (6)This approximate expression is valid provided L / π D = t min . t . t max = (cid:0) L /π D (cid:1) ln { / [ π (1 +3 c )] } where t min is the time when the n = 1 termin the sum (5) becomes negligible with respect to the n = 0 term, while t max represents a rough estimate ofthe time it takes the instability front to reach the end( x = L ) of the tube. The function given by Eq. (6) isshown in the upper panel of Fig. 1, and one can see thatthe asymptote is an excellent approximation past thecrossover point P .The corresponding asymptote for the velocity of thefront has the remarkable property that, when expressedin scaled variables and in terms of the position of thefront, it becomes independent even of the initial concen-tration c , LD v f = π π (cid:16) − x n L (cid:17) . (7)The above expression is displayed in the lower panel ofFig. 1 and one notices again that the approximation isvery good past the crossover point. IV. RESULTS
The coupled non-autonomous CH (1) and guidingfield (2) equations have been solved numerically for dif-ferent values of the initial density c , diffusion constant D , and length L of the tube. Figure 2 illustrates theearly stages of the cooling process, with the profiles of theconcentration c ( x ), guiding field ε ( x ), and spinodal line ± c s ( x ) = ± p ε ( x ) / t < t max (before theinstability front reaches the end of the tube). The con-centration field inside the high and low-density emergingbands relaxes rather rapidly to the instantaneous, local equilibrium values ± p ε ( x, t ), respectively.The pattern initiated by the instability front evolvesafterwards till reaching a stationary profile, made of al-ternate regions of c = ± stationary profile is still evolving through coars-ening and band coalescence, as predicted e.g. in [36].However (except eventually for some very closely-spacedbands, see below the comments on the plug ), its char-acteristic evolution time is usually well-beyond any rea-sonable experimental time [37]; from a practical point ofview one can therefore safely assume its stationarity.Three typical stationary patterns of the C -particle con-centration field are represented in Fig. 3.Before going into a more detailed analysis, let us enu-merate some general qualitative features of the emergingpatterns:(i) The total number of bands increases as D increases, x -1-0.500.51 c ( x ) , ε ( x ) ε (x)c(x)c s (x) x -1-0.500.51 ε ( x ) . c x f . FIG. 2: Early stage of pattern formation: snapshots of theconcentration field c ( x ) (continuous line), guiding field ε ( x )(dashed line), spinodal lines ± p ε ( x ) / equilibria ± p ε ( x ). The big dotrepresents the position of the instability front. The parame-ters are c = − . L = 1000, D = 4, and t = 4000. Theinset shows various possible profiles of the guiding field, seeSec. VI, namely: the usual diffusive configuration (dashed-dotted line); a step-like profile (dashed line); and a rigidparabola (continuous line). for fixed c and L .(ii) For fixed D and L , however, the number of bands de-creases with increasing | c | (approaching the spinodal).(iii) The first part of the pattern displays regular band-ing (i.e. increasing distance between consecutive bands),while a second part displays inverse banding . It is impor-tant to note that the transition from one type of patternto the other is related to the change in the behavior ofthe velocity of the guiding field, namely from the initialdiffusive-like motion, to the later-time accelerated one,see Fig. 1.(iv) In some situations, the pattern contains an initial plug , i.e. a rather wide initial region of constant concen-tration, see e.g. the second panel of Fig. 3. This effect hasalready been encountered in the usual Liesegang-patternformation [10, 11]. The plug may sometimes result fromthe coalescence, on a time scale of the order O ( t max ), of acertain number of very closely-spaced bands [37]. A plugcan also form at the end of the pattern, where the bandscan be again close enough to each other. Contrary tothe standard Liesegang pattern whose spatial extensionis only limited by the length of the tube, in our case thelength of the patterned region can thus be limited by thisband-coalescence effect.Let us consider now the characteristics of the patternsfrom a more quantitative perspective. Figure 4 shows aplot of the band positions x n (which are taken, conven-tionally, to be the points where c = 0, with an ascendentslope, dc ( x n ) /dx >
0, and are enumerated in the order x -1-0.500.51 c ( x ) x -1-0.500.51 c ( x ) x -1-0.500.51 c ( x ) FIG. 3: The numerical solution c ( x ) of the non-autonomousCahn-Hilliard equation (1) in the long-time limit t ≫ t max ,and for system size L = 4000. Upper panel: c = − .
05 and D = 1; middle panel: c = − .
05 and D = 8; and lower panel: c = − . D = 1. of their appearance, starting from the x = 0 end of the tube) as a function of n ; different values of the diffu-sion constant D were considered, for fixed c = − . L = 4000. The presence of a large initial plug mayhave some important effects on the n -dependence of x n for small n values. Accordingly, a simple and experimen-tally measurable functional expression is only expectedfor sufficiently large values of n , precisely as in the caseof the usual Liesegang patterns.
20 40 60 80 n x n D=1D=2D=4D=8D=16
FIG. 4: The position x n of the n -th band with respect to itslabel n for c = − . L = 4000, and different values of D . For the initial, regular-banding part of the pattern, ifenough bands present, one can fit the positions of thebands reasonably well with a geometric series, x n ∼ exp( n ˜ p ), as for a standard Liesegang pattern [10, 11].This is obviously related to the initial diffusive-like mo-tion of the instability front, that does not differ qualita-tively from the motion of the reaction front in the usualLiesegang configuration [12]. However, a power-law fit-ting cannot be excluded either, and further detailed workmeant to clarify this point is in progress and will be pub-lished elsewhere.For the second, inverse-banding part of the pattern, thepositions of the bands for large n -s can be fitted equallywell as x n ∼ ln n or with a power law x n ∼ n β , wherethe exponent β ≈ . . D . Since the corresponding distance λ n = x n +1 − x n between consecutive bands behaves like λ n ∝ n β − , theinequality β < λ n = x n +1 − x n as a function of n , for the same parameter values as in Fig. 4. One no-tices clearly the initial region of regular banding (with theeventual spurious initial plug) and the inverse banding,with the final plug. The tails of these plots for large- n values do not allow to discriminate further between thetwo above-suggested fittings of the relation between x n and n in the inverse-banding region.Finally, in Fig. 6 we plot the width w n of the n -th high-
20 40 60 80 n λ n D=1D=2D=4D=8D=16
FIG. 5: The distance λ n = x n +1 − x n between two successivebands as a function of n , for c = − . L = 4000, anddifferent values of D . One can notice outlier points at thebeginning and end of the lines. They correspond to the initialand final plug regions. density band as a function of n . It is remarkable that,except for a crossover region between direct and inversebanding, one can fit throughout, with a good approxima-tion: w n ≈ W x n + U . (8)For the regular-banding region
W >
0, and one can eas-ily justify this result simply by using mass-conservationarguments for the C -particles, as well as the geometricprogression of band positions, see [10, 11]. However, forthe inverse banding region W < P ofthe motion of the instability front, see Sec. III. V. THEORETICAL ARGUMENTS
Our goal is to devise a simple theoretical approach ableto explain the characteristics of the patterns observedin the numerical simulation of the non-autonomous CHequation (1), and to be used further on for predic-tive purposes. A basic element of our approach is thenumerical finding that the characteristics of the pat-terns are directly related to the motion of the instabilityfront. In particular, the local wavelength of the pattern λ n = x n +1 − x n (see Figs. 5-6) is related to the velocity v f = v f ( x f ) of the front (see the second panel of Fig. 1),as discussed below.Our approach is based on several assumptions, the va-lidity of which is verified a-posteriori by comparison of x n D=1D=2D=4D=8D=16 P PSfrag replacements w n FIG. 6: The width w n of the n -th band as a function of theband position x n , for c = − . L = 4000, and differentvalues of D . The vertical dashed line indicates the positionof the crossover point P (see Sec. III). The jumps at theendpoints of the lines are due to the plugs. the theoretical findings with the results of the numeri-cal simulations. Our first hypothesis is that the guidingfield moves faster than the diffusing C particles; there-fore, the phase-separation does not take place ahead theinstability front, but only behind it. Note, however, thatthe velocity of the front should not be too high either,since otherwise our second hypothesis about the quasi-stationarity may not be fulfilled. The meaning of thesecond hypothesis is that although the local value of thespinodal concentration c s ( x, t ) = p ε ( x, t ) / pulled by themotion of the guiding front, and consequently we can usethe standard results of the pulled-front theory [38, 39, 40]to establish the characteristics of the emerging pattern.Let us recall here the main results of the standardtheory. Consider an autonomous CH equation (1) with ε = constant throughout the system, and a uniform un-stable concentration c , | c | < c s = p ε/
3. A sharply-localized perturbation of this state will then evolve intoan instability front, with a well-defined velocity, leadingto phase-separation behind it and to the appearance ofa pattern of well-defined wavelength. Using linear sta-bility analysis arguments, one can easily compute boththe wavelength λ ∗ of the most unstable mode and the asymptotic velocity v ∗ of the instability front as a func-tion of the distance between the initial concentration andthe spinodal value. Namely, λ ∗ = 16 π √ √ √ / a − / , (9) v ∗ = 2( √ √ / a / , (10)where a ≡ c s − c ) = ε − c . Except for the cases whenone has band-coalescence (coarsening), this wavelengthprovides the wavelength of the asymptotic emerging pat-tern. By eliminating the parameter a between these twoexpressions, one obtains a direct relationship between theasymptotic wavelength of the pattern and the asymptoticvelocity of the instability front, λ ∗ = 9 . v ∗ ) / . (11)Note however that the relaxation of the system to thisasymptotic state goes rather slowly, like (1 /t ), bothfor the wavelength of the pattern and for the velocityof the instability front. Moreover, the transient effectstend to increase the wavelength of the pattern aboveits asymptotic value λ ∗ , see [38, 39, 40] for further details.Using the above results of the pulled-front theory, wemake now the Ansatz that the relationship (11) remainsvalid for our non-autonomous CH equation. More pre-cisely, we assume that the local wavelength of the patternis determined by the instantaneous/local velocity of theinstability front as λ n ≈ . v f ( x n )] / . (12)The physical picture underlying the above assumptionis the following. The instantaneous pulled-front veloc-ity v ∗ = v f ( t ) dictates , see Eq. (10), an instantanousvalue of the parameter a , let us call it a f ( t ). Thismeans that the local concentration in the vicinity of thequasi-stationary instability front adjusts rapidly to thevalue c f ( t ) corresponding to the parameter a f , namely a f ( t ) = ε ( x = x f ( t ) , t ) − c f ( t ).The comparison of the theoretical findings based onthe above Ansatz with the results of the numerical simu-lations is displayed in Fig. 7, where the local wavelengthof the pattern λ n is plotted versus x n for different valuesof D and L . This figure provides a double-check of theAnsatz. Namely,(i) If Eq. (12) is valid, then, since v f ( x n ) is universalunder appropriate scaling of space, time, and velocities(according to Sec. III), then the plots from the numer-ical results should merge when applying the rescaling λ n → λ n ( D/L ) / and x n → x n /L . This is, indeed,the case, as illustrated by both panels of Fig. 7.(ii) All the rescaled plots should fit the theoretical for-mula (12) shown by solid lines in Fig. 7.The agreement between our simple theoretical predic-tions and the results of the simulations is surprisinglygood. The only exceptions are a few outlier points corre-sponding, respectively, to the early band formation (ini-tial plug) and to the last bands close to the boundary(final plug). There is also a systematic initial and final x n /L λ n ( D / L ) / D=1, L=1000D=2, L=2000D=4, L=4000theory x n /L λ n ( D / L ) / D=4, L=500D=8, L=1000D=16, L=2000theory
FIG. 7: Comparison of the local wavelengths of the pat-tern as determined from numerical simulations (symbols) withthe theoretical results based on Eq. (12) (continuous line).The wavelengths λ n and the band positions x n , are rescaledas discussed in the text. Upper panel: c = − .
05 and
L/D = 1000. Lower panel: c = − .
05 and
L/D = 125.The vertical dashed lines indicate the position of the crossoverpoint P where the instability front has a minimal velocity (seeSec. III). The outlier endpoints are due to the plugs. mismatch, that may be due to the high acceleration ofthe instability front at the very beginning and the veryend of its motion along the tube (see Fig. 1), and thus tothe breaking of the quasi-stationarity hypothesis that liesat the basis of our Ansatz. Another origin of discrepancycan be the dynamics of the guiding field profile, that may,under some circumstances, fail the quasi-stationarity hy-pothesis.We note that the agreement between numerics and the-ory is better for large values of the velocity of the insta-bility front, i.e., for smaller values of L/D , as shown inthe upper panel of Fig. 7 as compared to the lower panel.This is probably related to a better, respectively worseadequacy of the basic hypothesis of a fast-moving frontas compared to the diffusion of the C particles. Finally,the fact that the numerical wavelengths are systemati-cally larger that the theoretically-estimated ones may bethe combined effect of slow relaxation to the asymptoticstate and the quasi-stationary nature of our configura-tion during the onset of the pattern (see the commentsabove on the effects of transients on the wavelength, inthe autonomous case). VI. PATTERN CONTROL
We address now the problem of controllability of thecharacteristics of the emerging pattern. It is obvious fromthe above results that both the qualitative (i.e., regularor inverse banding) and quantitative features (like totallength of the pattern, pattern local wavelength, width ofbands, etc.) can be controlled in the described configu-ration through an appropriate choice of the parameters L , c , and (to a less extent, as more difficult to manip-ulate) D . Moreover, these results can be described the-oretically in the frame of the pulled-front approximationthus providing a method for estimating the parametersof the patterns. However, this method of control, al-though very simple, is somewhat rigid, since the above-mentioned control parameters cannot be changed duringthe process, while, ideally, one requires an easily tuned,flexible, external tool of control. One can then thinkabout moving the tube with the gel (or maybe a thinfilm of gel) in a prescribed temperature profile, with avelocity that can be changed at any moment accordingto the needs. One achieves therefore a guiding field ε ( x, t )that can be externally tuned at any moment and point.For example, the simplest configuration one can imag-ine is an abrupt, step-like temperature profile that moveswith velocity v f , such that ε ( x, t ) = − x − x f ( t ))(0 x L ), where Θ(˙) designates the Heaviside stepfunction and x f ( t ) is the instantaneous position of thestep. If the motion is uniform v f = const . , then oneobtains equidistant banding . If the motion of the stepis accelerated or decelerated, then the pattern presents inverse-banding , respectively regular banding , with char-acteristics that depend on the details of v f = v f ( t ).Another simple option is to propagate a smooth, giventemperature profile along the tube, such that ε ( x, t ) = F ( x − x f ( t )). Now, the characteristics of the emergingpattern do not depend only on the velocity v f ( t ) of thepropagating rigid guiding field profile, but also on theshape of this profile.In order to illustrate these points, we considered, forcomparison, the emerging pattern in three situations(also illustrated in Fig. 2), namely for:(i) The diffusive guiding-field profile, as discussed in theprevious Sections, for a given set of parameters D , L , and c . Recall that the instability front moves with a velocity v f ( t ) described in Sec. III;(ii) A step-like profile of the guiding field that moves withthe same velocity v f ( t );(iii) Finally, a parabolic profile of guiding field, ε ( x, t ) = (cid:2) − A ( x f ( t ) − x − x ) Θ( x f ( t ) − x − x ) (cid:3) . One has ε = − x = x f + x and the parameter A is de-termined such that for x = x f one has ε = 1 − c . Asbefore, this rigid profile moves with the same velocity v f ( t ).The results of the numerical simulation are representedin Fig. 8, together with the theoretical result based onour Ansatz. One can notice that:(i) The pattern can be effectively controlled by the pro-posed methods. The effects are qualitatively the same asfor our usual configuration, but these new methods allowfor more flexible control.(ii) The pattern obtained for the parabolic-like profile(with x = 70) is closer to the pattern obtained for ourusual configuration, as well as to the theoretical predic-tions based on the pulled-front approximation; the pat-tern obtained for the step-like profile is much different.This convincingly illustrates the importance of the quasi-stationarity hypothesis for the pulled-front theory. In-deed, this basic ingredient is a good approximation bothfor our usual configuration and for the parabolic profile,but it is definitely absent in the case of the step-like pro-file, for which the associated abrupt jump in the localvalue of ε forbids any possibility of quasi-stationarity. x n /L λ n ( D / L ) / diffusivestep-like rigid-parabolatheory FIG. 8: Local wavelengths of the pattern λ n versus bandpositions x n , in appropriate rescaled variables. Symbols: nu-merical simulations for three different profiles of the guidingfield (see the text). Continuous line: theoretical calculationsbased on Eq. (12). The parameters are c = − . D = 16,and L = 2000. The vertical dashed line indicates the position,along the cylinder’s axis, of the crossover point P where theinstability front has a minimal velocity (see Sec. III). Theoutliers at the beginning and end of the lines correspond tothe plugs. VII. CONCLUSION
We have discussed the problem of how to control pre-cipitation patterns by bringing a system described by theCahn-Hilliard equation into an unstable state using a pre-scribed guiding field. It was shown that simple, physi-cally realizable fields, such as a temperature field gen-erated by a temperature jump at the boundary, is suf-ficient to generate rather complex precipitation patternseven in one dimension. The spacing characteristics ofthe patterns were determined numerically for the case ofa diffusive guiding field, and we developed a quantitativetheory for explaining the simulation results. The theoryis based on relating the velocity of the instability frontgenerated by the guiding field to the natural, pulled-frontvelocity of the phase-separation process which, in turn,controls the lengthscale of the pattern left in the wake ofthe moving front.From a theoretical point of view, our results suggestthat the inverse-banding phenomena observed in someLiesegang experiments may have an explanation in termsof a diffusive guiding field. This guiding field is perhapsnot a temperature field, but may be generated by thediffusion of some chemical species which do not take part in the reactions and the precipitation but may change e.g.the local pH value and thus influences the precipitationthresholds.As far as the technological applications are concerned,it appears that the problem of microfabrication of bulkstructures by chemical reactions and precipitation [2, 3,4, 5, 6, 7] is just in the first stages of its development. Theusefulness of this field will be decided on the possibilityof creating flexible ways to ensure controllability. Ourresults suggest experimentally feasible solutions for thecontrol of a particular precipitation process (formation ofLiesegang bands). Clearly, further studies are necessaryto develop new methods of control and to sort out thequestion of controllability in more complex cases. Acknowledgments
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