Spiral precipitation patterns in confined chemical gardens
Florence Haudin, Julyan H. E. Cartwright, Fabian Brau, A. De Wit
SSpiral precipitation patterns in confined chemicalgardens
Florence Haudin ∗ , Julyan H. E. Cartwright † , Fabian Brau ∗ and A. De Wit ∗ ∗ Nonlinear Physical Chemistry Unit, Facult´e des Sciences, Universit´e Libre de Bruxelles (ULB), CP231, 1050 Brussels, Belgium, and † Instituto Andaluz de Ciencias de laTierra, CSIC-Universidad de Granada, Campus Fuentenueva, E-18071 Granada, Spain.Submitted to Proceedings of the National Academy of Sciences of the United States of America
Chemical gardens are mineral aggregates that grow in three dimen-sions with plant-like forms and share properties with self-assembledstructures like nanoscale tubes, brinicles or chimneys at hydrother-mal vents. The analysis of their shapes remains a challenge, as theirgrowth is influenced by osmosis, buoyancy and reaction-diffusionprocesses. Here we show that chemical gardens grown by injec-tion of one reactant into the other in confined conditions featurea wealth of new patterns including spirals, flowers, and filaments.The confinement decreases the influence of buoyancy, reduces thespatial degrees of freedom and allows analysis of the patterns bytools classically used to analyze two-dimensional patterns. Injectionmoreover allows the study in controlled conditions of the effects ofvariable concentrations on the selected morphology. We illustratethese innovative aspects by characterizing quantitatively, with a sim-ple geometrical model, a new class of self-similar logarithmic spiralsobserved in a large zone of the parameter space.
Chemical gardens | Pattern formation | Self-similarity | C hemical gardens, discovered more than three centuriesago [1], are attracting nowadays increasing interest indisciplines as varied as chemistry, physics, nonlinear dynam-ics and materials science. Indeed, they exhibit rich chemical,magnetic and electrical properties due to the steep pH andelectrochemical gradients established across their walls dur-ing their growth process [2]. Moreover, they share commonproperties with structures ranging from nanoscale tubes incement [3], corrosion filaments [4] to larger-scale brinicles [5]or chimneys at hydrothermal vents [6]. This explains theirsuccess as prototypes to grow complex compartmentalized orlayered self-organized materials, as chemical motors, as fuelcells, in microfluidics, as catalysts, and to study the origin oflife [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. However, despitenumerous experimental studies, understanding the propertiesof the wide variety of possible spatial structures and develop-ing theoretical models of their growth remains a challenge.In 3D systems, only a qualitative basic picture for the for-mation of these structures is known. Precipitates are typicallyproduced when a solid metal salt seed dissolves in a solutioncontaining anions such as silicate. Initially, a semi-permeablemembrane forms, across which water is pumped by osmosisfrom the outer solution into the metal salt solution, furtherdissolving the salt. Above some internal pressure, the mem-brane breaks, and a buoyant jet of the generally less denseinner solution then rises and further precipitates in the outersolution, producing a collection of mineral shapes that resem-bles a garden. The growth of chemical gardens is thus drivenin 3D by a complex coupling between osmotic, buoyancy andreaction–diffusion processes [19, 20].Studies have attempted to generate reproducible micro-and nano-tubes by reducing the erratic nature of the 3Dgrowth of chemical gardens [10, 11, 13, 15, 21]. They havefor instance been studied in microgravity to suppress buoy-ancy [22, 23], or by injecting aqueous solutions of metallicsalts directly into silicate solutions in 3D to dominate osmoticprocesses by controlled flows [10, 11]. Analysis of their mi- crostructure has also been done for different metallic salts,showing a difference of chemical composition on the innerand the outer tube surfaces [24, 25]. The experimental char-acterization and modeling of the dynamics remains howeverdauntingly complex in 3D, which explains why progress inquantitative analysis remains so scarce.We show here that growing chemical gardens in a confinedquasi-2D geometry by injecting one reagent solution into theother provides a new path to discover numerous original pat-terns, characterize quantitatively their properties and explaintheir growth mechanism. A large variety of structures in-cluding spirals, filaments, worms, and flowers is obtained in ahorizontal confined geometry when varying the reagent con-centrations at a fixed flow rate. The patterns differ from thosein 3D as the growth methodology decouples the different ef-fects involved in the formation of classical chemical gardens.The buoyancy force is reduced by the vertical confinementwhile injection decreases the influence of osmotic effects. Experimental results
Experiments are conducted in a horizontal Hele-Shaw cell con-sisting in two transparent acrylate plates separated by a smallgap initially filled by a solution of sodium silicate. A solutionof cobalt chloride (CoCl ) is injected radially from the cen-ter of the lower plate at a fixed flow rate. Upon contact anddisplacement of one reactant solution by the other one, pre-cipitation occurs and various dynamics and patterns are ob-served when the concentration of sodium silicate and CoCl are varied (see Fig.1).The global trend of the phase diagram is that, if onereagent is much more concentrated than the other one, arather circular precipitation pattern is obtained. This pre-cipitate is concentrated at the outer rim as dark petals of flowers out of which viscous fingers grow if the sodium sili-cate is the more concentrated reagent. At the beginning ofthe injection, precipitation occurs inside the viscous fingerswhereas at longer times, the precipitate lags behind them. If,on the contrary, the metal salt is much more concentratedthan the sodium silicate, a compact circular pink precipitategrows radially. Above a critical radius, we observe a desta-bilization of the circle rim towards small scale hairs growingradially with a characteristic wavelength.When the concentration of both reactants is in the largerange of Fig.1, thin filaments growing with complex turnarounds are observed. Along the fifth column of the phase dia-gram, the transition from flowers to filaments occurs smoothlywhen the concentration of CoCl is increased at this fixed largeconcentration in sodium silicate.Intermediate concentrations are characterized by worms as seen in the lower middle zone of the diagram. These wormpatterns are reminiscent of structures observed in micellar sys-tems when a viscous gel product is formed in-situ at the in-terface between reactive aqueous solutions [26]. At later time,terrace-like precipitate layers grow along the first structure. a r X i v : . [ c ond - m a t . s o f t ] D ec .25 0.625 1.25 3.125 6.25 . . . . FilamentsHairs Spirals FlowersWorms S S S S S S S Lobes C oba l t c h l o r i de c on c en t r a t i on C ( M ) Sodium silicate concentration C2(M)
Fig. 1.
Experimental patterns . Classification of confined gardens in a parameter space spanned by the concentration C of the injected aqueous solution ofcobalt chloride and the concentration C of the displaced aqueous solution of sodium silicate. The diagram is divided in different colored frames referring to the various classesof patterns observed: lobes (dotted dark blue), spirals (solid red), hairs (dashed black), flowers (dashed-dotted green), filaments (long dashed-dotted purple) and worms (longdashed blue). The separation between each domain is not sharp as, for example, worms are sometimes delimited by curly boundaries that are reminiscent of spirals. The spiralcategory is divided into sub-categories S i analyzed in Fig.3 and in the SI Text. The injection rate is Q = ×
15 cm, shown 15 s afterinjection starts. Movie S1 shows some of the dynamics.
Along the lower line of the phase diagram, i.e. at a largeconcentration in CoCl , the transition upon an increase of thesodium silicate concentration from hairs to worms and even-tually filaments transits at intermediate values via the mostcompact precipitate structure from all those observed.In the upper middle zone (squares S to S in Fig.1), spi-rals growing upon successive break-ups of precipitate walls areobserved. A zoom in on some of these spirals (Fig.1) shows that, after a longer time, other precipitates with a rich varietyof colors start growing diffusively out of their semi-permeablewalls. A preliminary test with Raman microscopy has shownthat some green precipitates (like in Fig.1 for instance) ismade of Co(OH) and Co O , compounds also found in theinner surface of 3D tubes [25]. This highlights the fact thatdifferent types of solid phases can form in the reaction zonebetween the two solutions [27]. BC D
Fig. 2.
Spiralling precipitates . Panels A and B feature spirals andthe subsequent precipitates with various colors growing diffusively out of the semi-permeable spiral walls when cobalt chloride is injected into sodium silicate ( A : case S and B : case S of Fig. 1). Pictures are taken a few minutes after the end of theinjection. Panels C and D shows examples of spirals in a reverse chemical gardenobtained when sodium silicate in concentration 0.625 M is injected into pink cobaltchloride 0.25 M (inverse of case S , C : t = 24 s, Q = 0 . mL/s and D : fewminutes after the end of injection, Q = 0 . mL/s). The scale bars correspond to1 cm. We note that friction phenomena with the cell plates donot seem to be crucial to understand the properties of thepatterns as no stick/slip phenomena are observed during thegrowth of the patterns. Preliminary data also show that thearea A enclosed by the visible spiral, worm and filament pat-terns scales linearly with time with an angular coefficient pro-portional to the flow rate Q divided by the gap b of the Hele-Shaw cell, i.e. A = Qt/b . This suggests that the relatedpatterns hence span the whole gap of the reactor. Additionalexperiments must be conducted to obtain statistical informa-tion on such growth properties for these patterns and for theother structures observed.In the following, we will focus on the spiral shape precip-itate, since it is a robust pattern existing over a quite largerange of concentrations and is also formed in reversed gardens, i.e. when sodium silicate is used as the injected fluid (Fig.1Cand D).
Geometrical model
The spiral formation mechanism can be understood with aminimal geometrical argument. Consider a reagent 1 injectedradially from a source point S into reagent 2 in a 2D system(Fig.3A). The contact zone between the reactive solutions, inwhich precipitation occurs, initially grows as a circle of ar-bitrarily small radius r . If, because of further injection, thislayer of precipitate breaks at a critical value r = r c , the newprecipitate, formed as the bubble of reagent 1 expands, pushesthe already existing solid layer. As a result, the branch of solidprecipitate starts being advected out of the growing bubbleand rotates as a whole around the breaking point which lateron identifies to being the tip of the spiral (see also Movies S2and S3 of growing spirals). An arc of spiral is hence observed to develop with its tip moving in the fixed frame of referencecentered on S . It further grows by precipitation at its taillocated on the circle of growing radius r ( t ).The equation of the resulting curly-shaped precipitationlayer is obtained by considering two infinitesimally close timesteps as shown in Fig.3A. At time t + dt , the precipitate layerformed at time t is pushed away by the newly added materialand rotates by an angle dθ . The precipitate layer produced be-tween t and t + dt is created from the reaction occurring at thenew section of the contact line between the two reagents [28].Since this new section of contact line is generated by the bub-ble expansion, its length ds is proportional to the increase ofperimeter ( ∼ dr ) of the expanding bubble, namely ds = θ dr ( t ) [1] where θ is a constant growth rate controlling the length ofthe precipitate layer created as the bubble radius increases.A large value of θ implies that the length ds of the new seg-ment of precipitate created at t + dt is large compared to theincrease of length dr along the radial direction and leads to amore coiled spiral.Upon integration of Eq.(1) we get s = θ ( r − r c ), where s is the length of the curve from the tip to the tail. The con-stant of integration was fixed by considering that the tip ofthe curve ( s = 0) is generated from a circle of radius r c . Inaddition, the radius r of the expanding bubble coincides withthe radius of curvature R ( s ) of the tail since the bubble isalso the osculating circle (see Fig.3A). Therefore, we obtain R ( s ) = r c + s/θ which is the Ces`aro equation of a logarithmicspiral giving the evolution of the radius of curvature along thecurve as a function of the arclength [29, p. 26] (see SI Textfor more details).Alternatively, we can also proceed as follows to obtain theexpression of the spiral curve in polar coordinates. To derivethis equation, the relation existing between dr and dθ shouldbe known everywhere along the curve in the fixed frame ofreference centered around S . For non moving curves, this re-lation is usually found by analyzing how dr varies in a sectorof constant central angle dθ which rotates to span the entirecurve. In our case, the growing arc of spiral is moving in afixed system of coordinates centered on the source point S .Its r ( θ ) equation is therefore alternatively obtained by con-sidering a non moving sector of constant central angle dθ asin Fig.3A and analyzing how dr varies in it when the curvespans this angular sector thanks to its motion (see SI Text).As seen in Fig.3A, the length ds of the precipitate layeradded at t + dt in the fixed sector of angle dθ is given by ds = r ( t + dt ) dθ which, at first order in dt and dθ , reduces to ds = r ( t ) dθ. [2] Combining (1) and (2), we get dθ = θ drr [3] which is readily integrated to yield r = r e θ/θ . [4] r is a constant of integration that can be computed for eachspiral if we note that θ = 0 fixes the starting point of thespiral i.e . its tip. Reminding that the tip originated from thebreaking by injection of the initial small circle of radius r c , theradius of curvature of the spiral at θ = 0 is nothing else than r c . Therefore, r c = r (cid:112) θ /θ which relates the constantof integration r to r c . It is of interest to note that Eq.(4)exactly describes the (blue) segments of spiral of Fig.3A in θ dr dsr ( t ) r ( t + dt ) A dsS r r θ / θ S S S S S S S Logarithmic spiral B Cs ( t ) r c r TailTip
Fig. 3.
Spiral growth mechanism and scaling of logarithmic spiral-shaped precipitates. A,
Schematic of the growth mechanism of thecurly-shaped precipitates during an infinitesimal interval of time where r ( t ) is the radius of the expanding bubble of injected reagent and S is the point source. B, Scalinglaw for the experimentally measured evolution of the radial distance r/r as a function of the scaled angle θ/θ for 173 spirals (9 experiments) corresponding to the sevendifferent categories S i in Fig. 1. The inset shows a superimposition of a logarithmic spiral on a spiraled precipitate (the scale bar corresponds to 2 mm). The procedures forthe measurements of r ( θ ) and the distributions of r and θ are given in the SI Text. the frame of reference centered on S provided this segment istranslated and rotated such that the center C S of the spiralcoincides with S (see SI Text).Equation (4) constructs a logarithmic spiral in polar co-ordinates and quantitatively describe structures in many nat-ural systems such as seashells, snails, or the horns of animalswhere the growth mechanism preserves the overall shape bythe simple addition of new material in successive self-similarsteps [30]. It appears thus logical that a similar logarithmicshape is recovered here in the case of regular additional pre-cipitation at the tail of a growing arc of solid precipitate uponfurther injection of reactants at a fixed flow rate. Equation(4) quantitatively describes the spiral structures observed ina large part of our phase diagram as demonstrated in the nextsection. Comparison between experiments and model
To test our geometrical model, the radii of 173 spirals ob-served in 9 experiments for 7 pairs of concentrations (sectors S i of Fig.1) have been measured as a function of the polarangle (see Fig.3B and SI Text). We have selected precipitatelayers which were not deformed by other structures growingin their neighborhood or fractured. Among them, we havealso selected sufficiently coiled segments such that their po-lar angle span the interval [0 , θ max ] with θ max > .
44 (140 ◦ )in order to obtain significant constraints on the model, i.e. θ/θ large enough in Fig.3B. Indeed, segments which are notsufficiently coiled cannot be used to determine the type ofspiral emerging in the system. If the interval spanned by θ is too small, Eq.(4) is hardly distinguishable from the equa-tion r = r (1 + θ/θ ) describing an archimedean spiral. Thedistribution of θ max among the analyzed spirals is shown inFig.S7. As seen in Fig.3B, where the radial distance and thepolar angle are rescaled by r and θ respectively, all the ana-lyzed spiral profiles collapse onto an exponential master curve,illustrating that indeed they are all logarithmic to a good ac-curacy.The dispersion occurring at low θ (near the spiral center)is related to the fact that the experimental spiral structuresemerge from an arc of an initial tiny circular section of radius r c . 63 other spirals corresponding to the inverted cases S and S ( i.e. sodium silicate injected into cobalt chloride) have alsobeen analyzed and prove to follow the same logarithmic scal-ing law (see Fig.S4H). We note that, in our confined gardens,the spiral growth continues only as long as the precipitate canpivot within the system. When it becomes pinned by encoun-tering a solid wall or another precipitate, the spiral growthceases, the membrane breaks, and a new radial source is pro-duced, leading regularly to a new fresh spiral. This behavioris reminiscent of the periodic pressure oscillations reported inthe growth of some 3D chemical gardens [13, 31].Thus self-similar logarithmic spirals emerging in a signifi-cant part of the phase diagram can be quantitatively analyzedon the basis of a simple geometrical argument. The model,which describes an isolated source producing a circular pre-cipitation zone that then ruptures, proves to be robust evenwhen spirals are interacting, as in the current experiments. Conclusions
This work shows that new insight into the complexity ofself-assembled chemical garden structures can be obtained bygrowing them by injection of reactive solutions in confinedgeometries. The control of the concentrations of the reagentsand of the flow rate will allow the study of phase diagramsin reproducible experimental conditions [15, 32], as well asthe switch from dominant reaction–diffusion processes to flow-driven ones. Moreover, the quasi-2D nature of the precipitateswill permit, as done here, an easier characterization of the pat-terns using tools of classical 2D pattern selection analysis [33].Modeling and numerical simulations of this injection-drivenaggregation process will thus be simplified. This will facilitatethe analysis of growth mechanisms and of the relative effectsof reactions, hydrodynamics and mechanics in the resultingstructure; a prerequisite to the rational design of complex,hierarchical microarchitectures [10, 34].
Materials and Methods
The experimental set-up is a horizontal Hele-Shaw cell consisting of two transparentacrylate plates of size 21.5 cm x 21.5 cm x 0.8 cm separated by a gap of 0.5 mm andinitially filled with a sodium silicate solution of concentration C . A cobalt chloridesolution of concentration C is injected into the sodium silicate solution of concen-tration C . Acrylate was chosen instead of glass to avoid interactions between the · O)(Sigma-Aldrich). The sodium silicate solution is prepared from a commercial aqueoussolution (Sigma-Aldrich), with formula Na (SiO ) x × H O and the compositionSiO ≈ O ≈ ACKNOWLEDGMENTS.
We thank P. Borckmans and C.I. Sainz-D´ıaz for discus-sions. We are very grateful to K. Baert and I. Vandendael from Vrije UniversiteitBrussel (VUB) for the Raman analysis. A.D., F.B. and F.H. acknowledge PRODEXand FRS-FNRS (FORECAST project) for financial support. J.H.E.C. acknowledgesthe financial support of MICINN grant FIS2013-48444-C2-2-P.
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Intrinsic equation.
We give here an alternative and equiva-lent derivation of the equation describing the curve generatedby the geometrical model. This derivation is independent onthe coordinate system and leads to a so-called natural (or in-trinsic) equation giving the evolution of the local radius ofcurvature of the curve as a function of the arclength [29, p.26].We still consider the system at two infinitesimally closetime steps such as the length of the curve has increased by anamount ds given by Eq.(1) of the main text, namely ds = θ dr b = θ dr b dt dt, [5] where r b ( t ) is the radius of the bubble of injected reagent attime t (see Fig.4A). The curve is parametrized by its arclength s such that the value of s of an arbitrary point P is equal tothe length of the curve measured from the first point P to P (see Fig.4A). As explained in the main text, P is generatedfrom an initial small circle of radius r b (0) = r c . The radius ofcurvature at s = 0 is thus given by R (0) = r c . At time t , theradius of curvature of the last point of the curve at its tail isgiven by R ( s ) = r b ( t ) . [6] Similarly, at time t + dt , the radius of curvature of the lastpoint of the curve is given by R ( s + ds ) = r b ( t + dt ) = r b ( t ) + dr b dt dt, [7] where we used a first order expansion in dt . Therefore, thedifference between Eq.(7) and Eq.(6) leads to R ( s + ds ) − R ( s ) = dRds ds = dr b dt dt, [8] where we used a first order expansion in ds . Using the expres-sion (5) of ds , we obtain dRds = 1 θ . [9] Integration of this last equation leads to R ( s ) = R (0) + sθ = r c + sθ . [10] The radius of curvature is thus a linear function of the ar-clength. This is the Ces`aro equation of a logarithmic (equian-gular) spiral [29, p. 26]. Let us now obtain the equation ofthe curve in polar coordinates.
Polar equation.
We have thus obtained above the curvature κ as a function of the arclength s : κ ( s ) = 1 R ( s ) = θ θ r c + s . [11] The relation between the Ces`aro equation and the Cartesiancoordinates is [29, p. 26] x ( s ) = (cid:90) cos ¯ κ ( s ) ds = θ r c + s θ (cos ¯ κ ( s ) + θ sin ¯ κ ( s )) , [12] y ( s ) = (cid:90) sin ¯ κ ( s ) ds = θ r c + s θ (sin ¯ κ ( s ) − θ cos ¯ κ ( s )) , [13] where, ¯ κ ( s ) = (cid:90) κ ( s ) ds = θ ln( θ r c + s ) + K, [14] and K is a constant of integration which fixes the orientationof the curve in space and is fixed below. The constants ofintegration in Eq.(12) correspond to translations of the curveand are set to zero.Using Eqs. (12) and (14), the polar equation is obtainedas follows r ( s ) = (cid:112) x ( s ) + y ( s ) = θ r c + s (cid:112) θ , [15] and θ ( s ) = arctan (cid:16) yx (cid:17) = arctan (cid:18) sin ¯ κ ( s ) − θ cos ¯ κ ( s )cos ¯ κ ( s ) + θ sin ¯ κ ( s ) (cid:19) = arctan (cid:18) tan ¯ κ ( s ) − θ θ tan ¯ κ ( s ) (cid:19) = arctan [tan(¯ κ ( s ) − arctan θ )]= ¯ κ ( s ) − arctan θ = θ ln( θ r c + s ) + K − arctan θ . [16] Eliminating s between Eqs. (15) and (16) we have r = e (arctan θ − K ) /θ (cid:112) θ e θ/θ . [17] The constant K can be fixed by imposing that s = 0 cor-responds to θ = 0 meaning that the radius of curvature ofthe curve (17) at θ = 0 should be equal to r c . This leads to K = arctan θ − θ ln( θ r c ) and Eq.(17) reduces to r = r c θ (cid:112) θ e θ/θ ≡ r e θ/θ . [18] Discrete algorithm
The curves generated by the proposed mechanism and de-picted schematically in Fig.4A can also be directly constructedusing the following discrete algorithm (Fig.4B). We start froma circle of radius r c with a point P at the top which representsthe first point of generated curve.1. The radius of the circle is increased by a given ∆ r . Theordinate of P is increased by ∆ r and then rotated by anangle ∆ θ . A point P is added at the top of the circle andrepresents a new amount of precipitate.2. The radius of the circle is increased by a constant ∆ r . Theordinates of P and P are increased by ∆ r and then areboth rotated by an angle ∆ θ . A point P is added at thetop of the circle.The procedure is then iterated to generate the next points.The discrete angle used is provided by Eq.(3) of the maintext: ∆ θ = θ ∆ r/r . Figure 4B shows the generation of thefirst few points using a large value of ∆ r for clarity. Fig-ure 4C shows the resulting structures emerging from the 2000and 4000 iterations of the algorithm using a smaller value of∆ r . Figure 4D shows that these two curves, once properlyrotated and translated, are exactly described by Eq.(4) usingthe same value of θ and a value of r such that the radius ofcurvature at P is equal to the radius r c of the initial smallcircle, namely r = r c θ / (1 + θ ) / . r r c = r b (0) P P P P P P − − n = 4000 n = 2000 P P − − θ r P n = 2000 n = 4000Eq. (13) − −
10 0 10 20 30
A B C D dr b r b ( t ) r b ( t + dt ) s ss+ds ( s = 0) P ( s = 0) P Fig. 4. A, Schematic of the growth mechanism of the curly-shaped precipitates during an infinitesimal interval of time. B, First few points generated by this mechanismand obtained from a discrete algorithm using ∆ r = 0 . , θ = 2 and r c = 1 . The red straight lines are added to help visualize the emerging structure. Due to symmetry,the generation of only one curve is shown. C, Curves generated from a discrete algorithm using ∆ r = 0 . , θ = 2 and r c = 1 . The curves obtained after n = 2000 and n = 4000 iterations are shown and correspond to a bubble of injected reagent having a radius of r c and r c respectively ( n ∆ r + r c ). D, If the emerging curvesare logarithmic spirals, the position of the spiral center can be deduced from the position of the center of curvature of P (see section “Spiral analysis” below). In the contextof this algorithm, the position of the center of curvature of P is tracked during the growth process. The total angle of rotation of the curves during the growth is simplyequals to the sum of all ∆ θ applied. Therefore, these two curves can be properly rotated and translated such that their centers coincide with the origin of coordinates. Thesecurves are exactly described by Eq.(4) with θ = 2 and r = r c θ / (1 + θ ) / (cid:39) . , such as its radius of curvature at P ( θ = 0 ) is equal to r c . Spiral analysis
Each experiment is recorded by taking photos at regular timesteps adapted according to the injection rate. Typically thetime interval between two photos is 1 s for the injection ratesused. The pattern is then analyzed at one given time for eachexperiment when the spirals are sufficiently developed and/orwhen the overall pattern is as large as the field of view. Toanalyze the spiral observed in our experiments, we measurethe evolution of the spiral radii, r , as a function of the polarangle θ . By convention, the starting point P of the spiral,which is the closest to the spiral center, is characterized by θ = 0 and r = r (Fig.5). However, the exact position of thespiral center, C S , from which the radii should ideally be mea-sured, is not known. In this section, we explain the methodused to overcome this difficulty. System of coordinates centered on the spiral center.
Theequation of a logarithmic spiral written in a system of co-ordinates centered on the spiral center, C S , is given by r = r e θ/θ , [19] where r and θ are constant parameters. If the spiral radii, r , are measured from the exact position of the spiral center, C S , then the evolution of r as a function of the polar angle θ can be fitted using Eq.(19). The measured spiral radii andthe polar angle can then be rescaled by r and θ respectivelyto produce the graph displayed in Fig.3B of the main text. Arbitrary system of coordinates.
The exact position of the spi-ral center is generally not known a priori and the radii aremeasured in an arbitrary system of coordinates whose origindoes not coincide with C S . In such a system of coordinatescentered on say C A , the expression of a logarithmic spiral isno longer given by the simple form (19). In this section, wederive the general expression describing a logarithmic spiraloff centered with regard to C S which is used to fit the data.The approximate position of the spiral center, C A , used tomeasure the spiral radii is obtained from the osculating circle passing through P . Figure 5A shows this construction onan exact logarithmic spiral to illustrate and test the proposedprocedure. Figure 5B shows the evolution of the spiral radii, r , as a function of the polar angle, θ , when the radii are mea-sured from the exact ( C S ) and approximate ( C A ) positions ofthe spiral center respectively. The polar coordinates of thespiral obtained from the approximate position of the centeris noted ( r (cid:48) , θ (cid:48) ) whereas the polar coordinates of the spiralobtained from the exact position of the center is noted ( r, θ ).These two curves are obviously equivalent and describethe same spiral. As shown in Fig.4, they are just measured intwo different systems of coordinates related by a rotation anda translation as (cid:18) xy (cid:19) = (cid:18) cos ϕ − sin ϕ sin ϕ cos ϕ (cid:19) (cid:18) x (cid:48) y (cid:48) (cid:19) − (cid:18) x y (cid:19) , [20] with ϕ = arcsin (cid:18) y r (cid:48) (cid:19) , x (cid:48) = r (cid:48) cos θ (cid:48) , y (cid:48) = r (cid:48) sin θ (cid:48) . [21] x and y are the Cartesian coordinates of C A in the ( x, y )system of coordinates whose origin coincide with the spiralcenter C S . Since r (cid:48) , θ (cid:48) and r (cid:48) are the quantities measuredin practice, the curve ( r, θ ) obtained from the exact positionof the center can thus be reconstructed once x and y areknown by using r = (cid:112) x + y , θ = arccos (cid:16) xr (cid:17) . [22] We show below how x and y can be obtained.The spiral parameters, r and θ , together with the trans-lation parameters of the center, x and y , are obtained all atonce by fitting the data by the general expression of a loga-rithmic spiral valid in an arbitrary system of coordinates. Notice that if the spiral is exactly logarithmic and if the approximate center is constructed usingthe osculating circle passing through P , then x = 0 as seen on Fig.5A. In practice, the spiralsare obviously never exactly logarithmic and the position of the osculating circle is never perfect.Consequently, one needs to allow for a non vanishing value for x in the procedure. θr r P C S C A Measured radiiLogarithmic spiralOsculating circleSpiral center, C S Approximate center, C A −5−4−3−2−1012−4 −2 0 2 4 6 8 B θr Measured from the spiral center, C S Measured from the approximate center, C A Reconstructed012345678 0 1 2 3 4 5 6 7
Fig. 5. A, Logarithmic spiral together with the osculating circle passing throughthe point P which is the closest to the spiral center. The center of the osculatingcircle, C A , is used as approximate center to measure the spiral radii. The black dotesshow where the radii are measured. C S indicates the position of the spiral center. B, Plot of the radius r of the spiral as a function of the polar angle θ using the exactand the approximate positions of the spiral center. The curve reconstructed from thedata obtained with the approximate position of the spiral center is also shown (orangedots) and agrees well with the spiral curve measured directly from the exact centerposition C S . This general expression is simply obtained by consideringthe reverse of the transformation (20) (cid:18) x (cid:48) y (cid:48) (cid:19) = (cid:18) cos ϕ sin ϕ − sin ϕ cos ϕ (cid:19) (cid:18) x + x y + y (cid:19) , [23] with x = r e t/θ cos t, y = r e t/θ sin t, [24] P x'y'y xr' φr Logarithmic spiralSpiral center, C S Approximate center, C A Fig. 6.
Representation of the two systems of coordinates. The system of coordi-nates ( x, y ) is centered on the spiral center C S whereas the systems of coordinates ( x (cid:48) , y (cid:48) ) is centered on C A and is used to measure the spiral radii. and ϕ is given by Eq.(21). The parametric equations, where t is the parameter, describing a logarithmic spiral in an arbi-trary system of coordinates is finally given by r (cid:48) ( r , θ , x , y ; t ) = (cid:112) x (cid:48) + y (cid:48) [25a] θ (cid:48) ( r , θ , x , y ; t ) = arccos (cid:18) x (cid:48) r (cid:48) (cid:19) . [25b] The values of the parameters r , θ , x and y are obtained foreach spiral by fitting the expression (25) to the measured val-ues of r (cid:48) and θ (cid:48) using a nonlinear regression procedure (Math-ematica).At this stage, since x and y are known, we apply thetransformation (20) to the data in order to obtain the mea-sured curve in a system of coordinates centered on the spiralcenter. The result is shown in Fig.5B with a very good agree-ment compared to the measurements performed directly in asystem of coordinates centered on the spiral center. This illus-trates the correctness of the procedure proposed to treat thedata. Finally, the transformed data are rescaled by r and θ to produce the graph displayed in Fig.3B of the main text. Results from the analysis
The results of the analysis described in the previous sectionare gathered in Fig.3B of the main text. The same resultsare presented here separately for each categories S i defined inFig.1 of the main text. The spirals are logarithmic in goodapproximation in each of the seven sectors S i of the phase dia-gram (Fig.7 below). Only the sector S displays more disper-sion for low values of θ/θ but at larger values of the rescaledpolar angle, the spirals follow closely the evolution of a log-arithmic spiral. For information, we also show in Fig.7H theresults obtained for the inverted case, where sodium silicateis injected into cobalt chloride, corresponding to the sectors S and S . Those spirals are also logarithmic.In Fig.8, we show the distributions of the values of r and θ characterizing all the analyzed spirals. Those distributionsshow also the contributions of each sector S i . Notice thatthose distributions have a meaning only if r and θ are bothindependent on the reagent concentrations (or if the depen-dence is weak). It seems that this is roughly the case byinspecting the contributions of each sector S i . However, thenumber of analyzed spirals per sector is not large enough todraw definitive conclusions. The distributions of r and θ areboth rather well fitted by a log-normal distribution f ( x ; µ, s ) = λx e − (ln x − µ )22 s . [26] The expectation value E is then given by E = e µ + s / , [27] and the standard deviation σ is σ = (cid:16) e s − (cid:17) / E. [28] We find r = (0 . ± .
20) mm , θ = 1 . ± . . [29] Within our minimal geometric model, the radius of cur-vature of the spiral at its starting point P should be close to The only exception concerns the sector S which is characterized by larger values of r ( r (cid:38) he radius r c of the initial circle of solid precipitate before thesolid layer breaks. The radius of curvature R of a logarithmicspiral is given by R = r e θ/θ (cid:112) θ θ . [30] Consequently, at the point P ( θ = 0), the radius of curvature R P ≡ r c is given by r c = r (cid:112) θ θ . [31] The distribution of r c in our experiments is displayed inFig.9A and also follows a log-normal distribution. We find r c = (0 . ± .
24) mm . [32] Three typical spirals constructed by considering a constantvalue of r c equal to its expectation value and θ varying byone standard deviation around its expectation value are shownin Fig.9B. Those three spirals are generated from the sameexpanding bubble of reagent having a radius equals to 40 r c .Consequently, they all have the same couple of radii of cur-vature at their end points. An animation showing the growthof these spirals can be found as Supporting Information (seeMovie S2). Movie S3 shows a qualititative comparison be-tween the growth a spiral observed experimentally and a spiralobtained from the geometrical model.Finally, Fig.10 shows the distribution of the maximalvalue, θ max , of the polar angle characterizing each analyzedspiral. As explained in the main text, we choose θ max > . ◦ ) such that the maximal value of θ/θ for each spiral islarge enough to obtain a relevant comparison with the model. S Logarithmic spiral r / r q /q S Logarithmic spiral r / r q /q S Logarithmic spiral r / r q /q S Logarithmic spiral r / r q /q S Logarithmic spiral r / r q /q S Logarithmic spiral r / r q /q S Logarithmic spiral r / r q /q S invertedS invertedLogarithmic spiral r / r q /q A BC DE FG H
Fig. 7.
A-G , Evolution of the rescaled spiral radii, r/r as a function of the rescaled polar angle, θ/θ , for each category S i identified in Fig.1 of the main text. H , Evolution of the rescaled spiral radii, r/r as a function of the rescaled polar angle, θ/θ , for the inverted case, where sodium silicate is injected into cobalt chloride,corresponding to the sectors S and S . (mm) Log-normal S S S S S S S A θ B Log-normal S S S S S S S Fig. 8. A, Distribution of r and B, distribution of θ for all analyzed spirals. The contributions of each sector S i is indicated. The log-normal distributions (26) arecharacterized by µ = − . and s = 0 . for r and µ = 0 . and s = 0 . for θ . r c (mm) Log-normal S S S S S S S A r c = 0.5 mm B r = 0.38 mm, θ = 1.15 r = 0.43 mm, θ = 1.67 r = 0.45 mm, θ = 2.19−50510152025−20 −15 −10 −5 0 5 10 15 20 Fig. 9. A, Distributions of r c and for all analyzed spirals. The contributions of each sector S i is indicated. The log-normal distribution (26) is characterized by µ = − . and s = 0 . . B, Three typical spirals constructed by considering a constant value of r c equal to its expectation value and θ varying by one standard deviation aroundits expectation value. The spirals are generated from the same expanding bubble (in gray) of reagent having a radius equals to r c . Consequently, they all have the samecouple of radius of curvature at their end points. The graph graduations are in millimeters. θ max Fig. 10.
Distribution of θ max defined as the maximal value of the polar angle describing each analyzed spiral.defined as the maximal value of the polar angle describing each analyzed spiral.