Role of Electrolyte in Liesegang Pattern Formation - Langmuir (ACS

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Role of Electrolyte in Liesegang Pattern Formation Masayo Matsue, Masaki Itatani, Qing Fang, Yushiro Shimizu, Kei Unoura, and Hideki Nabika Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b02335 • Publication Date (Web): 27 Aug 2018 Downloaded from http://pubs.acs.org on August 28, 2018

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Role of Electrolyte in Liesegang Pattern Formation Masayo Matsue,† Masaki Itatani,† Qing Fang,‡ Yushiro Shimizu,† Kei Unoura,§ and Hideki Nabika*,§

†

Graduate School of Science and Engineering, ‡Department of Mathematical Science, Faculty of

Science, and §Department of Material and Biological Chemistry, Faculty of Science, Yamagata University, 1-4-12 Kojirakawa, Yamagata 990-8560, Japan * [email protected]

ABSTRACT

Pattern formation based on the Liesegang phenomenon is considered one of the useful models for gaining a mechanistic understanding of spontaneous spatiotemporal pattern formations in nature. However, for more than a century, the Liesegang phenomenon in chemical systems has been investigated by using electrolytes as both the reaction substrate and aggregation promoter, which has obfuscated the role of the electrolyte. Here, we distinguish the electrolyte (Na2SO4) from the reaction substrates (Ag+ ion and citrate), where Na2SO4 does not participate in the reaction step and acts as an aggregation promoter. The addition of Na2SO4 in Ag+-citrate-type

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Liesegang rings gave well-resolvable clear bands with a larger spacing coefficient. The observed changes were discussed by using the classical DLVO (Derjaguin-Landau-Verwey-Overbeek) theory, where the role of the electrolyte is to shield the electrostatic repulsive interaction among the reaction products. Furthermore, the numerical simulation of the reaction-diffusion equation with different aggregation thresholds reproduced the salt-dependent change in the spacing coefficient. We expect that an understanding of the exact role of the electrolyte as the aggregation promoter reported here will offer novel insight into how nature spontaneously forms beautiful spatiotemporal patterns.

INTRODUCTION. Concentric ring patterns are ubiquitous in nature, ranging from microscopic biological patterns seen as a concentric bacterial colony, macroscopic patterns of agate rocks, to cosmic-scale concentric orbits of solar system planets. These concentric patterns form spontaneously in nature. A similar pattern, which is known as the Liesegang phenomenon, can be obtained with simple chemical equipment such as test tubes or Petri dishes. For example, a few drops of AgNO3 (outer electrolyte) solution are put on a gelatin thin film doped with K2Cr2O7 (inner electrolyte) in a Petri dish. Then, Ag+ ion diffuses into the gelatin thin film, where the reaction between Ag+ and Cr2O72- forms Ag2Cr2O7 salt. During the continuous diffusion of Ag+ and continuous salt formation reaction, a large amount of precipitate grows in the gelatin film, which occasionally forms concentric patterns surrounding the droplet of the outer electrolyte solution put on the gelatin film. Since the first discovery of the Liesegang phenomenon,1 various patterns consisting of hydroxide salts,2-5 chromate and dichromate,6-9 phosphates,10,11 and oxalate12 have been reported.


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In the long history of the research on the Liesegang phenomenon, several empirical rules have been discovered. One of the most important rules is known as the spacing law, which is described by the following equation:

→ 1+

(1) as n is large

where xn and p are the position of the nth band (or orbit or ring) and the spacing coefficient, respectively. The spacing law implies that the ratio between two consecutive bands (xn/xn+1) converges to 1+p. Furthermore, the spacing coefficient depends on the initial concentrations a0 and b0 of the outer and inner electrolytes, respectively. = +

(2)

or = +

(3)

where F and G are monotonously decreasing functions of b0. This is known as the MatalonPackter law for the spacing coefficient of Liesegang patterns,13,14 which enables us to design the pattern periodicity through chemical conditions a0 and b0. This implies that the outer and inner electrolytes, acting as reaction substrates, play an important role in the formation mechanism of the Liesegang patterns in a chemical system. However, the mechanism underlying the pattern formation in the Liesegang phenomenon is not understood completely, although various models have been proposed so far. The first model—the supersaturation model—was proposed by Wilhelm Ostwald in 1897.15 This model considers a simple reaction between the outer (A) and inner (B) electrolytes to form precipitate P,


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i.e., A(aq) + B(aq) → P(s), where the precipitation occurs only when their ion product (ab) exceeds a certain threshold K1 that is above the solubility product of salt AB. Later, the importance of rapid change in the nucleation rate as a function of the supersaturation was proposed to be essential for pattern formation.16 This model is known as the nucleation and growth model and separates the reaction process A(aq) + B(aq) → C(aq) and precipitation process C(aq) → P(s) by introducing an intermediate state C, in which both former and latter processes have concentration thresholds K1 and K2, respectively, to trigger each step. The difference from the supersaturation model is to take two reaction processes and two concentration thresholds into account. K1 corresponds to the solubility product of salt AB, while K2 corresponds to the nucleation threshold.17,18,19 The concentration of C should exceed K2 to trigger the precipitation when there are no precipitates or impurities, whereas the presence of precipitates shifts the threshold from K2 to K1.17,18 Thus, threshold K1 exists only on the initial nucleation, and usually K1 is smaller than K2. Some papers exclude K1 and construct the reactiondiffusion equations by taking only K2 into account.8,20,21 Indeed, the assumption that the presence of K1 is not essential for the appearance of Liesegang patterns was experimentally proved by an experiment based on a reduction reaction, in which K1 does not exist in the reaction.22 In this system, the important role of K2 has been experimentally and numerically clarified by controlling the nucleation energy.23 In the nucleation and growth model, processes of nucleation, crystal growth, and precipitation are all incorporated into the second process C(aq) → P(s) that is triggered when the concentration of C(aq) exceeds K2. On the other hand, the sol-coagulation model considers the precipitation process independently by introducing a third threshold K3 in addition to the thresholds K1 and K2.24,25 In this model, the precipitation proceeds by an aggregation of formed


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crystals in the presence of electrolytes above K3. Thus, the process is divided into three steps: reaction step A(aq) + B(aq) → C(aq) above K1, nucleation and crystal growth C(aq) → C(s) above K2, and aggregation and precipitation from the colloidal state of crystals C(s) → P(s) above K3. It is generally known that the aggregation behavior can be treated in the framework of the DLVO theory. A colloidal suspension aggregates and forms precipitates when the electrolyte concentration increases to a threshold value that is kinetically sufficient to shield electrostatic repulsive forces between crystals, which acts as the third threshold K3 in the sol-coagulation model. For most experimental systems based on salt formation reactions, the outer and inner electrolyte concentrations affect the salt formation reaction (K1), nucleation reaction (K2), and aggregate and precipitate formation (K3). Therefore, the electrolyte concentrations control not only the pattern geometry according to the Matalon-Packter law, but also a structural bifurcation (band, spiral, and tree-like)26,27 and the response to an electric field.28,29 To understand how the electrolyte is involved in the pattern formation mechanism, it is necessary to isolate the contribution of the electrolyte to the salt formation reaction (K1), nucleation reaction (K2), and aggregate and precipitate formation (K3). However, the use of salt formation reactions makes this impossible, because the change in the electrolyte concentration affects both kinetics and thermodynamics of these three steps simultaneously. To overcome this limitation, an alternative experimental system, in which the effect of the electrolyte on each step can be discussed independently, is required. Since the roles of K1 and K2 have already been clarified as mentioned above, the remaining task to be solved is the role of K3 in the pattern formation.


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For this purpose, in the present work we have designed an experimental setup that can control K3 independently through the control of the electrolyte concentration. To eliminate the influence of the electrolyte on K1, the pattern formation was conducted, not with the conventional salt formation reaction, but with a reduction reaction, where Ag+ ions diffuse into a gelatin gel doped with citrate. In the gelatin matrix, the reduction reaction between Ag+ and citrate forms Ag nanoparticles. Because of the capping ability of citrate to Ag nanoparticles, they are usually stable and do not aggregate due to the electrostatic repulsion of each Ag nanoparticle.30-32 However, the addition of the electrolyte shields the electrostatic repulsive force, which induces the aggregation of nanoparticles. Therefore, we added Na2SO4 as an additional electrolyte in the gelatin matrix, fixing the concentration of the reaction substrates to control the shielding effect, which enabled an independent control of the shielding effect and K3. Because of the presence of citrate as the capping agent with much higher affinity for Ag species, the addition of Na2SO4 does not interfere with the nucleation process and K2. Thus, the obtained experimental results evaluated via the spacing coefficient were discussed exclusively based on the influence of Na2SO4 on K3. We also performed a calculation of the interparticle interaction based on the DLVO theory and a numerical simulation based on a reaction-diffusion equation, in order to clarify the effect of the electrolyte on the aggregation process and pattern geometry in conjunction with the experimental results.

Experimental Procedure To prepare a reaction matrix, 0.835 g of gelatin was dissolved in 15 mL of purified water. After stirring and heating at 75 ℃ for 25 min, 0.20 g of trisodium citrate dihydrate and required amount


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of Na2SO4 were added to the gelatin solution. The mixed gelatin solution was stirred and heated at 90℃ for 25 min. Then, 1.5 mL of the mixed gelatin solution was poured into a glass Petri dish (inner diameter = 59 mm) and stored in an incubator at 18 ℃ overnight to yield a gelatin gel matrix. The relative viscosity of the gelatin solution at various Na2SO4 concentrations was measured with an Ostwald viscometer at 30 ℃. To prepare an agarose stamp, 1.37 g of agarose was dissolved in 15 mL of purified water and degassed in vacuum for 5 min. Then, the degassed agarose solution was heated by microwave for 40 s. Immediately after heating, this solution was poured into a silicon tube (inner diameter = 7 mm) and left for 20 min at room temperature. Then, agarose stamps were made by cutting the agarose gel into pieces with a length of 3 cm and soaked in silver nitrate aqueous solutions (1.0 M) for more than two weeks. Then, the agarose stamp doped with Ag+ ion was placed on the surface of the gelatin gel formed in the Petri dish. Ag+ ions diffused from the agarose stamp into the gelatin gel matrix, where Ag+ ions were reduced by citrate and form silver nanoparticles.22 The sample was stored at 18 ℃ for 6 h and the obtained patterns were observed using an optical microscope (BX-43, Olympus Co., Ltd., Japan). For the experiment on the gelatin-gel concentration dependence, we prepared gelatin gels with difference gelatin concentrations (3–5 %), fixing other experimental conditions to the experiment described above.

Results and Discussion A brownish color, which is the characteristic of the surface plasmon resonance of silver nanoparticles, was observed in any condition for cases both with and without Na2SO4, indicating


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that silver ions diffusing from the agarose stamp into the gelatin matrix were reduced by the citrate and silver nanoparticles formed (Figure 1). However, the addition of Na2SO4 significantly changed the appearance of Liesegang patterns, i.e., well-resolvable clear bands appeared only in the presence of Na2SO4. Without Na2SO4, a continuous precipitation band grew around the agarose stamp and slightly resolvable periodic bands appeared at the edge of the precipitation front (Figure 1a). With the addition of Na2SO4, the periodic bands became clearer and the region between two adjacent rings contained less precipitate compared to the sample without Na2SO4. Based on the sol-coagulation model, the reaction proceeds with the following three steps: reaction: A(aq) + B(aq) → C(aq)

(4)

nucleation and crystal growth: C(aq) → C(s)

(5)

aggregation and precipitation: C(s) → P(s)

(6)


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Figure 1. Optical microscope images of Liesegang patterns obtained at different Na2SO4 concentrations. The presence of Na2SO4 does not affect reaction step (4) because the reaction in the present system is the reduction of silver ion by citrate. Then, the reduced silver atom is stabilized by the gelatin matrix and forms complex nucleates and silver nanoparticles above K2. This process is also independent of the presence of Na2SO4. On the other hand, the aggregation and precipitation processes denoted by (6) are highly affected by the presence of salt, as expected from the DLVO theory. Since the addition of salt induces aggregation kinetics of citrate-protected silver nanoparticles,33 it is highly possible that the addition of Na2SO4 promotes process (6) and subsequent pattern formation, in which our experiments revealed that 0.13 M of Na2SO4 was


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enough for obtaining well-resolved Liesegang patterns consisting of citrate-protected silver nanoparticles. Further increase in Na2SO4 did not affect the appearance of Liesegang patterns. For further analysis concerning the effect of salt on the geometry of Liesegang patterns, we evaluated the spacing coefficient p as a function of Na2SO4 concentration. Although the image exhibited no significant difference in the Liesegang patterns in the presence of Na2SO4 above 0.13 M (Figure 1), p exhibited gradual increase as a function of Na2SO4 concentration (Figure 2a). If Na2SO4 is assumed to be an inner electrolyte in the salt-formation reaction systems, it can be said that our results show a similar tendency to Liesegang patterns for saltformation reactions, where the increase in the inner electrolyte concentration led to the increase in p. However, the inner electrolyte in the salt-formation system acts as both reaction substrate and aggregation promoter, and its dominant role in determining p is still ambiguous. On the other hand, the present system separates the reaction substrate (Ag+ and citrate) and aggregation promoter (Na2SO4); the independent control of the aggregation promotor determines the aggregation and pattern formation behavior, as shown in Figure 1. Thus, it is expected to clarify the role of Na2SO4 in the present system from the perspective of the DLVO theory.


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Figure 2. (a) Spacing coefficient and (b) relative viscosity of gelatin solutions at 30 ℃ as a function of Na2SO4 concentration. However, the addition of Na2SO4 would alter the mechanical properties of gelatin gel, because the relative viscosity of gelatin solution at 30 ℃ varied depending on the Na2SO4 concentration (Figure 2b). The relative viscosity first increased with the Na2SO4 concentration from 0 to 0.13 M, whereas further increase reduced the relative viscosity. Such salt concentration dependence was reported for a gelatin solution containing various salts, in which the appearance of a maximum has been explained by competitions of short-rang/long-range interactions, saltingin/salting-out effects, or expanding/contraction effects.34-36 Variation in the mechanical properties is important for the Liesegang pattern formation, because the viscosity changes the


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diffusivity of a reaction substrate according to the classical Stokes-Einstein equation D=kT/cπηr, where D, k, T, c, η, and r are the diffusion coefficient, Boltzmann’s constant, temperature, constant, viscosity, and radius of diffusing species, respectively. Furthermore, it has been reported that the changes in the diffusion coefficient of reaction substrates affected the spacing coefficient p.17 Thus, before a discussion from the viewpoint of the DLVO theory, it is necessary to evaluate the effect of viscosity of the gelatin matrix on p in our experimental system. For this purpose, we have conducted additional pattern formation experiments using gelatin gels with different gelatin concentrations (3–5%) so that their viscosities cover the viscosity range observed in Figure 2b, where the concentration of Na2SO4 was fixed as 0.13 M. Similar to the results shown in Figure 1, well-resolved Liesegang patterns were obtained for samples under any gelatin gel concentration (Figure 3). From a numerical analysis of the obtained patterns, it was found that the spacing coefficient p was not dependent on the gelatin gel concentration, and thus, the relative viscosity of the gelatin matrix (Figure 3f). Although it has been reported that the gel concentration changes the pattern formation depending on the experimental system,37,38 our system based on the reduction of silver ion by citrate was found to be less sensitive to the gel concentration that modulates the gel viscosity and the diffusion coefficient of the reaction substrate.


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Figure 3. (a)–(e) Optical microscope images of Liesegang patterns obtained at different gelatin concentrations. (f) Spacing coefficient as a function of relative viscosity. Since it was clarified that the relative viscosity did not affect the spacing coefficient in the present system, our discussion is concentrated on the effect of salt on the inter-particle interaction. According to the classical DLVO theory, the total interaction energy between two adjacent nanoparticles (Wtot) can be estimated by a simple summation of the van der Waals (WvdW) and the electrostatic double layer (WDL) interactions:39

Wtot = WvdW + WDL.

(7)

Assuming two spherical particles with radius R (=5 nm),22 WvdW can be expressed as


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vdW = −

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(8)

where A and D are the Hamaker constant and separation distance between two particles, respectively. The Hamaker constant A of metals (Au, Ag, Cu) is known to be 25–40 ×10−20 J;39 we used 30 ×10−20 J as A of a silver nanoparticle in our calculation. To evaluate WDL between two silver nanoparticles, the following equation was used: &' *+, ./

DL = 4!"# $ % ) ( $=

/

67 ) 89: 67 0;. tanh< ? 89: =>

Role of Electrolyte in Liesegang Pattern Formation - Langmuir (ACS

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