Liesegang Mechanism with a Gradual Phase Transition - The Journal

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Liesegang Mechanism with a Gradual Phase Transition Yushiro Shimizu, Jun Matsui, Kei Unoura, and Hideki Nabika J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b01275 • Publication Date (Web): 10 Mar 2017 Downloaded from http://pubs.acs.org on March 12, 2017

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The Journal of Physical Chemistry

Liesegang Mechanism with a Gradual Phase Transition Yushiro Shimizu, Jun Matsui, Kei Unoura, Hideki Nabika* Department of Material and Biological Chemistry, Faculty of Science, Yamagata University, 14-12 Kojirakawa, Yamagata 990-8560, Japan

ABSTRACT. We report a novel and generalized chemical model for the Liesegang mechanism that involves gradual phase transitions of macromolecules, unlike previous models that involved definite and discontinuous phase transitions. As a model system, an agarose gel medium doped with an initiator was contacted with a solution containing monomer 2-methoxyethyl acrylate (MEA), in which monomers diffused into the gel medium and initiated a polymerization reaction. As there is a monomer concentration gradient along the diffusion direction, the degree of polymerization and the corresponding degree of insolubility showed a similar spatial gradient in the reaction medium, because molecular solubility gradually changes from soluble to insoluble with an increase in the degree of polymerization. Under this condition with such a spatial gradient in the degree of polymerization, multiple bands satisfying the spacing law were formed by repetitive precipitation and depletion of insoluble poly(2-methoxyethyl acrylate) (PMEA). In contrast to the Liesegang mechanism for salt formation reactions with definite transition points,

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such as the solubility product and nucleation threshold, our model involves a gradual transition with a non-zero transition width. The width of the transition point would be the governing parameter for the appearance and characteristics of periodic pattern formation. This model with a non-zero transition width is largely generalized and applicable to various processes involving any solubility transition and thus can be used as a generalized Liesegang model that can be applied to any system with both definite and gradual transitions.

Introduction Diverse periodic structures occur in nature, ranging from the periodic stripe pattern of a zebra to the periodic concentric orbits in a solar system. Understanding the formation mechanism of periodic structures is both of fundamental interest to obtain insight into self-organization processes in nature and of technological interest to aid in fabricating nature-mimetic materials with controlled periodic patterns. For example, black silicon with periodic nanofeatures similar to the surface of the wing of the dragonfly Diplacodes bipunctata demonstrated biomimetic bacterial activity.1 To achieve comprehensive understanding of the formation of periodic structures in nature, the construction of appropriate chemical and mathematical models is useful, as they are easier to handle than natural systems. A typical example is Turing patterns, production of which is well-established in chemical,2-4 biological,5-7 and numerical experiments,8-10 where the key process is understood as the interplay between activator and inhibitor species. Contrary to the Turing mechanism, in which a periodic pattern is formed from an initially homogeneous medium, in the Liesegang mechanism, periodic patterns are formed from a


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medium with a concentration gradient in one of reactants.11 Although the periodic patterns formed by Turing and Liesegang mechanisms resemble each other, the Liesegang pattern has a characteristic periodicity described by a geometric series, known as the spacing law, which indicates that Turing and Liesegang mechanisms contribute independently to pattern formation in nature. However, interdisciplinary discussions on the Liesegang mechanism from chemical, biological, and numerical viewpoints are less vigorous than those on the well-understood Turing patterns, which limits comparative understanding of Liesegang mechanisms in animate12-18 or inanimate systems.19 To achieve a comprehensive understanding of the Liesegang mechanism, modification of the conventional chemical model to a more generalized model would be helpful. From the experimental viewpoint, chemical Liesegang patterns have long been formed by insoluble salts, such as hydroxides,20-23 chromates,24-26 dichromates,27-29 phosphates,30-32 carbonates,33 iodides,34-36 oxalates,37 and oxinates.38 In these systems, the reaction can be written as A+(aq) + B－(aq) → AB(aq) → AB(s)↓ (1) In the initial step, salt AB(aq) is formed, and the reaction proceeds when [A+][B－] exceeds the solubility product Ksp. Then, AB(aq) forms precipitate AB(s) via nucleation when [AB(aq)] exceeds the nucleation threshold concentration C*. As a result, numerical models for these salttype chemical Liesegang systems involve step functions concerning Ksp and C* such as: [A ]

[B ]

= ∇ [A ] − [A ][B ][A ][B ] − (2) = ∇ [B ] − [A ][B ][A ][B ] − (3)


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[AB(s)]

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= [AB(aq)]([AB(aq)] − ∗ ) (4)

where the step functions θ are 1 when [A+][B－] > Ksp or [AB] > C*, and otherwise are 0.29,36,39,40

As these chemical and numerical models can only be applied for a system with two concentration thresholds, such as salt formation reactions, these models may fail to explain Liesegang-like phenomena in nature. To overcome this limitation and achieve a generalized understanding of Liesegang phenomena, systems that do not rely on the salt formation reaction have been explored. One example is the Liesegang patterns formed by oppositely charged nanoparticles, denoted as A+ and B－, through the reaction A+ + B－ → AB.41 It has been reported that the pattern formation mechanism for this reaction is not governed by Ksp but by the threshold of aggregation, which is related to C*. Similarly, a system involving the reduction reaction of metal ions has also been shown to yield Liesegang patterns.42,43 As the reduction reaction proceeds at any concentration owing to the absence of a concentration threshold, the mechanism is also not governed by Ksp, but by C* for nucleation and precipitation. These efforts successfully eliminate the initial concentration threshold Ksp from the minimal prerequisites for Liesegang pattern formation. Although this finding expands the potential of Liesegang phenomena to chemical reactions other than salt formation, most systems without Ksp are limited to inorganic materials bearing definite and discontinuous phase transitions from atom to nuclei with C*.41-43 However, as Liesegang patterns in biological tissues involve biological macromolecules, it is of interest, although challenging, to develop a novel Liesegang model that can be constructed by macromolecules with continuous phase transitions owing to their chemical or physical properties other than discontinuous nucleation at C*.


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Herein, we propose a novel mechanism for Liesegang patterns formed by a polymerization reaction with a gradual transition from the soluble hydrophilic monomeric state to the insoluble hydrophobic polymeric state as a function of the degree of polymerization. With the progress of polymerization, the formed polymers turn into a globular state accompanied by dehydration, followed by the formation of aggregates and precipitates owing to hydrophobic interactions. As a model system, we chose the polymerization of 2-methoxyethyl acrylate (MEA) in agarose gel medium, where molecular solubility gradually changes from soluble to insoluble with an increase in the degree of polymerization Pn. We observed periodic precipitation bands that satisfy a spacing law, implying that the mechanism underlying band formation is related to a Liesegang-type mechanism. Furthermore, the periodic band formation was found to result from the precipitation of insoluble polymer under the Pn gradient in the agarose medium. Our model does not rely on concentration thresholds Ksp and C*. Instead, the gradual transition from soluble to insoluble states, which could be characterized by Pn, played a critical role in achieving Liesegang patterns. From these results, we propose a mechanism for the polymerization Liesegang phenomenon in terms of the gradual transition in the reaction medium.


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Experimental MEA was purchased from Tokyo Chemical Industry. Acrylic acid (AA), initiator (VA-044, 2,2'azobis[2-(2-imidazolin-2-yl)propane]dihydrochloride), and agarose were purchased from Wako Pure Chemical Industries. Agarose gel was prepared by dissolving the required amount of agarose in hot Milli-Q water under Ar flow, followed by further heating by microwave for 60 s. Then, the solution was heated again on a hot plate at 100 °C for 10 min under Ar flow. After cooling to room temperature for 4 min, the required amount of purified initiator was added. Then, the solution was poured into a glass test tube with a diameter of 1.5 cm and stored at 18 °C overnight (depicted as “agarose with initiator” in Figure 1). Subsequently, 6.0 mL of a solution containing the appropriate amount of monomer (MEA or AA) was poured onto the agarose gel containing initiator. Then, the test tube was sealed under Ar and stored in an Ar-purged incubator at 50 °C. The time course of precipitation formation was recorded using a digital camera (EXZR850, Casio) by placing a LED tracing light as a light source behind sample so that the precipitate appears in the images as a dark shadow.

Fig. 1: Illustration of experimental setup for pattern formation. A solution containing monomer was poured onto an agarose gel doped with initiator. The test tube was sealed under Ar and stored in an Ar-purged incubator at 50 °C.


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Results and Discussion After pouring a solution containing monomer (M) onto an agarose gel doped with initiator (I), monomers diffuse into the agarose medium. In the agarose gel, thermal cleavage of initiators results in spontaneous formation of radicals (I・) (5), which react with monomer (6) and trigger the polymerization reaction (7). The polymerization is stopped by a bimolecular termination reaction (8). I → 2I・ (5)

I・ + M → IM・ (6)

IM・ + (n － 1)M → IMn・ (7) IMn・ + IMm・ → IMnMmI (8)

Monomers diffusing from the solution undergo successive polymerization in the agarose gel medium. In the case of MEA, insoluble dark precipitates appeared in the agarose medium (Figure 2). The precipitate was confirmed by NMR and IR to be poly(2-methoxyethyl acrylate) (PMEA) (Figure S1 and S2), which is the polymerization product of MEA. At a low initiator concentration of 0.083 mM (Figure 2a), precipitates formed at the topmost region of the agarose medium after


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Fig. 2: Photographs of agarose gels after precipitation under various monomer/initiator concentration conditions. Time noted above each image is the reaction time in an incubator at 50 °C. Under conditions (a) and (c), two bands formed, whereas under condition (b), multiple bands formed.

24 h. This is because the polymerization reaction proceeds and propagates from the solution/gel interface. If we assume that the monomer undergoes simple diffusion obeying Fick’s law, the monomer concentration would have a continuous gradient along the tube depth direction, which would lead to a continuous gradient in the amount of polymer and precipitate. However, longer reaction times did not lead to a continuous gradient but the formation of a second precipitation band after 144 h with an intervening depleted layer without precipitates. Further reaction resulted in thickening of the second band. Similar behavior was observed at a higher initiator concentration of 0.293 mM (Figure 2c). On the other hand, a fine structure with periodic precipitation bands was obtained at an intermediate initiator concentration of 0.22 mM (Figure 2b). Although the same initial behavior was observed, with the first and second bands appearing by ~150 h, further reaction at this initiator concentration resulted in the formation of periodic bands, with the number of bands increasing with the reaction time. Similar multiple bands were observed for a limited set of concentration conditions (Figure 3a). Under the present conditions,


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a maximum of eight bands was observed to form in the agarose gel medium. For the Liesegang mechanism, it is well-known that the distance of the nth band from the interface (xn) follows the spacing law xn+1/xn = 1 + p when n is large, where p is the spacing coefficient.25 As the present systems with n = 8 obeys the spacing law at larger n (Figure 3b), the observed periodic bands should be formed by a Liesegang mechanism.

Fig. 3: (a) Phase diagram for band formation. 〇: multiple bands, ×: single or double bands. Red numbers in parentheses indicate the number of bands observed at each condition. (b) Spacing law for the sample with n = 8.

From these results, the periodic bands were assumed to form spontaneously owing to the formation of insoluble PMEA by polymerization of MEA that diffused from the interface over the reaction time of more than 500 h. To verify this working hypothesis, we must clarify several points: (1) Does it take hundreds of hours for monomers to diffuse into the agarose gel medium? (2) Is the initiator active in the agarose gel medium for hundreds of hours? (3) Is insolubilization by polymerization required for precipitation?


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To address the first point concerning diffusion of monomers in the medium, we roughly estimated the monomer concentration distribution in the agarose gel medium using the following equation:

( , ") =

√$%

& '

( ) *%

(8)

where D is the diffusion coefficient of the monomer in the agarose gel medium. Although the exact value of D is unknown, we assumed D = 8 × 10-6 cm2/s, which is the diffusivity of a oligo(ethylene glycol) molecule with a similar molecular weight to MEA in 1 wt% agarose gels.44 As shown in Figure 4, this calculation shows that monomer diffusion from the interface (position = 0) into the agarose medium takes a long time. The monomer diffuses up to 6 cm after ~150 h, which covers the region of multiple bands. Although this is just a rough estimation, it clearly shows that there is

Fig. 4: Theoretical estimation of the monomer concentration gradient from 24 to 520 h. The inset depicts the definition of the position in the calculation.


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a monomer concentration gradient in the agarose medium throughout the experiment (>500 h). As the monomer concentration affects both the polymerization rate Rp and degree of polymerization Pn, a monomer concentration gradient causes gradients in both Rp and Pn along the depth direction of the agarose gel.45 However, the gradient on Rp is negligible because polymerization finishes within a few microseconds, which is much faster than monomer diffusion (a few hundreds of hours), i.e., the system is diffusion-limited. Thus, we have examined the effect of monomer concentration gradient on Pn. By comparing Pn at different depths from the interface using the same experimental configuration as shown in Figure 1, it was found that Pn decreases in deeper regions where the monomer concentration is expected to be lower (Figure S3). This result strongly supports the presence of a monomer concentration gradient in the agarose gel medium. Thus, the periodic band structure is the result of a Pn gradient that was caused by a monomer concentration gradient owing to significantly low diffusivity in the reaction medium. Next, we addressed the question of the lifetime of radicals in the agarose gel medium. As seen in Figure 2, band formation continues to proceed, even after 500 h. On the other hand, ten hour half-life temperature of VA-044 is 44 °C, which indicates that all initiators will have decomposed to form radicals after a few hundred hours. Furthermore, the lifetimes of initiator radicals are usually a few microseconds. Therefore, there are two possible explanations for such long reaction lifetimes. One is that the radicals are completely consumed in the initial few tens of hours and no chemical reactions are occurring. In this case, the bands are formed by polymers produced during the first tens of hours. The other possibility is that radicals are stabilized in the gel and both polymerization and precipitation reactions occur throughout the entire reaction time. To determine whether the radicals were deactivated, we carried out a simple control experiment


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(Figure 5a). Unlike the experiment shown in Figure 1, pure water was poured onto the agarose gel doped with initiator. After incubation at 50 °C for a given period twater, monomer was added to the aqueous phase and the sample was incubated at 50 °C for a further 504 h. If the initiator were deactivated during twater, no polymerization and precipitation would be observed after monomer addition. On the other hand, as shown in Figure 5b, dark precipitates appeared in the agarose when the monomer was added at twater = 24 h, indicating that radicals are active after incubation in agarose at 50 °C for 24 h. In this case, the precipitates were not formed near the interface, which is due to dissolution of the initiator from the agarose medium into the pure water phase. Even after much longer incubation periods (twater = 432 h), polymerization was still observed. This result strongly suggests that radicals with certain active/dormant equilibria have quite long lifetimes (>400 h) under the present conditions and are active for the polymerization reaction. Thus, periodic band formation proceeds through the interplay between continuous polymerization and monomer diffusion, indicating that the bands are the result of reactiondiffusion phenomena.


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Fig. 5: (a) Experimental protocol for evaluating radical lifetimes. Pure water was poured onto agarose gel doped with initiator. The test tube was sealed under Ar and stored in an Ar-purged incubator at 50 °C for a given period twater. After twater, monomer was added to the aqueous phase, and the test tube was sealed under Ar and stored in an Ar-purged incubator at 50 °C for 504 h. (b) Photographs taken at 504 h for samples with different twater. Precipitates appeared in all samples, even at twater = 432 h, indicating that the radicals are active for longer than a few hundred hours.

Finally, we examined the importance of the solubility of the polymer for precipitate formation. To compare the solubility of polymerization products, we conducted the experiment using AA instead of MEA (Figure 6a). Unlike PMEA, poly(acrylic acid) (PAA) is soluble in water.


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Fig. 6: (a) Illustration of experimental setup for pattern formation. Instead of MEA, a solution containing AA was poured onto an agarose gel doped with initiator. All other experimental conditions are the same as in Figure 1a. (b) Photographs of agarose gels after 432 h under various monomer/initiator concentration conditions.

The experimental results shown in Figure 6b clearly indicate that no precipitation was observed with PAA under the same conditions where precipitation was observed with PMEA. As the formation of PAA was confirmed by NMR and IR (Figure S4 and S5), we can conclude that the gradual transition from a soluble state to an insoluble state during polymerization is essential for precipitation. Considering the above three points, we can say that the formation of periodic bands results from the precipitation of insoluble polymer under a Pn gradient in the agarose medium,


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where polymerization occurs between radicals with long lifetimes and monomers with slow diffusivity. Based on these results, we propose a mechanism underlying Liesegang pattern formation by polymerization in this system (Figure 7). First, monomers diffuse from the interface, forming a concentration gradient in the agarose gel medium (Figure 7a, black). Then, radical polymerization is initiated and PMEA forms in the agarose medium. In accordance with the gradient in monomer concentration, Pn also has a gradient as a function of distance from the interface (Figure 7b, red). As Pn and solubility (Figure 7b, blue) are negatively correlated, PMEA with higher Pn near the interface undergoes a transition from a soluble state to an insoluble state, i.e., coil–globule transition. The coil–globule transition occurs gradually depending on a stiffness of the polymer.46 Although stiff polymers, such as DNA, have shown discontinuous soluble– insoluble transitions,47 flexible polymers, such as PMEA used in the present system, could show continuous transitions. Thus, PMEA undergoes a gradual transition to a globule state during the polymerization reaction with a continuous increase in Pn. The formed globules act as nuclei for aggregation and precipitation by a diffusion-limited colloidal aggregation (DLCA) process,48 thus forming aggregates and a precipitation band consisting of globules through hydrophobic interactions (Figure 7c, green). This growth mode is different from the growth of a crystalline polymer that is initiated when the polymer concentration exceeds the nucleation threshold C*.49 As aggregate formation at the globule nuclei consumes free polymers, neighboring polymers diffuse to this region and the aggregates further grow by the DLCA process. As a result, a depleted layer is formed with a lower polymer concentration and no precipitate next to the region with the first precipitation band, similar to the model for a salt formation system (Figure 7c, red).35 As the monomers diffuse slowly but continuously into the agarose medium during these


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processes, the amount of monomer and thus Pn increases throughout the entire agarose medium. When Pn beyond the first depleted layer increases enough to cause the transition to the insoluble state, precipitation

Fig. 7 Suggested mechanism for Liesegang pattern formation by polymerization-induced solubility transition. (a) Diffusion of monomer from solution causes a monomer concentration gradient in the agarose gel. (b) Polymerization under the monomer concentration gradient leads to a gradient in the degree of polymerization Pn. The inset in the lower panel depicts the relation between solubility and Pn. (c) Polymers with higher Pn have lower solubility in agarose gel and


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therefore precipitate. Neighboring polymers diffuse to and adsorb onto the precipitate instead of forming a new precipitate, which leaves a depleted layer. (d) Repetition of (a) to (c) forms periodic multiple bands.

occurs to form a band separated by the depleted layer (Figure 7d). Repetition of precipitation and depletion results in the formation of multiple bands. Obviously, this model does not rely on a concentration threshold for the initial reaction, i.e., polymerization reaction (6) in the present system, indicating the absence of Ksp. Furthermore, a second concentration threshold C* for nucleation is not involved. Instead of C*, the transition point between soluble and insoluble states, which is governed by Pn, plays a critical role in exhibiting a Liesegang pattern (Figure 7b). This observation indicates that Liesegang pattern formation in macromolecular systems can be governed by the transition to an insoluble state, which is characterized by a system-dependent specific parameter, such as Pn in the case of a polymerization reaction. Although the transition point (Pn*) of PMEA is unknown, poly(2-hydroxyethyl methacrylate) (PHEMA), which has a similar molecular structure, has a solubility transition at Pn* ~ 40.50 Furthermore, the solubility of PHEMA gradually decreases from 100% to less than 20% with an increase in Pn from 40 to 100, indicating that the transition is denoted by Pn* and a nonzero transition width δPn as Pn* ± δPn, which is 70 ± 30 in the case of PHEMA. Therefore, the coil to globule transition as a

function of Pn can be described by a gradual function such as a sigmoid function, +,- (/((0∗ )), 1

where the gain a corresponds to the transition width δPn. This situation is quite different from nucleation with a definite concentration threshold C* and step function θ(C – C*). Note that a limit to zero of a or δPn converges this system with a nucleation-type system, whereas a limit to infinity leads to a no-transition system, which would not exhibit Liesegang patterns. Thus, the


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width of the transition point is the governing parameter for the appearance and characteristics of periodic pattern formation. From the above suggested model, it has been succeeded in the construction of chemical Liesegang model that do not rely on concentration thresholds Ksp and C*. Similar to our model, a phase separation model based on the Cahn-Hilliard equation is also known to be independent of Ksp and C*.51 However, in the phase separation model, the density at the spinodal point plays the role of threshold densify for the nucleation C*.51 Thus, our model is largely generalized and applicable to various processes involving any solubility transition, i.e., transitions from mobile to immobile species in a given medium, which is triggered not only by polymerization but also by changes in folding state, molecular clouding state, and intermolecular and intercellular interactions. Further chemical, biological, and mathematical experiments will reveal a generalized Liesegang model that is constructed in terms of a transition width, not a definite transition point, as the governing parameter.

Conclusion We have examined the formation of Liesegang patterns by polymerization under a monomer concentration gradient. Our experiments revealed that multiple bands consisting of insoluble polymer were formed under a limited set of monomer/initiator concentrations. As the formed bands satisfied the spacing law, the mechanism underlying the formation of multiple bands was related to the Liesegang phenomenon. Furthermore, the formation of multiple bands was found to result from precipitation of an insoluble polymer under a Pn gradient in the agarose medium, where band formation occurred in a reaction-diffusion system with radicals with long lifetimes


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and monomers with slow diffusivities. Our model does not rely on concentration thresholds Ksp and C*. Instead, the gradual transition between soluble and insoluble states, characterized by Pn* ± δPn, plays a critical role in forming a Liesegang pattern. Further interdisciplinary experiments involving chemistry, biology, physics, and mathematics might lead to a generalized Liesegang model constructed based on a gradual transition as the governing parameter.

ASSOCIATED CONTENT Supporting Information. The Supporting Information is available free of charge on the ACS Publications website. NMR and IR data for PMEA and PAA; depth-dependent GPC data

AUTHOR INFORMATION Corresponding Author


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*E-mail: [email protected] Notes The authors declare no competing financial interest.

ACKNOWLEDGMENT This work was supported by JSPS KAKENHI Grant Number 16H04092 and 26520203.

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