Interplay between two radical species in the formation of periodic patterns during a polymerization reaction

Daisuke Sato a, Masaki Itatani a, Jun Matsui b, Kei Unoura b and Hideki Nabika *b
aGraduate School of Science and Engineering, Yamagata University, Japan
bFaculty of Science, Yamagata University, 1-4-12 Kojirakawa, Yamagata 990-8560, Japan. E-mail:

Received 9th June 2020 , Accepted 24th August 2020

First published on 24th August 2020

Periodic patterns are ubiquitous in nature and spontaneously form on molecular to cosmic scales by the interplay between reaction and diffusion. Understanding how these patterns form is important to understand the construction rules of nature and apply them in the synthesis of functional artificial materials. This work clarifies how radical (R˙) species affect pattern formation in periodic precipitated and depleted zones during a polymerization process in an agarose gel. When a monomer (Mon) solution was poured on top of the gel doped with an initiator (In) in a test tube, periodic and continuous precipitation occurred near and far away from the solution/gel interface, respectively. In contrast, a system without In exhibited only a continuous band of precipitates beyond a depleted zone without precipitates at a certain distance from the interface. In the depleted region, an inhibitor (Q) added to the solution limited the polymerization triggered by R˙ formed thermally from Mon. With the addition of enough In to overcome the quenching effect of Q, periodic bands appeared near the solution/gel interface. These results suggest the involvement of two independent polymerization processes: (i) polymerization triggered by R˙ formed from In, which is the dominant process up to 100 h and yields periodic structures near the interface. After 100 h, the dominant process is the polymerization triggered by R˙ generated thermally from Mon, which yields a continuous precipitation zone. These two R˙ species compete and generate periodic bands near the interface (<100 h) and a continuous band far away from the interface (>100 h).


Periodic patterns form owing to the interplay between reaction and diffusion processes in nature.1 One such structure formed through the interplay of reaction/diffusion is the Turing pattern represented by the body surface pattern of fish or zebras, which is characterized as a structure with a constant periodic wavelength formed by a uniform reaction field. Chemical2–4 and mathematical5,6 studies on the patterns have been performed extensively, where the interaction between activators and inhibitors and the differences between their diffusion coefficients are the key conditions for the formation of a pattern.7 Furthermore, theoretical studies8 to understand the self-organized structures that are similar to Turing patterns in nature have been performed using the reaction–diffusion equation. Thus, by chemically and mathematically clarifying the formation mechanism, a fundamental knowledge of periodic structures in nature is gained.

In contrast, the origin of the Liesegang pattern that is characterized by periodic bands and rings that obey a geometric series is still under debate, and different formation mechanisms have been proposed using both chemical and mathematical models.9 Liesegang-like structures are formed in the presence of concentration gradients under chemical and/or physical conditions, and are observed not only in chemical systems but also in petrology,10 biology,11 and various other fields in nature. To gain knowledge of the formation mechanism of these structures, it is necessary to construct a unified model from both experiments and theories that can be applied to similar patterns in several other fields, as in the study of Turing patterns. Since the first discovery of Liesegang pattern formation, the most widely used reaction of a chemical system is the salt formation reaction that produces chromate,12–16 hydroxide,17–20 carbonate,21 and phosphate.22,23 The use of the salt formation reaction in the Liesegang study involves essentially two thresholds to drive the reaction. One is the solubility product, which drives the reaction between cation and anion to form the corresponding molecular salt. The other one is the nucleation threshold to overcome the activation energy to form solid-like nuclei from salts in their molecular states. To describe these thresholds in the salt formation reaction, most theoretical models incorporate a step function in the reaction–diffusion equation to trigger each reaction only when the constituent concentration exceeds the threshold.

Although most experimental and theoretical studies have used salt formation reactions with some exceptions such as systems involving nanoparticle formation,24–26 the Liesegang pattern with polymerization reaction was reported in recent years.27 Different from the salt formation reaction with definite thresholds, each step in the pattern formation with the polymerization reaction is essentially independent of the threshold. As a typical experimental procedure, an agarose gel containing an initiator (In) was prepared in a test tube and an aqueous monomer (Mon) solution was poured on top of the gel. The initiator spontaneously generates radical species (R˙) by thermal decomposition in the gel (eqn (1)). The polymerization of the monomer proceeds by the reaction between R˙ and monomers diffusing from the upper solution (eqn (2) and (3)). The monomer can also form a radical under heat (eqn (4)) and act as an initiator in the system.28 The polymerization stops when two R˙ collide (eqn (5)) or R˙ reacts with a quencher (Q) (eqn (6)).

In → 2R˙(1)
Mon + R˙ → RMon˙(2)
RMon˙ + (n − 1)Mon → RMonn˙(3)
Mon → 2R˙(4)
RMonn˙ + RMonm˙ → RMonn+m(5)
R˙+ Q → X(6)

These reactions proceed without concentration thresholds. When the reaction yields an insoluble polymer from a soluble monomer, the solubility of the polymerizing molecule decreases gradually as the reaction progresses.29 This gradual transition is known as a coil-to-globule transition that occurs gradually according to the stiffness of the polymer (when the temperature and solvent are the same).30 The formed globular polymers then acts as nuclei for aggregation and precipitation induced by a diffusion-limited colloidal aggregation (DLCA) process.31 These three steps, i.e., polymerization, coil-to-globule transition, and DLCA, are the elementary processes for the Liesegang pattern formation with the polymerization reaction, which is a new class of Liesegang systems that proceeds with non-definite but gradual thresholds. Furthermore, the Liesegang patterns from the salt formation reaction proceed with a monotonous decrease in the inner electrolyte, except in a specially designed experiment with a continuously stirred tank reactor (CSTR) that keeps the system in a non-equilibrium state and maintains the inner electrolyte concentration constant.32 In the polymerization, although In and R˙ in the gel serving as the inner electrolyte are consumed with the progression of the polymerization reaction, R˙ can also be supplied by the diffusion of the monomer and subsequent R˙ formation reaction (eqn (4)). Therefore, the polymerization system can be regarded as a pseudo non-equilibrium system, similar to the salt formation system in CSTR configuration.

Thus, the polymerization system is a new model with a gradual threshold, and a pseudo non-equilibrium system that can be applied to clarify the formation mechanism of similar Liesegang-like patterns in nature. However, the details of the formation mechanism are not clear. In particular, the role of each component such as the monomer, initiator, quencher, oxygen, and gravitational force is still ambiguous. In this study, we experimentally verified the formation mechanism of the Liesegang pattern with a polymerization reaction. Understanding how each component interacts to form the periodic precipitate will provide insights into the construction of a mathematical model for the polymerization system to compare it with conventional salt-formation systems.

Materials and methods

Mon (2-methoxyethyl acrylate) and Q (4-methoxy phenol) were purchased from Tokyo Chemical Industry Co. Ltd (Tokyo, Japan). Agarose and In (VA-044) were acquired from Fujifilm Wako Pure Chemical Corporation (Osaka, Japan). In was purified by recrystallization from methanol. Agarose (0.449 g) was dissolved in degassed hot water (30 mL) under Ar flow. Then, the solution was heated with a microwave for 60 s and stirred for 10 min under Ar flow. After heating, the solution was poured into a test tube (15 mm in diameter and 165 mm in length) and left at room temperature. For an agarose gel doped with In, purified In was added to the agarose solution with heating on a hot stirrer. After gelation in the silicon tube, the Mon solution (1.5 M) was poured under Ar flow. The sample was incubated under an Ar flow at 50 °C. As the gelation temperature of the gelatin sol (20–30 °C) is much lower than the reaction temperature of the present system (50 °C), we used the agarose gel in the present study. The obtained patterns were observed with an optical microscope (SZ-61, Olympus, Tokyo, Japan). The polymer was characterized by gel permeation chromatography using a GPC-101 system with RI detector (Shodex, Japan). Polystyrene standards were used for calibration.

Results and discussion

To identify the R˙ species that are responsible for the periodic pattern formation in the polymerization system, we compared two experimental systems; one is the diffusion of Mon into a gel containing In, and the other is the diffusion of Mon into a gel without In (Fig. 1). In the presence of In, a dark region appeared just below the solution/gel interface. Although white precipitates formed, they are observed as a dark shadow because the image was acquired with a transmitted white light. The precipitated region expanded with time and the front moved about 4 cm away from the solution/gel interface at 312 h. This time scale is almost consistent with the time scale of the diffusion of Mon in the gel matrix (Fig. S1, ESI). Although periodic precipitated and depleted zones appeared near the solution/gel interface region, it does not look like a structure that follows the empirical rules for Liesegang patterns such as the spacing law and time law. Although the formation of a periodic pattern was observed at 72 h, only a non-periodic, continuous precipitation occurred after 72 h. GPC analysis revealed that both precipitates in the periodic and continuous regions contained a polymer with a degree of polymerization of ca. 400, indicating that the polymerization reaction proceeds in both regions. However, the concentration of the initiator used in the present study (VA-044) halved in 10 h at 44 °C, indicating that [In] decreased to below 7 × 10−3 times the initial concentration in 72 h at 50 °C. Thus, the polymeric precipitates that formed after 72 h were potentially caused by a component different from In.
image file: d0cp03089a-f1.tif
Fig. 1 Formation of precipitates in the presence (a) and the absence (b) of In in the agarose gel. The images were captured with the transmitted white light. Asterisks in the grey scale intensity line profiles depict the position of depleted bands. The solution contains 1.5 M of Mon and 156 μM of Q. [In] in (a) is 350 μM.

To investigate the reaction after 72 h, we carried out a similar experiment in the agarose gel without In (Fig. 1b). In the absence of In, no precipitates appeared until 144 h. The precipitates appeared after 216 h, 2 cm away from the solution/gel interface. The precipitated compounds were confirmed by NMR and infrared (IR) spectroscopy to be poly(2-methoxyethyl acrylate) (PMEA) (Fig. S2, ESI). Because In was not added to the gel, it can be suggested that thermally generated R˙ from Mon initiated the polymerization and PMEA was precipitated. After 216 h, along with the growth of the precipitated region downward, its growth toward the interface was also observed. Although the downward growth is along the direction of the diffusion of Mon, the upward growth can be explained by the catalytic effect of the pre-existing precipitated PMEA that promotes the formation of precipitates at much lower component concentrations.9 In the present case, the precipitates formed at 216 h catalyzed the precipitation of dispersed polymers in both directions, above and below. From the comparison between the two experiments with and without In, it can be clarified that (i) periodic precipitates appear in the presence of In, (ii) a continuous precipitated region forms away from the solution/gel interface in the absence of In, and (iii) a continuous monomer supply from the solution also plays a role in the continuous supply of R˙ in the gel.

As for the position that the precipitates appeared at, it is reasonable that they formed just below the interface because there is enough Mon and In in the presence of In in the gel. However, it is difficult to explain why the precipitates appeared away from the interface in the absence of In, although the concentration of the initiator radicals formed from Mon that trigger the polymerization reaction is the highest near the interface. To explain this issue, we examined three possibilities, viz., the effects of oxygen inhibiting the polymerization, the gravitational force pulling the precipitates downward, and the addition of Q to inhibit the polymerization (Fig. 2). It is well-known that oxygen acts as an inhibitor of polymerization reactions.33

image file: d0cp03089a-f2.tif
Fig. 2 The effects of (a) oxygen, (b) gravitational force, and (c) [Q] on the precipitate formation. The experiments were carried out under (a) Ar and air, (b) parallel and anti-parallel, and (c) at various [Q] conditions.

Although our samples were handled under Ar flow throughout the experiments, oxygen could potentially diffuse from the solution into the gel during the long incubation time of 300 h. To confirm the effect of oxygen, we carried out two independent experiments under Ar and air (Fig. 2a). In both cases, a depleted region with no precipitates was formed about 2 cm away from the interface, indicating that the oxygen in air does not generate the depleted zone below the interface. Similarly, no difference was observed upon changing the configuration of the samples. If the gravitational force acts on the precipitates and moves them along the force, the precipitate would stay near the interface when the test tube is inverted (Fig. 2b). After 504 h at the given configuration, we compared the position of the precipitated regions. From the intensity line profiles, it was clear that there was no effect of the gravitational force on the position of the depleted zone below the interface. The last possibility investigated was the effect of a trace amount of Q added. As the purchased monomer reagent contains a small amount of Q (ca. 156 μM) and it was used without purification, not only Mon but also Q diffuses from the solution into the gel and inhibits the polymerization near the interface. However, as the concentration of Mon (1.5 M) was much higher than that of Q (ca. 1.5 × 10−4 M), enough Q was supplied not far away from the interface. Furthermore, the rate constant of the reaction R˙ to Q is much larger than that of R˙ to R, which suggests that the PMEA polymerization is considerably hindered at the interface region. Therefore, a detectable amount of polymer precipitates appeared at a distance not reached by Q. To confirm this hypothesis, we investigated the precipitate formation at various concentrations of [Q] (Fig. 2c). It was clear that the precipitates appeared just below the interface for [Q] = 0, indicating that Q inhibited the formation of precipitates near the interface. With the increase in [Q], the depleted region without precipitates expanded almost linearly. This result strongly supported our hypothesis that Q diffuses from the solution into the gel inhibiting the polymerization reaction.

Although it was observed from the three control experiments that the inhibition by Q proceeds near the interface, there is still the possibility of polymerization triggered by a small amount of R˙ that was not consumed by Q. To investigate the existence of polymers, we characterized the polymeric components in both the depleted and precipitated zones by GPC (Fig. 3). For the depleted zone, two separate peaks were detected, whereas only one peak appeared for the polymer in the precipitated zone. Data obtained from the analysis of each peak are summarized in Table 1. Peaks A1 and B1 exhibited the same order of molecular weight (Mn ≈ 100[thin space (1/6-em)]000) and degree of polymerization (Pn ≈ 1000), indicating that polymers were formed in both the depleted and precipitated zones. The slightly lower Mn of B1 compared to that of A1 resulted from the concentration difference between Mon and R˙. As the consumption of R˙ by Mon is significantly higher in the vicinity of the interface, the polymerization reaction that is initiated from a smaller amount of R˙ proceeds near the interface, resulting in a higher Mn near the interface region as the reaction rate of the polymerization is proportional to the monomer concentration. Although the coil-to-globule transition point (Pn*) of PMEA is unknown, poly(2-hydroxyethyl methacrylate) (PHEMA), which has a similar molecular structure, has a solubility transition at Pn* of about 40.34 Thus, it seems that insoluble globular polymers form in both regions. However, the peak intensity of A1 is two orders of magnitude smaller than that of B1. Thus, it can be concluded that the formation of a detectable amount of precipitates was not achieved in the depleted region because the In density was too low. The appearance of peak A2 corresponding to the polymer with low Pn confirms the role of Q to inhibit the polymerization at the initial stage of reaction at the depleted zone. These processes are depicted in Fig. 4. In the agarose gel without In, R˙ species formed from Mon trigger the polymerization reaction, which produces insoluble globular polymers with Pn higher than Pn* over the whole region. However, the polymer concentration is low near the interface due to the presence of Q diffusing from the solution. Far away from the interface where no Q is present, enough globular polymers are formed and precipitated by the DLCA process. This indicates that precipitates are formed when two conditions are satisfied: (i) the polymerization proceeds, triggering the coil-to-globule transition of each polymer and (ii) the number of globular polymers becomes high enough to undergo DLCA.

image file: d0cp03089a-f3.tif
Fig. 3 (a) Two regions in the agarose sample after the reaction without In. GPC data for (b) region A and (c) region B.
Table 1 Assignment of each peak for the characterization of the polymers. Mn and Mw were determined using polystyrene as a standard
Peak A1 Peak A2 Peak B1
M n 1.81 × 105 3.38 × 103 1.09 × 105
M w 3.62 × 105 3.75 × 103 2.81 × 105
M w/Mn 1.99 1.10 2.57
P n 1390 26 837

image file: d0cp03089a-f4.tif
Fig. 4 (a) The agarose sample after the reaction without In. (b) Degree of polymerization (Pn) with a coil-to-globule transition point (Pn*), and the density of polymer that triggers detectable aggregation. (c) Schematic illustration of polymers in the gel.

Although radical species formed thermally from monomers and generated a detectable amount of precipitates far away from the interface, their concentration was not high enough. To further investigate the role of the radical initiator, the dependence of [In] on the precipitate formation was studied (Fig. 5). At 72 h, there were no precipitates formed in the absence of In. At [In] = 40 μM, precipitates appeared ca. 1 cm away from the interface. The depleted region is the region where the addition of In (40 μM) was not sufficiently high to satisfy the two previously discussed conditions to form precipitates. However, because [Q] decreased as the distance from the interface increased, an [In] of 40 μM was enough to initiate the polymerization, satisfying the two conditions mentioned above. Furthermore, it was noticed that some periodic depleted bands appeared inside the precipitated zone. With a further increase in [In], the position satisfying the two conditions moved closer to the interface. Although the position where precipitates started to appear varied depending on [In], the precipitated bands included a few periodic depleted bands. Although [In] increased from 40 to 350 μM, the precipitated zone did not expand to the same extent. This is related to the decrease in [In] by 7 × 10−3 times after 72 h, indicating that the effect of the increase in [In] weakened as the reaction progressed.

image file: d0cp03089a-f5.tif
Fig. 5 Effect of [In] on the precipitate formation. (a) Pictures taken at 72 h and for various values of [In]. (b) Grey scale intensity line profiles.


In this study, it was identified that at least two polymerization processes occurred in the studied system. One is the polymerization reaction triggered by R˙ formed from In that generates periodic structures, which is the dominant process up to 100 h. After 100 h, the dominant process is a polymerization reaction triggered by R˙ formed from Mon, which yields a continuous precipitated zone. These two processes compete and generate periodic bands near the interface (<100 h) and a continuous band far away from the interface (>100 h), as shown in Fig. 1a. Owing to the differences in the radical density, radical lifetime, and radical reactivity that originate from various reactions of In and Mon, a difference in the reaction rate constants and diffusion equation of this system should be expected.

Periodic precipitated and depleted zones form at conditions where enough In is present in the reaction system, although polymerizations proceed without In. In the absence of In, R˙ species thermally formed from Mon played the role of In and triggered the polymerization reaction. However, near the solution/gel interface, Q diffused from the solution and quenched R˙ generated from Mon. Consequently, a transparent region was formed near the interface in which globular polymers were dispersed without aggregation. At a position away from the interface where [Q] was low, R˙ created globular polymers that aggregated and precipitated according to the DLCA process. The addition of In promoted the polymerization and precipitation, generating periodic precipitated and depleted zones. However, as the lifetime of In is rather short compared to the reaction time, the polymerization triggered by In did not extend far from the interface. The pattern formation along with the polymerization reactions, therefore, proceeded according to two independent reactions triggered by R˙ formed from either Mon or In.

Conflicts of interest

There are no conflicts to declare.


This work was supported by JSPS KAKENHI Grant Number 19H02668 and JSPS Fellows Grant Number 19J23178.

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Electronic supplementary information (ESI) available. See DOI: 10.1039/d0cp03089a

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