Viscosity-Change-Induced Density Fingering in Polyelectrolytes

J. Phys. Chem. B 2008, 112, 14593–14596

14593

Viscosity-Change-Induced Density Fingering in Polyelectrolytes† ´ gota To´th* Tama´s Rica, Dezsö Horva´th, and A Department of Physical Chemistry, UniVersity of Szeged, Rerrich Be´la te´r 1., Szeged, H-6720, Hungary ReceiVed: March 20, 2008; ReVised Manuscript ReceiVed: June 13, 2008

We have studied the density fingering of an acid-catalyzed autocatalytic reaction in the presence of carboxylate containing polyelectrolyte. The decrease in viscosity as a result of the change in the ionic character of the polymer due to the pH-change during the reaction is the major driving force for the spatiotemporal pattern formation. The front evolution is quantitatively characterized by dispersion curves. 1. Introduction Spatiotemporal patterns resulting from the interaction of an autocatalytic reaction with various types of transport processes abound in nature and hence they have been studied extensively.1 The simplest form is a propagating reaction-diffusion front with diffusion acting as the sole transport process. The most common, however, originates from the coupling of a chemical front with convection, since density generally changes in the course of the reaction, which can give rise to fluid motion.2,3 This phenomenon where the geometry of the reaction front is distorted by the convective mixing of the reactant and product solution is called density fingering of a reactive interface via the Rayleigh-Taylor instability. Density change can also be affectedbytheexothermicityofthereactionviatheRayleigh-Be´nard instability where heat evolution can either stabilize or destabilize planar fronts, depending on the sign of the compositional density change.4,5 The effect of buoyancy in autocatalytic reactions has recently been the focus of several experimental6-9 and numerical10-13 studies of vertically oriented, thin solution layers. Density fingering is favored when density change is increased by increasing the reactant concentrations14 or by decreasing the temperature rise with more efficient heat removal through conducting walls.13 Convective motion, however, slows down upon increasing solution viscosity by chemically inert polyacrylamide solution15 or upon changing the orientation by tilting the reaction vessel horizontally.9 Among the wide variety of autocatalytic reactions, the acidcatalyzed reactions with hydrogen ion being the autocatalyst are of special interest because of their use for pH-oscillators or in chemomechanical and biomimetic devices. In a closed system, they have been shown to give rise to chemical fronts.16 Initially planar fronts may lose stability, and cellular structure develops if the diffusive flux of hydrogen ion is decreased by immobile binding of the autocatalyst using carboxylate containing hydrogel17 or polymethacrylate.18 The advantage of faster diffusion rate of hydrogen ion can also be utilized for pattern formation in open systems. When an acid-catalyzed autocatalytic reaction is carried out in open spatial reactors—where the reactants are fed from one side—spatial bistability and excitable behavior are observed.19 Therefore, the use of a carboxylate-containing matrix in acid-catalyzed autocatalytic reactions yields interesting novel patterns. For example, chemomechanical instabilities in † Part of the “Janos H. Fendler Memorial Issue”. * Corresponding author. E-mail address: [email protected].

pH-responsive gels of copolymers of N-isopropylacrylamide and acrylic acid have been observed in the form of several millimeter amplitude deformations of the gel caused by the change of pH induced by a chemical reaction.20 The manifestation of selfreplicating spots and labyrinthine patterns in the ferrocyanideiodate-sulfite reaction21,22 also involves an ionic polymer, since hydrogen ion is immobilized by binding it to polyacrylate, as reported by Szalai and De Kepper.23 Ionic polymers have therefore played an important role in the spatiotemporal pattern formation of reaction-diffusion systems; and similar importance is anticipated for convective systems. In this work we have designed experiments where an autocatalytic chemical reaction with hydrogen ion as the autocatalyst will lead to a change in the ionic character of an otherwise inert polyelectrolyte, resulting in a decrease of solution viscosity. We are going to show that this change gives rise to convective instability and hence to cellular pattern formation. 2. Experimental Study Reagent-grade materials (Sigma, Aldrich, Reanal) were used throughout the work, except NaClO2 (Aldrich), which was recrystallized as described previously.24 Polyacrylamide and acrylamide-sodium methacrylate copolymer solutions were synthesized for the reaction medium. Polyacrylamide solutions were prepared by adding 6.0 g of acrylamide and 0.9 mL of 30 v/v % triethanol-amine solution to 41.1 mL of water. The solution was then cooled to 0 °C and degassed for 30 min. The polymerization was initiated by adding 3 mL of 0.11 M K2S2O8, and the solution was injected into a 100 mL beaker. After 30 min of polymerization, the solution was washed with distilled water several times. The polymer content of the final solution was determined by thermogravimetry. The carboxylate containing polyelectrolyte was prepared similarly: to 2.282 g of acrylamide, 10.55 mL of sodium methacrylate (1.0 M), 0.45 mL of 30 v/v % triethanol-amine solution, and 10.5 mL of water were added. The further procedures of the preparation of the polyelectrolytewerethesameasthosedescribedforpolyacrylamide. The solution with composition in Table 1 was mixed and injected into a vertically positioned 16 cm wide, 12 cm long, and 1.0 mm thin reaction vessel (so-called Hele-Shaw cell) with 8 mm thick Plexiglas walls at 25 ( 1 °C. Planar reaction fronts were then initiated electrochemically by applying a 2.8 V potential difference between a Pt-wire electrode (0.25 mm in diameter) and a Cu-wire for 9-20 s. The color of the selected indicator changes from the initial blue to bright yellow as a result of the pH drop in the course of the reaction; therefore,

10.1021/jp802450r CCC: $40.75  2008 American Chemical Society Published on Web 08/12/2008

14594 J. Phys. Chem. B, Vol. 112, No. 46, 2008

Rica et al.

TABLE 1: Composition of Reactant Solutiona [K2S4O6]/mM [NaClO2]/mM [NaOH]/mM [Bromophenol blue]/mM Polyacrylamide/g/dm3 Polyelectrolyte/g/dm3 and hence [COO-]/mM

A

B

5.00 20.00 1.00 0.16 13.6 0 0

5.00 20.00 1.00 0.16 0 7.20 24.0

a The polyelectrolyte is an acrylamide sodium-methacrylate copolymer solution with a molar ratio of acrylamide/methacrylate ) 3:1.

Figure 2. Dispersion curves of planar fronts for upward (2) and downward (b) propagating fronts with composition B and in the inset for upward (∆) and downward (O) propagating fronts with composition A of Table 1. Lines are drawn to guide the eye.

Figure 1. Images of upward (a) and downward (b) propagating fronts with composition A and upward (c) and downward (d) propagating fronts with composition B of Table 1. Dark color corresponds to the reactant, while light corresponds to the product mixture. Field view: 57.3 mm × 8.4 mm.

the traveling fronts were monitored by a monochrome CCD camera connected to an MVdelta imaging card (Matrix Vision). The images were then processed by applying standard imaging procedures. The solution densities were measured by a density meter (AP PAAR DMA 58) with 10-5 g/cm3 precision, while the viscosity was measured by an Ostwald viscometer with 0.01 cP precision. The dispersion curve, defined as the growth rate as a function of the wavenumber, characterizes the initial evolution of the pattern formation. From the time evolution of the Fourier amplitudes obtained from a one-dimensional Fourier transformation of the front profiles, the growth rates are calculated for each mode as the slope of the linear regime, as described in detail previously.9 3. Results and Discussion We have shown earlier that the addition of polyacrylamide solution to the reactant mixture of the chlorite-tetrathionate reaction basically increases the solution viscosity without any effect on the isothermal density change arising from the change in chemical composition in the course of the reaction.15 In our system with composition A of Table 1, therefore, the product solution has greater density than the reactant, which should result in stable upward propagating planar fronts as shown in Figure 1a. By considering only this solutal density change, one would anticipate that downward propagating planar fronts should lose stability, giving rise to cellular fronts, which—by looking at Figure 1b—is clearly contrary to that observed in the experiments. This apparent discrepancy lies in the amount of polymer applied in the reaction mixture, resulting in an increase in solution viscosity by a factor of 11 compared to the reference

system without polyacrylamide. In this very viscous solution, the substantial slowing down of fluid motion does not lead to the deformation of reaction fronts within the time frame of the experimental observation. Furthermore, the heat evolved from the highly exothermic reaction9 dissipates more slowly in viscous solution, which also yields an additional stabilizing effect.25,26 The pattern formation can be described quantitatively by the dispersion curves shown in the inset of Figure 2. The upward propagating front is definitely stable, since all the modes of the corresponding dispersion curve have negative growth rates, i.e., random perturbations decay in time, and the front maintains its planar symmetry. For the downward propagating front, there is a mode with a slightly positive growth rate at a moderate wavenumber, in agreement with the positive solutal density change; however, its absolute value is so small that microscopic perturbations cannot evolve into macroscopic distortion of the front. The negative growth rates at low wavenumbers corroborate the stabilizing effect of temperature rise on density fingering as described previously.25,26 The marginal wavenumber separating the stable and unstable modes is around 0.3 mm-1. The behavior observed is entirely different when polyelectrolyte solution with composition B in Table 1 is added to adjust the viscosity to the same initial value of 12.17 cP. The upward propagating fronts retain their stability (see Figure 1c and the appropriate dispersion curve in Figure 2), but from the planar initiation, cellular patterns, illustrated in Figure 1d, develop when fronts propagate downward. The most unstable mode of the corresponding dispersion curve in Figure 2 is around 1 mm-1 wavenumber, which represents an average wavelength around 6 mm for the initially evolving cellular structure, in agreement with that seen in Figure 1d. The appearance of the initial fingers is significantly faster, since the growth rate is now at least a magnitude greater. The marginal wavenumber is 2.8 mm-1, which is close to the value of 3.5 mm-1 obtained for purely aqueous solutions in the absence of polymers.9,14 The density change of 2.7 × 10-4 g/cm3 for composition B in Table 1 is similar to that for composition A (3.5 × 10-4 g/cm3); the main difference is that the polyelectrolyte solution contains immobile carboxylate groups that bind hydrogen ions reversibly to the polymer. The lowering of the concentration of free hydrogen ions and hence their diffusive flux, however, is not sufficient to induce diffusive instability,17,18 since both upward and horizontally propagating planar fronts are observed to maintain their planar symmetry, as shown in Figure 1c.

Density Fingering in Polyelectrolytes

J. Phys. Chem. B, Vol. 112, No. 46, 2008 14595 the possibility of stronger convective motion and hence to density fingering if the less viscous solution lies on top of the more viscous one like at the downward propagating fronts in Figure 1d. The extent of convective instability may be characterized by the Rayleigh number—which represents the ratio of gravitational and viscous forces—as

Ra )

Figure 3. Viscosity of polyelectrolytes relative to water as a function of pH with [COO-]0 ) 29.2 mM (b), [COO-]0 ) 14.6 mM (9), [COO-]0 ) 0 mM (O), reactant solution (∆), and product solution of composition A (2), reactant solution ()) and product solution of composition B (().

Figure 4. Viscosity of polyelectrolyte relative to water as a function of ionic strength with [COO-]0 ) 29.2 mM (b).

Although the introduction of carboxylate groups into the polymer has only negligible or no effect on the density change accompanying the reaction, it leads to the conformational change of the polymer chain brought about by the protonation of the basic groups in the course of the reaction. The decrease in ionic character due to the bonding of hydrogen ions significantly alters the viscosity of the solution, illustrated in Figure 3, where pure polymer solutions with different carboxylate content are considered at various pH adjusted by an acid or a base, appropriately. Viscosity remains independent of pH when the polymer does not contain carboxylate groups; however, it decreases by more than a magnitude as pH is lowered when the carboxylate content is increased to 29.2 mM. The pH in the course of the reaction decreases from 7.9 to 2.0 and 2.5, corresponding to composition A and B, respectively, which would yield a magnitude decrease in the viscosity for composition B according to Figure 3. It is important to notice, however, that the decrease in viscosity accompanying the reaction is not as great as expected from Figure 3 because there is a difference between the viscosity of polyelectrolyte solution in water and in the reactant mixture due to the presence of reactant ions. Figure 4 illustrates that as ionicstrength—adjustedbytheadditionofpotassiumchloride—increases, the viscosity of polyelectrolyte solution used in composition B decreases significantly, as expected from the change in the surrounding ionic environment. Figures 3 and 4 clearly indicate that the autocatalytic reaction producing hydrogen ion significantly changes the ionic nature of the copolymer applied in composition B, which results in a substantial decrease in viscosity. In our case, it then leads to

ga2∆F 12ηU

(1)

where g is the standard acceleration, a is the thickness of the Hele-Shaw cell, ∆F is the change in density at the reaction front, η is the solution viscosity, and U is the velocity of propagation for reaction-diffusion fronts in the absence of convection.27 Its value is only 0.18 for the very viscous system of composition A, resulting in the very small growth rates in the dispersion curves. With the introduction of the polyelectrolyte leading to the decrease in viscosity results, the Rayleigh number changes from 0.73 to 5.6 across the reaction front, which yields the substantial increase in the range of the unstable modes and in the growth rates, despite the large viscosity of the reactant solution. Although viscosity decreases in the course of the reaction, the pattern formation is still driven by the density difference arising in the gravitational field; unlike viscous fingering that would otherwise appear in all directions. Our experiments also prove a recent numerical study on cubic autocatalysis, which reports that viscosity change may either hinder or enhance convective instability.28 In conclusion, we have shown experimentally that in an autocatalytic reaction where the pH changes in the course of the reaction, cellular structures evolve from planar reaction fronts because of the significant decrease in solution viscosity. It results from the drastic change in the ionic properties of the polymer—appliedtoadjusttheinitialviscosityofthesolution—induced by the drop in pH across the reaction front. The convective rolls in the less viscous solution behind the front will lead to the distortion of planar fronts by density fingering in the otherwise excessively viscous reaction medium. This is a new example where the change in polyelectrolyte properties serves as the basis for spatiotemporal pattern formation, which may also be significant in biological systems. Acknowledgment. This work was supported by ESA (PECS 98036). References and Notes (1) Epstein, I. R.; Pojman, J. A. An Introduction to Nonlinear Dynamics: Oscillations, WaVes, Patterns, and Chaos; Oxford University Press: Oxford, 1998. (2) Nagypa´l, I.; Bazsa, Gy.; Epstein, I. R. J. Am. Chem. Soc. 1986, 108, 3635. (3) Pojman, J. A.; Epstein, I. R. J. Phys. Chem. 1990, 94, 4966. (4) Pojman, J. A.; Epstein, I. R.; McManus, T. J.; Showalter, K. J. Phys. Chem. 1991, 95, 1299. (5) Legawiec, B.; Kawczyn´ski, A. L. J. Phys. Chem. A 1997, 101, 8063. (6) Masere, J.; Vasquez, D. A.; Edwards, B. F.; Wilder, J. W.; Showalter, K. J. Phys. Chem. 1994, 98, 6505. (7) Komlo´si, A; Nagy, I. P.; Bazsa, Gy.; Pojman, J. A. J. Phys. Chem. A 1998, 102, 9136. (8) Bo¨ckmann, M.; Mu¨ller, S. C. Phys. ReV. Lett. 2000, 85, 2506. ´ . J. Chem. Phys. 2002, 117, (9) Horva´th, D; Ba´nsa´gi, T., Jr.; To´th, A 4399. (10) Vasquez, D. A.; Littley, J. M.; Wilder, J. W.; Edwards, B. F. Phys. ReV. E 1994, 50, 280. (11) De Wit, A. Phys. ReV. Lett. 2001, 87, 054502. (12) Yang, J.; D’Onofrio, A.; Kalliadasis, S.; De Wit, A. J. Chem. Phys. 2002, 117, 9395. (13) D’Hernoncourt, J.; Kalliadasis, S.; De Wit, A. J. Chem. Phys. 2005, 123, 234503.

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