SCIENCE CHINA Technological Sciences • RESEARCH PAPER •
July 2010 Vol.53 No.7: 1862–1868 doi: 10.1007/s11431-010-3114-5
Nonlinear dynamics of self-oscillating polymer gels WANG PengFei1, ZHOU JinXiong1, LI MeiE2, XU Feng1,3 & LU TianJian1* 1
Biomedical Engineering and Biomechanics Center, SV Laboratory, Xi’an Jiaotong University, Xi’an 710049, China; 2 School of Materials Science and Engineering, Xi’an Jiaotong University, Xi’an 710049, China; 3 HST Center for Biomedical Engineering, Department of Medicine, Brigham and Women’s Hospital, Harvard Medical School, Boston, 510660, USA Received August 27, 2009; accepted January 21, 2010
Self-oscillating polymer gels driven by Belousov-Zhabotinsky (BZ) chemical reaction are a new class of functional gels that have a wide range of potential applications (e.g., autonomously functioning membranes, actuate artificial muscles). However, the precise control of these gels has been an issue due to limited investigations of the influences of key system parameters on the characteristics of BZ gels. To address this deficiency, we studied the self-oscillating behavior of BZ gels using the nonlinear dynamics theory and an Oregonator-like model, with focus placed upon the influences of various system parameters. The analysis of the oscillation phase indicated that the dynamic response of BZ gels represents the classical limit cycle oscillation. We then investigated the characteristics of the limit cycle oscillation and quantified the influences of key parameters (i.e., initial reactant concentration, oxidation and reduction rate of catalyst, and response coefficient) on the self-oscillating behavior of BZ gels. The results demonstrated that sustained limit cycle oscillation of BZ gels can be achieved only when these key parameters meet certain requirements, and that the pattern, period and amplitude of the oscillation are significantly influenced by these parameters. The results obtained in this study could enable the controlled self-oscillation of BZ gels system. This has several potential applications such as controlled drug delivery, miniature peristaltic pumps and microactuators. Belousov-Zhabotinsky reaction, self-oscillating gels, numerical simulation, limit cycle, parameter analysis Citation:
Wang P F, Zhou J X, Li M E, et al. Nonlinear dynamics of self-oscillating polymer gels. Sci China Tech Sci, 2010, 53: 18621868, doi: 10.1007/s11431-010-3114-5
Polymer gels are mixtures of a large number of small solvent molecules dissolved into a polymer network [1]. Some polymer gels can swell or deswell, in a controlled manner, in response to external stimuli (e.g., pH [2], temperature [3], magnetism [4], electricity [5] and light [6]). Because of the distinctive characteristics of these gels (i.e., polymer gels with controlled response driven by external stimulations), they have been used for applications such as periodic precision devices (e.g., micropumps, smart control valves [7, 8]) and environmental sensors (e.g., magneticfield sensor, pHresponsive device[9,10]). Besides external stimulation-controlled gels, self-oscil-
*Corresponding author (email:
[email protected]) © Science China Press and Springer-Verlag Berlin Heidelberg 2010
lating polymer gels have also been developed which do not require external stimulations to rhythmically produce a dynamic response in volume (i.e., swelling and deswelling) [11, 12], driven by Belousov-Zhabotinsky chemical reaction [13] (i.e., BZ gels). In BZ gels, catalyst ruthenium (i.e., Ru(bpy)3) of the BZ reaction was chemically bundled on cross-linked polymer chains (Figure 1), and the BZ solution without the catalyst was dissolved into the polymer network. During the reduction-oxidation (redox) process of the BZ reaction in the gel-solution system, Ru(bpy)3 continuously and transforms between two valences, i.e., Ru(bpy)2+ 3 Ru(bpy)2+ 3 . This reversible valence transformation is sustained by the coupling between gel swelling/deswelling and BZ reaction: BZ gels swell during the process of Ru(bpy)2+ 3 ĺRu(bpy)2+ and deswell during the opposite process due to 3 tech.scichina.com
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system parameters.
1 Model of BZ gels and key system parameters The BZ reaction found by Belousov [35] in 1959 consists of more than 70 basic and coupled chemical sub-reactions which in general can be described using the following three chemical equations [36]: BrO3 2Br 3H o 3HOBr Figure 1 Chemical structure of BZ polymer Chains [15]. The catalyst of BZ chemical reaction, Ru(bpy)3, is bounded on the polymer chain of NIPAAm. 3+ 3
the different hydrophilic capability of Ru(bpy) and , Figure 2; furthermore, BZ reaction is affected by Ru(bpy)2+ 3 the volume fraction changes of polymer chains, because the concentration ratios of reactants of the BZ reaction are changed [14]. As a result, in the absence of external stimulation, BZ gels exhibit the unique autonomous dynamic volume changes (i.e., self-oscillation). Significant advances have been made in the processing of BZ gels [16–19] and these gels have been applied to biomimetic materials, chemical robotics, and peristaltic actuator[18, 20–22]. However, further application of BZ gels has been hindered by the limited understanding of BZ gels kinetics, although the BZ reaction in a pure solution has been extensively studied [13, 23–26]. Besides a number of experimental investigations (e.g., influences of reactant concentration, temperature and light [24, 27] on BZ reaction), the FKN [28] and Oregonator [13] models have been proposed to predict the BZ reaction in a pure solution. The BZ reaction occurring in the gel-solution system is much more complicated due to the interplay between the volume deformation of a responsive medium and the nonlinear chemical dynamics within the BZ gels. In recent years, several theoretical models of BZ gels have been proposed. The existing theoretical models [29–34] have explained some interesting features of BZ gels. However, there are few studies on the influences of key system factors (e.g., initial reactant concentration, oxidation and reduction rate of catalyst, and response coefficient to BZ reaction) on the self-oscillating behavior of the BZ gels. To address the challenges, in this paper we employ the nonlinear dynamics theory and the Oregonator model to analyze the self-oscillating behavior of BZ gels, focusing on the influences of
BrO3 HBrO2 2 M red 3H o 2HBrO2 2M ox H 2 O (1) 2M ox MA BrMA o fBr 2M red other products
where Mox and Mred represent a certain kind of metal-ions in oxidation state and reduction state, respectively; MA is an acid, M is a chemical catalyst in various BZ reactions (e.g., , Ru(bpy)2+/3+ [12, 28, 37]) and is, in the Ce3+/4+, Fe(phen)2+/3+ 3 3 present study, M = Ru(bpy)3. The chemical coefficient f represents the number of Br oxidized by two ions of Mox in the redox process. Note that Br is produced in a number of mutually coupled basic reactions, and hence the quantity of Br produced in the BZ reaction can not be uniquely determined by the chemical valence change of Mox in the process of reduction. In other words, f is not only related to the ion species of Mox, but also affected by other factors including the concentration of initial reactants and temperature of the system [27]. So far, there are few studies quantifying the relations between f and the above system factors. It has been demonstrated that the value of f lies in the range of 0.5–2.4 for a sustained oscillating BZ reaction [38]. The Oregonator model proposed by Field and Noyes [38] has been widely applied to study of BZ reaction. This model enables the elucidation of the general mechanism of various BZ reactions. In the Oregonator model there are two important parameters, i.e., f and H{cB/(HA), where A, H and B are the concentrations of three reactant species of the BZ reaction and c is a parameter related to the BZ reaction rate. These two parameters represent the influences of several factors upon the characteristics of BZ reaction [39], including BZ reaction mechanisms, initial reactant concentration, oxidation and reduction rate of the catalyst, response coefficient of polymer chains to the BZ reaction, and system temperature. However, the original Oregonator model for BZ reaction in a pure solution is not suitable for BZ reaction in a coupled gel-solution system, due to the interplay between polymer chains of gels and solution. To address this, a modified Oregonator model was proposed to describe BZ reaction in a coupled gel-solution system, i.e., [31, 36] du dt
Figure 2
Schematic of the self-oscillation of BZ gels.
dv dt
§ dI · u(1 I ) 1 ¨ ¸ F (u, v, I ) , © dt ¹
(2)
§ dI · vI 1 ¨ ¸ H G(u, v, I ) , © dt ¹
(3)
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and then substituting v(I) into eqs. (2) and (3), the model of BZ gels can be simplified into two ordinary differential equations: du § dI · u(1 I ) 1 ¨ ¸ F (u, I ) , (5) dt © dt ¹
where F (u, v, I )
(1 I )2 u u2 (1 I ) fv
G(u, v, I )
u q(1 I )2 , u q(1 I )2
(1 I )2 u (1 I )v ,
u, v are the dimensionless values of the dynamic concentrations of HBrO2 and Ru(bpy)3+ 3 , respectively; q is a system temperature-related constant; and I represents the volume fraction of polymer chains in a gel. It is hypothesized that the initial volume V0 and volume fraction I0 of a gel become V(t) and I(t) after a period of time t, due to the swelling/deswelling of the gel. Therefore, one can obtain V(t) = V0I0/I(t), where the dynamic deformation of a gel can be formulated in terms of changes in the value of I. The diffusion of solution and polymer chains in a smallsized self-oscillating BZ gel occurs instantaneously [15] (i.e., the initial radius r of a spherical gel particle is less than the wavelength of BZ chemical reaction, with r 0.67, the system is not able to exhibit autonomously sustained oscillations whatever the value of f. In addition, the range of H is initially widened and then narrowed as the value of f increases in the range of 0.5–2.4. The range of H is maximized at f = 0.86. The influence of f on the period of the system for selected values of H is shown in Figure 5(b). With H= 0.08 as an example, the period of a self-oscillating BZ gel decreases slowly as f increases from 0.52; when f reaches
Figure 5 Influences of key parameters H and f on the period of BZ gels (F= 0.105). (a) 3D relationships among H, f and T; (b) influence of f on the period with different values of H; (c) influence of H on the period with f = 0.7.
the critical value of 1.86, the period of the system decreases sharply as f increases further. That the BZ gel system becomes very sensitive when the parameter f reaches the critical value may be attributed to the fact that the quantity of Br produced in the process of oxidation of two Mox reaches a critical value. For the BZ gel system studied here, its periodic limit cycle oscillations disappear when f > 2.12. In Figure 5(b), the curves corresponding to H= 0.04 and H= 0.12 exhibit trends similar to the one of H= 0.08. However, noticeable differences amongst the curves still exist: (i) with the value of f fixed, the oscillation period and its rate of change both increase with decreasing H, as shown in Figure 5(a); (ii) for small values of H, the sudden change of the oscillation period at the critical point is more obvious, as can be seen from Figure 5(b). In addition, similar to f, the selected value of H should be smaller than 0.52 to realize the
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steady-state limit cycle oscillation of BZ gels. Besides the oscillation period, the parameters H and f also have noticeable influences on the oscillation amplitude I of a self-oscillating BZ gel, as shown in Figure 6. With the value of H fixed, Figure 6(b) demonstrates that I decreases linearly with increasing f until f reaches a critical value beyond which the amplitude sharply decreases. When f is relatively small ( f 0.3 on the amplitude can be determined from Figure 6(a). 3.2 Influences of response coefficient F on period and amplitude of oscillation
Figures 7 and 8 demonstrate that the predicted period and
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amplitude of BZ gels both increase as the response coefficient F increases if f is smaller than its critical value ( f = 1.86), which is consistent with experimental observations on the interaction between chemical reaction and polymer dynamics [42, 44, 45]. The period and amplitude of BZ gels are only slightly affected by changes of F when f exceeds the critical value; see Figures 8(a) and (b). In addition, the amplitude is sensitive to F when it is relatively small (F0.2, the amplitude increases slowly with increasing F. In the following, the value of F is selected between 0 and 1 considering the predicted influence of F on the system shown in Figure 8(b). 3.3 Influences of chemical parameters f and H on selfoscillation pattern
Based on the results presented above, a limit cycle oscillat-
Figure 6 Influences of H and f on oscillation amplitude of BZ gels (F= 0.105). (a) 3D relationships among H, f and amplitude; (b) influence of f on amplitude for selected values of H; (c) influence of H on amplitude with f = 0.7.
Figure 7 Influences of F on the period of BZ gels (H= 0.08). (a) 3D relationships among F, f and period; (b) influence of f on the period for selected values of F; (c) influence of F on the period with f = 0.7.
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Figure 9 Influences of f and H on self-oscillation pattern. (a) f = 0.51 (H= 0.08, F= 0.105); (b) H= 0.53 ( f = 0.7, F= 0.105).
4 Conclusions
Figure 8 Influences of F on the amplitude of BZ gels (H= 0.08). (a) 3D relationships among F, f and amplitude; (b) influence of f on the amplitude for selected values of F; (c) influence of F on the amplitude with f = 0.7.
ing BZ gel system can be achieved only when the chemical parameters of the system f and H have values falling in certain ranges, except for the parameter F. It has already been established that the maximum range of f is 0.5 f 2.4 for a sustained oscillating BZ reaction occurring in pure solution. The present results (Figure 5(b)) demonstrate that the range of f, with F= 0.105 and H= 0.08, is 0.52 f 2.12 for sustained oscillating BZ reaction in a gel-solution system. Correspondingly, the range of H is 0
