Achieving synchronization with active hybrid materials: Coupling self-oscillating gels and piezoelectric films

Lightweight, deformable materials that can sense and respond to human touch and motion can be the basis of future wearable computers, where the material itself will be capable of performing computations. To facilitate the creation of “materials that compute”, we draw from two emerging modalities for computation: chemical computing, which relies on reaction-diffusion mechanisms to perform operations, and oscillatory computing, which performs pattern recognition through synchronization of coupled oscillators. Chemical computing systems, however, suffer from the fact that the reacting species are coupled only locally; the coupling is limited by diffusion as the chemical waves propagate throughout the system. Additionally, oscillatory computing systems have not utilized a potentially wearable material. To address both these limitations, we develop the first model for coupling self-oscillating polymer gels to a piezoelectric (PZ) micro-electro-mechanical system (MEMS). The resulting transduction between chemo-mechanical and electrical energy creates signals that can be propagated quickly over long distances and thus, permits remote, non-diffusively coupled oscillators to communicate and synchronize. Moreover, the oscillators can be organized into arbitrary topologies because the electrical connections lift the limitations of diffusive coupling. Using our model, we predict the synchronization behavior that can be used for computational tasks, ultimately enabling “materials that compute”.


B. Kinetics of the BZ reaction in gel
The kinetics of the BZ reaction is described by a modification of the Oregonator model [S1], formulated in terms of the dimensionless concentrations of the key reaction intermediate u here). The modified model [S2,S3] accounts for the dependence of the BZ reaction rates on the volume fraction of polymer  , and on the total concentration of catalyst grafted to the network.
The reaction rates BZ F and BZ G in eqs.
(1) and (2) are determined as follows: The above reaction rates depend on the dimensionless concentrations of the reduced catalyst BrO , which are denoted by r and w , respectively [S2]. The where Ru c and 0  are the catalyst concentration and volume fraction of polymer in the undeformed gel, respectively. The value of w , the concentration of the radical, is found from the following equation: where  is a dimensionless parameter (see ref. 15 for further details). Finally, the stoichiometric factor f and the dimensionless parameter q have the same meaning as in the original Oregonator [S1] model, and we use the notation BZ  for the Oregonator parameter  [S1].
We assume that the chemical composition of the BZ substrate and the volume fraction of polymer in the undeformed gel are the same as used in the experiments described in ref. [S3]

C. Gel swelling under the action of external force
The stresses acting on the swollen polymer network in the absence of an external force are described by the following general stress-strain equation [S4,S5]: where osm  is the osmotic pressure of the polymer, Î is the unity tensor, and el σ is the contribution to the stress tensor from the elasticity of the cross-linked polymer network. The osmotic pressure of the polymer is calculated as: Here, . The gel elasticity contribution to the stress tensor is calculated using the Flory model of rubber elasticity to obtain [S4]: Here, c 0 is the cross-link density, 0  and  are the volume fractions of polymer in the undeformed and deformed gel, respectively, and B is the Finger strain tensor [S6].
The equilibrium degree of swelling of the small gel sample shown in Fig Note that for isotropic swelling, the volume fraction of gel  and the degree of swelling  are related as . Hence, solving eq. (S6) yields the degree of swelling  as a function of the concentration of oxidized catalyst, v .
If the external force acts on the gel as shown in Fig. 1f, the gel deformation is described by the strain tensor where  and   are the degrees of swelling in the longitudinal and transverse directions, respectively. The equilibrium degree of swelling is determined by balancing all the forces in the two directions, namely, Here, 0 h is the undeformed gel size (see Fig. 1e), the osmotic pressure osm  is calculated according to eq. (S4), and the volume fraction of polymer depends on the degrees of swelling  and   as . Therefore, eqs. (S7) and (S8)  , which is known to describe the PNIPAAm-water interaction at C 20  [S7]. The interaction parameter   , which accounts for the hydrating effect of the oxidized catalyst, is an adjustable parameter of the model; we set in this study. The same value of   was used in our previous publications [S2,S4].
Finally, the undeformed gel size is mm The values of . In Fig. 4

D. Properties of a bending piezoelectric bimorph plate
The coefficients 11 m , 12 m , and 22 m in eqs. (4) and (5)
We now discuss how the piezoelectric coupling affects the waveforms in Figs. 3a and 3b.
For this purpose, we use eqs. (S16) and (S17) to obtain the equation for the deflections ) of the cantilevers, which are connected in parallel and experience forces ). In particular, we find that: . Each force is assumed to be positive and vary from approximately zero to some maximal value. In the synchronized state, the forces have the same wave form, and are shifted in phase relative to each other. In the case of 2  n shown in Fig. 3, the deflection of cantilever 1 is calculated as Here, the ) ( and ) ( signs in the parentheses correspond to the respective force polarity sets , the system displays antiphase synchronization (Fig. 3a). Correspondingly, the force 2 F exhibits a maximum when the force 1 F is at the minimal value, 0 1  F , and hence, the deflection of cantilever 1 exhibits a "dip" 2 /

F. Equations of phase dynamics
The phase dynamics approach is an approximation used for describing the behavior of interacting oscillators in the limit of weak coupling [S12,S13]. The system of n identical oscillators is assumed to be governed by the following set of equations where i x is the set of variables describing the i -th oscillator, ) , ( determines the interaction between oscillators i and j , and the parameter  controls the strength of the interaction. In the case of no interaction ( 0   ), all oscillators exhibit the same T -periodic . If the coupling is weak ( 1   ), the interaction results primarily in a deviation of the phase of oscillation, and the solution of eq. (S18) takes the The evolution of the phase in time is described by the following equation [S13]: The T -periodic function (vector function) ) ( Q is known as the phase response curve (PRC) [S13]; it is also called the phase projection vector (PPV) [S14,S15]. The PRC can be determined using Malkin's method [S12,S13]. Namely, the function ) ( Q is the solution of the "adjoint" under the initial condition 1 )) 0 ( is the Jacobian matrix for a free-running oscillator (see (S18)) calculated along the limit cycle.
Solutions of eq. (S19) exhibit both the fast dynamics on the time scale of the period of oscillation T and the slow dynamics on the time scale of 1   . Note that the synchronization phenomena take place on the slow time scale. To extract the slow dynamics, time averaging is applied to eq. (S19) to obtain [S12]: To apply the above formalism to the gel-piezoelectric units, we introduce the partial variables for the BZ reactants , and write eqs. (1) and (2) for a freerunning unit as is the solution of the equilibrium swelling equation (see eqs. (3) and (S7)), conveniently written in the following form: Here ) ( When the gel-piezoelectric units are electrically connected, the force exerted on a gel is , where F  is the perturbation resulting from the interaction due to the piezoelectric effect (here, for simplicity, we dropped the subscript index labeling units). On the right-hand side of eqs. (6) and (7), F  is given by the second terms, and the factor  controls the interaction strength. To represent the effect of interaction in the form of eq. (S18), eqs. (S23) and (S24) are expanded into a Taylor series to obtain: The explicit forms of eq. (S18) and hence the function ij g are obtained from eqs. (S26) and (S27) after taking into account eqs. (6) and (7) that describe interaction between the units connected in series and in parallel, respectively. For example, for the parallel connection, we obtain the following equation: On the r.h.s. of eq. (S29), the column-vectors having the subscript " i " depend on u and v in the unit i , and ) 0 ( j F depends on the volume fraction of polymer (see the paper) and thus, on v in the unit j . Note that in eq. (S29), the parameter  plays the role of the interaction parameter  in eq. (S18).
The limit cycle solution of eqs. (S23) and (S24) shown in Fig. 4a is obtained numerically and then used to solve eq. (S20) to determine the PRC functions u Q and v Q . The Jacobian K is defined by eq. (S28), and the subscripts denote the partial derivative with respect to the corresponding variable. Numerical integration of eq. (S20) is performed backwards as