# Multiphysics Simulator for the IPMC Actuator: Mathematical Model, Finite Difference Scheme, Fast Numerical Algorithm, and Verification

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

- Ionic polymer-metal composite is considered to be two-phase and includes the solid phase, which is a polymer porous structure, fixed negative charge and metal electrodes, and the liquid phase, which includes cations and water molecules, redistributed under an electric field and/or a mechanical load.
- The liquid phase flux consists of two components: diffusion (including electromigration) and convective. The diffusion fluxes of ions and water molecules are determined by the potential gradient, the concentration gradients of ions and water molecules, and the hydrostatic pressure gradient created by redistribution of ions and water molecules in polymer nanopores. The solid phase influences the diffusion fluxes through the nanopore structure and the electric field of fixed negative ions in the membrane. The convective fluxes are determined by the elastic force of the solid phase.
- In a short time interval, the hydraulic pressure and the inherent mechanical stress are balanced with the elastic stress of the composite solid phase.

## 3. Discretization of the Mathematical Model, Numerical Simulation Technique

- Subsystem 1, which includes Poisson Equation (15) with boundary conditions (17), (18), was solved by a direct method
- Subsystem 2, which includes modified Nernst-Planck Equations (7) and (8) with initial conditions (9) and (12) and boundary conditions (10), (11), (13), and (14), was solved using the Newton-Raphson method
- Subsystem 3, which includes cantilever beam mechanical oscillation Equation (26) with initial conditions (27) and (28), was solved using an explicit scheme

- Subsystem 1 (15), (17), (18)$$\frac{{\phi}_{j+1}^{m}-2{\phi}_{j}^{m}+{\phi}_{j-1}^{m}}{\Delta {y}^{2}}=-\frac{q{Z}_{I}F}{\epsilon {\epsilon}_{0}}\left({C}_{I}{}_{j}^{m}-{C}_{-}\right);$$$${\phi}_{1}^{m}=0;$$$${\phi}_{J}^{m}={U}_{m},$$
- Subsystem 2 (7)–(14)$$\begin{array}{ll}\phantom{\rule{1.em}{0ex}}\hfill & {{C}_{I}}_{j}^{m+1}-\frac{{D}_{II}\Delta t}{\Delta {y}^{2}}\left[{{C}_{I}}_{j+1}^{m+1}-2{{C}_{I}}_{j}^{m+1}+{{C}_{I}}_{j-1}^{m+1}\right.+\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & +\frac{{Z}_{I}F}{2RT}\left(\left({{C}_{I}}_{j+1}^{m+1}+{{C}_{I}}_{j}^{m+1}\right)\left({\phi}_{j+1}^{m+1}-{\phi}_{j}^{m+1}\right)-\left({{C}_{I}}_{j}^{m+1}+{{C}_{I}}_{j-1}^{m+1}\right)\left({\phi}_{j}^{m+1}-{\phi}_{j-1}^{m+1}\right)\right)+\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & +\frac{{n}_{dW}}{2}\left(\left({{C}_{I}}_{j+1}^{m+1}+{{C}_{I}}_{j}^{m+1}\right)\left(\mathrm{ln}{{C}_{W}}_{j+1}^{m+1}-\mathrm{ln}{{C}_{W}}_{j}^{m+1}\right)-\left({{C}_{I}}_{j}^{m+1}+{{C}_{I}}_{j-1}^{m+1}\right)\left(\mathrm{ln}{{C}_{W}}_{j}^{m+1}-\mathrm{ln}{{C}_{W}}_{j-1}^{m+1}\right)\right)+\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & +\frac{{\eta}_{I}{V}_{I}}{2}\left(\frac{{V}_{I}+{n}_{dW}{V}_{W}}{RT}+\frac{K}{{D}_{II}}\right)\left(\left({{C}_{I}}_{j+1}^{m+1}+{{C}_{I}}_{j}^{m+1}\right)\left({{C}_{I}}_{j+1}^{m+1}-{{C}_{I}}_{j}^{m+1}\right)\right.-\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & -\left.\left({{C}_{I}}_{j}^{m+1}+{{C}_{I}}_{j-1}^{m+1}\right)\left({{C}_{I}}_{j}^{m+1}-{{C}_{I}}_{j-1}^{m+1}\right)\right)+\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & +\frac{{\eta}_{W}{V}_{W}}{2}\left(\frac{{V}_{I}+{n}_{dW}{V}_{W}}{RT}+\frac{K}{{D}_{II}}\right)\left(\left({{C}_{I}}_{j+1}^{m+1}+{{C}_{I}}_{j}^{m+1}\right)\left({{C}_{W}}_{j+1}^{m+1}-{{C}_{W}}_{j}^{m+1}\right)\right.-\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & -\left.\left.\left({{C}_{I}}_{j}^{m+1}+{{C}_{I}}_{j-1}^{m+1}\right)\left({{C}_{W}}_{j}^{m+1}-{{C}_{W}}_{j-1}^{m+1}\right)\right)\right]-{{C}_{I}}_{j}^{m}=0;\hfill \end{array}$$$$\begin{array}{ll}\phantom{\rule{1.em}{0ex}}\hfill & {{C}_{W}}_{j}^{m+1}-\frac{\Delta t}{\Delta {y}^{2}}\left[{D}_{WW}\left({{C}_{W}}_{j+1}^{m+1}-2{{C}_{W}}_{j}^{m+1}+{{C}_{W}}_{j-1}^{m+1}\right)\right.+\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & +\frac{{\eta}_{W}{V}_{W}}{2}\left({D}_{WW}\frac{{V}_{W}}{RT}+K\right)\left(\left({{C}_{W}}_{j+1}^{m+1}+{{C}_{W}}_{j}^{m+1}\right)\left({{C}_{W}}_{j+1}^{m+1}-{{C}_{W}}_{j}^{m+1}\right)\right.-\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & -\left.\left({{C}_{W}}_{j}^{m+1}+{{C}_{W}}_{j-1}^{m+1}\right)\left({{C}_{W}}_{j}^{m+1}-{{C}_{W}}_{j-1}^{m+1}\right)\right)+{n}_{dW}{D}_{II}\left({{C}_{I}}_{j+1}^{m+1}-2{{C}_{I}}_{j}^{m+1}+{{C}_{I}}_{j-1}^{m+1}\right)+\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & +{\eta}_{I}{n}_{dW}\frac{{D}_{II}}{2}\frac{{{V}_{I}}^{2}}{RT}\left(\left({{C}_{I}}_{j+1}^{m+1}+{{C}_{I}}_{j}^{m+1}\right)\left({{C}_{I}}_{j+1}^{m+1}-{{C}_{I}}_{j}^{m+1}\right)-\left({{C}_{I}}_{j}^{m+1}+{{C}_{I}}_{j-1}^{m+1}\right)\left({{C}_{I}}_{j}^{m+1}-{{C}_{I}}_{j-1}^{m+1}\right)\right)+\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & +{\eta}_{W}{n}_{dW}\frac{{D}_{II}}{2}\frac{{V}_{I}{V}_{W}}{RT}\left(\left({{C}_{I}}_{j+1}^{m+1}+{{C}_{I}}_{j}^{m+1}\right)\left({{C}_{W}}_{j+1}^{m+1}-{{C}_{W}}_{j}^{m+1}\right)-\left({{C}_{I}}_{j}^{m+1}+{{C}_{I}}_{j-1}^{m+1}\right)\left({{C}_{W}}_{j}^{m+1}-{{C}_{W}}_{j-1}^{m+1}\right)\right)+\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & +\frac{{\eta}_{I}{V}_{I}}{2}\left({D}_{WW}\frac{{V}_{W}}{RT}+K\right)\left(\left({{C}_{W}}_{j+1}^{m+1}+{{C}_{W}}_{j}^{m+1}\right)\left({{C}_{I}}_{j+1}^{m+1}-{{C}_{I}}_{j}^{m+1}\right)-\left({{C}_{W}}_{j}^{m+1}+{{C}_{W}}_{j-1}^{m+1}\right)\left({{C}_{I}}_{j}^{m+1}-{{C}_{I}}_{j-1}^{m+1}\right)\right)+\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & +\left.{n}_{dW}\frac{{D}_{II}}{2}\frac{{Z}_{I}F}{RT}\left(\left({{C}_{I}}_{j+1}^{m+1}+{{C}_{I}}_{j}^{m+1}\right)\left({\phi}_{j+1}^{m+1}-{\phi}_{j}^{m+1}\right)-\left({{C}_{I}}_{j}^{m+1}+{{C}_{I}}_{j-1}^{m+1}\right)\left({\phi}_{j}^{m+1}-{\phi}_{j-1}^{m+1}\right)\right)\right]-{{C}_{W}}_{j}^{m}=0;\hfill \end{array}$$$${C}_{I}{}_{j}^{1}={C}_{+}\frac{{\rho}_{SPN}\left(1-{P}_{WN}\right)}{{\left(1+\alpha \right)}^{3}h};$$$${C}_{W}{}_{j}^{1}={K}_{w}\frac{{\rho}_{SPN}\left(1-{P}_{WN}\right){P}_{WS}}{{M}_{W}{\left(1+\alpha \right)}^{3}h};$$$$\begin{array}{l}{C}_{I}{}_{2}^{m+1}-{C}_{I}{}_{1}^{m+1}+\frac{{Z}_{I}F}{RT}{C}_{I}{}_{1}^{m+1}\left({\phi}_{2}^{m+1}-{\phi}_{1}^{m+1}\right)+{n}_{dW}{C}_{I}{}_{1}^{m+1}\left(\mathrm{ln}{C}_{W}{}_{2}^{m+1}-\mathrm{ln}{C}_{W}{}_{1}^{m+1}\right)+\\ +{\eta}_{I}{V}_{I}\left(\frac{{V}_{I}+{n}_{dW}{V}_{W}}{RT}+\frac{K}{{D}_{II}}\right){C}_{I}{}_{1}^{m+1}\left({C}_{I}{}_{2}^{m+1}-{C}_{I}{}_{1}^{m+1}\right)+\\ +{\eta}_{W}{V}_{W}\left(\frac{{V}_{I}+{n}_{dW}{V}_{W}}{RT}+\frac{K}{{D}_{II}}\right){C}_{I}{}_{1}^{m+1}\left({C}_{W}{}_{2}^{m+1}-{C}_{W}{}_{1}^{m+1}\right)-\Delta y\frac{{\gamma}_{I}}{H}{C}_{I}{}_{1}^{m+1}=0;\end{array}$$$$\begin{array}{l}{C}_{I}{}_{J}^{m+1}-{C}_{I}{}_{J-1}^{m+1}+\frac{{Z}_{I}F}{RT}{C}_{I}{}_{J}^{m+1}\left({\phi}_{J}^{m+1}-{\phi}_{J-1}^{m+1}\right)+{n}_{dW}{C}_{I}{}_{J}^{m+1}\left(\mathrm{ln}{C}_{W}{}_{J}^{m+1}-\mathrm{ln}{C}_{W}{}_{J-1}^{m+1}\right)+\\ +{\eta}_{I}{V}_{I}\left(\frac{{V}_{I}+{n}_{dW}{V}_{W}}{RT}+\frac{K}{{D}_{II}}\right){C}_{I}{}_{J}^{m+1}\left({C}_{I}{}_{J}^{m+1}-{C}_{I}{}_{J-1}^{m+1}\right)+\\ +{\eta}_{W}{V}_{W}\left(\frac{{V}_{I}+{n}_{dW}{V}_{W}}{RT}+\frac{K}{{D}_{II}}\right){C}_{I}{}_{J}^{m+1}\left({C}_{W}{}_{J}^{m+1}-{C}_{W}{}_{J-1}^{m+1}\right)+\Delta y\frac{{\gamma}_{I}}{H}{C}_{I}{}_{J}^{m+1}=0;\end{array}$$$$\begin{array}{cc}\phantom{\rule{1.em}{0ex}}\hfill & {D}_{WW}\left({{C}_{W}}_{2}^{m+1}-{{C}_{W}}_{1}^{m+1}\right)+{\eta}_{W}{V}_{W}\left({D}_{WW}\frac{{V}_{W}}{RT}+K\right){{C}_{W}}_{1}^{m+1}\left({{C}_{W}}_{2}^{m+1}-{{C}_{W}}_{1}^{m+1}\right)+\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & +{\eta}_{I}{V}_{I}\left({D}_{WW}\frac{{V}_{W}}{RT}+K\right){{C}_{W}}_{1}^{m+1}\left({{C}_{I}}_{2}^{m+1}-{{C}_{I}}_{1}^{m+1}\right)+{D}_{II}{n}_{dW}\left({{C}_{I}}_{2}^{m+1}-{{C}_{I}}_{1}^{m+1}\right)+\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & +{\eta}_{W}{n}_{dW}{D}_{II}\frac{{V}_{I}{V}_{W}}{RT}{{C}_{I}}_{1}^{m+1}\left({{C}_{W}}_{2}^{m+1}-{{C}_{W}}_{1}^{m+1}\right)+{\eta}_{I}{n}_{dW}{D}_{II}\frac{{{V}_{I}}^{2}}{RT}{{C}_{I}}_{1}^{m+1}\left({{C}_{I}}_{2}^{m+1}-{{C}_{I}}_{1}^{m+1}\right)+\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & +{n}_{dW}{D}_{II}\frac{{Z}_{I}F}{RT}{{C}_{I}}_{1}^{m+1}\left({\phi}_{2}^{m+1}-{\phi}_{1}^{m+1}\right)-\Delta y{\gamma}_{W}\frac{{D}_{WW}}{H}{{C}_{W}}_{1}^{m+1}=0;\hfill \end{array}$$$$\begin{array}{cc}\phantom{\rule{1.em}{0ex}}\hfill & {D}_{WW}\left({{C}_{W}}_{J}^{m+1}-{{C}_{W}}_{J-1}^{m+1}\right)+{\eta}_{W}{V}_{W}\left({D}_{WW}\frac{{V}_{W}}{RT}+K\right){{C}_{W}}_{J}^{m+1}\left({{C}_{W}}_{J}^{m+1}-{{C}_{W}}_{J-1}^{m+1}\right)+\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & +{\eta}_{I}{V}_{I}\left({D}_{WW}\frac{{V}_{W}}{RT}+K\right){{C}_{W}}_{J}^{m+1}\left({{C}_{I}}_{J}^{m+1}-{{C}_{I}}_{J-1}^{m+1}\right)+{D}_{II}{n}_{dW}\left({{C}_{I}}_{J}^{m+1}-{{C}_{I}}_{J-1}^{m+1}\right)+\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & +{\eta}_{W}{n}_{dW}{D}_{II}\frac{{V}_{I}{V}_{W}}{RT}{{C}_{I}}_{J}^{m+1}\left({{C}_{W}}_{J}^{m+1}-{{C}_{W}}_{J-1}^{m+1}\right)+{\eta}_{I}{n}_{dW}{D}_{II}\frac{{{V}_{I}}^{2}}{RT}{{C}_{I}}_{J}^{m+1}\left({{C}_{I}}_{J}^{m+1}-{{C}_{I}}_{J-1}^{m+1}\right)+\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & +{n}_{dW}{D}_{II}\frac{{Z}_{I}F}{RT}{{C}_{I}}_{J}^{m+1}\left({\phi}_{J}^{m+1}-{\phi}_{J-1}^{m+1}\right)+\Delta y{\gamma}_{W}\frac{{D}_{WW}}{H}{{C}_{W}}_{J}^{m+1}=0,\hfill \end{array}$$
- Subsystem 3 (26)–(28) was discretized on the time grid (32) and the extended non-uniform coordinate grid ${G}_{yH}$$${G}_{yH}=\left\{{y}_{j}\left|j=1,\dots ,J+2\right.\right\},$$$${s}_{1}=0;$$$$\frac{{s}_{2}-{s}_{1}}{\Delta t}=0;$$$$\begin{array}{cc}\phantom{\rule{1.em}{0ex}}\hfill & {s}_{m+1}=2\left(\frac{2-\Delta {t}^{2}{\omega}_{0m}^{2}}{\beta \Delta t+2}\right){s}_{m}+\left(\frac{\beta \Delta t-2}{\beta \Delta t+2}\right){s}_{m-1}+\frac{{\omega}_{0m}^{2}\Delta {t}^{2}{L}^{2}}{4\left(\beta \Delta t+2\right)}\times \hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & \times \left[{\eta}_{I}{V}_{I}{\displaystyle \sum _{j=2}^{J}}\phantom{\rule{0.166667em}{0ex}}\left(\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \left({{C}_{I}}_{j+1}^{m}-\frac{1}{2h}\underset{j=2}{\sum ^{J}}\phantom{\rule{0.166667em}{0ex}}\left({{C}_{I}}_{j+1}^{1}+{{C}_{I}}_{j}^{1}\right)\Delta {y}_{j}\right)\left({y}_{j+1}-{y}_{\frac{J}{2}}\right)+\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& +\left({{C}_{I}}_{j}^{m}-\frac{1}{2h}\underset{j=2}{\sum ^{J}}\phantom{\rule{0.166667em}{0ex}}\left({{C}_{I}}_{j+1}^{1}+{{C}_{I}}_{j}^{1}\right)\Delta {y}_{j}\right)\left({y}_{j}-{y}_{\frac{J}{2}}\right)\hfill \end{array}\right)\Delta {y}_{j}\right.+\hfill \\ & \left.+{\eta}_{W}{V}_{W}{\displaystyle \sum _{j=2}^{J}}\phantom{\rule{0.166667em}{0ex}}\left(\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \left({{C}_{W}}_{j+1}^{m}-\frac{1}{2h}\underset{j=2}{\sum ^{J}}\phantom{\rule{0.166667em}{0ex}}\left({{C}_{W}}_{j+1}^{1}+{{C}_{W}}_{j}^{1}\right)\Delta {y}_{j}\right)\left({y}_{j+1}-{y}_{\frac{J}{2}}\right)+\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& +\left({{C}_{W}}_{j}^{m}-\frac{1}{2h}\underset{j=2}{\sum ^{J}}\phantom{\rule{0.166667em}{0ex}}\left({{C}_{W}}_{j+1}^{1}+{{C}_{W}}_{j}^{1}\right)\Delta {y}_{j}\right)\left({y}_{j}-{y}_{\frac{J}{2}}\right)\hfill \end{array}\right)\Delta {y}_{j}\right]:\hfill \\ & :\frac{1}{2}{\displaystyle \sum _{j=2}^{J+1}}\phantom{\rule{0.166667em}{0ex}}\left({E}_{j+1}^{m}{\left({y}_{j+1}-{y}_{\frac{J}{2}}\right)}^{2}+{E}_{j}^{m}{\left({y}_{j}-{y}_{\frac{J}{2}}\right)}^{2}\right)\Delta {y}_{j};\hfill \end{array}$$$${\omega}_{0m}=\frac{\lambda}{{L}_{S}{}^{2}}\sqrt{\frac{{w}_{S}{{\displaystyle \sum}}_{j=1}^{J+1}\left({E}_{j+1}^{m}{\left({y}_{j+1}-{y}_{\frac{J}{2}}\right)}^{2}+{E}_{j}^{m}{\left({y}_{j}-{y}_{\frac{J}{2}}\right)}^{2}\right)\cdot \Delta {y}_{j}}{2{m}_{Lm}}};$$$${m}_{Lm}=\frac{{m}_{D}}{{L}_{S}}+2{\rho}_{M}{w}_{S}H+\frac{{M}_{W}{w}_{S}}{2}{\displaystyle \sum}_{j=2}^{J}\left({C}_{W}{}_{j+1}^{m}+{C}_{W}{}_{j}^{m}\right)\cdot \Delta {y}_{j},$$

## 4. Model Verification, Results, and Discussion

#### 4.1. Experimental Setup

#### 4.2. Numerical Simulation Results and Their Discussion—Comparison with Experimental Data

_{A}= 0–5 V) and frequencies (f = 0.5–50 Hz) of the control voltage were obtained. The values of the physical constants and parameters of the IPMC actuator model used in the calculations are given in Table 1.

_{A}= 5 V, harmonically changing in time with a frequency f = 1 Hz (Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11) and f = 10 Hz (Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20).

_{II}, D

_{WW}for the liquid phase; elastic moduli E

_{S}, E

_{M}, densities and geometric dimensions of the beam layers for the solid phase; filtration coefficient K).

_{R}≈ 36 Hz), which corresponds to the characteristic maximum in Figure 28. In this case, the calculated dependence of the beam tip displacement on the DC control voltage level (Figure 27) gives a good agreement with the experiment only at control voltage U

_{A}≤ 1.5 V, which is probably due to the influence of factors not considered in the model (34)–(50).

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Spatial distributions of the concentrations of ions C

_{I}(y) and water molecules C

_{W}(y) in the IPMC polymer membrane at U

_{A}= 5 V and f = 1 Hz at a time point t

_{m}= 3 ms.

**Figure 4.**Spatial distributions of the concentrations of ions C

_{I}(y) and water molecules C

_{W}(y) in the IPMC polymer membrane at U

_{A}= 5 V and f = 1 Hz at a time point t

_{m}= 27 ms.

**Figure 5.**Spatial distributions of the concentrations of ions C

_{I}(y) and water molecules C

_{W}(y) in the IPMC polymer membrane at U

_{A}= 5 V and f = 1 Hz at a time point t

_{m}= 42 ms.

**Figure 6.**Spatial distributions of the concentrations of ions C

_{I}(y) and water molecules C

_{W}(y) in the IPMC polymer membrane at U

_{A}= 5 V and f = 1 Hz at a time point t

_{m}= 57 ms.

**Figure 7.**Spatial distributions of the concentrations of ions C

_{I}(y) and water molecules C

_{W}(y) in the IPMC polymer membrane at U

_{A}= 5 V and f = 1 Hz at a time point t

_{m}= 72 ms.

**Figure 8.**Spatial distributions of the concentrations of ions C

_{I}(y) and water molecules C

_{W}(y) in the IPMC polymer membrane at U

_{A}= 5 V and f = 1 Hz at a time point t

_{m}= 87 ms.

**Figure 9.**Spatial distributions of the concentrations of ions C

_{I}(y) and water molecules C

_{W}(y) in the IPMC polymer membrane at U

_{A}= 5 V and f = 1 Hz at a time point t

_{m}= 102 ms.

**Figure 10.**Spatial distributions of the concentrations of ions C

_{I}(y) and water molecules C

_{W}(y) in the IPMC polymer membrane at U

_{A}= 5 V and f = 1 Hz at a time point t

_{m}= 117 ms.

**Figure 11.**Spatial distributions of the concentrations of ions C

_{I}(y) and water molecules C

_{W}(y) in the IPMC polymer membrane at U

_{A}= 5 V and f = 1 Hz at a time point t

_{m}= 132 ms.

**Figure 12.**Spatial distributions of the concentrations of ions C

_{I}(y) and water molecules C

_{W}(y) in the IPMC polymer membrane at U

_{A}= 5 V and f = 10 Hz at a time point t

_{m}= 3 ms.

**Figure 13.**Spatial distributions of the concentrations of ions C

_{I}(y) and water molecules C

_{W}(y) in the IPMC polymer membrane at U

_{A}= 5 V and f = 10 Hz at a time point t

_{m}= 6 ms.

**Figure 14.**Spatial distributions of the concentrations of ions C

_{I}(y) and water molecules C

_{W}(y) in the IPMC polymer membrane at U

_{A}= 5 V and f = 10 Hz at a time point t

_{m}= 9 ms.

**Figure 15.**Spatial distributions of the concentrations of ions C

_{I}(y) and water molecules C

_{W}(y) in the IPMC polymer membrane at U

_{A}= 5 V and f = 10 Hz at a time point t

_{m}= 12 ms.

**Figure 16.**Spatial distributions of the concentrations of ions C

_{I}(y) and water molecules C

_{W}(y) in the IPMC polymer membrane at U

_{A}= 5 V and f = 10 Hz at a time point t

_{m}= 15 ms.

**Figure 17.**Spatial distributions of the concentrations of ions C

_{I}(y) and water molecules C

_{W}(y) in the IPMC polymer membrane at U

_{A}= 5 V and f = 10 Hz at a time point t

_{m}= 18 ms.

**Figure 18.**Spatial distributions of the concentrations of ions C

_{I}(y) and water molecules C

_{W}(y) in the IPMC polymer membrane at U

_{A}= 5 V and f = 10 Hz at a time point t

_{m}= 21 ms.

**Figure 19.**Spatial distributions of the concentrations of ions C

_{I}(y) and water molecules C

_{W}(y) in the IPMC polymer membrane at U

_{A}= 5 V and f = 10 Hz at a time point t

_{m}= 24 ms.

**Figure 20.**Spatial distributions of the concentrations of ions C

_{I}(y) and water molecules C

_{W}(y) in the IPMC polymer membrane at U

_{A}= 5 V and f = 10 Hz at a time point t

_{m}= 27 ms.

**Figure 21.**Transients in the IPMC actuator at U

_{A}= 5 V and f = 0.5 Hz: green lines are the control voltage U(t); pink lines are the beam tip displacement s(t); blue lines are the force F(t).

**Figure 22.**Transients in the IPMC actuator at U

_{A}= 5 V and f = 1 Hz: green lines are the control voltage U(t); pink lines are the beam tip displacement s(t); blue lines are the force F(t).

**Figure 23.**Transients in the IPMC actuator at U

_{A}= 5 V and f = 10 Hz at intervals t = 0–10 s (

**a**) and t = 7–8 s (

**b**): green lines are the control voltage U(t); pink lines are the beam tip displacement s(t); blue lines are the force F(t).

**Figure 24.**Transients in the IPMC actuator at U

_{A}= 5 V and f = 36 Hz: green lines are the control voltage U(t); pink lines are the beam tip displacement s(t); blue lines are the force F(t).

**Figure 25.**Transients in the IPMC actuator at U

_{A}= 5 V and f = 40 Hz: green lines are the control voltage U(t); pink lines are the beam tip displacement s(t); blue lines are the force F(t).

**Figure 26.**The calculated and experimental dependences of the beam tip displacement amplitude on the peak-to-peak control voltage at a frequency f = 1 Hz.

**Figure 27.**The calculated and experimental dependences of the beam tip displacement amplitude on the DC control voltage.

**Figure 28.**The calculated and experimental amplitude-frequency characteristics (AFC) of the IPMC actuator.

Parameter | Symbol | Value | Unit |
---|---|---|---|

Polymer trademark | – | Nafion N117 | – |

Length of the dry beam ^{1} | L | 15 | mm |

Width of the dry beam ^{1} | w | 5 | mm |

Thickness of the dry beam ^{1} | H | 183 | μm |

Thickness of metal electrodes | H | 5 | μm |

Temperature | T | 293 | K |

Diffusion coefficient of cations | D_{II} | 5.3 × 10^{−6} | cm^{2}⋅s^{−}^{1} |

Diffusion coefficient of water molecules | D_{WW} | 3.87 × 10^{−6} | cm^{2}⋅s^{−}^{1} |

Concentration of ions in the polymer | C _{+} | 0.9 | mol⋅kg^{−}^{1} |

Molar volume of ions | V_{I} | −5.4 | cm^{3}⋅mol^{−}^{1} |

Molar volume of water | V_{W} | 18 | cm^{3}⋅mol^{−}^{1} |

Filtration coefficient | K | 3.4 × 10^{−14} | cm^{2}⋅Pa^{−}^{1}⋅s^{−}^{1} |

Elementary charge | q | 1.6 × 10^{−19} | C |

Faraday constant | F | 96,485 | C⋅mol^{−}^{1} |

Gas constant | R | 8.31 | J⋅K^{−1}⋅mol^{−1} |

Permittivity of vacuum | ε_{0} | 8.85 × 10^{−14} | F⋅cm^{−}^{1} |

Relative permittivity of water | ε | 81 | – |

Expansion coefficient of the membrane at maximum humidification | α | 0.1 | – |

Relative charge of ion | Z_{I} | 1 | – |

Number of water molecules associated with one cation | n_{dW} | 1 | – |

Mass fraction of water in the dry polymer ^{1} | P_{WN} | 0.05 | – |

Mass fraction of water in the humidified polymer | P_{WS} | 0.38 | – |

Young’s modulus of the dry polymer ^{1} | E_{N} | 249 | MPa |

Young’s modulus of the humidified polymer | E_{S} | 114 | MPa |

Young’s modulus of metal electrodes | E_{M} | 23 | GPa |

Empirical coefficient | η_{I} | 200 | MPa |

Empirical coefficient | η_{W} | 200 | MPa |

Coefficient depending on the mode of beam bending oscillations | λ | 3.52 | – |

Coefficient characterizing dissipative processes | β | 19 | s^{−}^{1} |

Empirical coefficient determining the evaporation rate of cations into the external environment | γ_{I} | 0 | – |

Empirical coefficient determining the evaporation rate of water molecules into the external environment | γ_{W} | 0 | – |

Layer density of the dry membrane ^{1} | ρ_{SPN} | 3.6 × 10^{−2} | g·cm^{−2} |

Density of the electrode material | ρ_{M} | 21.5 | g·cm^{−3} |

Molar mass of water | M_{W} | 18.01528 | g⋅mol^{−}^{1} |

^{1}At normal air humidity.

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## Share and Cite

**MDPI and ACS Style**

Broyko, A.P.; Khmelnitskiy, I.K.; Ryndin, E.A.; Korlyakov, A.V.; Alekseyev, N.I.; Aivazyan, V.M.
Multiphysics Simulator for the IPMC Actuator: Mathematical Model, Finite Difference Scheme, Fast Numerical Algorithm, and Verification. *Micromachines* **2020**, *11*, 1119.
https://doi.org/10.3390/mi11121119

**AMA Style**

Broyko AP, Khmelnitskiy IK, Ryndin EA, Korlyakov AV, Alekseyev NI, Aivazyan VM.
Multiphysics Simulator for the IPMC Actuator: Mathematical Model, Finite Difference Scheme, Fast Numerical Algorithm, and Verification. *Micromachines*. 2020; 11(12):1119.
https://doi.org/10.3390/mi11121119

**Chicago/Turabian Style**

Broyko, Anton P., Ivan K. Khmelnitskiy, Eugeny A. Ryndin, Andrey V. Korlyakov, Nikolay I. Alekseyev, and Vagarshak M. Aivazyan.
2020. "Multiphysics Simulator for the IPMC Actuator: Mathematical Model, Finite Difference Scheme, Fast Numerical Algorithm, and Verification" *Micromachines* 11, no. 12: 1119.
https://doi.org/10.3390/mi11121119