1 Introduction

In a solid or liquid ionic conductor, both the drift of ions under an electric field and the diffusion of ions due to concentration gradients contribute to the current^[1-3]. Therefore, these materials can be good conductors with various conventional applications such as energy-storage systems and electrowetting devices^[4-5]. Recently, because of the development of flexible and stretchable electronics as well as soft machines^[6-7] with relatively new applications in actuation^[8-9], sensing^[10], electronics^[11], and biomedical engineering^[12-14], there is a strong need for highly compliant and stretchable conductors for the electrodes of these devices. It has been found that some ionic conductors, e.g., polyacrylamide (PAM) hydrogel containing sodium chloride (NaCl), are good candidates for these applications^[15-16].

To better understand the fundamental behaviors of conduction and diffusion in ionic conductors, we develop a rigorous theoretical model to study the thin-film ionic conductors in this paper. The widely used Poisson-Nernst-Planck (PNP) theory for ionic conductors is employed here, which consists of the charge equation of electrostatics and the conservation of charge for both positive and negative ions. The PNP theory describes both drift and diffusion currents. It is inherently nonlinear because of the drift current term, which is the product of the ion concentration with the electric field. We linearize the PNP theory for small perturbations from reference ion concentrations. The linearized two-dimensional (2D) equations for thin ionic conductor films are obtained from the three-dimensional (3D) equations systematically in the manner of Mindlin by power series expansions^[17-21] in the film thickness coordinate and retaining the zeroth- and first-order equations, which can be used to predict the properties of ionic-conductor devices. In addition, sandwich structures, which consist of two ionic films and one dielectric layer, are very common in the practical applications of ionic-conductor devices, e.g., stretchable ionic cable^[22]. Furthermore, we obtain the 2D equations for the thin sandwich films of a dielectric layer between two ionic conductor layers by combining the thin-film 2D equations for ionic conductors with similar equations for a thin dielectric film under the interlayer continuity conditions. As an example for the application of the obtained equations, we analyze the sandwich film as an ionic cable proposed in Ref. [22]. The numerical results are consistent with those in Ref. [22]. Since the theoretical model developed in this paper includes both conduction and diffusion of ionic conductors, it is suitable for analyzing the performance of ionic-conductor devices with any frequency.

2 3D equations

In this section, we will summarize the basic equations for an ionic conductor and linearize them. We follow most researchers^[23-30], and employ the PNP theory. The index notation is also employed. A pair of repeated indices indicates a summation with respect to the repeated indices through 1, 2, and 3. An index following a comma represents a partial derivative with respect to the coordinate associated with the index. A superimposed dot is a time derivative. The 3D equations for an ionic conductor are

(1)

(2)

(3)

where D_i is the electric displacement vector. C⁺ and C^- are the concentrations of the positive and negative ions, respectively. J_i⁺ and J_i^- are the currents associated with the positive and negative ions, respectively. e is the elementary charge, and z is the valence of the ions. Equation (1) is the charge equation of electrostatics (Gauss's equation). Equations (2) and (3) are the conservation of charge for the ions (continuity equations). The constitutive relations for the electric displacement and currents are as follows:

(4)

(5)

(6)

where E_i is the electric field. ε is the dielectric constant. μ^± and D^± are the mobility and diffusion constants of the positive and negative ions, respectively. The drift current terms in Eqs. (5) and (6) consist of products of the electric field, and the ion concentrations are nonlinear. The electric field is related to the electric potential φ through

(7)

In the reference state, C⁺=C₀⁺, C^-=C₀^-, and all other fields vanish. We consider the case when C₀⁺ and C₀^- are constants, and assume charge neutrality at the reference state, i.e., C₀⁺ =C₀^-.

For small perturbations around the reference state, let^[31-32]

(8)

where and are the perturbations of the ion concentrations. Substituting Eq. (8) into Eqs. (1)-(3), (5), and (6) yields

(9)

(10)

where the ion concentration gradients are denoted by

(11)

and the drift current terms in Eqs. (5) and (6) are linearized for small and .

3 Hierarchy of 2D equations

The coupled equations in the previous section can be mathematically challenging when they are applied to device problems. Many devices of ionic conductors are made from thin films. For thin films, 2D equations can be derived, which are simpler than the 3D equations, and allow more theoretical studies. In this section, we will derive 2D equations for thin ionic conductor films systematically in the manner of Mindlin^[17-21]. Consider a thin ionic conductor film with its normal along the x₃-axis (see Fig. 1). The x₁- and x₂-axes are in the middle plane of the plate. The film may be electroded on its top or bottom surfaces as shown in the figure or on its side faces. In addition, the film is assumed to be thin in the sense that its thickness is much smaller than the in-plane dimensions.

To develop a 2D theory, we begin with the following expansions:

(12)

(13)

(14)

Substituting Eqs. (12)-(14) into Eqs. (7) and (11), we obtain

(15)

where

(16)

Substituting Eqs. (12)-(14) into the right-hand sides of the equations in Eq. (9), multiplying both sides of the equations by x₃ⁿ, and integrating the resulting equations along the plate thickness, we have

(17)

where we have introduced a 2D summation convention that the indices a and b range from 1 to 2 only, and

(18)

In Eq. (17), the thin film moments of the electric displacement and currents of various orders are defined by

(19)

The surface loads at the film top and bottom are represented by

(20)

(21)

(22)

The following 2D constitutive relations are obtained by integrating Eqs. (4)-(6) and (10) through the film thickness and using Eq. (15):

(23)

Thus, a complete system of 2D equations for ionic films has been developed.

4 Zeroth- and first-order equations

In applications, the zeroth- and first-order equations are used more often than the higher-order ones. In this section, we will truncate the hierarchy of the equations in the previous section into the zeroth- and first-order equations by keeping the B₀₀=2h and B₁₁=2h³/3 terms only in the summations. Both B₀₁ and B₁₀ are equal to zero according to Eq. (18). Therefore, we can summarize the zeroth- and first-order equations as follows:

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

With successive substitutions from Eqs. (30)-(32), we can rewrite Eqs. (24)-(29) as six equations for φ⁽⁰⁾, φ⁽¹⁾, C⁽⁰⁾⁺, C⁽¹⁾⁺, C^(0)-, and C^(1)-. At the boundary of a 2D domain with an in-plane unit exterior normal n and an in-plane unit tangent s (see Fig. 1), we may prescribe

(33)

The above equations are for an unelectroded ionic film. Within electrostatics, the electric potential is at most a function of time on an electrode. If the film is electroded at its upper or lower surface or both, the charge equations of electrostatics need to be modified accordingly. We discuss various cases separately below. When both the top and bottom surfaces of the film are electrodes, we can write

(34)

Within the first-order theory, since

(35)

we have

(36)

In this case, φ⁽⁰⁾ and φ⁽¹⁾ are determined from Eq. (36). The charge equations of electrostatics are not needed.

Substituting Eq. (20) into Eqs. (24) and (27), we obtain

(37)

If the film is electroded at its upper surface only, we can eliminate the unknown D₃(h) from Eq. (37). This gives

(38)

Similarly, if the film is electroded at its lower surface only, we eliminate the unknown D₃(-h) from Eq. (37), and obtain

(39)

5 Equations for thin sandwich films

Sandwich films of two ionic conductor layers with a dielectric layer in between (see Fig. 2) are often used in applications^{[15, 22, 33-34]}. The structure is symmetric about its middle plane, but the applied electric loads maybe symmetric, antisymmetric, or neither. The thickness of the middle dielectric layer is 2h, and its dielectric constant is ε. The thickness of both the ionic conductor layers is 2h, and their dielectric constant is ε. In this section, we will consider the case of thin ionic conductor films, in which the ion concentrations are nearly uniform along the film thickness and the zero-order equations are sufficient.

The equations in the previous section are applicable to each ionic conductor layer in Fig. 2, and the dielectric layer too as a special case when conduction is neglected. However, because of the unknown normal electric displacement components related to D₃ at the interfaces in the charge equations of electrostatics, in their present form, the equations are not convenient for sandwich films. In the following, we will eliminate the unknown electric displacements at the interfaces under interface continuity conditions, and obtain the charge equations for the sandwich film in a manner similar to what has been done in Refs. [35]-[39] for elastic plates. Consider the upper ionic conductor film first. We use an over bar for the fields in this film. In the local thickness coordinate measured from the middle plane of the upper film, the following surface loads at its top and bottom surfaces vanish:

(40)

Then, Eqs. (24)-(26) take the following forms with the use of Eqs. (24)-(26) and (40):

(41)

(42)

(43)

From Eqs. (30) and (32), we have the constitutive relations relevant to Eqs. (41)-(43) as follows:

(44)

For the lower ionic conductor film, we use a hat for its fields. Similar to Eqs. (40)-(44), we have

(45)

(46)

(47)

(48)

(49)

For the middle dielectric layer, because of the difference of the electric potentials in the two ionic conductor layers and hence the thickness variation of the electric potential in the dielectric layer, we need both the zeroth- and first-order equations. Since there are no ions in the dielectric layer, only the charge equations of statics are needed. From the relevant equations in Eqs. (20)-(22) and (24)-(31), we have

(50)

(51)

(52)

(53)

(54)

From electrostatics, we have the following continuity conditions between the ionic conductor and dielectric layers:

(55)

We use Eq. (55) to eliminate the interface electric displacement components in Eqs. (41), (46), (50), and (51). Then, we have

(56)

For the zeroth- and first-order equations of the dielectric layer, from the first equation in Eq. (12), we have the approximate electric potential as follows:

(57)

From the continuity of the electric potentials between the ionic conductor and dielectric layers, we have

(58)

In summary, the system of equations for a thin sandwich film consists of six equations, i.e., Eqs. (42), (43), (47), (48), (56), and (58). The relevant constitutive equations are Eqs. (44), (49), and (52)-(54). Together, Eqs. (42), (43), (47), (48), (56), and (58) are eight equations for φ⁽⁰⁾, φ⁽¹⁾, φ⁽⁰⁾, C⁽⁰⁾⁺, C^(0)-, , Ĉ⁽⁰⁾⁺, and Ĉ^(0)-.

6 Numerical examples and discussion

As an example for the applications of the equations for sandwich films, we consider the following ionic cable^[22] with the length 2a (see Fig. 3). At the left end, a time-harmonic input voltage V₁ =exp (iωt) is used. At the right end, the output circuit has a load impedance Z in harmonic motions. We assume that the mobilities and diffusion coefficients of the positive and negative ions are the same, and their initial concentrations are the same, i.e.,

We consider the case, where the ionic layers are very thin, and use the equations in Section 5. The fields mainly vary along the x₁-direction, and the dependence on x₂ is neglected. Let the dimension of the structure along the x₂-direction be one. From Eqs. (41), (43), (46)-(48), (56), and (58), we have that the eight one-dimensional (1D) governing equations are the conservation of charge for the upper ionic conductor film, i.e.,

(59)

the conservation of charge for the lower ionic conductor film, i.e.,

(60)

the charge equations of electrostatics for the combined sandwich film, i.e.,

(61)

and the two continuity equations in Eq. (58). The following constitutive equations are from Eqs. (44), (49), and (50)-(54):

(62)

(63)

(64)

(65)

(66)

From Eqs. (62)-(66), we can rewrite the eight equations in Eqs. (59)-(61) and the two continuity equations in Eq. (58) as follows:

(67)

(68)

(69)

(70)

(71)

(72)

(73)

(74)

which are eight equations governing φ⁽⁰⁾, φ⁽¹⁾, φ⁽⁰⁾, C⁽⁰⁾⁺, C^(0)-, , Ĉ⁽⁰⁾⁺, and Ĉ^(0)-.

The structure in Fig. 3 is symmetric but the input voltage is antisymmetric about x₃=0. Physically, we expect the fields to be antisymmetric about x₃=0, i.e.,

(75)

Mathematically, it can be verified that, under Eq. (75), the two equations in Eqs. (69) and (70) for the conservation of charge in the lower ionic conductor layer become identical to Eqs. (67) and (68) for the upper layer, Eq. (71) is trivially satisfied, and Eqs. (73) and (74) become identical. We use Eq. (73) to eliminate φ⁽⁰⁾ in Eq. (72). Then, Eqs. (67)-(74) can be deduced to the following three equations for φ⁽¹⁾, C⁽⁰⁾⁺ and C^(0)-:

(76)

(77)

(78)

Equations (76)-(78) are the system of governing equations for the ionic cable in Fig. 3. At this point, we make a comparison with the theoretical model in Ref. [29] for the same cable. The model in Ref. [22] is simple. It does not consider diffusion, and does not distinguish positive and negative ions. Therefore, we add together Eqs. (76) and (77), which are for positive and negative ions separately, multiply the resulting equation by the upper ionic conductor film thickness 2h, and set the diffusion coefficient D to zero. Then, we have

(79)

where

(80)

In Eqs. (79) and (80), with the notation of Ref. [22], i_q is the total current in the upper conductor film in the x₁-direction with contributions from both the positive and negative ions, q is the total charge per unit length of the upper conductor film from both the positive and negative ions, r is the resistance per unit length of the upper and lower conductor films together, w is the dimension along the x₂-direction, and v is the voltage between the upper and lower conductor films. In addition, in the third equation in Eq. (78), for slowly varying fields with long wavelengths in the x₁-direction, which is an approximation for thin structures, we drop the second derivatives of φ⁽¹⁾, and obtain

(81)

where

(82)

which is the capacitance per unit length of the cable in the notation of Ref. [22]. Equations (79) and (81) are the same as Eqs. (1) and (2) of Ref. [22]. Therefore, our model in Eqs. (76)-(78) can be deduced to that in Ref. [22] as a special case.

For the time-harmonic motions governed by Eqs. (76)-(78), we use the complex notation. All fields have the same (iωt) factor, which will be dropped below. The boundary conditions at the left and right ends are

(83)

where we have assumed that the ionic conductors and the electrodes do not exchange charge carriers^[22]. Therefore, the currents in the ionic conductor layer vanish at the left end. Similarly, the boundary conditions at the right end are

(84)

where V₂ is the output voltage, which is known. The additional condition for determining V₂ is the output circuit condition. We have

(85)

where Q is the free charge on the electrode at the right end of the upper ionic conductor film, and I is the current I that flows out of the electrode. Then, the output circuit condition can be written as follows:

(86)

Before solving Eqs. (76)-(78), we simplify Eqs. (76) and (77) by subtracting them from each other and adding them together to obtain

(87)

(88)

The first equation in Eq. (87) is uncoupled to Eqs. (78) and (88), and will be studied separately first. For a solution of Eq. (87), we let

(89)

The substitution of Eq. (89) into Eq. (87) gives the following equation for λ:

(90)

We denote the two roots of Eq. (90) by λ⁽¹⁾ and λ⁽²⁾. Then, the general solution of the time-harmonic form of the Eq. (87) can be written as follows:

(91)

Subtracting Eqs. (63) and (64) from each other, we have

(92)

From Eqs. (83) and (84), we have

(93)

Equations (91)-(93) imply that

(94)

Therefore,

(95)

For the solution of Eqs. (78) and (88), we let

(96)

The substitution of Eq. (96) into Eqs. (78) and (88) results in the following system of linear homogeneous equations for B₁ and B₂:

(97)

For nontrivial solutions, the determinant of the coefficient matrix of Eq. (97) has to vanish. This leads to the following equation for λ²:

(98)

We denote the two roots of Eq. (98) by λ₁² and λ₂², and let

(99)

For each λ^(m) (m=3, 4, 5, 6), Eq. (97) implies

(100)

Then, the general solution of the time-harmonic form of Eqs. (78) and (88) can be written as follows:

(101)

(102)

where A^(m) (m=3, 4, 5, 6) are four undetermined constants. The substitution of Eq. (102) into the voltage boundary conditions in Eqs. (83) and (84) leads to

(103)

From Eq. (62), we have

(104)

From Eq. (85), we have

(105)

With Eq. (105), the output circuit condition in Eq. (86) becomes

(106)

Adding Eqs. (63) and (64), we have

(107)

where φ⁽⁰⁾=hφ⁽¹⁾ has been used. From Eqs. (83) and (84), we also have

(108)

Using Eq. (107), the substitution of Eqs. (101) and (102) into Eq. (108) yields

(109)

where

(110)

In Eqs. (103), (106), and (109), there are five equations for A^(m) (m=3, 4, 5, 6) and V₂.

For the numerical results, we consider the ionic cable in Ref. [22]. The geometric parameters are

The dielectric constant of the middle dielectric layer of VHB4905 is ε =4.2×10^-11 F/m. For LiCl ions, z=1. There are two cases for the reference ion concentration with C₀ =0.01 and 8 mol/litre. The corresponding values of the resistivity are ρ =2 Ωm and ρ =1.6×10^-2 Ωm, respectively. Then, the mobility can be calculated from

and the diffusion constant can be calculated from De=kμT. For the load impedance, we consider two cases as in Ref. [22] with a real Z for a resistive load or a pure imaginary Z for an inductive load.

First, for comparison with Ref. [22], which does not describe diffusion, we neglect the diffusion in our equations and use Eqs. (79)-(82) to calculate the normalized output voltage. The results are shown in Fig. 4(a) for a resistive load and in Fig. 4(b) for an inductive load. The same results are also shown in Fig. 1(c) and Fig. 1(e) of Ref. [22], where the frequency range is for ω >10³ rad/s only. Our results agree with those in Ref. [22] identically in the common frequency range as expected because Eqs. (79)-(82) are exactly the same as the equations in Ref. [22]. The output voltage is bounded from above by the input voltage as expected. At very high frequencies, the ions cannot follow the drastic reversal of the electric field. Moreover, there is little conduction, and the output voltage is low.

Next, we will show the results of our full model including diffusion. The value of the dielectric constant for the ionic films was not provided in Ref. [22] because it was not needed. However, ε is present in Eqs. (105) and (106). According to Refs. [40]-[43], we use three values of ε in the usual range with ε=15ε₀, 50ε₀, and 100ε₀. The output voltage for the case of a resistive load is shown in Fig. 5. For these three values of ε, the results are not sensitive to ε. The output voltage in Fig. 5 is close to what is in Fig. 4(a). Most importantly, our results in Fig. 5 show a qualitatively behavior in the low-frequency range for which Ref. [22] did not provide. In the low-frequency limit, there is enough time for the diffusion to take place. The ions are less concentrated near the output electrodes at the right end. Therefore, the electrons are less concentrated on the output electrodes, and the output voltage is reduced.

Figure 6 shows the output voltage for the case of an inductor load. The low- and high-frequency behaviors are as expected. Again, the results are not sensitive to ε when ε is between 15ε₀ and 100ε₀.

7 Conclusions

2D equations for thin ionic conductor films are obtained from the corresponding 3D equations. The equations can describe both ionic conduction and diffusion in these films. The usefulness of the equations is demonstrated through the analysis of an ionic cable. The numerical results for the cable show that the effect of diffusion is important in the low-frequency range. In the mid-frequency range, diffusion can be neglected, and the ionic conductors can be treated as usual conductors with ohmic conduction. At very high frequencies, the ions cannot follow the rapidly changing electric field, and there is little conduction.