Analytical modeling of a multilayer,multimorph lithium-ion battery actuator

Abstract

Silicon-anode based lithium-ion batteries can provide large actuation strain and actuator strain energy through their novel electrochemical actuation mechanism. Battery actuators have strong potential as multifunctional actuators due to their ability to store energy, actuate, and self-sense. This paper presents a generic multilayer model that can predict the following actuator metrics for a multimorph: free deflection, blocked deflection, blocked force, actuator force, and actuator strain energy. Two case studies are explored including an experimentally validated case study with a 13-layer multimorph and a 33-layer multimorph. These case studies are investigated for a range of coating layer thicknesses and for two prevailing assumptions. The first assumes perfect bonding of all layers within the multimorph, and the second is a novel method where some layer interfaces in a multimorph are assumed be in slip. Results under the perfect bonding assumption indicate that the stiffer 33-layer performs better than the 13-layer in blocked force and actuator strain energy per active material volume. The 13-layer multimorph performs better in free deflection due to its larger compliance. When some layers are assumed to slip, the free and blocked deflection increase, due to the increased compliance, relative to all the perfectly bonded case studies.

Keywords

Multimorph lithium-ion batteries smart structures multilayer actuator cantilevered elastica

1. Introduction

The demand for multifunctional and customizable smart actuators is growing in the field of smart devices. Multilayer actuators are likewise a field of growing interest. Multiple active layers are needed to maintain low actuation voltages in piezoelectric and electroactive polymer (EAP) actuators (Ahmed et al., 2017; Chin et al., 2006; Crawley and Anderson, 1990). Recently developed actuators that take advantage of the volume change associated with lithiation of silicon anodes (Ma et al., 2020; Shan et al., 2021; Xia et al., 2019) also need multiple layers to increase energy storage capacity. This paper focuses on the development of an analytical model for the prediction of useful actuator metrics for a multimorph. Specific contributions of this paper include introducing a model with generality of $N_{Layer}$ layers and accounting for interfacial slip for multilayer laminates.

A multimorph is defined here as the full class of multilayer beam-like structures with an arbitrary number of active and passive layers which is used for actuation and sensing. Subsets of multimorphs include unimorphs (single active layer) and bimorphs (double active layers), usually allowing for bi-directional actuation (Shan et al., 2021). Each layer is comprised of a uniform, isotropic material that may be segmented in length with varying width, thickness, and induced actuation strain.

The predicted actuator metrics for a multimorph include free deflection, blocked force (the force necessary to achieve zero tip deflection), blocked deflection (the resulting deflection along the length from the blocked force applied at the tip), actuator force for nonzero prescribed tip displacements, and actuator strain energy.

Timoshenko’s (1925) classic work derives closed-form expression for the deflection of single and double clamped thermal bimorphs. Weinberg (1999) derives the deflection of thermal, piezoelectric, and shape-memory alloy (SMA) multimorphs. DeVoe and Pisano (1997) present a model to predict the static behavior of piezoelectric multimorphs (multilayer unimorphs).

A similar analytical model is presented by Pal and Xie (2012), again for free deflection of a thermally actuated multimorph. They perform experimental validation for a unimorph with an embedded tungsten resistor (Pal and Xie, 2011). Their work modeled both the transverse deflection and twisting of a curved multimorph (unimorph).

Seminal thesis work for modeling of piezoceramic unimorph actuator deflections and dynamic analyzes is conducted by Wang (1998). Much work has also been done on unimorph and bimorph configurations of electroactive polymers (Ahmed et al., 2017; Frecker and Aguilera, 2004; Zhang et al., 2017) and piezoelectric ceramic materials (Cappelleri et al., 2002; Crawley and Anderson, 1990; Low and Guo, 1995; Robbins and Reddy, 1991; Vel and Batra, 2001). An advantage lithium-ion batteries (LIBs) share with ceramic piezoelectrics and electroactive polymers (EAP) is the reduced operating voltages that result from stacking thin active layers (Crawley and Anderson, 1990; Low and Guo, 1995).

Previous work with multilayer, multifield actuation is conducted by Erol et al. (2020). They investigate the multi-objective optimization of the actuated shape of a segmented, multilayer, and multi-field actuator. However, they focus on free deflection and do not consider other actuator metrics which are addressed in this paper (e.g. blocked force, blocked deflection, actuator force, and actuator strain energy).

The model presented in this paper uses a novel electrochemical-based actuation mechanism due to its multifunctionality and advantages over other actuation mechanisms. These advantages include simultaneous energy storage, self-sensing, and actuation (Goodenough and Park, 2013; Sethuraman et al., 2010; Xia et al., 2019). The multifunctionality of silicon-anode based lithium-ion batteries and their promise as actuators drive the development of analytical models because self-sensing, actuating, energy storing actuators have the components necessary to be self-contained, after an initial charging.

LIB actuators have the advantage of being self-contained and self-powered. They do not require continuous external power input, like other actuation mechanisms do (e.g. electroactive, magnetoactive, pneumatic, etc.). Electroactive materials require an externally applied electric field (Pelrine et al., 2000); magnetoactive materials require a magnetic field (Erol et al., 2020); soft pneumatic actuators (SPAs) require pumps and tubes (Huang et al., 2020); and liquid crystalline elastomers require stimulation by temperature, light, or electric fields (Ohm et al., 2010). This ability to not require continuous external input, but to also power an external load, makes energy storage based actuators strong candidates for mobile applications and applications with minimal accessibility. Relevant application spaces for LIB actuators include minimally invasive surgical tooltips (MIS) and active mobile robotics applications. LIB actuators are suitable for applications that require portability and that are hindered by excess cables. This self-containment and the ability to be tailored to individual applications necessitate models that allow for the prediction of actuator metrics for widely applicable multilayer geometries.

Cannarella et al. (2014) investigate the stress-potential coupling of silicon, which is then harnessed in self-sensing experiments conducted by Ma et al. (2020). Xia et al. (2019) showcase silicon anode-based lithium-ion batteries as a reconfigurable material through both simulation and experiment. Chin et al. (2006) show the effectiveness of graphite-anode based lithium-ion batteries as actuators. Though, graphite proves to generate a poorer actuation strain than silicon due to its smaller volumetric expansion of 13.1% compared to silicon’s over 300% (Baranchugov et al., 2007; Chin et al., 2006).

Ma et al. (2017) model the free deflection of a silicon anode based unimorph. This model is further developed by the authors for predicting the free deflection (Gonzalez et al., 2018), as well as blocked force and blocked deflection of segmented unimorphs (Gonzalez et al., 2019, 2021). The model is then extended into a bimorph configuration to predict free deflection of a segmented bimorph (Gonzalez et al., 2020).

This paper contributes an analytical model that can predict actuator metrics for a wide array of multilayer actuators of varying geometries and actuation mechanisms. The previous derived perfect bonding model (Gonzalez et al., 2021) is compared with a model that allows slip between layers. The model predictions for two representative multimorphs are compared with experimental results.

2. Methods

Models of unimorph (Gonzalez et al., 2021) and bimorph (Gonzalez et al., 2020) LIBs have been published previously by the authors and are reviewed briefly here. A schematic for an $N_{Layer}$ , $N_{Seg}$ multimorph is shown in Figure 1. A centroidal coordinate system (x, y, z) has been defined at the centroid where $z = 0$ , such that a global coordinate system (x’, y’, z’) is located at the bottom left corner of the beam at a distance of $h_{j}$ away from the neutral axis. All modeled deflection is assumed to occur at the neutral axis. The layers are numbered beginning at the top and the segments begin from the base (on the left of the figure). The geometric parameters are the thickness $t_{i, j}$ , width $w_{i, j}$ , and length $L_{i, j}$ , for $i$ th layers and $j$ th segments. Additional parameters are defined in the Nomenclature.

Figure 1.

Schematic for an $N_{Layer}$ , $N_{Seg}$ multimorph.

The equations of equilibrium of a multimorph are solved to obtain the free curvature and axial strain at the neutral axis. All layers are assumed to be perfectly bonded such that a strain mismatch developed between active and passive layers drives actuation.

In the current paper, previous models are extended to account for multimorph configurations of any number of layers and segments for both perfectly bonded multimorphs, and multimorphs where some layers slip. Slip in this case means that there is no interlayer friction, and that the respective layers can slip freely. Slip cases or interfacial slip cases are defined to be those cases that assume some specified interfaces are in slip. This approximation is developed to predict the behavior of actuators with interfaces in slip. These slip cases are investigated to better understand the actuator metrics when layers are not bonded to each other as in multilayer battery actuators.

2.1. Analytical model

The analytical unimorph model derived primarily in prior work (Gonzalez et al., 2021), is discussed in brief here. The actuation strain $(S_{1}^{*} = β_{Eff} {\hat{C}}_{Li})$ is assumed to be the product of a linear strain constant $β_{Eff}$ and the average normalized concentration of lithium ions ${\hat{C}}_{Li}$ . $β_{Eff}$ is derived from experimental results and connects the actuation strain in the analytical model to experimental volumetric expansion. Although the variation of the effective linear strain constant $β_{Eff}$ with SOC is ignored for simplicity, the variation of $β_{Eff}$ with SOC is what differentiates it from the linear strain constant $β$ typically presented when the expansion of silicon is considered.

The linear strain constant or expansion coefficient $β$ , is reported for both silicon anode coatings and silicon nanoparticles. Xie et al. (2017) find an expansion coefficient or linear strain constant $β = 33 %$ for a silicon-anode unimorph. Some analysis on the mechanical properties is conducted, notably finding the maximum actuation strain of silicon as 59% (Berla et al., 2015; McDowell et al., 2013; Shenoy et al., 2010).

The average, normalized lithium concentration $({\hat{C}}_{Li} = SOC)$ ranges between 0 and 1 and is equal to state of charge (SOC). While some irreversible lithium loss in LIBs is inevitable, we assume that the minimal residual Li in the anode is negligible as we investigate the first cycle of charge and discharge after formation. In typical low-capacity electrodes, a negative to positive electrode capacity ratio, or N/P ratio, of 1.1 to 1.2 is typically used. Chen et al. (2019) state that while capacity fade occurs in both anode and cathode, over 90% occurs in the anode and thus the N/P ratio actively changes with time. Thus, for N/P ratios lower than 1, no capacity limiting constant is needed and ${\hat{C}}_{Li}$ is directly equal to SOC. In previous work, a capacity limiting constant of 0.9680 is used to account for a higher N/P ratio of 1.1 to 1.2 (Chen et al., 2019).

2.1.1. Assumptions

To execute the analytical models discussed, each element of the beam is considered to be in quasistatic equilibrium due to the relatively quick propagation of lithium into silicon (∼0.4 h for particles in the tens of nanometer diameter range) compared to the rates at which the battery is charged (20 h) (Gonzalez et al., 2021). In dynamic cycling of the battery actuator, the reaction kinetics and diffusion dynamics play a role in the displacement response. Actuation is directly linked to Si expansion with Li insertion, so the reaction kinetics of the anode contribute significantly to the overall system time constants. However, it is assumed that the experimental results are sufficiently slow to compare with the theoretical quasistatic results.

Small deflections are assumed for blocked force calculations. The length of the multimorph is assumed to be at least an order of magnitude greater than its width and thickness such that actuation is assumed to only occur in the longitudinal direction. The variation in width of the layers is accounted for when calculating the equivalent stiffness; however, the origin of the global coordinate system lies at the base of the beam (x-axis), the side of the anode (y-axis), and the bottom of the bottom ( $N_{Layer}$ ) layer (z-axis). The LIB is assumed to be fully charged (100% SOC) at the 4.2 V cutoff voltage and fully discharged (100% SOC) at the 2 V cutoff voltage.

The bending moment due to self-weight is considered negligible relative to the bending moment due to induced actuation. For the perfectly bonded case, there is no slip at all interfaces. For the slip case, slip is assumed to occur only on the interlayer separators.

2.1.2. Equations of equilibrium

To find the free curvature of a multilayer cantilever multimorph, the equations of equilibrium must be solved. The sum of forces $(\sum_{i = 1}^{N_{Layer}} F_{i})$ is shown in equation (1) and the sum of moments ( $\sum_{i = 1}^{N_{Layer}} M_{i})$ is shown in equation (2) for a multimorph of $N_{layer}$ layers. It is important to note that the equations of equilibrium must be solved for each $j$ th segment, as the location of the neutral axis may change as the geometric and material properties of the layers change. Thus, all geometric and material properties used hereon (e.g. thickness, width, elastic modulus) are specifically for the $j$ th segment being evaluated.

Figure 2 shows a cross-sectional diagram of the $j$ th segment of a multilayer of $N_{Layer}$ layers. Relevant terms in equations (1)–(5) are shown and feature the integration limits $z_{uppe r_{i = 1}}$ and $z_{lowe r_{i = 1}}$ , the neutral axis $h_{j}$ , and the force $F_{i = 5}$ and moment $M_{i = 5}$ for the top active layer $(i = 5)$ .

Figure 2.

Multilayer cross-section.

The induced stress of the $i$ th layer ( $σ_{i}$ ) due to the induced actuation stress ( $E_{i} S_{i}^{*}$ ) and resulting free curvature $(κ_{j}$ ) for the $j$ th segment is shown in equation (3). The axial or extensional strain at the neutral axis for the $j$ th segment is $S_{j}$ .

\sum_{i = 1}^{N_{Layer}} F_{i} = \sum_{i = 1}^{N_{Layer}} \int_{z_{lowe r_{i}}}^{z_{uppe r_{i}}} σ_{i} d A_{i} = 0

(1)

\sum_{i = 1}^{N_{Layer}} M_{i} = \sum_{i = 1}^{N_{Layer}} \int_{z_{lowe r_{i}}}^{z_{uppe r_{i}}} σ_{i} zd A_{i} = 0

(2)

σ_{i} = E_{i} (S_{j} + κ_{j} z - S_{i}^{*})

(3)

where,

d A_{i} = w_{i} dz .

The limits of integration in equations (1) and (2) are defined in equation (4). These limits are the respective positions of the upper and lower geometric location of the $i$ th layer of thickness $t_{i}$ , relative to the neutral axis $h_{j}$ of the $j$ th segment shown in equation (5). The total thickness of the multilayer is shown as $\sum_{i = 1}^{N_{Layer}} t_{i}$ . The location of the neutral axis of the $j$ th segment for a multilayer beam is shown in equation (5). The thickness, elastic modulus, and width of the $i$ th layer as well as the axial rigidity are $t_{i}, E_{i}, w_{i}$ and $E A_{j}$ (shown in equation (6)) respectively. The first upper limit applies to the first layer $(z_{uppe r_{i = 1}})$ and is used to find the first lower limit $(z_{lowe r_{i = 1}}),$ after which the two are calculated for all layers until $i = N_{Layer}$ :

\begin{matrix} z_{uppe r_{i = 1}} = \sum_{i = 1}^{N_{Layer}} (t_{i}) - h_{j}; \\ z_{lowe r_{i}} = {z_{upper}}_{i} - t_{i}; \\ z_{uppe r_{i}} = {z_{lower}}_{i - 1;} \end{matrix}

(4)

h_{j} = \sum_{i = 1}^{N_{layer}} (t_{i}) - \frac{\sum_{i = 1}^{N_{layer}} (E_{i} t_{i} w_{i} a_{i})}{E A_{j}}

(5)

E A_{j} = \sum_{i = 1}^{N_{Layer}} E_{i} w_{i} t_{i}

(6)

The constants $a_{i}$ in equations (7) and (8), are the centroid of the $i$ th layer.

a_{i > 1} = \frac{t_{i} + t_{i - 1}}{2} + a_{i - 1}

(7)

a_{1} = \frac{t_{1}}{2}

(8)

where,

1 < i \leq N_{layer} .

Equation (1) can be rewritten as the sum of the forces due to the transverse and axial loading of each $i$ th layer, as shown in equation (9).

\sum_{i = 1}^{N_{Layer}} F_{i} = F_{{transverse}_{i}} + F_{{axial}_{i}} = 0

(9)

The axial or longitudinal forces can be solved to find the axial strain at a reference axis $S_{j}$ of the $j$ th segment. The reference axis is the neutral axis.

The transverse force naturally sums to zero for an actuator based on increasing internal actuation strain. $F_{{transverse}_{i}}$ as shown in equation (10), is trivial as the neutral axis of the multimorph is located where the stress is zero, and the force due to the transverse bending above the neutral axis is balanced by the force due to transverse bending below the neutral axis.

${F_{axial}}_{i}$ , shown in equation (11) is used to find $S_{j}$ , the axial strain at the neutral axis. Solving the sum of axial forces in equation (11) allows for the axial strain at the neutral axis $S_{j}$ to be determined, which, in turn, allows for determination of the free curvature of the multimorph $κ_{j}$ such that

{F_{transverse}}_{i} = \int_{z_{lowe r_{i}}}^{z_{uppe r_{i}}} w_{i} E_{i} κ_{j} zdz = 0;

(10)

{F_{axial}}_{i} = \int_{z_{lowe r_{i}}}^{{z_{upper}}_{i}} w_{i} E_{i} (S_{j} - S_{i}^{*}) dz = 0 .

(11)

Equation (11) can be evaluated, the integral solved, and the results rearranged to find equation (12).

\sum_{i = 1}^{N_{Layer}} E_{i} w_{i} S_{j} (z_{uppe r_{i}} - z_{lowe r_{i}}) - \sum_{i = 1}^{N_{Layer}} E_{i} w_{i} S_{i}^{*} (z_{uppe r_{i}} - z_{lowe r_{i}}) = 0

(12)

The terms accompanying $S_{j}$ on the left-hand side of the equation can be grouped and divided from the right hand side to get $S_{j}$ as shown in equation (13).

S_{j} = \frac{\sum_{i = 1}^{N_{Layer}} E_{i} w_{i} S_{i}^{*} (z_{uppe r_{i}} - z_{lowe r_{i}})}{\sum_{i = 1}^{N_{Layer}} E_{i} w_{i} (z_{uppe r_{i}} - z_{lowe r_{i}})}

(13)

The difference in the upper and lower limits of the $i$ th layer can be simplified to the thickness of the $i$ th layer, as shown in equation (14).

t_{i} = (z_{uppe r_{i}} - z_{lowe r_{i}})

(14)

Equation (14) can be substituted into equation (13) to get the simplified expression for axial strain at the neutral axis for the $j$ th segment ( $S_{j}$ ) for an $N_{Layer}$ multilayer actuator as shown in equation (15).

S_{j} = \frac{\sum_{i = 1}^{N_{Layer}} S_{i}^{*} E_{i} t_{i} w_{i}}{\sum_{i = 1}^{N_{Layer}} E_{i} t_{i} w_{i}}

(15)

The sum of moments in equation (2) can be expanded by substituting equation (3) to get equation (16). The integral can be solved and the term including the free curvature can be subtracted from the right-hand side of the equation as shown in equation (17).

\sum_{i = 1}^{N_{Layer}} \int_{z_{lowe r_{i}}}^{z_{uppe r_{i}}} E_{i} w_{i} (S_{j} - S_{i}^{*} + κ_{j} z) zdz = 0

(16)

\begin{matrix} \sum_{i = 1}^{N_{Layer}} E_{i} w_{i} [\frac{(S_{j} - S_{i}^{*}) (z_{uppe r_{i}}^{2} - z_{lowe r_{i}}^{2})}{2}] \\ = - \sum_{i = 1}^{N_{Layer}} E_{i} w_{i} \frac{κ_{j} (z_{uppe r_{i}}^{3} - z_{lowe r_{i}}^{3})}{3} \end{matrix}

(17)

The free curvature of a multilayer can then be found by dividing the left-hand side of equation by the right-hand side variables to isolate free curvature. This can be further simplified to get the free curvature of the $j$ th segment of a multilayer actuator as shown in equation (18).

κ_{j} = - \frac{3}{2} \frac{\sum_{i = 1}^{N_{Layer}} E_{i} w_{i} (S_{j} - S_{i}^{*}) (z_{uppe r_{i}}^{2} - z_{lowe r_{i}}^{2})}{\sum_{i = 1}^{N_{Layer}} E_{i} w_{i} (z_{uppe r_{i}}^{3} - z_{lowe r_{i}}^{3})}

(18)

2.1.3. Strain throughout the multilayer

The strain $ϵ_{i}$ for $i$ layers throughout a multilayer is shown in equation (19). $S_{i}^{*}$ is subtracted but once from the strain (at each $i$ th layer) while the strain due to the bending curvature $κ_{j} z$ is a gradient throughout the $i$ th layer. $S_{j}$ will be constant throughout the $j$ th segment being evaluated but may vary segment to segment.

ϵ_{i} = S_{j} + κ_{j} z - S_{i}^{*}

(19)

2.1.4. Actuator metrics

2.1.4.1. Free and blocked deflection

The authors have previously reported the derivation of the actuator metrics here for a unimorph and the free deflection of a bimorph (Gonzalez et al., 2020, 2021). The free deflection and blocked deflection are found by taking the free curvature $κ_{j}$ and finding the local deflections, then using a transformation of coordinates to find the global deflections. The blocked deflection is found by using beam theory and the equivalent end moments used in finding the free deflection. The derivation is discussed at length in the appendices. Further actuator metrics, namely: blocked force and actuator force have been derived previously (Gonzalez et al., 2021), and are reviewed here.

2.1.4.2. Blocked force

The blocked force of a segmented multimorph is shown in equation (20) and is further generalized from previous work (Gonzalez et al., 2021) where $N_{Seg}$ refers to the number of segments.

F_{b} = \frac{\frac{N_{Seg}}{2 L} \sum_{j = 1}^{N_{Seg}} κ_{j} (2 N_{Seg} + 1 - 2 j)}{\sum_{j = 1}^{N_{Seg}} \frac{1}{E I_{j}} [{(j - 1)}^{2} - j + N_{Seg} + \frac{1}{3}]}

(20)

2.1.4.3 Actuator force and actuator strain energy

The actuator force $F_{a}$ and the actuator strain energy $U$ are shown in equations (21) and (22) respectively where $Z_{tip}$ is the prescribed transverse tip deflection. The normalized actuator strain energy per volume is defined here as ${UV}_{active}^{- 1}$ , where $V_{active}^{- 1}$ is the total volume of active material.

F_{a} = \frac{\frac{N_{Seg}^{3}}{L^{3}} Z_{tip} + \frac{N_{Seg}}{2 L} \sum_{j = 1}^{N_{Seg}} κ_{j} (2 N_{Seg} + 1 - 2 j)}{\sum_{j = 1}^{N_{Seg}} \frac{1}{E I_{j}} [{(j - 1)}^{2} - j + N_{Seg} + \frac{1}{3}]}

(21)

U = \frac{F_{b} | Z_{tip} |}{2}

(22)

2.2. Slip cases

Slip is mathematically defined such that slip occurs in one displacement coordinate direction only. The axial or longitudinal $(u$ ) displacements at the interface on the $i$ th and $(i - 1)$ th side are not equal. As in work by Heuer (2006), applying the Euler Bernoulli theory to each layer separately, the transverse (lateral) displacement $w_{i}$ , and longitudinal (horizontal) $u_{i}$ at transverse distance $z_{i}$ from the neutral axis of the $i$ th layer can be described in equation (23) as:

w_{i} = w; u_{i} = u_{i}^{(0)} - z_{i} w_{, x}

(23)

where $u_{i}^{(0)}$ denotes the axial displacement of the neutral axis of the $ith$ layer at $z_{i} = 0 .$ $w_{, x}$ denotes differentiation of $w$ with respect to x. Whereas a composite structure in perfect bond would have no slip at the interface $(Δ_{12} = Δ_{23} = 0)$ , a composite structure with interlayer slip allows layers to slide against each other.

For the example shown in Figure 3, the slip between the top and middle layer is denoted $Δ u_{12}$ while the slip between the central and bottom layer is shown as $Δ u_{23}$ . This process is described in more detail in work by Adam and Furtmüller (2019) as this work prioritizes using an approximation to predict the actuator metrics of a multilayer with interfaces in slip. Accounting for interfaces in slip allows for a beam with a much higher compliance, and therefore, deflection.

Figure 3.

Schematic showing effect of perfect bonding and slip at the interface of active (top) and passive (bottom) layers.

An equivalent passive layer is used to approximate layers with interfaces in slip as shown in Figure 4(a). Equivalent passive layer is defined here to mean a single passive layer that is sandwiched between two active layers. This equivalent passive layer is used to approximate the combined bending stiffness of all the individual passive layers of the multimorph being analyzed. It is comprised of the following layers (NCM and Al, Separator, and Cu). The NCM and Al are considered together because they are assumed to be perfectly bonded. Perfectly bonded layers add bending stiffness. This added bending stiffness must be reflected in the equivalent bending stiffness of the equivalent passive layer. The total thickness of the equivalent passive layer $(t_{Eq, passive})$ is equal to the summed thickness of all the passive layers (Cu) that are bonded to the active layers (Si), in the multimorph slip cases.

Figure 4.

(a) Schematic of equivalent passive layer (e.g. 13-layer bimorph (passive layers in color) and (b) Schematic of equivalent top and bottom active layers (active layers in white).

Although the elastic modulus of the Si coating layer is assumed to be the same for simplicity, the thickness of equivalent top active layer $(t_{Eq, top active})$ is equal to the summed thickness of all active layers (Si) above the neutral axis as shown, in white, in Figure 4(b). The thicknesses of the equivalent bottom active layer $(t_{Eq, bottom active})$ is equal to the summed thickness of all active layers (Si) below the neutral axis.

Thus, a 3-layer bimorph that consists of an equivalent passive layer, sandwiched by and perfectly bonded to two equivalent active layers can be used to approximate interfacial slip between specified layers of an $N_{Layer}$ multimorph.

The elastic modulus, $E_{Eq}$ (equation (24)), of this equivalent passive layer is modified to account for the equivalent bending stiffness of all other passive layers of the multimorph where interfacial slip is being approximated. The area moment of inertia of the equivalent passive layer $(I_{Eq})$ is shown in equation (25). Note that the area moment of inertia must be calculated for all passive layers. The equivalent bending stiffness of all passive layers of the multimorph with interfacial slip $E I_{Total Passive}$ , is shown in equation (26). If a layer is comprised of active material (whether activated or not), it is excluded from this summation of passive layer equivalent bending stiffness.

If interfacial slip is assumed between all layers, $E I_{Total Passive}$ in equation (26) will be equal to the sum of the bending stiffness of each individual passive layer. If interfacial slip is assumed at the interfaces between some layers, $E I_{Total Passive}$ will equal the sum of the bending stiffness of each bonded set of layers plus the bending stiffness of any individual passive layers with interfacial slip.

E_{Eq} = \frac{E I_{Total Passive}}{I_{Eq}}

(24)

I_{Eq} = \frac{1}{12} w_{Eq} {(t_{E q_{passive}})}^{3}

(25)

E I_{Total Passive} = \sum_{i = 1}^{N_{Layer}} E {I_{passive}}_{i}

(26)

This equivalent elastic modulus $E_{Eq}$ (equation (24)) can then be used in a simplified bimorph configuration (Figure 4) where the active material is summed up on either side of the equivalent passive layer. A 3-layer bimorph with equivalent passive layer $(E_{Eq}, t_{E q_{passive}})$ and equivalent active layers $(t_{Eq, active top}, t_{Eq, active bottom})$ is then used to approximate the actuator metrics of an $N_{Layer}$ multimorph actuator with interfacial slip.

2.3 Interfacial slip approximation

Two case studies are considered, a 13-layer and a 33-layer multimorph as examples of batteries that are comprised of multiple individual cells. These two configurations are shown in Figure 5. The 13-layer is studied because it is composed of two complete Li-ion battery cells (one top and one bottom). The 33-layer is investigated because it is composed of six Li-ion battery cells. Both studies are investigated for when interfacial slipping at the separator interfaces is allowed (slip), and for when interfacial slip is not allowed (perfect bond between all layers). All case studies consider a single segment, $N_{Seg} = 1$ .

Figure 5.

Schematic of the 3-layer approximation of both the 13-layer and 33-layer case studies (not to scale).

For the LIB studied, both electrodes are assumed to consist of perfectly bonded coating and metal current collector. The two electrodes are a positive electrode (cathode) comprised of an NMC (Nickel Manganese Cobalt Oxide) coating on an aluminum current collector, and a negative electrode (anode) comprised of a composite silicon coating coated on a copper foil current collector. The coated current collectors are separated by a plastic separator that prevents short-circuiting between electrodes. The interfaces of the separator are assumed to be in slip with all other layers. As seen in the 13-layer case study shown in Figure 5, five of the separator layers separate two unimorph anodes and two passive cathodes.

The two outer separators outside the cathodes simulate the packaging material. The 13-layer case study shown in Figure 5 is comprised of several layers of 3.5 cm length $l$ and 4 mm width $w$ . The widths of the separator, the NMC coating layer, and the aluminum current collector are broader than the anode (increased width pictured, but not to scale) for a value of $w_{1, 4, 7, 10, 13} = 8 mm$ (numbered from the top layer downward) and $w_{2, 3, 11, 12} = 6 mm$ . The separator is wider than the anode to prevent electrical shorting between the electrodes while the cathode is broader than the anode to provide sufficient lithium for full lithiation of the anode.

The thicknesses of layers are the separator of $t_{Sep} = t_{1, 4, 7, 10, 13} = 20 μ m$ , the aluminum of $t_{Al} = t_{2, 12} = 15 μ m$ , the NMC cathode coating of $t_{NCM} = t_{3, 11} = 44 μ m$ , the silicon coating layer of $t_{Si} = t_{5, 9} = 19 μ m$ , and the copper foil thickness of $t_{Cu} = t_{6, 8} = 17 μ m$ .

Equations (24)–(26) are applied to a 13-layer bimorph. Equation (24) is used to find the effective elastic modulus of the equivalent 3-layer bimorph that is necessary to achieve the same bending stiffness as all the passive layers combined. This includes the increased bending stiffness of the cathode due to the shifting of the neutral axis from the perfect bonding assumption between coating and current collector. Equation (25) is used to find the area moment of inertia for an equivalent passive layer where $t_{Eq, passive} = 2 t_{Cu} = 2 t_{6, 8} = 34 μ m$ .

The bending stiffnesses of the passive layers—five separators, two perfectly bonded cathodes (with Al and NCM layers), and two passive copper foil layers—are summed as in equation (26) to get $E I_{Total}$ . The additional bending stiffness provided by the cathode dominates the resulting $E I_{Total}$ . For the 13-layer case study shown in Figure 5, $2 E I_{Cathode} = 1.31 e^{- 6} N m^{2}$ and $E I_{Total} = 1.70 e^{- 6} N m^{2}$ . So, while the copper foil elastic modulus is $E_{Cu} = E_{6, 8} = 120 GPa$ with two layers of $t_{Cu} = 17 μ m$ , the 3-layer approximation shown in Figure 5 has an equivalent passive layer of $t_{Eq, passive} = 34 μ m$ and an $E_{Eq} = 130 GPa$ .

Also shown in Figure 5 is the schematic for a 33-layer bimorph along with an example 3-layer approximation. The approximation holds, though the thicknesses must change. The equivalent 3-layer bimorph for the 33-layer sums up the thickness of all the copper foil current collectors (6 layers, $E_{Cu} = 120 GPa$ ) $t_{Eq, passive} = 6 t_{Cu} = 54 μ m$ and finds an equivalent passive layer with an elastic modulus of $E_{Eq} = 2.07 TPa$ for $t_{Cu} = 9 μ m$ and $t_{Si} = 36 μ m$ (not to scale) in the 33-layer multimorph. The coating layers are summed up on either side of the equivalent passive layer for a total of $3 t_{Si} = 108 μ m$ on both sides.

Using the perfect bonding assumption, all the active layers above the neutral axis in the 33-layer are activated and held at a nonzero state of charge to achieve transverse deflection downward. Mathematically this equates to ${\hat{C}}_{L i_{i}} = {\hat{C}}_{Li}_{5, 9, 15} \neq 0$ . A result of this approximation is that, if the multimorph has interfacial slip occurring at the separators, the top anode of each of the six cells would need to be actuated, such that ${\hat{C}}_{L i_{i}} = {\hat{C}}_{Li}_{5, 15, 25} \neq 0$ . A Li-ion battery cell in this case is defined as a multilayer composed of anode, cathode, and separator between the two. The purpose of the 13-layer is to investigate how a 2-cell behaves and the purpose of the 33-layer is to investigate how a 6-cell behaves.

2.4. 13-layer experimental validation case study with perfect bonding

First, we analyze the 13-layer example shown in Figure 5, where all layers are assumed to be perfectly bonded. The 13-layer is arrayed in such a manner that the central bimorph is effectively split into two unimorphs by a plastic separator, as seen in Figure 5, to allow for practical control of each complete battery cell independently in experiment.

The thickness of layers are the separator of $t_{Sep} = t_{1, 4, 7, 10, 13} = 20 μ m$ , the aluminum of $t_{Al} = t_{2, 12} = 15 μ m$ , the NMC cathode coating of $t_{NCM} = t_{3, 11} = 44 μ m$ , the silicon coating layer of $t_{Si} = t_{5, 9} = 19 μ m$ , and the copper foil thickness of $t_{Cu} = t_{6, 8} = 17 μ m$ .

The experimental setup, adapted from work by Shan et al. (2021), is shown in Figure 6. The tip displacement of the 13-layer multimorph is captured every 3-minutes by macro-lens digital camera and then analyzed to obtain the deflection data using image processing software.

Figure 6.

Experimental setup.

As seen in Figure 6, the thirteen layers are not perfectly parallel, and the exterior separators bulge out. Due to fabrication limitations, the actuator twists and the tip displacement, which is tracked by the green tape, has some inherent error that shows up in the range of longitudinal tip displacement. Notably, the N/P ratio of this experimental validation 13-layer case study is 0.956 to ensure there is adequate lithium hosting capacity in the cathode to fully charge the anode. As a result, ${\hat{C}}_{L i_{i = 5}}$ is directly equal to $SOC$ . SOC is calculated via current integration, assuming that steady state cell voltages of 2V and 4.2V correspond to 0% and 100% SOC, respectively.

2.5. 13-layer experimental validation case study with interfacial slip

The relationship between $β_{Eff}$ and SOC is investigated and validated against the experimental tip displacement data shown in Figure 7. For a range of $β_{Eff}$ values (5%–45%) with a given increment (0.5%), the deflection resulting from a best fit $β_{Eff}$ for various SOC is shown in Figure 7. In previous work for a two layer unimorph (Gonzalez et al., 2021) or three layer bimorph (Gonzalez et al., 2020) and for the case studies of this work, $β_{Eff}$ is assumed to be a linear strain constant of $β = 17 %$ , however, the validity of this assumption is tested here. The best $β_{Eff}$ is defined to be the value of $β_{Eff}$ with the smallest root mean squared error (RMSE) between the experimental deflection tip deflection and model deflection at each SOC.

Figure 7.

Experimental validation for a 13-layer multilayer split unimorph case study with perfect bonding assumption.

The variation of $β_{Eff}$ with SOC for the 13-layer experimental validation case study is shown in Figure 8 along with linear fit and a 95% prediction interval. It is found that $β_{Eff}$ fits the experimental data well with a coefficient of determination of 0.912 for a linear fit. It is found that $β_{Eff}$ is a function of SOC as seen in Figure 8.

Figure 8.

An experimentally validated $β_{Eff}$ functional versus SOC for a linear fit for a 13-layer perfectly bonded multilayer.

The linear fit shown in Figure 8 is $β_{Eff} (SOC) = 1.3004 SOC - 0.0177$ . It is found that $β_{Eff} (SOC = 100 %) = 1.28$ 27, which we round to $β_{Eff} = 128 %$ for all further predictions. This effective linear strain constant $(β_{Eff})$ is used for all perfectly bonded case studies. Note that this is much larger than the previously assumed $β_{Eff} = 17 %$ . When there is no interfacial slip, a higher $β_{Eff}$ may more accurately predict the experimental results at higher SOC. It is worth noting that the actuation strain found when there is no interfacial slip is almost double the maximum actuation strain of silicon found in the literature (59%) (Berla et al., 2015; McDowell et al., 2013; Shenoy et al., 2010), and this indicates that the 13-layer multimorph is likely experiencing at least some interfacial slip.

If we assume slip at the separator interfaces, then the predicted tip deflections are much larger for a given $β_{Eff}$ . If we optimize $β_{Eff}$ to minimize error between experiment and theory in this case, then the $β_{Eff}$ varies from 0.15% to 0.6%. Due to the range of the $β_{Eff}$ , the case where perfect bonding $(β_{Eff} ≅ 128 %)$ between layers is assumed to be closest to the experiment. While the 13-layer multimorph may be experiencing some interfacial slip, the $β_{Eff}$ for the perfect bonding case is closest to both the reported literature values (Berla et al., 2015; McDowell et al., 2013; Shenoy et al., 2010; Xie et al., 2017) and previous work by the authors Gonzalez et al. (2021), by several orders of magnitude (compared to the cases where interfacial slip is assumed).

The 13-layer case study shown in Figure 5 is further explored for a range of coating layer thickness variables. All other parameters remain the same. This is reflected in the schematics embedded in Figure 9. The thickness of the silicon coating layer variable ranges from $9 μ m$ to $36 μ m$ , and each layer of copper foil is held constant at a thickness of $t_{Cu} = t_{6, 8} = 9 μ m$ . The actuation strain is applied in the fifth layer (silicon coating) from the top and is $S_{i = 5_{Bonding}}^{*} = 1.28$ or $S_{i = 5, 9, 15_{Slip}}^{*} = 0.0168$ for the perfect bonding or slip assumption, respectively. We define

γ = \frac{t_{Cu}}{t_{Si}}

(27)

as the ratio of Cu to Si thickness. The maximum free deflection for a 13-layer multilayer multimorph is shown in Figure 9. The ratio $γ$ increases from one quarter to one in increments of one quarter ( $\frac{1}{4} \leq γ \leq 1)$ , to showcase the effect of decreasing the amount of active material.

Figure 9.

Normalized longitudinal free deflection versus normalized transverse free deflection for a 13-layer multimorph cantilever.

2.6. 33-layer theoretical case study

The 33-layer case study (schematic shown in Figure 5 and embedded in the Figure 9 and Figure 14) is comprised of several layers of 3.5 cm length $l$ and 4 mm width $w$ . The width of the separator and both the NMC coating layer and the aluminum current collector are broader (increased width not pictured) for a value of $w_{Sep} = w_{1, 4, 7, 10, 14, 17, 20, 24, 27, 30, 33} = 8 mm$ (numbered from the bottom layer upward) and $w_{NCM} = w_{Al} = w_{2, 3, 11, 12, 13, 21, 22, 23, 31, 32} = 6 mm$ . There are six cells within the multimorph, complete with anode, cathode, and separator.

The thickness of layers are the separator of $t_{Sep} = t_{1, 4, 7, 10, 14, 17, 20, 24, 27, 30, 33} = 20 μ m$ , the aluminum of $t_{Al} = t_{2, 12, 22, 32} = 15 μ m$ , the NMC cathode coating of $t_{NCM} = t_{3, 11, 13, 21, 23, 31} = 44 μ m$ , the silicon coating layer of $t_{Si} = t_{5, 9, 15, 19, 25, 29} = 36 μ m$ , and the copper foil thickness of $t_{Cu} = t_{6, 8, 16, 18, 26, 28} = 9 μ m$

3. Results and discussion

A 13-layer case study is investigated and experimentally validated both when all layers are assumed to be perfectly bonded and when some layers are assumed to slip at the interface. Similarly, a 33-layer theoretical case study is shown to demonstrate the benefits of a LIB actuator with more Li-ion cells (6 cells) than the 2-cell, 13-layer configuration.

3.1. 13-layer case study for range of coating thickness

The 13-layer case study under the interfacial slip assumption shows larger maximum free deflection than the 13-layer under the perfect bonding assumption. For nearly two orders of magnitude less actuation strain based on the $B_{Eff}$ , larger free deflection can be achieved relative to the perfectly bonded case.

The larger compliance of the 13-layer with interfacial slip compared to the perfect bonding assumption can be seen in the blocked deflection results in Figure 10. Whereas the 13-layer case study achieves a maximum transverse blocked deflection of over 3 mm, the perfectly bonded beam achieves only about 2.5 mm.

Figure 10.

Normalized longitudinal curvature shortening versus normalized transverse blocked deflection for a 13-layer multimorph.

The comparatively large bending stiffness of the perfectly bonded case shows in Figure 11, in the higher blocked force per active material volume for all ratios of $γ$ . The slip case performs much poorer in the blocked force metric. In comparison to the blocked force of the perfectly bonded 13-layer, the interfacial slip case is barely visible on the plot.

Figure 11.

Blocked force normalized by active material volume versus SOC for a 13-layer multimorph. The values for slip lie on the x-axis and are approximately zero for all abscissa values.

Both the blocked force (y-intercept) and the free deflection (x-intercept) are shown in Figure 12 where the actuator force normalized by volume of active material is plotted against the prescribed tip displacement. The area under the curve is the actuator strain energy, and the case with the most area under the curve is the perfectly bonded case for $γ = 1 / 4$ .

Figure 12.

Actuator force normalized by active material volume versus prescribed tip displacement for a 13-layer multimorph. The values for slip lie on the x-axis and are approximately zero for all abscissa values.

Figure 13 shows that even with eightfold more active material, the normalized actuator strain energy still increases exponentially as the $γ$ ratio is increased. This is in part due to two active layers that are both increased for a total of eightfold increase in thickness of active material.

Figure 13.

Actuator strain energy normalized by active material volume versus SOC for a 13-layer multimorph. The values for slip lie on the x-axis and are approximately zero for all abscissa values.

3.2. 33-layer case study for range of coating thickness

The free deflection for a 33-layer multimorph is shown in Figure 14 for $\frac{1}{4} \leq γ \leq 1$ , where $γ = \frac{t_{Cu}}{t_{Si}}$ and $t_{Cu} = 9 μ m$ . The deflection for both the perfectly bonded and slip case approximation are shown. The free deflection of the $γ$ = 1/4 perfectly bonded case achieves the largest tip deflection. The larger compliance of the perfectly bonded case is clearly shown because, for all ratios, the slip case achieves less deflection than all ratios of the perfectly bonded case. The perfectly bonded case achieves a max transverse tip deflection of about 50% multimorph length, while the slip case achieves approximately 10% multimorph length transverse tip deflection.

Figure 14.

Normalized longitudinal free deflection versus normalized transverse free deflection for a 33-layer multimorph.

The actuation strain for the perfectly bonded and slip cases are $S_{i = 5, 9, 15_{Bonded}}^{*} = 1.28$ and $S_{i = 5, 9, 15_{Slip}}^{*} = 0.0168$ respectively. The perfectly bonded case for the 33-layer multimorph achieves a larger free deflection than the slip case due to the nearly two orders of magnitude increase (relative to the actuation strain applied in the slip case) in applied actuation strain. Additionally, the assumption for interfacial slip results in an equivalent elastic modulus for the 33-layer of $E_{Eq} = 2.07 TPa$ , while the equivalent elastic modulus for the 13-layer slip case is $E_{Eq} = 130 GPa .$

Figure 15 shows the blocked deflection for a 33-layer multimorph. As in Figure 14, what should be the larger compliance of the slip case, is overcome by the increased actuation strain of the perfectly bonded case and the increased equivalent elastic modulus of the slip case. The maximum blocked deflection is less than 10% of the length at less than 0.2 mm for the $γ$ = 1/4 slip case, while more than 1 mm for the $γ$ = 1/4 perfectly bonded case.

Figure 15.

Normalized longitudinal curvature shortening versus normalized transverse blocked deflection for a 33-layer multimorph.

The effect of the much larger actuation strain and bending stiffness of the perfectly bonded case is evident in Figure 16. Of note is that, although the total thickness of the summed coating layers is increased by twenty-four times (compared to $γ = 1$ ) for the $γ = \frac{1}{4}$ case (both with six active layers), only the activated layers (top three active layers) are considered when normalizing the blocked force (three active layers, four times the active material volume compared to the $γ = 1$ case). The blocked force is normalized by the volume of the activated active material layers in the top or three times the volume of one active layer (as they are all uniform). The same is true for the actuator force and actuator strain energy plots.

Figure 16.

Blocked force normalized by active material volume versus state of charge (SOC) for a 33-layer multimorph. The values for slip lie on the x-axis and are approximately zero for all abscissa values.

The largest blocked force normalized per volume of active material is over $2.5 e 8 N m^{- 3}$ which is over an order of magnitude larger than the maximum blocked force per active material volume for the 13-layer multimorph (∼1 $e 8 N m^{- 3}$ ), both for the $γ = \frac{1}{4}$ case). The blocked force for all cases where interfacial slip is assumed is negligible compared to the perfectly bonded cases.

Figure 17 shows the actuator force normalized by active material for prescribed tip displacement. The perfectly bonded 33-layer case achieves a large blocked force (y-intercept) and large free deflection (x-intercept) (both relative to the slip case) and has a relatively large area under the curve (actuator strain energy) relative to the comparatively negligible slip case. There is some inaccuracy of the predicted actuator force (due to the small deflection assumption inherent in Castigliano’s theorem) for large free deflection, but the difference in actuator force per volume for the bonded case and the slip case is so large, the inaccuracy is assumed negligible.

Figure 17.

Actuator force normalized by active material volume versus prescribed tip displacement for a 33-layer multimorph. The values for slip lie on the x-axis and are approximately zero for all abscissa values.

The actuator strain energy is predicted as the product of half the blocked force and free deflection (equation (22)) as is shown in Figure 18. The actuator strain energy for the 33-layer case study in slip maxes out at approximately ${UV}_{active}^{- 1} = 2.5 e 6 J m^{- 3}$ which is over double the actuator strain energy predicted for the 13-layer perfect bonding case study ( ${UV}_{active}^{- 1} = 1.2 e 6 J m^{- 3}$ ) shown in Figure 13. Both case studies are for a ratio of $γ$ = 1/4. This implies that at least for the 33-layer case, more active layers (and importantly all the requisite layers that must accompany these in a LIB) gives better actuator performances. Furthermore, while the slip case achieves the free deflection and blocked deflection metrics at very low strain, expectedly, there is negligible actuator strain energy that can be used.

Figure 18.

Actuator strain energy normalized by active material volume versus SOC for a 33-layer multimorph. The values for slip lie on the x-axis and are approximately zero for all abscissa values.

4. Summary and conclusions

An analytical model has been developed to predict the free deflection, blocked force, blocked deflection, actuator force, and actuator strain energy for an infinitely layered multimorph. An approximation for cases with interfacial slip has been developed in parallel. This approximation for interfacial slip between specified layers allows for prediction of the actuator metrics for a multilayer actuator. For both bonded and slip cases, an equivalent passive layer is sandwiched by two equivalent active layers to allow for bimorph actuation. The analytical model developed is experimentally tuned with a case study wherein the free deflection of a 13-layer multimorph is compared against its analytical model equivalent. $β_{Eff}$ versus SOC is tuned to limit model error.

It is found that $β_{Eff}$ for the perfectly bonded case study is 1.28 whereas the $β_{Eff}$ for the slip case is 0.0168. Both achieve the desired free deflection shown in experiment, but in both cases the $β_{Eff}$ required to match the experimental data is far from the previously published values of $β_{Eff}$ around $0.17$ . Although the previously published value of $β_{Eff} = 0.17$ lies between the perfectly bonded case (1.28) and the slip case (0.0168), it is closer to the perfectly bonded case. The $β_{Eff}$ required in the slip cases to match the free deflection of the experiment is much smaller than the reported literature value, so friction between the layers makes them stick together and the perfectly bonded assumption is justified.

Two case studies are investigated using the experimentally tuned model. The actuator metrics for both a 33-layer case study and a small parametric study of the 13-layer case are both shown for a range of coating layer thickness. It is found that the normalized actuator strain energy ${UV}_{active}^{- 1}$ of the $γ$ = 1/4 ratio for the 33-layer is greater than the 13-layer by a factor of twenty.

It is found that the blocked force and actuator strain energy increases for increasing numbers and thicknesses of active layers despite normalization by volume of active material. More layers are more effective, as has been previously seen in multilayer piezoelectric actuators. For all cases investigated here, the normalized actuator continues to increase with more active layers despite the added bending stiffness from both the added active material, and the necessary passive materials for the battery to work (additional separators, cathodes, and copper foil). Many LIB actuators can be stacked to achieve more actuator strain energy than any one LIB actuator could achieve alone. Six stacked LIB actuators has substantially more actuator strain energy than two stacked actuators, and so it is possible to stack many more actuators to achieve even greater actuator strain energy.

A primary benefit of the electrical energy storage functionality of LIB actuators is the ability to store charge while also providing actuation. While using active Li-ion cells does complicate actuator design, the tradeoff is considered worthwhile for potential applications in the fields of minimally invasive surgery (MIS) and active rehabilitation aids. In MIS, surgical space is limited and reduction of wiring by introducing battery power stored in the actuator could be very beneficial. Likewise, actuation and energy storage can be decoupled by simultaneously charging the top and bottom electrode of a bimorph. Such Si-anode based LIB actuators can improve both battery capacity and actuator performance by coupling or decoupling actuation and energy storage depending on the application.

Some critical future implementation challenges for this work include development of a flexible pouch cell to allow for more movement of the battery actuator. Cycling remains a difficulty that may be well tackled by a flexible current collector to eliminate the plastic deformation of the metal current collectors. Delamination of the Si composite presents a challenge that may be tackled by an increase in binder percentage, a thinner active material layer, and a flexible pouch cell that maintains a pressure on the electrodes to prevent delamination.

Footnotes

Appendices

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors gratefully acknowledge the support of the National Science Foundation under Grant No. 1662055.

ORCID iDs

Cody A. Gonzalez

Shuhua Shan

Christopher D. Rahn

References

Adam

Furtmüller

(2019) ‘Moderately Large Deflections of Composite Beams with Interlayer Slip’, in Altenbach

Irschik

Matveenko

V.P.

(eds) Advanced Structured Materials. Cham: Springer International Publishing, pp. 1–17. doi: 10.1007/978-3-030-21251-3_1.

Ahmed

Ounaies

Arrojado

EAF

(2017) Electric field-induced bending and folding of polymer sheets. Sensors and Actuators A Physical 260: 68–80.

Baranchugov

Markevich

Pollak

, et al. (2007) Amorphous silicon thin films as a high capacity anodes for Li-ion batteries in ionic liquid electrolytes. Electrochemistry Communications 9(4): 796–800.

Berla

Lee

Cui

, et al. (2015) Mechanical behavior of electrochemically lithiated silicon. Journal of Power Sources 273: 41–51.

Cannarella

Leng

Arnold

(2014) On the coupling between stress and voltage in lithium-ion pouch cells. In: Energy harvesting and storage: Materials, devices, and applications V, p. 91150K. SPIE. DOI: 10.1117/12.2055152.

Cappelleri

Frecker

Simpson

, et al. (2002) Design of a PZT bimorph actuator using a metamodel-based approach. Journal of Mechanical Design 124(2): 354–357.

Chen

Zhang

, et al. (2019) Effect of N/P ratios on the performance of LiNi0.8Co0.15Al0.05O2||SiO/graphite lithium-ion batteries. Journal of Power Sources 439: 227056.

Chin

Rhyner

Koyama

, et al. (2006) Lithium rechargeable batteries as electromechanical actuators. Electrochemical and Solid-State Letters 9(3): A134.

Crawley

Anderson

(1990) Detailed models of piezoceramic actuation of beams. Journal of Intelligent Material Systems and Structures 1(1): 4–25.

10.

DeVoe

Pisano

(1997) Modeling and optimal design of piezoelectric cantilever microactuators. Journal of Microelectromechanical Systems 6(3): 266–270.

11.

Erol

von Lockette

Frecker

(2020) Multi-objective optimization of a multi-field actuated, multilayered, segmented flexible composite beam. Smart Materials and Structures 29(2): 024001.

12.

Frecker

Aguilera

(2004) Analytical modeling of a segmented unimorph actuator using electrostrictive P(VDF-TrFE) copolymer. Smart Materials and Structures 13(1): 82–91.

13.

Gonzalez

Frecker

, et al. (2018) Analytical modeling of a multifunctional segmented lithium ion battery unimorph actuator, In:Volume 2: Mechanics and behavior of active materials; Structural health monitoring; Bioinspired smart materials and systems; Energy harvesting; Emerging technologies. San Antonio, TX: American Society of Mechanical Engineers. DOI: 10.1115/SMASIS2018-8123.

14.

Gonzalez

Frecker

, et al. (2019) Analytical modeling and simulation of the blocked force and large deformation of multifunctional segmented lithium ion battery unimorph actuator. In: ASME 2019 conference on smart materials, adaptive structures and intelligent systems, SMASIS 2019. Louisville, KT: American Society of Mechanical Engineers. DOI: 10.1115/SMASIS2019-5560.

15.

Gonzalez

Frecker

, et al. (2021) Analytical modeling and simulation of a multifunctional segmented lithium ion battery unimorph actuator. Smart Materials and Structures 30(1): 015039.

16.

Gonzalez

Shan

Frecker

, et al. (2020) Analytical modeling of a segmented bimorph lithium ion battery actuator.In: ASME 2020 Conference on smart materials, adaptive structures and intelligent systems. New York: American Society of Mechanical Engineers. DOI: 10.1115/SMASIS2020-2328.

17.

Goodenough

Park

(2013) The Li-ion rechargeable battery: A perspective. Journal of the American Chemical Society 135: 1167–1176.

18.

Heuer

(2006) Thermo-piezoelectric flexural vibrations of viscoelastic panel-type laminates with interlayer slip. Acta Mechanica 181(3–4): 129–138.

19.

Huang

Xiao

(2020) A variable structure pneumatic soft robot. Scientific Reports 10(1): 18778.

20.

Low

Guo

(1995) Modeling of a Three-Layer Piezoelectric Bimorph Beam with Hysteresis. Journal of Microelectromechanical Systems 4(4): 230–237.

21.

Gonzalez

Huang

, et al. (2020) Multifunctional Li(Ni0.5Co0.2Mn0.3) O2-Si batteries with self-actuation and self-sensing. Journal of Intelligent Material Systems and Structures 31(6): 860–868.

22.

Rahn

Frecker

(2017) Multifunctional NMC-Si batteries with self-actuation and self-sensing, In: Volume 1: Development and characterization of multifunctional materials; Mechanics and behavior of active materials; Bioinspired smart materials and systems; Energy harvesting; Emerging technologies. Snowbird, UT: American Society of Mechanical Engineers. DOI: 10.1115/SMASIS2017-3886.

23.

McDowell

Lee

Nix

, et al. (2013) 25th anniversary article: Understanding the lithiation of silicon and other alloying anodes for lithium-ion batteries. Advanced Materials 25: 4966–4985.

24.

Ohm

Brehmer

Zentel

(2010) Liquid crystalline elastomers as actuators and sensors. Advanced Materials 22(31): 3366–3387.

25.

Pal

Xie

(2011) A curved multimorph based electrothermal micromirror with large scan range and low drive voltage. Sensors and Actuators A: Physical 170: 156–163.

26.

Pal

Xie

(2012) Analysis and fabrication of curved multimorph transducers that undergo bending and twisting. Journal of Microelectromechanical Systems 21(5): 1241–1251.

27.

Pelrine

Kornbluh

Pei

, et al. (2000) High-speed electrically actuated elastomers with strain greater than 100%. Science 287(5454): 836–839.

28.

Robbins

Reddy

(1991) Analysis of piezoelectrically actuated beams using a layer-wise displacement theory. Computers & Structures 41(2): 265–279.

29.

Sethuraman

Chon

Shimshak

, et al. (2010) In situ measurements of stress evolution in silicon thin films during electrochemical lithiation and delithiation. Journal of Power Sources 195(15): 5062–5066.

30.

Shan

Gonzalez

Rahn

, et al. (2021) Experimental study of NCM-Si batteries with bi-directional actuation. In: ASME 2021 conference on smart materials, adaptive structures and intelligent systems. New York, NY: American Society of Mechanical Engineers. DOI: 10.1115/SMASIS2021-67596.

31.

Shenoy

Johari

(2010) Elastic softening of amorphous and crystalline Li–Si Phases with increasing Li concentration: A first-principles study. Journal of Power Sources 195(19): 6825–6830.

32.

Timoshenko

(1925) Analysis of bi-metal thermostats. Journal of the Optical Society of America 11(3): 233–255.

33.

Vel

Batra

(2001) Analysis of piezoelectric bimorphs and plates with segmented actuators. Thin-Walled Structures 39(1): 23–44.

34.

Wang

Q-M

(1998) Piezoelectric Ceramic Actuators and Composite Structure for Active Noise Control Applications. University Park, TX: Pennsylvania State University.

35.

Weinberg

(1999) Working equations for piezoelectric actuators and sensors. Journal of Microelectromechanical Systems 8(4): 529–533.

36.

Xia

Afshar

Yang

, et al. (2019) Electrochemically reconfigurable architected materials. Nature 573(7773): 205–213.

37.

Xie

Zhang

Song

, et al. (2017) Modeling and in situ characterization of lithiation-induced stress in electrodes during the coupled mechano-electro-chemical process. Journal of Power Sources 342: 896–903.

38.

Zhang

Ahmed

Masters

, et al. (2017) Finite element analysis of electroactive polymer and magnetoactive elastomer based actuation for origami folding. Smart Materials and Structures 26(10): 105032.