Variable Rigidity Module with a Flexible Thermoelectric Device for Bidirectional Temperature Control

Abstract

Dynamic stiffness tuning is a promising approach for shape reconfigurable systems that must adapt their flexibility in response to changing operational requirements. Among stiffness tuning technologies, phase change materials are particularly promising because they are size scalable and can be powered using portable electronics. However, the long transition time required for phase change is a great limitation for most applications. In this study, we address this by introducing a rapidly responsive variable rigidity module with a low melting point material and flexible thermoelectric device (f-TED). The f-TED can conduct bidirectional temperature control; thereby, both heating and cooling were accomplished in a single device. By performing local cooling, the phase transition time from liquid to solid is reduced by 77%. The module in its rigid state shows 14.7 × higher bending stiffness than in the soft state. The results can contribute to greatly widening the application of phase transition materials for variable rigidity.

Introduction

Most electrical and mechanical devices have a specific mechanical stiffness depending on the mechanical properties of the constituent materials and their geometrical configurations. Devices with such invariant stiffness can only be used for specific applications that require soft or rigid properties. However, if the stiffness can be adjustable so that the soft and rigid states can be implemented within a single device, the field of applications can be greatly expanded.^1,2 In recent years, devices that can change the stiffness under the required conditions have been increasing in many fields, including automotives,³ medical devices,^4,5 aeronautics,^6,7 and bioinspired robots.^8,9

Previous efforts in implementing variable rigidity devices have been based on gel hydration,¹⁰ jamming,^11,12 smart fluids such as electrorheological or magnetorheological fluids,^13,14 and motors for adjusting spring stiffness.¹⁵ However, these implementations typically use bulky external hardware for controlling air pressure, applying electromagnetic fields, and mechanical actuation, which could lead to a complicated structure with limitations on scalability and portability.

As an alternative to methods that depend on bulky and complex hardware, attention has been increasingly directed toward utilizing materials that can change their mechanical stiffness by applying specific stimulations.¹⁶ Among these approaches, methods that utilize phase change materials such as shape memory materials or low melting point materials (LMPMs) represent a simple and effective way to get a large and reversible stiffness change.^17–20 The stiffness of these materials can be varied by controlling temperature, which enables change in the phase or state of the materials. Usually these materials also require additional equipment to control the temperature of the materials. However, it can be made as a single module with attaching a thin flexible heater¹⁷ or by inducing the current directly if the material is electrically conductive.¹⁸

For variable rigidity devices that utilize phase transition materials, the phase transition from the rigid to soft state requires excitation energy, usually in the form of thermal energy. The rigid to soft transition can be rapidly completed by increasing the input current for heating. However, the transition from the soft to rigid state, which means that the absorbed thermal energy should be removed, has typically depended on air convection or liquid circulation requiring an external bulky cooling system. Due to this, the overall phase transition cycle takes more than a few minutes and reducing phase transition time from the rigid to soft state by improving cooling performance is the biggest challenge in applying this type of variable rigidity device in practical applications.

Thermoelectric devices, which can transfer the heat from one side to the other, have the ability to raise or lower the temperature of the local area through electrical stimulation.²¹ In particular, the flexible thermoelectric device (f-TED) presented here, which has been actively studied in recent years,^22–27 is highly bendable and can be attached to arbitrary shapes. In this way, the f-TED can be an efficient temperature controller in both functions of the heating and cooling to implement dynamic stiffness tuning of the phase change materials. In particular, it can be a solution for cooling as an alternative to combining a flexible heater (which can only perform a heating function) with a liquid circulation system (which requires an external hydraulic pump).

In this study, we fabricated such a device—a variable rigidity module composed of an f-TED, LMPM, and flexible polymer mold. In addition, the slot-type heat sink, which can be mounted on the electrode of the f-TED as an island form, was developed for enhancing the heat dissipation from the hot side of the f-TED. For the heating mode, it has been shown that the thermoelectric device can apply heat more efficiently than a Joule heater within a certain range of input power. In terms of the cooling mode, with the 2 A of current input, the phase transition time of the LMPM from soft to rigid state was shortened to <20 s, which is 77% less than without the Peltier effect. Besides, the stiffness of the fabricated structure in the soft state and the rigid state was measured by the customized bending stiffness measuring device, and the experimental results were verified with theoretical predictions obtained from an analytical model and finite element method (FEM) simulation. The fabricated structure has about 14.7 times higher rigidity in the rigid state than in the soft state.

Materials and Methods

Design and fabrication

Figure 1A illustrates the overall schematic diagram of the variable rigidity module with the f-TED. As shown in the figure, the variable rigidity part consists of a polymer mold engraved with an empty channel with rectangular cross section in which the LMPM is embedded. Among various LMPMs, gallium is selected because of its biocompatibility and safe melting temperature (29.8°C) for wearable and medical applications. On the top side of the variable rigidity part, the f-TED is attached. The photographs of the variable rigidity module are presented in Figure 1B. The f-TED is made of thermoelectric legs connected electrically in series and thermally in parallel, like a conventional rigid thermoelectric device. However, the rigid top and bottom plates are eliminated to make the device flexible, and the empty space between the legs is filled with a customized polyurethane foam which has very low thermal conductivity (0.03 W m⁻¹ K⁻¹).²⁸ The fabricated f-TED has a rectangular shape with an area of 2 × 4 cm².

FIG. 1.

f-TED design and fabrication process. (A) Overall schematic diagram of the variable rigidity module with f-TED. (B) Photograph of the variable stiffness module. (C) Side view of the module with the weight. The device can support the weight of 20 times more mass when in the rigid state. (D) Fabrication steps for the variable rigidity module. f-TED, flexible thermoelectric device. Color images are available online.

To enhance the efficiency of heat exchange between the f-TED and the air, a slot-type heat sink that is attachable to the electrode is also developed. The slot-type heat sinks are attached to the electrodes distributed in an island form, so that the heat sinks minimize its effect on the flexibility of the f-TED. Figure 1C demonstrates the concept of rigidity tuning operation by showing a sample that is able to hold up a weight of 20 times heavier than the module itself when the module is rigid. The module is easily bent by the same load when the module is in the soft state.

Figure 1D shows the fabrication process flow of the variable rigidity module. The process starts with the polymer mold fabrication. The base material of the polymer mold is from a commercially available urethane rubber (Clear Flex™ 95; Smooth On), and the channel embedded polymer mold is produced by the following processes—first, mixing Part A and B of the Clear Flex 95 Kit with a ratio of 1.5:1 and pouring the mixed liquid polymer onto the prepared 3D printed plastic mold, then curing the polymer at room temperature for 20 h. After it has adequately set, the polymer was separated from the mold, and the f-TED was attached to the top surface of the polymer mold.

Before the gluing process, a 30 μm thickness graphite sheet was attached to the bottom surface of the f-TED to increase the heat exchange area with the LMPM. As shown in Figure 1D, the liquid polymer mixture was uniformly coated on the surface of the f-TED in a pot life by a spin-coating method. After coating the mixture on the surface, the polymer mold with a channel was attached, and the coated polymer mixture was cured for about 24 h.

Finally, the variable rigidity module was fabricated using a syringe to inject gallium (99.999%; RND Korea, South Korea) into the embedded channels. Optionally, the customized slot-type heat sink can be mounted on the electrodes with metal adhesive. The heat sink blocks were manufactured by machining aluminum blocks and anodizing. Each heat sink has the same base area with the electrodes (2 × 5 mm²), and the height of the heat sink was designed to 4 mm.

Thermoelectric properties

The Seebeck coefficient and the electrical conductivity of the thermoelectric material were measured using a thermoelectric property measurement system (ZEM-3; ULVAC). The thermal conductivity $κ$ of the thermoelectric material was determined based on the relation $κ = λ C_{p} D$ , where the thermal diffusivity $λ$ , the heat capacity C_p, and the density D were measured using the laser flash method (LFA 457; Netzsch), differential scanning calorimetry (DSC 204F1; Netzsch), and the Archimedes' method, respectively. All of the thermoelectric properties are represented in Table 1.

Table 1.

Room Temperature Material Properties of the Thermoelectric Materials

Material type	$σ$ (S cm⁻¹)	$α$ (μV K⁻¹)	$κ$ (W m⁻¹ K⁻¹)
p-Type	993.5	176.9	1.2
n-Type	947.8	−187.8	1.2

Fabrication of f-TED

The fabrication process begins with patterning copper electrodes onto the polyimide substrate by wet etching process. After the Sn_96.5Ag_3.0Cu_0.5 solder paste (Phoenix materials) was printed onto the copper patterns, N and P types of thermoelectric legs were placed onto the solder material, followed by a bonding process at 250°C for 10 min with a reflow machine (551–15; SEF GmbH). After the bonding process, the empty space between the legs was filled with a proprietary polyurethane foam having a very low thermal conductivity (0.03 W m⁻¹ K⁻²).²⁸ For the thermoelectric legs, 36 couples of a commercially available Sb-doped p-type and Se-doped n-type BiTe-based thermoelectric material with an area of 1 mm² and heights of 0.7 mm were used. The fabricated f-TED has the device fill factor of 9%, which is defined as the ratio of the overall area of the thermoelectric legs to the total area of the device. The device figure of merit (ZT) of the f-TED was 0.71. The ZT value was measured using a Z-meter from RMT Ltd.

Thermal measurements

To measure the temperature, a thin K-type thermocouple (∼20 μm) was attached to the bottom surface of the polymer mold using a thermal paste (Prolimatech) with a thermal conductivity of 8.1 W m⁻² K⁻¹. The temperature was monitored using a Keithley 2700 multimeter. The air velocity under forced convection was controlled by an electrical fan and measured using a thermal anemometer (Testo 405i).

Results

Heating function of the f-TED

To change the phase of the LMPM reversibly, the variable rigidity module should have a function of not only cooling but also heating. Most reported LMPM-based variable rigidity modules supply heat to the LMPM from the directly applied current or the flexible heaters with a hot wire embedded into the polymer film.^17,18 The f-TED can play a role of not only a cooler but also a heater, so that it can also induce heat to the LMPM. As illustrated in Figure 2A, for the heating mode, the total heat, $Q_{t o t}$ , transferred to the LMPM from the f-TED can be calculated by the following equation²⁹:

FIG. 2.

Characterization of f-TED with respect to heat transfer performance on gallium. (A) Schematic diagram of thermoelectric heating. In the diagram, Q_J, Q_P, and Q_F represent the Joule heating, Peltier heating, and Fourier heating mechanisms, respectively. (B) Normalized transferred heat from the f-TED as a function of input current. (C) Transferred heat from the Joule heater and the Thermoelectric device for the same input power. (D) Temperature versus operating time with different input power. (E) Change in temperature as a function of time. The blue line shows the first derivative of the measurement results. (F) Phase transition time as a function of input power. Color images are available online.

Q_{t o t} = Q_{J} + Q_{P} + Q_{F} = \frac{1}{2} I^{2} R + N α I T_{h} - k A (T_{h} - T_{c})

(1)

where $Q_{J}$ is the Joule heat, $Q_{P}$ is the Peltier heat, $Q_{F}$ is the Fourier heat, I is the applied current to the f-TED, R is the total resistance of the f-TED, N is the number of the thermoelectric legs, $α$ is the Seebeck coefficient of the thermoelectric legs, k is the effective thermal conductivity of the f-TED in z-direction, and $T_{h}$ and $T_{c}$ are hot and cold side surface temperatures.

Figure 2B shows the normalized transferred heat of each component during the thermoelectric heating. The parameters for calculation are chosen to be the same values as with the fabricated device. The hot side temperature is fixed to 29.8°C, which is the same with the melting point of the gallium. As shown in the graph, the heat produced by the Peltier heater is larger than that from Joule heating when the input current is <8.8 A. The Fourier heat increases from 0 to 3.7 A of the input current and then decreased when the current becomes larger due to the temperature rise of the f-TED top surface.

Figure 2C shows a comparison graph of the transferred heat to the gallium by Joule heating ( $I^{2} R$ ) and Peltier heating ( $Q_{t o t} = \frac{1}{2} I^{2} R + N α I T_{h} - k A (T_{h} - T_{c})$ ). For comparison, it was assumed that the amount of generated heat is completely transferred to the heating target without loss. The derivation process for each parameter of Peltier heating is represented in Supplementary Information (see note S1). Up to 70 W of input power, with the device parameters of our f-TED, the Peltier heater transfers heat more to the LMPM because the sum of Peltier and Fourier heat is larger than half of the Joule heat when the current is less than a certain value as shown in Figure 2B.

Figure 2D illustrates the temperature change over time with different input powers of the f-TED. The temperature continues to increase until the LMPM reaches near its melting point and remains constant during the phase transition. Since the thermocouple was attached to the bottom side of the polymer mold, the temperature in the phase transition section was measured to be slightly lower than the melting point of LMPM due to the temperature gradient across the supporting polymer. As shown in the graph, the phase transition time decreases as the input power applied to the f-TED increases.

To define the phase transition time, the measured temperature change was differentiated over time. The blue line in Figure 2E shows the first derivative of the temperature versus time graph with input power 1 W. Even if the phase transition of the LMPM has started, the starting point of the section where the temperature is maintained is somewhat delayed because it takes time for the thermal mass of the polymer located between the thermocouple and the LMPM to follow the temperature of the LMPM.⁵ Therefore, we set the point where the slope begins to decrease rapidly as the starting point of the phase transition, and likewise the point when the slope increases from around zero is the ending point of phase transition. The time between two points is set as phase transition time.³⁰ As shown in the Figure 2F, the phase transition time decreases from 66.8 to 14.4 s as the input power increases from 1.0 to 4.0 W. If more power is applied to the f-TED, the phase transition time will be shorter.

Cooling function of the f-TED

The biggest advantage of utilizing f-TED in the variable rigidity module is that it enables active cooling even below 0°C theoretically. Unlike in the heating mode, the cooling efficiency increases up to a specific current defined as optimum current, and then efficiency decreases above the optimum current. This is due to the rise of the heat due to Joule heating and Fourier heating acting in a direction of suppressing cooling.

Figure 3A shows the cooling performance of the variable rigidity module without heat sink. The ambient temperature was set to 22°C for cooling tests. The heating for the solid to liquid phase transition was stopped at the point where the phase transition had just finished (0.3°C temperature rise in 5 s), and Peltier cooling began about 4 s later. In this process, we have to consider the supercooling effect of gallium. Gallium is a representative material that occasionally shows a supercooling effect that starts nucleation for solidification at a lower point than the melting point when cooling, so it prevents transition from liquid to solid at the melting point.³¹ However, if the final reaching temperature after gallium melts is close to the melting point of the gallium, the supercooling effect can be suppressed.³² As shown in the figure, when the input current is increased to 0.7 A, the phase transition time gradually decreases, but when more than 0.7 A of current was applied, the transition time increases again.

FIG. 3.

Cooling performance of variable rigidity module. (A) Change in temperature as a function of time with different current input for the variable rigidity module with no heat sink and (B) with the slot-type heat sink. The air velocity is set to 3 m/s. P1, L, P2, S in the plot indicate the state of gallium in the solid to liquid phase transition, liquid phase, liquid to solid phase transition, and solid phase with no input current, respectively. (C) Phase transition time as a function of input power with and without a heat sink. (D) Equivalent heat transfer coefficient (h_eqv) versus air velocity. (E) Normalized z-directional heat flux. Blue means that the f-TED rejects the heat from the gallium, and Red means that the f-TED induces the heat from the gallium. (F) Total heat rejection of the gallium with the f-TED. Points are measurement results, and dotted lines are the simulation results with the FEM simulator (COMSOL). FEM, finite element method. Color images are available online.

Figure 3B shows the temperature change over time of the variable rigidity module with slot-type heat sink. The phase transition time reduces 50% compared to that of the variable rigidity module without heat sink when no current is applied, and the phase transition time is significantly decreased when the f-TED is operated. In addition, with the heat sink, even though a large current is applied, the heat accumulated on the top side can be efficiently removed by air convection, allowing much larger optimum current to be applied compared to the case without a heat sink. As a result, the time for phase transition can be reduced effectively.

Figure 3C shows the phase transition time with and without the heat sink according to the input current. In case of the variable rigidity module without a heat sink, it takes about 141.2 s to change from the liquid phase to the solid phase, and the phase transition time is reduced by about 26.7% to about 103.5 s when the input current increases to 0.7 A. If the input current exceeds the optimum current of 0.7 A, the phase transition time was slightly increased to 128.1 s. With the slot-type heat sink, the phase transition time without current input is about 73.5 s, which is 47.9% lower than the phase transition time without heat sink. The phase transition time is reduced by 77% to 16.9 s as the input current increases to 2.0 A. Supplementary Table S1 shows the liquid-to-solid phase transition times from previously reported articles. As represented in Supplementary Table S1, the f-TED effectively promotes a liquid-to-solid phase transition without additional bulky cooling equipment.

The active cooling performance of the f-TED varies greatly with several factors such as an environment temperature that f-TED operates at, the efficiency of the heat dissipation, and the input current. In general, when measuring the cooling performance of the conventional TED, the temperature of the hot side is fixed to about 25°C using an external device like a chiller, and the opposite surface is cooled by the Peltier effect with an applied current.³³ In this method, the optimum current with maximum cooling capacity and the induced maximum temperature difference can be determined.

However, in the case of the f-TED in the variable rigidity module, the cold side temperature is fixed and the hot side is exposed to the air, not the fixed temperature condition. In this configuration, the cooling performance of the f-TED is determined by how efficiently the accumulated heat on the hot side is removed, which is influenced by various factors, including the device fill factor, thermal conductivity of filler, thermal properties of the thermoelectric legs, input current, and the heat transfer coefficient between a heat sink and air.

Since these factors affect cooling performance independently, the effect of the parameters on cooling performance was analyzed by heat transfer and thermoelectric model in the FEM simulation tool (COMSOL Multiphysics). The simulated structure, governing equations, and boundary conditions are represented in Supplementary Information (see note S2 and Supplementary Fig. S1).

The heat transfer coefficient on the top surface of the f-TED is one of the most important parameters for cooling performance.³⁴ For a fair comparison between the device with slot-type heat sink and with no heat sink, the heat sink structure is simplified to a flat plate with an equivalent heat transfer coefficient. Teertstra et al. reported the range of heat transfer coefficient of the slotted fin heat sink by deriving the equations of lower and upper bound^35,36 (see Supplementary Information note S3).

Figure 3D shows the equivalent heat transfer coefficient of the devices related to the air velocity. In the case of the f-TED without heat sink, it has the 8.1 W m⁻² K⁻¹ of the heat transfer coefficient with 0.1 m⁻¹ air velocity, and the coefficient increases to 44.3 W m⁻² K⁻¹ with 3 m s⁻¹ of air velocity. With slot-type heat sink, the range of the equivalent heat transfer coefficient increases from the range of 84.4 to 166.8 W m⁻² K⁻¹ to the range of 402.2 to 760.8 W m⁻² K⁻¹ as the air velocity increases to 3 m s⁻¹.

Figure 3E shows the normalized heat rejection from the LMPM according to the change of the input current and the equivalent heat transfer coefficient. In the graph, the darker blue color means that more heat is removed from the LMPM. The area of red color means that the total heat is added to the LMPM from the f-TED because the summation of the Joule heat and Fourier heat transferred to the LMPM dominates the Peltier cooling. As shown in the graph, the optimum input current, which can remove the maximum heat from the LMPM, increases as the equivalent heat transfer coefficient increases because the accumulated heat from Joule and Peltier heating on top surface can be removed effectively.

Figure 3F shows the removed heat from the LMPM as a function of the air velocity. The dotted line represents the results from FEM simulation with the equivalent heat transfer coefficient of the device with slot-type heat sink and with no heat sink. The experimental results were calculated by dividing the total heat needed to change the phase of the LMPM, which is the product of heat of fusion (80.17 J g⁻¹) and total mass (0.79 g) by the phase transition time with and without the heat sink. As shown in the graph, the trends of the experimental and simulation results are in good agreement. When the f-TED was operated at 2.0 A, 4.35 times more heat was removed from the LMPM than when the f-TED was not operated.

Figure 4A shows the temperature change over time without cooling by f-TED. Since the thermocouple was attached to the bottom side of the polymer mold, the temperature difference across the mold increases, which results in the measured temperature during phase transition becoming lower than the actual temperature of the gallium. The measured temperature decreases as the air velocity increases, because the lowered thermal resistance between the bottom surface and air increases the temperature difference across the polymer mold. Even when the f-TED is not in operation, as expected, the phase transition time largely decreases as the air velocity increases but >70 s is required for phase transition even with an air velocity of 3 m s⁻¹.

FIG. 4.

Cooling performance of the f-TED in a liquid-to-solid phase transition region according to air velocity. (A) Temperature versus time with no current input. (B) Temperature versus time with 1.0 A of input current. (C) Phase transition time as a function of the air velocity. Color images are available online.

Figure 4B shows the phase transition under cooling by f-TED with the input current of 1.0 A. The phase transition time is much shorter than that without f-TED cooling as expected. The phase transition times for the air velocity with and without current are compared in Figure 4C. Without input current, the result shows that the phase transition time under forced convection with 3 m s⁻¹ of air velocity decreases by about 51.7% compared to that time under natural convection without input current and decreases by about 39.3% with input current of 1 A. With the same air velocity, when the f-TED operates, the phase transition time decreases by about 66.8% from 73.5 to 24.4 s with 3 m s⁻¹ air velocity and 73.5% from 152.1 to 40.3 s under natural convection. Through the experimental results, it was verified that the time required for stiffness conversion was drastically shortened using the f-TED.

Bending stiffness

To verify the difference in structural stiffness when the gallium is in the fully solid or liquid states, we measured the bending stiffness of the variable rigidity module with and without activation of the f-TED. When the current is applied to the f-TED, the heat induced by f-TED melts gallium and the module is softened. On the contrary, when the direction of the current is switched for cooling, the liquid gallium transitions to the solid state and the module is stiffened.

The experiment setup for bending of the variable rigidity module is shown in Supplementary Figure S2. To minimize gravitational effects at initial state, the longitudinal axis of the sample module was set perpendicular to the ground and one end of it was clamped at a manual stage. A pulling rod made of paper clip was attached on the surface of the sample with a distance L from the clamped end and the rod was connected with a universal test machine (QM100S; QMESYS, Republic of Korea) by an alloy wire (Supplementary Fig. S2B, C). To induce a phase transition of gallium from solid to liquid and vice versa, the electric power cables were connected to the f-TED of the mechanism. For each solid and liquid state, we conducted 8 times bending using a single sample, respectively.

The experiment result was compared with the results from the theoretical model and the FEM simulation. To calculate the theoretical bending stiffness of the module, the analytical beam bending model used assumes linear elastic deformation under small deflection, which follows Euler beam bending equations³⁷ for the case of fully bonded layers. For a prismatic beam, which is simplified representation of the variable rigidity module, fixed at one end and subjected to an external load P at a distance $x = L$ , the bending stiffness of the beam, k_b, can be calculated by $k_{b} = \frac{P}{Δ z} = \frac{3 E I}{L^{3}}$ (2)

where $Δ z$ is deflection, E is the elastic modulus, and I is the second area moment of inertia of the beam.

The area moment of inertia can be calculated by converting the original cross-sectional layers of different materials in the module to equivalent geometries for a single material. As a result, the widths of the rectangles with different materials consisting of the beam cross-section are reconfigured, and they are defined by multiplying each original width and the ratio of the Young's moduli. In our module, there were three materials: Clear Flex 95, gallium, and f-TED. The f-TED was made of rigid thermoelectric semiconductors embedded in a soft polyurethane foam and connected with thin copper circuit (70 μm). The bending deflection of f-TED mainly occurs at the polyurethane foam sheet and the stiffness of thin copper circuit can be negligible; therefore, we assumed that the structure of f-TED is simplified as the polyurethane foam sheet. Therefore, we used the Young's modulus of polyurethane foam as that of f-TED in the analytical model.

The original and modified cross-section of the module for the analytical modeling is illustrated in Figure 5A and B, respectively. If the parts of gallium and polyurethane foam are transformed into Clear Flex 95, the equivalent widths of gallium and polyurethane foam parts considering the geometries in Figure 5B are given as,

FIG. 5.

Cross-section of variable rigidity mechanism for analytical modeling and comparison of bending response for experimental, analytical, and FEM analysis. (A) Original cross-section. (B) Modified cross-section with equivalent widths. (C) Load-displacement curves from the experiment for bending of the variable rigidity module. Black square line is for the solid state, and red square line is for the liquid state. Red dotted line is the estimated data extrapolated from curve fitting to the liquid state line. Black dashed box is the region for linearly elastic deformation. (D) Enlarged graph of black dashed box in (C). Symbol lines, dashed lines, and dash dotted lines are from the experiment, analytical modeling, and FEM simulation, respectively. The gray area in (C, D) indicates standard deviation. (E, F) are graphical images of FEM simulation result under 2 mm bending deformation in SolidWorks for solid state and liquid state, respectively. Color images are available online.

a_{2} = n \times \frac{E_{G}}{E_{C}} \times b + (n - 1) \times c + 2 d

(3)

a_{3} = \frac{E_{P}}{E_{C}} \times a

(4)

where n is the number of gallium reinforcements, and $E_{C}$ , $E_{G}$ , $E_{P}$ are Young's modulus of Clear Flex 95, gallium, and polyurethane foam, respectively.

The calculation of the area moment of inertia can proceed using the centroid of the cross-section and the parallel axis theorem.³⁷ At first, the centroid of the cross-section of the beam with the n number of gallium reinforcements is, $\bar{z} = \frac{\sum_{i} z_{i} A_{i}}{\sum_{i} A_{i}}$ (5)

$\begin{matrix} A_{1} & = a_{1} \times (t_{1} - t_{2}) \\ A_{2} & = a_{2} \times t_{2} \\ A_{3} & = a_{3} \times t_{3} \end{matrix}$ (6)

where A_i is the cross-section area of the ith rectangle, and $\bar{z}$ and z_i are the height of the centroid and the distance to the center of the rectangle from the reference line of the beam, respectively. Based on the centroid, the area moment of inertia is given as, $I = \sum_{i} \{I_{i} + A_{i} {(z_{i} - \bar{z})}^{2}\}$ (8)

where $I_{i} = a_{i} \times t_{i}^{3} / 12$ is the area moment of inertia for the ith rectangular section with its width a_i and thickness t_i. Using this value, we can obtain the stiffness of variable rigidity module at solid state. For the liquid state, the gallium reinforcement can be considered as an empty space because of the negligible stiffness of melting gallium according to the reference.¹⁸ Therefore, the bending stiffness at liquid state was calculated by considering only two materials except gallium.

For the FEM simulation, we performed a nonlinear static analysis in SolidWorks using the Newton–Raphson method and assuming perfect bonding contact and gravity. The Young's moduli of Clear Flex 95 (18.13 MPa) and polyurethane foam (0.53 MPa) were obtained from tensile test according to ASTM D412-A (Supplementary Fig. S3), and the Young's modulus of gallium (9.8 GPa) was used from the database on Gallium of AZO Materials (AZoM).³⁸ All geometries of cross-section of sample were measured by a surface profiler from Keyence as illustrated in Supplementary Figure S4.

The results from experiment, analytical modeling, and FEM simulation are shown in Figure 5C and D. In Figure 5C, the slope of load-displacement curve at solid state in experiment (black square symbol line) is changed at 2 mm displacement so that the deflection under 2 mm can be considered as elastic region (black dashed box) of variable rigidity module. In the liquid state, the slope over 2 mm displacement shows almost linear dependency because the stiffness of the structure was affected by elastomeric materials and the whole deformation still remains in the elastic region. However, due to the limitations on the load capacity of the force sensor in the tensile machine, extrapolation was required to estimate the load response for displacements below 2 mm (Fig. 5C, red dot, Liquid-EXP_est). To compare the experimental results with predictions from analytical modeling and FEM simulation, we focused on the linear elastic deformation regime. As shown in Figure 5D, the data plots from analytical modeling and FEM simulation show a good agreement with the experiment curves.

Figure 5E and F shows stress distribution of the structure under 2 mm bending deflection. In the solid state, gallium reinforcements supported most of the bending stress when in the solid state, while the Clear Flex 95 supported stress when the gallium was in the liquid state. The stiffness, which is the slope of the plot in the graph, is summarized in Table 2. The bending stiffness for solid state is 0.28 N mm⁻¹, which is 14.7 times higher compared with 0.019 N mm⁻¹, which is the bending stiffness in the liquid state.

Table 2.

Stiffness of Variable Rigidity Mechanism at Solid and Liquid State Under Linear Elastic Deformation

	Solid state [N/mm]	Liquid state [N/mm]
Experiment^a	0.28 ± 0.03	0.019 ± 0.003
Analytical modeling	0.30 (7.1%)	0.015 (−21%)
FEM simulation	0.29 (3.6%)	0.017 (−11%)

The experimental values indicate the mean and standard deviation.

FEM, finite element method.

Conclusion and Future Work

The use of the f-TED in variable rigidity module solves the conventional limitation of time-consuming cooling for the phase transition and also greatly increases the possibility that the module can be used in practical applications requiring real-time operation. The variable rigidity module, including the f-TED, is manufactured based on flexible materials that make it possible to adapt to any required shape for target applications. The gallium reinforcement manufacturing method using the 3D printed mold can be used to produce customized patterns for the gallium reinforcement that are suited for a desired bending response. The difference in bending stiffness between the soft and rigid states is one of the most important factors in application selection, and the range of stiffness tuning can also be adjusted to the desired range by adjusting the configuration of the gallium reinforcement.

Furthermore, the presented cooling solution using the f-TED can be applied not only to control for discrete rigidity states of solid and liquid in LMPM but also continuous rigidity tuning of smart materials actuated by thermal stimulation. For example, the bidirectional active temperature control from the f-TED to the device using shape memory polymer could enable us to obtain multilevel stiffness according to fine-tune control of temperature, rather than a binary transition between rigid and soft states.

The remaining issue for practical use is the need for a more effective heat dissipation design that can enhance the thermoelectric performance during the heating and cooling process. To solve this issue, further research is underway to develop a completely new type of a flexible heat sink with high thermal conductivity. In addition, the current variable rigidity module is flexible but not stretchable, so there are limitations in implementing structural deformation for multicurvature. To solve the problem, it is considered to use stretchable electrodes in the current structure.³⁹ If these issues are solved, the improved variable rigidity module could be used to further advance the use of stiffness tuning for applications in wearable electronics, soft robotics, and medical devices.

Footnotes

Author Disclosure Statement

No competing financial interests exist.

Funding Information

This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MSIP) (NRF-2015R1A5A1036133) and by the Korea Institute of Science and Technology intramural grants (2E30260).

Supplementary Material

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