Improving the Efficiency of Soft Phase-Change Actuators Using Thermodynamic Analysis

Abstract

Actuation is a key challenge for the field of soft robotics. One method of actuation, thermally driven liquid-to-vapor phase change heat engines, is particularly compelling due to its high forces, large strokes, and relative simplicity. However, this form of actuation suffers from a very low efficiency, making its impact for practical applications limited. Here we apply thermodynamic analysis to these phase-change actuators to identify major inefficiencies and offer three key insights that soft roboticists can leverage to improve efficiency: (1) maximize the ratio of input power to heat loss, (2) operate at an intermediate temperature, and (3) maximize volumetric expansion. We confirm the validity of these insights via benchtop experiments and show efficiencies nearly two orders of magnitude higher than previously reported. We demonstrate the usefulness of these insights by applying them to the design and construction of a compliant roller powered directly by sunlight and capable of rolling every 16 s. Our results guide the design of more efficient phase-change actuators for soft robots and more generally, demonstrate the potential of applying thermodynamic analysis to improve the efficiency of soft actuators.

Keywords

phase change actuator thermo-pneumatic soft heat engine thermoelectric pneumatic actuator photothermal soft actuator thermopneumatic actuator

Introduction

A major challenge of soft robotics is actuation. Researchers have explored a wide range of transduction technologies, which create movement from the energy available in compressed air, electricity, heat, light, or magnetic fields, for example.¹ One particularly compelling actuator is the soft, thermally activated phase-change actuators^2–31—we refer to these as STAPAs for convenience. We focus on STAPAs because of their low complexity and promising ability to achieve large strokes and high forces.² STAPAs typically comprise a soft constraint (e.g., a thin plastic or silicone case), a low-boiling-point fluid (e.g., ethanol or Novec 7000), and a heat source (e.g., nickel chromium wire and a voltage source). For untethered operation, wireless liquid-to-vapor phase change has been demonstrated via magnetic induction^2,4,28 and light stimulus.^6,7,23,26

For both tethered and untethered STAPAs, however, little attention has been given to their efficiency. Indeed, most literature on STAPAs neither focuses on nor reports the thermal efficiency of a complete work cycle (i.e., expansion and compression),^8–31 favoring other methods of efficiency reporting instead (e.g., efficiency of expansion alone).^2–4,6,7 Even when work cycle efficiency is accurately reported, the highest reported value in the literature is presently ∼0.0018%.⁵ For STAPA-driven soft robots to become a practically relevant technology, thermodynamic analysis that offers design guidelines for improving efficiency is needed. The unique combination of constraints inherent to small-scale soft heat engines—i.e., relatively low temperatures and pressures of operation, construction from soft polymeric materials, and cyclic processes in a deformable body—is not adequately addressed in the literature.

Herein, we apply thermodynamic theory to this new class of heat engine (Fig. 1A) to identify dominant sources of energy loss and illuminate key design principles for minimizing them (Fig. 1B). Experimental application of these theoretical insights yields actuators with an efficiency of up to 0.15% (Fig. 1C), which is nearly two orders of magnitude higher than previously reported.⁵ In turn, we demonstrate through a proof-of-concept compliant roller that application of these insights can enable a STAPA-powered soft robot to locomote using only the modest flux of sunlight (∼870 W m⁻²) (Fig. 1D). Our work offers critical insights for improving the efficiency of soft heat engines and demonstrates how this solution might inform the next generation of untethered, mobile, and adaptable soft robots. More generally, our work demonstrates the power of applying thermodynamic analysis to soft robotics, which, outside of sparse examples (e.g.,^32,33) is rare in the field.

Fig. 1.

Our thermodynamic analysis and the key insights led to significant efficiency improvements that enable new possibilities, such as our compliant phase-change solar-driven roller. (A) The key characteristics of soft, thermally activated, phase-change actuators (STAPAs) that distinguish them from large, rigid, industrial phase-change processes are: small geometries (<50 × 50 × 50 cm), low temperatures (<100 °C), low pressures (<50 kPa), large volume changes, and heat-loss-dominated performance. (B) Our analysis identifies three key insights (left to right): Increase ratio of power input to heat loss, operate at intermediate temperature, and maximize expansion. (C) Incorporating these insights into a soft actuator, we find an efficiency improvement over prior work (*see ref.⁵) by nearly two orders of magnitude. (D, E) With these actuators, we demonstrate a compliant roller capable of actuating every 16 s, using only the flux of sunlight.

Results

Theory: Modeling inefficiencies

In this section, we use simplified models to show that the three primary thermal sources of loss for STAPAs are small operating temperature ranges, high heat losses, and non-ideal cycle designs. We keep the discussion at a high level, because STAPA designs are varied. Figure 2A depicts our model of a STAPA. Additionally, we focus our discussion on optimizing for efficiency. Other metrics such as speed, while important, are outside the scope of this analysis.

Fig. 2.

Visualization of model and key results. (A) Representation of the STAPA model used in analysis. $P_{L}$ is the pressure exerted by the load, against which the actuator does useful work. The STAPA also does non-useful work to deform its walls and the atmosphere; this is represented by the spring and damper elements. The minimum and maximum specific volumes are $v_{2}$ and $v_{3}$ , respectively. The working fluid has a temperature, $T$ , and pressure, $P$ . Heat is inputted at a rate of ${\dot{q}}_{i n}$ . The ambient has a temperature of $T_{\infty}$ . (B) The rectangular cycle assumed by the model for a STAPA, drawn in pressure-volume (PV) space. A representative “Realistic PV curve” is drawn in the dotted-dashed line. (C) Modeled efficiency of rectangle cycle with heat transfer as the max specific volume and temperature are varied. We input to the model representative experimental values. (D) Modeled allocation of energy use, again with representative experimental values.

The first major limiter of STAPA efficiency is the achievable operating temperature range. For a heat engine operating between a high temperature of $T_{h}$ and a low temperature of $T_{c}$ , the Carnot efficiency, $η_{c}$ , is the upper bound of efficiency and is given by $η_{c} = 1 - T_{c} / T_{h}$ . Thus, according to this idealized equation, $T_{h}$ should be maximized and $T_{c}$ should be minimized.

At first glance, it seems that the melting or degradation temperature of the materials limits $T_{h}$ ; however, low burst pressures may be more of a limiter for some designs. The near exponential rise of vapor pressure with respect to temperature (i.e., the Clausius-Clapeyron relation)³⁴ can cause STAPAs to reach burst pressures before reaching melting or degradation temperatures. For example, consider a STAPA made of silicone (a common soft robotic material³⁵) that bursts at 50 kPa (a typical value for elastomeric actuators³⁶) and uses Novec 7000 as the working fluid. Novec 7000 reaches that vapor pressure at approximately 45.7 °C, which is only approximately 11.5 °C above boiling point and well below the material’s maximum serviceable temperature of about 220°C.³⁷ Thus, at best (i.e., at Carnot efficiency), a heat engine operating between 20 and 45.7 °C has an efficiency of 8.1%. In contrast, modern steam turbines, industrial heat engines that uses liquid-to-vapor phase change, operate at temperatures over 600 °C and pressures over 30 MPa.³⁴

Additionally, the boiling point of the working fluid effectively sets the lower bound of $T_{c}$ . This is because STAPAs typically operate in expansion. Unless compensated for (e.g., by an internal spring), the temperature inside of the STAPA must at least reach the boiling point, $T_{b}$ , of the working fluid to expand against the atmosphere. Thus, for an uncompensated STAPA operating expansionally, the maximum efficiency, $η_{x}$ , is $η_{x} = 1 - T_{b} / T_{h}$ . Returning to the example STAPA from above, including the boiling point limit on the low temperature, the best-case efficiency drops by more than half to approximately 3.7%. In contrast, the rigid design of steam turbines allows for operations below ambient pressures, resulting in increased efficiencies.³⁴

The second major limiter of STAPA efficiency is heat loss during the heating process. Given a heat input rate of ${\dot{q}}_{i n}$ and heat loss coefficient of $h$ , the efficiency considering heat loss can be written as the product of a heat loss prefactor and $η_{x}$ (see Supplementary Text S1):

η_{heat} = (1 - \frac{1}{{\dot{q}}_{i n} / [h (T_{h} - T_{\infty})]}) η_{x}

(1)

T_{\infty}

is the ambient temperature. This relation shows that a large heat input rate

{\dot{q}}_{i n}

and a small heat loss coefficient

h

are ideal. However, the thin-walled, flexible nature of STAPAs tend to make the heat loss coefficient large, and at the same time, the maximum practical heat input rate is often limited by the power supply.

The third major limiter for STAPA efficiency is the shape of the thermodynamic cycle. While there are many possible cycle shapes, we propose a rectangular-shaped cycle in pressure-volume space as an approximation of a STAPA cycle (Fig. 2B), because it is straightforward to implement in comparison to the Rankine cycle (i.e., the idealization of a steam engine³⁴) with a piston-like setup; simply lifting and lowering weights can achieve such a cycle shape. Furthermore, a rectangular-shaped cycle maximizes the work done per cycle given volume and pressure constraints. Maximizing work per cycle is a reasonable metric to optimize for many engineering applications.

From state 1–>2, the STAPA undergoes isochoric (constant volume) heating to build pressure to overcome the ambient atmosphere, the actuator resistance, and the load. From step 2–>3, the STAPA undergoes isobaric (constant pressure) expansion and does work against the load. For single component phase change, this is an isothermal (constant temperature process), which is thermodynamically favorable. From step 3–>4, the STAPA undergoes isochoric cooling as the pressure drops. From step 4–<1, the STAPA undergoes isobaric compression as the atmosphere and actuator structure compresses it. The efficiency of this rectangular cycle within the vapor dome and without heat loss (see Supplementary Text S2) is given by:

η_{rect} = \frac{(P_{2} - P_{1}) (v_{3} - v_{2})}{h_{3} - h_{2} + {(u}_{2} - u_{1})}

(2)

Here, $P$ is the pressure, $v$ is the specific volume, $h$ is specific enthalpy, and $u$ is specific internal energy. Subscripts correspond to thermodynamic states in the cycle depicted in (Fig. 2B). The isochoric heating (i.e., the $u_{2} - u_{1}$ term) that is involved in this rectangular cycle (but not in the Carnot cycle) is a major source of inefficiency, because the energy input for that process goes only to raising the temperature of the fluid without producing work.

Note that the real cycle differs from the proposed idealization (Fig. 2B) due to factors such as wall material elasticity and unwrinkling. While these deviations obviously decrease the accuracy of the model, they are not unusual. For example, the measured pressure volume relations of a combustion engine in³⁸ differ from that of an ideal Otto cycle. Nonetheless, these models still broadly explain the observed trends.

Heat loss can significantly impact performance. It can be shown (see Supplementary Text S3 and S4) that with heat loss accounted for, the full thermal efficiency can be approximated by:

η_{heat, rect} ≅ (1 - \frac{1}{{\dot{q}}_{i n} / [h (T_{h} - T_{\infty})]}) \frac{(P_{2} - P_{1}) (v_{3} - v_{2})}{h_{3} - h_{2} + (u_{2} - u_{1})}

(3)

We note that we chose to use this approximation here because of the intuitive insights its terms offer. We use Eq. S18 for the plots in Figure 3 because it is more accurate, but its terms offer less insight. Figure 2C plots the numerically evaluated $η_{heat, rect}$ as the maximum operating temperature and specific volume is varied.

Fig. 3.

Three insights for improving efficiency: theory and experiment. (A) A typical experimentally measured pressure-volume curve that is approximately rectangular. This curve comes from the point marked with a cross in B, with efficiency of 0.15%. (B) The first insight from the model (shown as dashed line) suggests increasing the ratio of input power to heat loss rate, especially at low values of this ratio. Experimental data (circle markers) confirm this trend. (C) The second insight from the model suggests that an intermediate value of the maximum cycle temperature should be used. Data confirms this trend. (D) The third model insight suggests increasing the expansion volume of the actuator, again confirmed by experimental data. The cross in C shows the point that the plots in Supplementary Figure S9 reference. Dashed lines represent trends, and not specific fits to the model.

Finally, we note that beyond thermal losses, the STAPA also faces losses from factors such as friction and hysteresis. Not all the pressure-volume work goes into the useful work against the load. We term the ratio between useful work and pressure-volume work the mechanical efficiency, $η_{m}$ . This can be empirically determined. We lump all other unmodeled losses and nonidealities into a single scaling factor, $η_{f}$ ; such losses include heating of actuator walls, deviations from the ideal cycle shape, and non-idealities of the working fluids (e.g., impurities). The total efficiency (thermal and non-thermal), $η$ , is given by:

η = η_{f} η_{m} η_{heat, rect}

(4)

Figure 2D shows the allocation of energy in a typical STAPA.

Three insights for improving thermal efficiency: Theory verified by experiments

By analyzing our full thermal model (Eq. 3) we suggest three methods for improving efficiency: (1) increase the ratio of power input to heat loss beyond a threshold, (2) operate at an intermediate temperature, and (3) maximize fluid vaporization. To verify these theoretical insights, we tested a bellows-like STAPA under a variety of thermal and mechanical conditions using a purpose-built testing apparatus that created an approximately rectangular cycle shape (Fig. 3A) (see Materials & Methods). After accounting for the mechanical efficiency ( $η_{m}$ ) and scaling by a constant fitted scaling factor ( $η_{f}$ ), the data confirms the trends of the three insights predicted by model. See Materials & Methods for further details.

The first insight suggests increasing the ratio of input power to heat loss rate, ${\dot{q}}_{i n} / [h (T_{h} - T_{\infty})]$ , especially for relatively low values of this ratio, in order to increase efficiency (Eq. 3). Increasing input power can be done by increasing the electrical input power to a Joule heater or solar flux for a light-harvesting STAPA. As long as the rate of output work can increase, increasing ${\dot{q}}_{i n}$ should not affect $T_{h}$ . This is a reasonable assumption of STAPAs, because they expand until mechanical equilibrium is reached.

Decreasing heat loss rate for a given temperature range can be accomplished by increasing insulation or decreasing the exposed area; however, doing so might be challenging. For example, finding suitable materials with significantly lower thermal conductivities is difficult, because polymeric materials already have low thermal conductivities.³⁹ Additionally, increasing insulation by increasing wall thickness or adding an insulating layer might decrease compliance. Furthermore, increased insulation might increase cycle times due to the reduced cooling rates. The development of new materials or using nonlinear thermal elements⁴⁰ might help address some of these challenges. The predicted effect of increasing this ratio on efficiency is shown as a curve in Figure 3B, and confirmed experimentally by varying the average supplied power to an actuator while maintaining the average heat loss rate as it completed a work cycle. Both theory and experiment show that for relatively low ratios of input power to heat loss rate, small increases in the ratio result in very large improvements in efficiency. In contrast, for high ratios, efficiency plateaus. See Supplementary Figures S5, S6 and S7 for additional visualizations of experimental results. Note that Supplementary Figure S5D shows that $T_{h}$ does not systematically increase with increasing ${\dot{q}}_{i n}$ .

The second insight suggests operating at an intermediate value for the high temperature ( $T_{h})$ to maximize efficiency. This is because heat loss increases with increasing temperature (left-hand term of Eq. 3), but the efficiency of the ideal rectangular cycle also increases with increasing temperature (right-hand term of Eq. 3). This is illustrated by the parabolic-shaped curve in Figure 3C. We experimentally tested our second insight by varying the applied load to the actuator while maintaining constant power input, which ultimately varies the high temperature of the actuator during a work cycle, and we confirm a peak efficiency at an intermediate value of the high temperature (∼43–44°C). A potential explanation for the deviation between the experimental data and the model at higher temperatures is an underestimation of the heat loss coefficient ( $h$ ), which determines the maximum operable temperature. For the model, we used a fixed $h$ value corresponding to the half-open state. See Supplementary Figures S8 and Figure S9 for additional visualizations of experimental results.

However, we note that this insight could be challenging to implement, because the optimal temperature depends on $T_{\infty}$ . If $T_{\infty}$ is too low, then actuation might not be possible because the heat loss is too high. In contrast, if $T_{\infty}$ is too high, then the cycle might not fully return to the starting state (i.e., $T_{\infty} > T_{b})$ , or the cycles might be too long due to the slower cooling process. Using working fluids with a lower or higher boiling point might help if one expects $T_{\infty}$ to remain, respectively, low or high.

The third insight suggests maximizing a STAPA’s expansion ratio, or the ratio of final to initial volume. This insight arises solely from the ideal rectangular cycle term in Eq. 3. Here, increasing fluid vaporization (i.e., increasing the $v_{3} - v_{2}$ term) increases the amount of output work done during the cycle. This improves efficiency because a fixed amount of energy is required to heat the fluid to the boiling point (i.e., the $u_{2} - u_{1}$ term), and this fixed input cost is only amortized when output work is large. This is illustrated by the curve in Figure 3D. Thus, a STAPA should always be designed to vaporize as much of its fluid as possible and produce a large stroke. To confirm experimentally, we used the cycle that corresponds to the high temperature of approximately 44 °C found above (Fig. 3C) and calculated the cycle efficiency at different expansion volumes. Note that for systems with strongly volume-dependent heat loss, increasing expansion may not result in a monotonic increase in efficiency.

From insight to implementation: STAPA-driven compliant roller

To demonstrate the impact of the above thermodynamic insights, we sought to design a STAPA-driven system that could only complete its goal task if the efficiency of STAPAs were improved beyond the state of the art. Accordingly, we targeted locomotion of a hexagonal compliant roller with rolling transitions occurring at least every 40 s, powered only by the modest flux of sunlight (i.e., ≤ 900 W m⁻²). We chose this speed because for the size and shape of the roller and the input power of the sun, it requires an efficiency at least an order of magnitude greater than previous work⁵ (see Supplementary Text S17). Here, we introduce a STAPA-driven compliant roller capable of locomotion exceeding this target, completing rolling transitions every 16 s. The sections that follow detail thermodynamically informed decisions made in its design, along with more general design details.

Working principle

The roller has six sides, each with an actuator (Fig. 4A), like previous hexagonal rollers (e.g., powered by a heated surface⁴¹ or powered pneumatically⁴²) Each STAPA actuator is sandwiched between a hinged leg and the roller body (Fig. 4B). To complete a flip (illustrated in Fig. 4C), sunlight heats the STAPA, pressing the leg into the ground.

Fig. 4.

Roller working principles and thermodynamically informed design. (A) We present a six-sided compliant STAPA-driven roller capable of locomotion using only sunlight. (B) To perform a rolling transition, sunlight is directed onto a STAPA pouch that contains an absorber and low-boiling-point fluid. See Supplementary Fig. S15 for schematic of actuation cycle. (C) After the fluid is heated to its boiling point, sufficient pressure is generated to expand the STAPA pouch, press the leg into the ground, and flip the roller to its next side. Panels show only the rolling phase, not the heating phase prior to it. (D) A labeled 3D rendering illustrates key components of one of the roller sides. To maximize input power from the sun flux while minimizing heat loss rate (Insight 1), we maximize the size of a concentrating Fresnel lens, filling the entire side of the roller, while not increasing the size of the STAPA (larger STAPAs have larger heat loss). Further, foam is placed under the STAPA to insulate it. The second insight suggests the size for the STAPA to achieve an optimal maximum cycle temperature. The third insight suggests the volume of working fluid incorporated.

Applying the three key insights

With the three insights as guidelines, we made the following decisions for the design of the roller (Fig. 4D). The first insight states that the ratio of power input to heat loss rate should be increased (Fig. 3B). However, this is not straightforward for solar-powered STAPAs: for a given light flux, increasing STAPA surface area will increase power input, but at the same time, will increase heat loss rate (due to more area for convective cooling). We solved this paradox by decoupling the light collection area from the STAPA pouch area by using a concentrating lens. While minimizing the pouch size, we maximized the size of the collector (and power input), filling most of each side of the roller with a Fresnel lens. Further, we added magnetic latches to the STAPAs to keep the pouches closed during heating. This both minimizes heat loss by keeping the exposed area small and maximizes input power by maintaining the sun’s alignment on the pouch. Finally, we incorporated foam insulation under the STAPAs.

The second insight states that the high temperature should be set to an intermediate value. For our outdoor implementation, the exact value of this optimal high temperature varies continually with changing environmental conditions. This means that there is not a specific temperature to design for, as would be the case in more controlled environments. Instead, we designed the combination of pouch size, magnet strength, and actuation moment arm (for the given roller mass) to create a high temperature that was always above the no-motion temperature (far left-hand point in Fig. 3C) and below the equilibrium temperature (far right-hand point in Fig. 3C).

The third insight states that larger specific volume changes result in higher efficiencies (Fig. 3D.). For a given final volume, a larger volume change can be realized by decreasing the initial volume. As such, we identified and used the minimum liquid volume required to reliably complete a rolling transition.

Further details regarding design decisions are included in Materials and Methods: Roller Design and Fabrication.

Compliance

We also designed load-dependent compliance into the roller frame. Under its body weight, the combination of composite structures (carbon fiber beams, fiberglass plates) and dynamic hinges (Fig. 4D) render the roller functionally stiff (see Supplementary Fig. S17), enabling robust locomotion. However, under higher compressive loads, these hinges can either buckle flat (i.e., ∼0°) or open (i.e., 180°) to accommodate the differing angular deflections in a compressed hexagon (see Supplementary Fig. S17). We demonstrated that the roller can be compressed nearly flat, after which it immediately springs back to its original shape (see Supplementary Text S15).

Roller locomotion

We demonstrated our roller completing more than a full rotation using only sunlight (Fig. 5, Supplementary Video S1). A full rotation consists of six rolling transitions—requiring that each of the six legs open and close sequentially because of vaporization and condensation in the irradiated and non-irradiated states, respectively. Thereafter, the roller was allowed to locomote autonomously and completed seven flips at an average rate of one flip per 15.7 s using ∼870 W m⁻² of sunlight.

Fig. 5.

Demonstration of solar-powered locomotion. The roller was placed outdoors in Santa Barbara, CA and allowed to move autonomously as it harvested sunlight. This composite figure illustrates the roller completing seven rolling transitions, with an average time per transition of 16 s, while using approximately 870 W m⁻² of sunlight (see Supplementary Video S1).

Furthermore, because of the built-in compliance, the roller can endure substantial mechanical loading and continue its operation. We demonstrate this (1) by manually manipulating the roller mid-cycle and illustrating its ability to continue operation (Supplementary Video S2), (2) by kicking the roller down a set of stairs, after which the roller is able to continue rolling (Supplementary Video S3), and finally (3) by demonstrating a drop from 92 cm that the roller is able to sustain and then roll away from (Supplementary Video S4).

Discussion

Herein, we offered a model describing the dominant thermal inefficiencies of soft phase-change heat engines, identified accessible methods for improving thermodynamic efficiency, validated these methods through experiment, and finally demonstrated the utility of these insights via a sun-powered compliant roller. In developing the theory, several simplifying assumptions had to be made (see Materials and Methods, Model). These assumptions mean that the model does not predict exact values for efficiencies, but rather offers insights into trends. Future work might include more detailed theoretical models that include a higher number of parameters that more accurately predict the experimental system. However, the congruence of trends predicted by theory and observed in experiment confirm the validity of the three insights we propose for minimizing losses in soft phase-change heat engines. Additionally, future work might explore ways to optimize for both speed and efficiency.

We also demonstrated the utility of these insights via a compliant roller capable of locomotion using only the modest flux of sunlight. The roller should be considered a proof-of-concept rather than a ready-to-deploy robot. At present, the roller includes no electrical sensors, uses no feedback, and only operates when the sun is within a 10° window of elevation angle and aligned with the direction of travel. As such, long-range locomotion is not currently possible but could be a goal of future work. However, if consistent illumination could be maintained, we hypothesize that the roller would roll slightly faster (and consequently more efficiently) after the initial cycles, because the temperature of each pouch would not be starting from ambient like it did in Supplementary Video S1, so less energy would be spent raising the pouches to the working temperatures.

Instead of a ready-to-deploy robot, this device is a demonstration that our proposed insights can enable STAPAs to actuate in the real-world with performance levels far beyond what was previously possible. While still relatively slow (one actuation per 16 s), the STAPAs in our roller showed efficiency improvements of over 30 times compared to previously published results (0.056% versus ∼0.0018%⁵) which, given a fixed input power from the sun, corresponds to improvements in available output power, or speed, of a similar amount. We note that our benchtop results were another factor of 2.7 higher than these outdoor tests (0.15%), possibly due to an overestimation of the sunlight flux entering a pouch, compliance and hysteresis in the roller structure, and thermal losses due to wind. As such, the benchtop results show a nearly two-orders-of-magnitude improvement in efficiency over previous results.

For comparison, utility scale steam engines achieve efficiencies of about 33%,⁴³ but they operate with much higher temperatures and heat fluxes. A potentially fairer comparison would be with thermoelectric generators (TEG). Although they use the thermoelectric effect instead of liquid-to-vapor phase change, they, like STAPAs, are used for low temperature power generation.⁴⁴ A typical TEG, operating between 20 and 38 °C (i.e., temperatures similar to that of the maximum efficiency test), has a maximum possible efficiency of about 0.78% (see Supplementary Text S6).

We believe that with proper design, it is possible to achieve levels of efficiency similar to TEGs. To do so requires developing better insulation techniques either through new mechanisms or materials. Ideally, the new materials would have lower thermal conductivities, high compliance, low hysteresis, and low gas permeability. Such a material will not only improve STAPA performance, but could be valuable for other applications such as clothing and insulation.

While there is still work to reach TEG-level performance, insights from our analysis, more immediately, have the potential to catalyze a new generation of soft robots that interface with the human body as well as unpredictable natural environments. For example, minimizing losses in active prostheses and smart textiles that leverage soft phase-change heat engines (e.g., see ref.19) in turn minimizes the size of the requisite energy sources (e.g., batteries) and can extend operation times. In another example, soft robots with minimized losses have the potential to increase their power output (e.g., lifting) or can operate for longer time periods (e.g., locomotion). Furthermore, two of our insights broadly apply to soft heat engines (e.g., liquid crystal elastomers) in general. The first and second insights can be derived from a Carnot-based model (Eq. S2).

Beyond the implications on STAPAs, our work shows more generally the power of applying rigorous thermodynamic analysis to the field of soft robotics. Our work may inspire a whole range of future studies for soft robotic actuators and systems much different than the one we studied herein, resulting in important advances in efficiency and thus applicability throughout the field of soft robotics.

Materials and Methods

Actuator fabrication

Bellows-like pouches were created by heat sealing Mylar film (WeVac, Vacuum Sealer Bags) with a disk of black conductive fabric (EonTex, NW170-PI-20) and working fluid (3M, Novec 7000) inside (see Supplementary Text S8). Mylar was chosen because it is transparent to visible wavelengths. Black conductive fabric was chosen to serve as a flexible/soft heater with high wetted surface area (see Supplementary Text S7). Novec 7000 was chosen for its low boiling point at ambient conditions (i.e., 34 °C at 1 atm), low toxicity and broad compatibility with soft polymer materials.

Actuator testing setup

A custom testing apparatus was fabricated to quantify the efficiency of a soft pouch under a variety of loading conditions (Fig. 6). As the working fluid is vaporized by the heater, the STAPA does pressure-volume work against a loaded hinge whose mechanical load can be programmed via the removal/addition of masses. Different loading conditions (i.e., expansion and compression curves observed in the pressure-angle cycles) could be programmed via a combination of an “offset mass” and a “constant-torque mass” (Fig. 6A, see Supplementary Text S9). To create work cycles in which positive work is done, the hinge is loaded with more mass during the expansion than during the compression step.

Fig. 6.

Details of the experimental setup. (A) A schematic illustrating the testing setup mechanism. A soft pouch containing a conductive fabric heater and low-boiling point fluid (Novec 7000) does pressure-volume work against a loaded hinge. Voltage is supplied to the heater to incite liquid–vapor phase change, and thus pressure-volume work against the hinge. The combined load of the offset ( $m_{1}$ ) and constant-torque ( $m_{2}$ ) masses dictates the pressure-volume conditions of the loaded pouch. Varying the magnitude of the masses prior to expansion or compression steps of a work cycle results in a broad variety of curves. Pressure, temperature, power, and angle are measured using a pressure sensor, thermocouples, a power meter, and a potentiometer coupled to hinge shaft, respectively. (B) A 3D rendering of the setup. (C) An isometric rendering of the setup.

Experimental protocol

To characterize the thermodynamic state of our system, we measure pressure, hinge deflection angle, electrical energy supplied, and temperature (Fig. 6B,C). Pressure quantifies the internal vapor pressure of the pouch. We used the angle data to estimate volume (see Supplementary Text S10). Electrical energy input is monitored and quantified using a power sensor. Unless otherwise noted, reports of efficiency are quantified by calculating the ratio of useful mechanical work output to heat input. The useful work is calculated by the net weight lifted (see Supplementary Text S9). The pressure-volume work is estimated by computing the integral of the pressure-volume relationship. Mechanical efficiency is computed by dividing useful mechanical work by the pressure-volume work.

Experimentally, we found that the STAPA recompresses at a pressure about 0 kPa gauge pressure (see Supplementary Figures S5A and S8A) across all sets of tests. We estimate the heat required for a cycle by integrating the power once the gauge pressure exceeds 0 kPa. We believe that this is a slight overestimate of heat input (and thus, underestimate of the efficiency) because the temperature at a similar pressure is higher during the recompression phase (see Supplementary Figures S5 and Figure S8). We use a pressure threshold rather than a temperature threshold because the air content inside the STAPA varied, which changes the pressure-temperature relationship.

Each test involves the completion of one work cycle.

The temperature inside of the pouch describes the state of the working fluid and the temperature outside provides information about thermal gradients. Unless otherwise noted, all experiments were performed in a temperature-controlled room at ∼20 ± 0.25 °C. A lumped convection coefficient of the setup was calculated and incorporated into our theoretical calculations (see Supplementary Text S11).

Theory-experiment comparison

To calculate the theoretical prediction, we used nine experimental parameters: initial specific volume ( $v_{1})$ , final specific volume ( $v_{3}$ ), cold temperature ( $T_{c})$ , maximum temperature ( $T_{h})$ , ambient temperature ( $T_{\infty})$ , heat loss coefficient ( $h$ ), heat input rate ( ${\dot{q}}_{i n}$ ), mechanical efficiency ( $η_{m}$ ), and $η_{f}$ . The mass of Novec 7000, $m$ , was measured to convert extensive to intensive quantities.

The first eight experimental parameters are used to compute, $η_{heat, rect}$ (Eq. S18). The parameters vary for each set of tests. The mass of Novec 7000 is the amount of fluid used in the test. To accommodate for the initial volume uncertainty (e.g., due to wrinkling) we set the initial specific volume to 7.5 × 10⁻⁴ m³/kg, approximately the saturated liquid specific volume for Novec 7000 at 50 °C. We then estimate the change in volume by averaging the expansion and compression volume-angle correlations for varying load configurations (see Supplementary Text S10). The final volume is the average estimated final volume for the tests that varied input power and maximum temperature; it varied for the tests that varied expansion volume. The cold temperature is the average temperature inside the pouch when it first reaches 0 kPa gauge pressure. For the tests that varied input power and final volume, we used the average maximum temperature reached as the high temperature.

For the set of tests that varied high temperature, we computed the efficiency for each of the recorded high temperatures. The ambient temperature is the average ambient temperature. The convection coefficient used corresponds to the fitted convection coefficient when the pouch is half open ( $h = . 0464 W$ /K) (see Supplementary Text S11). For the heat input rate, we used the average heat input rates for the different tests in the set of tests in which power was varied; we used the average of average heat input rates for the tests that varied maximum temperature and expansion volume. For the test that varied input power, we used the average mechanical efficiency for that set of tests. For the test that varied high temperature, we used the average mechanical efficiency for the pulley weight that corresponded to a given high temperature. For the test that varied expansion volume, we use the average mechanical efficiency for that set of tests. We then used a least-squares fit to determine $η_{f}$ for each of the set of tests.

We inputted $v_{1} = 7.50 \times 10^{- 4} m^{3} / kg$ , $v_{3} = 4.07 \times 10^{- 3} m^{3} / kg$ , $T_{c} = 29.27 ° C$ , $T_{h} = 38.52 ° C$ , $T_{\infty} = 16.61 ° C$ , $h = 14.10 W / kg / ° C$ , $η_{m} = 0.79$ , and $η_{f} = 0.32$ for the model for the test that varied power input.

We inputted $v_{1} = 7.50 \times 10^{- 4} m^{3} / kg$ , $v_{3} = 3.69 \times 10^{- 3} m^{3} / kg$ , $T_{c} = 33.39 ° C$ , $T_{\infty} = 19.77 ° C$ , $h = 12.78 W / kg / ° C$ , ${\dot{q}}_{i n} = 371.51 W / kg$ , $η_{f} = 0.48$ for the model for the test that varied maximum temperature.

We inputted $v_{1} = 7.50 \times 10^{- 4} m^{3} / kg$ , $T_{c} = 33.63 ° C$ , $T_{h} = 43.75 ° C$ , $T_{\infty} = 19.75 ° C$ , $h = 12.78 W / kg / ° C$ , $η_{m} = 0.74$ , and $η_{f} = 0.43$ for the model for the test that varied expansion volume.

Roller design and fabrication

The roller body is composed of six 12 cm × 15 cm panels framed by thin (4.5 × 1.1 mm cross section) carbon fiber beams. According to the first insight, we maximized the area of the polyvinyl chloride asymmetrical Fresnel lenses within each size. With the frame and a cutout for the actuator, the area of each lens is 0.013 m², resulting in 11.3 W of collected sunlight (based on 870 W/m² measured solar flux). These lenses are made asymmetrical by cutting a commercially available Fresnel lens (Cz Garden Supply Store, 8.3″ × 11.75″ Fresnel Lens, Amazon.com) in half. The panels are joined by flexible hinges and form a hexagon-shaped roller (side length, 12 cm). These hinges were 3D-printed from flexible filament (Ultimaker 2, NinjaFlex) and equipped with 28.6 N neodymium magnets (see “hinge magnets” in Fig. 4, K&J Magnetics B662). The dynamic hinges create maximum stiffness in the hexagon conformation, but compliance when deformed into other shapes (see Supplementary Text S14).

Each side is equipped with a 2.8 cm × 2.8 cm (7.6 cm²) Mylar STAPA pouch (see Supplementary Text S13). The first insight suggests minimizing the heat loss rate which can be accomplished by minimizing the pouch size. Accordingly, we designed the pouch to be as small as feasible given manufacturing limitations. Each pouch is backed by a 30 × 30 mm × 3.4 mm pad of polystyrene foam to decrease the heat loss coefficient when the leg is closed, as suggested by the first insight. Inside the pouch is a disc of black fabric, chosen for its broadband light absorptivity that maximizes input power.

During vaporization and condensation of the fluid, a hinged fiberglass leg deflects—flipping the roller as part of a rolling sequence. A thin piece of Nylon fabric serves as the hinge connecting the fiberglass leg to the roller body. A combination of latex bands and 10.3 N neodymium magnets (K&J Magnetics, B841) provide a restoring force (see “hinge magnets” and “latex bands” in Figure 4). All components of the roller are fastened to one another using cyanoacrylate glue (LocTite 40140), save for the flexible hinges which are adhered to the Fresnel lenses using flexible adhesive (LocTite 1360694).

Experiment: Roller locomotion test

The roller was placed outdoors in Santa Barbara, CA at a time of day such that the collector (i.e., Fresnel lens) was perpendicular to the collimated rays of the sun. Thereafter, the roller was allowed to roll autonomously until completion of the test. Supplementary Text S12 describes the cycle that the roller undergoes. Weather conditions (i.e., wind, solar flux, air temperature) were measured using a weather station (Ambient Weather Falcon WS-8480, see Supplementary Text S16 for details). The video and stills were taken using a mobile phone camera (iPhone 13 Pro).

The work required to complete one rolling transition of the device was measured as 0.1J. This was estimated by measuring the center of mass work and the magnet work. The center of mass work is the product of the roller weight (3 N) and the center of mass vertical excursion (1.65 cm), or 0.05J. This work is done during the first half of a rolling transition, and is then lost as the roller falls forward and settles. The magnet work is the work the STAPA does against the magnet that locks the position during heating. The total magnet force is 26 N, and the work is the area under the force-distance curve for the rectangular magnets used (0.05 J). We estimate that the STAPAs powering the roller achieved an average efficiency of 0.056% during operation (see Supplementary Text S17).

Authors’ Contributions

Conceptualization: L.F.G., C.X., M.T.V., and E.W.H. Methodology: L.F.G., C.X., M.T.V., E.W.H., and B.L. Investigation: L.F.G., C.X., and A.H. Visualization: L.F.G. and C.X. Funding acquisition: M.T.V. and E.W.H. Project administration: M.T.V. and E.W.H. Supervision: Y.Z., B.L., M.T.V., and E.W.H. Writing—original draft: L.F.G., C.X., M.T.V., and E.W.H. Writing—review and editing: L.F.G., C.X., Y.Z., B.L., M.T.V., and E.W.H.

Footnotes

Acknowledgements

The authors acknowledge use of the Microfluidics Laboratory within the California NanoSystems Institute, supported by the University of California, Santa Barbara and the University of California, Office of the President. Authors thank Michael Gordon for insightful discussions.

Author Disclosure Statement

Authors declare that they have no competing interests.

Funding Information

National Science Foundation grant EFMA 1935327 (E.W.H., M.T.V., C.X., L.F.G.). National Science Foundation Graduate Research Fellowship under grant # 2139319 (C.X.).

Data and Materials Availability

All data are available in the main text or the supplementary materials.

Supplemental Material

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Supplementary Material

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