Letter pubs.acs.org/NanoLett
Elastocaloric Effect in Carbon Nanotubes and Graphene Sergey Lisenkov,*,† Ryan Herchig,† Satyanarayan Patel,‡ Rahul Vaish,‡ Joseph Cuozzo,† and Inna Ponomareva† †
Department of Physics, University of South Florida, Tampa, Florida 33620, United States School of Engineering, Indian Institute of Technology, Mandi, Himachal Pradesh 175005, India
‡
ABSTRACT: Carbon nanotubes are famous for their many extraordinary properties. We use a thermodynamical approach, experimental data from the literature, and atomistic simulations to reveal one more remarkable property of the carbon nanotubes that has so far been overlooked. Namely, we predict the existence of very large elastocaloric effect that can reach up to 30 K under moderate loads. Potentially even larger values could be achieved under extreme loads, putting carbon nanotubes in the forefront of caloric materials. Other remarkable features of the elastocaloric effect in carbon nanotubes include linearity of elastocaloric temperature change in applied force (compressive or stretching), very weak dependence on the temperature, and an absence of hysteresis. Such features are extremely desirable for practical applications in cooling devices. Moreover, a similarly large elastocaloric effect is predicted for the graphene. The prediction of a large elastocaloric effect in carbon nanotubes and graphene sets forward an unconventional strategy of targeting materials with moderate caloric responses but the ability to withstand very large loads. KEYWORDS: Carbon nanotubes, graphene, elastocaloric effect, atomistic simulations
C
looking for materials that exhibit very large intrinsic caloric responses, we will look for materials with intrinsically moderate caloric responses but that allow application of very large fields. CNTs emerge as promising candidates for large ECE owing to their exceptional mechanical properties5,6 that include very large Young’s modulus (in the TPa range7,8), the ability to withstand loads of up to 52 GPa,9 and deformations of up to 12%.10 In addition, the absence of a phase transition is likely to eliminate hysteretic losses and the associated irreversibly. Because CNTs share their exceptional mechanical properties with the planar graphene we also explore ECE in this widely studied material. The ECE can be estimated using the thermodynamical expression derived with the help of the Maxwell relations
arbon nanotubes (CNTs) captivated researchers throughout the 1990s and the early 2000. Many remarkable properties have been predicted and discovered thanks to the unprecedented research effort.1,2 With the number of publications per year leveling off, it may appear that the scientific community has reached a thorough understanding of these extraordinary nanostructures. However, could there still remain some overlooked properties associated with CNTs? In this Letter, we predict the existence of an exceptionally large elastocaloric effect (ECE) in CNTs, which could open a way to their potential applications in nanoscale cooling devices. Caloric effects, such as magnetocaloric, electrocaloric, and elastocaloric effects, have received a lot of attention recently owing to the discovery of giant caloric effects in ferroics.3 It is presently believed that giant caloric effects occur mostly in materials that undergo structural first-order phase transitions. Although these materials indeed exhibit the largest changes in entropy or temperature under application of external fields, the changes usually occur only in the vicinity of the phase transition, which is a large disadvantage for their potential application in cooling devices.4 To overcome this critical drawback, it is desirable to look for materials that exhibit good caloric response in a wide temperature range. It is believed that materials that undergo second-order phase transitions could satisfy this requirement. Unfortunately, often times, the caloric responses from such materials are rather small. Could there be a way to overcome these fundamental limitations? Here, we propose a novel strategy with which to “bypass” such limitations and achieve the desired properties. Instead of © 2016 American Chemical Society
ΔT = −
1 ρ
∫0
σ
Tασ dσ Cσ
(1)
where ρ, T, and C are the density, temperature, and specific heat, respectively. σ is the applied stress, ασ =
( l∂∂Tl )σ is the
thermal expansion coefficient at a given stress, and l is the nanotube length. In the elastic regime, lσ = l0(1 + η), where l0 is the initial length, and η = σ/Y is the strain under the stress σ. Y Received: July 28, 2016 Revised: October 4, 2016 Published: October 6, 2016 7008
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Figure 1. Time dependence of the force applied to the (8,8) CNT at room temperature (a), of the associated strain (b), of the elastocaloric temperature change (c). Elastocaloric change in temperature as a function of the applied force (d).
is the Young’s modulus. Substitution of lσ into the expression for ασ and subsequent substitution of the result into eq 1 gives 1 ΔT = − ρ
∫0
σ
⎛ σ T⎜ α0 − ⎜ 2 Cσ 2Y 1 + ⎝
(
σ Y
)
⎞ ∂Y ⎟ dσ ∂T ⎟ ⎠
main advantage of CNT is that they can easily withstand this and much larger values of applied stress. It was shown experimentally that the CNT ropes can withstand a tensile load of up to 52 GPa.9 Motivated by this promising estimate, we investigate the ECE in CNTs using atomistic molecular dynamics. Molecular dynamics has been previously used to study both structural and mechanical properties of CNT6,15 and caloric effects in ferroics.23,24 To describe the interatomic interactions between the carbon atoms in CNTs, we use the REBO potential.25 This potential has a long history of successful application for the studies of hydrocarbons, CNTs, graphene, and others (see the more than 2400 articles citing ref 25.). In this work, we focus on two sets of CNTs, namely zigzag CNTs [(8,0), (10,0), (12,0), (16,0), (20,0)] and armchair CNTs [(8,8), (10,10), (12,12), (16,16), (20,20)]. All CNTs are modeled by a short segment of about 33 Å in length. One representative CNT from each set [(17,0) and (10,10)] is also modeled by a longer segment of 108 Å to cross-check the results. These lengths model well CNTs used in experimental studies of mechanical properties.8 To equilibrate CNTs at the desired temperatures, we carry out heating simulations. In such simulations the CNT is initially fully optimized at zero Kelvin using the conjugate gradient technique and then slowly heated from 50 to 1000 K in steps of 50 K. At each temperature, the CNTs are equilibrated during 16 ps using an Evans−Hoover thermostat. The results of the heating simulations were also used to estimate the computational heat capacity by taking the slope of the dependence of the total energy on the temperature. The associated computational value of the specific heat is 2070 J/kg × K and nearly independent of the temperature and the type of the CNT. It corresponds to the classical Dulong−Petit limit
(2)
where α0 is the thermal expansion coefficient in the absence of stress. For relatively small ΔT, eq 2 can be integrated approximately ΔT ≈ −
⎛ σ2 T ⎜ α0σ − ⎜ ρCσ 2Y 2 1 + ⎝
≈−
σ 2 ∂Y ⎞ T ⎛ ⎟ ⎜α0σ − ρCσ ⎝ 2Y 2 ∂T ⎠
(
σ Y
)
⎞ ∂Y ⎟ ∂T ⎟ ⎠ (3)
To estimate the ECE in CNTs at room temperature, we use the experimental values of the thermal expansion coefficient of 1.9 × 10−5 K−1,11 a specific heat of 645 J/kg K,12 a density of 1740 kg/m3,13 and a Young’s modulus for the nanotube wall of 5.6 TPa.14 We are not aware of experimental values for ∂Y , and ∂T so a computational value of −6.0 × 10−4 TPa/K for (10,10) CNT from ref 15 can be used. Substitution of these values into eq 3 gives the elastocaloric change in temperature of 12.7 K under 2.5 GPa compressive stress (53 ± 10 nN compressive force) applied to the CNT wall. Note that under given conditions the second term of eq 3 makes a negligible contribution to ΔT and can be omitted. The estimated elastocaloric ΔT is comparable to the values reported for ferroics with first-order phase transitions.16−22 However, the 7009
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Figure 2. Dependence of the elastocaloric ΔT on the applied force (a−c) and associated strain (d−f) at different temperatures. The temperatures are indicated in the graphs’ titles.
and compares well with this limit in graphite (2100 J/kg × K).26 The equilibrated CNTs are used in subsequent elastocaloric simulations. In such simulations one end of the CNT is kept fixed by constraining the end atoms, while the other end is subjected to a periodic force F = F0 sin(ωt), which causes periodic compression and elongation of the nanotube. The thermostat is turned off to simulate the adiabatic application of the force. A similar approach has been previously used to study the electrocaloric effect in ferroelectrics.23 Two different angular frequencies, 8 and 78 GHz, are used. The higher frequency allows more accurate numerical integration due to the use of a smaller integration time step of 0.02 fs. Lower frequency is simulated using the integration step of 0.2 fs and is used to establish if the ECE shows frequency dependence. We did not find significant differences between the data obtained for the two frequencies. The data in the remainder of the Letter correspond to the simulations with the angular frequency of 78GHz. The amplitude of the force applied in simulations is in the range of 15−30 nN and can be achieved experimentally by the application of an AFM tip.14 For each CNT and each temperature the force amplitude is chosen to achieve large deformations in the elastic regime (typically up to 4%) but avoid plastic regime or collapse. The time dependence of the periodic force applied to a (8,8) CNT at room temperature is given in Figure 1a. Figure 1b,c gives the associated strain and elastocaloric temperature change as a function of time. Because the Debye temperature for CNT is high (determined on the basis of the in-plane value of 2100−2300 K for graphite),27,28 the quantum corrections to the classical specific heat in computations are necessary to obtain reliable estimates for the ECE. We introduce such corrections empirically through rescaling the elastocaloric ΔT obtained in classical computa-
tions by a factor of Ccomp/Cexp(T), where Ccomp and Cexp(T) are the computational and experimental values of the specific heat at a given temperature. The experimental values for CNT specific heat below 300 K are taken from the ref 12, while above room temperature, we use the experimental values for graphite.26 The corrections were applied to all the data reported in this Letter, including the ones in Figure 1c. This figure predicts the existence of very large (up to 20 K) ECE in the CNT induced by the elastic deformations of up to 3%. This value is in agreement with the value, 12.7 K, estimated using eq 3 and experimental data in the literature. Figure 1d shows the dependence of the elastocaloric ΔT on the applied force. At room temperature, the dependence is mostly linear, indicating that the second term in eqs 2 and 3 makes a negligible contribution to the ECE and the latter is dominated by the thermal expansion coefficient at zero stress or force. The linearity of the elastocaloric ΔT in applied force was previously reported for ferroelectrics and antiferroelectrics below the Curie point21,29,30 and could be useful for optimizing the solidstate refrigeration cycle as proposed in ref 30. Figure 2 combines the data for two types of CNTs (zigzag and armchair) and for three different temperatures: 100, 300 and 900 K. In particular, we show the dependence of the elastocaloric ΔT on the applied force (top row) and on the resultant strain (bottom row). The reason for reporting ΔT as a function of strain is the ambiguity in the definition of the stress applied to CNT. Because the strain can be defined unambiguously in both computations and experiments, it provides a better basis for comparison. At high temperatures, ΔT is a linear function of the force and strain (see panels c and f of Figure 2). At 300 K, (12,0) CNT still exhibits linear behavior, while (8,8) deviates from it very slightly under the tensile load (positive F). Finally, for the lowest temperature 7010
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Figure 3. Dependence of the elastocaloric coefficient
dT dF
(a) and its normalized counterpart
dT dF ′
(b) on the temperature for different CNTs.
It should be noted that for the representative CNTs [(17,0) and (10,10)], we obtained very similar data from the simulation of 33 and 108 Å long tube segments. The main difference is that at higher temperatures, the longer tubes can withstand smaller compressive load than the shorter tubes due to bending deformations that are energetically too costly for shorter tubes. Because, for practical applications, CNT arrays are more feasible than isolated CNTs, the bending deformations are unlikely to make a significant impact. Therefore, shorter CNT segment calculations model CNT arrays more closely than the longer ones. It is also important to comment on the frequency of the external force application used in simulations. Because in simulations we do not find any hysteretic behavior or difference in the results from simulations with different frequencies (8 and 78 GHz), our predictions can be assumed to be frequency independent below 78 GHz. Therefore, for experimental confirmation or practical applications of the ECE much lower frequencies or even quasistatic application of the external force could be utilized. We have also estimated the ECE in graphene using eq 3. Using the specific heat of 8.7 J/mol × K,26 a density of 2250 kg/m3,26 a thermal expansion coefficient of 7.8 × 10−6K−1,31 a Young’s modulus of 362 GPa, and a ∂Y/∂T of −6.9 × 10−4 TPa/K,32 we obtained elastocaloric change in a temperature of −3.6 K under applied compressive stress of 2.5 GPa. In summary, we predicted the existence of very large ECE in CNTs. At room temperature, the estimates from experimental data yield the value of 12.7 K under an applied stress of 2.5 GPa, and computations predict that a value of up to 30 K can be attained by the application of a larger load in the elastic regime. The remarkable features of the ECE in CNTs is the linearity in the applied force (compressive or stretching) above the room temperature, very weak dependence of the elastocaloric coefficient on the temperature, and absence of hysteresis. These features are highly desirable for practical applications of ECE for solid-state refrigeration. A large electrocaloric effect of 3.6 K under compressive load of 2.5 GPa is predicted for the planar counterpart of CNTs graphene. The nanoscale size and low-density make CNTs and graphene promising candidates for nanoscale coolers. The unconventional approach adopted in this study is to look for materials that have moderate caloric response but that allow the application of unusually large fields. Success of such an approach is likely to open a new direction in the search of caloric materials.
reported in Figure 2, both CNTs exhibit nonlinear behavior. To understand the origin of the nonlinearity, we turn to eq 3, which predicts that the nonlinearity originates from the temperature dependence of the Young’s modulus. Indeed, we find from computations that for the (8,8) CNT the Young’s modulus increases by about 2% in the temperature interval from 50 to 300 K. Above the room temperature, the Young’s modulus remain constant within our computational resolution. Thus, our computational data from direct ECE simulations are in agreement with the predictions from eqs 2 and 3. The bottom row of Figure 2 suggests that a CNT strained up to 4% exhibits an elastocaloric temperature change of up to 25 K at temperatures of practical interest. Given that CNTs can withstand deformations of up to 12%10 the elastocaloric ΔT could potentially triple putting CNTs into the forefront of caloric materials. Note that computational data show no hysteresis, which is highly desirable for practical applications of ECE. Analysis of our computational data for all investigated CNTs suggest that ΔT remains a linear function of applied force for the temperatures at and above the room temperature. Therefore, in this temperature range, we can compute the elastocaloric coefficient dT/dF. Figure 3a presents the computational values as a function of temperatures. The data suggest that all investigated CNTs exhibit strong ECE. However, zigzag CNTs appear to have larger absolute values of the elastocaloric coefficient. To further investigate this, we factor out the CNT diameter by normalizing the applied force by the length of the CNT circumference l as follows: F′ = F/l. The resultant elastocaloric coefficient, dT/dF′, is given in Figure 3b. The data indicate that zigzag CNT still exhibit better elastocaloric coefficient. Furthermore, the elastocaloric coefficient decreases in magnitude as the CNT diameter increases. Both of these features are likely to be attributed to the differences in the zero-stress thermal expansion coefficient, as suggested by eq 3. While we attempted to compute thermal expansion coefficients, the values appear to be qualitative in nature due to the rather large thermal noise. A remarkable feature of the ECE in CNTs is its weak dependence on the temperature. It originates mostly from the fact that in the temperature range of practical importance, the ratio T depends C
on temperature only weakly, leading to only weak temperature dependence of elastocaloric ΔT according to eq 3. Such a feature differentiates ECE in CNTs from the ones reported for ferroics and is extremely desirable for practical applications. 7011
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(26) Pop, E.; Varshney, V.; Roy, A. K. MRS Bull. 2012, 37, 1273− 1281. (27) Tohei, T.; Kuwabara, A.; Oba, F.; Tanaka, I. Phys. Rev. B: Condens. Matter Mater. Phys. 2006, 73, 064304. (28) Tewary, V. K.; Yang, B. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 79, 125416. (29) Barr, J. A.; Beckman, S. P.; Nishimatsu, T. J. Phys. Soc. Jpn. 2015, 84, 024716. (30) Lisenkov, S.; Mani, B. K.; Cuozzo, J.; Ponomareva, I. Phys. Rev. B: Condens. Matter Mater. Phys. 2016, 93, 064108. (31) Shao, T.; Wen, B.; Melnik, R.; Yao, S.; Kawazoe, Y.; Tian, Y. J. Chem. Phys. 2012, 137, 194901. (32) Mirnezhada, M.; Modarresib, M.; Ansaria, R.; Roknabadib, M. R. J. Therm. Stresses 2012, 35, 913.
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS Financial support for this work provided by the National Science Foundation grant nos. DMR-1250492 and MRI CHE1531590. The authors acknowledge the use of the services provided by Research Computing at the University of South Florida.
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