Abstract
Mechanocaloric materials undergo sizable temperature changes during stressinduced phase transformations and hence are highly sought after for solidstate cooling applications. Most known mechanocaloric materials, however, operate at nonambient temperatures and involve firstorder structural transitions that pose practical cyclability issues. Here, we demonstrate large roomtemperature mechanocaloric effects in the absence of any structural phase transformation in the fastion conductor Li_{3}N (ΔS ~ 25 J K^{−1 }kg^{−1} and ΔT ~ 5 K). Depending on whether the applied stress is hydrostatic or uniaxial the resulting caloric effect is either direct (ΔT > 0) or inverse (ΔT < 0). The dual caloric response of Li_{3}N is due exclusively to stressinduced variations on its ionic conductivity, which entail large entropy and volume changes that are fully reversible. Our work should motivate the search of large and dual mechanocaloric effects in a wide variety of superionic materials already employed in electrochemical devices.
Introduction
Caloric materials undergo large temperature variations as a result of fieldinduced phase transformations involving sizable entropy changes^{1}. Solidstate cooling based on caloric materials represents an alternative to traditional refrigeration technologies based on compression cycles of greenhouse gases, which in addition to the obvious environmental threats cannot be scaled to small sizes. A major breakthrough took place more than two decades ago with the discovery of giant magnetocaloric effects in the intermetallic compound Gd_{5}(Si_{2}Ge_{2})^{2}. Magnetocaloric materials, however, are mostly based on rareearth elements, thereby are scarce, and require the generation of ultrahigh magnetic fields. Meanwhile, mechanocaloric (MC) effects produced by moderate mechanical stresses recently have attracted a lot of attention owing to their large isothermal entropy and adiabatic temperature shifts, which may surpass by an order of magnitude those achieved in magnetocaloric materials with commercially available permanent magnets^{1,3}. To date, giant MC effects near structural phase transitions have been experimentally observed in a number of relatively expensive magnetic materials^{4,5,6}, shapememory alloys^{7,8}, fluoridebased materials^{9,10,11,12}, polar materials^{13,14}, a polymer^{15}, organic—inorganic hybrid perovskites^{16,17}, and the archetypal superionic conductor AgI^{18}. We note that giant MC effects in AgI and other similar superionic compounds like CaF_{2}, PbF_{2}, and Li_{3}OCl were theoretically predicted prior to the experiments^{19,20}. Actually, computational approaches are increasingly being applied with success to the study of MC materials (see, for instance, refs.^{21,22}) since they are reliable and can be used in an effective and inexpensive manner. Yet, in spite of their promise, giant MC effects normally occur at temperatures that are distant from ambient conditions and involve firstorder transitions that in practice lead to severe thermal hysteresis problems. New caloric materials displaying large roomtemperature MC effects and nonirreversible atomic processes, therefore, are highly desirable for the development of solidstate cooling devices.
We present here an original strategy for achieving large and reversible MC effects at ambient temperature based on fastion conductors, and illustrate it in Li_{3}N. Rather than focusing on the triggering of a structural phase transition, which only serendipitously will occur at room temperature, our starting point is a material that is already superionic at ambient conditions. Large and reversible MC effects then can be obtained through the application of mechanical stresses since these affect significantly and sustainedly the ionic conductivity, and in turn the entropy and volume, in fastion conductors^{19,20,23}. (We note that application of the same strategy to other families of MC materials, for instance ferro/ferrielectrics and magnetic compounds, is likely to fail as due to fieldinduced saturation of the relevant order parameters at thermodynamic conditions other than transition points.) In Li_{3}N, we estimate a giant roomtemperature isothermal entropy change ΔS of about −25 J K^{−1 }kg^{−1} at a maximum hydrostatic stress of 1 GPa. The corresponding roomtemperature adiabatic temperature shift ΔT is ≈ + 3 K, which is large but not giant due to the huge heat capacity of such a lightweight material. Interestingly, when Li_{3}N is subjected to uniaxial tensile loads similarly large ΔS and ΔT shifts are obtained but with opposite signs (ΔS > 0). The disclosed large and dual, that is, direct (ΔT > 0) and inverse (ΔT < 0), MC effects can be understood in terms of simple arguments based on stressinduced variations on the volume an iontransport properties of superionic materials^{19,20,23}. We also discuss straightforward chemical strategies for increasing the ΔT values estimated in Li_{3}N by more than a factor of two.
Results
Direct barocaloric effects
Figure 1 shows the two common polymorphs found in commercially available samples of bulk Li_{3}N. The α phase (hexagonal, space group P6_{3}/mmm) has a layered structure composed of alternating planes of hexagonal Li_{2}N and pure Li^{+} ions (Fig. 1a). The β phase (hexagonal, space group P6_{3}/mmc) presents an additional layer of lithium ions intercalated between the Li_{2}N planes that is accompanied by a doubling of the unit cell (Fig. 1b). Exceptionally high ionic conductivities of the order of 10^{−4}–10^{−3} S cm^{−1} have been experimentally observed in Li_{3}N at room temperature^{24,25,26}. In our molecular dynamics (MD) simulations based on classical interaction potentials and firstprinciples methods (Methods, Supplementary Methods, and Supplementary Table 1), we find that stoichiometric αLi_{3}N displays high ionic conductivity at T = 300 K (i.e., of the order of 10^{−4} S cm^{−1}), which is in consistent agreement with the experimental observations^{24,25,26}. On the other hand, it is necessary to consider a small concentration of extrinsic Li^{+} vacancies (~1%) in βLi_{3}N to render similar roomtemperature superionic features.
Figure 2 shows our estimation of the barocaloric (BC) effects induced by hydrostatic stresses of up to 1 GPa in Li_{3}N at room temperature and T = 400 K. In the β phase, we obtain a maximum roomtemperature entropy shift of −24 J K^{−1 }kg^{−1} that increases slightly in absolute value at higher temperatures (Fig. 2a). The resulting BC effects, therefore, are direct (ΔT > 0, Methods). In view of previous reports, see for instance^{4,5,6} and Supplementary Table 2, the magnitude of such ΔS can be regarded as “giant” (that is, ΔS~10^{1} J K^{−1 }kg^{−1}). The accompanying adiabatic temperature shifts (Fig. 2b), however, are large but not giant (that is, ΔT < 10 K) owing to the huge heat capacity of Li_{3}N (C_{0} ~ 4.10^{3} J K^{−1 }kg^{−1} at T = 300 K, Supplementary Fig. 1), which is a very lightweight compound. We note that for a fixed temperature the pressureinduced increase of ΔS and ΔT are practically constant, which suggests a sustained change in the physical properties of the material under pressure (in contrast to the abrupt ΔS and ΔT changes observed during structural phase transitions^{20}). Analogous results are found in αLi_{3}N (Fig. 2c, d) although in this case the estimated isothermal entropy and adabatic temperature changes are slightly larger (for instance, ΔS = −32 J K^{−1 }kg^{−1} and ΔT = +2.8 K at T = 300 K and σ = 1 GPa), and the corresponding pressureinduced shift increments fluctuate somewhat.
The large and direct BC effects found in superionic Li_{3}N are not driven by any structural phase transformation (i.e., the crystal symmetry of the system remains the same over the whole studied pressure range). This is explicitly shown in Fig. 3a–f, where we plot the radial distribution function calculated for all ionic couples at T = 300 K and different σ conditions: The corresponding g(r) profiles present almost identical traits independently of the applied pressure within the interval 0 ≤ P ≤ 1 GPa. Further evidence for the absence of any σinduced structural phase transformation is provided by our coordination number (Supplementary Fig. 2), position correlation function^{27,28} (Supplementary Fig. 3), and enthalpy energy (Supplementary Fig. 4) results. We note that our findings are fully consistent with previous experimental as well as theoretical reports^{29,30,31}, which provides further confidence in the employed computational approach (Methods and Supplementary Methods).
The absence of any structural phase transformation in Li_{3}N (within the pressure range P ≤ 1 GPa) is in stark contrast to what is observed in other BC materials, in which most of the caloric response is concentrated near phase transition points^{1}. Then, which is the principal mechanism behind the disclosed giant ΔS in Li_{3}N? It is wellknown that hydrostatic pressure depletes significantly the ionic diffusivity, and therefore the entropy, in most fastion conductors^{19,27,28,32}. Essentially, the available volume to interstitial ions is effectively reduced under compression and as a consequence the kinetic barriers and formation energy of defects governing ion migration increase drastically^{19,28} (Supplementary Fig. 5). We illustrate such a σinduced reduction in the ionic conductivity of β – Li_{3}N in Fig. 4. Therein, we plot two different lithium density plots constructed as inplane Li^{+} position histograms (see Fig. 3g for plane notation) calculated at room temperature and σ = 0 and 1 GPa: The diffusivity of lithium ions at equilibrium (Fig. 4a) is larger than under pressure (Fig. 4b) as shown by the higher presence of highdensity areas between crystal lattice sites and on vacancy positions (highlighted with white boxes in the figure). Specifically, the estimated roomtemperature Li^{+} diffusion coefficient D_{Li} and corresponding ionic conductivity ϕ_{Li} (Methods) amount respectively to 8.6×10^{−7} cm^{2 }s^{−1} and 4.7×10^{−4} S cm^{−1} at σ = 0 GPa, and to 5.3×10^{−7} cm^{2 }s^{−1} and 2.8×10^{−4} S cm^{−1} at σ = 1 GPa (Supplementary Fig. 6). We note that in our ab initio and classical MD simulations the Li^{+} diffusivities calculated along the hexagonal c–axis and other perpendicular directions do not present significant differences (Supplementary Fig. 7).
Inverse elastocaloric effects
Uniaxial and biaxial mechanical stresses can be used to enhance the ionic conductivity in superionic materials^{19,20,23,33,34,35}. An enhancement of ionic conductivity normally is accompanied by an increase in the entropy of the system (ΔS > 0), which suggests the possibility of realizing also inverse MC effects (ΔT < 0) in fastion conductors. We have tested such an hypothesis in Li_{3}N by performing MD simulations under constrained uniaxial tensile loads. (We note that α–Li_{3}N is found to be vibrationally unstable in our uniaxial tensile stress simulations, hence we focus on the β–Li_{3}N polymorph in the following analysis.)
Figure 5 shows the elastocaloric (EC) effects induced by uniaxial tensile stresses applied along the hexagonal caxis in βLi_{3}N at room temperature and T = 400 K. An arbitrary maximum tensile stress of −8 GPa has been applied in our calculations, which corresponds to a maximum strain deformation along the hexagonal caxis of +5.8% (see inset in Fig. 5). A giant isothermal entropy change of ΔS = +20 J K^{−1 }kg^{−1} is estimated at the highest simulated uniaxial load and room temperature, which increases up to +25 J K^{−1 }kg^{−1} at T = 400 K (Fig. 5a). The resulting EC effects, therefore, are inverse (ΔT < 0). Also in this case, the accompanying adiabatic temperature shifts are large but not giant (that is, ΔT = −2.0 and −3.4 K at T = 300 and 400 K, respectively, Fig. 5b) due to the huge heat capacity of Li_{3}N (Supplementary Fig. 1).
In analogy to the BC case, none σinduced structural phase transition is responsible for the observed EC effects (Supplementary Figs. 2, 4, 8). The inverse EC response of superionic Li_{3}N can be entirely understood in terms of σinduced enhancements of its ionic conductivity. Uniaxial tensile stresses produce an effective increase in the volume of the system, which in turn provokes the kinetic barriers and formation energy of defects governing ion migration to decrease significantly^{19,34,35} (Supplementary Fig. 5). Such a σdriven enhancement of the Li^{+} ionic conductivity is illustrated in Fig. 6, where we show that the number of highdensity lithium ion regions appearing between crystal lattice positions and on vacancy sites is appreciably higher than found at σ = 0 GPa and same temperature (Fig. 4a). Likewise, for an uniaxial tensile stress of −7 GPa the estimated roomtemperature lithium diffusion coefficient D_{Li} and corresponding ionic conductivity ϕ_{Li} (Methods) amount to 10.2×10^{−7} cm^{2 }s^{−1} and 5.5×10^{−4} S cm^{−1}, respectively, to be compared with 8.6×10^{−7} cm^{2 }s^{−1} and 4.7×10^{−4} S cm^{−1} calculated at T = 300 K and σ = 0 GPa (Supplementary Fig. 6).
Discussion
The large MC roomtemperature effects disclosed in superionic Li_{3}N pose great prospects in the context of refrigerationcycle reversibility as due to the absence of mechanical hysteresis effects deriving from the nucleation of orderparameter domains (in contrast to what occurs in ferro/ferrielectrics and magnetic compounds where such issues may actually turn out to be critical). Furthermore, it has been experimentally demonstrated by means of ultrafast Xray spectroscopy that the characteristic time scale of superionic switching is just of few picoseconds^{36}, which is consistent with our simulations (Supplementary Fig. 7). Consequently, refrigeration cycles based on fastion conductors may be conducted efficiently at high repetition rates. In order to assess the reversibility features of superionicbased MC cycles, we have performed additional classical MD simulations in which we have varied at a constant rate of 10^{−3} GPa/ps the applied pressure on βLi_{3}N, first from 0 to 1 GPa and then backwards from 1 to 0 GPa, while monitoring the accompanying changes on the volume and energy of the system. Our simulation results reassure the highly reversible and responsive nature of σdriven changes on the physical properties of Li_{3}N, as it is demonstrated by the smallness of the volume and energy differences estimated between the 0→1 GPa and 1→0 GPa runs (Supplementary Fig. 9).
Arguably, the only disadvantage of superionic Li_{3}N for developing new MC solidsate cooling applications is that the corresponding adiabatic temperature shifts are not giant (ΔT < 10 K). As we have mentioned in previous sections, this shortcoming is due to the fact that the heat capacity of Li_{3}N C_{0} is huge owing to its light weight (note that giant isothermal entropy changes are already obtained under moderate stress variations, see Eq. (5) in Methods). A straightforward strategy for drastically enhancing the ΔT values obtained in Li_{3}N may consist in substituting the “static” sublattice of nitrogen ions by isoelectronic and largermass largerradius elements, namely, P and As, in order to effectively reduce the C_{0} of the system while keeping its superionicity^{28}. Certainly, Li_{3}P and Li_{3}As are isostructural to Li_{3}N and they are known to be better superionic conductors at room temperature^{26}. Consequently, one can reasonably expect that the σinduced ΔS shifts appearing in Li_{3}P and Li_{3}As will be at least as large as estimated in Li_{3}N^{37}, and then trivially estimate a lower bound for the increase in cooling performance due to their decrease in heat capacity. We have performed firstprinciples quasiharmonic calculations of the heat capacity (Methods) in Li_{3}P and Li_{3}As as a function of temperature (Supplementary Fig. 1), and found that at T = 300 K their C_{0}’s are respectively 1.3 and 2.2 times smaller than the estimated in Li_{3}N. In addition, the Tinduced volume expansion \(\left( {\frac{{\partial V}}{{\partial T}}} \right)_\sigma\) in Li_{3}As is about 1.4 times larger than the estimated in Li_{3}N at room temperature (Supplementary Fig. 10). These results, namely, smaller C_{0}’s and larger \(\left( {\frac{{\partial V}}{{\partial T}}} \right)_\sigma\)'s, suggest that giant adiabatic temperature changes (ΔT > 10 K) actually may be realized in lithiumbased fastion conductors (see Eqs. (3) and (5) in Methods).
Our MC findings on Li_{3}N should stimulate the development of roomtemperature cooling devices based on fastion conductors, whose energy efficiency and refrigerant performance are good as compared to magneto and electrocaloric materials. Unlike other caloric materials, the MC effects revealed in Li_{3}N are not driven by any σinduced structural phase transition but instead by changes on its ionic conductivity. As a result either direct or inverse reversible MC effects can be achieved at room temperature, just depending on how the mechanical stresses are applied. Biaxial stresses are likely to render similar MC effects than obtained with uniaxial stresses, adding the particularity that compressive loads also can be accomplished^{20} (Supplementary Fig. 11). The original strategy introduced in this study for realizing large and reversible MC effects at room temperature is not exclusive of lithiumbased fastion conductors, it should work equally well for any other family of materials exhibiting superionicity and robust structural stability at ambient conditions (e.g., solid oxide fuel cell materials).
Methods
Classical molecular dynamics simulations
MD (N,P,T) simulations are performed with the LAMMPS code^{38}. The pressure and temperature in the system are kept fluctuating around a setpoint value by using thermostatting and barostatting techniques in which some dynamic variables are coupled to the particle velocities and simulation box dimensions. The interactions between atoms are modeled with rigidion BornMayerHuggins potentials^{39}. We employ large simulation boxes, typically containing up to 8000 atoms, and apply periodic boundary conditions along the three Cartesian directions. Defective βLi_{3}N systems are generated by randomly removing a specific number of cations and anions from the system in order to reproduce the experimentally observed ionic conductivity data^{24} and to maintain the condition of charge neutrality. Newton’s equations of motion are integrated using the customary Verlet’s algorithm with a timestep length of 10^{−3} ps. The typical duration of a MD run is of 200 ps. A particle–particle particle–mesh kspace solver is used to compute longrange van der Waals and Coulomb interactions beyond a cutoff distance of 12 Å at each time step. We explicitly have checked that by creating point defects in an ordered manner and/or by increasing the total duration of the MD simulations up to 500 ps, our results remained invariant within the statistical uncertainties^{23}. Further details of our classical MD simulations (e.g., interatomic potential models and parameters) can be found in the Supplementary Methods and Supplementary Table 1.
Density functional theory calculations
Firstprinciples calculations based on density functional theory (DFT) are performed to analyze the energy, structural, vibrational, and ionic transport properties of Li_{3}N. We perform these calculations with the VASP code^{40} by following the generalized gradient approximation to the exchangecorrelation energy due to Perdew et al.^{41}. Possible dispersion interactions are captured with the D3 correction scheme developed by Grimme and coworkers^{42}. The projector augmentedwave method is used to represent the ionic cores^{43}, and the electronic states 1s2s of Li and 2s2p of N are considered as valence. Wave functions are represented in a planewave basis truncated at 650 eV. By using these parameters and dense kpoint grids for Brillouin zone integration, the resulting energies are converged to within 1 meV per formula unit. In the geometry relaxations, a tolerance of 0.01 eV·Å^{−1} is imposed in the atomic forces. Ab initio MD (AIMD) simulations are carried out to assess the reliability of the interatomic potential models employed in the classical MD simulations. Details of our AIMD tests can be found in the Supplementary Methods.
We also perform ab initio phonon frequency calculations with the smalldisplacement method in order to assess the vibrational stability of the analyzed systems and estimate their (vibrational) heat capacity. In the smalldisplacement method the forceconstant matrix is calculated in realspace by considering the proportionality between atomic displacements and forces^{44,45,46}. The quantities with respect to which our phonon calculations are converged include the size of the supercell, the size of the atomic displacements, and the numerical accuracy in the sampling of the Brillouin zone. We find the following settings to provide quasiharmonic free energies converged to within 5 meV per formula unit^{46:} 3 × 3 × 3 supercells (where the figures indicate the number of replicas of the unit cell along the corresponding lattice vectors), atomic displacements of 0.02 Å, and qpoint grids of 14 × 14 × 14. The value of the phonon frequencies are obtained with the PHON code developed by Alfè^{45}. In using this code we exploit the translational invariance of the system, to impose the three acoustic branches to be exactly zero at the center of the Brillouin zone, and apply central differences in the atomic forces.
Finally, we perform ab initio nudgedelastic band (NEB) calculations^{47} to estimate the kinetic energy barriers for lithium ion diffusion in Li_{3}N under different hydrostatic pressure conditions (Supplementary Fig. 5). The calculations are performed in a 2 × 2 × 1 supercell contianing 32 atoms. We use a qpoint grid of 7 × 7 × 10 and an energy planewave cutoff of 650 eV. Six intermediate images are used in our NEB calculations and the geometry optimizations are finalized once the total forces on the atoms are smaller than 0.01 eV·Å^{−1}.
Estimation of key quantities
Ionic diffusion coefficients are calculated with the formula:
where R_{i}(t) is the position of the migrating ion labeled as i at time t, t_{0} an arbitrary time origin, and \(\langle \cdots \rangle\) denotes average over time and particles. The mean squared displacement of each ionic species is defined as \(\left\langle {\Delta R_i^2\left( t \right)} \right\rangle \equiv \left\langle {\left {R_i\left( {t + t_0} \right)  R_i\left( {t_0} \right)} \right^2} \right\rangle\). Likewise, ionic conductivities are obtained with the wellknown NernstEinstein relationship^{48:}
where N and e are the number density and charge of Li^{+} ions, respectively.
Isothermal entropy changes associated to the BC effect are estimated as^{1:}
where σ_{f} represents the applied hydrostatic pressure, and V the volume of the system. In the case of EC effects, the same quantity is calculated as^{1}:
where σ_{f} now represents the applied mechanical stress along the Cartesian z direction, ε the strain deformation that the system undergoes along the same direction (i.e., \(\varepsilon \left( {\sigma ,T} \right) \equiv \frac{{L_z\left( {\sigma ,T} \right)  L_z\left( {0,T} \right)}}{{L_z\left( {0,T} \right)}}\) where L_{z} corresponds to the length of the simulation box along the Cartesian z direction), and V_{0} the equilibrium volume of the system. Finally, the resulting adiabatic temperature shifts are estimated with the formula:
where \(C_0\left( T \right) = \left( {\frac{{{\rm{d}}U}}{{{\rm{d}}T}}} \right)_{V_0}\) is the heat capacity of the crystal calculated at zerostress conditions. Within the quasiharmonic approximation^{44,45,46}, this latter quantity can be calculated as:
where N_{q} is the total number of wave vectors used for integration in the Brillouin zone, the summation runs over all wave vectors q and phonon branches s, and ω_{qs} are the phonon frequencies of the material at the equilibrium volume. Further technical details on our calculations can be found in the Supplementary Methods.
Data Availability
The data that support the findings of this study are available from the corresponding author (C.C.) upon reasonable request.
Additional information
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Change history
06 September 2018
This Article was originally published without the accompanying Peer Review File. This file is now available in the HTML version of the Article; the PDF was correct from the time of publication.
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Acknowledgements
This research was supported under the Australian Research Council’s Future Fellowship funding scheme (No. FT140100135). Computational resources and technical assistance were provided by the Australian Government and the Government of Western Australia through Magnus under the National Computational Merit Allocation Scheme and The Pawsey Supercomputing Centre.
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School of Materials Science and Engineering, UNSW Sydney, Sydney, NSW, 2052, Australia
 Arun K. Sagotra
 , Dewei Chu
 & Claudio Cazorla
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Contributions
A.K.S, D.C., and C.C. conceived the study and planned the research. A.K.S. and C.C. performed the theoretical calculations. Results were discussed by A.K.S., D.C., and C.C. The manuscript was written by A.K.S. and C.C. with substantial input from D.C.
Competing interests
The authors declare no competing interests.
Corresponding author
Correspondence to Claudio Cazorla.
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