Introduction

Proposals for environmentally friendly solid-state cooling devices have been inspired by the discovery of both giant magnetocaloric (MC) effects in a variety of magnetic materials1,2,3,4, and giant electrocaloric (EC) effects in a variety of ferroelectric materials4,5,6, but the need to generate large driving fields is problematic. Large magnetic fields are expensive to generate, while large electric fields can lead to electrical breakdown. By contrast, it is easy to generate the hydrostatic pressures required to drive larger4, 7 and more energy efficient8, 9 barocaloric (BC) effects non-destructively. BC materials have therefore sparked interest from both academia and industry, but materials selection remains rather limited.

To date, giant BC effects have only been experimentally demonstrated near room temperature in a polymer10, a small number of relatively expensive magnetic materials11,12,13,14,15,16, a number of fluorites17,18,19,20, a hybrid perovskite21 and a small number of ferro/ferrielectric materials22, 23. Following the recent prediction of giant BC effects in fluoride-based superionic conductors at very high temperatures24, we demonstrate here giant BC effects nearer to room temperature in a powder of AgI, which is the prototypical solid electrolyte that was shown to display fast ionic conduction over one century ago25, 26.

Above the superionic transition temperature T0 ~ 420 K, AgI exists as the α polymorph, in which the iodine anions adopt a body-centred-cubic \({\mathrm{Im}}\bar 3m\) structure27, 28, while the interstitial silver cations are disordered across a fraction of the tetrahedral interstices to yield a very large ionic conductivity that it is comparable with the conductivity of the molten state26. Each time the sample is cooled after heating just through the transition, the β and the γ polymorphs29 are formed in the same ratio30 due to a first-order reconstructive phase transition that is accompanied by a large 5% increase in volume31,32,33,34. The β polymorph has iodine anions in a hexagonal-close-packed \(P\bar 6_3mc\) structure27, 29, 35, the γ polymorph has iodine anions in a cubic-close-packed \(F\bar 43m\) structure27, 36, 37. The interstitial silver cations in γ-AgI and β-AgI are relatively ordered, whereas in α-AgI they are disordered across a fraction of the tetrahedral interstices, such that the entropy difference |ΔS0| = 63 ± 4 J K−1 kg−1 for the α ↔ β + γ transition is large38,39,40.

Here we show that AgI displays peak isothermal entropy changes of |ΔS| ~ 60 J K−1 kg−1, corresponding to adiabatic temperature changes of |ΔT| ~ 18 K, due to moderate changes of applied pressure |Δp| ~ 2.5 kbar (where we assume ambient pressure to be zero such that |Δp| ~ p). These giant and reversible inverse BC effects may be driven over a wide range of temperatures below the transition, yielding values of refrigerant capacity1 (RC) that exceed all values previously recorded for BC materials. The large BC effects in AgI are associated with a pressure-driven phase transition between the coexisting β and γ polymorphs at low pressure, and the α polymorph at high pressure. On increasing pressure at starting temperatures that lie below the thermally driven transition, the pressure-driven β + γ → α transition increases sample entropy by melting the interestitial sublattice of silver cations41, 42. This melting arises as a consequence of reducing the cation hopping distances, and increasing the number of vacant interstitial sites42.

Results

X-ray diffraction confirmed that our as-received AgI powder comprised a mixture of β and γ polymorphs, but it could not resolve their relative proportions. At atmospheric pressure, calorimetry data obtained on heating and cooling confirmed both T0 ~ 420 K and a large thermal hysteresis of ~25 K (Fig. 1a). The large 5% increase in volume31,32,33,34 for the α → β + γ transition implies large values of dT0/dp, tending in the p → 0 limit to −14.0 ± 0.5 K kbar-1 on heating and −12.8 ± 0.5 K kbar−1 on cooling (right inset, Fig. 1a), as expected31, 43.

Fig. 1
figure 1

Superionic transition in AgI under pressure. a Heat flow dQ/dT on heating and cooling through the transition at different values of increasing pressure p, after baseline subtraction. At pressure p, we show (left inset) the thermally driven entropy change |ΔS0| and (right inset) transition temperature T0, on heating (red) and cooling (blue) (lines denote fits). b Entropy S′(T,p), constructed by plotting S′(T) at p ~ 0 and 2.5 kbar, on heating and cooling as indicated via arrows (S′ denotes entropy with respect to absolute entropy at 340 K and p ~ 0). Adiabatic (isothermal) trajectories in red (blue) are irreversible (1 → 2), or reversible (2 ↔ 3) if wholly within the reversibility region (grey)

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At each measurement pressure, integrating the calorimetry data over temperature yields the entropy change for the thermally driven transition alone \(\left| {\Delta S_{\mathrm{0}}} \right| = \left| {{\int}_{T_{\mathrm{1}}}^{T_{\mathrm{2}}} {({\mathrm{d}}Q/{\mathrm{d}}T){\mathrm{d}}T/T} } \right|\)7, 22 (left inset, Fig. 1a) (the calculation that yields the temperature and pressure dependence of the absolute entropy (Fig. 1b) is explained later). The value of |ΔS0| = 64 ± 2 J K−1 kg−1 at zero pressure is in good agreement with values previously obtained by experiment38,39,40, while the small decrease in |ΔS0(p)| with increasing pressure may be quantitatively understood in terms of ‘additional’ isothermal entropy changes22 ΔS+(p) = −(∂V/∂T)p=0 × p away from the transition (this expression for ΔS+(p) assumes Maxwell relation (∂S/∂p) T  = −(∂V/∂T) p ). Plots of V(T)33, 34, 44, 45 imply that these ‘additional’ entropy changes ΔS+(p) are negligible at temperatures below the transition, while at temperatures above the transition they are small and conventional such that ΔS+(0 → p) < 0.

Given that we identify the same zero-pressure value of |ΔS0| from a number of heating and cooling runs at zero pressure, we infer that the ratio of the β and γ polymorphs on cooling through the transition is likely to be constant, as expected30. Given also that the pressure dependence of |ΔS0| (left inset, Fig. 1a) can be explained purely in terms of the finite additional entropy ΔS+(p) at temperatures lying above the transition, we infer that the ratio of polymorphs also remains constant at finite pressure. The nominally constant ratio of β and γ polymorphs at any temperature and pressure (where the β + γ phase exists) implies that it is reasonable to use the quasi-direct method4, 7 later in order to evaluate BC effects associated with the α ↔ β + γ transition. Even if there were a change in this ratio, it should have little influence on these BC effects, as phonon spectra imply that the two polymorphs are separated in entropy by a small amount (~1%) that varies little with pressure (see Methods and Fig. 2).

Fig. 2
figure 2

First-principles calculations of AgI polymorph entropy. For the a β and b γ polymorphs in the β + γ phase, we show phonon dispersion spectra at the equilibrium volume (solid black and red), and on reducing the volume by 2% (dashed blue) (ω denotes the angular frequency of vibration on passing between the labelled points of the Brillouin Zone). c Vibrational entropy S for the β (black) and γ (red) polymorphs at 400 K, normalized by formula unit (f.u., left axis) and mass (right axis). Experimentally, a pressure of 1.25 kbar drives the transition at 400 K (vertical blue line)

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Sample entropy S′(T, p) was evaluated with respect to the absolute entropy at a given temperature TR below the transition as,

$$S\prime \left( {T{\mathrm{,}}\,p} \right)\,{\mathrm{ = }}\,S\left( {T{\mathrm{,}}\,p} \right) - S\left( {T_{\mathrm{R}}{\mathrm{,}}\,p} \right){\mathrm{ = }}\mathop {\int }\nolimits_{T_{\mathrm{R}}}^T \left( {c\left( T \right) + \frac{{{\mathrm{d}}Q}}{{{\mathrm{d}}T}}} \right){\mathrm{d}}T/T.$$
(1)

Given that below the transition (∂V/∂T) p is very small, S(TR, p) ≃ S(TR, 0), and taking TR = 340 K, the results for zero and maximum pressure (2.5 kbar) are shown in Fig. 1b. When integrating over temperatures lying inside the transition region, we used our measured values of dQ(T, p)/dT (Fig. 1a), when integrating over temperatures lying outside the transition region we set dQ/dT = 0, and at all temperatures of integration we used the specific heat capacity data c(T) measured39 at atmospheric pressure (p ~ 0). At temperatures lying above the transition region, the decrease of S′ with increasing pressure (Fig. 1b) arises as a consequence of the finite additional entropy ΔS+(0 → p) < 0 which, as previously mentioned, is due to (∂V/∂T) p  > 0.

Inverse BC effects driven using our maximum pressure change of 2.5 kbar are only reversible in thermally anhysteretic regions of parameter space (see refs. 46,47,48), e.g. in the region of (S′,T) space that is bounded by S′(T,p ~ 0) on cooling and S′(T,2.5 kbar) on heating (grey, Fig. 1b). By following adiabatic (isothermal) trajectories denoted red (blue) in Fig. 1b, we see that an irreversible BC effect (1 → 2), whose trajectory starts outside the reversibility region, is larger than the corresponding reversible BC effect (2 ↔ 3), whose trajectory lies wholly within the reversibility region.

By likewise obtaining S′(T,p) for our other measurement pressures, we use trajectories such as those described above to identify the maximum values of ΔS(T) (Fig. 3a, c) and ΔT(T) (Fig. 3b, d) that may be achieved irreversibly (Fig. 3a, b) and reversibly (Fig. 3c, d) on both applying (0 → p) and removing (p → 0) pressure. The maximum values of |ΔS| ~ 62 J K−1 kg−1 and |ΔT| ~ 36 K that can be achieved irreversibly with |Δp| ~ 2.5 kbar are reduced to values of |ΔS| ~ 60 J K−1 kg−1 and |ΔT| ~ 18 K when the constraint of reversibility is imposed. The magnitude of irreversible (Fig. 4a, b) and reversible (Fig. 4e) isothermal entropy changes compares favourably with the values recorded for the best BC materials11,12,13,14,15,16, 20,21,22 whether assuming normalization by mass or volume, especially because some of these literature values contain an irreversible component owing to the fact that hysteresis was not taken into account11,12,13, 15, 16, 20. Separately, our large reversible BC changes in entropy compare favourably with recently theoretically predicted49 large mechanocaloric changes in entropy in thin films of γ-AgI driven by biaxial stresses up to 10 kbar.

Fig. 3
figure 3

Giant inverse BC effects in AgI. a, c Isothermal entropy change ΔS and b, d adiabatic temperature change ΔT for applying and removing pressure p, under a, b irreversible and c, d reversible conditions. Data for p = 2.5 kbar deduced from Fig. 1b, data for other pressures deduced from analogous constructions

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Fig. 4
figure 4

Comparison of BC effects. For AgI and other BC materials11,12,13,14,15,16, 20,21,22,23, we show for pressure changes |Δp| of magnitude ~p both a, b peak values of the temperature-dependent isothermal entropy change |ΔSpeak|, and c, d the corresponding refrigerant capacity RC = |ΔSpeak| × [FWHM of ΔS(T)], as normalized by a, c mass and b, d volume. Solid lines represent fits. e The largest values |ΔSpeakrev| from a, b that may be achieved reversibly (solid columns represent mass normalization on left axis, shaded columns represent volume normalization on right axis). For clarity, (NH)4SnF6 alone represents the fluoride compounds20. The colour code for BC materials is common to all panels

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The ability to drive large BC effects over a large ~60 K temperature span (Fig. 3a, b), which arises because of the large shift in T0 with pressure (right inset, Fig. 1a), yields values of refrigerant capacity RC = |ΔSpeak| × [FWHM of ΔS(T)] that exceed the values reported for all known BC materials11,12,13,14,15,16, 20,21,22,23, again whether normalizing by mass (Fig. 4c) or volume (Fig. 4d). Here we have used our larger values associated with irreversible BC effects, in order to achieve a fair comparison with the available literature data. However, the value of RC = 1.1 kJ kg−1 (6.2 J cm−3) for p = 2.5 kbar computed from our reversible value of ∆S(T) (Fig. 3c) is also large.

In summary, our observation of giant and reversible inverse BC effects in the prototypical solid electrolyte AgI, near its superionic phase transition, should inspire the wider study of BC effects in similar materials. In future, one may decrease the transition temperature of AgI by chemical substitution42 or nanostructuring50. More generally, it would be attractive to reduce the thermal hysteresis of any given superionic transition in order to increase the magnitude of BC effects, and widen the temperature range of reversibility.

Methods

Sample characterization

Powders of AgI (99.999%) from Sigma-Aldrich were characterized using a commercial TA Q100 differential scanning calorimeter, and a high-resolution X-ray Bruker D8 reflection diffractometer with Cu Kα1 = 1.5406 Å radiation.

Pressure-dependent calorimetry

AgI powder mixed with an inert perfluorinated liquid was hermetically encapsulated by Sn. Measurements of heat flow under hydrostatic pressure were performed at approx. ±4 K min−1, using a bespoke differential thermal analyser whose resistive heater operates between room temperature and 473 K, and an Irimo Bridgman pressure cell that operates up to 3 kbar with a pressure-transmiting medium of Therm 240 (Lauda).

Phonon dispersion curves

First-principles density functional theory calculations were performed using VASP51, 52 and the so-called direct method53, where components of the force-constant matrix are obtained in real-space within the small displacement approximation. For our calculations, we used dense k-point grids for integration within the Brillouin Zone, large supercells with 256 atoms to guarantee negligible force-constant components at their boundaries, and we considered both positive and negative atomic displacements in order to obtain null values of acoustic phonons at the Brillouin zone centre54, 55.

Entropy of the β and γ polymorphs at 400 K

We first treat thermal effects within the quasi-harmonic (QHA) approximation55, 56 by writing the vibrational contribution to the Helmholtz free energy as:

$$F_{{\mathrm{vib}}}\left( {V,T} \right)=k_{\mathrm{B}}T\mathop {\sum }\nolimits_{\rm{q,s}} {\mathrm{ln}}\left[ {{\mathrm{2sin}}{\mathrm{h}} \left( {\hbar \, \omega _{{\mathrm{qs}}}/2k_{\mathrm{B}}T} \right)} \right],$$
(2)

where ωqs represent the vibrational phonon frequencies of the crystal calculated at fixed volume V, and the subscripts q and s run over wave-vectors that span the Brillouin Zone and phonon branches, respectively.

The resulting Helmholtz free energy of the crystal is given by:

$$F\left( {V,T} \right)\,{\mathrm{ = }}\,E_{{\mathrm{static}}}\left( V \right) + F_{{\mathrm{vib}}}\left( {V,T} \right),$$
(3)

where the zero-temperature energy contribution Estatic(V) is calculated by considering the atoms fixed in their crystalline lattice sites. Calculating F over a grid of volume and temperature points yields an estimate of pressure p(V,T) = −∂F(V,T)/∂V and entropy S(V,T) = −∂F(V,T)/∂T, which permits the entropy of the crystal to be expressed as a function of pressure and temperature. We assume that this vibrational entropy represents total entropy, as the electronic entropy is negligible in light of the fact that AgI is a non-magnetic wide band-gap semiconductor.

Data availability

All relevant data are available from the authors.