Matteo Oldoni

Fabien Seyfert

Steven Caicedo Mejillones

Stefano Moscato

Giuseppe Macchiarella

Passivity and Maximum Quality Factor Assessment in Lossy 2-port Transfer Functions

Keywords: Lossy network, predistortion, quality factor, passivity, microwave filters

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I. INTRODUCTION

In the theory of electric networks, relevant research has been and currently is dedicated to lossless transfer functions. However, within the field of RF and microwave filters, losslessness is not a correct assumption as physical passive components are in fact strictly lossy.

As a matter of fact, the lossless approach remains a handy tool in everyday work [START_REF] Cameron | Microwave Filters for Communication Systems[END_REF], and often the designer chooses this robust and consolidated method by introducing losses only in the last steps and minimizing their impact by optimization.

Substantially different routes are instead the "lossy" techniques, dedicated to transfer functions exactly implementable by passive components [START_REF] Guyette | The design of microwave bandpass filters using resonators with nonuniformq[END_REF]- [START_REF] Senior | Synthesis of lossy filters[END_REF] which suit disparate technological possibilities, e.g. [START_REF] Miraftab | Generalized lossy microwave filter coupling matrix synthesis and design using mixed technologies[END_REF]- [START_REF] Qiu | A flat-passband microstrip filter with nonuniform-q dual-mode resonators[END_REF]. In this area, most of the properties supporting the designer [START_REF] Fettweis | Scattering properties of real and complex lossless 2-ports[END_REF] (transmission zeros in paraconjugate pairs if not on the imaginary axis, guaranteed minimum McMillan degree...) do not hold any more and open up several degrees of freedom; their adjustment dramatically affects implementability, but this often becomes clear only at the end of the synthesis, thus demanding more iterations. This paper proposes in section II an analytical technique to verify if a given 2 × 2 rational scattering function is passive.

After facing this issue, another need arises: how to compare the synthesizability of lossy functions? The practical answer proposed by this work is to evaluate the maximum required quality factor for synthesis. Section IV describes an analytical method relying on the solution of two bivariate polynomials.

II.

PASSIVITY REQUIREMENTS A 2-port reciprocal system is described in the complex domain s = σ + jω by the symmetrical scattering matrix S(s). As often the case, the involved functions are moreover rational, i.e. each is a ratio of polynomials in s and share the denominator D(s) which holds the N poles of the system as its roots and which is assumed monic (i.e. with unitary leading coefficient) without loss of generality:

S(s) = S 11 (s) S 21 (s) S 21 (s) S 22 (s) =   N 11 (s) N 21 (s) N 21 (s) N 22 (s)   D(s) (1)
All the involved polynomials are considered to be known, i.e. computed according to the requirements. A common further assumption is that the network can be implemented with N resonators, thus not requiring redundancy. This implies that the transfer function has minimum McMillan degree, which means that there exist a polynomial P (s):

N 11 (s)N 22 (s) -(N 21 (s)) 2 = D(s)P (s). (2)
The minimum McMillan degree property can be checked by computing the complex roots of the left-hand member and verifying that all the poles are among them. If so, the P (s) polynomial can be computed as taking all the other (not of D(s)) roots of the left-hand member and with a leading coefficient equal to that of the left-hand member.

The involved polynomials are classically with real coefficients in the s variable, guaranteeing that no constant reactances are needed. However, most synthesis techniques are applied in a lowpass domain where constant reactances are allowed, as they approximately translate to physical components in bandpass/bandstop domains. As a consequence, we consider complex polynomial coefficients to be likewise allowed, as they do not affect passive realizability in such cases.

For a passive reciprocal transfer function, two simultaneous conditions on the scattering matrix must be verified, [START_REF] Triverio | Stability, causality, and passivity in electrical interconnect models[END_REF]:

1) Each element of S(s) is analytic in Re (s) > 0, thus all the poles lie strictly in the left half-plane (Hurwitz); 2) A(s) = I -S * (s)S(s) = I -S(-s)S(s) is positive semi-definite for every s = jω where ω ∈ R, implying that the two eigenvalues of A(jω) are always both positive for whatever real value of ω. The complex conjugate of a complex x is denoted as x; the * operator is here the classical paraconjugate of complex rational functions, that is S * (s) = (S(-s)) t = S(-s). Practically, here the * operation amounts to conjugate the polynomial coefficients of the matrix rational function and to change the variable s in -s.

Condition 1 is immediate to check, hence not studied further. Condition 2 instead requires to evaluate A(jω) over the whole ω axis and check the eigenvalues at each step: this represents a cumbersome task whose outcome depends on the choice of finite sampled values of ω. To avoid such bruteforce attempts, the following section III develops a novel formulation which requires only three root computations.

III. PASSIVITY ASSESSMENT

Several approaches can be devised to verify the positive semi-definiteness of A(s) by exploiting algebraic properties and the inner structure of S(s). For any of these, first the matrix A(s) = I -S * (s)S(s) must be developed:

(3) For s = jω, A(s) is Hermitian and its positive definiteness on the whole imaginary axis is equivalent to the system passivity, provided condition 1) is met.

A(s) = 1 - N 11 N *

A. Sylvester's criterion

For a Hermitian matrix like A(jω) to be positive semidefinite, the Sylvester's criterion gives a set of conditions equivalent to the unpractical inspection of eigenvalues in ω:

A 1,1 (jω) ≥ 0 A 2,2 (jω) ≥ 0 Det (A(jω)) ≥ 0
The first two conditions are equivalent to:

C i (s) = DD * -N ii N * ii -N 21 N * 21 ≥ 0 for i ∈ {1, 2} (4
) By moving into the Fourier domain s = jω, the expression becomes a real polynomial in ω: if any of the two conditions has a real root ω with odd multiplicity, then A(jω) is not always positive semi-definite and S is not a passive network.

For the third condition we make use of the formula ( 5)

Det (I + B) = 1 + Tr(B) + Det (B)
which holds for any matrix B of size 2×2 and where Det and Tr denote respectively determinant and trace of their matrix arguments. This yields: ( 6)

Det (A(s)) = 1 -Tr(S * S) + Det (S * S) = DD * -N 11 N * 11 -N 22 N * 22 -2N 21 N * 21 + P P * DD *
where Det (S * S) has been computed using [START_REF] Guyette | The design of microwave bandpass filters using resonators with nonuniformq[END_REF]. Therefore,

C 3 (s) = DD * -N 11 N * 11 -N 22 N * 22 -2N 21 N * 21 + P P * ≥ 0. (7)
Checking the passivity of a 2×2 transfer function with poles in the left-hand plane thus amounts to assess the non-negativity of three real valued polynomials, C i (s), on the imaginary axis.

Proposition 3.1: Let q(ω) be a real polynomial of degree 2N . Denote by q k , k = 0 . . . n the coefficient of q(ω) in ω k . The polynomial q is non-negative on the real line if and only if q 0 ≥ 0 and q 2N ≥ 0 and either one of the following condition holds: 1) Any positive real root of q is of even multiplicity.

2) If ω i > 0 are the positive real roots of q , the derivative of q, then q(ω i ) ≥ 0 holds for all i. The two criteria necessitate, when applied to the passivity testing of our system of McMillan degree N , the computation of roots of polynomials of degree 2N and 2N -1.

Criterion 1) can be moreover tested using a purely algebraic procedure, that is without recurring to a numerical root solving procedure and by using Sturm sequences. To do so what is first needed is a square free decomposition of the polynomial q, that is the determination of the square free factors q i such that q = n i=1 (q k ) i . Here all the q k are coprime and have simple roots; these factors can be obtained via greatest common divisor calculations. For example q 1 = q/gcd(q, q ) and all other factors can be obtained in a similar fashion. Once the factors q k have been obtained, criterion 1) amounts to test for the presence of positive zeros in the q k s for every odd values of k. This can be done by first removing, if any, the possible roots of q k at zero, and then computing the sign changes in the Sturm signed reminder sequences [START_REF] Basu | Algorithms in Real Algebraic Geometry, ser. Algorithms and Computation in Mathematics[END_REF] at 0 and +∞.

The burden of the proposed procedures is lower than sampling the ω axis and they are thus more robust and convenient.

B. Numerical Example

An example is given for a 4-th degree transfer function with negative real parts of poles and N 22 (s) = -N 11 (s): The magnitude response of this scattering transfer function is shown in fig. 1 with the "predistorted" label (its meaning will be clearer in a successive example), and no non-passivity areas are evident from it. However, the evaluation of C i (jω) highlights two small regions where eq. ( 4)< 0 and two larger ones where eq. ( 7)< 0: the network is in fact not passive. 4) and of C 3 (jω) of eq. ( 7) of a non-passive transfer function, observed by C 3 (jω) being negative at some real values of ω.

To verify the proposed method, the ω roots of eqs. ( 4) and ( 7) (all with positive leading coefficients and constant term) can be respectively computed as: {4 cplx roots; ±6.9389; ±6.7069} {4 cplx roots; ±7.5540; ±6.0998}

The computed real roots correspond to the sign changes in fig. 2 and, as there exist at least one real ω root (there are four for each C i here), the proposed method successfully detects this transfer function as not passive.

As alternative confirmation, the real roots of the derivative of C 3 (jω) are {±6.9849; 0}, and the values of C 3 (jω) at these points are -6.25 • 10 4 ; 3.1 • 10 5 , which contain negative values hence making the network not passive. Similarly, the Sturm's theorem detects 4 distinct real roots to each condition.

IV. FINDING REQUIRED Q

After having established in a binary way whether a given transfer function is passive or not, the related problem is to quantify its passivity, for comparing different functions. The classical tool here proposed to this end is predistortion, which creates a function Ŝ(s) by shifting the complex plane:

Ŝσ (s) = S(s + σ), with real σ (8)
A negative predistortion σ < 0 creates Ŝσ with all its poles and zeros shifted toward the right of the s-plane. If S(s) is passive, Ŝ(s) σ has poles closer to the imaginary axis, hence sharper peaks in the magnitude response of fig. 1. By making σ more and more negative, Ŝ(jω) σ will become no longer the method of section III can be used for this check. The maximum predistortion violating passivity (σ 0 ), i.e. making the distorted function lossless at a real frequency, is computable for any transfer function and enables comparisons.

The practical usefulness of this proposed measure of passivity is that -σ 0 is inversely related to the maximum quality factor required by a circuit implementation. This relation, in common experience, holds generally although arbitrary lossy synthesis techniques often involve approximations, optimizations or no guarantees that components are all real; the relation is instead clearer when performing synthesis by even-odd mode decomposition [START_REF] Oldoni | A new approach to the synthesis of microwave lossy filters[END_REF] (requiring S 11 = S 22 ), in which the synthesis is reduced to two subnetworks with prescribed input reflection functions: at each step, a frequency where the 7) via numerical evaluation on the complex plane. Their apexes are denoted as σ 3,1 and σ 3,2 . Another value of σ = -0.25, used in fig. 1 is also depicted along with the two corresponding ω intersections network is lossless is needed (as the one obtained by applying σ 0 ) to then extract and remove one lossless resonator, i.e. by section extraction. Further predistortion σ n = σ n-1 +δ n can be cumulated after each step and the network is so synthesized by lossless resonators. It is then converted to implement the original non-distorted lossy response by applying the opposite σ n to all frequency-dependent components, i.e. resonators, which therefore become lossy with a quality factor Q n ∝ -1 σn , in general non-uniform across the whole network. The maximum required quality factor hence is given by the least negative value, which is σ 0 .

The following sections outline an analytical approach to determine σ 0 which relies on bivariate polynomials.

A. Passivity as predistortion

The rationale for this approach is that the predistorted polynomials defining Ŝσ (s), when evaluated along the imaginary axis jω, actually represent the values of the original polynomials along a shifted axis. As assessment of passivity must take place in the predistorted domain, the predistorted polynomials ( N11 (s), N21 (s), N22 (s), D(s) and P (s)) must first be paraconjugated and evaluated along jω. For instance:

N * 11 (jω) = N11 (-jω) = N 11 (-jω + σ). (9)
Passivity conditions built on the predistorted functions require that, for a given σ, D(s) be Hurwitz and neither of eqs. ( 4) and ( 7) recomputed on the σ-predistorted polynomials and denoted as C i,σ have any odd-multiplicity imaginary root jω.

Considering only one of the 3 conditions, the locus of points σ + jω where C i,σ (jω) = 0 is represented by curves (0-level contours) in the complex plane, as in fig. 3. These curves will lie entirely in a half-space σ < σ i,j , each curve j with its own σ i,j . Considering only one of such curves, with an excessive predistortion σ < σ i,j the curve will cross σ at least twice (say at ω i,j,1 or 2 (σ)): thus one of the eigenvalues of the matrix A changes sign at ω i,j,1,2 (σ), hence being negative in an ω range and representing a non-passive predistorted function.

For the numerical example considered, the value of σ = -0.25, used to predistort the response in fig. 1, intersects the 0-level contour at ω 3,1,1 (-0.25) 6.08 and ω 3,1,2 (-0.25) = 6.75. Therefore, that amount of predistortion is excessive, as it causes violation of passivity by the distorted transfer function.

When σ > σ i is chosen, instead, the curve does not intersect σ, and thus both eigenvalues are strictly positive, as in a passive predistorted function. When several curves are present, σ > σ i = max j σ i,j guarantees that condition i is fulfilled and σ i is the allowed predistortion. From this pictorial view, the determination of σ i can be turned into an algebraic problem.

This approach must be applied to all conditions C i,σ (jω) and the maximum σ 0 = max σ i among the three constitutes the allowed predistortion σ 0 of the transfer function (possibly limited by the poles remaining in the left half-plane).

B. Bivariate polynomials approach

Notably, the C i,σ (jω) do not coincide with either a mere predistortion applied to the polynomials eqs. ( 4) and [START_REF] Muller | Lossy coupling matrix filter synthesis based on hyperbolic reflections[END_REF] or their evaluation along s = σ + jω. The resulting expressions are in fact no longer polynomials in a single complex variable: conditions C i,σ (jω) are made by terms in the form f (s) f * (s) evaluated along jω with f (s) any of the σ-predistorted polynomials. The non-paraconjugated terms f (s) evaluated in jω are still polynomials in s = σ + jω:

(10) f (s) jω = f (s + σ)| jω = f k (s + σ) k jω = f k (σ + jω) k
However, the paraconjugated terms f * (s) when evaluated in jω are instead polynomials in σ -jω = s:

f * (s) jω = f (-s + σ) jω = f k (-s + σ) k jω = f k (σ -jω) k
Although the C i,σ are not polynomials in a single complex variable but actually involve both s and s (from terms paraconjugated and not), each of these can be converted into real bivariate polynomials. For a general polynomial f (s), a function f (σ, ω) can be defined on real parameters, σ and ω:

f (s) = f k s k = f (σ + jω) = f (σ, ω) = g n,m σ n ω m (11)
where f (σ, ω) is a bivariate polynomial. The conversion from univariate complex to bivariate real is merely the binomial expansion of the variable in f (s):

f (s) = f k (σ + jω) k = f k n k n j k-n σ n ω k-n = g(σ, ω) (12
) Similarly, for its paraconjugate part (as given in eq. ( 9)), f * (s) will involve the conjugated coefficients and the oddpower terms of ω will receive a further sign change.

Therefore, once the equivalent bivariate forms for D(s), D * (s), N 11 (s), N * 11 (s), N 21 (s), N * 21 (s), N 22 (s), N * 22 (s), P (s), P * (s) have been computed, the required products and sums create three large bivariate polynomials C i (σ, ω).

To determine the allowed predistortion of condition i, one can state that it lies within the roots of C i (σ, ω), i.e. it is onto the curves in the complex plane as in fig. 3, which are however rather complicated to compute. More specifically, in fact, the interest is in determining the apexes of those curves and selecting the right-most one. Thus a further algebraic condition is introduced: the partial derivative of the bivariate polynomial with respect to ω in the desired solution must be 0, which specifies that the relevant points are only those where the curves are tangent to the σ axis.

A system of two bivariates is hence built for each C i :

{C i (σ, ω) = 0} ∩ {∂C i (σ, ω)/∂ω = 0} , i ∈ {1, 2, 3} . (13)
This system has a 0-size solution, i.e. a finite number of points, determinable by algebraic techniques or dedicated solvers. Concerning the former, for the two large bivariate polynomials to have a common root, their resultant must be 0. Hence values of σ i,j can be obtained by computing the real roots of the determinant of the Sylvester's matrix:

R(σ) = Res ω {C i (σ, ω); ∂C i (σ, ω)/∂ω} (14)
Although R(σ) can be of high degree even for low-order functions, efficient algorithms dedicated to real roots are available. To this end, the polysol2 tool, part of the TensorLab package [START_REF] Vervliet | [END_REF], computes the resultant and solves a generalized eigenvalue problem, whose results are then selectively discarded and the rest improved by local optimization.

Of the obtained finite solutions σ i,j (the corresponding ω i,j are discarded), the one with the largest σ must be taken as allowed predistortion for condition i: σ i = max σ i,j .

The maximum of those is the least predistortion which makes the network lossless, hence its inverse is the largest required quality factor: the more negative the allowed predistortion is, the "more lossy" the transfer function is, the poorer resonators are needed for its synthesis.

C. Numerical Example

The numerical example is the lossy set of polynomials obtained by scaling by 3 dB a 4th-degree elliptic response with 15 dB return loss in the ±2π passband and more than 30 dB attenuation beyond ±3π: N11(s) = -N22(s) = (0.9995s (side note: these polynomials, predistorted excessively with σ = -0.25, give those of the example of section III-B, which are detected to be non-passive).

Their bivariate version is computed; for instance: By multiplying the bivariate forms of non-paraconjugated and paraconjugated polynomials according to eqs. ( 4) and ( 7), the bivariate polynomials C i (σ, ω) are finally obtained. The polysol2 tool yields, applied to this lossy scattering parameters set, for condition C 1 of eq. ( 4) and its derivative in eq. ( 13) (the same for C 2 ):

D(σ, ω) = ω 4 -4jσω 3 -10.196jω 3 -6σ 2 ω 2 -30.588σω
(15) σ1,1 = -0.2520 with ω1,1 = ±6.82 σ1,2 = -28.9658 with ω1,2 = 0 whereas, for C 3 of eq. ( 7) and its derivative in eq. ( 13): The maximum values for each condition are σ 1 = σ 2 = -0.252 and σ 3 = -0.1698, hence the maximum allowed predistortion is σ 0 = -0.1698, as also confirmed by fig. 3. A synthesis of the corresponding lumped-element circuit with passband edge at ±1 Hz is carried out by means of the evenodd decomposition technique in [START_REF] Oldoni | A new approach to the synthesis of microwave lossy filters[END_REF]. After removing a unitary admittance inverter on port 2, the circuit shown in fig. 4 is obtained, whose response exactly corresponds to the "lossy" one in fig. 1. The quality factor of the involved resonators at the passband edge is found to be 37 and 9.3. As expected, the maximum quality factor involved satisfies 37 = 2π•1 Hz σ0 .

V. CONCLUSION

This work addresses two closely-related issues arising when dealing with non-lossless transfer functions. One is the determination of whether a given 2-port scattering matrix is passive, whose proposed solution is based on a simple rootfinding operation over 3 independent polynomials, which is much more effective than inspecting the sampled responses; alternative root-counting algorithms are also proposed to the same purpose and a numerical example proves the approaches.

The other is the identification of the maximum allowed predistortion for passivity comparisons, which is also directly related to the required quality factor for resonators. The solution to this problem is provided as an analytic procedure with bivariate polynomials, also validated by numerical examples.

These tools can prove to be effective for the synthesis of transfer functions with practical lossy components, where many degrees of freedom with unclear effects must be handled.

M

  . Oldoni, S. Caicedo and S. Moscato are with R&D Dept., SIAE Microelettronica; via Buonarroti 21, IT-20093, Cologno Monzese (MI), Italy. F. Seyfert is with the Functional analysis for conception and analysis of systems (FACTAS) Team, INRIA; Sophia Antipolis cedex, BP 93, France G. Macchiarella and S. Caicedo are with the Electronics, Information and Bioengineering Dept., Politecnico di Milano; via Ponzio 34/5, IT-20133, Milan, Italy Manuscript received...; revised....

Fig. 1 .

 1 Fig. 1. Lossy scattering parameters obtained by scaling by 3dB a Cauer (elliptic) 4th degree function (blue and red traces) and predistorted version with σ = -0.25 (green and orange traces). The predistorted version is the one considered for the example of section III-B

Roots

  {N11(s)} = {0.2547 ± j5.9220; 0.2547 ± j2.7191} ; Roots {N21(s)} = {0.2547 ± j21.5363; 0.2547 ± j9.8883} ; Roots {D(s)} = {-0.7698 ± j6.8332; -3.8187 ± j3.7327} ; Roots {P (s)} = {1.2793 ± j6.8332; 4.3281 ± j3.7327} ; N11(s) = 0.7076s 4 -0.7210s 3 + 30.3223s 2 -15.3545s + 185.421; N21(s) = 0.022387s 4 -0.02281s 3 +12.5811s 2 -6.4066s+1016.1; D(s) = s 4 + 9.1769s 3 + 87.558s 2 + 405.03s + 1348.3; P (s) = -0.5012s 4 + 5.6207s 3 -51.6931s 2 + 251.556s -791.215.

Fig. 2 .

 2 Fig. 2. The evaluation of C 1 (jω) and C 2 (jω) of eq. (4) and of C 3 (jω) of eq. (7) of a non-passive transfer function, observed by C 3 (jω) being negative at some real values of ω.

Fig. 3 .

 3 Fig. 3. Locus of points (σ, ω) solving C 3,σ (jω) = 0 of the lossy function in fig. 1. Two curves are shown (others appear symmetrically for ω < 0 due to the symmetry of the example) as obtained by drawing the 0-height contour of eq. (7) via numerical evaluation on the complex plane. Their apexes are denoted as σ 3,1 and σ 3,2 . Another value of σ = -0.25, used in fig.1is also depicted along with the two corresponding ω intersections

Fig. 4 .

 4 Fig. 4. Synthesized lumped-element circuit for the lossy response of fig. 1 and section IV-C. Solid lines represent admittance inverters with the prescribed admittance inversion constant (J); solid circles represent unitary resonators with the prescribed quality factor Q. Hollow circles denote non-resonant nodes with shunt susceptance B. Barred circles are lossy non-resonant nodes with shunt admittance G + jB. Triangles denote external ports with unitary reference impedance.

  (16) σ3,1 = -0.1698 with ω3,1 = ±6.8305 σ3,2 = -0.6723 with ω3,2 = ±3.6777 σ3,3 = -0.7286 with ω3,3 = ±4.8611 σ3,4 = -1.1304 with ω3,4 = 0 σ3,5 = -1.4572 with ω3,5 = ±2.3139 σ3,6 = -2.8467 with ω3,6 = ±10.7140 σ3,7 = -29.122 with ω3,7 = ±1.9

  = s 4 + 10.1959s 3 + 94.9604s 2 + 451.491s + 1457.34; P (s) = -s 4 + 10.1959s 3 -94.9604s 2 + 451.491s -1457.34.

	4 + 42.4423s 2 + 259.1558)/ N21(s) = (0.03166s 4 + 17.7801s 2 + 1435.8135)/ √ 2;	√ 2
	D(s)

  2 -94.960ω 2

	+ 4.000jσ 3 ω + 30.588jσ 2 ω + 189.921jσω + 451.491jω
	+ σ 4 + 10.196σ 3 + 94.960σ 2 + 451.491σ + 1457.34
	and the bivariate form of the D * (s) is found by conjugating
	all coefficients of D(s), then applying bivariate expansion and
	further changing sign of the coefficients of odd-powers of ω:
	D(σ, ω) = ω 4 + 4jσω 3 + 10.196jω 3 -6σ 2 ω 2 -30.588σω 2 -94.960ω 2
	-4jσ 3 ω -30.588jσ 2 ω -189.921jσω -451.491jω
	+ σ 4 + 10.196σ 3 + 94.96σ 2 + 451.491σ + 1457.34


Lars Fritzsche
email: lars.fritzsche@imk-automotive.de

Christian Gärtner

Michael Spitzhirn

Pavel E Galibarov

Michael Damsgaard

Pauline Maurice

Jan Babič

Assessing the efficiency of industrial exoskeletons with biomechanical modelling -Comparison of experimental and simulation results

Keywords: Musculoskeletal modelling, exoskeleton, biomechanical simulation, industrial exoskeletons, AnyBody Modelling System, Paexo Shoulder

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Introduction

Industrial exoskeletons can potentially be used for supporting workers in heavy physical tasks that may be associated with high risks for developing musculoskeletal disorders (MSDs). Laboratory experiments and field studies are common approaches to evaluate feasibility and effects of industrial exoskeletons (de Looze et al., 2015). This kind of research is necessary to get valuable and real-world insights into objective measures as well as subjective comfort evaluation and acceptance. However, they often require complex sensor technologies like EMG electrodes that are difficult to use in practice. They also do not allow conclusions about internal joint loads or compensatory mechanisms and they are costly or may disrupt ongoing work in field studies. Biomechanical simulations can tackle these challenges because they can be prepared on a computer and safely analyzed using digital human models. They also allow to calculate forces or joint moments inside the body and they can be used for investigating various use cases without expensive experimental set up.

In this paper we present an approach to extent laboratory and field studies by using the biomechanical software AnyBody Modelling System provided by AnyBody Technology A/S, Denmark.

Methods

This study applies the simulation framework firstly presented in Galibarov et al.

(2019) using a set of data recorded at a laboratory experiment and compares simulation and experimental outcomes. The laboratory experiment included 12 participants (all male) performing an overhead drilling task with a hand-held tool (Figure 1). Detailed descriptions are presented in [START_REF] Maurice | Objective and Subjective Effects of a Passive Exoskeleton on Overhead Work[END_REF], the data set is available on Zenodo (http://doi.org/10.5281/zenodo.1472214). All participants performed the task while wearing the Paexo Shoulder exoskeleton (WE), and without wearing it (NE). Whole body kinematics were retrieved using an Xsens MVN inertial motion tracking suit. Moreover, muscle activities of the right anterior deltoid and right erector spinae longissimus were recorded with Biometrics EMG system. Additionally, heart rate and oxygen consumption were measured as indicators of metabolic effort. Using the motion capturing data that were recorded at the laboratory experiment, simulations were created and analyzed with AnyBody Modeling System v.7.3.2 (AMS). AMS uses a biomechanical human model comprised of most of the muscle elements, bones, and joints in the body. The system computes muscle activations, joint moments and reactions forces necessary to generate the specified motions by recruiting muscles in an optimal way [START_REF] Damsgaard | Analysis of musculoskeletal systems in the AnyBody Modeling System[END_REF]. In this study, a total of 2.880 trials (12 participants, 2 conditions NE/WE, 5 sets of 24 trials) were simulated and processed using a Python script [START_REF] Lund | AnyPyTools: A Python package for reproducible research with the AnyBody Modeling System[END_REF]. The processing model used anthropometric measurements available in the recorded Xsens files to scale corresponding model body parts to consistently represent body size of the participants (Figure 2). Inverse dynamics analysis was then carried out to compute estimations of muscle activities and joint reaction forces.

Based on the repeated-measures design in the laboratory experiment, simulation data was grouped into two conditions "with exoskeleton" (WE) and "without exoskeleton" (NE). Data analysis was also done using a Python script to automatically analyze the 2.880 trials for each simulation variable. Single outlier values beyond three standard deviations above or below the mean were excluded. Descriptive data analysis included calculating box-plots and histograms for each variable. Wilcoxon signed-rank test was used to test for significant differences between the two conditions, since most of the data was not normally distributed according to Kolmogorov-Smirnov-Test. Results were considered as statistically significant with α < .05 (two-sided).

Results

Results of the AMS biomechanical simulation demonstrate that wearing the exoskeleton reduces muscle activation in the three deltoid shoulder muscles (anterior, posterior, lateral) by 74 % to 87 % compared to the baseline activation with no exoskeleton. Effects were stronger for the right deltoid muscles than for the left deltoid muscles because the drilling tool was used in the right hand, whereas the left hand was only needed to stabilize the body (Figure 1). Similarly, infraspinatus muscle at the shoulder front showed a high baseline activation (NE) at the right side that is reduced by 42 % while wearing the exoskeleton (WE). Other relevant muscles in the shoulder/arm area, such as triceps, biceps and trapezius, showed a very low baseline activation (< 10% of maximal possible activation in both conditions on both sides). This indicates that these muscles were not very much involved in the overhead drilling task, although some of them also showed significant decreases of muscle activation on a low level. Joint reaction forces in the shoulder were also analyzed by means of AMS simulation. The glenohumeral joint forces were reduced between 56 % and 80 % in all three force directions while wearing the exoskeleton. Similarly, forces in the acromioclavicular joint and in the sternoclavicular joint were reduced between 54 % and 68 % in all directions while wearing the exoskeleton, with one small opposite effect in the sternoclavicular medio-lateral force on a very low force level (4.5 N increase in WE condition). Overall, results on joint reaction forces are very consistent across different joints and force directions indicating that the exoskeleton is substantially reducing strain in the shoulder joints. Finally, muscle activities and joint forces in the spine were analyzed with the simulation. Results showed a medium activation pattern in erector spinae muscles on left and right side (approx. 20 %), but there were no significant differences with or without exoskeleton use. Moreover, compression forces in the L5/S1 disc area were unchanged in two directions or even slightly decreased (approx. 12 %) while wearing the exoskeleton. Overall, wearing the exoskeleton did not significantly influence muscle activities or joint forces in the lumbar spine.

Discussion

Results of the AMS simulation study suggest that the Paexo Shoulder exoskeleton is an effective device to reduce biomechanical strain in overhead drilling tasks. Muscle activations of the shoulder complex are reduced, which should decrease the fatigue level of the workers. Similarly, it reduces reaction forces in the shoulder joint supposedly leading to a decrease in shoulder joint cartilage degeneration rates. The device does not redistribute the arm loads onto the lumbar spine, indicating that no adverse side effects for the lumbar spine have to be expected. Results of the laboratory experiment and the simulation are quite similar for most parameters: (1) in the lab, measured EMG activity in the anterior deltoid muscle is decreased by 54 % in average; in the simulation, the same muscle activity is reduced by 74 % while wearing the exoskeleton. (2) Muscle activity of the erector spinae does not show any difference for with/without exoskeleton conditions in both laboratory measurement and simulation. (3) Metabolic parameters in the lab, such as decreased oxygen consumption (-33 %) and heart rate (-19%), also confirmed that the use of the exoskeleton is related to reduced strain for the entire body. In summary, these results are suggesting that the simulation framework is a valid approach for investigating the effects of exoskeletons supplementing experimental studies by providing insights into changes inside the human musculoskeletal system. This framework could be extended by analyzing a variety of basic movements/tasks wearing the exoskeleton with different human populations. It also allows investigating intended main effects as well as side effects of exoskeletons (and possibly other wearable devices). Such analysis can not only be used for product evaluation, but also for improving the design and functionalities of exoskeletons in the development phase.

Limitations include the exoskeleton fit to the body, because motion of the exoskeleton was not tracked during the experiment (virtual fitting of the exoskeleton was done for each participant according to the manufacturer's guidelines, but may contain some discrepancies with reality). Furthermore, experimental and simulation data can only be compared to some extent, because computed muscle activations represent a percentage of a maximum muscle force needed to perform motions and do not necessarily match EMG signals on an absolute scale. Future research should take into account that the presented study and the simulation framework was developed based on the evaluation of passive exoskeletons with mechanical components (springs, etc.). It seems more difficult to apply for soft exoskeletons with mainly textile components and active exoskeletons with external power supply. Another field of future research is the question how biomechanical simulations can be prepared without any motion capturing data. AMS already allows to do that, but it still requires to a lot of expertise and effort to create realistic simulations. Other digital human models, such as ema Work Designer (Fritzsche et al., 2019) use algorithms for generating artificial motions with lower effort, which potentially could be used as an input for AnyBody biomechanical simulations in order to make such studies independent from lab recordings and allow evaluation of virtual prototypes.

Fig. 1 .

 1 Fig. 1. Experimental set up at the laboratory experiment (adapted from Maurice et al, 2020).

Fig. 2 .

 2 Fig. 2. AnyBody Model with "Paexo Shoulder" exoskeleton (left); application of Paexo Shoulder model in overhead drilling task (right) (adapted from Fritzsche et al., in press).

  The simulation is applied to the "Paexo Shoulder", a commercially available exoskeleton provided by Ottobock SE & Co. KGaA, Germany. It weighs approx. 1.9 kg and is specifically designed to support overhead work.

Detailed study results are published in Fritzsche et al. (in press).


Araceli Aznar

Philippe Negrier

Antoni Planes

Lluís Mañosa

Enric Stern-Taulats

Xavier Moya

María Barrio

Josep-Lluís Tamarit

Pol Lloveras
email: pol.lloveras@upc.edu

Reversible colossal barocaloric effects near room temperature in 1-Xadamantane (X=Cl, Br) plastic crystals

Keywords: Barocaloric effects, plastic crystals, 1-bromoadamantane, 1chloroadamantane, pressure, entropy

Plastic crystals undergo phase transitions with unusually large volume and entropy changes related to strong molecular orientational disordering. These features have led to a resurgent interest in these materials because recently they have shown great potential in solid-state cooling applications driven by pressure. Here we demonstrate that two plastic crystals derived from adamantane -1-Br-adamantane and 1-Cl-adamantane-undergo colossal reversible barocaloric effects under moderate pressure changes in a wide temperature span near room temperature thanks to a relatively small hysteresis and very high sensitivity of the transition temperature to pressure. In particular, 1-Cl-adamantane displays an optimal operational temperature range covering from ~40 K below and up to room temperature, and under a pressure change of 1 kbar this compound outperforms any other barocaloric material known so far. Our work gives strong support to plastic crystals as best candidates for barocaloric cooling. We also provide insight into the physical origin of the entropy changes through the analysis of the disorder on the involved phases.

Introduction

One of the numerous urgent environmental challenges posed by the global warming is related to the need to replace the widely extended use of vapor compression in billions of air conditioners and refrigerators [1,[START_REF]Energy savings potential and rd&d opportunities for non-vapor-compression hvac technologies[END_REF]. This mature technology shows moderate efficiencies and uses greenhouse fluids that contribute significantly in terms of CO2equivalent emissions. One of the best-positioned alternative methods is based on the exchange of the latent heat at solid-state first-order phase transitions driven by external fields such as magnetic, electric or mechanic, leading to magnetocaloric, electrocaloric and mechanocaloric effects, respectively [START_REF] Mañosa | [END_REF][4][5]. In addition to the use of Global Warming Potential-free refrigerants, these methods may also exhibit additional advantages as higher efficiencies or scalability, allowing other applications such as wearables, solutions for the overheating of electronic elements that restrict their miniaturization, or for heat pumping [6] and energy harvesting and conversion. In the last years, intense research has led to significant advances as for magnetocaloric (MC), electrocaloric (EC) and elastocaloric (eC) materials and prototypes [7]. However, a major limitation for a competitive implementation is given by the available nontoxic materials, which exhibit small refrigerant capacity and may suffer from irreversibility. These issues may be overcome by applying high fields, but this is technologically challenging, energetically costly and may be destructive, leading to mechanical failure and/or dielectric breakdown. Pressure-induced (barocaloric, BC) effects have been less investigated but on the one hand they promise better efficiencies than vapor compression [8] and on the other hand the abundance of potential BC materials [START_REF] Lloveras | Advances and obstacles in pressure-driven solid-state cooling: A review of barocaloric materials[END_REF] has allowed the identification of BC effects that surpass in many aspects best MC, EC and eC effects reported so far.

Recently, plastic crystals have been identified as outstanding barocaloric (BC) agents [10][11][12] because they may display enormous entropy changes related to orientational disorder released across solid-to-solid phase transitions that are sensitive to mechanical stress due to associated large volume changes. In the case of plastic crystal neopentylglycol, it has been shown that, at the atomic scale, a large compressibility facilitates the shortening of the hydrogen bond length by application of pressure, thus leading to an increase of the orientational energy barrier and, therefore, to the stabilization of the ordered phase [13].

While the extremely high fragility of these compounds prevents the cyclic application of uniaxial stress without compromising their mechanical integrity and hinders very much the application of electric fields on their polar molecules, in form of powder plastic crystals can be easily subjected to changes in hydrostatic pressure. Therefore, these compounds may offer opportunities in energy-related applications based on controlling by pressure the absorption and release of large amounts of latent heat associated with their solid-state transitions, such as the mentioned BC effects for cooling or heat pumping.

To date, among plastic crystals only neopentane derivatives [10][11][12] and C60 [14] have been analyzed for BC applications and so there are still many compounds of this kind that remain unexplored. In this work we open up the BC analysis to two plastic crystals derived from adamantane (C10H16): 1-bromoadamantane (C10H15Br, 1-Br-ada for short) and 1-chloroadamantane (C10H15Cl, 1-Cl-ada for short). 1-Br-ada and 1-Cl-ada are obtained via substitution of an H atom linked to a 3-coordinated carbon site by a Br or Cl atom, respectively [15]. They are widely used as precursors in the synthesis of other adamantane derivatives used in pharmaceutical industry as antiviral drugs and also for the synthesis of polymers and other materials [16,17]. At low temperatures both 1-Br-ada and 1-Cl-ada are isostructural and exhibit a fully ordered monoclinic (M) structure (𝑃𝑃2 1 /𝑐𝑐, Z=4) [18]. At high temperatures, they are also isostructural, arranging in an FCC lattice (𝐹𝐹𝐹𝐹3 � 𝐹𝐹, Z=4) with orientational disorder that has been suggested to consist of quasi-free rotation [19]. In 1-Cl-ada, the orientational disorder fully develops in a single M→FCC transition that displays a giant entropy change ∆𝑆𝑆 𝑀𝑀→𝐹𝐹𝐹𝐹𝐹𝐹 = 135 J K -1 kg -1 and volume change ∆𝑉𝑉 𝑀𝑀→𝐹𝐹𝐹𝐹𝐹𝐹 ~ 47 • 10 -6 m 3 kg -1 (corresponding to a relative volume change . These giant values from literature for the transition entropy change and 𝑑𝑑𝑑𝑑/𝑑𝑑𝑑𝑑 at the M → FCC transition for 1-Cl-ada and at the O → FCC transition for 1-Br-ada suggest an excellent BC potential for these two transitions, but the transition hysteresis and highpressure transition properties in and out of thermodynamic equilibrium, which remain unknown, could confute this supposition. In this work we study these features and find colossal BC effects well above a hundred J K -1 kg -1 for the two transitions, that are reversible at moderate pressures and span over few tens of K near room temperature. We also calculate the refrigerant capacity and the coefficient of refrigerant performance, confirming the great potential of these materials in pressure-driven heat exchange. Our method combines variable-pressure quasi-direct isobaric heat flow measurements across the temperature-induced transition and indirect estimation of thermal expansion-related BC effects from x-ray measurements. We also report the unknown structure of the O phase of 1-Br-ada, that confirms its partial orientational disorder.

Experimental details

Powdered samples of 1-Br-ada (99 wt\% purity) and 1-Cl-ada (98 wt\%) were purchased from Sigma-Aldrich and used as received. Experimental details of high-resolution X-ray powder diffraction measurements can be found in Ref. [15]. Molecular energy minimization with the Forcite module using the Dreiding forcefield available in Materials Studio Program [20] has been used to build up a rigid body molecule. It is well-known that molecular parameters of adamantane derivatives, in particular those consisting in 1-X-or 2-X-adamantane derivatives, are virtually the same [21][22][23][24][25][26][27]. For the final structural solution, rigid molecules were placed in a randomly oriented general position and through Powder Solve using Monte-Carlo approach, both the position and orientation of molecules within the unit lattice were refined. The ultimate structural solution was obtained by means of Rietveld refinement [28], in which the position and orientation of the molecule, within the rigid-body constraint, with a single overall isotropic displacement parameter and the preferred orientation (using the Rietveld-Toraya function [29]) were refined. The obtained R factors were 𝑅𝑅 𝑤𝑤𝑑𝑑 = 8.11%, 𝑅𝑅 𝑑𝑑 = 5.61% for the O phase of 1-Br-ada. To ensure the robustness of the structural refinement of the orthorhombic phase we have applied the same method and experimental conditions to the low-temperature monoclinic phase, previously reported from a study of single-crystal Xray diffraction [18]. By doing so, the obtained structure for this monoclinic phase is virtually the same and the aforementioned quality factors being 𝑅𝑅 𝑤𝑤𝑑𝑑 = 8.53%, 𝑅𝑅 𝑑𝑑 = 6.18%, very close to those of the O phase.

Calorimetry at normal pressure was carried out using a conventional Differential Scanning Calorimeter Q100 from TA Instruments. Pressure-dependent differential thermal analysis was performed using two homemade high-pressure cells that operate in a pressure range up to 3 kbar and use Bridgman K-type thermocouples (cromel-alumel, ±2.2 K of tolerance, from Pyromation) as thermal sensors for both temperature and heat flow measurements. For 1-Br-ada, the used cell works within a temperature range from room-temperature to 473 K controlled by a resistive heater. Heating ramps were carried out at 3 K min -1 and coolings were performed with an air stream system at an average rate of ~-2.5 K min -1 . For 1-Cl-ada the used cell works within a temperature range from 200 K to 393 K controlled by an external thermal bath (Lauda Proline 1290). Heating and cooling ramps were performed at ~4 K min -1 and ~-2 K min -1 , respectively. A Pt-100 thermometer (RS components; accuracy ±0.30 K at 273.15 K, ±0.80 at 373.15 K) was used to monitor the temperature ramps by means of a temperature controller. In both cells, few hundreds of mg of each sample were encapsulated in tin capsules along with a perfluorinated inert fluid (Galden Bioblock Scientist) to remove air. It was checked that this fluid does not affect the thermodynamic transition properties of the sample. A hole was drilled in the tin capsules to insert the Bridgman thermocouples. The pressuretransmitting fluid used in the pressure circuit was DW-Therm M90.200.02 (Huber). A high-pressure transducer Model HP from Honeywell (range 0-5.2 kbar, 0.5% accuracy) was used as pressure sensor. A manual pressure pump was used to change pressure. Calorimetric signals were recorded under nearly isobaric conditions using a custom-built software.

Results and discussion

Characterization of the phase transitions

Figure 1a shows the Rietveld refinement of the 1-Br-ada O phase in which all H atoms and all but one C atom exhibit an occupancy of 50%, due to

= 𝑅𝑅𝑀𝑀 -1 ln(𝑁𝑁 𝑂𝑂 /𝑁𝑁 𝑀𝑀 ) = 𝑅𝑅𝑀𝑀 -1 ln 2 ~27 J K -1 kg - 1 .
This value is of the order of the aforementioned total transition entropy change ∆𝑆𝑆 𝑀𝑀→𝑂𝑂 ~ 21 J K -1 kg -1 [15], which indicates that there are no other significant contributions to the latter, such as the entropic contribution concerning the volume change.

Calorimetry measurements at constant pressure, as shown in Fig. 2a for 1-Cl-ada and Fig. 2b for 1-Br-ada, yielded positive and negative 𝑑𝑑𝑑𝑑 |𝑑𝑑𝑇𝑇| ≡ 𝑑𝑑𝑄𝑄 ̇/|𝑑𝑑𝑑𝑑 ̇| peaks in temperature corresponding to endothermic and exothermic phase transitions, respectively. Transition enthalpy changes ∆𝐻𝐻 𝐼𝐼𝐼𝐼→𝐼𝐼 and ∆𝐻𝐻 𝐼𝐼→𝐼𝐼𝐼𝐼 , and transition entropy changes ∆𝑆𝑆 𝐼𝐼𝐼𝐼→𝐼𝐼 and ∆𝑆𝑆 𝐼𝐼→𝐼𝐼𝐼𝐼 , were determined from integrations in 𝑑𝑑 of 𝑑𝑑𝑄𝑄/|𝑑𝑑𝑑𝑑| and of � 1 𝑇𝑇 � 𝑑𝑑𝑑𝑑 |𝑑𝑑𝑇𝑇| peaks after baseline subtraction, respectively. Here, II stands for the M phase for 1-Cl-ada and for the O phase for 1-Br-ada, whereas I stands for the FCC phase for the two compounds. At atmospheric pressure, for 1-Cl-ada we found ∆𝐻𝐻 𝐼𝐼𝐼𝐼→𝐼𝐼 = 32 ± 1 kJ kg -1 , ∆𝐻𝐻 𝐼𝐼→𝐼𝐼𝐼𝐼 = 31.5 ± 1.0 kJ kg -1 , ∆𝑆𝑆 𝐼𝐼𝐼𝐼→𝐼𝐼 = 132 ± 4 J K -1 kg -1 and ∆𝑆𝑆 𝐼𝐼→𝐼𝐼𝐼𝐼 = 136 ± 4 J K -1 kg -1 whereas for 1-Br-ada, we found ∆𝐻𝐻 𝐼𝐼𝐼𝐼→𝐼𝐼 = 32 ± 1 kJ kg -1 , ∆𝐻𝐻 𝐼𝐼→𝐼𝐼𝐼𝐼 = 31 ± 1 kJ kg -1 , ∆𝑆𝑆 𝐼𝐼𝐼𝐼→𝐼𝐼 = 102 ± 3 J K -1 kg -1 and ∆𝑆𝑆 𝐼𝐼→𝐼𝐼𝐼𝐼 = 104 ± 3 J K -1 kg -1 .

For the two compounds under study, sources for these very large entropy changes ∆𝑆𝑆 𝐼𝐼𝐼𝐼→𝐼𝐼 can be found in the configurational entropy change ∆𝑆𝑆 𝑐𝑐 associated with the release of molecular orientational degrees of freedom and in lattice strain at the transition. The latter can be decomposed in a volumic entropy change ∆𝑆𝑆 𝑉𝑉 associated with the huge expansion occurring across the transition, and in a nonvolumic strain entropy change. The latter cannot be calculated due to the lack of data on temperature-dependent elastic constants (which are difficult to measure due to the brittleness of the compounds) and the reconstructive nature of the transition, but it is expected to be small [12]. Therefore, the nonvolumic strain contribution to the total entropy change will be negligible compared to the volume contribution and thus not be considered in the following discussion.

On the one hand, the configurational entropy change can be calculated from the ratio between the number of accessible configurations in the high-and low-temperature phases (𝑁𝑁 𝐼𝐼 and 𝑁𝑁 𝐼𝐼𝐼𝐼 , respectively) as ∆𝑆𝑆 𝑐𝑐 = 𝑅𝑅𝑀𝑀 -1 ln � 𝑁𝑁 𝐼𝐼 𝑁𝑁 𝐼𝐼𝐼𝐼 �. For the two compounds, the tetrahedral molecular symmetry along with the lattice symmetry in the plastic phase enables 2 and 8 nonequivalent orientations associated with the point group Td and subgroup C3v, respectively. Then, taking into account that there is a single conformation for each compound due to their rigid molecular character, 10 possible configurations are obtained [30]. As for the low-temperature phase, for 1-Cl-ada M phase is completely ordered and therefore 𝑁𝑁 𝐼𝐼𝐼𝐼 = 1 whereas for 1-Br-ada 𝑁𝑁 𝐼𝐼𝐼𝐼 = 2 as discussed previously. This yields a configurational entropy change ∆𝑆𝑆 𝑐𝑐 = 112.1 J K -1 kg -1 for 1-Cl-ada and ∆𝑆𝑆 𝑐𝑐 = 62.2 J K -1 kg -1 for 1-Br-ada. On the other hand, the volume entropy change can be estimated as [31] ∆𝑆𝑆

𝑉𝑉 = 𝛼𝛼 � 𝜒𝜒 � ∆𝑉𝑉 𝐼𝐼𝐼𝐼→𝐼𝐼 (1)
where 𝛼𝛼 � and 𝜒𝜒̅ are the thermal expansion and isothermal compressibility averaged over the two phases close to the transition. For 1-Cl-ada, 𝛼𝛼 � = 3.7 • 10 -4 K -1 and for 1-Br-ada 𝛼𝛼 � = 5.9 • 10 -4 K -1 [15], whereas 𝜒𝜒̅ = 0.31 GPa -1 [32][33][34] is used for the two compounds. This yields ∆𝑆𝑆 𝑉𝑉 ~ 51 J K -1 kg -1 for 1-Cl-ada and ∆𝑆𝑆 𝑉𝑉 ∼ 76 J K -1 kg -1 for 1-Br-ada. Alternatively, assuming a harmonic elastic Helmholtz free energy for the volume change, ∆𝑆𝑆 𝑉𝑉 can also be estimated by using the expression

∆𝑆𝑆 𝑉𝑉 ≃ 1 2𝑉𝑉 〈 𝜕𝜕𝜕𝜕 𝜕𝜕𝑇𝑇 〉 ∆𝑉𝑉 𝐼𝐼𝐼𝐼→𝐼𝐼 2 ( 2
)
where 𝐾𝐾 is the bulk modulus and the average is taken over the two phases close to the transition. Using 〈 𝜕𝜕𝜕𝜕 𝜕𝜕𝑇𝑇 〉 ≃ 3.2 • 10 7 Pa K -1 for 1-Cl-Ada calculated from literature data [32], we obtain ∆𝑆𝑆 𝑉𝑉 ≃ 44 J K -1 kg -1 which is in good agreement with the value obtained previously.

While these values are intended to give only a rough estimate (notice that we obtain ∆𝑆𝑆 𝑐𝑐 + ∆𝑆𝑆 𝑉𝑉 > ∆𝑆𝑆 𝐼𝐼𝐼𝐼→𝐼𝐼 ), they indicate that for these materials ∆𝑆𝑆 𝑉𝑉 and ∆𝑆𝑆 𝑐𝑐 are of the same order and that vibrational entropy changes at the transition are mainly contributed by volume changes, whereas other contributions are minor. On the other hand, these values are consistent with the suggested trend for adamantane derivatives and observed in some cyclohexanes according to which ∆𝑆𝑆 𝑐𝑐 ~ 0.6∆𝑆𝑆 𝐼𝐼𝐼𝐼→𝐼𝐼 [19,[35][36][37]. For 1-Br-ada, the relative contribution of ∆𝑆𝑆 𝑐𝑐 compared to ∆𝑆𝑆 𝑉𝑉 is minor due to the partial disorder surviving in the O phase. On the other hand, this set of values for ∆𝑆𝑆 𝑐𝑐 and ∆𝑆𝑆 𝑉𝑉 could be also a signature that the orientational disorder in the plastic phase is actually restricted to a smaller number of configurations than the 10 that are nominally achievable for the specific symmetries of these materials. It is worth mentioning here that vibrational entropy changes outside the transition, contributing to changes in the heat capacity (i.e. not associated with the transition latent heat), were analyzed in detail in Ref. [19].

From integration of the calorimetric peaks at high pressures, we obtained that the transition entropy change decreases with pressure for the two compounds (see Fig. 2d,f), which is a behavior observed in other substituted adamantanes [38]. In particular, we obtain

𝑑𝑑|∆𝑆𝑆 𝐼𝐼𝐼𝐼→𝐼𝐼 | 𝑑𝑑𝑑𝑑 = -13 ± 2 J K -1 kg -1 kbar -1 and
𝑑𝑑|∆𝑆𝑆 𝐼𝐼→𝐼𝐼𝐼𝐼 | 𝑑𝑑𝑑𝑑 = -14 ± 2 J K -1 kg -1 kbar -1 for
1-Cl-ada and

𝑑𝑑|∆𝑆𝑆 𝐼𝐼𝐼𝐼→𝐼𝐼 | 𝑑𝑑𝑑𝑑 = 𝑑𝑑|∆𝑆𝑆 𝐼𝐼→𝐼𝐼𝐼𝐼 | 𝑑𝑑𝑑𝑑
= -14 ± 2 J K -1 kg -1 kbar -1 for 1-Br-ada. Taking into account the decrease of ∆𝑆𝑆 𝐼𝐼𝐼𝐼→𝐼𝐼 with pressure and the constant 𝑑𝑑𝑑𝑑/𝑑𝑑𝑑𝑑 in the Clausius-Clapeyron equation, we obtained that the transition volume change also decreases with pressure, which is also a common behavior observed in plastic crystals in general [37,39,40], and in adamantane in particular [26,41].

Endothermic and exothermic transition temperatures, 𝑑𝑑 𝐼𝐼𝐼𝐼→𝐼𝐼 and 𝑑𝑑 𝐼𝐼→𝐼𝐼𝐼𝐼 , respectively, were determined from the position of the maximum of the calorimetric peaks. Transition temperatures as a function of pressure are displayed in Fig. 2(c) and (e) for 1-Cl-ada and 1-Br-ada, respectively. The shadowed area indicates the transition peak width, which is narrower for the exothermic transitions because their out of stable equilibrium behavior triggers the transitions abruptly. For 1-Cl-ada, at atmospheric pressure, 𝑑𝑑 𝐼𝐼𝐼𝐼→𝐼𝐼 = 254 ± 1 K and 𝑑𝑑 𝐼𝐼→𝐼𝐼𝐼𝐼 = 245 ± 1 K, which renders a hysteresis Δ𝑑𝑑 𝐼𝐼𝐼𝐼→𝐼𝐼 ~ 9 K that is maintained at high pressure because endothermic and exothermic transition temperatures shift similarly with pressure:

𝑑𝑑 𝑟𝑟𝑟𝑟𝑟𝑟 ≃ ∆𝑑𝑑 𝐼𝐼𝐼𝐼↔𝐼𝐼 � 𝑑𝑑𝑇𝑇 𝐼𝐼→𝐼𝐼𝐼𝐼 𝑑𝑑𝑑𝑑 � -1 , (3)
which yields values as small as ~0.3 and ~0.35 kbar for 1-Cl-ada and 1-Br-ada, respectively. The aforementioned transition thermodynamic data obtained from variablepressure calorimetry are summarized in Table 1.

Determination of pressure-and temperature-dependent entropy

Temperature-and pressure-dependent entropy 𝑆𝑆(𝑑𝑑, 𝑑𝑑) is constructed with respect to a reference entropy 𝑆𝑆 0 ≡ 𝑆𝑆(𝑑𝑑 0 , 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 ) where 𝑑𝑑 0 is a reference temperature chosen below but close to the transition temperature at atmospheric pressure. In particular, 𝑑𝑑 0 is set to 290 K for 1-Br-ada and 220 K for 1-Cl-ada. At 𝑑𝑑 0 , 𝑆𝑆(𝑑𝑑 0 , 𝑑𝑑) is determined as

𝑆𝑆(𝑑𝑑 0 , 𝑑𝑑) -𝑆𝑆(𝑑𝑑 0 , 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 ) = -∫ � 𝜕𝜕𝑉𝑉 𝐼𝐼𝐼𝐼 𝜕𝜕𝑇𝑇 � 𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 ≃ -� 𝜕𝜕𝑉𝑉 𝐼𝐼𝐼𝐼 𝜕𝜕𝑇𝑇 � 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 ∆𝑑𝑑, (4)
where we have assumed that � = (2.63 ± 0.04) m 3 K -1 kg -1 are taken from literature [16]). This assumption is reasonable given the data provided by similar compounds in Ref. [39]. Notice that this procedure corresponds to an indirect method [4] of calculation of BC effects at 𝑑𝑑 = 𝑑𝑑 0 .

Above 𝑑𝑑 0 , 𝑆𝑆(𝑑𝑑, 𝑑𝑑) -𝑆𝑆(𝑑𝑑 0 , 𝑑𝑑) is calculated following the quasi-direct method, using our measured variable-pressure isobaric heat flow across the temperature-induced transition and heat capacity data 𝐶𝐶 𝑑𝑑 (𝑑𝑑, 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 ) at atmospheric pressure from literature (ref. [42] for 1-Cl-ada and Ref. [19] for 1-Br-ada) for the involved phases. Mathematical details can be found elsewhere [10] and entail that outside the transition regions, 𝐶𝐶 𝑑𝑑 (𝑑𝑑, 𝑑𝑑) ≃ 𝐶𝐶 𝑑𝑑 (𝑑𝑑, 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 ) within the narrow pressure range under study. Notice that this relation is valid to a very good approximation because the linear dependence of volume on temperature for both 1-Br-ada and 1-Cl-ada relatively close to the transition [15] entails that 𝐶𝐶 𝑑𝑑 (𝑑𝑑, 𝑑𝑑) is independent of pressure due to the thermodynamic relation

� 𝜕𝜕𝐹𝐹 𝑝𝑝 𝜕𝜕𝑑𝑑 � 𝑇𝑇 = -𝑑𝑑 � 𝜕𝜕𝑉𝑉 2 𝜕𝜕 2 𝑇𝑇 � 𝑑𝑑 = 0. (5)
This is further supported by pressure-dependent measurements of heat capacity for adamantane [43], which shows that for this compound 𝐶𝐶 𝑑𝑑 is approximately independent or very weakly dependent on pressure in the pressure range under study. It is worth mentioning here that the linear 𝑉𝑉 -𝑑𝑑 behavior is commonly observed among organic crystals [35].

To combine the pressure independence of 𝐶𝐶 𝑑𝑑 with the temperature shift of the transition temperature with pressure, 𝐶𝐶 𝑑𝑑 of phase II is extrapolated towards the pressure-dependent transition temperature 𝑑𝑑(𝑑𝑑). The resulting 𝑆𝑆(𝑑𝑑, 𝑑𝑑) 𝐻𝐻,𝐹𝐹 -𝑆𝑆 0 functions are shown in Fig. 3, where the subscripts H and C stand for curves obtained on heating and on cooling, respectively.

Once the entropy curves are determined, we can obtain insight about their reliability by using two different methods to calculate the isothermal difference between the entropy curves at different pressures at the high temperature phase, 𝑆𝑆(𝑑𝑑, 𝑑𝑑) -𝑆𝑆(𝑑𝑑, 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 ), and compare the obtained results. On the one hand, method 1 consists of directly subtracting the curves. On the other hand, method 2 consists of applying the thermodynamic equation used previously to determine 𝑆𝑆(𝑑𝑑 0 , 𝑑𝑑) -𝑆𝑆(𝑑𝑑 0 , 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 ) but after replacing 𝑑𝑑 0 by 𝑑𝑑 [> 𝑑𝑑 𝐼𝐼𝐼𝐼→𝐼𝐼 (𝑑𝑑)] and 𝑉𝑉 𝐼𝐼𝐼𝐼 by 𝑉𝑉 𝐼𝐼 , so that we can write 𝑆𝑆(𝑑𝑑, 𝑑𝑑) -𝑆𝑆(𝑑𝑑, 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 ) = ∫ � = (5.66 ± 0.09) • 10 -4 cm 3 K -1 g -1 for 1-Br-ada). For 1-Cl-ada, under ~1 kbar (~2 kbar), method 1 yields ~46 J K -1 kg -1 (~90 J K -1 kg -1 ) and method 2 yields ~42 J K -1 kg -1 (~99 J K -1 kg -1 ). For 1-Br-ada, under ~1 kbar (~2 kbar), method 1 yields ~58 J K -1 kg -1 (~134 J K -1 kg -1 ) and method 2 yields ~62 J K -1 kg -1 (~153 J K -1 kg -1 ). Therefore, we obtain values that are in agreement within a relative error of ~10%, which contributes to an error over the total BC entropy changes (see next section) of ~3% at ~1 kbar and ~5-8% at 2 kbar. Notice that, as such, these isothermal entropy changes correspond to the BC effects arising outside the transition in the plastic phase, and confirm that their magnitude is typically very large in plastic crystals [10,12].

Barocaloric effects and performance

To determine the BC effects, we use the quasi-direct method for which ∆𝑆𝑆 and ∆𝑑𝑑 are determined as differences between entropy curves at different pressures following isothermal and adiabatic paths, respectively [4]. Since the compounds under study exhibit positive 𝑑𝑑𝑑𝑑/𝑑𝑑𝑑𝑑, endothermic transitions occur on heating or decreasing pressure while exothermic transitions occur on both cooling or increasing pressure. This entails that irreversible isothermal entropy changes occurring when changing pressure between atmospheric pressure 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 and a given pressure 𝑑𝑑 > 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 , are calculated as

∆𝑆𝑆(𝑑𝑑, 𝑑𝑑 → 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 ) = 𝑆𝑆 𝐻𝐻 (𝑑𝑑, 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 ) -𝑆𝑆 𝐻𝐻 (𝑑𝑑, 𝑑𝑑) (6a) ∆𝑆𝑆(𝑑𝑑, 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 → 𝑑𝑑) = 𝑆𝑆 𝐹𝐹 (𝑑𝑑, 𝑑𝑑) -𝑆𝑆 𝐹𝐹 (𝑑𝑑, 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 ) (6b)
(see Fig. 4a,c for 1-Cl-ada and 1-Br-ada, respectively). In turn, irreversible adiabatic temperature changes are obtained as

∆𝑑𝑑(𝑑𝑑 𝑠𝑠 , 𝑑𝑑 → 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 ) = 𝑑𝑑(𝑆𝑆 𝐻𝐻 , 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 ) -𝑑𝑑 𝑠𝑠 (𝑆𝑆 𝐻𝐻 , 𝑑𝑑) (7a) ∆𝑑𝑑(𝑑𝑑 𝑠𝑠 , 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 → 𝑑𝑑) = 𝑑𝑑(𝑆𝑆 𝐹𝐹 , 𝑑𝑑) -𝑑𝑑 𝑠𝑠 (𝑆𝑆 𝐹𝐹 , 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 ), ( 7b
)
where ∆𝑑𝑑 is conveniently plotted as a function of the starting temperature 𝑑𝑑 𝑠𝑠 of the compression or decompression process (see Fig. 4b,d for 1-Cl-ada and 1-Br-ada, respectively).

Reversible ∆𝑆𝑆(𝑑𝑑) [∆𝑆𝑆 𝑟𝑟𝑟𝑟𝑟𝑟 (𝑑𝑑)] are obtained as the overlap between |∆𝑆𝑆(𝑑𝑑, 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 → 𝑑𝑑)| and |∆𝑆𝑆(𝑑𝑑, 𝑑𝑑 → 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 )| whereas reversible ∆𝑑𝑑 are obtained as

∆𝑑𝑑 𝑟𝑟𝑟𝑟𝑟𝑟 (𝑑𝑑 𝑠𝑠 , 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 → 𝑑𝑑) = 𝑑𝑑(𝑆𝑆 𝐹𝐹 , 𝑑𝑑) -𝑑𝑑 𝑠𝑠 (𝑆𝑆 𝐻𝐻 , 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 ) (8a) ∆𝑑𝑑 𝑟𝑟𝑟𝑟𝑟𝑟 (𝑑𝑑 𝑠𝑠 , 𝑑𝑑 → 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 ) = 𝑑𝑑(𝑆𝑆 𝐻𝐻 , 𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 ) -𝑑𝑑 𝑠𝑠 (𝑆𝑆 𝐹𝐹 , 𝑑𝑑). (8b)
Results are shown in Fig. 5. Maximum values for irreversibe and reversible BC effects are plotted in Fig. 6.

In addition to ∆𝑆𝑆 and ∆𝑑𝑑, we calculate two parameters for comparison between different materials. On the one hand, the refrigerant capacity (RC) (see Figs 6c,g), which gives an estimation of the transferable heat between hot and cold reservoirs and corresponds to the area beneath the ∆𝑆𝑆 vs. 𝑑𝑑 peaks displayed in Figs 4(a,c) and 5(a,c). On the other hand, the coefficient of refrigerant performance (CRP) is approximately defined for BC materials as

𝐶𝐶𝑅𝑅𝑃𝑃 = ∆𝑆𝑆×∆𝑇𝑇 𝑟𝑟𝑟𝑟𝑟𝑟 1 2 𝑑𝑑∆𝑉𝑉 (9)
so that it takes into account the work needed to drive the transition (see Figs 6d,h). Finally, for practical applications it is desired that materials undergo giant BC effects when in contact with both hot and cold ends, which entails that such giant BC effects should be obtained in a relatively wide temperature interval spanning the whole cycle. In this respect, Fig. 7 shows the temperature spans where minimum values of BC effects are obtained under a given pressure change.

Discussion

As it can be seen in Figs 567, our results show that both 1-Cl-ada and 1-Br-ada display outstanding reversible BC effects at moderate pressures, reaching ~150 J K -1 kg -1 isothermally and ≳16 K adiabatically, over a temperature span of ~10 K under a pressure change of ~1 kbar. A comparison of ∆𝑆𝑆 𝑟𝑟𝑟𝑟𝑟𝑟 and ∆𝑑𝑑 𝑟𝑟𝑟𝑟𝑟𝑟 obtained upon a pressure change of ~1 kbar between different BC materials available in literature (see Fig. 8) reveals that the joint values for the compounds studied in this work outperform any other BC material reported so far [START_REF] Lloveras | Advances and obstacles in pressure-driven solid-state cooling: A review of barocaloric materials[END_REF]. The values obtained in this work compare also favorably to maximum reversible values driven by other fields, that for best MC materials reach about ~10 J K -1 kg -1 and ~5 K under 2 T created by permanent magnets, and ~15 K and 19 J K -1 kg -1 under 5 T, for best eC materials reach ~32 K and ~45 J K -1 kg -1 under 700 MPa and for best EC materials reach ~5 K and ~6 J K -1 kg -1 [49,50]. The keys for this excellent behavior can be found in typical thermodynamic transition values for plastic crystals (colossal latent heat and volume change) and a small hysteresis. This lower pressure at which colossal barocaloric effects are obtained compared to other compounds, permits narrower walls for the pressurized chamber, which will facilitate the heat transfer with the environment in a future barocaloric cooling device.

Moreover, the temperature range for 1-Cl-ada is optimal because it can span from ~255 K to room temperature, which matches the operational temperature range of fridges, freezers and air conditioners for most applications, ranging from household devices to industrial or medical applications, such as the storage of medical drugs and vaccines. Notice also that the ~10 K of temperature span for colossal entropy changes obtained upon a pressure change of ~1 kbar (see Fig. 8) can be shifted to high temperatures at will (and therefore brought to room temperature) by displacing the pressure range (i.e. increasing both the lower and upper working pressures) while maintaining the pressure change at ~1 kbar. In this respect, a device operating between variable pressure limits can be conceived, to adapt actively to the temperature of the working body. This feature provides 1-Cl-ada with a key advantage with respect to other plastic crystals such as NPG [10,11] or PG [12], that show colossal BC effects from slightly above room temperature. Nonetheless, 1-Br-ada (as NPG or PG) could be more appropriate for cooling at higher temperatures, such as applications in industry or refrigeration of lithium-ion batteries, that degradate above ~330 K [51].

1-Cl-ada and 1-Br-ada, as plastic crystals in general, suffer from low thermal conductivities due to orientational disorder [START_REF] Lloveras | Advances and obstacles in pressure-driven solid-state cooling: A review of barocaloric materials[END_REF], which in the case of the plastic phase of adamantane reaches ~0.18 W m -1 K -1 under normal conditions [43]. Here, several aspects should be pointed out. On the one hand, in the ordered phases and/or higher pressures, the thermal conductivities experience a significant increase [43,52]. On the other hand, powdered materials display lower thermal conductivities than their single crystal counterparts because the free path is shortened due to phonon scattering at the grain boundary. However, powder can be combined with a small percentage of excellent conducting nanoparticles [53] or embedded in highly conductive porous matrix [54][55][56][57][58] resulting in a composite with notably improved thermal conductivity. Finally, as an added value for technological applications, it must be noted that both compounds under study are of low cost, low toxicity (irritant) and nonhazardous [59,60].

Conclusions

In this work we have studied the barocaloric performance of two plastic crystals belonging to the family of adamantane derivatives: 1-bromoadamantane and 1chloroadamantane. In the two cases, the combination of high orientational molecular disordering, very large volume changes across a phase transition and a small to moderate hysteresis allow colossal isothermal entropy changes and giant adiabatic temperature changes upon cyclic application and removal of moderate pressures. Interestingly, colossal BC effects in 1-chloroadamantane span from below to room temperature, making it very suitable for real applications. These materials are outstanding among barocaloric systems and should stimulate similar studies in other adamantane derivatives. Structure refinement of the orthorhombic phase of 1-bromoadamantane has confirmed the uniaxial (around the C3v molecular axis) disorder in this phase. Reversible adiabatic temperature changes upon 0 → ~1 kbar pressure increase as a function of temperature. Literature data taken from: PG: [12]; C60: [13];

𝑑𝑑∆𝑆𝑆 𝑎𝑎 /𝑑𝑑𝑑𝑑 K K kbar -1 J g -1 J K -1 kg -1 J K -1 kg -1 kbar -1 𝐼𝐼𝐼𝐼 → 𝐼𝐼 𝐼𝐼 → 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 → 𝐼𝐼 𝐼𝐼 → 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 → 𝐼𝐼 𝐼𝐼 → 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 → 𝐼𝐼 𝐼𝐼 → 𝐼𝐼𝐼𝐼 𝐼𝐼𝐼𝐼 → 𝐼𝐼 𝐼𝐼 → 𝐼𝐼𝐼𝐼 1-Cl- ada
[TPrA][Mn(dca)3]: [44]; [(CH3)4N]Mn[N3]3: [45]; MnCoGeB0.03 (asterisk): [46]; (MiNiSi)0.60(FeCoGe)0.40 (diamond) and (MiNiSi)0.61(FeCoGe)0.39 (triangle): [47]; Ni50Mn31.5Ti18.5 (circle): [48]. •

  ∆𝑉𝑉 𝑀𝑀→𝐹𝐹𝐹𝐹𝐹𝐹 𝑉𝑉 𝑀𝑀 ~ 6%). These two quantities entail a sensitivity of the transition temperature to pressure 𝑑𝑑𝑇𝑇 𝑀𝑀→𝐹𝐹𝐹𝐹𝐹𝐹 𝑑𝑑𝑑𝑑 ~27 K kbar -1 . Differently, in 1-Br-ada the orientational disorder arises across two transitions M → O → FCC, where O refers to an intermediate semiordered orthorhombic phase (Pmcn, Z=4). In this phase, disorder is believed to consist of a 6-fold molecular rotation about the three-fold molecular axis C3v[19]. Although both M → O and O → FCC transitions take place close to each other near room temperature, only the O → FCC raises interest for BC cooling applications as it involves by far most of the total entropy change associated with full orientational disorder and a much larger transition volume change. In particular, ∆𝑆𝑆 𝑀𝑀→𝑂𝑂 ~ 21 J K -1 kg -1 and ∆𝑉𝑉 𝑀𝑀→𝑂𝑂 ~ 6•10 -6 m 3 kg -1 � ∆𝑉𝑉 𝑀𝑀→𝑂𝑂 𝑉𝑉 𝑀𝑀 ~0.9%� whereas ∆𝑆𝑆 𝑂𝑂→𝐹𝐹𝐹𝐹𝐹𝐹 ~ 107 J K -1 kg -1 and ∆𝑉𝑉 𝑂𝑂→𝐹𝐹𝐹𝐹𝐹𝐹 ~ 40 • 10 -6 m 3 kg -1 In turn, the values for the O → FCC transition yield

𝜋𝜋 3

 3 rotations about the C-Br three-fold axis (cif number: CCDC 2042237). Unit cell representations of the O cell are displayed in Figs 1b,c projected onto the (010) and (100) planes, respectively. Since this disorder leads to 𝑁𝑁 𝑂𝑂 = 2 possible molecular configurations and the M phase is completely ordered, the contribution to the M → O transition entropy change due to the increase in the molecular configurations is ∆𝑆𝑆 𝑀𝑀→𝑂𝑂 𝑐𝑐

  0.2 K kbar -1 . For 1-Brada, at atmospheric pressure, 𝑑𝑑 𝐼𝐼𝐼𝐼→𝐼𝐼 = 316 ± 1 K and 𝑑𝑑 𝐼𝐼→𝐼𝐼𝐼𝐼 = 308 ± 1 K, which renders a hysteresis Δ𝑑𝑑 𝐼𝐼𝐼𝐼↔𝐼𝐼 ∼ 9 K. However, it increases up to Δ𝑑𝑑 𝐼𝐼𝐼𝐼↔𝐼𝐼 ∼ 12 K just above atmospheric pressure and increases slightly with pressure because 3 ± 0.8 K kbar -1 . From the hysteresis and 𝑑𝑑𝑇𝑇 𝐼𝐼→𝐼𝐼𝐼𝐼 𝑑𝑑𝑑𝑑 we can determine the minimum pressure to achieve reversible BC effects as[12]

  .43 ± 0.07) • 10 -4 cm 3 K -1 g -1 for 1-Cl-ada and

3 - 14 ± 2 - 14 ± 2 Figure1.

 3142142 Figure1. (a) X-ray diffraction pattern taken at 𝑑𝑑 = 295 K of the orthorhombic phase in 1-Br-ada. The inset shows an enlargement of the 2𝜃𝜃 range (50,80)°. Red symbols, blue and black lines stand for experimental data, refined pattern and the corresponding difference, respectively. Green symbols indicate the position of Bragg peaks. (b,c) Orthorhombic unit cell of 1-Br-Ada, projected onto (b) the (010) and (c) the (100) planes. Red, gray and white stands for Br, C and H atoms, respectively. Degenerated positions indicate occupational disorder.

Figure2.

  Figure2. (a,b) Pressure-dependent calorimetry measurements for (a) 1-Cl-ada and (b) 1-Br-ada. (c,e) Temperature-pressure phase diagram and (d,b) transition entropy change for the two compounds under study. In panels (c-f), red and blue symbols correspond to measurements on heating and cooling, respectively an shadowed areas in (c) and (e) indicate the transition peak width. Lines are linear fits to the data.

Figure3.

  Figure3. Isobaric temperature-dependent entropy curves at different pressures obtained on heating (a,c) and on cooling (b,d) for 1-Cl-ada (a,b) and 1-Br-ada (c,d).

Figure 4 .

 4 Figure 4. (a,c) Isothermal entropy changes and (b,d) adiabatic temperature changes upon first application or removal or pressure BC effects for different applied pressures for (a,b) 1-Cl-ada and (c,d) 1-Br-ada.

Figure 5 .

 5 Figure 5. Reversible (a,b) entropy and (c,d) temperature changes upon application and removal of pressure under isothermal and adiabatic conditions, respectively, for (a,b) 1-Cl-ada and (c,d) 1-Br-ada.

Figure 6 .

 6 Figure 6. (a,e) Maximum Isothermal entropy changes and (b,f) adiabatic temperature changes, (c,g) Refrigerant Capacity and (d,h) Coefficient of Refrigerant Performance as a function of pressure changes for (a-d) 1-Cl-ada and (e-h) 1-Br-ada. Red, blue and green data correspond to first compression, first decompression and reversible data. Lines are fits to the data.

Figure 7 .

 7 Figure 7. Temperature spans where different magnitudes of (a,c) reversible isothermal entropy changes and (b,d) reversible adiabatic temperature changes are obtained, as a function of applied pressure change, for (a,b) 1-Cl-ada and for (c,d) 1-Br-ada.

Figure 8 .

 8 Figure 8. Comparison between different BC materials: (a) Reversible isothermal entropy changes upon 0 ↔ ~1 kbar pressure change as a function of temperature. (b)Reversible adiabatic temperature changes upon 0 → ~1 kbar pressure increase as a function of temperature. Literature data taken from: PG:[12]; C60:[13];[TPrA][Mn(dca)3]:[44]; [(CH3)4N]Mn[N3]3:[45]; MnCoGeB0.03 (asterisk):[46]; (MiNiSi)0.60(FeCoGe)0.40 (diamond) and (MiNiSi)0.61(FeCoGe)0.39 (triangle):[47]; Ni50Mn31.5Ti18.5 (circle):[48].

  Figure 1

  Figure 6

Table 1 .

 1 Transition thermodynamic properties at variable pressure. 𝑑𝑑 𝑎𝑎 (𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 ) 𝑑𝑑𝑑𝑑/𝑑𝑑𝑑𝑑 ∆𝐻𝐻 𝑎𝑎 (𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 ) ∆𝑆𝑆 𝑎𝑎 (𝑑𝑑 𝑎𝑎𝑎𝑎𝑎𝑎 )

Acknowledgements

This work was supported by the MINECO projects FIS2017-82625-P and MAT2016-75823-R (Spain), the DGU project 2017SGR-42 (Catalonia), the UK EPSRC grant EP/M003752/1, and the ERC Starting grant no. 680032. X. M. is grateful for support from the Royal Society.

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.