Multifunctional Molecular Magnets: Magnetocaloric Effect in Octacyanometallates

Abstract

Octacyanometallate-based compounds displaying a rich pallet of interesting physical and chemical properties, are key materials in the field of molecular magnetism. The [M(CN)₈]ⁿ⁻ complexes, (M = W^V, Mo^V, Nb^IV), are universal building blocks as they lead to various spatial structures, depending on the surrounding ligands and the choice of the metal ion. One of the functionalities of the octacyanometallate-based coordination polymers or clusters is the magnetocaloric effect (MCE), consisting in a change of the material temperature upon the application of a magnetic field. In this review, we focus on different approaches to MCE investigation. We present examples of magnetic entropy change ΔS_m and adiabatic temperature change ΔT_ad, determined using calorimetric measurements supplemented with the algebraic extrapolation of the data down to 0 K. At the field change of 5T, the compound built of high spin clusters Ni₉[W(CN)₈]₆ showed a maximum value of −ΔS_m equal to 18.38 J·K⁻¹ mol⁻¹ at 4.3 K, while the corresponding maximum ΔT_ad = 4.6 K was attained at 2.2 K. These values revealed that this molecular material may be treated as a possible candidate for cryogenic magnetic cooling. Values obtained for ferrimagnetic polymers at temperatures close to their magnetic ordering temperatures, T_c, were lower, i.e., −ΔS_m = 6.83 J·K⁻¹ mol⁻¹ (ΔT_ad = 1.42 K) and −ΔS_m = 4.9 J·K⁻¹ mol⁻¹ (ΔT_ad = 2 K) for {[Mn^II(pyrazole)₄]₂[Nb^IV(CN)₈]·4H₂O}_n and{[Fe^II(pyrazole)₄]₂[Nb^IV(CN)₈]·4H₂O}_n, respectively. MCE results have been obtained also for other -[Nb(CN)₈]-based manganese polymers, showing significant T_c dependence on pressure or the remarkable magnetic sponge behaviour. Using the data obtained for compounds with different T_c, due to dissimilar ligands or other phase of the material, the ΔS_m ~ T_c^−2/3 relation stemming from the molecular field theory was confirmed. The characteristic index n in the ΔS_m ~ ΔHⁿ dependence, and the critical exponents, related to n, were determined, pointing to the 3D Heisenberg model as the most adequate for the description of these particular compounds. At last, results of the rotating magnetocaloric effect (RMCE), which is a new technique efficient in the case of layered magnetic systems, are presented. Data have been obtained and discussed for single crystals of two 2D molecular magnets: ferrimagnetic {Mn^II(R-mpm)₂]₂[Nb^IV(CN)₈]}∙4H₂O (mpm = α-methyl-2-pyridinemethanol) and a strongly anisotropic (tetren)Cu₄[W(CN)₈]₄ bilayered magnet showing the topological Berezinskii-Kosterlitz-Thouless transition.

1. Introduction

In the quest for novel materials which could be used in modern technologies, molecular substances play an important role, as they may show properties not available in conventional materials. Molecule-based compounds attract much attention as they combine interesting magnetic, electronic, and optical properties together with low weight, transparency, and chemical sensitivity. Molecular magnets represent a vast and still growing family of compounds based on molecular buildings blocks, where organic groups mediate magnetic interactions between metal ions or may also carry their own magnetic moment. Research into molecule-based materials is motivated by their potential functionality due to the properties of the molecular building blocks or the specific character of the resulting molecular network. Thanks to the rational design and advanced chemical syntheses, it is possible to obtain systems with different magnetic dimensionalities: molecular ferro-, ferri-, or antiferromagnets with the substantial ordering temperature T_c, magnetic molecular layers, magnetic molecular chains, or magnetic molecular clusters, the latter two regarded as molecular nanomagnets. In many cases, properties of the systems, like T_c, coercive field, magnetic moment, and colour, may be changed and even controlled with different factors, such as temperature, irradiation with light, external pressure, or sorption of guest molecules. Molecular magnets which show spin crossover transition, photomagnetism, magnetic sponge behaviour, or optical activity are potential candidates for efficient sensors and switches. On the other hand, superparamagnetic character, slow relaxation, and quantum tunneling in anisotropic high-spin clusters, may be used in high-density magnetic storage, spintronics, or quantum computing. Results of extensive studies on molecular magnetism have been reviewed in books and monographies [1,2,3,4].

The relatively later explored functionality of molecular magnets has been the magnetocaloric effect (MCE), consisting in change of the material temperature when a magnetic field is applied or removed. MCE is an intrinsic thermodynamic property of all magnetic solids: it occurs in paramagnets, ferro-, and ferrimagnets [5] but also in antiferromagnets at the metamagnetic transition [6]. Investigations of MCE are significant also for the basic reason that its dependence on the magnetic field is related to the critical behaviour of the material. The microscopic description of MCE has been presented in [7,8,9,10]. Of particular interest are “magnetic coolers”, i.e., substances for which an adiabatic demagnetization provokes the substantial temperature decrease. The value of the effect depends on the temperature derivative of magnetization, thus for paramagnets it is strongest at the lowest temperatures, while for ordered magnets it achieves maximum at the magnetic ordering temperature T_c. The aim is to replace standard, non-ecological refrigeration techniques, studies on magnetocaloric effect in conventional magnetic solids are concentrated on cooling in the room temperature range.

The most suitable candidates for magnetic refrigeration are here systems like Gd₅Si₂Ge₂, Tb₅Si₂Ge₂, MnAs, or Ni-Mn-In Heusler alloys [11]. Materials with first order magnetic transitions at T_c may be considered as most suitable for large MCE but thermal and magnetic losses appearing in transitions of this type may impede the practical applications. Consequently, materials showing the second order magnetic phase transition are also taken into account. Interest in the magnetocaloric effect in molecular materials and “chilling with magnetic molecules” [12] started about a decade ago. At first, it has been investigated for slowly relaxing molecular nanomagnets with large spin, Single Molecule Magnets (SMM) [2]. Because of the large ground-state spin S of SMM, the entropy associated with the magnetic degrees of freedom, S_m = Rln(2S+1) (R is the gas constant) should be sizeable. However, molecular anisotropy, which is essential in SMMs as it decides on long relaxation time and blocking temperature, delays magnetization and demagnetization of the system, thus brings about weak MCE. As stated by Evangelisti and Brechin, the ideal molecular refrigerant should have large spin, negligible magnetic anisotropy, prevailing ferromagnetic coupling, and large magnetic density [13]. These conditions are fulfilled in metal–organic frameworks with densely packed magnetic ions, like [Gd(HCOO)₃] [14]. Large values of MCE observed in frameworks are not related to transition at T_c as these systems remain paramagnetic or show the long-range order only in the sub-Kelvin range. MCE originates here from the Schottky anomaly consequent on splitting of the energy levels in the field. Very large MCE values have been also observed in the ferromagnetic acetate tetrahydrate Gd³⁺ dimer [15], in the high nuclearity Gd₄₂Co₁₀ cluster [16], and in a 24-Gd capsule-like cluster at temperature 2.5 K [17]. An interesting increase in the magnetocaloric effect in Mn^II glycolate on transition between the three-dimensional coordination polymer and discrete mononuclear phase induced by water molecules was reported in [18]. It follows that MCE in molecular clusters will play an important role in cooling in the sub-Kelvin temperature range. Advances in the design of magnetic molecules for use as cryogenic magnetic coolants were reviewed in [19].

This paper presents investigations on the magnetocaloric effect performed by us for molecular magnets based on octacyanidometallates. As it has been known from long-lasting studies of Prussian blue analogs, M_A^II[M_B^III(CN)₆]_2/3·zH₂O or A^IM_A^II[M_B^III(CN)₆] (M_A^II and M_B^III are 3d metal ions and A^I is the alkali ion), as well as of other hexacyanidometallates, the cyanobridge is able to mediate strong antiferromagnetic or ferromagnetic interactions between the metal moments [20]. Besides high temperatures of magnetic order reaching room temperature, Prussian blue analogs (PBAs) may show many fascinating properties, like light- and pressure switchable magnetism, magneto-optical effects, or chemically controlled growth of nanosized systems [21]. Unlike the hexacyano- [M(CN)₆] blocks of octahedral symmetry, which set up the cubic structure of PBAs, bimetallic octacyanometallates are based on more flexible building blocks offering eight CN-bridges to link the M and M’ metal ions. Most often used [M(CN)₈] blocks are 5d [W^V(CN)₈] or 4d [Nb^IV(CN)₈] and [Mo^V(CN)₈] complexes, all of spin S = 1/2. Eight possible coordination sites, the proper choice of other metal M’ and of additional organic ligands resulted in a variety of obtained structures and magnetic properties [22,23,24]. Among octacyanometallates one can find slowly relaxing systems [25,26,27], guest-molecule absorptive porous networks [28,29], photomagnets [30,31], and magneto-optically active compounds [32], as well as molecular sponges, which change, in a reversible way, the ordering temperature T_c and the coercive field H_c upon hydration/dehydration process [33] and other functional materials [34].

Below, we discuss the magneto-thermal properties of the octacyanometallates showing the different types of crystal architecture. The most numerous group is the family of 3D octacyanoniobate-based networks with different nonmagnetic organic ligands which significantly affect the structure and overall behaviour of the material. Another subject under study is the high-spin dodecanuclear cluster compound Ni₉[W(CN)₈]₆, a possible candidate for cryogenic magnetic cooling. Two experimental methods for measuring the MCE data, i.e., calorimetry and/or magnetometry were used. Moreover, the new approach, consisting in measuring MCE for two perpendicular sample orientations (so called Rotating Magnetocaloric Effect RMCE), is reported for a low anisotropy 2D {Mn^II(R-mpm)₂]₂[Nb^IV(CN)₈]}∙4H₂O ferrimagnet, as well as for an anisotropic bilayered 2D Cu₄[W(CN)₈]₄ molecular crystal showing the topological phase transition. Conclusions regarding the scaling and critical behaviour in some systems under study are also included.

2. Deriving Magnetocaloric Effect from Calorimetric Data

2.1. Thermodynamic Setup

The magnetocaloric effect (MCE) is quantified either by the isothermal entropy change ΔS_m or the adiabatic temperature change ΔT_ad due to the external field change H_i→H_f. While the former quantity may be derived from magnetometric data by using the integral version of the Maxwell relation $\partial S(T,H)/\partial H=\partial M(T,H)/\partial T$, the latter one can be obtained solely from calorimetric data. In this way, the heat capacity measurements performed without an external magnetic field as well as in a nonzero field represent a more complete set of characteristics allowing one to arrive at both MCE quantities. The step of pivotal importance in such a derivation is the determination of the temperature and field dependence of the entropy thermodynamic function S(T,H). Then, the isothermal entropy change ΔS_m is obtained by a simple subtraction:

$$\Delta S_{\mathrm{m}}(T,\Delta H=H_{\mathrm{f}}-H_{\mathrm{i}})=S(T,H_{\mathrm{f}})-S(T,H_{\mathrm{i}})$$

(1)

while the calculation of the adiabatic temperature change ΔT_ad is based on the formula:

$$\Delta T_{\mathrm{ad}}(T,\Delta H=H_{\mathrm{f}}-H_{\mathrm{i}})={[T(S,H_{\mathrm{f}})-T(S,H_{\mathrm{i}})]}_S$$

(2)

which requires the inversion of the S(T,H) dependence with respect to variable T. However, the above procedure is not as straightforward as it might seem. The problem lies in the experimental limitations where we can measure the heat capacity C_p(T,H) down to a possibly small but finite temperature T_L never reaching the limit of 0 K, while for the correct determination of the S(T,H) function one needs to know the C_p(T,H) function from zero absolute temperature: $S(T,H)={\displaystyle {\int }_0^T{C}_p(T^{\prime },H)/T^{\prime }}\mathrm{d}{T}^{\prime }$. A natural way to solve the problem is using a plausible extrapolation scheme. In what follows we will present two such schemes employed in the investigations of molecular magnets. The first scheme is more complete and involves the determination of the so called baseline incorporating the lattice contribution to the heat capacity. Having the baseline, one can extract the magnetic excess heat capacity ΔC_p and extrapolate it down to 0 K by assuming a two parameter algebraic function $\Delta C_p(T,H)=A(H)T^{B(H)}$ which is believed (there are no theoretical accounts of that) to work well for the nonzero-field case as it is well-known to do in the zero-field case (the Bloch law for ferromagnets). The extrapolation scheme consists in two independent steps. Firstly, the algebraic function is fitted to the magnetic excess heat capacity data ΔC_p in the narrow temperature range [T_L,T_max] for the different applied field values. Next, the resultant fits are complemented with the extrapolated baseline to yield the final form of the sought-for low temperature extrapolation of C_p(T,H). The other scheme is a simplified one as it obviates the need to determine the normal heat capacity (baseline). However, it does require the extrapolation of the C_p data down to 0 K. The extrapolation is performed assuming again a two parameter function $C_{p,\mathrm{LT}}(T,H)=A(H)T^3+B(H)T^{3/2}$, where the first term corresponds to the lattice contribution, while the second term represents the Bloch law for the magnonic contribution.

The above schemes will be exemplified by three compounds belonging to the class of molecular magnets. The first compound (1) with the chemical formula {Ni[Ni(4,4′dtbpy)(H₂O)]₈[W(CN)₈]₆}∙17H₂O (4,4′dtby − 4-4-di-tert-butyl-2,2′-bipyridine, C₂₀H₂₄N₂O₄) (Ni₉[W(CN)₈]₆ in short) consists of unique clusters. The main component of the compound is a fifteen-center cyanido-bridged Ni(II)₉W(V)₆ molecule forming a cube-like framework to which the spacious tert-butyl substituted bipyridine ligands are connected. The spin carried by the W(V) ion is S_W = 1/2, while that of the Ni(II) ion is S_Ni = 1. This allows one to calculate the maximal molar magnetic entropy of the cluster as equal to $S_{\max ,1}=R\ln {(2S_{\mathrm{Ni}}+1)}^9{(2S_{\mathrm{W}}+1)}^6\approx$ 116.79 J·K⁻¹∙mol⁻¹, which makes one anticipate a considerable magnetocaloric response. However, the spins constituting the cluster are ferromagnetically coupled through the cyanide linkages giving rise to a relatively high spin S = 12 [35], which leads to the entropy content amounting to $R\ln (2S+1)\approx 26.76\mathrm{J}\cdot {\mathrm{K}}^{-1}\cdot {\mathrm{mol}}^{-1}\approx S_{\max ,1}/5$. It is this reduced value that sets the order of magnitude of MCE at low temperatures, where the intracluster interactions are at play. The detailed study of MCE for this compound was reported in [36]. The second compound (2) is a bimetallic molecular magnet {[Mn^II(pyrazole)₄]₂[Nb^IV(CN)₈]·4H₂O}_n (pyrazole is a five membered ring ligand C₃H₄N₂) [37].The Mn^II ions carry the spin of S_Mn = 5/2, while the Nb^IV ions possess the spin of S_Nb = ½, which implies that the maximal molar magnetic entropy of the system is $S_{\max ,2}=R\ln {(2S_{\mathrm{Mn}}+1)}^2(2S_{\mathrm{Nb}}+1)\approx$ 35.56 J·K⁻¹ mol⁻¹. This, together with the fact that the compound exhibits the second-order phase transition, makes one anticipate a considerable magnetocaloric response. Yet, an antiferromagnetic coupling between the Mn^II and Nb^IV ions was suggested by the analysis of the magnetometric data [37], which implies that a more representative spin value per formula unit (at least in the magnetic fields below the decoupling threshold) is S = 2S_Mn − S_Nb = 9/2. The corresponding entropy content is reduced down to $R\ln (2S+1)\approx$ 19.14 J·K⁻¹ mol⁻¹ but still substantial. The detailed analysis of MCE for 2 was reported in [38]. The third compound (3) is a bimetallic coordination polymer {[Fe^II(pyrazole)₄]₂[Nb^IV(CN)₈]·4H₂O}_n, isostructural with 2 [37]. The spin of the Fe^II ion is S_Fe = 2, while the spin carried by the Nb^IV ion is S_Nb = ½, which sets the maximal molar magnetic entropy of the system as equal to $S_{\max ,3}=R\ln {(2S_{\mathrm{Fe}}+1)}^2(2S_{\mathrm{Nb}}+1)\approx$ 32.53 J·K⁻¹ mol⁻¹. The compound is known to display the second-order phase transition and thus a considerable magnetocaloric effect may be anticipated. On the other hand, the preliminary analysis of the magnetometric data implied an antiferromagnetic coupling between the constituent ions [37], which suggests that the total spin of S = 2S_Fe − S_Nb = 7/2 is a more representative spin value per formula unit, at least in the low temperature and low field regime. The entropy content associated with this spin value amounts to $R\ln (2S+1)\approx$ 17.29 J·K⁻¹ mol⁻¹, which is reduced by half but still substantial. The detailed report of MCE for this compound is given in [39].

2.2. Cluster Compound Ni₉[W(CN)₈]₆

The compound crystallizes in a triclinic system, space group $P\overline{1}$. The unit cell comprises one centrosymmetric cluster, the structure of which is shown in Figure 1. Due to the relatively large size of the tert-butyl substituted bpy (bpy = 2,2′-bipyridine = C₁₀H₈N₂) the distances between the cluster centers exceed 20 Ǻ and the π–π interactions in the system are practically absent. Hence the inter-cluster superexchange interactions may be completely neglected in the studied temperature range of 2–300 K.

Heat capacity of the compound was detected on cooling with the PPMS instrument without a field and for an array of applied field values in the temperature intervals 0.4–20 K with the ³He probe and 1.8–100 K with the standard probe cooled with liquid ⁴He. Due to the different sensitivities of the probes in different temperature regimes, we used the data provided by the former system below 15 K and those provided by the latter system above 15 K. In the zero applied field a broad anomaly of width about 2 K with a very flat maximum around 1.9 K was revealed, see Figure 2. An increasing applied field is apparent to gradually suppress the anomaly leaving no trace of it for the maximal field value of 9T.

The zero-field heat capacity of 1 is depicted in Figure 3 in the range of 0.4–100 K. To construct the baseline an approximate approach was employed. The observed C_p values within the specific range 30–63 K above the anomaly, which we shall elucidate in what follows, are assumed to comprise two separate contributions. The first contribution involves the lattice degrees of freedom C_p(lattice) and the second contribution C_mag(H,T) is due to the Schottky-like anomaly anticipated for spin clusters coupled via the exchange interactions. While a polynomial approximation involving cubic and higher-order terms is employed to describe the former, the latter is introduced with the term proportional to $T^{-2}$. Thus the experimental heat capacity values are assumed to be represented within the range of 30–63 K by the following formula:

$$C_p=C_p(\mathrm{lattice})+C_{\mathrm{mag}}(\mathrm{HT})={\displaystyle \sum _{i=3}^m{a}_i{T}^i+b{T}^{-2}}$$

(3)

Fitting Equation (3) with m = 6 to the experimental heat capacity values yielded a₃ = 6.07(7) × 10⁻² J·K⁻⁴ mol⁻¹, a₄ = −2.08(4) × 10⁻³ J·K⁻⁵ mol⁻¹, a₅ = 2.76(9) × 10⁻⁵ J·K⁻⁶ mol⁻¹, a₆ = −1.33(6) × 10⁻⁷ J·K⁻⁷ mol⁻¹, and b = 45,355.9(1) J∙K∙mol⁻¹. The baseline is defined by C_p(lattice) thus determined and extrapolated down to 0 K (green solid curve in Figure 3). The subtraction of the baseline from the detected heat capacity values yields the magnetic excess heat capacity ΔC_p, see Figure 4.

To calculate the entropy content associated with the excess heat capacity ΔC_p in zero field the low temperature behaviour of ΔC_p was additionally analyzed. A function $f{T}^{\mu }$ was fitted to the magnetic excess heat capacity in the narrow range of 0.41−0.62 K (10 experimental points). The best fit yielded f = 11.1(2) J∙K^−(1+μ)mol⁻¹ and μ = 1.04(2) (see Inset of Figure 3). The low-temperature algebraic best-fit function was next used to estimate the entropy contribution in the temperature range 0–0.62 K. The entropic contribution in the temperature range 0.62–63 K was obtained by numerical integration of the area under the excess heat capacity signal $\Delta S={\displaystyle \int \Delta C_p\mathrm{d}\ln T}$. Finally, the high temperature excess heat capacity $b{T}^{-2}$ was used to estimate the entropic contribution above 63 K. The total entropy content amounts to 116.97 J·K⁻¹·mol⁻¹ and is slightly higher than the value of S_max,1 = 116.79 J·K⁻¹·mol⁻¹. Such a good level of agreement of the calculated magnetic entropy with the theoretically predicted value is not accidental. The particular choice of the temperature interval used to determine the baseline (30–63 K) was made so as to reproduce the total magnetic entropy associated with the reported spin cluster. The above procedure was automatized within a specially designed notebook of the Mathematica 8.0 environment.

Equations (1) and (2) were used to determine temperature dependences of ΔS_m and ΔT_ad. The entropy thermodynamic function S(T,H) was calculated upon the algebraic extrapolation of the magnetic excess heat capacity ΔC_p down to 0 K using the data in the interval 0.40–0.63 K (10 experimental points) for each field value. The resultant extrapolating functions were next appended with the baseline to yield the final forms of C_p(T,H).

Temperature dependence of ΔS_m for several indicated field change values (μ₀ΔH = μ₀(H-0) = μ₀H) is depicted in Figure 5. The −ΔS_m vs. T curves display maxima placed in the range 2.4−6.5 K in addition to becoming broader and higher with increasing field change values. The peak values of |ΔS_m| and positions are given in

Table 1. Peak values of ΔS_m and ΔT_ad of 1.

μ0ΔH (T)	Tpeak (K)	$|\mathbf{\Delta }{\mathit{S}}_{\mathbf{m}}^{\mathbf{peak}}|(\mathbf{J}·{\mathbf{K}}^{-1}·{\mathbf{mol}}^{-1})$	Tpeak (K)	$\mathbf{\Delta }{\mathit{T}}_{\mathbf{ad}}^{\mathbf{peak}}(\mathbf{K})$
1	2.4	7.18	2.9	1.6
2	3.0	11.42	2.6	2.8
3	3.5	14.48	2.4	3.6
4	3.9	16.62	2.3	4.2
5	4.3	18.38	2.2	4.6
7	5.3	20.86	2.0	5.2
9	6.5	22.77	2.0	5.6

. $|\Delta S_{\mathrm{m}}^{\mathrm{peak}}|$ for μ₀ΔH = 5 T of other magnetic molecules was reported to range from 25 to 45 J∙K⁻¹∙kg⁻¹ [19] which is considerably higher than $|\Delta S_{\mathrm{m}}^{\mathrm{peak}}|$ = 3.36 J∙K⁻¹∙kg⁻¹ (18.38 J∙K⁻¹∙mol⁻¹) recorded for 1. This is most probably due to the relatively higher molecular weight of 1. On the other hand, the observed $|\Delta S_{\mathrm{m}}^{\mathrm{peak}}|$ values lie close to the low temperature physical threshold defined by $R\ln (2S+1)\approx 26.76$ J∙K⁻¹∙mol⁻¹ with S = 12. In the whole experimental range, the −ΔS_M values remain positive exceeding 3 J∙K⁻¹∙mol⁻¹, although it is apparent they steeply drop below their peaks.

One usually looks at the parameter n ($=\mathrm{d}\ln |\Delta S_{\mathrm{m}}|/\mathrm{d}\ln H$) conveniently quantifying the local sensitivity of the isothermal entropy change ΔS_m to the external field amplitude $H=\Delta H=H_{\mathrm{f}}$ ($H_{\mathrm{i}}=0$). The value of n means that in the vicinity of a given thermodynamic point (T,H) the entropy change is approximately given by the power function $H^n$. At high temperatures, where the magnetization is directly proportional to the field (the Curie–Weiss law), the integral version of the Maxwell relation implies a quadratic field dependence of ΔS_m, giving rise to n = 2. Figure 6 shows the temperature dependence of the field-averaged value of parameter n for 1. The field-variation of n differs in intensity at different temperatures, which is reflected by the size of the error bars. The behaviour of n is consistent with the above high-temperature paradigm, showing a steady increase with temperature and approaching the limiting value of 2. At the experimental low-temperature threshold parameter n attains a value as low as 0.16.

In Figure 7 the adiabatic temperature change ΔT_ad is depicted. All the ΔT_ad(T) curves can be seen to display well-defined peaks. The peak amplitudes of ΔT_ad and the corresponding peak positions are listed in Table 1. The value of $\Delta T_{\mathrm{ad}}^{\mathrm{peak}}$ for μ₀ΔH = 1, 3, 5, 7 and 9 T amounts to 1.6, 3.6, 4.6, 5.2, and 5.6 K, respectively. The record beating Gd³⁺ dimer with $\Delta T_{\mathrm{ad}}^{\mathrm{peak}}\approx$ 3.5, 9.0, 12.7 K for μ₀ΔH = 1, 3, and 7 T, respectively [40], exceeds the $\Delta T_{\mathrm{ad}}^{\mathrm{peak}}$ values of 1 more than twice. However, they fall closer to those reported for the Mn₃₂ cluster ($\Delta T_{\mathrm{ad}}^{\mathrm{peak}}\approx$ 2.2, 4.5, 6.7 K for μ₀ΔH = 1, 3, and 7 T, respectively) [41]. The values of $\Delta T_{\mathrm{ad}}^{\mathrm{peak}}$ reported for the compound {[Ni^II(pyrazole)₄]₂ [Nb^IV(CN)₈]∙4H₂O}_n, 2.0 K for μ₀ΔH = 5 T, and 2.9 K for μ₀ΔH = 9 T [42], are significantly lower than for 1. The still lower value was recorded for molecular magnet Mn₂-pyridazine-[Nb(CN)₈] (1.5 K for μ₀ΔH = 5 T), but this is probably due to the relatively higher transition temperature of 43 K [43].

2.3. Coordination Polymer {[Mn^II(pyrazole)₄]₂[Nb^IV(CN)₈]·4H₂O}_n

Compound 2 crystallizes in the tetragonal space group I4₁/a [37]. Its unique structure consists of a 3D skeleton, where each Mn^II center is bridged to only two Nb^IV ions through the cyanido linkages Mn^II-NC-Nb^IV, while each Nb^IV centre is linked to four Mn^II ions. The remaining part of the pseudooctahedral coordination sphere of Mn^II is substituted with pyrazole molecules, while four further terminal cyanide ligands are linked with the Nb^IV ion. It is an interesting and unique structural feature that a three-dimensional (3D) extended network should emerge from such low connectivity indices. Figure 8 shows the crystal structure of the compound for instant reference.

The magnetic data imply that at T_c ≈ 23.8 K compound 1 undergoes a phase transition to the long-range magnetically ordered state [37]. The mean-field approach was employed to analyze the isothermal magnetization and the dc susceptibility, revealing the antiferromagnetic character of the coupling between the Mn^II and Nb^IV centers, which gives rise to an overall ferromagnetic behaviour. The coupling constant was estimated to amount to −6.8 cm⁻¹ (−9.8K) [37].

The relaxation calorimetry technique implemented in the PPMS Quantum Design instrument was used to measure the heat capacity of 1. The zero-field measurements carried out in the temperature range of 1.9‒101 K revealed a well-defined λ-shaped anomaly at 22.8 K (slightly lower than the magnetometric counterpart) providing evidence of the presence of a magnetic continuous transition, see Figure 9. Additionally, the sample was measured in the applied field of μ₀H = 0.1, 0.2, 0.5, 1, 2, 3, 4, 5, 7, and 9T in the range of 1.9–40.4 K in the cooling direction. To estimate the normal heat capacity (baseline), the observed C_p values within the specially selected range 31–55 K above the transition temperature (justified in what follows) were considered to involve two independent contributions. The first contribution originates from the lattice degrees of freedom C_p(lattice) and the second contribution is magnetic and due to the short-range order C_mag (short-range order). While the latter is introduced by the term proportional to T⁻², a polynomial approximation with cubic and higher order terms is used to model the former. Thus the experimental values of the heat capacity within the range of 31−55 K are assumed to be given by the formula:

$$C_p=C_p(\mathrm{lattice})+C_{\mathrm{mag}}(\mathrm{short}-\mathrm{range}\mathrm{order})={\displaystyle \sum _{i=3}^n{a}_i{T}^i+b{T}^{-2}}$$

(4)

Fitting Equation (4) with n = 6 to the experimental heat capacity values yielded a₃ = 1.8674 × 10⁻² J·K⁻⁴·mol⁻¹, a₄ = −7.0801 × 10⁻⁴ J·K⁻⁵·mol⁻¹, a₅ = 1.0546 × 10⁻⁵ J·K⁻⁶·mol⁻¹, a₆ = −5.7808 × 10⁻⁸ J·K⁻⁷·mol⁻¹, and b = 8345.94 J·K mol⁻¹. The lattice heat capacity C_p(lattice) thus determined extrapolated down to 0 K defines the baseline (solid curve in Figure 9).

The subtraction of the lattice heat capacity from the detected heat capacity values yields the magnetic excess heat capacity ΔC_p, see Figure 10. Increasing magnetic field is apparent to suppress the anomaly peak. Moreover, consistently with a system with dominating antiferromagnetic coupling it slightly moves the anomaly toward lower temperatures. The calculation of the entropy associated with zero-field ΔC_p proceeded in three separate steps. Firstly, the contribution to the entropy above 55 K was estimated by considering the high temperature excess heat capacity bT⁻². Secondly, the contribution in the temperature range 1.9–55 K was calculated using the formula $\Delta S={\displaystyle \int \Delta C_p\mathrm{d}\ln T}$. And finally, to obtain the low temperature contribution function fT^μ was fitted to ΔC_p in the range of 1.9–3.0 K, yielding f = 0.45(1) J·K^−(1+μ) mol⁻¹and μ = 1.45(3) (see Inset of Figure 9). Next, the fitted function was logarithmically integrated in the interval 0–1.9 K. The total magnetic entropy is received by summing the above three contributions giving 35.31 J·K⁻¹ mol⁻¹. This value compares perfectly well with S_max,2 = 35.56 J·K⁻¹·mol⁻¹, which is attributable to the specific choice of the temperature interval (31–55 K) used to determine the baseline.

Similarly as for 1 using Equations (1) and (2), the isothermal entropy change ΔS_m and the adiabatic temperature change ΔT_ad were determined. The entropy thermodynamic function S(T,H) was calculated using the baseline and the algebraic extrapolation of ΔC_p based on the data in the temperature range of 1.9–3.0 K. Figure 11 shows the temperature dependence of ΔS_m for μ₀ΔH = 0.1, 0.2, 0.5, 1, 2, 3, 4, 5, 7 and 9T (H_i = H, H_f = 0). It is apparent that the corresponding curves display a peak. In

Table 2. Peak values of ΔS_m and ΔT_ad of 2.

μ0ΔH (T)	Tpeak (K)	$|\mathbf{\Delta }{\mathit{S}}_{\mathbf{m}}^{\mathbf{peak}}|(\mathbf{J}·{\mathbf{K}}^{-1}·{\mathbf{mol}}^{-1})$	Tpeak (K)	$\mathbf{\Delta }{\mathit{T}}_{\mathbf{ad}}^{\mathbf{peak}}(\mathbf{K})$
0.1	23.3	0.29	23.3	0.06
0.2	23.8	0.68	23.3	0.14
0.5	23.8	1.50	23.8	0.30
1	23.8	2.44	23.8	0.50
2	24.3	3.85	23.8	0.80
3	24.3	4.99	23.8	1.04
4	24.3	5.97	23.8	1.24
5	24.3	6.83	23.8	1.42
7	24.3	8.30	23.8	1.73
9	25.5	9.49	23.8	1.97

the peak values of ΔS_m together with the peak positions are provided. In addition to the ΔS_m data inferred from the calorimetric measurements Figure 11 shows also the ΔS_m values obtained by using the Maxwell relation from the magnetometric data [42]. Except for the two highest field change values (4 and 5 T), where the magnetometric data are slightly scattered around the smoother calorimetric data, both sets agree strikingly well. The lack of smoothness is probably a consequence of the problems of the particular instrument with temperature stabilization.

The value of $\Delta S_{\mathrm{m}}^{\mathrm{peak}}$ = 6.83 J·K⁻¹·mol⁻¹ detected for μ₀ΔH = 5T slightly exceeds that reported in [42] (6.7 J·K⁻¹·mol⁻¹). The isostructural compound {Ni^II(pyrazole)₄]₂[Nb^IV(CN)₈]·4H₂O}_n reveals at the same time the lower entropy change of 6.1 J·K⁻¹·mol⁻¹ [42], which most probably can be attributed to the smaller spin value of the Ni^II ion (S_Ni = 1). Figure 12 shows the thermal dependence of the field-averaged exponent n for 2. On heating parameter n it is apparent that it smoothly decreases down to the minimal value of 0.62 attained at 23.6 K (slightly above the transition temperature), and subsequently increases toward the high temperature value of 2. At the transition temperature T_N = 22.8 K parameter n takes on the value of 0.64. It can be compared to that calculated using the relationship: ${n|}_{T=T_{\mathrm{c}}}=1+(\beta -1)/(\beta +\gamma )$[8,44]. The value of 0.6424(4), obtained with the above equation and the theoretical estimates of β = 0.3689(3) and γ = 1.3960(9) for the 3D Heisenberg universality class [45], is very close to that obtained for 2.

The ΔT_ad(T) curves are shown in Figure 13. A two-peak structure of the curves is apparent. In addition to the expected peak located near the transition temperature they reveal the second peak at the lowest experimentally accessible temperatures. In Table 2 the primary peak values of ΔT_ad are provided. The positions of the primary peaks are practically independent of the applied field change. Moreover, there is their slight shift off the transition temperature toward higher temperatures. By contrast, the increasing magnetic field change seems to move the secondary peaks toward lower temperatures. The isostructural compound {Ni^II(pyrazole)₄]₂[Nb^IV(CN)₈]·4H₂O}_n revealed higher values of ΔT_ad, i.e., 2.0 K for μ₀ΔH = 5 T, and 2.9 K for μ₀ΔH = 9 T [42]. The amplitudes of ΔT_ad reported for Mn₂-pyridazine-[Nb(CN)₈] (1.5 K for μ₀ΔH = 5 T) [43] and hexacyanochromate Prussian blue analogues (1.2 K for μ₀ΔH = 7 T) [46] are at the same time comparable to those for 2.

The occurrence of the secondary peaks in the ΔT_ad(T) signal around 2 K points to the possibility that beyond the experimentally accessible temperature range an additional magnetic transition is concealed. This conjecture is neither confirmed by the shape of the ΔS_m(T) signal nor it was confirmed by additional heat capacity measurements in the temperature range 0.8–10 K [38]. Therefore it seems plausible that the reason for the presence of the secondary peaks is merely the proximity effect to the natural boundary of the temperature scale (T = 0 K). In order to shed some light onto that point a hypothetical paramagnetic medium was considered which consists of entities carrying the effective spins of S = 9/2 (2 S_Mn − S_Nb = 5/2 + 5/2 − 1/2). ΔT_ad of this medium may be found solving the following differential equation implied by the Maxwell thermodynamic relation:

$$\frac{\mathrm{d}T}{\mathrm{d}H}=-\frac{{(\frac{\partial M(T,H)}{\partial T})}_H}{C_{\mathrm{L}}(T)+C_H(T,H)}$$

(5)

where M(H,T) denotes the molar magnetization, C_L stands for the lattice contribution to the total heat capacity, C_H is the magnetic contribution. The baseline determined for 2 may serve as C_L. The paramagnetic Hamiltonian corresponding to an isolated spin S subject to applied magnetic field provides the basis to calculate C_H and $\partial M(T,H)/\partial T$. Let us note that in [38] an integral version of Equation (5) was used, which is incorrect due to the explicit T-dependence of its right-hand side. Solving Equation (5) with initial conditions T(H_i ≠ 0) = T_i in the field interval [H_i,H_f = 0] for several arbitrary values of the initial temperature T_i yields the system of adiabats shown in Figure 14 in the (H,T) plane.

Using Equation (5) the ΔT_ad signal was calculated for the magnetization processes H_i = 0 → H_f, with H_f assuming the integer field values, within the temperature interval 0 −55 K. The result is shown in Figure 15 together with the ΔT_ad signal of 2 (cf. Figure 13). Two points may be raised here. Firstly, the ΔT_ad(T) curves of the hypothetical medium are peaked at 0 K, which is the direct vicinity of the secondary peaks in the ΔT_ad signal of 2. Secondly, the magnitude of ΔT_ad of 2 is about four times smaller than that of the hypothetical medium. Strikingly enough, it may be concluded that the presence of a magnetic phase transition at some finite temperature has an adverse effect on MCE.

2.4. Coordination Polymer {[Fe^II(pyrazole)₄]₂[Nb^IV(CN)₈]·4H₂O}_n

Compounds 2 and 3 share the crystallographic structure. The graphical representation of the crystal structure of 3 is shown in Figure 8 with the Mn^II ions replaced by the Fe^II ions. The magnetic data implied that the compound undergoes a transition to a long-range magnetically ordered state at T_c ≈ 7.8 K [3,16]. Moreover, the muon spin rotation spectroscopy (μSR) strongly suggested that the compound may be assigned to the universality class of the 3D Heisenberg model [40]. The calculations based on the mean field-model and the magnetometric data pointed to the antiferromagnetic character of the superexchange coupling between the Fe^II and Nb^IV centres [37].

The relaxation calorimetry technique implemented in the PPMS Quantum Design instrument was employed to measure the heat capacity of 3. The measurements were carried out both without applied field and in nonzero field in the range 0.36–20.2 K in the cooling direction. The temperature dependences of ΔS_m and ΔT_ad were determined by employing Equations (1) and (2). In this case the entropy thermodynamic function S(T,H) was calculated without determining the normal heat capacity. However, the calculation required the extrapolation of C_p down to 0 K. Two independent contributions were considered in the extrapolation scheme involving two field-dependent parameters: $C_{p,\mathrm{LT}}(T,H)=A(H)T^3+B(H)T^{3/2}$. The first contribution accounts for the lattice degrees of freedom, whereas the second one is due to the magnonic excitations (the Bloch law).

Figure 16 shows the temperature dependence of ΔS_m for μ₀ΔH = 0.1, 0.2, 0.5, 1, 2, 5, and 9 T (H_i ≠ 0, H_f = 0).

Table 3. Peak values of ΔS_m and ΔT_ad of 3.

μ0ΔH (T)	Tpeak (K)	$|\mathbf{\Delta }{\mathit{S}}_{\mathbf{m}}^{\mathbf{peak}}|(\mathbf{J}·{\mathbf{K}}^{-1}·{\mathbf{mol}}^{-1})$	Tpeak (K)	$\mathbf{\Delta }{\mathit{T}}_{\mathbf{ad}}^{\mathbf{peak}}(\mathbf{K})$
0.1	8.9	0.3	8.8	0.1
0.2	8.9	0.5	8.8	0.2
0.5	8.9	1.0	8.9	0.4
1	9.3	1.6	8.8	0.6
2	9.3	2.7	8.8	1.1
5	10.3	4.9	8.9	2.0
9	6.9	6.9	8.8	2.8

lists the values of ΔS_m attained at the maximum. The isostructural compounds {[M^II(pyrazole)₄]₂[Nb^IV(CN)₈]·4H₂O}_n with M = Ni and M = Mn reveal $\Delta S_{\mathrm{m}}^{\mathrm{peak}}$ for μ₀ΔH = 5 T equal to 6.1 J·K⁻¹·mol⁻¹ and 6.7 J·K⁻¹·mol⁻¹, respectively, which is higher than the value of 4.9 J·K⁻¹·mol⁻¹ detected for 3 [42]. It is surprising that it should be so for the Ni congener as the Ni^II ion carries lower spin (S_Ni = 1 < S_Fe = 2). However, it is consistent with the ferromagnetic superexchange present in the Ni compound [37] and the relatively stronger anisotropy of the Fe^II centre [13].

The field-averaged exponent n of 3 vs. temperature is depicted in Figure 17. Its estimation drew on the data in Figure 16. The n vs. T curve reveals a minimum of 0.63 slightly above T_c = 8.3 K (implied by the peak in the zero-field heat capacity). On further increase of temperature, it increases toward the value of 2, consistent with the fact that in the paramagnetic phase the susceptibility is independent of field (the Curie-Weiss law). At T_c parameter n takes on the value of 0.64, which, like in the case of 2, indicates that 3 belongs to the universality class of the 3D Heisenberg model.

Temperature dependence of ΔT_ad of 3 calculated using Equation (2) is shown in Figure 18. All the ΔT_ad vs. T curves reveal peaks in the vicinity of T_c. The peak values together with the peak positions are listed in Table 3 which indicates that the peak positions are slightly moved off T_c toward higher temperatures and are practically independent of the applied field change value. The presence of a field induced effect is suggested by the occurrence of an inflection point at about 4 K (most apparent for higher fields) followed by a sharp drop. The values of ΔT_ad are comparable to those reported for its Ni congener (2.0 K for μ₀ΔH = 5 T, and 2.9 K for μ₀ΔH = 9 T) [42], and larger than those found for Mn₂-pyridazine-[Nb(CN)₈] (1.5 K for μ₀ΔH = 5 T) [43], hexacyanochromate Prussian blue analogues (1.2 K for μ₀ΔH = 7 T) [45], and the Mn congener (1.42 K for μ₀ΔH = 5 T, and 1.97 K for μ₀ΔH = 9 T) [38]. The last case might seem surprising as the Mn ion carries the maximal spin value (S_Mn = 5/2) among the 3d ions. However, the Mn congener (2) displays the transition to the magnetically ordered phase at the temperature (22.8 K) which is more than twice as high than for 3, and the ratio of the magnetic excess heat capacity to the lattice heat capacity is expected to decrease with temperature. Thus the magnetic entropy for 2 may be admittedly higher than for 3, but it is faced with a relatively higher heat capacity at the transition temperature, which suppresses the magnitude of ΔT_ad.

2.5. Final Remarks of Section 2

In this Section we have demonstrated how one can employ the calorimetric data to determine two main characteristics of the magnetocaloric effect, i.e., the isothermal entropy change ΔS_m and the adiabatic temperature change ΔT_ad. The pivotal step in the corresponding procedure is the calculation of the entropy thermodynamic function S(T,H) in the temperature interval starting from 0 K and for the given applied field values. Due to the experimental limitations, where the heat capacity measurements go down to a possibly small but finite temperature T_L never reaching the limit of 0 K, the procedure is out of necessity only approximate. The approximation consists in the extrapolation of the low-temperature C_p(T,H) values from T_L down to 0 K by using the heuristic algebraic function. Two approaches were considered, the simplified approach obviating the need to determine the normal heat capacity (baseline) and the more comprehensive approach, where the baseline is constructed using the polynomial approximation. Both approaches were exemplified by three instances of molecular magnets, the cluster compound involving 15 ferromagnetically coupled spins (1) and two isostructural 3D coordination polymers 2 and 3 including Mn^II and Fe^II ions, respectively. While 2 and 3 display the transitions to the magnetically long-range ordered phases, compound 1 does not order magnetically above 0.4 K. For the sake of comparison of the MCE signals of the studied compounds Figure 19 shows the temperature dependences of the isothermal entropy change ΔS_m and the adiabatic temperature change ΔT_ad corresponding to the field change μ₀ΔH = 5 T. It is apparent that at the lowest temperatures the cluster compound 1 substantially exceeds 2 and 3 in terms of the ΔS_m and ΔT_ad signals. The peak values of the ΔS_m quantity seem to monotonically depend on the constituent spin value with that for 1 being the highest (S = 12), that for 2 being intermediate (S_Mn = 5/2) and that for 3 being the smallest (S_Fe = 2). This hierarchy breaks down for the ΔT_ad peak values with the peak value of 3 exceeding that of 2, which can be rationalized by remembering that higher temperatures involve higher heat capacity values and a concomitantly lower heating or cooling effect. Compounds 2 and 3 can compete with 1 in terms of the ΔT_ad signal only in the intermediate temperature range. One can thus conclude that in order to optimize the MCE parameters one should strive to organize separate spin carriers in ferromagnetically coupled clusters rather than in 3D extended networks.

3. Magnetocaloric Properties and Critical Behaviour in Magnetically Ordered Compounds Investigated by Magnetometry

3.1. {[M^II(H₂O)₂]₂[Nb^IV(CN)₈]·4H₂O}_n(M = Fe, Mn) Molecular Compounds

{[Mn^II(H₂O)₂]₂[Nb^IV(CN)₈]·4H₂O}_n (4) and {[Fe^II(H₂O)₂]₂[Nb_IV(CN)₈]·4H₂O}_n (5) are isostructural molecular magnets, which crystallize in tetragonal space group I4/m. Despite the same structure these compounds show different types of magnetic moments order. Mn-based compound (4) is a ferrimagnet with the coercive field equal to 0 Oe and critical temperature T_c equal to 50 K. Fe-based analogue (5) is an example of a ferromagnet with the coercive field equal to 145 Oe and T_c equal to 43 K [47,48]. For both compounds the magnetocaloric effect was determined using an indirect method involving the measurements of a series of isothermal magnetization curves as a function of external magnetic field M(H) in a wide temperature range: above and below the critical temperature (Figure 20a). The magnetic entropy change |ΔS| was calculated based on the integrated Maxwell’s relation given by the equation:

$$\Delta S_M(T,\Delta H)={{\displaystyle \int }}_{H_1}^{H_2}{(\frac{\partial M(T,H)}{\partial T})}_{H}dH$$

(6)

As expected, a maximum ΔS value occurs at temperatures corresponding to T_c, and |ΔS| increases with ΔH. The replacement of Fe by Mn in isostructural {[M^II(H₂O)₂]₂[Nb^IV(CN)₈]·4H₂O}_n network causes an increase of |ΔS|^max as well as the temperature at which the maximum value of the entropy change occurs [49]. The maximum values of |ΔS| are equal to 4.82 J mol⁻¹K⁻¹ (8.65 J kg⁻¹K⁻¹) and 5.07 J mol⁻¹K⁻¹ (9.09 J kg⁻¹K⁻¹) for the field change of 0–5 T for 5 and 4, respectively. Slightly higher values of the magnetic entropy change observed for 4 may be related to the spin value of the Mn^II ion (S_Mn = 5/2), being higher than that of the Fe^II ion (S_Fe = 2) and the typically soft magnetic character.

A useful parameter describing the efficiency of a magnetocaloric material is the relative cooling power (RCP) defined as $RCP= {|\Delta S|}^{max}{\delta }_{FWHM}$, where δ_FWHM is the full width at half maximum of the magnetic entropy change curve. The RCP measured for an applied magnetic field of 5 T is equal to 118.40 and 125.43 J mol⁻¹ (212.61 and 225.59 J kg⁻¹) for 4 and 5, respectively. These values of RCP make about 50% of that of pure gadolinium-prototype magnetocaloric material.

An important aspect of the analysis of MCE is the construction of a universal curve of magnetic entropy change, a so-called master curve. Franco et al. [44,50] proposed that such universal entropy curve can be successfully used for determination of phase transition order. For the substances undergoing the second order phase transition, the temperature dependences of the magnetic entropy changes obtained at different applied magnetic fields may overlap after rescaling. The phenomenological universal curve can be constructed by normalizing all the ΔS(T) curves using their maximum value ΔS^max and, subsequently, re-scaling the temperature axis according to the following expression:

$$\theta =\frac{-(T-T_{\mathrm{c}})}{(T_{\mathrm{r}1}-T_{\mathrm{c}})},T\le T_{\mathrm{c}};\theta =\frac{(T-T_{\mathrm{c}})}{(T_{\mathrm{r}2}-T_{\mathrm{c}})},T>T_{\mathrm{c}}$$

(7)

The values of reference temperatures T_r1, T_r2 were calculated based on $\Delta S(T_{\mathrm{r}})/\Delta S^{\max }=h$, where h is a reference level with value from the range 0–1. For both studied materials: 4 and 5, the rescaled ΔS(T)curves, create one master curve in a wide temperature range, and thus the type of the magnetic order does not affect their construction. Figure 21 shows the comparison of universal entropy curves obtained for samples 4 and 5.

3.2. {[M^II(pyrazole)₄]₂[Nb^IV(CN)₈]₃∙4H₂O}_n(M = Ni, Mn) Molecular Compounds

{[Mn^II(pyrazole)₄]₂[Nb^IV(CN)₈]₃∙4H₂O}_n (6) and [Ni^II(pyrazole)₄]₂[Nb^IV(CN)₈]₃∙4H₂O}_n (7) are subsequent examples of isostructural bimetallic compounds, where pyrazole is the five membered C₃H₄N₂ ring. MCE results for 6 determined by calorimetry, including both the entropy change and the adiabatic temperature change, have been already discussed in paragraph 2.3. Below, we proceed the study to compare the effect in the Mn-based compound with that in the Ni-based compound. Both substances show a sharp phase transition, from a high- temperature paramagnetic to a low-temperature ordered state and while 6 is a ferrimagnet with critical temperature equal to 23.8 K, 7 is a ferromagnet with T_c = 13.4 K [37]. The higher values of the magnetic entropy change were observed for sample 6, despite the ferrimagnetic coupling between magnetic ions (see Figure 22) [42]. The higher value of magnetic entropy change observed for 6 may be related to the spin value of the Mn^II ion (S_Mn=5/2) being higher than that of the Ni^II ion (S_Ni = 1). For Ni-based compound, the magnetic entropy change was also determined by means of the heat capacity measurements. Obtained result is in good agreement with the previous estimation inferred from M(H, T) data. The similar reasonable agreement of the magnetically and thermally determined values of the magnetic entropy change was previously observed for some inorganic alloys [51]. Based on the heat capacity data for 7, we estimate the maximum value of the adiabatic temperature change ΔT_ad upon the applied magnetic field change of 9 T was equal to 2.9 K [42].

3.3. {Mn^II₂(imH)₂(H₂O)₄[Nb^IV(CN)₈]∙4H₂O}_nMolecular Magnetic Sponge

{Mn^II₂(imH)₂(H₂O)₄[Nb^IV(CN)₈]∙4H₂O}_n (8) is a molecular magnet, where imidazole is a bridging ligand. This compound is an example of magnetic sponge, because it allows an easy and reproducible control of the amount of water molecules absorbed in this material. During one-step dehydration, one new CN⁻bridge is formed in this material. The changes in the compound structure generated by the loss of water molecules cause the shift of the magnetic ordering temperature from 25 K (8) to 68 K (8_deh) [52]. The dehydration process is responsible for the change of magnetocaloric properties of the system: the maximum value of magnetic entropy change determined for 8_deh is 40% lower than for the as-synthesized compound 8 [53] (see Figure 23). Using the molecular field approximation (MFA) an attempt was made to explain the origin of the magnetic entropy changes in the system, for both forms 8 and 8_deh. As shown in Figure 23, in comparison to the experimental MCE results, the calculated entropy change is underestimated above the transition temperature, which can be attributed to the fact that MFA does not account for the short-range correlations. However, below the critical temperature, the calculated entropy change is overestimated in relation to the experimental data [53]. This furthermore, may be attributed to the lack of thermal and quantum fluctuations in the mean field model. These fluctuations impede the magnetic moments reorientation caused by the change of magnetic field, thus the magnetic susceptibility is higher than in normal paramagnets.

3.4. [{[Mn^II(pydz)(H₂O)₂][Mn^II(H₂O)₂][Nb^IV(CN)₈]}∙2H₂O]_nTwo-Step Molecular Magnetic Sponge

[{[Mn^II(pydz)(H₂O)₂][Mn^II(H₂O)₂][Nb^IV(CN)₈]}∙2H₂O]_n(9) is an example of multifunctional molecular magnet, where a pyridazine,C₄H₄N₂, was chosen as a bridging ligand. One of the unique properties of this compound is related to its two-step switchable magnetic sponge behaviour [33]. Within 6 h of dehydration of 9, the loss of four water molecules is observed, and 9 is transformed into dehydrated form—9_deh. This process is responsible for destabilization of hydrogen bonds and causes intraskeletal molecular rearrangement: formation of a new Nb–CN–Mn bond as well as migration of the pyridazine ligand between two Mn centres. The structural changes occurring during the dehydration induce a significant increase of critical temperature from 43 K, observed for 9, to 68 K recorded for 9_deh. In the second step of dehydration (of about 24 h) two remaining water molecules are removed and new stable, anhydrous phase 9_anh is formed. During this process the material structure does not change but some bonds are shortened. The critical temperature of anhydrous sample is equal to 98 K. The transformation from the initial, as synthesized 9 to the anhydrous form 9_anh is reversible [33].

As depicted in Figure 24, for all three forms of the compound: 9, 9_deh, and 9_anh, a maximum ΔS value occurs at temperatures corresponding to consecutive T_c, and |ΔS| increases with ΔH. The highest value of magnetic entropy change is revealed for 9. For dehydrated and anhydrous samples values of magnetic entropy change are similar—lower by 40% than that observed for the as-synthetized material. This result is perfectly correlated with AC susceptibility data: the decrease in ΔS by a factor of 0.6 for 9_deh and 9_anh relative to 9 corresponds to the same reduction in the χ’(T) magnitude of 9_deh and 9_anh [43].

For the as-synthesized sample 9 the heat capacity measurements were performed, first at zero applied field and then at 0.2, 0.5, and 1 T. The data obtained from this experiment were used for the evaluation of magnetocaloric effect based on the magnetothermal method. The values of the magnetic entropy change calculated based on the heat capacity and magnetic data are in very good agreement. Furthermore, it was also possible to estimate, from the heat capacity data, the adiabatic entropy change associated with the change of magnetic entropy. The maximum value of the adiabatic temperature change determined for 9, corresponding to the change of magnetic field from 0 to 1 T, is equal to 1.5 K [43].

3.5. [{[Mn^II(pydz)(H₂O)₂][Mn^II(H₂O)₂][Nb^IV(CN)₈]}∙2H₂O]_nMolecular Compound under Pressure

The uniqueness of compound 9 is also related to the possibility of the change of the magnetic properties by the application of hydrostatic pressure [54]. The application of hydrostatic pressure of 1.8 GPa results in the compression of the unit cell (ca. 7.6% in total volume) and a noticeable bending of the Mn-NC-Nb linkages. These structural modifications are accompanied by the change in magnetic properties of examined compound. The increase of the applied pressure increases the ordering temperature up to 48 K, 50.5 K and 52.5 K for pressure of 0.27 GPa, 0.57 GPa and 1.19 GPa, respectively. The shift of T_c towards higher temperatures due to application of mechanical stress is the result of the strengthening of Mn- and Nb-sublattice coupling due to the enhanced overlap of magnetic orbitals. The analysis of MCE for sample under pressure of 1.19 GPa (9_HP) showed the reduction of |ΔS|^max value in relation to as-synthesized compound 9 by 13.6% (see Figure 25) [55]. The summary of the magnetocaloric properties of all forms of molecular magnet 9 as well as results obtained for other octacyanoniobate- based compounds are presented in

Table 4. Comparison of MCE data obtained for samples discussed in Section 3. |ΔS|^max and RCP were determined for µ₀ΔH = 5 T; MCE: magnetocaloric effect; RCP: relative cooling power

Sample	Tc (K)	|ΔS|max (J mol−1 K−1)	|ΔS|max (J kg−1 K−1)	RCP (J mol−1 K−1)	RCP (J kg−1 K−1)
4	50.0	5.07	9.09	118.40	212.61
5	43.0	4.82	8.65	125.43	225.59
6	23.8	6.70	6.50	136.9	132.9
7	13.4	6.10	5.90	75.6	73.1
8	25.0	6.70	8.95	186.2	248.9
8deh	60.0	4.02	7.73	152.8	293.8
9	43.0	5.36	8.95	160.8	268.0
9deh	68	3.33	5.82	109.9	192.1
9anh	98	3.38	6.88	101.4	206.5
9HP	52.5	4.63	7.73	138.9	231.8

.

3.6. T_c^−2/3 Dependence of the Maximum Entropy Change

Oesterreicher and Parker predicted that the maximum value of magnetic entropy change is proportional to T_c^−2/3 [7]. This relation has been proved for the series of intermetallic samples [9]. Taking into account that in the compounds based on manganese and octacyanoniobate, the structural changes caused by the external stimuli or selection of a bridging ligand did not affect magnetic moment of Mn and Nb sublattices, while the critical temperature was changed, it was possible to verify the relation: |ΔS|^max~T_c^−2/3.

Figure 26a shows the comparison of the temperature dependences of magnetic entropy upon changing the magnetic field from 0 to 3 T, measured for samples: 9, 9_deh, 9_anh, 9_HP, 8, 8_deh, 4 and 6. The clear reduction of the height of the ΔS(T) peak with the increase of the critical temperature is observed. The slight deviation from this behaviour occurs for 9_deh, because the maximum value of magnetic entropy change is smaller than expected. It can be explained by instability of partially dehydrated sample. Figure 26b presents the values of magnetic entropy change as a function of T_c^−2/3 in all Mn-Nb based compounds under study obtained at the change of magnetic field from 0 to 1 T, 3 T and 5 T. The linear character of all determined relations confirms the proportionality of |ΔS|^max to T_c^−2/3.

3.7. The Critical Behaviour of the 3 D Octacyanoniobate- Based Compounds

The MCE is an intrinsic thermodynamic characteristic and in the vicinity of the ferro- and ferrimagnetic transition reflects the critical behaviour of the system. Thus, the critical exponents can be extracted by use of the isothermal magnetization curves measured with the aim to determine the magnetic entropy change. According to the scaling hypothesis, the spontaneous magnetization M_s below T_c, the inverse initial susceptibility χ₀ above T_c, and the magnetization dependence on magnetic field M(H) measured at temperature T_c are characterized by the set of critical exponents β, γ and δ. These exponents are defined by the following relations:

$$M_S(T)=M_0{(\epsilon )}^{\beta },T<T_{\mathrm{c}}$$

(8)

$${\chi }_0^{-1}(T)=(h_0/M_0){(\epsilon )}^{\gamma },T>T_{\mathrm{c}}$$

(9)

$$M=D{H}^{\frac{1}{\delta }},T=T_{\mathrm{c}}$$

(10)

where ε is a reduced temperature defined as

$$\epsilon =(T-T_c)/T_c$$

According to the mean field theory, which predicts in the vicinity of critical temperature T_c, β = 0.5 and γ = 1, the isotherms M²(H/M), so called Arrot plots, should consist of series of parallel straight lines, and the isotherm measured at T_c should pass through the origin of coordinate system. Arrot plots can be used for the determination of the order of phase transition occurring in the material under study. According to the Banerjee criterion [56], a positive slope of the M²(H/M) indicates a magnetic phase transition of the second order, while a negative slope corresponds to the first order phase transition. For all the octacyanoniobate-based samples the positive slope of M²(H/M) plots was observed, indicating the second order of the magnetic phase transition. The non-linear shape of M²(H/M) Arrot for all discussed samples plots indicates that mean field approach does not describes properly the critical properties of these materials.

The critical exponents β and γ were determined by using the Kouvel–Fisher method [57]. In the Kouvel–Fisher method two new variables X and Y are defined as

$$X(T)={\chi }_0^{-1}{(\frac{\mathrm{d}{\chi }_0^{-1}}{\mathrm{d}T})}^{-1}=\frac{T-T_{\mathrm{c}}}{\gamma },$$

(11)

$$Y(T)=M_0{(\frac{\mathrm{d}{M}_0}{\mathrm{d}T})}^{-1}=\frac{T-T_c}{\beta }.$$

(12)

X(T) and Y(T) are the linear functions of temperature with slope of 1/γ and 1/β, respectively. The intercepts of X(T) and Y(T) with the temperature axis correspond to the critical temperatures. Figure 27a shows an exemplary result obtained for sample 4 with the Kouvel–Fisher method, which next was used for the determination of the critical exponents β and γ for samples 9, 9_deh, 9_anh, 9_HP, 8, 8_deh, 4, and 6 [42,43,53,55,58]. The obtained results are presented in

Table 5. Comparison of the critical exponents determined for samples discussed in Section 3 and for the mean field and 3D Heisenberg, 3D Ising, and 3D XY model. n_MCE—the critical exponent n determined based on the MCE data, n_theor—the critical exponents n determined based on Equation (13).

Sample	β	γ	δ	nMCE	ntheor
4	0.41	1.32	4.39	0.69	0.66
5	0.37	1.33	4.37	0.67	0.63
6				0.64
7				0.59
8	0.37	1.35	4.48	0.65	0.63
8deh	0.37	1.40	4.95	0.67	0.64
9	0.38	1.35	4.69	0.66	0.64
9deh	0.43	1.38	4.23	0.68	0.69
9anh	0.39	1.37	4.49	0.69	0.65
9HP	0.37	1.40	4.48	0.67	0.64
Mean field model	0.500	1	3		0.66
Heisenberg model	0.365	1.385	4.8		0.64
Ising model	0.325	1.24	4.82		0.61
XY model	0.346	1.316	4.81		0.57

.

The value of critical exponent δ was estimated using the M(H) isotherm measured at temperature T_c. The magnetization curve measured at T_c presented in log–log plot is expected to be a straight line with a slope 1/δ. The example result of the analysis of the critical isotherm measured for 4 is shown in Figure 27c.

In the materials undergoing the second order phase transition, the magnetic entropy change is related to the change of the external magnetic field by the relation:$|\Delta S|(T, H)\propto H^n$ where n is a successive critical exponent. The mean field theory predicts the exponent n = 1 for the temperatures well below T_c, and n = 2 well above T_c. At critical temperature n is equal to 2/3. The analysis of temperature dependence of exponent n shows that for all octacyanoniobate-based compounds, shapes of the n(T) plot are very similar and the values of n at T_c are very close to 2/3. The example of dependence n(T) obtained for 4 is presented in Figure 27d.

At temperature T_c, critical exponents β and γ are related to exponent n by the formula [8]:

$$\mathrm{n}({\mathrm{T}}_{\mathrm{c}})=1+\frac{\mathsf{\beta }-1}{\mathsf{\beta }+\mathsf{\gamma }}$$

(13)

Thus, having the values of the critical exponents β and γ obtained with the Kouvel–Fisher method, the value of the critical exponent n can be estimated. For all octacyanoniobate-based samples we obtained a very good agreement between the values of critical exponent n determined using Equation (13) (in Table 5 denoted as n_theor) with the experimentally determined values based on MCE data (in Table 5 denoted as n_MCE).

The analysis of the critical exponents determined for the series of octacyanoniobate-based samples showed that except sample (7), all the examined compounds can be assigned to the universality class of the 3D Heisenberg model. It means that neither structural change due to the insertion of bridging ligands into the structure nor the influence of the external stimuli significantly affect the critical behaviour of the octacyanometallates-based materials. The result obtained for sample 7 is more consistent with XY model.

We have also compared the described above values of the critical exponents with the values predicted from the scaling hypothesis. In critical region, the magnetic equation of state can be written as:

$$M(H,\epsilon )={\epsilon }^{\beta }{f}_{\pm }(\frac{H}{{\epsilon }^{\beta +\gamma }})$$

(14)

where f_± is the scaling function: f₊ for T > T_c, f₋ for T < T_c. Equation (14) suggests, that M |ε|^−β plotted as a function of H·|ε |^−(β+γ) should give two different curves: one corresponding to the temperatures below the ordering temperature, and the other for temperatures above T_c. For all the studied samples for which the critical exponent β and γ were determined with Kouvel–Fisher method, M |ε|^−β vs. H·|ε |^−(β+γ) plots exhibit two independent branches, indicating that the values of the critical exponents are reasonably accurate. Figure 27b presents the example of logarithmic scaling plot of M |ε|^−β vs. H·|ε |^−(β+γ) obtained for sample 4.

3.8. Final Remarks of Section 3

In this section we have investigated the magnetocaloric effect and critical behaviour in molecular magnets showing long-range magnetic order. The most important issue of this study was the test of the influence of bridging ligands and external stimuli on magnetocaloric properties of examined samples.

We have demonstrated that for the manganese octacyanoniobate-based samples with different brigding ligand—pyridazine, pyrazole, imidazole, for which the magnetocaloric effect was determined for as-synthetized samples and modified by external stimuli—the maximum value of magnetic entropy change ΔS^max is proportional to T_c^−2/3.

We have also proved, that 3D Heisenberg model is the most adequate for the description of critical behaviour of octacyanoniobate-based molecular magnets. The analysis of critical exponents β, γ, δ, and n showed that neither structural changes due to the insertion of bridging ligands into the structure nor the influence of external stimuli such as hydrostatic pressure and dehydration/hydration process do not significantly affect the critical behaviour of octacyanometallate- based materials. Moreover, the value of critical exponent n describing the field dependence of MCE according to $|\Delta S|(T, H)\propto H^n$ is consistent with other critical exponents obtained from the magnetization data.

4. Rotating Magnetocaloric Effect in Anisotropic Two-Dimensional Molecular Magnets

The rotating magnetocaloric effect (RMCE) is a new issue in the magnetic cooling research. In contrast to conventional magnetocaloric effect, in RMCE the change of entropy is obtained not by changing the external magnetic field, but with rotation of a single crystal in a constant magnetic field [59,60,61,62]. If the compound reveals a substantial magnetic anisotropy, then the magnetic entropy will depend on the crystal orientation in the magnetic field. The rotation is changing the crystal orientation with respect to the applied field direction, therefore a change of magnetic entropy is observed and can be used in cooling cycle. Recently, Balli et al. [63,64] have introduced a realization of refrigerator based on RMCE. This approach has several advantages: simple construction, high efficiency [63,64,65] (cycles in higher frequency than the conventional MCE), or working in constant field (lower power consumption, possibility of use permanent magnets).

Most of the research concerning rotating magnetocaloric effect deals with inorganic materials [59,60,61,63,64,66,67,68] and there are only few examples related to molecular magnets [69]. In our research we have focused on two-dimensional molecular compounds which reveal magnetic anisotropy and transition to long-range ordered phase. Single crystal studies allowed us to explore the anisotropy of MCE. The dependence of MCE on the orientation was used to study the RMCE in case of low (Mn^II(R-mpm)₂]₂[Nb^IV(CN)₈]}∙4H₂O) [70] and high ({(tetren)H₅)_0.8Cu^II₄[W^V(CN)₈]₄·7.2H₂O}_n) [71] magnetic anisotropy. In particular, we have shown that inverse magnetocaloric effect can be used to enhance the RMCE up to 51% in respect to conventional MCE.

4.1. Low Anisotropy Case: {Mn^II(R-mpm)₂]₂[Nb^IV(CN)₈]}∙4H₂OCrystal

{Mn^II(R-mpm)₂]₂[Nb^IV(CN)₈]}∙4H₂O (10), where mpm = α-methyl-2-pyridinemethanol, is a two-dimensional coordination ferrimagnet [70]. The separation between layers of square grid topology is 7.5 Å. Nevertheless, the compound reveals a phase transition to 3D long-range magnetic ordered state at T_c = 23.5 K due to intermolecular dipole-dipole interactions. The magnetic single crystal measurements of 10 showed an easy-plane type anisotropy within the layers, whereas the perpendicular direction was a hard axis [72]. The observed anisotropy was not significant, since above 0.35 T applied field both orientations was magnetically undistinguishable. The magnetocaloric effect was obtained by the indirect method from magnetization measurements in two orientations: bc||H (easy plane) and a*||H (hard axis).

In high magnetic fields (above 1.0 T) the difference between magnetic entropy change in both orientations is modest (Figure 28a), as a consequence of low magnetic anisotropy. In lower fields the value of −ΔS_m is small, but the relative change between easy plane and hard direction is more significant (Figure 28b). Moreover, in the hard axis orientation an inverse magnetocaloric effect can be noticed. The magnetic entropy change related to rotation (ΔS_RMCE) by 90° from hard direction (a*||H, hard axis) to easy axis (bc||H) can be calculated by

$$-\Delta S_{RMCE}=-(\Delta S_{easy}-\Delta S_{hard})$$

(15)

where ΔS_easy and ΔS_hard stands for magnetic entropy change in easy plane and hard axis orientations, respectively. Figure 28 and Figure 29 show the obtained values of −ΔS_RMCE. In high magnetic fields (Figure 28a) both conventional MCE, for hard axis and easy plane, are greater than the RMCE in whole temperature range. The situation is changing in lower fields μ₀H < 0.2 T (Figure 28b), for which the RMCE can have higher output even than the MCE for easy plane. This excess is a consequence of rotation from hard axis (higher entropy) to easy plane (lower entropy) orientation and the inverse MCE in hard axis. Figure 30 shows the ratio between the entropy change from RMCE and conventional MCE. Depending on the applied field and the temperature, the RMCE can be more efficient than MCE in an easy plane up to 51%.

4.2. High Anisotropy Case: (tetren)Cu₄[W(CN)₈]₄ Crystal

The high anisotropy case was studied [71] with (tetren)Cu₄[W(CN))₈]₄ (11)(full formula: {(tetren)H₅)_0.8Cu^II₄[W^V(CN)₈]₄·7.2H₂O}_n, tetren = tetraethylenepentamine), a 2D cyanido-bridged network with significant 2D XY magnetic anisotropy (ac is the easy plane, b is the hard axis) and the Berezinskii-Kosterlitz-Thouless [73] topological phase transition at T_BKT ≈ 33 K [74,75]. The anisotropy for 11 is so high, that even μ₀H = 7.0 T magnetic field is too weak to merge magnetization curves of two orientations at 2.0 K [71]. The conventional and rotating magnetocaloric effects were studied in a similar procedure to the low anisotropy case (10). However, in this case the difference between magnetic entropy change for hard and easy orientations was relevant up to the highest measured field μ₀H = 7.0 T (Figure 31). Therefore, the absolute values of −ΔS_RMCE for the RMCE were one order of magnitude higher than for 10 (Figure 32). The temperature dependences of ΔS_RMCE for 11 have peculiar shapes with two peaks for fields below 7.0 T and a single peak for the highest field. The double peaks are related to different dependences of ΔS maxima for easy and hard orientation. The study also showed that it is possible to obtain an inverse RMCE (Figure 31 and Figure 32), as a consequence of temperature region where the −ΔS_m for hard axis is higher than for the easy plane.

Figure 33 shows the ratio between the conventional and rotating MCE for 11. Similar to the low anisotropy case of 10, there is a region where ΔS_RMCE is more efficient than the MCE for easy plane. However, the ΔS_RMCE/ΔS_easy ratio shows also negative values for lower fields (μ₀H < 3.0 T) and temperatures around 30 K. These negative values correspond to the conditions, where the sign of the ΔS_RMCE is negative, in other words, where the inverse RMCE occurs.

4.3. Final Remarks of Section 4

The rotational magnetocaloric effect is an alternative method for magnetic cooling. In our works we were studying 2D molecular magnets with low and high anisotropy. We have proven that the RMCE can be, in specific circumstances, more efficient than the conventional MCE and that the inverse RMCE is also possible. In our opinion, the rotating magnetocaloric effect in molecular magnets has potential application as cryogenic refrigerator. However, to compete with other types of low temperature refrigerators (e.g., He-3 type), the good candidate for a molecular RMCE refrigerant should be characterized by the following features [71]: (1) strong magnetic anisotropy, (2) T_c around 2 K, (3) large magnetic moments with ferromagnetic interactions, (4) easy plane anisotropy to achieve the RMCE enhancement due to the inverse MCE.

5. Final Conclusions

In this review, the magneto-thermal properties of octacyanometallate-based molecular magnets showing the different types of crystal architecture have been discussed. The investigation of magnetocaloric effect has been performed by two experimental methods: calorimetry and magnetometry. The highest value of magnetic entropy change and the change of adiabatic temperature was observed for the high-spin dodecanuclear cluster Ni₉[W(CN)₈]₆ making this compound a suitable candidate for application in cryogenic magnetic cooling. The systematic study of magnetocaloric effect in the ligand tunable coordination compounds is based on manganese and octacyanoniobate, showing the wide temperature range in which T_c occurs, allowing us to confirm the ΔS_m ~ T_c^−2/3 relation stemming from the molecular field theory. The set of critical exponents obtained for these compounds series belong to the 3D Heisenberg universality class. Finally, we have presented the study of rotating magnetocaloric effect (RMCE) in two 2D molecular magnets: ferromagnetic {Mn^II(R-mpm)₂]₂[Nb^IV(CN)₈]}∙4H₂O and strongly anisotropic (tetren)Cu₄[W(CN)₈]₄ bilayered magnet showing the topological Berezinskii- Kosterlitz-Thouless transition.