Negative thermal expansion and metallophilicity in Cu$_3$[Co(CN)$_6$]

TL;DR

This study synthesizes and characterizes Cu$_3$[Co(CN)$_6$], revealing its unique thermal expansion properties and weaker metallophilic interactions compared to similar materials, providing insights into designing highly responsive frameworks.

Contribution

It provides the first detailed analysis of Cu$_3$[Co(CN)$_6$]'s thermal expansion and metallophilic interactions using experimental and computational methods, highlighting the counterintuitive effects of interaction strength.

Findings

Cu$_3$[Co(CN)$_6$] exhibits anisotropic thermal expansion with opposing signs.

Weaker Cu...Cu interactions lead to moderate structural flexibility.

Strong interactions can be used to design ultra-responsive materials.

Abstract

We report the synthesis and structural characterisation of the molecular framework copper(I) hexacyanocobaltate(III), Cu$_3$[Co(CN)$_6$], which we find to be isostructural to H$_3$[Co(CN)$_6$] and the colossal negative thermal expansion material Ag$_3$[Co(CN)$_6$]. Using synchrotron X-ray powder diffraction measurements, we find strong positive and negative thermal expansion behaviour respectively perpendicular and parallel to the trigonal crystal axis: $\alpha_a$ = 25.4(5)\,MK$^{-1}$ and $\alpha_c$ = $-$43.5(8)\,MK$^{-1}$. These opposing effects collectively result in a volume expansivity $\alpha_V$ = 7.4(11)\,MK$^{-1}$ that is remarkably small for an anisotropic molecular framework. This thermal response is discussed in the context of the behaviour of the analogous H- and Ag-containing systems. We make use of density-functional theory with many-body dispersion interactions (DFT+MBD) to demonstrate that Cu$\ldots$Cu metallophilic (`cuprophilic') interactions are significantly weaker in Cu$_3$[Co(CN)$_6$] than Ag$\ldots$Ag interactions in Ag$_3$[Co(CN)$_6$], but that this lowering of energy scale counterintuitively translates to a more moderate---rather than enhanced---degree of structural flexibility. The same conclusion is drawn from consideration of a simple lattice dynamical model, which we also present here. Our results demonstrate that strong interactions can actually be exploited in the design of ultra-responsive materials if those interactions are set up to act in tension.

Tables

Table 1. Table 1:  Structural details for Cu 3 [Co(CN) 6 ] obtained by Pawley/Rietveld refinement against X-ray powder diffraction data collected at 300 K and estimated 0 K values extracted from linear fits to 100–598 K refinements. Atom positions are Co ( 0 , 0 , 0 ) 0 0 0 (0,0,0) , Ag ( 1 2 , 0 , 1 2 ) 1 2 0 1 2 (\frac{1}{2},0,\frac{1}{2}) , C ( x C , 0 , z C ) subscript 𝑥 C 0 subscript 𝑧 C (x_{\textrm{C}},0,z_{\textrm{C}}) , N ( x N , 0 , z N ) subscript 𝑥 N 0 subscript 𝑧 N (x_{\textrm{N}},0,z_{\textrm{N}}) .

	300 K (experimental)	0 K (estimated)
Crystal system	Trigonal	Trigonal
Space group	$P\overline{3}1m$	$P\overline{3}1m$
$a$ (Å)	6.9085(10)	6.8552
$c$ (Å)	6.7077(16)	6.7970
$V$ (Å3)	277.25(8)	276.66
$x_{\text{C}}$	0.2177(15)	0.2167
$z_{\text{C}}$	0.1566(14)	0.1533
$x_{\text{N}}$	0.3161(15)	0.3182
$z_{\text{N}}$	0.2920(14)	0.2887
$B_{\text{iso}}$ (Å2)	3.91(14)	–

Table 2. Table 2:  Experimental coefficients of thermal expansion for A 3 3 {}_{\textrm{3}} [Co(CN) 6 6 {}_{\textrm{6}} ] systems.

A	${\alpha }_a$	${\alpha }_c$	${\alpha }_V$	$\Delta T$	Ref.
	(MK-1)	(MK-1)	(MK-1)	(K)
H	12.0(4)	$-$8.8(3)	15.1(6)	4–300	34
Cu	25.4(5)	$-$43.5(8)	7.4(11)	100–598	This work
Ag	145.9(6)	$-$122.1(3)	169.8(9)	16–500	23

Table 3. Table 3: Comparison between experimental and ab initio lattice parameters for Cu[Co(CN) 6 ]. The difference term Δ Δ \Delta corresponds to the sum of absolute cell strains ∑ i | ( x i , calc − x i , exp ) / x i , exp | subscript 𝑖 subscript 𝑥 𝑖 calc subscript 𝑥 𝑖 exp subscript 𝑥 𝑖 exp \sum_{i}|(x_{i,\textrm{calc}}-x_{i,\textrm{exp}})/x_{i,\textrm{exp}}| .

	exp. (0 K)	PBE	TS	MBD
$a$ (Å)	6.8552	7.267	7.130	6.495
$c$ (Å)	6.7970	6.365	6.432	6.978
$V$ (Å3)	276.62	291.06	283.00	254.98
$\Delta$ (%)	0	18.4	13.4	13.2

Table 4. Table 4: The PBE+TS free-atom and in-solid vdW parameters for A atoms in A 3 [Co(CN) 6 ] (A = Ag, Cu, H) at experimental lattice constants.

	$C_6$ (hartree bohr6)		$R_0$ (bohr)
	free-atom	in-solid	free-atom	in-solid
Ag3[Co(CN)6]	339.00	295.73	3.82	3.73
Cu3[Co(CN)6]	235.00	207.03	3.76	3.64
H3[Co(CN)6]	6.50	4.28	3.10	2.89

Table 5. Table 5:  Lattice dynamical model parameters and comparison between calculated and observed lattice parameters. Refined parameters are shown in bold.

	H3[Co(CN)6]	Cu3[Co(CN)6]	Ag3[Co(CN)6]
$k_{\text{harm}}$ (eV/Å2)	400	400	400
$r_0$ (Å)	4.319	4.867	5.070
$k_{\text{angle}}$ (eV/rad2)	47	47	47
${\theta }_0$ (∘)	89	89	89
$C_6$ (eV Å6)	0	8810	14400
$a$ (Å)	6.450	6.901	6.812
$a_{\text{expt}}^{0\text{K}}$ (Å)	6.409	6.855	6.740
$\Delta a/a$ (%)	+0.6%	+0.7%	+1.1%
$c$ (Å)	5.749	6.842	7.474
$c_{\text{expt}}^{0\text{K}}$ (Å)	5.713	6.797	7.390
$\Delta c/c$ (%)	+0.6%	+0.7%	+1.1%

Equations

$$r_{\text{Cu…Cu}}=\frac{a}{2}.$$

$$r$$

$$\theta$$

$$E_{\text{harm}}=\frac{1}{2}{k}_{\text{harm}}(r_{\text{A–X}}-r_0{)}^2;$$

$$E_{\text{angle}}=\frac{1}{2}{k}_{\text{angle}}({\theta }_{\text{A–X–A}}-{\theta }_0{)}^2;$$

$$E_{\text{disp}}=-\frac{C_6}{(r_{\text{A…A}}{)}^6}.$$

$$E_{\text{latt}}=E_{\text{Coulomb}}+E_{\text{harm}}+E_{\text{angle}}+E_{\text{disp}}.$$

$${\alpha }_i\simeq \frac{C_T}{V}\hat{\gamma }{K}_i.$$

Full text

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††footnotetext: a* Department of Chemistry, University of Oxford, Inorganic Chemistry Laboratory, South Parks Road, Oxford OX1 3QR, U.K. Tel: +44 (0)1865 272137; E-mail: andrew.goodwin@chem.ox.ac.uk

b State Key Laboratory of Mechanics and Control of Mechanical Structures, Key Laboratory for Intelligent Nano Materials and Devices of Ministry of Education, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

c Diamond Light Source, Chilton, Oxfordshire OX11 0DE, U.K.

d Physics and Materials Science Research Unit, University of Luxembourg, L-1511 Luxembourg*††footnotetext: † Electronic Supplementary Information (ESI) available: X-ray powder diffraction measurements and Rietveld fits, lattice parameter data and structural parameter data, ab initio van der Waals parameters and energies, GULP input files. See DOI: 10.1039/b000000x/

1 Introduction

The development of responsive materials often exploits weak interactions as key design elements because lower interaction energies heighten the sensitivity of a material to external perturbations.1, 2, 3, 4 It is no accident, for example, that the weak inter-molecular forces in molecular crystals generally allow more extreme responses to changes in temperature5, 6 and pressure7, 8 than is possible in conventional inorganic ceramics, the structures of which are held together by strong ionic and covalent bonding networks. In this context, supramolecular interactions assume a particular importance, given that their energy scales are so much lower than those of electrostatic or covalent interactions. Hence the prevalence of hydrogen-bonding,9 halogen-bonding,10 $\pi$–$\pi$,11 van der Waals (vdW),5 host–guest,12, 13 and metallophilic14 interactions amongst many of the important materials in the field.

Thermal expansion behaviour is a straightforward measure of responsiveness: it quantifies the effect of temperature on the linear dimensions of a material.15 Compounds with large thermal expansion coefficients often show extreme and counterintuitive responses to pressure,16, 17 and may harbour various other anomalous elastic properties, such as negative Poisson’s ratios18 or thermosalient19, 20 effects. So it is perhaps unsurprising that some of the most extreme (‘colossal’) thermal expansion known has been observed in framework materials whose lattice dimensions are a function of weak metallophilic interactions.14, 17, 21 The canonical system of this type is Ag3[Co(CN)6], which adopts a lattice structure22 that can flex in such a way as to vary argentophilic Ag$\ldots$Ag separations without affecting covalent interactions within the lattice itself.23, 24 A geometric consequence of this flexing behaviour is that the positive thermal expansion (PTE) of argentophilic interactions (i.e. increase in separation with increasing temperature) is translated into a negative thermal expansion (NTE) effect in a perpendicular direction [Fig. 1]. The same mechanism operates under application of hydrostatic pressure, such that volume compression actually results in linear expansion for a particular set of directions17—so-called negative linear compressibility (NLC).25, 26, 27 NTE and NLC are valuable material properties, exploitable in the design of athermal composites used in optical devices and next-generation pressure sensors.

In seeking to design even more responsive analogues of Ag3[Co(CN)6], we considered the possibility of replacing Ag by Cu. Metallophilic interactions involving Cu+ ions are perhaps less well studied than argentophilic and aurophilic interactions, but are expected to be weaker given the reduced polarisability of the $3d$ shell.28, 29 Hence, by the arguments discussed above, Cu3[Co(CN)6] has always been an obvious candidate for extreme thermomechanical response. To the best of our knowledge, this system has never previously been reported: the difficulty of preparing the phase is likely a consequence of the propensity for Cu+ to disproportionate under the aqueous reaction conditions used to prepare the family of materials related to Ag3[Co(CN)6].23 We have recently exploited the Cu2+ reduction protocol developed in Ref. 30 to allow access to otherwise unrealisable Cu(I)-containing frameworks,31 suggesting that a similar synthetic approach may provide an alternative synthetic entry point to Cu3[Co(CN)6].

Here we validate such an approach, reporting the synthesis, crystal structure, and thermal expansion behaviour of Cu3[Co(CN)6]. Using a combination of high-resolution synchrotron X-ray diffraction measurements and Rietveld refinement, we show the system to be isostructural with Ag3[Co(CN)6] and H3[Co(CN)6].22, 32, 33, 34 Variable-temperature (100–598 K) diffraction measurements allow determination of the corresponding coefficients of thermal expansion $\alpha_{\ell}=(\partial\ln\ell/\partial T)_{p}$, which we find to be substantially less extreme than those of Ag3[Co(CN)6] (even if they remain large in the context of the behaviour conventional inorganic solids35). In particular, our data give $\alpha_{a}=25.4(5)$ MK*-1* and $\alpha_{c}=-43.5(8)$ MK*-1*; cf $\alpha_{a}=144(9)$ MK*-1* and $\alpha_{c}=-126(4)$ MK*-1* for Ag3[Co(CN)6].23 In order to rationalise this more moderate thermomechanical response in terms of the relative strengths of Cu${}^{+}\ldots$Cu+ and Ag${}^{+}\ldots$Ag+ metallophilic interactions, we carry out a series of ab initio calculations. Our analysis suggests (i) that cuprophilic interactions are indeed weaker than argentophilic interactions in this family, and (ii) the more extreme thermomechanical response of the Ag-containing compound is a result of the balance of metallophilic and electrostatic interaction energies rather than a signature of particularly weak argentophilicity. Lattice dynamical calculations using a highly simplified interaction model relevant to the entire A3[Co(CN)6] structural family lead to the same conclusions. Our results suggest that competing interactions—rather than low-energy interactions per se—might be key in the design of ultra-responsive materials.

2 Methods

All reagents were obtained from commercial suppliers and used as received.

2.1 Copper(I) hexacyanocobaltate(III)

We prepared polycrystalline samples of copper(I) hexacyanocobaltate(III) following a modification of the reduction protocol reported in Refs. 30, 31. A saturated aqueous solution of copper(II) sulfate (Sigma Aldrich, 99%; 0.17705 g) was added dropwise to a concentrated aqueous solution of sodium bisulfite (Sigma Aldrich, 0.05771 g), present in stoichiometric excess, to afford a mint-green solution. The solution was stirred for 30 min, after which time an aqueous solution of potassium hexacyanocobaltate(III) (Sigma Aldrich, 97%, 0.12288 g; stoichiometric with respect to copper) was added dropwise to afford a pale blue precipitate. The solution was stirred for a further 2 h, and the pale-blue solid product formed was isolated by filtration, washing (H2O) and drying under vacuum. The solid contained a mixture of copper(I) hexacyanocobaltate(III) and Prussian-blue-structured potassium copper(II) hexacyanocobaltate(III), a seemingly inescapable by-product of this synthetic strategy.

Copper(I) hexacyanocobaltate(III) could also be obtained in impure form using mechanochemical synthesis. Stoichiometric quantities of solid tetrakis(acetonitrilo)copper(I) hexafluorophosphate (Chem Cruz, 98%, 0.41346) and potassium hexacyanocobaltate (Sigma Aldrich, 97%, 0.12288g) were combined in an agate mortar, and intimately mixed via solid-state grinding for 30 min. An obvious colour change from white to pale blue occurred during this process. The resulting solid was washed (H2O) and dried to afford a mixture of copper(I) hexacyanocobaltate(III), potassium copper(II) hexacyanocobaltate(III) and at least one further unidentified product. Given the reduced purity of this product, the solution-phase product described above was used for all diffraction measurements carried out in this study.

2.2 Powder X-ray diffraction

High-resolution synchrotron powder diffraction measurements were carried out using the I11 beamline at the Diamond Light Source. A finely-ground sample of copper(I) hexacyanocobaltate(III), prepared as above, was loaded into a borosilicate capillary (0.5 mm diameter) and mounted on the diffractometer. Diffraction patterns were recorded using the Mythen2 point sensitive detector over the angular range $2\theta=2$–$92^{\circ}$, using an X-ray wavelength $\lambda=0.826210$ Å calibrated by refinement of a silicon NIST 640c standard. Each measurement consisted of two scans of 5 s exposure, offset relative to one another by $\Delta 2\theta=0.25^{\circ}$. The sample temperature was controlled using an Oxford Cryostream (100–500 K) and a Cyberstar hot air blower (523–598 K). Diffraction patterns were measured at intervals of 25 K between 100 and 500 K and again between 523 and 598 K.

Both Pawley and Rietveld refinements were carried out using TOPAS Academic (version 4.1).36 We employed a modified Thompson–Cox–Hasting pseudo-Voigt (TCHZ) peak shape, combined with a simple axial divergence correction and a Stephens anisotropic peak broadening term.37 The potassium copper(II) hexacyancobaltate(III) impurity phase was modelled using Pawley refinement of the $Fm\bar{3}m$ double-metal cyanide cell ($a\sim$10 Å).38 Rietveld refinement of the Cu3[Co(CN)6] phase made use of a starting model based on the known structure of Ag3[Co(CN)6].22 Refinement was stable for all temperature points, provided that Co–C/C–N bond distance restraints and a single isotropic displacement parameter for all atom types were used in the Rietveld model. Sequential (seed-batch) Rietveld refinements, where the starting structural parameters for each temperature point were those used at the preceding temperature, provided structural models with physically-sensible temperature dependencies for $T\leq 450$ K. For the temperature regime $450\leq T\leq 598$ K, we found that the positional coordinates of the C and N atoms and the value of $B_{\textrm{iso}}$ showed strong covariance, and hence we have reduced confidence in the absolute values of these parameters. This regime corresponds to the temperature range over which decomposition of the KCu[Co(CN)6] phase appears to set in.

2.3 Thermal expansivity determination

Thermal expansivities were calculated using the PASCal software.39 We employed estimated temperature uncertainties of 5 K and fitted the principal axis expansivities using linear functions. For internal consistency with the uniaxial expansivities, the volume expansivity was determined using the trace of the expansivity tensor40 rather than via the direct $V$–$T$ fit given by PASCal.39

2.4 Ab initio calculations

Ab initio calculations were performed within the FHI-aims code,41 using the numeric atom-centred orbital tier 1 basis set for the wavefunction and a $5\times 5\times 5$ $k$-point mesh for the Brillouin zone sampling. The Perdew-Burke-Ernzernhof (PBE) functional42 was used to model the semilocal exchange-correlation energy. To describe the non-local dispersion energies, we used both the interatomic pairwise Tkatchenko-Scheffler (TS) method,43 as well as the many-body dispersion (MBD) method, which includes many-body dipolar interatomic interactions to all orders in perturbation theory.44, 45 The lattice constants were obtained from unit cell relaxations with cell angles fixed to experimental values. Full a posteriori relaxation of the unit cell proved the reliability of this scheme.

2.5 Lattice dynamical calculations

Lattice dynamical calculations made use of the GULP software (version 4.4).46 Cell optimisations were carried out under constant pressure conditions $p=0$ and at $T=0$, with strains constrained by symmetry. Dispersion interactions were modelled using a Buckingham potential with vanishingly small repulsive term, and the ‘c6’ flag was activated to employ Ewald-type summation. For all calculations, checks were carried out to ensure optimisation convergence and to verify the conservation of angle terms in the parameterisation.

3 Results and discussion

3.1 Crystal structure of Cu${}_{\textbf{3}}$[Co(CN)${}_{\textbf{6}}$]

The ambient-temperature X-ray powder diffraction pattern of our Cu3[Co(CN)6] sample is shown in Fig. 2, where it is compared to that of Ag3[Co(CN)6] as reported in Ref. 23. The structural similarity between the two phases is immediately evident, as is the presence of a substantial quantity of an impurity phase. We could account for the entire diffraction pattern using two components, one based on the Ag3[Co(CN)6] structure-type (space group symmetry $P\bar{3}1m$) and one with the cubic Prussian blue structure (space group symmetry $Fm\bar{3}m$). This second phase would be consistent with the formation of KCu[Co(CN)6] during synthesis, which is certainly feasible on chemical grounds.47, 48 A Pawley fit using this two-phase model confirms our assignment of space group symmetries and rules out the presence of any additional crystalline phases [Fig. 2(d)]. We note that there is good (if fortuitous) distinction between the diffraction profiles of the two phases present, which allows us to clearly distinguish the corresponding lattice parameters and their thermal expansion behaviour (see SI).

Having established the space group symmetry of Cu3[Co(CN)6] we proceeded to carry out a Rietveld refinement, employing a starting model based on the lattice parameters obtained during Pawley fitting and the published atom coordinates of Ag3[Co(CN)6].22 We continued to model the KCu[Co(CN)6] phase using a Pawley fit—indeed this is the case for all subsequent refinements and is not discussed further. We found good stability in the refinement of our structural model of Cu3[Co(CN)6], obtaining a $R$-value of $3.029\%$; the corresponding fit is shown in Fig. 2(e) and the relevant structural details are summarised in Table 1. A representation of the crystal structure itself is given in Fig. 3. All refined bond lengths are chemically sensible: we find a Co–C distance of 1.832(11) Å, which is similar to that in Ag3[Co(CN)6] ($d($Co–C$)=1.895$ Å);22 likewise the Cu–N separation of 1.887(10) Å is comparable to that found in CuCN ($d($Cu–C/N$)=1.839(9)$–$1.872(12)$ Å).49

A property of the particular space group symmetry of Cu3[Co(CN)6] is that the Cu$\ldots$Cu separation is directly related to the lattice parameters:

[TABLE]

Hence we find $r_{\textrm{Cu\ldots Cu}}=3.4543(5)$ Å, which lies at the very upper bound of Cu$\ldots$Cu separations for which cuprophilic interactions are considered relevant.50 One crude measure of the strength of metallophilic interactions is to consider the ratio of the observed interatomic distance to the sum of the corresponding vdW radii.28 Using our room-temperature lattice parameters and the vdW radii given in Ref. 51 we obtain a ratio of 1.00 for Cu3[Co(CN)6], which is remarkably similar to the corresponding value for Ag3[Co(CN)6] (0.99).23 So, at face value, one might expect comparable metallophilic interaction strengths for the two systems.

3.2 Thermal expansion behaviour

Having collected a series of X-ray diffraction patterns over the temperature range 100–598 K, we carried out Rietveld refinements for each data set using the same approach described above. We obtained satisfactory refinements for all temperatures, albeit with some signs of increased uncertainties at the very highest temperatures—i.e. close to the onset of decomposition of the Prussian blue phase. The temperature dependence of the lattice parameters, illustrated in Fig. 4(a), was observed to be approximately linear over this entire temperature range. As in nearly all members of this structural family, Cu3[Co(CN)6] exhibits an NTE effect parallel to the $c$ crystal axis, and PTE effects in perpendicular directions (i.e., including the $a$ and $b$ crystal axes). Hence the basic thermomechanical response of this system can be understood in terms of the same ‘wine-rack’ mechanism illustrated in Fig. 1. The remaining structural parameters $x_{\textrm{C}},z_{\textrm{C}},x_{\textrm{N}},z_{\textrm{N}},B_{\textrm{iso}}$ also show linear temperature dependencies [Fig. 4(b–d)]; taken together these values allow us to estimate a set of 0 K structural parameters that may prove useful for comparison against e.g. ab initio studies [Table 1].

Coefficients of thermal expansion were determined using linear fits to the lattice parameter data,39 and are given in Table 2. What is immediately apparent is that the magnitudes of both PTE and NTE effects in Cu3[Co(CN)6] are substantially smaller than those in the Ag-containing system. Consequently, Cu3[Co(CN)6] is not a colossal thermal expansion material, and its thermomechanical response shares more in common with other Cu-containing networks such as $\alpha$-Cu[C(CN)3] (Ref. 31) and CuCN (Ref. 52) than with Ag3[Co(CN)6] and Ag3[Fe(CN)6].14 We will come to rationalise the differences in behaviour of the copper(I) and silver(I) hexacyanocobaltates below, but include first some additional analysis of the trends in lattice parameters we observe using our newly-measured data.

The ‘wine-rack’ mechanism that is thermally activated in this system can be interrogated directly using the so-called mechanical building unit (XBU) approach.2 We make use of the pair of transformations

[TABLE]

which relate the unit cell dimensions to the framework strut length $r$ and framework angle $\theta$ [Fig. 3]. Using these same relationships, we can recast the lattice expansivities in terms of XBU expansivities, obtaining $\alpha_{r}=-8.2$ MK*-1* and $\alpha_{\theta}=43.1$ MK*-1*. Hence the bulk of the thermal expansion response can be accounted for by changes in the framework geometry ($|\alpha_{\theta}|\gg|\alpha_{r}|$); the lattice expansivities attributable to this flexing mechanism alone are $\alpha_{a}^{\prime}=33.5$ MK*-1* and $\alpha_{c}^{\prime}=-35.7$ MK*-1*, where we use the prime notation to indicate calculation from $\alpha_{\theta}$. The observation $\alpha_{r}<0$ indicates that the Co–CN–Cu–NC–Co ‘struts’ from which the framework structure of Cu3[Co(CN)6] is assembled actually contract with increasing temperature. This behaviour is likely due to thermal activation of transverse vibrational modes where lateral displacements of the chain (maximal at the Cu site) require shortening of the Co$\ldots$Co vector.53, 54 Such a mechanism is implicated in the uniaxial NTE of CuCN itself ($\alpha_{\textrm{chain}}=-32.1$ MK*-1*, Ref. 52, 49), and is presumably tempered here somewhat relative to that system by the increased strength of Co${}^{\textrm{III}}$–C vs Cu${}^{\textrm{I}}$–C bonds.47

One consequence of the negative value of $\alpha_{r}$ is that the volume coefficient of thermal expansion of Cu3[Co(CN)6] is unusually small for systems in this particular family. Formally, this situation arises because of the fortuitous equivalence $\alpha_{r}\simeq-\frac{1}{3}|\alpha^{\prime}|$, which is the geometric requirement for $\alpha_{V}\rightarrow 0$.***Note that $\alpha_{i}=\alpha_{i}^{\prime}+\alpha_{r}$, and hence $\alpha_{V}\sim\alpha_{a}^{\prime}+3\alpha_{r}$. Hence this material has the unusual property of (approximately) temperature-independent density despite its relatively large linear thermal expansivities. At face value, this property may be expected to result in unusually extreme uniaxial compressibilities under application of hydrostatic pressure, since small changes in volume would appear to be linked to large changes in lattice dimensions. However, we anticipate by analogy to related systems that the XBU compressibility $K_{r}$ is actually positive rather than negative, and so a small $\alpha_{V}$ need not require a large bulk modulus.55, 2 Nevertheless we expect the particular uniaxial compressibility corresponding to the $c$ crystal axis to be negative, and so investigation of the NLC behaviour of Cu3[Co(CN)6] could prove a fruitful avenue of future research.

3.3 Ab initio calculations

The observation of more moderate thermal expansion behaviour in Cu3[Co(CN)6] relative to that in Ag3[Co(CN)6] poses a simple question: does this situation arise because cuprophilic interactions are actually stronger than argentophilic interactions, and hence less susceptible to changes in temperature?

In order to answer this question, we turn to ab initio calculations, which if carried out so as to include consideration of vdW interactions allow direct quantification of the metallophilic interactions in both compounds. We begin by reporting the 0 K structure for Cu3[Co(CN)6] obtained computationally and demonstrate that the inclusion of dispersive interactions is necessary to obtain good consistency with our experimental results. By mapping out the potential energy surface (PES) for all three A3[Co(CN)6] systems (A = H, Cu, Ag) across a variety of lattice strains and then taking into account the variation in vdW energies at each point, we extract the free-atom and in-solid (effective) $C_{6}$ coefficients. The value of these coefficients for each atom type A acts as a measure of the strength of metallophilic interactions in the corresponding A3[Co(CN)6] system.

The unit cell dimensions obtained in our DFT + vdW calculations are given in Table 3. The influence of dispersion energy on the lattice constants is large, just as is now known to be the case for Ag3[Co(CN)6].56 Our PBE calculation without vdW interactions overestimates $a$ and underestimates $c$. Upon including dispersion interactions the lattice constants move closer to the experimental values. We note that the enhanced cohesive MBD energy for Cu3[Co(CN)6] arises from the collective effect of vdW interactions and the self-consistent polarisation in the unit cell.56 In Figure 5 we show a representative section of the PES for the three calculation regimes, and Figure 6 shows the TS and MBD vdW energies as a function of the individual $a$ and $c$ lattice constants. Our results make clear that the vdW energy depends more strongly on $a$ than it does on $c$. Since the framework strut length $r$ is more rigid than the framework angle $\theta$, then to lower the total energy the lattice simply contracts along $a$ (and $b$) while expanding along $c$. Hence the same mechanism explains the qualitative change in lattice constants observed both as a result of using different vdW calculation methods and as a result of an increase in the polarisability of atom A. Indeed because the MBD energy depends almost linearly on the lattice constants it behaves as an effective pressure on the lattice, equivalent to 1.22 GPa along $a$ and 1.76 GPa along $c$.

To compare the strength of cuprophilic interactions in Cu3[Co(CN)6] with that of argentophilic interactions in Ag3[Co(CN)6] we further analysed our DFT+vdW results. Our basic approach was to parameterise the vdW contribution to the TS-vdW energy in terms of dispersion coefficients $C_{6}$ and vdW radii $R_{0}$ for each atom type. In the PBE+TS calculations, the free-atom $C_{6}$ coefficient and vdW radii $R_{0}$ are used as the initial input parameters. The effect of the local chemical environment is taken into account by calculating the effective in-solid $C_{6}$ and $R_{0}$ as described in Ref. 43. Table 4 lists our results for the free-atom vdW parameters and the effective parameters for A3[Co(CN)6] (A = Ag, Cu, H) at the experimental lattice constants. We find that the argentophilic interactions are indeed stronger than cuprophilic interactions in these systems, as both the free-atom and effective $C_{6}$ values are larger by $\sim 40$% for Ag relative to Cu. For completeness we note that the effect of the local chemical environment on the $C_{6}$ coefficients is to reduce the dispersion coefficients.

3.4 Lattice dynamical calculations

We supplement these high-level ab initio results with a series of extremely simple lattice-dynamical calculations that also allow us to estimate the relative strengths of metallophilic interactions in Cu3[Co(CN)6] and Ag3[Co(CN)6]. The approach we use is to develop the very simplest abstraction of all three A3[Co(CN)6] systems (A = H, Cu, Ag) that captures the key interactions responsible for their thermomechanical response. We parameterise this model with sufficiently few variables that six experimental observables (the two independent lattice parameters for each of the three systems) can be used to estimate the relative magnitudes of the metallophilic interactions in the A = Cu, Ag compounds.

The same structural model is used for all three systems: $P\bar{3}1m$ crystal symmetry, with a single anion (mass $m=m({\textrm{Co}})+6m({\textrm{C}})+6m({\textrm{N}})$) of charge $-1.5e$ at position $(0,0,0)$ and a cation ($m=m(\textrm{A})$) with charge $+0.5e$ at position $(\frac{1}{2},0,\frac{1}{2})$ [Fig. 7(a)]. These charges reflect the approximate Mulliken charges determined for H3[Co(CN)6] and Ag3[Co(CN)6] in Ref. 57 and are consistent with the Hirshfeld and Bader charges obtained in our own ab initio calculations (see SI). We refer to the anion using the symbol X (formally this corresponds to the [Co(CN)6]3- ion), giving the unit cell composition A3X. This structural model is then decorated with three interaction potentials: first, a harmonic bond potential between neighbouring A and X sites

[TABLE]

second, a harmonic bond angle potential governing A–X–A triplets

[TABLE]

and, third, (in the case of Cu and Ag systems) dispersive interactions between neighbouring A sites intended to reflect the empirical $\frac{1}{r^{6}}$-dependence of metallophilic interactions58

[TABLE]

In this way the lattice energy is given by the sum

[TABLE]

In order to reduce the number of parameters involved in this model, we make the following assumptions. First, we take the effective charges at X and A sites to be system-independent. We justify this assumption by noting that the Mulliken charges reported for H3[Co(CN)6] and Ag3[Co(CN)6] vary more greatly by calculation method than they do between systems;57 the A = Cu case is intermediate to the A = H and A = Ag cases (see SI). Second, we take the flexing stiffness $k_{\textrm{angle}}$ and equilbrium angle $\theta_{0}$ also to be system-independent. This is probably reasonable given that both terms will be governed by the chemistry of the [Co(CN)6]3- anion, which is common to all three systems. Third, we take the (system-dependent) values of $r_{0}$ as the sum of bond lengths $d($Co–C$)+d($C–N$)+d($N–A$)$ determined crystallographically: we use the values from Ref. 34 for A = H, from Ref. 17 for A = Ag, and from our present study for A = Cu. Fourth, we assume that the $E_{\textrm{disp}}$ term is negligible for the A = H case, which is likely given the small electron density expected at the A site for this system (and is certainly supported by our own ab initio results as given in Table 4).

Our starting point is to determine a set of parameters $k_{\textrm{harm}},k_{\textrm{angle}},\theta_{0}$ that, when used to drive a lattice-dynamical geometry optimisation, result in the closest possible agreement between 0 K (derived from experiment) and relaxed cell parameters for A = H. We make the additional requirement that $\theta_{0}$ should be as close to $90^{\circ}$ as possible. The parameter set we obtain is listed in Table 5, together with a comparison of the experimental and simulated lattice parameters; the corresponding match in framework geometry is illustrated in Fig. 7(b).†††We found the quality of fit was relatively insensitive to changes in $k_{\textrm{harm}}$ of up to ca 25% of its value. Variations in this parameter did affect the absolute values of the compressibilities determined subsequently; however the same trend in magnitudes of compressibilities shown in Fig. 8 was found in all cases. We note that we do not attach any particular physical meaning to the parameter values in our model, since (in particular) the charge distribution we use is a heavily simplified representation of reality. Nevertheless it is reassuring that even this simple model allows robust geometry optimisation to a physically-sensible state.

Having used the geometry of the A = H system to determine all of the system-independent parameter values, we proceeded to optimise the geometry of analogous models for A = Cu and Ag. In each case the value of $r_{0}$ was updated according to the experimental bond lengths, and only the value of $C_{6}$ was varied in order to obtain the closest match between calculated and experimental (0 K extrapolated) lattice parameters. The corresponding parameter values and optimised cell dimensions are again summarised in Table 5; we note that the level of agreement ($<2\%$) is encouraging given the simplicity of the lattice dynamical model we have used. Also encouraging is that, for both compounds, the $a$ lattice parameters are overestimated in the absence of a metallophilic contribution to the lattice enthalpy. This mirrors the results of vastly higher-level ab initio geometry optimisations,57, 59 and indicates that the electrostatic contribution to the free energy (the single component of our model acting to increase $a$) operates in tension with the metallophilic interactions. While we do not attach any importance to the absolute values of the $C_{6}$ parameters that emerge from our calculations, what we do think is meaningful is the observation that $C_{6}$ is larger for A = Ag than for A = Cu. In other words, the experimental unit cell dimensions for Cu3[Co(CN)6] and Ag3[Co(CN)6] are consistent with stronger argentophilic interactions in the latter than cuprophilic interactions in the former. Moreover, the ratio of cuprophilic:argentophilic interaction strengths we deduce from our simple lattice-dynamical model is essentially the same as that obtained in our ab initio calculations: $C_{6}$(Cu)/$C_{6}$(Ag) = 61% vs 70%, respectively.

3.5 Flexibility from competing interactions

So our various calculations converge on the same scenario whereby cuprophilic interactions in Cu3[Co(CN)6] are weaker than argentophilic interactions in Ag3[Co(CN)6] by 30–40%. One obvious question remains: how is this observation consistent with the more moderate thermal expansion behaviour of the Cu-containing compound?

To address this question we exploit the approximate proportionality between thermal expansivities and isothermal compressibilities noted in Refs. 60, 17, 27:

[TABLE]

Here $C_{T}$ is the isothermal specific heat, $V$ the molar volume, $\hat{\gamma}$ the mean effective Grüneisen parameter and $K_{i}$ the uniaxial compressibilities. We estimate that the pre-factor $C_{T}\hat{\gamma}/V$ varies by not more than $\sim 25\%$ between the A = Cu and A = Ag systems,‡‡‡Here we have made use of three relationships: first, that $\hat{\gamma}$ appears to be relatively system-independent;57 second, that the ratio of the $C_{T}$ values for A = Cu and Ag will be approximately equal to the ratio of the $\sqrt{m}$ terms, since the low-energy phonon dispersion will be dominated by heavy-atom displacements; and third, we use the experimental molar volumes. such that a comparison of compressibilities for the two compounds provides a reasonable first-order approximation to the relative thermal expansivities. We concern ourselves with compressibilities rather than expansivities since the former are obtainable directly from the calculations (both ab initio and lattice-dynamical) described above. The relative compressibilities for all three compounds are illustrated graphically in Fig. 8. What is evident is that the Cu-containing compound exhibits intermediate behaviour to the H- and Ag-containing systems, despite its relatively weaker metallophilic interactions. The qualitative similarity to the relative thermal expansivities is striking, particularly given the (necessary) omission of anharmonic contributions from our calculations which likely contribute substantially to the experimental behaviour.61

4 Concluding remarks

We are led to the counterintuitive conclusion that stronger interactions can actually make a material more compliant: Ag3[Co(CN)6] exhibits colossal thermomechanical responses but Cu3[Co(CN)6] does not, despite the energy scale associated with metallophilic interactions being larger in the former than in the latter. Of course the key here is that metallophilic interactions are net attractive, and act in tension with the (repulsive) electrostatic component.59, 62 Any effective harmonic potential can be made increasingly shallow by the addition of attractive $r^{-6}$ terms, as illustrated in Fig. 9. This is the nub of the physics at play in this family: in the absence of metallophilic interactions, the frameworks are not especially mechanically responsive but they do become so as metallophilicity is introduced.

Hence the conventional materials design rules are reversed, and we anticipate that the member of the A3[Co(CN)6] family likely to show the most extreme thermomechanical response is actually the as-yet-unrealised compound Au3[Co(CN)6]. It was shown in Ref. 57 that this system is likely to have a particularly compliant structure, although the degree of compliance will depend heavily on the strength of the aurophilic interaction contribution to the lattice enthalpy. Given the notorious difficulty of accessing aqueous Au(I) chemistry, it is not yet clear how Au3[Co(CN)6] might be accessed synthetically. A viable alternative is the (also unrealised) compound Fe[Au(CN)2]3—i.e., with Co(III) replaced by Fe(III) and the CN ion orientations reversed—which by analogy to Fe[Ag(CN)2]3 should in principle be accessible via reaction of aqueous Fe3+-containing solutions with KAu(CN)2.63 The observation14 of qualitatively similar ‘colossal’ thermal expansion in Ag3[Co(CN)6] and Ag3[Fe(CN)6] suggests that chemical substitution at the trivalent metal site is unlikely to influence the degree of thermomechanical response observed.

From a computational perspective, one key implication of our study is the importance of obtaining accurate descriptions of vdW interactions in compliant framework materials. This importance is particularly acute for systems such as Cu3[Co(CN)6] and Ag3[Co(CN)6] where the PES is anomalously shallow as a result of competition between vdW and electrostatic contributions. A key challenge in this regard is the treatment of finite-temperature effects; i.e. anharmonicity. We anticipate that the discovery of anomalous mechanics in increasingly many systems based on vdW-type interactions64, 65 will motivate further research effort along precisely these lines.

Acknowledgements

The synchrotron diffraction measurements were carried out at the Diamond Light Source (I11 Beamline). We are extremely grateful for the award of a Block Allocation Grant, which made this work possible, and for the assistance in data collection provided by S. J. Cassidy (Oxford) and the I11 beamline staff. A.F.S., A.R.O. and A.L.G. gratefully acknowledge funding through the European Research Council (Grant 279705). C.S.C. and A.L.G. acknowledge funding through the Leverhulme Trust (Grant RPG-2015-292). A.R.O. thanks the Diamond Light Source for a Studentship. This project received funding from the European Union (EU) Horizon 2020 Research and Innovation Programme under Marie Sklodowska-Curie Grant Agreement 641887 (project acronym DEFNET).