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Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

Density Functional Studies Revealing Anomalous Lattice Behavior in Metal Cyanide, AgC8N5 Baltej Singh,†,‡ Mayanak K. Gupta,† Ranjan Mittal,*,†,‡ and Samrath L. Chaplot†,‡ †

Solid State Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India Homi Bhabha National Institute, Anushakti Nagar, Mumbai 400094, India

‡

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S Supporting Information *

ABSTRACT: We have investigated anomalous lattice behavior of metal−organic framework compound AgC8N5 on application of pressure and temperature using ab initio density functional theory calculations. The van der Waals dispersion interactions are found to play an important role in structural optimization and stabilization of this compound. Our ab initio calculations show negative linear compressibility (NLC) along the c-axis of the unit cell. The ab initio lattice dynamics and molecular dynamics simulations show large negative thermal expansion (NTE) along the c-axis. The mechanism of NLC and NTE along the c-axis of the structure is governed by the dynamics of Ag atoms in the a−b plane. The NLC along the c-axis drives the NTE along that direction.

1. INTRODUCTION The understanding of atomic-level mechanisms responsible for various functional properties1−8 of materials is very important to improve their performance. By intuition, a material should contract (or expand) on application of hydrostatic pressure (or temperature), yet a small number of crystals show opposite behavior in a few directions. This type of anomalous behavior with pressure (or temperature) is called negative linear compressibility (or negative thermal expansion) behavior.9 Materials with negative linear compressibility (NLC) and negative thermal expansion (NTE) could have interesting technological applications9−11 in body muscle systems (as actuators), marine optical telecommunication, components of aviation, sensors, and devices working in high pressure− temperature surroundings. So far, a few studies have been performed to realize the origin of NLC and NTE in crystalline solids.9,12 This type of behavior is predicted in an open framework structure with high unit cell volume per atom and low density (Table 1). Many cyanide-based metal−organic flexible framework structures,4,5,13−15 such as ZnAu 2(CN) 4, M 3Co(CN)6, and MAuX2(CN)2, where M = H, Au, Ag, Cu, etc. and X = CN, Cl, Br, etc., show anomalous NLC and NTE behavior. It seems that the occurrence of NLC is closely related to the NTE behavior in these compounds.2 Moreover, there is a very strong link among NLC, NTE, and geometry of the material.3,4,8,16 The NLC and NTE in ZnAu2(CN)4 arise from anharmonic nature of low-energy optic phonon modes involving bending of the −Zn−NC−Au−CN−Zn− linkage.2 This bending produces the effects of a compressed spring upon heating and an © XXXX American Chemical Society

Table 1. Experimental Unit Cell Volume per Atom, Density, and Linear Thermal Expansion Coefficients (αl), at 300 K, of Metal Cyanides from the Literaturea αl × 10−6 K−1 compound

V/atom (Å3)

density (g/cm3)

αa

αb

αc

AgCN40 AgC4N316 AgC8N5 Ag3Co(CN)63 ZnAu2(CN)44 KMnAg3(CN)623

13.6 15.5 16.3 19.1 19.3 19.8

4.07 2.63 2.00 2.93 4.37 2.83

66 −48 105 132 37 61

66 200 32 132 37 61

−24 −54 −98 −130 −58 −60

The values of αl(l = a, b, c) for AgC8N5 are obtained from our ab initio calculations.

a

extended spring17 under hydrostatic pressure in specific springlike topology of ZnAu2(CN)4. In many cases, such as MFM133(M) (M = Zr, Hf), L-tartrate, and M3Co(CN)6 (M = H, Ag, Cu), the anomalous lattice behavior has been observed to arise from deformation mode of wine-rack-like geometries, which is contributed from molecular strut compression and the angle opening mechanism3,5,6,9,13,14,18,19 in an anisotropic framework. In M3Co(CN)6, the intense and quicker response of phonon group velocity along the c-axis than in the a−b plane upon heating and compression facilitates the c-axis to be a carrier of anomalous lattice behavior.20 In a framework Received: May 9, 2018 Revised: June 18, 2018 Published: June 21, 2018 A

DOI: 10.1021/acs.jpcc.8b04398 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C material, NLC can be effectively tuned by varying the inorganic component of the framework without changing the network topology and structure.18 For two-dimensional layered framework compounds like Co(SCN)2(pyrazine)2, the layer sliding mechanism is found to be responsible for the observed NLC behavior.21 The framework hinging mechanism leads to an extreme NLC in InH(BDC)2, which is much higher than in CN-based metal−organic framework compounds.21,22 Large NLC is observed in frameworks composed of rigid linear ligands and flexible framework angles.2,4,23 The NTE behavior along the c-axis in a linear chain structure like MCN (M = Au, Ag, Cu) is caused by the chain sliding phonon modes and −CN− bond flipping in the chain.24 On the basis of current understanding of the mechanisms responsible for NLC, different approaches to designing and fabricating new structures with NLC and NTE behaviors are being studied.25,26 High-pressure and high-temperature X-ray and neutron diffraction techniques are used to experimentally determine the anisotropic linear compressibility and linear thermal expansion coefficients of crystalline materials.3−5,7−9,16,23 However, ab initio quantum mechanical calculations are well established to understand the microscopic mechanism governing these phenomena.2,12,24,27−29 The compressibilities and expansion coefficients as calculated using ab initio density functional theory (DFT) and phonon calculations are found to reproduce the experimental values fairly well. These calculations, from the analysis of eigen vectors, provide insight into the anharmonic phonon modes responsible for anomalous lattice behavior in the material.2,24,27,29 The metal−organic framework compound AgC8N5 has a comparatively lower density30 (Table 1) than that of the compounds of its family16,24 like AgCN and AgC4N3. As per our knowledge, there are no temperature-/pressure-dependent experimental or theoretical studies reported on this compound regarding its anomalous lattice behavior. This motivated us to study the anomalous lattice behavior of AgC8N5 using ab initio density functional theory. The details of the calculations are given in the Supporting Information.31

Figure 1. (Top) Crystal structure of AgC8N5 containing two types of AgN4 (poly-1: black and poly-2: blue) tetrahedra connected through C (red) and N (green) atoms. (Bottom) The calculated displacement pattern giving rise to negative linear compressibility along the c-axis of AgC8N5. The arrow represents the displacement vector for Ag as obtained from the difference in atomic coordinates of all of the atoms corresponding to ambient pressure and 4 GPa structure.

Table 2. Comparison of Calculated and Experimental Lattice Parameters of AgC8N5 optimization scheme

a (Å)

b (Å)

c (Å)

V (Å3)

DFT-GGA (0 K) DFT-GGA + vdW (0 K) experimental (300 K)

12.46 12.26 12.43

15.91 13.25 13.62

32.27 32.61 32.30

6392 5297 5466

the equilibrium lattice. The elastic constants are used to get the bulk modulus and elastic compliance matrix SC−1 (in 10−3 GPa−1 units) as

2. RESULTS AND INTERPRETATIONS The crystal structure (Figure 1) of AgC8N5 contains the C8N5 planar ligand and AgN4 tetrahedral units. Both the planar and polyhedral units are distorted. The planar sheets of C8N5 are vertically placed along the c-axis with small tilting in the a−b plane. These sheets are well separated along the b-axis (distance > ≈3 Å) in the a−b plane. Therefore, these sheets must be weakly interacting through weak dispersion interactions to make a stable structure. The structure optimizations done without considering these weak interactions are found to highly overestimate (Table 2) lattice parameter b. However, when van der Waals interactions are considered between these planar sheets, the calculated structure is found to match with the experimental structure,30 within the limitations of GGA (Table 2). It seems that van der Waals interactions play a very important role in governing the structure stability of AgC8N5. The weak dispersion interactions acting in the a−b plane (especially along the b-axis) make the structure flexible in the a−b plane as compared to that along the c-axis. The structure with the presence of van der Waals dispersion interactions is considered for all further calculations. The elastic constants of AgC8N5 are derived from the strain−stress relationships obtained from finite distortions of

ij162.95 − 99.12 − 42.81 0 yz 0 0 jj zz jj−99.12 143.22 7.97 zz 0 0 0 jj zz jj zz jj−42.81 7.97 zz 30.81 0 0 0 jj zz jj zz jj 0 zz 0 0 435.16 0 0 jj zz jj 0 zz 0 0 0 202.80 0 jj zz jj zz 0 0 0 0 0 203.79 k { For negative compressibility along the crystallographic axes,32 in an orthorhombic crystal, the following inequalities should hold Xa = S11 + S12 + S13 < 0, Xb = S12 + S22 + S23 < 0, and Xc = S13 + S23 + S33 < 0

where Xi (i = a, b, c) are the compressibilities of the crystal along various crystallographic axes. It is observed that only the last inequality holds, implying that the compound exhibits negative linear compressibility along the c-axis. To quantify this property, the crystal structure is relaxed under the application B


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The Journal of Physical Chemistry C

Figure 2. Calculated pressure dependence of lattice parameters (l), unit cell volumes, bond lengths, and total energy/atom for AgC8N5.

unit, attached to the corresponding Ag atoms. We found that CN units only connected to Ag atoms are displaced in the a−b plane. Therefore, a hinging movement of rigid CN units occurs about Ag atoms. On application of pressure, this gives rise to an expansion of the framework along the c-axis and contraction in the a−b plane. As pressure increases above 4 GPa, the structure undergoes an unusual change. Around this pressure, there are sudden jumps in lattice parameters, total energy, and bond lengths of the compound (Figure 2). A sudden decrease in volume (Figure 2e) gives a signature of some high-pressure phase transition, which could also be amorphization or decomposition. However, detailed high-pressure diffraction would be needed to identify the resultant structure of the new phase. We found an anomalous change in the Ag atomic coordinate at this pressure. There are two different types of AgN4 polyhedral units (poly-1 and poly-2) corresponding to the two different types of Wyckoff sites of Ag atoms present in this compound. Figure 2c,d shows the pressure dependence of Ag−N bond lengths in the two polyhedral units. Poly-1 has four different values of Ag−N bond lengths, whereas poly-2 has only two different bond lengths. The bonds that lie in the a−b plane of the crystal show very little pressure dependence. These bonds are strained very little with an increase in pressure because of the movement of Ag atoms in the a−b plane. However, large strain is developed in these bonds above 4.0 GPa, which results in abrupt changes in these bonds at the phase transition (or

of isotropic pressures and corresponding lattice parameters are calculated. The calculated lattice parameters as a function of pressure are shown in Figure 2. It is observed that lattice parameters “a” and “b” decrease with the increasing pressure and show normal behavior. Lattice parameter b shows a larger decrease as compared with lattice parameter a. This arises from the soft nature of van der Waals dispersion interaction acting among the planar sheets of C8N5 along the b-axis. However, the “c” lattice parameter shows an increase with an increasing pressure. This confirms the negative linear compressibility along the c-axis. The overall volume is found to decrease with the increase in pressure. Bulk modulus is calculated from the pressure dependence of unit cell volume. The PV equation of state is fitted with the well-known Birch−Murnaghan (second ordered) isothermal equation of state to get the value of bulk modulus. The calculated bulk modulus using this approach has a value of 14.5 GPa. This is consistent with that calculated from the elastic compliance (14.48 GPa) matrix. At ambient pressure, the calculated linear compressibilities are found to have the values of Xa = 21.0 × 10−3 GPa−1, Xb = 52.1 × 10−3 GPa−1, and Xc = −4.0 × 10−3 GPa−1. We observe that atomic coordinates of Ag atoms show a major change on application of pressure. The displacement corresponding to this change is indicated as vectors in Figure 1. The displacement vector shows that the Ag atoms displace along the a-axis of the crystal. Furthermore, we found significant displacement of the CN unit, as a single rigid C


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the vibrational modes associated with Ag atoms would be populated at low temperatures and would give rise to interesting structural and dynamical properties. This also confirms, on application of pressure, the dominant role of Ag in giving rise to NLC and phase transition in AgC8N5. The stress dependence of phonon energies is used for the calculation of anisotropic Grüneisen parameters of AgC8N5. An anisotropic stress of 5 kbar is implemented by changing only one of the lattice constants and keeping the others fixed. The calculated mode Grüneisen parameters as a function of phonon energy along different crystal directions are shown in Figure 4a.

dissociation/amorphization). Above 4.0 GPa, AgN4 tetrahedra (Figure 2c) are found to change the coordination and convert to AgN5. The calculated structure as a function of pressure shows that the CN bond remains unchanged with a value of about 1.16 Å. Moreover, various C−C bonds (Figure 4) in −C8N5− planar units do not show any significant changes (less than 1− 2%) with an increase in pressure. The Ag−N bonds of AgN4 tetrahedral units show (Figure 4) a variation of 5−6%. This change would be due to the flexible nature of Ag−N bonds. There are significant changes (up to 6% of original bond angle) in the C−CN bond angle (Figure 4) present on the periphery of C8N5 structural units. To understand these structural changes and the mechanism of negative and positive linear compressibility along the c-axis and in the a−b plane, respectively, we have calculated the difference in atomic coordinates of all of the atoms corresponding to the ambient pressure structure and 4 GPa structure. The primitive unit cell of AgC8N5 contains 168 atoms and has 504 phonon modes of vibrations. The calculation of complete phonon spectra in the entire Brillouin zone is computationally expensive for such a large system. We have calculated the zone-center phonon spectra in a conventional unit cell with 336 atoms using linear response density functional perturbation theory. These 1008 phonons at the zone center of the conventional unit cell correspond to 504 phonons each at the zone center and the (111) zone-boundary point of the Brillion zone of the body-centered orthorhombic structure. Leaving three acoustic branches, we have calculated the phonon spectrum for 1005 phonon modes. The calculated partial densities of states of C, N, and Ag atoms show that these atoms contribute in different energy regions (Figure 3)

Figure 4. (a) Calculated energy dependence of anisotropic Grüneisen parameters, Γ, for AgC8N5. (b) Contribution of phonon mode of energy E to the linear thermal expansion coefficients. (c, e) Calculated temperature dependence of linear and volume thermal expansion coefficients, respectively. (d, f) Calculated temperature dependence of lattice parameters and unit cell volume, respectively.

The Grüneisen parameters along the a and b axes show normal positive behavior. However, the Grüneisen parameters along the c-axis have large negative values. The calculated anisotropic Grüneisen parameters and elastic compliance matrix are used to calculate the anisotropic thermal expansion coefficient. The calculated temperature dependence of anisotropic thermal expansion coefficients and lattice parameters is shown in Figure 4c,d. Negative thermal expansion is found along the caxis, whereas positive along the a and b axes. The change with temperature along the a-axis is more pronounced than that along the b-axis. This is in contrast to the calculated behavior as a function of pressure. The volume shows normal positive thermal expansion behavior. The quantitative thermal expansion behavior is obtained from the calculated temperature dependence of linear thermal expansion coefficients (Figure 4). At 300 K, the values of linear and volume thermal expansion coefficients are found to be αa = 105.4 × 10−6 K−1, αb = 32.3 × 10−6 K−1, αc = −98.1 × 10−6 K−1, and αV = 39.6 ×

Figure 3. Calculated total and partial phonon densities of states for various atoms of AgC8N5.

of the spectra. The very high energy peaks around 270 meV in the spectra of C and N are related to the vibrational stretching modes associated with very strong CN bonds. The spectra in the 100−200 meV range are highly contributed by C atoms and are related to stretching vibrations of strong C−C bonds in the −C8N5− planar structural units. The lower-energy modes in spectra of C and N atoms are associated with bending vibrations of constructing −C8N5− units. It is interesting to note that the Ag atoms contribute only to the vibrational spectra at very low energy up to 40 meV. Hence, D


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Figure 5. Displacement pattern of optic phonon modes projected in the a−b plane of AgC8N5. The arrow represents the displacement vector for Ag atoms. The displacement vectors for C and N atoms are negligible and are not shown for clarity.

10−6 K−1, respectively. These values are comparable to those reported for the highly anomalous thermal expansion cyanides3,7,16 such as AgC4N3, Ag3Co(CN)6, ZnAu2(CN)4, etc. The quasiharmonic approximations have been proven very good for studying the NTE materials such as2,12,24,28,29 LiAlSiO4, V2O5, MCN (M = Ag, Au, Cu), ZrW2O8, Ag3Co(CN)6, and ZnAu2(CN)4. The available data in the literature show3−5,7,16 that the compounds exhibiting NLC along a certain crystallographic axis also show the NTE along the same axis. However, the reverse is not always true.24,29,33−36 Therefore, it seems that NLC drives NTE, although there is no specific thermodynamic relation connecting NLC and NTE. Thermal expansion behavior is a consequence of anharmonicity of crystal potential energy. The anharmonicity of phonons arises due to implicit as well as explicit contributions. The implicit anharmonicity arises from the volume dependence of bond lengths and is implemented via quasiharmonic lattice dynamics (LD) calculations. On the other hand, the explicit anharmonicity arises from the temperature-dependent thermal amplitudes of atoms. Both the explicit and the implicit parts are implemented in molecular dynamics simulations. The explicit contribution is found to give significant contribution in a few compounds.28,37−39 Therefore, we have calculated the temperature dependence of lattice parameters using ab initio molecular dynamical (MD) simulations (Figure 4c). These calculations reproduce the anomalous thermal expansion behavior as calculated from the quasiharmonic lattice dynamics (LD). The very good agreement between LD and MD results for thermal expansion behavior suggests that the contribution from the explicit part is insignificant in this compound. To understand the mechanism of anomalous thermal expansion behavior in AgC8N5, we have calculated the contribution of individual phonon modes to the linear thermal expansion coefficients (Figure 4b). It is observed that the lowenergy phonon modes around 5 meV are responsible for the observed negative thermal expansion along the c-axis and positive thermal expansion along the a and b axes. These modes show highly anisotropic Grüneisen parameters (Figure 4a). The calculated phonon spectra show (Figure 3) that the low-energy phonon modes have large contributions from Ag atoms in comparison to those from C and N atoms. Moreover,

the contribution of phonons of low energies to mean square displacements of Ag is highly anisotropic with large values in the a−b plane (Figure S5), whereas N/C has almost negligible contributions. This also confirms the criticality of Ag motion to negative thermal expansion along the c-axis of AgC8N5. The displacement patterns of two modes with energies 3.34 and 5.32 meV are shown in Figure 5. The low-energy phonon modes have dominant contributions from the Ag atoms. These two modes (assuming them as Einstein modes with 1° of freedom each) give the linear thermal expansion of αa = 4.3 × 10−6 K−1, αb = −1.0 × 10−6 K−1, αc = −3.1 × 10−6 K−1 and αa = 0.3 × 10−6 K−1, αb = 0.4 × 10−6 K−1, αc = −0.2 × 10−6 K−1 at 300 K, respectively. The mode at 3.34 meV involves the rotation of Ag atoms around the connecting −C8N5− structural units in the a−b plane. On the other hand, the mode at 5.32 meV involves the displacement of Ag atoms along the b-axis. The motion of Ag produces the effect of a closing hinge exactly opposite to that observed on application of pressure. This gives rise to contraction along the c-axis and expansion along the a and b axes.

3. CONCLUSIONS In conclusion, our ab initio DFT calculations reveal large anomalous lattice behavior in AgC8N5, which is a metal− organic framework material with very low crystal density. Extensive calculations as a function of pressure reveal negative linear compressibility along the c-axis. Moreover, the pressuredependent phonon calculations performed using linear response density functional perturbation theory methods show anomalous thermal expansion behavior in this compound. The temperature dependence of lattice parameters with (MD) and without (LD) explicit anharmonic effects agrees very well. The NLC and NTE along the c-axis of the structure are governed by the dominant dynamics of Ag atoms in the a− b plane, which give rise to a hinge-like mechanism. The AgC8N5 compound may be very useful for strong armor applications because of its anomalous lattice behavior. E


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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.8b04398. Computational details, elastic properties, and thermal expansion calculations (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Ranjan Mittal: 0000-0003-3729-9352 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS S.L.C. would like to thank the Department of Atomic Energy, India, for the award of Raja Ramanna Fellowship. The use of ANUPAM super-computing facility at BARC is acknowledged.

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