Correlation between Uniaxial Negative Thermal Expansion and

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The Correlation Between Uniaxial Negative Thermal Expansion and Negative Linear Compressibility in Ag[Co(CN)] 3

6

Lei Wang, Cong Wang, Hubin Luo, and Ying Sun J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b09944 • Publication Date (Web): 19 Dec 2016 Downloaded from http://pubs.acs.org on December 24, 2016

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The Journal of Physical Chemistry

The Correlation between Uniaxial Negative Thermal Expansion and Negative Linear Compressibility in Ag3[Co(CN)6] Lei Wang,∗,† Cong Wang,∗,† Hubin Luo,‡ and Ying Sun† Center for Condensed Matter and Materials Physics, School of Physics, Beihang University, Beijing 100083, China, and Key Laboratory of Magnetic Materials and Devices, Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo 315201, China E-mail: [email protected]; [email protected]

∗

To whom correspondence should be addressed Center for Condensed Matter and Materials Physics, School of Physics, Beihang University, Beijing 100083, China ‡ Key Laboratory of Magnetic Materials and Devices, Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo 315201, China †

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Abstract Density functional theory (DFT) calculations are used to investigate the correlation between the uniaxial negative thermal expansion (NTE) and negative linear compressibility (NLC) behaviors in Ag3 [Co(CN)6 ]. First, we reproduce the uniaxial-NTE and NLC behaviors under temperature- and pressure-field. And then the temperature dependence of elasticity is studied. The abnormal nature of elastic constants C 33 and C 11 +C 12 as the function of temperature is predicted.

The

hardening of phonon modes (below 568.7 cm−1 ) with increasing temperature can be as an indicative for abnormal physical properties of Ag3 [Co(CN)6 ].

Through

analyzing the vibration mode with the strongest phonon hardening, the deformation of wine-rack motif in anisotropic framework can be identified as the mechanism that leads to the coexistence of uniaxial-NTE and NLC in Ag3 [Co(CN)6 ]. The response of phonon group velocity along c-axis is intense and quicker than that of a-axis on heating and on compression, facilitating c-axis itself to be the carrier of abnormal uniaxial-NTE and NLC properties.

1. Introduction The interest in materials with abnormal physical properties has been driven during the past decades. Typically, the strange response to temperature- or pressure-field attracts gradually the attention of material scientists, for instance, the negative thermal expansion (NTE) 1,2 and negative linear compressibility (NLC) 3,4 effects. The vast majority of materials expand on heating due to the asymmetric shape of a typical interatomic potential well. There are, however, certain categories of materials show the unusual property of NTE, i.e., the volume contracts on heating. The overall expansion coefficients of composites using NTE materials as thermal-expansion compensators can be precisely tailored to a specific positive, negative or even zero value. 5 Or the expansion coefficients of the NTE materials can be controlled by doping. 6 Hence, the NTE materials have important potential applications in 2 ACS Paragon Plus Environment

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various areas, such as optical, electronic, biomedical applications and so on. 7,8 Another unique property is the NLC behavior. Most materials are compressed in all directions with increasing hydrostatic pressure, while the NLC materials expand along a specific direction with total volume reduction under these conditions. The NLC materials are remarkably rare and of great application value. For example, owing to stretch densification along the NLC orientation, the NLC materials can be applied in the ultrasensitive pressure detectors and robust shock-absorbing composites etc. 9,10 Normally, the NTE and NLC properties exist separately in some materials. 11–13 If these abnormal properties can coexist in materials, it will make materials themselves have a more flexible application in temperature- and pressure-field. This has always been the pursuit of material scientists. Recently, some studies show that NLC can be found in framework structures that exhibit very anisotropic NTE behavior. Among them, the most striking material is the metallic cyanide framework Ag3 [Co(CN)6 ] reported by Goodwin et al. 14,15 The Ag3 [Co(CN)6 ] shows colossal uniaxial-NTE (a c = -130×10−6 K

−1

) and NLC

(K c = -75 TPa−1 ) behaviors along the c-axis of its trigonal lattice. After a phase transition from a trigonal to a monoclinic space group (at 0.19 Gpa), the NLC turns into K c = -5 TPa−1 . Rational design of materials with extreme NLC on the basis of Ag3 [Co(CN)6 ] is also put forward. This can be achieved by substitution of Mn2+ for Co3+ while balancing charge with extra-framework K+ ions, resulting in the mechanically stable compound KMn[Ag(CN)2 ]3 . 16 Experimental Raman spectra and Gr¨ uneisen parameters of the Raman modes have been reported by Rao et al. for both Ag3 [Co(CN)6 ] phases as a function of temperature and pressure. 17 Mittal et al. have shown the experimental and calculated inelastic neutron scattering spectra of the Ag3 [Co(CN)6 ] trigonal phase. 18 The lattice dynamics and thermal expansion of Ag3 [Co(CN)6 ] have been investigated by Hermet et al. from calculations and infrared spectroscopy. 19 The Ag-Ag dispersive interaction and physical properties of Ag3 [Co(CN)6 ] have been introduced by Fang et al.

from

first-principles calculations. 20 In spite of the above reports, a very critical and important



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research on the correlation between unixial-NTE and NLC behaviors in Ag3 [Co(CN)6 ] is still absent. Therefore, the study on the coexistence mechanism of unixial-NTE and NLC behaviors can not only fill the theoretical blank but also provide basis for further experimental research. In this work, within the framework of density functional theory (DFT) combined with the quasi-harmonic approximation (QHA), our study aims to the following aspects: i. investigating the unixial-NTE and NLC behaviors in Ag3 [Co(CN)6 ]; ii. exploring the factors that can lead to the two strong negative behaviors and the correlation between unixial-NTE and NLC behaviors in Ag3 [Co(CN)6 ]; iii. analyzing the intrinsic characteristics of a-axis and c-axis and explaining why c-axis itself can be as the carrier of uniaxial-NTE and NLC behaviors.

2. Theoretical method 2.1. First-principles calculations The plane-wave-based DFT calculations are performed using the VASP program with the core orbital replaced by ultrasoft pseudopotentials, and a kinetic energy cutoff of 500 eV. The generalized gradient approximation (GGA) with the PBE exchange correlation functional is adopted. 21,22 A 11×11×11 Monkhorst-Pack scheme is used for k -point sampling in the first irreducible Brillouin zone. All the electronic structures are calculated on the corresponding optimized crystal geometries. In order to determine the equilibrium lattice structure, the total energy is minimized simultaneously with respect to all of the lattice parameters. In our work, the equilibrium volume is obtained by fitting the total energy as a function of volume to a Murnaghan equation of state. At the fixed volume, all atoms are full relaxed using the conjugate gradient (CG) algorithm. The optimization is stopped when forces on all relaxed atoms are less than 0.01 eV/˚ A.


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2.2. Lattice dynamics calculations Lattice dynamical properties are obtained from frozen phonon method under the quasi-harmonic approximation (QHA). 23 In this method, phonon frequencies are calculated from Hellmann-Feynman (HF) forces generated by nonequivalent atomic displacement in a supercell for a crystal structure. In the present study, the dimension of supercell is 2×2×2 of their unit cells for trigonal structures. For this calculations of supercell system, k -point sampling is limited to 9×9×9. A dynamical matrix is constructed from HF forces acting on all atoms with a displaced atom, and the displacement amplitude is 0.01 ˚ A. The phonon frequencies are calculated by solving the eigenvalue problem for the dynamical matrix.

2.3. Effects of finite temperature and pressure The equilibrium state of a crystal at specific temperature and pressure is determined by the condition that the Gibbs free energy G has the lowest value, where G = U + P V − T S,

(1)

and U is the internal energy, P is the pressure, V is the volume, T is the temperature, and S is the entropy. At zero pressure, the effect of temperature can refer to the Helmholtz free energy F (V, T ), defined as F (V, T ) = U − T S.

(2)

At zero temperature, the thermodynamically stable phase at pressure P is the one with the lowest enthalpy H, given by H = U + P V.

(3)

In study of the thermodynamic properties, it is obviously crucial to include the effects of changing temperature. One effect is that it becomes easier to pass over energetic barriers to other structures with increasing temperature, so that hysteresis is reduced. Another important effect is that the vibrational contribution to Gibbs free energy may alter the 5 ACS Paragon Plus Environment


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equilibrium coexistence pressure and volume. In the thermodynamic calculations, the QHA has been proved to be both accurate and computationally efficient. Within the QHA, the vibrational contribution to the Gibbs free energy is written as a sum over the harmonic modes (v, q) and the anharmonic modes where the frequencies of the modes are allowed to depend on the volume, ωv (q, V ). The vibrational contribution to the Helmholtz free energy is expressed as

Fvib = kB T

∑

[

(

log 2 sinh

v,q

¯hωv (q, V ) 2kB T

)] ,

(4)

which includes the effects from both zero-point motion and finite temperature. The total Gibbs free energy is then G = U + P V + Fvib ,

(5)

where U is the internal energy at T = 0 K. According to the definition of Helmholtz free energy F = U + Fvib , V and T are variable. At the given temperature T , the equilibrium volume results from a minimization of F (V, T ). One scheme for obtaining the equilibrium structure at pressure P is as follows. First, minimize the Helmholtz free energy F with respect to the structural degrees of freedom xi , while holding the temperature and volume fixed. This gives the values of the structural degrees of freedom as a function of volume and temperature, xi (V, T ), and the Helmholtz free energy F (V, T ). The most stable structure at temperature T and pressure P is the one with the lowest value of G(T, P ). Various lattice dynamics parameters as the function of temperature and pressure can be derived analytically from Gibbs free energy. Herein, the derivative process of the phonon group velocity is presented. 24 Within the framework of phonon anharmonic interactions, an atomic displacement operator can be written as


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( uα (lκ) =

h ¯ 2N mκ

×e

iq·r(lκ)

)1/2 ∑

−1/2

wqj [ˆ aqj + a ˆ†−qj ]

qj

Wα (κ, qj),

(6)

where uα (lκ) is the atomic displacement of the κth atom in the lth unit cell. N is the number of unit cells in the crystal and mκ is the atomic mass of type κ. h ¯ is the reduced Planck constant. The α, β, and γ are Cartesian indices. r (lκ) is the equilibrium atomic position. a ˆ†−qj and a ˆqj are the phonon creation and annihilation operators of normal mode of band index j and wave vector q. The frequency ωqj and polarization vector W (κ, qj) are obtained from the eigenvalue problem of a dynamical matrix D(q) ∑

2 Dαβ (κκ′ , q)Wβ (κ′ , qj) = wqj Wα (κ, qj),

(7)

κ′ β

with Dαβ (κκ′ , q) = √

∑ 1 ′ ′ Φαβ (lκ, l′ κ′ )eiq·[r(l κ )−r(lκ)] . ′ mκ mκ l ′

(8)

Φαβ (lκ, l′ κ′ ) denotes the second-order force constant. l, l′ , . . . are the indices of unit cells, and κ, κ′ , . . . are the indices of atoms in a unit cell. The phonon group velocity vα (qj) can be obtained directly from the eigenvalue equation, ∂wqj ∂qα ∂Dβγ (κκ′ , q) 1 ∑ Wβ (κ, qj) = Wγ (κ′ , qj). 2wqj κκ′ βγ ∂qα

vα (qj) ≡

(9)

2.4. The elasticity calculations The elastic constants determine the stiffness of a crystal against an externally applied strain. A trigonal Ag3 [Co(CN)6 ] crystal is classified as Laue class ¯3m and features six independent



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elastic constants, 25



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

C14  C11 C12 C13   · C11 C13 −C14     · · C33 C=   · · C44    C44 C14   · C66

              

(10)

where C66 = (C11 − C12 )/2. These six independent elastic constants can be determined by applying a small strain to the equilibrium lattice and computing the resultant change in its total energy. 26 The distorted lattice vectors can be obtained via a matrix multiplication ′

′

R = RD, where R and R are the 3×3 matrices containing the components of the distorted and undistorted lattice vectors, respectively. D is a symmetric distortion matrix: 



ε5 /2  1 + ε1 ε6 /2  D=  ε6 /2 1 + ε2 ε4 /2  ε5 /2 ε4 /2 1 + ε3

    

(11)

where εi is the ith component of the strain tensor Voigt notation (xx=1, yy=2, zz=3, yz=4, xz=5, and xy=6). Here it indicates that 1-3 are associated with normal strains, and 4-6 are associated with shear components. According to linear elasticity theory, the distortion energy can be written as a function of strain as 6 V0 ∑ ∆E = E − E0 = Cij εi εj + O(ε3 ), 2 i,j=1

(12)

where E and E0 are the total energies of the distorted and equilibrium structures, respectively.

V0 is the equilibrium volume.

The six distortion matrices necessary for

computing the six elastic constants of a trigonal structure are shown in Table 1. At each given strain, we fully relaxed all atoms to their energetically more favorable lattice 8 ACS Paragon Plus Environment

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according to strain-induced forces while holding the distorted lattice vectors fixed. 27,28 We then fitted the distortion energies calculated at δ = ±0.005 and ±0.01 to the function ∆E (δ) = k2 δ 2 + k3 δ 3 , and obtained the elastic constant Cij from the quadratic coefficient k2 (see Table 1 for relationships). Table 1: Deformation matrices used to calculate the six independent elastic constants of a trigonal crystal. Strain I 1 2 3 4 5 6

Parameters εi = 0 if not listed ε1 = ε2 = δ ε6 = δ ε3 = δ ε4 = ε5 = δ ε1 = ε2 = ε3 = δ ε5 = ε6 = δ

Relationship between k2 and elastic constants k2 = (C11 + C12 )V0 k2 = 14 (C11 − C12 )V0 k2 = 21 C33 V0 k2 = C44 V0 k2 = (C11 + C12 + 2C13 + 12 C33 )V0 k2 = C14 V0

The temperature dependence of bulk modulus B0 is obtained from ) ∂ 2 F (V, T ) ∂V 2 ( 2T ) 2 ∂ E ∂ Fvib (ω, T ) = V +V . ∂V 2 ∂V 2 T (

B0 (T ) = V

(13)

3. Results and discussion 3.1. The uniaxial-NTE and NLC behaviors The calculated axial thermal expansion behavior of Ag3 [Co(CN)6 ] is shown in Fig. 1. The experimental results determined from x-ray powder diffraction by Goodwin et al. are shown for comparison. Due to the complete decomposition of the sample by 500 K, 14 the temperature range in this study is set to 400 K. In our work, through the full lattice and atomic relaxation, the calculated c-axis and a-axis of Ag3 [Co(CN)6 ] at 0 K are 7.14 ˚ A and 6.97 ˚ A respectively, which are smaller and larger respectively than experimental data at 9 ACS Paragon Plus Environment


low temperature.

And the calculated volume at 0 K is larger than the existing

experimental measurement (see Fig. 3.(a)). This is typical for the GGA approximation to DFT calculations. Additionally, we see that the c-axis of Ag3 [Co(CN)6 ] exhibits the NTE behavior (7.14 ˚ A → 7.11 ˚ A), while the a-axis shows positive thermal expansion behavior (6.97 ˚ A → 7.09 ˚ A). There is a qualitative agreement between calculations and experimental measurements.

However, compared with the experimental quantities, the ab initio

calculations within QHA framework underestimate the variation of lattice parameters, and there is no cross of the c-axis and a-axis curves. The QHA contains harmonic part, and anharmonic cubic term in Hamiltonian operator. Neglect of the high-order anharmonic effects (≥ the quartic term) at fixed volume is often a good approximation at room temperature, but at higher temperature anharmonic effects can be important. 29–32 This can be the reason for the discrepancy between our calculations and experimental results, 14 where the uniaxial-NTE behavior (-130×10−6 K−1 < ac < -120×10−6 K−1 ) from the x-ray diffraction data is colossal and far more than other typical framework NTE materials such as ZrW2 O8 , MOFs and so on. 1,33,34 7.4

7.3

)

Expt c

Å Lattice parameter (

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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.

Calc c

7.2

.

7.1

7.0

Calc a

Expt a

.

6.9

.

6.8

6.7 0

50

100

150

200

250

300

350

400

Temperature (K)

Figure 1: (Color online) The calculated temperature dependence of c-axis and a-axis of Ag3 [Co(CN)6 ]. The experimental lattice parameters as a function of temperature from Ref. 14 is also shown. The calculated pressure dependent behavior of Ag3 [Co(CN)6 ] is exhibited in Fig. 2. It is found that, besides the NTE behavior, the c-axis has the peculiar NLC effect as shown 10 ACS Paragon Plus Environment

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in solid blue tag (7.14 ˚ A → 7.29 ˚ A). By contrast, the pressure dependent behavior of aaxis are normal presented as solid red tag (6.97 ˚ A → 6.55 ˚ A). Our calculations reproduce the experimental trend of pressure dependence of lattice parameters that are marked as the hollow tag in Fig. 2. 15 However, the experimentally observed a phase transition from Ag3 [Co(CN)6 ]-I (P 31m) to Ag3 [Co(CN)6 ]-II (C2/m) at 0.19 GPa has not been harvested in this DFT calculations. The displacements of Ag atoms in alternative rows might be involved in the high-pressure phase with an interdigitated structure. 20 The NLC behavior of c-axis is qualitatively unaffected by this transition. Compared with other NLC materials, for example, elemental Se (K c ≈ -2.5 TPa−1 ), 35 TeO2 (K c ≈ -5.1 TPa−1 ), 36 CH3 OH·H2 O (K c ≈ -2.7 TPa−1 ), 37 BiB3 O6 (K c ≈ -6.7 TPa−1 ) 38,39 etc., the NLC behavior of Ag3 [Co(CN)6 ] detected by the high-pressure neutron powder diffraction is colossal (0 GPa ≤ P ≤ 0.19 GPa, K c ≈ -75 TPa−1 ; 0.19 GPa < P ≤ 7.65 GPa, K c ≈ -5 TPa−1 ). 15 Previous calculations once propose that only the GGA+D calculations including a correction for the dispersive interaction can reproduce the pressure-induced phase transition of Ag3 [Co(CN)6 ]. 20 Relevant theoretical correction can give an improved agreement between calculations and experimental data. In our opinion, the underlying physical mechanism can not be fade within the framework of classical theory. Here, although the adoption of classical DFT method can not give a direct observation to the high-pressure phase, we will discuss the possible phase instability by prediction of unusual phonon response under the external field in the later content. The effects of temperature and pressure on unit cell volume of Ag3 [Co(CN)6 ] are presented in Fig. 3.(a) and (b), respectively. The experimental results are also shown for comparison. 14,15 In spite of uniaxial-NTE and NLC behaviors in Ag3 [Co(CN)6 ], from Fig. 3 we see that the unit cell volume increases on heating and reduces on compression, behaving the normal thermal expansion and compression natures.

The overall volume thermal

expansion within the framework of DFT calculations is from 300.4 ˚ A3 to 310.3 ˚ A3 in the temperature range 0 K ∼ 400 K, which is smaller than experimental results. The values of red line in Fig. 3.(a) is the post-processing based on the temperature dependent lattice



7.6

Lattice parameter (

Å

)

7.4

Calc. c

7.2

Calc. a 7.0

6.8

6.6

Expt. c

6.4

Expt. a 6.2

6.0

5.8 0

1

2

3

4

5

6

7

8

Pressure (GPa)

Figure 2: (Color online) The calculated pressure dependence of lattice parameters of Ag3 [Co(CN)6 ]. The experimental measurements from Ref. 15 are added in comparison. parameters in experimental results. The quantitative difference is understandable due to the adoption of QHA method. In Fig. 3.(b), the calculated pressure dependence of unit volume presents a trend of monotonous decrease, while the phase transition is missed due to the limitation of DFT calculations. The pressure-induced phase transition at 0.19 GPa results in a structural collapse in the experimental measurements. The new high-pressure phase exhibits a normal volume compression property as well. 312

320 (b)

Å

3

)

308

Unit cell volume (

Å

3

)

(a)

Unit cell volume (

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 26

304 Calc.

300

Expt.

296

300 280 260 Calc. Expt.

240 220

292

200 0

100

200

300

400

500

0

1

2

Temperature (K)

3

4

5

6

7

8

Pressure (GPa)

Figure 3: (Color online) (a) The calculated temperature dependence of unit volume of Ag3 [Co(CN)6 ] (black tag line), compared with the experiment data (red tag line) (Ref. 14 ) ; (b) The calculated pressure dependence of unit volume of Ag3 [Co(CN)6 ] (black tag line) and the experimental results from Ref. 15 (red tag line). The effects of temperature- or pressure-field on lattice parameters of Ag3 [Co(CN)6 ] are 12 ACS Paragon Plus Environment

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always considered separately in experiments. Herein, the variation of lattice parameters under the combination of temperature- and pressure-field is explored.

In Fig.

4, the

calculated lattice parameters are exhibited for four different pressures, 2, 4, 6, 8 GPa, as a function of temperature. The performances of c-axis and a-axis are opposite. The c-axis behaves the peculiar thermal contraction and hydrostatic pressure elongation, and the a-axis has normal response under the combination of temperature- and pressure-field. Table 2: Elastic constants of Ag3 [Co(CN)6 ] at T = 0 K and P = 0 GPa obtained from DFT calculations, along with other theoretical values by different exchangecorrelation functional LDA 19 and GGA+D 20 method. Elastic constants (GPa) This work Ref. 19 C11 37.68 28.57 C33 129.94 165.62 C44 38.41 23.60 C12 13.74 12.46 C13 47.65 45.51 C14 8.15 8.96

Ref. 20 34.38 139.61 32.75 13.12 47.17 8.52

(a)

(b)

7.32

7.00

7.30

6.95

2 Gpa 4 Gpa

a-axis

Å

)

6 Gpa

)

7.28

2 Gpa

c-axis

Lattice parameter (

Å Lattice parameter (

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4 Gpa

7.26

6 Gpa 8 Gpa

7.24

7.22

7.20

7.18

8 Gpa

6.90

6.85

6.80

6.75

6.70

6.65

6.60

7.16

6.55

7.14

6.50 0

50

100

150

200

250

300

350

400

0

50

100

Temperature (K)

150

200

250

300

350

400

Temperature (K)

Figure 4: (Color online)(a) The calculated c-axis changed with temperature at four different pressures; (b) the calculated a-axis changed with temperature at the same four pressures.



3.2. The temperature dependence of elasticity The elastic constants of solids give important information on their mechanical and dynamical properties. They may be used as a means of probing the inter-atomic forces. Particularly, they provide information on the stability and stiffness of materials. In this work, we investigate the elastic behavior of trigonal Ag3 [Co(CN)6 ]. The Results of elastic constants calculations (at T = 0 K, P = 0 GPa) are presented in Table 2.

Other

theoretical values from related literatures are also listed for comparison. 19,20 The overall observation is that there is a good agreement between our calculated results and other theoretical values with different exchange-correlation functional LDA and GGA+D. Additionally, Fang et al. in Ref. 20 lists out detailedly the pressure dependence of elastic constants, bulk modulus and compliance, which can be as a supplement to the data we have presented.

Lately, Gaillac et al.

reported that linear compressibility can be

represented as a 3D parametric surface and linear compressibility of Ag3 [Co(CN)6 can be negative from 2D projections on the (xz), (yz) planes. 40 145 Elastic Constant (GPa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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140 135

C C C 33

130

+

11

12

55

50

0

100

200

300

400

Temperature (K)

Figure 5: (Color online) The calculated elastic constants of C 33 (green tag line) and C 11 +C 12 (blue tag line) in Ag3 [Co(CN)6 ]. Moreover, the temperature dependence of elastic constants C 33 and C 11 +C 12 of trigonal Ag3 [Co(CN)6 ] are calculated, which are the representatives of stiffness for c-axis and a-axis. 41 In this procedure, it is assumed that the temperature dependence of elastic constants are solely caused by thermal expansion. A stress-strain approach at 0 K can be combined with 14 ACS Paragon Plus Environment

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the phonon theory of thermal expansion. 42 Trigonal Ag3 [Co(CN)6 ] is composed of covalent chains of Co-CN-Ag-NC-Co linkages. The primary effect of temperature upon this structure is to pull the chains away from each other decreasing the interchain interactions and thus increasing the anisotropy. From Fig. 5, we observe the abnormal elastic behavior where the elastic constants of C 33 and C 11 +C 12 increase with increasing temperature, indicating that the stiffness of c-axis and a-axis is enhanced on heating. This unique behavior is in contrast with other materials. 42 To the best of our knowledge, related data on finite temperature elastic constants of Ag3 [Co(CN)6 ] is vacant, so the experimental verification is expected. At 0 K, the theoretical value of C 33 (129.94 GPa) is larger than that of C 11 +C 12 (51.42 GPa), implying that intrachain bonding is much stronger than interchain bonding. The increase rate of longitudinal modulus of the chain structure dC 33 /dT (0.023 GPa/K) is about twice as large as that of isotropic modulus in the basal plane d(C 11 +C 12 )/dT (0.011 GPa/K), suggesting that the increase in stiffness of c-axis is much quicker than that of aaxis with increasing temperature. It can be indicated indirectly that there will be difference performances of c-axis and a-axis under external temperature- or pressure-field. The calculated temperature dependence of bulk modulus of Ag3 [Co(CN)6 ] is shown in Fig. 6. Other theoretical values using LDA and GGA exchange-correlation functional 19,43 and the experimental value 14 are also attached. We find that the bulk modulus of Ag3 [Co(CN)6 ] decreases gradually on heating. At 0 K, our calculated bulk modulus is about 9.6 GPa lower than other theoretical values (11.4 GPa from Ref. 19 and 9.9 GPa from Ref. 43 ). At room temperature, there is a good agreement between our result (6.6 GPa) and the experimental value (6.5 GPa from Ref. 14 ).

3.3. The phonon hardening on heating The phonon frequencies as the function of temperature of Ag3 [Co(CN)6 ] under the anharmonic effect are studied. Here, one parameter S is used to express the change of phonon frequencies with temperature, which is defined as S = −dω/dT , ω is the phonon 15 ACS Paragon Plus Environment


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Present

Calc Calc Expt

Calc

.

. (LDA) . (GGA)

9

. (300K)

8 7 6 5

0

100

200

300

400

Temperature (K)

Figure 6: (Color online) The calculated temperature dependence of bulk modulus of Ag3 [Co(CN)6 ]. Other theoretical values using LDA and GGA exchange-correlation functional from Ref. 19 and Ref. 43 are represented as blue and green solid square, respectively. The experimental value at room temperature from Ref. 14 is also attached for comparison (red solid square). frequency and T is temperature.

For S > 0 the phonon frequencies decrease with

increasing temperature, whereas for S < 0 the phonon frequencies increase with increasing temperature. From Fig. 7, the hardening (S < 0) and softening (S > 0) of phonon modes can be observed. The softening of phonon modes with increasing temperature is always common. 44 By contrast, the hardening of phonon modes on heating is abnormal and can be connected with some peculiar physical properties such as the phase instability, the NTE behavior and so on. 19,45,46 In our calculations, the frequencies of high-frequency phonons in the range of 623.2 cm−1 - 2502.3 cm−1 (No.

37 - No.

48, blue lines) decrease with

increasing temperature, behaving the normal softening. The modes below 568.7 cm−1 (No. 1 - No. 36, brown lines) show the hardening of phonons. Especially the hardening of low-frequency optical phonon modes below 97.8 cm−1 (No. 4 - No. 10) is strong and the phonon frequencies increase by approximately 0.3 - 0.45 cm−1 /K. For Ag3 [Co(CN)6 ], the hardening of phonons on heating can be connected with peculiar uniaxial-NTE and NLC behaviors, and the lowest-frequency optical phonon mode (No. 4, 44.7 cm−1 ) with the strongest hardening has the most important contribution to these peculiar negative behaviors.


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0.1 -1 S (cm /K)

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0.0

-0.1

-0.2

-0.3

-0.4

-0.5 A

L

M

G

A

H

K

G

Figure 7: (Color online) The calculated temperature dependence of phonon frequencies curves of Ag3 [Co(CN)6 ] along the high-symmetry directions in the Brillouin zone. The parameter S is defined as S = −dω/dT . The eigendisplacement vectors corresponding to the lowest-frequency optical phonon at A point (0, 0, 0.5) are displayed in Fig. 8A. The unit cell of Ag3 [Co(CN)6 ] consists of three interpenetrating Co-CN-Ag-NC-Co linkages. Through the 1×1×2 supercell model shown in Fig. 8A looking from the [1 1 0] direction, we can observe the structural characteristics of a three-dimensional wine-rack pattern. In the approximately linear bridges Co-CN-AgNC-Co, the Co and Ag atoms are stationary, and the C and N atoms vibrate transversely away from the linkages. This vibrational mode can be classified as translational motions. In Ag3 [Co(CN)6 ], the coexistence of uniaxial-NTE and NLC behaviors can be linked to the deformations of the wine-rack frameworks hinged by strong covalent bond. 47 With increasing temperature, the volume expansion is accompanied with the deformation of the wine-rack motif, involving the simultaneous expansion along a- and b-axis and a contraction along c-axis. When the hydrostatic pressure is increased, the space in the crystal compresses persistently. There is an opposite deformation of wine-rack motif, i.e., the elongation of caxis orientations and the shrinkage of a- and b-axis. The schematic diagram can be visualized in Fig. 8B. Not all NTE materials have the NLC nature, whereas one strategy for identifying NLC candidates is to target the framework materials that exhibit anisotropic NTE. That is because materials that contract in one direction on heating are likely to expand in the same direction on compression. For abnormal materials with simultaneous uniaxial-NTE and 17 ACS Paragon Plus Environment


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NLC behaviors, some characteristics are indispensable including the flexible framework, lowdensity structure, extreme mechanical anisotropy, unusual phonon response under applied temperature- or pressure-field, and a specific topology such as wine-rack motif etc.

2.0

1.5

Figure 8: (Color online) A. The eigendisplacement vector is corresponding to the lowestfrequency optical phonon mode at A point (0, 0, 0.5) looking from the [1 1 0] direction. Green and blue balls are on behalf of the Ag and Co atoms, respectively. Black and purple balls represent the C and N atoms, respectively. Arrows are proportional to the amplitudes of atomic motion. B. Schematic representations of the deformations of wine-rack framework with increasing temperature and hydrostatic pressure. 1.0

0.5

3.4. Phonon group velocities The uniaxial-NTE and NLC behaviors in Ag3 [Co(CN)6 ] is favored by the wine-rack topology. 0.0

Meanwhile, it is worth thinking deeply that why the specific c-axis in crystal structure is the carrier of abnormal behaviors under the applied temperature- or pressure-field. Based on this, we will discuss the intrinsic differences of a-axis and c-axis. The phonon group velocity is an important characteristic that determines the propagation speed of phonon energy. Here, we calculate the phonon group velocities along a-axis and c-axis as the function of temperature and pressure. It can be seen from Fig. 9 that the phonon group velocities have significant differences between a-axis and c-axis. The whole trend is that phonon group 18 ACS Paragon Plus Environment

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18

Å

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a c

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(b)

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Å

)

22

Group velocity (THz

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16 14 12 10 8

35

a c

30 25 20 15

6

10

4 0

50

100 150 200 250 300 350 400

1

2

3

4

5

6

7

8

Pressure (GPa)

Temperature (K)

Figure 9: (Color online) The phonon group velocities along a-axis (black line) and c-axis (red line) as the function of temperature (a) and pressure (b). velocities decrease on heating but increase on compression. Compared with the moderate variation of phonon group velocity along a-axis, the response of phonon group velocity along c-axis are intense and quicker facing external temperature- or pressure-field. This feature facilitates c-axis itself to undertake the unique role in the wine-rack framework structure.

4. Conclusion In this paper, the correlation between the uniaxial-NTE and NLC behaviors in Ag3 [Co(CN)6 ] has been investigated by DFT calculations. The uniaxial-NTE and NLC behaviors are reproduced. The abnormal nature of elastic constants C 33 and C 11 +C 12 as the function of temperature is predicted. Within the framework of QHA, we find that low-frequency optical phonons (below 568.7 cm−1 ) show the unusual hardening with increasing temperature, which are considered to be relevant to the uniaxial-NTE and NLC behaviors. Through the analysis of vibration mode with the strongest phonon hardening, the deformation of wine-rack motif in the flexible framework can be identified to result in the coexistence of uniaxial-NTE and NLC behaviors in Ag3 [Co(CN)6 ]. The intrinsic differences between a-axis and c-axis have also been studied. Facing external temperature- and pressure-field, it is found that the response of phonon group velocity along c-axis are intense and quicker compared with that 19 ACS Paragon Plus Environment


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of a-axis. This characteristic facilitates c-axis itself to as the carrier of uniaxial-NTE and NLC behaviors.

Acknowledgement This work was supported partly by National Natural Science Foundation of China (NSFC) ( No. 51502010, No. 51572010 and No. 51401227), and partly by the China Postdoctoral Science Foundation Funded Project (No. 2016M590028) and Fundamental Research Funds for the Central Universities (Grand No. YWF-16-JCTD-B-03).

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Correlation between Uniaxial Negative Thermal Expansion and

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