PHYSICAL REVIEW B 71, 014112 ͑2005͒

Normal mode approach for predicting the mechanical properties of from first principles: Application to compressibility and thermal expansion of zeolites

R. Astala,* Scott M. Auerbach,† and P. A. Monson‡ Department of and Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003, USA ͑Received 1 July 2004; revised manuscript received 1 October 2004; published 25 January 2005͒

We present a method for analyzing the mechanical properties of solids, based on normal modes and their coupling to lattice strains. This method was used to study elastic compression and thermal expansion of zeolites, with parameters calculated from density functional theory. We find in general that the bulk modulus can be divided into two contributions: a positive term arising from compression without internal relaxation, and a negative term from coupling between compression and internal vibrational modes. For silica polymorphs, the former term varies little among the phases studied, reflecting the intrinsic rigidity of SiO4 tetrahedra. In contrast, the latter term varies strongly from one polymorph to the next, because each polymorph exhibits different symmetry constraints on internal vibrations and their couplings to lattice strains. Typically only a few normal modes contribute to the bulk modulus. To facilitate parametrization of this normal mode model, we constructed a simplified classical spring-tetrahedron model for silica. After fitting to properties of silica so- dalite, this model reproduces cell volumes and predicts bulk moduli of ␣-cristobalite and silica zeolites CHA, LTA, and MFI. We incorporated anharmonic effects into the theory, allowing the calculation of the thermal expansion coefficient. The resulting expression provides a generalization of classical Grüneisen theory, taking into account additional anharmonicities. This method was used to study thermal expansion of fcc aluminum and an aluminosilica sodalite, yielding good agreement with experiment.

DOI: 10.1103/PhysRevB.71.014112 PACS number͑s͒: 62.20.Dc, 65.40.De, 71.15.Mb, 82.75.Ϫz

I. INTRODUCTION ing a classical harmonic potential energy in terms of these two kinds of distortions. Although our classical treatment The discovery of new nanostructures with increasingly breaks down at sufficiently low temperatures, where the complex architectures challenges our understanding of the quantum nature of vibrations becomes important, the bulk relationship between a material’s atomic structure and its modulus turns out to be most sensitive to low-frequency vi- mechanical properties.1 An important example involves the brations, which are adequately modeled under ambient con- crystalline polymorphs of silica, SiO2. These include dense ditions by classical theory. Below we show that the bulk phases such as quartz and cristobalite,2 as well as nanopo- modulus can be written as a positive term arising from sud- rous ͑zeolitic͒ polymorphs such as sodalite and silicalite.3 den elastic response, and a negative term controlled by cou- The mechanical properties of zeolites are particularly impor- plings between vibrations and lattice strains. When param- tant because channel deformations can radically change a etrized for several silica polymorphs by density functional zeolite’s capacity for adsorption, diffusion, and reaction. theory ͑DFT͒ calculations, we find that the sudden elastic These dense and nanoporous phases share common structural response is remarkably uniform from one polymorph to the features, including Si-O bond lengths ͑ϳ1.6 Å͒, O-Si-O next, like the cohesive energy discussed above. However, angles ͑ϳ109°͒, and networks with low coordination and normal vibrations and their couplings to lattice strain are found to depend sensitively on the symmetry of a given strong association.4 These phases differ in Si-O-Si angles ͑ ͒ polymorph. 140°–180° , densities and symmetries. The cohesive energy By augmenting the harmonic potential energy with anhar- appears to be relatively insensitive to these structural differ- 5 monicity, we have developed a theory for the thermal expan- ences, varying by only 0.1 eV per SiO2. However, the bulk sion of crystalline solids. Our approach provides a generali- modulus varies significantly from one polymorph to the zation to classical Grüneisen theory,9 which accounts for 5 next. Despite progress in understanding the mechanical anharmonicity by tracking the volume dependence of phonon properties of networks,6 this variation in bulk modulus frequencies. Instead, we construct a potential with a com- among silica phases remains poorly understood. In this ar- plete set of cubic anharmonicities, and derive analytically the ticle, we investigate a theory of compression and ex- thermal expansion coefficient showing the importance of pansion based on normal vibrations and their coupling to each kind of anharmonicity. We apply this approach to the lattice strains, which elucidates the relationship between a thermal expansion of aluminum metal and aluminosilica so- crystal’s structure and its mechanical properties. dalite zeolite. In the former case, we find that our theory The elastic response of a solid to compression can be reproduces the success of classical Grüneisen theory. In the decomposed into two parts: ͑i͒ atomic vibrations at fixed latter case, we find that our approach gives good agreement volume,7,8 and ͑ii͒ unit cell volume fluctuations for a fixed with experiment, while Grüneisen theory accounts for only atomic configuration. In what follows, we denote the latter 22% of the thermal expansion coefficient. motion as a “sudden” elastic response, because the vibrations To facilitate the parametrization of this normal mode po- do not relax during compression. Our theory begins by writ- tential function, we develop a classical spring-tetrahedron

1098-0121/2005/71͑1͒/014112͑15͒/$23.00014112-1 ©2005 The American Physical Society ASTALA, AUERBACH, AND MONSON PHYSICAL REVIEW B 71, 014112 ͑2005͒ model for silica. This approach was inspired by rigid-unit small atomic displacements. This procedure gives a nondi- models developed by Dove and co-workers to study soft vi- agonal force constant matrix, which is then diagonalized to 10 ͕ ͖ brational modes in silica. In our model, SiO4 units are obtain Di and x. The parameter K controls the energy cost treated as semirigid tetrahedra, while Si-O-Si angles are of a sudden elastic response, i.e., isotropic expansion or con- relatively flexible. Our spring-tetrahedron model is similar in ͕ ͖ traction of the solid. The parameters Li describe the cou- spirit to valence bond force fields developed for simulating 11,12 pling between normal mode displacements and lattice strain. zeolite framework vibrations. These force fields were de- This kind of potential energy function is a starting point for signed to accurately reproduce infrared spectra of zeolites. In activated rate theories,13 and has also been used to study contrast, the spring-tetrahedron model was constructed to displacive phase transitions in ferroelectric materials within provide a simple, qualitative picture of zeolite compression. the effective Hamiltonian approach.14–16 We show below that, despite its simplicity, the spring- We note that normal mode analysis applied to small mol- tetrahedron model reproduces cell volumes and predicts bulk moduli of several silica phases. In the end we find that the ecules usually involves mass weighting the force constant matrix, which upon diagonalization yields normal mode normal mode picture of compression and expansion, param- 17 etrized by DFT calculations and the spring-tetrahedron force constants with units of square frequency. In contrast, model, provides a powerful new approach for understanding we use fractional coordinates and omit the mass weighting the mechanical properties of nanostructured materials. for simplicity, so that Di has units of energy. The remainder of this paper is organized as follows: in The bulk modulus is expressed as V −1⌬ץ V −1ץ pץ Sec. II we present the normal mode theory of crystalline compression and expansion, deriving equations for the bulk B =−Vͩ ͪ =−Vͩ ͪ =−Vͩ ͪ , ͑2͒ pץ pץ Vץ modulus and thermal expansion coefficient. In Sec. III we n,T n,T n,T ⌬ describe the DFT and spring-tetrahedron calculations re- where V=V−V0 is the volume change relative to the equi- ͑ ͒ quired to parametrize the new theory. In Sec. IV we give the librium volume V0 at T=0 K. Because the energy in Eq. 1 results for a variety of materials including several silica poly- depends explicitly on volume through the strain l, we aver- morphs, aluminum metal, and an aluminosilica sodalite. In age over volume fluctuations in the npT ensemble to obtain Sec. V we discuss our results and offer concluding remarks. ͗V͘ and hence ͗⌬V͘, according to Finally, we present an Appendix to clarify our derivation of ϱ ץ ץ .the bulk modulus for anisotropic crystals ͗⌬V͘ =− ln Q͑n,p,T͒ =− ln ͵ dx dx dl p 1 ¯ nץ␤ pץ␤ II. THEORY −ϱ ϫ ͓ ␤͑ ⌬ ͔͒ ͑ ͒ Here we present an approach for studying mechanical exp − E0 + p V . 3 properties of solids based on classical normal mode analysis. In the limit of small strains for cubic lattices, we can substi- As discussed above, elastic responses are usually dominated ⌬ Х 2 tute V 3a0l. This approximation, which is consistent with by lower frequency vibrations, which are adequately treated the harmonic energy expression in Eq. ͑1͒, leads to a product with classical theory. The theory presented below is appli- of Gaussian integrals that can be evaluated successively over cable to the mechanical properties of general anisotropic lat- normal modes. After one such integration, we obtain tices. For pedagogical purposes, we begin by deriving an ϱ ץ expression for the bulk modulus in the simple case of a cubic ͗⌬ ͘ ͵ lattice. We then proceed to the more general case of aniso- V =− ln dx1 dxn−1 dl ¯ p ϱץ␤ tropic lattices. We end this section by deriving an expression − for the thermal expansion coefficient for cubic lattices. n−1 ϫ ͭ ␤͚ͫ ͑ 2 ͒ exp − Dixi + Lixil A. Harmonic approximation for cubic lattices i=1 The harmonic potential energy of the lattice, E0, can be L2 expressed in terms of small displacements of the lattice pa- ͩ n ͪ 2 ⌬ ͬͮ ͑ ͒ + K − l + p V . 4 3 4Dn rameter given by l=a−a0, where V0 =a0 is the equilibrium unit cell volume at T=0 K. The energy also depends on an As such, coupling to the nth normal mode effectively re- n-dimensional vector of fractional Cartesian displacements duces the sudden elastic force constant K to K−L2 /4D . Af- q=͑r−r ͒/a from an equilibrium geometry r . We use frac- n n 0 0 ter integrating over all degrees of freedom we obtain tional coordinates to describe vibrations so that varying the ␤ ץ strain l, while keeping q fixed, produces isotropic ͗⌬V͘ =− lnͭexpͫ ͑3a2p͒2ͬͮ, ͑5͒ p 4KЈ 0ץ␤ -compression/expansion of the system. By performing an or thogonal transformation to normal modes x=UTq,weex- press the harmonic lattice energy according to where n n L2 ͑ ͒ ͑ 2 ͒ 2 ͑ ͒ i ͑ ͒ E0 x1, ...,xn,l = ͚ Dixi + Lixil + Kl . 1 KЈ = K − ͚ . 6 i=1 i=1 4Di ͕ ͖ Here Di are the normal mode force constants, which can be By substituting this result into Eq. ͑2͒ and taking the zero computed by mapping the potential energy, or the forces, for pressure limit, we obtain

014112-2 NORMAL MODE APPROACH FOR PREDICTING THE… PHYSICAL REVIEW B 71, 014112 ͑2005͒

n 2 ͑ T Ј−1 ͒−1 ͑ ͒ 2 2 2 L B =2V0 c K c , 12 B = KЈ = ͩ Kͪ − ͩ ͚ i ͪ ϵ B − B . 9a 9a 9a 4D sud vib 0 0 0 i=1 i where the matrix KЈ is defined by ͑7͒ n Thus, the present theory shows that for cubic lattices, the LijLijЈ KЈ = K − ͚ . ͑13͒ bulk modulus can be expressed as a sudden elastic response jjЈ jjЈ i=1 4Di Bsud which is softened by an amount Bvib through coupling to vibrations. B is the bulk modulus in a fictitious situation in sud The sudden elastic response for anisotropic systems can be which the fractional coordinates of atoms are fixed. Coupling extracted by setting the L couplings to zero in Eq. ͑13͒, to vibrations softens this response by an amount B so that ij vib giving B =2V ͑cTK−1c͒−1. If the system were cubic, i.e., if a normal mode is either floppy, i.e., small D , or strongly sud 0 i with only one independent lattice parameter, then K, KЈ, and coupled, i.e., large Li, it decreases the bulk modulus to a greater extent. c reduce to scalars, and we recover the previous results in Eq. ͑7͒. B. Harmonic approximation for anisotropic lattices Silica polymorphs typically exhibit anisotropic lattices. If C. Anharmonic treatment of cubic lattices: Simple case there are m independent lattice parameters, e.g., m=3 for an Silica polymorphs often exhibit low frequency normal orthorhombic cell, the energy expression generalizes to modes due to flexible Si-O-Si angles, which give rise to n m interesting behavior such as negative thermal expansion ͑ ͒ ͚ ͩ 2 ͚ ͪ coefficients.18,19 Soft mode behavior is encountered in other E0 x1, ...,xn,l1, ...,lm = Dixi + Lijxilj 14 i=1 j=1 systems such as ferroelectric materials. To treat such phe- m nomena we consider anharmonic perturbations expressed in normal mode coordinates. In what follows, we consider the + ͚ K l l . ͑8͒ jjЈ j jЈ thermal expansion of cubic lattices; the generalization to an- Ј j,j =1 isotropic systems will be reported in a forthcoming publica- These normal modes are obtained from fractional Cartesian tion. We augment the harmonic energy E0 with a complete displacements whose x, y, and z components are scaled by set of third-order anharmonicities. To ensure the convergence their respective lattice parameters. In this subsection, the in- of partition functions, we also add fourth-order anharmonici- dex i runs over vibrational modes, while the indices ͑j, jЈ͒ ties; the corresponding fourth-order constants are set to zero run over lattice parameters. Using Eq. ͑8͒ to compute the after integration. For pedagogical purposes, we begin by npT partition function again leads to a product of Gaussian treating the simple case of only a single uncoupled normal integrals. However, in this case the anisotropic p⌬V term is mode. This will turn out to yield a result essentially identical slightly more complicated. We again focus on the limit of to classical Grüneisen theory.9 In the next section, we gener- small volume changes, and define an m-dimensional vector c alize this to the multi-dimensional coupled case for cubic so that the volume change takes the form lattices. The harmonic energy of one normal mode uncoupled m from cubic lattice strain is given by E ͑x,l͒=Dx2 +Kl2.In ⌬V = ͚ c l . ͑9͒ 0 j j this simple case, we augment E with the anharmonicity ⑀x2l, j=1 0 and also with fourth-order convergence factors. The energy For example, the volume of an orthorhombic unit cell is thus becomes given by ͑ ͒ ͑ ͒ ⑀ 2 ͑␭͑x͒ 4 ␭͑l͒ 4͒ ͑ ͒ ͑ ͒͑ ͒͑ ͒ ͑ ͒ E x,l = E0 x,l + x l + x + l . 14 V = a1 + l1 a2 + l2 a3 + l3 , 10 ͑ ͒ where a1 ,a2 ,a3 are the T=0 K lattice parameters, and To facilitate evaluating the partition function, we assume that ͑ ͒ l1 ,l2 ,l3 are the anisotropic strains. To lowest order in these the anharmonic effects are relatively small and expand the strains, the volume change is given by anharmonic contribution to the Boltzmann factor to first or- ⌬ der, yielding V = V − V0 = V − a1a2a3 = a2a3l1 + a3a1l2 + a1a2l3. ϱ ץ 11͒͑ ͗⌬V͘ =− ln ͵ dx dl exp͕− ␤͓E ͑x,l͒ + p⌬V͔͖ 0 ץ␤ ͒ ͑ ͒ ͑ Comparing Eq. 11 with Eq. 9 suggests that the vector c is p −ϱ given by c =a a , c =a a and c =a a . This approach can 1 2 3 2 1 3 3 1 2 ϫ͑ ␤⑀ 2 ͒ ͑ ͒ also be used for systems with nonorthogonal lattice vectors, 1− x l + ¯ . 15 as long as their angles remain fixed. In such a case, the vector c depends on the unit cell shape. The evaluation of the At this stage the fourth-order terms may be discarded. We ␣ ␥ ⌬ multidimensional Gaussian integral is presented in the Ap- add auxiliary terms x and l to E0 +p V, which allows us pendix. We obtain the following equation for the bulk modu- to write the partition function using derivatives of ␣ and ␥ by lus: applying the Leibniz integration rule according to

014112-3 ASTALA, AUERBACH, AND MONSON PHYSICAL REVIEW B 71, 014112 ͑2005͒

⑀ ϱ a 3ץ ץ ͗⌬V͘ = ͯ − lnͭͩ1+␤−2⑀ ͪ͵ dx dl ␥ =− 0 . ͑24͒ G ␥ץ2␣ץ ץ␤ p −ϱ 6D

ϫexp͕− ␤͓E ͑x,l͒ + p ⌬V + ␣x + ␥l͔͖ͮͯ . The equation for thermal expansion within Grüneisen theory 0 9 ␣,␥→0 is given by ͑16͒ ␥ C C = G V , ͑25͒ The Gaussian integral can be evaluated as was done for the VB cubic harmonic case discussed above, leading to where CV is the heat capacity at constant volume. Using the 3ץ ץ ͗⌬ ͘ ͯ ͭͩ ␤−2⑀ ͪ V = − ln 1+ classical Dulong–Petit value CV =nkB with n=1, and substi- ͒ ͑ ␥ץ2␣ץ pץ␤ tuting Eq. 7 for the uncoupled bulk modulus B=2K/9a0, ␤ ␣2 we recover the result in Eq. ͑20͒. The fact that we recovered ϫ ͫ ͑ 2 ␥͒2 ␤ ͬͮͯ ͑ ͒ exp 3a0p + + . 17 classical Grüneisen theory without coupling between vibra- 4K 4D ␣,␥→0 tion and strain, and with only one of many possible anhar- Differentiating with respect to ␣ and ␥ and setting ␣=0=␥ monicities, suggests that a more complete theory of thermal yields the equation of state: expansion is possible, which we develop below.

−9a4 −3a2⑀ ͗⌬V͘ = ͩ 0 ͪp + ͩ 0 ͪ␤−1. ͑18͒ D. Anharmonic treatment of cubic lattices: General case 2K 4KD We now consider thermal expansion for a cubic system The first term on the right-hand side of Eq. ͑18͒ yields the with n normal modes coupled to strain, and with the com- sudden elastic response found above for cubic lattices. The plete set of third-order anharmonicities. We find below that second term gives the thermal expansion coefficient, C, the presence of vibrations coupled to strain leads to qualita- which is defined by tively different thermal expansion behavior. The energy now takes the form V⌬ץ V 1ץ 1 C ϵ ͩ ͪ = ͩ ͪ . ͑19͒ n ץ ץ V T n,p V T n,p ͑ ͒ ͑ ͒ ⑀͑3͒ E x1, ...,xn,l = E0 x1, ...,xn,l + ͚ ijkxixjxk Evaluating the derivative and setting the normalizing volume i,j,k=1 3 to its zero-point value of a0 gives n n ͑ ͒ ͑ ͒ + ͚ ⑀ 2 x x l + ͚ ⑀ 1 x l2 + ⑀͑0͒l3 1 −3a2⑀k 3⑀k ij i j i i ͩ 0 B ͪ ͩ B ͪ ͑ ͒ i,j=1 i=1 C = 3 =− , 20 a0 4KD 4a0KD n ͩ ␭͑x͒ 4 ␭͑l͒ 4ͪ ͑ ͒ + ͚ i xi + l . 26 where kB is Boltzmann’s constant. We note that the sign of C i=1 is controlled by and opposite to the sign of the anharmonicity ⑀ parameter, . Here E is the harmonic potential energy in Eq. ͑1͒ with The result in Eq. ͑20͒ is analogous to the classical Grü- 0 9 vibrations coupled to strain. Once again, the anharmonic neisen theory of thermal expansion, which can be seen by contribution to the Boltzmann factor is expanded to first or- writing the energy as der, yielding E͑x,l͒ = ͑D + ⑀l͒x2 + Kl2 + ͑␭͑x͒x4 + ␭͑l͒l4͒. ͑21͒ ϱ ץ ͗⌬ ͘ ͵ ͕ ␤͓ ͑ ͒ Equation ͑21͒ clearly shows how anharmonicity can manifest V =− ln dx1 ...dxn dl exp − E0 x1, ...,xn,l p ϱץ␤ itself through normal mode frequencies that depend on vol- − ume. Indeed, the Grüneisen theory is cast in terms of the n n ␥ ⌬ ͔͖ͩ ␤ ͚ ⑀͑3͒ ␤ ͚ ⑀͑2͒ parameter G given by + p V 1− ijkxixjxk − ij xixjl i,j,k=1 i,j=1 V d␻ n ␥ =− . ͑22͒ ͑ ͒ ͑ ͒ G ␻ ␤͚ ⑀ 1 2 ␤⑀ 0 3 ͪ ͑ ͒ dV − i xil − l − ¯ . 27 i=1 Assigning the normal mode an arbitrary mass m leads to The fourth-order terms are included merely as convergence 2͑D + ⑀l͒ ␻ ͱ ͑ ͒ factors for the integral, and can be discarded once the expan- = , 23 ͚n ␣ ␥ m sion is made. We now add auxiliary terms i=1 ixi and l to the exponential, which again allows us to write the partition which gives the following Grüneisen parameter: function using derivatives according to

014112-4 NORMAL MODE APPROACH FOR PREDICTING THE… PHYSICAL REVIEW B 71, 014112 ͑2005͒

−1 3 n 3 tional to ␤ . These terms scale as a , while the four other 0 ץ ͒ ͑ ץ ͗⌬V͘ =− lnͩ1+␤−2 ͚ ⑀ 3 -terms on the right remain constant. The first term propor ␣ץ ␣ץ ␣ץ ijk ץ␤ p i,j,k=1 i j k ␤−1 ⑀͑2͒ tional to , controlled by ii , is responsible for shifting n 3 n 3 vibrational frequencies with cell volume, and as such is ץ ͒ ͑ ץ ͒ ͑ + ␤−2 ͚ ⑀ 2 + ␤−2͚ ⑀ 1 equivalent to classical Grüneisen theory for thermal expan- 2␥ץ ␣ץ i ␥ץ ␣ץ ␣ץ ij i,j=1 i j i=1 i sion. The remaining terms generalize classical Grüneisen theory for complex nanostructures that involve coupling be- ϱ 3ץ + ␤−2⑀͑0͒ ͪ͵ dx dx dl tween vibrations and strain. n ¯ 1 3␥ץ −ϱ E. Analysis of the coupling ϫ ͭ ␤ͫ ͑ ͒ ⌬ exp − E0 x1, ...,xn,l + p V As we shall show below, it turns out that most of the Li couplings vanish. The nonvanishing couplings still exert n strong effects on bulk moduli and thermal expansion coeffi- ͚ ␣ ␥ ͬͮ ͑ ͒ cients. Nonetheless, the fact that most couplings vanish is + ixi + l . 28 i=1 important for both computational efficiency and conceptual understanding. With relatively few couplings to compute, our After evaluating the Gaussian integrals, we obtain theory can be parametrized from a tractably small number of density functional theory calculations. Moreover, insights 3ץ n ץ ͗⌬ ͘ ͯ ͭͩ ␤−2 ͚ ⑀͑3͒ V = − ln 1+ into the compression and expansion of complex materials ␣ץ ␣ץ ␣ץ ijk ץ␤ p i,j,k=1 i j k may be gleaned by visualizing the small number of vibra- n 3 n 3 tions that actually couple to strain. Before describing the ץ ͒ ͑ ץ ͒ ͑ + ␤−2 ͚ ⑀ 2 + ␤−2͚ ⑀ 1 computational methods used to parametrize our theory, we 2␥ץ ␣ץ i ␥ץ ␣ץ ␣ץ ij i,j=1 i j i=1 i pause to reflect on why most of the couplings vanish. n 2 This result follows from crystallographic symmetries and L ␤ 3ץ + ␤−2⑀͑0͒ ͪexpͫ ͩ3a2p + ␥ − ͚ i ␣ ͪ the stability of a given phase with respect to small lattice Ј 0 i 3␥ץ 4K i=1 2Di deformations. To elaborate on this, we start with a minimum energy ͑T=0 K͒ structure. Next, we assume phase stability n ␣2 ␤͚ i ͬͮͯ ͑ ͒ so that the space group symmetry of a given phase is con- + , 29 served even if the lattice is deformed by a small strain ␦l.We i=1 4Di ␣ ␥→ i, 0 further impose the constraint that the deformation ␦l con- where KЈ is given by Eq. ͑6͒. Differentiating with respect to serves the unit cell shape. Using Eq. ͑1͒, we calculate the ␣ ␥ ␣ ␥ i and , setting i =0= , and assuming the limit of low force per normal mode in the deformed configuration as ץ temperature and pressure yields an equation of state of the ͗⌬ ͘ ͑ ͒ E form V =V0 −p/B+TC , where B is the bulk modulus and = L ␦l + O͑x ͒. ͑31͒ i i ץ C is the thermal expansion coefficient. These manipulations xi ͗⌬ ͘ yield the following expression for V : For the phase to be stable at T=0 K, this force must vanish n ͑ ͒ n ͑ ͒ unless the normal mode xi conserves the space group sym- 9a4 3a2 ⑀ 2 3L ⑀ 3 ͗⌬ ͘ 0 ␤−1 0 ͚ͩ ii ͚ j iij ͪ metry. Otherwise, the respective Li must also vanish! This V =− Јp − Ј − 2K 4K i=1 Di i,j=1 2Dj Di relatively strong condition causes most of the Li couplings to n n vanish. For example, in fcc and bcc metals, all such cou- 9a2 L L L ␤−1 0 ͩ⑀͑0͒ ⑀͑1͒ i ⑀͑2͒ i j plings vanish and the thermal expansion comes only from the − − ͚ + ͚ ͑ ͒ Ј2 i ij ⑀ 0 l3 anharmonicity and the frequency shift with volume as 4K i=1 2Di i,j=1 4DiDj controlled by ⑀͑2͒x2l terms. For such systems, the classical n ͑ ͒ L L L limit of Grüneisen theory is applicable. However, in zeolites − ͚ ⑀ 3 i j k ͪ. ͑30͒ ijk 8D D D and other complex nanostructures, which exhibit anharmonic i,j,k=1 i j k modes coupled to lattice strain, these additional terms may Equation ͑30͒ shows that the bulk modulus is not influ- have nontrivial and rather interesting contributions. enced by anharmonicity at this level of theory. The terms proportional to ␤−1 describe thermal expansion, which van- III. COMPUTATIONAL METHODS ishes in the limit of all ⑀→0. This expression is similar to that of the bulk modulus in Eq. ͑7͒, in that it contains a Here we describe the density functional theory calcula- “sudden anharmonic” term ⑀͑0͒ that is corrected by terms tions we have performed to both parametrize and test the arising from coupling between the lattice strain and normal normal mode theory. We then discuss the spring-tetrahedron modes. Our numerical calculations below reveal that the model used to generate normal modes of silica polymorphs. various terms in Eq. ͑30͒ exhibit different scaling behaviors with system size. In particular, in the limit of large system A. Density functional theory calculations sizes, i.e. many normal modes, the thermal expansion coef- We have performed density functional theory ͑DFT͒ cal- ficient becomes dominated by the first two terms propor- culations using plane wave sets, pseudopotentials, and

014112-5 ASTALA, AUERBACH, AND MONSON PHYSICAL REVIEW B 71, 014112 ͑2005͒ periodic supercells20 as implemented in the Vienna Ab-Initio lattice distortions become large enough so that anharmonic- Simulation Package ͑VASP͒.21–23 We used the local density ity with respect to lattice parameters, or symmetry-breaking approximation ͑LDA͒ of electron exchange-correlation based unit cell deformations become important. The Im¯3 m and on Ceperley–Alder data.24 In previous work, we5,25 and ¯ 26,27 I 4 3 m symmetries of Si-SOD allow one and two coupled others have shown that LDA reproduces unit cell struc- ¯ tures and volumes of silica polymorphs with surprising ac- vibrations, respectively. The P 4 3 n symmetry of Al-SOD curacy. Ion cores were represented by Vanderbilt-type ultra- gives four coupled vibrations, the two additional modes com- soft pseudopotentials available within VASP.28,29 ing from counterion motion and Si–O–Al bond asymmetry. Convergence tests were performed ͑data not shown͒ to en- To test our normal mode theory for the bulk modulus, we ␣ sure sufficient completeness of Brillouin zone sampling and calculated the bulk moduli of -cristobalite, Si-SOD, and plane wave basis sets. Below we give the results of these Al-SOD using the conventional technique of varying unit convergence tests for each system studied. To reduce the cell volume, relaxing atomic coordinates, and fitting ele- computer time, each structure was constrained to its putative ments of the elastic tensor to the resulting energies as out- space group symmetry during DFT calculations. lined in Ref. 5. In what follows, we denote these bulk moduli We have used DFT to analyze the normal modes of as Bdir because they arise directly from DFT-based optimiza- ␣-cristobalite, silica sodalite zeolite ͑Si-SOD͒ and an alumi- tions, in contrast with the bulk moduli Btheo, which are com- ͑ ͒ nosilica sodalite ͑Al-SOD͒. The sudden elastic force con- puted with our normal mode theory through Eqs. 7 and ͑ ͒ stants of silica zeolite structures CHA, LTA, and MOR30 12 . We used DFT to calculate direct bulk moduli of ␣ have also been computed with DFT. As a simple test of our -cristobalite, Si-SOD, and Al-SOD. Below we discuss a anharmonic theory, the thermal expansion of fcc Al metal spring-tetrahedron model, from which direct and theoretical was studied. And finally, we have used DFT to parametrize bulk moduli were also computed for various silica polymor- our theory for the thermal expansion of Al-SOD. phs.

1. ␣-Cristobalite, silica SOD, and aluminosilica SOD 2. Silica zeolites CHA, LTA, and MOR ͑ ͒ We studied ␣-cristobalite because of ample experimental We used DFT to calculate the force constants K or KjjЈ 30 data on its bulk modulus,31,32 and because its tetragonal lat- for silica zeolite structures CHA, LTA, and MOR, to com- tice provides a good test of our theory for anisotropic solids. pare their sudden elastic responses. We imposed the follow- Our interest in Si-SOD was motivated by our recently re- ing space group symmetries for CHA, R ¯3 2/m; for LTA, ported DFT calculations finding that Si-SOD’s bulk modulus Pm¯3 m; for MOR, Cmcm. These calculations were per- is extremely sensitive to symmetry constraints.5 Also, Al- formed using a plane wave cutoff of 420 eV. For CHA and SOD provides a good test of our anharmonic theory because LTA, we used a 2ϫ2ϫ2 Monkhorst–Pack grid, while for of experimental thermal expansion data on the composition MOR we used a 1ϫ1ϫ2 grid. High-silica LTA, which has 33,34 37 Na8Al6Si6O24Cl2. recently been synthesized, has a cubic structure, and hence ␣-cristobalite has a tetragonal unit cell, while Si-SOD and has only one independent lattice parameter and force con- Al-SOD both exhibit cubic unit cells. We studied I ¯4 3 m stant K. With rhombohedral CHA, we assumed that the lat- ␥ and Im¯3 m symmetries of Si-SOD. The transition between tice vector angle does not change. This is a reasonable ␥ the two phases was the subject of an earlier periodic DFT assumption since has an equilibrium value close to 90° ͑ ͒ study.35 In our DFT calculations, the latter is a saddle point 94° , which does not change significantly as the volume is 5 on the potential energy surface, unstable at T=0 K. We varied. With orthorhombic MOR, we made an isotropic as- found in our previous study that the bulk moduli for these sumption, i.e., that the lattice parameters change uniformly phases are 18 and 93 GPa, respectively.5 Below, we elucidate and that the sudden elastic response is described by a single this remarkable sensitivity of the bulk modulus to symmetry parameter K. This was necessary because of the computa- constraints with our normal mode theory. We studied the tional expense of these DFT calculations. P ¯4 3 n symmetry of Al-SOD. The composition 3. Calibrating anharmonic theory with fcc Al metal Na8Al6Si6O24Cl2 of this Al-SOD features a highly symmetric arrangement of Na and Cl counterions.33,34 The symmetry of As a simple test of our anharmonic theory, we computed ␣ -cristobalite was constrained to P 41 21 2. the thermal expansion coefficient of fcc Al metal. For the The Brillouin zone was sampled using 2ϫ2ϫ2 monatomic fcc lattice, none of the internal degrees of free- Monkhorst–Pack grids for the sodalites, and by using a 3 dom is coupled to strain, and hence all the Li /2Di-type terms ϫ3ϫ2 grid for ␣-cristobalite.36 For the sodalites, a plane in Eq. ͑30͒ vanish. As such, we are left with the ⑀͑0͒ and ␣ ͚ ⑀͑2͒ wave cutoff of 420 eV was used, while for -cristobalite, a i ii /Di terms. These were calculated by varying the lattice cutoff of 460 eV was used to reduce noise in the total energy parameter and estimating the force constant matrix for dif- due to small energy differences. ferent values of l. Because all atoms in this structure are As discussed above, the constants Li vanish for normal equivalent, and the structure is isotropic, the force constant mode displacements that break a given space group symme- matrix can be determined by displacing just a single atom in try. Symmetry constraints can thus be used to dramatically one Cartesian direction and computing the Hellmann– reduce the number of coupled modes that need to be consid- Feynman forces. System size effects were determined by ered. We note that this reduction does not necessarily hold if comparing results from supercells containing 4 and 32 at-

014112-6 NORMAL MODE APPROACH FOR PREDICTING THE… PHYSICAL REVIEW B 71, 014112 ͑2005͒ oms, to check convergence with respect to the wavelength ͑x,y,z͒. As such, this symmetry allows four coupled internal cutoff for normal modes. With the smaller cell, a 6ϫ6ϫ6 degrees of freedom. The anharmonic terms were calculated Monkhorst-Pack grid was used, while with the larger cell a as follows. Anharmonic constants for coupled modes were 4ϫ4ϫ4 grid was sufficient. We found essentially identical obtained by simultaneously displacing the lattice parameter results from these two system sizes, suggesting a minimal and normal modes, and fitting the energies to a multivariable influence from system size effects. Because the ultrasoft polynomial using least squares. The anharmonicities in pseudopotential for Al is significantly softer than that for ⌺ ⑀͑2͒ ⌺ ⑀͑3͒ i ii /Di and Lj /2Dj i iij /Di are less convenient to evalu- oxygen, the plane wave cutoff can be reduced to 140 eV. ate, because they require repeated calculations of the force The partial occupancies of Kohn-Sham eigenstates were constant matrix: for different values of the lattice parameter, treated by using a Gaussian smearing of 0.05 eV. and for displacements along the mode xj, respectively. For- tunately, Al-SOD possesses enough symmetry to make sev- 4. Parametrizing normal modes eral force constants interdependent. The full force constant matrix was evaluated with these symmetry properties as fol- The sudden elastic force constants K were computed by lows. We made ͑⌬,0,0͒ displacements for a single Na and a performing a series of DFT calculations on isotropically ex- single Cl atom; we made ͑⌬,0,0͒, ͑0,⌬,0͒, and ͑0,0,⌬͒ panded and contracted solids. Polynomials were then fitted to displacements for a single Al, Si, and O atom. These calcu- the energies obtained. We calculated the force constants Di lations suffice to construct the full force constant matrix, by first computing elements of the nondiagonal force con- Ј because all atoms in a given sublattice are symmetry equiva- stant matrix, Dij, in some convenient representation. This lent and the structure is isotropic. was achieved by displacing atoms by small amounts, and In ␣-cristobalite, Si atoms are at ͑u,u,0͒ positions while calculating the resulting Hellmann-Feynman forces. A prac- O atoms are at ͑x,y,z͒. As such, there are four internal de- tical way to generate displacements consistent with symme- grees of freedom. The tetragonal unit cell has two indepen- try is to vary the Wyckoff special positions of atoms.2 The dent lattice parameters. The coupling terms were calculated nondiagonal force constant matrix is then computed from the by making small displacements of the ith normal mode, and relation varying either the a lattice parameter, yielding Li1-type Ј ͑ ͒ terms, or the c parameter, yielding Li2-type terms. The terms Fi =−2͚ Dijqj, 32 j K11, K12, and K22 were calculated by varying the a and/or c lattice parameters while keeping atomic fractional coordi- ͕ ͖ where Fi is the Hellmann-Feynman force, and qj are Car- nates fixed. tesian atomic displacements. By diagonalizing DЈ, we obtain ͕ ͖ the normal modes x and force constants Di . The coupling B. Spring-tetrahedron model constants Li were obtained by simultaneously varying the One difficulty with computing the normal modes of silica lattice parameter and displacing atoms along normal mode polymorphs with DFT is its high computational cost. This is directions, and fitting a polynomial to the resulting DFT en- especially true when one wishes to relax symmetry con- ergies. straints. It is therefore worthwhile to pursue a simpler In the I ¯4 3 m phase of Si-SOD, O atoms occupy method that still captures the essential physics of elastic re- ͑x,x,y͒-type Wyckoff special positions while Si atoms are sponse in silica materials. Here we present such a model, located at ͑0.25, 0.5, 0.0͒-type positions. As such, there are based on semirigid SiO4 tetrahedra connected with flexible two internal degrees of freedom for this symmetry. The non- springs. We show that this “spring-tetrahedron” model can diagonal force constant matrix is calculated by making small indeed reproduce the elastic properties of a variety of silica displacements of O atoms of type ͑⌬,⌬,0͒ and ͑0,0,⌬͒.We polymorphs. We used this method to study the normal modes found that stable convergence is obtained for these force and bulk moduli of silica zeolites SOD, CHA, and LTA.30 constants using ⌬=10−2 Å. Diagonalizing the force constant The bulk modulus and orthorhombic-to-low symmetry phase matrix leads to two normal modes: one associated with rela- transition of silica MFI was also studied. As discussed in the Introduction, our approach was in- tive rotations of several SiO4 tetrahedra, and one with SiO4 unit deformation. We thus expect the force constants for spired by rigid-unit models developed by Dove and co- these normal modes to be significantly different. In the workers to study soft vibrational modes in silica.10 From a Im¯3 m phase of Si-SOD, O atoms occupy ͑x,x,0.5͒-type topological perspective, silica polymorphs can be viewed as networks of corner-sharing SiO tetrahedra. The rigid-unit positions, thus giving only one internal degree of freedom, 4 model assumes rigid tetrahedra connected with two-body which turns out to be associated with SiO unit deformation. 4 springs.10 Using this rigid-unit approach, we have found that The fact that relative rotations of SiO4 tetrahedra are sym- ¯ some silica polymorphs became unstable at nonzero pres- metry forbidden in the Im3 m phase of Si-SOD already sures because of barrierless deformation pathways. Further- explains qualitatively why its bulk modulus is more than five more, the bulk moduli of some zeolites are significantly times higher than that of I ¯4 3 m Si-SOD.5 Below we show overestimated with this approach, and calculating full normal that our normal mode theory accounts for this fact semiquan- mode spectra becomes impossible because deforming tetra- titatively as well. hedra is forbidden. In Al-SOD, Cl atoms are at ͑0,0,0͒-type positions, Na at A simple remedy is to allow tetrahedra to become some- ͑u,u,u͒,Alat͑0.25, 0.5, 0͒,Siat͑0.25, 0, 0.5͒,andOat what flexible using two-body springs, and to add a three-

014112-7 ASTALA, AUERBACH, AND MONSON PHYSICAL REVIEW B 71, 014112 ͑2005͒

tion, we used the spring-tetrahedron model to generate the normal modes of SOD, CHA, and LTA, for use in decom- posing their bulk moduli into sudden and vibrational contri- butions. These constitute predictions of properties that have not yet been measured. Unit cell shapes were held fixed dur- ing these calculations: tetragonal for ␣-cristobalite, cubic for SOD and LTA, and rhombohedral for CHA. No additional symmetry was imposed on atomic coordinates. To avoid a 2 FIG. 1. A schematic showing the spring-tetrahedron system and divergence when calculating terms of the form Li /4Di,we its response to strain. removed the three normal modes corresponding to the center-of-mass translation, whose normal force constants vanish. body term that penalizes the bending of Si-O-Si angles ͑Fig. ͒ To further test the spring-tetrahedron model, we computed 1 , which we found restores mechanical stability. Although the bulk modulus of silica MFI zeolite. With MFI, an addi- such an energy function with two-body and three-body 11,12 tional complexity stems from a structural phase transition springs begins to resemble valence bond force fields, our between monoclinic and orthorhombic phases.38 Because our approach is substantially different. Indeed, our spring- normal mode theory assumes that unit cell shapes remain tetrahedron model only explicitly accounts for the positions preserved despite strains and vibrations, this theory cannot of oxygen atoms in a given silica structure. For a unit cell be applied to study the transition of MFI between different with no oxygens, the spring-tetrahedron energy is given by unit cell shapes. Instead, we performed direct optimizations n to determine the bulk modulus. However, local optimization 1 o k V͑r , ...,r ͒ = ͚ ͚ S ͉͑r − r ͉ − r ͒2 algorithms yielded unsatisfactory results because of a rugged 1 no j k 0 2 i=1 k෈nn 2 energy landscape. As such, we annealed the structure using j 39 n Metropolis Monte Carlo simulations. The bulk modulus o k ͚ A ͑ ␪ ␪ ͒2 ͑ ͒ was extracted by adding a pV term to the energy, and map- + cos i − cos 0 , 33 ping the equilibrium volume as a function of pressure. All i=1 2 internal coordinates and lattice vectors were allowed to vary. We show below that the spring-tetrahedron model exhibits a where rj is the three-dimensional location of the jth oxygen, phase transition for MFI from orthorhombic to lower sym- and “nnj” are its six nearest neighbor oxygens. The factor of 1 metry at small negative pressures. 2 in the first term corrects for double counting. The param- eter r0 is the typical distance between oxygens in a given ͑ ͒ SiO4 tetrahedron. The second term in Eq. 33 applies a IV. RESULTS ␪ spring to each Si-Oi -Si angle i, i.e., to each shared vertex of adjacent tetrahedra ͑Fig. 1͒. Without explicit Si locations, Here we present the numerical results of our study on the ␪ we compute i as the angle formed by the ith oxygen and the mechanical properties of nanostructured silica. We begin by centers of mass of the two tetrahedra it joins. The spring- comparing sudden elastic responses from various silica poly- tetrahedron model thus replaces rigid tetrahedra with semi- morphs calculated from DFT. We then present the best-fit rigid ones connected to each other by flexible springs, as parameters and predictions of the spring-tetrahedron model ͑ ␪ ͒ controlled by the parameters kS ,kA ,r0 , 0 . for a comparison with DFT data. Next we discuss the normal The spring-tetrahedron parameters were fitted to DFT- modes of Si-SOD generated by DFT and spring-tetrahedron LDA data for Si-SOD ͑I ¯4 3 m͒, which serves as a bridge energies, as well as the normal modes of Al-SOD and ␣-cristobalite obtained from DFT. We then apply the spring- between dense and zeolitic silica polymorphs. The parameter tetrahedron model to explore the bulk moduli of CHA, LTA, r contains much of the chemistry in Si-O bonding. Using a 0 and MFI zeolites. We close this section with the thermal Si-O bond length of 1.6 Å and an O-Si-O bond angle of ͱ ϫ expansion coefficients of fcc Al metal and Al-SOD zeolite 109.47° gives r0 = 8/3 1.6 Å=2.61 Å. The reference Si ␪ computed from DFT. -O-Si angle 0 was set to 155°, a value close to that found in Si-SOD ͑I ¯4 3 m͒, ␣-cristobalite, and in relaxed silica chain 25 A. Sudden elastic responses from DFT polymers. The force constant kS uniquely determines the sudden elastic response K, thereby allowing kS to be fitted to Sudden elastic response constants computed from DFT the energy dependence of isotropic volume changes. The an- for various silica polymorphs are shown in Table I. For cubic gular force constant kA was fitted to the direct bulk modulus lattices, these are converted to units of GPa according to ¯ 5 ͑ ͒ of Si-SOD ͑I 4 3 m͒ calculated from DFT-LDA data. This Bsud= 2/9a0 K. For noncubic lattices, Bsud is computed from ͑ ͒ was achieved by calculating the direct bulk modulus of Si- Eq. 12 by setting the couplings Lij to zero. The correspond- SOD using spring-tetrahedron potential energies, and vary- ing direct bulk moduli are also shown in Table I. While the ing kA until this bulk modulus matched the DFT value. bulk moduli in Table I show a significant variation from one We benchmarked the spring-tetrahedron model by calcu- polymorph to another, the sudden elastic response constants lating equilibrium volumes and bulk moduli of ␣-cristobalite exhibit remarkable uniformity. Indeed, the spread in Bdir val- and silica zeolite structures SOD, CHA, and LTA. In addi- ues amounts to 60% of its average, whereas that for Bsud is

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TABLE I. Sudden elastic responses and direct bulk moduli TABLE III. Direct bulk moduli ͑GPa͒ and equilibrium unit cell ͑GPa͒ of various silica polymorphs computed with DFT. Sudden volumes ͑Å3͒ computed using DFT and the spring-tetrahedron ͑ ͒ ͑ ͒ elastic responses given as Bsud= 2/9a0 K for cubic systems SOD model STM . ␣ and LTA. For anisotropic lattices -cristobalite and CHA, Bsud is ͑ ͒ computed from Eq. 12 setting couplings Lij to zero. For MOR, Bdir, DFT Bdir, STM V0,DFT V0, STM Bsud is approximated by changing the lattice parameters uniformly and assuming a scalar K. SOD I ¯4 3 m 18 18 57 57 CHA 68 67 65 66 Bsud Bdir LTA 46 54 69 70 ␣-cristobalite 8 13 46 44 SOD I ¯4 3 m 118 18 SOD Im¯3 m 112 93 LTA 92 46 tetrahedron model with no imposed symmetry, which yields CHA 103 68 a pattern of couplings essentially identical to the DFT results ␣ -cristobalite 76 8 for Si-SOD ͑I ¯4 3 m͒. An inspection of Table IV shows that MOR 117 57 the spring-tetrahedron model performs well in reproducing the results of DFT calculations. High symmetry Si-SOD ex- hibits a normal vibration corresponding to the deformation of only 15%. This uniformity among sudden elastic responses individual SiO4 tetrahedra, denoted mode 1. Low symmetry reflects the intrinsic rigidity of SiO4 tetrahedra shared by all Si-SOD allows this mode as well as another, denoted mode silica polymorphs. The unformity of sudden elastic response 2, corresponding to rotations of adjacent SiO4 tetrahedra constants is well reproduced by the spring-tetrahedron relative to each other. These two vibrations are shown sche- model. matically in Fig. 2. The sudden elastic response constants of the two symme- B. Spring-tetrahedron model parameters tries of Si-SOD are quite similar, as discussed above. The extent to which the bulk modulus is decreased from mode 1 The spring-tetrahedron force constants kS and kA in Eq. ͑ ͒ is also quite similar between the forms of Si-SOD, as dem- 33 were fitted to the sudden elastic force constant and bulk 2 ͑ ¯ ͒ onstrated by the values of L1 /4D1 in Table IV. However, modulus of Si-SOD I 4 3 m . As shown in Table I, these because mode 2 involves relative rotations of SiO tetrahe- take the values 118 and 18 GPa, respectively. The resulting 4 dra, the force constant D2 is much smaller than D1, leading best-fit parameters are given in Table II. To test the spring- to a much greater diminution of the bulk modulus from mode tetrahedron model, we computed the bulk moduli and unit 2 in Si-SOD ͑I ¯4 3 m͒. This explains why the bulk moduli ͑ ¯ ͒ cell volumes of Si-SOD I 4 3 m , CHA, LTA, and are so different, even though these two forms of Si-SOD ␣ -cristobalite. Table III shows the spring-tetrahedron results have nearly identical atomic configurations. alongside the corresponding DFT data. The agreement is ex- cellent, considering the simplicity of the spring-tetrahedron potential function. For the dense polymorph ␣-cristobalite, TABLE IV. Sudden elastic responses and vibrational coupling the unit cell volume agrees well while the bulk modulus is terms ͑GPa͒ for two symmetries of Si-SOD, from DFT and the somewhat overestimated. Spring-tetrahedron energy differ- spring-tetrahedron model ͑STM͒. High symmetry ͑Im¯3 m͒ STM ences between different silica polymorphs are small ͑data not calculations were not performed. Results are multiplied by 2/9a0 to shown͒, which is in qualitative agreement with experimental obtain GPa units. Direct and theoretical bulk moduli, Bdir and Btheo, data40 and DFT calculations.5 are also shown for comparison.

¯ C. Normal modes of Si-SOD from DFT and the spring- Si-SOD I 4 3 m DFT STM tetrahedron model K 118 112 2 The normal mode analyses of Si-SOD from DFT with L1 /4D1 16 14 ͑ ¯ ͒ ͑ ¯ ͒ 2 lower symmetry I 4 3 m and higher symmetry Im3 m L2 /4D2 79 80 are presented in Table IV. Also shown in Table IV is the normal mode analysis of Si-SOD based on the spring- Btheo 23 18 Bdir 18 18

TABLE II. Parameters for the spring-tetrahedron model. Si-SOD Im¯3 m DFT

2 K 112 kS 8.82 eV/Å 2 L1 /4D1 14 kA 2.35 eV r0 2.61 Å Btheo 98 ␪ 0 155° Bdir 93

014112-9 ASTALA, AUERBACH, AND MONSON PHYSICAL REVIEW B 71, 014112 ͑2005͒

TABLE VI. Sudden elastic force constants KjjЈ and coupling ͑ 2͒ ␣ terms LijLijЈ/4Di eV/Å for -cristobalite from DFT. Direct and theoretical bulk moduli, Bdir and Btheo in GPa, are also shown for comparison with experimental values Bexp.

K11 14.936 K12 3.699 K22 3.360 2 L11/4D1 11.271 L11L12/4D1 3.822 2 L12/4D1 1.292 2 L21/4D2 0.210 L21L22/4D2 0.125 2 L22/4D2 0.074 2 L31/4D3 0.595 L31L32/4D3 0.514 2 L32/4D3 0.382 2 L41/4D4 0.784 L41L42/4D4 −0.708 2 L42/4D4 0.640

Btheo 11 Bdir 8 ͑ ͒ Bexp Ref. 31 16.0 ͑ ͒ Bexp Ref. 32 11.5

and counterions in the structure, the couplings in Al-SOD are FIG. 2. Schematic representation of the soft ͑up͒ and hard split into four vibrations coupled to strain. As with the other ͑down͒ normal modes in Si-SOD, in which tetrahedra represent solids discussed above, the sudden elastic response constant SiO4 units. In the upper picture the six ring is viewed along the for Al-SOD is near 100 GPa. Normal vibrations 1 and 3 ͓111͔ axis, while in the lower picture the four ring is viewed down decrease the bulk modulus by about 40%, with another 10% the ͓100͔ axis. Normal mode displacement vectors are projected decrease coming from vibrations 2 and 4. onto corresponding planes. In contrast to Si-SOD, Al-SOD exhibits stronger mixing between deformations and relative rotations of tetrahedra. In ͑ ͒ Also shown in Table IV are bulk moduli computed from the softest mode mode 1 , the motion of framework O atoms direct optimizations and normal mode theory. These are in and Na counterions is in phase, while in the next more rigid ͑ ͒ excellent agreement, signaling the first numerical success of mode mode 2 they are in antiphase. The third softest mode ͑ ͒ our normal mode theory of elastic response. mode 3 consists of a mixture of rotations and deformations, while the fourth and most rigid mode ͑mode 4͒ has a strong component of O atom displacement along the Si–Al internu- D. Normal modes of Al-SOD and ␣-cristobalite from DFT clear axis. These modes couple to strain in such a way that The normal mode analysis of Al-SOD ͑Na Al Si O Cl ͒ modes 1 and 3 exert the greatest diminution on the bulk 8 6 6 24 2 modulus. In contrast to Si-SOD, none of these modes is very from DFT is presented in Table V. Due to the presence of Al soft, leading to a bulk modulus of 55 GPa. This likely arises from counterions in Al-SOD stiffening of the relative rota- TABLE V. Sudden elastic response and vibrational coupling tions of SiO tetrahedra. ͑ ͒ 4 terms GPa for Al-SOD from DFT. Results are multiplied by 2/9a0 Studying ␣-cristobalite allows us to test our normal mode to obtain GPa units. The theoretical bulk modulus is also shown. theory on an anisotropic structure. However, the effects of individual modes on the bulk modulus is blurred because the ͑ ͒ K 107 matrix inverse in Eq. 12 mixes the terms LijLijЈ/4Di among L2 /4D 20 the coupled modes i=1, ... ,4. Due to the tetragonal unit cell 1 1 ␣ L2 /4D 5 of -cristobalite, varying the lattice parameter a requires 2 2 changing b by the same amount. The total effect on the unit L2 /4D 23 3 3 cell volume is greater compared to that from varying c alone, L2 /4D 4 4 4 making the force constants associated with the lattice param- eter a larger. As shown in Table VI, the L L /4D terms for Btheo 55 ij ijЈ i the softest mode ͑mode 1͒ are again the largest. This mode

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TABLE VII. Sudden elastic force constants and coupling terms TABLE VIII. Sudden elastic force constants and coupling terms ͑GPa͒ for silica CHA from the spring-tetrahedron model ͑STM͒. ͑GPa͒ for silica LTA from the spring-tetrahedron model ͑STM͒. ͑ ͒ Each term is converted to GPa units using Eq. 12 , setting either Results are multiplied by 2/9a0 to obtain GPa units. Direct and 2 KЈ=K or KЈ=Li /4Di, and setting all other terms to zero. Direct and theoretical bulk moduli from STM potential energies, Bdir and Btheo, theoretical bulk moduli from STM potential energies are also shown are also shown for comparison. for a comparison. K 92 K 97 2 L1 /4D1 14 L2 /4D 2 2 1 1 L2 /4D2 8 L2 /4D 2 2 2 2 L3 /4D3 5 L2 /4D 1 2 3 3 L4 /4D4 11 2 L4 /4D4 8 2 Btheo 54 L5 /4D5 1 2 Bdir 54 L6 /4D6 16

Btheo 67 ¯ Bdir 67 by vibrations as much as was found for Si-SOD ͑I 4 3 m͒ ͑see Table IV͒. The direct and normal mode bulk moduli of CHA and LTA are in perfect agreement, which might be consists mostly of rigid-body rotations and translations of expected when applying a harmonic theory to a spring po- SiO4 tetrahedra. The remaining three modes consist of SiO4 tential. tetrahedral deformations of increasing magnitude from modes 2 to 4. Also shown in Table VI are the direct and normal mode F. Mechanical properties of MFI from the spring-tetrahedron bulk moduli of ␣-cristobalite from DFT. These are 8 and model 11 GPa, respectively, diminished from a sudden elastic re- We investigated the applicability of the spring-tetrahedron sponse of 76 GPa. Experimental values of ␣-cristobalite’s model to study the elastic properties and phase behavior of bulk modulus are 16.0 GPa31 and 11.5 GPa,32 which are in silica MFI ͑silicalite͒. We annealed the structure at various reasonable agreement with our direct and theoretical results, pressures to obtain the T=0 K equilibrium volume as a func- considering the spread in experimental values. This level of tion of pressure. The behavior of the unit cell volume versus agreement bodes well for our normal mode theory of aniso- pressure is shown in Fig. 3. Under compression at positive tropic lattices. pressures, the bulk modulus takes the value 12–13 GPa. A unit cell slightly distorted from an orthorhombic shape is obtained. Due to very small distortions in the lattice vector E. Normal modes of CHA and LTA from the spring- angles, the exact symmetry of this phase is difficult to iden- tetrahedron model tify. We also considered the hypothetical possibility of nega- The normal modes of silica zeolite structures SOD, CHA, tive pressures, i.e., when the structure is under tension. Such and LTA were analyzed with the spring-tetrahedron model by a situation may effectively occur when the lattice is loaded keeping the respective unit cell shapes fixed, but without with guest molecules, which pull inward into the pores additional symmetry constraints. Despite this relaxation of through host–guest attractions. In this case the bulk modulus symmetry, most of the Li couplings were still found to van- takes the value 65±5 GPa. At the transition point the unit ish. This is expected to occur for crystal structures whose cell deforms from orthorhombic to a lower symmetry phase. symmetry is nearly preserved by the spring-tetrahedron model, because the symmetric structure is close to a local minimum of the spring-tetrahedron potential. Such a result was found above for Si-SOD ͑I ¯4 3 m͒, but would not be expected for the higher symmetry Im¯3 m structure, because this is a saddle point of the spring-tetrahedron potential.5 The spring-tetrahedron model applied to CHA predicts that six vibrations are coupled to lattice strain, as shown in Table VII. Starting with a sudden elastic response constant of 97 GPa, only two modes significantly influence the elastic response. When applied to LTA, the spring-tetrahedron model identifies four modes coupled to strain. Each of these modes diminishes the bulk modulus significantly from its sudden elastic value of 92 GPa, as shown in Table VIII. It is FIG. 3. Behavior of the unit cell volume of Si-MFI as the struc- remarkable that, even though SOD and LTA are both built ture is annealed at different pressures. The two lines highlight the from SOD cages, the elastic response of LTA is not softened transition from low symmetry to an orthorhombic structure.

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Our previous DFT study on orthorhombic MFI gave a bulk modulus of 41 GPa.5 Although the predictions of the spring- tetrahedron model are not too far from this value, a much more careful analysis is required before the elastic properties and phase behavior of MFI are well understood.

G. Thermal expansion of fcc Al metal from DFT We now turn our attention to the thermal expansion coef- ficient, beginning with a test of our theory on fcc Al metal with parameters computed by DFT. We first calibrate the accuracy of DFT-LDA with ultrasoft pseudopotentials for fcc Al, by discussing its lattice parameter and bulk modulus. Our calculated lattice parameter is 3.98 Å, which is in good FIG. 4. Comparing experimental and theoretical data on the agreement with the experimental room temperature value of temperature dependence of Al-SOD unit cell volume. Experimental points are circles ͑Ref. 33͒ and squares ͑Ref. 34͒. Theoretical points 4.05 Å, and in excellent agreement with a previous DFT- are computed from Eq. ͑30͒ parametrized by DFT. LDA result of 3.97 Å obtained with norm-conserving pseudopotentials.41,42 Our calculated bulk modulus for fcc Al with a 4-atom unit cell is 84 GPa; with a 32-atom unit cell next bigger cubic cell for Al-SOD contains over 300 atoms. we obtain 81 GPa. This agrees well with the experimental However, a reasonably converged result for Al metal was value of 75.9 GPa, and with the previous DFT-LDA value of obtained using a supercell that is much smaller than the one 79.4 GPa.41,42 for Al-SOD, suggesting that our results for Al-SOD are We obtain a thermal expansion coefficient C=62 likely valid. In addition, the fact that our thermal expansion ϫ10−6 K−1 for fcc Al using a 4-atom supercell. This agrees results for Al-SOD agree well with experiments further sug- well with the experimental room-temperature value of 65 gests that the accuracy of our computations has not been ϫ10−6 K−1, and with earlier DFT results based on Debye- compromised by finite size effects. ϫ −6 −1 Grüneisen theory.9,42 Effects from quantized vibrations are We note that only about 8 10 K of the thermal ex- ⌺ ⑀͑2͒ negligible at room temperature for fcc Al, because the ther- pansion coefficient stems from the term i ii /Di, while ⌺n ͑ ͒⑀͑3͒ mal expansion coefficient and specific heat have already ij Lj /2Dj iij /Di accounts for the remaining 29 reached plateaus at this temperature. Using the 32-atom su- ϫ10−6 K−1. This suggests that Grüneisen theory would only percell, we obtain a value of 67ϫ10−6 K−1, indicating that account for 22% of the thermal expansion coefficient of convergence with respect to normal mode wavelength cutoff Al-SOD, because it ignores the coupling between strain and has been reached. With the smaller supercell, more than 90% anharmonic vibrations. of the thermal expansion coefficient stems from the ⌺n⑀͑2͒ V. DISCUSSION AND CONCLUDING REMARKS i ii /Di term. In the case of the larger supercell, this term contributes more than 99%, confirming the scaling behavior We have presented a method for analyzing the mechanical ͑ ͒ discussed above for Eq. 30 . properties of solids, based on normal modes and their cou- pling to lattice strains. This method was used to study elastic H. Thermal Expansion of Al-SOD from DFT compression and thermal expansion of silica zeolites and re- lated materials, with parameters calculated from density To explore how coupling between vibrations and strain functional theory ͑DFT͒ calculations. We have found in gen- influences thermal expansion, we now consider Al-SOD eral that the bulk modulus can be divided into two contribu- ͑ ͒ Na8Al6Si6O24Cl2 with anharmonicities computed from tions: a positive term arising from compression without in- DFT. Though in principle the thermal expansion of Al-SOD ternal relaxation, which we call the sudden elastic response, depends on several anharmonic terms, in practice the greatest and a negative term from coupling between compression and ⌺n⑀͑2͒ contributions come from terms of the form i ii /Di and internal vibrational modes. For silica polymorphs, the sudden ⌺n ͑ ͒⑀͑3͒ ͑ ͒ ij Lj /2Dj iij /Di in Eq. 30 . In the temperature range elastic response term varies little among the phases studied, 250–750 K, very good agreement with experimental volume taking values near 100 GPa because of the intrinsic rigidity data is seen in Fig. 4. Our calculated value of the thermal of SiO4 tetrahedra. In contrast, the latter term varies strongly expansion coefficient is 37ϫ10−6 K−1, which compares well from one polymorph to the next, because each polymorph with the experimental values at 500 K of 39.5ϫ10−6 K−1,33 exhibits different symmetry constraints on internal vibrations and 33.9ϫ10−6 K−1.34 The systematic underestimation of and their couplings to lattice strains. Numerical results of equilibrium volume by 3–4% evident in Fig. 4 is character- this approach agree well with experiment for the dense silica istic of the LDA functional. polymorph ␣-cristobalite. The normal mode theory gives re- We have not systematically tested the convergence with sults that agree well with previous DFT calculations for respect to system size for Al-SOD, and therefore should put silica zeolite structures SOD, CHA, and LTA. We hope that fairly large error bars on these thermal expansion results. these developments will spawn new experiments to test our Additional calculations with larger Al-SOD systems would predictions on the mechanical properties of nanostructured be nearly computationally impossible at present, because the silica.

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We have found that, for a given structure, only a few frequency vibrations to soften the elastic response ͑e.g., normal modes influence the bulk modulus. This follows from ␣-cristobalite and SOD͒, and materials with network and crystallographic symmetries and the stability of a given symmetry constraints that preclude such softening ͑e.g., phase with respect to small lattice deformations. The modes CHA and LTA͒. When applying the spring-tetrahedron model that do couple exert strong effects on bulk moduli and ther- to silica MFI zeolite, we observed a transition between these mal expansion coefficients. Nonetheless, the fact that most two regimes involving a transformation from low-symmetry couplings vanish is important for both computational effi- to an orthorhombic phase. ciency and conceptual understanding. With relatively few In the end we find that the normal mode picture of com- couplings to compute, our theory can be parametrized from a pression and expansion, parametrized by DFT calculations tractably small number of DFT calculations. In addition, in- and the spring-tetrahedron model, provides a powerful new sights into the compression and expansion of complex mate- approach for understanding the mechanical properties of rials may be gleaned by visualizing the small number of nanostructured materials. Despite this progress, much future vibrations that actually couple to strain. work is suggested by the present study. We plan to develop When comparing the properties of normal modes in Si- SOD and Al-SOD, we found that the more complex chemical the theory of thermal expansion for anisotropic solids. We composition, i.e., going from Si-O to Si-O-Al-Na-Cl, will also pursue a quantum version of our normal mode tends to mix the vibrations. The presence of ionized species theory. Treating strain as a slow mode within the adiabatic Na and Cl appears to stiffen the vibrations. For a given com- approximation should yield a simple theory for the bulk position, structures with lower symmetry generally permit a modulus. However, developing a quantum theory for thermal larger number of coupled modes, but in such structures the expansion remains quite challenging. Finally, we plan to mixing between soft Si-O-Si bending modes and hard O study the phase transition observed in MFI in more detail to -Si-O bend and Si-O stretch modes may reduce the effect of determine the connection between our calculations and ex- such couplings. perimental data. ͑ ͒ We augmented our harmonic theory with a perturbative Note added in proof. Equation 7 also shows that our treatment of anharmonic terms, allowing the calculation of theory predicts a bulk modulus that does not depend on tem- the thermal expansion coefficient. The resulting expression perature. Even when accounting for anisotropy and anharmo- provides a generalization of classical Grüneisen theory, nicity, the resulting formulas for the bulk modulus do not wherein phonon frequencies are calculated at different vol- depend on temperature either. Experimental data for bulk umes to obtain the free energy as a function of volume. We moduli often exhibit very weak temperature dependencies. found that Grüneisen theory ignores bilinear couplings be- Indeed, over the temperature range 300–1800 K, bulk moduli ␣ tween lattice strain and anharmonic vibrations. We have ap- of -Al2O3 and SiC only change by 12.5% and 9.4%, 43 plied our approach to the thermal expansion of fcc aluminum respectively. metal and an aluminosilica sodalite zeolite. In the former case, we found that our theory reproduces the success of ACKNOWLEDGMENTS classical Grüneisen theory. In the latter case, our approach gives excellent agreement with experiment, while Grüneisen The authors thank the National Science Foundation for theory accounts for only 22% of the thermal expansion co- their generous support through a Nanoscale Interdisciplinary efficient. This result shows that in general, including cou- Research Team ͑NIRT͒ grant ͑CTS-0103010͒. plings between vibrations and strain will be important for predictive models of thermal expansion. To facilitate the parametrization of this normal mode APPENDIX: DERIVATION OF THE ANISOTROPIC BULK model, we constructed a simplified classical spring- MODULUS, EQ. (12) tetrahedron model for silica. This is especially useful when Beginning with Eq. ͑8͒, we write the equilibrium volume one wishes to relax symmetry constraints. Our spring- change singling out the nth normal mode according to tetrahedron model is based on semirigid SiO4 tetrahedra con- ϱ ץ nected with flexible springs. After fitting to properties of silica sodalite determined from DFT, this model reproduces ͗⌬V͘ =− ln͵ dx dx dl dl n 1 ¯ m ¯ 1 ץ␤ experimental cell volumes and predicts bulk moduli of p −ϱ ␣ -cristobalite and silica zeolites CHA, LTA, and MFI. The n−1 m spring-tetrahedron model also captures the fact that the sud- ϫexpͭ− ␤͚ͫ ͩD x2 + ͚ L x l ͪ i i ij i j den elastic response varies little from one silica polymorph i=1 j=1 to the next. The spring-tetrahedron model was used to analyze the m m + ͚ K l l + p͚ c l ͬͮ normal modes of silica zeolite structures SOD, CHA, and jjЈ j jЈ j j LTA. When comparing with DFT calculations on SOD, we j,jЈ=1 j=1 found that the spring-tetrahedron model generates the correct m normal modes, and perhaps more remarkably, the model ϫ ͭ ␤ͫͩ 2 ͚ ͪͬͮ ͑ ͒ exp − Dnxn + Lnjxnlj . A1 quantitatively reproduces the couplings of these modes to j=1 lattice strain. We have found that silica polymorphs can be divided in two categories: structures that allow low- After integrating out the nth normal mode we obtain

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ϱ ץ ϱ ץ ͗⌬V͘ =− ln͵ dx dx dl dl ͗⌬V͘ =− ln͵ dl dl m ¯ 1 ץ␤ n−1 1 ¯ m ¯ 1 ץ␤ p −ϱ p −ϱ n−1 m m m m ϫexp − ␤ ͚ ͩD x2 + ͚ L x l ͪ + ͚ K l l ϫexpͫ− ␤ͩ ͚ KЈ l l + p͚ c l ͪͬ, ͑A3͒ ͫ j j j j ͭ Ј i i ij i j jjЈ j jЈ jjЈ i=1 j=1 j,jЈ=1 j,jЈ=1 j=1 m m 2 where + p͚ c l ͬͮexpͫ͑␤/4D ͚͒ͩ L l ͪ ͬ. ͑A2͒ j j n nj j n L L j=1 j=1 Ј ij ijЈ ͑ ͒ KjjЈ = KjjЈ − ͚ . A4 i=1 4Di This can be evaluated using standard techniques of multidi- In a fashion analogous to the cubic case, coupling to mensional Gaussian integration, yielding p2␤ ץ mode n effectively reduces the sudden elastice force constants from K to K −L L /4D . However, in ͗⌬V͘ =− ln expͫ ͑cTKЈ−1c͒ͬ. ͑A5͒ p 4ץ␤ jjЈ jjЈ nj njЈ n the anisotropic case the Lnj couplings become mixed. After integrating over the remaining normal modes, we By taking the derivatives and inserting the result into Eq. ͑2͒, obtain we obtain Eq. ͑12͒.

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