Quantum, classical, and multi-scale simulation of silica ... - Springer Link

Journal of Computer-Aided Materials Design (2006) 13:161–183 DOI 10.1007/s10820-006-9009-x

© Springer 2006

Quantum, classical, and multi-scale simulation of silica–water interaction: molecules, clusters, and extended systems HAI-PING CHENGa,∗ , LIN-LIN WANGa,b , MAO-HUA DUa,c , CHAO CAOa , YING-XIA WANa , YAO HEa , KRISHNA MURALIDHARANa , GRACE GREENLEEa and ANDREW KOLCHINa a

Department of Physics and Quantum Theory Project, University of Florida, Gainesville, FL, 32611, USA b National Renewable Energy Research Laboratory, 1617 Cole Blvd., Golden, CO, 80401-3393, USA c Deptartment of Materials Science and Engineering, University of Illinois at Urbana-Champaign, 1304 W. Green St., Urbana, IL, 61801, USA Received 26 September 2005; Accepted 5 January 2006; Published online 25 July 2006 Abstract. Over the past 6 years, we have engaged in a multi-faceted computational investigation of water–silica interactions at the fundamental physical and chemical level. This effort has necessitated development and implementation of simulation methods including high-accuracy quantum mechanical approaches, classical molecular dynamics, finite element techniques, and multi-scale modeling. We have found that water and silica can interact via either hydration or hydroxylation. Depending on physical conditions, the former process can be weak (J Q |RI Q − RJ Q | IQ |PI C |2 + (2.3) + U {RI C } + U RI Q , RI C , 2mI C IC where Eelec = Te + EeI + Eee , is the sum of the electron kinetic energy, electron-ion interaction, and electron–electron interaction. I Q and I C are indices for particles in ∗ the QM and CM regions, respectively. Rm , r∗ are nuclear and electron coordinates for the link-atoms or pseudo-atoms, respectively, and m is the index of the link/pseudo atoms. In our first attempt, a link atom is placed on a straight line between a QM O atom and a CM Si atom at the QM–CM interface. The distance between the O and link atoms is held at a constant of 1.82 a0 (Bohr), a value obtained from reproducing the equilibrium structure of a training molecule, H6 Si2 O7 . The first three terms of Eq. (2.3) form the energy of the QM region in the presence of a group of linkatoms. The fourth and fifth terms are the kinetic and potential energies of the CM

Multi-scale simulation of silica–water interaction

165

region particles. U ({RI C }) is calculated by summing up all pair interactions between any two CM particles (and three-body terms if they exist in the CM potential). The sixth term U RI Q , RI C is the sum of all interactions between QM and CM ions. The CM–QM ion interactions are chosen to be the same classical potential functions as for the CM–CM ion interactions. In other words, a CM ion does not distinguish between ions in the QM and CM regions. In our work, a pair-wise potential function of the van Beest, Kramer, and van Santen (BKS)/TTAM form [82] as discussed at length in other articles of this collection, φI,J (|RI − RJ |) =

cI J qI qJ + aI J e−bI J RI J − 6 RI J RI J

(2.4)

is chosen to describe the classical interactions. Clearly this is not a necessity. The detailed specification of a QM–CM interface always involves a certain degree of arbitrariness depending on the specific nature of the application. In our work, we adopt the principle of optimizing the continuity of forces across the interface as the way of addressing that arbitrariness. We also neglect the constraint forces (that come from the constraints imposed on the link atoms) on QM atoms at the boundary. This approximation causes the energy associated with Eq. (2.3) to be a non-conserving quantity but provides correct forces and dynamics within the limit of a classical force field. The link atoms in the system do not carry kinetic energy; they are used only to terminate the wave functions of a finite subsystem. Unlike some other methods, in which link atoms play a role in the QM–CM interaction directly, our approach eliminates the direct influence they would assert on the dynamics of the system had they been allowed to participate as real particles. In a way, one can view an O–H unit as a special pseudo-atom whose mass is all on the O atom site. ∗ ∗ In BO–MD–DFT, the gradients ∇RI Q Eelec RI Q , Rm ; ρ [r, r ] can be evaluated straightforwardly by taking the gradients of the Hamiltonian matrix elements with respect to the ionic position without basis-set corrections. Since Te and Eee are independent of the ion coordinates, the forces are from ∇RI Q EeI , which contains derivatives of both local and non-local terms in the pseudo-potentials, i.e., lc nlc + EeI , (2.5a) ∇RI Q EeI = ∇RI Q EeI lc EeI = d 3 rρ (r) VIlc (|r − RI |), (2.5b) I nlc EeI =

j,σ

fj,σ

FlI

2

3 I

d rK r − R ψj,σ (r) , I lm

(2.5c)

I,l,m

I where ρ is the total electronic charge density VIlc is the local potential, FlI and Klm are the coefficients and the kernel of the Kleinman–Bylander expansion [83], fj,σ is the Fermi distribution function, and ψj,σ is the KS orbital with spin index σ . The parameters in the classical potential for SiO2 are the BKS values [82], discussed at length in earlier papers of this collection. We have tested this idea on H6 Si2 O7 , a molecule often used to represent a prototype defect structure on silica surfaces. This system has two isomeric forms, D2d and C2v , both having bridging oxygen atoms in the center. We performed both QM and

166

Hai-Ping Cheng et al.

Figure 1. Hydroxylated silica clusters with a water molecule: (a) and (b) and (d)–(f) are ring structures (c) is a chain-structured cluster.

CM simulations to minimize the total energies. Consistent with the findings of our colleagues, who used more sophisticated quantum chemical methods [84], the QM calculations yield nearly the same energy for the two isomers while the classical force field gives a slightly lower energy to D2d . The subsequent studies are based on the D2d isomer (Figure 1b). The QM–CM model also was applied to a H6 Si3 O9 ring cluster (Figure 1(d)– (f)). The optimized structure demonstrates again the transparency of the interface. The structure parameters, generated by the QM–CM hybrid method, are in excellent agreement with the results from QM calculations in the quantum region as well as with the results from classical calculations in the classical region. 2.3. Interface between finite element and MD The reason to do atomic-continuum modeling is twofold. First, there are multiple phenomena that couple strongly on different physical scales. For example, in crack propagation, the bond breaking at the crack tip depends on the deformation of the surrounding material, which in turn depends on the long-range strain field. Conversely, the dissipation of strain energy is through dynamical processes at the crack tip including bond breaking, plastic deformation, and emission of elastic waves. All of these processes happen at the same time when the crack propagates. So a successful description of crack propagation requires simultaneous resolution at both atomistic and continuum length scales. Second, it is not possible to compute all the relevant dynamical processes in the most accurate and intensive model with a reasonable computational cost. The essential perspective of multi-scale modeling is to find a suitable balance between accuracy and efficiency, i.e., let the most dramatically changing


167

region be dealt with via the most accurate method and treat the broad, surrounding regions by less accurate and less computationally costly methods. The multi-scale combination of MD and the finite element (FE) method is muchstudied as a means of investigating fracture and crack propagation in macroscopic materials. One group of studies focuses on deriving the FE method from the underlying atomistic model rather than from the traditional continuum model. In the quasicontinuum technique proposed by Tadmor and co-workers [85–88], the energy of each element is computed from an underlying atomistic Hamiltonian, such that nonlinear elastic effects can be included. In the coarse-grained MD proposed by Rudd and Broughton [89], a similar idea is pursued. The interpolation functions in the FE mesh are assembled from the atomistic model. The other group of studies focuses on combining MD and FE through an interface. Kohlhoff et al. [90] introduced an interface plane between the MD and FE regions to pass the displacements as boundary conditions for the two regions. Abraham et al. [91, 92] and Broughton et al. [93] used a scheme based on coupling through forces. In it, the FE elements sitting at the interface plane can have forces of MD nature. Smirnova et al. [94] extended the imaginary interface plane to a finite size. In our work, we proposed an improved MD/FE interface with gradual coupling of force and used it to study the mechanical behavior of a SiO2 nano-wire. Amorphous silica is the major constituent in optical fiber. Quartz is the material for timing in electronic circuits. Other crystalline silica, such as cristobalite, can be found in Si and SiO2 interfaces in microchips [95, 96]. Since the discovery of carbon nano-tubes, different types of nano-tubes and nano-wires have been studied both in theory and experiment. For example, a SiSe2 nano-wire has been proposed and studied with MD [97–99]. In experiments, both SiO2 sheathed Si nanowires [100] and pure SiO2 nano-wires [101–103] have been found. SEM and TEM images show that the nano-wires are several µm long with diameters of 10–50 nm. Electron diffraction on SiO2 sheathed Si nano-wires shows that the core is crystalline Si and the cover is amorphous SiO2 . In a pure SiO2 nano-wire, however, the structure is also amorphous. While the structures of crystalline silica are well understood as different arrangements of corner-sharing SiO4 tetrahedral, the structure of the amorphous silica surface is still an open problem [104, 105]. Since we do not have a well-defined structure for the amorphous SiO2 nano-wire to start with, a closed crystalline structure should be used. Besides amorphous structures, silica can have as many as 40 crystalline structures in nature [106]. Among these different silica polymorphs, only quartz (α, β) and cristobalite (α, β) are stable at atmospheric pressure. As remarked above and elsewhere in this collection, the density of β-cristobalite is closest to that of amorphous silica. So β-cristobalite often is used as a preliminary model for amorphous silica. As the first step to understand the mechanical properties of an amorphous SiO2 nano-wire, we constructed an α-cristobalite nano-wire and used the combined MD/FE method to study its amorphization and fracture under tensile stretch. As we will demonstrate, during the initial part of tensile stretching, a phase transition occurs from α-cristobalite to β-cristobalite. With further tensile stretching, the nano-wire become amorphous before it starts to fracture.

168


2.3.1. Finite element method The FE method is a general approach to solve a differential equation approximately [107]. A continuous system has infinite degree of freedom. The FE method uses a finite number of degrees of freedom to approximate the continuous solution. When a continuous system is divided into a FE mesh, the displacement field uc (r) within an element can be interpolated by the local displacement on the nodes of that element uie as uc (r) =

Ns

Hi (r)uie ,

(2.6)

i=1

where Hi (r) is the interpolation or shape function and 1 ≤ i ≤ Ng , with Ng , the number of nodes in each element. The continuous strain within an element can be defined symmetrically as [108]: Ng 1 ∂Hi (r) e ∂Hi (r) e c εµν ui,µ + ui,ν , (r) = 1 − δµν 2 ∂rν ∂rµ

(2.7)

i=1

where all quantities are written in Cartesian components and, 1 ≤ µ, ν ≤ Nf , with Nf the number of degrees of freedom of each node. These equations usually are expressed in matrix form uc (r) = H (r) ue

(2.8)

and ε c (r) = D (r) ue ,

(2.9)

where the local displacement ue is written as a vector of Ng × Nf dimensions. The matrix D (r) is the strain–displacement matrix as defined from Eq. (2.7). In solid mechanics, the FE method is introduced as a minimization of the total potential functional [107] 1 T = ε d + u p d + uT q dS, (2.10) 2 S where σ is the stress, p the body force per unit volume, and q is the applied surface force per unit area. The FE method often is used in the elastic regime, which means small strain, harmonic response, and no plasticity. The corresponding stress–strain relation is linear, σ = Cε = CDu,

(2.11)

where C is the elastic matrix. As the system is divided into a FE mesh, the elastic potential energy functional can be written as = e 1 e

=

e

2

e

T

u D CDu d + eT

u H p d +

e

eT

e

u H q dS .

T

eT

Se

T

(2.12)


169

Energy minimization to determine the equilibrium configuration of the system gives

∂ T e D CD) d u + HT p d + TT q dS (2.13) 0= e = ∂u ∫ from which Ke ue + Fe = 0, where

(2.14)

Ke =

D T CD d

(2.15)

is the local stiffness matrix and Fe is the nodal force that results from the last two terms in Eq. (2.13). There are as many as Ne equations like the one in Eq. (2.14), since Ne is the total number of elements in the system. These equations are coupled through ue . They can be assembled as one global matrix equation, Ku + F = 0,

(2.16)

where K is the global stiffness matrix and u is the generalized displacement matrix. To this point, we have only elastic statics, perhaps the most common application of the conventional FE method. To consider elastic dynamics, we have to introduce the kinetic energy for an element, 1 e T = ρ (r) u˙ e (r)2 d 2 e e T 1 = H˙u ρ (r) H˙ue d (2.17) 2 e 1 = u˙ eT Me u˙ e , 2 where

M =

ρ (r)HT H d

e

(2.18)

e

is the local mass matrix and ρ (r) is the density of the material. If we construct the Lagrangian and use the variational principle as before, a global dynamical equation can be obtained M u¨ + K u + F = 0.

(2.19)

With the condition of no external force and free boundary condition, the dynamical equation becomes that of free response, M u¨ + K u = 0. The harmonic solution u = u(0)eiωt gives an eigenvalue equation

2

−ω M + K = 0,

(2.20)

(2.21)

170


which is the same as the dynamical matrix equation for phonons in a crystal lattice [109]. With the condition of no external force and strain applied on the boundary of the system, the dynamical equation is now u¨ = −M−1 Ku.

(2.22)

Generally M, the consistent-mass matrix, is non-diagonal. With a set of interpolation functions satisfying HT H = I,

(2.23)

where M is diagonal and is called the lumped-mass matrix. The dynamical equation can be solved in the central difference method (Verlet) or Newmark’s method [107]. The former method is simple and very computationally efficient when combined with the lumped-matrix approximation. Each individual node vibrates linearly. The latter method is more stable numerically but involves implicit solution of a matrix equation, hence is more computationally demanding. 2.3.2. Hybrid MD/FE: new gradual coupling In our approach, the system is partitioned into three regions, to wit, the core MD (CMD), dilute FE (DFE) and transition (TRN) regions. The mesh of FE nodes in the TRN region matches the crystal lattice sites. The total Hamiltonian of the system is Htot =HCMD ( {r, r˙ } ∈ CMD) + VCMD/TRN ( {r} ∈ CMD/TRN) +HDFE ( {u (r) , u˙ (r)} ∈ DFE) + VDFE/TRN ( {u (r)} ∈ DFE/TRN) +HTRN ( {u (r) , u˙ (r) , r} ∈ TRN) .

(2.24)

In HCMD , HDFE and HTRN , we include the kinetic energy from each region and the contribution of the potential energy between any two particles or connecting nodes if they both are in the same region. In VCMD/TRN and VDFE/TRN , we include the interaction between two particles or connecting nodes which are in adjacent regions. Inside the TRN region, we have HTRN ( {u (r) , u˙ (r) , r} ∈ TRN) =TTRN (˙u (r)) + w (|r|) V MD (r) + [1 − w (|r|)] V FE (u (r)) ,

(2.25)

where the weight function, w (r), is determined by the distance of the nodes from the CMD and DFE regions. The forces between two FE nodes (or two particles) in the TRN region are calculated by both FE and MD according to a chosen weight function. The relative weight is determined by the distance of the nodes/particles from the CMD and DFE regions. So the hybrid force in the TRN region can change from the long-ranged MD interaction (in the case that the nodes are very close to the CMD region), to the nearest neighbor FE force (in the case that the nodes are close to the DFE region). For the studies discussed here, we also used the BKS inter-atomic potential; see above. We used a two-dimensional FE grid formed by dividing the system into iso-parametric triangular elements with linear interpolation functions, Ng = 3 and Nf = 2. The


171

third dimension is treated as uniform, continuum medium. The total numbers of elements and nodes are Ne and Nd , respectively. The potential energy is V FE =

Ne 6 1 m m um p Kpq uq 2 m

(2.26)

p,q=1

and the stiffness matrix is L [K]m = m [D m ]T [C] [D m ] , (2.27) 4A where L is the thickness in the third dimension and Am is the area of the element. For such a system the elastic constant [C], is a (3 × 3) matrix with the third dimension decoupled from the other two,   C11 C12 0 0 . [C] =  C12 C11 (2.28) 0 0 C14 The strain–displacement matrix [D] consists of coordinate differences of the nodes on each element. It is of dimension (3 × 6). The force on each element is m m Fpm = Kpq uq ,

(2.29)

where the indices p and q run from one to six. For obtaining the dynamics of the FE nodes, it is necessary to project the forces in Eq. (2.29) onto each FE node and combine them with forces from atomistic region, Fin =

Ne 3 m

δnml Fpm ,

(2.30)

l=1

where the index i runs from one to two. The kinetic energy is T FE =

Ne 6

2 1 m m m u˙ pm Mpq u˙ q = n = 1 u˙ n

2 m N p,q=1

(2.31)

d

and the mass matrix is M n = ρL

Ne 6 m p,q=1

δnm

Am , 3

(2.32)

where ρ is the bulk density of the material and we have used the lumped-mass approximation. After the forces are calculated from different contributions, we use the Verlet algorithm to integrate the dynamical equations for both MD and FE. The FE equation is written as Ne 3 ∂ m m M u¨ = −w (|rn |) φ (rn − rs ) + 1 − w (|rn |) δnml Kpq uq , ∂rn m n n

s∈TRN s=n

where φ is the inter-atomic potential defined in Eq. (2.4).

l=1

(2.33)

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Table 1. Hydration energy of SiO and SiO2 with 1–4 water molecule (all values in eV) Clusters

SiO+H2 O

SiO+2(H2 O)

SiO+3(H2 O)

SiO+4(H2 O)

SiO SiO2

0.30 1.04

0.80 0.82

0.30 0.42

0.05 Hydroxylation 2.16

3. Results 3.1. SiO and SiO2 molecule and their interactions with water In gas phase, both SiO and SiO2 can be observed. The comparison between SiO and SiO2 molecules is very interesting. Both can interact with water either via hydration or hydroxylation. In former process, a water molecule does not dissociate, in the latter, it does. From calculations, SiO is more stable than SiO2 when interacting with water. Both hydration and hydroxylation energies for one SiO2 are higher than those for one SiO. Table 1 lists the calculated energies of these two molecules using BO–MD–DFT with the PBE GGA; recall prior discussion. One sees that the hydration energy of a SiO with one to four waters is always higher than the corresponding energy for one SiO2 . When hydrated by four water molecules, a SiO2 will react spontaneously with water, behavior which was not observed in the SiO system. The hydroxylation gain of SiO–H2 O is 1.67 eV, an energy gained by dissociation of a water and formation of two O-H bonds. This energy is consistently over 2 eV in pure SiO2 systems. 3.2. SiO2 and Sin Om Hl clusters: size dependence and interaction with water We have performed a systematic investigation on the size dependence of (SiO2 ) n Om Hl systems, which can take many isomeric forms [31]. These clusters can be classified into two groups: hydrophobic ones and hydrophilic ones. We emphasize here that this specific classification only applies to hydration and not hydroxylation, i.e., the process that does not involve water dissociation. Figure 1 depicts a water interacting with a special family of silica clusters that are already hydrated. For dry silica, i.e., clusters that do not have an O-H group (one may construct such clusters by removing an O–H group from a Si atom and an H atom from one oxygen atom), the hydration energy is often close to 1 eV (see Table 2), with some exceptions for “magic number” clusters. For example, Figure 2 shows various positions at which a water can interact with a (SiO2 )36 nano-rod (the 108 nano-rod); the highest hydration energy is less than 0.2 eV (Table 3). Hydrophobic clusters often have hydration energies under 0.3 eV, or even lower. This is the case for all clusters in which oxygen bonds are saturated in the Si–O network or by terminal O–H units. Tables 2 and 3 give detailed energetic information on both clusters and the 108-atom nano-rod. 3.3. SiO2 crystalline structures and surface studies To obtain a comparison between clusters and bulk systems, we have performed a thorough investigation of a few common crystal structures (Figure 3). The


173

Table 2. Cohesive energy Eb and hydration energy Ehyd (4th and 4th columns), all in eV. The two numbers in parenthesis in the last column are energies for dissociated water states. Strictly speaking, they are not hydration energies but reaction energies. Note that, for a given n in the third column, there can be more than one m n

(SiO2 )n Eb /atom (SiO2 )n (OH2 )m Eb /atom (SiO2 )n + H2 O (SiO2 )n (OH2 )m + H2 O

1 2 3 3 108 nano-rod

4.35 5.02 5.20 5.30 6.01

4.17 4.63 4.97 4.82

(m = 2) (m = 2) (m = 2) (m = 3)

1.04 0.90 0.90 0.95

0.20 (1.20) 0.26 0.25 (−0.08) 0.18

Table 3. Hydration energy of rings and 108-atom nano-rod for various sites (as shown in Figure 2) (SiO2 )n (OH2 )n

n=2

n=3

n=4

n=5

n=6

Eb /atom (eV) Hydration E H.E. for rod (a)–(e)

4.6 0.20 0.03

4.82 0.25 0.05

4.84 0.12 0.11

4.83 0.27 0.18

4.85 0.20 0.019

Figure 2. Interaction of a water molecule with a nano-rod at various sites (see Table 3 also).

configuration of atoms in a unit cell as well as lattice constants are fully optimized to minimize the cohesive energy. Table 4 gives the calculated structure parameters and cohesive energies for various plane–wave pseudo-potential codes in comparison with experimental data and with the all-electron LDA calculation of our colleagues on α-quartz [110]. The calculated cell parameters do not vary more among various theoretical treatments any more than is common for GGA versus LDA DFT exchangecorrelation (XC) models and compare reasonably well with experimental value. The cohesive energies from PWSCF-GGA calculations are also in good agreement with the experimental values, with a discrepancy of less than 2% compared with 3–4% for VASP-GGA. With LDA, the errors are around 16%, consistent with the wellknown over binding tendency of LDA. Notice that both the VASP and PWSCF LDA cohesive energies for α-quartz are in close agreement with the all-electron result from Ref. [109]. As is often the case, LDA gives the right relative energies among different

174


Figure 3. (a)–(e) are α-quartz, α-cristobalite, β-quartz, and β–cristobalite

silica crystalline structures, a result that also has been found by other groups [114, 115]. It should be pointed out that Demuth et al. [114] found much higher silica crystal binding energies using the VASP code, e.g., 23.83 eV/SiO2 with GGA for α–quartz. We have obtained a similar value before including the correction for the atomic spin state. As discussed in the cluster section, the true ground states for Si and O atoms are triplets. The spin-state corrections to cohesive energies in PWSCF GGA and LDA are 3.99 and 3.68 eV/SiO2 , respectively. In VASP, the corrections are 3.85 and 3.69 eV/SiO2 for GGA and LDA, respectively. After adding the corrections, the binding energies are much closer to the experimental values. It should also be pointed out that the cohesive energies from SIESTA are not as good those from PWSCF or VASP. Thus, the correct energy ordering of crystalline phases given by SIESTA (GGA) could be misleading. Among the SIESTA, VASP, and PWSCF calculations, the PWSCF results should be most accurate, since the Troullier–Martin pseudo-potential [60, 61] with a large cut-off energy was used. With the GGA XC approximation, both VASP and PWSCF yield an incorrect sequence of structures but by very small energy differences. The correct ordering given by SIESTA could very well be accidental, and one should be very cautious in drawing any conclusion based on these numbers. Furthermore, the differences in energy among all the crystalline structures are so small that it is very possible that none of the DFT codes with present XC approximations can actually make prediction with such precision even though numerical convergence has been achieved in the calculations. One of our major efforts in recent years is to study the water–SiO2 surface using the multi-scale model described in Section 2.2 [32]. Figure 4a depicts a physical model of an amorphous surface. That surface is prepared by a well-documented method [55] using classical molecular dynamics. We then divide the surface into a classical and a quantum region. The quantum region is described, as before, by BO-MD-DFT [58] with the GGA XC approximation. Three different sizes of quantum region are used to examine the energy convergence with respect to what is a methodological choice. For the investigation of reaction barriers, we have used a 31-quantum-atom region (plus link atom) embedded in a 104 particle classical matrix (see Figure 4b). The results indicate a zero-barrier reaction when we use more than one water molecule. For a single water molecule, the reaction barrier is found to be 0.4 eV. Compared to an isolated cluster model (using the same 31-atom system but without the matrix), which shows a very small hydration energy, the multi-scale model provides a more realistic description of the boundary condition of the quantum region and thus a better estimation


175

Table 4. Calculated structure parameters and cohesive energies, with the PBE GGA, for four crystalline structures of silica. Values in parenthesis are from LDA calculations. Experimental values are as found in previous publications [111–113]

a(˚A) EXPTL a(˚A) SIESTA a(˚A) PWSCF a(˚A) all-electron LDAa c/a EXPTL c/a SIESTA c/a all-electron LDAa c/a PWSCF Ec (eV/SiO2 ) EXPTL SIESTA VASP PWSCF all-electron LDAa a

α-quartz

α-cristobalite

β-quartz

4.92 5.02 5.06 (4.88) (4.931) 1.10 1.10 (1.087) 1.11 (1.10) 19.23 21.34 19.98 (22.27) 19.56 (22.37) (21.74)

4.96 4.93 5.13 (5.00)

5.00 5.18 5.13 (5.02)

1.39 1.41

1.09 1.09

1.40 (1.39) 19.20 21.30 20.01 (22.25) 19.58 (22.35)

1.09 (1.10) 19.18 21.29 19.97 (22.25) 19.55 (22.34)

β-cristobalite

21.13 19.99 (22.21) 19.55 (22.31)

Ref. [110].

Figure 4. Model for multi-scale simulations: (a) is a cross-sectional view of the silica surface and (b) is a close-up of part of the quantum region that reacts with two water molecules. A concurrent proton transfer is observed.

of hydration energies. Table 5 shows the results of water binding/dissociation energies for various model surfaces. Note that two potential energy function BKS and Watanabe [116] have been used; the latter gives better energetics while the former focuses on better forces. 3.4. Stress–strain relation for bulk silica and silica nano-wires To understand size effects upon mechanical strength in silica, we have conducted two groups of simulations comparing bulk silica and nano-wires. The first group was done in conjunction with FE embedding, while the second used periodic conditions

176


Table 5. Hydration or dissociation energies E = Etotal − Ewater − Esurface , calculated using different models. Esurface is the energy of an unperturbed dry surface and Ewater is the energy of an isolated water molecule. In the cluster model Esurface = Ecluster , and “fix-H” means all H atoms are held fixed during relaxation. “BKS” and “Watanabe” refer to classical potentials (see text) Model used

31-atom cluster

Si2 O6 H4

31-QM–CM BKS

31-QM–CM Watanabe

209-QM–CM Watanabe

Hydration energy (eV) Dissociation energy (eV)

−0.12 (free) 0.05 (fix-H)

−0.5

N/A

−1.5

−1.24 −5.80 (1 H2 O) −5.88 (2 H2 O)

0.69 −1.45 (2 H2 O) −1.12 (1 H2 O)

−0.03 −1.06 (1 H2 O) −1.60 (2 H2 O)

Figure 5. Geometry of the α-cristobalite (SiO2 ) nano-wire projected on (a) xy and (b) yz planes. The CMD region is in the center. The two ends are the DFE regions. The region between the CMD and DFE regions are the TRN region. The two-dimensional FE mesh is projected on the xy plane from the three-dimensional crystal lattice. In the TRN region, FE nodes match with the positions of crystal lattice.

along the direction of wire. The main purpose of the first group is to test the MD–FE interface, while the second one was focused on comparison with bulk matter. The first nano-wire sample that we use is shown in Figure 5. The dimensions are 19.91, 27.79, and 119.47 ˚A in the x, y, and z-directions, respectively. The strain is applied in the y-direction. The projection from MD particles to FE nodes in the transition (TRN; recall notation above) region is in the z-direction as shown in Figure 3d.1(b). To construct the nano-wire, we use the 12-basis unit cell for α-cristobalite with dimensions of 4.978 × 4.978 × 6.948 ˚A. The CMD region has 4 × 8 × 4 unit cells with 1536 particles. Each of the TRN regions has 4 × 4 × 4 unit cell, thus has dimensions larger than the cut-off distance (∼25 ˚A) used with the BKS potential. Each FE node in the TRN region corresponds to four MD particles in different layers. The TRN regions have 384 FE nodes, hence correspond to 1536 MD particles. The two DFE regions have 32 FE nodes. In total, there are 408 FE nodes and 782 FE triangle elements in the sample. 3.4.1. Interface test After the code was developed, we did a few simple tests. First, the force generator from MD was turned off so that only FE forces determined the dynamics. As seen in Figure 6a, the kinetic energy and potential energy compensate each other and the total energy is conserved. If the FE force generator is turned off and only MD forces are used, the total energy is also conserved. When both the MD and FE forces are used, the total energy calculated from the two parts is shown in Figure 6b. To match the magnitude of the energy calculated from FE with that from MD, we have shifted


177

Figure 6. Energy conservation test with respect to time for (a) FE only, and (b) both FE and MD.

the energy of the FE with an average potential energy density. The two curves are compensating. So the total energy is conserved very well. The main goal of the hybrid FE/MD interface is to let elastic waves propagate from the CMD region to DFE region. To test if the interface works appropriately, we squeeze two central layers in the y-direction in the middle of the CMD region to make a pulse and let it propagate. As seen in Figure 7, at time zero only the two central layers feel the stress. As the dynamics evolves, we plot the distribution of velocity in they direction in different colors corresponding to different instants. Initially the pulse is in the center of the wire. Later on, the pulse is propagated into the whole wire. At 0.1 ps, the pulse reaches the TRN region. At t = 0.24 ps, the pulse arrives at the DFE region. Finally, at t = 0.6 ps, the pulse spreads all over the nano-wire and causes some local distortion. 3.4.2. Stretch simulation using the combined MD–FE model Figure 8 shows the stress–strain relation and a set of snapshots for a uniaxial stretch applied in the y-direction of the nano-wire at the speed of 0.035 ps−1 at 300 K. The nano-wire breaks at a strain of 0.12. During the first stretching interval, the stress actually decreases from 10 to 5 GPa. That corresponds to the phase transition of α-cristobalite to β-cristobalite because the stretch in the y-direction makes the unit cell change from tetragonal to cubic. Under more strain at t = 2.0 ps, we see, in Figure 8, that the tetrahedral units around the surface of the nano-wire start to rearrange themselves, to increase the Si–O–Si bond angles in the direction of the stretch, and to cause local amorphization on the surface. The neighboring tetrahedra along the x and z-directions get closer and closer with more applied strain. When the Si–O bond between neighboring tetrahedra becomes shorter than the corresponding bond in the tetrahedron, the latter bond will break. A fracture tip will be formed if the bond breaking happens on the nano-wire surface or a fracture void will be formed if the bond breaking happens inside the nano-wire. The latter case is seen in Figure 8 at t = 4.0 ps, when the applied strain is around 0.12. The fractures propagate and move under further applied strain, and cause more amorphization as seen in Figure 9 for the times t = 6.0 and 8.0 ps. Eventually, the two ends of the nanowire detach around t = 10.0 ps. 3.5. Comparison with bulk SiO2 For the second group of simulations, we generated an amorphous bulk and an amorphous nano-wire sample sequentially. A bulk sample of 3000 atoms first was melted

178


Figure 7. Distribution of velocity of atoms in the y-direction during a pulse propagation test for the MD/FE interface. Top panel shows the distribution of force in the y-direction at time zero. Two atomic layers in the center of the x–y plane have been pushed together. The distribution of velocity in the y direction at t = 0.1, 0.6, and 1.2 ps are show in panel 2–4, respectively.

at 8000 K and cooled down to 300 K, just as the sample used in surface studies in the previous section was prepared. We then use three unit cells in the z-direction and one unit cell in the x and y-direction to comprise the periodically bounded box. This rectangular box was heated again to 2000 K, then cooled to 300 K using NPT ensemble simulation. The result is a new bulk sample. To generate the nano-wire sample, we remove the periodic boundary condition in the x and y-directions and equilibrate the wire at 300 K using the NVE ensemble.


179

Figure 8. Left: Five snapshots from the tensile stretch applied in the y-direction of the nano-wire at speed of 0.035 ps−1 . The nano-wire is viewed from the x-direction at times of 2.0, 4.0, 6.0, 8.0, and 10.0 ps. The FE region is uniform in the z-direction; Above: The stress–strain relation for a uniaxial stretch. Stress-Strain for Bulk and Nano-wire (0.1%/ps, 300K, 1atm) 200.00 Bulk Wire

150.00

Stress

100.00

50.00

0.00 0 -50.00

0.05

0.1

0.15

0.2

0.25

0.3

Strain

Figure 9. Stress-strain curve of SiO2 bulk (blue) and nanowire (red).

The two systems are then stretched at a rate of 0.1% per pico-second. Figure 9 depicts the stress–strain curves at 300 K. Ten initial configurations for each sample were chosen and the results averaged over corresponding the 10 simulations. It is clear that the two curves are different in a fundamental way. The bulk stress–strain curve indicates a brittle material while the nano-wire is a very ductile one. This difference is also reflected by the snapshots at the breaking point (see Figure 10).

180


Figure 10. Snapshot of bulk (upper) and wire (bottom) at breaking point.

Note that exactly the same potential energy function was used for the two systems. The strong contrast between the two is predominately a size effect. Nano-scale systems have very large surface-to-volume ratio. Consequently, a nano-wire can release stress from the surface easily hence behaves mechanically like a metal. Finally, the differences between the two wires, cristobalite and amorphous, are mainly from the initial structures and stretching speed. The cristobalite wire breaks faster at faster stretching speed than the amorphous one at slower speed. Further investigations are underway to gain better understanding of the differences. 4. Summary Over last 5–6 years, we have made substantial progress in understanding the interaction between water and silica using high-level QM calculations to operate in conjunction with powerful MD tools. We have also developed interfaces for both QM/CM and MD/FE hybrid simulation. It should be pointed out that the multi-scale methods can be very useful when a system or a process has to be described at different length scales concurrently. There is always a price to pay at the interface. To illustrate what can be achieved with purely classical simulation, we have present a study on the fracture of SiO2 wires and bulk systems. It is evident that MD alone can still provide very valuable information that is hard to obtain (because of computational cost) using methods that involve high accuracy quantum modeling. Acknowledgments This work was supported in substantial part by NSF ITR (medium) grant DMR0325553 and, prior to that, by NSF KDI grant DMR-9980015. References 1. 2. 3.

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