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Molecular dynamics simulation of twinning in devitrite, Na2Ca3Si6O16 Bin Li
a b
& Kevin M. Knowles
b
a
State Key Laboratory of Advanced Ceramic Fibers and Composites , College of Aerospace and Materials Engineering, National University of Defense Technology , Changsha , 410073 , China b
Department of Materials Science and Metallurgy , University of Cambridge , Pembroke Street, Cambridge , CB2 3QZ , UK Published online: 06 Feb 2013.
To cite this article: Bin Li & Kevin M. Knowles (2013): Molecular dynamics simulation of twinning in devitrite, Na2Ca3Si6O16 , Philosophical Magazine, 93:13, 1582-1603 To link to this article: http://dx.doi.org/10.1080/14786435.2012.748989
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Philosophical Magazine, 2013 Vol. 93, No. 13, 1582–1603, http://dx.doi.org/10.1080/14786435.2012.748989
Molecular dynamics simulation of twinning in devitrite, Na2Ca3Si6O16 Bin Lia,b and Kevin M. Knowlesb* a
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State Key Laboratory of Advanced Ceramic Fibers and Composites, College of Aerospace and Materials Engineering, National University of Defense Technology, Changsha 410073, China; b Department of Materials Science and Metallurgy, University of Cambridge, Pembroke Street, Cambridge, CB2 3QZ, UK (Received 8 October 2012; final version received 7 November 2012) Reports of Type II twins are quite rare for most crystal structures. When they do occur, they are usually one of a number of possible twinning modes observed in a particular material. However, for the triclinic phase devitrite, Na2Ca3Si6O16, which nucleates from commercial sodalimesilica float glass subjected to suitable heat treatments, the only reported twinning mode to date is a Type II twinning mode. In this study, this Type II twinning mode is first examined by molecular dynamics simulation to determine the lowest energy configuration of perfect twin boundaries for the twin mode. This is then compared with the lowest energy configurations of perfect twin boundaries found for six possible Type I twinning modes for devitrite for which the formal deformation twinning shear is less than 0.6. The most favourable twin plane configuration for the Type II twinning crystallography is shown to produce reasonably low twin boundary energies and sensible predictions for the optimum locations of the twin plane, K1, and the [1 0 0] rotation axis, η1, about which the 180° Type II twinning operation takes place. By comparison, all the Type I twinning modes were found to have very energetically unstable atomic configurations, and for each of these twinning modes, the lowest energy configurations found all led to high effective K1 twin boundary energies relative to perfect crystal. These results therefore provide a rationale for the experimental observation of the particular Type II twinning mode seen in devitrite. Keywords: boundary; devitrite; molecular dynamics; simulation; twinning
1. Introduction During crystal growth, or when crystals are subjected to a particular stress state, or particular temperature or pressure conditions, two or more intergrown crystals can be formed in a symmetrical fashion. These symmetrical intergrowths of crystals are twinned crystals. Twinning is an important field for investigation because twin boundaries are one of the main constituents of the microstructure and nanostructure of crystalline materials [1–7]. The selection of twin boundaries and twin modes is relevant for both crystal growth and deformation behaviour, as a consequence of which the presence or absence of twins can influence the properties of crystalline materials [8–13]. *Corresponding author. Email:
[email protected] Ó 2013 Taylor & Francis
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Geometrically, there are three distinct types of twinning, Type I twinning, Type II twinning and compound twinning [14]. Twins whose formal deformation twinning elements K1 and η2 are rational, while K2 and η1 are irrational, are called Type I twins or reflection twins. Twins whose K2 and η1 twinning elements are rational, while K1 and η2 are irrational, are called Type II twins or rotation twins. More commonly in crystals of high, or relatively high, symmetry, all four of these elements can be rational, in which case the twinning is termed compound twinning [14,15]. For most materials, observations of Type II twins are much more rare than those of Type I twins and compound twins, a striking exception being the Type II twinning mode seen in nickeltitanium shape memory alloys [16]. This Type II twinning mode is the dominant twinning mode seen in the monoclinic martensitic phase in such alloys [17]. Devitrite, Na2Ca3Si6O16, is a centrosymmetric triclinic silicate phase [18–20]. This phase nucleates heterogeneously on the surfaces of suitably heat treated sodalimesilica float glass, growing as spherulites consisting of thin needles, within which a single twinning mode has been observed [21,22]. Analysis of this twinning mode has shown that it has a rotation axis, η1, of [1 0 0] and that, with respect to the triclinic unit cell of devitrite, the indices of the twin plane, K1, are (0, 2bcosγ/a, 1 (2cosβ/a)), i.e. a plane which does not have rational indices [21]. However, for devitrite, this formally irrational twin plane is actually the plane (0 1 0:0004), 0.02° from (010), using the unit cell parameters for devitrite in the form defined by Kahlenberg et al. [20]. Knowles and Ramsey [21] considered the geometry of other possible twinning modes for devitrite and showed that, in principle, there were a number of possible twinning modes for which the magnitude of the deformation twinning shear, s, is less than 0.6 and the quantity Σ = hu + kv + lw is 2, where for Type I twinning K1 is (h k l) and η2 is [u v w], and for Type II twinning K2 is (h k l) and η1 is [u v w]. From the observation that the surfaces of the needles of devitrite have prominent (0 1 0) planes, Knowles and Ramsey rationalized that the Type II twinning mode they characterized was favoured because K1 had a low interfacial energy. In this study, we have used molecular dynamics simulation to study twinning in devitrite, with the aim of understanding on a more quantitative basis the competition between a number of plausible twinning modes in devitrite. For this first ever molecular dynamics simulation study on devitrite we have chosen only to consider geometries for which the interface under consideration is K1, and so we have therefore not attempted to examine other aspects of twins such as step defects and twin nucleation and growth in this work. Specifically, we have considered the observed Type II twinning mode in devitrite and the six postulated Type I twin modes for which s < 0.6 and R 2. 2. Possible twin modes for devitrite 2.1. Crystal structure Devitrite is centrosymmetric with the space group P1 (No. 2) [19,20]. The lattice parameters for single crystal devitrite determined by Kahlenberg et al. [20] were used for this study: a = 7.2291 Å, b = 10.1728 Å, c = 10.6727 Å, α = 95.669°, β = 109.792°, γ = 99.156°. Data for the atomic coordinates of the constituents of this unit cell are given in Table 2 of [20] with the centre of symmetry of the crystal structure at the origin.
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2.2. Possible twin modes The observed Type II twinning mode and the six Type I twin modes for which s < 0.6 and R 2 are able to be specified using the data in Table 4 of [21]. For ease of computation, the 0.02° deviation of the formally irrational K1 twin plane from (0 1 0) for the Type II twinning mode was ignored when setting up the initial configurations for the simulations, the justification being the same as that used recently by Ostapovets in his study of a Type II twin boundary in 2H CuAlNi martensite, i.e. that this physical difference between (0 1 0) and ð0 1 0:0004Þ is negligible [2]. For the Type I twins, the six twin planes are (0 1 0), (0 0 1), (0 1 1), (0 1 1), (2 0 1) and (2 1 1). For the Type II twinning mode, the direction of the rotation axis η1 is specified, i.e. [1 0 0], but not where it enters the unit cell of devitrite. In addition, the Miller indices of K1 are clearly defined, but not the position of this plane relative to the unit cell of devitrite. Hence, a number of possibilities for the positions of K1 and η1 have to be considered when simulating the twin boundary by molecular dynamics. We have chosen to consider in principle 10 possible K1 planes parallel to (0 1 0) as shown in Figure 1, labelling the possible planes as P0, P1 , … , P9, with the final plane P10 being the same as P0 because the two planes are translated with respect to one another by [0 1 0]. Likewise, because of the choice of centre of symmetry as (0, 0, 0) in the description of the crystal structure, and as a useful check during the simulations, the initial atomic structures on P1 and P9 are related by the centre of symmetry, as are those yD
P 10 P9 P8 P7 P6 P5 P4 P3
o P2
xD
P1 P0
zD
Figure 1. (Colour online). Possible (0 1 0) twin plane positions in a single unit cell of devitrite for type II twins. The atom positions with the centrosymmetric triclinic unit cell are those taken from the work of Kahlenberg et al. [20], with the centre of symmetry at (0, 0, 0). The subscript ‘D’ denotes that the x-, y- and z-axes in this figure are defined with respect to the conventional devitrite unit cell.
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yD
o
xD
zD
Figure 2. Possible positions of [1 0 0]D rotation axes in each reconstructed unit cell of devitrite for a (0 1 0)D type II twin with the twin interface at y = 0 with respect to the conventional unit cell. The subscript ‘D’ denotes that the x-, y- and z-axes in this figure are defined with respect to the conventional devitrite unit cell.
on (i) P2 and P8, (ii) P3 and P7 and (iii) P4 and P6. Hence, in practice only the six positions P0 … P5 need detailed consideration. The subscript ‘D’ is used for the x-, y- and z-axis directions in Figure 1 to denote ‘devitrite’ and to distinguish these axis directions from a Cartesian coordinate axis system used in Section 3 in the description of the construction of the various twin models. Within each of these six possible K1 planes for the Type II twinning mode, 10 possible axis positions of η1 were considered, X0 … X9, as shown in Figure 2. It is evident that the axis positions X0 and X10 are equivalent, since they are translated relative to one another by the vector [0 0 1]. For the six possible Type I twins considered, the choice of position of the K1 planes was similar to that shown in Figure 1, in that, for each K1 plane under consideration, the possibilities P0 … P5 were considered as a possible initial position for K1 within the unit cell. Centre of symmetry considerations meant that P6 … P9 were equivalent to P4 … P1 as for the Type II twinning. The mirror relationship between atomic positions in the matrix and twin once K1 is chosen then enables atomic positions to be specified in the twin knowing atomic positions in the matrix. Hence, in contrast to the Type II twinning mode where we have examined 60 initial configurations, as defined above, we needed to examine just six for each of the six Type I twinning modes for essentially the same degree of unit cell subdivision. 3. Construction of initial twinning configurations for the computer simulations The detailed steps of the construction of the initial twinning configurations for models of the Type II twins are described here. The methods for constructing the initial twinning
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configurations for the Type I models were similar; clearly defined differences in the constructions of the initial twinning configurations are specified where they occur. 3.1. Conversion of atomic coordinates and vectors for the observed Type II twinning mode The first step in constructing the initial twinning configuration was to impose a suitable orthonormal Cartesian coordinate system in units of Å to enable conversion of coordinates between fractional atomic coordinates within a single unit cell of devitrite, (uD vD wD) to the orthonormal Cartesian coordinates (uC vC wC). For convenience, the subscript ‘C’ has been chosen to denote the Cartesian set of axes. Choosing the two coordinate systems to have the same origin, as shown in Figure 3, with the a-axis of the devitrite unit cell parallel to [1 0 0]C, and the c-axis of the devitrite unit cell in the (0 1 0)C plane, a position [uD vD wD] described with respect to the triclinic unit cell becomes [uC vC wC] when defined with respect to the Cartesian coordinate system, where
yc
yD B
E
A
o
F
zD C
xc xD
D
zc
Figure 3. Schematic for the conversion from the triclinic coordinate axis system defined by xD, yD and zD to the orthonormal Cartesian coordinate axis system defined by the orthogonal vectors xC, yC and zC. E is the intersection of the axis yC with the top (0 1 0)D surface, while F is the intersection of the axis zC with the [1 0 0]D vector linking zD to D, so that the vector OD is [1 0 1]D.
Philosophical Magazine 3 2 a uc 4 vc 5 ¼ 6 40 wc 0 2
b cos c bsinxb
b cos c Þ bðcos acos sin b
32 3 c cos b uD 0 7 5 4 vD 5 wD c sin b
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ð1Þ
with
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x¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 cos2 a cos2 b cos2 c þ 2cosa cosb cosc
ð2Þ
Hence, to four decimal places, [1 0 0]D = [7.2291 0 0]C, [0 1 0]D = [1.6187 9.9066 1.6505]C and [0 0 1]D = [3.6138 0 10.0422]C. Point E in Figure 3 is the intersection of the Cartesian coordinate y-axis with the upper surface of the unit cell, while F is the intersection of the Cartesian coordinate z-axis with the vector linking the vector zD to the point D in the (0 1 0)D plane in which xD and zD lie. Defining OA = a′, OE = b′ and OF = c′, it is apparent that, in units of Å, a′ = 7.2291, b′ = 9.9066 and c′ = 10.0422. 3.2. Construction of the twin model 3.2.1. Overall description In this study, we used a personal computer operating in a Windows XP environment to run DL_POLY_4. This is a general purpose parallel molecular dynamics simulation package developed at Daresbury Laboratory in the UK by Smith and Todorov [23,24]. In seeking a balance between the number of atoms in the simulations and the time taken for the simulations to show stable twin structures, simulations were run for not more than 105 time steps when it was evident that they were achieving a stable twin structure. Such simulation runs took a total of almost 11 h for the equivalence in volume of 32 unit cells (1728 atoms) of devitrite. For those geometries where it was evident that a stable configuration could not be obtained, and where instead running the simulations produced highly unstable, highly energetic and highly unlikely atomic configurations, with significant atomic movements, simulations were usually terminated after 2000 time steps, again using a sample size equivalent in volume to 32 unit cells of devitrite. For a number of these unstable simulations, simulations were continued for 104 – 105 time steps, but all of them remained unstable, in the sense that they remained of high energy and with significant atomic movements that did not readjust themselves into a lower energy state with smaller atomic movements after more time steps had been taken. For each initial twin boundary configuration prior to the subsequent molecular dynamics simulation runs, the construction of the twin models was broken down into three separate procedures. Firstly, the origin of the unit cell of devitrite was moved so that the position of the twin plane under consideration (either the (0 1 0)D plane for the Type II twinning mode or the six different Type I planes specified in Section 2.2)) was a surface of the newly redefined unit cell. This newly defined unit cell was then replicated using translational symmetry to construct a grouping of 4 2 2 unit cells representing the matrix and containing 864 atoms in total. Finally, atom positions in the twin were obtained from those in the matrix using the appropriate geometric transformation of either a rotation of 180° around the [1 0 0]D position vector defined as in Figure 2 for the Type II twinning, or reflection in the twin plane for the Type I twins. A schematic of the combination of these three procedures is shown in Figure 4 for the Type II twins. We
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Matrix
[1 0 0] C,D Rotation of 180°
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Twin
Figure 4. Schematic showing the production of the twin block of atoms from the matrix block of atoms in the simulation models for the Type II twins.
chose to have four unit cells projected along [1 0 0]D as shown in Figure 4, with blocks of 2 2 unit cells either side of the twin boundary then defining the (1 0 0)D plane, from a consideration of the limitations imposed by how many unit cells we could realistically incorporate into our initial twinning configurations to have sensible computation times to reach stable configurations after the molecular dynamics runs. Periodic boundary conditions were imposed at the boundaries of the grouping of ions used to model the various twin boundary configurations considered. Thus, while in the individual grouping, the twin plane ran through the middle, the periodic boundary conditions meant that twin boundaries were also produced along the two external planes of the grouping which were parallel to the twin plane. Thus, for example, for the Type II twinning mode, periodic boundary conditions applied were ‘parallelepiped boundary conditions’ as defined by DL_POLY_4, with the ‘unit cell’ being repeated being one where the basis vectors of the ‘unit cell’ were [4 0 0]D, [0 4 0]D and [0 0 2]D with respect to the ‘matrix’ side of this ‘unit cell’. Similar parallelepiped boundary conditions were applied to the simulations of the Type I twins, for which the basis vectors of the ‘unit cell’ are given in Section 4.4. 3.2.2. Translation of the origin The translation of the origin is easy to appreciate (Figure 5). For each twinning mode considered, six possible positions of twin plane, P0 … P5, as in Figure 1, were considered. The fractional coordinates of the atoms (ions) within the conventional description of a single centrosymmetric unit cell can be defined with respect to the Cartesian coordinates using (1), with the two origins of the Cartesian coordinate system and unit cell coordinate system at O1 in Figure 5. Periodic continuation then enables the Cartesian coordinates of the positions of the atoms in the entire 4 2 2 unit cell grouping to be determined, all with O1 as the origin. Suppose we now wish to move the origin of the unit cell to O2 in Figure 5(a) so that the twin plane position under consideration passes through O2. Let this be the plane
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O3 O3
O2
xC
xC
O2
O1 O1
zC
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zC
Figure 5. Schematic for the reconstruction of the structure of the matrix. For the Type II twins, the block of the matrix represented by the solid lines in (a) with O1 as its origin is four unit cell lengths along the xD axis, two unit cell lengths along the yD axis and two unit cell lengths along the zD axis.
Pi where 1 i 5 and let O2 have a position vector [u2 v2 w2]C relative to the Cartesian set of coordinates in Figure 5(a). With respect to O2 as origin, as in Figure 5(b), it is evident that, for atom coordinates with Cartesian coordinate positions [u0 v0 w0]C in the 4 2 2 unit cell grouping above or at the plane Pi, their new coordinate positions with O2 as origin become ½u v wC ¼ ½u0 v0 w0 C ½u2 v2 w2 C
ð3Þ
For atom coordinates in the 4 2 2 unit cell grouping below Pi with Cartesian coordinate positions [u0 v0 w0]C, their new coordinate positions with O2 as origin are translated by [0 2 0]D to be in the 4 2 group of unit cells immediately above O3 in Figure 5(a), so that they become ½u v wC ¼ ½u3 v3 w3 C þ ½u0 v0 w0 C ½u2 v2 w2 C
ð4Þ
where [u3 v3 w3]C [3.2374 19.8132 3.3010]C is the representation in the Cartesian set of coordinates in Figure 5(a) of the vector [0 2 0]D, i.e. the vector joining O1 to O3 in Figure 5(a). 3.2.3. Construction of an orthorhombic ‘supercell’ It is evident that the block of 4 2 2 unit cells in Figure 5 containing the 864 ions with O2 as the origin is triclinic in shape. While it is possible in principle to use this shape and rotate it by 180° about [1 0 0]C || [1 0 0]D to produce the coordinates of atoms in the twin for the Type II twinning operation, it is much easier to do this computationally using an orthorhombic shape, such as the ‘supercell’ outlined by dashed lines in Figure 6 with lengths 4a′, 2b′ and 2c′ as shown along the positive xC, yC and zC directions, respectively.
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Figure 6. Orthogonal unit cell with sides of length 4a′, 2b′ and 2c′ along the xC, yC and zC axes, respectively, identified within the block of 4 2 2 unit cells of the triclinic structure of the matrix in Figure 5.
Just as when discussing the shift of origin in Section 3.2 and how it is accomplished, periodic continuation in the matrix enables atom positions to be specified within this orthorhombic shape. Clearly, for atoms (ions) with coordinate positions [u v w]C determined through the procedure described in Section 3.2, where u, v and w are all positive, no adjustment is required. However, while all the v coordinate positions are necessarily positive, both u and w can be < 0 for atoms within the triclinic block. Dealing with the coordinate positions for which w < 0 first, a shift by [0 0 2]D = [7.2276 0 20.0844]C ensures by periodic continuation that equivalent positions can be established for which v and w are both > 0 in the Cartesian coordinate system. Secondly, for all those coordinate positions where now u < 0, a shift by [4 0 0]D = [28.9164 0 0]C then determines the required coordinate positions of all atoms (ions) within the ‘supercell’ defined by the lengths 4a′, 2b′ and 2c′ shown in Figure 6. 3.2.4. Formation of the twins by rotation of the matrix ‘supercell’ for Type II twinning It is evident from Figure 2 that, in general, the position of the [1 0 0]D rotation axis for Type II twinning intersects the z-axis of the unit cell of devitrite away from the origin. Such a position can be represented by the large dot in the diagram shown in Figure 7, in which the axis xC || xD, about which the rotation is taking place, points into the plane of the paper, and so xC points out of the plane of the paper. The ‘matrix’ from which the ‘twin’ is created is shown as the darker shaded block of 2 2 supercells, i.e. the 4 2 2 supercells seen in projection along xC. Suppose we consider the rotation about xC || xD about the Xj axis, as in Figure 2 where 0 j 9. This axis intersects the zC axis at a position jc′/10. A rotation of 180° of the matrix supercell about this axis then produces the twin supercell in (a), seen
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xC
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yC
zC
xC
(a)
zC
(b)
Figure 7. Schematic showing the formation of the twin by a rotation of the matrix of 180° about the axis Xj for Type II twins parallel to xC. In both (a) and (b), the axis xC points into the plane of the paper. Lattice translation periodicity is invoked in the conventional triclinic unit cell after the rotation in (a) to produce the lightly shaded block defining the twin to populate the atom (ion) positions in the twin shown in (b).
lightly shaded. It is apparent that a feature at a position [u v w]C in the matrix supercell produces a feature at its twin position [ut vt wt]C, where ut ¼ u; vt ¼ v; wt ¼ w þ jc0 =5
ð5Þ
The translational periodicity of the triclinic crystal structure can then be used to define the contents of the twin at positions shown in Figure 7(b) for which 0 vt 2b0 and 0 wt 2c0 . We now have starting coordinates of atoms (ions) in the matrix and twin for our choice of twin plane and rotation axis positions to model the observed Type II twinning operation by molecular dynamics simulation. 3.2.5. Formation of the twins for Type I twinning For Type I twinning the procedures followed in Sections 3.2.2 and 3.2.3 enable the relevant coordinates of atoms (ions) in the twin to be found relatively straightforwardly: for a particular [u v w]C, the twin position generated was [ut vt wt]C, where ut ¼ u; vt ¼ v; wt ¼ w
ð6Þ
enabling the starting coordinates of atoms (ions) in both matrix and twin to be readily established for the subsequent molecular dynamic simulation runs.
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3.2.6. Matrix and twin combined cell sizes for DL_POLY_4 The procedure described to model the observed Type II twinning mode in Sections 3.2.1 – 3.2.4 produced a final combined cell size of four unit cell lengths along the xDaxis, four unit cell lengths along the yD-axis and two unit cell lengths along the zD-axis relative to the crystal axes in the matrix. Thus, with respect to a unit cell chosen to be the matrix, the vectors defining the volume of material subjected to molecular dynamics simulations were [4 0 0]D, [0 4 0]D and [0 0 2]D, i.e. relative to the Cartesian coordinate system defined in Figure 3, the vectors [28.9164 0 0]C, [6.4749 39.6265 6.6020]C and [7.2277 0 20.0845]C, respectively. Likewise, we defined the row vectors with respect to the crystal and Cartesian coordinate axis systems defining the volume of material subjected to molecular dynamics simulation for the six Type I twinning modes that we have considered. As-constructed and relaxed configurations of these twinning modes are discussed in Section 4.4 where the dimensions of the cells used to model these twinning modes are specified. 4. Molecular dynamics simulation One of the advantages of molecular dynamics is that it gives a route to dynamical properties of the system, especially about time-dependent responses to perturbations [25–29]. This was used here to evaluate the stability of different ionic configurations for the various twin modes considered in this study. In addition, the energies of the postulated twin boundaries, cb , can be calculated from the difference between the computed energy of the perfect crystal and the computed energy of the crystal with a twin plane: cb ¼
ET E0 Ab
ð7Þ
where, ET is the configurational energy of the structure with a twin plane, E0 is the configurational energy of the structure without any twin plane and Ab is the calculated area of the twin boundary. Depending on the specific location of the twin plane relative to the axes for the conventional centrosymmetric unit cell, ion–ion distances could be unacceptably close for the initial starting configurations for the simulation runs. Under these circumstances, the molecular dynamic simulation quickly became unstable after only a few time steps, leading to clearly unrealistic movements of ions supposedly in an attempt to find a stable twin boundary structure. In order to try and achieve stable twin boundary structures for both the Type II twin and the Type I twin simulation runs, slight local expansions were introduced at the twin boundary, so that starting coordinates of all the ions in the twin were each moved no more than 2.0 Å perpendicular to the twin boundary. Reassuringly, and significantly, for the Type II twin boundary structures with the lowest energy, essentially the same relaxed atom structure could be obtained after a suitable number of iterations with or without a small initial local expansion at the twin boundary. However, it was always the case that the introduction of an initial small local expansion at the twin boundary produced the lowest energy configuration for a given PiXj. Translations of the twin relative to the matrix parallel to η1 were not studied for the various Type II twin boundary structures.
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Interatomic potential parameters used in the simulations [34,35].
Interaction
A (kJ mol1)
ρ (Å)
C (kJ mol1 Å6)
Ca2+O2 Na+O2 Si4+O2 O2O2
105,207 122,231.1 123,878 2,196,384
0.3437 0.3065 0.3205 0.1490
0.0 0.0 1028 2690
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4.1. Potential The potential used in the current study to model short range forces was the Buckingham pair potential used by others to study ionic materials such as oxides and silicaceous minerals [30–32]: rij C U ðrij Þ ¼ A exp ð8Þ 6 rij q where A, ρ and C are variables and where rij is the distance r between the ith and jth ions in the material. In this model, the first term is a Born-Mayer repulsion term, while the second term is an attractive term representing the relatively weak dispersion forces [33]. The relevant interatomic potential parameters of A, ρ and C for Ca2+O2, Na+O2, Si4+O2 and O2O2, taken from the work of Purton et al. [34,35] are listed in Table 1. While we have retained the nomenclature ‘Si4+O2’and ‘O2O2’ for the silicon–oxygen and oxygen–oxygen potentials as used by workers in the minerals field [34,35], it should be noted that these two potentials derive from the work of Sanders et al. on quartz [36]. Sanders et al. were careful to state that, although their model used formal ionic charges as a convenient way of describing correctly the cohesive properties of quartz, the use of formal ionic charges in a potential model does not require that the electron distribution in the solid should correspond accurately to this description, a point also reinforced by Catlow and Stoneham [37]. For this reason, Sanders et al. simply used the terms ‘SiO potential’ and ‘OO potential’ when describing the variables they determined for the Buckingham potential, recognizing that the silicate network in quartz is generally considered to be covalently bonded, with some degree of ionicity, as has been discussed by Catlow and Stoneham. The robustness of these parameters for devitrite was tested by molecular dynamics simulation of a block of perfect crystal with a cell size of four unit cell lengths along the xDaxis, four unit cell lengths along the yD-axis and two unit cell lengths along the zD-axis. This was used subsequently as a reference structure with which to compare twin structures. As we have already mentioned in Section 3.2, the simulation work was performed using DL_POLY_4 [23,24]. Conditions chosen for the simulation work were atmospheric pressure (0.1 MPa) and a temperature of 298 K, with a time step, t, in the calculations between 0.1 and 1 fs. 4.2. Simulation for a perfect devitrite crystal The model for a block of 4 4 2 triclinic unit cells of perfect devitrite crystal without any twin was constructed by the replications of single unit cells. The structures before and after 105 time steps of the molecular dynamics simulation are shown in Figure 8.
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After 10 5 time steps
As-constructed
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Ca 2+ -
Na + -
O 2- -
Si 4+ -
Figure 8. (Colour online). The perfect crystal without a twin before and after simulation. In this simulation run and the simulations in Figures 9 and 10, the projection direction is [1 0 0]D || [1 0 0]C. [0 0 1]C is a vector parallel to the short side of each of the two parallelograms shown here. The direction labelled ‘c’ in the caption parallel to the short side of the parallelogram in the plane of the paper is the projection onto the (1 0 0)C plane of [0 0 1]D. The direction labelled ‘b’ in the caption parallel to the long side of the parallelogram is the projection onto the (1 0 0)C plane of [0 1 0]D.
P 0X0 as-constructed
P 0X0 after 105 time steps
P0X5 as-constructed
P 0X5 after 105 time steps
Ca 2+ -
Na + -
O 2- -
Si 4+ -
Figure 9. (Colour online). Structures for Type II twinning with stable twin modes for devitrite before and after molecular dynamics simulation. The twin plane is (0 1 0)D and is shown by the dashed line in the two as-constructed models and the two models after 105 time steps. Periodic boundary conditions ensure that the two external boundaries of the atom grouping parallel to this twin plane are also (0 1 0)D twin boundaries. It should be noted that the choice of ‘unit cell’ for the molecular dynamics simulation does not produce an outer surface of this ‘unit cell’ perpendicular to the projection direction. The effect of this is to produce some apparent breaks in symmetry in the above diagrams, and also the diagrams in Figure 10 and Figures 12–17, as artefacts arising from projection of the contents of the ‘unit cell’ onto the plane of the page.
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Philosophical Magazine
1595
P 0X1 as-constructed
P 0X1 after 2000 time steps
P 5X0 as-constructed
P 5X0 after 2000 time steps
Ca 2+ -
Na + -
O 2- -
Si 4+ -
Figure 10. (Colour online). Structures for Type II twinning with unstable twin modes for devitrite before and after molecular dynamics simulation. The twin plane is (0 1 0)D and is shown by the dashed line in the two as-constructed models and the two models after 2000 time steps. Periodic boundary conditions ensure that the two external boundaries of the atom grouping parallel to this twin plane are also (0 1 0)D twin boundaries.
In this figure, [1 0 0]D || [1 0 0]C is pointing out of the plane of the paper, with four unit cells along this direction. [0 0 1]C is parallel to the short side of each of the two parallelograms shown here. The direction labelled ‘c’ in the caption is the projection onto the (1 0 0)C plane of [0 0 1]D and the direction labelled ‘b’ in the caption is the projection onto the (1 0 0)C plane of [0 1 0]D. Therefore, from Section 3.1 and Figure 3, the direction labelled ‘c’ is parallel to [0 0 1]C, while the direction labelled ‘b’ is parallel to [0 9.9066 1.6505]C, so that the angle between these two projections of [0 1 0]D and [0 0 1]D is 99.46°, rather than α. It is apparent that, although there are very small (
