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J. Phys. Chem. C 2007, 111, 6724-6731
Pressure-Driven Phase Transitions in Crystalline Nanoparticles: Surface Effects on Hysteresis Benjamin J. Morgan* School of Chemistry, Trinity College, UniVersity of Dublin, Dublin 2, Ireland
Paul A. Madden† School of Chemistry and Centre for Science at Extreme Conditions, UniVersity of Edinburgh, Edinburgh, UK, EH9 3JJ ReceiVed: February 21, 2007; In Final Form: March 15, 2007
Single domain nanocrystals in a dense rock salt-structured (B1) polymorph of a material that adopts the wurtzite (B4) structure under ambient conditions may be prepared by pressurizing a suspension of wurtzite nanoparticles. Here the way the particle surface affects the reVerse transition (B1 f B4) on depressurization is examined in molecular dynamics simulations: it is shown to affect the degree of hysteresis in the phase transition and the possibility of recovering metastable single domain nanocrystals of the dense polymorph at low pressures. B1-structured nanocrystals with a hexagonal prismatic shape, which have been formed by previous pressurization simulations of B4 f B1 transitions, show a single surface nucleation event and subsequent ripening to form single domain products with a retention of the crystallographic orientation. Cubic B1-nanocrystals with only {100}B1 surfaces exposed, which represent well-annealed particles of the highpressure phase, transform in a two-step process with multiple surface nucleation events and delayed growth into the particle interior to form complex multigrain products. The local atomic rearrangement mechanism and the domain growth process are described in detail, and how these interact with the morphology of the starting crystal to determine the internal geometry and morphology of the product low-pressure particles is examined.
I. Introduction Many materials of technological interest, such as II-VI and III-V semiconductors, have tetrahedrally coordinated crystal structures under ambient conditions. Nanocrystalline samples of such materials are of particular interest because the choice of particle size allows controlled modification of electronic, optical, and thermodynamic properties.1 Materials with tetrahedral coordination readily undergo phase transitions to denser six-coordinate structures under the application of pressure. In particular, many sphalerite (zinc blende) and wurtzite (denoted B3 and B4) structured materials have been observed to undergo such pressure-driven phase transitions to the rock salt structure (B1). These phase transitions are also observed in samples consisting of nanocrystals contained in an inert pressurization medium and have been studied extensively, notably by Alivisatos and co-workers.2-5 The nanoparticle transitions observed are the consequence of single nucleation events, in contrast with bulk transitions which are often initiated at grain boundaries etc. Furthermore, the thermodynamic and kinetic factors governing the transitions are seen to be strongly influenced by surface effects.5,6 These observations make the relationship between nanoparticle and bulk transitions an interesting topic for fundamental studies. At a more practical level, it is also of interest to examine the factors which might allow one to prepare nanocrystalline samples of the high-pressure structure at ambient pressure, where they would be metastable. Rock salt-structured * Address correspondence to this author. E-mail:
[email protected]. † E-mail:
[email protected].
nanoparticles of a semiconductor would have different optical (and other) properties to tetrahedrally coordinated structures of the same material and could have distinct technological applications. Decremps et al. have demonstrated the practicality of this suggestion in preparing nanoparticles of rock saltstructured ZnO which are metastable under ambient conditions.7 It has been found that the transition pressures observed for nanoparticles exceed the bulk transition pressures and this has been attributed to surface effects.8 Nanocrystals are expected to adopt morphologies which minimize their total free energies which, for a given crystal structure, prescribes some equilibrium distribution of low-energy surfaces. During a solid-solid phase transition atoms follow particular trajectories, according to the transition mechanism appropriate to that crystal structure and material. This means that during such a transformation it is highly likely that the product crystallite will be formed with a nonoptimum set of surfaces, and is unable to optimize its shape. Thus the surface contribution to the free energy of the as-formed nanocrystal will be greater for the product polymorph than for the original, equilibrium particle. This results in a greater pressure being required before the transition is thermodynamically favorable, relative to the bulk. However, this may be only part of the reason for the variation in the observed transition pressure. Even though a transition is feasible at the (elevated) thermodynamic transition pressure, there is typically a kinetic barrier to be overcome, so the transition only occurs on an observable time scale if the applied pressure is elevated beyond the thermodynamic transition pressure. Similarly the return transition only occurs at pressures below the thermodynamic
10.1021/jp0714670 CCC: $37.00 © 2007 American Chemical Society Published on Web 04/14/2007
Phase Transitions in Crystalline Nanoparticles transition pressure, resulting in a hysteresis loop in any real experiment. In observations on CdSe nanoparticles, Alivisatos et al.3,4 reported a hysteresis loop with a width of ∼6 GPa relative to a limiting thermodynamic transition pressure of 2.3 GPa. This hysteresis loop narrowed at elevated temperatures and the position of the hysteresis loop was found to be controlled by the size of the nanocrystals.5 The hysteresis might allow the formation of nanoparticles in metastable structures following a pressurization-depressurization cycle. Metastable B1 CdSe crystals in a specific size range were recovered at ambient conditions in the Alivisatos et al. studies.3,5 If a sample was to be annealed at high pressure following the pressure-driven transition, the particle morphology may change so that an equilibrium arrangement of low-energy surfaces for the denser product phase is achieved. In the case of the B1 structure this is a cube with six equivalent {100}B1 surfaces. Now the thermodynamic surface considerations discussed above apply to the downstroke transition, and the thermodynamic pressure for the reverse transition will be lowered. If the hysteresis loop is wide enough the practical reverse transition pressure may become negative, so that the nanoparticles will remain in the high-pressure structure for a very long time under ambient conditions.3 Furthermore, a different distribution of surfaces may lead to an otherwise favored transition pathway (such as the reverse of the path adopted on the upward transition) being blocked, resulting in an alternative pathway being adopted. For example, in the case of the B4 f B1 transition, high-energy B1 surface regions are formed (see below) and are good candidates as sites of nucleation events for the reverse transition, but are likely to be absent in an annealed structure. This interesting interplay of thermodynamic and kinetic factors, as well as surface and bulk free energies, has attracted a number of simulation studies,9-11 which have so far focused largely on the upward transition. In these studies nanocrystals in the size range of experimental interest (2-5 nm, containing several thousand atoms), described with more-or-less realistic interaction potentials, are embedded in an atomic pressurization medium and subjected to an increase in applied pressure until a transition is observed. The simulations have confirmed, with a high degree of consistency, that the local atomic rearrangement mechanisms seen in the nanocrystalline phase transitions are the same as those found in the corresponding bulk transitions and have detailed the consequences for the morphology of the nanoparticles of the product phase. In particular, we have studied both the B3 f B112 and the B4 f B111 transitions for the same interaction potential and have traced the very different morphologies of the product particles to the sequence of elementary steps occurring in the atomic rearrangement mechanism.13 Although we use a simple ionic model (which favors a wurtzite bulk crystal structure) for the interactions we find the mechanisms of the transitions are the same as those found with more sophisticated potentials designed to model real materials such as CdSe and BeO.9,14 In accordance with experiment, we find an increase in the transition pressure (by about 4 GPa) relative to the thermodynamically predicted bulk transition pressure (about 5 and 6 GPa for B3 T B1 and B4 T B1, respectively). Despite the very rapid rate at which pressure is applied in the simulations relative to real experiments, the increase in the nanoparticle transition pressure over the bulk is therefore not much greater than that observed in experiments. In the present paper we will discuss the reverse transitions which occur as the applied pressure is decreased. In particular, we will examine the effect of the surface morphology on the
J. Phys. Chem. C, Vol. 111, No. 18, 2007 6725 characteristics of the reverse transition by contrasting the behavior of ideally annealed rock salt particles with rock salt particles prepared by pressurizing wurtzite particles and not allowing long annealing times. We will describe how these different surface preparations influence the nucleation of the reverse transition and trace the consequences for the morphologies and structure of the product particles. Since the free energies of the sphalerite and wurtzite structures are so similar, some tendency to return to a sphalerite-structured particle, or at least to form stacking faults, might be expected. For the unannealed particles, we will be interested in the extent to which the pathway of the depressurization transition is the reverse of that of the upward transition. If this were the case, one would expect a symmetrical hysteresis loop and a restoration of the wurtzite particle in its original, as prepared shape. The upstroke transition in B4 f B1 transitions was observed to take place via surface nucleation and subsequent growth in several directions throughout the nanocrystal. A downstroke transition pathway, the exact reverse of this upstroke, would require nucleation in the bulk, with subsequent growth, terminating at a surface, or multiple simultaneous surface nuleation events which converged in a specific manner. Singular surface nucleation would prevent an exact reverse pathway, but the transition could still occur via a sequence of the same local atomic rearrangements. II. Background The simulation methodology and interaction potential have been described in the previous publications.11,12 In short, a crystalline nanoparticle is suspended in a binary Lennard-Jones atomic fluid, which constitutes the pressurization medium, and the external pressure is gradually changed by using the thermostatted/barostatted MD algorithm due to Martyna et al.15 In the present runs, the thermostat temperature is set at 500 K and the pressure is periodically decreased by -0.074 GPa every 24 ps. This rate of pressure variation is extremely rapid compared to experimental rates, but is slow enough that thermal and hydrostatic equilibrium between the particle and pressurization fluid are achieved before each successive pressure decrease. During the course of the simulations nanocrystal configurations were periodically selected and used for structural analysis. To reduce any thermal disorder that might obscure underlying structural features, these configurations were thermally quenched. This consisted of performing a few steps of a conjugate gradient minimization until an energy of (3/2)NkT had been removed (equivalent to the harmonic thermal contribution to the potential energy). The interaction potential is a simple ionic model that has a wurtzite (B4) ground-state crystal structure with a sphalerite (B3) phase at a slightly higher energy. The zero temperature bulk transition pressures between these tetrahedral phases and rock salt (B1) are 5.7 GPa for B4 T B1 and 4.8 GPa for B3 T B1.16 We consider two generic starting particle morphologies: several runs have been performed on particles of different size from these two families with results very similar to the particular trajectories we describe in detail below. The first morphology is a distorted hexagonal prism that was found to be the product morphology in the upstroke B4 f B1 simulations.9-11,17 Such post-transition nanocrystals all have two low-energy (001)B1 surfaces terminating the z direction, and perpendicular surfaces ranging from (100)B1 to (110)B1 with intermediate indices corresponding to a range of stepped surfaces. xy cross-sections of such particles are illustrated in Figure 3 below. The starting configurations were taken from the outputs of the upstroke
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Figure 2. Time evolution of the ion-ion potential energy and number of octahedrally coordinated cations as a 3360 ion nanocrystal undergoes a B1 f B4 transition. The time origin in this figure is the same as in Figure 3. The vertical red and blue lines broadly contain the actual transition, and correspond to the same times as the identically colored diffraction curves in Figure 1.
Figure 1. Calculated powder diffraction patterns obtained from periodically quenched configurations from a 3360 ion nanocrystal. The nanocrystal transforms from rock salt back to wurtzite at 2.80 GPa. The red and blue patterns have been obtained at a pressure of 2.8 GPa and indicate the boundaries of the time period marked by the same colored lines in Figure 2.
simulations, and the precise distribution of surfaces of each nanocrystal is a consequence of the previous trajectory. The upstroke transition occurred at ∼11.0 GPa and we started the depressurization simulations at an applied pressure of 12 GPa after allowing 100 ps for equilibration following a reassignment of all particle velocities and thermo/barostat parameters: no reorganizations of the particles were observed at these pressures during this period and direct annealing, in the manner suggested by Decremps et al.,7 proved impracticable in much longer simulations even at elevated temperatures. The second morphology we consider is that of cubic B1 nanoparticles which have only low-energy {100}B1 surfaces, which represent ideally annealed structures. These particles were constructed from an ideal rock salt structure with a density appropriate to a pressure above the transition and then embedded in the pressurization fluid and equilibrated at this pressure. There are therefore two major differences between the two particle families. First, as noted, the high-pressure particles produced via a simulated B4 f B1 transition have high-energy surface facets, which are absent in the idealized nanocrystals with cubic morphologies. Second, the high symmetry of the rock salt crystal structure is broken in these particles because of their overall shape and the differing nature of each set of exposed surfaces. The posttransition particles retain a “memory’’ of the orientation of the original hexagonal wurtzite structure, with the [001]B1 direction inequivalent to the perpendicular [100]B1 and [010]B1, whereas the second particle type has the full cubic symmetry of the underlying crystal. III. Results: Hexagonal Prismatic B1 Morphology Figure 1 shows a series of calculated powder diffraction patterns for a 3360 ion B1 nanocrystal created by pressurizing a B4 structured nanocrystal, as described above. The diffraction patterns are from a segment of the depressurization run (initiated at 12 GPa) in which the external pressure is varying over the range of 2.94 to 2.71 GPa and during which the actual transition
occurs. These diffraction patterns have been calculated from quenched ion configurations obtained every 6.04 ps throughout the simulation. The patterns nearest the onset and completion of the transition are highlighted in red and blue, and correspond to time delays of 36.3 and 60.0 ps within this segment. The lowermost pattern indexes to a B1 structure indicating that at the start of the segment the cluster is still in the rock salt structure, despite now being below the thermodynamically predicted transition pressures (5.7 GPa for B1 to B4 and 4.8 GPa for B1 to B3 for the bulk crystals at 0 K). At ∼2.8 GPa this pattern changes to give a curve that indexes to the B4 structure. As the pressure is further reduced, the peaks move to smaller scattering angles as the lattice continues to expand. Comparing the peak widths in the product B4 curve with those for the pre-transition B1 nanocrystal shows that the crystalline domain size is unchanged, indicating that a perfect B4 nanocrystal has been formed without any stacking faults or grain boundaries. Figure 2 shows the ion-ion potential energy and number of octahedrally coordinated cations, n(Oh), through a longer portion of the same simulation. The region to the left of the vertical dashed line covers the same period as the diffraction patterns shown in Figure 1, while the vertical red and blue lines broadly contain the actual transition, and correspond to the identically colored diffraction curves in the same figure. At around 36 ps the potential energy and the number of octahedrally coordinated cations both begin to fall. By 60 ps n(Oh) has also fallen to zero, and the rate at which the ion-ion potential energy decreases has fallen. After this point, although the transition is complete, this energy continues to decrease at a more gradual pace, as the newly formed B4 structure relaxes. As indicated by the calculated diffraction patterns, the transition occurs at a pressure of 2.8 GPa, which is well below the pressure required for the forward wurtzite to rock salt nanocrystalline transition for the same potential model that was observed in the range 10.0-11.6 GPa.11 The thermodynamically predicted bulk B4 T B1 transition pressure for the potential model used here is 5.7 GPa16 at 0 K. This indicates considerable hysteresis, as has been seen in the experimental studies of this behavior.3,5 If the midpoint of the up- and downstroke transitions is considered then this gives a thermodynamic transition pressure for this 3360 ion system that is elevated relative to the bulk thermodynamic pressure, in agreement with experimental results.8
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Figure 4. An example of differences in bond opening and closing in the six-membered ring a 2 × 3 square net transition producing different structures and dangling bonds. The blue arrows denote the bonds that are forming (in the case of the first step) or breaking (in the second).
Figure 3. Sequence of molecular dynamics snapshots showing a slice through a 3360 ion B1 nanocrystal, previously formed from the compression of a B4 crystal, as the pressure is gradually reduced. The pressure during this sequence is decreased from 2.9 to 2.7 GPa. The structures indicated by the red and blue rectangles correspond to the times marked by the vertical lines in Figure 2 and to the diffraction patterns of the same colors in Figure 1.
A. Nucleation and Growth: Consequences for Nanocrystal Morphology. Figure 3 shows a series of molecular dynamics snapshots of two of the (001)B1 layers from the 3360 ion nanocrystal as it undergoes the reverse transition, viewed along the [001]B1 direction. The upper and lower right-hand images show the slices after 36.3 and 60.0 ps, respectively, and correspond to the red and blue curves and lines seen in the previous figures. Broadly speaking, the pattern of local atomic rearrangements we see is the reverse of those seen in the forward B4 f B1 transition.11 The latter has been discussed in detail by Saitta and Decremps18 and consists of two orthogonal sets of motions. The first of these is a flattening of the rumpled layers of the 6-membered “chair’’ conformation rings which make up the basal planes of the wurtzite structure, and constitutes ionic motion entirely along the [0001]B4 direction. The second, orthogonal, set of motions consists of the closing up of pairs of opposing vertices within these six-membered rings, forming pairs of square rings which together make a 2 × 3 rectangle. This motion in the ab plane converts the unit cell from hexagonal to orthorhombic, and these square rings form the (001)B1 planes in the product B1 structure. These motions can take place simultaneously, or with one preceding the other, depending on the precise local pathway adopted by the material in question. In all the nanocrystal compression simulations conducted thus far the transition process occurs via the so-called h-MgO intermediate, corresponding to flattening of the ab chairs before the ab angle closes from 120° to 90°. The reverse transition (B1 f B4) involves the reopening of the (001)B1 2 × 3 pairs of square rings to form hexagons, and this process can clearly be seen from the atomic motions in the sequence of images shown in Figure 3. This is accompanied by a puckering motion in the direction perpendicular to the plane of the figure, to restore the basal plane chair conformations of the B4 structure. This local pattern of atomic rearrangements is topologically the same as that predicted by Tolbert and
Alivisatos8 in their studies of CdSe nanoparticles, where they considered possible transition pathways as symmetry breaking phenomena analogous to a Peierls distortion.19 The consequences for the shape and structure of the product nanoparticle depend on the particular point in the crystal from which this sequence of rearrangements was initiated. In the figure, the first 2 × 3 rectangle that opens to form a hexagon appears on the (100)B1 surface of the nanocrystal that faces the top of the page when viewed in this orientation in the 32.0 ps snapshot. This ring opening occurs at a [110]B1 step in this surface, as can be seen by comparing with the 0.6 ps image. The opening of this six-membered ring can be clearly seen to produce a large strain across an adjacent 2 × 3 rectangle, in the [110]B1 direction. By 36.3 ps four rings along this vector away from the step edge have opened up and a fifth can be seen to be in the process of opening. In addition a further [110]B1 row of six-membered rings has opened up on the right-hand side of the first row. It should be noted that at 0.6 ps there are two open hexagons in the lower left corner of the image, but that these then quickly close up again and do not play a nucleation role in the transition. This can be viewed as a failed nucleation event: successful nucleation is rare on the time scale of the simulation and so only single nucleation events are observed, which leads to transformation without the formation of grain boundaries. The formation of each [110]B1 line of six-membered rings produces considerable strain in the adjacent bonds, which in turn causes the next layer of 2 × 3 rectangles (above and below the one shown in the figure) to open up. This process propagates across the whole nanocrystal from a single point, until it has completed after 60.0 ps. The complete reverse transition is slower than the forward B4 f B1 transition, taking approximately 30 ps compared with 7 ps.11 This pattern of isolated nucleation at a [110]B1 step at the surface of the nanocrystal, followed by growth as ring opening spreads from this region, has been observed generally for all of the reverse transitions of this type that were simulated. Although the observed local atomic rearrangement can be described as the reverse of that involved in the forward B4 f B1 transformation, the low-pressure B4 nanocrystal that is produced does not have the same hexagonal prismatic shape as the original cluster, which was cleaved from a bulk lattice to have only {0001}B4 and {12h10}B4 surfaces in an approximation of an equilibrium morphology.20 For the original B4 shape to reform, all of the 2 × 3 square nets must open up by separating the same pairs of counterions that came together in the original B4 f B1 transition. In general this is unlikely as a priori a pair of square rings can open up with motions along either the [100]B1 or [010]B1 axes. The consequences for the morphology of the nanoparticle recovered from a pressurization-depressurization cycle are illustrated as a 2D schematic in Figure 4. The first (left-hand side) transition illustrates the transformation of a basal plane of the wurtzite structure from the original B4
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Figure 5. The upper panel shows the ion-ion potential energy throughout the simulation of a 2744 ion cubic B1 nanocrystal as the applied pressure is reduced from 2.9 to 0.8 GPa. The lower panel shows the number of octahedrally coordinated cations in the nanocrystal during the same period. The two vertical lines indicate the end and beginning, respectively, of the two periods of time covered by the diffraction patterns in Figure 6.
particle on the upstroke transition into the square net structure of the (001)B1 via the ring closure step. The second transition indicates the consequence of opening a different set of bonds in the downstroke transition to return to B4. It can be seen that a different set of surfaces, now containing dangling bonds, terminate the basal plane. Such surfaces are evident in the final frame of Figure 3. Despite a change of shape, the B4 product nanocrystals have the same crystallographic orientation as the starting crystal. Even though bulk B1 crystals are cubic, the high-pressure particles retain a memory of the hexagonal structure from which they were formed in the properties of their surfaces. Nucleation takes place at the stepped (110)B1 surfaces, which are present as a result of the mechanism for the earlier B4 f B1 transition. There are no stepped (011)B1 or (101)B1 surfaces, which means the (001)B1 surfaces of the crystal are nonequivalent to the perpendicular (100)B1 and (010)B1 surfaces. This nonequivalence remains through the reverse transition, because the hexagonal close-packed direction that appears from the nucleation lies in a plane orthogonal to the stepped surface containing the nucleation site. This results in recovered B4 crystals having the same c direction orientation as the initial low-pressure crystals. IV. Results: Cubic B1 Nanoparticle To see how changing the character of the exposed surfaces of the B1 nanocrystal affects the transition pressure and the morphology of the product particle we additionally simulated cubic particles, considered to represent ideally annealed particles of the high-pressure structure. In the previous section we saw that the transition nucleated at a step edge, and such defects will now be absent ,which ought to block such nucleation events, thus decreasing the pressure at which a transition is observed. The sample consisted of a 2744 ion rock salt particle, formed by cleaving a bulk configuration along six equivalent {100}B1 surfaces to produce a perfect cube consisting of 7 × 7 × 7 unit cells. This nanocrystal was embedded in the binary LennardJones fluid, which had already been equilibrated at a pressure of 3 GPa, i.e., above the pressure at which the hexagonal prismatic nanocrystals described above underwent the B1 f B4 transition. The system was equilibrated in a long run at 3 GPa and the crystal showed no tendency to transform. The rate of pressure change for these simulations was -0.074 GPa every 14.51 ps, and the methodology used was the same as described above.
Figure 6. Diffraction patterns calculated during the simulation of a 2744 ion nanocrystal as the pressure is reduced from 2.9 to 0.8 GPa. The lower panel shows diffraction patterns from the first 130.6 ps, i.e., the period covered in Figure 5, and the upper panel shows the final 141.5 ps. The subscripts cub and hex refer to assignments made using the expressions given for patterns due to cubic and hexagonal patterns respectively. In both panels the separation in time of adjacent diffraction patterns is 15.5 ps.
The variation in the ion-ion potential energy and number of octahedrally coordinated cations throughout the simulation are both shown in Figure 5. This figure covers the period of the simulation in which the pressure was reduced from 2.9 to 0.8 GPa. At the start of this period the number of octahedrally coordinated cations is approximately constant at around 864. This does not include all the cations: the total number in the nanocrystal is 1372 since those that lie on the {100}B1 surfaces are discounted in this analysis due to being undercoordinated. This value remains unchanged until after 80 ps when, at a pressure of 2.42 GPa, there is a sudden drop to a value of ∼400. After this point there is a continuing more gradual decrease in n(Oh), which falls to around 200 by 360 ps, at which stage the pressure is 0.96 GPa. n(Oh) then begins to fall more rapidly, reaching zero by 375 ps. As in the previous simulations the ion-ion potential energy shows the same general trend as n(Oh) with two fairly rapid decreases at around 80 and 370 ps. It is clear from the comparison of these figures to the corresponding ones for the hexagonal prismatic nanocrystals that the presence of different surfaces of the B1 nanocrystal has had a substantial effect on the six- to four-coordinate transition. A further difference is reduction in the pressure at which the transition takes place. The B1 f B4 transition seen earlier occurred in a concerted process over a pressure range of 2.9 to 2.7 GPa, whereas the {100}B1 faced cubic B1 nanocrystal is unchanged down to 2.4 GPa. Furthermore, there are still octahedrally coordinated cations present in the nanocrystal at a pressure of 0.96 GPa. The shapes of the curves in Figure 5 also appear to indicate a more complex process than seen previously,
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Figure 8. Schematic showing how 2 × 3 square nets opening on the {100}B1 faces of the cubic nanocrystal lead to the surface curvature observed in the simulations.
Figure 9. A sequence of molecular dynamics snapshots showing a [100]B1 slice through the center of the 2744 ion cubic nanocrystal over the first 140 ps of the simulation discussed in the text. These images should be compared to those in Figure 7 to provide a comparison between the behavior of the surfaces and the interior of the nanocrystal during this period.
Figure 7. A sequence of molecular dynamics snapshots showing one of the {100}B1 surfaces during the first 90 ps of the simulation of the 2744 ion cubic nanocrystal discussed in the text.
suggesting two distinct events taking place. Again we stress that the particular trajectories we have discussed are representative of the results of a much larger set of simulations for both starting structures. Figure 6 shows two series of calculated diffraction patterns, both taken from the same simulation trajectory, covering the two transition regions evident from Figure 5; the patterns are calculated at intervals of 15.5 ps. The lower panel shows the first 139.5 ps (corresponding to the period of time up to the left-hand vertical dashed line in Figure 5) and the bottom pattern of these (at 2.9 GPa) indexes to the B1 structure. When a pressure of 2.42 GPa is reached, the [200] peak broadens and the [220] peak decreases in intensity. This new pattern does not then change substantially until 280 pssa pressure of 1.3 GPaswhich corresponds to the bottom pattern in the upper panel. As the pressure is further decreased the structure of the broad peak at ∼2.2 Å-1 is seen to split into two peaks, one of which moves to slightly lower values of k and increases slightly in intensity. At a pressure of 0.96 GPa the pattern undergoes another substantial change and can now be indexed to either a B3 or B4 structure. The single peak at low k at first suggests a B3 structure, since the recovered B4 structures seen earlier under the same simulation conditions had a visible splitting of the [100] and [002] peaks. However, the [102] peak seen here would be absent in a cubic pattern and its presence indicates a hexagonal crystallographic cell. The peaks here are broadened with respect to the initial pattern and also to the single domain B4 pattern seen in Figure 1, corresponding to a decrease in the size of the crystalline domains. This broadening makes an exact measurement of the c/a ratio difficult since the [100] and [002] peaks may be smeared on top of one another. However, the single peak seen here is still narrower than the splitting of the two peaks seen in the relaxed single domain B4 nanocrystals
(Figure 1), suggesting that this structure, at some level, has a greater degree of equivalence between the three crystallographic directions. A. Mechanistic Details. Figure 5 suggests a two-step mechanism for the transition from the cubic nanoparticle. The two stages can be elucidated by analyzing the ionic trajectories throughout the simulation, and will be discussed in turn in what follows. 1. Surface Ring Opening. Figure 7 shows one of the {100}B1 surfaces of the nanocrystal during the first 90 ps of the period of the simulation described above, i.e., until approximately 10 ps after the first rapid decrease in both the ion-ion potential energy and n(Oh). After a period of 82.8 ps the square net structure has begun to open up to give a hexagonal net. This is the same process as seen in the transition of the hexagonal prism, except that, in the absence of a nucleation site like the [110]B1 step seen in Figure 3, the process first occurs in the center of the {100}B1 faces. After a further 8 ps nearly the entire surface has transformed in this fashion. Each surface of the nanocrystal has a choice of two equivalent directions in which bonds can open up, and whichever orientation for the ring opening is chosen leads to an elongation along that direction and a narrowing across the perpendicular direction, which combine to give a waisted “hourglass’’ shape, as is illustrated schematically in Figure 8. Snapshots of a (100)B1 slice containing two atomic layers lying parallel to the surface discussed above, taken through the center of the nanocrystal during the same period of the simulation, are shown in Figure 9. Like the surface layer, this plane has also developed a distorted shape, although here the structure can still be described as a distorted square net, since it has not yet undergone bond-breaking. In contrast to the hexagonal prismatic nanocrystal seen earlier, the initial decrease in Uion-ion does not correspond to a complete B4 f B1 transition; here there is only ring opening at the {100}B1 surfaces and the strain generated by this change does not seem to be sufficient to promote a rearrangement of the internal bonds. In addition to atomic motion in the plane of each of the surfaces the initial surface rearrangement also consists of motion perpendicular to the surface plane. Figure 10 shows how one of the {001}B1 surfaces of the nanocrystal has buckled into a
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Figure 10. Molecular dynamics snapshots of the bottom (100)B1 surface of the cubic nanocrystal. The upper panel shows the surface at a pressure of 2.9 GPa under which conditions the rock salt structure is still stable. By the time the pressure has been reduced to 2.3 GPa the square net that characterizes the surface has opened up to give the hexagonal net seen in the lower figure. This process takes place on all six faces and causes the surfaces to adopt “saddle-shaped’’ geometries.
Figure 11. Schematic of how the changes in shape of the nanocrystal {100}B1 surfaces due to ring opening are related, producing saddle shaped surfaces in the intermediate structure. The blue arrows form convex surfaces and the pairs of opposing red arrows indicate those surfaces that develop a concave curvature.
“saddle’’ shape. The other surfaces of the nanocrystal undergo the same shape change with a definite relationship between the direction of ring opening on connected surfaces, as illustrated in Figure 11. This allows all six surfaces to undergo ring opening simultaneously, while minimizing the distortion of the interior of the crystal. Consideration of the strain generated by these coupled surface deformations on the interior of the crystal allows one to understand the change in shape of the internal planes illustrated in Figure 9. The shape of an interior plane is dictated by the distortions of the perpendicular surfaces. 2. Ring Opening in the Nanocrystal Interior. Figure 12 shows the same internal (100)B1 plane as seen in Figure 9, covering the period with the second drop in n(Oh). Even at the beginning of this sequence, the bounding surfaces of these planes have already transformed into hexagonal nets in the process described above, and the strain on the individual bonds close to the surfaces is evident. At approximately 300 ps bonds that lie near the nanocrystal surface begin to break, in doing so producing
Morgan and Madden
Figure 12. A sequence of molecular dynamics snapshots showing the (100)B1 slice through the center of the nanocrystal, as seen in Figure 9, at a later time. This sequence corresponds to the time period covered to the right of the second dashed line in Figure 5 and by the upper panel of diffraction patterns in Figure 6. By this point the external pressure is low enough that bonds lying perpendicular to the nanocrystal surfaces begin to break. This happens nearly simultaneously in two different regions, resulting in a mismatch between the two grains in the center of the final image.
six-membered rings. Two such regions can be seen to form: one in the top right corner of the slice and one in the lower left. As the simulation continues each of these regions grows, seemingly with continuous [110]B1 rows opening at the same time, and with the same local orientation, as was seen in the noncubic B1 nanocrystal discussed in section III.A. The development of these two regions results in the formation of two grains and the boundary between them is made up of a combination of four- and eight-membered rings. This grain boundary is similar in bond geometry to the d-BCT boundary structure observed in relaxed B4 nanocrystals.21 In a situation such as this, where crystal ripening develops from two points, it is possible for a single coherent grain to form but only if there is a particular relationship between the sets of bonds which are involved in the initial bond-breaking sequences. Otherwise isolated four-membered rings are left which can either undergo ring-opening with an adjacent six-membered ring to give an 8-ring or are stuck as they are. When considering the entire three-dimensional nanocrystal the situation becomes even more complicated. The first stage of ring opening at the surfaces produces a template for the subsequent growth of hexagonal regions into the center of the nanocrystal. At each individual surface the [001]B1 direction becomes a local [0001]B4 direction, which then grows into the nanocrystal. This takes place at all six of the surfaces producing hexagonal regions with three different close-packed orientations, and leads to the formation of areas with complex bonding geometries between the more easily defined product closepacked regions. It is this granular structure of the transformed nanoparticles that leads to the broadening of the diffraction peaks illustrated in Figure 6. Since the surface ring opening dictates the orientation of these grains, the local [0001]B4 direction within them is perpendicular to the nearest surface.
Phase Transitions in Crystalline Nanoparticles V. Concluding Remarks To illustrate the microscopic events which underlie the influence of the surface preparation of nanoparticles on the degree of hysteresis observed in pressure-driven phase transformations and on the morphology of product particles we have compared representative simulation trajectories for decompression of two families of nanoparticles initially in the high-pressure rock salt (B1) structure. Decompression of the rock salt-structured nanocrystals with a hexagonal prismatic morphology, obtained from the forward B4 f B1 transitions,11 results in a reverse transition to single domain B4 nanocrystals at a pressure of around 2.8 GPa. A single nucleation event with subsequent growth of the B4 region results in the crystals transforming as single grains. The particles modeled here are small enough that a new phase can propagate across the whole nanocrystal on a time scale that is shorter than the time between successive nucleation events. The crystallographic orientation of these product particles is the same as those with which the pressurization-depressurization sequences were begun, but the particle shape has in general changed. Examination of the transition pathway suggests that a return to the exact starting particle shape is unlikely. However, conservation of the crystallographic orientation is a consequence of the fact that the exposed surfaces of the B1 particle retain a memory of the starting wurtzite crystal orientation due to the absence of (101) or (011) surfaces (even after annealing on the simulation time scale) and that the local atomic rearrangement on the downstroke is the reverse of the upstroke mechanism. The reverse transition pressure is well below both the upstroke transition pressure of 10.7 GPa11 and the thermodynamically predicted bulk pressure of 5.7 GPa,16 showing considerable hysteresis. The midpoint of the forward and reverse transition pressures is thus at 6.7 GPa, which indicates an increase in thermodynamic transition pressure on going from bulk to these finite systems, in agreement with the trends seen experimentally.8 Alivisatos et al. observed a hysteresis loop of ∼6 GPa width in studies of CdSe particles in the same size range, where the thermodynamic transition pressure is 2.3 GPa.3,4 Since hysteresis is a kinetic effect, associated with the long waiting time for a nucleation event, it is not surprising that we observe a broader hysteresis loop than in the experimental studies since our (de)pressurization rate is much larger than the experimental one: nevertheless, in all other respects, the phenomena we observe in the simulation studies are in accord with the experimental findings. In the absence of annealing (as above) the reverse transition was found to nucleate at a (110)B1 surface exposed by the forward transition process. To simulate the effect of annealing the particles in the high-pressure phase so as to eliminate such unstable surface features, we also prepared and depressurized ideal cubic B1 nanoparticles on which only relatively low-energy {100}B1 surfaces are exposed. In this case the B1 phase was seen to remain metastable to lower pressures until an alternative pathway became accessible, mirroring the experimental observations of Decremps et al.,7 who were able to prepare metastable rock salt-structured ZnO nanoparticles at ambient pressure following high-pressure annealing. In our case, the cubic particles did not complete the reverse transformation to the four-
J. Phys. Chem. C, Vol. 111, No. 18, 2007 6731 coordinate phase until the pressure had dropped to 0.96 GPa, considerably lower than that of the unannealed particles. The alternative nucleation pathway involves the {100}B1 surfaces; because the surfaces are all equivalent this leads to all six trying to expand in the plane simultaneously to form hexagonal nets which seem to be able to coexist with a rock salt-structured particle interior. The hexagonal nets increase the total surface area of the nanocrystal, but the interior remains unchanged until a considerably lower pressure, and so each surface has to adopt a curved “saddle-shape’’, which allows each transformed surface to fit with all of its neighbors, and minimizes the expansion of the interior. The stability of this structure gives the transition from the cubic particle a two-step character and, as the pressure is further reduced, the interior of the nanocrystals finally transforms at 0.96 GPa where it opens up to form fourcoordinate regions. These appear as local extensions of the B4 regions already formed at the crystal surfaces. Because the surface regions have three orthogonal B4 orientations this results in a product four-coordinate nanocrystal with different [0001]B4 orientations and mismatched grain boundaries in the interior giving a complex multidomain product. Thus both the degree of metastability and the transition mechanism (as it affects the product particle morphology) have been shown to be affected by the character of the exposed surfaces on the high-pressure nanocrystals. Acknowledgment. This work was supported by EPSRC grant GR/R57584/01. References and Notes (1) Alivisatos, A. P. J. Phys. Chem. 1996, 100, 13226-13239. (2) Tolbert, S.; Alivisatos, A. P. Science 1994, 265, 373-376. (3) Chen, C.-C.; Herhold, A. B.; Johnson, C. S.; Alivisatos, A. P. Science 1997, 276, 398-401. (4) Jacobs, K.; Zaziski, D.; Scher, E. C.; Herhold, A. B.; Alivisatos, A. P. Science 2001, 293, 1803-1806. (5) Jacobs, K.; Wickham, J.; Alivisatos, A. P. J. Phys. Chem. B 2002, 106, 3759-3762. (6) Zhang, H.; Gilbert, B.; Huang, F.; Banfield, J. F. Nature 2003, 424, 1025-1029. (7) Decremps, F.; Pellicer-Porres, J.; Datchi, F.; Itie´, J. P.; Polian, A.; Baudelet, F.; Jinag, J. Z. Appl. Phys. Lett. 2002, 81, 4820-4822. (8) Tolbert, S. H.; Alivisatos, A. P. J. Chem. Phys. 1995, 102, 46434656. (9) Gru¨nwald, M.; Rabani, E.; Dellago, C. Phys. ReV. Lett. 2006, 96, 255701. (10) Lee, N. J.; Kalia, R. K.; Nakano, A.; Vashishta, P. Appl. Phys. Lett. 2006, 89, 093101. (11) Morgan, B. J.; Madden, P. A. Phys. Chem. Chem. Phys. 2006, 8, 3304-3313. (12) Morgan, B. J.; Madden, P. A. Nano Lett. 2004, 4, 1581-1585. (13) Wilson, M.; Hutchinson, F.; Madden, P. A. Phys. ReV. B 2002, 65, 094109. (14) Cai, Y.; Wu, S.; Xu, R.; Yu, J. Phys. ReV. B 2006, 73, 184104. (15) Martyna, G. J.; Tobias, D. J.; Klein, M. L. J. Chem. Phys. 1994, 101, 4177-4189. (16) Wilson, M.; Madden, P. A. J. Phys.: Condens. Matter 2002, 14, 4629-4643. (17) Gru¨nwald, M.; Dellago, C. J. Mol. Phys. 2006, 104, 3709-3715. (18) Saitta, A. M.; Decremps, F. Phys. ReV. B 2004, 70, 035214. (19) Burdett, J. K.; McLarnan, T. J. J. Chem. Phys. 1981, 75, 57645773. (20) Hamad, S.; Cristol, S.; Catlow, C. R. A. J. Phys. Chem. B 2002, 106, 11002-11008. (21) Morgan, B. J.; Madden, P. A. Phys. Chem. Chem. Phys. 2007, DOI: 10.1039/b701267e.
