Nanocrystals as Model Systems for Pressure-Induced Structural Phase Transitions

1529-6466/01/0044-0002$05.00 DOI:10.2138/rmg.2001.44.02

2 Nanocrystals as Model Systems for Pressure-Induced Structural Phase Transitions

Keren Jacobs and A. Paul Alivisatos

Department of Chemistry University of California–Berkeley

Berkeley, California 94720

INTRODUCTION

An understanding of first-order pressure-induced structural transitions is relevant to many research areas in geological and planetary sciences because it involves the study of materials exposed to high pressures. For example, solid-solid transitions in silicates are responsible for the seismic discontinuities in the earth's mantle (Chudinovskikh and Boehler 2001) and may play a role in plate tectonics and deep earthquakes (Kirby et al. 1991). In geological applications, models of structural transition kinetics simulate rock formation taking place over millions of years (Shekar and Rajan 2001). Despite their importance in earth science applications, the microscopic processes of solid-solid phase transitions are difficult to study in the bulk solid for several reasons (Putnis 1992). In extended solids, the transformation nucleates at defects, which are present at equilibrium even in the highest quality crystals. As a transformed region of the crystal grows larger, mechanical forces generate new defects, which in turn act as new nucleation sites. The difficulties in bulk are compounded by the irreversibility of the kinetics, which depends strongly on the preparation of the sample and its history. The study of first-order phase transitions can be greatly simplified in nanocrystal systems because small crystals can behave as single structural domains and reproducibly cycle through multiple transitions (Wickham et al. 2000). To illustrate the advantages of the simple kinetics in nanoscale systems in exploring fundamental questions of structural phase transitions, this chapter focuses on the CdSe nanocrystal system. Nanocrystalline solid-solid transitions in geologically relevant material such as Fe2O3 nanocrystals are also now being studied (Rockenberger et al. 1999), and a comparable understanding of these materials is a current research goal.

The CdSe nanocrystals are a particularly well-controlled and characterized test system (Tolbert et al. 1995). The nanocrystals consist of hundreds to thousands of covalently bonded atoms, synthesized as nearly defect-free crystals with narrow size distribution and controlled shape. At low pressure, the atoms are tetrahedrally bonded and form a hexagonal network of stacked sheets with wurtzite and zinc blende (its mineral name is sphalerite) stacking sequences. At high pressures, the nanocrystals contract by 18% as they transform to a denser six-coordinate rocksalt structure. As in most inorganic solid-solid transitions, there is a substantial barrier to the structural transition at room temperature. To overcome the energetic barrier, the pressure must be raised beyond the thermodynamic transition pressure in the forward transformation, and lowered in the reverse. The result is an experimental hysteresis loop, which reflects the time needed for the system to relax to the equilibrium configuration. In the bulk solid, this barrier is usually related to the fragmentation of the bulk solid into multiple domains during the transition. Nanocrystals are substantially smaller than these domains and are essentially defect free, so that the barrier arises from the microscopic motions of atomic reorganizations. The area of the hysteresis loop provides an initial idea of the energies involved in the transformation process, because it represents the irreversible energy

60 Jacobs & Alivisatos

released in a PV transition cycle. For nanoparticles with a hysteresis width of ~6 GPa, the energy release is estimated to be several orders of magnitude larger than thermal energy at room temperature. This is an indication that in structural transformations, the energy barrier is much larger than the thermal energy accessible in most laboratory experiments.

SINGLE-STRUCTURAL DOMAIN

Experiments on nanocrystal transitions are usually done on ensembles of nearly monodisperse nanocrystals coated with a surfactant allowing them to be dissolved in a solvent (Murray et al. 1993, Peng et al. 1998). The solvent, such as ethylcyclohexane, also acts as the pressure-transmitting medium. The nanocrystals are spatially isolated and separated, and ideally each nanocrystal transforms independently of the others. Pressures are applied using a diamond anvil cell (DAC) and are measured using standard ruby florescence techniques (Barnett et al. 1973) (see Fig. 1). Static temperature can be achieved by placing the cell in a resistively heated ceramic oven, and measured with a thermocouple in contact with the diamond. In the CdSe nanocrystal system, a static temperature limit of 550 K exists because at higher temperatures interparticle diffusion can occur. This limit is close to the synthesis temperature, but should in general depend on the capping group and solvent, and can be determined by the narrowing of X-ray diffraction peaks signifying crystal growth and aggregation.

Figure 1. Schematic of an experimental setup for high-pressure X-ray diffraction (XRD) experiments on nanocrystals under pressure of tens of GPa (1 GigaPascal ≅ 10,000 atmospheres). Nanocrystal powder is dissolved in a pressure medium and placed between two opposing diamonds, which are clear to X-rays and visible absorption. The pressure is applied by bringing the diamond closer together, and measured using pieces of ruby chips placed inside the pressure cell. The XRD diffraction peaks are Debye-Scherrer broadened by the finite size of the particles, yielding information on the shape and size of the nanocrystals before and after the transition, such as shown in Figure 3.

The structural transition is directly monitored using X-ray diffraction (XRD) (see Fig. 1), by observing the disappearance and appearance of the characteristic peaks. The limits of this method are mainly from the small sample size of ~100 microns from a solution of a few nanoliters, compressed between two diamonds of ~1-mm thickness. The small physical size of the nanocrystal produces Debye-Scherrer broadening of the diffraction peaks (Cullity 1956), and very concentrated samples are required (optical

Nanocrystals: Models for Pressure-Induced Structural Phase Transitions 61

density of 3-4) for sufficient signal-to-noise. Since XRD requires the focused, bright X-rays available only from a synchrotron light source, a more practical method is often desired. The transitions can often be correlated to optical or vibrational changes in UV-vis or Raman spectroscopy. Samples prepared for these methods are at much lower concentrations (optical density of 0.1-1), allowing nanocrystals to be separated by several lengths.

Several independent observations have led to the conclusion that nanocrystals, at least for the CdSe system, transform as a single structural domain. Single domain behavior produces a macroscopic shape change of the nanocrystal during the transition, discussed in the next section; and is consistent with the observation of simple kinetics behavior, discussed in the Kinetics section. A direct method for observing the microscopic structure of nanocrystals is high-resolution transmission electron microscopy (HRTEM). Electron microscopy is routinely used in viewing samples after synthesis, but cannot currently be done inside the pressure cell. However, recovered samples can be retrieved from the pressure cell after they have undergone phase transition cycles and returned to the initial four-coordinate phase (Jacobs et al. 2001). It must be ensured that the recovered sample is from inside the pressure cell and is not residue from the initial loading.

Figure 2 shows HRTEM images of large representative particles before and after a transition cycle. The nearly isoenergetic wurtzite and zinc blende structures are arranged as stacks in the nanocrystal. The higher fraction of zinc blende stacks in the recovered sample is consistent with interpretations of X-ray diffraction simulations (Wickham et al. 2000). The stacking faults are a particularly simple type of grain boundary, but do not interrupt the connectivity of the atoms in the nanocrystal. Well-defined microscopic structures and volumes are involved in the nanocrystal transition, since the tetrahedral bonding between Cd and Se atoms remains uninterrupted even after pressure cycling.

Figure 2. High-resolution transmission electron microscope image of representative CdSe nanocrystals in the four-coordinate structure before and after pressure cycling. The arrow indicates the unique c-axis, along which (001) layers are stacked. Two possible stacking sequences, ABAB- and ABCABC-, yield the nearly isoenergetic wurtzite (hex-agonally close-packed) and zinc blende (cubic close-packed—f.c.c.) structures within the nanocrystal. Wurtzite stacking is indicated with solid lines; zinc blende stacking with dashed lines. After pressure cycling, the nominally spherical shape recovers and the nanocrystal has an increased fraction of zinc blende layers. The stacking faults do not interrupt the connectivity of the atoms in the nanocrystal [Used by permission of the editor of Science, from Jacobs et al (2001), Fig. 1].


SURFACE EFFECTS AND SHAPE CHANGE

The surface energy can play a dominant role in determining the relative thermodynamic stability of two structural nanocrystal solids, as discussed in the chapters by Banfield and Zhang and Navrotsky (this volume). Differences in surface energy between two polymorphs are important in the study of nanocrystalline phase transitions. For example, in melting studies on a wide variety of nanocrystal materials, a depression in melting temperature is observed with decreasing size (Goldstein et al. 1992, Coombes 1972, Buffat and Borel 1976). The data is explained by the notion that the liquid phase is stabilized relative to the solid because surface energy contributions are minimized in liquids. This idea also explains the crossovers in thermodynamic stability at the nanoscale from one structure to another in Navrotsky and Zhang chapter (this volume). Smaller crystallites preferentially favor polymorphs with surfaces of relatively low energy, as illustrated in the nanocrysaline alumina systems (McHale et al. 1997). In solid-solid transitions in nanocrystals, the differences in surface energy between the two crystal structures produces a shift in the thermodynamic transition pressure as a function of nanocrystal size. For example, an observed shift to lower pressure with increasing nanocrystal size in CdSe nanocrystals is explained by higher surface energy in the high-pressure rocksalt structure (Tolbert et al. 1995; Jacobs, in preparation). High-energy surfaces (different from those in an annealed particle) are exposed as a result of the macroscopic shape change accompanying the transformation process, because the experimental temperatures used were too low for surface rearrangements to occur after the transformation. The resulting thermodynamic shift can be significant and should be taken into consideration in interpreting experimental transition pressures as a function of crystal size and in comparison to bulk.

The actual macroscopic shape change has been determined to our knowledge in only CdSe and Si nanoparticles. Experimental determination of shape changes is inferred from XRD patterns because the crystallographic lengths of the crystallite are given by the Debye-Scherrer broadening of diffraction peaks. Si nanoparticles of 500 Å in diameter were found to transform from spherically shaped to an ellipsoidal shape along the (001) axis (Tolbert et al. 1996). In the case of CdSe and InP and CdS, the high-pressure phase is cubic and each diffraction peak is a convolution of the different crystallographic axis. By simulating the XRD patterns, the shape change in CdSe was inferred from the relative widths and intensities in the XRD patterns before and after the transformation. The reason this method works is because the simulated patterns show strong sensitivity towards structural parameters, most importantly to the crystal shape. From anomalously high intensities of the 111 reflection in rocksalt, it was determined that the CdSe nanocrystals covert from a nominally spherically shaped four-coordinate crystal to a slab-shaped rocksalt structure with one set of (111) planes parallel to the long axis (Wickham et al. 2000) (see Fig. 3). The shape change exposes high-energy {111} faces in rocksalt and accounts for the relative instability of the rocksalt structure in small CdSe crystallites.

THERMODYNAMIC SMEARING

It has been implied that the surface of the nanocrystal acts as a perturbation of the total nanocrystal free energy. The assumption has been that, from a thermodynamics viewpoint, the nanocrystal behaves merely as a fragment of the extended solid. However, evidence suggests that structure throughout a nanoparticle in very small particle size may be perturbed relative to the bulk (see chapters by Banfield and Zhang and Waychunas, this volume). Although the thermodynamic properties of the nanocrystal ensemble are well defined, structural phase transitions are rigorously defined only in an infinite medium. A lower limit must exist for the number of atoms required for the crystal in


Figure 3. Shape change in CdSe nanocrystals found by comparing simulated powder X-ray diffraction (XRD) patterns to experimental data (Wickham et al. 2000).

practice to undergo a structural phase transition, rather than more closely resembling the limit of a molecular isomerization. A general discussion of this issue is presented from a theoretical standpoint in Thermodynamics in Small Systems (Hill 1963). We will outline a parallel interpretation and provide some experimental numbers (Jacobs et al. 2001) showing that CdSe nanocrystal particles contain enough atoms for the study of structural phase transition, despite their finite size.

In small systems, smearing of the thermodynamic transition pressure can arise from thermal fluctuations between the structures (see Fig. 4). In the smallest CdSe nanocrystals in the studies (25 Å in diameter, ~300 atoms), the total volume change during the transition is ~ 1500 Å3. Because the differential of free energy with respect to pressure is volume, the corresponding thermal fluctuations are estimated to smear the nanocrystal transition by less than 0.1 GPa at 500 K (Jacobs et al. 2001). This smearing is insignificant compared to the kinetics-induced hysteresis width in CdSe nanocrystals of ~6 GPa, and implies that the transition is observed “far” from equilibrium even in nanocrystals. The insignificance of the smearing in sizes >20 Å should also apply to geophysically relevant nanoparticles such as Fe2O3, which although have smaller volume changes ~7%, have very large hysteresis >30 GPa. The simplest way to view the above calculation is in terms of the Gibbs equilibrium condition, more commonly used for molecular reactions:

ΔG = −k BT ln K( ) (1)

where ΔG is the free energy difference between the two structural phases, kBT is thermal energy (Boltmann’s constant times temperature), and K is the equilibrium constant for the population ratio in the two phases at equilibrium. The transformation of reactants to products in molecular reactions occurs over a range of equilibrium conditions, because substantial thermal population of both phases is possible with finite free energy differences between the phases. Equation 1 is used to estimate the thermal smearing in the nanocrystal transition by substituting the free energy term with its pressure-volume


Figure 4. Schematic showing a hysteresis loop for the CdSe nanocrystals with the smearing of the thermodynamic transition pressure caused by the finite nature of the nanocrystal particle. The thermodynamic transition pressure is offset from the hysteresis center to emphasize that in first-order solid-solid transformations, this pressure is unlikely to be precisely centered. The lower plot shows the estimated smearing for CdSe nanocrystals as inversely proportional to the number of atoms in the crystal, at two temperatures, as discussed in the text. Note that nanocrystals are not ordinarily synthesized or studied in sizes smaller than 20 Å in diameter. This figure shows that this thermal smearing is insignificant compared to the large hysteresis width in the CdSe nanocrystals studied (25-130 Å in diameter), such that the transition is bulk-like from this perspective. This means that observed transformations occur at pressures far from equilibrium, where there is little probability of back reaction to the metastable state once a nanocrystal has transformed. In much smaller crystals or with larger temperatures, the smearing could become on the order of the hysteresis width, and the crystals would transform from one structure to the other at thermal equilibrium.

equivalent (∂ΔG = ΔV∂ΔPt, ΔV is extensive) and defining smearing as the pressure change relative to the thermodynamic transition pressure (ΔPt) needed to convert 90% of the nanocrystal population to the thermodynamically stable phase. With this substitution, the smearing scales inversely with the number of atoms in the particle (proportional to the absolute volume change) and increases with thermal energy. In a bulk phase transition, the (extensive) free energy difference diverges except at a single pressure where there is equal population of both phases (K = 1). The estimate for minimal smearing in the CdSe nanocrystal transition is consistent with the large energies released in a hysteresis cycle at room temperature, mentioned in the Introduction. If a crystal were still definable for smaller sizes where the smearing is appreciable relative to the hysteresis, the structures


would flip back and forth at thermal equilibrium. Structural fluctuations of this type are thought to occur in some gas-phase inorganic cluster species (Berry and Smirnov 2000).

TRANSFORMATION KINETICS: ACTIVATION VOLUME

The hysteresis loop is considered a dynamic property because the kinetics of activated processes should depend on the observation time. The limit of long observation times is important in geophysics for simulating processes occurring over geological time scales. Kinetics experiments are the most important tool in this respect because the study of hysteretic transitions is otherwise limited to measurements taken under non-equilibrium conditions.

Elementary transition state theory provides a general framework for investigating simple chemical reactions and transformations. Transition state theory is based on assigning quasi-equilibrium variables to a transition state with the highest energy along the mechanistic pathway. The most relevant parameter for pressure-induced transitions is the activation volume, or the volume change between the transition state and the reactant. This parameter gives information on the structural changes that occur during a transition, and so is particularly useful for reactions involving significant atomic rearrangements. Experimentally, the activation volume, ΔV‡, is determined from the pressure-dependence of the rate constant:

∂ln(k)∂P

⎛ ⎝

⎞ ⎠ T

=−ΔV‡

kBT (2)

where kBT is the thermal energy, P is pressure, and k is the rate constant for the relax-ation. This expression is the thermodynamic expression for Gibbs free energy with the Arrhenius rate law, and was first used by Evans and Polyani (1935) to explain the intrinsic pressure-dependence of rate constants for activated molecular processes. The activation volume can have an appreciable effect on transition kinetics, as even small microscopic deformations on the order of ~Å3 can change the rate constant by several orders of magnitude over relatively small pressure changes (<1 GPa). The activation volume is routinely measured for molecular processes, although very little is known about this parameter in structural phase transitions. Since there is only one transformation event per nanocrystal, the activation volume is simpler to interpret in nanosystems than in extended solids (Onodera 1972). (The volume change is for one transformation—or Avagadro’s number of transformations, using the gas constant) Nanocrystals also allow for direct comparison of the forward and reverse transformation directions of the same sample because the transition is reproducible.

Extensive measurements of the kinetics to determine rate constants for the nano-crystal transition have been made only on the CdSe system (Chen et al. 1997, Jacobs et al. 2001). Both the forward and reverse transition directions have been studied in spherically shaped crystallites as a function of pressure and temperature. The time-dependence of the transition yields simple transition kinetics that is well described with simple exponential decays (see Fig. 5). This simple rate law describes the transformation process in the nanocrystals even after multiple transformation cycles, and is evidence of the single-domain behavior of the nanocrystal transition. Rate constants for the nanocrystal transition are obtained from the slope of the exponential fits. This is in contrast to the kinetics in the extended solid, which even in the first transformation exhibits complicated time-dependent decays that are usually fit to rate laws such as the Avrami equation.

The measurements of activation volumes and activation energies in both transition directions reveal that the forward and reverse transition kinetics differs. (These


Figure 5. Time-dependence of the transformation in 25-Å CdSe crystals at 463 K at different pressures. (A) The forward transition at Δ 5.2 GPa, 5.7 GPa, ◊ 6.9 GPa. (B) The reverse transition at ◊ 0.7 GPa, 1.0 GPa, Δ1.2 GPa. (1 GPa ≅ 10,000 atm) The abscissa is the intensity of the four-coordinate electronic absorption feature. Fits are single-exponential decays. Rate constants were obtained from the slope of the fitted lines, and are equivalent to relaxation times (= ln 2/k) in the forward transition of: Δ 21 min, 3.6 min, ◊ 20 s. In the reverse transition: ◊ 24 min, 3.8 min, and Δ 16 s. Each crystal in the ensemble transforms instantaneously relative to the experiment time such that the relaxation is a measure of the average time required to overcome the kinetic barrier. Once a nanocrystal transforms to the stable structure, it is statistically unlikely that it will fluctuate back unless the pressure is changed accordingly, because the transition is measured far from equilibrium. [Used by permission of the editor of Science, from Jacobs et al (2001), Fig. 2.]


parameters are calculated for experi-mental pressures, namely, “far” from the thermodynamic transition pressure.) For the case of 25-Å CdSe nanoparticles, the activation volumes are -17 Å3 and +65 Å3 in the forward and reverse transition directions, respectively (see Fig. 6A). The acti-vation energies (enthalpic barrier) are 0.9 and 0.2 eV/nanocrystal (88 and 19 kJ/mol of nanocrystals) in the forward and reverse transition directions, respectively. This is in line with the work of Rubie and Brearly (1994), who observed differ-ences in the two transition directions in bulk Mg2SiO4 by studying the transformation in samples synthe-sized as either one of the two structures. From the different activa-tion volumes, a smaller deformation during the forward transition in CdSe nanocrystals implies that the transi-tion state more closely resembles the low-pressure four-coordinate struc-ture than the high-pressure six-coordinate structure. A transition state that is not precisely midway between the two stable minima is reasonable in transformations be-tween phases with differing volumes. The observed asymmetry has profound consequences for the hysteresis loop because it implies that the thermodynamic transition pres-sure, which can be extrapolated to at very long observation times, is not in the center of the loop. By default, this thermodynamic pressure for struc-tural transitions is frequently taken as the center of the hysteresis (Li and Jeanloz 1987), under the assumption that the two transition directions are symmetric and equivalent (Shekar

and Rajan 2001). However, this pressure is observed to be markedly lower than the hysteresis center as the direct result of different activation volumes in the two transition directions. This result was also consistent with the different activation energies in the two transition directions in this CdSe nanocrystal transition, as evident from the greater sensitivity of the forward transition to temperature (see Fig. 6B). The asymmetry is reasonable considering the two structural phases are distinguishable, such that the forward and reverse transition can not be expected to be identical. Nevertheless, it is still difficult to say precisely where this pressure is in systems of different composition or

Figure 6. Hysteresis loop dependence on (A) relaxation time and (B) temperature for the transformation in 25-Å diameter CdSe nanocrystals. In (A), the rate constants are determined from time-dependent decays at 463 K, such as those shown in Figure 5. Activation volumes determined from the slopes using Equation (2) are -17 Å3 and +65 Å3 in the forward and reverse transitions, respectively. The activation volumes are assumed to be constant over the pressure ranges under study to simplify the analysis. In (B), the temperature data are compiled from measurements of the rate constant for ln (k) = -3.5 at the given temperatures. The inset is a sample hysteresis loop measured for 35 Å nanocrystals with the hysteresis width indicated. The loop starts at low pressure and proceeds in the direction of the arrows, as the normalized ratio of sample transformed is monitored [Used by permission of the editor of Science, from Jacobs et al (2001), Fig. 3].


how this relative position within the hysteresis loop depends on crystal size.

Additional information regarding the transition state was determined from opposite signs of the activation volumes in the two transition directions, positive in the reverse transition and negative in the forward transition (from the corresponding slopes in Fig. 6A and Eqn. 2). The opposite signs imply that the transition state in the CdSe nanocrystal transition must be intermediate in volume between that of the two structures, unlike a system that expands in both directions to accommodate the atomic rearrangements (see Fig. 6A schematic). This result is not expected for all solid-solid transitions. For example, the graphite-diamond transition in carbon has a fluid-like transition state (Shekar and Rajan 2001) that hinders the transition with increasing pressure and should therefore produce positive activation volumes in both transition directions. However, the case of an intermediate transition state volume observed for the CdSe nanocrystal transition is probably more common for transitions that exhibit conventional hysteresis loops (the forward transition is facilitated with increasing pressure). It is inherently implied in classical nucleation models for structural transitions (Christian 1965), in which the transition state is an average volume of both structures. In such models, the activation volume is proportional to the critical nucleus size because it is equivalent to the volume change that accompanies its formation. Nucleation theory also predicts that the critical nucleus size should vary with pressure, but is largest at the thermodynamic transition pressure. This premise is consistent with the experimental activation volumes, which are expected to increase in magnitude “near” the thermodynamic pressure in order for the forward and reverse activation volumes to add up suitably to the 18% total volume change across the transition.

The activation volumes in both transition directions is relevant to understanding the dependence of hysteresis loops on observation time, as indicated in Figure 6A. The hysteresis loop narrows with increasing observation time to a half-life corresponding to >10 years at room temperature, at the intercept of the two fitted lines. This is a lower limit because the assumption of constant activation volumes does not hold over larger pressure ranges (see the end of the previous paragraph). The actual relaxation time at the thermodynamic transition pressure may be much longer than 10 years even in the small nanocrystals. Nevertheless, in terms of geologically relevant transitions, this analysis shows that the hysteresis can depend on the observation time, and that this dependence can be understood in terms of magnitudes of the activation volumes.

TRANSFORMATION MECHANISM

The comparison of the CdSe transition to the diamond-graphite transition illustrates that there are several types of transformation mechanisms by which solids transform between polymorphs. How nanocrystals of a particular composition transform can be important by providing insight into the nature of the barrier and the dynamics in the extended solid. For example, if the nanocrystal were small enough to transform between solids in a coherent deformation mechanism, involving coordinated motion of the entire crystal lattice, this would imply a critical nucleus in the bulk larger than several nanometers. The transformation mechanism for CdSe nanocrystals has been the subject of many experimental and theoretical investigations (Moletini et al. 2001, Kodiyalam et al. 2001, Leoni and Nesper 2000). The major issue has been whether the transition is initiated locally in just part of the crystal or involves the entire nanocrystal in a coherent deformation event (Brus et al. 1996).

The microscopic transformation mechanism was recently determined for the CdSe nanocrsytalline transition, based on the kinetics measurements discussed in the previous section and the shape change discussed earlier in this chapter (Jacobs et al. 2001;


Figure 7. Hysteresis width versus nanocrystal size at room temperature and a relaxation time of ~3 minutes. The width for the largest sample is a lower limit because the 126-Å diam-eter nanocrystals remain trapped in the high-pressure rocksalt structure as the pressure is released (Jacobs, in preparation). In the inset, ΔS‡ is the entropic barrier and kB is Boltz-mann’s constant. [Used by permis-sion of the editor of Science, from Jacobs et al. (2001), Fig. 4.]

Wickham et al. 2000). The nanocrystal was determined to transform by a local nucleation event—a sliding-planes model involving the shearing motion of the (001) crystal planes (shown in Fig. 2) relative to one another. This model explains the variation in the stacking sequence along the c-axis before and after the transition in the TEM in Figure 2, because the transition is directionally dependent and not sensitive to the stacking along the c-axis (Osugi et al. 1966). The transition nucleates without preference at any of the planes along the c-axis, as inferred from the strongly increasing frequency factor (entropic factor) with size (see Fig. 7). Although none of the experimental observations alone is conclusive, the sliding planes model, established in the study of martensitic transformation (Porter and Easterling 1992), is the only model under consideration that is consistent with the entire set of data. The following is an outline of the kinetics data supporting the sliding planes mechanism. Activation volume

The microscopic mechanism in CdSe nanocrystals from which the hysteresis originates is constrained by the small magnitude of the activation volume of ~0.2%, in comparison to the 18% total volume change. The simplest model is a coherent deformation mechanism, as described above, in which the entire nanocrystal transforms in a single coordinated motion. However, this model is inconsistent with the activation volume, because 0.2% distributed throughout the nanocrystal is even smaller than the volume change caused by thermal vibrations of the crystal lattice. A nucleation mechanism involving just part of the crystal is more plausible because it requires only a modest structural deformation. A spherically shaped nucleus is excluded because the activation volume would correspond to a nucleus smaller than the size of a unit cell. (This size was calculated directly from the activation volume magnitude, given that the activation volume is proportional to the critical nucleus size and the overall 18% volume change between the two phases.) In addition, this case is inconsistent with the observation of an increasing activation volume magnitude from 17 to 34 and 124 Å3 in the forward direction for 25-, 30-, and 35-Å diameter nanocrystals, respectively. The sliding planes mechanism is consistent with the small magnitude of the activation volume, as well as with the increasing activation volumes with size, as the shearing of planes with larger diameter produces larger structural deformations during the transition.


Activation energy (enthalpic barrier)

Increasing activation energies of 0.5, 0.9, 1.2, 1.7, and 2.4 eV/nanocrystal have been reported for CdSe nanocrystal diameters of 20, 25, 27, 34, and 43 Å, for the forward direction (Chen et al. 1997, Jacobs et al. 2001). This increasing trend can be understood as arising from additional bonds that must break as the planes in larger particles slide. The measured activation energies are in remarkable agreement with estimates of the energy necessary to shear the (001) plane, using the stacking fault energy of 14 meV/atom (Takeuchi et al. 1984). For example, the activation energy of 0.9 eV in 25 Å nanocrystals (~50 atoms per plane) compares well with an estimate of 0.7 eV based on the transformation model. Although martensitic transformations are generally considered to be athermal, the shearing appears to be activated in this nanocrystal transition, perhaps because the role of defects has been eliminated. Entropic factor (entropic barrier)

The size dependence of the hysteresis width at room temperature has been measured in nanocrystal sizes up to 130 Å in diameter (see Fig. 7). The nearly constant hysteresis width of ~6 GPa over this size range suggests the corresponding free energy barrier responsible for the hysteresis is insensitive to nanocrystal size. The increasing entropic factor as a function of size accounts for the observed behavior because it offsets the increasing activation energy with size. This is uncharacteristic of a coherent deformation mechanism, for which the hysteresis width should strongly broaden with size without increasing entropic factors. If present, the entropic factors in a coherent deformation are expected to decrease with size because, the likelihood that there is enough thermal energy in a “breathing” mode should decrease with the number of atoms in the system. Instead, the data are indicative of a nucleation mechanism with multiple sites from which the transition can initiate. In the sliding planes model, this corresponds to greater probability of nucleation in nanocrystals with larger numbers of planes since only one nucleation event occurs per nanocrystal transformation. If high enough temperatures were experimentally feasible in this system, the hysteresis should ultimately narrow with increasing crystal size due to the greater influence of the entropic factor.

The nearly constant hysteresis width is important because it means that, at least in CdSe, the size-dependence of the hysteresis width is not sensitive to the kinetics. The upstroke transition pressure can nevertheless shift with nanocrystal size due to thermodynamic effects, as discussed earlier in this chapter (see SURFACE EFFECTS… above). Some geophysically-relevant systems may not exhibit full hysteresis loops because the high-pressure phase remains metastable as the pressure is released. Thus, special care should be taken to consider both the thermodynamics and kinetics in interpreting hysteresis measurements on such systems.

TEMPERATURE DEPENDENCE

Temperature is an important parameter that has been studied but not fully explored. Figure 6B shows temperature dependent data obtained for 25-Å diameter CdSe nanoparticles. The hysteresis loop systematically narrows with increasing temperature, as expected for an activated process. The larger activation energy discussed earlier is apparent in this figure, as the forward transition is significantly more sensitive to temperature compared to the reverse transition. A considerable shift of the hysteresis loop to lower pressure accompanies the narrowing with increasing temperature. Hysteresis shifts are usually assumed to directly correspond to shifts of the thermodynamic transition pressure (although the pressure is not expected to be precisely in the center of the loop). This assumption was used earlier in this chapter for the shift of the


thermodynamics with size attributed to surface effects (see SURFACE EFFECTS… above). Based on this assumption, the shift of the hysteresis with temperature in Figure 6B can be interpreted using the Clausius-Clapeyron equation for the slope of the phase boundary. The hysteresis shift of -0.07 kbar/K corresponds to an entropy change of +65 eV/nanocrystal in the 25-Å nanocrystal, roughly two factors larger than the -0.001 kbar/K shift in bulk CdSe (Jayaraman et al. 1963). A similarly large shift was also observed for 35-Å diameter nanocrystals. An explanation for such a large shift is an additional entropic contribution in the rocksalt phase in nanocrystal relative to the four-coordinate structure. Additional disorder of the surface in the rocksalt nanocrystal may reflect the shape change the nanocrystal undergoes during the transition, and the high-energy faces that are exposed (see Fig. 3). This interpretation suggests that the entropic differences between the phases may be large and are thus important in understanding the nanocrystal structural transition. The shift of the hysteresis to lower pressure with increased temperature is also experimentally related to the observation of different activation energies in the two transition directions. The larger sensitivity to temperature in the forward transition is responsible for the hysteresis shift to lower pressure with temperature (see Fig. 6B). This suggests that one could predict relative difference in activation energies in the forward and reverse transitions in bulk transitions from measurements for the Clausius-Clapeyron slope.

Extending the current state of knowledge could involve measurements at high temperature with shock experiments to access transition pressures closer to the thermodynamic limit. Progress is currently being made to study the transition in individual nanocrystal particles, to both eliminate the ensemble statistics and allow for individual transitions to be observed on ~femtosecond time scales. The study of nanocrystal solid-solid phase transitions to oxide nanosystems should also prove to be useful in understanding the microscopic process of solid-solid transitions relevant to geophysically important systems.

ACKNOWLEDGMENTS

This work was supported by the Director, Office of Energy Research, Office of Science, Division of Materials Sciences, of the U. S. Department of Energy, and the Air Force Office of Scientific Research, Air Force Material Command, USAF.

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Nanocrystals as Model Systems for Pressure-Induced Structural Phase Transitions

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