J . Phys. Chem. 1992,96, 6756-6762
6756
Arrested Solid-Solid Phase Transition in 4-nm-Diameter CdS Nanocrystals M. Haaset and A. P.Alivisatos* Department of Chemistry, University of California. Berkeley, Berkeley, California 94720 (Received: January 7 , 1992; In Final Form: April 23, 1992)
CdS crystallites of 4-nm diameter have been studied under high pressure up to 10 GPa by optical absorption and resonance Raman scattering. The solid-solid phase transition from the zinc blende to the rock salt phase is observed at pressures far in excess of the bulk phase transition pressure of 3 GPa. The pressure of the phase transition depends on the nature of the moiety used to derivatize the surface. In addition, because the compression of the lattice with pressure is the same as in the bulk crystal, it is possible to observe the dependence of the zinc blende crystallite properties up to pressures far higher than in the bulk. The elevated phase transition pressure and its dependence on the surface stabilizer can be explained by a higher value of the surface tension for the rock salt phase nanocrystals compared to the zinc blende.
Introduction The surface of an inorganic nanocrystal can play a dominant role in determining the relative stability of the phases of the system. In metal nanocrystals, it is well-established that the melting point is reduced from the bulk value, and this is explained in terms of the influence of surface tension on the relative stability of the solid and liquid phases.' In nanocrystals of cr-Fq03, the lattice spacing is increased, relative to the bulk, and the ferromagnetic-antiferromagnetic phase transition temperature is shifted lower, as one would expect for a system under negative pressure? In both instances,the inherent properties of the phases of the bulk material are maintain&, only the phase transition point is altered. Thus, the possibility exists to systematically adjust the phase diagram of a system by controlling the size and surface composition of the crystallites which comprise it. In addition, observations of the values of phase transition points vs size can be used to obtain the surface energy of a nanocrystal. Methods have recently been developed to prepare highly crystalline, monodisperse nanocrystals of semiconductorsin the condensed phase. These new materials have many interesting and size-dependent ~roperties.~One important feature of these methods is the presence of a chemical moiety on the surface of the nanocrystal, which is required in order to make the nanocrystals stable against dissolution or aggregation and in order to confer solubility. For a complete understanding of semiconductor nanocrystals, we need to know how the surface influences the phase diagram. CdS crystallizes in a four-coordinate zinc blende or wurtzite structure. Bulk wurtzite crystals transform to the rock salt structure at 2.7-3.0 GPa (Figure 1)."'* Experiments by Cor11 on CdS powders indicate that the phase transition pressure is very similar for zinc blende (3.1 GPa), wurtzite (2.5 GPa), and zinc blende-wurtzite mixes (2.8 GPa).I3 Substantiallattice contraction is observed in X-ray and electron diffraction measurements on nanocrystals of CdS smaller than 3 nm in diameter.I4J5These contractions are due to reconstruction at the nanocrystal surface. Thermodynamically, the surface tension, 7 , exerts a radial pressure on the nanocrystal 2Y
P, = r
Since nanocrystals of CdS appear to be under a positive pressure from surface tension to begin with, one might expect the solidsolid phase transition pressure to be depressed in CdS nanocrystals. Confirmation of this point of view has been provided by a recent X-ray diffraction study which shows that 20-A-diameter CdS crystallites synthesized by yeast are in the rock salt crystal structure with no application of external pressure.Is In this case, *To whom correspondence should be addressed. DGF Postdoctoral Fellow, 1989-1990.
0022-365419212096-6756$03.00/0
lattice contractions as large as 8% are observed, which would certainly correspond to a sufficiently high pressure to reverse the relative stability of the zinc blende and rock salt crystal structures. On the other hand, a recent high-pressure study of nanocrystals of CdSe found no evidence for the solidsolid phase transition, even at pressures as high as twice the bulk phase transition pressure.16 The question still remains, then, at what pressure do semiconductor nanocrystals undergo the expected solid-solid phaae transition? How does surface energy influence the transition pressure? In this paper, we present experimental observations of 4-nmdiameter CdS nanocrystals under pressures up to 10 GPa (at room temperature). The phase transition pressure is identified by abrupt changes in the optical absorption and resonance Raman spectra of the nanocrystals. The role of the surface in determining the phase transition pressure is studied by varying the stabilizrrs bound to the nanocrystal surface. In addition to determining the value of the phase transition pressure, we also have determined the change in the longitudinal optical vibration frequency and the change in the electronic absorption onset in the nanocrysals. These measurements show that the bonding in the nanocrystal interior closely resembles the bonding in the bulk solid.
Experimental Section Prepprntion of CdS Nanocrystrrls. In order to study the effect of hydrostatic pressure on the nanocrystals, it is necessary to prepare nanocrystals which are directly soluble in the pressuretransmitting medium. For pressures below 10 GPa, 4:l methanol/ethanol mixtures are commonly used in diamond anvil cells. CdS nanocrystals homogeneously soluble in a variety of organic solvents have been reported previously; however, methanol is not among these. Since the preparation involves some nonstandard methods, we provide protocols here. CdS nanocrystals were prepared in aqueous solution in the presence of either sodium polyphosphate or tetrabutylammonium-EDTA as the stabilizing agent. Nanocrystals stabilized with tetrabutylammonium-EDTA are directly soluble in 4:l methanol/ethanol mixtures (and many other polar organic solvents). Solubility of the polyphosphatestabilized particles was obtained after the initial preparation by partly exchanging the sodium ions against tetrabutylammonium ions. 1. Polyphosphate-Stabfized4.5-nm CdS Nanocrystals. Polyphosphate (330 mg) was dissolved in 2 L of deionized water in a three-necked vessel fitted with septum, stopcock, and pH electrode. Sixteen milliliters of a 0.1 M CdC12solution was added, the pH adjusted to 8.0, and the whole solution cooled down to 0 OC. Next, 18 mL of H2S was added by injection of 1-mL aliquots using a gas-tight syringe. The HIS ws always injected into the air volume above the vigorously stirred solution, and the pH was allowed to stabilize between the addition of successive aliquots. The pH was kept between 8.0 and 6.0 using 0.2 M NaOH. Two minutes after the last injection of H2S, 8.8 g of Q 1992 American Chemical Society
Arrested Solid-Solid Phase Transition
I
2.7 GPa
. .
-Cd-s-Cd-S-Cd-S-Cd-S-
I I I I I I I I
S-Cd-S-Cd-S-Cd-S-Cd-
I I I I I I I I I I
S-Cd-S-Cd-S-Cd-S
Figure 1. Schematic illustration of the phase transition under investigation. One slice through a CdS crystal is shown transforming to the rock salt phase. The bulk phase transition pressure is 2.7 GPa.
sodium diphosphate (Na4P207.10H20,Aldrich, ACS grade) was dissolved in the solution. The solution was heated to 85 OC for 2 h in a water bath, concentrated to 400 mL in a rotary evaporator, and dialyzed overnight against 15 L of deionized water. Variant 1. One gram of polyphosphate was dissolved in the dialyzed solution and HClO, added until the pH reached 2.3. The acidic colloid was further concentrated by evaporation until a volume of 150 mL was obtained. This was subsequently dialyzed for 18 h against 15 L of water brought to pH 1.3 with HC104. The excess of HCIO, was removed by dialyzing against water for 4-5 h. Finally, the colloid was brought to pH 10.5 with tetrabutylammonium hydroxide and rotavaped down to a highly viscous residue. The latter was dried for 24 h at 0.05 Torr and room temperature. Variant 2. Mercaptoacetic acid (HSCH,COOH, 0.5 mL) was dissolved in about 60 mL of deionized water and subsequently brought to pH 10.5 with tetrabutylammonium hydroxide. The resulting solution of tetrabutylammonium mercaptoacetate was added to the dialyzed colloid, and by using a rotary evaporator, the mixture was concentrated to a volume of 100 mL. This was dialyzed for 10 h against a mixture consisting of 4 L of water and 0.2 mL of mercaptoacetic acid adjusted to pH 10 using tetrabutylammonium hydroxide. After 5 h the dialysis bath was replaced by a new one of the same recipe. Finally the solution was rotavaped down to a thick residue and dried at 0.05 Torr and room temperature. 2. EDTA-Stabilized4nm Nanocrystals. Three-hundred thirty milligrams of EDTA (1,2-ethylenediamine-N,N,N'JV-tetraacetic acid; H4Y) and 5 mL of a 40% solution of tetrabutylammonium hydroxide were dissolved in 1 L of deionized water. While the solution was stirred vigorously, the pH was adjusted to 11.4 with tetrabutylammonium hydroxide, and then 10 mL of 0.1 M CdClZ solution was added. Finally, 5.5 mL of H2S was injected into the gas volume. During the last step, the pH was kept between 11.O and 11.4 by injecting tetrabutylammonium hydroxide solution. After all the H2S was dissolved, 0.1 M HCl was poured rapidly into the colloid until the pH dropped to 3.2. The solution was stirred for 3 min and subsequently purged with Nz in order to
The Journal of Physical Chemistry, Vola96, No. 16, 1992 6151 remove unreacted traces of H2S. Next, the colloid was concentrated by a factor of 2 using a rotary evaporator and dialyzed against 15 L of water brought to pH 9 with tetrabutylammonium hydroxide. Finally, the colloid was rotavaped until a viscous film was obtained. The latter was dissolved in about 100 mL of methanol and the resulting solution concentrated to 5 mL and mixed 4:l with ethanol. Apparatus. Raman and optical absorption measurements on the nanocrystals were performed in a gasketed diamond anvil cell (High Pressure Diamond Optics, Model B-1). The 301 temperad stainless steel gasket had a hole of 150 or 200 pm in diameter; the thickness of the gasket near the hole was 30-60 pm. The pressure inside the cell was measured by the ruby fluorescence method.I7 The ruby fluorescence was excited with the continuous wave (cw)light of an argon ion laser (Coherent, Innova 200), using the 458-nm or the 514-nm line. The optical absorption of the CdS nanocrystals inside the diamond anvil cell was determined in a microoptic setup described as follows. The light of a 100-W Osram tungsten iodine lamp was focused on the entrance slit of a SPEX 1681 0.22-m monochromator, with a 1200 groove/" grating blazed at 500 nm. By using a 150-mm lens, the outcoming light was focused at a position 10 cm in front of a Hammamatsu R 955 phototube. The photocurrent was measured by using a Keithley 485 picoammeter, interfaced to a personal computer. This computer also controlled the wavelength of the monochromator. The diamond anvil cell was located in the 11-cm space between the lens and the aperture. Hence, at the position of the gasket, the image of the exist slit was defocused and much larger than the gasket hole, making the spectral distribution of the light passing through independent of the size and shape of the hole. At 550 nm, 15 pA of photocurrent at 600 V on the PMT were obtained at a spectral resolution of 1.8 nm (slit width 0.5 mm). At the beginning of a set of measurements, the cell was filled with methanol only and the light density behind the cell was measured in step of 1 nm. Later on, those blank values were used to compute the optical density of the colloid under pressure. Since the hole in the gasket deforms under pressure, the values obtained for the optical density need to be corrected. As long as the hole diameter does not change, no corrections for the optical density need to be made, because in this case the pressureinduced decrease in optical path length is exactly compensated by the increase in concentration of the nanocrystals. If, on the other hand, the diameter of the hole does change under pressure, the resulting spectrum has an offset and the optical density is rescaled at all wavelengths by a wavelength-independent factor. The change in hole area was measured from the change in transmission at wavelengths where the nanocrystals showed no absorption. By using this one number, the offset and the rescaling of the spectrum were simultaneously determined Resonance Raman Experiments. In the resonance Raman experiments, the light of the 458-nm line of the argon ion laser was used. The light was fmt expanded by a telescope and then focused into the diamond anvil cell by using a microscope objective. The final spot size was 5 pm. The lsaer line width was 0.03 cm-*. A 45O scattering geometry was used. The Raman scattered light was isolated by using a SPEX Triplemate with an 1800 groove/mm grating blazed at 500 nm. The spectrum was collected by using a Photometrics CCD camera with a PM-512 chip. The resonance Raman spectra of different sized CDS nanocrystals at atmospheric pressure have been described earlier.'* Briefly, when the incident is resonant with the cluster HOMO-LUMO transition, the first two Stokes lines can be observed at about 300 and 600 cm-' respectively. These Raman peaks have been assigned to a totallly symmetric longitudinal optical vibration and its first overtone. Results The nanocrystals prepared in these experiments exhibit quantum-confined optical spectra, in accord with previous results. The nanocrystals were sized by using the optical spectra and the calculated size dependence of Lippens and Lannoo (tight binding
Haase and Alivisatos
6758 The Journal of Physical Chemistry, Vol. 96, No. 16, 1992
//r - 0
2
2.2
2.4
2.6
Energy (ev)
2.3 2.5
2.7 2.9 3.1 Photon Energy (ev)
3.3 3.5
Figure 2. Optical absorption spectra of the CdS nanocrystals at various pressures. The curves are vertically offset for clarity. The numbers at the left indicate the pressure in GPa. The bar of 0.2 OD unit applies to all the curves.
Figure 4. Square root of the optical density vs photon energy for the 8.5-GPa scan of Figure 3 above. The straight line is a least-squares fit, yielding an intercept of 2.1 eV.
i 0
2.3
Wavelength (nm)
Figure 3. Optical absorption spectra at various pressures, under conditions of extremely high nanocrystal density in the cell. In this case, no light is transmitted once the photon energy exceeds the HOMO-LUMO transition energy. The oscillations a t lower photon energies come from the interference between the two diamond faces. The numbers next to the curves are the pressures in GPa. Note that in (a) the HOMOLUMO transition shifts uniformly to higher energy with increasing pressure. Above 8.3 GPa, a new feature to the red appears in the absorption spectrum.
calc~lation),'~ as well as directly, by transmission electron microscopy. In accord with the prior observations of Lippens and Lannoo, we find the calculated and observed sizes were in reasonable agreement. The pressure-induced change in the optical absorption spectrum of the polyphosphate-stabilized CdS nanocrystals is shown in Figure 2. As a function of pressure, the optical absorption spectrum smoothly shifts to higher energies, up to a pressure of 8.0 GPa, where an abrupt transformation in the optical spectrum is observed. No change in the shape of the spectrum is observed at 2.1 GPa. Figure 3 shows the optical absorption spectrum of an extremely high optical density (OD) solution of the CdS nanocrystals. Due to the enlarged scale, oscillations in each spectrum become visible. These are caused by interference between the two parallel diamond faces. (In Figure 2, the interference fringes cannot be seen because their amplitude is low compared to the range of OD in the whole absorption spectrum.) As the pressure is increased, the absorption onset shifts to higher energy, up to 8.0 GPa, where a new, weak feature in the absorption spectrum appears. A plot of the optical density of this new feature vs energy yields a straight line with intercept of Eo = 2.1 0.1 eV. When the deformation of the gasket with increasing pressure is taken into account, we find that below 8.0 GPa there is no change in the shape of the optical absorption spectrum with in-
*
2.5
2.7 2.9 3.1 3.3 Photon Energy (eV)
3.5
Figure 5. Illustration that the optical spectrum shifts to higher energy with increasing pressure, but there is no detectable change in shape. (a) Data were taken at roughly 0.5-GPa intervals. (b) Curves have all been shifted horizontally to overlap. 0.24I
-I
0 161
0.08
I
0.04l/ 0
0
,
2
,
, 4
,
,
6
,
,
8
Pressure (GPa)
Figure 6. Change in the HOMO-LUMO transition energy with increasing pressure.
creasing pressure, only a uniform shift of the entire spectrum to higher photon energy. The magnitude of the shift with pressure can be measured by shifting the high-pressure absorption curve until it completely superimposes with the atmospheric pressure curve. This result can be seen most clearly in data obtained by releasing the pressure (Figure S), because in this case no deformation of the gasket occurs, and so no corrections need to be applied. The optical absorption spectrum responded reversibly to pressures lower than 8.0 GPa. The transformation in the spectrum at 8.0 GPa was reversible, but one this pressure was exceeded, the low-pressure spectrum was permanently broadened and reduced in intensity.
The Journal of Physical Chemistry, Vol. 96, No. 16, 1992 6759
Arrested Solid-Solid Phase Transition
Wavelength (nm)
(b)
Pressure (GPa)
Figure 9. Shift of the fundamental and overtone vs pressure. The straight line fits are described in the text.
\
OJ
8
,
3.6
,
9
r
4
8
'
'
'
4.4
'
'
'
',
4.8
I.
Volume/Unit Cell (m3 x 10.~)
sE -
K O -
...... 5.0 GPa
0.5 GPa
c .
-c
-
2 P)
0 . (D
-
5 8
)w .i
, :,
i
,
4'
,d2vfl
Figure 10. P-Vcurves for the bulk (solid) and nanocrystal (dashed) of CIS. Equation 8 and the parameters shown in Table I were used to generate these curves.
match well with all the others; e.g., no hysteresis is observed. Above 8.0 GPa (polyphosphate) and 6.5 GPa (EDTA), the resonance Raman spectrum abruptly disappears (Figure 10).
&Nfl
Discussion Origin of the Elevated Phase Transition Pressure. From the smooth and reversible behavior of the optical spectra up to 8.0 GPa (polyphosphate stabilized) or 6.5 GPa (EDTA stabilized) and from the Raman data with the 1LO and 2LO peaks persisting far above 3.0 GPa, we conclude that no solidsolid phase transition occurs in 4-nm CdS nanocrystals up to pressures substantially higher than in the corresponding bulk. The shift of the vibrational frequencies of the CdS nanocrystals with increasing pressure exactly matches the shift of the bulk. Therefore, it is possible to reduce the unit cell volume in the nanocrystals to a greater degree than in the bulk, before any transformation occurs. The abrupt changes in the optical absorption spectra at 8.0 and 6.5 GPa are accompanied by the formation of a new absorption, the optical density of which increases quadratically away from the band gap. The same phenomenon is observed at 3.0 GPa for bulk CdS at its phase transition from the zinc blende to the rock salt phase. The rock salt phase has an indirect band gap for which this quadratic behavior is typical. We therefore assign the transformations in the CdS nanocrystals at 8.0 and 6.5 GPa to the zinc blenderock salt solidsolid phase transition. Why would the zinc blende to rock salt solidsolid phase transition take place at much higher pressure in the nanocrystals than in the bulk, when the zinc blende nanocrystals are already under substantial internal pressure from surface tension? To understand this, we need to first consider the thermodynamic conditions for the phase transition in the bulk and then how these transitions will be modified in a finely divided sample. In the bulk, the internal energy of each phase separately is given by Ul = TS1 - PVI + wNI Uz = TS2 - PVZ + "2 (3) The necessary condition for a phase transition is then U,(T,P) - TSl(T,P) + PVI = Uz(T,P) - TS,(T,P) + PVz (4)
Haase and Alivisatos
6160 The Journal of Physical Chemistry, Vol. 96, No. 16, 1992
When considering a solidsolid phase transition of the type under study here, the change in entropy between the two phases is small. For bulk CdSe, where variable-temperature high-pressure data are available, the T dependence of the phase transition is found to be weak (-0.0012 GPa/K) and the change in enthalpy at the phase transition is only 0.5 k c a l / m ~ l . For ~ ~ this reason, it is common to compare room-temperature high-pressure data with 0 K theoretical calculations of the relative stability of the phases. Following this assumption, we find that the phase transition pressure is given by
Thus, the phase transition pressure is given by the line of common tangent between the internal energy vs volume curves for the two phases (seeFigure 9). For the bulk, the slope of this curve is 2.7 GPa. We will return to this shortly. When the bulk solid is finely divided into nanocrystals, the equation for the internal energy is altered to take into account the presence of the surface: UI = TS1- PVI + PNI U2 TS2 - PV2 + pN2 - 72A2
(6)
where A, is the area of phase i and yi is the surface tension of that phase. Substituting this revised expression into eq 3, we see that the phase transition pressure for nanocrystals of a given size will depend on the difference in the surface tensions between the two solid phases, in this case zinc blende and rock salt. Examination of this relationship allows us to understand the observed increase in the pressure of the solidsolid phase transition in nanocrystals, compared to the bulk. In general, the surface tension of a solid is difficult to define. Surfacetension applies to a finely divided phase at thermodynamic equilibrium. In a liquid, where the atoms are free to move, a spherical droplet shape is automatically assumed under the influence of surface tension. This tension imposes a force equally from all sides and results in a compression of the droplet in accord with the Laplace law (eq 1). In a solid, the equilibrium shape of the crystallite is not spherical. Rather, the surface tension of each low index plane must be estimated separately, and the equilibrium shape of a solid crystallite is the one in which each face is present in inverse proportion to the magnitude of its surface tension. This shape is given by the Wulff construction. At the simplest level, the surface tension of each low index plane is given by (7)
where Ecohis the cohesive energy per atom in the bulk, N, is the number of bonds per surface atom, Nb is the number of bonds in the interior, and ps is the surface atom density. On the basis of such arguments, we would expect the equilibrium shape to have predominantly (1 1 1 ) planes, leading to a cuboctahedron, since this produces the smallest number of broken bonds. From eq 7, we can see immediately that the rock salt phase crystallites have a larger surface tension than the zinc blende phase, because a larger fraction of bonds at the surface will be broken in the rock salt phase. Figure 1 illustrates schematically how atoms completely in the interior of the nanocrystals in the zinc blende phase are forced to the surface in the rock salt phase. This necessarily produces a larger fraction of broken surface bonds and hence a greater surface tension. Even if colloidal stabilizers change the value of the surface tension from the bare nanocrystal value, we still expect the same general trend. Qualitatively, this aocounts for the increased phase transition pressure in the nanocrystals. Estimating surface tensions theoretically is rather difficult, however, and some experimental measure seems requisite. To the extent that a lattice contraction is observed,we can relate this contraction to a pressure, and to a surface tensicn? via the Laplace law, eq 1. This approach is approximate. When a solid
TABLE I: Parametera Used To Generate Figures 10 and 11 bulk 4-nm diameter zinc blende2’ rock salt28 zinc blende rock salt v,, x ~ O - * m3 ~ 4.9 4 4.8 3.9 Bo, GPa Bd 79 N/m PT,GPa
62 0.420
87
2.1
2.1
62
81
0.436 1.7
8.0
2.0 8.0
is finely divided into nanocrystals, shear forces can be sustained at the interface, yielding displacementsin the surface and forces that do not point inward radially, as they do in a liquid droplet. Detailed study of the each of the line positions and peaks in the electron diffraction patterns may yield a more detailed picture of the stress and strain in nanocrystals. In order to make some initial progress on this rather complex problem, we assume the surface energy can be described in terms of a single surface tension for one face of the material and that the resulting pressure is hydrostatic. Furthermore, we derive the value of this pressure, and hence the surface tension, from the lattice contraction. The change in stability as a function of size is a central one is cluster science. The preceding discussion illustrates that in the case of colloidal nanocrystals, it is through measurements of the surface tension that such information can best be obtained. In what follows, we demonstrate how it is possible to use pressuredependent information to determine quantitativelythe difference in the surface tensions between two nanocrystal phases. For the purpose of this discussion, we will focus on the polyphosphatestabilized nanocrystals only, but it is important to bear in mind that other chemical stabilizers will yield different results. In the final section, we discuss how it is possible to determine the absolute values of the surface tension for each phase separately and how such numbers can be used.
Equation of State for CdS Nanocrystals The pressurevolume curve for CdS bulk and for the polyphosphate-stabilized CdS nanocrystals is shown in Figure 10. For each phase, the P-V curve is assumed to have the form given by Mumaghan’s equation of state? V, v= 1+-P
.m
where V, is the unit cell volume at atmospheric pressure, Bo is the bulk modulus, and B,,’ is the change in bulk modulus with pressure. For bulk CdS, all the parameters are known, for both the zinc blende and the rock salt phases (Table I). At 2.7 GPa, the bulk transforms from the zinc blende to the rock salt phase. The values of Bo and Bd are assumed to be the same in the nanocrystals as in the bulk solid. This assumption is justified because the shift of the LO mode vibrational frequency and the HOMGLUMO transition energy with pressure are the same as the shifts for the bulk, indicating that the compressibility is unchanged. The fractional contraction of the bond length in the nanccrystals because of surface tension is not so large as to change the compressibility. This is the only assumption required to determine the difference in surface tension between the two phases. The total energy stored in the nanocrystalsas pressure is increased is given by the integral of the pressure over the change in the unit cell volume. Unlike the bulk, the nanocrystals can be compressed all the way to 8.0 GPa, before the solidsolid phase transition occurs to the rock salt phase. The excess energy required to induce the phase transition in the polyphosphate-stabilized nanocrystals compared to the bulk is thus equal to the difference in surface energy between the two phases, and this in turn is equal to the integral of the pressure over the volume between 2.7 and 8.0 GPa. This yields a value of 0.3N/m for the difference in surface tension between the two phases. To understand the meaning of this number, we need a handle on the surface tension of each phase separately, rather than just the difference of surface tensions. X-ray and electron diffraction
The Journal of Physical Chemistry, Vol. 96, No. 16, 1992 6761
Arrested Solid-Solid Phase Transition
7
8 OPO
3
2.7 GPa
34
4
5
Votume/Unit Cell (ma x IO Figure 11. Energy vs volume curves for the nanocrystals, wnstructcd by integrating the PYcurves of Figure 10. Note that the solid-solid phase transition takes place. at the prcasure given by the line of wmmon tangent between the zinc blende and the rock salt phases (eq 5). For the nanocrystals, the surface tension parameters in Table I were used.
studies show that the volume of the unit cell in the nanocrystal is less at atmospheric pressure and decreases with size as l/r. X-ray powder diffraction experiments on CdS nanocrystals stabilized with a variety of thiols (thiophenol, mercaptoacetic acid) and electron diffraction studies of bare, supported nanocrystals that have been melted and recrystallized all show a lattice contraction that yields a surface tension near 1.7 i 0.4 N/m. This is comparable to surface tensions that have been measured for other types of inorganic nanocrystals?6 For the polyphosphatestabilized particles used here, it was not possible to obtain good enough X-ray diffraction data to get the surface tension. This is because excess polyphosphate in the solution cannot be completely removed by dialysis, and the salt peaks interfere with the measurements of the CdS peaks. To illustrate the information that is in principle available, we assume a value of 1.7 N/m for the surface tension of the low-pressure phase, since this is very close to the measured value for three different colloid stabilizers. Under these assumptions, the surface tension of the rock salt phase polyplmphatbstabilized CdS nanocrystah would be 2 N/m. Only a modest difference in the surface tension (17%) between the two phases is enough to cause a 3-fold increase in the phase transition pressure. It is interesting to compare the value of the surface tension fnnn our experiments to the values we get from eq 7 above. We find the calculatedvalues of the surface tension to be 2.2 and 3.0 N/m for the bare (1 11) zinc blende and rock salt surfaces, respectively. The calculated value in general provides an upper limit on the tension. The approach outlined above also allows us to answer one of the most important questions about colloids. On a per unit cell basis, how does the binding of stabilizers to the colloid surface alter the energy, compared to the bulk. The P-Vcurves of Figure 10 can be integrated to yield energy-volume curves, as shown in Figure 11. The equation for each phase is
For all the curves, the zero of energy has been set to the value of the energy/unit cell of bulk zinc blende CdS under no pressure. If we make the additional common assumption that the entropy does not change between the two phases, the solid-solid phase transition pressure on such a diagram is the line of common tangent between the E-V curves for the two phases. The point of tangency, which is the critical volume at which a phase transition occurs is given by
The offset of the bulk rock salt E-Vcurve is obtained from the requirement that it be tangent to the zero offset bulk zinc blende curve. Surface tension increases the offset of the E-Vcurve for
the zinc blende nanocrystals. The amount it is offset from the bulk curve is determined from the observed lattice contraction when no external pressure is applied and eq 9. Surface tension raises the energy of the zinc blende phase of the nanocrystals relative to zinc blende bulk, but it raises the energy of rock salt phase nanocrystals even more relative to rock salt phase bulk so that the slope of the line of common tangent increases from 2.7 to 8.0 GPa. The rock salt nanocrystal phase E-Y curve offset was determined by the requirement that it be tangent to the 8.0-GPa curve that is also tangent to the zinc blende nanocrystal E- V Curve. We have outlined a method by which thermodynamic concepts can be applied to colloidal nanocrystals, in order to gain information about the nanocrystal phase diagram. The advantage of this approach is it allows us to deal with complex systems and to make predictions about how the phase diagram will change as the size of the nanocrystal changes. Numerous tests of the a p proach we have outlined can be performed, the most important being a study of the phase transition pressure as a function of nanocrystal size,for one type of surface derivatization. However, at present the requirement of methanol-soluble nanocrystals has limited the range of sizes we can study to a very narrow region. Previous studies on quantum confinement effects in CdS nanocrystals have emphasized the shift in the direct allowed optical transitions with size. In this experiment, we have observed the indirect band gap in a semiconductor nanocrystal whose optical properties are dominated by quantum confiiement. At the phase transition pressure,the indirect band gap is shifted to higher energy by 0.4 eV compared to the indirect band gap of the bulk crystal at its phase transition pressure. This is not surprising, since indirect transitions are also af€ectedby quantum confiiement. As yet there is no model for how these transitions should vary with size. In this paper, we have shown that 4nm CdS nanocrystals have the same compressibility as the bulk solid but that they can be compressed to a higher degree than the bulk solid withcut undergoing a solid-fid transition to the rock salt crystal structure. This change in the phase diagram of the nanocrystals depends on the detailed bonding of the stabilizer to the nanocrystal surface. Future work on this phase transition will require a direct study of the structure of the surface of the n a n q t a l s in both the lowand high-pressure phases. Acknowledgment. We thank our colleagues Vicki Colvin, Avery Goldstein, Joseph Shiang, and Sarah Tolbert for usehl discussions. Dr. Haase thanks the DRG for a fellowship. This work was funded by the National Science Founation, under Grant DMR 9057186. Registry No. Cadmium sulfide, 1306-23-6.
References and Notes (1) Buffat, Ph.; B o d , J.-P. Phys. Rev. A 1976, 13, 2287.
(2) Scbroeer, D.; Nininger, R. C., Jr. Phys. Rev. Lerr. 1967,19, 632. (3) Reviews: (a) Brus, L. E. IEEE J. Quanrum Electron. 1986, 22, 1909. (b) Henglein, A. Topics in Currenr Chemistry; Springer: Berlin, 1988; p 113. (4) Edwards, A. L.; Drickamer, H. G . Phys. Rev. 1961, 122, 1149. (5) Safnara, G.A,; Drickamer, H. G. J. Phys. Chem. Solids 1962,23,457. (6) Minomura, S.;Samara, G.A.; Drickamer, H. G. J. Appl. Phys. 1962, 33. 3 196. (7) Owen, N.B.;Smith, P. L.; Martin, J. E.; Wright, A. J. Phys. Chem. Solids 1%3, 24, 1519. (8) Rooymans, C. J . M. Phys. Lert. 1963, 4, 186. (9) Samara, G.A.; et al. Phys. Rev. 1965, 140, IA, A388. (10) Batloan. B.: Javaraman.. A.:. Van Cleve.. J. E.:. Maines. R. G . Phvs. Rev. B 1983,17, 3920.(1 1) Venkateswaran, U.; Chandrasekhar, M. Phys. Rev. B 1985,31, 1219. Schrccdcr,J.; Bilcdeau, T. G.; Hwa, L.-G. Phys. Rev. (12) Zhao, X.-S.; B 1989,40, 1257. (13) Cork J. A. J . Appl. Phys. 1964,35, 3032. (14) Goldstein, A. N.;Echer, C. M.; Alivisatos, A. P. Science, in press. (IS) Dameron, C.;Reese. R.; Mehra, R.; Kortan, A.; Carroll, P.; Steigerwald, M.; Brus, L.; Winge, D. Narure 1989, 338, 546. (16) Alivisatos, A. P.; Harris, T. D.; Brus, L. E.; Jayaraman, A. J. Chem. Phys. 1989.89, 5979. (17) Bamett, J. D.;Block, S.;Piermani, G. J. Reu. Sci. Insrrum. 1973,44, 1. (18) Shiang, J. J.; Goldstein, A. N.; Alivisatos, A. P. J . Chem. Phys. 1990, 92, 3232. (19) Lippcns, P. E.;Lannoo, M. Phys. Reo. B 1989,39, 10935.
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(20) hndolt-B6rnstein. Numerical Data and Functional Relationships in Science and Technology; New Series; Madelung, O., Ed.; Springer: Berlin, 1987; Vol. 22a. (21) Rcaetti, R.; Nakahara, S.;Brus, L. E. J. Chem. Phys. 1983,79,1086. (22) Venkateswaran, U.; Chandrasekhar,M.; Chandrasekhar,H. R. Phys. Rev. B 1984, 30, 3316. (23) Variano, B. F.; Schlotter, N. E.; Hwang, D. M.;Sandroff, C. J. J . Chem. Phys. 1988,88,2848. (24) Onodera, A. Rev. Phys. Chem. Jpn. 1969, 39, 65.
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Importance of Orientational Rearrangement during Vitrification of Hydrocarbons: Dependence on Molecular Shapet Aparna Chakrabarti, S. Yashonath, and C. N. R. Rao* Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560012, India (Received: January 27, 1992; In Final Form: April 20, 1992)
On the basii of Monte Carlo calculations of 2,2-dimethylpropane (neopentane), n-pentane, and 2,2ðylbutane
(neohexane) at several temperatures, thermodynamic properties and radial distribution functions as well as dimerization and bonding energy distribution functions are reported for both liquid and glassy states. Changes in the radial distribution functions on cooling depend on whether the groups are accessible (peripheral) or inaccessible. Peaks in the radial distribution functions corresponding to peripheral groups do not shift to lower distances on cooling and at times display a large increase in the intensity of the first peak. The peaks due to inaccessiblegroups, on the other hand, shift to lower distances on cooling. The magnitude of the reorientational contribution in determining the resulting structure of the glass is estimated for the different hydrocarbon molecules investigated. The reorientational contribution is highest for neopentane (26%) followed by isopentane (24%), neohexane (22%), and n-pentane (0%). It appears that molecular geometry has an important role in determining the magnitude of the reorientational contribution to the structure of the glass.
Introdwtion The difficulty in obtaining information on the structural changes accompanying the glass transition and our inadequate understanding of the nature of the glass transition are closely related.' One of the most useful techniques in the study of glasses undoubtedly is computer simulation which yields microscopic information pertaining to the structure as well as relaxation processes. During the past few years, molecular dynamics calculations by Andersen and -workers2 on monatomic glasses have shown that they possess properties quite similar to those of laboratory glasses. Thermal effects such as the dependence of the glass transition temperature on the cooling rate have been found in computer-simulated glasses. Further investigations have indicated that the structural motif that predominantly occurs in glasses compared to the liquid is the ico~ahedron.~ Unlike monatomic glasses where only translational degrees of freedom exist, in polyatomic or molecular systems, structural relaxation during the rapid oooling of the liquid involves both positional and orientational degrees of freedom. It has been suggested that the latter play an important role in the determination of the properties of the glass.4 Our earlier studies on 2-methylbutane (ipentane) indicate the orientational degrees of freedom to be of crucial importance in determining the properties of the glass5 We have now carried out detailed Monte Carlo simulations of the liquid and glassy phases of 2,2-dimethylpropane (neopentane), n-pentane, and 2,2-dimethylbutane (neohexane) with the p u v e of calculating the properties of glasses and more importantly to obtain insight into the structural changes occurring during vitrification. The results of these simulation studies have shed light on the role of reorientation in the determination of the structure of the glass and its dependence on molecular shape. Computational Details Standard geometrical parameters used for hydrocarbons are C(s$)
