Size-Dependent TemperaturePressure Phase Diagram of Carbon

Nov 9, 2007 - developed technologies.4 One of the most comprehensive carbon phase diagrams that was proposed by Bundy et al. (solid lines in Figure 1)...
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J. Phys. Chem. C 2008, 112, 1423-1426

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Size-Dependent Temperature-Pressure Phase Diagram of Carbon C. C. Yang* and S. Li School of Materials Science and Engineering, The UniVersity of New South Wales, NSW 2052, Australia ReceiVed: July 30, 2007; In Final Form: NoVember 9, 2007

The polymorphic behavior of carbon in consideration of surface stress contributions to the internal pressure of nanocrystals is investigated through nanothermodynamics. It was found that nanodiamond is more stable than nanographite when the crystal size approaches the deep nanoscale. With theoretical and experimental data, a size-dependent temperature-pressure phase diagram of carbon is developed and established. It shows that the diamond/graphite/liquid triple point shifts toward lower temperature and pressure regions with decreasing nanocrystal size. This study also reveals a possible origin of metastable structure formation on the nanometer scale.

1. Introduction Carbon, the lightest Group IV element, is one of the most versatile, interesting, and useful chemical elements. Recently, interest in the synthesis of diamond (D) has been stimulated by its unique physical and chemical properties, such as high mechanical hardness, high thermal conductivity, excellent chemical corrosion resistance, excellent optical transparency, and more.1-3 The phase diagram is an essential reference for materials fabrication and applications. In the past decades, the temperature-pressure phase diagram of bulk carbon has been continually updated with the information obtained from newly developed technologies.4 One of the most comprehensive carbon phase diagrams that was proposed by Bundy et al. (solid lines in Figure 1) shows that (1) graphite (G) undergoes a wellcharacterized first-order transition upon pressurization to the D, and (2) both phases melt to form the liquid phase (L) as the temperature increases. On the other hand, recent advances in the research of carbon phase diagram provide new experimental data to make a great improvement in the accuracy of G-L phase boundary (the dash line in Figure 1).5-9 In this figure, the G in sp2 hybridization is the stable macroscale phase while the D in sp3 hybridization is merely metastable under ambient temperature and pressure. However, recent experimental results,10-16 computer simulations,17-21 and theoretical analyses22-29 indicate that Bundy’s diagram is not suitable for nanoscaled carbon. For example, nanodiamonds with diameters of less than 5 nm are more stable than their graphitic counterparts. This suggests that the size effect on the stability of D in the deep nanometer scale is highly significant. Therefore, the establishment of a sizedependent temperature-pressure phase diagram for carbon is essential to provide useful information for the fabrication of advanced materials on the nanoscale. Because the nanophase equilibrium is metastable in nature, it is difficult to be determined accurately, especially at high temperatures and under high-pressure environments. As a result, theoretical modeling presents itself as an attractive alternative approach for the establishment of a nanoscale phase diagram. Recently, several theoretical methods have been proposed to develop the size-dependent temperature-pressure phase diagrams of carbon.30-35 Unfortunately, the established phase * Correspondingauthor.Fax: +61-2-93855956.E-mail: [email protected].

Figure 1. T-P phase diagram of bulk carbon as proposed by Bundy et al.4 (solid lines) where the G-L phase boundary has been corrected with recent experimental results (the dash line).5-9 The symbol O denotes characterized points.

diagrams are either tentative or inconsistent with recent experimental results. In particular, the position of G/D/L triple point remains controversial, and the physical origin of the stability of nanodiamonds is not clear. It is necessary to establish a complete and accurate size-dependent temperature-pressure phase diagram of carbon. The knowledge gained from the studies of nanocarbon stability will play an important role in the establishment of the phase diagram. In this work, the phase equilibrium between G and D, G and L, as well as D and L were considered individually. The results calculated with the model developed during this study are consistent with experimental results in the open literature. Moreover, a complete and accurate size-dependent temperaturepressure phase diagram of carbon was established based on the obtained information. 2. Methodology On the basis of the information obtained from Bundy’s carbon phase diagram with the correction of recent experimental data in Figure 1, the corresponding pressure-dependent

10.1021/jp076049+ CCC: $40.75 © 2008 American Chemical Society Published on Web 01/16/2008

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transition temperature T(P) functions for phase boundaries are given as

TGL(∞, P) )

{

TGD(∞, P) ) -825.243 + 485.437P 185.246P + 4798.148 f 0.01 e P e 5.5 -125.692P + 6508.304 f 5.5 < P e 12 TDL(∞, P) ) 4905.263 + 7.895P

(1a) (1b) (1c)

where P is the pressure, T is the absolute temperature, ∞ denotes the bulk material, and the subscripts GD, GL, and DL denote the above transitions, respectively. Note that the TGL(∞, P) function in eq 1b was obtained from the corrected G-L phase boundary (the dash line in Figure 1). 2.1. Size-Dependent G-D Transition. For a spherical or quasi-isotropic nanocrystal with a diameter x, the internal pressure Pin induced by the curvature can be expressed as Pin ) 4f/x using the Laplace-Young equation, where f denotes the surface stress.36,37 Thus, the total pressure Ptot can be expressed as Ptot ) Pin + P. In this case, we have (1) when P ≈ 0, Ptot ) Pin, which is the case of the size-dependent solid-solid-phase transition, and (2) when Pin ≈ 0 with x f ∞, Ptot ) P, which is the normal situation of pressure-dependent solid-solid-phase transition for bulk materials. Therefore, Ptot ) Pin + P is applicable for both bulk materials and nanocrystals. For the transition between nanodiamond and nanographite, TGD(x, P) ) -825.243 + 485.437(P + Pin) can be expressed as TGD(x, P) ) -825.243 + 485.437(P + 4f/x)

(2)

At equilibrium, nanodiamond and nanographite should have the same pressure, or the same f. As a first order approximation, f ) (fD + fG)/2 is taken in eq 2 where fD and fG denote surface stresses of D and G, respectively. 2.2. Size-Dependent G-L Transition. From Lindemann’s melting criterion,38 a crystal melts when the root of mean square amplitude of the atoms reaches a certain fraction of the equilibrium atomic distance h. This criterion has been verified experimentally and is valid for both bulk materials and nanocrystals.39 With this criterion and Mott’s equation, the sizedependent melting temperature Tm(x) can be obtained. Therefore, for an isolated nanoparticle, the Tm(x, P) function can be expressed as Tm(x, P) ) Tm(∞, P)exp{-[2Svib(∞)/(3R)]/[x/(6h) - 1]} (3)

where Svib(∞) is the vibrational part of the overall melting entropy Sm(∞), and R is the ideal gas constant. As a result, the corresponding TGL(x, P) function is given as TGL(x, P) )

{

(185.246P + 4798.148) exp{-[2SVib(∞)/(3R)]/[x/(6h) - 1]} f 0.01 e P e 5.5 (-125.692P + 6508.304) exp{-[2SVib(∞)/(3R)]/[x/(6h) - 1]} f 5.5 < P e 12 (4)

2.3. Size-Dependent D-L Transition. When TGL(x, P) is equal to TGD(x, P), the G/D/L triple point (Tt, Pt) therefore can be obtained. The slope of the phase boundary between nanosized D and L is expressed as dTDL(x)/dP ) 7.895exp{-[2Svib(∞)/ (3R)]/[x/(6h) - 1]}. As a result, the TDL(x, P) function is given as TDL(x,P)) Tt + 7.895exp{-[2Svib(∞)/(3R)]/[x/(6h) - 1]}(P - Pt) (5)

3. Results and Discussions In Figure 1, Bundy’s carbon phase diagram exhibits that the point (5000 K, 0.011 GPa) is the most probable graphite/liquid/

Figure 2. TGD(x) function of nanocarbon. The solid line denotes the model prediction on the basis of eq 2. The dash line denotes the model prediction from f ) γ. The symbols [10 and b13 denote experimental results, and the symbol 4 denotes other theoretical results from the charge lattice model.24,25 The symbols ×40 and +41 in the inset are the experimental results of transition size between nanodiamond and fullerene at 1300 K or onionlike carbon at 1400 K, respectively.

gas triple point.4 It also shows (1) TGL(∞, P) increases with increasing P to a maximum 5-6 GPa, (2) TGL(∞, P) then decreases toward the G/D/L triple point (5000 K, 12 GPa), and (3) the achievable maximum temperature on the G-L phase boundary is 5200 K that is corresponding to the point (5200 K, 5.5 GPa) in Figure 1. However, the recent researches experimentally evidenced that the graphite/liquid/gas triple point is at 4800 ( 100 K and 0.01-0.1 GPa, providing an accurate correction on Bundy’s result.5-9 As a result, the point (4800 K, 0.01 GPa) was used as the graphite/liquid/gas triple point in our considerations. Moreover, the slope of the G-L phase boundary is dTGL(∞, P)/dP ≈ 185 K GPa-1, which corresponds to the latest experimental findings that the molar transition enthalpy is 120 kJ mol-1 and the molar volume expansion during the melting is 70% at a pressure 0.01-0.1 GPa for the graphite melting.9 Therefore, the maximum temperature on the G-L phase boundary can be roughly determined as 5817 K, which is corresponding to the point (5817 K, 5.5 GPa) in Figure 1. For comparison, Figure 2 plots (i) the results of TGD(x) for nanocarbon that were calculated from eq 2, (ii) the experimental results, and (iii) the results from other theoretical works. In eq 2, f ) (fD + fG)/2 ) 3.6 J m-2 with fD ) 6.1 J m-2 and fG ) 1.1 J m-2,28 and P ≈ 0 as the experimental results are obtained under the atmospheric pressure.10,13 From the calculation, it can be seen that the equilibrium size of nanodiamond and nanographite decreases from 8.5 nm at 0 K to 3 nm at 1500 K. As the size decreases, the stability of nanodiamond increases compared with that of nanographite. The predictions of our model are in a better agreement with the experimental results than the charge lattice model.24,25 The inset in Figure 2 shows the experimental results in which the transition size of nanodiamond and the fullerene or onionlike carbon is 2 nm at 1300 or 1400 K.40,41 It clearly indicates that the nanodiamond is stable above 2 nm, supporting the analysis of our model. As shown in the figure, the model predictions based on f ≈ γ are also consistent with the experimental results where γ is the surface energy. Note that γ ) (γD + γG)/2 ) 3.485 J m-2 with γD ) 3.7 J m-2 and γG ) 3.27 J m-2.28 Thus, f ≈ γ for carbon. However, this criterion is not suitable for the other materials in most cases. From a thermodynamic viewpoint, the fundamental difference between a solid surface and a liquid surface is the distinction between f and γ. Essentially γ describes

Temperature-Pressure Phase Diagram of Carbon

Figure 3. T-P phase diagrams of bulk and nanocrystalline carbon on the basis of eqs 1, 2, 4, and 5. The solid, dash, and dot lines denote the bulk, x ) 5 nm, and x ) 2 nm, respectively.

a reversible work per unit area to form a new surface, while f denotes a reversible work per unit area to elastically stretch the surface, which corresponds to the derivative of γ with respect to the strain tangential to the surface.36,37 For the fluid, γ ) f while for the solid γ * f. Thus, for a solid particle with a diameter of x and an isotropic surface stress covered by the surrounding fluid, the Laplace-Young pressure Pin ) 4f/x. It is noted that the above curvature-induced internal pressure Pin for a solid is different from the Laplace pressure for a spherical fluid droplet surrounded by other fluid in the equilibrium, which is expressed as Pin ) 4γ/x. As a result, the phase stability of nanodiamond is dominated by the surface stress-induced internal pressure, not the surface energy difference between G and D. In fact, the surface stress is a vital factor in promoting sp3 bonding in the synthesis of nanodiamond by low-temperature and low-pressure methods.42 The size-dependent temperature-pressure phase diagrams of carbon obtained from the eqs 2, 4, and 5 are plotted in Figure 3. For G, Svib(∞) ) 4.75 J mol-1 K-1 and h ) 0.142 nm;28 thus from eq 3, we have Tm(x,P)/Tm(∞,P) ) 0.925 for x ) 5 nm and Tm(x,P)/Tm(∞,P) ) 0.754 for x ) 2 nm. For D, Svib(∞) ) 6.37 J mol-1 K-1 and h ) 0.154 nm;28 therefore, Tm(x,P)/Tm(∞,P) ) 0.891 for x ) 5 nm and Tm(x,P)/Tm(∞,P) ) 0.645 for x ) 2 nm. Similarly, under the ambient pressure, Tm(∞) ) 3723 K for D;28 as a result, Tm(x) ) 3190 K for x ) 4 nm. This result is consistent with the observation of the melting of ∼4 nm diamond nanoparticles at 3000 K,43 verifying the accuracy of eq 3. As shown in Figure 3, the position of the G/D/L triple point (Tt, Pt) of carbon nanoparticles is different from that of the bulk materials. Although similar phenomena have been calculated by the other theoretical works, the results in the triple point displacement, which was induced by the particle size, were completely divided into two opposite directions.30-34 With consideration of the surface energy, a theoretical method was proposed to estimate the displacement of the phase equilibrium lines by equating the Gibbs free energy of the corresponding phases of carbon nanoparticles.30,31 It was shown that the G/D/L triple point (Tt, Pt) displaces toward higher pressure and lower temperature with decreasing x. A T-P-x three-dimensional phase diagram of nanocarbon was derived on the basis of the properties of detonation diamond.32 It was found that (1) D is more stable than G when x < 3 nm, (2) D diminishes at 1.8 nm or below, and (3) the triple point (Tt, Pt) displaces toward lower pressure and temperature as x decreases. On the basis of these findings, another three-dimensional phase diagram of nanocarbon was suggested by tentatively introducing a change of the

J. Phys. Chem. C, Vol. 112, No. 5, 2008 1425 slope of the G-D equilibrium line with decreasing x.33 This change reflects the higher stability of nanodiamond over nanographite in ambient conditions. Recently, the size-dependent T-P phase diagram of nanocarbon was also constructed by taking into account the pressure induced by the surface energy.34 It is found that the triple point (Tt, Pt) first displaces toward higher pressure and lower temperature with decreasing x when x > 1.2 nm, while it displaces toward lower pressure and temperature when x < 1.2 nm. From eq 3, it can be seen that the melting temperature of isolated carbon nanoparticles decreases as x decreases, resulting in that the triple point (Tt, Pt) displaces toward lower temperature with decreasing x. Available experimental results and the above theoretical methods are all in good agreement with this principle. From eq 2, it can be seen that the nanodiamond has higher stability than the nanographite in ambient conditions when x < 6 nm, resulting in the G-D transition line shifts toward lower temperature and pressure as x decreases. Recent experimental data indicate that nanodiamonds with sizes of 3-6 nm can be prepared under ambient conditions.15 It proves our finding on the stability trend of nanocarbon in ambient conditions. Moreover, the latest highpressure-high-temperature experimental results of size-dependent phase transition between nanodiamond and nanographite also suggest that the triple point (Tt, Pt) displaces toward lower temperature and pressure with decreasing x.16 Our findings and the latest experimental data disagreed with some reported theoretical works, which were based on the surface energies of carbon nanoparticles γ(x).30,31,34 However, an accurate value of such surface energy is one of the most critical issues in the carbon research community.33 This results in a number of uncertainties on the surface energies of D, G, and liquid carbon thus failing to make an accurate conclusion. Note that a change of the slope of G-D phase boundary dP/ dTGD was suggested in the three-dimensional phase diagram of nanocarbon.33 On the basis of Clausius-Clapeyron equation, dP/dTGD ) ∆S/∆V where ∆S and ∆V are the solid-solidtransition entropy and volume between G and D, respectively. As the crystal size has a very limited influence on the atomic diameter, the size effect on ∆V can be neglected.39 Because G and D are polymorphous solid phases of the same substance, it is conceivable that the size effect on ∆S should be limited and dP/dTGD is approximately a constant with decreasing x. This is consistent with most theoretical results, and further experimental works will be implemented to investigate this issue. From the discussion above, it could be derived that G-D, G-L, and D-L transition lines should shift toward lower temperature and pressure as x decreases, which results in the displacement of the triple point (Tt, Pt) from (5000K, 12 GPa) to (4968 K, 9.05 GPa) at 5 nm and further (4000 K, 2.74 GPa) at 2 nm in Figure 3. Therefore, our calculation clarifies the argument in the triple point displacement trend of nanocarbon phase diagram. Moreover, as shown in Figures 3, the slope dTDL(x)/dP for the melting line of nanodiamond is negative. This is analogous to that of the bulk D but different from silicon and germanium although they all have diamondlike structures. This difference is attributed to the absence of p-electron in the atomic core of diamond, which allows the p-character sp3 bonding electrons to close to the nucleus.44 As shown in Figures 1 and 3, the triple point (Tt, Pt) can also be determined with TDL(x, P) ) TGD(x, P) or TGL(x, P) ) TDL(x, P). However, the D-L transition line is located in both high-temperature and high-pressure regions and the available experimental data needs to be further verified. Different from the size-dependent D-L transition, the G-D and G-L transition

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lines are located in either low-temperature-low-pressure or lowpressure-high-temperature regions. As a result, the triple point (Tt, Pt) determined by TGL(x, P) ) TGD(x, P) in our work is remarkably accurate. In general, exp{-[2Svib(∞)/(3R)]/[x/(6h) - 1]} ≈ 1 - 4hSvib(∞)/(Rx) when x approaches 60h (≈8-9 nm for nanocarbon). In this case, eq 3 can be rewritten as Tm(x,P) ≈ Tm(∞,P)[1 - 4hSvib(∞)/(Rx)]

(6)

As shown in eqs 2 and 6, under the given pressure both solidsolid transition temperature and solid-liquid transition temperature obey the thermodynamic rules of low-dimensional materials in which the alternation of size-dependent quantity is associated with the surface/volume ratio, or 1/x.39 It is known that for a number of substances, the metastable high-pressure phases and even some more dense packing phases do not exist in the bulk state. Examples of such substances include BN, CdSe, ZnO, ZnS, Fe, Co, Cr, etc. These phases are, however, easily formed at the ambient pressure when the material size decreases to the nanometer scale.3 Such instances of phase stability at the nanoscale support the conclusions of this work. It indicates that the structural modification could be easily controlled by adjusting the crystalline size and the temperature. An understanding of this phenomenon may provide new insights into the fundamental mechanisms of polymorphism in nanocrystals. The thermodynamic approach proposed in this work may also open an avenue to understand the physical origin of metastable structure formation on the nanometer scale and to establish nanophase diagrams for practical applications. Moreover, the developed general model could also be used to predict different phase transformations in other nanomaterials if the relevant thermodynamic parameters are available. 4. Conclusions Thermodynamic quantitative models were developed to describe the polymorphism of nanocarbon and its melting temperature. The size-dependent phase stability of nanodiamond and nanographite as well as the temperature-pressure phase diagrams of the nanocarbon were established with the models developed in this study. It was found that the transition size of nanodiamond and nanographite is in the range of 3-8.5 nm at 0-1500 K and the G/D/L triple point of the size-dependent temperature-pressure phase diagrams shifts toward lower temperature and pressure with decreasing the crystal size. The results obtained from the developed models are in good agreement with experimental findings. The detailed thermodynamic considerations on the phase stability of nanocarbon demonstrate an easy way to establish similar nanophase diagrams. Acknowledgment. This project is financially supported by AustraliaResearchCouncilDiscoveryProgram(GrantDP0666412). References and Notes (1) Angus, J. C.; Hayman, C. C. Science 1988, 241, 913.

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