ARTICLE pubs.acs.org/jchemeduc
Complexities of One-Component Phase Diagrams Andrea Ciccioli† and Leslie Glasser*,‡ † ‡
Department of Chemistry, Sapienza - University of Rome, p.le Aldo Moro 5 00185 Roma, Italy Nanochemistry Research Institute, Department of Applied Chemistry, Curtin University, GPO Box U1987, Perth, WA 6845, Australia ABSTRACT: For most materials, the solid at and near the triple-point temperature is denser than the liquid with which it is in equilibrium. However, for water and certain other materials, the densities of the phases are reversed, with the solid being less dense. The profound consequences for the appearance of the pVT diagram of one-component materials resulting from such “anomalous” volume changes in solidliquid transitions are discussed. We discuss and illustrate how the 3D pVT phase diagram changes for this case. A more complex case occurs in systems where the solid þ liquid field displays continuous density reversal at high pressure, making the phase diagrams of some elements unexpectedly complex. The controversial case of graphite is presented as an example of the difficulties of interpretation. A current version of the carbon pT phase diagram is provided, in a 2D pT representation as well as in a virtual 3D version. The phase diagram of sodium, newly determined to extremely high pressures and illustrated here, shows both melting maxima and minima as well as a number of phase transitions as pressure increases. KEYWORDS: Upper-Division Undergraduate, Graduate Education/Research, Physical Chemistry, Materials Science, Phases/ Phase Transitions/Diagrams, Thermodynamics
’ STANDARD AND “ANOMALOUS” VOLUME CHANGES UNDER MELTING To lend clarity to the complexity of the region around the triple point, where solid, liquid, and vapor or gas are in equilibrium, it has been customary to offer diagrams of generalized models, such as that in Figure 1. Generalized models such as this depict the properties of “standard” materials, for which the density of the solid is greater than that of the equilibrium liquid. However, water (together with germanium, silicon and carbon,3 which are also tetrahedrally coordinated as in ice, together with a few other materials) is anomalous in that the solid, ice, is less dense than its liquid, water, at the triple-point temperature. This results in such pVT diagrams as Figure 1 being nonrepresentative and potentially confusing. Consider now how the diagram in Figure 1 is altered in the anomalous density case. Thermodynamic phase transition behavior is described by the Clapeyron equation dp Δtrans H Δtrans S ¼ ¼ dT trans Ttrans Δtrans V Δtrans V
P
hase diagrams depict the relations among the various phases of a material, generally within the space of pressure, volume, and temperature (pVT); that is, phase diagrams depict the equilibrium pVT conditions under which phase transitions occur. Since pVT is a space of three dimensions, it is most appropriate to display the diagram either as a physical model or in virtual 3D space,1 but the difficulties inherent in such constructions have led to a general preference for projections onto the pT, pV, and VT planes. Most introductory textbooks of physical chemistry tend to present thorough discussions of pT planes, where a clear correspondence between degrees of freedom and dimensionality emerges. Owing to their nature as thermodynamic potentials, the p and T variables assume the same value in phases coexisting at equilibrium, and so invariant points, monovariant lines, and bivariant areas are displayed in the pT plane. pV representations are more often presented as experimental isothermal condensation curves (that is, isothermal sections of the 3D diagram), whereas little attention is usually devoted to the VT plane. Recent computational developments, however, permit extension to interactive virtual 3D representation,2 making it easier for students to appreciate the shape of the pVT surfaces and their relation to 2D projections. In this article, we consider some of the complexities that may occur in one-component phase diagrams. These are seldom mentioned, even beyond the educational literature, although they are of considerable chemical and technical concern. Copyright r 2011 American Chemical Society and Division of Chemical Education, Inc.
where ΔtransH, ΔtransS, and ΔtransV represent the enthalpy, entropy, and volume change, respectively, of an equilibrium phase transition at the absolute temperature, Ttrans. This equation Published: March 11, 2011 586
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Figure 1. pVT phase diagram model of a generalized material, emphasizing the region (yellow) surrounding the “triple-point” line (the line connecting the molar volumes of the three phases in equilibrium at the triple point) and showing the usual case where the solid is denser than its equilibrium liquid, so that the slope of the melting curve is positive according to the Clapeyron equation (see text). Three surface types are depicted in white: T, p, V (solid) surface; T, p, V (liquid) surface, continuing into the T, p, V (vapor or gas) surface.
describes how pressure (temperature) changes as a result of temperature (pressure) alterations when two phases are in mutual equilibrium; that is, it describes the slope of the pressuretemperature curve. At the equilibrium condition, ΔtransG = Δtrans H - TtransΔtransS = 0, hence, ΔtransS = ΔtransH/Ttrans. Because Ttrans is necessarily positive, the sign of the pressuretemperature slope (whether it is positive or negative) depends on the ratio of the signs of the enthalpy (entropy) and volume changes involved in the phase transition. More than a century ago, Gustav Tammann4 discussed in detail all the possible cases associated with the positive and negative sign of both numerator and denominator in the Clapeyron equation, in melting and amorphization processes, resulting in a loop-shaped melting line in the pT plane (also termed a Kauzmann curve5), with the solid inside and the liquid surrounding the loop.6 In principle at least, melting processes with negative ΔH and ΔS values could not be ruled out. Although a few examples of exothermic melting have actually been presented in the literature in more recent times,7 in the overwhelming majority of cases enthalpies (and entropies) of materials increase in the sequence solid < liquid , gas as the molecules in the phases become successively increasingly independent of each other. Hence, in a melting process, where solid is converted to liquid, the enthalpy change is usually positive. For most materials, the volume change on melting is also positive (because densities generally decrease in the reversed sequence, solid > liquid . gas), so that the pressure-temperature slope is positive. This results in the solid þ liquid two-phase “ribbon” in Figure 1 (labeled “s þ l”) facing the viewer. But ice is less dense than its equilibrium liquid at the triple-point temperature, so that the pressure-temperature slope for water (and the other above-mentioned anomalous materials) is negative. This reversed slope in the pT projection is simply the 2D image of a profound change in the appearance of the full pVT phase diagram near the triple point (Figure 2A). The solid plus liquid phase gap in this circumstance has become twisted around,
Figure 2. pVT phase diagram model of a generalized material, emphasizing the region surrounding the “triple-point” line and showing the anomalous case where the liquid is denser than its equilibrium solid, so that the slope of the melting curve is negative. (B) The temperature axis of the diagram is reversed with respect to panel A to better display the “s þ l þ vapor” areas (and thus corresponds to a mirror image of panels A).
to face the increasing temperature regime, whereas the phase equilibrium lines surrounding the gap now slope toward higher pressure as the temperature decreases, that is, they are reversed from the standard situation and are not easily visualized. To have the “s þ l” ribbon face the viewer for easier comprehension, the temperature axis should be reversed, as shown in a mirror image version in Figure 2B. In contrast with early texts, discussion and presentation of similar 3D representations are seldom found in recent textbooks of physical chemistry, the focus being now preferentially directed toward pT projections only. However, these 3D diagrams have considerable value from a pedagogical point of view. Not only do they show the source of the pT projections depicted in the 2D phase diagram, but consideration of their isothermal and isobaric sections permits discussion of the role of the compressibility8 and thermal expansion coefficients (and their respective derivatives) in 587
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Figure 3. The conventional pT phase diagram of carbon, displaying a maximum in the melt curve of graphite, shown as the broken convex solid-liquid equilibrium curve. The diamond-liquid curve is shown with a metastable extension into the graphite region. (Adapted from ref 3c. See additional note for the data.) The solid near-vertical graphite melt line is currently considered to be more plausible (see text). Figure 4. pT phase diagram of sodium from ambient conditions to high pressures. As pressure increases, the crystal structure commences as body-centerd cubic (bcc), passes through a melting-point maximum at about 31 GPa and 990 K, and converts to face-centered cubic (fcc) at about 65 GPa and 700 K (with virtually no volume change; see Figure 6). As pressure increases, the melting point decreases to about 300 K at 118 GPa, while the crystal structures change through a progression of complex structures19a,c,d,21 (see text and Table 1).
determining the slopes (and curvatures) of solid and liquid single phase pVT domains. For example, the negative thermal expansion coefficients of hexagonal ice and water in some temperature ranges can be visualized on a properly expanded T scale.
’ MELTING CURVES WITH MAXIMA AND MINIMA For some materials, the phase diagrams can display interesting behavior if the pressure scale is extended to higher pressures. This is particularly useful because students are often intrigued by the fate of the melting line at high pressure, especially as to whether it ends with a critical point, as is the case for the pT liquid-vapor line (i.e., the liquid þ vapor field in the 3D diagram) or otherwise. Consider the situation when the solidliquid equilibrium curve turns over from a positive slope to a negative slope (as in the case of graphite in the current standard carbon phase diagram;3 broken convex curve in Figure 3) or vice versa, from negative to positive. This corresponds to a maximum (or minimum when the slope changes from negative to positive) in the melt curve with pressure; for graphite, such a maximum is reported to occur at about 6 GPa and 4300 K, as depicted by the broken convex curve in Figure 3.a This behavior may be read in the light of the Clapeyron equation. Indeed, the change of sign in the slope of a solid-liquid equilibrium curve can reflect the fact that, with changing pressure and temperature along the curve, the volumes of the two phases change progressively to different extents, so that their volume difference becomes zero at some point, and the dp/dT slope becomes infinite (corresponding to division by zero in the Clapeyron equation) or, equivalently, the dT/dp slope becomes zero, corresponding to a maximum or minimum in the Tp plane. With some caution, comparison may be made with the more familiar, but profoundly different, situation occurring at the critical point where the liquid-vapor line terminates. Here, with increasing temperature and pressure along the two-phase line, the liquid and vapor molar volumes tend to approach each other and eventually become equal at the critical point. However, the analogy between the two situations begins and ends here because, at the critical point, the two phases “merge” into a single supercritical phase. Not only the volumes, but also the enthalpies assume the same value, so the Clapeyron slope in this case becomes indeterminate (0/0). Gibbs’s phase rule, F = C - P þ
2, where P is the number of phases in thermodynamic equilibrium and C is the number of independent components, determines that beyond the critical point the one-phase fluid system has two degrees of freedom, F = 2, to be represented on the phase diagram by a surface, whereas the liquid-gas curve terminates at a critical point with zero variance.9 The unusual behavior of an apparent continuation for carbon of the phase equilibrium curve through the melt maximum has evoked an alternative suggestion that there is actually an obscured liquid-liquid phase transition (LLPT) at this point, where one form of liquid carbon is converted into another,10 each in equilibrium with solid. If correct, then F = 1 - 3 þ 2 = 0, and this corresponds to an invariant, triple point. While this is an unusual condition, phosphorus under pressure does undergo such an LLPT transition.11 High-pressure melt experiments are exceedingly difficult to perform, and the reported maximum for carbon has only been found indirectly. Moreover, no direct studies have found this invariant point for graphite melt, while careful simulations show a small positive slope throughout the melting range.12 It is currently considered that the melt maximum is an experimental artifact that should be rejected, whereas the graphite melt curve correctly has a slight positive slope, as depicted by the solid line in Figure 3. This would result in a “standard” pVT diagram in the high-pressure range for carbon (cf. Figure 1). For diamond, other simulations have suggested a maximum in the melt curve13 while very recent experimental measurements14 indicate a negative Clapeyron slope between 0.60 and 1.05 TPa with a complex, conducting fluid state produced at higher pressures and temperatures. There are a number of reports in the literature of other materials having maximum melting points with pressure, such as the early report by Bridgman on the transition between HgI2 solids I and II15 and the melting behavior of several block s (rubidium, cesium, barium) and block p (bismuth, tellurium) 588
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Figure 5. pVT phase diagram model of a generalized material, showing the situation when the initial slope of the “s þ l” curve is positive, passes through the vertical, and then becomes negative. This progressively developing situation is shown as a rising ribbon, with width initially decreasing to zero to the point of equality of solid and liquid molar volumes, then expanding again as the solid molar volume progressively exceeds that of the liquid.
Figure 7. pVT phase diagram of sodium, solid and liquid only, from ambient conditions to high pressures. The molar volume of the liquid has been exaggerated for clarity of presentation. The broken line represents the transition between the bcc and fcc crystalline phases; the dots represent further crystalline phase transitions into a series of complex solids. The pressure ranges from 0 to 120 GPa, the temperature from 200 to 1000 K, and the molar volume from 0 to 40 Å3.
In the case of an unidentified phase change, the triple point appears as a maximum or minimum in the melting temperature, with the derivative passing discontinuously from positive to negative or vice versa. This behavior appears as a turnover point in the pVT surface, as may readily be seen1b,c for water at the liquid-ice I-ice III transition point (251.165 K, 0.2099 GPa), where the molar volume of liquid is 16.52 cm3 mol-1, that of ice I is greater at 19.4 cm3 mol-1 (hence a negative slope before the transition), and that of ice III is less at 15.7 cm3 mol-1 (yielding a positive slope after the transition).
Figure 6. pV relationship of the solid phases of sodium at T = 298 K from ambient to high pressure; the volume difference between fcc and bcc phases is not discernible on the scale of this diagram. There is a more-or-less continuous change in volume even as the crystal structure undergoes a number of increasingly complex phase transitions with increasing pressure.20.
elements.16 Although some of these observations may be associated with obscured triple points,17 where a new solid or liquid phase becomes stable at high pressure without detection, others may indeed involve the volumes of liquid and solid phases passing through a point of equal density (i.e., equal volume).18
’ PHASE DIAGRAMS OF ALKALI METALS The alkali metals are of technological interest as liquid coolants for nuclear reactors and of chemical interest as they are generally anticipated to be simple representatives of metals. However, it turns out that they exhibit quite complex phase behavior as pressure increases,16 undergoing a number of melting maxima and minima together with changes of crystal structure, from the simple cubic ambient structure to increasingly complex structures. As an example of the case of a smooth, genuine maximum in the melting curve, let us consider sodium. According to recent firstprinciples simulations,18 the maximum displayed by the melting temperature of sodium (about 990 K at 31 GPa; Figure 4) would be caused by the larger compressibility of the liquid phase compared to the solid. With increasing pressure, the (positive) difference Vliquid - Vsolid is reduced by this effect and eventually 589
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passes through zero to negative values at some pressure. This causes a smooth maximum in the pT plane, without any accompanying first-order liquid-liquid phase transition. Figure 5 illustrates how the width of the solid-liquid phase gap in the pVT diagram passes through zero as the Clapeyron slope goes from positive to negative through the point of volume (density) equality. Figure 4 displays the experimental pT phase diagram of sodium in all its complexity,19 whereas Figure 6 shows the simple pV behavior of the phases at T = 298 K as determined by a firstprinciples density functional theory study.20 In this figure, the bcc-fcc phase change is not distinguishible owing to the very small volume change accompanying the transition, so that a single continuous curve fits the pV data for both solid phases rather well. The 3D pVT diagram of sodium, liquid and solid only, is reported in Figure 7. This shows that at low pressures the less dense liquid is more compressible than the solid, with the two phases reaching equality of density at a maximum melting point at about 31 GPa and 990 K, whereafter the melting point decreases to a bcc-fcc phase transition. As pressure continues Table 1. pVT Data on the High-Pressure Phase Relations of Sodium Crystal Structurea,b or Transition Point p/GPac T/Kc V/Å3 (estimated) Hexagonald
ambient
