Are Crystallization and Melting the Reverse Transformation of Each

Research: Science and Education

Are Crystallization and Melting the Reverse Transformation of Each Other?

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Hermínio P. Diogo Centro de Química Estrutural, Complexo I, IST, Av. Rovisco Pais, 1049-001 Lisboa, Portugal Joaquim J. Moura Ramos* Centro de Química-Física Molecular, Complexo I, IST, Av. Rovisco Pais, 1049-001 Lisboa, Portugal; *[email protected]

Despite the importance of crystal technology and of epilayer fabrication, these topics are not addressed in the typical undergraduate curriculum. Chemistry students are usually exposed to crystallization only as an equilibrium phase transition that is part of a phase diagram. Phase transitions occur in nature and in technological processes under nonequilibrium conditions, where crystallization is not simply the reverse of melting. For example, liquids can be cooled below their freezing point without crystallizing, and substances can crystallize when heated. In this article we will show that crystallization is not simply the reverse of melting, and we will explain apparently anomalous crystallization behavior by examining crystallization as a two-step process: nucleation and crystal growth.

Four Examples of Crystallization

Melting and Crystallization Process Melting is most often a single-step process determined solely by thermodynamic factors. The melting of a crystal may be a first-order thermodynamic transition according to the Ehrenfest (1) classification. Such phase transitions are typical for low molecular weight crystals, they occur at a definite temperature, and are accompanied by a discontinuous change in thermodynamic properties such as enthalpy and density. Sometimes, however, complex features may occur in the melting peak obtained by differential scanning calorimetry (DSC). So-called “pre-melting” or surface melting can occur several degrees below the melting point of the bulk solid (1). For a hydrogen-bonded crystal, the progressive destruc-

caffeine

phenyl salicylate

Four substances (Figure 1) showing various crystallization properties are examined. The DSC thermograms are shown in Figures 2 to 5. The background of the technique of DSC and the significance of the information it provides is clearly explained in several textbooks (3). Figure 2 shows caffeine with linear heating and cooling ramps of 2, 5, 10, and 15 ⬚C兾min. The endothermic peaks, (downwards) obtained on heating, correspond to the melting process. The exothermic peaks, (upwards) obtained on cooling, correspond to the crystallization process. The onset temperatures of the two processes are very similar (ton = 235–236 ⬚C), they are not very dependent on the heating or cooling rate, the peak shapes are similar, and thus the enthalpy changes

p-cresol

salicyl salicylate

Figure 1. Chemical structures of the compounds.

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tion of those bonds on heating prior to the melting point can cause a broadening of the melting peak. The melting process, while not uncomplicated, is still much simpler than the crystallization process. Crystallization is a combination of two processes: nucleation and crystal growth (2). Crystallization requires the presence of a nucleus on which the crystal will subsequently grow. In the absence of nuclei, the material will form a glass upon cooling. The nucleus for crystallization may be either homogeneous (forming spontaneously in the melt) or heterogeneous (forming at a pre-existing surface of an impurity, for example). In this article, we are concerned only with homogeneous nucleation.

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Figure 2. Results of DSC experiments performed on caffeine. The experimental protocol was a sequence of heating and cooling linear ramps with different rates: 2, 5, 10, and 15 ºC兾min.

Vol. 83 No. 9 September 2006

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Journal of Chemical Education

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for crystallization and melting are equal in magnitude but opposite in sign. Figure 2 corresponds to so-called normal behavior. Figure 3 shows p-cresol with the same experimental protocol as Figure 2. The results are decidedly different. The onset of the melting peak (ton = 25 ⬚C) and of the crystallization peak (ton from ᎑30 to ᎑25 ⬚C) are markedly displaced, the onset of crystallization is variable in a way not dependent of

Figure 3. Results of DSC experiments carried out on p-cresol. The experimental protocol was a sequence of heating and cooling linear ramps with different rates: 2, 5, 10, and 15 ºC兾min.

the cooling rate, and the shapes of the peaks also vary. This shows supercooling (4) of p-cresol, where the melt enters a metastable state at temperatures less than 25 ⬚C. This supercooled state persists for over 50 ⬚C, even at the lowest cooling rate of 2 ⬚C兾min. The third sample is phenyl salicylate, which requires a different procedure. Salol shows a lower tendency to crystallize on cooling when compared to p-cresol. It is easily supercooled, and it is relatively easy to vitrify. After melting at 60 ⬚C, the melt is cooled to a glassy state at ᎑90 ⬚C. Figure 4 shows the subsequent heating and cooling cycle for three different rates. All show three common features: (i) a glasstransition temperature, Tg, at about ᎑52 ⬚C; (ii) an exothermic (upwards) signal occurring during heating over a temperature interval from ᎑25 ⬚C to the melting point; and (iii) the endothermic (downwards) signal located between 32 ⬚C and 42 ⬚C according to the heating rate. The variable endothermic melting peak is the result of crystal polymorphism (5). The exothermic signal is the result of crystallization occurring during the heating cycle, also called “cold crystallization”. This counterintuitive phenomenon will be discussed later. A final example, salicyl salicylate, is examined in Figure 5. It melts over a wide range (139 ⬚C to 151 ⬚C) and cools without crystallization to form a supercooled liquid that can be molded and stress fractured. No normal means of inducing crystallization, including crystal seeding, are effective in this material. Only dissolution and recrystallization yields the crystalline solid (6). A glass-transition signal was observed at about 6 ⬚C, but no cold crystallization is observed. Description of the Nucleation Process

Figure 4. Results of DSC experiments performed on phenyl salicylate. The sample was heated from the glassy state at t = ᎑90 ⬚C to above the melting temperature. The heating rates were: (A) 2 ºC兾min; (B) 5 ºC兾min; (C) 20 ºC兾min.

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In any equilibrated macroscopic system, local variations of the values of the macroscopic properties occur and are called fluctuations. These fluctuations correspond to a local increase of the thermodynamic potential, so that they are less probable than the mean (macroscopic) state of the system. In a homogeneous molecular system (liquid or vapor) there are always small fluctuations of the density, that is, small molecular aggregates that are well compatible with the actual state of aggregation of the system. The concentration of these fluctuations is higher near the conditions of phase equilibrium, where the chemical potentials of the two phases are equal. The fluctuations in an equilibrated phase (the most stable phase at some specified temperature) are ephemeral in the sense that their sizes are negligible and they decay without revealing any tendency to grow. However, in a metastable phase (the supercooled liquid or the superheated liquid, for example) the tendency of the clusters of the most stable phase to grow prevails after exceeding a certain critical size. These density fluctuations or clusters are the so-called critical nuclei of the new phase, and, as will be seen, some Gibbs energy must be expended to form such clusters. These clusters are continuously being formed and dissociated in the liquid, and this molecular accretion process occurs based on localized energy fluctuations. Clustering is a random event where molecules come together, interact, and then dissociate again on some time scale. If conditions are


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correct (sufficient undercooling), the rate of association is greater than the rate of dissociation, and a cluster of critical size is formed that converts into a crystal lattice. Thermodynamics of Nucleus Formation The theory of homogeneous nucleation is presented in several specialized books (7–9). As shown in the Supplemental Material,W the Gibbs energy change for Z monomeric liquid molecules aggregating to form a spherical cluster (nucleus) is given by ∆G =

4 3 π r ∆GV + 4π r 2 σ 3

Jgr = a

∆mS ∆ S ∆U exp − m exp − k BT kB kBT

∆T

(4)

where a is a constant, ∆mS is the change in entropy for melting, ∆U is the energy barrier to flow owing to viscosity, and ∆T is the degree of undercooling, that is, the difference between T (the temperature of the supercooled melt) and Tm

(1)

where ∆GV is the Gibbs energy of formation of the solid phase per unit volume, σ is the surface tension between the solid and the liquid, and r is the radius of the cluster. A graphical solution of ∆G as a function of r is shown in Figure 6. The maximum in Figure 6 corresponds to the critical size of the nucleus, rc, above which crystal growth is spontaneous. The value of the Gibbs energy change making a cluster of this critical size, ∆Gc, is the nucleation thermodynamic barrier. Since the entropy change of liquid molecules becoming the critical-sized cluster, ∆Sc, is negative, the magnitude of ∆Gc decreases with a decrease in temperature; that is, ∂ ∆G c ∂ T

= −∆Sc

(2)

P

W

The Supplemental Material shows in more detail how ∆Gc, and rc both decrease with decrease in temperature. Thus, from a thermodynamic viewpoint, nucleation is more efficient at low temperatures.

Figure 5. Results of DSC experiments performed on salicyl salicylate. The crystalline sample was heated at a rate of 10 ºC兾min and the endothermic melting peak was observed at ton = 144 ºC. Subsequently, the melt was cooled at 10 ºC兾min to t = ᎑50 ºC: no crystallization is observed but the fingerprint of the glass transition on cooling is clearly shown. Finally, the sample was heated from the glassy state at 10 ºC兾min: the glass-transition signal is again observed (onset at ton = 6 ºC), and no cold crystallization takes place.

Kinetics of Nucleus Formation As shown in the Supplemental Material,W the kinetic rate of nucleus formation is given by a Boltzmann distribution:

Jnucl =

NV kB T 3

3 πη λ

exp −

∆Gc k BT

(3)

where NV is the number density of molecules in the nucleus, η is the viscosity coefficient, and λ is the mean free path length (approximated as the molecular diameter). At a temperature approaching the glass-transition temperature, Tg, the viscosity is too high and the resulting motion of molecules is too sluggish for a reasonable kinetic rate of aggregate cluster formation. At a temperature approaching the melting point, the thermodynamics become unfavorable for nucleus formation. Thus there is an inevitable tradeoff between nucleation at a lower temperature (thermodynamically favored via a lower ∆Gc) and a nucleation at a higher temperature (kinetically favored via a lower viscosity). Crystal Growth As shown in the Supplemental Material,W the kinetic rate of crystal growth is given by (7):

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Figure 6. Representation of the Gibbs energy change associated with the formation of a stable nucleus, ∆G, as a function of the radius of the nucleus, r. The increasing and decreasing curves correspond respectively to the second and first term in the right hand side of eq 1. The critical radius, rc, and the barrier ∆Gc are indicated.


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Figure 7. Schematic representation of the nucleation rate, Jnucl, and of the rate of crystal growth, Jgr, as a function of the temperature. The rate units are arbitrary. Tg is the glass-transition temperature and Tm is the melting temperature. The gray area, where the curves overlap, defines the temperature range where both nucleation and growth can occur with significant probability.

(the normal melting point). Here again we see a dual role of temperature in the rate expression. For a melt that is significantly supercooled, ∆T is large, and the crystal growth rate increases proportionately. But this also implies a low absolute value of T, which in the second exponential term will make for a more negative exponent, and thus a smaller rate of crystal growth. Like the nucleus formation, crystallization will occur between Tg and Tm. Equations 3 and 4 show that the optimum temperature for nucleus formation and the optimum temperature for crystal growth are not necessarily the same. One version of this is shown graphically in Figure 7. Qualitative Interpretations of the Four Examples Development of structured clusters is recognized to be more enhanced as the temperature is decreased down to the glass-transition temperature. The maximum rate of growth process, on the other hand, has been observed for many systems rather in the middle of the interval between Tg and Tm, or at temperatures closer to Tm. The maximum rates of the homogeneous nucleation and of the crystal growth are expected to appear, in many cases, at considerably different temperatures (10). For caffeine in Figure 2, the rate of nucleation and the rate of crystal growth must both be efficient at a temperature very near Tm (i.e., the two curves in Figure 7 significantly overlap and are shifted to the right-hand extreme of Tm). For p-cresol in Figure 3, the crystallization peak was strongly displaced to temperatures below Tm. This indicates that the nucleation curve and crystal growth curve overlap only at a temperature shifted left from Tm in Figure 7. For phenyl salicylate, the normal cooling curve results in no overlap between the nucleation curve and the crystal growth curve. As temperature is lowered from Tm, a temperature range is reached where crystal growth is efficient, but no nu-

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clei have been formed. At yet lower temperatures, nuclei begin to form, but crystal growth has become kinetically impractical. In the case of Figure 4, the melt was quenched to temperatures below Tg. On heating, the nuclei are formed first and, if thermally stable, survive up to a temperature range where crystal growth can occur. This explains the anomalous result of cold crystallization, the phenomenon of crystals forming in an exothermic process while heating the supercooled melt. For salicyl salicylate in Figure 5, we see a strong resistance to crystallization both on heating as well as cooling; there is no cold crystallization. This could be the result of the nuclei formed at lower temperatures not having sufficient thermal stability to reach the temperatures required for crystal growth. Alternatively, one could propose very restrictive stereochemical requirements that make crystallization inefficient. Summary In teaching phase transitions, we should avoid giving students the impression that crystallization is simply the reverse of melting. While melting is a single-step process controlled by thermodynamics, crystallization is a two-step process (nucleation and crystal growth) of which both steps reflect thermodynamic and kinetic control. W

Supplemental Material

A more complete presentation of classical nucleation and crystal growth theories is available in this issue of JCE Online. Literature Cited 1. Papon, P.; Leblond, J.; Meijer, P. H. E. The Physics of Phase Transitions: Concepts and Applications; Springer–Verlag: Berlin, 2002. 2. Shelby, J. E. Introduction to Glassy Science and Applications; Royal Society of Chemistry: Cambridge, United Kingdom, 1997. 3. Principles of Thermal Analysis and Calorimetry; Haines, P. J., Ed.; Royal Society of Chemistry: Cambridge, United Kingdom, 2002. 4. Debenedetti, P. G. Metastable Liquids: Concepts and Principles; Princeton University Press: Princeton, NJ, 1996. 5. Moura Ramos, J. J.; Correia, N. T.; Diogo, H. P. Phys. Chem. Chem. Phys. 2004, 6, 793–798. 6. Greener, B.; Archibal, S. J.; Hodkinson, M. Angew. Chem. Int. Educ. 2000, 39, 3601–3604. 7. Markov, I. V. Crystal Growth for Beginners: Fundamentals of Nucleation, Crystal Growth and Epitaxi; World Scientific Publishing Co.: Singapore, 1998. 8. Hartel, R. W. Crystallization in Foods; Aspen Publishers, Inc.: Gaithersburg, MD, 2001. 9. Gutzow, I.; Schmelzer, J. The Vitreous State: Thermodynamics, Structure, Rheology and Crystallization; Springer–Verlag: Berlin, 1995. 10. Hikima, T.; Hanaya, N.; Oguni, M. Solid State Comm. 1995, 93, 713–717.


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Are Crystallization and Melting the Reverse Transformation of Each

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