Communication pubs.acs.org/crystal
Isobaric Cooling or Isothermal Compression? Unveiling the Effect of Path Dependence on Crystallization Karolina Adrjanowicz,*,†,§ Kajetan Koperwas,†,§ and Marian Paluch†,§ †
Institute of Physics, University of Silesia, ulica Uniwersytecka 4, 40-007 Katowice, Poland SMCEBI, ulica 75 Pulku Piechoty 1a, 41-500 Chorzow, Poland
§
S Supporting Information *
ABSTRACT: In this work, we have investigated the effect of path dependence on the crystallization tendency of a van der Waals liquid. The sample, dimethyl phthalate, was crystallized at carefully selected (Tc, pc) state points approached via two alternative routes. The first involved (i) isothermal compression to the crystallization pressure pc, followed by isobaric cooling to the crystallization temperature Tc; for the second (ii) isobaric cooling to Tc was followed by isothermal compression to pc. Here, the final (Tc, pc) point is located in the undercooled liquid state, but far above the glass transition. From changes in the dielectric response as crystallization proceeds, we show that the two pathways are not equivalent. A clear difference was observed in the induction period, defined as the time required to produce a crystal nuclei of a sufficient size to grow. We interpret this finding based on predictions of the classical theory of nucleation and growth, as well as results of molecular dynamics simulations of a Lennard-Jones system.
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history plays a crucial role in determining the dynamics properties of glass. The pathway to an out-of-equilibrium state, whether by isobaric cooling or isothermal compression, results in two glassy materials of entirely different densities and secondary relaxation dynamics.13,14 A question that arises is whether the alternative ways of approaching the final state are equivalent for a complex and multivariable phenomenon such as crystallization. Intuitively, we expect this to be the case because even at atmospheric pressure crystallization/glassforming tendency varies with cooling or heating rates.15−20 We are unaware of any previous studies, showing that crystallization behavior in the T-p plane depends on the way to approach the final state (either via isobaric cooling or isothermal compression). What is certain is that the key factors responsible for crystallization,21 i.e., molecular mobility and thermodynamic driving force, are assumed to depend only on the final state point, not on the pathway.22 To study the effect of path dependence on crystallization, we followed the crystallization kinetics of a van der Waals liquid, dimethyl phthalate, at selected crystallization conditions (Tc, pc) approached via two routes, involving either isobaric cooling or isothermal compression. We employed the classical expressions from nucleation and growth theory to analyze the data, along with molecular dynamics simulations of a Lennard-Jones material. We start by providing an overview of the present study. Figure 1 shows a schematic T-p phase diagram of a glassforming liquid. Assuming the sample is in the equilibrium liquid
he physical and chemical properties of matter may drastically change when subjected to high pressure. Among these, pressure-induced changes in the crystallization behavior of various materials are of great scientific and technological import. Crystallization under high pressure plays a central role in geology and mineralogy.1−3 It also serves as a versatile tool to produce new types of materials with improved hardness, superconductivity, or optoelectronic properties.4−7 As revealed by numerous experimental observations, crystallization at increased pressure can be quite different from that observed when cooling a liquid at atmospheric pressure. The difference arises from the fact that temperature and pressure are not equivalent thermodynamic variables, although both produce superficially similar effects. Specifically, temperature and pressure act differently on the internal energy of the system. Temperature changes the kinetic energy, whereas pressure modifies the intermolecular distances. Only by exploring both thermodynamic variables is it possible to get a complete understanding of crystallization and vitrification phenomena.8−12 The motivation for the present study is to examine the effect of path dependence on the crystallization of molecular liquids. Considering the experimental difficulties when using high pressure, the particular approach to T-p coordinates is typically less important than the behavior of the material at a given thermodynamic condition. In high-pressure research, compression of a system at fixed temperature is more convenient, because pressure takes less time to stabilize than temperature. This is not a problem for experiments on liquids far above the glass transition temperature because relaxation does not exceed experimentally available times. In contrast, thermodynamic © XXXX American Chemical Society
Received: April 7, 2017 Revised: May 5, 2017 Published: May 18, 2017 A
DOI: 10.1021/acs.cgd.7b00493 Cryst. Growth Des. XXXX, XXX, XXX−XXX
Crystal Growth & Design
Communication
Figure 1. Schematic illustration of the general idea of this study. In the T-p phase diagram of a modeled glass-forming liquid, we set an initial thermodynamic condition (T0, p0). A studied liquid is transferred to crystallization point (Tc, pc), either via isobaric cooling (1) or isothermal compression (2). Then, we have performed timedependent dielectric measurement to verify if the crystallization tendency at the final (Tc, pc) state depends on the chosen path.
Figure 2. Real part of the dielectric permittivity ε′ (at 2 kHz) plotted as a function of time for dimethyl phthalate held at 282 K and 100 MPa. Ongoing changes in the dielectric signal signify crystallization progress. Red circles and blue squares denote data recorded at a given crystallization conditions reached by moving in T-p space via path 1 and path 2, respectively. In the inset, we show the T-p phase diagram of dimethyl phthalate which includes the initial and final points, as well as the two alternative routes of approaching selected (Tc, pc) conditions. Tg(p) and Tm(p) dependences were taken from ref 23. The final (Tc, pc) point is located in the undercooled liquid state, but far above Tg (Tc ≅ Tg + 80 K).
state initially at (T0, p0), we wish to transfer it to the state point (Tc, pc) located in the metastable supercooled liquid regime, but still above the glass transition temperature Tg. With two thermodynamic variables available, this can be accomplished by at least two different pathways. One entails isothermal compression to pc, then isobaric cooling to Tc, and the second, isobaric cooling to Tc followed by isothermal compression to pc. Crystallization is thermodynamically favored only below the melting line, Tm(p). Therefore, moving from (i) (T0, p0) → (T0, pc) or (ii) (T0, p0) → (Tc, p0) does not induce or affect crystallization. However, the liquid can be undercooled, with consequent crystallization, either by lowering the temperature at constant pressure (path 1) or increasing the pressure at fixed temperature (path 2). When the former pathway is chosen, the melting point is always approached from higher temperature and higher pressure than for path 2. The experimental protocol requires that the time for transferring a system to the crystallization condition is approximately the same in both cases. Unfortunately, the equilibration period for stabilization of temperature versus pressure conditions is usually not the same. Entering the supercooled state via isobaric cooling takes more time than for isothermal compression. This can result in earlier crystal formation for the latter, distorting the results. A critical aspect of the present study was to make these two time scales comparable. We achieved this by combining high-pressure equipment utilizing an automated pressure pump (Unipress) and highly dynamic temperature control system (Presto W85, Julabo). Details of the instrumentation and experimental procedures can be found in the Supporting Information. The sample chosen for this study is dimethyl phthalate, a low-molecular-weight van der Waals liquid (Tm = 272.5 K at 0.1 MPa). In the studied T-p range, the material crystallizes in a very reproducible manner, with no indications of polymorphic transitions. The inset in Figure 2 illustrates the Tm(p) and Tg(p) dependences of dimethyl phthalate.23 In the T-p phase diagram, we included the initial (T0 = 308 K, p0 = 0.1 MPa) and final (Tc = 283 K, pc = 100 MPa) state points. As can be seen, the final (Tc, pc) point is located in the undercooled liquid state, but far above the glassy state (at 100 MPa Tc ≅ Tg + 80 K which is also Tc = Tm − 14 K). Changes in the real part (ε′) of
the complex dielectric permittivity at 2 kHz accompanying crystallization are presented in Figure 2. As can be seen, after some time at (Tc, pc), ε′ starts to decrease. This effect is due to reduction in the number of reorienting dipoles. We found that the crystallization of dimethyl phthalate depends strongly on the path taken to the final conditions. The crystallization halftime, t1/2, defined as the time for changes in crystallinity of 50%, is almost two times shorter for path 1; to wit, t1/2 ≅ 27 min, versus 51 min for path 2. As we suppose, shifting the (Tc, pc) point closer to Tg should not qualitatively change the behavior observed by us experimentally at low undercooling. As seen in Figure 2, the difference in crystallization behavior of the liquid is mostly in the induction period, tind. For path 2, tind ≅ 36 min, which is more than twice the time for path 1, tind ≅ 16 min. The induction time reflects the nucleation process, being inversely proportional to the nucleation rate. Formally, the induction period is composed of three parts: the relaxation time tr required for a system to achieve quasi-steady-state conditions; the time tn needed to form stable nuclei; and tg, the time for nuclei to grow to detectable size.24 Experimentally, we are not able to separate these contributions to tind. However, at selected (Tc, pc) conditions the molecular motions have time scales of a few nanoseconds (we are far above Tg), and thus will not depend on the thermodynamic pathway. Therefore, for these starting conditions the contribution from the relaxation time is negligible. The time needed to grow nuclei to a detectable size is expected to be the same for both routes. This size depends on the resolution of the experimental technique employed to follow the early stages of the crystallization process. Here, we use only dielectric spectroscopy as a probe. It follows that any differences in the induction period at (Tc, pc) must be related with the nucleation time. From the above it can be concluded that the two alternative routes of approaching B
DOI: 10.1021/acs.cgd.7b00493 Cryst. Growth Des. XXXX, XXX, XXX−XXX
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final conditions modify primarily the very early stages of the crystallization. It is worth noting that the dielectric permittivity of the fully crystallized material also depends on the thermodynamic path. The sample crystallized via path 1 has a slightly higher value of the dielectric constant than for path 2. This indicates that the former material has more and/or larger defects. A more irregular alignment of molecules within the crystal will change the charge polarization and affect the dielectric response. Below we show that the probability of having a less perfect crystalline structure is indeed obtained for path 1. In order to understand better the effect of the thermodynamic pathway on crystallization, we have estimated the nucleation and crystal growth rates from the classical expressions. The classical theory of crystallization is often criticized for a number of simplifications and predictions of inaccurate nucleation and crystal growth rates.25,26 Nevertheless, it is a source of fundamental information on the crystallization of many materials.18,27 The theory has not formally addressed T-p path dependent effects as studied herein. More details regarding the calculation of nucleation and growth rates for different thermodynamic conditions can be found in the literature.28−31 We have also demonstrated that in the Supporting Information for the reader’s convenience. The analysis requires a number of basic physical parameters describing the properties of our material, such as the change of the melting point, α-relaxation time (or viscosity), density and specific heat capacity (for the liquid and crystalline phases), and their evolution as a function of both temperature and pressure.23 In Figure 3 we present the dependences of the nucleation and growth rates at two pressures, 0.1 and 100 MPa. As can be seen, the nucleation rate increases and shifts to higher temperature with increasing pressure. The growth rate curve shifts toward higher temperature, but its magnitude remains essentially the same. In order to compare changes in the nucleation rate for our two pathways, we plotted the predicted
dependences as a function of the reduced specific volume quantity (Vm − V)/(Vm − Vc), where Vm and Vc are the specific volumes of the liquid phase at the melting point and crystallization point, respectively. The inset of Figure 3 shows changes in the nucleation rate along an isobar (100 MPa) and an isotherm (283 K). As we cannot compare directly isothermal and isobaric dependences, we have analyzed them as a function of density. For that we have made use of pressure−volume− temperature data (measured in the liquid state). The results indicate that path 1 gives rise to a higher nucleation rate curve. When path 2 is selected, dimethyl phthalate enters the supercooled regime with a lower rate of nucleation. At the final state, the nucleation rates are the same, irrespective of the chosen path. In agreement with the classical theory, crystal formation is a dynamic process that involves attachment or detachment of single molecules to clusters of different size. However, only those with a sufficient (overcritical) size grow, with the nucleation continuing. The smaller clusters dissolve to single molecules, which may subsequently develop into larger clusters.18,32 With increasing the temperature, the cluster size distribution decreases, and only the growth of large clusters is favored.19 Some time ago, Kashchiev predicted that the initial cluster size distribution affects primarily the time lag, but not the nucleation rate.33,34 In this regard, changes in crystallization behavior of dimethyl phthalate with T-p path might be ascribed to formation of somewhat different cluster ensembles. The effect of pressure on the cluster size distribution remains to be investigated. Finally, we performed molecular dynamics simulations of a Lennard-Jones system. In analogy to the experimental study, the simulation was carried out at a thermodynamic condition (120 K, 10 MPa) approached either by isothermal compression or by isobaric cooling. The starting point in the liquid phase was (150 K, 100 bar) for the isobaric path and (120 K, 10 bar) for isothermal path. Each starting configuration was equilibrated for 500 000 time steps, followed by cooling or compression to the final crystallization condition (also located far above Tg). The transformation from the liquid to the crystalline phase was followed by analyzing changes in the density over 2 million time steps. This procedure was repeated 20 times for each path. Details regarding the MD simulations can be found in the Supporting Information. Figure 4a summarizes the results. As can be seen, three different scenarios are possible. Only a few samples did not crystallize within the allotted time, whereas the remainder formed phases of two different and clearly distinguishable densities. For samples subjected to isobaric cooling, 65% of the material formed a denser, more packed structure, while 35% formed the less dense one. In contrast, for isothermal compression the denser crystalline phase was more favored (80%). The structural properties of the two crystals having different densities were characterized by the radial distribution function (RDF). In the inset of Figure 4a, we show the RDF that corresponds to a standard fcc lattice. What distinguishes the two crystalline systems is the more random arrangement of the less dense sample. In this case, we expect a larger population of overcritical clusters which results in more random orientation of each crystallite with respect to the others. The consequence is a slight broadening of the RDF peaks. This is also seen by comparing the final configurations (Figure 4b,c). Therefore, from the MD simulations we conclude that it is statistically of higher probability to get uniformly formed
Figure 3. Evolution of the nucleation rate as a function of pressure as predicted for dimethyl phthalate from the classical nucleation theory. Upper inset: Estimated temperature dependence of the crystal growth rate at 0.1 and 100 MPa. Lower inset: Nucleation rate (normalized by Nc, i.e., nucleation rate at the final state) along isobar 100 MPa and isotherm 283 K plotted as a function of the reduced volume (Vm − V)/(Vm − Vc). C
DOI: 10.1021/acs.cgd.7b00493 Cryst. Growth Des. XXXX, XXX, XXX−XXX
Crystal Growth & Design
Communication
Figure 4. (a) Changes in the density as a function of time collected upon simulation runs at 120 K and 100 bar. Desired conditions (T = 120 K, p = 100 bar) were reached via isobaric cooling or isothermal compression. Inset shows radial distribution functions for the crystal phases of the different densities. Panels (b) and (c) show final configurations of the simulated systemsthe denser and less dense crystals, respectively.
project (Grant No. DEC 2014/15/B/ST3/00364). K.A. acknowledges financial support from Ministry of Science and Higher Education within “Iuventus Plus” project (0001/IP3/ 2016/74). The authors would like to thank Dr. Juern W. P. Schmelzer from Rostock University for helpful discussion and Dr. C. M. Roland from Naval Research Laboratory for providing language assistance.
material with denser structure when isothermal compression is chosen rather than isobaric cooling. In summary, we have demonstrated that crystallization is T-p path dependent. The pathway to the (T, p) condition affects primarily the initial stages of the phase transition. For dimethyl phthalate, isobaric cooling and isothermal compression affect the induction period on the time scale of minutes. Of course, these changes would be more substantial at higher temperature or in the gigapascal regime where the density effects are amplified. Thus, the present results contribute to a general understanding of high-pressure crystallization. One example of the applicability of the effect of the thermodynamic path to crystallization is to estimate or reconstruct the transformation kinetics of natural minerals formed in the deep interior of the Earth or other planets.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.cgd.7b00493. Description of the procedure for calculation of nucleation and crystal growth rates from the classical expressions; experimental and computational methods (PDF)
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REFERENCES
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Karolina Adrjanowicz: 0000-0003-0212-5010 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors are grateful for the financial support from the National Science Centre within the framework of the Opus D
DOI: 10.1021/acs.cgd.7b00493 Cryst. Growth Des. XXXX, XXX, XXX−XXX
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Communication
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DOI: 10.1021/acs.cgd.7b00493 Cryst. Growth Des. XXXX, XXX, XXX−XXX
