Special Issue "Crystallization Thermodynamics"

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Thermodynamics".

Deadline for manuscript submissions: 1 July 2021.

Special Issue Editor

Dr. Jürn W.P. Schmelzer
Website
Guest Editor
Institute of Physics, University of Rostock, Albert-Einstein-Strasse 23–25, 18059 Rostock, Germany
Interests: glass and the glass transition: thermodynamics, structure, rheology, relaxation, and crystallization kinetics; thermodynamics and kinetics of first-order phase transitions: theory and applications; crystal nucleation and interplay of crystallization and glass transition

Special Issue Information

Dear colleagues,

The theoretical description of crystallization processes has a long history. Nevertheless, it remains an actively developing area of research with a variety of unresolved and newly evolving problems and a wide spectrum of applications. The classical theory of nucleation and growth processes assumes that crystallization proceeds via the formation of small crystallites, with properties being essentially the same as those of the finally evolving macroscopic phases. This assumption allows one to describe theoretically nucleation and growth processes in a qualitatively adequate way but leads to significant problems in reconciling theory and experiment quantitatively. New theoretical developments and advanced experimental analysis are required to fill this gap, which is the topic proposed for this Special Issue. Particular problems in this direction are, for example, (i) the further elaboration of methods to determine the thermodynamic driving force of crystallization by advancing the knowledge of phase diagrams of multi-component systems and the methods of theoretical predictions of the properties of the melts, (ii) the further development of methods of specification of the curvature dependence of the surface tension in the classical theory of nucleation and growth and its generalizations, (iii) the specification of both bulk and surface properties of critical crystallites in dependence on the degree of deviation from equilibrium, (iv) the interplay of crystal nucleation and glass transition in cooling processes, (v) the qualitative change of the response of the ambient phase on crystallization near to the glass transition temperature, (vi) crystallization and growth in inhomogeneous media, and (vii) new methods of specification of the crystallization activity of heterogeneous nucleation cores. In the present Special Issue, it is proposed to concentrate on thermodynamic aspects being one of the essential ingredients of the theory of crystallization processes.

Dr. habil. Jürn W. P. Schmelzer
Guest Editor

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Keywords

  • crystal nucleation
  • crystal growth
  • general theory of phase transition
  • phase diagrams
  • prediction of properties of glass-forming melts
  • glass and glass transition
  • thermodynamics of nucleation
  • surface thermodynamics
  • surface energies in surfaces and interfaces

Published Papers (16 papers)

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Research

Open AccessArticle
Statistical Approach to Crystal Nucleation in Glass-Forming Liquids
Entropy 2021, 23(2), 246; https://doi.org/10.3390/e23020246 - 20 Feb 2021
Viewed by 165
Abstract
In this work, methods of description of crystal nucleation by using the statistical approach are analyzed. Findings from classical nucleation theory (CNT) for the average time of formation of the first supercritical nucleus are linked with experimental data on nucleation in glass-forming liquids [...] Read more.
In this work, methods of description of crystal nucleation by using the statistical approach are analyzed. Findings from classical nucleation theory (CNT) for the average time of formation of the first supercritical nucleus are linked with experimental data on nucleation in glass-forming liquids stemming from repetitive cooling protocols both under isothermal and isochronal conditions. It is shown that statistical methods of lifetime analysis, frequently used in medicine, public health, and social and behavioral sciences, are applicable to crystal nucleation problems in glass-forming liquids and are very useful tools for their exploration. Identifying lifetime with the time to nucleate as a random variable in homogeneous and non-homogeneous Poisson processes, solutions for the nucleation rate under steady-state conditions are presented using the hazard rate and related parameters. This approach supplies us with a more detailed description of nucleation going beyond CNT. In particular, we show that cumulative hazard estimation enables one to derive the plotting positions for visually examining distributional model assumptions. As the crystallization of glass-forming melts can involve more than one type of nucleation processes, linear dependencies of the cumulative hazard function are used to facilitate assignment of lifetimes to each nucleation mechanism. Full article
(This article belongs to the Special Issue Crystallization Thermodynamics)
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Open AccessFeature PaperArticle
Anisotropic Nucleation, Growth and Ripening under Stirring—A Phenomenological Model
Entropy 2020, 22(11), 1254; https://doi.org/10.3390/e22111254 - 04 Nov 2020
Viewed by 338
Abstract
The anisotropic formation of elongated metal-oxide aggregates in water under intensive stirring is analyzed. It is treated in terms of anisotropic ballistically mediated aggregation kinetics in open systems. The basic kinetic equations describing the stages of homogeneous nucleation, independent growth, and ripening of [...] Read more.
The anisotropic formation of elongated metal-oxide aggregates in water under intensive stirring is analyzed. It is treated in terms of anisotropic ballistically mediated aggregation kinetics in open systems. The basic kinetic equations describing the stages of homogeneous nucleation, independent growth, and ripening of the aggregates are formulated for the open system under the external influence with the stirring intensity as the main parameter governing the process. The most significant elongation of the aggregates is shown to evolve at the ripening stage. Full article
(This article belongs to the Special Issue Crystallization Thermodynamics)
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Open AccessArticle
Effects of Glass Transition and Structural Relaxation on Crystal Nucleation: Theoretical Description and Model Analysis
Entropy 2020, 22(10), 1098; https://doi.org/10.3390/e22101098 - 29 Sep 2020
Cited by 2 | Viewed by 674
Abstract
In the application of classical nucleation theory (CNT) and all other theoretical models of crystallization of liquids and glasses it is always assumed that nucleation proceeds only after the supercooled liquid or the glass have completed structural relaxation processes towards the metastable equilibrium [...] Read more.
In the application of classical nucleation theory (CNT) and all other theoretical models of crystallization of liquids and glasses it is always assumed that nucleation proceeds only after the supercooled liquid or the glass have completed structural relaxation processes towards the metastable equilibrium state. Only employing such an assumption, the thermodynamic driving force of crystallization and the surface tension can be determined in the way it is commonly performed. The present paper is devoted to the theoretical treatment of a different situation, when nucleation proceeds concomitantly with structural relaxation. To treat the nucleation kinetics theoretically for such cases, we need adequate expressions for the thermodynamic driving force and the surface tension accounting for the contributions caused by the deviation of the supercooled liquid from metastable equilibrium. In the present paper, such relations are derived. They are expressed via deviations of structural order parameters from their equilibrium values. Relaxation processes result in changes of the structural order parameters with time. As a consequence, the thermodynamic driving force and surface tension, and basic characteristics of crystal nucleation, such as the work of critical cluster formation and the steady-state nucleation rate, also become time-dependent. We show that this scenario may be realized in the vicinity and below the glass transition temperature, and it may occur only if diffusion (controlling nucleation) and viscosity (controlling the alpha-relaxation process) in the liquid decouple. Analytical estimates are illustrated and confirmed by numerical computations for a model system. The theory is successfully applied to the interpretation of experimental data. Several further consequences of this newly developed theoretical treatment are discussed in detail. In line with our previous investigations, we reconfirm that only when the characteristic times of structural relaxation are of similar order of magnitude or longer than the characteristic times of crystal nucleation, elastic stresses evolving in nucleation may significantly affect this process. Advancing the methods of theoretical analysis of elastic stress effects on nucleation, for the first time expressions are derived for the dependence of the surface tension of critical crystallites on elastic stresses. As the result, a comprehensive theoretical description of crystal nucleation accounting appropriately for the effects of deviations of the liquid from the metastable states and of relaxation on crystal nucleation of glass-forming liquids, including the effect of simultaneous stress evolution and stress relaxation on nucleation, is now available. As one of its applications, this theoretical treatment provides a new tool for the explanation of the low-temperature anomaly in nucleation in silicate and polymer glasses (the so-called “breakdown” of CNT at temperatures below the temperature of the maximum steady-state nucleation rate). We show that this anomaly results from much more complex features of crystal nucleation in glasses caused by deviations from metastable equilibrium (resulting in changes of the thermodynamic driving force, the surface tension, and the work of critical cluster formation, in the necessity to account of structural relaxation and stress effects) than assumed so far. If these effects are properly accounted for, then CNT appropriately describes both the initial, the intermediate, and the final states of crystal nucleation. Full article
(This article belongs to the Special Issue Crystallization Thermodynamics)
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Open AccessArticle
Analytical Determination of the Nucleation-Prone, Low-Density Fraction of Subcooled Water
Entropy 2020, 22(9), 933; https://doi.org/10.3390/e22090933 - 25 Aug 2020
Viewed by 607
Abstract
Subcooled water is the primordial matrix for ice embryo formation by homogeneous and heterogeneous nucleation. The knowledge of the specific Gibbs free energy and other thermodynamic quantities of subcooled water is one of the basic prerequisites of the theoretical analysis of ice crystallization [...] Read more.
Subcooled water is the primordial matrix for ice embryo formation by homogeneous and heterogeneous nucleation. The knowledge of the specific Gibbs free energy and other thermodynamic quantities of subcooled water is one of the basic prerequisites of the theoretical analysis of ice crystallization in terms of classical nucleation theory. The most advanced equation of state of subcooled water is the IAPWS G12-15 formulation. The determination of the thermodynamic quantities of subcooled water on the basis of this equation of state requires the iterative determination of the fraction of low-density water in the two-state mixture of low-density and high-density subcooled water from a transcendental equation. For applications such as microscopic nucleation simulation models requiring highly frequent calls of the IAPWS G12-15 calculus, a new two-step predictor-corrector method for the approximative determination of the low-density water fraction has been developed. The new solution method allows a sufficiently accurate determination of the specific Gibbs energy and of all other thermodynamic quantities of subcooled water at given pressure and temperature, such as specific volume and mass density, specific entropy, isothermal compressibility, thermal expansion coefficient, specific isobaric and isochoric heat capacities, and speed of sound. The misfit of this new approximate analytical solution against the exact numerical solution was demonstrated to be smaller than or equal to the misprediction of the original IAPWS G12-15 formulation with respect to experimental values. Full article
(This article belongs to the Special Issue Crystallization Thermodynamics)
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Open AccessArticle
Crystallization of Supercooled Liquids: Self-Consistency Correction of the Steady-State Nucleation Rate
Entropy 2020, 22(5), 558; https://doi.org/10.3390/e22050558 - 16 May 2020
Cited by 5 | Viewed by 827
Abstract
Crystal nucleation can be described by a set of kinetic equations that appropriately account for both the thermodynamic and kinetic factors governing this process. The mathematical analysis of this set of equations allows one to formulate analytical expressions for the basic characteristics of [...] Read more.
Crystal nucleation can be described by a set of kinetic equations that appropriately account for both the thermodynamic and kinetic factors governing this process. The mathematical analysis of this set of equations allows one to formulate analytical expressions for the basic characteristics of nucleation, i.e., the steady-state nucleation rate and the steady-state cluster-size distribution. These two quantities depend on the work of formation, Δ G ( n ) = n Δ μ + γ n 2 / 3 , of crystal clusters of size n and, in particular, on the work of critical cluster formation, Δ G ( n c ) . The first term in the expression for Δ G ( n ) describes changes in the bulk contributions (expressed by the chemical potential difference, Δ μ ) to the Gibbs free energy caused by cluster formation, whereas the second one reflects surface contributions (expressed by the surface tension, σ : γ = Ω d 0 2 σ , Ω = 4 π ( 3 / 4 π ) 2 / 3 , where d 0 is a parameter describing the size of the particles in the liquid undergoing crystallization), n is the number of particles (atoms or molecules) in a crystallite, and n = n c defines the size of the critical crystallite, corresponding to the maximum (in general, a saddle point) of the Gibbs free energy, G. The work of cluster formation is commonly identified with the difference between the Gibbs free energy of a system containing a cluster with n particles and the homogeneous initial state. For the formation of a “cluster” of size n = 1 , no work is required. However, the commonly used relation for Δ G ( n ) given above leads to a finite value for n = 1 . By this reason, for a correct determination of the work of cluster formation, a self-consistency correction should be introduced employing instead of Δ G ( n ) an expression of the form Δ G ˜ ( n ) = Δ G ( n ) Δ G ( 1 ) . Such self-consistency correction is usually omitted assuming that the inequality Δ G ( n ) Δ G ( 1 ) holds. In the present paper, we show that: (i) This inequality is frequently not fulfilled in crystal nucleation processes. (ii) The form and the results of the numerical solution of the set of kinetic equations are not affected by self-consistency corrections. However, (iii) the predictions of the analytical relations for the steady-state nucleation rate and the steady-state cluster-size distribution differ considerably in dependence of whether such correction is introduced or not. In particular, neglecting the self-consistency correction overestimates the work of critical cluster formation and leads, consequently, to far too low theoretical values for the steady-state nucleation rates. For the system studied here as a typical example (lithium disilicate, Li 2 O · 2 SiO 2 ), the resulting deviations from the correct values may reach 20 orders of magnitude. Consequently, neglecting self-consistency corrections may result in severe errors in the interpretation of experimental data if, as it is usually done, the analytical relations for the steady-state nucleation rate or the steady-state cluster-size distribution are employed for their determination. Full article
(This article belongs to the Special Issue Crystallization Thermodynamics)
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Open AccessEditor’s ChoiceArticle
Exploring the Phase Space of Multi-Principal-Element Alloys and Predicting the Formation of Bulk Metallic Glasses
Entropy 2020, 22(3), 292; https://doi.org/10.3390/e22030292 - 02 Mar 2020
Cited by 1 | Viewed by 1077
Abstract
Multi-principal-element alloys share a set of thermodynamic and structural parameters that, in their range of adopted values, correlate to the tendency of the alloys to assume a solid solution, whether as a crystalline or an amorphous phase. Based on empirical correlations, this work [...] Read more.
Multi-principal-element alloys share a set of thermodynamic and structural parameters that, in their range of adopted values, correlate to the tendency of the alloys to assume a solid solution, whether as a crystalline or an amorphous phase. Based on empirical correlations, this work presents a computational method for the prediction of possible glass-forming compositions for a chosen alloys system as well as the calculation of their critical cooling rates. The obtained results compare well to experimental data for Pd-Ni-P, micro-alloyed Pd-Ni-P, Cu-Mg-Ca, and Cu-Zr-Ti. Furthermore, a random-number-generator-based algorithm is employed to explore glass-forming candidate alloys with a minimum critical cooling rate, reducing the number of datapoints necessary to find suitable glass-forming compositions. A comparison with experimental results for the quaternary Ti-Zr-Cu-Ni system shows a promising overlap of calculation and experiment, implying that it is a reasonable method to find candidates for glass-forming alloys with a sufficiently low critical cooling rate to allow the formation of bulk metallic glasses. Full article
(This article belongs to the Special Issue Crystallization Thermodynamics)
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Open AccessArticle
Ice-Crystal Nucleation in Water: Thermodynamic Driving Force and Surface Tension. Part I: Theoretical Foundation
Entropy 2020, 22(1), 50; https://doi.org/10.3390/e22010050 - 30 Dec 2019
Cited by 3 | Viewed by 1254
Abstract
A recently developed thermodynamic theory for the determination of the driving force of crystallization and the crystal–melt surface tension is applied to the ice-water system employing the new Thermodynamic Equation of Seawater TEOS-10. The deviations of approximative formulations of the driving force and [...] Read more.
A recently developed thermodynamic theory for the determination of the driving force of crystallization and the crystal–melt surface tension is applied to the ice-water system employing the new Thermodynamic Equation of Seawater TEOS-10. The deviations of approximative formulations of the driving force and the surface tension from the exact reference properties are quantified, showing that the proposed simplifications are applicable for low to moderate undercooling and pressure differences to the respective equilibrium state of water. The TEOS-10-based predictions of the ice crystallization rate revealed pressure-induced deceleration of ice nucleation with an increasing pressure, and acceleration of ice nucleation by pressure decrease. This result is in, at least, qualitative agreement with laboratory experiments and computer simulations. Both the temperature and pressure dependencies of the ice-water surface tension were found to be in line with the le Chatelier–Braun principle, in that the surface tension decreases upon increasing degree of metastability of water (by decreasing temperature and pressure), which favors nucleation to move the system back to a stable state. The reason for this behavior is discussed. Finally, the Kauzmann temperature of the ice-water system was found to amount T K = 116 K , which is far below the temperature of homogeneous freezing. The Kauzmann pressure was found to amount to p K = 212 MPa , suggesting favor of homogeneous freezing on exerting a negative pressure on the liquid. In terms of thermodynamic properties entering the theory, the reason for the negative Kauzmann pressure is the higher mass density of water in comparison to ice at the melting point. Full article
(This article belongs to the Special Issue Crystallization Thermodynamics)
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Open AccessFeature PaperArticle
Application of the Nucleation Theorem to Crystallization of Liquids: Some General Theoretical Results
Entropy 2019, 21(12), 1147; https://doi.org/10.3390/e21121147 - 25 Nov 2019
Cited by 2 | Viewed by 739
Abstract
Different aspects in applying the nucleation theorem to the description of crystallization of liquids are analyzed. It is shown that, by employing the classical Gibbs’ approach in the thermodynamic description of heterogeneous systems, a general form of the nucleation theorem can be formulated [...] Read more.
Different aspects in applying the nucleation theorem to the description of crystallization of liquids are analyzed. It is shown that, by employing the classical Gibbs’ approach in the thermodynamic description of heterogeneous systems, a general form of the nucleation theorem can be formulated that is valid not only for one-component but generally for multi-component systems. In this analysis, one basic assumption of classical nucleation theory is utilized. In addition, commonly employed in application to crystallization, it is supposed that the bulk properties of the critical clusters are widely identical to the properties of the newly evolving crystal phase. It is shown that the formulation of the nucleation theorem as proposed by Kashchiev [J. Chem. Phys. 76, 5098-5102 (1982)], also relying widely on the standard classical approach in the description of crystal nucleation, holds for multi-component systems as well. The general form of the nucleation theorem derived by us is taken then as the starting point for the derivation of particular forms of this theorem for the cases that the deviation from equilibrium is caused by variations of either composition of the liquid phase, temperature, or pressure. In this procedure, expressions recently developed by us for the curvature dependence of the surface tension, respectively, its dependence on pressure and/or temperature are employed. The basic assumption of classical nucleation theory mentioned above is, however, in general, not true. The bulk and surface properties of the critical crystal clusters may differ considerably from the properties of the evolving macroscopic phases. Such effects can be incorporated into the theoretical description by the application of the generalized Gibbs approach for the specification of the dependence of the properties of critical crystal clusters on the degree of metastability of the liquid phase. Applying this method, it is demonstrated that a similar formulation of the nucleation theorem, as derived based on classical nucleation theory, holds true also in cases when a dependence of the state parameters of the critical clusters on the degree of deviation from equilibrium is appropriately accounted for. Full article
(This article belongs to the Special Issue Crystallization Thermodynamics)
Open AccessArticle
Estimating the Nucleation Ability of Various Surfaces Towards Isotactic Polypropylene via Light Intensity Induction Time Measurements
Entropy 2019, 21(11), 1068; https://doi.org/10.3390/e21111068 - 31 Oct 2019
Cited by 3 | Viewed by 742
Abstract
Crystallization of isotactic polypropylene (iPP) at the interface with crystalline films of two commercially employed nucleating agents (sodium benzoate (NaBz) and sodium 2,2’-methylene bis-(4,6-di-tert-butylphenyl)phosphate (NA-11)) and with a glass fiber (GF) was investigated using a polarized optical microscope. The analysis [...] Read more.
Crystallization of isotactic polypropylene (iPP) at the interface with crystalline films of two commercially employed nucleating agents (sodium benzoate (NaBz) and sodium 2,2’-methylene bis-(4,6-di-tert-butylphenyl)phosphate (NA-11)) and with a glass fiber (GF) was investigated using a polarized optical microscope. The analysis of the light intensity evolution during the crystallization process enabled the successful estimation of the time at which the crystal growth began, i.e., the induction time (ti), at various crystallization temperatures. Meaningful differences in the ti values were observed between the investigated systems. Moreover, the ti data have been analyzed according to different nucleation models proposed in the literature, which consider either the time to form the first crystalline layer in contact with the substrate or the time required to grow a cluster of critical size. It has been found that the two models are applicable in different temperature ranges depending on the efficiency of the given substrate. Therefore, in order to obtain the value of the surface free energy difference function, Δσ, which is directly related to the nucleation energy barrier and useful for the definition of a universal nucleating efficiency scale, a model that considers both the above-mentioned times was fitted to the overall data. The values of Δσ for the nucleation of iPP on the surface of the different substrates are thus obtained and discussed in the framework of the literature results. Full article
(This article belongs to the Special Issue Crystallization Thermodynamics)
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Open AccessArticle
Critical Size of Secondary Nuclei Determined via Nucleation Theorem Reveals Selective Nucleation in Three-Component Co-Crystals
Entropy 2019, 21(11), 1032; https://doi.org/10.3390/e21111032 - 24 Oct 2019
Cited by 1 | Viewed by 660
Abstract
The critical size of the secondary nuclei plays an important role in determining the crystal growth rate. In the past, the Nucleation Theorem has been applied to determine the number of molecules in the critical nuclei of a single-component crystal via variation of [...] Read more.
The critical size of the secondary nuclei plays an important role in determining the crystal growth rate. In the past, the Nucleation Theorem has been applied to determine the number of molecules in the critical nuclei of a single-component crystal via variation of the crystal growth rate with dilution by the non-crystallizable component. In this work, we extend the method to the three-component co-crystal poly (ethylene oxide)/urea/thiourea inclusion compound. The theoretical crystal growth kinetics were deduced and the dependence of the radial growth rate of the inclusion compound spherulites on the mass fraction of urea in urea/thiourea was measured. The results reveal that the secondary nuclei of the poly (ethylene oxide)/urea/thiourea inclusion compound consist mainly of ethylene oxide repeating units and urea molecules. We propose that only urea molecules and ethylene oxide repeating units are selected to form the secondary nuclei while co-crystallization of the three components happens at the lateral spreading stage. As a result, the composition of the critical secondary nuclei is different from that of the bulk inclusion compound crystals. The work is expected to deepen our understanding of the nucleation of multi-component co-crystals. Full article
(This article belongs to the Special Issue Crystallization Thermodynamics)
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Open AccessFeature PaperArticle
The Criterion for the Crystallization Ability Assessment as Applied to Borate Glass Powders and Monoliths
Entropy 2019, 21(10), 994; https://doi.org/10.3390/e21100994 - 12 Oct 2019
Cited by 1 | Viewed by 584
Abstract
The glasses of three borate systems, Na2O-B2O3, K2O-B2O3 and BaO-B2O3, were studied over a wide range of the compositions by differential thermal analysis (DTA) and X-ray powder diffractometry [...] Read more.
The glasses of three borate systems, Na2O-B2O3, K2O-B2O3 and BaO-B2O3, were studied over a wide range of the compositions by differential thermal analysis (DTA) and X-ray powder diffractometry (XRPD). The thermal parameters obtained by DTA method (the glass transition temperature, Tg, the crystallization onset temperature, Tx, and the melting temperature, Tm) were used to calculate the criteria (coefficients) characterizing glass stability against crystallization. The Lu–Liu, Weinberg and Hrubý coefficients were tested for verification of their consistency with several simple requirements. Since each of the criteria has its drawbacks, the coefficient of glass crystallization ability, Kcr, which meets all of the requirements, was also used. The advantage of this coefficient is demonstrated on the example of the glass powders and the monolithic glasses of the mentioned above borate systems. Full article
(This article belongs to the Special Issue Crystallization Thermodynamics)
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Open AccessArticle
Enthalpy Relaxation of Polyamide 11 of Different Morphology Far Below the Glass Transition Temperature
Entropy 2019, 21(10), 984; https://doi.org/10.3390/e21100984 - 10 Oct 2019
Cited by 8 | Viewed by 1268
Abstract
Polyamide 11 (PA 11) samples of different supermolecular structure, including the crystal-free glass and semi-crystalline PA 11 of largely different semi-crystalline morphology, were prepared by fast scanning chip calorimetry (FSC). These samples were then annealed at different temperatures well below the glass transition [...] Read more.
Polyamide 11 (PA 11) samples of different supermolecular structure, including the crystal-free glass and semi-crystalline PA 11 of largely different semi-crystalline morphology, were prepared by fast scanning chip calorimetry (FSC). These samples were then annealed at different temperatures well below the glass transition temperature Tg. The main purpose of the low-temperature annealing experiments was the calorimetric detection of mobility of chain segments at temperatures as low as −40 °C (≈Tg − 80 K) where still excellent impact resistance is predicted. It was found that annealing PA 11 at such low temperature, regardless the thermal history and supermolecular structure including crystallinity as well as crystal shape and size, permits distinct enthalpy relaxation at rather short time scale with the structural changes reverting on subsequent heating as detected with pronounced sub-Tg-enthalpy-recovery peaks. The main glass transition, associated to large-amplitude segmental mobility, as well as relaxations at temperatures only slightly below Tg are even more distinctly sensitive to the crystal morphology. In contrast to spherulitically grown lamellar crystals, presence of high-specific-surface area nanometer-sized ordered domains causes a shift of the glass transition temperature of the amorphous phase to higher temperature, proving stronger coupling of ordered and amorphous phases than in case of lamellae. In addition, the increased coupling of the crystalline and amorphous phases slows down the cooperative rearrangements on annealing the glass slightly below Tg. The performed study contributes to further understanding of the spectrum of structural relaxations in PA 11 including the effect of presence of crystals. Enthalpy relaxation and consequently the reduction of entropy at temperatures slightly below Tg strongly depends on the semi-crystalline morphology, while an only minor effect is seen on low-temperature annealing at Tg − 80 K, possibly indicating different molecular mechanisms for the processes occurring in both temperature ranges. The low-temperature process even seems proceeding in the crystalline fraction of the material. Full article
(This article belongs to the Special Issue Crystallization Thermodynamics)
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Open AccessArticle
Crystallisation in Melts of Short, Semi-Flexible Hard-Sphere Polymer Chains: The Role of the Non-Bonded Interaction Range
Entropy 2019, 21(9), 856; https://doi.org/10.3390/e21090856 - 01 Sep 2019
Cited by 1 | Viewed by 854
Abstract
A melt of short semi-flexible polymers with hard-sphere-type non-bonded interaction undergoes a first-order crystallisation transition at lower density than a melt of hard-sphere monomers or a flexible hard-sphere chain. In contrast to the flexible hard-sphere chains, the semi-flexible ones have an intrinsic stiffness [...] Read more.
A melt of short semi-flexible polymers with hard-sphere-type non-bonded interaction undergoes a first-order crystallisation transition at lower density than a melt of hard-sphere monomers or a flexible hard-sphere chain. In contrast to the flexible hard-sphere chains, the semi-flexible ones have an intrinsic stiffness energy scale, which determines the natural temperature scale of the system. In this paper, we investigate the effect of weak additional non-bonded interaction on the phase transition temperature. We study the system using the stochastic approximation Monte Carlo (SAMC) method to estimate the micro-canonical entropy of the system. Since the density of states in the purely hard-sphere non-bonded interaction case already covers 5600 orders of magnitude, we consider the effect of weak interactions as a perturbation. In this case, the system undergoes the same ordering transition with a temperature shift non-uniformly depending on the additional interaction. Short-range attractions impede ordering of the melt of semi-flexible polymers and decrease the transition temperature, whereas relatively long-range attractions assist ordering and shift the transition temperature to higher values, whereas weak repulsive interactions demonstrate an opposite effect on the transition temperature. Full article
(This article belongs to the Special Issue Crystallization Thermodynamics)
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Open AccessEditor’s ChoiceArticle
Distinguishing between Clausius, Boltzmann and Pauling Entropies of Frozen Non-Equilibrium States
Entropy 2019, 21(8), 799; https://doi.org/10.3390/e21080799 - 15 Aug 2019
Cited by 2 | Viewed by 2047
Abstract
In conventional textbook thermodynamics, entropy is a quantity that may be calculated by different methods, for example experimentally from heat capacities (following Clausius) or statistically from numbers of microscopic quantum states (following Boltzmann and Planck). It had turned out that these methods do [...] Read more.
In conventional textbook thermodynamics, entropy is a quantity that may be calculated by different methods, for example experimentally from heat capacities (following Clausius) or statistically from numbers of microscopic quantum states (following Boltzmann and Planck). It had turned out that these methods do not necessarily provide mutually consistent results, and for equilibrium systems their difference was explained by introducing a residual zero-point entropy (following Pauling), apparently violating the Nernst theorem. At finite temperatures, associated statistical entropies which count microstates that do not contribute to a body’s heat capacity, differ systematically from Clausius entropy, and are of particular relevance as measures for metastable, frozen-in non-equilibrium structures and for symbolic information processing (following Shannon). In this paper, it is suggested to consider Clausius, Boltzmann, Pauling and Shannon entropies as distinct, though related, physical quantities with different key properties, in order to avoid confusion by loosely speaking about just “entropy” while actually referring to different kinds of it. For instance, zero-point entropy exclusively belongs to Boltzmann rather than Clausius entropy, while the Nernst theorem holds rigorously for Clausius rather than Boltzmann entropy. The discussion of those terms is underpinned by a brief historical review of the emergence of corresponding fundamental thermodynamic concepts. Full article
(This article belongs to the Special Issue Crystallization Thermodynamics)
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Open AccessFeature PaperArticle
Heterogeneous Nucleation in Solutions on Rough Solid Surfaces: Generalized Gibbs Approach
Entropy 2019, 21(8), 782; https://doi.org/10.3390/e21080782 - 09 Aug 2019
Cited by 4 | Viewed by 1189
Abstract
Heterogeneous nucleation of new phase clusters on a rough solid surface is studied. The ambient phase is considered to be a regular supersaturated solution. In contrast to existing studies of the same problem, the possible difference between the state parameters of the critical [...] Read more.
Heterogeneous nucleation of new phase clusters on a rough solid surface is studied. The ambient phase is considered to be a regular supersaturated solution. In contrast to existing studies of the same problem, the possible difference between the state parameters of the critical cluster and the corresponding parameters of a newly formed macroscopic phase is accounted for. This account is performed within the framework of the generalized Gibbs approach. Surface imperfections are chosen in the form of cones. The model allows us to simplify the analysis but also to obtain the basic results concerning the defect influence on the nucleation process. It is shown that the catalytic activity factor for nucleation of the cone depends both on the cone angle and the supersaturation in the solution determining the state parameters of the critical clusters. Both factors considerably affect the work of critical cluster formation. In addition, they may even lead to a shift of the spinodal curve. In particular, in the case of good wettability (macroscopic contact angle is less than 90°) the presence of surface imperfections results in a significant shifting of the spinodal towards lower values of the supersaturation as compared with heterogeneous nucleation on a planar solid surface. With the decrease of the cone pore angle, the heterogeneous spinodal is located nearer to the binodal, and the metastability range is narrowed, increasing the range of states where the solution is thermodynamically unstable. Full article
(This article belongs to the Special Issue Crystallization Thermodynamics)
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Open AccessArticle
Entropy and the Tolman Parameter in Nucleation Theory
Entropy 2019, 21(7), 670; https://doi.org/10.3390/e21070670 - 09 Jul 2019
Cited by 13 | Viewed by 1231
Abstract
Thermodynamic aspects of the theory of nucleation are commonly considered employing Gibbs’ theory of interfacial phenomena and its generalizations. Utilizing Gibbs’ theory, the bulk parameters of the critical clusters governing nucleation can be uniquely determined for any metastable state of the ambient phase. [...] Read more.
Thermodynamic aspects of the theory of nucleation are commonly considered employing Gibbs’ theory of interfacial phenomena and its generalizations. Utilizing Gibbs’ theory, the bulk parameters of the critical clusters governing nucleation can be uniquely determined for any metastable state of the ambient phase. As a rule, they turn out in such treatment to be widely similar to the properties of the newly-evolving macroscopic phases. Consequently, the major tool to resolve problems concerning the accuracy of theoretical predictions of nucleation rates and related characteristics of the nucleation process consists of an approach with the introduction of the size or curvature dependence of the surface tension. In the description of crystallization, this quantity has been expressed frequently via changes of entropy (or enthalpy) in crystallization, i.e., via the latent heat of melting or crystallization. Such a correlation between the capillarity phenomena and entropy changes was originally advanced by Stefan considering condensation and evaporation. It is known in the application to crystal nucleation as the Skapski–Turnbull relation. This relation, by mentioned reasons more correctly denoted as the Stefan–Skapski–Turnbull rule, was expanded by some of us quite recently to the description of the surface tension not only for phase equilibrium at planar interfaces, but to the description of the surface tension of critical clusters and its size or curvature dependence. This dependence is frequently expressed by a relation derived by Tolman. As shown by us, the Tolman equation can be employed for the description of the surface tension not only for condensation and boiling in one-component systems caused by variations of pressure (analyzed by Gibbs and Tolman), but generally also for phase formation caused by variations of temperature. Beyond this particular application, it can be utilized for multi-component systems provided the composition of the ambient phase is kept constant and variations of either pressure or temperature do not result in variations of the composition of the critical clusters. The latter requirement is one of the basic assumptions of classical nucleation theory. For this reason, it is only natural to use it also for the specification of the size dependence of the surface tension. Our method, relying on the Stefan–Skapski–Turnbull rule, allows one to determine the dependence of the surface tension on pressure and temperature or, alternatively, the Tolman parameter in his equation. In the present paper, we expand this approach and compare it with alternative methods of the description of the size-dependence of the surface tension and, as far as it is possible to use the Tolman equation, of the specification of the Tolman parameter. Applying these ideas to condensation and boiling, we derive a relation for the curvature dependence of the surface tension covering the whole range of metastable initial states from the binodal curve to the spinodal curve. Full article
(This article belongs to the Special Issue Crystallization Thermodynamics)
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