On the theoretical description of nucleation in confined space

Schmelzer, Jürn W. P.; Abyzov, Alexander S.

doi:10.1063/1.3664905

Cited by 7 publications

(8 citation statements)

References 15 publications

(21 reference statements)

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“…The set of relations in Equation ( 15 ) with

represents a system of linear equations for the determination of the steady-state values of the functions,

. Employing, in addition, Equation ( 22 ), the steady-state cluster-size distribution in this range of cluster sizes can be expressed as [ 2 , 81 , 88 ]:

…”

Section: Comparison Of the Different Theoretical Treatmentsmentioning

confidence: 99%

Crystallization of Supercooled Liquids: Self-Consistency Correction of the Steady-State Nucleation Rate

Abyzov

Schmelzer

Fokin

et al. 2020

Entropy

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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.

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“…The set of relations in Equation ( 15 ) with

represents a system of linear equations for the determination of the steady-state values of the functions,

. Employing, in addition, Equation ( 22 ), the steady-state cluster-size distribution in this range of cluster sizes can be expressed as [ 2 , 81 , 88 ]:

…”

Section: Comparison Of the Different Theoretical Treatmentsmentioning

confidence: 99%

Crystallization of Supercooled Liquids: Self-Consistency Correction of the Steady-State Nucleation Rate

Abyzov

Schmelzer

Fokin

et al. 2020

Entropy

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“…As also can be traced from the quotation given above, Kauzmann connected the rate of crystallization with the average time of formation of the first supercritical nucleus at steady-state conditions with a steady-state nucleation rate,

. Only in this case is the average time,

, of formation of the first supercritical crystallite correlated with the work of critical cluster formation (the energy barrier to crystal nucleation),

, via [ 127 , 128 , 129 ]:

…”

Section: Is the Kauzmann Paradox Really In Conflict With Basic Lawmentioning

confidence: 99%

Glass Transition, Crystallization of Glass-Forming Melts, and Entropy

Schmelzer¹,

Тропин

2018

Entropy

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A critical analysis of possible (including some newly proposed) definitions of the vitreous state and the glass transition is performed and an overview of kinetic criteria of vitrification is presented. On the basis of these results, recent controversial discussions on the possible values of the residual entropy of glasses are reviewed. Our conclusion is that the treatment of vitrification as a process of continuously breaking ergodicity with entropy loss and a residual entropy tending to zero in the limit of zero absolute temperature is in disagreement with the absolute majority of experimental and theoretical investigations of this process and the nature of the vitreous state. This conclusion is illustrated by model computations. In addition to the main conclusion derived from these computations, they are employed as a test for several suggestions concerning the behavior of thermodynamic coefficients in the glass transition range. Further, a brief review is given on possible ways of resolving the Kauzmann paradox and its implications with respect to the validity of the third law of thermodynamics. It is shown that neither in its primary formulations nor in its consequences does the Kauzmann paradox result in contradictions with any basic laws of nature. Such contradictions are excluded by either crystallization (not associated with a pseudospinodal as suggested by Kauzmann) or a conventional (and not an ideal) glass transition. Some further so far widely unexplored directions of research on the interplay between crystallization and glass transition are anticipated, in which entropy may play-beyond the topics widely discussed and reviewed here-a major role.

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“…A correct treatment of these correlations is of significant importance for the interpretation of both real and computer experiments but not always properly implemented into the analysis (cf. also our previous analysis of related problems).…”

Section: Discussionmentioning

confidence: 99%

“…The time required to observe experimentally the first crystal after initiation of the nucleation process is commonly denoted as incubation time and sometimes also as lag‐time . In comparing theory with the experimental lag‐time, one has to account for both the average time required to form the first critical nucleus and the time for its growth to measurable sizes . Consequently, the average incubation or lag‐time, taken similarly to the time of formation of the first supercritical nucleus as the average over a sufficiently large number of experiments, has to be generally written as

{italicτ}_{lag - time} = false⟨ italicτ false⟩ + {italicτ}_{normalgrowth} .

In Eq.…”

Section: Characteristic Time‐scales In Nucleation: Isothermal and Isomentioning

confidence: 99%

“…35,36 In comparing theory with the experimental lag-time, one has to account for both the average time required to form the first critical nucleus and the time for its growth to measurable sizes. 1,37 Consequently, the average incubation or lag-time, taken similarly to the time of formation of the first supercritical nucleus as the average over a sufficiently large number of experiments (cf. 38 ), has to be generally written as…”

Section: Time Of Occurrence Of the First Supercritical Nucleus And mentioning

confidence: 99%

See 1 more Smart Citation

Time of Formation of the First Supercritical Nucleus, Time‐lag, and the Steady‐State Nucleation Rate

Schmelzer

Abyzov

Байдаков

2016

Int J of Appl Glass Sci

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General equations are derived for the description of correlations between average time of formation of the first supercritical nucleus, time‐lag in nucleation, and steady‐state nucleation rate. In their implementation, a set of representative equations is employed modeling the time‐dependence of the nucleation rate as proposed first by Zeldovich and advanced by a variety of other authors. The analysis is performed for both isothermal–isobaric and time‐dependent process conditions. It is shown that, in general, the time of formation of the first supercritical nucleus depends both on time‐lag and the average time of formation of a supercritical nucleus at steady‐state conditions. The conditions are specified at which one of these characteristic times may dominate. In contrast to alternative statements it is also demonstrated that for isothermal–isobaric conditions, due to the Poisson character of nucleation, steady‐state nucleation rates can be determined without having at one's disposal a detailed knowledge of the time‐lag and the kinetic laws of approach of steady‐state nucleation conditions. Results of molecular‐dynamics simulations of crystal nucleation in Lennard‐Jones liquids are presented for an illustration of the general conclusions. Several applications of the results to the interpretation of experimental data are discussed.

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On the theoretical description of nucleation in confined space

Cited by 7 publications

References 15 publications

Crystallization of Supercooled Liquids: Self-Consistency Correction of the Steady-State Nucleation Rate

Crystallization of Supercooled Liquids: Self-Consistency Correction of the Steady-State Nucleation Rate

Glass Transition, Crystallization of Glass-Forming Melts, and Entropy

Time of Formation of the First Supercritical Nucleus, Time‐lag, and the Steady‐State Nucleation Rate

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