Critical Nucleus Size Research Articles

Numerical calculation of free-energy barriers for entangled polymer nucleation.

The crystallization of entangled polymers from their melt is investigated using computer simulation with a coarse-grained model. Using hybrid Monte Carlo simulations enables us to probe the behavior of long polymer chains. We identify solid-like beads with a centrosymmetry local order parameter and compute the nucleation free-energy barrier at relatively high supercooling with adaptive-bias windowed umbrella sampling. Our results demonstrate that the critical nucleus sizes and the heights of free-energy barriers do not significantly depend on the molecular weight of the polymer; however, the nucleation rate decreases with the increase in molecular weight. Moreover, an analysis of the composition of the critical nucleus suggests that intra-molecular growth of the nucleated cluster does not contribute significantly to crystallization for this system.

Direct evaluation of attachment and detachment rate factors of atoms in crystallizing supercooled liquids.

Kinetic rate factors of crystallization have a direct effect on formation and growth of an ordered solid phase in supercooled liquids and glasses. Using the crystallizing Lennard-Jones liquid as an example, in the present work, we perform a direct quantitative estimation of values of the key crystallization kinetic rate factors-the rate g+ of particle attachments to a crystalline nucleus and the rate g- of particle detachments from a nucleus. We propose a numerical approach, according to which a statistical treatment of the results of molecular dynamics simulations was performed without using any model functions and/or fitting parameters. This approach allows one to accurately estimate the critical nucleus size nc. We find that for the growing nuclei, whose sizes are larger than the critical size nc, the dependence of these kinetic rate factors on the nucleus size n follows a power law. In the case of the subnucleation regime, when the nuclei are smaller than nc, the n-dependence of the quantity g+ is strongly determined by the inherent microscopic properties of a system, and this dependence cannot be described in the framework of any universal law (for example, a power law). It has been established that the dependence of the growth rate of a crystalline nucleus on its size goes into the stationary regime at the size n > 3nc particles.

Dynamical aspects of nanoparticle formation by wire explosion process

Copper nanoparticles (NPs) were produced by wire explosion process (WEP) and it was noted that the amount of energy (E) deposited on the wire and the ambient pressure play a major role on the size of particles formed. Dynamic diffusion and condensation processes of NPs formation by WEP were modelled. Calculations of critical size of embryo, activation energy and nucleation rate of the formation of NPs in WEP were made considering classical homogeneous nucleation theory. Decrease in critical size of nuclei and activation energy, increase in nucleation rate with high E (540 J) and low operating pressure (10 kPa) confirm the formation of small size NPs (26 nm). Different cooling rates due to unsymmetrical shape of the vapour cloud has been identified as the cause for generating mixed particle sizes. The qualitative analysis conducted in this work validates the obtained experimental results and can be used as a design tool for industrial apparatus to produce NPs in bulk.

Propane and Water: The Cooperativity of Unlikely Molecules to Form Clathrate Structures.

Many unanswered questions still exist at the molecular level to understand the nucleation process and mechanism of clathrate hydrates, especially for larger guest molecules that would result in the structure II crystal. Here, we report on molecular dynamics simulations for propane and water to describe the molecular mechanism leading to a structure II system. Through a large number (30) of long (5 μs) and coupled annealing (20 μs) simulations, we detail the prenucleation, nucleation, growth, and annealing of propane clathrate hydrate structures at 250 K and 1800 bar. The results demonstrate the equal importance of the empty and occupied cages in the nucleation of propane hydrates. The critical nucleus size is identified to be eight cages. While separate distinct clusters may exist during the prenucleation period, only one survives to grow beyond the critical nucleus size, with the others remaining subcritical. From the annealing simulations, it is clear that solid rearrangement is a very slow process, and 20 μs is still not long enough to capture long-range ordering resembling the structure II crystal. These results, along with the developed analysis method, have a significant impact in advancing our understanding of the nucleation process for unlike molecules.

Estimation of the Size of Critical Secondary Nuclei of Melt-Grown Poly(l-lactide) Lamellar Crystals

Recently, we proposed a simple method to derive the size of critical secondary nuclei formed on the growth front of lamellar crystals via the nucleation kinetics of random copolymers and polymer bl...

Evaluation of the critical nucleus size without using interface free energy

Abstract The size of the critical nucleus serves as a benchmark in the nucleation theories. But its thermodynamic calculation requires knowledge of the interface free energy, which is size-dependent. A theoretical approach avoiding the need of interfacial energy data for evaluating the size of the critical nucleus is described in this paper. The point at which the volume term of the free energy change during new-phase nucleation balances the surface term is used. The size of the new-born crystals that survive up to this point (denominated ‘stable nuclei’) enables evaluation of the critical nucleus size.

Equilibrium Shape of 2D Nuclei Obtained from Spiral Hillocks on {010} Form of Gypsum

Growth patterns on the {010} form of gypsum crystals were investigated, under controlled temperature and supersaturation, by atomic force microscope. With the morphology of hillocks formed by growth spirals generated by screw dislocations, the two-dimensional (2D) equilibrium morphology on the {010} form of gypsum was calculated. Edge energies of the [001], [102], [101], [201], and [100] steps were obtained by measuring the spacing between hillocks steps running in the same direction. Their values range between 8 × 10–7 and 2.3 × 10–6 erg/cm. The calculated critical size of 2D nucleus on {010} form is quite large (∼3400 nm2) and witnesses the low supersaturation value at which experiments were done.

Probing the critical nucleus size for ice formation with graphene oxide nanosheets.

Water freezing is ubiquitous and affects areas as diverse as climate, the chemical industry, cryobiology and materials science. Ice nucleation is the controlling step in water freezing1-5 and has, for nearly a century, been assumed to require the formation of a critical ice nucleus6-10. But there has been no direct experimental evidence for the existence of such a nucleus, owing to its transient and nanoscale nature6,7. Here we report ice nucleation in water droplets containing graphene oxide nanosheets of controlled sizes and show that they have a notable impact on ice nucleation only above a certain size that varies with the degree of supercooling of the droplets. We infer from our experimental data and theoretical calculations that the critical size of the graphene oxide reflects the size of the critical ice nucleus, which in the case of sufficiently large graphene oxides sits on their surface and gives rise to ice formation behaviour consistent with classical nucleation theory. By contrast, when the graphene oxide size is smaller than that of the critical ice nucleus, pinning at the periphery of the graphene oxide deforms the ice nucleus as it grows. This gives rise to a much higher free-energy barrier for nucleation and suppresses the promoting effect of the graphene oxide11. The results provide experimental information on the existence and temperature-dependent size of the critical ice nucleus, which has previously only been explored theoretically and through simulations12-16. As pinning of a pre-critical nucleus at a nanoparticle edge is not specific to the ice nucleus on graphene oxides, we expect that our approach could be extended to probe the critical nuclei in other nucleation processes.

Chemical Control over Nucleation and Anisotropic Growth of Two-Dimensional Covalent Organic Frameworks.

Two-dimensional covalent organic frameworks (2D COFs) are composed of structurally precise, permanently porous, layered polymer sheets. 2D COFs have traditionally been synthesized as polycrystalline aggregates with small crystalline domains. Only recently have a small number of 2D COFs been obtained as single crystals, which were prepared by a seeded growth approach via the slow introduction of monomers, which favored particle growth over nucleation. However, these procedures are slow and operationally difficult, making it desirable to develop polymerization methods that do not require the continuous addition of reactants over days or weeks. Here, we achieve the rapid growth of boronate ester-linked COFs by chemically suppressing nucleation via addition of an excess of a monofunctional competitor, 4-tert-butylcatechol (TCAT), into the polymerization. In situ X-ray scattering measurements show that TCAT suppresses colloid nucleation, which enables seeded growth polymerizations in the presence of high monomer concentrations. Kinetic Monte Carlo simulations reveal that TCAT limits oligomers to sizes below the critical nucleus size and that in-plane expansion is restricted compared to out-of-plane oriented attachment of oligomers. The simulations are consistent with transmission electron micrographs, which show that the particles grow predominantly in the stacking direction. This mechanistic insight into the role of the modulators in 2D polymerizations enables the size and aspect ratio of COF colloids to be controlled under operationally simple conditions. This chemically controlled growth strategy will accelerate the discovery and exploration of COF materials and their emergent properties.

Critical Size of Secondary Nuclei Determined via Nucleation Theorem Reveals Selective Nucleation in Three-Component Co-Crystals

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.

Determination of the Critical Size of Secondary Nuclei on the Lateral Growth Front of Lamellar Polymer Crystals

Nucleation is the rate-limiting first step of crystallization. The energy barrier related to form the critical nucleus predominantly affects the nucleation rate. Determination of the size of the cr...

Role of physical nucleation theory in understanding conformational conversion between pathogenic and nonpathogenic aggregates of low-complexity amyloid peptides

Amyloid-forming proteins aggregate within and between neuronal cells, and the resulting deposits are associated with neurodegenerative diseases. The amyloidogenic property of these proteins, in fact, arises from a relatively short part of the whole amino acid sequence. Recent studies have also shown that some short subsequences drive amorphous aggregation; the resulting aggregates are structurally distinct from amyloid aggregates and may thus play different functional and pathogenic roles than amyloid aggregates. Although the process of conformational conversion between amyloid and amorphous aggregates has attracted much attention, the detailed molecular mechanism underlying such a conformational conversion is not yet clear. In this mini-review, I review some aggregation studies that employ the concept of nucleated polymerization to describe early stage on-pathway protein aggregation. I specifically look into one of the most important aggregation properties, critical nucleus size, for a variety of amyloid proteins/fragments and demonstrate that this quantity can be used to help understand the molecular mechanism of early stage protein aggregation. I argue that a similar nucleated polymerization scheme can be applied to study functional/amorphous aggregates without a fundamental difference from a theoretical perspective. I hypothesize that the physical principle underlying the conformational conversion between pathogenic and functional aggregates is associated with several morphological properties (e.g., lateral alignment and intrinsic polymorphism) that can be modulated through proline-mediated conformational rigidity of the sequence. This phenomenon may thus be responsible for the length-dependent amyloidogenesis of amyloid-forming proteins. This notion may shed light on predicting amyloidogenic propensity by correlating the change of a protein’s mechanical properties with the resulting protein morphologies at the sequence level.

Nucleation kinetics in supercooled Ni50Ti50: Computer simulation data corroborate the validity of the Classical Nucleation Theory

Understanding the crystal nucleation mechanism and kinetics is essential for predicting and controlling numerous natural and industrial processes. However, it is extremely difficult to experimentally determine certain key parameters, such as the critical nucleus size, the interfacial free energy, and the diffusion coefficient in multicomponent liquids. In this work, we used MD data on nucleation rates and diffusion coefficient to analyze the nucleation kinetics in a Ni50Ti50 alloy in the framework of the Classical Nucleation Theory (CNT). Our analysis validates the CNT as a good descriptor of the crystal nucleation rates in this supercooled alloy, corroborating recent results of MD simulation of supercooled Lennard-Jones and BaS liquids.

Temperature-dependent nucleation kinetics of Guinier-Preston zones in Al–Cu alloys: An atomistic kinetic Monte Carlo and classical nucleation theory approach

Guinier-Preston (GP) zones, which form in Al–Cu alloys, exhibit characteristic patterns of Cu-rich, disk-shaped atomic clusters along the {100} planes. GP zones have received much attention not only for their precipitation-hardening effect on the Al matrix, but also from fundamental interest in the physics and chemistry of the nanoscale organization of solute atoms. In this study, we established an atomistic kinetic Monte Carlo (kMC) modeling technique for exploring the nucleation and formation processes of GP zones in Al–Cu alloys by constructing an effective on-lattice multibody potential for a dilute Al–Cu-vacancy system by density functional theory calculations. Our model can describe the clustering of Cu atoms via successive exchanges with vacancies and reproduce the characteristic planar nanoprecipitates along the {100} planes in a manner consistent with the crystallographic nature of the GP zones in the early-stage aging of Al–Cu alloys. The time evolution and critical nucleus size of Cu clusters were characterized at various temperatures based on the kMC results. Consequently, we predicted the existence of an optimum temperature (i.e., nose temperature) for the formation of Cu clusters at which the cluster growth was maximized, which was attributable to the interplay between the critical nucleation barrier and the diffusion rate. In addition, the critical nucleus size and temperature for cluster formation were examined based on classical nucleation theory along with the developed multibody potential. These provided an insight into the competition between the enthalpic and entropic effects on the formation of GP zones in the Al–Cu system.

Plasma-enhanced chemical vapor deposition of graphene optimized by pressure

Fast growth of graphene on non-catalytic substrates by plasma-enhanced chemical vapor deposition (PECVD) usually results in poor crystalline quality and sometimes is accompanied by the formation of amorphous carbon. In this study, we slow down the graphene growth on quartz substrates by elevating pressure in a PECVD setup. The lower growth rate is caused by slower diffusion of gaseous species in the plasma to the substrate surface at higher pressure. The slower growth at high-temperature is crucial for larger critical nucleus size, higher nucleation barrier, and better electrical properties. Better graphene quality under higher pressure growth is verified by Raman spectroscopy. A hole mobility up to 89.6 cm2·V−1·s−1 is acquired with a growth rate of 1.34 nm h−1 at 950 °C, 1.2 Torr.

Studies on phase diagram, thermodynamic stability and interfacial structure of solid dispersions of phenothiazine-adipic acid drug system

In the present communication phase diagram, thermodynamic quantities, stability and interface structure of phenothiazine (PT)-adipic acid (AA) binary system have been reported. Eutectics (E1, E2) at 0.631 and 0.868 mole fraction of PT and addition compounds (A 1:1, A 4:1) are formed in the system. Partial and integral thermodynamic quantities such as excess Gibbs energy (gE), excess enthalpy (hE) and excess entropy (sE) of eutectic and non-eutectic mixtures were also calculated using activity coefficient data. The positive value of excess Gibbs free energy indicates deviation from ideal behavior which refers stronger association between like molecules during formation of binary mix. The positive value of Gibbs free energy of mixing (ΔGM) suggests the mixing for eutectic and non-eutectic is non-spontaneous. The size of critical nucleus at different undercoolings has been found in nanoscale, which may lead a big challenge in pharmaceutical world.

Study of the Crystal Growth Mechanism and Critical Secondary Nucleus Size of Poly(ethylene oxide)/Urea Inclusion Compound

This study focuses on the crystal growth mechanism and the critical nucleus size of the inclusion compound formed between poly(ethylene oxide) and urea (PEO-U-IC) β spherulites crystallized from the melt. The surface nucleation mechanism during the crystal growth process is proposed based on the exponential dependence of the spherulite radial growth rate G on the negative reciprocal of product of the isothermal crystallization temperature multiplied by the degree of supercooling (−1/(TcΔT)). The radial growth rates G of PEO-U-IC β spherulites crystallized from the mixed melt with the gradual addition of the diluting agent N,N′-dimethyl urea (2MeU) were measured under a polarized optical microscope. On the basis of the first nucleation theorem proposed by Kashchiev, the critical secondary nucleus size n* during the surface nucleation is determined from the slope of linear fitting line of ln G versus ln x, where x is the mass fraction of urea in U/2MeU. The critical secondary nucleus consists of 3–9 entities with each entity of the inclusion compound crystal containing 1 urea and 1.5 ethylene oxide repeating units when the crystallization temperature ranges from 65 to 85 °C. The relationship that n* is proportional to 1/(ΔT)2 confirms that the critical secondary nuclei at the solid–melt interface are two-dimensional.

Critical Nucleus Size at Electrocrystallization

A new interpretation of critical nucleus, which is based on the kinetic analysis of nucleation, is proposed. By critical size of nucleus is meant the size that corresponds to the smallest flux of transition to the next size class. The relationship between the critical size and overpotential and equilibrium surface coverage with adatoms or between the critical size and the specific surface energy and adhesion energy is determined.

Prediction of PEKK properties related to crystallization by molecular dynamics simulations with a united-atom model

Semicrystalline polyetherketoneketone (PEKK) is widely used as the matrix in carbon-fiber composites. Understanding and predicting the crystallization thermodynamics and kinetics of this class of polymers is, thus, of great interest. This paper uses molecular dynamic (MD) simulations using Dreiding united-atom (UA) potentials to characterize a wide range of thermos-physical properties of PEKK. We characterized the effect of terephthaloyl chloride to isophthaloyl chloride (T/I) on an extensive set of properties, including the lattice parameters and stability of PEKK crystal structures, glass transition temperature, melting temperature, crystal/amorphous interfacial energy, and enthalpy of fusion. We find good overall agreement between predicted properties and experimental values and the simulations help clarify inconsistencies in the literature. In combination with classical nucleation theory, nucleation barriers and critical nucleus size at different temperature are predicted.

Structure and morphology of crystalline nuclei arising in a crystallizing liquid metallic film

Control of the crystallization process at the microscopic level makes it possible to generate the nanocrystalline samples with the desired structural and morphological properties, that is of great importance for modern industry. In the present work, we study the influence of supercooling on the structure and morphology of the crystalline nuclei arising and growing within a liquid metallic film. The cluster analysis allows us to compute the diffraction patterns and to evaluate the morphological characteristics (the linear sizes of the homogeneous part and the thickness of the surface layer) of the crystalline nuclei emergent in the system at different levels of supercooling. We find that the liquid metallic film at the temperatures corresponded to low supercooling levels crystallizes into a monocrystal, whereas a polycrystalline structure forms at deep supercooling levels. We find that the temperature dependence of critical size of the crystalline nuclei contains two distinguishable regimes with the crossover temperature T/Tg≈1.15 (Tg is the glass transition temperature), which appears due to the specific geometry of the system.