A mathematical model for nanoparticle melting with size-dependent latent heat and melt temperature

Ribera, Helena; Myers, T.G.

doi:10.1007/s10404-016-1810-6

Cited by 25 publications

(51 citation statements)

References 38 publications

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“…This suggests a runaway process which, once started, will only increase in speed. This may be seen in all published results for the radius of a melting nanoparticle where R t → −∞ as R → 0, see [4,5,3,20,21,35,43,52] for example. Similarly the analysis of [28], using the Maxwell-Cattaneo heat equation, shows 'supersonic melting', where the speed of the melt front is faster than the speed of heat propagation.…”

Section: Non-dimensionalisationmentioning

confidence: 60%

“…Sun and Simon [49] concluded that the melt temperature is well approximated by the Gibbs-Thomson relation while the latent heat decrease is much greater than theoretically predicted. Ribera and Myers [43] review theoretical predictions for latent heat reduction and conclude that none of the models examined match the experimental data. Decreases in surface tension are usually less significant than those observed in the latent heat and melt temperature, typically of the order 15% below the bulk value for R = 5 nm [50].…”

Section: Introductionmentioning

confidence: 99%

“…Surface energy is also included in the model of [15]. In [43] it is shown that the formula of [33] underestimates the value of latent heat near the bulk value (where the data is most reliable). They also demonstrate that other formulations, such as those of [46,54], also provide a poor match with experimental data.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Stefan problem with variable thermophysical properties and phase change temperature

Myers

Hennessy

Calvo-Schwarzwälder

2020

International Journal of Heat and Mass Transfer

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In this paper we formulate a Stefan problem appropriate when the thermophysical properties are distinct in each phase and the phase-change temperature is size or velocity dependent. Thermophysical properties invariably take different values in different material phases but this is often ignored for mathematical simplicity. Size and velocity dependent phase change temperatures are often found at very short length scales, such as nanoparticle melting or dendrite formation; velocity dependence occurs in the solidification of supercooled melts. To illustrate the method we show how the governing equations may be applied to a standard one-dimensional problem and also the melting of a spherically symmetric nanoparticle. Errors which have propagated through the literature are highlighted. By writing the system in non-dimensional form we are able to study the large Stefan number formulation and an energy-conserving one-phase reduction. The results from the various simplifications and assumptions are compared with those from a finite difference numerical scheme. Finally, we briefly discuss the failure of Fourier's law at very small length and time-scales and provide an alternative formulation which takes into account the finite time of travel of heat carriers (phonons) and the mean free distance between collisions. * tmyers@crm.cat 1 arXiv:1904.05698v1 [physics.comp-ph]

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Section: Non-dimensionalisationmentioning

confidence: 60%

Section: Introductionmentioning

confidence: 99%

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The Stefan problem with variable thermophysical properties and phase change temperature

Myers

Hennessy

Calvo-Schwarzwälder

2020

International Journal of Heat and Mass Transfer

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“…In practice, other forms of boundary conditions should be applied, such as the Newton cooling condition. This has been used in the modelling of melting of nanoparticles [44] and nanowires [45], and leads to a significant increase of the melting times. The cooling condition has also been used in a recent study by Hennessy et al [33], where a thorough asymptotic analysis of the Guyer-Krumhansl-Stefan problem is performed.…”

Section: Boundary and Initial Conditionsmentioning

confidence: 99%

The one-dimensional Stefan problem with non-Fourier heat conduction

Calvo-Schwarzwälder

Myers

Hennessy

2020

International Journal of Thermal Sciences

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We investigate the one-dimensional growth of a solid into a liquid bath, starting from a small crystal, using the Guyer-Krumhansl and Maxwell-Cattaneo models of heat conduction. By breaking the solidification process into the relevant time regimes we are able to reduce the problem to a system of two coupled ordinary differential equations describing the evolution of the solid-liquid interface and the heat flux. The reduced formulation is in good agreement with numerical simulations. In the case of silicon, differences between classical and nonclassical solidification kinetics are relatively small, but larger deviations can be observed in the evolution in time of the heat flux through the growing solid. From this study we conclude that the heat flux provides more information about the presence of non-classical modes of heat transport during phase-change processes.

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“…An additional observation to make about (2) is that our porous media flow model is also relevant for Stefan problems which describe certain melting or freezing phenomena; in that context, the boundary condition (2) is appropriate (here p would need to be interpreted as temperature) as it describes the Gibbs-Thomson condition that relates melting/freezing temperature to the curvature of the solid-melt interface [30][31][32][33][34].…”

Section: Introductionmentioning

confidence: 99%

Interfacial dynamics and pinch-off singularities for axially symmetric Darcy flow

2019

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We study a model for the evolution of an axially symmetric bubble of inviscid fluid in a homogeneous porous medium otherwise saturated with a viscous fluid. The model is a moving boundary problem that is a higher-dimensional analogue of Hele-Shaw flow. Here we are concerned with the development of pinch-off singularities characterised by a blow-up of the interface curvature and the bubble subsequently breaking up into two; these singularities do not occur in the corresponding two-dimensional Hele-Shaw problem. By applying a novel numerical scheme based on the level set method, we show that solutions to our problem can undergo pinch-off in various geometries. A similarity analysis suggests that the minimum radius behaves as a power law in time with exponent α = 1/3 just before and after pinch-off has occurred, regardless of the initial conditions; our numerical results support this prediction. Further, we apply our numerical scheme to simulate the time-dependent development and translation of axially symmetric Saffman-Taylor fingers and Taylor-Saffman bubbles in a cylindrical tube, highlighting key similarities and differences with the well-studied two-dimensional cases.

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A mathematical model for nanoparticle melting with size-dependent latent heat and melt temperature

Cited by 25 publications

References 38 publications

The Stefan problem with variable thermophysical properties and phase change temperature

The Stefan problem with variable thermophysical properties and phase change temperature

The one-dimensional Stefan problem with non-Fourier heat conduction

Interfacial dynamics and pinch-off singularities for axially symmetric Darcy flow

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