**Title**: Kinetics of phase transformations in the peridynamic formulation of continuum mechanics

**Auhors**: Kaushik Dayal and Kaushik Bhattacharya

**Source**: Journal of Mechanics and Physics of Solids, 54, 9, September 2006, pp. 1811-1842.

**Abstract**:

We study the kinetics of phase transformations in solids using the peridynamic formulation of continuum mechanics. The peridynamic theory is a nonlocal formulation that does not involve spatial derivatives, and is a powerful tool to study defects such as cracks and interfaces.

We apply the peridynamic formulation to the motion of phase boundaries in one dimension. We show that unlike the classical continuum theory, the peridynamic formulation does not require any extraneous constitutive laws such as the kinetic relation (the relation between the velocity of the interface and the thermodynamic driving force acting across it) or the nucleation criterion (the criterion that determines whether a new phase arises from a single phase). Instead this information is obtained from inside the theory simply by specifying the inter-particle interaction. We derive a nucleation criterion by examining nucleation as a dynamic instability. We find the induced kinetic relation by analyzing the solutions of impact and release problems, and also directly by viewing phase boundaries as traveling waves.

We also study the interaction of a phase boundary with an elastic non-transforming inclusion in two dimensions. We find that phase boundaries remain essentially planar with little bowing. Further, we find a new mechanism whereby acoustic waves ahead of the phase boundary nucleate new phase boundaries at the edges of the inclusion while the original phase boundary slows down or stops. Transformation proceeds as the freshly nucleated phase boundaries propagate leaving behind some untransformed martensite around the inclusion.

**Notes**: Via iMechanica.

What is peridynamics? Here is an answer (that I got by googling):

Peridynamics is a theory of continuum mechanics that is formulated in terms of integral equations rather than partial differential equations. It assumes that particles in a continuum interact across a finite distance as in molecular dynamics. The integral equations remain valid regardless of any fractures or other discontinuities that may emerge due to loading. In contrast, the differential equations of classical theory break down when a discontinuity appears. Peridynamics predicts the deformation and failure of structures under dynamic loading, especially failure due to fracture. Cracks emerge spontaneously as a result of the equations of motion and material model and grow in whatever direction is energetically favorable. The implementation does not require a separate law that tells cracks when and where to grow.

I know that in the pre-spinodal decomposition days, there was some talk about how spinodal region is actually a region of instability, since the barrier to nucleation goes to zero as one approaches the spinodal points resulting in infinite nuclei. However, it is not clear to me if something similar is meant by “nucleation as dynamic instability” in this paper. On the other hand, the phase boundary as a travelling wave sounds similar to what one finds in phase field models. A paper that I have to read more carefully and understand!

## Two dimensional mass spectroscopy

### May 15, 2007

**Title**: Single-molecule mass spectroscopy in solution using a solitary nanopore

**Authors**: Joseph W F Robertson, Claudio G Rodrigues, Vincent M Stanford, Kenneth A Rubinson, Oleg V Krasilnikov, and John J Kasianowicz

**Source**: PNAS, May 15, 2007, Vol. 104, No. 20, pp. 8207-8211

**Abstract**: We introduce a two-dimensional method for mass spectrometry in solution that is based on the interaction between a nanometer-scale pore and analytes. As an example, poly(ethylene glycol) molecules that enter a single {alpha}-hemolysin pore cause distinct mass-dependent conductance states with characteristic mean residence times. The conductance-based mass spectrum clearly resolves the repeat unit of ethylene glycol, and the mean residence time increases monotonically with the poly(ethylene glycol) mass. This technique could prove useful for the real-time characterization of molecules in solution.

## In Nature this week

### May 11, 2007

A monolayer of manganese on a tungsten substrate, spin sensitive scanning tunneling microscopy and first principle electronic structure calculations result in the unveiling of chiral magentic structurs, which, apparently mix electronic, optical, magnetic and structural properties. Here is the paper; here is the commentary.

## Role of fluctuations in fluid mixing

### May 8, 2007

It is well known that fluctuations play a crucial role in fluid mixing in turbulent flows, and that continuum models have difficulty in capturing the fluid mixing in such flows. In the latest issue of PNAS, Kadau et al report on their atomistic simulation results of fluid mixing in turbulent systems. In addition, they also use magnetic levitation Rayleigh-Taylor instability experimental results to show that their atomistic results are in better qualitative and quantitative agreement with experiments. Here is the abstract of their paper:

A ubiquitous example of fluid mixing is the Rayleigh–Taylor instability, in which a heavy fluid initially sits atop a light fluid in a gravitational field. The subsequent development of the unstable interface between the two fluids is marked by several stages. At first, each interface mode grows exponentially with time before transitioning to a nonlinear regime characterized by more complex hydrodynamic mixing. Unfortunately, traditional continuum modeling of this process has generally been in poor agreement with experiment. Here, we indicate that the natural, random fluctuations of the flow field present in any fluid, which are neglected in continuum models, can lead to qualitatively and quantitatively better agreement with experiment. We performed billion-particle atomistic simulations and magnetic levitation experiments with unprecedented control of initial interface conditions. A comparison between our simulations and experiments reveals good agreement in terms of the growth rate of the mixing front as well as the new observation of droplet breakup at later times. These results improve our understanding of many fluid processes, including interface phenomena that occur, for example, in supernovae, the detachment of droplets from a faucet, and ink jet printing. Such instabilities are also relevant to the possible energy source of inertial confinement fusion, in which a millimeter-sized capsule is imploded to initiate nuclear fusion reactions between deuterium and tritium. Our results suggest that the applicability of continuum models would be greatly enhanced by explicitly including the effects of random fluctuations.

Take a look!

## Method of manufactured solutions

### May 7, 2007

In any computational code, the verification and validation of the code is crucial; Biswajit at iMechanica describes one such verification process called the method of manufactured solutions, with specific reference to the elasticity problem.

While I wrote a code for the study of microstructural evolution in elastically stressed solids, I used the inclusion/inhomogeneity solutions of Eshelby, the hole in a circular plate under uniaxial stress, and the homogeneous strain in a system with misfitting, homogeneous precipitates as test cases to check for the correctness of my elasticity code.

The verification process that Biswajit describes might not be easy to implement in a code of the type I wrote, I suppose–I do not think it is relevant even. However, for people who study elastic stresses in a macroscopic system, it would be relevant, and Biswajit describes the method rather nicely. Take a look!

Most nanoparticle synthesis methods result in nanoparticles bounded by low-index, low-energy faces such as the {111} or {100} atomic planes. This makes intuitive sense, as any high-energy face should grow itself out of existence, leaving particles bound by more stable faces. Unfortunately, particles with mostly low-energy surfaces contain a low percentage of atomic edge and corner sites. The synthetic method of Tian et al. produces particles capped by {730} faces, a surface structure that contains a relatively high density of atomic step edges (see the right panel of the figure). The authors calculated that 43% of the total number of surface atoms reside along steps, which can be compared to 6%, 13%, and 35% for 5-nm-diameter platinum cubes, spheres, and tetrahedral particles, respectively.

From this perspective article of David L Feldheim; the synthetic method of Tian et al in question is an electrochemical method (electrodeposition). Some of the particle shapes shown in the paper of Tian et al reminded me of the particle shapes seen in the simulations of Saswata Bhattacharyya et al (See pp.18-19 of this pdf file for example) — ~~I will try and get a preprint of their paper uploaded on the net, and link here~~. Here is the draft of the paper by Saswata Bhattacharyya et al on roughening transitions.

## Growth above and below eutectic temperatures

### May 4, 2007

The perspective article by Volker Schmidt and Ulrich Goesle summarises the results rather neatly:

As expected, above the eutectic temperature, nanowire growth involves a liquid droplet on top of the germanium nanowires (…). However, when the authors reduced the temperature to below the eutectic temperature while keeping the supply of germanium constant, they observed two distinctly different phenomena (…). Some gold nanodroplets remained liquid even though the temperature was, in one case, more than 100°C below the of 361°C. The authors observed this VLS-type growth mostly for nanowires with relatively large diameters.

In contrast, for nanowires with relatively small diameters, the gold droplet became solid as the temperature fell below . The nanowires continued to grow, but did so much more slowly than in the case of VLS growth (…). Further cooling experiments showed that the transformation of the gold caps from liquid to solid at temperature below could be delayed for tens of minutes. Kodambaka et al. show that this delay depends on various parameters, such as the vapor pressure, the temperature, and the diameter of the nanowires.

The bibliographic details of the paper referred to above are as follows:

**Title**: Germanium nanowire growth below the eutectic temperature

**Authors**: S Kodambaka, J Tersoff, M C Reuter, and F M Ross

**Source**: Science May 4 2007. Vol. 316, No. 5825, pp. 729-732.

**Abstract**: Nanowires are conventionally assumed to grow via the vapor-liquid-solid process, in which material from the vapor is incorporated into the growing nanowire via a liquid catalyst, commonly a low–melting point eutectic alloy. However, nanowires have been observed to grow below the eutectic temperature, and the state of the catalyst remains controversial. Using in situ microscopy, we showed that, for the classic Ge/Au system, nanowire growth can occur below the eutectic temperature with either liquid or solid catalysts at the same temperature. We found, unexpectedly, that the catalyst state depends on the growth pressure and thermal history. We suggest that these phenomena may be due to kinetic enrichment of the eutectic alloy composition and expect these results to be relevant for other nanowire systems.

Here are a few papers that discuss the renormalization group asymptotic analysis of PDEs in general, and phase field methods in particular:

- Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory. L-Y Chen, Nigel Goldenfeld, and Y Oono. Phys Rev E, 54, 1, 376, July 1996.

Perturbative renormalization group theory is developed as a unified tool for global asymptotic analysis. With numerous examples, we illustrate its application to ordinary differential equation problems involving multiple scales, boundary layers with technically difficult asymptotic matching, and WKB analysis. In contrast to conventional methods, the renormalization group approach requires neither ad hoc assumptions about the structure of perturbation series nor the use of asymptotic matching. Our renormalization group approach provides approximate solutions which are practically superior to those obtained conventionally, although the latter can be reproduced, if desired, by appropriate expansion of the renormalization group approximant. We show that the renormalization group equation may be interpreted as an amplitude equation, and from this point of view develop reductive perturbation theory for partial differential equations describing spatially extended systems near bifurcation points, deriving both amplitude equations and the center manifold.

- Renormalization group approach to multiscale simulation of polycrystalline materials using the phase field crystal model. Nigel Goldenfeld, Badrinarayan P Athreya, and Jonathan A Dantzig. Phys Rev E, 72, 020601(R), 2005.

We propose a computationally efficient approach to multiscale simulation of polycrystalline materials, based on the phase field crystal model. The order parameter describing the density profile at the nanoscale is reconstructed from its slowly varying amplitude and phase, which satisfy rotationally covariant equations derivable from the renormalization group. We validate the approach using the example of two-dimensional grain nucleation and growth.

- Renormalization-group theory for the phase-field crystal equation. Badrinarayan P Athreya, Nigel Goldenfeld, and Jonathan A Dantzig. Phys Rev E, 74, 011601, 2006.

We derive a set of rotationally covariant amplitude equations for use in multiscale simulation of the two-dimensional phase-field crystal model by a variety of renormalization-group (RG) methods. We show that the presence of a conservation law introduces an ambiguity in operator ordering in the RG procedure, which we show how to resolve. We compare our analysis with standard multiple-scale techniques, where identical results can be obtained with greater labor, by going to sixth order in perturbation theory, and by assuming the correct scaling of space and time.

- Reductive use of renormalization group. K Nozaki, Y Oono, and Y Shiwa. Phys Rev E, 62, R4501, 2000.

It has been recognized that singular perturbation and reductive perturbation can be unified from the renormalization group (RG) theoretical point of view. However, the recognition has been only formal in the sense that it has not given us any new insight nor provided any new technical advantage over the usual RG approach. With our approach, the proto RG method proposed here, we can clearly show that system reduction is the key to singular perturbation methods. The approach also makes the calculation of singular perturbation results more transparent than the conventional RG approach. Consequently, for example, a consistent and easy RG derivation of the rotational covariant Newell-Whitehead-Segel equation is possible.

Of the three, the first one, is the earliest and is also a good read. Further, it also puts several asymptotic analysis techniques–methods of multiple scales, boundary layers, asymptotic mathcing, WKB–in perspective, and discusses them in the renormalization group (RG) framework. And, the last one gives an algorithm with five steps to obtain RG equations (Hat tip: KuoAn Wu for the pointer).

Some other links of interest: Prof. Goldenfeld’s homepage; Prof. Jonathan A Dantzig’s homepage; Wiki page on renormalization.

## In Nature this week

### May 3, 2007

- An editorial urges scientists to keep e-notebooks and share them;
- Joseph Mazur reviews Ian Stewart’s
*Why beauty is truth: the history of symmetry*; - Dirk M Guldi writes about the recent report by Simmons et al of controlling the electrical conductivity of single walled nanotubes using light; such light sensitivity is apparently achieved by filling the nano tubes with photosensitive dyes;
- Tim D White pays his tributes to a paleoanthropologist, F Clark Howell.