Distribution of plates׳ sizes tell the thermal history in a simulated martensitic-like phase transition
Introduction
The temperature memory effect, as manifested by shape memory alloys, remarkably shows that a thermodynamic system can remember not only the sense of a transformation, as in magnetic hysteresis, but also quasi-precise temperatures at which certain actions were performed in the past, such as the interruption of a previous phase transition. Naturally, the subject attracted a number of experimental efforts (see, e.g. [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]) towards a clear description, but surprisingly few theoretical scenarios have been proposed (see e.g. [2], [8], [9] ), an unanimously accepted phenomenological model being so far rather elusive.
Succinctly, the temperature memory effect (also called “thermal arrest”) consists in the system remembering one – or several – previous incomplete phase transition(s), as described in the following four-steps process: at step I, a complete direct phase transition (parent to product) is performed, then at step II the reverse (product to parent) transformation is “arrested” – stopped before completion – at a temperature “TA”. Next, at step III, another complete direct transformation is performed and finally at step IV a complete reverse one. In this final reverse transformation, the calorimetric signal shows a dip at a temperature close to TA, at which the transformation was previously arrested at step II.
Efforts are on-going towards understanding of the thermal memory effect. Previous theoretical works suggest that the release of stress from the untransformed martensite can increase its transformation back temperature, which results in a separated second peak [2], [7], [8]. Rodriguez-Aseguinolaza et al. [9] propose a more quantitative model also relaying on the redistribution of stress amongst martensite fractions and the parent matrix after one or multiple incomplete reverse transformations.
In this paper we propose a different mechanism possibly relevant for the thermal memory effect, namely that incomplete thermal circles influence the distribution of the product phase plates׳ sizes, making the latter a “witness” of the thermal history – which should be readable by a calorimetric scan.
The idea is rather simple and is based on the existing microscopic evidence that the martensitic transformation takes place with formation of plates, which do not grow indefinitely, but rather stop at a certain maximum size; then, the phase transition proceeds by formation of other plates. There is also evidence that this intrinsic maximum size decreases with temperature (see, e.g. [11]). Keeping in mind that the size of a formed plate possesses such an intrinsic limitation, it is a simple next step to assume that geometrical constraints may impose supplementary limitations: the new plates can be limited in growth by the existing puzzle of neighboring plates. As such, the distribution of plates׳ sizes becomes history dependent and it will be shown that this property can generate thermal memory.
The outline of the paper is as follows: in Section 2, the general assumptions of the model are described, then a numerical illustration is given in 3 Numerical simulations assuming isothermal direct transformation, 4 Reading thermal memory by reverse transformation (toy model illustration); Section 5 discusses briefly the relevance and limitations of the calculations, and Section 6 concludes the paper.
Section snippets
Model
In this section we present the basic assumptions within a simplified phenomenological model for a solid state phase transition inspired by certain properties of the martensitic transformation. The basic features of the proposed model are:
- (i)
Finite size plates: The product phase, once nucleated, grows in the form of plates only up to some intrinsic maximum size where the growth stops. Then, the phase transition proceeds by formation of new plates (rather than further growth of existing ones). In
Numerical simulations assuming isothermal direct transformation
In our numerical simulation, the product phase will be assumed to nucleate in random positions in the parent matrix and to grow in the form of squares which gradually fill the available surface. If a square, during its growing process, encounters neighboring (already existing) squares, or the system borders, it will stop growing, remaining at that respective size.
While the 2D model is justified by the plates-like growth of martensite germs (also, the alloys can be obtained as thin films, etc.),
Reading thermal memory by reverse transformation (toy model illustration)
Performing reverse transformations is a key part in the arrest experiments. In this section we assume a phenomenological transformation rate to simulate DSC curves for the reverse transformation.
The simplest way to introduce the property (iii) is to assume that the interface energy has an elastic component, which can actually be responsible for initiation of the reverse product-parent transition. Let us denote the elastic fraction of the border energy with (necessarily, one must have
Relevance and limitations of the model
A number of experimental papers (see, e.g. [10], [11]) present microscopic evidence that, when increasing the temperature, the martensite plates transform back to austenite in reversed order of their formation. More general, even if some alloys undergo morphologic changes during cooling, the reverse transformation takes place through the same stages as the direct one – but in reverse order [27]. Also in the paper [11], chosen as an example, the author comments on the fact that during the direct
Conclusions
We propose a simple phenomenological model for a solid state phase transition, inspired by some experimentally observed features of the martensitic transformation. Then it is shown that the model naturally exhibits thermal memory.
First, the key assumptions are described, namely that the direct phase transition takes place by formation of finite plates, whose sizes are temperature dependent (with bigger plates formed first) and also – possibly – geometrically restricted by existing surrounding
Acknowledgments
We acknowledge support from the Romanian Ministry of National Education, Grant PN-II-ID-PCE-2012-4-0516 and Core program 45N/2009.
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