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Understanding Glassy Phenomena in Materials

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Disorder and Strain-Induced Complexity in Functional Materials

Part of the book series: Springer Series in Materials Science ((SSMATERIALS,volume 148))

Abstract

A basis for understanding and modelling glassy behaviour in martensitic alloys and relaxor ferroelectrics is discussed from the perspective of spin glasses.

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Notes

  1. 1.

    Caveat: The author is not a materials scientist, but a theoretical statistical physicist concerned with modelling and understanding complex cooperative behaviour in disordered and frustrated many-body systems in idealized contexts in a number of application areas. He makes no claim to expertise in the literature of the materials systems discussed in this article, but hopes that his complementary perspective can be stimulating.

  2. 2.

    The style will be tutorial/expository rather than attempting to give all historical originality credits.

  3. 3.

    Rejuvenation refers to a situation in which, after a perturbation, a system starts a process anew as though previous events had not occurred. In spin glasses, it is observable in χ”, which decays with time, where a sudden reduction in the temperature after decay at the higher temperature causes it to return quickly to a higher value (closer to the original) and then start to decay again. See [15] and also E. Vincent in [11].

  4. 4.

    Memory refers to a system storing knowledge of its history. For example, in the previous footnote, a further sudden resumption of the earlier higher temperature makes χ” jump to the value it had just before the temperature was reduced. See [15] and also Vincent in [11].

  5. 5.

    In general, the distribution depends on the separation of the relevant sites and hence on the subscript sep.

  6. 6.

    Note that, in accord with a common practice, we use the expression ‘spin glass’ to describe both a material exhibiting a spin glass phase and the phase itself.

  7. 7.

    The rejuvenation and memory effects observed in spin glasses are explainable in terms of the hierarchical yet evolving metastable state structure, with the free energy acquiring more and more nested metastability as the temperature is lowered but melting as it is raised again.

  8. 8.

    In fact, our main interest for glassiness is in alloys.

  9. 9.

    The first recognition that there should be a spin glass analogue in martensitic alloys was by Kartha et al. [20], looking for an explanation of ‘tweed’, with similarities of ideas to those discussed here, but without the direct mappings and specificity reported in this chapter, which the present author believes provide conceptual and quantitative underpinning.

  10. 10.

    We shall briefly discuss extension to three dimensions later, but note at this time that since we are employing mean-field theory considerations of critical dimensions caused by fluctuations are irrelevant.

  11. 11.

    This is the usual Ginzburg ( ∇ ϕ)2 term in a spatially continuous formulation.

  12. 12.

    There has been much interest recently in stripe ordering in systems with a combination of short-range ferromagnetic and long-range power-law-decreasing anti-ferromagnetic interactions and it has been proved that the preferred order is of stripes for d < p ≤ d + 1, where d is the spatial dimensionality and ( − p) is the power of the long-range decay [52]. Stripe widths are determined by the relative strengths of the two types of interaction. Here, \(p = d = 2\) and the system is marginal with relevant boundary size L and it has been shown that the average twin stripe width depends on L (as the square root); see [53].

  13. 13.

    Indeed, even with specified interactions, its evaluation is surely NP-hard [54].

  14. 14.

    See also X Ren’s chapter in this book [51].

  15. 15.

    That is random-site Ising with the same J(R).

  16. 16.

    Note that y measures the density of defects compared with the pure case Ti50Ni50, whereas x earlier was the density of normal (host) atoms.

  17. 17.

    Such “re-entrance” has been called “inverse freezing” and has been receiving much attention in other contexts; e.g. [55].

  18. 18.

    Note: The transition curves shown in Fig 4.4 between pure twinned and mixed phase and between twinned and strain glass are qualitative, but both are swung towards the twinned state compared with their SK counterparts.

  19. 19.

    Within mean field theory, Potts spin glasses of Potts dimension greater than 2 show additional re-entrance from spin glass to ferromagnet as the temperature is reduced [56].

  20. 20.

    This was also the initial motivation that led to [19].

  21. 21.

    Reference [19] also assumed that the origin of tweed was quenched disorder but, as above, locally in a system with frustrated but not necessarily disordered exchange. In fact, the consequence is strain glass.

  22. 22.

    Another example is \({\mathrm{PbZn}}_{1/3}{\mathrm{Nb}}_{2/3}{\mathrm{O}}_{3}\) (abbreviated to PZN).

  23. 23.

    Relaxors are so-called because of this significant frequency-dependent permittivity peak behaviour.

  24. 24.

    The feature of frequency dependence of the peak in the real part of the dielectric permittivity or the magnetic susceptibility as a function of temperature decreasing with decreasing frequency is interpretable as reflecting the range of characteristic barrier penetration times, with higher barriers taking longer to surmount.

  25. 25.

    The usual convention in the field is to write PMN-PT, but we shall use PT-PMN here in keeping with our perspective of disordering a pure matrix.

  26. 26.

    Note that this is in contrast to the case of the martensitic materials discussed above, and implies a continuous transition, although one could easily modify to a negative u and include a positive sixth-order term to bound the Hamiltonian if one wished to allow the possibility of a first-order transition.

  27. 27.

    Recall that most experimental spin glasses have site disorder but frustration in their exchange.

  28. 28.

    Effective random fields from further B atoms have intermediate orientations.

  29. 29.

    Also, in this connection it is probably appropriate to draw attention to recent studies of dipolar glasses with local anisotropic preferred-axis disorder [57,  58,  59].

  30. 30.

    For example, by simulated annealing or extremal optimization [60].

  31. 31.

    Indeed some have been studied, but no attempt at a survey is made here.

  32. 32.

    Or Potts.

  33. 33.

    The present author is inclined to the belief that both effects are likely to be playing a role in martensitic systems, while random fields are dominant for relaxor ferroelectrics.

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Acknowledgments

The author is grateful to Avadh Saxena and Turab Lookman for introducing him to martensitic shape-memory alloys and for useful discussions on the topic over many years of visits to Los Alamos National Laboratory, whose hospitality he also acknowledges. Also, in connection with martensitic alloys, he has appreciated correspondence with Jim Sethna and Xiaobing Ren. He thanks Roger Cowley, his colleague at Oxford, for introducing him to relaxor ferroelectrics, for informing him of several results and comparisons and for useful discussions on how to model and understand, and also Wolfgang Kleemann for very helpful comments on a draft of this paper and for drawing his attention to other relevant works on relaxors. Finally, he apologises again to the experts in martensites and relaxors whose work he has not acknowledged and indeed much of which he is insufficiently familiar with, but if he waited until he had had an opportunity to read, absorb and understand everything that has been done and written about, this article would not have been completed. Hopefully it will stimulate reactions, even if only of correction and objection.

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Authors and Affiliations

  1. Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM, 87501, USA

    David Sherrington

  2. Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Rd., Oxford, OX1 3NP, UK

    David Sherrington

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Editors and Affiliations

  1. , Graduate School of Engineering, Osaka University, Yamada-oka, Suita 2-1, Osaka, 565-0871, Japan

    Tomoyuki Kakeshita

  2. , Graduate School of Engineering, Osaka University, Yamada-oka, Suita 2-1, Osaka, 565-0871, Japan

    Takashi Fukuda

  3. Theoretical Division, T-11, MS B262, Los Alamos National Lab, Los Alamos, 87545, New Mexico, USA

    Avadh Saxena

  4. Dept. d'Estructura i Constituents, de la Matèria, Universidad de Barcelona, Diagonal 647, Barcelona, 08028, Spain

    Antoni Planes

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Sherrington, D. (2012). Understanding Glassy Phenomena in Materials. In: Kakeshita, T., Fukuda, T., Saxena, A., Planes, A. (eds) Disorder and Strain-Induced Complexity in Functional Materials. Springer Series in Materials Science, vol 148. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20943-7_10

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