Exploring the link between crystal defects and nonaffine displacement fluctuations

Abstract

We generalize, and then use, a recently introduced formalism to study thermal fluctuations of atomic displacements in several two and three dimensional crystals. We study both close packed as well as open crystals with multi atom bases. Atomic displacement fluctuations in a solid, once coarsegrained over some neighborhood may be decomposed into two mutually orthogonal components. In any dimension d there are always d 2 affine displacements representing local strains and rotations of the ideal reference configuration. In addition, there exists a number of non-affine localized displacement modes that cannot be represented as strains or rotations. The number of these modes depends on d and the size of the coarse graining region. All thermodynamic averages and correlation functions concerning the affine and non-affine displacements may be computed within a harmonic theory. We show that for compact crystals, such as the square and triangular in d = 2 and the simple, body-centered and face-centered cubic crystals in d = 3, a single set of d−fold degenerate modes always dominate the non-affine sub-space and are separated from the rest by a large gap. These modes may be identified with specific precursor configurations that lead to lattice defects. In open crystals, such as the honeycomb and kagome lattices, there is no prominent gap although soft non-affine modes continue to be associated with known floppy modes representing localized defects. Higher order coupling between affine and non-affine components of the displacements quantify the tendency of the lattice to be destroyed by large homogeneous strains. We show that this coupling is larger by almost an order of magnitude for open lattices as compared to compact ones. Deformation mechanisms such as lattice slips and stacking faults in close packed crystals can also be understood within this framework. The qualitative features of these conclusions are expected to be independent of the details of the atomic interactions.

Elasticity in crystals with high density of local defects : insights from ultra-soft colloids

“…Previous studies have employed spatial projection operator formalism 36 , within a statistical mechanics framework, to segregate length-scale dependent particle displacement modes responsible for (i)the elastic response or the affine deformations and (ii)particle rearrangements or non-affine deformations. A description of dislocation pre-cursors in an equilibrated ideal solid at finite temperature, emerges as a consequence [37][38][39] of this analysis. This approach provides the foundation for understanding the onset of plastic response 40 and the origin of shear rate dependence of the yield-point 41 , in an ideal crystal, using the language of discontinuous phase transition.…”

Nonaffine Displacements Encode Collective Conformational Fluctuations in Proteins

“…Here the minimisation is over choices of the deformation matrix D. 15 Thus N AP (i) quantifies the "local structural changes" of i-th particle of the biomolecule. The central quantity of our current work, the global non-affine parameter (GNAP), is the average of NAP over the macromolecule (Figure 1b): GNAP = N −1 N i=1 NAP(i) One can show 15,17 that the thermal average of the local non-affine parameter is given by the trace of the displacement co-variance matrix projected onto the non-affine subspace viz.…”

“…The copyright holder for this preprint (which was this version posted November 20, 2019. ; https://doi.org/10.1101/840850 doi: bioRxiv preprint Theory Non-affine displacements in a bio-macromolecule are derived by filtering out routine homogenous displacements (known as 'affine deformation') from reference structure. Figure 1a-b schematically introduces us with non-affine displacements as well as the local and global non-affine parameter [15][16][17] (NAP and GNAP respectively). Figure 1a illustrates affine and non-affine deformation for a crystal.…”

Non-affine displacements encode collective conformational fluctuations in proteins

“…number of particles in the neighbourhood of particle 0 Non-affine displacements in a bio-macromolecule are derived by filtering out routine homogenous displacements (known as 'affine deformation') from reference structure. Figure 1a-b schematically introduces us with non-affine displacements as well as the local and global non-affine parameter [15][16][17] (NAP and GNAP respectively). Figure 1a illustrates affine and non-affine deformation for a crystal.…”