Non-affine projection formalism and lattice defect precursors

Mar 31, 2021

Atomic fluctuations are generally classified as either “smooth", consisting of long wavelength phonons, or “singular", arising from the nucleation of defects. This classification has led to a deeper understanding of many observed properties of crystalline solids, such as melting and failure as a response to external loads. In contrast to crystals, a lack of long-range structural order makes it hard to distinguish between smooth and singular displacements in amorphous solids. Similarly, constructing a precise set of phonons which tantamount to the dominant lattice defect in crystals is also a non trivial task. Motivated by such questions, we studied defect precursors in crystalline systems using a formulation that explicitly decomposes particle displacements into affine and non-affine components.

Particle displacements \(\{\mathbf{u}_i\}\), away from some specified reference configuration \(\{\mathbf{R}_i\}\), can be projected onto two mutually orthogonal subspaces, affine and non-affine repectively. These parts of the displacements are linearly independent and form a complete basis. In this way of classification, the affine subspace consists of homogeneous linear transformations of the reference configuration \(\mathbf{u}_j-\mathbf{u}_i=\mathrm{D}(\mathbf{R}_j-\mathbf{R}_i)\) such as shear, uni-axial strain, rotation and a volume change. Since in general, not all the deformations of the reference configuration can be described in terms of linear transformations, a non-affine component remains and is quantified by, \[\chi_i = \underset{\mathrm{D}}{\text{min}}\sum_{j\in\Omega}\left[(\mathbf{u}_j-\mathbf{u}_i) - \mathrm{D}(\mathbf{R}_j-\mathbf{R}_i)\right]^2.\] Here the sum over \(j\) extends over all the particles contained in the coarse grained region \(\Omega\) around particle \(i\). We study the development of defects (on a given lattice site \(i\)) within in this coarse-grained region \(\Omega\). The non-affine displacements, in contrast to affine, lead to particle rearrangement and hence may not preserve the local neighborhood connectivity between the particles.

We next used this projection formalism to investigate the statistics of non-affine fluctuations in several two and three dimensional crystals in thermal equilibrium. In each case we showed that the non-affine components of atomic fluctuations act as a precursor to lattice defects. We analytically computed the relative probability (\(\sigma_i\)) of different defect modes. Our analysis showed that the prevalent defects in crystalline systems are dipolar in nature (see Fig.), a result that has important implications for the stability of crystalline solids. The proliferation of such non-affine defect precursors leads to plastic deformations of the crystals. Furthermore, the susceptibility of a crystalline solid to any particular defect can also be estimated from the excitation spectra of these non-affine modes. In addition we showed that the zero temperature deformation mechanism such as slips or stacking faults can be described within the framework of non-affine modes arising from thermal fluctuations in the crystals.

Read at Jou. »

Translationally invariant colloidal crystal template

Mar 31, 2021

Stable assemblies of colloidal particles are nowadays readily available. Ordered colloidal crystals have provided insights into the behaviour and physical properties of the solids at a fundamental level. In order to generate such crystals in the laboratory, one often uses either of the two easily accessible techniques, namely (a) using patchy/tethered colloids (b) using a template, designed by an optical laser or etched onto a suitable surface. However, crystals produced using these techniques are of limited usability. In the former case, once the interactions are defined, the lattice structure obtained is fixed and cannot be changed further, whereas, the latter yields a crystal which lacks the translational invariance, therefore, zero-energy modes are absent.

We proposed a method involving video microscopy and spatial light modulation technology to stabilize a system of colloidal particles with any pairwise or many body interaction into a desired lattice symmetry. This is accomplished in an energy efficient way by suppressing non-affine fluctuations using the extended Hamiltonian \(\mathcal{H}=\mathcal{H}_0-h_{\chi}X\). Here \(h_{\chi}\) is a thermodynamic field conjugate to a collective positive scalar variable \(X=\sum_i\chi_i\) and \(\mathcal{H}_0\) defines inter-particle interactions. Since affine and non-affine modes are orthogonal, “non-affine forces" calculated using the above Hamiltonian alters the particle’s arrangement in order to minimize non-affine fluctuations and allows only affine transformations. These allowed affine transformations preserve the local neighborhood connectivity.

The colloidal lattice thus produced is translationally invariant and retains all the low energy modes. We further note that non-affine forces depends upon the reference structure therefore the symmetry of the lattice can be changed at will. We showed this using our theory and Monte-Carlo simulation by stabilizing a square lattice (2D) in a Gaussian Core potential (\(V = \exp(-\mathbf{r}_{ij}^2)\)), (See Fig.). Our results can be extended to any interactions and higher dimensions.

Read at Jou. »

Stabilizing patterns in active robotic swarms

Mar 31, 2021

Flocks of birds or schools of fish naturally organize themselves into ordered swarms as a result of active forces which can counteract the destabilization tendency of embeding medium noise. Similarly, patterns of drones or robotic agents are useful for many purposes such as surveying unknown territory, taking measurements of scientifically or economically important quantities over a large area. Stabilizing any given pattern in such a swarm is energetically expensive and requires extensive computation as well as communication overheads. Recently, we showed that suppressing the non-affine displacements compared to a reference pattern, while allowing affine deformations such as translations and rotations can indeed stabilize such patterns.

We demonstrated this by considering a finite system of equally spaced active particles in a ring geometry. The particles considered are point-like, have mass \(m\) and are confined to a \(2d\) box. Due to the presence of a flow field in the background, the position \(\mathbf{r}_i\) and velocity \(\mathbf{v}_i\) of the \(i^{th}\) particle is determined by the following set of coupled equations \[\begin{aligned} \frac{d\mathbf{r}_i}{dt}=\mathbf{v}_i,\hspace{10pt} m\frac{d\mathbf{v}_i}{dt}=-\gamma\left(\mathbf{v}_i-\mathbf{U}(\mathbf{r}_i,t)\right)+\mathbf{F}_i,\end{aligned}\]

where \(\gamma\) is the constant drag coefficient and \(\mathbf{U}(\mathbf{r}_i,t)\) is a continuous spatio-temporal turbulent field measured at particle position \(\mathbf{r}_i\) at time \(t\). These active particles (such as robotic agents or drones) are capable of self propulsion which can alter their position in space. In the above equation, \(\mathbf{F}_i\) represents such active forces which are chosen so as to minimize the non-affine displacements i.e. \(\mathbf{F}_i=-\frac{\partial h_{\chi}X}{\partial \mathbf{r}_i}\).

The pattern obtained is stable (See Fig.) and as a whole can be translated without interfering with the stabilization algorithm. The agents are not forced to sense, difficult to measure, environmental parameters such as local velocity of air or water in order to stabilize the swarm. A novel outcome of our study is that by maintaining the structure of such a robotic swarm, the statistics of the underlying flow field can be determined solely from “non-affine" forces. As these forces are a-priori known, no extra measurement on the turbulent field is required to obtain the statistics. Therefore, such techniques will be useful in studying the turbulent flow where direct measurement of flow velocities are difficult.

Read at Jou. »