What do we have?

As a reminder, we have a an active nematic material on the surface of a torus. This means our nematic lives on a two dimensional curved surface embedded in a three-dimesional space. An example confocal dataset of the nematic on a torus surface can be seen in the video in Figure 1. The video rotates the dataset to show the shape of the surface. For each nematic torus, we have a time series of the these datasets allowing us to look at the time evolution of the nematic.

active nematic on a the surface of a torus

Figure 1: A an example confocal dataset of an active nematic on the surface of a torus. The video rotates the dataset to give perspective on the surface.

What do we want?

For this project, we are going to treat variables like the curvature of the substrate, the activity of the material, the size of the torus, etc., as processes that potentially have an impact on the structure of the nematic. In this context, we are looking for a process-structure relationship.

The microstructure function

Since we are dealing with a nematic material, we need a function that can characterize not only the director “n”, but also the degree of alignment “S.” As mentioned in the Project Introduction, the director is a bivector that characterizes the local average orientation of the nematogens and can take values from [0,180) degrees. If “n” can be seen as the mean of an orientation distribution, “S” can be thought of as the width of that distribution. It characterizes how well the individual molecules align along “n”. These two quantities are encapsulated in the nematic order parameter “Q”, a traceless, symmetric rank 2 tensor defined in 3 dimensions as: $\mathit{Q}_{ij} = S \left ( n_i n_j-\frac{1}{3}\delta_{ij}\right ) = \left \langle u_i^{\alpha} u_j^{\alpha}-\frac{1}{3}\delta_{ij} \right \rangle_{\alpha},$ where u is the orientation of an individual nematogen and the ensemble average is carried out over a local region. We will use Q as our microstructure function. Thus, if we can find the orientation of our nematic everywhere on the surface, we can compute Q, obtaining our microstructure function.

This leads to the next question: how do I get the nematic orientation from my data?