Estimating physical properties from liquid crystal textures via machine learning and complexity-entropy methods

Imaging techniques are essential tools for inquiring a number of properties from different materials. Liquid crystals are often investigated via optical and image processing methods. In spite of that, considerably less attention has been paid to the problem of extracting physical properties of liquid crystals directly from textures images of these materials. Here we present an approach that combines two physics-inspired image quantifiers (permutation entropy and statistical complexity) with machine learning techniques for extracting physical properties of nematic and cholesteric liquid crystals directly from their textures images. We demonstrate the usefulness and accuracy of our approach in a series of applications involving simulated and experimental textures, in which physical properties of these materials (namely: average order parameter, sample temperature, and cholesteric pitch length) are predicted with significant precision. Finally, we believe our approach can be useful in more complex liquid crystal experiments as well as for probing physical properties of other materials that are investigated via imaging techniques.

Figure 1

Examples of a nematic liquid crystal texture at different temperatures and phases. These textures are generated by using the Monte Carlo method described in Appendix pp2. Each texture corresponds to a different reduced temperature $T_r$ (as indicated within the plots), and this system presents a nematic to isotropic phase transition at the critical temperature $T_c=1.1075$. We note that textures from different phases are easily distinguishable, while nematic textures $\left(T_r<T_c\right)$ from different reduced temperatures are very similar to each other as well as the ones from the isotropic phase $\left(T_r>T_c\right)$.

Figure 2

Dependence of the image quantifiers and the order parameter on the temperature. (a) Values of the permutation entropy $H$ and (b) the statistical complexity $C$ as a function of the reduced temperature $T_r$. The solid curves represent average values over 50 realizations, and the shaded areas are the $95\%$ bootstrap confidence intervals. (c) Dependence of the order parameter $p$ versus the reduced temperature $T_r$. In all panels, the dashed line indicates the critical temperature $\left(T_c=1.1075\right)$ of the nematic-isotropic phase transition. We note that the phase transition is clearly and properly identified by the extreme values of the complexity measures.

Figure 3

Predicting the order parameter $p$ with a machine learning algorithm. (a) Training and cross-validation scores of the $k$-nearest neighbors algorithm as a function of the number of neighbors $k$. The scores represent the coefficient of determination $\left(R^2\right)$ of the relationship between the predicted and true values and can be interpreted as the percentage (or fraction) of data variation that is explained by the model. We note that the algorithm overfits the data when $k<10$, whereas for $k>20$ it starts underfitting the data. (b) Learning curves, that is, training and cross-validation scores as a function of the training size (fraction of the whole data used for training the model) with $k=15$. We observe no significant improvement in the cross-validation score when more than $35\%$ of the data is used. The shaded areas in both plots represent the $95\%$ confidence intervals obtained in a three-fold-cross-validation splitting strategy. We note the high accuracy achieved by the algorithm $\left(\approx 99.2\%\right)$. (c) True and predicted order parameter $p$ as a function of the reduced temperature $T_r$. These predictions were generated by exposing the trained regressor (with $k=15)$ to a set of textures never presented before to the algorithm.

Figure 4

Examples of E7 liquid crystal textures at different temperatures and phases, and the dependence of the complexity measures on the temperature. (a) Experimental textures of an E7 liquid crystal sample at different temperatures. We observe practically no visual change until the critical temperature $T_c\approx 58{}^{\circ }\mathrm{C}$. (b) Dependence of the permutation entropy $H$ and (c) the statistical complexity $C$ on the temperature $T$. The solid curves represent the average of the quantities calculated over the results of six samples. The shaded areas are $95\%$ bootstrap confidence intervals. The vertical dashed lines indicate the critical temperature $T_c$. We note that the phase transition is properly identified by the extreme values of the complexity measures.

Figure 5

Predicting the temperature with a machine learning algorithm. (a) Training and cross-validation scores of the $k$-nearest neighbors algorithm as a function of the number of neighbors $k$. The scores represent the coefficient of determination $\left(R^2\right)$ of the relationship between the predicted and true values and can be interpreted as the percentage (or fraction) of data variation that is explained by the model. Note that for $k=1$ the algorithm overfits the data, while for $k>3$ the model starts to underfit the data. (b) Learning curves, that is, the training and cross-validation scores as a function of the training size (fraction of the whole data) for $k=2$. The shaded areas in both plots represent the $95\%$ confidence intervals obtained in a three-fold-cross-validation splitting strategy. We observe practically no significant improvement in the cross-validation score when more than $80\%$ of the data is used to train the model. (c) True versus predicted temperatures obtained by exposing the trained regressor to a set of experimental textures never presented before to the algorithm. The dashed line represents the 1:1 relationship. We observe an excellent agreement between the true and predicted temperatures, reinforcing the great accuracy achieved by the method $\left(\approx 93\%\right)$.

Figure 6

Discriminating among cholesteric textures with different pitches via the complexity-entropy plane. Each colored marker represents the values of $H$ and $C$ for five realizations of the cholesteric textures with different pitches (as indicated by the different markers). We note that the values of $H$ and $C$ for each cholesteric pitch are localized over a small region in the complexity-entropy plane. We further observe that the textures from small pitches are localized in a high-entropy region, while those from large pitches are in a low-entropy region. This result indicates that textures from large pitches are more ordered than those obtained for small pitches (as illustrated by the insets).

Figure 7

Predicting the cholesteric pitch with statistical learning algorithms. Training and cross-validation scores (fraction of correct classifications) of the $k$-nearest neighbors algorithm as a function of the number of neighbors in (a) and as a function of the training size in (b). The shaded areas in both plots are the $95\%$ confidence intervals obtained in a five-fold-cross-validation splitting strategy with the number of neighbors equal to 20. Note that for the number of neighbors smaller than 15 the algorithm overfits the data, but for more than 20 neighbors it starts underfitting the data. We also notice that when more than $60\%$ of the data is used to train the model (training size), there is no significant score improvement. (c) True and predicted cholesteric pitch. This confusion matrix shows the good performance achieved by the algorithm represented by the high fractions of right predictions in the diagonal. In some cases, the algorithm underestimates the pitch. This results from the overlap observed in the complexity-entropy plane in Fig. 6.