Structural heterogeneity: a topological characteristic to track the time evolution of soft matter systems

We introduce structural heterogeneity, a new topological characteristic for semi-ordered materials that captures their degree of organisation at a mesoscopic level and tracks their time-evolution, ultimately detecting the order-disorder transition at the microscopic scale. Such quantitative characterisation of a complex, soft matter system has not yet been achieved with any other method. We show that structural heterogeneity can track structural changes in a liquid crystal nanocomposite, reveal the effect of confined geometry on the nematicisotropic and isotropic-nematic phase transitions, and uncover physical differences between these two processes. The system used in this work is representative of a class of composite nanomaterials, partially ordered and with complex structural and physical behaviour, where their precise characterisation poses significant challenges. Our newly developed analytic framework can provide both a qualitative and a quantitative characterisations of the dynamical behaviour of a wide range of semi-ordered soft matter systems. Many soft composite and biological materials present very complex phase dynamics [1–9] that require powerful analytic methods for their accurate and quantitative characterisation. Persistent homology, a tool from topological data analysis [10–12], has recently emerged as an effective, quantitative method to reveal structural and morphological features in various soft materials, such as granular samples [13, 14], silica glasses [15], glassy polymers [16], living cells, tissues [17–22] and biological objects [23, 24]. Although this topological method has been successfully used to characterise these kinds of materials, to the best of our knowledge, neither persistent homology nor any other analytic method have been used to characterise the dynamical behaviour of structured soft matter systems using optical experimental data. In this paper we introduce structural heterogeneity, a persistent homology-based characteristic for semi-organised soft matter systems. Structural heterogeneity allows one to measure the deviation of a soft matter system from being in a homogeneous or uniform state at a mesoscopic scale. In this work, we use structural heterogeneity to analyse the time-evolution of a nematic liquid crystal doped with gold nanoparticles [25] in phase transition processes. This system has an intrinsically complex dynamics that makes it a non-trivial example of an evolving soft-matter system: the plasmonic nanoparticles are embedded in a non-isotropic complex fluid that organises them in a periodic manner as it evolves from the nematic to the isotropic phase, and retains them in this configuration when the process is reversed. In addition, this composite fluid is Corresponding author 1 ar X iv :2 10 6. 13 16 9v 1 [ co nd -m at .s of t] 2 4 Ju n 20 21 constrained by the cylindrical geometry of the capillary that contains it. At the same time, the system is quasi one-dimensional, which simplifies data acquisition and data analysis. Furthermore, we obtain a representation of each topological descriptor, associated to a particular state of the system, as a point in a Euclidean space. This representation allows us to characterise algorithmically the dynamical behaviour of the system. We show that our topological methods detect distinct temperature-induced macroscopic states, with different degrees of order. An important outcome of our analysis is the development of a persistent homology-based framework to quantify the structural changes experienced by a soft matter system during a thermodynamical process. The novelty of our framework is that persistent homology is used to reveal the structural organisation at the mesoscopic scale, when the analysed system consists of a number of micron-size molecular ensembles with different order parameter [25]. Even though persistent homology has been used in previous works to characterise the order phase transitions in spin lattices [26–28], in none of them persistent homology was used to analyse the dynamical behaviour of the materials at the supramolecular level. In the context of liquid crystals, our results can potentially lead to more elaborate free energy landscapes for nanoparticle-loaded materials and to provide a new perspective on the Landau-de Gennes theory of phase transitions between nematic and isotropic phases. The importance of our pipeline is, however, far more general: it shows that persistent homology can be used as a powerful quantitative method to characterise the complex phase dynamics of non-homogeneous soft materials. Structural heterogeneity The goal of this section is to define structural heterogeneity (SH) as a new topological characteristic for soft matter systems. SH is based on persistent homology, a topological tool used to extract meaningful information from the shape of the data. To help with the discussion, we start by describing the physical properties and the topological features observed in a toy model of a soft matter system. Fig. 1 a shows a cross section of a confined LC system placed between two cross-polarisers; a grey-scale picture of the whole system is showed in Fig. 1 b, where each pixel (square) represents a 2D projection of Fig. 1 a. In this set-up, light passes through the polariser at the top, interacts with an ensemble of LC molecules, to finally cross the polariser at the bottom. The higher the order of the molecular ensemble is, the higher the intensity value of the pixel. A magenta arrow and a bar represent the polarisation and the intensity of the light. In the picture, there are pixels of high intensity values surrounded by loops of pixels of lower intensity, which are generated by highly ordered molecular ensembles surrounded by loops of highly disordered molecules. This picture might indicate, for instance, that the system is in a state where two distinct mesophases coexist. From this analysis, one can see that the quantification of the topological features present in the images of a semi-organised system might be used to characterise it. Before defining SH, we give a brief introduction to persistent homology (PH). A more detailed exposition of PH is found in the Supplementary Material (SM). Starting with a grey-scale picture X as in Fig. 1 b, and a number i, with 0 ≤ i ≤ 255, we define a partial picture X (i) as the union of pixels in X with light intensity not greater than i; changing the value of i creates a set of partial pictures parametrised by light intensity. The topological features of X , are analysed by keeping track of the appearance and disappearance of the topological features in the partial pictures X (i) as i increases, Fig. 1 c. There are two types of topological features to analyse: connected pieces, or 0-cycles, and loops of pixels, or 1-cycles. Each topological feature α observed in the partial pictures is represented by a point (bα, dα) in the Euclidean plane, where bα is the value i when α appears for the first time, and dα when it disappears. The values bα and dα are called