Bulk-boundary correspondence in soft matter

Abstract

Bulk-boundary correspondence is the emergence of features at the boundary of a material that are dependent on and yet distinct from the properties of the bulk of the material. The diverse applications of this idea in topological insulators as well as high energy physics prove its universality. However, whether a form of bulk-boundary correspondence holds also in soft matter such as gels, polymers, lipids and other biomaterials is thus far unknown. Aerosil-dispersed liquid crystal gels (LC+aerosil) provide a good testing ground to explore the relation between the controlled variations of the aerosil density within the liquid crystal host bulk and the surface topography of the sample.Here we report on one of the earliest if not the first direct observation of such a correspondence where the controlled strength of random disorder created by aerosil dispersion in the bulk liquid crystal is correlated with the fractal dimension of the surface. We obtained the surface topography of our gel samples with different quenched random disorder strengths by using atomic force microscope techniques, and computed the fractal dimension for each sample. We found that an increase of the aerosil gel density in the bulk corresponds to an increase in the fractal dimension at the surface. From our results emerges a new method to acquire the bulk properties of soft matter such as density, randomness and phase merely from the fractal dimension of the surface.The connection between a material's bulk and its boundary has been one of the guiding principles in several branches of physics in the last decade. The main idea is that the boundary of the system would feature excitations that do not occur in the bulk, yet the physics on the boundary is still determined by the properties of the bulk. For example in topological insulators, the index theorem relates the Chern number quantifying the topology of the insulating bulk to the spectrum of the edge states at the boundary [1][2][3][4]. The holographic principle in high energy physics, also known as gauge/gravity duality, is another example of the bulk-boundary correspondence where the spectrum of the strongly interacting gauge theory in four spacetime dimensions is connected to the weakly interacting theory on the three dimensional boundary via duality [5,6].Here we report on the first ever test of whether bulkboundary correspondence holds in soft condensed matter systems [7,8], particularly in aerosil dispersed liquid crystals ( Fig. 1a). We prepared liquid crystal+aerosil (LC+aerosil) gel mixtures with varying amount of aerosil within, and observed that the aerosil gel density ρ s = m SiO2 /V LC in the bulk is correlated with the fractal dimension of the surface. This experimental verification of the bulk-boundary correspondence in soft matter is the main goal of this study.Liquid crystals (LCs) are not only utilized in screens and TVs, but also used to study phase transitions [9]. Since they possess a rich spectrum of different phases with different types of phase transitions, they stand ou...

Topological edge state assisted dynamically tunable microwave propagations in photonic crystals

“…Amongst these two edge states, only one can transmit in the forward (K ′ valley-locked) or backward (K valley-locked) direction depending on the valley dependence [10,34]. The edge state creation is important for the topological study because it confirms that the proposed design is topologically protected according to the bulk-edge correspondence theory [35,36].…”

A New Probe: AFM Measurements for Random Disorder Systems

“…The random pinning effect of nematic director and short range disordered smectic phases were studied extensively for this material by x-ray diffraction techniques. [5,6,[9][10][11][12]15] According to these experiments, the percolation limit was observed somewhat smaller than 𝜌 s = 0.025 g/cm 3 . This is the reason why many previous experiments on these gel samples were started with a density of 𝜌 s = 0.025 g/cm 3 .…”

“…This also confirms the very short random correlation lengths observed for the high aerosil mass density samples from x-ray scattering experiments. [9] Especially starting with 𝜌 s = 0.347 g/cm 3 , 𝜉 RD becomes very short and saturates at ∼400, ∼500 Å for four different LCs; 8CB, 8OCB, 10CB and 408. [5,6,8,9−11] The saturated values calculated for 𝜉 RD do not vary much over these different LC molecules.…”

“…The 𝜉 RD values were obtained from the previous 8CB+Aerosil and 8OCB+Aerosil x-ray diffraction experiments. [5,6,9,15] The 𝜉 RD results given in the last column belong to the 8CB+Aerosil data except the last two which correspond to the 𝜌s=0.49 g/cm 3 and 𝜌s=0.647 g/cm 3 samples. These are from the 80CB+Aerosil and 10CB+Aerosil experiments, respectively.…”

“…These are from the 80CB+Aerosil and 10CB+Aerosil experiments, respectively. [9,10,18] The average aerosil group size is measured on the pictures manually using the AFM analysis software. [13] The roughness values given for these two samples contain both AFM and profilometer results.…”