# Geometric Confinement of 3D Smectic Liquid Crystal Foams
## 1 Introduction  
Foams are materials produced by injecting air into a solid or liquid [1; 2]. These systems are unique and significant for many fundamental questions and industrial applications [3]. Some examples include the use of foam in cosmetic products [4], food industries [5], packaging [6], noise absorption [7], and firefighting [8; 9]. Foams have also inspired many other fields, such as the sectors of design, art, and architecture, where new constructs are inspired by bubble assembly and structure. Some recent examples are the water cube swimming pool built for the 2008 Beijing Olympic Games, which became an iconic structure [8]; Rotterdam's solar-powered floating bubble pavilion; and Amazon's bubble office building in Seattle. Moreover, foams are significant for multiple fundamental aspects. For example, mathematicians have strived to comprehend foam arrangements and how they maintain particular mathematical allures [10; 11; 12], whereas physicists have attempted to understand foam assemblies and predict their properties [13]. For these reasons, scientists and engineers are always looking for new ways to fabricate foams using different materials with various properties [14; 15]. We can classify foams into two different categories based on their interfacial systems: foams that are separated by hard boundaries, called solid foams (air and solid) [16], and those that are separated by fluid boundaries, called liquid foams (air and liquid) [17]. In this paper, we focus on liquid foams that can be dry or wet depending on their volume fraction. The equilibrium configuration of these systems is determined by the minimization of their surface energy [13]. For millimetric foams made of simple fluids, the pressure difference inside them is given by the Young-Laplace law \(\Delta P=2\gamma/R\), where \(\gamma\) is the surface tension of the film, and \(R\) is the radius of the bubble. This difference is in the order of \(\sim\)100 Pa,which is very small compared to the atmospheric pressure (\(P_{atm}\approx 10^{5}\) PA). For this reason, liquid foams are considered incompressible. Their total energy is given by \(E=\gamma\sum_{i=1}^{n}S_{i}\), where \(n\) is the number of bubbles in the foam, and \(S_{i}\) is the surface area of the bubble \"\(i\)\". This energy takes into account only the effect of the surface. Little is known about how other parameters, such as bulk elasticity and viscosity, could affect the properties of foams. Liquid foams can assemble into different 3D structures as a function of various parameters, such as the system's temperature [18], the flow rate of the injected air [19], the geometry of the given confinement [20], and the system's drainage properties [21]. Among these well-organized assemblies, we can find the body-centered cubic (BCC or Kelvin) or the face-centered cubic (FCC) structures. These structures can affect foam stabilities and can be controlled via different experimental methods; by changing the volume fraction of the foam [21], for example, or by controlling the geometry of the confining boundaries [20], most of the previous studies have focused on the properties and structures of foams created by simple fluids. However, little is known about how foams arrange and pack when they are made by an ordered material such as a liquid crystal (LC). LCs are condensed materials with properties between those of conventional liquids and those of traditional solids. Their molecules can be oriented like crystals; simultaneously, they can flow like liquids [22]. Depending on the arrangements of these molecules, we distinguish different LC phases, such as nematics, smectics, and cholesterics. These materials are characterized by their anisotropic properties and delicate sensitivity to external stimuli, such as deformations, electric, and magnetic fields. For this reason, they are used in various applications, particularly in display devices, such as TVs, monitors, screens, smartphones, and watches [23; 24].