The isolation of a hot tube is a standard task in industry. Here we derive the fully time dependent solution of a tube with an aerogel cover whcih shows the importance of thermal diffusivity in instationary isolation tasks.

Gelation describes the transformation from a liquid or fluid state to a somewhat solid state, well known from daily experience, making, for instance, a jelly pudding. Gels and gelation are studied quite extensively in chemistry and physics, and especially over the past three decades theoreticians have discussed the formation of gels and the relation between structure and properties quite extensively (percolation theory, fractals). In this chapter, we will first discuss the viscosity of solutions and how it changes during gelation. For an understanding of modern equipment, to analyse gelation, it is important to briefly discuss viscoelasticity andsimple models for a viscous fluids. We thendescribe how gelation is measured, from very simple methods to more elaborate ones. The chapter closes with a survey of theoretical models for gelation such as percolation, diffusion-limited cluster aggregation, mean field theory using the Smoluchowski equation, scaling analysis and polymerisation-induced phase transformation (PIPS). For all these models, predictions of gel time can be made, showing how the composition of the solution, the viscosity and the temperature affect it.

The seemingly simplest property of any material and aerogels especially is the density, defined as the ratio of mass and volume. For any regularly shaped body such as a cube, sphere or cylinder, the volume is readily determined and the mass obtained by simply weighing the body. For a porous material, especially if the shape is not regular, the density is not that easy to determine. For aerogels, two different values are usually determined: the so-called envelope densityandthe skeletal density. The envelope density is defined as the massdivided by the total volume enclosing the porous structure. The skeletal density instead is the density of the solid backbone of the aerogel, i.e., the sum of the volume of all nanoparticles making up the aerogel. The chapter discusses techniques to measure both densities and all aspects of these techniques and closes with a section discussing how to estimate the final aerogel density from the known composition of the monomeric precursor solution or in the case of biopolymers that of a polymeric solution.

In synthesis, processing and applications of aerogels transport of liquids and gases in and through a wet gel or aerogel quite often determine the time scale and the properties. The transport of molecules is a diffusive process, meaning that the molecules move randomly in the solvent biased by concentration or chemical potential gradients. They typically diffuse from points of higher potential (mostly also higher concentration) to regions of lower potential or concentration. Similarly, a diffusion process occurs when a wet gel is washed, for instance, in an ethanol bath to exchange the pore fluid after gelling and ageing to one which is miscible with, for instance, carbon dioxide. In such a situation, the wet gel is overlaid by ethanol, which then diffuses into the pore space. Inasmuch as it diffuses inwards, the gel fluid moves outwards into the ethanol layer. During adsorption and desorption studies on aerogels, nitrogen diffuses into the pore system and will then be adsorbed or desorbed there. This transport takes time, and wediscuss characteristic times for such a process. This chapter discusses concepts of diffusion of species in general and in aerogels especially.

The book would not complete if its readers would not be able to make aerogels by themselves. If one is interested in doing so, however, a chemical lab is needed and anyone doing it should have a bit of experience working in a lab. If supercritical drying is needed, and a lot of aerogels ask for that, a suitable facility should be available. This chapter gives recipes and explains how aerogels are made in the chemical lab and the procedures, that is, how they are made.

The synthesis of aerogels need not to start with monomers, but also can startwith bundles of polymers of crystalline or amorphous nature. If such polymers are dissolved in a suitable medium down to their single polymeric strands, the solution can be rearranged to form an open, porous, nanostructured network by various methods, such as temperature change, pH inversion orthe addition of a suitable cross-linking salt. In this chapter, we discuss two types of aerogels made from biopolymers: cellulose and alginates. Their chemistry is explained as well as synthesis routes for wet gel preparation.

In this appendix, we briefly review the concept of a multi-component system exhibiting a miscibility gap, and define the concept of the binodal and spinodal lines explain phase separation process once the system moves from a single-phase field into a two-phase field.

Aerogels starting with monomeric solutions quite often from polymers by polycondensation reactions. We give in the appendix a model of polycondensation based on an approach made more than 100 years ago by Smoluchowski, and derive from that the standard equations, such as the Carothers one and the Flory–Schulz distribution. We also present a volume fraction of polymers and explain how it depends on the degree of polymerisation or time.

Pores are in aerogels essential. Experimentersoften usethe nitrogen adsorption measurement technique and derive from the desorption curve the pores' size distribution assuming cylindrical pores and the Kelvin equation to be applicable.A description of the pores is difficult and the situation is not comparable with, for instance, closed cell foams. Scanning electron microscopy gives an imagination of the particles or fibrils and thus also the pores. Nevertheless, there are simple measures for pore sizes possible, which are well defined in stereology, namely the mean free distance between particles or fibrils in a network and the next nearest neighbour distance. In addition, scattering methods allow us to extract chord lengths in pores and the solid phase assuming a suitable model of the two-phase structure. The experimental techniques such as the BJH model and thermoporosimetry are discussed and the basic equations derived. The theoretical models are compared with experimental results for different aerogels.

This chapter describes the Bose--Einstein condensate (BEC) interferometry. We first introduce the optical interferometer, briefly discussing the role of fringe contrast in observing interference patterns. Next, we examine a BEC interferometer in a Michelson geometry that consists of a BEC in a trapping potential well. The BEC cloud in the potential well is put into a linear superposition of two clouds that travel along different paths in the trap and are recombined using the same light beams. By studying the population of atoms in the different motional states of the BEC clouds that emergeafter recombination, we obtain information about the relative phase shift accumulated by the two traveling superposed condensates during the interferometry. We then characterize the atom distribution that is found in the output of the interferometer by finding the probability density distribution and calculating the features of the probability density such as the mean, the variance, and the interference fringe contrast. Finally, we parameterize the effect of two-body collisional interactions on the observed interference fringes in a way that can be directly controlled in an experiment.

This chapter introduces the basic theoretical tools for handling many-body quantum systems. Starting from second quantized operators, we discuss how it is possible to describe the composite wavefunction of multi-particle systems, and discuss representations in various bases. The algebra of Fock states is described for single and multi-mode systems, and how they relate to the eigenstates of the Schrodinger equation. Finally, we describe how interactions between particles can be introduced in a general way, and then describe the most common type of interaction in cold atom systems, the s-wave interaction

This chapter discusses the spin degrees of freedom in an atom. We first review how atomic energy levels can be classified in terms of the electron spin and orbital angular momenta and how this couples to the nuclear spin. We then describe how atoms interact with each other, and how the spins affect this interaction. The effect of electromagnetic radiation on the energy levels of an atom is described, and the Hamiltonian for energy levels transitions . After briefly describing how the important phenomena of the ac Stark shift and Feshbach resonances occur, we then turn to describing how dissipative dynamics affect atomic systems. Specifically, we examine master equations for spontaneous emission and atom loss, and look at these can be solved. Finally, we consider an alternative framework for solving such open systems using the quantum jump method, which allows for a stochastic approach to solving the dynamics