Ebook: Physics of Complex Colloids

Colloids, i.e. systems comprised of particles of mesoscopic size suspended in a liquid, have been attracting increasingly more attention from scientists and engineers alike. This is partly due to the fact that nowadays colloids are present in many industrial products such as paints, oil additives, e-paper, and drugs. Apart from being a prominent segment of soft condensed matter, colloids also serve as versatile model systems for phenomena and structures known from solid-state physics, surface science, and statistical mechanics. Due to their large lengthscales and slow timescales they can be readily studied using tabletop experiments, providing insight into processes which are not easily accessible in atomic systems.

Organized within the Marie-Curie Initial Training Network COMPLOIDS, the School was conceived to present the diversity of research in the area of colloidal systems in a comprehensive style to young scientists and to indicate novel trends in the field. Colloidal science is distinguished by an intimate connection between experiments and theory, and the School addressed experimental, theoretical, numerical results and methods. The topics of the lectures covered a broad spectrum of aspects starting from the synthesis of colloids and their use in commercial products. Particular attention was paid to the different types of colloidal interactions, how they can be measured and how they lead to ordered and disordered structures. Colloidal systems do not only allow us to address equilibrium states but also provide appealing options to investigate non-equilibrium phenomena, some of which were discussed in lectures on active Brownian motion and on stochastic thermodynamics.

The School was designed as a series of minicourses, most of which consisted of five or three lectures. The minicourses were complemented by seminars addressing selected recent advances in the field and by poster presentation of the participants. We did our best to accept as many participants as possible, giving priority to Ph.D. students. The complete list of participants is included in this volume.

The School took place at the magnificent location of the Villa Monastero in Varenna, which stimulated an intense scientific exchange between the lecturers, the students, and the directors. The local organizer, Barbara Alzani, together with Ramona Brigatti and Marta Pigazzini, did a truly remarkable job in making this meeting memorable and unique. Their mindfulness and readiness in organizational issues were instrumental for the overall success of the School. Also essential for the organization was the assistance of Roberta Comastri (administrative office), Monica Bonetti (editorial office) and Marcella Missiroli (production office). Finally, we thank the Italian Physical Society (SIF) for making this event possible. In these proceedings, the authors review the state of the art of colloidal science in a pedagogical way, discussing both the basics and the latest results. We hope that the volume will serve as a reference both for students entering this rapidly growing field and for the experts.

C. Bechinger, F. Sciortino and P. Ziherl

In this paper we present an overview of some key theoretical tools employed for the investigations of the equilibrium structure and thermodynamics of complex fluids. We place our focus on the most common systems of soft-matter science, namely dispersions of colloidal particles, which may be hard or soft, and which may carry electric charge. The concept of the effective Hamiltonian, which plays the key role in these coarse-grained approaches is introduced, and it is given a precise mathematical definition. Its properties as well as key approximations involved in deriving effective potentials are discussed. Thereafter, the ways in which the effective interactions can be combined with tools from the theory of fluids and statistical mechanics are presented, putting emphasis on Density Functional Theory. The general principles are combined with extensive exposures from currently active research fields, and applications to interfacial and wetting behavior of complex mixtures as well as to crystallization and cluster formation are discussed. Appendix A serves as a mini-course on functionals and functional differentiation.

By a length scale analysis, we study the equilibrium interactions between two like-charge planes confining neutralising counterions. At large Coulombic couplings, approaching the two charged bodies leads to an unbinding of counterions, a situation that is amenable to an exact treatment. This phenomenon is the key to attractive effective interactions. A particular effort is made for pedagogy, keeping equations and formalism to a minimum.

Recent progress in approaches to determine the elastic constants of solids starting from the microscopic particle interactions is reviewed. On the theoretical side, density functional theory approaches are discussed and compared to more classical ones using the actual pair potentials. On the experimental side, video microscopy has been introduced to measure the elastic constants in colloidal solids. For glasses and disordered systems, the theoretical basis is given for this novel technique, and some challenges and recent advances are reviewed.

In these notes we will examine the main forms of arrested states that are encountered in colloidal and soft matter systems. The two main categories of arrested states will be discussed, namely glasses and gels. While glasses can be attributed to the effect of either dominant repulsive (e.g. excluded-volume or electrostatic) or dominant attractive (e.g. depletion) interactions, gels can only arise by means of attraction. However, the type of attraction, being isotropic or directional, or the presence of additional repulsion can significantly alter the properties of the resulting gel state. Another aspect that we will address is the role of softness of the particles in the formation of the arrested state, a crucial way to tailor the rheology of the macroscopic materials. As an example of a glass-forming soft system, which has been recently investigated in a joint experimental and theoretical effort, we will focus on star polymers. Finally we will also examine the case of competing interactions, e.g. short-range attraction and long-range repulsion, which give rise to arrested states mediated by finite-size clusters rather than by particles. Our description is largely based on the role of effective colloidal interactions and mainly focused on recent theoretical and numerical results, with strong reference to related experimental observations.

The main purpose of statistical mechanics is to give a microscopic derivation of macroscopic laws, including in particular the celebrated second law of thermodynamics. In recent years, there have been spectacular developments in this respect, including the integral and detailed work fluctuation theorems and the theory of stochastic thermodynamics. Here we give a brief introduction to these developments. In the first step, we derive the first and second law of thermodynamics for a Markovian stochastic process at the ensemble level, including two major advances: 1) the theory can be applied to small-scale systems including the effect of fluctuations, 2) the theory is not restricted to near-equilibrium dynamics. As an application, we evaluate the efficiency at maximum power of a two-state quantum dot. We also briefly discuss the connection to information-to-work conversion (Landauer principle) and the splitting of the second law into an adiabatic and non-adiabatic component. In a second step we formulate stochastic thermodynamics at the trajectory level, introducing stochastic trajectory-dependent quantities such as stochastic entropy, energy, heat, and work. Both the first and the second law can be formulated at this trajectory level. Concerning the second law, the crucial observation is that the stochastic entropy production can be written as the logarithm of the ratio of path probabilities. This in turn implies a detailed and integral work and fluctuation theorem, linking the probability to observe a given stochastic entropy production to that of observing minus this entropy change in a reverse experiment. The usual second law, stipulating the increase on average of the stochastic entropy production, follows as a subsidiary consequence.

This paper discusses the Monte Carlo Molecular Dynamics methods. Both methods are, in principle, simple. However, simple does not mean risk-free. In the literature, many of the pitfalls in the field are mentioned, but usually as a footnote—and these footnotes are scattered over many papers. The present paper focuses on the “dark side” of simulation: it is one big footnote. I should stress that “dark”, in this context, has no negative moral implication. It just means: under-exposed.

In order to predict the equilibrium phase behaviour of colloidal particles, one should first identify the “candidate” structures in which the particles may assemble. Subsequently, the free energy of the candidate structures should be determined to establish the thermodynamically stable phases and to map out the bulk phase diagram. Here, we describe a simple method based on a simulated annealing approach to predict candidate structures and several techniques to calculate the free energy of a thermodynamic system and to map out the phase diagram. Exemplarily, we present phase diagrams of several shape-anisotropic hard particles, e.g., hard dumbbells, hard bowl-shaped particles, and hard oblate spherocylinders.

The past few years have seen a conspicuous development of novel optical methods for investigating the structural and dynamic behavior of soft matter, where scattering and imaging concepts are often strongly intermixed. Grasping the working principle of these techniques and the interrelation between them requires a solid background in statistical optics. Rather than entering into technical details or discussing specific applications, this review aims to provide such a general framework.

An active colloid is a suspension of particles that transduce free energy from their environment and use the energy to engage in intrinsically non-equilibrium activities such as growth, replication and self-propelled motility. An obvious example of active colloids is a suspension of bacteria such as Escherichia coli, their physical dimensions being almost invariably in the colloidal range. Synthetic self-propelled particles have also become available recently, such as two-faced, or Janus, particles propelled by differential chemical reactions on their surfaces driving a self-phoretic motion. In these lectures, I give a pedagogical introduction to the physics of single-particle and collective properties of active colloids, focussing on self-propulsion. I will compare and contrast phenomena in suspensions of “swimmers” with the behaviour of suspensions of passive particles, where only Brownian motion (discovered by Robert Brown in granules from the pollen of the wild flower Clarkia pulchella) is relevant. I will pay particular attention to issues that pertain to performing experiments using these active particle suspensions, such as how to characterise the suspension's swimming speed distribution, and include an appendix to guide physicists wanting to start culturing motile bacteria.

Increasingly functional colloidal particles (nanoparticles) are finding new applications in a wide range of both established and emerging technologies. This article describes the synthesis of such particles, from classical and naturally occurring colloidal particles to more recent, highly designed particles having some particular structure and physical properties, usually with a specific application in mind. The systems described range from inorganic particles (ionic salts and metal particles, through to modern graphene-based particles), polymer latex particles (including non-spherical, liquid and electrically conducting polymer particles), composite particles (in particular core/shell particles) and finally porous and swellable particles (microgels). Throughout, various applications are considered, in particular, one that I have been strongly involved with, namely controlled uptake and release from functional colloidal particles.

This paper is intended to describe modern applications of colloids. We will focus on two topics, the versatility of colloids in industry due to tuning of the interactions and assembly, and the use of particle shape to obtain new properties. In the section dealing with interactions, we will focus on naturally occurring colloids that must be dispersed, jammed soft particles, tunable and programmable interactions between colloids, reinforcement of polymers with colloidal aggregates, and conducting nanogels. In addition, we will discuss how anisotropy can be used to bring about new materials for applications.

We explore the influence of particle shape on the behavior of evaporating drops. A first set of experiments discovered that particle shape modifies particle deposition after drying. For sessile drops, spheres are deposited in a ring-like stain, while ellipsoids are deposited uniformly. Experiments elucidate the kinetics of ellipsoids and spheres at the drop's edge. A second set of experiments examined evaporating drops confined between glass plates. In this case, colloidal particles coat the ribbon-like air-water interface, forming colloidal monolayer membranes (CMMs). As particle anisotropy increases, CMM bending rigidity was found to increase, which in turn introduces a new mechanism that produces a uniform deposition of ellipsoids and a heterogeneous deposition of spheres after drying. A final set of experiments investigates the effect of surfactants in evaporating drops. The radially outward flow that pushes particles to the drop's edge also pushes surfactants to the drop's edge, which leads to a radially inward flow on the drop surface. The presence of radially outward flows in the bulk fluid and radially inward flows at the drop surface creates a Marangoni eddy, among other effects, which also modifies deposition after drying.

The formation of crystals starts with nucleation and control of nucleation is crucial for the control of the number, size, perfection, polymorph modification and other characteristics of the crystalline population. Recently, there have been significant advances in the understanding of the mechanism of nucleation of crystals in solution. The most significant of these is the two-step mechanism of nucleation, according to which the crystalline nucleus appears inside pre-existing metastable clusters of size several hundred nanometers, which consist of dense liquid and are suspended in the solution. While initially proposed for protein crystals, the applicability of this mechanism has been demonstrated for small-molecule organic and inorganic materials, colloids, and biominerals. This mechanism helps to explain several long-standing puzzles of crystal nucleation in solution: nucleation rates which are many orders of magnitude lower than theoretical predictions, nucleation kinetic dependencies with steady or receding parts at increasing supersaturation, the role of heterogeneous substrates for polymorph selection, the significance of the dense protein liquid, and others. More importantly, this mechanism provides powerful tools for control of the nucleation process by varying the solution thermodynamic parameters so that the volume occupied by the dense liquid shrinks or expands.

The present lecture notes survey the combined inertia-free motion of mesoscale particles and dispersing fluid under low Reynolds number conditions. The survey starts with a discussion of relevant time scales, progressing then to methods to obtain flow solutions of the fundamental Stokes equation for isolated rigid particles such as spheres and rods. Kinematic reversibility is discussed, as well as the peculiar hydrodynamic propulsion of active microswimmers by non-reciprocal, periodical changes of their body shape. The three-beads model and a simple sperm model are analyzed as examples, respectively, of an artificial and biological microswimmer. The fluid-mediated hydrodynamic interactions (HIs) between moving particles are thoroughly discussed, in particular regarding the settling of two-, three- and many-particle clusters under gravity. HIs between three and more settling particles give rise to horizontal mixing, the appearance of chaotic particle trajectories, and a mean settling velocity of the cluster, which for an infinite fluid increases with increasing number of cluster particles. It is shown how the mobility matrices characterizing the HIs for a given particle configuration can be calculated for spherical particles. These matrices are then used for the calculation of the effective viscosity, and the mean sedimentation velocity in homogeneous suspensions of colloidal particles. The interplay of HIs and Brownian motion leading to non-Newtonian flow behavior of colloidal hard-sphere suspensions is analyzed. The hindered particle sedimentation in a homogeneous suspension is shown to arise from the effect of fluid backflow, in combination with hydrodynamic and direct particle interactions. While the difference between the steady shear viscosity and its (short-time) high frequency part becomes large for concentrated suspensions, the difference between long-time and short-time sedimentation coefficients remains very small.