Chapter 1: Introduction to Colloidal Particles

In the field of chemistry and materials science, colloids represent a subject of profound scientific interest. These systems, characterized by the dispersion of minuscule particles within a continuous medium, have played a pivotal role in understanding matter’s behavior and have found applications in various scientific and industrial domains. Frequently used definitions for the term ‘colloid’ include ‘a homogeneous non-crystalline substance (or first phase) dispersed within a second, continuous phase’. The Encyclopedia Britannica specifies that the first phase can be ‘any substance consisting of particles substantially larger than atoms or ordinary molecules but too small to be visible to the unaided eye […], having at least one dimension in the general size range of 10⁻⁷ to 10⁻³ cm’.⁴  The particles do not settle over time and cannot be separated by conventional filtration or centrifugation (unlike those in suspension), although this criterion and the size limits are not to be applied too strictly. This nomenclature was introduced by Thomas Graham in 1861, who is also regarded as the father of colloidal chemistry.⁵  The terminology stems from the greek ‘colla’ which translates to glue and ‘eidos’, meaning ‘species’ or ‘forming’. Colloidal materials are available in a large variety of compositions, as shown in Figure 1.1.

In light of the book’s thematic focus, this introductory chapter will primarily concentrate on colloidal particles, where the dispersed phase is solid. It is important to note that this scope intentionally omits discussions on polymers, micelles, and other soft colloidal systems to offer a more focused introduction to the materials forming active colloidal systems. Please note that this chapter serves as an initial overview and will provide pertinent references and suggested reading materials for a comprehensive understanding of the existing literature.

Due to the multitude of available methods for colloidal synthesis, different classifications are necessary to didactically elaborate and systematize the approaches. To this end, a variety of factors can be used for differentiation, which will also be used as subsections in this part of the book.

• Particle formation methods can be broadly divided into dispersion and condensation methods. The dispersion or top-down methods primarily involve physical processes like grinding, milling, or laser ablation, where an energy intake leads to an overall reduction of particle size. Conversely, bottom up, or condensation methods encompass most chemical synthesis schemes and are more prevalent in colloidal chemistry. Therein, some of the frequently used synthetic approaches include precipitation,⁶  emulsion polymerization,⁷  sol–gel syntheses,⁸  and template-based replication methods.⁹  Precipitation is probably the easiest approach, which refers to the crystallization of soluble substances in contact with an anti-solvent. A number of factors, from concentration to temperature and mixing, can influence the primary precipitation, allowing for the synthesis of a wide variety of materials.^10,11  The one sol–gel synthesis that every chemist and material scientist associates with this type of reaction is the fabrication of SiO₂ particles. This method gained prominence through the pivotal work by Stöber,¹²  and has since been the subject of numerous impactful follow-up studies.^13,14  Broadly considered, a sol–gel synthesis is a procedure where a precursor solution is transformed into a colloidal network, finally precipitating to discrete particles.¹⁵  This method offers clear advantages over traditional glass melting or powder based methods such as higher homogeneity and purity of the products and notably, also decreased energy usage. Various metal oxides and aero- or cryogels (i.e. gels after a supercritical or freeze-drying process) can be synthesized using related methods. Hydro- or solvothermal synthesis usually takes advantage of the increased solubility at elevated temperatures and higher pressures to obtain (usually crystalline) particles. To perform this type of synthesis, normally an autoclave setup is required. A few literature examples of colloidal syntheses are listed in Table 1.1.

• This flexibility gives rise to a wide range of materials, ranging from surfactants and polymers to inorganic or hybrid materials. Nano-sized noble metal colloids have attracted considerable attention,¹⁶  largely because of their wide range of applications, ranging from their historic use in ancient glasses¹⁷  and bridging to innovative catalytic approaches¹⁸  and plasmonics.¹⁹  More reactive materials often form oxidic derivatives, leading to metal oxides or semiconducting materials.²⁰  While their often colorful appearance has been used in various dyes since ancient times, the seminal work of Honda and Fujishima on TiO₂ colloids for light-driven water splitting sparked interest in heterogeneous photocatalysis.²¹  A variety of magnetic materials can also be prepared with nanoscale dimensions.²²  Small dimensions can lead to interesting phenomena such as paramagentism, which often results in interesting properties related to medical applications (e.g. local heating in a magnetic field). When considering a whole bulk fluid made up of individual magnetic particles, the magnetic properties can be imparted to the colloidal solution as a whole, resulting in fascinating ferrofluids.^23,24  Colloidal particles are not limited to inorganic materials. Polymeric nano- and microstructures have been widely used in many industrial applications,²⁵  often as fillers or to achieve enhanced abrasive properties. However, they have now been replaced in many parts of the world by other materials to reduce the emission of microplastics into water bodies. There has also been a surge in the discovery of organic nanomaterials,^26,27  ranging from biopolymers and dendrimers to DNA nanostructures and covalent organic frameworks. Many proposed applications involve biomedical targets or components in novel energy devices.

• The sizes of colloidal materials can often range for a single material from the lower nanoscale to several micrometers. For example, gold particles can range from the well-known nanoparticles (AuNP)¹⁶  to microscale spheres, frequently used in active matter.⁴⁵  This diversity in sizes not only shows how well we have mastered colloidal synthesis, but also highlights the flexibility of many bulk materials on the small scale. It is well established that size strongly affects the properties at the nanoscale, ranging from light interactions and colloidal stability to bio-interactions (see display of colloidal objects of different sizes in Figure 1.2).

• A particularly intriguing aspect of colloidal particles is their shape. The largest variation of shapes has probably been achieved for gold nanoparticles⁴⁶  and captivating additions such as chiral properties are constantly being added to the synthetic toolbox.⁴⁷  Interest in the modification of microscale colloidal shapes has lagged behind its smaller cousin,⁴⁸  most likely due to the decrease in plasmonic properties. Of course, most synthetic approaches yield either spherical particles due to surface energy minimization or random shapes. Manipulation of the growth conditions can often be tuned by using templated approaches or by achieving preferential growth in certain directions through specific capping agents. The field of active materials has mainly focused on spherical morphology, not least because of the much easier interaction with theoretical approaches. Recently, however, rod morphologies have become increasingly popular.

  Table 1.1

  Overview over a variety of colloidal materials and their syntheses.

  Material .	Method .	Ref. .
  SiO2	Sol–gel	28 and 29
  TiO2	Sol–gel	30 and 31
  ZnO	Sol–gel	32 and 33
  Fe2O3	Hydrothermal	34–36
  BiVO4	Hydrothermal	37–39
  BiIO3	Hydrothermal	40
  AgCl	Diverse	41 and 42
  Polystyrene	Diverse	43
  Polymethyl methacrylate	Diverse	44
  Au	Diverse	45

  Material .	Method .	Ref. .
  SiO2	Sol–gel	28 and 29
  TiO2	Sol–gel	30 and 31
  ZnO	Sol–gel	32 and 33
  Fe2O3	Hydrothermal	34–36
  BiVO4	Hydrothermal	37–39
  BiIO3	Hydrothermal	40
  AgCl	Diverse	41 and 42
  Polystyrene	Diverse	43
  Polymethyl methacrylate	Diverse	44
  Au	Diverse	45

  View Large

  Figure 1.2

  

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  Scale representation of nanomaterials.

  Figure 1.2

  

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  Scale representation of nanomaterials.

  Close modal

The following section gives a brief introduction on analytical techniques that reveal specific material properties. Examples of some analytical devices are given in Figure 1.3.

• Optical or light microscopy uses a combination of light and one or multiple lenses to magnify small scale objects. More in-depth information on microscopy will be given in Chapter 2. In colloidal sciences, optical microscopy is used to characterize samples using still images, called micrographs. The dynamics of various colloidal processes are frequently documented using image sequences or videos. Depending on the scope of the imaging and the detection, different combinations and setups can lead to different image configurations. If a sample is directly irradiated, this is referred to as bright field microscopy, while indirect illumination around the object is dark field microscopy. Practically, the resolution of an optical microscope is limited by light diffraction, which makes two points indistinguishable when located too close to each other. Therein, the resolution d was defined by Ernst Abbe as

  $d= λ ( 2 NA )$

  (1.1)

  where λ is the wavelength of the light and NA is the numerical aperture of the objective lens. This limits the resolution of conventional optical microscopy to a few hundred nm, so in order to evaluate material features on the particles we need other methods that allow visualization beyond visible light.

• Similar to optical microscopy, electron microscopy also uses a wave to produce an image of a sample. The main difference is the nature of the employed waves, which are of electronic nature. Depending on the interaction with the sample, we can distinguish between the two main types: transmission and scanning electron microscopy, (TEM and SEM, respectively) along with combinations and other subtypes that usually focus more on surface and material properties. More detailed descriptions can be found in a variety of textbooks.^49,50  Using electrons as a means of imaging in combination with electron optics, electron microscopy can reach much lower values for resolution d and therefore allows precise characterization of much smaller features and morphologies. Examples for SEM and TEM micrographs can be seen in Figure 1.2. The use of alternative detectors also allows for coupling the optical representation of materials with detection of certain material properties. A back scattered detector for example collects the electrons after an elastic collision (scattered ‘back’). These elastic collisions happen more frequently for atoms with a greater atomic number and a larger cross-sectional area. This allows the correlation of the mean atomic number distribution over the sample, and allows high-resolution compositional mapping of a sample. When other radiation types are considered, access to further compositional information is unlocked. Among the most common examples is energy dispersive X-ray spectroscopy (EDS), using the produced X-rays upon irradiation with the electron beam to obtain elemental information.⁵¹ 

• X-ray diffraction (XRD), is a non-destructive method to characterize regular distances within materials, which allows evaluation of the material composition of crystals by comparison with reference patterns, as well as the determination of the crystallinity and crystal sizes. Given the wavelength of X-rays (0.01–10 nm), this method is adequate to interfere with crystal planes in a wide range of materials. X-rays are produced by the impact of electrons on a target (often molybdenum or copper). The resulting gap in the electron orbit can be filled and cause X-ray emission and are characteristic of those materials. In order to measure XRD, a monochromatic beam is required. The most frequently used device is a single crystal diffractometer.⁵²  When the beam of X-rays constructively interferes with the sample, this signal can be collected by a detector (which can be either fixed or orbiting around the sample) and plotted as counts over the angle θ in a diffractogram. This allows for the determination of a unit cell and the subsequent calculation of atomic positions and bond lengths. For colloidal samples, a single particle is usually too small to be measured individually, so both the sample and the detector are rotated constantly and the reflected X-rays are collected and plotted as a so called powder diffractogram.

• Particle size and surface potential (zeta potential) measurements most commonly use the interaction of colloidal particles with light to infer other particle properties.⁵³  While imaging-based methods are available, dynamic light scattering clearly dominates the market. In this technique, the motion of colloids interferes with the light scattered by a particle, causing a Doppler effect that can be measured and correlated with the underlying particle velocity. Brownian motion in the absence of external fields is commonly used to measure the hydrodynamic size of particles. To assess the surface charge of a particle, we most commonly consider a modelled quantity called the zeta potential, which can be determined from measurements of the electrophoretic mobility in a suspension. Since the mobility depends on the charge conditions, the Stokes–Einstein equation can be used to relate the measured velocity to the amount of charge present in the double layer. The reverse effect, i.e. measuring the electrophoretic mobility and then using it to determine the diffusion coefficient, is also used to evaluate diffusion when the objects are too small to be measured optically, as is often the case in the study of enzymes.

• Gas and vapor adsorption, also referred to as BET (Brunauer–Emmett–Teller gas adsorption measurements),⁵⁴  make use of the different areas in a gas adsorption setup at low temperatures to evaluate properties such as surface area and pore sizes. In a typical measurement, the sample is cooled down to −196 °C and set under vacuum. An inert gas is flown over the sample with gas molecules initially adsorbing to isolated sites on the solid surfaces. With increasing gas pressure, they form a monolayer. Further increase in the pressure leads to multilayer adsorption and subsequent condensation. Plotting the volume of the adsorbed gas as a function of the gas pressure reveals the surface area of the solid sample.

• Further specific material properties such as intrinsic responses to external stimuli (light, magnetic fields etc.) can also be studied for colloidal materials using conventional techniques, with certain adaptations.

Brownian motion refers to the random movement of tiny particles suspended in a fluid or gas. Given its importance in the history of science, we want to start with a short anecdote about its discovery. Following the development of some of the first simple microscopes, the botanist Robert Brown discovered the random motion of particles when observing pollen in the 1820s. Given the general scientific framework at the time, he struggled to understand his observation.⁵⁵  Note that the existence of atoms and molecules had not been confirmed at that time, so possible explanations included curiosities ranging from the ether theory to the existence of a soul for pollen. It was only about 80 years later that Albert Einstein was able to explain what was already referred to as ‘Brownian motion’. In doing so, Einstein not only provided a coherent explanation for this phenomenon but also furnished compelling evidence for the existence of atoms themselves.^56,57  Brownian motion occurs due to collisions between the particles and the surrounding molecules, leading to their erratic motion. This phenomenon is driven by thermal energy and has applications in physics, chemistry, biology, and even finance. The discovery and subsequent developments of Brownian motion marked a significant leap in scientific understanding, offering key insights into particle behavior at the microscopic scale.

Translational Brownian motion refers to the random movement of particles suspended in a fluid due to collisions with the surrounding solvent molecules. This phenomenon is a result of the thermal energy possessed by particles at the molecular level. As the particles collide with the fluid molecules, they undergo erratic, unpredictable movements, leading to their diffusion throughout the fluid.

The phenomenon of translational Brownian motion, as described by the Stokes–Einstein relationship, is inherently linked to the process of diffusion. Fick’s First Law and Fick’s Second Law are fundamental principles in the field of diffusion and mass transfer. Together, they describe the diffusion process in a fluid.

• Fick’s first law was formulated by Adolf Fick in the mid-19th century and establishes a relationship between the rate of diffusion of a substance (D) and the concentration gradient of that substance within the medium with respect to distance (∇n). This means that substances will naturally move from areas of higher concentration to areas of lower concentration, and the rate of this movement is determined by the substance’s diffusion coefficient.

  $J → =−D∇n$

  (1.5)

  where J represents the flux of the substance (the amount of substance passing through a unit area per unit time).
• Fick’s second law describes how the concentration of a substance varies with time and space. It tells us that the rate of change in concentration is directly proportional to the diffusion coefficient and the curvature of the concentration gradient.

  $∂ n ∂ t =D ∇ 2 n$

  (1.6)

The behavior and stability of colloidal particles are influenced by the interplay of forces acting at the nanoscale. These forces dictate whether colloids will attract, repel, or remain suspended in a stable equilibrium. These behaviors result from the collective influence of various interaction forces, each exerting its unique effect on colloidal stability and dynamics. In the following sections, we will look at various electrostatic double layer models, van der Waals interactions and finally the DLVO theory. As we dissect these influences individually, it becomes evident that the stability and behavior of colloids emerge as a complex interplay of attractive and repulsive forces. However, it is essential to recognize the interdependence of these forces. The electrical double layer and van der Waals forces, while distinct, interact and overlap in their effects. The DLVO theory itself underscores the intricacies of these interactions, highlighting that a comprehensive understanding of colloidal behavior necessitates considering the combined influence of all contributing factors.

When a colloidal particle is added to an aqueous solution, it inadvertently becomes charged. This can be attributed to the preferential adsorption of ions from the solution to an uncharged surface or ionization/dissociation of surface groups on the particle’s surface.

Consequently, the charged state of the particle leads to the accumulation of charges in the form of a surface layer and the accumulation of counterions in the surrounding fluid. The entire system maintains overall electrical neutrality. The combined structure of these two layers of charges surrounding the colloidal particle is commonly denoted as the electrical double layer (EDL).

Understanding the EDL is essential for comprehending colloidal stability, particle aggregation, and the interactions between colloidal particles and other species in the solution. It influences the transport of particles and ions in colloidal systems, affecting phenomena such as electrophoresis and electrokinetics, which we will talk about in subsequent sections.

It is also vital for developing effective strategies for controlling and manipulating colloidal systems, with applications spanning fields like materials science, environmental engineering, and biotechnology.

Various models are used to explain EDL. We will start from the simplest model and gradually add complexity to it to get the full picture.

This is one of the simplest models to explain the electrostatic interactions. In this model, the counterions directly bind to the surface of the particle, such that they exactly counterbalance the charges on the surface of the particle, see Figure 1.4. In this sense, the double layer is compact and rigid and can be compared to a capacitor. However, this model is limited in its capability, and cannot explain most of the electrokinetic experiments.

The present depiction of the electrical double layer is deemed insufficient, as it considers the distribution of charges to be confined to two distinct planes. However, due to thermal motion of the molecules the ions cannot remain fixed and diffuse in the solution giving rise to a diffuse layer. In response to this dynamic behavior, the diffuse double layer model is introduced as a more accurate representation.

The diffuse double layer model provides a more comprehensive and realistic description of the electrical double layer compared to the classical Helmholtz model. The counterions in the electrolyte solution are not strictly confined to the immediate vicinity of the charged surface but are diffused in the solution, which extends to more than a molecular layer, see Figure 1.5. In this section, we will try to address questions such as what is the spatial distribution of ions and counterions in the solution? What is the thickness of the double layer?

• planar surface $ψ= ψ 0 exp(−kx)$

• spherical surface $ψ= ψ 0 R s r exp [ − k ( r − R s ) ]$

where, R_s represents the radius of the spherical particle, while r denotes the distance from the center of the particle to any point in the double layer. Here, κ⁻¹ (also denoted by λ_D) is known as the Debye length, given by eqn (1.20) and has units of length. λ_D is approximately 1 μm for water at room temperature. It characterizes the spatial extent of the electrostatic interactions within the double layer. The Debye length essentially represents the characteristic distance over which the electric potential decays to 1/e of its original value within the electrolyte.

Points to be noted:

• The Debye–Hückel approximation is strictly applicable only to the cases of ‘low potentials’. From eqn (1.14), for a monovalent ion that satisfies z_ieψ = k_BT at 25 °C, ψ becomes k_bT/e = 25.7 mV. Thus, a potential may be said to be low or high at 25 °C, if it is lower or higher than 25.7 mV.

• It is the limiting case to which all equations that are more general must reduce to at low potential.

The Gouy–Chapman–Stern (GCS) model encompasses both the static layer and diffuse model of ions. In the GCS model, the electrical double layer is divided into three regions: the Stern layer, the diffuse layer, and the bulk solution. The Stern layer, closest to the charged surface, consists of specifically adsorbed ions, while the ions are more spread out in a diffuse layer. Beyond the diffuse layer lies the bulk solution. One significant addition in the GCS model is the consideration of specifically adsorbed ions in the Stern layer. These ions are held tightly to the charged surface due to specific chemical interactions and do not contribute to the diffuse layer. While the GCS model simplifies certain aspects of the EDL, it forms the basis for more sophisticated models that consider factors like ion size, ion-specific effects, and surface roughness. Figure 1.6 shows a schematic of GCS model of EDL. The figure illustrates the Stern layer, diffuse layer and the bulk solvent.

One important parameter that has to be taken into account when talking about colloids is the zeta potential.

The zeta potential (ζ) is the electric potential at the shear plane or the slipping plane, which is the plane that separates the fixed charges on the surface of a colloidal particle from the mobile charges in the surrounding liquid medium. The magnitude of the zeta potential is typically measured in millivolts (mV). It directly reflects the net charge present on the particle’s surface and is a critical indicator of its electrostatic properties. It is a pivotal factor in colloidal stability. Particles with a high absolute zeta potential (either positive or negative) experience strong electrostatic repulsions among themselves, preventing aggregation and ensuring colloidal stability. Conversely, particles with low or near-zero zeta potential tend to aggregate due to attractive forces. Zeta potential can be determined using various techniques, including electrophoretic methods like dynamic light scattering, and phase analysis light scattering. Electrophoretic methods involve observing the particle’s movement under an applied electric field to calculate its electrophoretic mobility, from which the zeta potential is derived. Surface modifications, such as coating colloidal particles with charged molecules, changing the pH or ionic strength, or altering surface chemistry, can be used to control and manipulate the zeta potential. This can significantly impact the particle’s behavior, interactions, and applications. The zeta potential is relevant not only for colloidal suspensions but also for charged interfaces in complex fluids, such as emulsions, foams, and biological systems. It influences the stability, rheological behavior, and functionality of these systems. In industries like pharmaceuticals, food, and cosmetics, monitoring zeta potential is essential for quality control. Maintaining a specific zeta potential range ensures product stability, prevents sedimentation, and enhances shelf life.

In the previous section, we talked about electrostatic interactions in colloids. The stability and behavior of colloids are governed by a complex interplay of different forces, among which van der Waals (vdw) forces are particularly significant because of their ubiquitous nature. vdw forces are a subset of intermolecular forces that arise from the quantum mechanical fluctuations in the electron distribution around atoms and molecules. These fluctuations lead to transient electric dipoles, which induce dipoles in neighboring atoms or molecules, culminating in an attractive force. It is to be noted that vdw forces are intermolecular forces, i.e., forces acting between the molecules and are different from intramolecular forces that act between atoms inside the molecule (covalent, ionic, metallic).

Let us look into the different types of vdw forces with some examples:

• London dispersion: these forces exist between all atoms and molecules, regardless of polarity and are the weakest form of vdw interactions. These are the primary form of vdw forces in non-polar molecules. For example iodine exists as diatomic molecules in its elemental form. Each iodine molecule is made up of two iodine atoms connected by a covalent bond. The molecule itself is non-polar because the two atoms have equal electronegativity, resulting in no permanent dipole moment. However, the movement of electrons creates a temporary, fluctuating dipole moment. This induces a dipole in a neighboring iodine molecule separated by a distance d, leading to a weak, transient attraction between them

  $V=− C dispersion d 6 =− 2 3 α 1 α 2 I 1 I 2 ( I 1 + I 2 ) d 6$

  (1.23)

  where α₁ and α₂ are the polarizability volumes of the two molecules, and I₁ and I₂ are the ionization energies of the molecules respectively.
• Keesom force (dipole–dipole): these forces exist between polar molecules and are stronger than dispersion forces but weaker than hydrogen bonding. The interaction is characterized by the alignment of dipoles in such a way that the positive end of one molecule is close to the negative end of another. Consider a sample of HCl gas, each molecule consists of a hydrogen atom bonded to a chlorine atom. The chlorine atom is more electronegative, creating a permanent dipole in the HCl molecule. When multiple HCl molecules are close to each other, the positive end of one molecule (hydrogen) is attracted to the negative end of another molecule (chlorine). This permanent dipole–dipole interaction is known as a Keesom force. For two molecules separated by a distance d, electric dipole moments µ₁ and µ₂, the interaction at a temperature T is given by:

  $V=− C Keesom d 6 =− μ 1 2 μ 2 2 3 ( 4 π ε 0 ε r ) 2 k B T d 6$

  (1.24)
• Debye forces (dipole–induced dipole): these interactions occur between a polar and a non-polar molecule. Water is a polar molecule with a permanent dipole moment due to the difference in electronegativity between oxygen and hydrogen atoms. On the other hand, oxygen is a non-polar molecule because it consists of two oxygen atoms with equal electronegativity, resulting in no permanent dipole. When oxygen gas comes into contact with water, as in aquatic environments or in a glass of water left exposed to air, the permanent dipole in the water molecules can induce a temporary dipole in the oxygen molecules. This dipole–induced dipole interaction, or Debye force, allows oxygen molecules to be somewhat soluble in water. This solubility is crucial for the survival of aquatic life, as it provides the necessary oxygen for respiration. In this example, the Debye forces between the water’s permanent dipole and the induced dipole in the oxygen molecules facilitate the dissolution of oxygen in water, a phenomenon of significant biological and environmental importance

  $V=− C Debye d 6 =− μ 2 α ( 4 π ε 0 ε r ) 2 d 6$

  (1.25)

Now we can compare the order of energy for various forces desribed as:

• van der Waals forces: generally ≤kT

• Hydrogen bonding: often >kT

• Intramolecular forces (covalent, ionic): ≫kT

Since, the order of forces for vdw interactions is less than kT and scales very strongly with distance, why should we bother with it in the case of colloids? The strength of vdw forces becomes more significant (≈100kT) as the molecular forces scale up in the case of colloids. This is mainly because each of these particles contains a large number of atoms or molecules.

When molecules are far apart, a time lag or phase difference occurs between their vibrations, similar to light scattering by large particles. This ‘retardation’ effect becomes significant when the separation between particles is of the order of the wavelength of the propagating field (Table 1.2).

While effective for short-range interactions, Hamaker theory does not account for these retardation effects, which become significant at larger distances.

Lifshitz theory: this theory was developed by E. M. Lifshitz in the 1950s and provides a more comprehensive framework that incorporates quantum mechanical and electromagnetic effects. It is especially useful for long-range interactions. This theory considers the fluctuating electromagnetic fields arising from quantum mechanical fluctuations in the electron cloud. Unlike Hamaker theory, the Lifshitz approach accounts for the finite speed of light, which leads to changes in the force at large distances and also incorporates the dielectric properties of the materials and the intervening medium, providing a more versatile framework for different systems.

Both Hamaker theory and the Lifshitz approach have their own merits and limitations. While Hamaker theory provides a simpler, albeit less accurate, description suitable for short-range interactions, the Lifshitz approach offers a comprehensive and accurate framework for long-range interactions. Understanding these theories is crucial for predicting and manipulating the behavior of colloidal systems, from stabilizing colloidal dispersions to designing advanced materials with tailored properties.^58,59 

• Primary minimum: at very short distances, the attractive van der Waals forces are dominant, and this strong attraction leads to the formation of a primary minimum in the potential energy curve. This means that particles can reach a stable state of attraction, facilitating aggregation or even coalescence. It is typically the strongest attractive minimum and can lead to irreversible aggregation if particles cross this energy barrier.

• Energy barrier: at intermediate distances, if the particles carry a charge, the electrostatic repulsion due to the overlapping of electric double layers can dominate, creating an energy barrier. This barrier can prevent particles from coming closer and aggregates from forming.

• Secondary minimum: a secondary minimum can also form where van der Waals attraction competes with the weaker electrostatic repulsion, leading to a less stable, shallow minimum compared to the primary minimum.

• Beyond secondary minimum: at even larger distances, the electrostatic repulsion diminishes, and the vdw attraction dominates again, leading to a gradual decrease in potential energy.

The DLVO potential is affected by a number of factors, including the charge of the colloidal particles, the ionic strength of the solution, the Hamaker constant of the particles, the size of the particles, and the temperature of the solution to name a few.

• The charge of the colloidal particles: the greater the charge of the particles, the stronger the electrostatic repulsion between them and the more stable the dispersion will be.

• The ionic strength of the solution: the ionic strength of the solution is a measure of the concentration of ions in the solution. A higher ionic strength will compress the electrical double layer around the particles, which will reduce the electrostatic repulsion between them and make the dispersion less stable.

• The Hamaker constant of the particles: the Hamaker constant is a measure of the strength of the van der Waals attraction between the particles. A higher Hamaker constant will result in a stronger van der Waals attraction between the particles and make the dispersion less stable.

• The size of the particles: smaller particles will have a larger surface area to volume ratio, which means that they will have a higher charge density. This will result in a stronger electrostatic repulsion between the particles and make the dispersion more stable.

• The temperature: temperature affects the thickness of the electrical double layer, the strength of the van der Waals attraction, and the mobility of the ions in the solution.^60–62 

A thorough understanding of DLVO interactions has resulted in numerous practical applications in various fields including pharmaceuticals and drug delivery, food and beverage industry, cosmetics and personal care, paints and coatings to name a few.

The stability of colloidal particles refers to their ability to remain dispersed and evenly distributed within a medium, without the particles aggregating, settling, or separating over time (see Figure 1.8). When a colloid becomes unstable, the particles start to aggregate, leading to phenomena such as flocculation or coagulation, where particles clump together to form larger clusters and sediment.

The DLVO theory, as we introduced in Section 1.3.3, provides a comprehensive explanation for the stability of colloids by considering the interplay between electrostatic repulsion and van der Waals attraction. According to DLVO theory, the total potential energy of interaction between colloidal particles is the sum of the attractive van der Waals forces and the repulsive electrostatic forces. The theory predicts the existence of an energy barrier that prevents particles from coming close enough to aggregate under normal conditions. However, if the energy barrier is overcome, for instance, by altering the ionic strength of the solution or the pH, particles can aggregate.

Flocculation involves the aggregation of colloidal particles into larger flocs, initiated by the formation of doublets, which is a phenomenon called diffusion-limited aggregation (DLA) or Brownian aggregation. DLA occurs in conditions where repulsive forces between colloidal particles are minimal or absent, allowing individual particles to collide and adhere together due to Brownian motion, resulting in the formation of dimers.

The rate of DLA can be significantly influenced by the medium’s properties and the particles themselves. For instance, in environments with high salt concentrations, the Debye length, which measures the electrostatic screening effect, is reduced. This reduction in Debye length decreases the repulsive forces between charged particles, making them more prone to coming into contact with each other and aggregating. Similarly, particles with very small zeta potentials exhibit weaker electrostatic repulsion, which facilitates their aggregation as they are less capable of overcoming the attractive van der Waals forces.

Besides DLVO, steric stabilization and depletion forces are two mechanisms that significantly influence the stability of colloidal systems, each acting through different physical interactions.

Steric stabilization is achieved by adsorbing polymers onto the surfaces of colloidal particles, forming a physical barrier that extends outward and prevents the particles from coming into direct contact and aggregating. This polymer ‘brush’ is particularly effective in environments where electrostatic repulsion is reduced, such as in non-polar solvents or under high ionic strength conditions, thereby ensuring the colloids remain dispersed.

On the other hand, depletion forces come into play in mixtures of larger colloidal particles with smaller molecules or polymers. These smaller species create a zone of lower concentration around the larger colloidal particles due to their exclusion from these regions, leading to an osmotic pressure difference. When the depletion zones of two particles overlap, this pressure difference drives the particles together, promoting aggregation through an attractive force.

The study of colloidal systems requires a thorough understanding of their behavior under various conditions, including hydrodynamics. Colloidal particles are subject to Brownian motion, thermal fluctuations, and hydrodynamic interactions. The dynamics of colloidal suspensions are governed by fundamental equations of fluid mechanics, among which the Navier–Stokes equation plays a central role. We will see how the flow at such small dimensions leads to viscous forces dominating the inertial forces and how that leads to the Stokes equation, which is linear.

We began our journey into the world of colloids by establishing foundational definitions, introducing the intricate processes underlying synthetic strategies and the pivotal role of various characterization techniques, from SEM to zetasizer. Then we explored the Brownian motion and underlined its significant influence on colloidal behavior, distinguishing the nuances of both translational and rotational facets. The realm of interaction forces brought forth the complexities of electrostatic interactions, from the Helmholtz model to the more encompassing Gouy–Chapman–Stern model. Furthermore, the intricate dance of colloidal dynamics, governed by hydrodynamics and epitomized by Stokes law and phoretic motion, shed light on the very essence of colloidal movement.

Colloidal systems, especially active colloids, hold immense promise and relevance in today’s scientific landscape. Firstly, the potential applications of active colloids in various sectors, notably medicine, underscore their significance. These colloids offer innovative solutions for targeted drug delivery, diagnostic mechanisms, and even therapeutic interventions, presenting a revolutionary frontier in healthcare and biotechnology. Secondly, beyond their direct applications, active colloids serve as an invaluable model system for understanding physical interactions and behaviors. Their unique dynamics, interactions, and responses provide researchers with a tangible and observable platform to decode, analyze, and predict the intricate dance of physical forces in microscopic realms.

Armed with this foundational knowledge, we are poised to delve deeper into the riveting domain of active colloids. As we transition into the ensuing chapters, anticipate a richer, more intricate exploration of these dynamic entities – from their synthesis to their diverse behaviors and potential applications.

• Colloidal particles in aqueous solutions become charged, forming an electrical double layer.

• High zeta potential indicates strong electrostatic repulsions, ensuring colloidal stability.

• van der Waals forces result from transient electric dipoles due to electron fluctuations.

• The Hamaker constant measures the strength of vdw interactions, crucial for predicting colloidal stability.

• DLVO theory combines electrostatic and vdw forces to understand charged particle interactions in liquids.

• Navier–Stokes equations describe fluid motion in relation to velocity, pressure, and temperature.

• For colloids, viscous forces dominate over inertial forces due to a very small Reynolds number.

• Colloidal stability refers to particles’ ability to stay dispersed without agglomerating.

All authors acknowledge a the Volkswagen foundation for a Freigeist grant number 91619. JS is grateful for a Fulbright Cottrell Award, ZX appreciates a CSC fellowship for his PhD and we thank Wei Wang and William Uspal for careful proof-reading.