Capillary effects play a crucial role in colloidal systems, influencing particle behavior and organization. These forces arise from interactions between liquid-gas interfaces and solid surfaces, impacting stability and assembly of colloids.

Understanding capillary forces is key to controlling colloidal systems. From capillary rise and condensation to particle interactions and assembly, these effects have wide-ranging applications in microfluidics, particle synthesis, and separation technologies.

Capillary forces in colloidal systems

• Capillary forces play a crucial role in the behavior and organization of colloidal systems
• These forces arise from the interaction between the liquid-gas interface and the solid surfaces of the colloidal particles
• Understanding capillary forces is essential for controlling the stability, assembly, and properties of colloidal systems

Origin of capillary forces

• Capillary forces originate from the surface tension of the liquid and the curvature of the liquid-gas interface
• Surface tension is the result of the imbalance of intermolecular forces at the liquid surface, causing the liquid to minimize its surface area
• The curvature of the liquid-gas interface is determined by the pressure difference across the interface, known as the Laplace pressure

Laplace pressure

• Laplace pressure is the pressure difference between the inside and outside of a curved liquid-gas interface
• It is caused by the surface tension acting along the curved surface
• The Laplace pressure is given by the Young-Laplace equation, which relates the pressure difference to the surface tension and the curvature of the interface

Young-Laplace equation

• The Young-Laplace equation describes the relationship between the Laplace pressure ($\Delta P$), surface tension ($\gamma$), and the principal radii of curvature ($R_1$ and $R_2$) of the liquid-gas interface
• The equation is given by: $\Delta P = \gamma (\frac{1}{R_1} + \frac{1}{R_2})$
• For a spherical interface, the equation simplifies to $\Delta P = \frac{2\gamma}{R}$, where $R$ is the radius of the sphere

Contact angle

• The contact angle is the angle formed between the liquid-gas interface and the solid surface at the point of contact
• It is a measure of the wettability of the solid surface by the liquid
• The contact angle is determined by the balance of the interfacial tensions between the solid, liquid, and gas phases, as described by Young's equation: $\gamma_{SG} = \gamma_{SL} + \gamma_{LG} \cos \theta$

Wetting vs non-wetting

• Wetting refers to the ability of a liquid to spread on a solid surface, characterized by a contact angle less than 90°
• Non-wetting occurs when the liquid forms a droplet on the solid surface, with a contact angle greater than 90°
• The wettability of a surface depends on the relative strengths of the intermolecular interactions between the solid, liquid, and gas phases (hydrophilic vs hydrophobic surfaces)

Capillary rise in colloidal systems

• Capillary rise is the phenomenon where a liquid is drawn upwards against the force of gravity in a narrow space, such as a thin tube or between colloidal particles
• It is a consequence of the capillary forces arising from the surface tension and the curvature of the liquid-gas interface
• Capillary rise is important in understanding the behavior of colloidal systems, such as the wetting of powders, the flow of liquids in porous media, and the assembly of colloidal particles

Capillary rise in tubes

• In a thin tube, the liquid rises to a height where the capillary force balances the gravitational force
• The height of the capillary rise depends on the radius of the tube, the surface tension of the liquid, and the contact angle between the liquid and the tube wall
• The capillary rise in tubes is described by Jurin's law

Jurin's law

• Jurin's law relates the height of the capillary rise ($h$) to the radius of the tube ($r$), the surface tension of the liquid ($\gamma$), the contact angle ($\theta$), the density of the liquid ($\rho$), and the acceleration due to gravity ($g$)
• The equation for Jurin's law is: $h = \frac{2\gamma \cos \theta}{\rho g r}$
• This law assumes that the tube is sufficiently narrow and the liquid wets the tube wall (contact angle < 90°)

Capillary rise between particles

• Capillary rise also occurs between colloidal particles, where the liquid is drawn into the narrow spaces between the particles
• The height of the capillary rise between particles depends on the size and shape of the particles, the packing density, and the surface properties of the particles
• Capillary rise between particles is important in understanding the cohesion and flow properties of wet granular materials (sand castles, soil)

Capillary condensation

• Capillary condensation is the phenomenon where a vapor condenses into a liquid in a confined space, such as a pore or a capillary, at a pressure lower than the saturation vapor pressure
• It occurs when the curvature of the liquid-gas interface in the confined space leads to a decrease in the vapor pressure, as described by the Kelvin equation
• Capillary condensation is important in understanding the adsorption and desorption of vapors in porous materials, as well as the stability of colloidal systems

Kelvin equation

• The Kelvin equation relates the vapor pressure over a curved liquid-gas interface ($p$) to the saturation vapor pressure over a flat surface ($p_0$), the surface tension of the liquid ($\gamma$), the molar volume of the liquid ($V_m$), the radius of curvature of the interface ($r$), the ideal gas constant ($R$), and the temperature ($T$)
• The equation is given by: $\ln \frac{p}{p_0} = -\frac{2\gamma V_m}{rRT}$
• The Kelvin equation predicts that vapor condensation occurs at a lower pressure in smaller pores or capillaries

Capillary condensation in porous media

• Porous media, such as silica gel or activated carbon, have a large surface area and a network of pores with various sizes and shapes
• Capillary condensation in porous media leads to the formation of liquid bridges between the particles, which can affect the mechanical and transport properties of the material
• The adsorption and desorption of vapors in porous media exhibit hysteresis, due to the different pressures required for condensation and evaporation in the pores (ink-bottle effect)

Capillary interactions between particles

• Capillary interactions arise between colloidal particles when they are partially immersed in a liquid or when liquid bridges form between them
• These interactions can be attractive or repulsive, depending on the wetting properties of the particles and the geometry of the liquid-gas interface
• Capillary interactions play a crucial role in the stability, aggregation, and assembly of colloidal particles at liquid interfaces

Lateral capillary forces

• Lateral capillary forces act between particles that are partially immersed in a liquid, causing them to attract or repel each other
• These forces arise from the deformation of the liquid-gas interface around the particles, which minimizes the total interfacial energy
• Lateral capillary forces can lead to the self-assembly of particles into ordered structures at liquid interfaces (Cheerios effect, floating particles)

Immersion capillary forces

• Immersion capillary forces act between particles that are completely immersed in a liquid, but close to a liquid-gas interface
• These forces are caused by the curvature of the liquid-gas interface between the particles, which leads to a pressure difference and an attractive force
• Immersion capillary forces are important in the stability and rheology of particle-stabilized emulsions and foams

Flotation capillary forces

• Flotation capillary forces act between particles that are trapped at a liquid-gas interface, but not in direct contact with each other
• These forces arise from the deformation of the liquid-gas interface around the particles, which can lead to attractive or repulsive interactions, depending on the wetting properties and the surface chemistry of the particles
• Flotation capillary forces are exploited in the separation of minerals by froth flotation, where hydrophobic particles are selectively attached to air bubbles and separated from hydrophilic particles

Capillary assembly of colloidal particles

• Capillary assembly is a powerful technique for organizing colloidal particles into ordered structures using capillary forces
• It relies on the controlled evaporation of a liquid suspension of particles, which leads to the formation of capillary menisci between the particles and drives their assembly
• Capillary assembly can be used to create 2D or 3D structures with various symmetries and functionalities, depending on the size, shape, and surface properties of the particles

Capillary self-assembly

• Capillary self-assembly is a spontaneous process where particles organize themselves into ordered structures driven by capillary forces
• It occurs when a liquid suspension of particles is allowed to evaporate on a substrate, leading to the formation of capillary menisci between the particles and their close packing
• Capillary self-assembly can be used to create colloidal crystals, photonic structures, and patterned surfaces (coffee ring effect, colloidal lithography)

Directed capillary assembly

• Directed capillary assembly is a guided process where the assembly of particles is controlled by external fields, templates, or surface patterns
• It allows for the precise positioning and orientation of particles into desired structures, beyond the limits of spontaneous self-assembly
• Directed capillary assembly can be achieved by using magnetic or electric fields, patterned substrates, or microfluidic channels (capillary micromolding, field-assisted assembly)

2D vs 3D capillary assembly

• Capillary assembly can be used to create both 2D and 3D structures, depending on the geometry of the liquid-gas interface and the confinement of the particles
• 2D capillary assembly occurs when particles are trapped at a planar liquid-gas interface, such as at the surface of a drying droplet, leading to the formation of monolayer structures (colloidal monolayers, Langmuir-Blodgett films)
• 3D capillary assembly occurs when particles are assembled in the bulk of a liquid or in a confined space, such as in a drying colloidal suspension or a capillary bridge, leading to the formation of multilayer or three-dimensional structures (colloidal crystals, inverse opals)

Applications of capillary effects

• Capillary effects have numerous applications in various fields, ranging from microfluidics and particle synthesis to separation and assembly
• These applications rely on the ability to control and manipulate the flow of liquids and the organization of particles using capillary forces
• Understanding and harnessing capillary effects is crucial for developing new technologies and improving existing processes in colloidal science and engineering

Capillary-based microfluidics

• Capillary-based microfluidics utilizes capillary forces to drive the flow of liquids in microchannels without the need for external pumps or pressure sources
• It relies on the wetting properties of the channel walls and the geometry of the liquid-gas interface to control the flow rate and direction
• Capillary-based microfluidics is used in various applications, such as point-of-care diagnostics, drug delivery, and chemical synthesis (paper-based microfluidics, capillary electrophoresis)

Capillary-based particle separation

• Capillary-based particle separation exploits the differences in the capillary interactions between particles of different sizes, shapes, or surface properties to selectively separate them from a mixture
• It can be achieved by using capillary forces to drive the particles through a porous medium, a microfluidic channel, or a liquid-gas interface
• Capillary-based particle separation is used in various applications, such as mineral processing, environmental remediation, and biological assays (capillary chromatography, froth flotation)

Capillary-based particle synthesis

• Capillary-based particle synthesis uses capillary forces to control the nucleation and growth of particles from a precursor solution
• It relies on the confinement of the precursor solution in a capillary space, such as a porous template or a microfluidic channel, to control the size, shape, and composition of the particles
• Capillary-based particle synthesis is used to create various functional materials, such as catalysts, adsorbents, and drug delivery vehicles (mesoporous silica, core-shell particles)