¶ Macroscopic Description of Fluid Interfaces
¶ Surface Excess
Some solutes may not be uniformly distributed throughout a solution, as they can either accumulate at or avoid the interface
- In order to discuss this phenomenon in a quantitative manner, the concept of surface excess is introduced
Surface excess is a quantitative measure of the amount of a substance that accumulates at the interface, in excess of what would be expected based on its bulk concentration
- For a solute that accumulates at the surface, the surface excess is positive
- For a solute that avoids the surface, the surface excess is negative
We can write something similar for Gibbs energy as well
- Since the surface has no volume, we can express the differential of the surface Gibbs Energy as such at constant temperature
- The Euler's theorem allow us to write down , so its total differential is
- Equating both expressions of gives us the Gibbs-Duhem equation
- We shall define the surface excess per unit area as , the surface concentration
This equation can be interpreted as follows
- If adding more of solute (raising its chemical potential) decreases the surface tension, then is positive and solute will accumulate at the interface
- If adding more of solute (raising its chemical potential) increases the surface tension, then is negative and solute will avoid the interface
Similarly, we can express for a solute as a rate of change of surface tension with respect to its chemical potential
- We also refer to as the excess adsorption of a solute
¶ Dividing Surface
The interface between two phase is usually not a surface with zero thickness, in the sense that the surface is composed of one layer of molecules from each phases only
- The density profile is a measure of the distribution of molecules across the interfacial layer between two phases
- It provides information on how the density of the molecules changes as a function of distance from the interface
- The system under consideration can consist of either a liquid and a vapor phase, or two liquid phases
We define the location of the interface using the Gibbs Dividing Surface
- This conceptual boundary is chosen to be at a location such that the excess adsorption of a solvent at the interface, denoted as
The mass balance condition can be defined mathematically as
- The equation says that the excess mass in the region where the density transitions from to is zero
- In other words, the Gibbs dividing surface is placed such that the total "extra" mass on one side of the surface cancels out the "missing" mass on the other side.
To find the precise location for this dividing surface, we use
- is the gradient of the density across the interface, describing how the density changes with position.
- The integral gives a weighted average of the position , where the weight is the density gradient. This effectively gives the center of mass of the density transition across the interface.
- The factor normalizes this weighted average with respect to the difference in bulk densities of the two phases.
¶ Curved Interfacial Surface
To quantitatively describe curvature of a surface about a point, we shall do the following
- We draw a line that is normal to the surface of that point
- Randomly draw a circle ( circle 1 ) that has its radius aligned with the line and its perimeter just tangental to the surface
- Draw another circle ( circle 2 ) that also fit the same criteria but is also orthogonal to circle 1
The principle curvatures at that point and are defined as reciprocals of the radii of circle 1 and 2:
To define surface curvatures, we have to define two different curvatures
Mean curvature
It describes how the membrane bends in a local region, averaging the curvatures in orthogonal directions
- For small deviations from a plane, we can approximate it to
- Moreover, the pressure difference across the interface is also dependent on
Gaussian curvature
It measures how the membrane deforms globally, such as whether it twists or forms saddles.
- For small deviations from a plane, we can approximate it to
- The sign of the Gaussian curvature characterises the nature of a surface
Tolman Length
Surface tension is typically considered a constant for flat interfaces, but for curved interfaces, it can vary
- The Tolman equation describes how surface tension depends on the curvature of the interface:
- The Tolman length is a correction factor accounting for curvature, typically in the range of 0.01-0.1 nm
For small droplets or bubbles (on the nanometer scale), this curvature correction becomes significant
- As decreases, the surface tension decreases compared to the planar limit
¶ Capillary Wave Theory
Thermal fluctuations at interfaces cause small undulations, or "waves," that modify the interface's shape and thickness
- These fluctuations are particularly important in systems where surface tension and temperature play major roles in determining the interface's stability
- Capillary Wave Theory (CWT) is a powerful framework used to describe these undulations and predict the behavior of fluctuating interfaces.
The term "capillary waves" refers to undulations on the surface of a fluid that arise due to thermal fluctuations
- The wavelength of a capillary wave can vary from molecular scales (nanometers) to millimeters, with larger wavelengths corresponding to lower-energy fluctuations.
- The wavevector is given by q=\frac{2\pi}
- The position of the interface at any horizontal position is denoted as
While a perfectly flat interface might have a sharp boundary, fluctuations smear out this boundary, leading to a finite interfacial thickness.
- The thickness is given by the root mean square :
Spectrum of Capillary Waves
A key feature of CWT is that it predicts a continuous spectrum of waves, each contributing to the overall undulation of the interface.
- The energy associated with each wave depends on its wavelength and surface tension .
- Energy of a fluctuation decreases as the surface tension increases, and also decreases for longer wavelengths.
Free Energy and Capillary Waves
The free energy of an interface is minimized when it remains flat.
- Thermal fluctuations increase the free energy by introducing small deviations from the planar state
For small perturbations from a flat interface, the Hamiltonian (energy function) for capillary waves is given by:
- The free energy due to capillary is given by the difference in surface free energy of the undulated and flat state
Main result of the capillary wave theory
One of the most important results of CWT is that interface thickness increases with the system size
is the mean-square thickness
- When the its value is large,it indicates greater undulations and a thicker, more diffuse interface.
relates the magnitude of the thermal fluctuations to both temperature and surface tension.
- Higher temperatures increase the fluctuations because more thermal energy is available to disturb the interface.
- Higher surface tension dampens the fluctuations, as it acts to resist changes in surface area and restore the interface to its equilibrium state.
The logarithmic factor shows that the mean-square displacement grows as the system size increases relative to the microscopic cutoff length
- is the characteristic length scale of the system (e.g. the lateral size of the interface). The longer the lateral size, the greater the contribution of long-wavelength capillary waves to the overall fluctuations. In large systems, more fluctuations across different scales can occur, leading to a thicker interface.
- is a microscopic cutoff length, often taken as the molecular size or correlation length. It is needed to avoid the equation predicting unphysical behavior at very small scales (). This cutoff represents a lower limit on the length scales over which continuum theories (like Capillary Wave Theory) are valid.
Gravitational Cut-off for Capillary Waves
While the interface thickness theoretically increases without bound with increasing system size, gravity imposes an upper limit on the amplitude of the fluctuations
- This upper limit is governed by the capillary length , which is the length scale beyond which fluctuations are damped by gravitational forces
- (or ) defines the largest wavelength of capillary waves that can be supported by the system before gravity starts to significantly damp them.
- is the density difference between the two phases
Given that gravity imposes an upper limit, we must then modify our previous equation to
- Thus, the largest fluctuations that can exist are of the order of , preventing unbounded growth of the interface thickness.
- This equation accounts for the finite thickness of the interface, limited by gravity and molecular interactions.
¶ Fluctuations of Biological Membranes
Biological membranes are fluid and can deform in response to various forces.
- Membranes are assumed to be in a tensionless state, meaning that there is no external force causing them to stretch or compress.
- To achieve the equilibrium condition (tensionless state), we require that the free energy be at a minimum with respect to changes in the membrane are
Membrane Fluctuations
Membranes fluctuate due to thermal energy. These fluctuations involve undulations and bending of the membrane surface
- Unlike solid surfaces, membranes can undergo bending without significant stretching.
- A parameter for describing of a membrane's resistance to bending is the bending modulus (). A lower value implies the membrane is more flexible.
The Helfrich bending energy model describe the energy cost associated with bending a biological membrane
- The first term, , describes the energy cost due to deviations from the membrane's intrinsic curvature the membrane adopts without any external forces ().
- The second term, , represents the Gaussian curvature energy, which is concerned with the membrane's topological deformations (e.g. how the membrane bends and twists).
Symmetric Membrane and Fourier Transform
The bending energy of a symmetric membrane (with no spontaneous curvature, () is given by
- However, solving this equation directly is complex because of the spatial variations in . To simplify the problem, we express the membrane's thickness function in Fourier space.
In Fourier space, the Laplacian becomes multiplication by , meaning that:
- The bending energy depends on the amplitude of the fluctuations and increases rapidly with the wave vector (because of the term)
- Short-wavelength fluctuations (large ) are much more costly in terms of energy than long-wavelength fluctuations (small ).
Thermal Fluctuations and Equipartition of Energy
Thermal energy causes the membrane to fluctuate, and each wave mode receives a share of the thermal energy due to the equipartition theorem
- According to this principle, each mode contributes:
- The amplitude of the fluctuations decreases with , meaning that large wave vectors (small-wavelength fluctuations) are suppressed by the bending energy
- Larger wavelengths (small ) dominate the fluctuations because they incur a smaller energy cost.
To compute the total mean-square thickness of the membrane, we sum (or integrate) the contributions from all wave modes. This can be done by integrating the mean-square amplitude over all possible wave vectors:
- Hence, the thickness of the membrane grows with the membrane area () and scales inversely with the bending modulus ()
- In other words, larger membranes exhibit more significant overall fluctuations, and stiffer membranes have smaller fluctuations.
Bending Modulus: Flicker Spectroscopy and Simulations
The can be experimentally determined using methods such as flicker spectroscopy
- It measures the spontaneous fluctuations of a membrane in thermal equilibrium
The mean-square displacement of these fluctuations can be related to the bending modulus and the temperature
- Longer-wavelength fluctuations (small ) have larger amplitudes, but are damped by the membrane's bending rigidity.
¶ Wetting and Capillarity
¶ Interface Potential and Wetting Layer Thickness
The interface potential is a function of the wetting layer thickness , where is the thickness of the liquid film on the solid substrate
- The interface potential represents the free energy per unit area required to create and sustain a wetting layer of thickness .
In wetting theory, the wetting layer thickness describes how a liquid spreads over a solid surface
- The film starts with a thin layer, referred to as the precursor film, which has a thickness denoted by the adsorption thickness ( ). The precursor film spreads out uniformly across the surface and is much thinner than the height of the droplet.
- As we move along the surface, increases until it reaches the full droplet thickness.
The interface Potential depends on the relative size of and
- For small deviations of from (i.e. when ), the interface potential can be approximated as:
- For much larger wetting layers (), the interface potential is influenced by several factors, and the approximation becomes:
- This more general expression accounts for various types of forces acting on the wetting layer as its thickness increases
Contributions to the Interface Potential
Van der Waals forces act between the liquid and the substrate. The form of depends on whether the forces are non-retarded (dominant at smaller distances) or retarded (dominant at larger distances):
- Non-retarded van der Waals forces: These are dominant when the wetting layer thickness is small ( ) and can be approximated as:
- Retarded van der Waals forces: For larger thicknesses ( ), retardation effects become important, and the potential decays more rapidly:
- The transition from to occurs due to retardation effects, which weaken the interaction over larger distances as the speed of light influences how these forces propagate.
In systems where electrostatic interactions are present (e.g. ionic liquids, solutions with surface dissociation), Coulombic forces contribute to the interface potential.
- These forces are screened by the presence of dissolved ions, and the screening length ( ) determines how far the electrostatic interactions extend.
- The Coulombic potential is given by:
- Electrostatic forces decay exponentially as the distance increases, with the characteristic length scale determined by the screening effects (e.g., ionic strength in the solution).
¶ Wetting and Pre-Wetting Transitions
The equilibrium wetting layer thickness ( ) corresponds to the point where the system finds mechanical equilibrium for the wetting layer ( i.e. is minimized at that point )
- Since , performing the differentiation will give us
- As , the wetting layer thickness , meaning that the system transitions into a regime of complete wetting where the liquid forms a continuous film on the surface.
The spreading coefficient quantifies the driving force for the liquid to spread across the surface
- If , the liquid completely wets the surface.
- If , the liquid forms droplets on the surface (partial wetting).
- as and
Moreover, wetting transition occurs when
- At this point, the interfacial potential equals the original solid-vapor surface tension, meaning the solid is effectively "replaced" by the liquid, leading to the formation of a thick liquid film on the surface.
The Interfacial Phase Diagram
However, does not increase smoothly as the temperature drop, indicating the existence of phase transitions
- Complete wetting occurs when the liquid fully coats the surface, and the thickness of the film becomes large, often referred to as
- Prewetting is a first-order phase transition (involves a discontinuous jump in the film thickness) that can occur when the system forms a thin liquid film on the surface, even below the wetting transition temperature
The interfacial phase diagram is a tool used to describe the behavior of surfaces at different temperatures and compositions.
- Prewetting Line: Marks the temperature and chemical potential where a thin film starts to form on the surface. Below this line, the surface is essentially "dry."
- Wetting Transition Line: The temperature where the liquid completely wets the surface, forming a thick or infinite film.
- Surface Critical Point: Where the distinction between phases at the surface vanishes, leading to critical fluctuations in the wetting film.
¶ Three-Phase Line and Line Tension
In the wetting of surfaces, the balance of forces at the three-phase contact line determines the shape and stability of droplets or thin films. These forces include:
- Surface Tension (): The force acting along the surface of a liquid, which minimizes the surface area.
- Line Tension (): The force acting along the three-phase contact line, contributing to the wetting behavior at very small scales.
Free Energy and Line Tension
The total free energy of the system can be written in differential form, incorporating contributions from temperature (), volume (), chemical potential (), surface area (), and the length of the three-phase contact line ():
- represents the contribution of line tension to the system's free energy, where is the force per unit length along the three-phase contact line.
Line tension length ( ) is a characteristic length scale that relates the **line tension ( ) to the surface tension ( ). It provides a measure of how significant line tension effects are compared to surface tension:
- has units of meters (m), and it represents the distance over which line tension significantly affects the wetting behavior.
- When , line tension effects are important.
- For larger droplets, where , the influence of line tension becomes negligible, and surface tension dominates.
The theoretical value of line tension ( ) can be estimated using thermal energy ( ) and a characteristic molecular size ( )
Near Wetting Transitions
Near wetting transitions (when and ), the influence of line tension becomes particularly significant
- The characteristic length scale ( ) associated with line tension near wetting transitions is given by the ratio of the line tension and spreading coefficient:
- As , the approaches the micrometer scale.
- This suggests that as the contact angle decreases (near the point of complete wetting), line tension effects can become important over larger length scales
Contact angle dependence with line tension
The free energy change of the system is given by:
- is the height of the droplet
- is the radius of the droplet's base (contact line radius).
The Young-Laplace equation describes how the contact angle is influenced by surface tensions. However, when line tension is considered, a modified form is used:
- : Three phase line contracts + Doplet beads up
- : Three phase line expands + Doplet spreads out
Temperature plays a role in determining the sign and magnitude of line tension, which in turn affects the contact angle.