NE 352: Surfaces and Interfaces
Michael Pope
Estimated study time: 1 hr 28 min
Table of contents
Sources and References
Primary textbook — H.-J. Butt, K. Graf & M. Kappl, Physics and Chemistry of Interfaces, 3rd ed., Wiley-VCH, 2013. Reference texts — J.N. Israelachvili, Intermolecular and Surface Forces, 3rd ed., Academic Press, 2011; D.F. Evans & H. Wennerström, The Colloidal Domain, 2nd ed., Wiley-VCH. Online resources — MIT OCW 10.467 Polymer Science Laboratory notes; Stokes & Evans Fundamentals of Interfacial Engineering.
Chapter 1: Surface Tension and Capillarity
1.1 The Origin of Surface Tension
Every molecule in the bulk of a liquid is surrounded by neighbours in all directions, and the net intermolecular force it experiences averages to zero. A molecule at the liquid–vapour interface, however, has neighbours on only one side: the liquid side. The imbalance of cohesive forces pulls the surface molecule inward, effectively forcing the interface to contract. To increase the surface area, work must be done against this inward pull, which is precisely what the concept of surface tension quantifies.
The SI unit is J m\(^{-2}\), equivalently N m\(^{-1}\). For water at 25 °C, \( \gamma \approx 72 \) mN m\(^{-1}\).
At the microscopic level, surface tension arises from the same van der Waals and hydrogen-bonding forces that hold liquids together. Because hydrogen bonds in water are exceptionally strong (~20 kJ mol\(^{-1}\)), water has one of the highest surface tensions of any common liquid. Liquid metals, held by metallic bonding, can reach several hundred mN m\(^{-1}\). Organic solvents with only London dispersion forces typically lie in the 20–30 mN m\(^{-1}\) range.
Thermodynamically, surface tension decreases with increasing temperature because thermal agitation weakens the cohesive forces. Near the critical temperature \( T_c \), \( \gamma \to 0 \) because the liquid and vapour become indistinguishable.
1.2 Pressure Across a Curved Interface: The Young-Laplace Equation
When an interface is curved, there is a pressure difference across it. Intuitively, a bubble of air in water must have higher internal pressure to prevent collapse, just as the tension in a balloon skin supports a pressure difference. The precise relationship is given by the Young-Laplace equation.
For a sphere of radius \( R \), both principal radii are equal, giving \( \Delta p = 2\gamma / R \). For a cylindrical surface of radius \( R \), one principal radius is infinite, giving \( \Delta p = \gamma / R \).
The work done to increase the surface area is
\[ \delta W_{\text{surface}} = \gamma \cdot \delta A = \gamma \cdot 8\pi R \, \delta R \]At equilibrium the total virtual work vanishes:
\[ \Delta p \cdot 4\pi R^2 \, \delta R = \gamma \cdot 8\pi R \, \delta R \]Dividing both sides by \( 4\pi R^2 \, \delta R \):
\[ \Delta p = \frac{2\gamma}{R} \]For an arbitrary surface, the argument generalises by replacing the single-curvature contribution with the sum of the two principal curvatures \( 1/R_1 + 1/R_2 \), recovering the full Young-Laplace equation. \(\square\)
1.3 Capillary Rise
When a liquid wets the wall of a narrow tube, the meniscus is concave (for a contact angle \( \theta < 90^\circ \)) and the Young-Laplace pressure difference drives liquid up the tube until hydrostatic pressure balances the capillary pressure.
For a cylindrical tube of radius \( r \), the meniscus radius of curvature is \( R = r / \cos\theta \). The capillary pressure is
\[ \Delta p = \frac{2\gamma \cos\theta}{r} \]Balancing against the hydrostatic pressure of a column of height \( h \):
\[ \rho g h = \frac{2\gamma \cos\theta}{r} \]so the capillary rise height is
\[ h = \frac{2\gamma \cos\theta}{\rho g r} \]A glass capillary tube of radius \( r = 0.20 \) mm is dipped vertically into water at 25 °C. The contact angle of water on glass is approximately \( \theta = 0° \) (complete wetting), \( \gamma = 72.0 \) mN m\(^{-1}\), and \( \rho = 997 \) kg m\(^{-3}\).
\[ h = \frac{2 \times 0.0720 \times \cos 0°}{997 \times 9.81 \times 2.0 \times 10^{-4}} \]\[ h = \frac{0.1440}{1.956} \approx 0.074 \text{ m} = 7.4 \text{ cm} \]This result — nearly 7.4 cm — illustrates why capillarity is important in soil moisture transport, plant water uptake, and microfluidic devices where channel radii are of order micrometres.
1.4 Measuring Surface Tension
Several experimental methods exist, each suited to different material systems:
Pendant drop method. A droplet is allowed to hang from a capillary tip. Its equilibrium shape is determined by the balance of surface tension and gravity. Fitting the drop profile to the Young-Laplace equation yields \( \gamma \) with high accuracy and requires only a camera and image analysis software.
Wilhelmy plate method. A thin plate (usually platinum or paper) is partially immersed in a liquid. The downward force on the plate due to surface tension is measured with a microbalance. For a plate of width \( w \) and contact angle \( \theta \):
\[ F = 2w \gamma \cos\theta \]If \( \theta = 0 \) (complete wetting), the measurement is independent of immersion depth, making this method extremely precise.
Du Noüy ring method. A platinum ring is pulled through the interface. The maximum force needed to detach the ring, corrected for geometry, yields \( \gamma \). It is fast but requires careful correction factors for the finite wire diameter.
1.5 Classical Nucleation Theory and the Kelvin Equation
A supersaturated vapour or a supercooled liquid cannot remain in its metastable state forever; thermal fluctuations continuously create small clusters or droplets. Whether these embryos grow into stable nuclei or dissolve depends on the competition between the bulk free-energy gain (driving condensation) and the surface free-energy cost (opposing the creation of new interface).
For a spherical droplet of radius \( R \) forming from vapour:
\[ \Delta G(R) = -\frac{4}{3}\pi R^3 \frac{k_B T}{\Omega} \ln S + 4\pi R^2 \gamma \]where \( S = p/p_{\text{sat}} \) is the supersaturation ratio and \( \Omega \) is the molecular volume. The critical radius \( R^* \) is found by \( d(\Delta G)/dR = 0 \):
\[ R^* = \frac{2\gamma \Omega}{k_B T \ln S} \]This critical radius is intimately related to the Kelvin equation, which describes how the equilibrium vapour pressure over a curved surface differs from that over a flat surface.
where \( V_m \) is the molar volume of the liquid and \( R_g \) is the gas constant.
The Kelvin equation has profound consequences: small droplets have higher vapour pressure than large ones, so small droplets evaporate while large ones grow — the mechanism driving Ostwald ripening in aerosols and colloidal dispersions.
Estimate the fractional increase in vapour pressure for a water droplet of radius \( R = 10 \) nm at 25 °C. For water: \( \gamma = 72.0 \) mN m\(^{-1}\), \( V_m = 18.0 \times 10^{-6} \) m\(^3\) mol\(^{-1}\), \( R_g = 8.314 \) J mol\(^{-1}\) K\(^{-1}\), \( T = 298 \) K.
\[ \ln \frac{p_r}{p_\infty} = \frac{2 \times 0.0720 \times 18.0 \times 10^{-6}}{10 \times 10^{-9} \times 8.314 \times 298} \]\[ = \frac{2.592 \times 10^{-6}}{2.477 \times 10^{-5}} = 0.1047 \]\[ \frac{p_r}{p_\infty} = e^{0.1047} \approx 1.110 \]The vapour pressure is 11% higher than over a flat surface. For a 1 nm droplet, the Kelvin correction would exceed 100%, illustrating that nucleation from the vapour phase requires substantial supersaturation.
Chapter 2: Thermodynamics of Interfaces
2.1 Surface Free Energy and the Gibbs Dividing Surface
Real interfaces are not infinitely sharp two-dimensional planes; they have a finite thickness over which properties change continuously from bulk phase \( \alpha \) to bulk phase \( \beta \). Gibbs introduced a convenient mathematical device — the dividing surface — to handle this ambiguity rigorously.
The surface excess concentration \( \Gamma_i \) (mol m\(^{-2}\)) of component \( i \) is
\[ \Gamma_i = \frac{n_i - n_i^\alpha - n_i^\beta}{A} \]where \( n_i \) is the total moles of \( i \) in the system and \( n_i^{\alpha,\beta} \) are the hypothetical amounts that would be present if the bulk phases extended uniformly to the dividing surface.
The fundamental thermodynamic equation for the Gibbs surface excess (internal) energy \( U^s \) is
\[ dU^s = T \, dS^s + \gamma \, dA + \sum_i \mu_i \, dn_i^s \]2.2 The Gibbs Adsorption Isotherm
The Gibbs adsorption isotherm connects measurable changes in surface tension to the composition of the interface. It is one of the most powerful and general results in surface thermodynamics.
For an ideal (dilute) solution where \( \mu_2 = \mu_2^\circ + R_g T \ln c \):
\[ \Gamma_2 = -\frac{c}{R_g T} \frac{d\gamma}{dc} \]Differentiating and subtracting \( dU^s \) yields the surface Gibbs-Duhem equation:
\[ S^s \, dT + A \, d\gamma + \sum_i n_i^s \, d\mu_i = 0 \]At constant \( T \), dividing by \( A \):
\[ d\gamma = -\sum_i \Gamma_i \, d\mu_i \]With the convention \( \Gamma_1 = 0 \) this reduces to \( d\gamma = -\Gamma_2 \, d\mu_2 \). Substituting \( d\mu_2 = R_g T \, d\ln c \) gives the standard form. \(\square\)
The isotherm has two key implications. If \( d\gamma/dc < 0 \) (surface tension decreases as concentration increases), then \( \Gamma_2 > 0 \): the solute concentrates at the interface — it is surface-active. If \( d\gamma/dc > 0 \), the solute is depleted from the surface (negative adsorption), as observed for inorganic salts in water.
2.3 Surfactants and the Critical Micelle Concentration
As surfactant concentration increases, surface tension decreases because the interface becomes progressively more densely packed with surfactant. Above a critical concentration, the surface is saturated; instead of continuing to lower surface tension, the additional surfactant self-assembles into micelles — aggregates typically 3–10 nm in diameter with a hydrophobic interior and hydrophilic exterior.
Surface tension measurements of an aqueous SDS solution at 25 °C give the following data:
| \( c \) / mM | 0.5 | 1.0 | 2.0 | 4.0 | 8.0 | 12.0 | 20.0 |
|---|---|---|---|---|---|---|---|
| \( \gamma \) / mN m\(^{-1}\) | 68.2 | 64.5 | 59.0 | 50.7 | 42.3 | 37.3 | 37.2 |
Plot \( \gamma \) vs. \( \log c \). The data fall on a steeply declining straight line from 0.5 to 12 mM, then plateau at \( \gamma \approx 37.2 \) mN m\(^{-1}\). The breakpoint — where the linear decline meets the plateau — occurs at \( c_\text{CMC} \approx 8 \)–\(12\) mM, consistent with the literature value for SDS of ~8.3 mM. The slope of the linear region, combined with the Gibbs adsorption isotherm, yields the surface excess \( \Gamma_2 \) and from that the area per molecule at saturation: \( a_0 = 1/(\Gamma_2 N_A) \approx 0.4 \) nm\(^2\).
2.4 The Marangoni Effect
Surface tension is a function of temperature and composition. Whenever gradients in surface tension exist along an interface, they generate tangential stresses that drive flow — the Marangoni effect. In tear film thinning, the Marangoni flow stabilises the film. In wine “legs,” ethanol evaporates preferentially from thin film ridges, raising local surface tension and driving upward flow. In materials processing (e.g., laser welding and crystal growth), Marangoni convection profoundly influences heat and mass transport.
The Marangoni number \( \text{Ma} = -(\partial \gamma / \partial T)(\Delta T \cdot L)/(\eta \kappa) \) characterises the relative importance of surface-tension-driven flow versus viscous dissipation and thermal diffusion.
Chapter 3: The Electric Double Layer and Electrokinetics
3.1 Origin of Surface Charge
Most surfaces in aqueous solution acquire a net charge. The mechanisms include:
- Ionisation of surface groups. Silica (\(\text{SiOH}\)) and metal oxides undergo protonation/deprotonation (\(\text{SiOH} \rightleftharpoons \text{SiO}^- + \text{H}^+\)), so the surface charge depends strongly on pH.
- Preferential ion adsorption. Silver iodide surfaces develop charge by the preferential adsorption of either Ag\(^+\) or I\(^-\) from solution (the potential-determining ions).
- Isomorphous substitution. In clay minerals, Al\(^{3+}\) substitutes for Si\(^{4+}\), leaving a permanent negative framework charge independent of pH.
The surface charge is characterised by the surface potential \( \psi_0 \) (relative to the bulk solution taken as zero reference) and the surface charge density \( \sigma_0 \) (C m\(^{-2}\)).
3.2 The Poisson-Boltzmann Equation and Gouy-Chapman Theory
Counter-ions are attracted to a charged surface while co-ions are repelled, but thermal motion prevents complete collapse onto the surface. The result is a diffuse layer of ions extending from the surface into the solution, with an exponentially decaying potential profile in the linearised limit.
The electrostatic potential \( \psi(x) \) at distance \( x \) from a planar surface obeys the Poisson equation:
\[ \frac{d^2 \psi}{dx^2} = -\frac{\rho_e}{\varepsilon_0 \varepsilon_r} \]where \( \rho_e = \sum_i z_i e n_i \) is the local charge density and \( n_i \) is the local ion concentration. Assuming Boltzmann statistics for each ionic species:
\[ n_i(x) = n_i^\infty \exp\!\left( -\frac{z_i e \psi}{k_B T} \right) \]Substituting yields the Poisson-Boltzmann (PB) equation. For a 1:1 electrolyte (\( z = \pm 1 \)):
\[ \frac{d^2 \psi}{dx^2} = \frac{2 n^\infty e}{\varepsilon_0 \varepsilon_r} \sinh\!\left( \frac{e\psi}{k_B T} \right) \]with solution
\[ \psi(x) = \psi_0 \, e^{-\kappa x} \]where \( \kappa^{-1} \) is the Debye screening length.
For \( ze\psi / k_B T \ll 1 \), use \( \sinh(u) \approx u \):
\[ \frac{d^2 \psi}{dx^2} \approx \frac{2 z e n^\infty}{\varepsilon_0 \varepsilon_r} \cdot \frac{ze\psi}{k_B T} = \frac{2 z^2 e^2 n^\infty}{\varepsilon_0 \varepsilon_r k_B T} \psi = \kappa^2 \psi \]Identifying the Debye parameter:
\[ \kappa^2 = \frac{2 z^2 e^2 n^\infty}{\varepsilon_0 \varepsilon_r k_B T} = \frac{e^2}{\varepsilon_0 \varepsilon_r k_B T} \sum_i z_i^2 n_i^\infty \]The general solution satisfying \( \psi \to 0 \) as \( x \to \infty \) and \( \psi(0) = \psi_0 \) is \( \psi = \psi_0 e^{-\kappa x} \). \(\square\)
For a 1:1 electrolyte at 25 °C with concentration \( c \) in mol L\(^{-1}\):
\[ \kappa^{-1} \approx \frac{0.304}{\sqrt{c}} \text{ nm} \]Calculate \( \kappa^{-1} \) for 10 mM NaCl at 25 °C (\( \varepsilon_r = 78.5 \)).
Using the approximate formula for a 1:1 electrolyte:
\[ \kappa^{-1} = \frac{0.304}{\sqrt{0.010}} = \frac{0.304}{0.100} = 3.04 \text{ nm} \]At 100 mM NaCl, \( \kappa^{-1} = 0.96 \) nm; at 1 mM, \( \kappa^{-1} = 9.6 \) nm. Adding salt compresses the double layer, which has major consequences for colloidal stability (Chapter 5).
3.3 The Stern Layer
The Gouy-Chapman model overestimates the concentration of counter-ions immediately adjacent to the surface, because it treats ions as point charges and ignores finite ion size and specific adsorption. Stern (1924) corrected this by introducing a compact inner layer (the Stern layer or inner Helmholtz plane) of adsorbed ions separated by a distance \( \delta \) from the surface, within which the potential drops linearly from \( \psi_0 \) to the Stern potential \( \psi_d \). Beyond \( \delta \), the potential decays according to the Gouy-Chapman diffuse layer. The overall capacity of the double layer then becomes two capacitors in series (Stern + diffuse).
3.4 Electrocapillary Phenomena and the Lippmann Equation
At a polarisable metal–electrolyte interface, the surface tension depends on the electrode potential \( E \). This is electrocapillary behaviour.
where \( \sigma_0 \) is the surface charge density on the metal. Differentiating again gives the differential capacitance: \( C = -\partial^2 \gamma / \partial E^2 \).
The potential at which \( \gamma \) is maximum (the electrocapillary maximum) corresponds to the point of zero charge (PZC) where \( \sigma_0 = 0 \). The classic experiments of Lippmann with mercury electrodes produced parabolic \( \gamma \)–\( E \) curves, confirming the theory and providing early precise measurements of double-layer capacitance.
3.5 Zeta Potential
In practice, the Stern potential \( \psi_d \) is not directly measurable. What is accessible is the zeta potential \( \zeta \), defined as the electrostatic potential at the shear plane — the boundary between the fluid that moves with the particle and the fluid that remains stationary. The shear plane lies just beyond the Stern layer, at a distance of roughly one hydration shell from the surface. The zeta potential is the key experimental parameter characterising colloidal stability:
- \( |\zeta| > 30 \) mV typically indicates a stable, electrostatically stabilised dispersion.
- \( |\zeta| \approx 0 \) (at the isoelectric point) signals minimum stability and maximum coagulation tendency.
3.6 Electrokinetic Phenomena
When a charged surface is subjected to an applied field or pressure gradient, four related phenomena arise from the coupling between electrostatics and fluid flow:
Electrophoresis. Motion of charged particles through a stationary electrolyte under an applied electric field \( E_0 \). For a sphere in the limit \( \kappa a \gg 1 \) (thin double layer), the Smoluchowski equation gives the electrophoretic mobility:
\[ \mu_E = \frac{v}{E_0} = \frac{\varepsilon_0 \varepsilon_r \zeta}{\eta} \]For \( \kappa a \ll 1 \) (thick double layer), the Hückel limit gives \( \mu_E = 2\varepsilon_0 \varepsilon_r \zeta / (3\eta) \). The Henry function interpolates between these limits.
Electroosmosis. Motion of liquid through a porous membrane or capillary under an applied electric field. The flow velocity far from the wall is \( v_\text{eo} = -\varepsilon_0 \varepsilon_r \zeta E_0 / \eta \), independent of pore radius (for thin double layers). Electroosmosis is exploited in microfluidic chip-based separations.
Streaming potential. An electric potential that develops when liquid is forced through a charged capillary by an applied pressure \( \Delta p \). The streaming potential is
\[ \frac{\Delta V}{\Delta p} = \frac{\varepsilon_0 \varepsilon_r \zeta}{\eta \kappa_s} \]where \( \kappa_s \) is the solution conductivity. Measuring streaming potential provides a route to the zeta potential for surfaces that cannot be suspended as colloidal particles.
Streaming current. The convective current arising from the motion of the diffuse layer ions with the pressure-driven flow. Streaming current measurements are preferred over streaming potential in low-conductivity media.
Chapter 4: van der Waals and Intermolecular Forces
4.1 Components of van der Waals Forces
Van der Waals forces encompass three distinct quantum-mechanical mechanisms, all arising from interactions between fluctuating or permanent electric dipoles:
Keesom interaction (orientation force). Between two permanent dipoles of moments \( \mu_1 \) and \( \mu_2 \), the rotationally averaged interaction energy is
\[ w_\text{Keesom}(r) = -\frac{\mu_1^2 \mu_2^2}{3(4\pi\varepsilon_0)^2 k_B T} \cdot \frac{1}{r^6} \]Debye interaction (induction force). A permanent dipole polarises a neighbouring molecule, inducing a dipole. For molecule 1 (permanent dipole \( \mu_1 \)) acting on molecule 2 (polarisability \( \alpha_2 \)):
\[ w_\text{Debye}(r) = -\frac{\mu_1^2 \alpha_2}{(4\pi\varepsilon_0)^2} \cdot \frac{1}{r^6} \]London dispersion interaction (fluctuation force). Even for non-polar molecules, instantaneous fluctuations of the electron cloud create temporary dipoles. Using the London formula with characteristic frequencies \( \nu_i \) and ionisation energies \( h\nu_i \approx I_i \):
\[ w_\text{London}(r) = -\frac{3}{2} \cdot \frac{\alpha_1 \alpha_2}{(4\pi\varepsilon_0)^2} \cdot \frac{I_1 I_2}{I_1 + I_2} \cdot \frac{1}{r^6} \]All three contributions scale as \( r^{-6} \) and are always attractive for like molecules. Dispersion forces dominate in most non-polar systems.
4.2 The Hamaker Constant and Macroscopic van der Waals Forces
The pairwise additive summation of molecular interactions between macroscopic bodies leads to the concept of the Hamaker constant.
For two identical materials in vacuum, \( A_{11} > 0 \). Typical values range from \( \sim 10^{-21} \) J (soft organics) to \( \sim 10^{-19} \) J (metals). For material 1 interacting with material 2 across medium 3, the combining relation is \( A_{132} = (\sqrt{A_{11}} - \sqrt{A_{33}})(\sqrt{A_{22}} - \sqrt{A_{33}}) \).
The non-retarded Hamaker constant for silica across vacuum is \( A_{11} = 6.5 \times 10^{-20} \) J. For water \( A_{33} = 3.7 \times 10^{-20} \) J. The effective constant for silica–silica interaction across water is
\[ A_{132} = \left(\sqrt{A_{11}} - \sqrt{A_{33}}\right)^2 \]\[ = \left(\sqrt{6.5 \times 10^{-20}} - \sqrt{3.7 \times 10^{-20}}\right)^2 \]\[ = \left(2.55 - 1.92\right)^2 \times 10^{-20} = (0.63)^2 \times 10^{-20} = 4.0 \times 10^{-21} \text{ J} \]This effective constant is roughly an order of magnitude smaller than in vacuum, explaining why van der Waals forces between inorganic particles in water are significantly weaker than in air.
For two flat surfaces separated by distance \( D \), the van der Waals interaction energy per unit area is
\[ G_\text{vdW}(D) = -\frac{A}{12\pi D^2} \]and the interaction energy between two spheres of equal radius \( R \) in the Derjaguin approximation (\( D \ll R \)) is
\[ G_\text{vdW}(D) = -\frac{AR}{12 D} \]4.3 Retardation Effects
The London formula assumes instantaneous propagation of the electromagnetic fluctuation. For separations larger than \( \lambda/2\pi \sim 10 \)–\(20 \) nm (where \( \lambda \) is a characteristic UV wavelength), the time for the field to travel between molecules becomes comparable to the oscillation period. The fluctuating dipole of molecule 1 has partially de-correlated by the time its field reaches molecule 2 and returns, weakening the interaction. The retarded van der Waals force then scales as \( r^{-7} \) (pair potential) and \( D^{-3} \) (flat plates) rather than the non-retarded \( r^{-6} \) and \( D^{-2} \). Retardation is significant for forces at distances \( > 20 \) nm and must be accounted for in accurate DLVO calculations.
Chapter 5: Colloid Stability — DLVO Theory and Beyond
5.1 The DLVO Framework
Colloidal particles in suspension are subject to attractive van der Waals forces (always present) and repulsive electrostatic double-layer forces (when the particles are charged). Derjaguin, Landau, Verwey, and Overbeek independently developed a theory combining these two contributions to predict whether a colloidal dispersion would be stable (particles remain dispersed) or unstable (particles coagulate).
Stability is controlled by the height of the energy barrier \( G_\text{max} \) relative to thermal energy \( k_B T \).
In the Derjaguin approximation for two identical spheres of radius \( R \):
\[ G_\text{total}(D) = 2\pi R \left[ 64 n^\infty k_B T \kappa^{-2} \tanh^2\!\left(\frac{e\psi_0}{4k_BT}\right) e^{-\kappa D} \right] - \frac{AR}{12 D} \]5.2 Critical Coagulation Concentration
Adding electrolyte compresses the double layer (increases \( \kappa \)) and thus reduces \( G_\text{max} \). At the critical coagulation concentration (CCC), the barrier vanishes and coagulation becomes rapid.
At the CCC, both \( G_\text{total} = 0 \) and \( dG_\text{total}/dD = 0 \). Solving these two conditions simultaneously gives the Schulze-Hardy rule: the CCC scales as the inverse sixth power of the counter-ion valence \( z \):
\[ \text{CCC} \propto \frac{1}{z^6} \]This dramatic dependence explains why divalent ions (e.g., Ca\(^{2+}\)) are much more effective at coagulating negatively charged clay particles than monovalent Na\(^+\).
A colloidal silica dispersion at pH 9 has \( \psi_0 \approx -80 \) mV and \( A_{132} = 4 \times 10^{-21} \) J. Estimate the CCC for NaCl and CaCl\(_2\).
For NaCl (\( z = 1 \)), the CCC is characterised by \( \kappa_\text{CCC}^{-1} \) determined by setting barrier height to zero. Using the simplified result:
\[ \text{CCC(NaCl)} \approx \frac{(8.64 \times 10^4 \varepsilon_0^3 \varepsilon_r^3 (k_BT)^5 \gamma_0^4)}{e^4 A^2 N_A} \]where \( \gamma_0 = \tanh(e\psi_0/4k_BT) \). Numerically, for typical silica parameters this yields CCC(NaCl) \(\approx 200\) mM. For CaCl\(_2\) (\( z = 2 \)), the Schulze-Hardy rule gives CCC(CaCl\(_2\)) \(\approx 200/2^6 \approx 3\) mM — over 60 times lower, consistent with experimental observations.
5.3 Beyond DLVO: Additional Surface Forces
DLVO theory is remarkably successful but fails in several experimentally important situations:
Steric forces. Polymer chains grafted to or adsorbed on surfaces create a steric repulsion that is entropic in origin. As two coated surfaces approach, the polymer chains are compressed and lose conformational entropy, giving a strongly repulsive, monotonically increasing force. Steric stabilisation is robust to added salt (unlike electrostatic stabilisation) and is used in sterically stabilised latex paints, pharmaceutical dispersions, and polymer-coated nanoparticles.
Solvation (hydration) forces. At very short range (< 3 nm), water molecules are structured at hydrophilic surfaces. Disturbing this structured hydration layer requires extra work, manifesting as an oscillatory or purely repulsive force. Hydration forces explain why highly charged clay platelets do not coagulate even at high salt concentrations.
Hydrophobic forces. Between hydrophobic surfaces in water, an anomalously strong long-range attraction (stronger than van der Waals) pulls the surfaces together. The origin — whether cavitation, reduced water ordering, or correlated charge fluctuations — remains debated. Hydrophobic forces drive protein folding, membrane fusion, and the adsorption of surfactant tails onto hydrophobic surfaces.
Depletion forces. When non-adsorbing polymer or small colloidal particles (depletants) are excluded from the gap between two large particles, an unbalanced osmotic pressure pushes the large particles together. The depletion potential is
\[ G_\text{dep}(D) = -p_\text{osm} \cdot V_\text{overlap}(D) \]where \( V_\text{overlap} \) is the volume from which depletant is excluded.
Chapter 6: Polymers at Interfaces
6.1 Polymer Conformation in Solution
A polymer of \( N \) statistical segments (each of length \( b \)) adopts a random coil in dilute solution with a root-mean-square end-to-end distance \( R_0 = b\sqrt{N} \) and a radius of gyration \( R_g = b\sqrt{N/6} \). In a good solvent (Flory exponent \( \nu \approx 3/5 \)), the excluded-volume interactions swell the coil to \( R_F \approx b N^{3/5} \). In a theta solvent (\( \nu = 1/2 \)), excluded volume is exactly compensated and the ideal chain result is recovered.
6.2 Polymer Adsorption
Polymers adsorb strongly to surfaces when there is any attractive interaction between the segments and the surface, even if the adsorption energy per segment \( \chi_s k_B T \) is much less than \( k_B T \), because the total adsorption energy summing over many segments greatly exceeds \( k_B T \). Once adsorbed, polymers rearrange into trains (segments in contact with the surface), loops (arching into solution), and tails (dangling free ends). The thickness of the adsorbed layer, characterised by the hydrodynamic thickness \( \delta_h \), typically ranges from 1 to 50 nm depending on molecular weight.
Brush versus mushroom regime. When polymers are end-grafted to a surface at surface density \( \Sigma \) (chains per nm\(^2\)):
- If \( \Sigma^{1/2} < R_F^{-1} \) (low density), chains do not overlap and adopt mushroom-like conformations with extension \( R_F \).
- If \( \Sigma^{1/2} > R_F^{-1} \) (high density), chains overlap and are forced to stretch away from the surface into a polymer brush of thickness \( L \approx N b (\Sigma b^2)^{1/3} \).
6.3 Steric Stabilisation
Polymer brushes provide an excellent steric barrier. When two brush-coated surfaces approach to separation \( D < 2L \), the chains must compress, generating a repulsive pressure. The Alexander-de Gennes model gives the interaction free energy per unit area:
\[ G_\text{steric}(D) \approx k_B T \Sigma^{3/2} \left[ \frac{28}{11} \left(\frac{2L}{D}\right)^{5/4} + \frac{20}{77} \left(\frac{D}{2L}\right)^{7/4} - \frac{48}{11} \right] \]This is strongly repulsive for \( D < 2L \) and decays rapidly beyond \( 2L \), making steric stabilisation effective at short range. PEG-coated liposomes (stealth liposomes) exploit this principle to avoid phagocytic uptake in vivo.
6.4 Bridging Flocculation and Depletion Interaction
Bridging flocculation occurs when a polymer chain adsorbs simultaneously on two different particles, drawing them together. This is a kinetically complex process: if the polymer is added rapidly to a concentrated dispersion, each particle may be bridged to several others before the chains have time to rearrange and fully cover individual particle surfaces. Bridging is exploited in water treatment, where high-molecular-weight polyacrylamide flocculants destabilise colloidal silt.
Depletion flocculation (see also Section 5.3) arises when the concentration of non-adsorbing polymer exceeds the overlap concentration \( c^* \). The Asakura-Oosawa model predicts that the depletion attraction has a range equal to the polymer radius of gyration and a depth of order \( \Pi_\text{osm} V_\text{overlap} \). Depletion can cause reversible phase separation into colloid-rich and polymer-rich phases, with application in protein crystallisation and colloidal assembly.
Chapter 7: Wetting and Contact Angle
7.1 The Young Equation
When a liquid droplet rests on a solid surface in a vapour atmosphere, the contact angle \( \theta \) at the three-phase contact line is determined by the balance of interfacial tensions.
or equivalently
\[ \cos\theta = \frac{\gamma_{SV} - \gamma_{SL}}{\gamma_{LV}} \]where \( \gamma_{SV} \), \( \gamma_{SL} \), and \( \gamma_{LV} \) are the solid–vapour, solid–liquid, and liquid–vapour interfacial tensions, respectively.
When \( \gamma_{SV} - \gamma_{SL} > \gamma_{LV} \), no finite contact angle satisfies the Young equation — the liquid spreads completely. The spreading coefficient is defined as
\[ S = \gamma_{SV} - \gamma_{SL} - \gamma_{LV} \]Complete wetting occurs when \( S > 0 \); partial wetting when \( S < 0 \).
7.2 Contact Angle Hysteresis
Real surfaces are neither chemically homogeneous nor geometrically smooth. The advancing contact angle \( \theta_A \) (measured as the liquid front moves forward) is larger than the receding contact angle \( \theta_R \) (measured as the front retreats). The hysteresis \( \theta_A - \theta_R \) arises from surface roughness, chemical heterogeneity, and pinning of the contact line at defects. A large hysteresis (\( > 10° \)) indicates a rough or heterogeneous surface; a small hysteresis (\( < 5° \)) is characteristic of smooth, chemically homogeneous surfaces.
7.3 Wetting on Rough and Heterogeneous Surfaces
Wenzel model. If a liquid fills the surface texture completely (Wenzel state), the apparent contact angle \( \theta_W \) relates to the Young contact angle \( \theta_Y \) through the roughness factor \( r \) (actual area / projected area, \( r \geq 1 \)):
\[ \cos\theta_W = r \cos\theta_Y \]Roughness amplifies both hydrophilicity (\( \theta_Y < 90° \Rightarrow \theta_W < \theta_Y \)) and hydrophobicity (\( \theta_Y > 90° \Rightarrow \theta_W > \theta_Y \)).
Cassie-Baxter model. If the liquid does not penetrate the texture (Cassie state, air trapped beneath), the apparent contact angle is a weighted average:
\[ \cos\theta_{CB} = f_1 \cos\theta_1 + f_2 \cos\theta_2 \]For a surface with solid fraction \( f_s \) and trapped air fraction \( f_A = 1 - f_s \) (where \( \theta_\text{air} = 180° \)):
\[ \cos\theta_{CB} = f_s \cos\theta_Y - (1 - f_s) \]A hydrophobic wax coating has \( \theta_Y = 110° \) on a flat surface. Micro-pillars create a roughness factor \( r = 1.8 \) and solid fraction \( f_s = 0.2 \).
Wenzel prediction (if liquid fills the texture):
\[ \cos\theta_W = 1.8 \times \cos 110° = 1.8 \times (-0.342) = -0.616 \]\[ \theta_W = 128° \]Cassie-Baxter prediction (if air is trapped):
\[ \cos\theta_{CB} = 0.2 \times \cos 110° - (1 - 0.2) = 0.2(-0.342) - 0.8 = -0.068 - 0.8 = -0.868 \]\[ \theta_{CB} = 150° \]The Cassie-Baxter state predicts superhydrophobicity (\( \theta > 150° \)), consistent with lotus leaf surfaces. In the Cassie state, the contact angle hysteresis is also very low, enabling water droplets to roll off and carry away dirt particles — the self-cleaning effect exploited in lotus-inspired coatings.
Chapter 8: Adsorption Isotherms and Surface Analysis
8.1 Introduction to Adsorption
When a gas or dissolved species contacts a solid surface, molecules accumulate at the surface — the phenomenon of adsorption. The solid is the adsorbent; the adsorbing species is the adsorbate. Adsorption that involves only weak physical (van der Waals) interactions is physisorption, with characteristic adsorption enthalpies of 5–40 kJ mol\(^{-1}\). Chemical bond formation (chemisorption) involves 40–400 kJ mol\(^{-1}\). At a fixed temperature, the amount adsorbed at equilibrium as a function of the gas pressure (or solution concentration) defines the adsorption isotherm.
8.2 The Langmuir Isotherm
The Langmuir isotherm is the foundational model for monolayer adsorption on energetically equivalent, independent sites.
where \( K \) (units of pressure\(^{-1}\)) is the Langmuir equilibrium constant, equal to the ratio of adsorption to desorption rate constants. At low pressure \( \theta \approx Kp \) (Henry’s law regime); as \( p \to \infty \), \( \theta \to 1 \) (monolayer saturation).
Solving for \( \theta \):
\[ \theta = \frac{k_a p / k_d}{1 + k_a p / k_d} = \frac{Kp}{1 + Kp} \]where \( K = k_a / k_d \).
Adsorption rate: proportional to gas pressure and fraction of vacant sites,
\[ r_a = k_a p (1 - \theta) \]Desorption rate: proportional to fraction of occupied sites,
\[ r_d = k_d \theta \]At equilibrium \( r_a = r_d \):
\[ k_a p (1-\theta) = k_d \theta \]\[ k_a p - k_a p \theta = k_d \theta \]\[ k_a p = (k_a p + k_d)\theta \]\[ \theta = \frac{k_a p}{k_a p + k_d} = \frac{(k_a/k_d) p}{1 + (k_a/k_d) p} = \frac{Kp}{1+Kp} \qquad \square \]The Langmuir assumptions are: (i) a fixed number of identical adsorption sites; (ii) each site adsorbs at most one molecule; (iii) no interaction between adsorbed molecules; (iv) adsorption is reversible and reaches equilibrium.
For solution-phase adsorption, replacing pressure with concentration \( c \) and writing the adsorbed amount as \( q = q_m \theta \) (where \( q_m \) is the monolayer capacity):
\[ q = \frac{q_m K c}{1 + K c} \]Linearising (Langmuir linearisation):
\[ \frac{c}{q} = \frac{1}{q_m K} + \frac{c}{q_m} \]A plot of \( c/q \) vs. \( c \) yields a straight line with slope \( 1/q_m \) and intercept \( 1/(q_m K) \), from which both \( q_m \) and \( K \) are extracted.
Adsorption of methylene blue dye onto activated carbon at 25 °C:
| \( c_\text{eq} \) / (mg L\(^{-1}\)) | 5 | 10 | 20 | 40 | 80 |
|---|---|---|---|---|---|
| \( q \) / (mg g\(^{-1}\)) | 38 | 58 | 82 | 104 | 118 |
Construct the Langmuir linearisation (\( c/q \) vs. \( c \)):
| \( c \) | 5 | 10 | 20 | 40 | 80 |
|---|---|---|---|---|---|
| \( c/q \) | 0.132 | 0.172 | 0.244 | 0.385 | 0.678 |
Linear regression gives slope \( = 0.00716 \) g mg\(^{-1}\) and intercept \( = 0.095 \) g L\(^{-1}\) mg\(^{-1}\) (note consistent units).
\[ q_m = 1/\text{slope} = 1/0.00716 = 140 \text{ mg g}^{-1} \]\[ K = \text{slope}/\text{intercept} = 0.00716/0.095 = 0.075 \text{ L mg}^{-1} \]The monolayer capacity is 140 mg g\(^{-1}\) and the Langmuir affinity constant is 0.075 L mg\(^{-1}\). The dimensionless separation factor \( R_L = 1/(1+Kc_0) \) for an initial concentration of 80 mg L\(^{-1}\) is \( R_L = 1/(1+0.075 \times 80) = 0.14 \), indicating favourable adsorption (\( 0 < R_L < 1 \)).
8.3 Multilayer Adsorption: The BET Isotherm
The Langmuir model restricts adsorption to a single monolayer. At pressures approaching saturation (\( p/p_0 \to 1 \)), multiple layers of adsorbate condense on the surface. Brunauer, Emmett, and Teller (1938) extended the Langmuir kinetic argument to treat multilayer adsorption.
where \( n \) is the moles adsorbed, \( n_m \) is the monolayer capacity, \( p_0 \) is the saturation vapour pressure, and \( C \) is the BET constant related to the first-layer adsorption energy by \( C \approx \exp\!\left[(E_1 - E_L)/R_g T\right] \) with \( E_L \) the heat of liquefaction.
The surface is divided into fractions \( \theta_0 \) (bare), \( \theta_1 \) (monolayer), \( \theta_2 \) (bilayer), etc. In steady state:
\[ k_a p \theta_0 = k_d^{(1)} \theta_1 \]\[ k_a p \theta_i = k_d^{(L)} \theta_{i+1} \quad \text{for } i \geq 1 \]Normalisation: \( \sum_{i=0}^\infty \theta_i = 1 \). Setting \( x = p/p_0 \) and summing the geometric series yields the BET equation above.
Let \( y_i = \theta_i / \theta_0 \). From detailed balance:
\[ \theta_1 = K_1 p \,\theta_0 = Cx \,\theta_0 \]\[ \theta_i = (K_L p)^{i-1} \theta_1 = Cx^i \,\theta_0 \quad \text{for } i \geq 1 \]Total amount adsorbed (in units of monolayer capacity \( n_m \)):
\[ n/n_m = \sum_{i=1}^\infty i\,\theta_i = C\theta_0 \sum_{i=1}^\infty i\,x^i = \frac{Cx\,\theta_0}{(1-x)^2} \]Normalisation:
\[ 1 = \theta_0 + \sum_{i=1}^\infty \theta_i = \theta_0\left(1 + \frac{Cx}{1-x}\right) \]\[ \theta_0 = \frac{1-x}{1-x+Cx} \]Dividing:
\[ \frac{n}{n_m} = \frac{Cx}{(1-x)\left(1 - x + Cx\right)} \]Rearranging into the standard BET linear form:
\[ \frac{p/p_0}{n(1-p/p_0)} = \frac{1}{n_m C} + \frac{C-1}{n_m C}\cdot\frac{p}{p_0} \qquad \square \]8.4 BET Surface Area Measurement
The following N\(_2\) adsorption data were collected at 77 K for a mesoporous silica sample (mass \( m = 0.200 \) g). The cross-sectional area of N\(_2\) is \( a_m = 0.162 \) nm\(^2\).
| \( p/p_0 \) | \( n \) / (mmol g\(^{-1}\)) | \( (p/p_0)/\left[n(1-p/p_0)\right] \) |
|---|---|---|
| 0.05 | 1.21 | 0.0435 |
| 0.10 | 1.50 | 0.0741 |
| 0.15 | 1.73 | 0.1019 |
| 0.20 | 1.93 | 0.1294 |
| 0.30 | 2.30 | 0.1869 |
The BET plot (\( (p/p_0)/[n(1-p/p_0)] \) vs. \( p/p_0 \)) is linear with:
- Slope \( s = (C-1)/(n_m C) = 0.573 \) g mmol\(^{-1}\)
- Intercept \( i = 1/(n_m C) = 0.0146 \) g mmol\(^{-1}\)
The specific surface area per gram of adsorbent:
\[ S_\text{BET} = n_m \times N_A \times a_m = 1.701 \times 10^{-3} \times 6.022 \times 10^{23} \times 0.162 \times 10^{-18} \]\[ = 1.701 \times 6.022 \times 0.162 \times 10^{(-3+23-18)} \text{ m}^2 \text{g}^{-1} \]\[ = 1.659 \times 10^2 \text{ m}^2 \text{g}^{-1} \approx 166 \text{ m}^2 \text{g}^{-1} \]This is a typical surface area for mesoporous silica (compare: dense silica \(\approx\) 1 m\(^2\) g\(^{-1}\); activated carbons 500–3000 m\(^2\) g\(^{-1}\)).
8.5 Pore Size Analysis
8.5.1 Types of Adsorption Isotherms
The IUPAC classification recognises six isotherm types:
- Type I (Langmuir-like): Microporous solids; plateau at monolayer capacity.
- Type II: Non-porous or macroporous; multilayer adsorption with inflection at BET monolayer point.
- Type III: Weak adsorbate–surface interaction; isotherm convex throughout.
- Type IV: Mesoporous (2–50 nm pore diameter); Type II character with hysteresis loop due to capillary condensation.
- Type V: Weak interaction in mesoporous material; less common.
- Type VI: Stepped multilayer adsorption on highly uniform surfaces.
Hysteresis in Type IV isotherms reflects the different mechanisms for pore filling (condensation starting from the pore wall) and pore emptying (evaporation from the pore mouth). The H1 hysteresis loop (nearly vertical and parallel adsorption/desorption branches) is characteristic of cylindrical pores with uniform diameter, while H4 loops indicate slit-shaped pores.
8.5.2 BJH Pore Size Distribution
The Barrett-Joyner-Halenda (BJH) method extracts the pore size distribution from the adsorption or desorption branch of a Type IV isotherm. It applies the Kelvin equation to relate the relative pressure at which a pore of radius \( r_p \) fills by capillary condensation:
\[ \ln \frac{p}{p_0} = -\frac{2\gamma V_m}{R_g T r_k} \]where \( r_k \) is the Kelvin radius of the meniscus. The actual pore radius \( r_p = r_k + t \), where \( t \) is the thickness of the pre-adsorbed multilayer, estimated from the Harkins-Jura or Frenkel-Halsey-Hill equations. The BJH algorithm peels back pore volumes layer by layer as pressure is reduced, accumulating the volume contributed by each pore size class, yielding a differential pore volume distribution \( dV/dr \) vs. \( r_p \).
8.5.3 Mercury Porosimetry
Mercury is non-wetting on most materials (\( \theta \approx 130°–140° \)); it can only be forced into pores by applying external pressure. The Washburn equation relates the pressure \( p \) to the pore radius \( r \):
\[ r = -\frac{2\gamma_\text{Hg} \cos\theta}{p} \]By increasing pressure from \(\sim\)0.003 MPa (pore diameter 500 \(\mu\)m) to \(\sim\)200 MPa (pore diameter 3.6 nm), mercury porosimetry covers the macropore and large-mesopore range inaccessible to gas adsorption. The cumulative mercury volume intruded as a function of pressure yields the pore size distribution. A key limitation is that high applied pressures can compress or collapse delicate mesoporous frameworks.
Appendix: Key Equations Summary
The following equations represent the core quantitative framework of the course.
Surface tension (work definition):
\[ \gamma = \left(\frac{\partial G}{\partial A}\right)_{T,p,n_i} \]Young-Laplace equation:
\[ \Delta p = \gamma\left(\frac{1}{R_1} + \frac{1}{R_2}\right) \]Capillary rise:
\[ h = \frac{2\gamma \cos\theta}{\rho g r} \]Kelvin equation:
\[ \ln\frac{p_r}{p_\infty} = \frac{2\gamma V_m}{R \, R_g T} \]Gibbs adsorption isotherm:
\[ \Gamma_2 = -\frac{c}{R_g T}\frac{d\gamma}{dc} \]Debye length (1:1 electrolyte):
\[ \kappa^{-1} = \sqrt{\frac{\varepsilon_0 \varepsilon_r k_B T}{2 n^\infty e^2}} \approx \frac{0.304\text{ nm}}{\sqrt{c/\text{mol L}^{-1}}} \]Gouy-Chapman potential:
\[ \psi(x) = \psi_0 \, e^{-\kappa x} \]Lippmann equation:
\[ \left(\frac{\partial \gamma}{\partial E}\right)_{T,p,\mu_i} = -\sigma_0 \]Smoluchowski electrophoretic mobility:
\[ \mu_E = \frac{\varepsilon_0 \varepsilon_r \zeta}{\eta} \]Van der Waals between flat plates (non-retarded):
\[ G_\text{vdW}(D) = -\frac{A}{12\pi D^2} \]Hamaker combining relation:
\[ A_{132} = \left(\sqrt{A_{11}} - \sqrt{A_{33}}\right)\left(\sqrt{A_{22}} - \sqrt{A_{33}}\right) \]Young (wetting) equation:
\[ \cos\theta = \frac{\gamma_{SV} - \gamma_{SL}}{\gamma_{LV}} \]Wenzel and Cassie-Baxter equations:
\[ \cos\theta_W = r\cos\theta_Y \qquad \cos\theta_{CB} = f_s\cos\theta_Y - (1-f_s) \]Langmuir isotherm:
\[ \theta = \frac{Kp}{1+Kp} \]BET isotherm:
\[ \frac{p/p_0}{n(1-p/p_0)} = \frac{1}{n_m C} + \frac{C-1}{n_m C}\cdot\frac{p}{p_0} \]Washburn equation (mercury porosimetry):
\[ r = -\frac{2\gamma_\text{Hg}\cos\theta}{p} \]