The Electrical Double Layer, Zeta Potential and Coagulation

Add a pinch of salt to milk and nothing happens. Add the same salt to a dilute gold sol and it turns from wine-red to grey-blue within seconds, as particles clump and fall out of suspension. The difference lies in a thin charged skin around every colloidal particle: the electrical double layer. This layer controls whether colloids stay dispersed or coagulate, and it explains why some electrolytes are far more effective flocculants than others. Water treatment, food emulsions, blood stability, and clay soil behavior all hinge on this same physics. This article builds the double layer from first principles. You will see how surface charge attracts a diffuse ionic cloud. Then you will meet the zeta potential, the quantity actually measured in the lab. Finally, DLVO theory combines electrostatic repulsion with van der Waals attraction to predict coagulation, and the Schulze–Hardy rule falls out as a clean numerical consequence.

Key takeaways

A charged colloidal surface attracts a diffuse layer of counter-ions, forming the electrical double layer with characteristic thickness $1/\kappa$ , the Debye length.
The zeta potential, ζ, is the potential at the slipping plane, measured via electrophoretic mobility, not the true surface potential.
DLVO theory sums an attractive van der Waals term and a repulsive electrostatic term to give the total interaction energy versus separation.
Adding electrolyte compresses the double layer, lowers the repulsive barrier, and triggers coagulation once the barrier drops to about $kT$ .
The Schulze–Hardy rule states coagulating power scales roughly as $z^6$ with counter-ion charge, a direct prediction of DLVO theory.
Key idea: Colloidal stability is a competition between a short-range attractive force and a screened electrostatic repulsion whose range is set by the ionic strength of the medium.

The core insight

Every charged colloidal particle carries a surface charge balanced by a diffuse cloud of oppositely charged ions in solution; the resulting electrical double layer creates a repulsive energy barrier between approaching particles, and adding electrolyte compresses that layer, shrinks the barrier, and lets short-range van der Waals attraction win, causing coagulation.

The physical picture

A colloidal particle in water is rarely neutral. Ionizable surface groups, preferential ion adsorption, or lattice defects leave it with a net surface charge, $\sigma$ , typically negative for oxides, clays, and most hydrophobic sols above their point of zero charge. This fixed charge attracts a mobile cloud of counter-ions and repels co-ions, building the electrical double layer (EDL). The classical Gouy–Chapman–Stern model splits this layer into two regions. The Stern layer is a compact layer of ions and solvent molecules adsorbed directly onto the surface, roughly one to a few ionic diameters thick, where the potential falls linearly from the surface value $\psi_0$ to the Stern potential $\psi_d$ . Beyond it lies the diffuse layer, where thermal motion competes with electrostatic attraction, and ion concentration follows a Boltzmann distribution that decays smoothly into the bulk. The potential in this diffuse region falls off roughly exponentially with distance, over a characteristic length called the Debye length, $1/\kappa$ . When two particles approach, their diffuse layers begin to overlap before their surfaces touch. Overlapping layers of like-sign counter-ion clouds cost free energy, producing an electrostatic repulsion. Superimposed on this is the ever-present van der Waals attraction between the particle cores. The balance between these two, as a function of separation, is the heart of DLVO theory (named for Derjaguin, Landau, Verwey, and Overbeek). Experimentally, the diffuse layer cannot be probed directly. What can be measured is the potential at the slipping plane — the boundary between the fluid that moves with the particle under an applied field and the fluid left behind. This is the zeta potential, ζ, obtained from electrophoretic mobility measurements. Because the slipping plane sits slightly outside the true surface, ζ is somewhat smaller in magnitude than $\psi_0$ , but it tracks the same physics and is what governs measured stability.

Figure

Structure of the electrical double layer: surface, Stern layer, diffuse layer, and the potential profile with slipping plane

The second key picture is the interaction energy curve itself, obtained by adding the repulsive and attractive contributions.

Figure

DLVO interaction energy versus separation: primary maximum barrier, secondary minimum, and primary minimum

The core relations

The Debye screening parameter for a symmetric electrolyte of concentration $c$ (mol dm⁻³) and valence $z$ :

\kappa^2 = \frac{2 N_A e^2 z^2 c \times 1000}{\varepsilon_r \varepsilon_0 k T}

where: $N_A$ Avogadro's number, $e$ elementary charge, $\varepsilon_r\varepsilon_0$ solvent permittivity, $T$ temperature; $1/\kappa$ is the Debye length. The electrophoretic mobility related to zeta potential (Henry/Smoluchowski, thin double layer, $\kappa a \gg 1$ ):

u_e = \frac{\varepsilon_r \varepsilon_0 \zeta}{\eta}

where: $u_e$ electrophoretic mobility (velocity per unit field), $\eta$ solvent viscosity. The DLVO total interaction energy between two spherical particles of radius $a$ , separation $h$ :

V(h) = V_R(h) + V_A(h) = 2\pi \varepsilon_r \varepsilon_0 a \zeta^2 e^{-\kappa h} - \frac{A a}{12 h}

where: $A$ the Hamaker constant, $V_R$ electrostatic repulsion, $V_A$ van der Waals attraction (valid for $h \ll a$ ). The Schulze–Hardy rule, coagulation (critical) concentration $c_{ccc}$ scaling with counter-ion valence:

c_{ccc} \propto \frac{1}{z^6}

where: $z$ the charge number of the ion opposite in sign to the particle surface charge.

Derivation AFrom the Debye length to the DLVO stability barrier

Goal. Show how the ionic strength of the medium sets the range of electrostatic repulsion, and how this combines with van der Waals attraction to produce — and then destroy — an energy barrier against coagulation.

Step 1: linearized Poisson–Boltzmann equation. In the diffuse layer, the local potential $\psi(x)$ and local ion concentration are linked self-consistently. Poisson's equation in one dimension gives

\frac{d^2\psi}{dx^2} = -\frac{\rho(x)}{\varepsilon_r \varepsilon_0}

where $\rho(x)$ is the local charge density built from Boltzmann-weighted ion concentrations, $\rho = \sum_i z_i e c_i^0 \exp(-z_i e\psi/kT)$ . For a symmetric $z:z$ electrolyte, expanding the exponential for $ze\psi \ll kT$ (the Debye–Hückel approximation) gives a linear source term.

Step 2: solve for the exponential decay. Substituting the linearized charge density into Poisson's equation yields

\frac{d^2\psi}{dx^2} = \kappa^2 \psi

with $\kappa^2$ defined as above. The physically sensible solution decaying into the bulk is

\psi(x) = \psi_d \, e^{-\kappa x}

so $1/\kappa$ is literally the distance over which the potential falls to $1/e$ of its Stern-layer value. Because $\kappa \propto \sqrt{c}\,z$ , doubling the electrolyte concentration or squaring up the counter-ion charge sharply compresses the layer.

Step 3: repulsive energy from overlapping diffuse layers. When two flat double layers of potential $\zeta$ overlap at separation $h$ , the excess osmotic pressure of squeezed-in counter-ions integrates to an interaction free energy per unit area that, after converting to the sphere–sphere geometry via the Derjaguin approximation, gives

V_R(h) = 2\pi \varepsilon_r \varepsilon_0 a \zeta^2 e^{-\kappa h}

This is repulsive (positive), decays exponentially with the same Debye length $1/\kappa$ that sets the double-layer thickness, and grows with the square of the zeta potential.

Step 4: attractive energy from dispersion forces. Instantaneous dipole–induced dipole (London) attraction between all atom pairs in the two particles, summed over both particle volumes, gives for spheres at close approach ( $h \ll a$ )

V_A(h) = -\frac{Aa}{12h}

This attraction is short-ranged in the sense that it grows steeply as $h \to 0$ , is essentially independent of electrolyte concentration, and depends only on material properties through the Hamaker constant $A$ (typically $10^{-21}$ to $10^{-19}\,\text{J}$ ).

Step 5: superpose and locate the barrier. Adding the two contributions,

V(h) = 2\pi \varepsilon_r \varepsilon_0 a \zeta^2 e^{-\kappa h} - \frac{Aa}{12h}

At very small $h$ , the $1/h$ attraction dominates and diverges negatively (the primary minimum, where irreversible coagulation occurs). At moderate $h$ , if $\zeta$ is large and $\kappa$ small (dilute electrolyte), the exponential repulsion dominates, creating a primary maximum — an energy barrier $V_{max}$ that particles must overcome by Brownian collision to coagulate. At larger $h$ still, the repulsion has decayed exponentially while attraction persists as $1/h$ , sometimes producing a shallow secondary minimum responsible for reversible, loose flocculation.

Step 6: the coagulation condition. Stability requires $V_{max} \gg kT$ , so that thermal collisions essentially never cross the barrier; the sol is kinetically stable indefinitely. Coagulation becomes fast once added electrolyte raises $\kappa$ enough that $V_{max}$ falls to roughly $kT$ per collision — the barrier is no longer an effective obstacle. Since $\kappa^2 \propto z^2 c$ , and the condition for the barrier to vanish involves both $\zeta^2$ (which itself falls with increasing $\kappa$ through double-layer compression at the Stern plane) and the exponential term, detailed analysis (setting $dV/dh = 0$ and $V=0$ simultaneously at the critical point) shows the critical coagulation concentration must scale as

\boxed{c_{ccc} \propto \frac{1}{z^6}}

the Schulze–Hardy rule: trivalent counter-ions are roughly $3^6/1^6 \approx 729$ -fold more potent, per mole, than monovalent ones at destabilizing a sol of opposite charge.

Worked examples

Example 1 — Debye length and barrier estimate for a model sol. Consider an aqueous colloid at 298 K with a 1:1 electrolyte (NaCl) at $c = 1.0 \times 10^{-3}\ \text{mol dm}^{-3}$ , $\varepsilon_r = 78.5$ , particle radius $a = 100\ \text{nm}$ , and $\zeta = 40\ \text{mV}$ . Step 1: compute $\kappa$ . Using

\kappa^2 = \frac{2 N_A e^2 z^2 c \times 1000}{\varepsilon_r \varepsilon_0 k T}

with $N_A = 6.022\times10^{23}\,\text{mol}^{-1}$ , $e = 1.602\times10^{-19}\,\text{C}$ , $z=1$ , $c\times1000 = 1.0\ \text{mol m}^{-3}$ , $\varepsilon_0 = 8.854\times10^{-12}\,\text{F m}^{-1}$ , $k = 1.381\times10^{-23}\,\text{J K}^{-1}$ , $T = 298\ \text{K}$ : Numerator $= 2 \times 6.022\times10^{23} \times (1.602\times10^{-19})^2 \times 1.0 = 3.088\times10^{-14}$ . Denominator $= 78.5 \times 8.854\times10^{-12} \times 1.381\times10^{-23} \times 298 = 2.862\times10^{-30}$ .

\kappa^2 = \frac{3.088\times10^{-14}}{2.862\times10^{-30}} = 1.079\times10^{16}\ \text{m}^{-2}

\kappa = 1.039\times10^{8}\ \text{m}^{-1}, \qquad 1/\kappa = 9.6\ \text{nm}

This matches the familiar rule of thumb that a 1 mM 1:1 electrolyte gives a Debye length near 10 nm. Step 2: estimate the barrier height. Take a Hamaker constant $A = 5\times10^{-20}\,\text{J}$ (typical for oxide-in-water). At the barrier, take a representative separation of order $h \approx 1/\kappa = 9.6\ \text{nm}$ as a rough working estimate (the true maximum lies close to this scale for these parameters):

V_R = 2\pi(78.5)(8.854\times10^{-12})(100\times10^{-9})(0.040)^2 e^{-1}

V_R = 2\pi \times 6.951\times10^{-10} \times 1.6\times10^{-3} \times 0.368 = 2.57\times10^{-12}\ \text{J}

V_A = -\frac{(5\times10^{-20})(100\times10^{-9})}{12(9.6\times10^{-9})} = -4.34\times10^{-14}\ \text{J}

V(h) \approx 2.57\times10^{-12} - 0.043\times10^{-12} = 2.53\times10^{-12}\ \text{J}

Compare with thermal energy $kT = 1.381\times10^{-23} \times 298 = 4.12\times10^{-21}\ \text{J}$ :

\frac{V_{max}}{kT} \approx \frac{2.53\times10^{-12}}{4.12\times10^{-21}} \approx 6.1\times10^{8}

The barrier is enormous compared with $kT$ , so this sol is strongly kinetically stable at 1 mM NaCl — consistent with the well-known robustness of dilute charged sols in low-ionic-strength water.

Example 2 — Applying the Schulze–Hardy rule to predict a coagulation concentration. A negatively charged silver iodide sol is found experimentally to have a critical coagulation concentration (ccc) of $c_{ccc}(\text{Na}^+) = 140\ \text{mmol dm}^{-3}$ for NaCl (monovalent counter-ion, since the sol is negative, cations do the coagulating). Estimate the expected ccc for $\text{CaCl}_2$ ( $z=2$ ) and $\text{AlCl}_3$ ( $z=3$ ). Step 1: apply $c_{ccc} \propto z^{-6}$ , so

\frac{c_{ccc}(z)}{c_{ccc}(1)} = \left(\frac{1}{z}\right)^6

Step 2: for $\text{Ca}^{2+}$ ( $z=2$ ):

c_{ccc}(\text{Ca}^{2+}) = 140\ \text{mmol dm}^{-3} \times \frac{1}{2^6} = \frac{140}{64} = 2.19\ \text{mmol dm}^{-3}

Step 3: for $\text{Al}^{3+}$ ( $z=3$ ):

c_{ccc}(\text{Al}^{3+}) = 140\ \text{mmol dm}^{-3} \times \frac{1}{3^6} = \frac{140}{729} = 0.192\ \text{mmol dm}^{-3}

Sanity check: these predicted values (2.19 mmol dm⁻³ for Ca²⁺, 0.19 mmol dm⁻³ for Al³⁺) sit remarkably close to classic experimental ranges for AgI and As₂S₃ sols, where ratios of roughly 1 : 60–70 : 600–700 between mono-, di-, and trivalent coagulants are routinely observed — confirming the $z^6$ scaling is not just a formal curiosity but matches real measurements to within experimental factors of 2–3, since real ions also differ in specific adsorption and hydration, which the simple DLVO scaling ignores.

Limiting cases & common traps

Situation	Behavior	Common error
Very dilute electrolyte, high ζ	Large $1/\kappa$ , tall barrier, sol kinetically stable for years	Assuming "stable" means thermodynamically stable — it is only kinetically trapped in the secondary/no minimum regime
At or above ccc	Barrier collapses toward $kT$ , rapid (diffusion-limited) coagulation	Forgetting that coagulation rate changes sharply, not gradually, near the ccc
Counter-ion charge increase	ccc drops steeply, roughly as $z^{-6}$	Using the co-ion charge instead of the counter-ion charge in the Schulze–Hardy comparison
Very high electrolyte, specific adsorption	Charge reversal possible; ζ can change sign	Assuming ζ always has the same sign as the fixed surface charge
Secondary minimum only	Loose, reversible flocculation ("weak gel"), redispersible by shaking	Confusing secondary-minimum flocculation with true coagulation in the primary minimum

The chemist behind it

Boris Derjaguin's work in Soviet colloid science, alongside Lev Landau, and the parallel Dutch work of Evert Verwey and Jan Overbeek in the 1940s, unified electrostatic double-layer repulsion with van der Waals attraction into a single quantitative theory of colloidal stability — DLVO theory remains the standard framework taught today, over eighty years later, essentially unchanged in its core structure.

Origins · 1 minBoris Derjaguin and how this was discovered

Boris Derjaguin

1902-1994 · Soviet physical chemist who pioneered surface force measurement and co-founded DLVO theory of colloidal stability

Ask the tutor: How does the Hamaker constant itself get calculated or measured for a real material pair?

Why does adding salt make a colloid coagulate instead of stabilizing it further?

Added electrolyte increases the ionic strength, which raises the Debye parameter κ and compresses the diffuse double layer. Because the electrostatic repulsion term decays as $e^{-\kappa h}$ , a larger κ means the repulsion dies off over a much shorter distance. Van der Waals attraction is essentially unaffected by ionic strength. Once the double layer is compressed enough, the repulsive barrier height falls toward $kT$ , so thermal collisions can push particles past it into the deep primary minimum, causing irreversible coagulation.

Why does the coagulating power depend so strongly on the ion charge, roughly as z to the sixth power?

Two effects compound. First, κ itself depends on $z^2c$ , so higher-charge ions screen the surface charge at much lower concentrations. Second, the effective surface (or Stern) potential itself drops as more highly charged counter-ions adsorb closer to the surface, weakening ζ further. Combining both effects in the full DLVO stability analysis, where the barrier height and its position are set simultaneously by κ and ζ, produces the steep $z^{-6}$ dependence of the critical coagulation concentration observed experimentally by Schulze and Hardy.

Is the zeta potential the same as the surface potential of the particle?

No. The surface potential ψ0 exists right at the charged surface, while zeta potential is measured at the slipping plane, a small distance further out, past the Stern layer. Ions and bound solvent within the Stern layer move with the particle during electrophoresis, so ζ reflects conditions just outside that immobile layer. In practice ζ is always somewhat smaller in magnitude than ψ0, though they usually track each other and share the same sign under normal conditions.

What is the difference between the primary minimum and the secondary minimum in a DLVO curve?

The primary minimum sits at very small separation, where van der Waals attraction diverges and dominates any residual repulsion; particles trapped there are coagulated essentially irreversibly, held by strong short-range attraction. The secondary minimum, when present, is a much shallower dip at larger separation, arising because the exponential repulsion has already decayed while some weak attraction persists. Particles caught here form loose, easily redispersed flocs rather than permanent aggregates.

Can a colloid be restabilized by adding even more electrolyte after it has coagulated?

Sometimes, but only through specific mechanisms outside basic DLVO screening, such as charge reversal from strong specific ion adsorption at high concentration, or steric stabilization by added polymers or surfactants. Plain DLVO theory with a simple symmetric electrolyte predicts monotonic destabilization as concentration rises past the critical coagulation concentration; charge reversal effects require additional chemistry, such as multivalent or surface-active ions binding beyond simple electrostatic screening.