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Formation of an insoluble fraction by acidifying a chemical solution or colloid.

The Asakura-Oosawa model treats dilute nonadsorbing polymer coils as mutually interpenetrable ideal particles excluded from a layer around each colloid. For colloid radius $a$ and polymer exclusion thickness $R_g$, two exclusion spheres of radius $R_e=a+R_g$ overlap when $2a\le R<2R_e$. Adding their two spherical cap volumes gives $V_{\mathrm{ov}}=\pi (4R_e+R)(2R_e-R{)}^2/12$. Multiplication by $-{\Pi }_p$ gives the displayed pair free energy, with zero depletion interaction beyond $2R_e$. The range is controlled by polymer size and the depth by reservoir polymer concentration. At larger polymer concentration, polymer interactions and many-body overlap effects require care.

Like-charged colloids can resist aggregation because bringing their electric double layers together costs free energy. In DLVO theory, screened electrostatic repulsion competes with attractive Van der Waals forces. A repulsive barrier much larger than $k_{B}T$ suppresses thermally activated aggregation, although the dispersion can remain metastable. Lower ionic strength generally lengthens the Debye–Hückel screening length, while multivalent counterions or added salt shorten it and can lower the barrier. The surface charge, charge regulation and solvent-dependent attraction also matter; screening length alone does not guarantee stability.

A colloid contains dispersed particles in a continuous medium. A fine particle suspension can form a colloid; the particles in the Perrin sedimentation experiment illustrate this case.

The critical coagulation concentration is the approximate electrolyte concentration separating slow, barrier-limited aggregation from rapid aggregation of a charge-stabilized colloid. In the simplest DLVO theory, increasing salt shortens the Debye–Hückel screening length and lowers the repulsive barrier until rapid aggregation is possible. It depends on particle charge, solvent, temperature and particularly counterion valence. It is a kinetic crossover, not a universal salt concentration or necessarily an equilibrium phase boundary. Aggregation just below this crossover can be reaction limited; exactly at or above the rapid-aggregation threshold, proximity to the threshold alone does not imply a small sticking probability.

An electric double layer consists of an interfacial electric charge and the compensating distribution of mobile ions in the adjacent chemical solution. Counterions are enriched and coions depleted near a simple charged interface. A diffuse layer is described approximately by the Poisson-Boltzmann equation; a compact specifically adsorbed layer requires additional surface chemistry. Overlap of diffuse layers can produce a screened repulsion between like-charged colloids.

A hydrosol is a colloid whose continuous liquid medium is water. For example, small solid particles dispersed in water form a hydrosol when they remain dispersed on the relevant observational timescale. This differs from a molecular solution: the dispersed objects are particles with interfaces, rather than individual dissolved ions or molecules.

Surface charge and adsorbed layers can inhibit aggregation. Changing ionic strength can alter electrostatic screening and destabilize a charged hydrosol, while Brownian motion competes with settling. “Hydro” identifies the dispersion medium; it does not specify one particle composition or one stabilization mechanism.

Particle aggregation is association of initially separate particles into larger clusters. For colloids, both the rate of Brownian motion encounters and the probability that an encounter produces attachment matter. Strong attraction with negligible repulsive barriers can give diffusion-limited cluster aggregation; a barrier that makes attachment unlikely can give reaction-limited cluster aggregation. Reversible association and bond rearrangement require additional kinetic assumptions.

In the Asakura-Oosawa model, nonadsorbing polymers are treated as ideal, mutually penetrable particles, but their centers cannot enter a layer around a hard colloid. The polymer osmotic pressure is $\Pi =\rho k_{\mathrm{B}}T$ in the ideal dilute limit, with the appropriate number density in the space dimension considered.

For fixed polymer number, reducing the inaccessible volume increases the available volume and lowers the polymer Helmholtz free energy. Equivalently, at fixed polymer chemical potential, the polymer contribution to the effective colloid potential isWhen the excluded layers overlap, their union occupies less space than two separate layers. More polymer configurations become available, so the resulting depletion force attracts the colloids. It is an entropy-driven effective interaction, even though each microscopic colloid–polymer interaction is purely repulsive.

The critical coagulation concentration is the electrolyte concentration at which rapid, approximately diffusion-limited colloidal coagulation begins: the electrostatic barrier has become ineffective and nearly every sufficiently close encounter sticks. It is an operational threshold for the specified colloid and electrolyte, not a universal concentration or a guarantee of a mathematically exact zero barrier in every system.

Adsorbed or grafted polymers stabilize a colloid through steric stabilization when coverage is high and the solvent keeps the chains extended. Charged chains can add electrostatic repulsion, giving electrosteric stabilization.

They can destabilize it through polymer bridging flocculation when a chain adsorbs onto two particles, especially at incomplete coverage; through depletion force when nonadsorbing polymer is excluded from the gap and the surrounding solution's osmotic pressure pushes particles together; or through collapse of an adsorbed layer in a poor solvent. Polymer identity, molecular size, solvent quality, adsorption and concentration determine which mechanism dominates.

The sedimentation scale height simplifies because the same drag coefficient controls both drift and diffusion:Applying the given criterion $a\le z_0$ givesEquivalently, the gravitational Péclet number on the particle-radius scale is $a/z_0=m_{b}ga/(k_{B}T)$ and must be at most one.

At $300\mathrm{K}$ the limiting radius is $a_{\max }\simeq 8.42\times 10^{-7}\mathrm{m}=0.84\mu \mathrm{m}$, corresponding to diameter approximately $1.68\mu \mathrm{m}$. Viscosity cancels from this equilibrium criterion, although it affects the time to approach equilibrium.

For the $100\mathrm{nm}$ radius particle in part (i), $D\simeq 2.20\times 10^{-12}{\mathrm{m}}^2{\mathrm{s}}^{-1}$ and $z_0\simeq 0.50\mathrm{mm}$, about $5.0\times 10^3$ particle radii. It is therefore firmly diffusion-dominated on its own size scale. This does not mean uniform concentration over an arbitrarily tall vessel: the equilibrium density still decays on the finite scale $z_0$. The supplied size criterion is an approximate gravity-versus-diffusion criterion, not a universal definition of every stable colloid.

DLVO theory adds attractive Van der Waals forces and electric double layer repulsion to obtain an effective colloid pair potential. For two identical spheres of radius $b$ at a surface gap $h\ll b$, a useful schematic form isHere $A_H$ is the Hamaker constant. For a symmetric monovalent electrolyte with number density $n$ of each ionic species, ${\kappa }_D^2=2n{e}^2/(ϵk_{B}T)$. The exponential is a simplified screened repulsion; its amplitude depends on electric potential at the surface, charge regulation and geometry. Constant-charge and constant-potential boundary conditions need not produce the same close-approach law.

A primary attractive minimum near contact may be separated from dispersed configurations by an energy barrier; a shallower secondary minimum can produce loose reversible aggregation. A barrier much larger than $k_{B}T$ can give long-lived kinetic stabilization without making the dispersed state the global free-energy minimum. Adding salt shortens the Debye–Hückel screening length and can lower the barrier, allowing colloidal coagulation. Changing pH can alter surface ionization, so screening alone does not determine stability.

One can vary salt concentration while holding particle concentration, temperature and pH fixed, then follow particle aggregation using light scattering or microscopy. A solvent-only control and an independently measured surface-charge indicator help distinguish a change of screening from adsorption or charge reversal. Classical DLVO theory omits steric stabilization, hydration forces and strong ionic correlations; the formal attraction diverging at contact is cut off by molecular-scale physics.

Write $s$ for the number of grafted chains per unit area and $\Sigma =s{a}^2$ for its dimensionless form; this separates the physical grafting density from a dimensionless coverage. A free dilute chain has size $R_0\sim a{N}^{\nu }$, with $\nu =1/2$ in a theta solvent and approximately $3/5$ in a good solvent. An isolated end-grafted chain loses translations and configurations that cross the substrate, but retains the same leading size exponent if the surface is nonadsorbing. When the lateral graft spacing $D\sim s^{-1/2}$ exceeds $R_0$, the layer consists of polymer mushrooms. When coils overlap, crowding stretches the chains away from the surface into a polymer brush.

The polymer mushroom-to-brush crossover isFor the stipulated statistical-segment interpretation of $N$ and $a$, the scaling estimates areThus ${\Sigma }_{\ast }\simeq 2.50\times 10^{-3}$ and $7.54\times 10^{-4}$ respectively. The numerical constants are scaling conventions: using a radius of gyration rather than an end-to-end size changes order-one amplitudes, not the powers of $N$.

In the Alexander-de Gennes brush model, take blobs of lateral size $D$. Within a blob, $D\sim a{g}^{\nu }$; beyond it, the $N/g$ blobs are aligned away from the wall. ThusWith coarse-grained segment volume $a^3$, conservation gives the polymer-brush volume fraction through $\varphi H\sim N{a}^{3}s$, henceThe theta-solvent brush height includes three-segment repulsion even though the effective two-segment interaction vanishes. These are scaling estimates for a roughly uniform brush, not an assertion that every chain end lies at the same height in an actual brush.

The many-segment-blob requirement $g\gg 1$ gives $D\gg a$, or $\Sigma \ll 1$. Together with overlap, the regime is $N^{-2\nu }\ll \Sigma \ll 1$. At the dense-brush packing limit, $\varphi \sim 1$, $g\sim 1$ and $H\sim Na$. Here $s\sim a^{-2}=2.04\mathrm{n}{\mathrm{m}}^{-2}$ and $H\sim 280\mathrm{nm}$. If one formally continues a dense incompressible layer above this boundary, conservation would instead giveThis is the requested formal dense-layer expression, but for $\Sigma >1$ it gives $H>Na$, which violates finite extensibility of a polymer chain. There is no physically admissible continuation above this limit in the stated segment-volume model. A Kuhn length segment is not necessarily a cube of material: with an independently specified material volume $v_0$ per counted segment, the dense-layer law is $H=N{v}_{0}s$, and the packing bound is $s\le a/v_0$. The data do not determine $v_0$ separately. The simple model therefore establishes its own breakdown, rather than a universal microscopic maximum grafting density.

grafting-to uses preformed chains whose lengths and dispersities can be characterized before attachment. As coverage rises, each incoming coil must penetrate a crowded brush and lose conformational entropy to bring its end to the surface; the growing insertion barrier limits achievable coverage. grafting-from supplies small monomers to surface-bound growing chains and avoids insertion of a whole coil. Choose grafting-from for high grafting density. Initiator coverage and reaction control still matter, and inaccessible active sites, termination and unequal growth can broaden the resulting chain-length distribution. Neither route removes the elastic and packing costs of the final brush.

A generic application is steric stabilization of synthetic colloids in a good solvent. Bringing coated particles together compresses and interpenetrates their brushes, increasing osmotic pressure and conformational free-energy cost. This repulsion can oppose aggregation. The chains must remain attached and sufficiently cover the surface; poor-solvent collapse, sparse coverage or interparticle bridging can defeat stabilization.

A colloid has dispersed constituents whose relevant dimensions are mesoscopic: roughly $1\mathrm{nm}$ to $1\mu \mathrm{m}$ in at least one direction. Interfaces and Brownian motion are consequently important. The size range is a useful convention, not a sharp boundary separating all physical behaviour. A sol of fine silicon dioxide particles in water illustrates solid in liquid; an oil-in-water emulsion illustrates liquid in another immiscible liquid; and a foam illustrates gas dispersed in a liquid, with thin intervening films supplying the small structural dimension. A colloid need not be a thermodynamic equilibrium state.^[ref]

Unprotected particles attract through Van der Waals forces. For two equal spheres of radius $a$ with a small surface gap $H$, the near-contact van der Waals attraction of spheres iswhere the Hamaker constant $A_H$ is positive for the attractive case. The large number of interacting constituent molecules can give a well much deeper than $k_{B}T$, even though individual molecular attractions are weak. Brownian motion brings particles together, and a sufficiently deep attractive well makes separation unlikely. This gives particle aggregation; solid particles can remain distinct within a cluster rather than merging into one continuous solid. Emulsion droplets can additionally coalesce if the separating film ruptures.

Charge stabilization of colloids supplies a repulsive barrier. Like-charged surfaces carry compensating counterions and electric double layers; bringing the layers together costs free energy. DLVO theory combines this repulsion with Van der Waals forces. A barrier large compared with $k_{B}T$ can make aggregation slow, even when sticking would lower the final free energy. Adding salt generally shortens the Debye–Hückel screening length and can lower the barrier; surface charge regulation and specific ion interactions also matter. A negatively charged particle and a positively charged particle need not repel.

Steric stabilization instead coats each particle with adsorbed or grafted polymer chains. In a good solvent, overlap and compression of these layers raise the osmotic pressure and reduce chain entropy, opposing close contact. Electrosteric stabilization combines the two mechanisms. Dense, securely bound layers matter: incomplete adsorption can allow polymer bridging, while nonadsorbing dissolved polymers can create an attractive depletion force. Poor solvent quality can collapse a protective layer. Reducing the density mismatch slows sedimentation but does not remove the interparticle attraction, so uniform suspension and resistance to aggregation are different requirements.

The left panel is an original qualitative potential with arbitrary coefficients: it illustrates a barrier and its reduction by shorter-ranged repulsion, without fitting an experimental DLVO theory curve or extrapolating the continuum attraction to molecular contact. The right panel plots the derived quasi-steady concentration profile in dimensionless variables.

In diffusion-controlled growth, attachment is rapid and delivery through the surrounding medium is the bottleneck. The growing surface depletes the neighboring solute, so a substantial concentration gradient develops. In reaction-controlled growth, arrival is fast compared with incorporation; many arrivals fail to attach, and the surface concentration stays near the reservoir value. The analogous distinction for particle aggregation is between near-unit sticking probability in diffusion-limited cluster aggregation and barrier-suppressed sticking in reaction-limited cluster aggregation. Here the subsequent calculation concerns accretion onto one growing particle, rather than a collision kernel for two mobile clusters.

The printed time-dependent equation is the diffusion equation, not the usual Poisson equation. For constant diffusion coefficient $D$, no bulk sources and negligible advection, Fick's laws give the number flux $\mathbf{j}=-D\nabla c$. Local conservation of mass, expressed here as conservation of the accreting constituent, then givesAn arbitrary time-independent exterior profile does not solve the full moving-boundary problem exactly. The required elementary result uses a quasi-steady approximation: diffusion establishes the exterior profile much faster than the particle radius changes. At each instantaneous radius $a$, set $\partial c/\partial t\simeq 0$. Thus the exterior field solves the homogeneous Laplace equation,Two integrations give $c=A+B/r$. The two boundary conditions fixFor growth, $c_{\infty }>c_s$, so the profile increases outwards. Its radial flux is negative, and the positive inward arrival rate isA material density is needed to turn an arrival rate into a radial velocity. Introduce the constant constituent number density $c_p$ inside the particle, using the same units as $c_s,c_{\infty }$. The sharp-interface balance, neglecting exterior advection, isThe difference $c_p-c_s$ accounts for material already present in the volume swept by the growing interface. In the usual dilute-feed approximation $c_s\ll c_p$, it can be replaced by $c_p$. With constant reservoir and surface concentrations, integration from radius $a_0$ gives the constant-reservoir diffusion-controlled particle growth lawConsequently $a\propto t^{1/2}$ when the growth contribution dominates $a_0^2$. For a finite seed at the literal limit $t\to 0$, the increment $a-a_0$ is instead initially proportional to $t$. Here “early time” means before appreciable reservoir depletion or interactions with other particles, while already beyond the rapid initial establishment of the diffusion field. Indeed, the ratio of diffusion-adjustment time $a^2/D$ to radial-growth time $a/\dot{a}$ is $(c_{\infty }-c_s)/(c_p-c_s)$; small supersaturation relative to the particle density makes the quasi-steady approximation self-consistent. No initial concentration field was specified, so an exact transient solution is not uniquely determined.

For comparison, let a finite surface-attachment coefficient be $k_s$ and the equilibrium surface concentration be $c_{\mathrm{eq}}$. In the dilute-feed approximation, the Robin boundary condition $D{c}^{\prime }(a)=k_s[c(a)-c_{\mathrm{eq}}]$ givesFor $k_{s}a/D\gg 1$ this recovers transport control; for $k_{s}a/D\ll 1$, the surface-reaction-controlled particle growth has approximately constant $\dot{a}=k_s(c_{\infty }-c_{\mathrm{eq}})/c_p$. Thus reaction control gives $a-a_0\propto t$, in contrast to the diffusion-controlled square-radius law. No numerical $c_p$ or $k_s$ is supplied or needed for these scaling results.

For an ideally dissociated electrolyte, the Debye–Hückel screening length isHere $n_i$ is ionic number density, $z_i$ the charge in units of the elementary charge, and $c_i$ is molar concentration. Conversion between $I$ and number density cancels in a ratio. At fixed relative permittivity, ${\lambda }_D\propto \sqrt{T/I}$.

In $0.01\mathrm{M}$ sodium chloride, $c_{\mathrm{N}{\mathrm{a}}^{+}}=c_{\mathrm{C}{\mathrm{l}}^{-}}=0.01\mathrm{M}$, giving $I_1=0.01\mathrm{M}$. Complete dissociation of $0.1\mathrm{M}$ aluminium chloride gives $c_{\mathrm{A}{\mathrm{l}}^{3+}}=0.1\mathrm{M}$ and $c_{\mathrm{C}{\mathrm{l}}^{-}}=0.3\mathrm{M}$, soThe factor is sixty, not ten: the squared charge and the three counterions per formula unit both matter. Using the given $5\mathrm{nm}$ reference length directly, and $T_1\simeq 298\mathrm{K}$, $T_2=323\mathrm{K}$,The modest temperature increase raises the screening length by only about $4\%$ at fixed concentration; the large increase in ionic strength shortens it by almost an order of magnitude overall.

Under charge stabilization of colloids, the electrostatic double-layer repulsion now has a much shorter range. Attractive Van der Waals forces can then dominate at closer separations, reducing the barrier to particle aggregation in DLVO theory. Thus an electrostatically stabilized colloid is likely to be destabilized, with aggregation more likely than in the original dispersion.

This is a conditional physical prediction, not a proof of instability from a length alone. No particle radius, surface potential, attractive interaction strength or steric coating is specified. Steric stabilization could preserve stability, and trivalent-ion adsorption can alter the surface charge. Moreover, ideal screening theory is only an estimate at $I\simeq 0.6\mathrm{M}$, where ion association, hydrolysis and activities can matter. The numerical result uses the supplied reference length and the stipulated unchanged permittivity, rather than replacing either with an unprovided calibration.

Two stabilization mechanisms. Steric stabilization coats each colloid with adsorbed or grafted polymer layers. Bringing two particles close overlaps or compresses those layers, increasing their free energy through loss of chain conformations and, in a suitable solvent, increased local osmotic pressure. The resulting repulsion can keep the cores apart. A well-solvated nonionic coating can remain effective when added salt would screen electrostatic forces. Coatings can fail through desorption, poor solvent conditions or bridging, so simply adding arbitrary free polymer does not guarantee stabilization.

Charge stabilization of colloids instead uses similarly charged surfaces and their diffuse electric double layers. Their approach costs electrostatic and ionic free energy. In DLVO theory, that repulsion competes with attractive Van der Waals forces. It is particularly sensitive to salt concentration, counterion valence and surface ionization.

Bjerrum length. The standard Bjerrum length is the separation at which the Coulomb interaction energy of two elementary charges equals thermal energy:At $T\simeq 298\mathrm{K}$ and ${\epsilon }_r=78$, this isIt compares ionic electrostatic coupling with thermal agitation. For ions of signed valences $z_1,z_2$, $U/(k_{B}T)=z_1{z}_2{\ell }_B/r$. The standard reference length uses $e$, not the whole colloid charge: assigning a colloid charge $Ze$ does not change the solvent's tabulated ${\ell }_B$.

Screening and the printed normalization. In a symmetric $z:z$ electrolyte, $z$ is the ionic valence magnitude, $e$ the elementary electric charge, $n_{\infty }$ the bulk number density of each ionic species, ${\epsilon }_0$ the vacuum permittivity, ${\epsilon }_R$ the solvent's relative permittivity, $k_B$ the Boltzmann constant and $T$ absolute temperature. Mobile ions rearrange around a charged surface; the Debye–Hückel screening length ${\kappa }^{-1}$ sets the range of the resulting weak potential.

There is an SI normalization error in the original PDF. Combining the Boltzmann distribution $n_{\pm }=n_{\infty }{e}^{\mp ze\psi /(k_{B}T)}$ with the SI Poisson equation and linearizing at $|ze\psi |\ll k_{B}T$ givesThe extra $4\pi$ in the PDF denominator is inconsistent with those SI charge, permittivity and density conventions. Keeping the printed algebra would make its screening length $\sqrt{4\pi }$ times the SI value at the same ion density. No numerical screening length is required here, and the qualitative dependence ${\kappa }^{-1}\propto n_{\infty }^{-1/2}$ is the same in either normalization. Added salt shortens the range of repulsion, allowing the attractive potential to capture particles more readily.

Interaction-potential sketches. Let $h$ be the surface separation between two spheres of radius $a$. A useful schematic DLVO theory potential has the formaway from the microscopic contact region. Here $A_H>0$ is a Hamaker constant; excluded-core repulsion and molecular details regularize contact. The exponential is the double-layer repulsion and the inverse-gap term the short-gap Van der Waals force contribution. A fixed $B$ illustrates the changing screening range; actual charge regulation can change the repulsive amplitude as well.

At low salt there is a high positive barrier between a deep primary minimum near contact and the dispersed state at $U(\infty )=0$. A shallow secondary minimum can occur outside the barrier and support weak reversible association. Increasing salt narrows the repulsive range and lowers the barrier. Thermally activated primary aggregation becomes appreciable when the barrier is only a few $k_{B}T$; the idealized barrier-free condition defines the fast-aggregation threshold associated with the critical coagulation concentration. Above that threshold, collisions can result in rapid capture in the primary minimum.

Aggregate structures. Well above the critical coagulation concentration, the sticking probability is close to one. In diffusion-limited cluster aggregation, the first encounter attaches particles or clusters at accessible outer branches before they can explore many alternative configurations. The result is an open, tenuous structure. Typical ideal three-dimensional cluster models have mass fractal dimension $d_f\simeq 1.8$.

Near the threshold on its lower-salt, slow-aggregation side, a remaining barrier makes most encounters unsuccessful. Reaction-limited cluster aggregation then permits repeated approaches and orientations before attachment, commonly giving denser clusters with $d_f\simeq 2.1$. Since particle count scales as $N\propto (R_g/a{)}^{d_f}$, the larger exponent means more particles at a given large cluster size. This explanation does not require already formed bonds to rearrange: unsuccessful encounters themselves alter which configurations eventually bind.

The side of the threshold matters. At the ideal barrier-free threshold itself, or just above it, aggregation can already be diffusion limited; proximity to the critical coagulation concentration alone does not universally imply a denser reaction-limited structure. The quoted dimensions are typical of irreversible approximately isotropic cluster models, not exact values for every colloid. Strong bond rearrangement or anisotropic attractions can change this comparison.

Write the particle volume fraction as $\eta =(4\pi a^3/3)n_p$. In a stable, repulsion-dominated colloid, low $\eta$ gives a disordered fluid. Increasing $\eta$ makes the interaction between neighboring particles important and can produce an ordered crystal, separated from the fluid by a phase coexistence curve. Increasing the Debye–Hückel screening length at fixed charge strengthens the repulsion at a given separation and allows freezing at smaller geometrical volume fraction.

The geometric intuition is that particles behave as though they occupy more than their bare diameter: roughly, an effective exclusion diameter $d_{\mathrm{eff}}$ extends out to the separation where $U_{\mathrm{el}}$ is of order $k_{B}T$. A hard sphere comparison then uses ${\eta }_{\mathrm{eff}}\sim \eta [d_{\mathrm{eff}}/(2a){]}^3$. This explains the trend; it is not an exact freezing criterion.

At short Debye–Hückel screening length and negligible attraction, the diagram approaches hard sphere fluid–crystal coexistence. For longer-ranged soft repulsion, body-centered cubic lattice and face-centered cubic lattice crystals can both be stable, with the former often appearing towards the softer, lower-density freezing boundary. Screened-Coulomb simulations exhibit both these crystal structures.^[ref]

No universal quantitative diagram follows from $(\eta ,{\lambda }_D)$ alone: the charge, Hamaker constant and surface conditions must also be fixed. If salt suppresses the stabilizing barrier enough for Van der Waals force attraction to dominate, a colloid-poor/colloid-rich separation or aggregation can replace part of this repulsive reference diagram. Dense colloidal glasses or colloidal gels may also be observed after kinetic arrest; they should not be mistaken for additional thermodynamic equilibrium thermodynamic phases.

Use SI units, uniform absolute permittivity $\epsilon$ and a neutral bulk reservoir. Local equilibrium of ideal mobile ions gives $n_i=n_{i,\infty }{e}^{-z_{i}e\psi /(k_{B}T)}$. Combining their charge with the Poisson equation yieldsThe reference is $\psi =0$ in the bulk. Boundary data specify fixed charge, fixed potential or a surface reaction/charge regulation law. These are physically different problems. The mean-field theory assumes point ions, thermal equilibrium and a continuum solvent; ion-size packing, dielectric structure and strong electrostatic correlations can invalidate it.

For a symmetric monovalent electrolyte let $n_{\infty }$ count each sign and define $u=e\psi /(k_{B}T)$, ${\kappa }_D^2=2e^2{n}_{\infty }/(\epsilon k_{B}T)$. Outside fixed sources,For $|u|\ll 1$, the Debye-Hückel approximation is ${\nabla }^{2}u={\kappa }_D^{2}u$. A planar wall bounding electrolyte in $z>0$ has $\psi ={\psi }_0{e}^{-{\kappa }_{D}z}$; assuming zero field on the other side, its charge is $\sigma =\epsilon {\kappa }_D{\psi }_0$. A sphere of radius $a$ and total charge $Q$ instead givesThe factors arise from geometry and boundary conditions; planar screening cannot be substituted unchanged for a small sphere. The weak-potential condition must hold where the potential is largest, particularly at the surface.

The nonlinear planar Poisson-Boltzmann profile is also exactly integrable. Multiply $u^{\prime \prime }={\kappa }_D^2\sinh u$ by $u^{\prime }$ and use $u,u^{\prime }\to 0$ at infinity:The sign selects a profile decaying to the bulk for either sign of surface potential. Integration givesFor weak $u_0$ this reduces to the linear solution. Strong surface potentials have a bounded far-field amplitude $4\gamma$, while the near-wall ion density still grows strongly. There is no need to permit the negative ion densities produced by an invalid extrapolation of a linearized density formula.

The same first integral yields the Grahame equation and its diffuse-layer differential capacitance per unit area:These describe the diffuse layer. A compact interfacial layer adds a voltage drop and can place its capacitance in series with $C_D$; a measured electrode capacitance need not equal the ideal diffuse contribution.^[ref]

For comparison, a planar counterion-only double layer next to a negative plane has positive counterions and no salt reservoir. Its profile is algebraic:Integrating $en(z)$ over $z>0$ gives $|\sigma |$, checking charge neutrality. This problem cannot be obtained by putting a finite reservoir density into the symmetric-electrolyte answer without changing its boundary conditions.

Applications include screening around synthetic colloids, diffuse-layer electrode capacitance and forces between charged plates. In planar geometry, the bulk-subtracted normal stress is constant:At the midpoint of identical plates, symmetry sets $E=0$, so the remaining excess ionic osmotic pressure determines their repulsion. A comparison with force or capacitance measurements should vary salt and surface potential independently, test charge regulation and adsorption, and distinguish compact-layer effects from failures of diffuse screening. Nonlinear Poisson-Boltzmann theory retains finite-potential Boltzmann populations, but still omits correlations and finite-ion-volume saturation.

Polyethylene glycol is a water-soluble polymer with repeating ethylene oxide units. Nonadsorbing dissolved chains can cause a depletion force between larger colloids. Chains attached to a surface can instead give steric stabilization. Which effect occurs depends on surface affinity, chain size and concentration.