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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.

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.

DLVO theory combines screened electrostatic repulsion from overlapping electric double layers with attractive Van der Waals forces to model colloid stability. A repulsive barrier can inhibit aggregation even when a deep short-distance attractive minimum exists. Increasing electrolyte concentration reduces the screening length and can lower that barrier. Molecular surface forces and polymer-mediated interactions require additions to this description.

A stable charge-stabilized dispersion has an interaction barrier large compared with $k_{B}T$. In DLVO theory, the total pair potential combines attractive Van der Waals forces with repulsion from overlapping electric double layers. Similar surface charges raise the free energy of overlap; a sufficiently high positive barrier prevents thermally diffusing particles from reaching the deep primary minimum near contact.

Other stabilization mechanisms include steric stabilization by solvated surface chains and combined electrosteric protection by charged chains. Such contributions are additional to the simplest DLVO pair potential. Increasing salt shortens the Debye length, so double-layer repulsion becomes shorter-ranged and the protective barrier can decrease.

charge stabilization of colloids supplies repulsion when like-charged surfaces and their electric double layers approach. steric stabilization uses attached or adsorbed molecular layers whose overlap costs conformational and osmotic free energy. Combining both gives electrosteric stabilization; a solvated coating must remain attached and well solvated. A viscous medium slows encounters but does not by itself create an equilibrium repulsive barrier.

DLVO theory adds screened electrostatic repulsion to Van der Waals force. A representative close-gap form isfor equal spheres of radius $a$, with Hamaker constant $A_H$ and inverse Debye–Hückel screening length $\kappa$. The formula is schematic near molecular contact, where hydration, surface roughness and finite ion size matter. A repulsive barrier many $k_{B}T$ high produces kinetic stability even if the primary attractive minimum is deeper than the dispersed state. Entry into that primary minimum generally gives strongly bound aggregation; a shallow secondary minimum can give reversible flocculation.

Adding electrolyte increases $\kappa$, shortens the repulsion and can remove the barrier, enabling fast diffusion-limited aggregation. The critical coagulation concentration is the approximate salt concentration at which this rapid-coagulation regime begins under specified conditions. It is not a universal property of the salt alone. Test the prediction using time-resolved dynamic light scattering or turbidity at controlled particle concentration, pH and temperature; check that added ions do not specifically adsorb or change the surface chemistry.

DLVO theory adds short-range attractive Van der Waals forces to repulsive overlap of the electric double layer. Increasing salt concentration raises the ionic strength and shortens the Debye length, screening the repulsion. The repulsive barrier falls, so thermal collisions can carry particles into the primary attractive minimum and cause colloidal coagulation. An intermediate shallow secondary minimum can instead produce weak, potentially reversible association.

The figure's assignments are a: low salt; b: intermediate salt; c: high salt. Curve a has the largest protective barrier, b a lower barrier and secondary minimum, and c no effective repulsive barrier. This explanation assumes the surface charging chemistry does not independently reverse or radically change the surface potential.

For like-charged plates, DLVO theory combines an attractive dispersion contribution and a screened electrostatic repulsion. A useful far-field schematic model isAt high salt concentration, ${\lambda }_D={\kappa }_D^{-1}$ is short. The repulsion then disappears over a short distance, leaving attraction across much of the accessible range. The usual high-salt sketch has a greatly reduced or absent repulsive energy barrier, so approach into a primary attractive minimum is easier. At very small separations, short-range repulsion cuts off the unphysical divergence of the simple $-d^{-2}$ continuum expression.

The large-distance tail of the stated model is attractive because an exponentially decaying repulsion eventually becomes smaller than the algebraic attraction. A barrier is not ruled out by the words “high salt” alone: its existence also depends on surface charge or potential, charge regulation, $A_H$ and the boundary conditions that determine $B$. The figure shows one declared barrier-free high-screening example rather than claiming a universal threshold concentration.

The force is determined by the total normal stress of the ionic liquid. The ions contribute osmotic pressure because confining an excess number density costs translational entropy. The osmotic stress in a symmetric electric double layer, relative to the bulk reservoir, isThe second term is the normal electrostatic stress. The Poisson-Boltzmann equation makes this combination independent of position between planar surfaces. At the midpoint of identical equally charged surfaces, symmetry gives $E=0$, soCalculating only the repulsion of bare surface charges would omit both the screening ion cloud and its entropic stress. Evaluating the total stress at the midpoint packages their combined force in a particularly simple form.

For equal spheres, assume $h\ll a$, $\kappa a\gg 1$, weak surface potential and weak overlap, with prescribed ${\psi }_0$. Then ${\psi }_{\mathrm{mid}}\simeq 2{\psi }_0{e}^{-\kappa h/2}$ gives $\Pi \simeq 2\epsilon {\kappa }^2{\psi }_0^2{e}^{-\kappa h}$. Integrating once gives the planar interaction energy per area, $W={\int }_h^{\infty }\Pi (s)ds\simeq 2\epsilon \kappa {\psi }_0^2{e}^{-\kappa h}$, and the Derjaguin approximation gives the repulsive sphere force. Combining it with the near-contact van der Waals attraction of spheres yields the DLVO theory approximationPositive force is repulsive. The exponential term can produce a repulsive barrier in the corresponding pair energy; the short-range attraction leads to a primary minimum, while a shallower secondary minimum may occur farther out. Adding salt shortens the repulsion and can destroy the barrier, allowing particle aggregation. Molecular contact requires a repulsive-core correction, and changing the charge-regulation boundary condition changes the electrostatic prefactor.

DLVO theory combines the attractive Van der Waals force with repulsion between overlapping electrical double layers. A charged colloid attracts counterions; their electrostatic and osmotic response screens its field over the Debye length. Adding salt generally shortens this range and can lower the repulsive aggregation barrier, allowing particles to reach the strongly bound primary minimum. A shallow secondary minimum can instead give weak reversible association. Whether Brownian encounters cause rapid aggregation depends on barrier heights relative to $k_{B}T$, not just on the sign of the force at one separation.

Adsorbed or grafted polymer layers can provide steric stabilization: overlap costs chain entropy and increases the local osmotic pressure. Nonadsorbing smaller particles or polymers can create a depletion force, because bringing large surfaces together increases the volume accessible to the smaller species. Polymer bridging can be attractive when one chain binds more than one particle. A useful comparison experiment varies salt concentration while monitoring scattered light or aggregation, then repeats with a sterically protected suspension. Charge, solvent quality, polymer coverage and particle size determine which interaction dominates; none is a universal constant of all colloids.

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.

Balancing buoyant weight with Stokes drag givesThis needs isolated spherical particles, low Reynolds number and negligible wall corrections. The supplied settling measurement gives $\Delta \rho \simeq 99.1\mathrm{kg}{\mathrm{m}}^{-3}$, hence ${\rho }_p\simeq 1.10\times 10^3\mathrm{kg}{\mathrm{m}}^{-3}$. Its Reynolds number is about $7\times 10^{-5}$, consistent with the approximation.

Compare gravitational work over one radius with thermal energy, or use ${\mathrm{Pe}}_g=U_{s}r/D$ with $D=k_{B}T/(6\pi \eta r)$. The gravitational Peclet number givesBelow this sedimentation-Brownian crossover radius Brownian motion dominates settling on the particle-size scale. It does not eliminate gravity: the sedimentation scale height is ${\ell }_g=k_{B}T/[(4\pi r^3/3)\Delta \rho g]$. For negligible concentration gradients across a sample height $H$, the stronger condition is ${\ell }_g\gg H$. No universal permanently nonsedimenting size follows without a length/time criterion.

Brownian encounters bring small particles into contact; van der Waals attraction can then bind them. Surface charge can create a repulsive barrier through Debye screening of colloid repulsion. For a monovalent salt reservoir with number density $n$ of each ionic species, Poisson–Boltzmann theory hasFor $|e\psi |\ll k_{B}T$, linearization gives ${\nabla }^2\psi ={\kappa }^2\psi$, with ${\kappa }^{-1}=\sqrt{ϵk_{B}T/(2n{e}^2)}$. If $n$ instead means total ion density, the factor two is absent. The planar potential decays as $e^{-\kappa h}$; a sphere has an additional $1/r$ factor. This mean-field approximation neglects specific adsorption, ion correlations and finite ion size.

DLVO theory adds screened repulsion to dispersion attraction. At small surface separation $h$ for equal spheres of radius $r$, a useful schematic form iswith a short-range repulsive core needed outside the continuum model. Low salt can leave a barrier many times $k_{B}T$, giving kinetic stability. Higher salt shortens the repulsion and can lower or remove the barrier, allowing coagulation into a deep primary minimum. The primary and secondary DLVO minima have different consequences: a shallow secondary minimum permits looser, sometimes reversible aggregation. The surface boundary condition affects $C$; screening length alone is not a complete measured interaction law. Non-DLVO hydration or steric forces can matter at contact.

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.

Stabilization and screening. Charge stabilization of colloids means suppressing aggregation through repulsion between similarly charged particles and their electric double layers. In DLVO theory, this repulsion competes with attractive Van der Waals forces. A substantial repulsive barrier compared with thermal energy $k_{B}T$ can make aggregation slow, even when the dispersed state is only metastable.

The Debye–Hückel screening length ${\lambda }_D={\kappa }^{-1}$ is the characteristic range over which a weak electric potential is attenuated by rearrangement of mobile ions. It measures the thickness of a diffuse screening cloud, not a molecular diameter. For a plane, a weak potential decays as $e^{-\kappa x}$ into the bulk.

Derivation and the source convention. Put a planar interface at $x=0$, with electrolyte in $x>0$, bulk potential zero, and absolute permittivity $\epsilon ={\epsilon }_0{\epsilon }_r$. For a symmetric $z:z$ electrolyte, let $n_{\infty }$ be the bulk number density of each ionic species. The Boltzmann distribution givesso the mobile charge density isSubstitution into the SI Poisson equation, ${\nabla }^2\psi =-{\rho }_e/\epsilon$, gives the Poisson-Boltzmann equationFor $|ze\psi |\ll k_{B}T$, linearize the hyperbolic sine. The Debye–Hückel approximation is thenHere $z$ is the ionic valence magnitude, $e$ the elementary electric charge, $k_B$ the Boltzmann constant, $T$ absolute temperature, ${\epsilon }_0$ the vacuum permittivity, and ${\epsilon }_r$ the solvent's relative permittivity. For several ionic species, the numerator $2z^2{e}^2{n}_{\infty }$ is replaced by $e^2{\sum }_i{z}_i^2{n}_{i,\infty }$.

The original PDF includes an additional $4\pi$ in the denominator. That factor does not follow from SI Poisson equation with number densities and the stated permittivities. The Bjerrum length provides a useful check: ${\ell }_B=e^2/(4\pi \epsilon k_{B}T)$ and the consistent result is ${\kappa }^2=8\pi {\ell }_B{z}^2{n}_{\infty }$. A Gaussian-unit convention moves factors of $4\pi$ together with the charge and permittivity conventions; it does not justify inserting this extra factor while retaining all the given SI quantities. Both numerical interpretations are reported below rather than silently changing the source.

Conditions favoring stability. Increasing the surface-potential or surface-charge magnitude usually strengthens the electrostatic barrier. Lower ionic strength increases ${\lambda }_D$ and generally makes repulsion longer ranged; added salt and especially multivalent counterions shorten it and can promote aggregation. Suitable pH can keep an ionizable surface charged, while conditions near an isoelectric state weaken charge repulsion. The solvent-dependent Van der Waals forces, charge regulation and particle geometry also affect the barrier, so a long screening length is not by itself sufficient. At fixed density and valence, the linear formula gives ${\lambda }_D\propto \sqrt{\epsilon T}$, but actual temperature changes can also alter the permittivity and surface chemistry.

Surface charge at small potential. Take the usual one-sided planar-interface model, with negligible electric displacement on the nonaqueous side. The solution decaying into the solution isThe electric field is $E_x=-{\psi }^{\prime }=\kappa {\psi }_0{e}^{-\kappa x}$. The electrostatic interface boundary conditions then give the surface charge densityOne can also derive it by integrating the diffuse charge density:Thus the diffuse layer neutralizes the surface charge. This is the planar Debye-Hückel double layer. If instead the object is an isolated sheet wetted by identical electrolyte on both sides, each side screens half its total charge and ${\sigma }_{\mathrm{sheet}}=2\epsilon \kappa {\psi }_0$. The one-sided assumption is the conventional local model for the exterior of a large charged colloid; it must be stated because the question does not specify the material on the other side.

Numerical ion density. For $T=298.15\mathrm{K}$, $z=1$, $\kappa =10^9{\mathrm{m}}^{-1}$ and $\epsilon =80{\epsilon }_0=7.0834\times 10^{-10}\mathrm{F}{\mathrm{m}}^{-1}$, the consistent SI result isThis is $\boxed{0.0943\mathrm{mol}{\mathrm{L}}^{-1}}$ for each ion species, and hence also the salt concentration for a monovalent $1:1$ salt. The combined number density of both signs is $2n_{\infty }$.

Literal substitution into the formula printed in the PDF instead givesThese values are distinct consequences of the source's extra factor; they cannot both describe the same physical screening length in SI with the same definition of $n_{\infty }$.

Surface potential and validity. The requested linear estimate in the one-sided model isHowever, the thermal voltage is $k_{B}T/e=25.7\mathrm{mV}$, so $e{\psi }_{0,\mathrm{linear}}/(k_{B}T)\simeq 13.7$. Thus the small-potential approximation is not self-consistent for the supplied surface charge. The boxed value is its formal extrapolation, not a reliable prediction of the actual surface potential.

For a valid nonlinear comparison within the ideal one-sided planar Poisson-Boltzmann equation, multiply the differential equation by ${\psi }^{\prime }$ and use zero potential and field in the bulk. This givesUsing the surface boundary condition yields the Grahame equation and its inverse:With the SI screening length, the latter gives $\boxed{{\psi }_{0,\mathrm{PB}}\simeq 0.135\mathrm{V}}$. This is a nonlinear mean-field estimate under explicit assumptions, not a unique experimentally determined potential: compact interfacial layers, finite ion size and surface chemistry were not supplied. The source's normalization problem and its use of a linear formula far outside the small-potential regime are separate issues.

One method is to give every particle a sufficiently large surface charge density of the same sign. For example, attach ionizable sulfonate groups covalently to the surface. Dissociation leaves negatively charged surface groups and releases mobile positive counterions into the water. A surfactant with a strongly bound head group can provide another charging route, provided it does not desorb during the experiment.

Each charged particle is surrounded by an electric double layer. As two such particles approach, overlap of these ionic clouds raises the electrostatic and ionic free energy, producing repulsion. In a dilute electrolyte, the range is set by the Debye–Hückel screening length,For identical weakly charged spheres a representative screened Coulomb pair potential isIts sum with the Van der Waals force attraction is the DLVO theory pair potential. Choose the charging level so that the repulsive barrier is many $k_{B}T$ high; particles then rarely reach the primary attractive well during the observation time. This is kinetic stabilization rather than removal of all attraction.

Deionized water is useful because low added ionic strength gives a long-ranged repulsion. It does not mean that the charged dispersion contains no ions: the surface's own counterions are unavoidable and contribute to screening. Surface dissociation, contamination and particle concentration therefore matter even without added salt. An alternative valid route is a dense, firmly attached layer of water-soluble polyethylene glycol, producing steric stabilization without relying on a large electrostatic barrier.

Without stabilization, the potential energy is predominantly attractive outside the molecular core and drops into a deep primary minimum. With the charging treatment, the DLVO theory sum develops a positive barrier between distant particles and that primary minimum. The barrier must be large compared with $k_{B}T$ to inhibit aggregation on the experimental timescale. A shallow secondary minimum can remain at larger separation, depending on parameters.

The horizontal axis is the surface gap, so the molecular exclusion boundary is at its origin. Both interactions tend to zero at large separation. Raising salt concentration shortens the Debye–Hückel screening length and can reduce the barrier, allowing particles to stick again.

In a magnesium chloride chemical solution, the principal pair interactions are hard sphere exclusion, repulsion of overlapping electric double layers, and attractive Van der Waals forces. The positive surface attracts chloride counterions and repels magnesium coions; mobile ions screen the bare Coulomb's law interaction. The competition between screened electrostatic repulsion and Van der Waals force attraction is described by DLVO theory.

For well-separated identical weakly charged spheres, the approximate repulsive pair potential energy, at centre separation $R\ge 2a$, isIt is a screened Coulomb pair potential. An attractive Hamaker constant adds a negative short-range contribution, while overlap is forbidden. Very near contact, solvation and molecular surface structure can also matter. No polymer-induced interaction should be assumed before polymer is added in part (f).