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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.

A grafted polymer chain is tethered to a surface at one end. Let its isolated coil size be $R\sim b{N}^{\nu }$, where $N$ is the number of statistical units and $b$ their statistical segment length. For a good solvent, the Flory approximation gives $\nu \simeq 3/5$; an ideal chain has $\nu =1/2$. Let ${\sigma }_g$ be the number of tethers per unit area, distinct from the membrane tension in Question 1. Their typical lateral separation is $D\sim {\sigma }_g^{-1/2}$.

At low coverage, $D\gg R$, the chains hardly overlap. Each forms a polymer mushroom, retaining roughly the isolated-coil conformation apart from the constraint of the solid surface. Raising coverage until $D\sim R$ makes neighboring coils overlap. The crossover criterion isFor $D<R$ in a good solvent, maintaining isolated coil shapes would crowd segments into the same lateral regions, increasing excluded-volume and osmotic free energy. Chains stretch predominantly away from the surface to relieve that crowding, at the cost of their conformational entropy. The resulting dense tethered layer is a polymer brush.

A simple scaling estimate makes the balance explicit. For brush height $H$, the monomer concentration is of order $N{\sigma }_g/H$. The free energy per chain in units of $k_{B}T$ iswhere $v>0$ is the effective segment excluded volume. The first term is Gaussian stretching; the second is the cost of segment crowding in a good solvent. Minimizing with respect to height givesWhen $v\sim b^3$, this is $H\sim Nb({\sigma }_g{b}^2{)}^{1/3}$. Height increases with chain length and grafting density, while the lateral spacing is fixed by the attachment points. The estimate assumes stretching strong compared with an isolated coil but still much smaller than contour length $Nb$; it does not describe fully extended chains. Poor-solvent attraction can instead collapse the layer, so solvent quality must be specified before using the good-solvent formula.

The mushroom-to-brush change is a crowding crossover for finite chains, rather than automatically a sharp thermodynamic phase transition. A swollen brush can provide steric stabilization: bringing two coated surfaces together further compresses and crowds their chains, producing a repulsive free-energy cost. The controlling comparison is tether separation versus isolated coil size, and the brush height follows from competition between chain stretching and segment interactions.

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.

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.

For dissolved, nonadsorbing polyethylene glycol, the usual new interaction is a depletion force. Polymer centres are excluded from a layer of thickness of order their radius of gyration $R_g$ around each particle. When the particles approach, their exclusion layers overlap. This increases the volume accessible to polymer and its translational entropy, lowering the colloid pair free energy. Equivalently the surrounding polymer osmotic pressure pushes the particles together.

In the dilute-polymer Asakura-Oosawa model, let $h=R-2a$ be the surface gap and $R_e=a+R_g$. Thenfor $2a\le R<2R_e$, and $U_{\mathrm{dep}}=0$ for $R\ge 2R_e$. Here ${\Pi }_p\simeq n_p^{(\mathrm{polymer})}{k}_{B}T$ is the reservoir polymer osmotic pressure. The overlap-volume formula follows by adding two equal spherical cap volumes with height $R_e-R/2$. Thus the added attraction has range about $2R_g$ and grows with polymer concentration. The total interaction is approximately hard exclusion plus screened electrostatic repulsion plus Van der Waals force and depletion attractions.

Increasing polymer concentration therefore creates a family of phase diagrams, rather than a unique shift determined by polymer size alone. Where the attraction overcomes repulsion, colloid-poor fluid can coexist with a colloid-rich phase, and fluid–crystal coexistence generally broadens. For sufficiently long-ranged attraction, stable fluid–fluid separation can occur. For the stipulated small polymer, the attraction is short-ranged: fluid–fluid separation can be metastable to crystallization, and a rapid attraction increase can instead produce an arrested colloidal gel. Residual long-ranged repulsion can oppose bulk separation. Simulations and free-volume calculations confirm that longer-ranged electrostatic repulsion can counteract depletion-driven separation.^[ref]

The nonadsorption assumption matters. If the polyethylene glycol adsorbs to the particles, a dense attached layer can give steric stabilization, whereas chains attaching to two particles can produce polymer bridging. The paper does not specify surface affinity or polymer concentration, so depletion is the standard conditional answer, not a prediction of an inevitable particular final phase.

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.

Polymer bridging occurs when one polymer chain attaches to more than one particle, producing an effective attraction or network connection. Surface affinity and incomplete coverage are essential to this mechanism. It should be distinguished from both steric stabilization by densely coated particles and the depletion force produced by nonadsorbing chains.

Steric stabilization is resistance to particle aggregation supplied by attached or adsorbed molecular layers. Bringing two coated particles close compresses or overlaps the layers, increasing their free energy through conformational and osmotic effects. It differs from the depletion force of nonadsorbing dissolved chains.