Colloid

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.

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.

A dispersion of gas bubbles in a liquid or solid matrix. Thin intervening films and interfacial forces govern stability; bubble diameter need not itself lie below a micrometre.

A fluid colloidal system. A solid-in-liquid example is fine silicon dioxide dispersed in water. The term can also include other fluid colloidal systems, so the dispersed and continuous constituents should be specified.

A force caused by confinement or restructuring of solvent near interfaces. It can depend on molecular layering and specific surface chemistry and need not have the range or sign of an electrostatic double-layer force.

A short-range solvent-mediated interaction associated with water organization and hydration near surfaces. It is distinct from an ideal point-ion double layer and can contribute repulsion at separations where continuum far-field models fail.

A dispersion of droplets of one liquid in another immiscible liquid. Its interfacial energy and kinetic persistence depend on droplet size, interfacial tension and stabilizing species.

Merging of droplets after their separating film fails. This differs from clustering of still-distinct droplets. Interfacial protection can delay merging without making the two bulk liquid phases molecularly miscible.

A surfactant-stabilized dispersion can remain macroscopically uniform while its two liquids remain distinct phases. Lower interfacial tension reduces the work of making droplets, and repulsive interfacial films can slow droplet coalescence. Kinetic emulsion stability does not by itself establish equilibrium molecular miscibility or a thermodynamically stable microemulsion.

An emulsion stabilized by particles adsorbed at the liquid–liquid interface. A partially wetting sphere has a detachment-energy scale $\pi r^2\gamma (1-|\cos \theta |{)}^2$, which can greatly exceed $k_{B}T$ for nanoparticles; complete wetting and insufficient coverage prevent a universal stability guarantee.

Aggregation caused when an adsorbing polymer chain binds more than one particle. Incomplete surface coverage can favour bridges, whereas a densely covered, well-solvated surface can instead be sterically stabilized.

Aggregation of colloidal particles when encounters lead to persistent sticking. Lowering an electrostatic barrier can change a suspension from reaction-limited to approximately diffusion-limited aggregation.

The distribution of particle dimensions in an ensemble, with number-, volume- or intensity-weighting specified. Broad size distributions can broaden a melting transition, frustrate colloidal crystallization and bias scattering-based size estimates. A mean size alone is then insufficient.

A spread of particle or molecular dimensions rather than a single size. A quantitative measure must state the weighting and statistic; particle-size variation and polymer chain-length variation are related examples but use different common indices.

The weight-average molar mass divided by the number-average molar mass. Its conventional written symbol is Đ. With number weighting it equals $1+\mathrm{Var}(M)/⟨M{⟩}^2$, so it is at least one and equals one only for a uniform molar mass. It measures one aspect of distribution breadth, not the full shape of a molar mass distribution.

An ordered arrangement of colloidal particles. Nearly monodisperse hard-sphere-like particles can order through packing entropy at sufficient concentration, while weak reversible attractions or templates can guide ordering. An irreversibly aggregated solid is not necessarily a colloidal crystal.

Resistance of a dispersion to aggregation or irreversible separation. Electrostatic and steric interactions and solvent compatibility can all matter.

For a sphere of radius $a$ and buoyant density difference $\Delta \rho$, define ${\mathrm{Pe}}_g=v_{s}a/D=m_{b}ga/(k_{B}T)$ using radius as the transport length. With Stokes law and Stokes–Einstein diffusion, ${\mathrm{Pe}}_g=4\pi \Delta \rho g{a}^4/(3k_{B}T)$. Replacing radius by diameter changes the convention by a factor of two. This local criterion does not guarantee a uniform suspension over a macroscopic vessel.

Setting the gravitational Péclet number to one gives the displayed particle-scale estimate. A macroscopic sample instead requires its height to be small compared with the sedimentation scale height. These different observational criteria give different size thresholds; “no sedimentation” is not an exact universal cutoff.

For noninteracting particles with positive buoyant mass $m_b$ in a gravitational field, a dilute thermodynamic equilibrium profile is proportional to $e^{-z/{\ell }_g}$ with ${\ell }_g=k_{B}T/(m_{b}g)$. Uniformity over a vessel height $H$ requires $H\ll {\ell }_g$. Requiring only ${\ell }_g$ to exceed a particle radius is a weaker microscopic gravity-versus-diffusion condition.

For gently curved interacting surfaces whose range and closest gap are small compared with their radii, integrate the local planar interaction over nearly parallel surface elements. With $W(s)$ the planar interaction free energy per area and $\Pi (s)=-W^{\prime }(s)$, the local gap for two spheres is $s\simeq h+r^2/(2R_{\mathrm{eff}})$, where $R_{\mathrm{eff}}^{-1}=a_1^{-1}+a_2^{-1}$. Thus $F=2\pi {\int }_0^{\infty }r\Pi (s)dr=2\pi R_{\mathrm{eff}}{\int }_h^{\infty }\Pi (s)ds=2\pi R_{\mathrm{eff}}W(h)$ when $W(\infty )=0$. Equal spheres have $R_{\mathrm{eff}}=a/2$; a sphere and plane have $R_{\mathrm{eff}}=a$. Use consistent surface boundary conditions when supplying the planar interaction.

For two equal spheres of radius $a$ separated by a surface gap $h\ll a$, a positive Hamaker constant $A_H$ gives the nonretarded Van der Waals force approximation $U=-A_{H}a/(12h)$ and $F=-dU/dh=-A_{H}a/(12h^2)$. The Derjaguin approximation derives the geometry factor from the planar interaction energy per area $W=-A_H/(12\pi h^2)$ and effective radius $a/2$. A sphere and plane instead have effective radius $a$, doubling the force at the same gap. Molecular repulsion, roughness and retardation limit these continuum formulas.

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.

Association of dispersed particles into larger clusters, often after repulsive stabilization is weakened. Salt screening, reduced surface charge, polymer bridging or depletion force can promote it. Clusters may sediment rapidly; preventing aggregation and preventing gravitational separation are distinct stability requirements.

In the standard single-seed model, individual particles execute random walks until first contact with a growing cluster, where they stick irreversibly. Exposed tips intercept more particles than screened inner regions, producing branched clusters. This differs from diffusion-limited cluster aggregation, in which whole existing clusters move and join one another. The two models need not share the same mass fractal dimension.

In reaction-limited cluster aggregation, encounters occur frequently compared with successful attachment: an activation or interaction barrier gives a small sticking probability. Repeated unsuccessful approaches allow clusters to sample more relative configurations before bonding, commonly producing denser aggregates than diffusion-limited cluster aggregation. Typical three-dimensional isotropic irreversible cluster models give mass fractal dimension around $2.1$. This structural contrast does not require previously formed bonds to break and restructure; it can arise from unsuccessful encounters alone. ^[ref]

In diffusion-limited cluster aggregation, mobile clusters undergoing Brownian motion attach with probability close to one when they encounter another cluster. Irreversible first-contact sticking tends to trap particles in exposed outer branches, producing open aggregates. In the common three-dimensional isotropic cluster model, the mass fractal dimension is roughly $1.8$. This is different from depositing individual random walkers onto one fixed cluster. Rotational motion, anisotropic attraction and bond restructuring can change the morphology.

For two dilute spherical species with absorbing contact and negligible interaction barriers, relative diffusion has coefficient $D=D_1+D_2$ and contact radius $R=a_1+a_2$. The steady concentration outside one target is $c(r)=c_{\infty }(1-R/r)$, solving the radial Laplace equation with zero contact concentration. Its inward diffusive flux is $4\pi DR{c}_{\infty }$, proving the displayed bimolecular rate coefficient. For identical particles, converting this encounter coefficient into a rate of loss of cluster number additionally needs the factor that prevents counting the same pair twice.

A colloidal glass is a disordered dense particle assembly with structural relaxation too slow on the observation timescale. Crowding, repulsion or short-range attraction can impede rearrangement. Kinetic arrest must be distinguished from an equilibrium phase transition to a crystal.

A colloidal gel is a mechanically persistent, space-spanning particle network arrested on the observation timescale. Attractions can produce such a network during aggregation or arrested phase separation. A gel is a dynamical, often nonequilibrium state; its appearance does not establish an additional equilibrium thermodynamic phase.

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.

Repulsion associated with overlap of surface-bound layers, often through chain confinement and osmotic free-energy costs. Layer thickness, solvent quality and grafting/adsorption state matter; it is not simply a bare hard-sphere collision.

Stabilization by a surface-bound polymer layer that supplies both conformational/osmotic repulsion and electrostatic repulsion. Chain coverage, solvent quality and screening determine the resulting interaction barrier.

A depletion force is an effective interaction caused by exclusion of smaller particles or macromolecules from regions around larger objects. Overlap of the excluded regions increases the accessible volume for the smaller species. At fixed reservoir osmotic pressure ${\Pi }_p$, the associated pair free energy is $-{\Pi }_p$ times the overlap volume in the ideal dilute-depletant approximation. It is an entropic attraction for two large hard particles. Surface adsorption can introduce different interactions, so adding an arbitrary polymer does not by itself guarantee depletion attraction.

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.

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 deep short-distance minimum can bind contacting particles, while a more distant shallow minimum can permit loose reversible aggregates. A repulsive barrier between them controls kinetic access to contact. Salt can lower that barrier through screening. Molecular-scale steric or hydration forces lie outside the simplest continuum dispersion-plus-double-layer model.

For a symmetric monovalent reservoir with number density $n$ of each species, linearizing Poisson-Boltzmann theory gives the displayed inverse screening length. A planar potential decays exponentially; a spherical far field also has a geometrical inverse-distance factor. Increasing salt reduces the range of repulsion. Fixed surface potential, fixed charge and charge regulation need not give identical interaction amplitudes.

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.

Strong sensitivity of a lyophobic colloid's critical coagulation concentration to counterion valence. A high-surface-potential DLVO approximation gives $c_{\mathrm{ccc}}\propto z^{-6}$; specific adsorption and charge regulation can invalidate this simple scaling.

A screened Coulomb pair potential models the effective electrostatic interaction of particles in an electrolyte. For identical weakly charged spheres of radius $a$, charge $Q$, and Debye–Hückel screening length ${\kappa }^{-1}$, a common pairwise approximation is $U(R)=Q^2{e}^{-\kappa (R-2a)}/[4\pi \epsilon (1+\kappa a{)}^{2}R]$ for $R\ge 2a$. The near-contact interaction can require more detailed boundary conditions or nonlinear screening. Charge, screening length and temperature all matter to a resulting phase diagram.