= Solution
A <colloid> has dispersed constituents whose relevant dimensions are mesoscopic: roughly $1\,\mathrm{nm}$ to $1\,\mathrm{\mu m}$ 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.<Definition and classification of colloids by International Union of Pure and Applied Chemistry>[Definition and classification of colloids: approximate size range, colloidal dispersions, sols, emulsions and foams]{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> is
$$
U_{\rm vdW}(H)\simeq-\frac{A_Ha}{12H},\qquad H\ll a,
$$
where 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 $k_BT$, 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 $k_BT$ 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.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2016/paper-1-colloid-growth.png]
{title=An illustrative interaction barrier and the radial concentration field around a growing <sphere>}
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 \b[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 \b[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 \b[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 gives
$$
\frac{\partial c}{\partial t}+\nabla\cdot\mathbf j=0
\quad\Longrightarrow\quad
\frac{\partial c}{\partial t}=D\nabla^2c.
$$
An 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 $\partial c/\partial t\simeq0$. Thus the exterior field solves the homogeneous <Laplace equation>,
$$
\nabla^2c=\frac1{r^2}\frac{d}{dr}\left(r^2\frac{dc}{dr}\right)=0,
\qquad c(a)=c_s,\quad c(\infty)=c_\infty.
$$
Two integrations give $c=A+B/r$. The two <boundary conditions> fix
$$
\boxed{c(r)=c_\infty-(c_\infty-c_s)\frac ar,\qquad r\geq a.}
$$
For growth, $c_\infty>c_s$, so the profile increases outwards. Its radial <flux> is negative, and the positive inward arrival rate is
$$
\dot n=4\pi a^2D\left.\frac{dc}{dr}\right|_{a^+}
=4\pi aD(c_\infty-c_s).
$$
A 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>, is
$$
4\pi a^2(c_p-c_s)\dot a=4\pi aD(c_\infty-c_s).
$$
The 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> law
$$
\boxed{a^2(t)=a_0^2+\frac{2D(c_\infty-c_s)}{c_p-c_s}\,t.}
$$
Consequently \b[$a\propto t^{1/2}$ when the growth contribution dominates $a_0^2$]. For a finite seed at the literal limit $t\to0$, 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 $c_{\rm eq}$. In the dilute-feed approximation, the <Robin boundary condition> $Dc'(a)=k_s[c(a)-c_{\rm eq}]$ gives
$$
\dot a\simeq\frac{D(c_\infty-c_{\rm eq})}{c_p(a+D/k_s)}.
$$
For $k_sa/D\gg1$ this recovers transport control; for $k_sa/D\ll1$, the <surface-reaction-controlled particle growth> has approximately constant $\dot a=k_s(c_\infty-c_{\rm eq})/c_p$. Thus \b[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.
Back to article page
