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For an isolated growing sphere, a quasi-steady exterior diffusion equation, constant surface concentration $c_s$, and a maintained far-field concentration $c_{\infty }$ give $c(r)=c_{\infty }-(c_{\infty }-c_s)a/r$. A sharp moving boundary with interior concentration $c_p$ then obeys $(c_p-c_s)\dot{a}=D(c_{\infty }-c_s)/a$. The displayed integrated law assumes all concentrations and the diffusion coefficient are constant. It becomes $a\propto \sqrt{t}$ when the seed radius is negligible. This reservoir-fed law differs from curvature-controlled Ostwald ripening in a depleting many-particle system.

Replacing domain-scale viscous stresses by a local linear drag density $-\bar{\eta }\mathbf{v}$ gives the scales $L_1=(\sigma \rho /{\bar{\eta }}^2{)}^{1/3}$ and $t_1=\rho /\bar{\eta }$. The single-scale force model isFor a regular power-law asymptotic $g\sim C{u}^y$, $g^{\prime }\sim Cy{u}^{y-1}$, $g^{\prime \prime }=Cy(y-1)u^{y-2}+o(u^{y-2})$, $y>0$, the two inertial-to-drag ratios scale as $(y-1)/u$ and $y/u$. Thus drag dominates inertia at late time, and drag-capillary balance integrates to $g^3\sim 3(c_s/c_d)u$. This proves the displayed $1/3$ exponent and its independence of mass density. The scaling assumes a homogenized stationary network and dominant advective transport. Diffusive Ostwald ripening can have the same exponent, so its omission must be justified separately; order-parameter mobility can otherwise enter the prefactor.

Addition or rearrangement of material in a nanocrystal. Solute addition, Ostwald ripening and oriented attachment are distinct mechanisms that can coexist.

Use the intended regular scaling ansatz: $g(u)\sim C{u}^y$, $g^{\prime }(u)\sim Cy{u}^{y-1}$ and $g^{\prime \prime }(u)=Cy(y-1)u^{y-2}+o(u^{y-2})$, with $C>0$, $y>0$. Relative to the drag magnitude $g^{\prime }$, the two inertial contributions obeyThus both inertial-to-drag ratios tend to zero at large $u$, whatever the positive growth exponent; for $y=1$ the leading acceleration term vanishes identically. The long-time drag-limited hydrodynamic coarsening balance is thereforesoThe leading law is independent of mass density. Within the drag-capillary reduction, finite initial data give $L^3(t)-L^3(t_{\ast })=3(c_s/c_d)(\sigma /\bar{\eta })(t-t_{\ast })$.

The derivative hypotheses are important: bare $g(u)\sim u^y$ does not justify differentiating an asymptotic equivalence. For instance, $g(u)=u+\frac{1}{2}{\int }_1^u\sin (s^3)ds$ is smooth and increasing, with $g(u)/u\to 1$, but $g^{\prime \prime }=\frac{3}{2}{u}^2\cos (u^3)$ is not small compared with $g^{\prime }=1+\frac{1}{2}\sin (u^3)$. This is a counterexample to an inference from the bare asymptotic relation, not a solution of the coarsening equation. The physical claim uses the regular self-similar power law, for which the ratio calculation above applies.

This conclusion concerns the model's advective drag-capillary channel. Diffusive Ostwald ripening can also have a $1/3$ growth exponent; equality of exponents does not establish that diffusion is negligible. If it is retained, the prefactor and crossover function may also depend on order-parameter mobility, and one must compare transport mechanisms rather than infer the mechanism from the exponent alone.

For a convex droplet, the Kelvin equation gives $\ln (p_{\mathrm{eq}}(r)/p_{\mathrm{sat}})=2\gamma {\overline{V}}_l/(rRT)$. Smaller droplets therefore require a higher equilibrium vapor pressure. At a common ambient vapor pressure, sufficiently small droplets evaporate while larger ones can grow, transferring material through the vapor. Ostwald ripening decreases the droplet count and increases a characteristic surviving radius. Direct collision and coalescence can also reduce the count but are a distinct mechanism. In a closed container the final state depends on total liquid/vapor inventory and volume; an initially undersaturated, liquid-poor system can lose all droplets rather than indefinitely grow a large one.

The competition between crystallization, exposed surfaces, adhesion and elastic strain can be summarized byHere $\Delta g>0$ measures the bulk driving force from supersaturation, and the $\gamma$ values are interfacial free energies. For a free spherical nucleus, $\Delta G=4\pi r^2\gamma -4\pi r^3\Delta g/3$ gives $r_{\ast }=2\gamma /\Delta g$. A support changes the heterogeneous nucleation barrier through adhesion and geometry.

Precursor activities and supply rate, temperature, time, solvent and ligands control nucleation, attachment, detachment and surface diffusion. Many nuclei divide a fixed solute inventory into smaller particles; continued growth or Ostwald ripening can enlarge them. Substrate defects determine preferred nucleation sites; lattice mismatch and interface bonding constrain orientation and elastic relaxation. Facet-selective adsorption can change both surface energies and growth rates.

Near equilibrium, a free nanocrystal follows a Wulff construction; a supported particle can follow a Winterbottom construction. Under kinetic control, slowly advancing facets remain visible while rapidly growing ones disappear. Equilibrium facet energy and kinetic growth rate are different controls, so a crystal structure does not uniquely determine particle shape.

Three useful growth mechanisms are monomer addition, Ostwald ripening and oriented attachment. They can occur sequentially or compete; they are not necessarily three distinct experimental recipes.

In monomer-addition crystal growth, dissolved atoms, ions or molecular growth units attach to an existing nanocrystal. Supersaturation provides a chemical-potential driving force, approximately $\Delta \mu =k_{B}T\ln (c/c_{\mathrm{eq}})$ for an ideal solution. Transport through solution and incorporation at steps or facets determine the rate. Growth consumes the excess solute without requiring collisions between whole crystals.

In Ostwald ripening, smaller crystals dissolve while larger ones grow through solution. Curvature raises the surface contribution to chemical potential, approximately $\mu (r)={\mu }_{\infty }+2\gamma \Omega /r$ for a sphere with surface free energy $\gamma$ and atomic volume $\Omega$. Dissolved material therefore moves from high-curvature particles to lower-curvature ones, reducing total interfacial free energy.

In oriented attachment, whole crystallites align compatible lattice planes and join. Removing exposed surface and forming a coherent interface lowers free energy; rotation, ligand removal and interface healing can be rate limiting. Misalignment can leave twins or other defects. Simple aggregation without crystallographic registration is related particle attachment, but is not necessarily oriented attachment.