Into a Kink in the Case of Diffusion Limited Kinetics

We derive an expression for the kinetic coefficient of growth of crystals from solution as diffusion over an energy barrier U, followed by unimpeded incorporation (73,74). The barrier U may be of electrostatic origin (74); however, for the ferritin/apoferritin pair, it probably accounts for the need to expel the water molecules attached to the incoming molecules and to the growth site (82). Repulsive potentials owing to water structuring at hydrophobic and hydrophilic surface patches can have significant strength and range (83,84).

Fig. 12. Schematic illustration of the potential energy relief in front of growth interface. For details, see the text. (From ref. 40.)

In Fig. 12, the resulting potential relief is schematically depicted. The potential reaches its maximum value Umax at the crossing of the increasing branch, owing to the repulsion between the incoming solute molecules and the crystal surface at medium separations, and the receding branch, which corresponds to the short-range attraction required if the molecules should enter the growth site. We position the beginning of the coordinate axis x = 0 at the location of this maximum. We assign a finite curvature of U(x) about this maximum and link it with the expulsion of the last few solvent molecules as an incoming solute molecule joins the crystal. The finite curvature assumption follows previous solutions to similar problems (85,86). The distance d, used as the upper integration limit, is bound from below by the range of interaction of the solute molecules with the surface, which can be a few solute molecular sizes, and from above by the distances between the solute molecules in the solution bulk, n-1/3 ^ 0.2 mm. Thus, d can be chosen significantly longer than the molecular sizes and kink lengths, and using a one-dimensional model is justified. Since the rate of diffusion over a sharp barrier only depends on the curvature around the maximum (85,86), the choice of d does not affect the result

To calculate the flux J of molecules with concentration n that, driven by a concentration gradient, overcome a barrier to reach the surface, we orient the coordinate x perpendicular to a growing surface and denote the potential relief close to this surface as U(x). From the generalized Fick law, J = (nD/kBT)dyddx, with |(T,x) = |0(T) + kBT ln[yn(x)] + U(x) and 1 (10), J is linked to U(x), n(x) and the gradient of n as (73,86)

dx dx I

with D being the Stokes diffusion coefficient of the molecules. In search of a steady J = const, we integrate Eq. 17 with two sets of boundary conditions:

(1) that at a certain distance from the surface S, x >8, U = 0, and n = n8; and

Dividing by D and multiplying both sides by exp[U(x)/kBT], we get

J exp[U(x)/kT] = — n(x) exp[U(x)/k„T] D B dx\ B

Integrating from x = 0 to x = 8, using the boundary conditions at x = 0 and x >8, we get

an analog to equation 9.51 in ref. 74 and the Fuchs expression for coagulation of particles interacting through U(x).

If U(x) has a sharp maximum at x = 0, we can represent it with a symmetric function around the point of the maximum. As shown below, in many cases \d2U/dx2\ < a, and this justifies the assumption of a sharp maximum. We use only the first two members of the Taylor series: U(x) = Umax - 1/2\d2U/dx2\x2. The minus sign stems from d2U/dx2 < 0 at the maximum. Then, the integral

" kBT

" kBT


d2( U/kBT)



The approximate equality above is based on 5 >> [1/2\d2U/dx2\] 1/2, the half-width of the Gaussian function in Eq. 20 Finally,

" kBT

Note that only half of the flux J from Eq. 21 contributes to growth: on top of the barrier, the force driving the molecules into the crystal is zero, and a molecule has equal chances of getting incorporated, or going back to the solution (87). With this, and introducing the parameter L as the radius of curvature of U(x)/kBT at its maximum


d2( U/kBT)


the expression for J becomes n ( U

Equation 23 is essentially identical to the nucleation rate expression derived by Zeldovich (86) as a diffusion flux over a potential barrier in the space of cluster sizes.

If Umax is owing to the hydration of the incoming molecule and the site where it attaches, the radius of curvature of U(x) around Umax should be the size of a few water molecules, 2 to 4 Á, and the length A should be approx 510 Á. Note that in this evaluation, we apply discrete considerations to a continuous model. Still, we expect the estimate of A to be roughly correct.

If all molecules that overcome the barrier are incorporated into a kink, the incoming flux into a kink is j+ = JDSkink ~ Ja2, in which a2 is an effective surface area of a kink. If there are no solute transport constraints (kinetic growth regime), nd is equal to that in the solution bulk n. Furthermore, in equilibrium, when n equals the solubility, ne, j+ = j_. Since j_ does not depend on n in the solution, the step velocity v is a / • • \ a n I Umax i / \ /r\ a \

Because Brownian diffusion does not depend on the molecular mass, the above model yields a mass-independent kinetic coefficient (for details, see the supplementary information). The resulting step growth rate v is a / • • \ a n I Umax I / \

in which L contains the radius of curvature of U(x) around its maximum and, hence, is likely the size of a few water molecules, approx 5-10 Á.

With a3 = W, we can rewrite Eq. 25 in the typical form of Eq. 16 that is readily comparable to experimental data (40). This defines P as

The parameters in this expression, nk, D, A, and Umax, have a clear physical meaning and can be independently measured (40).

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