Molecular Pathway From Solution to Crystal

During crystal growth from solution, the solute molecules have two possible pathways between the solution and the kinks: they can be directly incorporated (49,71), or they can first adsorb on the terraces between the steps, diffuse along them, and then reach the steps (48,49).

If a crystal grows by the direct incorporation mechanism, the competition for supply between adjacent steps is mild (71). By contrast, competition for supply confined to the adsorption phase is acute (79) ; it retards step propagation and acts as a strong, effective attraction between the steps. This dramatically affects the stability of the step train; the appearance and evolution of step bunches (51); and, ultimately, the crystal's quality and utility (97).

The two mechanisms can be directly discerned by monitoring the adsorbed solute molecules on the crystal surface, similar to experiments with metal atoms at lowered temperatures (98). However, during solution growth at room temperature, the diffusivity of the adsorbed species is approx 10-8 cm2/s (22,94); that is, a molecule passes 100 nm in approx 0.01 s. With in situ AFM, this distance is covered by the scanning tip typically approx 0.1 s; that is, imaging is too slow to detect and monitor the adsorbed molecules. Electron microscopy of flash-frozen samples has, in several cases, revealed the presence of adsorbed solute molecules on the crystal's surface (99); however, their participation in growth cannot be confidently judged by this technique. Because direct tests appear impossible, indirect evidence for the growth mechanism of several systems has been sought.

For several solution-grown crystals, the growth mechanism has been deduced by comparing the velocities of isolated steps with those of closely spaced steps. Similar values of the two velocities for silver (100) and calcite (15,16) were taken as evidence for the direct incorporation mechanism. Conversely, slower growth of dense step segments was interpreted in favor of the surface diffusion mechanism for KDP/ADP (21,94), lysozyme (101), and canavalin (22). A known problem for such mesoscale data is that the data sets interpreted in favor of direct incorporation could also reflect a surface diffusion range shorter than the shortest step separation probed (15,16). Thus, critical evidence about the growth mode should be sought by studying the growth processes at the molecular level (10,25).

As shown in Subheading 7.3., in the case of direct incorporation from the solution (see also refs. 72 and 78), j - j_ = v C Qexp -

Here, Umax is the energy barrier for incorporation into the kinks (50,71); in the case of ferritin, it likely accounts for the need to expel the water molecules structured around hydrophilic patches on the surfaces of the incoming molecules and the molecules forming the kink (82). Q = 1.56 x 10-18 cm3 is the crystal volume per ferritin molecule; D = 3.2 x 10-7 cm2/s is the ferritin diffusivity (37); and A is the radius of curvature of the surface-molecule interaction potential around its maximum at Umax (73,74) and, hence, should be of the order of a few water molecule sizes, approx 5-10 A (95). The step velocity v for this growth mode is a ac D u a / ■ ■ \ e D I max v = — (j + - j ) = —--exp--

Analogous considerations for the case of growth via surface diffusion yield for the net flux into the step from the surface are

kBTl n

A a j n in which ns and ne are the surface concentration of adsorbed ferritin and its equilibrium value, respectively; Us0 is the energy barrier for incorporation into the kink from the surface; and Ds and As are, respectively, the surface diffusivity n and curvature of the surface Us. For the step velocity, one gets through v = a/n k(js+ - js ) an expression analogous to Eq. 25.

To evaluate the ratios of the fluxes in and out of step, we use that Fig. 9 reveals that for ferritin at C/Ce = 2, jjj_ < 1.105. For apoferritin, similar experiments in Fig. 8 show that at C/Ce = 3, j+lj_ < 25/22 = 1.14. For both proteins, these ratios represent gross violations of the last equality of Eq. 27. These violations cannot be attributed to depletion of the solution layer adjacent to the crystal. This factor becomes significant at approx 100 times higher growth rates (102) and suggests that the direct incorporation mechanism may not apply. In the case of Langmuir adsorption, ns = ns C(B + C)-1 (B - Langmuir constant) and ns/ne < C/Ce. Hence, the lower ratios of the in- to out-flux are compatible with a mechanism of incorporation from the state of adsorption on the surface.

For further tests of the growth mode, we examine the step velocity law in Eq. 28. The only unknown parameter here is the energy barrier U0. Determinations of v at four temperatures and two ferritin concentrations in Fig. 13 yield Etotal = 41 ± 3 kJ/mol. In Eq. 28, Ce (56) and Q do not depend on temperature, and A is about the size of a few water molecules and in a first approximation does not depend on T (95). For a molecule following the Stokes law, D = D0 exp(-Evisc/kBT), where Evisc is the temperature factor in an Arrhenius-type expression for the dependence of the solvent viscosity on temperature. For NaCl solutions in Na acetate buffer, Evisc = 7.4 kJ/mol (38). As shown in Eq. 28, nk has a weak near-exponential dependence on T through the kink energy w = 3.8 kJ/mol. This leaves U0 ~ 30 kJ/mol. This value is close to the 28 kJ/mol found as the average over systems ranging from inorganic salts, through organics, to proteins and viri (81).

Substituting into Eq. 28, we get at C/Ce = 2, v = 0.0014 nm/s, and at C/Ce = 3, v = 0.0028 nm/s. These values are more than two orders of magnitude lower than actually observed. The measured values of 0.20 and 0.31 nm/s would require Umax ~ 18 kJ/mol, beyond the range of the determination in Fig. 13. This discrepancy supports the assertion that the direct incorporation mechanism is inapplicable to the growth of ferritin. We conclude that a mechanism involving adsorption on the terraces better corresponds to the available data for ferritin. As noted, in the ferritin/apoferritin system, the steps only exhibit attraction at very short separations. We conclude that the characteristic surface diffusion length (49) must be shorter than a few lattice parameters to account for this. Note that an investigation limited to data on the mesoscale step kinetics would have concluded that the growth mechanism is direct incorporation.

A relevant question is: why do the energetics of the system select the surface diffusion mechanism over the direct incorporation? This question can only be addressed with the molecular-level data available for the system. We note that when the surface diffusion mechanism operates, the energy barrier determined

Fig. 13. Dependencies of step velocity v for growth of ferritin on temperature (A), and in Arrhenius coordinates (B). (■) at C/Ce = 4; (□) at C/Ce = 3. For each point, positions of advancing steps were compared in sequences of molecular resolution in situ AFM images; approx 20 such determinations of v were averaged. The error bars represent the 90% confidence interval of the average. (From ref. 31.)

Fig. 13. Dependencies of step velocity v for growth of ferritin on temperature (A), and in Arrhenius coordinates (B). (■) at C/Ce = 4; (□) at C/Ce = 3. For each point, positions of advancing steps were compared in sequences of molecular resolution in situ AFM images; approx 20 such determinations of v were averaged. The error bars represent the 90% confidence interval of the average. (From ref. 31.)

from the data in Fig. 13 is a function of the barriers of the elementary steps of this mechanism and should be denoted as Usum. As shown in refs. 79 and 95, Usum = Uads - Udesorb + USD + Ustep, which are the barriers, respectively, for adsorption, desorption, surface diffusion, and incorporation into the step (see Fig. 14). Since the energy effect of one intermolecular bond of ferritin should be equal to that of apoferritin, Q = 3kBT = 7.4 kJ/mol (25), we can safely assume that for adsorption - desorption on a (111) f.c.c. surface, Uads - Udesorb = DHads = -3Q = -22 kJ/mol. When interactions between the adsorbed molecules are ignored, the lowest possible value of USD occurs when only one bond is broken on passage between two adsorption sites; hence, USD > Q. This yields Ustep < 44 kJ/mol, similar to the ADP value (95). Since an equal number of bonds, three, are created during adsorption and incorporation into the step, we can roughly assume Uads ~ Ustep. Thus, the highest barrier encountered by a

Fig. 14. Energy landscape of surface diffusion mechanism. For notations, see the text. (Modified from ref. 109.)

Reaction Pathway

Fig. 14. Energy landscape of surface diffusion mechanism. For notations, see the text. (Modified from ref. 109.)

molecule en route to the kink is <44 kJ/mol. For direct incorporation into kinks, for which all of the six bonds are created simultaneously, Ukink ~ Uads + Ustep ~ 88 kJ/mol. A crude estimate yields that this would make growth via this pathway slower by a factor of ~exp[(88,000 - 44,000)/R7] ~ 108.

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