Nonequilibrium Boundary Condition t + div[J t] = 0,

Can Modern Statistical Mechanics Unravel Some Practical Problems . . . 21 accepting the argumentation given in the literature [39, 41], let us confine ourselves to choose a ballistic-type case in which for a protein-type system the matter flux is of mass- convective nature [39] J r, t = cr, tvr, t, 4 where v r, t represents the velocity vector field, acting mostly in the interfacial region that can be limited to the Stern-type macroion-depleted double layer enclosing the growing object, see Fig. 2 [39, 42]. Figure 2. Cartoon of the Stern-type double layer surrounding the SPC-A. The lysozyme- type proteins, performing their random walks two trials are depicted, would exceed by their mean-free paths contributed to by a λ B the width of the layer, λ DL , which makes mostly time-correlational contribution to the formation dynamics, cf. text for additional explanation. This influences the form of the deterministic kinetic equation that results from ap- plying, here without any substantial loss of generality, Eq. 3 with Eq. 4 together, to a spherical object of radius R, with C = const. too, assumed that now cr, t because of spherical symmetry finally becomes c r, t = c e R. The equilibrium concentration c e R C can then be taken, like in the Mullins-Sekerka MS instability concept, as the one given by Gibbs-Thomson GT condition c e R = c o [1 + Γ D × K 1 R] Γ D - capil- lary constant; K 1 R = 2R - the mean curvature but now applied for low-dimensional viz non-macromolecular crystals emerging from a supersaturated solution. The above yields the kinetic equation as follows d dt R = σ R e × v mi , 5 where the dimensionless supersaturation σ R e = c e RC − c e R, and v mi can be consid- ered as a parameter, herein an average constant velocity of the macroion near the spheroid surface. This is by no means the case of high-dimensional or protein crystal growth, cf. Fig. 3: Here a modification of the GT boundary condition has to be expected [39]. This and the stochasticity of v r, t thus, considered in a scalar form will be discussed below.

2.1.2. Nonequilibrium Boundary Condition

Modification of the GT equilibrium boundary condition for the protein crystal formation results in either some modified GT condition of equilibrium type originating from more refined consideration about the molecular nature of the surface tension, and the additional 22 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al. Figure 3. Protein crystallization by LBS thin film nanotemplate: LBS nanotemplate, used to modify classical hanging drop method [43], trigger the specific aggregates’ forma- tion in the protein drop solution and thereby helps to overcome the nucleation free energy barrier, see Fig. 13. e.g. curvature terms it may contain or gives rise to readily nonequilibrium effects at the boundary, especially the ones coming from elastic interactions [39] between the boundary proteins and its nearby electrolyte surroundings also, within the boundary when some addi- tional strain between the vacancies may arise, or when spherulites composed of fibrils tend to emerge, even with a tendency of lateral growth violating here somehow the continuous- growth mode invoked by Cahn which suits also our type of reasoning [41]. Thus, the overall nonequilibrium boundary condition can be proposed as follows [39, 42] c ne R = c o h 1 + Γ D × K 1 R + δ T 2 × K 2 R + Σ 3 i =1 α i x i R − β k × dR dt i , 6 where δ T Tolman length, K 2 r = 1R 2 Gaussian curvature, α i the elastic coefficients and x i R = R i − R i o R i o R o is the initially taken object’s radius, with the latter representing three main crystal-surface nucleation mechanisms [39]. β k stands for the kinetic coefficient, and dR dt points to the readily nonequilibrium character of the boundary condition [42], expected to occur for spherulites, especially those of lysozyme type. Because of stating Eq. 6 in such a general form one is then able to write down a more general kinetic equation [39] d dt R = σ R ne × v mi , 7 where σ R ne = c ne R C − c ne R 8 under a necessary restriction that all nonequilibrium mechanisms prescribed at the bound- ary, cf. Eq. 6, do not need to operate at the same time together. They can rather be switched onoff whenever physically legitimate [44, 40, 41, 45].

2.1.3. Protein-Involving nearby Velocity Field