MNET Applied to Micelles

48 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al. has D m =1 B = 2D m =4 B since the hydrodynamic radius of the tetramer is twice so big than that of the monomer. Therefore D m =4 B = D m =1 B 2 holds, when compared to that monomer-oriented case. Thus, to properly account for the overall rate increase andor decrease competition net effects expected in the modeled crystal growth, let use define ν comp = ν di f f ν mass . The quantity in the numerator points to deceleration effect, carrying from the bulk effect on the diffusion within the interface whereas its denominator counterpart accounts for the acceler- ating mass effect. It gives ν comp = 1 2 3 4 = 2 3 , so that D m =4 S = D m =1 B 100 × 23, which finally reads D m =4 S = D m =1 B 150, and which above all conforms in an excellent way to the experimental results, cf. [15, 16, 43] and figures therein. This is also indicated by Fig. 2 in Ref. [18] obtained by computer simulation. Note that all figures mentioned point to the crystal-growth acceleration rate of ca. 4 3 when compared to the non-template case. In com- parison to our previous study[18] the application of tetrameric growth units modifies growth rate by multiplying it by ν comp , because the simulation time t, involved in the equation which expresses the growth rate, see Eq. 4 in [18], is expressed by D m =4 S = D m =1 S × ν comp . Ultimately, the obtained growth rate of modeled structure for the tetrameric growth units V m =4 gr is about 4 3 times greater than that derived for the case when monomeric growth units yield the crystal V m =1 gr , cf. [43]. Summing up in part, it is now legitimate to conclude that all effect on the accelerated crystal formation likely comes from the contribution of masses of the tetramers relative to the masses of monomers, namely by realizing that the inverse of ν mass actually reads ν mass −1 ≡ 4 3 which is formally the experimentally confirmed factor[43] of interest realize that the coefficient D also depends on mass[55].

2.3. Towards Aggregates Micelles’ Formation in Close-to-Equilibrium

Conditions

2.3.1. MNET Applied to Micelles

Amphiphiles such as lipids and surfactants can assembly into a variety of nanostructures in aqueous solutions. These may transform from one to another by changing the solution con- ditions such as lipid or electrolyte concentration, temperature or pH. They self-assembly most frequently into simple micelles, inverted or reverse micelles, bilayers or bilayer vesi- cles, etc. [2, 24] E.g., the aggregates of lipid molecules emerge as a result of association of a certain number of monomers into an aggregate; since the lifetime of an amphiphilic molecule in a small aggregate, let us say a micelle, is very short, typically of the order of 10 −4 s, one can anticipate both association to and dissociation from the aggregate, experienced by a single lipid molecule. A phenomenon of special interest appears to be the restoration of quasi-equilibrium of the micellar nanostructure when a T -jump or p-jump, or finally the so-called shock- tube conditions occur [99]. If such conditions are becoming effective, an initial deviation from the quasi-equilibrium results. It can be viewed as a reversible chemical reaction with different associationdissociation constants. The resulting departure from the equilibrium can be measured as a relative time-dependent departure from the equilibrium micellar concentration. It apparently results in a matter flow emerging in the micellar system. This Can Modern Statistical Mechanics Unravel Some Practical Problems . . . 49 flow can be studied in terms of a diffusion in a tube [36]. This tube can be viewed as a dynamic geometrical object in which at different positions, measured along the tube’s axis, different micellar cross-sections appear - note that the problem is well justified by a good number of experiments [100]. The MNET approach, equipped with its basically close-to-equilibrium fluctuation-dis- sipation context, and being strongly motivated by having the objects’ viz micelles’ sizes, and shapes, thoroughly incorporated bear in mind the case of non-Kossel crystals and spherulites presented above, does offer a very good prerequisite as to reconsider the curvi- linear dynamic-tube problem primarily in terms of micellar quasi-equilibrium sizes, post- poning somehow their locations along the tube’s central axis, cf. [67]. By virtue of the robust practical approach carefully revealed and extensively discussed in [99], we can propose, see their Eq. 7, the following matter flux J in R–space R - micelle’s radius, namely J R, t = −DRAR ∂ ∂ R ρ R, t, 65 where D R - a diffusion coefficient of the micellar formation; ρ R, t is a nonequilib- rium concentration-mimicking probability density, being formally analogous to the PDF, abbreviated by PDF, ρ r, t from the Sec. 2.1.10. as taken in a position space, but here specified in a configurational R-space of micellar sizes, with R as above; and the most im- portant micellar excess quantity A R, giving a deficiency of quasiequilibrium N-monomer micelles’ concentration relative to the concentration of the monomers kept in a dynamic quasi-equilibrium state with the micelles from which the micelles are being eventually formed, can be defined as A R = c eq N −micelle R c eq monomer R , 66 since the monomer versus N-aggregate micelle reversible random elementary processes appear to be most decisive after an external loadshock, thereby attracting one’s attention to the restoration of a quasi-equilibrium in the micellar system. Note that the weighting function A R = 1 when there is no micellar state at quasi-equilibrium, and the system is purely homogeneous well-mixed. Moreover, let us realize that, by construction of A R, typically 0 AR 1 holds, which is very characteristic of inhomogeneous ill-mixed IFS environments. It implies that such a modification to the first and second Fick’s laws, i.e. Eqs. 65 and 67, respectively, introduced by means of the A R-departure from the quasi- equilibrium, cf. Eq. 66, defines the process to be: i slowly diffusing along the micelle-size axis R, cf. Eq. 65; ii slowly varying over time t, cf. Eq. 67. The above i-ii qualify the so-constructed process to be close to its stationary state but still slightly out of it. Then, let us assume that the monomers-exchanging diffusion-controlled process goes by a slightly modified i.e., weighted by A local micellar mass-conservation law A R ∂ρ R, t ∂ t + ∂ J R, t ∂ R = 0, 67 cf. Eq. 6 in Ref. [99]. 50 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al. The MNET-justified assumption on D R ∝ R d E −1 , where d E - space dimension, appears to be of special importance here [38]. This assumption simply tells us that the monomers’ exchange process goes via micelles’ surfaces in any space dimension d E . This can be intentionally described as the principal geometric assumption of the physics of such colloid systems [2]. Thus, it points to the exchange process as the most efficient when the micelle’s surface is sufficiently large, additionally viewing the problem as d E -dependent [101]. The equations 66 as well as 67, certainly with D R ∝ R d E −1 , enable someone to see the kinetics of micellar formations in terms of MNET, this way easily recovering, e.g., striking Ch. Tanford’s results on micelle shape and size [102], formally reconstructing in possibly general terms the micellar phase diagram, cf. Fig. 1 therein. From MNET, when anticipating the FPS scenario - it follows that the micellar, close-to-equilibrium, but distinct from equilibrium concentration of micelles, i.e., J = 0 stationary state, appears to be inversely proportional to the micelle’s active viz catalytic surface R d −1 , namely ρ R ∝ 1 R d E −1 , 68 where, in general, ρ R 6= c eq N −micelle R; the “singular” case of ρ R 6= c eq monomer R can be mentioned too. Summing up about the kinetics of the the micelles’ formation topic seen in terms of MNET , in any shock-involving experiments, e.g. that one of about 1kbar p-jump acting on a model lipid membrane, cf. [23], and refs. therein, one is left with an extremely complex problem, having its, so to say, ’own’ viz internal time scale: Therefore it seems plausible to offer a simple-minded extension by having legitimate the passage of D R → DR, t, with D being this time an explicit function of time t: This leaves the problem as a truly nonequilibrium problem, also advocating for having “additional” close-to-equilibrium or quite beyond it? states such as all the ones, represented by the inequality: ρ R 6= c eq R. This reasoning defines ultimately the problem as non-Markovian, again, or specified in mi- crorheological terms as some aging problem [103]. This all together sets a quite fundamen- tal property thereof, pointing to the fact that the problem is always properly catalyzed by biggersmaller micellar surface available for the matter exchange already mentioned. Last but not least, let us note that two-dimensional spontaneously formed aggregates, emerging in d E = 2, such as discs cylindrolites, in case when they additionally suffer from some de- gree of crystallinity or sheets say, model membranes [104] are also the naturally appear- ing objects under the MNET-involving context described [105]. Therefore, the approach proposed therein seems worth reconsidering in terms of A-weighted modification prop- erly compared to available computer simulations, see above, also revealing an adequately conveyed context coming from applying the first law of thermodynamics and the entropy production staying firmly behind it [13]. It has already found its seminal-literature based justification, see [2], chap. 16 therein. In addition, the explicit d E -dependence seen above points to the number of freedom degrees’ of the system - the problem closely related with close-packing conditions, also of very relevance for micellar systems [101, 2]. Can Modern Statistical Mechanics Unravel Some Practical Problems . . . 51

2.3.2. MNET in an Effectively 2D Computer Experiment on Model Biopolymer Ag-