Description of Dynamic Friction-Biolubrication Applied to mAC at a sub

80 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al. the SAPL in their squeezed, mixed and elongated states, characterized by the corresponding mechanical exponents, γ , where γ = 1 3 ; 1 2 ; 3 5 , respectively [23]. Note formally, that: i the function κ above includes another exponent, h, obeying [21] h = 1 − d S 2 ; 73 ii by a simple analytic inspection it is easy to show that the ODEs system 69-71 is possible to obtain in such a least-dissipation autonomous closed form only if h = 3 γ − 1. 74 Otherwise, it can be exclusively derived in its explicit time-dependent form, thus going easily out of control both mathematically and physically [21]. The temporal behavior, represented by system 69-71 can be explored mainly in a numerical way, cf. [21], leaving the system as proven to be quasi-periodic, which suits very well the specifics of on-AC and, mAC based dynamics of FL, which are also observed to be quasi-periodic, owing to their rest-activity periods, distributed often in a stochastic manner but sometimes expressing also some almost deterministic regularities [232].

3.5. Description of Dynamic Friction-Biolubrication Applied to mAC at a sub

Mesoscopic Level Anomalous Random Walk aRW In this section, we wish to explore the more microscopic than mesoscopic see the preceding section behavior of the mAC, with a hope that it helps elucidate some interstitial-fluid pressurization effects in any AC [124, 232]. First, from the above it follows that to have ODEs system 69-71 virtually at work, that is, in its least-dissipative autonomous form, cf. [233] and refs. therein, we have to accept its basic dynamic constraint, namely 3 γ + d S 2 = 2, 75 which truly reflects its synergistic structure-property relationship S-PR, where the ’struc- ture’ is given a microscopic contraction-elongation mechanical property [23], γ , while the ’property’ implies a RW type transport of the ions along the so designed structure, com- posed of SAPL-s. The spectral or, fracton dimension of the process, d S , can be, by means of a Alexander-Orbach AO conjecture [21], interconnected with two other parameters, the apparent fractal dimension of the SAPL in a given state, as well as the trajectory geomet- rical dimension of the RW along a SAPL, d F and d RW , respectively, by means of an AO type formula, d S = 2d F d RW . 76 It ultimately results in having Eq. 75 rewritten as 3 γ + d F d RW = 2, 77 yielding finally the desired S-PR. Can Modern Statistical Mechanics Unravel Some Practical Problems . . . 81 Let suppose, that transport of the ions after a load goes via possibly effective geometri- cal route. It comes out from both, the formulation of system 69-71 as least-dissipative [233], cf. [21], and refs. therein, as well as from the fact that the ion transport is also fa- cilitated by electrostatics [220], that d F = 1 is reasonable to demand, because the transport should go along possibly shortest, straight micro-trajectories as if it was within temporar- ily formed tubes, certainly along the SAPL-s, perhaps in a piece-wise way. This is as if it was performed along microscopic intermicellar, or sometimes intramicellar tubes, that could resemble ion channels to some extent [234]. Such physically motivated assumption gives Eq. 77 in a slightly simpler form 3 γ + 1 d RW = 2, 78 which enables to get, for the three distinguished mechanical states, γ = 1 3 ; 1 2 ; 3 5 , the RW exponents: d RW = 1; 2; 5, for the vigorous, “intermediate” Gaussian and slow random walkers [21], respectively, primarily the hydrogen ion, being the lightest, that is, much prone to undergo an appropriate action after the external load being applied. Note that the squeezed state appears naturally after external load’s action, and also naturally, the RW is vigorous, thus supplying the system with enough directional kinetic energy, E kin , which automatically increases the system’s momentum, p, since for each individual ion of mass m, the well-known relation E kin = p 2 2m applies. Certainly, the total momentum contributes univocally to the pressure of the mAC, thus in the γ = 1 3 state by increasing it strikingly due to presence of the vigorous RW for which the broad “avenues” or passages are most open. This microscopic mechanism revealed may contribute to explain the electrostatics- enhanced [220] interstitial-fluid pressurization as the main factor contributing to facilitate FL process in AC, herein rationalized formally by its more ideal abstracted counterpart, abbreviated by mAC. One may also say, when embarking on the ion-channel analogy [235], that the pressure-induced [23] channel is open in the γ = 1 3 state, neither open nor close in the γ = 1 2 state, or finally, definitely closed or, inactivated in the γ = 3 5 state, that is sup- posed to be the acid-base quasi-equilibrium state [20]. Since the structure of the complex SF, in the AC, changes systematically over time, suffering additionally from many loads applied, a certain degradation as well as aging effect can also be anticipated as associated to the biolubrication effect. Therefore, an analogy, readily based on non-Markovianity of the open-close dynamics of the ion channel [90], attributed to a flickering of the channel walls’ constituents, is also plausible to occur - this is in a close conceptual accord with what has been presented so far, and what is presented below. It is then only a matter of solving Eq. 71, which yields an algebraic function in time, as well as of applying an extended ES formula [2], eES, η t = k B T Dt, wherein Dt = d dt r 2 t , and r 2 t = Ct 2 dRW 79 with another constant, C [236] in order to estimate the dynamic time-dependent viscosity of the system. Note that Eq. 79 is a solution to a generally non-Markowian [8] diffusion equation; the case of d RW = 2 yields a standard linear behavior of r 2 t , characteristic of Einstein diffusion which is a Markovian process [237]. Thus, the viscosity given in terms of the extended, means here: explicitly time-dependent ES formula, behaves also 82 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al. inverse powerly in time. Bear in mind that this quantity is proportional to the interstitial- fluid pressure, cf. [124, 232]. The dynamic friction coefficient f t of the mAC during a load of duration t is naturally redefined in a non-CA form to be f t = 1 K t , 80 thus being an inverse of the tribomicellization kernel function of the mAC, K t, given above. It is then easy to recover, at least in a qualitative way, some experimental curves by Ateshian et al. [124, 238], displaying the dynamic microscopic behavior of our systems in a { η t, f t} plane, for a given time interval t. A detailed fitting of the curves displayed on Fig. 5 from [124] is proposed to be left for another study. It is noteworthy to underline that the model view presented designs both functions as algebraic functions of time power laws, excellent for really simple fitting procedure to be applied. A more realistic fitting can be done by departing from the plane, for a given time interval. A detailed fitting of the curves displayed on Fig. 5 from [124] is proposed to be left for another study. It is noteworthy to underline that the model view presented designs both functions as algebraic functions of time power laws, excellent for really simple fitting procedure to be applied. A more realistic fitting can be done by departing from the d F = 1 most effective behavior, that is, by relaxing the straight ion-channel assumption employed above, which is by the way fairly idealized towards realizing the purpose of this work; an additional option appears to be to vary surely, according to the constraints 77-78 γ and d RW exponents to reproduce well step by step the more realistic temporal behavior in { η t, f t} experimental plane, cf. [124]. It is noteworthy to explore further the already sketched above non-CA avenue of re- search on the AC, with interstitial-fluid pressurization as the appropriate mechanism for facilitated AC-biolubrication, because such channels involved, have been already detected to exist, and are directly named as the voltage-gated proton channels, being also of SAPL- involving nature [239]. Aggregation vs shear, a MNET-type picture - When a shear stress is applied laterally to the system through a boundary, as for example in the case of a Couette flow [88], it induces a velocity gradient ∇ v on the system. This is a typical situation in which the AC could be involved. To estimate the influence of the shear flow on the viscoelastic and even the non- Newtonian properties of the synovial liquid, one may propose different approaches, see for example Ref. [112]. In this case the description must be performed at smaller times since, in general, the shear stress introduces a new time scale associated with the magnitude of | ∇ v | = ˙ γ . In this physical situation, one must extend the space of variables accounting for the state of the system, by also considering the instantaneous velocity u of the particle. Hence, the PDF will be now of the form: f

r, u,t. Note a notation change with respect to