76 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al.
equilibrium of the adsorbing species. Surface-charge alterations, being closely related with pH changes modify transport conditions for the ions. We chose to alter the surface
charge by changing the pH of the solution used to assemble the bilayer since the pH affects the degree of dissociation of both polyelectrolytes if present and the charge density on
PLs bilayer. This liposome bilayer is a model for PL bilayers and will be applied for the investigation of lubricin and other macromolecules.
If acid-base quasi-equilibra are keptrecovered by the system, it is more resistive to wear under static CA-type friction; hydration of PLs makes the coagulation ineffective, and the
layers involving hydrated PLs, and being electrostatically adsorbed at the surfaces of AC, are more mechanically robust. The latter gives rise to weak-friction in the sense of CA [22]
promoting sliding effect, due to electrostatic repulsion, and opposes possible peptization, which, however, depends upon keeping a balance of salts within the system. If the balance
is not kept by the system, the coagulation effects may prevail, which leads to loosing one of the desired acid-base quasi-equilibra, thereby causing systemic disequilibrium. This may
spoil the quasi-periodic character of the process, which would imply an imbalance in the ions involving a prone-to-friction viscoelastic membrane, while also causing the ions to
flow [21]. This appear to be a good premise for studying the facilitated biolubrication of the AC.
3.4. aD3DS Description of Dynamic Friction-Lubrication FL Applied to
mAC at a Mesoscopic and Microscopic Level
In order to describe the dynamic conditions, a new lubrication model with capacity beyond the Hills-Hardy concepts which can account for wear in a mechano-chemical process is
required. Such a model will also incorporate the age-dependent characteristic of the SF which can be included in the form of electrical repulsive mechanism in accordance with
Roberts [220]. Moreover, we argue that such a model will be based on hydrophilic lubrica- tion [20] rather than the hydrophobic mechanism proposed by Hills [205]. SAPL-s can be
adsorbed on surface of cartilage membranes as bilayers or multilayers. They can aggregate and form hexagonal reverse micelles. Lubricins contain 86 proteoglycans and 12 of
SAPL-s with 2 remaining unspecified - about the role of lubricin and other biomolecules, see the preceding Subsections.
A physical stage before applying a hydrostatic pressure normal load - In pursuance
of our understanding of the dynamics of the AC, now transferred to the construction of the mAC, that we would like to propose, the articulation space essentially consists of two semi-
permeable membranes M, two bilayers BL, plus an interlayer IL, in which lubricin L, as well as hyaluronan molecules HA, and water H
2
O dipoles are dispersed, see Fig. 16. Our model of AC is a nonequilibrium-thermodynamics model of a complex-fluid based
shock absorbera complex-fluid cushion or sponge, working periodically under a normal or lateral load, or static-pressure effect on it. The BL-s consist of H
2
O in its dissociated state [20], and SAPL in an elongated state [23]. The L-s include a certain content of SAPL
too. Both BL-s are constituted as double layers say, of Stern type [20], in which the outer monolayer is amenable to some structural changes, see Fig. 32. The changes are
assumed to be defects, i.e., the creation of a SAPL-lacking hole with on average two H
2
O molecules surrounding the SAPL molecule in its elongated state as well as, those taking
Can Modern Statistical Mechanics Unravel Some Practical Problems . . .
77 place after applying a pressure load, leading to the annihilation of the hole by two L-s,
involving just one H
2
O molecule, perhaps engaging mutually the L-s in a pair. Notice that any annihilation-creation event may contain H
2
O in its dissociated state. Other ions, mostly those coming from the presence of salts in the system also contribute to the characteristics
of the structured fluid [20].
Figure 32. The mAC before load with surface active phospholipids SAPL, lubricin L and hyaluronan HA.
A physical stage after applying a hydrostatic pressure normal load - According to
[23], the elongated SAPL-s present in the monolayer get squeezed, with each carrying au- tomatically two H
2
O molecules on average with them. An excess of H
2
O in the interlayer gives rise to internal pressure actions resulting in back-flow of L-s, with each carrying on
average one molecule of water back into upper monolayer, thereby substituting the previ- ously desorbed and water-carrying SAPL, such that the hole annihilation effect prevails.
This ability of the bilayer to undergo partial destruction, since the first layer of the adsorbed bilayer is assumed to be undisturbed, makes it possible to maintain articular space
fluid with dispersed SAPL molecules. We argue that these dispersed molecular units have the capacity to aggregate under an applied pressure due to the minimization of their surface
energies to form micelles. In this respect, in the inter-articular fluid layer this aggregation of SAPL will result in them becoming partially hydrophobic and orientating their hydrophilic
heads inwards and exposing their tails outwards. In this condition, the relatively smooth surfaces created by the SAPL aggregates, namely reverse micelles, are altogether capable
of facilitating sliding motion on the surfaces of the lower and upper contacting AC. Because of this capacity to organize or aggregate themselves, the SAPL in their reverse micelles form
are capable of both load carriage and facilitation of the low friction that is associated with the mammalian joint [230, 116], see Fig. 33.
Given the above presented context, AC can be approached as complex fluid-based load processing and lubrication layer which mostly loses efficiency with age in the same man-
ner as a car’s shock absorber after frequent and extensive use. Consequently, the surface
78 A. Gadomski, I. Santamaria-Holek, N. Kruszewska et al.
Figure 33. Same as in Fig. 32, but after load when SAPL-s build reverse micelles. The upper and lower arrows indicate the load. Note that the load from above is slightly bigger
than from below.
tribochemical functional characteristics of the tissue can be modeled in terms of a dynamic mechano-chemical system that includes a dissipation mechanism as developed in [21]. This
will involve monolayer dynamics which can be described mathematically by Eq. 1 of [21]. It is well known in surface physics and is an analog of Smirnov’s equation.
In this present fairly idealized formulation, the SAPL concentration ρ
, is included as a function of time t, so that this Smirnov-type equation [21] reads:
d ρ
dt = b
S
+ c ρ
− δρ
2
, 69
where b
S
≡ b
S
[ φ
] is the external concentration source of lipids that are responsible for exchange of matter with the monolayer, as described in [20]. The coupling between
ρ and
φ is assumed to be linear in a similar manner to that of [21], according to the local
number-conservation law, which we argue also holds for our SAPL-water system. There- fore, b
S
[ φ
] = a + b φ
, in a similar manner to the variation of e
S
= e
S
[ ρ
] in [21], now trans- ferred to a similar first-order ordinary differential equation [23] of the form
d φ
dt = e
S
+ N
HA
K [1 −
φ ],
70 with e
S
≡ e
S
[ ρ
] = e + f ρ
. Eq. 70 reflects well the first-order kinetics of micelles forma- tions [23].
It is worth to note, that φ
denotes the concentration of micelles at time t. In a way simi- lar to that followed in [21] a
, e are designated as independent free-source SAPL-production parameters, with the sources at the surface and in the bulk, respectively. b
, f represent the main system coupling parameters, where the coupling is thought of as acting between all
surface and bulk SAPL-production sources in the system, including both internal and exter- nal supplying sources. N
HA
stands for the number of HA molecules - they are assumed,
Can Modern Statistical Mechanics Unravel Some Practical Problems . . .
79 because of their affinity to water, to facilitate the reverse-micellar formation. Finally, c
, δ
are quantitatively responsible for the creation-annihilation mechanism to work in a com- petitive manner, depending on whether the former is greater in value than the latter, or vice
versa. Note that the creation-annihilation sources in Eq. 69 are of opposite signs. Cer- tainly,
φ
nM
+ φ
= 1, where the quantity indexed by nM corresponds the the SAPL number concentration or, probability of finding SAPL in a non-micellar state. This also means
that preferentially only the concentration of SAPL in the micelles, φ
and possibly the con- centration of SAPL included in L-s contribute to the overall dynamics of the tissue model,
with other contributors, such as HA-s, etc. see above playing a secondary role, thus being not included formally in the mAC dynamics so described.
The φ
obeys the AK kinetics [23], thus resulting in Eq. 70, wherein the most interest- ing aspect, or equivalently, the most essential novelty of our modeling, is concerned with
the new time kernel, K ≡ Kt, contributing to behaviors that were not represented in the
formulation contained before, cf. [21], and Eqs. 13-15 therein. It is because it includes not only the subtleties of time changes of the viscosity in the interlayer, pointing to the SF
as a rheological one, but also anticipates the non-homogeneities formed during the process, emerging due to SAPL aggregation and reverse micelles’ creation in the interlayer, while
also including the possibility of the micelles varying in geometry geometrical thermody- namics to round, spherical or not, just in a manner similar to the development in [23].
Thus, in this present work, by completing, similarly as in [116] the system 69-70 by the third equation, here of the following form
dK dt
= const. × K[ κ
− κ
−1
], 71
we propose the following autonomous system of ordinary differential equations ODEs, Eqs. 69-71, with
κ , obeying
κ ≡
κ t =
Kt const
.
−
1 h
− 1, 72
to describe the overall FL dynamics in the mAC in terms of suitable phase-space {
ρ ,
φ , K}
relationships involved in the tribo-lubrication and fluid pressure characteristics [124] in the inter-articular space during joint loading. The only essential difference when compared
with the analysis carried out previously in [21] is that the third equation of the system is more complex, giving rise to some new FL sub-effects, especially to zero- [21] and
nonzero this is the novel part proposed stationary states, responsible for quick or slow decays of micelles in the system, respectively. Furthermore, the most interesting correla-
tion with regards to the earlier model developed for the ’tribopolimerization’ process which also involves dissipation mechanism is that the temporal characteristics of micelles forma-
tion can be qualitatively elucidated by means of a probability of system return, namely that the longer the ’cycles’ of micelles’ appearancedisappearance takes, the less probable
its return to its origin, such that P
ret
∼ t
−
dS 2
[231]. The of-first-importance parameter, d
S
, named the spectral Random Walk dimension [21] appears to be, under the condition K
t
dV
n
dt
−→ const. V
n
- the volume of a micellar nucleus, an informative parameter too. In the case of our newly developed model, it is also related to the mechanical behaviors of
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
