Scenario SOC abnormally slow relaxation of the system clusters the fluid in disordered nanoporous confinement environment

The relaxation of the system of the nanoporous medium with the nonwetting liquid is a self-organized criticality process characterized by waiting for fluctuation necessary for overcoming a barrier of local metastable state with the subsequent avalanche decay of local metastable configurations of pores. The dependence of the interaction between local configurations on the number of filled pores belonging to the infinite percolation cluster of filled pores serves as an internal feedback initiating the SOC process.


Introduction
Anomalously slow relaxation and hysteresis properties are a characteristic phenomena of many disordered media [1][2][3][4]. These media include atomic, molecular and polymeric glass of atomic particle size, colloid media, nanocomposites with nanoscale particles and loose material with a micron scale particles, as well as spin glasses.
The phenomenon of hysteresis is associated with the existence of long-lived metastable states and related spatial inhomogeneity of a disordered medium. The assumption of the existence of heterogeneous inhomogeneities and local configurations is central in phenomenological models of description of anomalously slow relaxation in disordered media. Analysis of these models performed in several recent reviews [1,2,5,6]. From this analysis it follows that the cause of the inhomogeneous of disordered media, and therefore the mechanism of anomalously slow relaxation are unclear until today.
A system of liquid clusters is a nonlinear dissipative systems. The relaxation of such system can be considered in the frameworks of model of self-organized criticality (SOC) [7,8]. This model was proposed in [9,10] to describe relaxation as an avalanche process of falling of a pile of sand. It is commonly accepted [7,8] that the SOC state appears in nonlinear dissipative systems in the critical state that relax without the external control parameter through rapid avalanche transitions between different metastable states of a system. The critical state is maintained not at a point, but in a wide region of the phase diagram of states through a nonlinear feedback mechanism because of the existence of the internal mechanism leading to the dynamic self-organization of transitions between intermediate metastable states of the system. Self-organized criticality is characteristic of the systems with fractal objects. However, physical mechanisms responsible for avalanche transitions between metastable states, as well as the reason for the appearance of the feedback mechanism leading to the dynamic self-organization of transitions between such states, are not yet understood. In particular, numerical studies [11] indicate that the properties of SOC with avalanche relaxation for spin-glass systems are not manifested at a finite number of neighbors and a divergent number of neighboring interacting spins are necessary.
In this work, we propose a new mechanism of anomalous relaxation of states of a liquid confined in a random nanoporous medium, which can be characterized as the SOC mechanism in revealed properties. The interaction energy of a system of clusters is calculated in the quasiparticle approach [12] as the sum of the energies of the local liquid cluster configurations interacting with clusters in neighboring pores. The process of relaxation involves overcoming the metastable state of the local configurations and the subsequent rapid hydrodynamic extrusion of the liquid. The dependence of the interaction between local configurations on the number of filled pores belonging to the infinite percolation cluster of filled pores serves as an internal feedback initiating the SOC process.
The calculations of the density of states and extrusion time distribution functions of liquid clusters give a power-law time dependence of the relative volume  of the confined liquid Such a dependence is characteristic of known self-organized criticality phenomena.

Main part
We assume that the disordered nanoporous medium includes pores with different sizes and the size of the porous medium is much larger than the maximum size of pores, so that the porous medium can be considered as infinite and the infinite percolation cluster of empty pores is formed in it.
According to [13,14], the metastable state of filled pores in the porous medium is formed at times t > 0  , relaxes slowly at times much larger than 0  , and then decays at times 0 tt  . It was shown in [35] that the relaxation of the metastable state in the Libersorb23-water system with a narrow pore size distribution with the relative width Therefore, such a state of the disordered porous medium with the confined nonwetting liquid can be described by local equilibrium distribution functions and can be considered thermodynamically with distribution functions in 5N -dimensional phase space of pores coordinates, pore radii and their occupation numbers i n of all N pores in the porous medium [13][14][15]. For the filling of pores in the porous medium, it should contain an infinite cluster through which the transport of the liquid is possible [13,14]. Consequently, the percolation cluster of filled pores is formed inside this percolation cluster of pores at filling [13]. It follows that the ground state of partially filled porous medium is a state containing a percolation cluster of pores with the percolation cluster of filled pores the inside of it. The state of the system containing the infinite cluster of filled pores can have many realizations; i.e., this state is degenerate. Consequently, this state can be characterized by the probability () P  for the pore to belong to the infinite cluster of filled pores.
The liquid is removed from pores of porous medium only if at least one of its neighboring pores belongs to the infinite cluster of filled pores through which the extrusion of the liquid from a granule of the porous medium is possible. According to [13][14][15]   characterized by the probability () P  for the pore to belong to the infinite cluster of filled pores and depends on the number of nearest neighbors z of depleted pore and the number of unfilled pores in its environment, composing the local configuration of depleted pore.
It was shown in [13][14][15] that a change in energy  of pore at the extrusion of the liquid from this pore can be represented as a sum of changes in energy of various local configurations containing n empty pores n  , divided by the number of nearest neighbors [13,14] Expression for the energy n  of the local configuration containing n empty pores was obtained in [13,14 ]: Here,  is the surface energy of the liquid, (3) To calculate this dependence following [13,14] can be done by the distribution function is the maximum energy barrier for the decay of the local metastable configuration in the nth state.
As follows from Eqs. (5), the exponents n a and times and specify the relaxation law of the state of the n th local configuration, are determined by the energy barrier To analyze this equation, we note that the terms in sum (9) Figure 2 shows the energy of the state of the local configuration n  corresponding to the nth mode for the configuration of the pore with the radius equal to the average radius R R  for a Gaussian pore volume distribution The calculations were performed with the same parameters as for Fig. 1.

Conclusion
To summarize, the relaxation of the system of the nanoporous medium with the nonwetting liquid is a self-organized criticality process characterized by waiting for fluctuation necessary for overcoming a barrier of local metastable state with the subsequent avalanche decay of local metastable configurations of pores. The dependence of the interaction between local configurations on the number of filled pores belonging to the infinite percolation cluster of filled pores serves as an internal feedback initiating the SOC process.
The model of the relaxation of the porous medium with the nonwetting liquid developed in this work demonstrates possible mechanisms and scenarios of SOC for disordered atomic systems.