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The article is devoted to modeling of the molecular microscopic journal bearing. The walls and the lubricant of the bearing are fullerene-like molecules. On the basis of similarity theory and analysis of the dimensions, the similarity criterion is proposed. This criterion characterizes the convergence of a numerical solution. The test calculation is also made to evaluate the quality of the proposed criterion.

Friction is an important component of the physical processes in nature and technology. It is, however, difficult and inconsistent in demonstrations: On the one hand it is impossible to walk without friction and on the other hand about a third of produced energy is lost on friction.

An uncircumscribed field of action in chemistry, biology and physics was discovered in 1985 together with discover the new form of carbon, named fullerene. Later, a set of materials with great potential importance in tribology was discovered, such as the nanotube, the fullerene and fullerene-like nano-onions and others. According to the Stribeck curve [

The hydrodynamic theory of lubrication, the main points of which were formulated over a century ago, uncovered one of the most efficient ways to decrease friction losses in the bearing. Those basics are permanent and simple [

This paper presents a theoretical study of the motion of microscopic particles of a lubricating layer of about 10 molecules thick in the journal bearing. The main objective is to define the basic parameters of a computational experiment which will provide satisfactory convergence of the result. The problem is solved by known methods of molecular dynamics [^{3}–10^{6} particles during a 10^{−6}–10^{−9} s in this stage of computer technology development. These restrictions prevent us from studying the macroscopic objects in the usual time slots, but help to clarify many important physical laws.

The two-dimensional model of the molecule of fullerene-based lubricant is formed as a hexagon with a radius of the circumscribed circle of 3.5 A, which corresponds to the radius of the fullerene C_{60}. At the vertices of the hexagon there are atoms that are similar in physical properties to carbon atoms. The central atom compensates stiffness and mass of the volumetric fullerene molecule C_{60} in its 2nd model. Such a two-dimensional model was used in the study of the motion of a monolayer fullerene lubricant between the parallel walls [

According to the molecular dynamics approaches, it is assumed that the interaction of any two atoms is determined only by the distance between them, and that the superposition principle holds true for interaction of many atoms [_{i}_{,j}—the distance between

ξ—characteristic energy that determines the depth of the potential well, J;

σ—characteristic length that determines the distance with zero energy, m.

The state with the lowest energy corresponds to the distance between the atoms _{m}^{1/6}σ. This is evident from the Equation (1).

The movement of the set of atoms in the molecule as a result of intramolecular and intermolecular interaction is determined by many of the classical equations of motion:
_{i}

The solution of the set of Equation (2) is convenient to carry out by the method of finite differences. The derivative of the left-hand side of the Equation (2) is approximated by the difference analogue of finite difference schemes [

Under the terms of the hydrodynamic lubrication theory, in order to create the effect of hydrodynamic lubrication the lubricant is required to have the property of viscous internal friction and the ability to “stick” to the surface of the bearing [

Two-dimensional model of journal bearing lubricated with fullerene.

The movement of all molecules or atoms except the central atoms of molecules on surfaces of the trunnion and the bearing bush is determined by numerically solving of the set of the Equation (2). At each point of time, acceleration, speed and position are defined for each atom. To obtain the correct result it is necessary that during the step Δ

It is convenient to consider the case of one-dimensional motion of two atoms along a straight line. This assumes a situation in which the atoms move in opposite directions and at the time _{0} the distance between them is equilibrium _{m}

It is further assumed that from time _{0} to time of atoms stop is one time step Δ

The Equation (4) shows that the value of the characteristic length σ does not affect the convergence of numerical solutions. Graphically the dependence of the time step on the dimensionless energy balance criterion is presented in

The proposed procedure allows identifying only the necessary conditions of correct solutions and helps to assess the computational complexity of the problem in advance, for example, an approximate value of time needed for a calculation.

The dependence of the time step

The character model with fullerene-bearing lubricant was investigated using a program developed by the authors in the environment “GNU-Octave” [

The flow chart of the simulation program is presented on

Block 1 “Input data” represents input source data, such as journal bearing sizes, trunnion position and its angular velocity, number of molecules and their properties, process duration and others.

Block 2 “Initial conditions” is a subprogram which determines initial position and velocity of all molecules.

Block 3 “Time event loop” is the FOR-loop of

Block 4 “Contact detecting” is a subprogram which determines the intramolecular interaction condition by means of comparison the distance between each pair of molecules and predefined no-acting distance (a multiple of _{m}

Block 5 “Inner reactions” is a subprogram which determines intermolecular interaction forces of atoms motion.

Block 6 “Atoms event loop” is the double FOR-loop with outer cycle of _{i}_{g}_{i}_{g}_{g}_{g}

Block 7 “Equations of motion solver” represents equations of motion (2) solver by various algorithms.

Block 8 “Output data” represents the results in matrix and graphics.

The flow chart.

The first set of numerical experiments was carried out to study of the convergence of solutions depending on the size of the time step and validation of an approximate estimate of step by the criterion of the energy balance and the Equation (4). We examined the journal bearing with the following characteristics: The radius of trunnion is ^{7} rad/s; the mass of the lubricant atom is _{a}_{m}_{a}_{A}

The second set of numerical experiments was carried out to study the shear viscosity effect appearance in fullerene-based journal bearing. The same bearing filled with fullerene lubricant was examined. The trunnion and the bush are coaxial.

The convergence of the calculation results.

According to the approaches of mechanics of continua [_{ρφ}

μ—shear viscosity coefficient (or shear viscosity);

ξ_{ρφ}

_{φ}

So, if the values of the shear stress and shear strain rate in a vicinity of the point (e.g., on the surface of the trunnion) are known, the shear viscosity can be found using Equation (5).

According to the Equation (1) dual atomic motion looks like undamped oscillation, and this means, firstly, that molecules cannot “stick” to each other, secondly, they cannot dissipate their energy. Both conclusions mean that the Lennard-Jones interaction (1) cannot simulate shear viscosity effect. It is commonly considere, that kinetic energy of the molecules is in correlation with shear viscosity, temperature and thermal conductivity [

The value of

The results of a series of numerical experiments are presented in

Fullerene-based journal bearing and the velocity profile.

Simulation results in terms of some integral characteristics.

Process time/(trunnion rotation angle), s/(radian) | Summary kinetic energy in fractions of its initial value, _{k}^{0} ~ 10^{-20}, J |
Average friction force on the trunnion, N | Shear viscosity on the trunnion, Pa·s | |
---|---|---|---|---|

Fullerene-based journal bearing | 10^{-7} /(1) |
(1…1.25)_{k}^{0} |
1.1×10^{-12} |
1.8×10^{-12} |

Fullerene-based journal bearing with damped walls | 10^{-7} /(1) |
(0.95…1.05)_{k}^{0} |
1.3×10^{-12} |
4×10^{-12} |

Modern chemistry describes the huge number of existing and theoretically possible materials based on fullerenes and their compounds. Investigation of the physical properties of these materials in specific tribological devices will probably help to find an alternative to liquid lubricants in such field of tribology as the hydrodynamic theory of lubrication. The proposed mathematical model of the journal bearing based on fullerene-like lubricant can be used in future to explore the possibility of new lubricants.

Significant computational complexity of applying of the methods of molecular dynamics is the size of the data matrices: For real physical objects the number of molecules is in the order of Avogadro’s number, and the relatively strong interatomic interactions at short distances require that the calculation of a second of real time takes about 10^{11} steps. Too small a number of steps leads to inadequate results, too large slows down the calculation. On the basis of similarity theory and dimensional analysis, the criterion of energy balance and the method of calculation of the indicative value of the time step were proposed. This technique allows us to determine in advance the computational complexity of numerical solutions of the problem and an approximate calculation of the value of computer time.

On the basis of mechanics of continua the macroscopic method of shear viscosity definition towards to microscopic object were proposed. The method is acceptable in order to compare various lubricants on a molecular (microscopic) level and it is unacceptable in order to extrapolate the calculated results on a macroscopic level.

The work was performed as part of the “Fundamentals of micropolar and hybrid nanomaterial’s granular lubricant” on the instructions of Ministry of Education of Russia No. 7.516.2011.

The authors declare no conflict of interest.

_{60}additives to lubricating oils

_{60}fullerenes inside a dimyristoylphosphatidylcholine lipid bilayer