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1 Molecular dynamics model
In classical molecular dynamics, the behavior of an individual particle is described by the Newton equations of motion Gould . For a simulation of particle interaction we use the Lennard-Jones potential LJ with and . It is the most used to describe the evolution of water in liquid and saturated vapor form. Equations of motion were integrated by Velocity Verlet method Verlet . Berendsen thermostat Berendsen is used for temperature equilibrating and control. Coefficient of the velocity recalculation at every time step depends on the so called rise time of the thermostat which belongs to the interval . The Berendsen algorithm is simple to implement and it is very efficient for reaching the desired temperature from far-from-equilibrium configurations.
2 Computer simulation of microscopic model
We made simulation for a pore of dimensions nm, nm, nm with integration time step ps and evolution time 65.3 ns. Otherwise, we have considered the following input data for the drying process:
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1000 molecules in the pore volume nm form a saturated water vapor at temperature and pressure ;
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1800 molecules in the outer area that form 20 % of the saturated water vapor
and input data for the wetting process:
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200 molecules in the pore volume nm are 20 % of saturated water vapor;
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9000 molecules in the outer area form a saturated water vapor at temperature and pressure .
The diffusion coefficients for drying process (left) and for wetting process (right) are shown in Fig. 1. The left figure depicts diffusion coefficients for pore (upper curve), for outer area (down curve), for mean value of previous (middle curve) and for constant value [nm/ps] (middle dashed line). The right figure shows diffusion coefficients for pore (upper curve), for outer area (down curve), for mean value of previous (middle curve) and for constant value [nm/ps] (middle dashed line).
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3 Macroscopic diffusion model
Let us denote the water vapor concentration as [] where are space independent variables and is time independent variable. Then, we consider the following macroscopic diffusion model
| (1) |
| (2) |
| (3) |
| (4) |
where is the diffusion coefficient []; are 3D pore dimensions []; are boundaries of 3D pore ( is free boundary while the rest boundaries are isolated; is the initial concentration of water vapor, for the drying process, and for the wetting process []; is the water vapor concentration in outer area []; is the coefficient of water vapor transfer from pore space to outer space, [].
We suppose that the outer area water vapor concentration is expressed as where is the relative humidity of outer space () and is saturated water vapor concentration at outer temperature .
The linear problem (1)–(4) can be solved exactly using the variables separation method Bitsadze and the result of the solution is
| (5) | |||
Here, are coefficients of unity expansion
and
are eigenvalues where
and are solutions of the equation
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The dynamics of water vapor concentration at different time moments
ps (cross section at and ), drying process (left) from top to bottom, and wetting process (right) from bottom to topResults of macro model calculations are presented in Fig. 2.
4 Comparison of microscopic and macroscopic models
Finally, we compare the space mean value
of water vapor concentration (5) for macro model with the density obtained by micro model. The results are in the Fig.3.
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Our investigations allow to affirm that an approach based on combination of diffusion coefficients determination by means of molecular dynamics and further application of these coefficients in macro model computations is useful for accuracy increasing of water-pore interaction description.
References
- (1) H. Gould, J. Tobochnik, W. Christian, An Introduction to Computer Simulation Methods (Addison Wesley, San Francisco, Chapter 8, Third edition, 2007) pp. 267-268
- (2) J. E. Lennard-Jones, Proc. Roy. Soc. A 106, pp. 463-477 (1924)
- (3) L. Verlet, Phys. Rev. 159 98-103 (1967)
- (4) H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. DiNola, J. R. Haak, J. Chem. Phys. 81 pp. 3684–3690 (1984)
- (5) A. V. Bitsadze and D.F. Kalinichenko A Collection of Problems on the Equations of Mathematical Physics (in Russian, Alians Publisher, Moscow, 2007), pp. 106-310