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**The present paper deals with motion of carbon nanotubes in a temperature gradient field. A determined-static theory of nanosized particles’ thermophores is developed. Analytical expressions for thermophoretic velocity and force of ultramicroheterogeneous particles in a gaseous atmosphere under near-normal conditions are provided. The calculations performed according to the suggested theory, as applied to closed carbon nanotubes, found the value of dimensionless velocity of thermophoresis. In accordance with the proposed hypothesis, Waldman’s limit is achieved, which is expressed in constancy of thermophoretic velocity within the interval of the Knudsen parameter change from 10 to 100. In addition, it is found out that under conditions defined below, velocity of thermophoresis is independent of the length of a carboxylic nanotube. A good agreement with experiments is reached, which makes it possible to assume correspondence of the theory to the physical truth****.**

Thermophoresis of fine, yet not nanosized particles, has been reported in a large number of studies among which it is possible to point out the following scientific works of general character [1-5].

In this work, we follow the approach described in [6,7], which is used to determine the velocity of thermophoretic displacement of a particle in a temperature-gradient field, as well as the thermophoretic force acting on the moving sample particle, which is balanced by the force of particle drug under equilibrium conditions.

Similar to the case of finding the force of particle drag [

Therefore we state a stepped temperature change in our theory, the size of a step in this gradation also being equal to λ.

For example,

Let us orient the tube along the temperature gradient and define the velocity of a particle moving under the action of thermophoretic force. Kinetic energy of translatory motion is distributed to three degrees of freedom, therefore

From this equation, we get

Let vector grad T be directed along the Oz-axis. Take a derivative with respect to the z-coordinates for both parts of Equation (2):

It is reasonable to assume that in the neighbourhood of an ultrafine or nanosized particle the behaviour of temperature variation is linear

Replace the left-hand part of Equation (3) with finite differences that correspond to the velocity variation during a transition from one isothermal layer to another. Thus instead of Equation (3) we can approximately write

When determining the action of molecules on the particle let us use a scheme of equalized actions [

Since within the path length shorter than λ there is no molecular collision, it is reasonable to assume the temperature in every of the layers marked out and the velocities of thermal motion to be equal υ(T).

We have limited the amount of surrounding molecules by their amount in the λ-layer. However, it is not enough to perform the simplest calculation of thermophoresis velocity. Let us pick an elementary interaction act out of the whole collection of molecules interacting with the particle, in which a particle and a counter-moving pair of molecules participate. Therefore, we substitute all actual double collisions by model triple ones which do not cause Brownian motions. If we accept a regular pattern of particle reflection, the final result would represent a simple sum of interaction acts between the counter-moving pair and the particle.

The top part of

The balance of momentum projection onto the Oz-axis in a laboratory reference system for the case of a regular reflection is given by

Hence

Here m, M are the masses of the molecule and the particle, respectively, and, are the particle velocities prior to and after the collision with the countermoving pair

Summing Equation (7) over all counter-moving pairs we obtain

Here n is the number of molecules that have collided with the particle and υ_{P} is the average thermophoresis velocity of the tube prior to counter-moving pairs collisions in the λ-layer, is the average thermophoresis velocity of the tube after counter-moving

pairs collisions which is equal to υ_{P} under equilibrium conditions.

Taking into account that the left-hand part (Equation (9)) is equal to zero we get

This velocity is the particle thermophoresis velocity.

In Equation (10), σ is found as

In cases of practical calculation we use the following formula for σ:

Thus σ is the average value of slope angle cosines modules of counter-moving pairs. Introducing Δυ from Equation (5) into Equation (10), we finally get

The λ-layer under consideration is found between the effective surface of the nanotube and the equidistant surface λ-spaced from it. Let us circumscribe a parallelepiped around the outer surface of the tube. Let us populate this parallelepiped with molecules of the gas surrounding the particle with the help of the random number generator used three times for every molecule, for the purpose of setting its spatial coordinates. Let us form a unit cube around every molecule and place a sampling particle into it in a random manner, i.e. use the random number generator three more times. Let us join this particle with the molecule, the resulting right line determining the direction of molecules motion in space. From here on we define N as the number of molecules initially present in the

λ-layer (

This value is included in the formulae used for defining particle resistance.

The value σ determining thermophoresis is found as the average of the slope angle cosines modules of molecules trajectories against the temperature gradient direction. Calculating δ and σ is the final stage of one test. The suggested number of such tests is 150, the values of δ and σ being averaged thereafter.

Motion of nanotubes in gas is governed by the principle of least action or the principle of least constraint which are the same under the conditions of equilibrium. Both principles state the fact that in a gradient medium of molecules a nanotube is fixed along the direction grad T, i.e. it is subject to least resistance.

The particle motion velocity found in Equation (13) is actually the velocity of thermophoresis. As seen, it is independent of the number of molecular collisions (provided their number is sufficient to ensure proper statistical data for determination of σ) and is weakly dependant on the particle size. Within the Knudsen number range, which corresponds to proper nanoparticles, the thermophoresis velocity does not depend on the particle size and is solely determined by the number of atoms in a gas molecule and the values of temperature gradient and geometrical parameter σ (see Equation (11)). Yu. V. Valtsyferov and S. M. Muradyan [

where f is the dimensionless coefficient that depends on the Knudsen number and varies within the range of 0.05 - 1.56 and υ is the coefficient of kinematic gas viscosity.

Comparing Equation (13) and Equation (14), we obtain

Statistical estimation for a spherical particle states σ = 0.515. Introducing this value into the previous formula we obtain that f = 0.535 which is in a good agreement with the measurement data and the theoretical results introduced in [

In order to take into account the occupied volume of the spherical particle in case, we take into consideration four temperature layers, as shown in

In case, we get that r_{p} = λ, in cases when the Knudsen number has a greater value, the r_{p}-layer is considered in the same way as the λ-layer.

Analogic complex distribution (the dashed line and the full curve line) can be obtained for the friction coefficient. In such a case, the dashed line will refer to Stockes distri-

bution and the full curve line will refer to CunninghamMilliken-Davis data which agree with calculations via monokinetic theory [

Thermophoretic force can be defined on the basis of Newton’s third law of motion, i.e. under the conditions of dynamic equilibrium

where υ_{p} is found from Equation (14) and the drag coefficient γ was reported in [6,7]:

Here υ is the velocity of thermal molecular motion, λ is the free lath length of a molecule, Kn is the Knudsen number, N_{L} is the Loschmidt number under standard condition and m is the molecular mass.

The radius of the tube can be estimated in terms of the Knudsen number:. The lengths of the tubes included in calculations are given in

Thus in case L > 0.5λ, we discover that neither the lengths of carboxylic tubes nor their diameters influence σ, and consequently, the thermophoresis velocity. It is found that within the calculation range the changes of parameters are σ = 0.455, which corresponds to the value of the dimensionless thermophoresis velocity of carboxylic tubes: f = 0.485.

The present paper shows that the thermophoresis velocity

of carboxylic tubes does not depend on the sizes of tubes but is solely determined by the number of atoms in molecules of a gas medium and the temperature gradient in it, and the thermophoretic force acting on a particle being dependent on each of the above mentioned parameters. The calculations define that in cases of carboxylic tubes oriented along the temperature gradient field, the dimensionless coefficient in terms of linear relationship between the thermophoresis velocity and the temperature logarithm gradient is f = 0.485.

The present scientific work was performed with financial support from Russian Fundamental Research Fund. Grant No 14-01-3165.