###### Abstract

The dependence of the chemical potential jump coefficient on the
evaporation coefficient is analyzed for the case in which the
evaporating component is a Bose gas. The concentration of the
evaporating component is assumed to be much lower than the
concentration of the carrier gas. The expression for the chemical
potential jump is derived from the analytic solution of the
problem for the case in which the collision frequency of molecules
of the evaporating component is constant.

PACS numbers: 05.20.Dd Kinetic theory; 05.30. Jp Boson systems; 71.35.Lk Collective effects (Bose effects, phase space filling, and excitonic phase transitions).

Chemical Potential Jump during Evaporation of a Quantum Bose Gas

E. A. Bedrikova^{1}^{1}1
A. V. Latyshev^{2}^{2}2

Faculty of Physics and Mathematics,

Moscow State Regional University, 105005,

Moscow, Radio str., 10–A

INTRODUCTION

Interest in problems associated with the behavior of gases under conditions in which quantum properties cannot be ignored has been growing in recent years [1]. The behavior of mixtures of such gases is also of considerable interest. The "quantum" gases that are studied most frequently are He и He. It should be noted that He is a Fermi gas, while He is a Bose gas. Such a combination of different quantum-mechanical statistics attracts considerable attention to the mixtures of these gases [2]. A large number of publications are also devoted to the properties of solutions of these gases [3, 4]. At the same time, boundary value problems for such mixtures have been studied insufficiently. Such problems include the behavior of a mixture of quantum gases in the vicinity of the evaporation boundary. Let us consider the most frequently encountered case of a dilute mixture. Let molecular concentration of one gas be much lower than molecular concentration of the other gas: . We will consider the problem of evaporation of the first gas to the gas mixture. We will henceforth consider the problem in a more general formulation, taking into account the possibility of other applications also. Let us suppose that we have a mixture of two quantum gases. We consider the problem of evaporation of a Bose gas to this mixture, provided that its concentration is much lower than the concentration of the other component.

1. FORMULATION OF THE PROBLEM AND BASIC EQUATIONS

Let us consider evaporation from a plane surface to a binary gas mixture. We assume that concentration of the evaporating component of the mixture is much lower than concentration of the nonevaporating component (; dilute mixture). It should be noted that this condition holds in the most important applications.

In the semiclassical approximation, the Boltzmann equation for a binary gas mixture has the form [5]

Here, is the distribution function for the th component of the mixture and and are the integrals of collisions of molecules of the th component with one another and with molecules of the th component, respectively.

It should be noted that and (since ). Quantity is a small parameter (), since . Obviously, . Consequently, quantity can be disregarded as compared to in the first approximation in . In addition, the action of the first component on the distribution function for the second component can also be disregarded in this approximation in . Therefore, under the conditions of the given problem, the distribution function for the second component of the gas mixture can be treated as an equilibrium function with mean velocity and constant temperature and concentration .

Quantity can be approximated by a kinetic model of the Bhatnagar—Gross—Krook (BGK) type [6, 7]. Taking into account Eq. (1), we can write the kinetic equation for the first component in the form

where

is the equilibrium Bose distribution function for the first component, is the Boltzmann constant, is the effective collision frequency for molecules of the first component, is the chemical potential of the first component, and is the temperature of the mixture, which is assumed to be constant.

Let us choose the Cartesian system of coordinates with the center on the surface from which evaporation takes place. We direct the axis along the normal to the surface. During evaporation from the surface, a constant gradient of the first component concentration

exists at a large distance from the surface.

Here

is the concentration (number density) of the first component and is the momentum of molecules of the first component.

We assume that evaporation is weak; in other words, we assume that the relative variation in the concentration of the first component over the mean free path of molecules is much smaller than unity:

Here, is the concentration of saturated vapor (gas) of the first component on the evaporation surface, which corresponds to surface temperature .

Under such conditions, the problem can be linearized. Preliminarily, we pass to dimensionless velocity , dimensionless coordinate and dimensionless chemical potential .

Here is the thermal velocity of the first component, , is the mean free thermal path of molecules of first component, is the mean time between two collisions of molecules of the first component.

We can now write Eq. (2) in the form

Here is a locally equilibrium Bose distribution function,

We will linearize the problem relative to the absolute Bosean

where is the value of the dimensional chemical potential corresponding to surface temperature and the concentration of the saturated vapor at this temperature.

We will linearize the dimensionless chemical potential relative to quantity ; i.e., we assume that . Linearizing the locally equilibrium Bose distribution function relative to , we obtain

where

We will seek the distribution function in the form

Using this expression (4), we can write Eq. (3) as:

We can now find the dimensionless relative deviation . of the chemical potential from its value at the wall using the law of conservation of the number of particles:

The law of conservation of the number of particles leads to the equation

which gives

We can easily find that

where

We transform the numerator of expression (6) as follows:

Consequently, in accordance with relation (6), the deviation of the dimensionless chemical potential is given by

Thus, kinetic equation (5) for the problem of evaporation, taking into account relation (7), has the form

2. FORMULATION OF THE BOUNDARY CONDITIONS

Let us consider the boundary condition at the evaporation surface for molecules of the first component taking into account the effect of the surface by introducing evaporation coefficient (see [8, 9]),

where

Quantity can be determined from the nonpercolation conditions for molecules reflected from the surface without being condensed on it (the probability of such a process is ),

where ) is the Heaviside function, for and for .

Taking into account the definition of the Heaviside function, we can transform the nonpercolation condition to

The boundary condition for function at the wall can be derived from condition (10). Substituting function (4) into (10), taking into account condition (9) as well as the results of the linearization

and carrying out the substitution, we obtain the following boundary condition at the wall:

At a large distance from the surface (outside the Knudsen layer having a thickness on the order of the mean free path of molecules), function has the form

where

here, is the gradient of the dimensionless chemical potential defined far away from the wall,

and is the chemical potential jump, viz., the unknown quantity to be determined from the solution of the problem.

Substituting function into definition (6) of the dimensional chemical potential, we obtain the asymptotic distribution of this potential:

It follows, hence, that , i.e., the chemical potential jump is defined as the difference between the extrapolated value of chemical potential at the wall and its value immediately at the wall.

Quantity can be determined from nonpercolation condition (10). This condition can be written in the explicit form

It should be noted that the sum of the integrals of the first and second terms in each square bracket is zero; therefore, we arrive at the following equation:

The second integral, which can be evaluated over the negative half-space, will be calculated using the conservation law for momentum (to be more precise, the component of the momentum). For this purpose, we replace from the first integral on the right-hand side of the obvious equality

by and from the second integral by from boundary condition (11). After evaluating the required integrals, we arrive at the equation, which gives

Here

Evaluating the first two of these integrals, we obtain

Consequently, equality (13) can be written in the explicit form

Now, boundary condition (11) is complete

Thus, the boundary value problem involves the obtaining a solution
to Eq. (8), which satisfies boundary conditions
(14) and (12). Quantity will be henceforth denoted as .

3. DIFFUSION COEFFICIENT AND THE MASS FLOW RATE

Using this function, we can calculate diffusion coefficient . Diffusion flux emerges due to the presence of a density gradient in the gas; consequently, for small values of , we have [10]. In accordance with the formulation of the problem, this gives

Here, we replace dimensionless coordinate by dimensional coordinate and pass to integration with respect to dimensionless velocity. This gives

It should be noted that this result is transformed into the classical one for [9]. Indeed, using the asymptotic form

we obtain the known result

It should be noted that, in the approach used here, diffusion coefficient is treated as an empirical quantity. For example, for the He – He mixture at a temperature of , we have [11] см/с.

Let us find the mass flow rate of the evaporating component of the binary gas in the direction of the axis. By definition, the mass flow rate in the direction is

Substituting this expression into the definition of diffusion coefficient, we obtain

It can easily be seen that

passing to dimensionless mass flow rate , we obtain

where

where is the thermal wave number.

The number density in the linear approximation is given by

where

Thus, we obtain

whence the mass flow rate is

4. SOLUTION OF THE PROBLEM

We will seek the solution to Eq. (8) in the form

where ц is the spectral parameter or the separation parameter.

Using these two equalities, we obtain from Eq. (8) the characteristic equation

For we obtain the eigenfunctions of the characteristic equation,

Here, the symbol indicates the principal value of the integral of , is the Dirac delta function, and is the dispersion function of the problem,

We will seek the solution to problem (8), (12), and (14) in the form of an expansion

where is an unknown function (coefficient of the continuous spectrum) and is an unknown constant (coefficient of the discrete spectrum). It will be shown below that coefficients and are connected by a linear relation,

in which function is the coefficient of the chemical potential jump.

Let us find the coefficients of the discrete and continuous spectra. The substitution of eigenfunctions into expansion (16) and of expansion (16) into boundary condition (14) leads to a singular integral equation with the Cauchy kernel:

We introduce an auxiliary function,

with upper and lower boundary values on the positive semiaxis related by the Sokhotsky formulas

and the last integral in which is treated as a singular integral in the sense of the Cauchy principal value.

Using the upper and lower boundary values of functions and on the cut , we can reduce the singular equation to the Riemann boundary value problem:

Let us consider the corresponding homogeneous boundary value problem

Using the homogeneous problem, we can reduce the inhomogeneous problem to determining the analytic function from its zero-point jump at the cut:

The solution to this problem has the form

Using the condition , we obtain the jump of the dimensionless chemical potential,

Coefficient of the continuous spectrum can be determined by substituting the solution to the boundary value problem into equality (18):

Thus, the unknown coefficients in expansion (16) are determined. This means that the distribution func-tion for the evaporating component is constructed completely and can be written in the form

Using the contour integration methods for boundary , we obtain

This expression shows that for , the distribution
function for molecules
reflected from the wall exactly satisfies boundary condition (14).

5. CHEMICAL POTENTIAL JUMP AND PROFILE

Let us compare equalities (17) and (19). Using the second equality in (14), we obtain the coefficient of the dimensionless chemical potential:

Figure 2 shows the dependence of coefficient on the evaporation coefficient ; curves 1 and 3 correspond to values of dimensionless chemical potential , and , respectively. The curve 2 correspods to case of Fermi–gases at .

Figure 3 shows the dependence of coefficient on the dimensionless chemical potential; curves and correspond to the values of evaporation coefficient and , respectively. The curve 2 correspods to case of Fermi–gases at .

Equality (17) describes the jump of the dimensionless chemical potential. Passing in this equality to dimensional quantities, we obtain the chemical potential jump in a quantum Bose gas:

where is the mean free path of molecules.

We express collision frequency in accordance with relation (15). Then the chemical potential jump (in terms of dimensional quantities) can be calculated by the formula

where is the coefficient of the chemical potential jump, defined as

Note that in the limit of high temperatures, when quantum properties of the gas can be ignored, result (20) is transformed into the following known result for the concentration jump in a classical gas [12]:

The chemical potential distribution in half-space (known as the chemical potential profile) is defined by equality (7). Substituting the expansion of distribution (known as the chemical potential profile) is defined by equality (7). Substituting the expansion of distribution function (16) into (7), we find that the chemical potential profile can be constructed using the formula

6. CONCENTRATION JUMP AND PROFILE

Concentration profile of the gas in half-space is given by the equality

where

We can easily find that , where

and the deviation of the concentration from concentration of saturated vapor is given by

where

where is the coefficient of the concentration profile, is the quantity of the concentration gradient (with respect to the dimensionless coordinate),

For the constant gas concentration, the chemical potential increases upon cooling [14], so that for , while for , .

Consequently, the curves plotted for small values of correspond to a lower effective gas temperature.

It follows from Eqs. (21) and (22) that the profiles of chemical potential and concentration are proportional to within the corresponding gradients.

In addition, it can easily be seen that the gradients of chemical potential and concentration are proportional to each other with a factor : .

The latter equality indicates that the chemical potential gradient is the gradient of the logarithm of concentration:

We can easily find that the relation between the deviations of concentration and chemical potential from the corresponding values at the wall is

Using this relation, we can find the concentration jump for ,

where is the coefficient of the concentration jump.

The behavior of the concentration jump coefficient from relation (24) is depicted in Figs. 4 and 5.

Figure 4 shows the dependence of the concentration jump coefficient of the evaporation coefficient; curves and correspond to dimensionless chemical potential and , respectively. The curve 2 corresponds to case of Fermi–gases at .

Figure 5 shows the dependence of the
concentration jump coefficient on the chemical potential ; curves
, and correspond to values of accommodation coefficients
, and . The curve 2
corresponds to case of Fermi–gases at .

7. CONCLUSIONS

The problem of evaporation of one of the binary gas components has been solved analytically. The evaporating component is a Bose gas. We studied the dependence of the chemical potential jump coefficient on the evaporation coefficient and on the chemical potential. On the basis of the analytic solution of the problem, an explicit representation is obtained for the distribution function, as well as the expressions for the chemical potential jump and the chemical potential distribution in the half-space (chemical potential profile). It is also shown that the concentration jump for the Bose gas and its distribution in the half-space are proportional to the chemical potential jump and distribution. On Figs. 2-5 graph comparison of the received results with similar results from work [15] is shown .

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