FEATURES OF EXCITATION OF LINEAR AND NONLINEAR THERMAL WAVES IN DIELECTRIC FILMS ON A SUBSTRATE UNDER IRRADIATION WITH A HARMONICALLY MODULATED ION BEAM

UDC:534.16: 535.341

FEATURES OF EXCITATION OF LINEAR AND NONLINEAR THERMAL WAVES IN DIELECTRIC FILMS ON A SUBSTRATE UNDER IRRADIATION WITH A HARMONICALLY MODULATED ION BEAM

Salikhov T.Kh., Abdurahmonov A.A.

Research Institute of the Tajik National University

In [1] we considered the features of the generation of linear and nonlinear thermal waves in free dielectric films in air under the action of a harmonically modulated ion beam with a frequency of [2-6]. However, the case when the film is attached to a substrate was not considered. The purpose of this work is to generalize the results of [1] taking into account the presence of a substrate.

In fairness, we note that a sufficient number of works are devoted to various aspects of the interaction of an ion beam with dielectrics (see, for example, [7-15]), and the features of the generation and propagation of thermal waves in condensed media are described in sufficient detail in reviews [16, 17] and the monograph [18]. We also note that in [19-29] similar problems in photoacoustics were solved and it turned out that in this case nonlinear temperature oscillations are generated in the medium, both at a frequency and at a doubled frequency (the second harmonic).

As in [1], we assume that the sample is irradiated in an air environment and take into account that the thickness of the dielectric is greater than the mean free path of ions in it. Then, to describe the phenomenon under study, we will proceed from the following system of nonlinear heat conduction equations:

, , (1)

, , (2)

. . (3)

where is the unit Heaviside function, and , is the heat capacity per unit volume and the thermal conductivity of the corresponding layers, respectively, and , , are the beam current density, the ion charge in units of electron charge, and the initial ion energy, respectively.

Taking into account that the initial value of the sample temperature is , the six boundary conditions required to solve system (1)-(3) are as follows [8]

, (4)

, (5)

, , (6)

, . (7)

In (4) and (5), , are the heat transfer coefficient of the sample surface and its emissivity, respectively [30-32]. Following [11-15], we will assume that the temperature dependence of thermophysical and optical quantities has the following form:

, , ,

where

, , ,

are thermal coefficients of these quantities.

We will represent the magnitude of the temperature perturbation as the sum of linear , nonlinear and locally equilibrium components, i.e. in the form

. (8)

Here and are nonlinear components of the temperature oscillations of the corresponding layer at the fundamental and second harmonics. Taking this circumstance into account, the system of equations (1)-(3) breaks down into the following systems of equations for , , and :

, , (9) , , (10)

, . (11)

, , (12) , , (13)

, , (14)

, (15)

, (16)

, (17)

, (18) . (19)

. (20)

In (9)-(20) , - thermal diffusivity of the corresponding layers at .

From (4)-(7) we obtain the following boundary conditions for :

, (21)

, (22)

, (23)

where . In [13-15] we obtained a solution to the system of equations (9)-(11), satisfying the boundary conditions (4)-(7), and also performed an analysis of the features of the formation of a stationary temperature field.

Solution of the system of equations for the linear component of temperature fluctuations

Taking into account that the incident ion beam is modulated according to a harmonic law, we represent the quantities in (12)-(14) in the form and we will have:

, (24)

, (25)

, (26)

Where , , , - thermal diffusion length.

Obviously, the solution to the system of equations (24)-(26) has the following form:

, (27)

. (28)

. (29)

Here , and are constants of integration. Using boundary conditions (21)-(23), we obtain a system of linear equations for finding the explicit form of and the solution of which has the following form:

, .

The following notations are used here:

, ,

, , , , , , , .

The terms in (27), (28) and (29) are divergent and contradict physical laws. On this basis, we will neglect them in what follows. The first term in (27) is constant, since it does not depend on frequency, so we will also neglect it. Then we will have

, , (30)

. . (31)

, . (32)

Expressions (30)-(32) describe the frequency dependence of the amplitude and phase of the excited thermal waves for the case under consideration.

Let us consider the case when , then and the equalities are valid

, , , .

Taking these equalities into account, we will analyze expressions (30)-(32). We will take into account that the value , and . We will assume that . For dielectrics and is significantly greater, then and . Also , and , , , . As a result we get that

, и .

These expressions show that in the irradiated layer the frequency dependence of the amplitude of the linear component of the excited thermal wave is , and for the non-irradiated layer .

Solution of the system of equations for the fundamental harmonic of nonlinear temperature oscillations

We proceed from the system of equations obtained above for :

, (33)

, (34)

. (35)

From (4)-(7) it follows that the six boundary conditions necessary for solving the system (33)-(35) have the following form:

, , (36) . (37)

. (38)

, (39)

.(40)

From expressions (37)-(40) it follows that in order to use these boundary conditions we need to have an equation for the function . From (15)-(17) we obtain the following equation for this function:

, . (41)

Then the boundary conditions (39) and (40) will take the form

, (42)

. (43)

Here . Considering that in (41), we will put and we will have

, ,3. (44)

The solution to equation (44) has the form

, (45)

. (46)

. (47)

The following notations are used here:

, , ,

, , (48)

, , (49)

, , (50)

, , .

To determine the quantities and , using boundary conditions (36)-(40), we obtain the following system of algebraic equations:

, (51)

, (52)

, (53)

, (54)

, (55)

. (56)

Using definitions

, we rewrite the system of equations (51)-(56) in the form

(57)

, (58)

, (59)

, (60)

, (61)

. (62)

From (59) and (61), which turned out to be independent, we find the following expressions for and :

и , (63)

where , , .

From equation (62) we find that

. (64)

In equation (57) there is no term with . Taking this circumstance into account, from (57) we find

(65)

Substituting expressions (64) and (65) into (58) and (60), we obtain the following system of equations for determining and :

(66)

. (67)

The following notations are used here:

, ,

The solution of system (66)-(67) can be represented as

, , (68)

where

Then for the quantities and we obtain the following expressions:

(69)

. (70)

The expressions (45)-(47) obtained above, together with (63) and (68)-(70), represent the solution to the formulated problem. In what follows, we will neglect the first and third terms in expressions (45)-(47), since they are divergent and are not physical. Then we will have

, (71)

, (72)

. (73)

Expressions (71)–(73) together with (63) and (68) describe all the features of excitation of the fundamental harmonic of a nonlinear thermal wave.

Solution of the system of equations for the second harmonic of the nonlinear temperature oscillation

We proceed from the system of equations for , which corresponds to the second harmonic of the temperature oscillation

, (74) . (75)

. (76)

It is obvious that to solve this system of differential equations it is necessary to have six boundary conditions. These conditions follow from condition (4)-(7) and have the following form:

, (77) , (78)

, (79) , (80)

.(81)

Тогда, очевидно, возникает необходимость иметь уравнение для функции . Из (74)-(76) для получим следующее уравнение:

Then, obviously, there is a need to have an equation for the function . From (74)-(76) for we obtain the following equation:

. (82)

From (80) and (81) we obtain the following boundary conditions for

, (83)

. (84)

As before, let us assume , then from (82) we will have

, , (85)

где , .

Solution (85) for all layers of the sample has the following form:

, (86)

, (87)

, (88)

where

, .

, ,

, .

To determine the quantities , , and from the boundary conditions (77)–(79), (83) and (84) we obtain the following system of algebraic equations:

, (89)

, (90)

, (91)

, (92)

, (93)

In (89)-(94) we use the following notations:

Then the system of equations (89)-(94) can be rewritten as

, (95)

, (96)

, (97)

, (98)

, (99)

. (100)

From (98) –(99) for and we find the following expressions:

, (101)

. (102)

Substituting these expressions into (100) and using the notations

for , we find the following expressions:

. (103)

From (97) we find that

. (104)

We substitute the expressions for and into (95)–(96) and we will have

, (105)

The following notations are used here:

, , , , .

, .

Then the solution of system (105) will take the following form:

, , (106)

where

, .

Substituting these expressions into (103) and (104) we get

, . (107)

The first and third terms in (86)–(88) are divergent and are not physical. In what follows we will neglect them and have

, (108)

, (109)

. (110)

Expressions (108)-(110) together with (102), (106) and (107) are the general solution of the formulated problem of excitation of the second harmonic of a nonlinear thermal wave in dielectric films attached to a substrate. We emphasize that the generation of this wave is caused by thermal nonlinearities of the thermophysical quantities of all layers of the sample, as well as the emissivity and heat transfer coefficient of the surfaces.

Thus, within the framework of this work, a theory of generation of linear and nonlinear thermal waves in dielectric films attached to a substrate by means of a harmonically modulated ion flux is proposed. It is established that the frequency dependence of the amplitude of the linear component of the excited thermal wave in the irradiated layer is , while for the non-trainable layers of the sample and substrate . General expressions for the fundamental and second harmonics of the nonlinear component of thermal waves are also obtained, taking into account the temperature dependence of the thermophysical and optical parameters of the sample.

Funding of the work: The study was carried out within the framework of the Republican target program (0121TJ1095).

REVIEWER: Komilov Q.,

Doctor of Physical and Mathematical Sciences,

Professor

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Feature of excitation of thermal waves in dielectric films under irradiation with a harmonically modulated ion beam

A mathematical model has been formulated for the problem of generating linear and nonlinear thermal waves in dielectric films attached to a substrate and in the air by means of a harmonically frequency-modulated ion beam. To solve the formulated problem, a system of nonlinear heat conduction equations is used for two layers of the sample and the substrate. The temperature dependence of the thermophysical quantities of both parts of the sample and , as well as the heat transfer coefficient and emissivity degree, is taken in linear form using thermal coefficients. Temperature disturbances are presented as the sum of stationary and oscillatory components, and the oscillatory part as the sum of linear and nonlinear. In turn, the nonlinear part consists of the sum of oscillations at the fundamental and second harmonics. Due to the fact that the temporary change in the heat source due to the absorption of the ion flow has a harmonic form, the time dependence of the temperature fluctuation in the original equations is also presented in a harmonic form. By solving the boundary value problem, an explicit form of expression was obtained for all parts of the oscillatory component of the temperature disturbance. It was discovered that the frequency dependence of the amplitude of the linear component of the excited thermal wave in the irradiated layer is , while for the non-irradiated layer and substrate .

Key words ion beam, harmonically modulated ion beam, irradiation, temperature field, thermal nonlinearity, thermal conductivity, dielectric films, substrate, thermal waves.

Information about the authors: Salikhov Tagaymurod Haitovich – Tajik National University, Research Institute, Doctor of Physical and Mathematical Sciences, Professor, Chief Researcher of the Department of «Condensed Matter Physics». Address: 734042, Dushanbe, Tajikistan, Rudaki Avenue, 17. Phone: (+992) 919-24-83-11. Е-mail: This email address is being protected from spambots. You need JavaScript enabled to view it..

Abdurahmonov Abdurahmon Abdulqadimovich - Tajik National University, Research Institute, senior researcher at the Department of «Condensed Matter Physics». Address: 734025, Dushanbe, Tajikistan, Rudaki Avenue, 17. Phone: 93-707-05-90. E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it. .

Article received 03.03.2024

Approved after review 08.05.2024

Accepted for publication 24.09.2024

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FEATURES OF EXCITATION OF LINEAR AND NONLINEAR THERMAL WAVES IN DIELECTRIC FILMS ON A SUBSTRATE UNDER IRRADIATION WITH A HARMONICALLY MODULATED ION BEAM