UDC:53+54(575.3)
EFFECT OF ANNEALING TEMPERATURE ON THE STRUCTURE AND THERMAL PROPERTIES OF GRAPHENE OXIDE-FILLED POLYETHYLENE
Rashidov D., Tabarov S.Kh., Ismatov Sh.P., Aknazarova Sh.I., Sodikov F., Dustov A.I.
Research Institute of Tajik National University
These days, 2D carbon structures - graphene structures - have attracted special attention from researchers in the formation of composite polymer materials [1, p. 5124]. However, the labor-intensive method of obtaining and the high cost of the resulting 2D structures practically exclude their use in the real practice of obtaining composite materials.

One of the promising ways of synthesizing 2D carbon structures is the carbonization of biopolymers under the conditions of the self-propagating high-temperature synthesis (SHS) process [2, p. 180; 3, p. 126]. The 2D carbon structures obtained by this method are an oxidized form of graphene - graphene oxide. Therefore, the problem of obtaining and studying materials with high physical and mechanical characteristics based on the use of graphene structures (graphene oxides) with ordered nanostructures is relevant..
It should be noted that many works have been devoted to the study of the structure, thermal and mechanical properties, and the nature of structural transformations in polymers modified with nanocarbon particles [4, p. 174; 5, p. 20; 6, p. 21; 7, p. 388; 8, p. 8; 9, p. 13; 10, p. 35; 11, p. 53]. However, the issues of the influence of both the annealing temperature and the concentration of fillers on the structure and thermophysical properties of nanographene oxide-containing polymers, their changes depending on the annealing conditions, remain incompletely understood.
In particular, the degree of crystallinity and the distribution of crystallites by their sizes can vary sharply depending on the preliminary heat treatment of the sample, which is manifested in the thermograms of composites in the form of multiplets or a doublet [12, p. 482].
In this regard, the goal of this work was to study the effect of thermal annealing on the structure and thermal properties of polymer composite films. Low-density polyethylene (LDPE) with a molecular weight of M = 6 x 104 was used in the polymer matrix [13, p. 64]. The filler was graphene oxide (GO) powders synthesized from natural lignin (LGO) in the SHS process.
The composite film samples were obtained by casting mixed solutions of the polymer and filler in bromobenzene. The concentration of graphene oxide was varied within 1-5% by weight. The formed films of the composite material had a thickness of 70-80 μm.
Preliminary thermal annealing of both the original non-oriented and composite film samples was performed in a free state in a thermal chamber with an air environment at three temperatures of T0t = 50, 70 and 85 °C for 3 hours.
The structure of the samples was studied on DRON-2 and KRM-1 X-ray diffractometers using copper radiation filtered by nickel. The thermal properties of the composites were studied on a DSC 204F1 device with a heating rate of 10 °C / min.
Naturally, for more in-depth structural features of composite systems, it is advisable to first of all give a separate characteristic of their components - the polymer matrix and filler particles, which was done.
Thus, Fig. 1 shows a wide-angle X-ray diffraction pattern (WAX) of lignin graphene oxide (LGO) powder. As can be seen from the wide-angle X-ray diffraction patterns of lignin powder, along with two wide amorphous halos with maxima at angles of 2θ=13° and 23°, crystalline reflections are also observed at angles of 2θ=21°, a doublet of 2θ=26.5° and 26.6°. This form of diffraction patterns is characteristic of multilayer graphene particles [14, p. 148; 15, p. 3021; 16, p. 3122]. We can associate the appearance of the doublet with terminal oxygen-containing groups, i.e. graphene oxides.

Figure 1. Wide-angle X-ray diffraction pattern of particles of lignin carbonized in the SHS process. The inset shows crystal reflections at angles (doublet) 2θ=26.50 and 26.60
Fig. 2 shows the BR (a) and MR (b) of LDPE-KGO composites depending on the concentration of nanoparticles in air. As can be seen, the BR of the original and filled samples (Fig. 2a) shows the main characteristic reflections 110 and 200, indicating a spherulitic structure of polyethylene with an average spherulite diameter of ~ 5 μm [17, p. 75]. Traces of graphene oxide nanoparticles, their clusters or agglomerates are not visible.
In the range of used concentrations of graphene oxide additives C = 1-5%, no noticeable changes in the angular positions 2θ, radial half-width Δ2θ and intensity Ic of the reflections are observed on the BR of the filled samples, the sizes of the crystalline formations are 5-6 nm.

Figure 2. High-angle (a) and low-angle (b) X-ray diffraction patterns of LDPE+LGO composites at different concentrations: 1-C=0; 2-1; 3-3; 4-5% LGO.
Therefore, as in previous studies for LDPE-fullerene C60 composites [18, p.889; 19, p.1285], LGO particles are not included in the LDPE crystal lattice, but are located in the interbeam and interspherulitic amorphous regions of the polymer.
In the MR of the initial and filled LDPE-LGO samples (Fig. 2b), weak discrete scattering with a tangential periodicity of ~ 30 nm is observed; with an increase in the filler concentration, an increase in the intensity of diffuse scattering occurs, which absorbs the discrete scattering, indicating the formation of microheterogeneities such as pores and cracks in the polymer matrix. Heat treatment (annealing) of the same samples at 850C leads to a 20% change in the intensity of Ic, invariance of the radial half-width Δ2θ and angular positions of the 2θ reflections, the size of the crystalline formations with an increase in the concentration of graphene oxide remains within 5 nm. Traces of nanoparticles – graphene oxides and their clusters are not visible.
Figure 3 shows thermograms of unannealed and annealed LDPE-LGO composites obtained in the first and second heating cycles, and the results of thermographic studies of the composites are summarized in Tables 1 and 2.

Figure 3. Thermograms of unannealed (a) and annealed (b) samples of LDPE-LGO composites in the first (a, b) and second (c) heating cycle.

Table 1.Thermal characteristics of LDPE-LGO composites
Sample 1 cycle 2 cycle
Тпл,°С Ткр, °С ΔНпл., Дж/г ΔНкр., Дж/г Тпл,°С Ткр, °С ΔНпл., Дж/г ΔНкр., Дж/г Ск,%
ДСК Ск,%
X-ray
LDPE outg. 107,4 94,8 -40,35 54,1 107,4 94,8 41,23 63,25 57,0 54
LDPE+0.5% LGO 107,1/109,9* 94,7 -35,64 62,3 107,1 94,7 -43,7 67,32 54 53
LDPE+1% LGO 107,2/109,6 94,8 -62,66 96,70 107,2 94,8 -54,38 85,60 56 51
LDPE+3% LGO 107,7/109,4 94,9 -45,37 67,9 107,5 94,9 -39,76 60,9 52 50
LDPE+5% LGO 108,0/110,2 94,9 -39,6 63,1 107,4 94,9 -42,9 64,27 50 47
* additional peak melting point

As can be seen, the samples of unannealed initial polyethylene and their composites containing 0.5-5% LGO give a thermogram that differs significantly from the thermogram of annealed (T0 = 85 °C) composites (see Fig. 3a (1), b (1)). The thermogram of the initial unannealed polyethylene is characterized by the presence of a sharp melting peak at a temperature of 107.40, which is 1.40 lower than the peak position of the same sample annealed at 85 °C (Table 2), and the thermogram of this same sample shows the presence of an additional low-temperature peak (90 °C), Fig. 3b (1), the appearance of which is most likely associated with the formation of less perfect crystalline structures, i.e., recrystallization during annealing. The area limited by the melting peak for the annealed initial sample is more than 25% greater than the same area for the unannealed sample.
Table 2.Thermal characteristics of LDPE-LGO composites annealed at 85°
Sample 1 cycle 2 cycle
Тпл,°С Ткр, °С ΔНпл., Дж/г ΔНкр., Дж/г Тпл,°С Ткр, °С ΔНпл., Дж/г ΔНкр., Дж/г Ск,%
ДСК Ск,%
X-ray
LDPE outg. 108,8 94,8 -44,55 70,2 107,8 94,8 -148,5 77,9 47 54
LDPE-1% LGO 107 (109,8)* 94,7 -57,07 81,8 108 94,7 -60,72 86,3 43 50
LDPE-3% LGO 107,8 (110,2) 94,7 -50,5 71,17 107,3 94,7 -47,52 67,8 40 46
LDPE-5% LGO 110,8 95,1 -55,9 80 108,5 95,3 -50,5 79 42 40
An increase in the filler concentration also has a certain effect on the crystalline structure of polyethylene and, consequently, on the shapes of thermograms (see Fig. 3a (1-5)). A similar effect was observed for LDPE and polyamide-6 (PA-6) samples containing small additives of fullerene C60, fullerene soot, nanodiamonds and nanotubes [20, p. The thermogram of LDPE samples with high contents of graphene oxide additives (C=2-5%) is characterized by the presence of a doublet and, upon reaching C=5%, a wider and flatter melting peak at a temperature of 108°C is observed on the thermograms (Fig. 3a (5)). An increase in the annealing temperature of the composites (up to 85°C) leads to an even greater change in the shape of the thermograms in the form of low-temperature peaks and multiplets or a doublet (Fig. 3b). The thermograms of filled and unfilled composite samples in the second heating-cooling cycle show one main melting peak (Fig. 3c), which indicates stabilization of the structure. All thermograms confirm the conclusion about the independence of the behavior of the crystalline phase of LDPE and graphene oxide, since the temperatures of the melting and crystallization peaks of LDPE only slightly change their temperature position with variations in the concentration of graphene oxide.
Studying the features of the thermograms obtained for the same sample (unfilled) polyethylene (Fig. 4), annealed at different temperatures (T0 = 300, 700, 800, 900 and 1040C) in the thermostat of the DSC 204F1 device for 2 hours with an accuracy of 0.10C, showed that all thermograms show one single melting peak, the temperature position of which changes with an increase in the annealing temperature, and the presence of a doublet is not detected. The observed low-temperature peak (79.50C) at T0=700C (see Fig. 4) increases its area with increasing T0 and shifts toward higher temperatures reaching 98.70C at T0=1040C. The observed changes are most likely associated with melting and recrystallization of fine-crystalline structures at the annealing temperature. The results of thermal characteristics of samples annealed at different temperatures are given in Table 3.

Fig. 4. Thermograms of the original LDPE samples annealed at different temperatures. 1 – То=30О; 2 - 1 – 70О; 3 - 80О; 4 – 90О; 5- 104ОС
As can be seen from the table, an increase in the annealing temperature stimulates the crystallization process, which is manifested in an increase in the melting and crystallization temperatures by more than 2 degrees compared to the initial samples (T0 = 300C). The degree of crystallinity of the samples with an increase in the annealing temperature takes on an extreme character, the maximum of which corresponds to Тann = 900C. The decrease in Ck at a temperature of 1040C is apparently associated with partial melting of imperfect crystallites, as evidenced by the separation of the low-temperature peak at a temperature of 98.70C. (see Fig. 4 term. 5).
Table 3. Thermal characteristics of LDPE samples annealed at different temperatures
Sample 1 cycle 2 cycle
Тотж.,°С Тпл,°С Ткр, °С ΔНпл., Дж/г ΔНкр., Дж/г Тпл,°С Ткр, °С ΔНпл., Дж/г ΔНкр., Дж/г Ск,%
ДСК Ск,%
X-ray
30°С 107,7 92,7 -51,56 75,62 108,3 92,7 -32,94 48,6 47 47
70°С 108 95 -45,63 67,28 108,3 92,5 -39,1 57,5 47,3 50
80°С 107,8 92,4 -49,33 73,20 108,4 92,6 -35,71 53,5 50 52
90°С 107,7 92,7 -43,46 69,1 108,3 92,8 -35,37 56,34 60 55
104°С 109,9 94,8 -30,23 46,0 108,8 92,8 -37,37 55,60 50 55
The observed changes can most likely be explained by the fact that during the formation of the initial and composite samples from a solution based on LDPE and graphene oxide additives, various crystalline structures of different shapes, sizes and perfection are formed. Therefore, due to the presence of a distribution of small and large crystallites by size, an expansion of the melting temperature range is observed, which causes the appearance of a wide diffuse peak on the thermogram (Fig. 3b (terms 3, 4)), and the appearance of low-temperature peaks at temperatures of 92O, 97OC, are apparently due to the separate melting of two or more crystallites of different sizes, surrounded by graphene oxide layers. As for the appearance of a doublet on the thermograms, as noted in [20, p. ] is associated with partial disordering of macromolecules in the crystalline regions of the polymer under the influence of thermal energy and the force field of nanocarbon particles.
The ratio between the two melting peaks can change depending on both the filler concentration and the annealing temperature. For example, the thermogram of unannealed LDPE-LGO composite samples consists of the main peak corresponding to a temperature of 107.1°C and the second peak corresponding to a temperature of 110.0°C (Fig. 3a(2)). An increase in the filler concentration will not change the temperature position of the peaks, but the shapes of the peaks change significantly, the main melting peak increases (Fig. 3, term. 2), they are leveled (term. 3), then the main melting peak increases (term. 5) and the area of the peaks expressed in the specific heat of melting () and crystallization () changes significantly (see Table 1). It is known that the total area under the peak corresponds to the heat of melting [12, p.]. Taking this into account, the degree of crystallinity of the original, doped and annealed LDPE-LGO composite samples during annealing was calculated. The degree of crystallinity of the polyethylene composite samples was calculated by comparing the area of the corresponding endothermic peak with the area of the exothermic peak [3]. These areas are numerically equal to the total amount of heat absorbed during melting Hpl or released during crystallization Hcr. Subtracting one from the other Hpl.-Hcr.=H/ we obtain the amount of heat H/ given off by that part of the polymer that was already in the crystalline state before the polymer was heated.
Next, using the HI number, the percentage ratio of the crystalline part to the total part was calculated. Dividing this number by the specific heat of fusion (Q/m=ΔH_пл ), we obtain mс.
т.с. m_c=Н^I/(ΔH_пл ) [г]
This is the total amount of polymer (in grams) that was crystalline at a temperature below Тс. By dividing this number by the mass of the sample mобр, one can obtain the fraction of the sample that was crystalline, and then the percentage of crystallinity:
С_к=m_c/m_обр ∙100"\%"
Thus, using the H/ number, the percentage ratio of the crystalline part to the total was calculated. The calculated values of the degree of crystallinity (Table 2) are in good agreement with the data obtained by other methods.
Reviewer: Sultonov N.,
Doctor of Physical and Mathematical Sciences,
Professor

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EFFECT OF ANNEALING TEMPERATURE ON THE STRUCTURE AND THERMAL PROPERTIES OF GRAPHENE OXIDE-FILLED POLYETHYLENE
The influence of thermal annealing on the structure and thermophysical properties of initial and graphene oxide-containing low-density polyethylene (LDPE) films was studied using X-ray diffraction and thermography methods. It has been shown that the introduction of graphene oxide powder synthesized from natural lignin into LDPE leads to changes in the structure and thermal properties, which appear in the thermograms of the images in the form of a multiplet or doublet. An increase in filler concentration leads to an increase in the intensity of diffuse scattering, which absorbs discrete scattering, which indicates the formation of microinhomogeneities such as pores and cracks in the polymer matrix. Based on thermophysical data, a method for determining the degree of crystallinity (Ск) of initial and composite LDPE samples is proposed. The value of Ск of the composites was calculated by comparing the area of the corresponding endothermic peak with the area of the exothermic peak. The obtained value of Ск, determined by the proposed method, is in good agreement with the data obtained by radiography methods.
Key words: structure, properties, composite, annealing, graphene oxide, thermography, radiography, endotherm, ectotherm.

Information about authors: Rashidov Jalil – Research Institute of the Tajik National University, Chief Researcher of the Laboratory of Condensed Matter Physics. Address: 734025, Dushanbe, Tajikistan, Rudaki Ave., 17. Phone: 988-57-89-11. E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it..
Tabarov Saadi Кholovich - Research Institute of the Tajik National University, Head of the Physics of Condensed medium Department. Address: 734025, Dushanbe, Tajikistan, Rudaki Ave., 17. Phone: 2-21-79-31.
Ismatov Shaboz - Research Institute of the Tajik National University, researcher of Department of Solid State Physics. Address: 734025, Dushanbe, Tajikistan, Rudaki Ave., 17.
Aknazarova Shafoat Ikboliddinovna - Research Institute of the Tajik National University, Senior Researcher, Laboratory of Condensed Matter Physics. Address: 734025, Dushanbe, Tajikistan, Rudaki Ave., 17. Phone: 933-33-80-99. E- mail: This email address is being protected from spambots. You need JavaScript enabled to view it..
Sodikov Firuz - Research Institute of the Tajik National University, leading researcher of the Department of Physics of the Condensed medium. Address: 734025, Republic of Tajikistan, Dushanbe, Rudaki Avenue 17. Phone: 918-21-69-60. E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.
Dustov Alisher Iskandarovich - Research Institute of the Tajik National University, Senior Researcher, Department of Physics of the Condensed medium. Address: 734025, Dushanbe, Tajikistan, Rudaki Ave., 17. Phone: 919-64-72-04. E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it..
Article received 05.04.2024
Approved after review 24.06.2024
Accepted for publication 16.10.2024

 
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.

2. 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 .
3. 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.

4. 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