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Magneto-Photo-Thermoelastic Excitation Rotating Semiconductor Medium Based on Moisture Diffusivity
1 Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, 44519, Egypt
2 Department of Mathematics, Faculty of Science, Taibah University, Madinah, 42353, Saudi Arabia
3 Department of Mathematics and Statistics, College of Science, Taif University, Taif, 11099, Saudi Arabia
4 Arab Academy for Science, Technology and Maritime Transport, Alexandria, 1029, Egypt
* Corresponding Author: A. M. S. Mahdy. Email:
Computer Modeling in Engineering & Sciences 2024, 141(1), 107-126. https://doi.org/10.32604/cmes.2024.053199
Received 27 April 2024; Accepted 21 June 2024; Issue published 20 August 2024
Abstract
In this research, we focus on the free-surface deformation of a one-dimensional elastic semiconductor medium as a function of magnetic field and moisture diffusivity. The problem aims to analyze the interconnection between plasma and moisture diffusivity processes, as well as thermo-elastic waves. The study examines the photo-thermoelasticity transport process while considering the impact of moisture diffusivity. By employing Laplace’s transformation technique, we derive the governing equations of the photo-thermo-elastic medium. These equations include the equations for carrier density, elastic waves, moisture transport, heat conduction, and constitutive relationships. Mechanical stresses, thermal conditions, and plasma boundary conditions are used to calculate the fundamental physical parameters in the Laplace domain. By employing numerical techniques, the Laplace transform is inverted to get complete time-domain solutions for the primary physical domains under study. Reference moisture, thermoelastic, and thermoelectric characteristics are employed in conjunction with a graphical analysis that takes into consideration the effects of applied forces on displacement, moisture concentration, carrier density, stress due to forces, and temperature distribution.Keywords
The study of how mechanical and thermal processes interact with one another in a material is known as thermoelasticity. The impact of rotation on material behavior is one of the fascinating features of thermoelasticity. In thermoelasticity, rotation is the result of mechanical or thermal forcing on a solid substance. The rotation might be calculated as part of the solution or specified, depending on the particular situation. For instance, in a thermally induced situation, the moment brought on by the non-uniform thermal strain might create a rotation because of the material’s differential expansion. A torque applied to the solid body in a mechanically generated situation has the potential to spin it. The behavior of materials, especially the distributions of stress and strain, is significantly influenced by rotation in thermoelasticity. Thermal stress, or stress resulting from a temperature differential in the material, is a key idea in this context. The thermal stress distribution of a rotating solid body varies, which may lead to non-uniform deformation and eventual collapse. The impact of rotation on natural frequencies and vibrational modes is a significant additional factor in thermoelasticity. The modes of vibration of the material vary as it rotates, potentially leading to instability or even deterioration of the material’s qualities. As a result, while assessing and creating thermoelastic systems, the influence of rotation must be taken into account.
Diffusion in thermoelasticity refers to the process where mass or heat is transferred between two or more materials due to the differences in temperature and pressure between them [1]. Recently, considerable research has been conducted on the diffusion processes in thermo-elastic materials. This research has focused on understanding the mechanisms that control the diffusion process, as well as the factors that affect the rate of diffusion. One area of intense study has been the effect of defects, such as cracks, pores, and dislocations, on the diffusion process in thermo-elastic materials. Several theoretical models have been proposed to describe the diffusion process in thermo-elastic materials. For example, the Arrhenius model assumes that the rate of diffusion is exponentially dependent on temperature and that the activation energy for diffusion can be estimated based on the energy barrier that must be overcome for diffusion to occur. Another model, the Fickian model, assumes that the speed at which diffusion occurs is directly related to the difference in concentration of the substance being diffused [2]. Overall, Diffusion in thermo-elasticity has improved knowledge of mechanisms. that control the transfer of mass and heat in these materials. The study of material properties and behavior under extreme conditions, as well as the design and development of energy conversion devices, are all significantly affected by this understanding. As research in this area continues, new models and techniques will likely be developed that further enhance our understanding of this complex phenomenon. Szekeres [1,2] investigated how generalized heat transport and moisture interact. According to Gasch et al. [3], changes in temperature and moisture levels might cause more damage than actual loads. To create equations that control the behavior of coupled hygrothermoelasticity. Povstenko [4] used the thermal stresses theories to study the time-fractional telegraph equations in the context of thermoelasticity theory.
In recent years, the photothermal (PT) technique has gained recognition for analyzing the electrical and thermal properties of semiconductor materials. These materials, increasingly important in industries like sensors, solar panels, and advanced medical equipment, are essential for renewable energy production. Semiconductors possess moderate conductivity, and when light energy interacts with them, electrons and holes are stimulated, creating electronic deformations (ED). Thermoelastic deformation (TED) refers to structural alterations caused by the excitation of electrons through light impact, leading to the creation of an electron cloud exhibiting characteristics akin to plasma waves. Heat generated in this process contributes to TED. Researchers have explored innovative methods to study how semiconductor samples react to a laser beam, including sensitivity analysis using photoacoustic spectroscopy. These explorations incorporate the application of thermoelasticity theory as well as the PT theory [5,6]. To ascertain the precise values of temperature, internal displacements, thermal diffusion, and other electrical properties of nano-composite semiconductor materials, PT techniques were applied in several physical experiments [7–10]. Electronic deformation is directly related to changes in the density of free carriers, which are in turn caused by the thermal wave generated by laser or sunlight pulses within the material’s internal structures [11]. In their research, Hobiny et al. [12] examined the phenomenon of PT waves in a limitless medium by utilizing a cylindrical cavity filled with semiconductor material. These topics have been explored in various studies [13,14]. Mahdy et al. [15–18] extensively investigated the concept of photo-thermoelasticity by studying the influence of external elements like magnetic fields and rotation on the interplay of thermal-plasma-elastic waves in semiconductor materials. Wave propagation in a photo-thermoelastic semiconductor medium is studied, together with the effects of humidity, magnetic field, and rotational field.
Sur et al. [19] studied the fibre-reinforced magneto-thermoelastic medium subjected to a moving heat source and took some applications according to the thermal therapy of cancer cells. Changing a material’s temperature can substantially affect its mechanical and thermal properties. Since the expanded theory of thermoelasticity has been the subject of multiple investigations, many researchers have studied the effect of the temperature gradient, which has been largely disregarded in the past [20–24]. Understanding how temperature affects material characteristics is extremely important because we cannot assume that the characteristics of a material remain constant as temperature changes [25]. Since earlier discussions on coupled and uncoupled theories did not match up with practical tests, the thermoelasticity theory was established. This issue was fixed by the novel theory of connected thermoelasticity suggested by Biot [26]. Biot [26] proposed a dynamic theory of thermoelasticity (CD theory) that utilized The Fourier law of heat. The behavior of thermal waves moving at limitless velocities was the focus of his theory. Adding a single relaxation period to the heat equation was an innovative approach introduced by Lord et al. [27]. This was expanded upon by Green et al. [28], who added two relaxation intervals to the heat conduction equation. Multiple studies [29,30] have made use of GL’s proposed generalized thermoelasticity theory. The theory explores how several types of waves behave and interact within a solid medium, including heat waves, electric fields, and mechanical vibrations. Lata et al. [31] investigated the impact of multi-dual-phase-lag heat for isotropic thermoelastic medium according to a couple stress model with two temperature theory. On the other hand, Lata [32] used fractional calculus to study the magneto-thermoelastic rotating material according to GN-II theory. Marin et al. [33] applied a new model according to Some results in Moore-Gibson-Thompson thermoelasticity with biological application for skin tissue. Roy et al. [34] studied the fractional heat order for a thermoelastic medium under the impact of voids according to three-Phase-Lag thermal memory.
This problem is investigated in the context of one-dimensional thermo-elasticity, with special attention paid to the role that mechanical force and the diffusivity of moisture play in the photo-thermal transport process. An answer is found in the research, but only at the free surface of a semi-infinite semiconducting medium. Important physical quantities have analytical solutions found by applying Laplace transforms to the governing equations. Using a powerful programming language, the inversion of the transforms is carried out numerically. The study is completed by numerically and graphically computing several properties, such as normal displacement, normal force stress, moisture concentration, carrier density, and temperature distribution.
Assume that the object being studied is rotating evenly. In such circumstances, the angular velocity can be denoted as
(1) Acceleration of Coriolis’s
(2) Centripetal acceleration,
In a problem that involves only one dimension, every variable is solely dependent on the coordinate x and time t. considered the angular velocity as
In addition, the magnetic field must be adjusted so that
In these equations, we have the magnetic permeability represented by a symbol
A medium’s magnetic intensity is a vector whose components are [15]:
Let’s pretend we have a homogeneous, transversely anisotropic semiconductor material with linear elastic characteristics. Here, we explore this material’s behavior throughout the PT transport stage of its life cycle. We also account for the fact that moisture and plasma-thermal diffusion within the semiconductor material often coincide. In this study, the displacement vector
In this scenario, the effects of Lorentz force and moisture diffusion are factored into the equations of motion for a plasma wave that is subject to both a rotating field and a magnetic field [15,35]:
The following equation describes the displacement and strain tensor:
The relationship between plasma temperature, displacement, stress, and moisture concentration using a tensor form can be represented as follows [37]:
The equation includes several parameters such as temperature diffusivity
We can define the basic physical quantities in one dimension (1D) using the following expressions [35]:
The form of the motion Eq. (9) is as follows:
where
The equation that defines the relationship in one dimension can be expressed as:
3 Mathematical Formulation of the Problem
To make it simpler, new simplified variables are introduced that have no units associated with them.
In Eqs. (13)–(17), the dashes are omitted for convenience. Using Eq. (18), when the strain in 1D is
The stress component in one dimension appears in the non-dimensional form in the following way:
where
The parameters
One way to approach solving an analytical problem involves taking into account certain initial conditions that exhibit homogeneity properties. These conditions can be expressed as follows:
Using Laplace transformation, which is a mathematical technique that can be utilized on any function
Using Eq. (25) for the basic five Eqs. (19)–(23), yields:
where
Eliminating
where
Using the factorization technique, the original ordinary differential equation (ODE) (31) may be solved, as shown below:
where
The other quantities’ solutions can also be stated as:
where
In the domain of unknown parameters
To specify
(I) When
Therefore,
(II) To evaluate the condition of the mechanical normal stress components on the surface when
Therefore,
(III) When the diffusion of carrier density and the photo-generation during recombination processes are being transported, the plasma boundary condition at the free surface (
In contrast, we found the following relationship:
(IV) Free-surface boundary conditions for displacement when
alternatively, the following relationship was derived:
The equations above use symbols
6 Transforming the Fourier-Laplace Transforms in Reverse
Laplace transform inversion can be used to derive the dimensionally-free physical fields in the time domain. To get an approximate estimate of the Laplace transform, one can use numerical approaches like the Riemann-sum approximation method [38]. Any function
In the case of
Extending the function over a closed interval
where
7 Numerical Results and Discussions
The physical quantities in this problem such as displacement, temperature, moisture concentration, stress distribution, and carrier density are represented by numerical values for a brief duration. Materials are utilized for numerical simulations and SI units and constants are utilized, with software such as MATLAB used to plot the physical constants. Table 1 provides the physical constants of the semiconductor media for Si and Ge, as indicated in references [39–42].
7.1 The Impact of Thermoelastic Coupling Characteristics
According to the principles of photo-thermoelasticity, the representation of physical fields concerning horizontal distance (
Figure 1: The variation of physical field distributions with distance with the effect of thermoelastic coupling parameter
7.2 The Effect of the Thermoelectric Coupling Parameter
According to the photo-thermoelasticity theory, Fig. 2 displays the main physical fields in connection to the horizontal distance (
Figure 2: The variation of physical field distributions with distance at different values thermo-electric coupling parameter
7.3 Influence of Rotational Filed
Fig. 3 illustrates the relationship between the horizontal distance and the main physical fields in the third category. It showcases different values of rotational field constants, taking into account the influence of magnetic and moisture diffusivity. All calculations are conducted within the framework of thermally induced mechanical deformations.
Figure 3: The variation of physical field distributions with distance with the effect of rotational field when
7.4 Influence of Reference Moisture
Fig. 4 displays the primary physical fields plotted against the horizontal distance. It depicts the variations in moisture constant along with the influence of magnetic and rotational fields in different scenarios. Thermal and elastic coupling are taken into account when conducting all calculations
Figure 4: Shows how physical field distributions vary at different distances. The steady state of moisture
7.5 Influence of Magnetic Field
The basic physical fields are displayed against a horizontal distance
Figure 5: The variation of physical fields distribution under the effect of magnetic field
7.6 The Comparison between Ge and Si Materials
Comparison of the elastic semiconductor materials (Si) and (Ge) is shown in Fig. 6. The values of the physical fields under inquiry have been numerically analyzed in this category. when
Figure 6: The comparison of physical field distributions with distance under the influence of a magnetic field in Si and Ge materials
This study investigates how various factors influence the transmission of PT-elastic waves in solid semiconductors, particularly focusing on the impact of electron deformation on the material’s deformation potential and wave generation. Using photo-thermoelasticity theory, the research analyzes wave dynamics in semiconductor media, exploring the effects of thermoelasticity, thermoelectricity, magnetic and rotational fields, and moisture concentration on wave propagation. The practical significance of photothermal theory lies in understanding material deexcitation and light absorption. The findings highlight the significant influence of medium properties on the studied aspects, benefiting physicists, material designers, thermal engineers, and geophysicists. The methodology developed offers a tool for addressing various photo-thermoelasticity and thermodynamic challenges. Understanding diffusion phenomena in thermoelasticity has wide implications in engineering, materials science, and biology. Further research in this area promises enhanced comprehension and practical applications. Additionally, studying rotational effects in thermoelasticity is crucial across industries like aerospace, energy, and biomechanics, offering safer solutions to real-world problems.
Acknowledgement: The authors extend their appreciation to Taif University, Saudi Arabia, for supporting this work through Project Number (TU-DSPP-2024-172).
Funding Statement: This research was funded by Taif University, Taif, Saudi Arabia (TU-DSPP-2024-172).
Author Contributions: Khaled Lotfy and A. M. S. Mahdy: Conceptualization, Methodology, Software. E. S. Elidy: Writing—Original draft preparation, Data curation. A. M. S. Mahdy and Alaa A. El-Bary: Supervision, Visualization, Investigation, Software, Validation. All authors: Writing—reviewing and editing. All authors reviewed the results and approved the final version of the manuscript.
Availability of Data and Materials: The data supporting this article are available upon request.
Conflicts of Interest: The study, writing, and publishing of this paper were all done without any possible conflicts of interest disclosed by the authors.
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