=Paper=
{{Paper
|id=Vol-3806/S_27_Petryk_Doroshenko_Mykhalyk_Yatsenko
|storemode=property
|title=
Modeling Diffusion Transport Mechanisms in Multilayer Nanofilms Using Computer Simulations
|pdfUrl=https://ceur-ws.org/Vol-3806/S_27_Petryk_Doroshenko_Mykhalyk_Yatsenko.pdf
|volume=Vol-3806
|authors=Mykhaylo Petryk,Anatoliy Doroshenko,Dmytro Mykhalyk,Olena Yatsenko
|dblpUrl=https://dblp.org/rec/conf/ukrprog/PetrykDMY24
}}
==
Modeling Diffusion Transport Mechanisms in Multilayer Nanofilms Using Computer Simulations
==
https://ceur-ws.org/Vol-3806/S_27_Petryk_Doroshenko_Mykhalyk_Yatsenko.pdf
Modeling Diffusion Transport Mechanisms in Multilayer
Nanofilms Using Computer Simulations
Mykhaylo Petryk1,†, Anatoliy Doroshenko2,3,†, Dmytro Mykhalyk1,†, and Olena Yatsenko2,*,†
1
Ternopil Ivan Puluj National Technical University, Ruska str. 56, build. 1, Ternopil, 46001, Ukraine
2
Institute of Software Systems of the National Academy of Sciences of Ukraine, Glushkov ave. 40, build. 5, Kyiv, 03187,
Ukraine
3
National Technical University "Ihor Sikorsky Kyiv Polytechnic Institute", Polytechnichna str. 41, build. 18, Kyiv, 03056,
Ukraine
Abstract
In this paper, the authors strive to integrate complex mathematical models with optimal software
development practices to tackle the challenge of computationally simulating diffusion transport processes
in multilayer nanofilms. Drawing on the experimental findings and using the proposed model,
identification was performed using the theory of state control for multicomponent systems. Utilizing
optimal control methods for multicomponent transport systems, the analytical solution of the model, and
experimental observation data, the diffusion coefficient distributions for the components of the nanofilms
(samples of aluminum, molybdenum, silicon) were reproduced. The numerical simulation results were
compared with experimental observations. The profiles generated from the simulations closely align with
the experimental profiles, particularly as the multilayer formation process approaches the final stages of
completing the protective nanofilm layers. The maximum observed deviation does not exceed 2–3 %,
confirming the reliability of the mathematical model and highlighting the practical value of the results
presented. A software framework was developed to automate the specified calculations, with the potential
for extension to other subject areas that involve similar tasks of identifying key process factors and further
numerical modeling of time-space characteristics using the obtained results.
Keywords 1
Coefficient identification, diffusion, mathematical model, multilayer nanofilm, numerical simulation, oxide
film
1. Introduction
The challenges of investigating diffusion in multilayer films necessitate advancing modern modeling
techniques and software frameworks to accurately depict phenomena while considering transitions
between neighboring layers [1, 2]. One of the most effective approaches to thoroughly address these
issues is the widely recognized integral transformation. These methods are utilized to derive solutions
for diverse boundary value problems in mathematical physics pertaining to homogeneous structures,
encompassing diffusion scenarios across various environments and enabling their mathematical
representation.
In addition to the indispensable role of advanced modeling techniques and software frameworks in
addressing the challenges of studying diffusion within multilayer films, it is crucial to recognize the
pivotal contribution of sophisticated computational approaches. By utilizing cutting-edge
methodologies in computer programming and software architecture, the capabilities of mathematical
approaches, such as integral transformations, can be elevated to unprecedented levels of precision and
efficiency. This synergy between mathematical modeling and computational innovation not only
14th International Scientific and Practical Conference from Programming UkrPROGР’2024, May 14-15, 2024, Kyiv, Ukraine
*
Corresponding author.
†
These authors contributed equally.
mykhaylo_petryk@tntu.edu.ua (M. Petryk); doroshenkoanatoliy2@gmail.com (A. Doroshenko); d.mykhalyk@gmail.com
(D. Mykhalyk); oayat@ukr.net (O. Yatsenko)
: 0000-0001-6612-7213 (M. Petryk); 0000-0002-8435-1451 (A. Doroshenko); 0000-0001-9032-695X (D. Mykhalyk); 0000-
0002-4700-6704 (O. Yatsenko)
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
enhances our ability to accurately represent diffusion phenomena within multilayer structures but also
opens new avenues for exploration and analysis.
Furthermore, the integration of modern modeling techniques with state-of-the-art computational
tools facilitates a more nuanced understanding of diffusion processes across diverse material
compositions and environmental conditions. By leveraging the power of numerical simulations and
data-driven analyses, researchers can extract valuable knowledge on complex diffusion mechanisms
that were previously inaccessible. This multidisciplinary approach enables scientists and engineers not
only to solve fundamental questions in materials science and engineering but also to devise innovative
strategies for optimizing the performance and functionality of multilayer films in various
technological applications.
Problem formulation. In the proposed research, the authors attempt to combine complex
mathematical models with the best practices of software engineering to solve the problem of computer
simulation of diffusion transport processes in multilayer nanofilms. The primary objectives are:
• modeling diffusion processes: to integrate complex mathematical models with contemporary
software development methodologies, in domain of the diffusion within multilayer films;
• parameter identification: to employ the theory of state control for multi-component systems to
identify diffusion coefficients. It includes using methods of optimal control to analyze
experimental data and accurately reproduce the distributions of diffusion coefficients for the
nanofilm components;
• numerical simulation and validation: to conduct numerical simulations and compare the
results with experimental observations. The target is to achieve a high degree of correspondence
between the modeled and experimental profiles, particularly as the duration of multilayer
formation approaches completion;
• development of a software framework: to create a software framework that automates the
specified calculations and can be extended to other subject areas with similar tasks. This
framework should facilitate the identification of key process factors and enable further numerical
modeling of time-space characteristics based on the obtained results.
By addressing these objectives, this research seeks to overcome the limitations of existing methods
and provide a robust tool for studying and optimizing the diffusion processes in multilayer films. The
ultimate aim is to enhance the efficiency of experimental studies and contribute to the development of
new nanomaterials with improved properties.
2. Mathematical model
We propose a physical problem and corresponding mathematical model for the diffusion mass
transfer process in multilayer films, based on the following multilayer medium comprised of n
double layers composed of two distinct media with differing properties (Figure 1).
According to this representation, at each interface within the formed multicomposite, there is a
mutual diffusion of components between two adjacent layers of the medium. The mechanisms
governing such mutual transfer are determined by the variable gradients and rates of concentration
change at the interface boundaries between layers. By incorporating changes in concentrations and
their gradients over time into the boundary and interface conditions, it becomes feasible to model the
mechanisms of this additional mutual transport alongside the fundamental transport equations.
When formulating a mathematical model for diffusion transfer within oxide films, a multilayer
structure is taken into account. Assuming that the diffusion of atoms of constituent components of
oxide films (aluminum, molybdenum, silicon) primarily drives system mixing, concentration profiles
for such a multilayer system can be derived from Fick’s equations, incorporating boundary conditions
at the outer layers and contact conditions between successive layers.
Figure 1: Schema of multilayer nanofilm.
This approach provides a mathematical model describing the diffusion transfer process within a
planar multilayered medium
∂ 2Ck ∂ 2Ck (1)
∂
Ck ( t,x,z ) + γ k2Ck = D0 + Dzk
∂z ∂x2 ∂z 2
at domain
t > 0, x ∈ (0, R ),
n +1
z : z∈ U (lk −1, lk ); l0 ≥ 0; ln+1 = ∞,
k =1
j = 1, 2 k = 1, n,
where Ck is diffusion distribution; Dk is diffusion coefficient; γ k2 is mass distribution
coefficient;
The corresponding initial and boundary conditions for the model:
Ck (t , x, z )⏐t =0 = C0k (x, z ) ≡ C0k (z ), (2)
⎡ 0 d 0⎤
⎢α11 dz + β11 ⎥ C1( t,x,z ) z =l0 = Cl0 ( t,x );
⎣ ⎦
∂Cn +1
⏐z =∞ = 0;
∂z
⎡⎡ k ∂ k ⎤ ⎤
⎢ ⎢α j1 ∂z + β j1 ⎥ Ck − ⎥
⎢⎣ ⎦ ⎥ = 0,
⎢ ⎡ k ∂ k ⎤ ⎥
⎢ − ⎢α j 2 + β j 2 ⎥ Ck +1 ⎥
⎣ ⎣ ∂z ⎦ ⎦ z =l k
And boundary conditions for variable x
∂Ck (3)
x =0 = 0; Ck x = R = C1k ( t,z ),
∂x
where α ijk , β ijk ; k = 0,n ; i, j = 1,2, are coefficients determining boundary conditions and contact
conditions; x, y, z are spatial coordinates.
The exact solution of the problem described by equations (1)–(3) is directly written out by
applying integral Fourier transforms [3]:
Ck (t , x, z ) =
tR
∫ ∫ Wl (t − τ ; x, ς ; z) Cl (τ , ς )dς dτ +
0 ,k 0
00
n +1 t r lk1
+ ∑ ∫∫ ∫ H k ,k (t − τ ; x, ς ; z, ξ ) ⋅
1
k1 =1 0 0 lk −1
1
⋅ C0k (ς , ξ ) ⋅ δ + (τ )σ k1 d ς d ξ dτ +
n +1 t lk1
+ ∑ ∫ ∫ WR ,k (t − τ ; x; z, ξ ) ⋅
k 1
k1 =1 0 lk −1
1
⋅ C1k (τ , ξ )σ k1 dξ dτ .
1
Here are Green’s functions:
Wl0 ,k (t , x; z,ξ ) =
2 ∞ m cos ηmξ
= ∑ Wl ,k (t − τ , z )(-1)m ;
R m =0 0 ηm
WRk ,k (t; x, ς ; z,ξ ) =
1
∞
2 2
= ∑ e− D0ηmt D0 (-1)m+1ηmε km,k (t; z,ξ )(-1)m ⋅
R m =0 1
⋅ cos ηm x.
Cauchy’s function
H k ,k1 (t ; x, ς ; z ,ξ ) =
2 ∞ m
= ε (t , z, ξ ) ⋅ cosηmξ ⋅ cosηm x.
∑
R m=0 k ,k1
3. Experimental data and coefficient identification
According to the results of experimental data and using the proposed model, identification was
carried out using the theory of state control of multicomponent systems (results obtained in [4])
(Figure 2).
Figure 2: Coefficients distribution at first sample
The distributions of diffusion coefficients are reproduced using methods of optimal control of the
state of multicomponent transport systems, analytical solution of the model, and data of experimental
observations, for the considered constituent components of nanofilms (samples of aluminum,
molybdenum, silicon) (Figure 3).
Figure 3: Coefficients distribution at the second sample
The diffusion coefficients identified in such a way, correspond to real experimental data and are
used as input parameters of the obtained mathematical solution of the model for modeling and
analysis of concentration distributions of the main components of nanofilms (aluminum,
molybdenum, silicon, etc.).
4. Numerical simulation
Numerical simulation results are compared with experimental observations depicting the sample
content. These concentration distributions are generated for varying time intervals during the
formation of the multilayer. The designated time frame corresponds to the experimental duration of
three weeks. The process of forming the technological multilayer through molecular diffusion of the
specified components is segmented into 5 periods, encompassing the inception of the protective
multilayer from the initial period to the concluding period (Figure 4).
Figure 4: Simulated concentrations for case 5
Continuous lines represent simulated data for different time durations. The dashed line is
experimentally measured data at the final duration.
As depicted in the figures above, the profiles derived from modeling closely align with the
corresponding experimental profiles, particularly as the duration of multilayer formation approaches
the completion period of the protective nanofilm multilayer formation. The maximum observed
deviation does not surpass 2–3 %, affirming the reliability of the mathematical model and indicating
the practical utility of the provided results (Figure 5 and Figure 6).
Figure 5: Simulated concentrations for case 3
Figure 6: Simulated concentrations for case 1
5. Software framework
The above results require complex resource-intensive calculations [5]. For this purpose, we
developed a framework that automates the specified calculations with the possibility of extension to
other subject areas with similar tasks of identifying the key factors of the process and further
numerical modeling of the time-space characteristics using the obtained results.
The software framework in focus is implemented in Java programming language. JVM-based
implementation allows us to rid of target-specific builds and follow the write-once-run-everywhere
idea. At the same time, the JIT-compilation feature of JVM optimizes code execution by converting
bytecode into native machine code at runtime. It allows to have runtime optimizations that are not
possible with ahead-of-time compilation and are beneficial for CPU-intensive applications. Also,
using Java language allows to implement performance-effective algorithms to speed up the calculus
process [6].
The software tools based on the framework are developed as a stand-alone, so it is self-sufficient
and uses the resources of the system on which it is running to perform calculations, and does not
require connection to external servers or systems.
The framework design follows an object-oriented approach, and the code base is divided into
major modules, such as Models, DataProcessing, Identification, and GUI. The input either can be read
from the file or provided in UI. Calculated results also can be visualized or saved to the file. All
components are connected with abstract interfaces, which gives the ability to easily replace one model
implementation with another implementation (separation of interface and implementation) [7].
The hierarchy of classes for the visualization of modeling results is based on the DrawArea class,
which contains the basic elements for displaying graphics (Figure 7). Consequently, the graphical user
interface can be adapted to the needs of the user and can show visualizations of the modeled
parameters in the desired directions.
Figure 7: Procedure for internal parameters identification
For modeling implementation, the other hierarchy of classes is designed and begins with the
DMTBase class (Figure 8). The hierarchy contains models of different complexity and is compatible
with visualization classes (Figure 7) through the implementation of the DMTDraweble interface. Such
an approach allows you to experiment, quickly replace one model with another, and find out how
changes in model parameters or features will affect changes in the dynamics of certain kinetic
characteristics of the modeled system.
Figure 8: Procedure for internal parameters identification
One example is the implementation of diffusion coefficient identification, which defines the
internal kinetic parameters of the process. To ascertain the coefficient distribution, we employ the
gradient methods, the mathematical underpinnings of which lay in the context of parametric
identification challenges in multicomponent distributed systems [8]. Tailoring our approach to the
nanofilms domain, we find the method of minimum errors particularly apt. So, we picked the
implementation of this method in simulations among others in the software framework. The
coefficient identification algorithm is based on a gradient-identification procedure wherein, to
ascertain the next approximation of the diffusion coefficient within the intraparticle space, we adhere
to the following protocol (Figure 9).
Figure 9: Procedure for internal parameters identification
The determined distributions of diffusion coefficients within the interparticle space enable the
modeling of concentration fields and integral mass distributions within the catalytic nanoporous layer
with a high level of precision. As evidenced by the concentration distributions presented (Figure 2 and
Figure 3), the profiles derived from the model closely match those obtained experimentally (Figure 4
and Figure 5), illustrating a noteworthy degree of consistency between the two sets of data. This
alignment is further emphasized by the complete coincidence observed between the model and
experimental graphs depicting the integral mass during benzene adsorption.
6. Conclusion
The difficulties associated with studying diffusion in multilayer films require the progress of
contemporary modeling methods and software platforms to precisely represent phenomena, taking
into account transitions between adjacent layers. Alongside the essential function of cutting-edge
modeling techniques and software frameworks in addressing the difficulties of examining diffusion
within multilayer films, it’s vital to acknowledge the significant input of sophisticated computational
methods.
In this paper, we endeavor to merge intricate mathematical models with optimal software
development methodologies to address the challenge of simulating diffusion transport processes in
multilayer nanofilms computationally. Based on the experimental findings and employing the
suggested model, identification is conducted utilizing the theory of state control for multicomponent
systems. Using the methods of optimal control of the state of multicomponent transport systems, the
analytical solution of the model, and the data of experimental observations, the distributions of
diffusion coefficients for the considered components of nanofilms (samples of aluminum,
molybdenum, silicon) were reproduced. Numerical simulation results were compared with
experimental observations. The profiles obtained from the modeling closely match the corresponding
experimental profiles, especially as the duration of multilayer formation approaches the final stages of
completing the protective nanofilm multilayer formation. The maximum observed deviation does not
exceed 2–3 %, confirming the reliability of the mathematical model and demonstrating the practical
value of the results provided.
A software framework is developed for the automation of the specified calculations with the
possibility of extension to other subject areas with similar tasks of identifying the key factors of the
process and further numerical modeling of the time-space characteristics using the obtained results.
References
[1] J. Kärger, F. Grinberg, P. Heitjans, Diffusion fundamentals, Leipziger Unviersite, Leipzig, 2005.
[2] M. R. Petryk, A. Khimich, M. M. Petryk, J. Fraissard, Experimental and computer simulation
studies of dehydration on microporous adsorbent of natural gas used as motor fuel, Fuel 239
(2019) 1324-1330. doi: https://doi.org/10.1016/j.fuel.2018.10.134
[3] M. Petryk, P. Vasylyuk, D. Mykhalyk, N. Babiy, O. Petryk, Models of diffusive transport
processes and methods of parameter estimation in multi-composite nanofilms, TNTU, Ternopil,
2015. [in Ukrainian]
[4] V. S. Deineka, M. R. Petryk, Identification of parameters of inhomogeneous diffusion problems
in nanomulticomposites using gradient methods, Computer mathematics 1 (2012) 41–51. [in
Russian]
[5] M. Petryk, A. Doroshenko, D. Mykhalyk, P. Ivanenko, O. Yatsenko, Automated parallelization
of software for identifying parameters of intraparticle diffusion and adsorption in heterogeneous
nanoporous media, in: Shkarlet, S., et al. (Eds.), Mathematical Modeling and Simulation of
Systems. MODS 2022, volume 667 of Lecture Notes in Networks and Systems, Springer-Verlag,
Cham, 2023, pp. 33–47. doi:10.1007/978-3-031-30251-0_3.
[6] O. Musch, Design Patterns with Java: An Introduction, Springer Vieweg, Wiesbaden, 2023.
[7] R. Sedgewick, K. Wayne, Algorithms (4th. ed.), Addison-Wesley Professional, Boston, 2011.
[8] V. Sergienko, V. S. Deineka, System analysis of multicomponent distributed systems, Naukova
Dumka, Kyiv, 2009. [in Russian]