A consistent methodology of the approach is proposed, which allows establishing the parameters of the Lennard-Jones potential for diffusion processes in mesoscale materials in the quantum-mechanical description of the activation energies of the phase transition. The developed method is based on the regression of data obtained by analyzing concentration distribution data and applying the machine learning method of the convolutional neural network. The proposed method allows establishing the parameters of the Lennard-Jones potential with reliable accuracy directly from the specified spatial concentration distributions and determining the substance to which these parameters correspond. Software implementation of data regression and machine learning of the neural network was carried out using specialized Python libraries, namely: Scikit-learn, TensorFlow, and PyTorch. The obtained results and the developed method will be useful for specialists in the field of physical chemistry and nanomaterials science.
Nanoscale and mesoscopic materials [1-4] are of significant practical interest for modern technologies. They are composite devices operating in various areas of electronics, physical and chemical technology, crystallography, and the development of new functional materials and compounds [5-7]. This is primarily due to the wide selection of existing materials from which a variety of nano- and
micro-scale structures can be created. For this purpose, metals, semiconductors, organic semiconductors, and various magnetic materials can be used.
One of the main directions concerning the use and research of mesoscopic materials is to increase their efficiency in practical applications and to ensure the achievement of stability of this work. This problem has a fairly broad nature and a variety of its manifestations, and in general it cannot be solved by a simple experimental or theoretical study of these materials, in particular by mathematical modeling of their properties and precision characteristics, but requires a broader and more detailed analysis.
Examples of such low-density structures are materials with micropores — the so-called zeolites, which are widely used both in catalytic processes and in electronics. The properties of these mesostructures in their kinetics are mainly determined by the energies of molecular transitions or so-called activation energies corresponding to molecules of volatile hydrocarbons or charged microparticles. At the same time, these energies depend significantly on the size of these particles and the properties of interaction in the environment of these mentioned materials. As a result, the construction of even microscopic models of such physical processes does not allow to fully identify the parameters of the kinetics of microparticles in cellular materials due to a fairly wide spread of their geometric parameters. Such a problem requires the use of additional methods, in particular, the analysis of a large number of concentration distributions together with the calculation of the activation energy and the subsequent application of information technology tools. Such methods can be machine learning methods and convolutional neural networks, which allow to systematize the obtained results and adequately identify material parameters in various materials with a porous structure.
We begin by considering a sample of a porous material that has a spherical shape of radius R and contains pores with an average radius r =r. The geometric structure of such a sample and an separate micropore is shown in Fig 1.
It is assumed that inside each micropore a particle (in our case, methane or propane molecules) interacts with the sample material. This interaction is described by the Lennard-Jones potential. Thus, the molecule falls into a potential trap, the geometric dependence of which U =U (r ) in the selected coordinate system can be represented as follows:
U (r )=4 ϵ [(σ / r )12−(σ / r )6 ],
(
Since in our problem we are dealing with hydrocarbon molecules, it is impossible to apply
formula (
∑ (ψ2 s (r )+di⋅(ψ2 px (r ) , ψ2 py (r ) , ψ2 pz (r ))) 2 i=1 where di=(± 1 ; ± 1 ; ± 1) - this is a vector that specifies the direction of the hybrid orbital and takes into account all four possible directions realized by combinations of signs. The functions directly sought look like as follows:
1 ψ1(r )= 2 (ψ2 s (r )+ψ2 px ( x , y , z )+ψ2 py ( x , y , z )+ψ2 pz ( x , y , z ))
1 ψ2(r )= 2 (ψ2 s (r )+ψ2 px ( x , y , z )−ψ2 py ( x , y , z )−ψ2 pz ( x , y , z ))
1 ψ3(r )= 2 (ψ2 s (r )−ψ2 px ( x , y , z )+ψ2 py ( x , y , z )−ψ2 pz ( x , y , z ))
1 ψ 4 (r )= 2 (ψ2 s (r )−ψ2 px ( x , y , z )−ψ2 py ( x , y , z )+ψ2 pz ( x , y , z )) .
The explicit form of the functions ψ2 s (r ) , ψ2 px (r ) , ψ2 py (r ) , ψ2 pz (r )is as follows: ψ2 px ( x , y , z )= ψ2 py ( x , y , z )= where aB - is the Bohr radius.
In the case of propane, we will have a chain of three carbon atoms in sp3 hybridization with four sp3 orbitals. For a single carbon atom (shifted to a point with coordinates ( xC , yC , zC )) in a propane molecule, we can write:
1
ψ(iC)( x , y , z )= 2 (ψ2 s ( x− xC , y − yC , z− zC )+ di , x ψ2 px ( x− xC , y − yC , z− zC ))
+ di , y ψ2 py ( x− xC , y − yC , z− zC )+ di , z ψ2 pz ( x− xC , y − yC , z− zC )
where ψ2 s , ψ2 px , ψ2 py , ψ2 pz - atomic orbitals calculated according to the relations (
4 ∑ ∑ ψ(iC)( x , y , z )
C∈C 1,C 2,C 3 i=1 or in a more visual form:
4 4 4 ψC3 H8( x , y , z )= ∑i=1 ψ(iC 1)( x , y , z )+ ∑i=1 ψ(iC 2)( x , y , z )+ ∑i=1 ψ(iC 3)( x , y , z )
C1=(−d ,0 ,0) , C2=(0,0,0) , C3=(d ,0 ,0)
(
a) b)
Next, we calculated the matrix elements of potential (
~ U CH4=4 ϵ CH4 ⟨ψCH4 ( x , y , z )||[(σ CH4 / r )12−(σ CH4 / r )6 ]|ψCH4 ( x , y , z )⟩ ~
U C3 H8=4 ϵ C3 H8 ⟨ψC3 H8( x , y , z )||[(σ C3 H8 / r )12−(σ C3 H8 / r )6 ]|ψC3 H8( x , y , z )⟩ .
As a result, the activation energies can be calculated depending on the temperature T as follows: Eac (CH 4)=U CH4+ iCH4 kT ; Eac (C3 H 8)=U C3 H8+ iC3 H8 kT ~ ~
2 2 where the number of degrees of freedom for methane and propane, respectively, is: iCH4=15 ; iC3 H8=33.
The main objective of our work is to provide automated, high-performance and reliable identification of methane and propane parameters based on a developed mathematical model of activation energy and experimentally obtained concentration distributions. For this purpose, we apply the basic system of kinetic equations inside the material and separately inside the pores. These equations are as follows: (8) (9) ∂ C (r , t ) ∂ t
= Deff r12 ∂∂r (r2 ∂ C∂(rr , t ) ), ∂ q (r , z , t ) ∂ t = D p( r ∂ r ∂ r 1 ∂ (r ∂ q (r , z , t ) )+ ∂2 q (r , z , t ) together with the Langmuir-type isotherm:
qmax b (T )C (r , t ) (12) q (T )= ; b (T )=b0⋅exp (− Eac / RT ),
1+b (T )C (r , t ) where it is taken into account that the dependence on temperature T is determined by the Van 't Hoff law, Deff and D p - is the diffusion parameters in bulk material and pores.
In order to be able to use the datasets of concentration distributions obtained from experimental measurements, the mathematical model (10)-(12) should be presented in discretized form on a three-dimensional grid:
ri=i Δ r ; z j= j Δ z ; t n=n Δ t ; Cn+1=Cin+ Deff Δ t (Cin+1−2 Cin+ Cin−1 + 2 i (Δ r )2 ri n
Cin+1− C2i−1 Δ r); Cn+1=C0n+6 Deff Δ t C1n− Cn0 ; 0 (Δ r )2
n n n n qin,+j1=qin, j+ D p Δ t (qin+1, j−2 qin, j+ (Δ r )2 ri 2
qi−1, j + qi+1, j − qi−1, j Δ r + qin, j+1−2 qin, j+ (qΔi , zj−)12 ); q0n +,j1=q0n , j+ D p Δ t (2 q1n, j−
n n q0 , j + q0n , j+1−2 q0n , j+ (qΔ0 , zj−)12 ); (Δ r )2 qin,,j(eq)=b0 exp (− Eac / R T n)(1+ qmax)Cin . (10) (11) (13) Since the parameters ϵ and σ are included in the difference scheme due to the dependence of the activation energy (8) and (9), they can be obtained for each dataset from the last equation (13).
Next, we develop a neural network in which the input is the concentration distributions C (r , t ); q (r , z , t ) obtained in the experiment (in total, we used 280 datasets taken from works [810] and related ones), and parameters ϵ and σ obtained using finite different scheme (13). An example of a concentration distribution used to train a neural network is shown in Fig. 3.
The network was trained using class labels, which were taken to be diffusion coefficients Deff , D p and parameters ϵ , σ . In this case, the network performs data regression, which ensures a fast and optimal approximation of the numerical solutions of the difference scheme (13). Regression for each difference scheme and class labels produces a stack of maps C ∈ R H×W for all its nodes. Now t the tensor X ∈ RT ×H×W is fed into CNNs, which are 3D convolutions over the spatial dimensions with temporal layers as channels over the parameters (t, x, y). The output of the network in the system is interpreted as an estimate of the set of parameters ( Deff , D p, ϵ , σ ), which is the probability of pore morphology classes. The architecture of the developed neural network and an illustration of its operation are presented in Fig. 4. The software implementation of the neural network and work with it was carried out using the Jupiter environment for the Python programming language and the Scikit-learn, TensorFlow, and PyTorch libraries.
The learned neural network was used to identify the parameters of the Lennard-Jones potential and diffusion coefficients based on given input data of concentration distributed in various cellular media. In all text datacenters, concentration distributions were used, for which diffusion processes involving methane or propane occurred. The results of the work of the neural network with concentration distributions taken from experimental works [11-16] are presented in Table 1.
It is accepted that the theoretically calculated values of the Lennard-Jones parameters are equal to σ =0.373 nm ; ϵ =12.7 meV for methane and σ =0.512 nm ; ϵ =20.3 meV for propane, respectively. Table 1 also shows the value of the prediction with which the neural network sets the parameters of the Lennard-Jones potential from the concentration distribution for each of the studied samples. As can be seen from Table 1, the value of prediction is from 92 to 98%. In general, this indicates that the developed neural network allows us to set the parameters of substances used in diffusion of porous samples, namely methane and propane, with sufficiently high accuracy.
An approach to identifying the parameters of the Lennard-Jones potential for methane and propane from the concentration distributions of these substances in cellular samples using a trained neural network is proposed. To solve this problem, a microscopic model of the activation energy of methane and propane is used, and a mathematical model of a certain difference scheme is developed based on the equations of diffusion transfer kinetics. The developed neural network is applicable to identifying the parameters of the Lennard-Jones potential in various porous samples. It has been established that the developed method allows determining these parameters with a probability of 92 to 98% only with a concentration distribution at the input. We see the development of the proposed method in its further improvement for application to identifying the parameters of a wider range of volatile hydrocarbons and other substances experimentally studied and technologically applied in porous materials.
The authors have not employed any Generative AI tools.