List of gas phase reactions and corresponding rate constants [24].
\r\n\t
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He is also the founder and director of Adaptive AgroTech Consultancy Int, a network of professional experts focused on technology adaptation for food security.",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"203413",title:"Dr.",name:"Redmond Ramin",middleName:null,surname:"Shamshiri",slug:"redmond-ramin-shamshiri",fullName:"Redmond Ramin Shamshiri",profilePictureURL:"https://mts.intechopen.com/storage/users/203413/images/system/203413.jpg",biography:"Dr. Redmond R. Shamshiri received M.Sc. Dr. Eng, and Ph.D. from the University of Florida and the Universiti Putra in control system and dynamics. He is currently a research scientist at the Leibniz Institute for Agricultural Engineering and Bioeconomy (ATB) in Potsdam, Germany. He is also the founder and director of Adaptive AgroTech Consultancy Int, a network of professional experts focused on technology adaptation for food security. Dr. Shamshiri's research focus is on digital agriculture for food security, involving high-tech control methods, embedded systems, LPWAN sensors, prediction models, and robust data acquisitions for smart farming. He has widely consulted with the industry and academics. 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Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"878",title:"Phytochemicals",subtitle:"A Global Perspective of Their Role in Nutrition and Health",isOpenForSubmission:!1,hash:"ec77671f63975ef2d16192897deb6835",slug:"phytochemicals-a-global-perspective-of-their-role-in-nutrition-and-health",bookSignature:"Venketeshwer Rao",coverURL:"https://cdn.intechopen.com/books/images_new/878.jpg",editedByType:"Edited by",editors:[{id:"82663",title:"Dr.",name:"Venketeshwer",surname:"Rao",slug:"venketeshwer-rao",fullName:"Venketeshwer Rao"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"4816",title:"Face Recognition",subtitle:null,isOpenForSubmission:!1,hash:"146063b5359146b7718ea86bad47c8eb",slug:"face_recognition",bookSignature:"Kresimir Delac and Mislav Grgic",coverURL:"https://cdn.intechopen.com/books/images_new/4816.jpg",editedByType:"Edited by",editors:[{id:"528",title:"Dr.",name:"Kresimir",surname:"Delac",slug:"kresimir-delac",fullName:"Kresimir Delac"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"19439",title:"Model Approximation and Simulations of a Class of Nonlinear Propagation Bioprocesses",doi:"10.5772/24131",slug:"model-approximation-and-simulations-of-a-class-of-nonlinear-propagation-bioprocesses",body:'It is well known that the biotechnology is one of the fields that over the last three decades has a very high and quick development. Therefore, due to their advantages, the control of industrial bioprocesses has been an important practical problem attracting wide attention. The main motivation in applying control methods to such living systems is to improve their operational stability and production efficiency. The operation in Stirred Tank Reactors (STR) has been and it is still a widely used technology in fermentation bioprocesses. But, other new technologies such as fixed bed, fluidized bed or air lift reactors, are considered for bioprocesses operation. These reactors present several advantages over the “classical” STRs. For instance, the fixed bed and fluidized bed reactors are characterized by higher production performance, i.e. larger production capacity and higher productivity (Bastin & Dochain, 1990; Bastin, 1991;\n\t\t\t\tBouaziz & Dochain, 1993).
From mathematical point of view, the dynamics of these processes are characterized by partial differential equations and therefore are classified as distributed parameter systems (Bastin & Dochain, 1990; Bouaziz & Dochain, 1993; Christofides, 2001; Dochain et al., 1992). For instance, the concentrations of the reactants and products are not anymore homogeneous in the whole reactor, like in STRs, but are characterized by a spatial profile along the reactor. It is clear that the distributed parameter feature of these systems makes the control problem even more difficult (Bouaziz & Dochain, 1993; Christofides, 2001; Dochain et al., 1992; Petre & Selişteanu, 2007a; Slotine & Li, 1991). Therefore, the modelling and simulation of them, which are the objectives of this chapter, are associated with the formulation of the process model using partial differential equations (PDEs), in general nonlinear, and with the computation of the solution. Since in the most of cases for this kind of systems, with the exception of a few simple cases, there are no analytical methods for finding the solution of the involved equations, the following alternatives are commonly employed (Christofides, 2001; Petre & Selişteanu, 2007a; Slotine & Li, 1991; Vilas, 2008):
To assume that these processes behave like lumped parameter systems (the states are only time dependent).
To use classical numerical methods like finite differences, finite elements or finite volumes.
These methods are based on discretization techniques which allow us to approximate the infinite set of numbers that represent a continuous function by means of a finite set of parameters (Bouaziz & Dochain, 1993; Christofides, 2001; Dochain et al., 1992; Petre & Selişteanu, 2007a; Slotine & Li, 1991; Vilas, 2008).
The first option is only valid when the spatial distribution is negligible as compared with the time evolution, for instance in reactors where the homogenization of the medium is achieved by means of stirring devices (Dochain & Vanrolleghem, 2001; Vilas, 2008). Nevertheless, in the remaining cases it is necessary to use the second alternative. Its main inconvenience is that the numerical solution is computationally involved (especially in 2D or 3D spatial domains) making the approach unsuitable for real time tasks like control or online optimization (Bouaziz & Dochain, 1993; Christofides, 2001; Dochain et al., 1992; Petre & Selişteanu, 2007a; Slotine & Li, 1991; Vilas, 2008).
An alternative to these classical numerical methods is the development of some techniques for the projection of the PDEs onto a low dimensional subspace. In accordance to these techniques, the original PDEs are transformed into a set of ordinary differential equations (ODEs) known as reduced order model (Aksikas et al., 2007; Americano da Costa Filho et al., 2009; Bouaziz & Dochain, 1993; Christofides, 2001; Dochain et al., 1992; Hoo & Zheng, 2001; Petre et al., 2007; Shvarstman et al., 2000).
As a result, the first objective of this chapter is to provide the mathematical tools, which are used for most of numerical methods, for solving PDEs and, on this basis, to give a brief outline of the most commonly employed techniques. Among the different alternatives, some of them based on Garlerkin scheme will be described and used in this chapter. In particular, the finite element method will be chosen on the basis of its flexibility and reduced order models since they are the most efficient (Vilas, 2008). These reduced order models obtained in this way can be used either for the process simulation or computation of their solution.
As we mentioned above, over recent years, a considerable research effort was concentrated on the design of control policies for distributed process systems (Christofides, 2001). Standard approaches to control this kind of systems are based on the spatial discretization of the original set of PDEs to obtain a set of ODEs. This allows us to employ standard finite-dimensional methods just described above to construct the controller (Christofides, 2001; Dochain et al., 1992; Hoo & Zheng, 2001; Shvarstman et al., 2000). However, this approach can result in a set of ODEs of high dimensionality which could make the approach unsuitable for real time applications. Also, the controllability and observability properties would depend on the number of discretization points as well as its location and may lead to a poor control quality (Christofides, 2001). Due to these disadvantages, new methods based on spectral decomposition techniques, which take into account the spatially distributed nature of these systems, have developed (Aksikas et al., 2007; Shi et al., 2006). This approach uses the Galerkin method so as to approximate the system by a low-dimensional set of ODEs to design the controller (Aksikas et al., 2007; Shvarstman et al., 2000). In (Americano da Costa Filho et al., 2009) this approach is used in combination with the Lyapunov\'s direct method to derive stabilizing controllers applied in the case of chemical processes that are carried out in tubular reactors.
This chapter is an extended work of the research achieved in some works of the authors: (Petre, 2003; Petre & Selişteanu, 2005, 2007a,\n\t\t\t\t2007b;\n\t\t\t\tPetre et al., 2007, 2008;\n\t\t\t\tPetre, 2008; Selişteanu & Petre, 2004), and deals with the approximation and simulations of the dynamical model for a class of nonlinear propagation bioprocesses.
First, the dynamics of a class of propagation bioprocesses involving n components and m reactions that are carried out in fixed bed reactors without dispersion is analyzed. Since the dynamics of these bioprocesses are described by partial differential equations, either for simulation but especially for their controlling, one method consists of approximation of these infinitely order models by finite order models. These approximate models are in fact a set of ordinary differential equations obtained here by orthogonal collocation method. More exactly, infinitely dimension of the initial parameter distributed model will be reduced by approximating the partial derivative equation of each reaction component by a finite number, equal to p+1, of ordinary differential equations at p+1 discrete spatial positions along the bioreactor. These points are chosen as zeros of some orthogonal polynomials. Since it is difficult to know the connections between the original distributed parameter model and its approximate version (Christofides, 2001), our objective is to analyze the behaviour of both models to observe their intrinsic dynamical properties. This is realized by simulations conducted in the case of a fixed bed bioreactor without diffusion (Dochain et al., 1992; Petre et al., 2007; Petre & Selişteanu, 2007a).
In the following the control problem of these classes of propagation bioprocesses is analyzed. Since the biotechnological processes have a nonlinear nature, to control these processes some nonlinear control techniques will be used. These techniques not only improve the linear control methods and allow the analysis of strong nonlinearities but it also allow us to deal with model uncertainties and even the controller design may result simpler than in its linear counterpart (Slotine & Li, 1991). A widely extended nonlinear control technique is the feedback linearization (Isidori, 1995; Khalil, 2002; Slotine & Li, 1991), which makes use of algebraic transformations to obtain a closed loop linear system in which the conventional control techniques can be applied. The main inconveniences of this technique are two: firstly, the tracking control problem may lead to complex transformations and secondly, model uncertainty may affect the control performance. Therefore, it is necessary to apply adaptive and robust-adaptive control techniques able to drive the system to the desired reference despite the presence of uncertainties.
Consequently, by using the obtained results in (Petre & Selişteanu, 2005, 2007a; Petre et al., 2007, 2008; Petre, 2008; Selişteanu & Petre, 2006), to control the mentioned propagation bioprocesses, in this chapter a class of nonlinear adaptive controllers are designed based on their finite order models. The nonlinear controller design is based on the input-output linearizing technique. The information required about the process is the measurements of the state variables and its relative degree. It must be noted that if for the analyzed process there are no accessible state variables, these will be estimated by using an appropriate state observer.
Numerical simulations conducted in the case of a fixed bed reactor are included to illustrate the performances of the presented adaptive control strategies.
All simulations are achieved by using the development, programming and simulation environment MATLAB (registered trademark of The MathWorks, Inc., USA).
The chapter is organized as follows. Section 2 introduces the distributed parameter dynamical model for the class of fixed bed reactors. Its reduction to an ordinary differential equation system by orthogonal collocation method is presented in Section 3. A detailed analysis of obtained results by application of this method in the case of a fixed bed reactor without diffusion is presented in Section 4. The adaptive control strategies of propagation bioreactors are developed in Section 5, the performances of the designed adaptive controllers being presented in Section 6. Finally, concluding remarks and further research directions are presented.
A fixed bed bioreactor is a reactor where the biomass is immobilized on fixed carriers such as polymers, porous glass or ceramics. Consider a fixed bed bioreactor without dispersion operating in plug flow conditions as shown in Fig. 1, in which takes place a single autocatalytic growth reaction
A schematic view of a fixed bed bioreactor
To achieve the model of this bioreactor consider a section of the reactor with length dz located at a distance z (0 ≤ z ≤ L) from the bioreactor input. Assuming that along the length L of the bioreactor the cross section is constant and equal to A, then, the volume of this section is dV = Adz.
The mass balance of substrate concentration S around this section is given by:
where the term
Similarly, the mass balance of the biomass concentration X in the volume dV is given by:
where the term
After some calculus, the relations (1) and (2) lead to:
The equations (3) constitute the distributed parameter dynamical model of the analyzed fixed bed bioreactor. For completeness, we must to define the limit and initial conditions as:
where Sin(t) is the influent substrate concentration and X0(z) is the initial immobilized biomass concentration.
Assume now that two reactions take place in bioreactor: (i) an autocatalytic growth reaction with one limiting substrate S and one biomass population X with a reaction rate
where kd is the death coefficient. The limit and initial conditions are defined as:
Let us define the state vector
If we denote by
where
with the limit and the initial conditions:
From the two above examples, one can deduce that in the case of a fixed bed bioreactor in which m biochemical reactions with n reactants take place, among which n1 are microorganisms fixed on some supports and which remain within the reactor, and n2 other components flow through the reactor, the distributed parameter dynamical model will be described as:
with the following limit conditions:
Usually in (11) and (12),
Since the model (11) is infinitely dimensional, in this section we will reduce the model order by approximating it by a set of ordinary differential equations. From (11) one can see that the state variables
where
The integer p in (13) corresponds to the number of interior collocation points determined by collocation method. The points z = z0 and z = zp+1 correspond to the input (z = 0) and the output (z = L) of the reactor.
It must be noted that the collocation method offers two important advantages: first, its implementation is easier and second, the nature and physical dimension of the state variables remain unchanged after the reduction procedure. Moreover, the orthogonal methods preserve mass balances (Christofides, 2001; Dochain et al., 1992).
According to collocation method, the partial derivative of
By introducing (13)-(15) into (11) and (12), each partial derivative equation is transformed into p+1 differential equations at the p interior collocation points and at the output of the reactor. Thus it is obtained the following n(p+1) order system of one order ordinary differential equations:
where:
with
In this section, the above presented collocation method will be applied in the case of the distributed parameter dynamical model of the fixed bed bioreactor described by (5)-(6). We will consider four interior collocation points, i.e. p = 4. Firstly, we define the values of the concentration of X, S and Xd and of the specific growth rate
For p = 4, the relations in (13) are particularized as follows:
where
Let us define the state vectors at collocation points as:
Then, the reduced order model which approximates the exactly infinitely dimensional model (5)-(6) will be described by the following ordinary differential equations:
where:
Numerous simulation experiments have been performed on the example presented above to illustrate the dynamical behaviour of the two classes of models (exactly and reduced). The values of bioreactor dimensions and of process parameters used in simulation are (Petre & Selişteanu, 2007a; Petre et al., 2008): L = 1 m, A = 0.02 m2, k1 = 0.4, kd = 0.05 h-1. For the specific growth rate
with
In fact, in simulations we compared the reduced order model obtained by orthogonal collocation method with another approximation of the original model (5) obtained by a finite difference method, where the derivatives of variable
where Δt and Δz are discretization intervals. The choice of the discretization intervals is a very important problem (Bouaziz & Dochain, 1993). In order to have a satisfactory accuracy, a high number of discretization points may be necessary, but this choice requires excessive computer time.
The graphics in Fig. 2 show the response of the bioprocess to a step of the influent substrate concentration Sin from 7.5 to 10 g/l at time t = 5 s, for three values of space discretization points (1: M = 100, 2: M = 200, 3: M = 400).
To choose the best approximation, for the simulation of the reduced model (16) different solutions have been tested. So, as orthogonal basis functions have been chosen Lagrange polynomials (14), which are assumed to be reliable and easy to compute. The collocation points have been chosen as the zeros of the Jacobi polynomials and of the Legendre polynomials. The Jacobi polynomials can be computed by using the following recursive expression (Corduneanu, 1981; Dochain et al., 1992):
with:
The Legendre polynomials can be computed by using the following recursive expression (Corduneanu, 1981):
Different values of the parameters
The simulation results performed in the same conditions as in the first experiment are presented in Fig. 3. These graphics show the behaviour of reduced order process model for four different sets of values of
Thus, we obtain a set of four Jacobi polynomials, whose zeros are given by the following four sets of values:
\n\t\t\tThe response of the original model
The behaviour of the reduced order model
Profiles of steady state regime of biomass X and substrate S
The behaviour of the two classes of models (original and reduced order model)
The initial simulation conditions have been chosen so as to correspond to a steady state obtained from:
with S(t, z = 0) = Sin(t), Xd(t, z = 0) = 0. From the first two equations in (25), the steady state regime for biomass X and substrate S is shown in Fig. 4.
Note that in all the simulations, the influent flow rate is constant, F = 2 l/s.
From Fig. 2 and 3 one can observe that the reduced order model performed by orthogonal collocation method using only p = 4 interior collocation positions obtained as solutions of the Jacobi polynomial with
In this section, the control problem of a class of propagation bioprocesses that are carried out in fixed bed reactors without dispersion is presented. The nonlinear adaptive controllers are designed based on the finite order model (16) obtained from exactly model (11) by using the orthogonal collocation method. It can be see that the model (16) may be rewritten as (Bastin & Dochain, 1990; Petre, 2008):
where
For the bioreactors described by the model (16) the control objective is to regulate the concentration of a single component at the bioreactor output, under the following conditions:
The control input is the influent flow rate
The controlled variable is measured not only at the bioreactor output, but also at every interior collocation point and at the reactor input (only in the case of external substrate).
The yield coefficients are positive constants (some of them being unknown).
For simplicity, we will denote by y the concentration of the controlled component, by
where
Using (16), the dynamics of
Consider that for the bioprocess described by the model (20), the controlled variable is the substrate concentration at the output of the bioreactor, that is
then the entries of the vector c in (27) will be:
The dynamics of the concentration
Using (29), the dynamics of
where:
It is easy to verify that the term
where
As a conclusion,
As it was mentioned above, the control objective is to regulate the concentration of variable
Controller design is achieved by using the input-output linearizing technique. Remember that the input-output linearizing principle (Isidori, 1995) consists in the calculus of a nonlinear control law such that the behaviour of closed loop system (controller + process) is the same as the behaviour of a linear stable system. Assume that for the closed loop system we wish to have the following first-order linear stable dynamics:
Firstly, we consider the ideal case, where maximum prior knowledge concerning the process is available. In particular we suppose that the parameters
The control law (36) leads to the following linear error model:
with
If the parameters
The estimates
where
The performances of the designed nonlinear adaptive controllers were verified by several simulation experiments performed upon the fixed bed bioreactor described by the model (5). The values of bioreactor dimensions and process parameters used in simulation are the same as in Section 4 (Petre & Selişteanu, 2007a; Petre et al., 2008). Also, for the specific growth rate
The interior collocation points of the reduced model (20) have been chosen as zeros of the Jacobi polynomials given in Section 4. For
The control objective is to regulate the substrate concentration
The exactly linearizing control law (36) takes the form:
The behaviour of the closed loop system in the ideal case, when all the parameters are completely known, is presented in Fig. 6.
The initial simulation conditions correspond to a process steady state regime. So, for the interior collocation points, the used values are: X1(0) = 44.1051 mg/l, X2(0) = 19.8101 mg/l, X3(0) = 9.8169 mg/l, X4(0) = 6.2634 mg/l; X5(0) = 5.6010 mg/l; S1(0) = 2.9403 g/l, S2(0) = 1.3207 g/l, S3(0) = 0.6545 g/l, S4(0) = 0.4176 g/l, S5(0) = 0.3734 g/l, and X0(0) = 0 mg/l, S0(0) = Sin(0) = 7.5 g/l.
To verify the regulation properties of the controller, for the reference variable a piece-wise constant variation was considered as:
The value of the gain parameter
The behaviour of the closed loop system with the exactly linearizing controller
From Fig. 6 one can observe that the controller (41) is efficiently both in regulation of controlled variable and in rejection of the perturbation
Assume now that the death parameter
where
Assume also that at the output of the bioreactor the only measured variable is the substrate concentration S5. It can be seen that the practical implementation of the control law (41) requires the knowledge of the state
Since in (41) the variable
The dynamic of
From (45) and (46), the estimated value
Then, the adaptive version of the control law (41) is given by:
where the estimates
The adaptive algorithm given by (48), (39) and (49) was implemented under the same conditions as in the first case. The values of the controller design parameters used in simulations are:
The simulation results are shown in Fig. 7. As in the first case, the system evolves in open loop starting from
The behaviour of the closed loop system with the adaptive controller
From the graphics in Fig. 7 one can deduce that even if the initialization of
Moreover, it was proven that the adaptive algorithm given by (48), (39) and (49) is robust, that is even though the process model (5) has uncertainty parameters, the behaviour of closed loop system is good. It was verified that if the death coefficient
The approximation of the infinitely order dynamical model for a class of nonlinear propagation bioprocesses described by partial differential equations was examined. These approximate models consist of a set of ordinary differential equations obtained by orthogonal collocation method. The results obtained by application of this method in the case of a fixed bed reactor showed that by an appropriately choosing of the collocation points along the reactor, the behaviour of the reduced order model is very close to the behaviour of original infinitely order model.
After that, the obtained reduced order model was used to design some control algorithms for these types of reactors. The controller design is based on the input-output linearization technique. The obtained algorithm was tested in the controlling problem of substrate concentration for a propagation bioprocess that is carried out in a fixed bed reactor.
The simulation obtained results demonstrated that the designed adaptive algorithms used in control of propagation bioreactors yield good results closely comparable to those obtained in the case when the process parameters are completely known and/or time invariable.
Moreover, these algorithms prove to be robust as well yielding good results even though the measurable variables are affected by noises and/or the model parameters suffer variations between wide limits. It must be also noted that these algorithms can relatively easily be extended to other types of distributed parameters bioreactors: fluidized bed, air lift reactors.
This work was supported by CNCSIS–UEFISCDI, Romania, project number PNII–IDEI 548/2008.
In statistical physics only a few problems can be solved exactly. For complex problems, numerical methods can give exact results for problems that could only be solved in an approximate way. Numerical simulation can be a way to test the theory. The numerical results can be compared to the experimental results. The numerical simulation is placed between the fundamental and the experimental treatment; it has a quasi-experimental character (numerical experience). For problems of statistical physics, the most widely used simulation methods are the Monte Carlo method and the molecular dynamics method.
The first Monte Carlo simulation (MCS) was proposed by Metropolis et al. in 1953 [1]. The second Monte Carlo simulation was proposed by Wood and Parker in 1957 [2]. The obtained results were in good agreement with the experimental results of Bridgman [3] and those of Michels et al. [4]. In this method we attribute a series of initial positions chosen randomly to a system of N particles interacting through a defined potential. A sequence of particle configurations is generated by giving successive displacements to particles; we only retain configurations to ensure that the probability density is that of the chosen.
Molecular dynamics simulation (MDS) has been first introduced to simulate the behavior of fluids and solids at the molecular or atomic level. MDS was used for the first time by Alder and Wainwright in the late 1950s [5, 6] to study the interactions of hard spheres. The principle is the resolution of equations of motion for a hard sphere system in a simulation cell. The basic algorithm is Verlet’s algorithm [7].
In this chapter, we will present techniques of numerical simulations using the Monte Carlo method. We will present an application on the gas phase during plasma-enhanced chemical vapor deposition (PECVD) of thin films. The application concerns collisions between particles. Particles are in Brownian motion. Collisions, elastic or inelastic, are considered to be binary. Non-elastic collisions result in effective chemical reactions.
In Section 2, we cite some MCS and MDS works on PECVD processes. Section 3 presents general rules on numerical simulation methods. Section 4 presents how to simulate a physical problem using MCS? We present the Metropolis algorithm as a scheme to trait random configurations and different modules related to elaborate an MCS code. In Section 5, we apply the MCS on SiH4/H2 gas mixture during a PECVD process. Finally the conclusion summarizes the contents of the chapter.
The PECVD is the most widely used technique to produce hydrogenated amorphous silicon thin films (a-Si:H) for solar cells and for film transistors and electronic devices [8, 9]. Reactions during plasma deposition are complex and are not understood completely.
Gorbachev et al. [10, 11, 12] have developed a model that is based on chemical reactions and different processes in a PECVD reactor. The model takes into account the formation of SinHm oligomers (n ≤ 5). It presents a simulation of the growth of the films. Gorbachev et al. found that Si2H5 and Si3H7 strongly influence the growth of the film [11].
Valipa et al. [13] calculated the β reactivity of the SiH3 radical on a surface of a silicon lattice plane during the growth of a-Si:H using MDS. The mechanisms of physical and chemical interactions of low temperature plasmas with surfaces can be explored using MDS [14].
For a CH4/H2 mixture, Farouk et al. used the Monte Carlo method (PIC/MC); they calculated the ionization rate of the plasma and the deposition rate of the thin layer [15]. Rodgers et al. [16] have developed three-dimensional Monte Carlo simulations of diamond (100) surface CVD. Other works on MCS are in [17, 18, 19].
In our previous works [20, 21, 22, 23, 24], we were interested in the study of the gas phase and the interaction of plasmas with the surface, for SiH4/H2 and CH4/H2 gas mixtures during PECVD processes. The used numerical simulation techniques were MCS and MDS. To complete the studies, we used the fluid model [25].
The starting point of numerical simulation is a physical phenomenon; its purpose is to obtain useful physical results. Between these two points, several steps can be identified. These steps are general and they are applicable for MCS. The steps can be summarized as follows:
The physical phenomenon must be defined by the description of the dominant domain of physics. The main assumptions and simplifying approximations are necessary to understand the physical phenomenon and the design of the first model.
Mathematical model requires a mathematical formulation of the problem. It may be a problem of elements or discrete object or a problem of a continuous medium; it may be a spatiotemporal problem or frequency problem and may be a deterministic or probabilistic problem.
It would be interesting to know the mathematical equations that govern the phenomenon:
The forces between particles and elements
The potential interaction
The determination of a time scale
The determination of a length scale
Definition of constant magnitudes of motion and equilibrium magnitudes
Continuity equations, balance equations, transfer equations, etc.
The MCS technique has been chosen for this work; knowing its basic algorithm is necessary for elaborating the simulation. This step requires some actions:
Validation of the model on simple cases
Simulation calculation on complex phenomena
The MCS is based on a probabilistic process with a random choice of configurations and samples of the situation of the physical system. The two pedagogical examples most cited in the literature are the integration of a single variable function and Ising’s model of spin. In the following subsection, we define the integration of a single variable function. We introduce the Ising model at the end of Section 4.2.2.
Calculation of the definite integral for a function f(x) of a single variable x on domain {a, b} has been proposed (Figure 1):
The integral of a function f(x).
Let:
Let xi and yi be real random numbers (i = 1, 2,…, N), and let H be a real number greater than the f(x) for x belonging to the domain {a, b} (or x ∈ {a, b}).
Let r1 and r2 be two random numbers belonging to the domain {0, 1} according to a uniform distribution law. Generators (e.g., Ran, RANDOM, RANDUM, or other IMSL mathematical libraries) of random numbers can be used:
where xi and yi are random numbers (xi ∈ {a, b} and yi ∈ {0, H}).
The Monte Carlo (MC) method is based on a probabilistic process. Let N be the total number of cases chosen (possible cases). It is necessary to count the number of favorable cases (or the number of points below the curve y = f(x)); let yi ≤ f(xi)). The number of favorable cases is Nfav. When N➔∞, the value I of the integral is [26]:
An example [26] is the calculation of the value π by calculating the integral I on a quarter circle of unit radius (R = 1.0). The pairs of random numbers (xi, yi) satisfying the condition: xi2 + yi2 ≤ 1. The function f(x) is equal to
We take a = 0.0, b = 1.0, and H = 1.0.
For different values of N, we show that the numerical solution tends to π = 4I.
Although this integral is simple, it shows the strength and simplicity of the method. The technique can be generalized for the integration of multivariate functions.
We note that integration by the MC method is based on:
The choice of random configurations according to a uniform distribution law
Each configuration chosen is either favorable or unfavorable (the “or” is exclusive).
For statistical physics problems, the probabilistic choice of configurations is not always deterministic; the favorable and unfavorable cases are not exclusive. According to the Metropolis algorithm [26, 27], the steps of the simulation are:
Choice of a simulation cell of adequate shape to the studied phenomena. The size of the simulation cell is related to a scale of length characteristic of the forces and interaction potential of the studied phenomenon. This cell may contain Npc particles (and/or elements).
Choice of an initial configuration that responds to some physical and thermodynamic properties. The total or internal energy of the system is Ei.
Infinitesimal random displacement of a particle (or element of the system) and calculation of the new internal energy of the system Ef. This displacement is related to the physical magnitudes: time scale and length scale. The physical system tends toward a minimization of the internal energy of the system with some fluctuation. Let ΔE = Ef-Ei the fluctuation.
If ΔE ≤ 0; the new configuration is retained (favorable) and the different averages can be obtained; go to step (c).
If ΔE > 0; a random number ε is chosen such that 0 < ε < 1. Let the probability Pr equal to: Pr = exp. (−ΔE/kBT) (where kB is the Boltzmann constant and T is the temperature).
If ε < Pr, accept the move and in any case go back to step (c) for a new choice of an infinitesimal displacement (new configuration). Note that if such a trial move is rejected, the old configuration is again counted in the averaging with probability Pr.
Figure 2 shows how to choose between the selected configurations. Let ε be a random number following a uniform law; If ε1 ≤ Pr the configuration is retained, and if ε2 > Pr the configuration is rejected.
Configuration choice according to Metropolis scheme.
Numerical simulation using the MC method is a very important tool for the study of static properties. The basic algorithm is based on probability notions. Understanding of the distribution function and/or interaction potentials is the heart of the calculation.
In equilibrium statistical physics, the system has a certain probability that can be in any states. The probability of being in a state μ with energy H(μ) is given by the Boltzmann distribution P(μ):
where T is the absolute temperature and kB is called Boltzmann’s constant. It is conventional to denote the quantity (kBT)−1 by the symbol β. The normalizing factor Z, or partition function, is given by:
The average of a quantity Q fora system in equilibrium is:
The internal energy U, is given by:
which can be written in terms of a derivative of the partition function:
From thermodynamics we have expressions for the specific heat C, the entropy S, and the Helmholtz free energy F:
or
and
and
We can calculate other parameters affecting the system.
The Monte Carlo method is an excellent technique for estimating probabilities, and we can take advantage of this property in evaluating the results. The simplest and most popular model of a system of interacting variables in statistical physics is the Ising model. It consists of spins σi which are confined to the sites of a lattice and which may have only the values (+1) and (−1). These spins interact with their nearest neighbors on the lattice with interaction constant J; they can interact with an external magnetic field B coupling to the spins. The Hamiltonian H for this model is [26]:
The Ising model has been studied in one and two dimensions to obtain results of thermal properties, phase transition, and magnetic properties [26, 27, 28]. For chosen values of J and/or B, different steps may be taken for the calculations (simulation cell, initialization, configurations, boundary conditions, calculation algorithms). For any configuration, each spin takes the two possible directions. The detail of the calculation procedure is not the purpose of this chapter.
We give a system of N particles (atoms, molecules, ions or particles) placed in a cell of fixed volume, generally of cubic form. The initial positions may, depending on the case, be distributed randomly according to a certain law (uniform or otherwise) or have a given symmetry. In a fluid, a gas, or a plasma, the particles may have random positions in general; in a solid or surface, with a crystal structure, the particles take ordered positions. The choice of random initial positions allows great freedom on the choice of the number of particles in the cell.
At the first step, the particles are given velocities that are generally selected to have a zero total momentum. If the system is in thermodynamic equilibrium, the initial velocities will be randomly chosen according to a Maxwell-Boltzmann law. In the general case, the velocity distribution is according to the problem dealt with. All other phase properties can be initialized to the particles; the main thing is the conservation of the total quantities of the system.
The particles interact with each other according to chosen interaction potentials. Since the interaction potentials are specific for each “numerical experiment,” the main part of the work consists in calculating the interaction energies for each proposed configuration.
The choice of interaction potentials is directly related to the mathematical formulation of the problem according to the state of the medium: fluid, gas, plasma, or solid. It can be Lennard-Jones potential, Coulomb potential, Debye potential, Morse potential, Stillinger-Weber potential, Born-Mayer potential, Moliere potential, or others.
In general, two main boundary conditions are used: periodic boundary conditions (PBC) and minimum image convention (MIC) [29].
To minimize the surface effect, periodic boundary conditions (PBC) [30] are invariably imposed. The simulation cell is reproduced throughout the space to form an infinite mesh. We can simulate the properties of an infinite system. The particles that we follow are in the central cell; if a particle crosses a wall with a certain velocity, its image returns with the same velocity by the opposite wall. Under these conditions, the number of particles in the central cell, and consequently the density, is constant. These conditions also allow the conservation of the energy and the momentum of the system and do not introduce periodic effects (because of the interaction between particles).
According to the hypotheses and according to the geometry of the problem, other boundary conditions are proposed [26]. For example, in order to model thin films, the simulation cells are longitudinal and parallel to the film; one uses PBC in the directions parallel to the film. In the direction normal to the film, free edge boundary conditions can be used. In such cases, it may be appropriate to also include surface fields and surface interactions. In this way, one can study phenomena such as wetting, interface localization-delocalization transitions, surface-induced ordering and disordering, etc.
The core of the program includes calculating the potential energies of particle configuration and particle collisions. The interactions and collisions between particles can be elastic or inelastic; they can be binary or collective. For computation, the interaction energy of a particle with its neighbors is carried out by refocusing a base cell on the particle. This particle only interacts with particles in this region. This is called the “minimal image convention” (MIC) [1].
Generally, a RANDOM generator of real random numbers ri belonging to the domain {0, 1} (or ri ∈ {0, 1} is available. This distribution law is uniform.
To have a real random number xi belonging to the domain {a, b} (or xi
To have a real random number xi belonging to the domain {a, b} (or xi ∈ {a,b}) according to a formula (or law) of nonuniform distribution f(x), a histogram technique is used. Let Nm be the number of intervals. If the mesh is regular (Figure 3):
Random number selection according to f (x) distribution.
We define:
We define the sequence:
and the sequence:
Hence each real random number ri belongs to the domain {0, 1} (where ri ∈ {0, 1}) (according to the uniform law); this number belongs to the domain {rxj-1, rxj}. It corresponds to a random value xran of the domain {xj-1, xj}; this number satisfies the formula (or the law) of nonuniform distribution f(x).
This technique can be generalized for a nonuniform distribution law f(x) with an irregular mesh Δxi, or with tabular data f(xi) with i = 1,…, m.
The technique can be generalized, too, for a discrete distribution law f(i) with i = 1,…, m.
In the literature, the reader can find simple algorithms for the choice of random numbers of some simple functions (Gaussian, etc.).
It is necessary to find some parameters allowing the control of the smooth course of the evolution of the system. We must look for the constants of movement. For example for an isolated system, we have the conservation of the total energy and the quantity of matter.
By using the numerical simulation, it is possible to calculate many spatiotemporal quantities F(r,t). These quantities can be positions, speeds, kinetic moments, particle energies, concentrations, transport coefficients, etc. It would then be possible to calculate all other quantities related to F(r,t).
For the calculation of the averages, one can note the quantities on the space, on the time or on both. The histogram methods can be used. Static or dynamic distribution functions and spatial or temporal correlation functions can be calculated. It should be noted that the SMC is much more adequate for static properties because of the probabilistic choice of configurations.
Any calculated function or parameter F(r,t) can be used for another application in another calculation program.
In the MCS model discussed extensively in this chapter, it’s more about collisions between particles. It’s particle-particle MCS or PP-MCS. In many problems of physics, the general idea is the same, but the applications and proposed models are numerous.
Other MCS models, named particle-in-cell MCS (PIC-MCS), are based on particle-cell interactions. In these last models, we also use a probabilistic choice of configurations and small variations in the state of the system (following the Metropolis algorithm); the interaction is between the particle with a cell, a mesh, or a drop. The parameters and variables of the cell, although local and instantaneous, are macroscopic. These parameters and variables can be thermodynamic, fluid, or electromagnetic. An example of the model based on PIC-MCS is described by Mattei et al. [31] for simulation of electromagnetic particle-in-cell collision in inductively coupled plasmas. Several works can be found in the literature on this same line of work. Other MCS models using particles may be considered. [32].
For statistical physics problem solving (such as thin film deposition problems), MCS models use experimental, numerical, or theoretical data from other methods and models. Models can be improved to hybrid models. In the hybrid models, connections between two modules can be realized. The first module is MCS; the second module is fluid, electromagnetic, or other. An example of a three-module hybrid model is presented by Mao and Bogaerts [33] to study gas mixtures in PECVD system. The three modules are MCS, fluid, and electromagnetic. The first module EM calculates the electromagnetic fields by solving Maxwell equations. These fields are used as inputs in the module MCS, where the electron density, electron temperature, electron energy distribution function, and electron impact reaction rates can be computed with a Monte Carlo procedure. Subsequently, the module fluid calculates densities and fluxes of the various plasma species (i.e., heavy particles and electrons) with continuity equations and the electrostatic field with Poisson’s equation. This electrostatic field is used as input again in the EM. This cycle is iterated until convergence. The schematic of the hybrid model is given in Figure 4.
Schematic of a hybrid model of three modules used to study gas mixtures in the PECVD [33].
To solve statistical physics problems with evolutions as a function of time, kinetic models of MCS (kMCS) are used. Using kMCS, Battaile and Srolovitz [17] described kinetic phenomena of the diffusive motion of a single interstitial atom in a close-packed metal crystal. The motion of the interstitial atom is usually limited to two types: vibration of the atom around the center of the interstitial hole in which it resides and hops to nearest-neighbor interstitial sites. The atom can hop into any of the nearest-neighbor interstitial sites; it executes a random walk. In an MC simulation of this diffusion process, the new position of the interstitial atom is chosen at random from a list of the adjacent interstitial sites.
Other CVD and PECVD works on MCS are presented in Ref.s [15, 34, 35, 36, 37, 38]. They show how MCS methods can study properties of gas mixtures and properties of the growth of thin films.
In this section, we present an example of PP-MCS of collisions and reactions in gas phase of SiH4/H2 mixture used in PECVD process. Some paragraphs have been treated in previous works [21, 24].
We use a MCS to study collisions and chemical reactions in gas phase of SiH4/H2 mixture used in the PECVD process. In this phase, important reactions have been identified that contribute to the production and the consumption of hydrogen (H), silylene (SiH2), and silyl (SiH3). The hydrogen consumption reactions SiH4 + H → SiH3 + H2 and SiH3 + H → SiH2 + H2 are found to play a central role in deciding the distribution of hydrogen [39].The plasma chemistry indicates that H atoms and SiH3 radicals play an important role in the a-Si:H deposition process [40]. Experimentally, it is generally accepted that SiH3 radicals dominate a-Si:H and μc-Si film growth from SiH4 plasmas in the PECVD; it is the key precursor of a-Si:H deposition [41]. The proposed MCS allowed to get the ratio SiH2/SiH3 and mean value of densities of species. It provides information on SiH4 dissociation and on the production of SiH3, H, SiH2, and Si2H6 and other important parameters.
The plasma in the PECVD reactor is weakly ionized. For our study, the mixture gas contains 22% of SiH4 and 78% of H2; the pressure is 100 mtorr, the temperature of the gas ranges from 373 to 723 K, the electron temperature is about 2.5 eV, and the electron density is 3. 108 cm−3. The process is considered to be stationary. We take into account electrons and eight neutral species (SiH4, SiH3, SiH2, H, H2, Si2H6, Si2H5, SiH). Reactions taken into account include seven electron-neutral and 14 neutral-neutral reactions. Table 1 shows the 21 reactions and rate constants Kreac. At low temperature, the neutrals interact occasionally with each other and move under the effect of thermal agitation; their velocity distribution function is Maxwell-Boltzmann distribution. Electrons have the mean velocity with kinetic energy Te.
Symbol | Reactions | Kreac (cm3/s) |
---|---|---|
R1 | SiH4 + e→SiH3 + H+e | k1 = 3 × 10−11 [42] |
R2 | SiH4 + e→SiH2 + 2H + e | K2 = 1.5 × 10−10 [42] |
R3 | SiH4 + e→SiH + H + H2 + e | K3 = 9.34 × 10−12 [42] |
R4 | SiH4 + e→SiH2 + H2 + e | K4 = 7.19 × 10−12 [42] |
R5 | H2 + e→2H + e | K5 = 4.49 × 10−12 [42] |
R6 | Si2H6 + e→SiH3 + SiH2 + H + e | K6 = 3.72 × 10−10 [42] |
R7 | Si2H6 + e→SiH4 + SiH2 +e | K7 = 1.1 × 1010× (1.(1./(1. + (0.63 × P)))) [43] |
R8 | SiH4 + H→SiH3 + H2 | K8 = 2.8 × 10−11 × exp.(−1250/T) [44] |
R9 | SiH4 + SiH2→Si2H6 | K9 = 1.1 × 1010 × (1.−(1./(1. + (0.63 × P)))) [43] |
R10 | SiH3 + SiH3→SiH4 + SiH2 | K10 = 0.45 × 1.5 × 10−10 [44] |
R11 | SiH4 + Si2H5→SiH3 + Si2H6 | K11 = 5 × 10−13 [42] |
R12 | SiH3 + H→SiH2 + H2 | K12 = 2 × 10−11 [44] |
R13 | SiH3 + Si2H6→SiH4 + Si2H5 | K13 = 4 × 10−10 × exp. (−2500/T) [44] |
R14 | SiH2 + H→SiH + H2 | k14 = 2 × 10−11 [44] |
R15 | Si2H6 + H→Si2H5 + H2 | K15 = 0.66 × 2.4 × 10−10 × exp. (−1250/T) [43] |
R16 | Si2H6 + H→SiH4 + SiH3 | K16 = 0.34 × 2.4 × 10−10 × exp. (−1250/T) [44] |
R17 | SiH + H2→SiH3 | K17 = 2 × 10−12 [43] |
R18 | SiH2 + SiH3→Si2H5 | K18 = 3.77 × 10−13 [43] |
R19 | SiH2 + H2→SiH4 | K19 = 3 × 10−12 × (1. + (1./1. + (0.03 × P))) [43] |
R20 | 2SiH3→Si2H6 | K20 = 0.1 × 1.5 × 10−10 [43] |
R21 | SiH4 + SiH→Si2H5 | K21 = (1.−(1./(1. + (0.33 × P)))) × (6.9 × 10−10) [43] |
List of gas phase reactions and corresponding rate constants [24].
Let
And chemical reaction for the production of A is as:
Rate production and consumption for any species A are taken as:
The MCS is based on binary collisions at the microscopic level. Elastic collisions are between all particles, and inelastic collisions (or effective collisions) are those that result in a chemical reaction. A chemical reaction needs a collision involving at least two particles (atoms, ions, electrons, or molecules). According to kinetic theory, gases consist of particles in random motion. These particles are uniformly distributed in a cell which has a parallelepiped form of sizes Lx, Ly, and Lz (Figure 5). These particles move in a straight line until they collide with other particles or the walls of their container. Dimensions and volume of Monte Carlo cell must take into consideration the mean free path of species.
Form of the simulation cell.
Let ni be the density of neutral spice i (i = 1,…, 8). The first particle i is randomly chosen according to a probability of neutral species Prsp,I (nonuniform discrete distribution) given by:
The chosen particle takes randomly three components of space in cell ri(xi, yi, zi) according to the normal distribution (nonuniform distribution). It takes also randomly three components of velocity vi (vxi, vyi, vzi) according to Maxwell-Boltzmann distribution.
Let ni and nj be the densities of species i and j in the gas and Vij the relative velocity between the two species i and j.
According to the kinetic theory of gases, we have for an incident particle i on a target particle j the average collision frequency νij as:
where <sij> is the cross section of the particle j.
The mean free path <λι> of species i is:
The time between two collisions τij is then:
For chemical effective reactions (inelastic collisions) between two reactive species i and j giving products i’ and j’, the rate constant reaction verifies [45]:
General rules of collision theory are applied:
The new velocities of the colliding particles are calculated using conservation of energy and momentum for elastic collisions.
Conservation of total energy as isolated system.
Movement of the center of mass and relative motion around the center of mass.
The reader can refer to some fundamental physics books that deal with general notions of collisions and corresponding parameters [45, 46, 47, 48].
The plasma in the PECVD reactor is weakly ionized. At low temperature, particles interact occasionally with each other and move under the effect of thermal agitation. In reality, only a small fraction of collisions are effective (result in a chemical reaction) [21].
In our MCS, after traveling a random walk given by a Gaussian distribution, the first chosen particle collides with a second particle (molecule, atom, radical, or electron). The last particle j is randomly chosen according to a (i-j) collision probability Prcol,j (nonuniform discrete distribution) given by:
where
The activation energy is given by:
where the pre-exponential factor is assumed to be the collision frequency factor and Kreac is the rate constant of the gas phase reaction.
The two colliding particles (e.g., the electron and SiH4 molecule) can interact by several reactions (R1, R2, R3, and R4 in Table 1); we choose randomly one of gas phase reactions occurring according to a, nonuniform discrete distribution reaction probability Prreac (i,j):
where
All chemical systems go naturally toward states of minimum Gibbs free energy [21, 24]. A chemical reaction tends to occur in the direction of lower Gibbs free energy. To determine the direction of the reaction that is taking place, we use the old and new values of Kreac and the equilibrium constant with reactants and product concentrations. Each set of binary collisions can be related or converted into time. As cited in section (a), Table 1 gives gas phase reactions and corresponding rate constants used in this MCS.
To continue the simulation, after the elastic collision, particle i takes new values of components velocity and new mean free path; mean free path is taken from a normal (nonuniform) distribution (Gaussian distribution). If the collision is inelastic, we have to take a new particle.
From Metropolis algorithm, the scheme of this MCS is as follows:
Choices of particle of spice i with random position, velocity, and mean free path; periodic boundary conditions are used to keep particles in the elementary cell.
Choices of random collision with a spice j.
Study of collision type (elastic, inelastic). If the collision is elastic the particle i move with a new velocity and mean free path, and we return to step (b). If the collision is inelastic particles i and j give new particles i’ and j’, according to Metropolis scheme, and we return to step (a) or (b). Periodic boundary conditions are used to keep particles in the elementary cell.
At each step, we can note the different statistics.
Once the species are selected for the simulation model, an estimate of species densities should be made. Following the model of interaction and collisions between particles (binary, collective, etc.), a first choice of the minimum number Ni of particles of each species is made. A first estimate of the sizes (Lx, Ly, Lz) of the elementary cell is made.
The study of the types of interaction potentials and the calculation of the approximate values of the force ranges, the kinetic energies, the internal energies, and the energies of activation make it possible to correct the minimal numbers Ni of particles and the sizes (Lx, Ly, Lz) of the elementary cell.
Let kp be the number of a species, kp = 1,…, 9. The minimal numbers Qnp(kp) and the sizes (Lx, Ly, Lz) have to be discussed for statistical calculations.
For numerical programming, according to the programming language used and according to the size (or the computational capacity) of the computer, it is necessary to find a judicious choice of the tables of integer or real values and which values would be useful to save all during simulation. Let Ncol,m be the maximum number of elastic collisions per particle, and let Ncycle be the number of cycles to average the simulation calculations.
For this MCS, the numerical chosen values are in Table 2.
Cell dimensions and steps for collisions | Number of species Kp | Initial number of particles in cell | ||
---|---|---|---|---|
Lx (m) | 4.68 10−6 | 1 | Qnp(SiH4) | Qnp1 |
Ly (m) | 4.68 10−6 | 2 | Qnp(SiH3) | 10 |
Lz (m) | 20.0 10−3 | 3 | Qnp(SiH2) | 10 |
4 | Qnp(H) | 10 | ||
Ncol,m | 500 | 5 | Qnp(H2) | Qnp5 |
Ncycle internal cycle | 2000 | 6 | Qnp(Si2H6) | 10 |
Ncycle external cycle | 200,000 | 7 | Qnp(SiH) | 10 |
8 | Qnp(Si2H5) | 10 | ||
9 | Qnp(e) | Qnp9 |
Used quantities and parameters in calculations for the gas temperature Tg = 520 K.
For radicals (e.g., SiH3), particle numbers Qnp(k) are very small; we take Qnp(k) = 10. These numbers cannot take value 1 or 0, even if a species k is in trace form in the gas. The value 0 for a species k means that any other species k’ does not make a collision with the species k; and the value 1 means that we have no collisions between particles of the same species in the cell.
Qnp1, Qnp5, and Qnp9 are calculated from the volume of cell, the pressure, the temperature, and the total number of particles in the cell (Qnp1 = 0.81187824 * 109; Qnp5 = 0.20296956 * 109; Qnp9 = 131).
As we have chosen a stationary regime, we must reach the values and properties at equilibrium. The results of the simulation show this trend. In MCS, averaged values, distribution functions, autocorrelation functions, and correlation functions can be calculated. To ensure rapid convergence of calculations, it would be useful to look for statistically symmetric (or stationary or unsteady) parameters [26, 50].
As an example for our MCS calculation, we have:
The number of Si2H6, SiH, and Si2H5 particles reaching the surface is negligible.
Let Ns,i and Ns, H2 be the densities of a species i and H2 reaching the surface. The ratios Ns,i/Ns, H2 are too small (Table 3).
Let Ns,i be the density of a species i reaching the surface and Nv,i the density of same species i in volume. The ratios Ns,i/Nv,i are too small (Table 4); the surface effect is negligible.
The reactions begin with the dissociation (consumption) of H2 and SiH4 by R5, R1, and R2 reactions.
The production of SiH3 is done by R8, and then there is production of SiH2 by R12.
The reaction R2: SiH4 + e → SiH2 + 2H + e plays the central role in SiH4 dissociation by electron impact [24]. This result is compatible with [39].
The second important chemical reaction in the SiH4 dissociation is R1: SiH4 + e → SiH3 + H + e [24]. This result is compatible with that of Perkins et al. [51] and that of Doyle et al. [52].
Type | H2 | SiH4 | H | SiH3 | SiH2 |
---|---|---|---|---|---|
Ns,i/Ns, H2 | 1 | 0.23 | 1.67 10−4 | 8.60 10−5 | 9.86 10−6 |
Ratios Ns,i/Ns, H2 of particles reaching the surface compared to H2.
Type | SiH4 | SiH3 | SiH2 |
---|---|---|---|
v, j | 6.695 10−6 | 7.965 10−6 | 775 10−6 |
Ratios Ns,i/Nv,i of particles reaching the surface compared to volume.
MCS is a widely used method in statistical physics to study thermodynamic, structural, or phase properties. It is based on random and probabilistic processes. The purpose of this chapter is to present the technique for general use in physics for the study of thin film deposition problems. The technique can be generalized to other fields of science: biology, economics, transportation, and social sciences.
We started by presenting general rules for numerical simulation methods. Metropolis algorithm has been considered as the basic algorithm. After, we presented the different steps for the realization of a MCS code. We chose the particle-particle model MCS (PP-MCS) to explain the different steps and procedures to be applied in the deposition of thin layers by PECVD processes. We have shown that this technique can be generalized to the particle-in-cell MCS (PIC-MCS) case or kinetic MCS (kMCS), as it can be joined with other modules to give hybrid models. It is important to know how to choose random configurations from the laws or probability distributions in the system.
A numerical application is presented for collisions in a SiH4/H2 gas mixture in the PECVD process. A preliminary work of determination of the chemical reactions between molecules and radicals is made. A choice of the simulation cell is made, and the definition of the probabilities of the collisions between peers is made. The Metropolis algorithm makes it possible to follow the various elastic and inelastic collisions; it also makes it possible to make the statistics of the interactions with the surface. The results are compatible with [39, 51, 52].
Other questions may be asked to account for molecular ions, surface and volume kinetics, or thin film formation. The techniques and different models of the MCS (PP-MCS, MCS-PIC, kMCS) allow taking care of these questions.
The interconnection of the MCS with other models (MDS, hybrid model, fluid model, electromagnetic model, etc.) would allow answering more questions. The methods can be applied to other specialties than the physical sciences.
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