process control strategies for the gas phase synthesis of silicon nanoparticles

Chemical Engineering Science 73 (2012) 181–194

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Chemical Engineering Science

0009-25

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Process control strategies for the gas phase synthesis of silicon nanoparticles

Michael Groschel a,n, Richard Kormer b, Maximilian Walther a, Gunter Leugering a, Wolfgang Peukert b,nn

a Institute of Applied Mathematics, Cauerstrasse 11, Friedrich-Alexander University, Erlangen-Nuremberg, Germanyb Institute of Particle Technology, Cauerstrasse 4, Friedrich-Alexander University, Erlangen-Nuremberg, Germany

a r t i c l e i n f o

Article history:

Received 9 August 2011

Received in revised form

16 December 2011

Accepted 10 January 2012Available online 28 January 2012

Keywords:

Aerosol

Silicon nanoparticles

Population balance

Process control

Parameter identification

Optimisation

09/$ - see front matter & 2012 Elsevier Ltd. A

016/j.ces.2012.01.035

esponding author. Lehrstuhl fur Angewand

er Universitat Erlangen-Nurnberg, Cauerstr. 1

9 9131 85 67134.

cipal corresponding author. Lehrstuhl fur Fes

technik, Friedrich-Alexander-Universitat Erla

8 Erlangen, Germany. Fax: þ49 9131 85 2940

ail addresses: [email protected]

[email protected] (W. Peukert).

a b s t r a c t

In this contribution the identification of new reaction conditions for the production of nearly

monodisperse silicon nanoparticles via the pyrolysis of monosilane in a hot wall reactor is considered.

For this purpose a full finite volume model has been combined with a state-of-the-art trust-region

optimisation algorithm for process control. Verified against experimental data, specific process

conditions are determined accomplishing a versatile range of prescribed product properties. The main

achievement of the optimisation is the possibility to control the different mechanisms in the particle

formation process by mainly adjusting the temperature profile. Due to a successful separation of the

nucleation and growth process, significantly narrower particle size distributions are obtained. More-

over, the presented optimisation framework establishes rate constants based on measured data.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Various kinds of gas phase processes have been developed forthe production of high purity silicon nanoparticles (SiNPs) in thepast three decades. In the early eighties, Cannon et al. (1982a,b)were one of the first groups who produced SiNPs within a laserheated reactor via silane pyrolysis. Besides other reactor conceptslike plasma reactors (Kortshagen, 2009) or microwave reactors(Knipping et al., 2004), particularly the hot wall reactor becamewell-established in the production of SiNPs. Alam and Flagan(1986) were the first to utilise this reactor and several otherresearch groups continued working in this field (Onischuk et al.,1994; Wiggers et al., 2001). Hot wall reactors show importantadvantages like high production rates, simple scalability, and aprecise control of the present reaction conditions. This helps toproduce well defined particulate building blocks requested, e.g.,in hierarchically ordered systems (Bishop et al., 2009).

The controlled production of nanoparticles with defined sizeand shape requires a deep understanding of the underlyingsystem dynamics. Typically, high temperatures and rapid chemi-cal reactions limit the experimental access to particle formation

ll rights reserved.

te Mathematik II, Friedrich-

1, 91058 Erlangen, Germany.

tstoff- und Grenzflachenver-

ngen-Nurnberg, Cauerstrasse

2.

(M. Groschel),

mechanisms or local reaction conditions. Despite huge progresshas been made, most aerosol reactors operate mainly underempirically obtained process conditions. In general, simulationtechniques currently represent the best way to achieve a sys-tematic approach to process control.

The first accurate numerical models for the description ofaerosols based on the so-called general dynamic equation weresuggested by Gelbard and Seinfeld (1978). Significant advance-ments were presented by Wu et al. (1988) who included fastchemical reactions into the aerosol model or by Kruis et al. (1993)who tracked simultaneously agglomeration and sintering pro-cesses. Several authors established different aerosol modellingapproaches over the years and monodisperse (Warren andSeinfeld, 1985a), moment (Pratsinis, 1988), and sectional meth-ods (Warren and Seinfeld, 1985b) have become extensively used.A comparison of the method of moments, the quadrature methodof moments, and the sectional method in the context of silanepyrolysis is stated in Talukdar and Swihart (2004). If a highnumerical efficiency and a more precise numerical descriptionof the particle size distribution are required simultaneously, thefinite element method represents a suitable technique (Appel,2001). Due to its inherent mass conservation property, the finitevolume method has become increasingly important in the lastyears. In the context of particulate processes Qamar andWarnecke (2007), extended by Kumar and Warnecke (2010),employ the framework of finite volume schemes besides themethod of characteristics in their simulations. The issue ofstructure formation in aerosol processing was theoretically inves-tigated by Schmid et al. (2004) using Monte Carlo simulations(see also Al Zaitone et al., 2009).

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M. Groschel et al. / Chemical Engineering Science 73 (2012) 181–194182

Further work on aerosol synthesis of silicon nanoparticles wasprovided by Swihart and Girshick (1999) who rigorously exam-ined the reaction kinetics and the molecular structure of theinitial clusters. This detailed reaction mechanism was thencoupled to an aerosol model to account for nucleation and particlegrowth under various reaction conditions (Girshick et al., 2000).Another sophisticated, fully coupled model was presented byDang and Swihart (2009a,b) which incorporates fluid dynamics,laser heating, gas phase and surface reaction as well as a thoroughdescription of the aerosol dynamics.

A detailed knowledge about the gas phase chemistry whichsignificantly influences particle formation and growth is animportant prerequisite towards multiscale modelling of gas phasesynthesis processes. Recently, Menz et al. (2011) presented amodel that simultaneously solves the reaction mechanism forsilane pyrolysis of Swihart and Girshick (1999) coupled to aspecific aerosol model with special focus on the sinteringprocesses.

A detailed comparison between experimentally obtained andcalculated particle size distributions (PSDs) of SiNPs at variousprocess conditions was presented for the thermal silane pyrolysisby Kormer et al. (2010a,b). Including simplified global reactionkinetics, nucleation, and growth rates, the applied populationbalance model proved to be consistent with the actual processand therefore constitutes a verified basis for this contribution.Genuinely, the next step aims towards the development ofoptimal process control strategies in order to meet specificpredefined product properties.

Techniques for the dynamic inversion of population balanceequations (PBE) on the level of moment methods have beenpresented by Vollmer and Raisch (2006), Vassilev et al. (2010),or recently by Bajcinca et al. (2011) using the framework of flatsystems. In Chittipotula et al. (2010), a genetic algorithm has beenused to optimise a CFD soot prediction model combined with thedirect quadrature method of moments. Doyle et al. (2002) alsowork with genetic algorithms to control the product quality inpolymerisation processes.

In particular, process control strategies for aerosol modelshave been presented by Christofides et al. (2008) who used amoment approximation of the PSD described by a lognormaldistribution. Simulation, estimation, and control of size distribu-tion in aerosol processes with simultaneous reaction, nucleation,condensation, and coagulation are considered in Shi et al. (2006).Adjoint methods for the inverse modelling of aerosol dynamicshave been presented by Sandu et al. (2005b) based on a piecewiselinear representation of the PSD in eight bins (Sandu, 2006) orcombined with a finite difference discretisation for data assimila-tion in local air quality models (Sandu et al., 2005a).

In all these contribution either a reduced moment or sectionalapproximation was used or the full complexity of the underlyingprocess has been disregarded. In this study a full finite volumemodel has been combined with a state-of-the-art trust-regionoptimisation algorithm for process control. In contrast to momentapproximations, the chosen discretisation of the particle sizedistribution is capable to distinguish multimodal PSDs causedby several nucleation bursts. The resolution of the applied finitevolume scheme is in general superior to sectional models andguarantees mass conservation (Gunawan et al., 2004).

The structure of the present paper is as follows: Section 2summarises the most important experimental findings which arerelevant for the production of nearly monodisperse SiNPs (Kormeret al., 2010a,b). Furthermore important features of the aerosolmodel and fundamental results are recapitulated. Besides a shortoutline of the simulation techniques, the framework of theinverse problem is presented in Section 3. The investigation ofoptimal process conditions and improved parameter estimates is

described and discussed in Section 4. Moreover, specific reactionconditions are subsequently determined to obtain predefinedparticle size distributions. The paper is concluded in Section 5and prospects for future work are outlined.

2. Prior study on the aerosol synthesis of silicon nanoparticles

2.1. Experimental results

SiNPs have been synthesised via pyrolysis of monosilane (SiH4)diluted in high-purity argon in a resistively heated, vertical hot-wall reactor. Typical operation conditions comprise temperaturesbetween 900 and 1100 1C, a total pressure of 25 mbar, a silanepartial pressure of 1.0–3.2 mbar and a residence time between 80and 420 ms. Aerosolised SiNPs were extracted by a hooked probewhich is mounted centred at the exit of the reactor. These SiNPswere deposited by a low pressure impactor and subsequentlycharacterised by image analysis on the basis of scanning electronmicroscopy pictures. Spherical SiNPs with narrow PSD (geometricstandard deviation (GSD) of r1:1) could be obtained in thespecified process parameter regime.

Identified prerequisites for the generation of narrow PSDs aremainly a low total pressure (25 mbar) and an initial silane partialpressure of 1 mbar or below. The proposed particle growthmechanism for the narrowly distributed SiNPs relies on a shortnucleation burst in the beginning, predominantly followed bysurface growth and condensation. The particle number concen-tration in the reactor is sufficiently low to avoid significantagglomeration. A particle synthesis process, which is dominatedby agglomeration and sintering, would lead to much broader PSD.The particle number concentration is essentially influenced by thetotal pressure of the reactor since the silane pyrolysis reaction isstrongly depending on the number of collision partners andtherefore on the argon partial pressure. A high number of collisionpartners leads to rapid silane pyrolysis and subsequently to asignificant broadening of the PSD. Further information concerningthe experiments can be found in Kormer et al. (2010a).

2.2. Review of the modelling approach

Experimental observations indicated that surface reaction andcondensation play a significant role in the growth of particleseven though a direct proof through measurements was notavailable. Therefore, an additional investigation of the underlyingmechanisms of silicon nanoparticle formation by means ofprocess simulation was initiated. The applied model includesmechanisms for a simplified gas phase as well as surface reactionof silane. Particles are incepted by homogeneous nucleation froma supersaturated vapour. Furthermore, the model accounts forcondensation on the surface of the particles and agglomeration ofthe particles.

In the prior work, the associated equations for each mechan-ism were implemented in the commercial software PARSIVAL(Wulkow, 2001) which solves the system of differential equa-tions. A brief description of the model is given below, for moredetails see Kormer et al. (2010b) and Artelt et al. (2005, 2006).

The common approach to describe multimodal particle sizedistributions which are altered by different process mechanismsis based on a population balance equation. Including the phe-nomena of growth and nucleation, the resulting partial differen-tial equation describes the evolution of the number distributionyðx,tÞZ0 in time. One specific realisation of the general PBE isgiven in the case where the particle diameter xARþ representsthe only internal variable (Friedlander, 2000). Assuming a con-stant feed rate in an ideally mixed system with regard to the

M. Groschel et al. / Chemical Engineering Science 73 (2012) 181–194 183

cross-section of the reactor, the position of a particle transfersaccordingly to its individual residence time t. This approach leadsto a reduced model for the pyrolysis of monosilane previouslydescribed in Section 2.1

@

@tyðx,tÞþ

@

@xðGðtÞyðx,tÞÞ ¼ Bðx,tÞ ð1Þ

The growth rate is denoted by G and the corresponding term tonucleation is represented by the source B on the right hand side.No initial seed particles yðx,0Þ ¼ 0, 8xARþ or influx boundarycondition yð0,tÞ ¼ 0, 8tA ½0,T� are assumed since all particles areformed at a critical cluster diameter through homogeneousnucleation.

The aerosol model accounts for SiNP formation via thepyrolysis of silane. Thereby, silane can undergo two kinds ofpossible reactions: A homogeneous gas phase reaction and asurface reaction on particles. The principally very complicatedreaction network of silane decomposition and the subsequentformation of silicon hydride species (Purnell and Walsh, 1966;Becerra et al., 1995; Chambreau et al., 2002; Swihart and Girshick,1999) is considerably simplified by taking an overall reactionequation into account. This equation is based on the studies ofPetersen and Crofton (2003) who investigated the rate limitingstep of the silane decomposition mechanism in the gas phase fortemperatures between 787 and 1457 1C and a total pressure of0.6–5.0 bar. The equation accounts for the formation of a silyleneradical due to the collision of a silane molecule with any third-body collision partner which is in the present case argon

SiH4þAr-SiH2þH2þAr ð2Þ

Therefore, the corresponding pressure dependent silane pyr-olysis kinetic is given by

d½SiH4�

dt¼�k1a � ½SiH4� � ½Ar� ð3Þ

in the isothermal case. The rate coefficient k1a in Eq. (3) is statedby

k1a ¼ F1 � 7:2� 109� e �22 710 K=Tð Þ m3

mol sð4Þ

including an additional scaling term F1. For the time being, it isreasonable to restrict the complex description of all possiblechemical species and reactions in the gas phase to the statedglobal reaction kinetic. For the particular case operating at1100 1C and a residual time of 80 ms, the fit parameter F1 hasempirically been determined. The choice allows for a reasonableapproximation of the chemical reaction as shown in Kormer et al.(2010b, Section 3) for the considered process conditions. In thepresent contribution this analysis will further be substantiated byparameter estimation techniques based on more than one experi-mental data set.

Besides the gas phase reaction, a heterogeneous deposition ofmonosilane on the surface of the SiNPs takes place (surfacereaction). According to Peev et al. (1991) the corresponding filmgrowth rate Rd is described by the following global equation:

Rd ¼ F2 �kp � KSiH4

� ½SiH4�

1þKSiH4� ½SiH4�

�MSi

rSi

ð5Þ

Again, the stated equation has been fitted to the present reactionconditions by introducing the parameter F2. The applicableheterogeneous rate coefficient kp and the equilibrium adsorptionconstant of silane KSiH4

in Eq. (5) are also adopted from Peev et al.(1991) (see Kormer et al., 2010b).

The theoretical nucleation study from Kruis et al. (1994)highlights that silicon clusters which contain more than one atomare typically present in the gas phase; therefore, the criticalcluster diameter xc can comprise up to 25 monomers depending

on the process conditions. According to the Kelvinequation (Kodas and Hampden-Smith, 1999) xc is given by

xc ¼4 � g � V1

kB � T � ln Sð6Þ

where g signifies the temperature dependent surface energy ofthe cluster, V1 the volume of a silicon atom, kB the Boltzmannconstant, T the current process temperature, and S the super-saturation. All quantities are defined according to the presenta-tion in Kormer et al. (2010b). Altogether, the homogeneousnucleation rate Bn,Si is calculated according to an extendedapproach of the kinetic nucleation theory of Girshick and Chiu(1990) given by

Bn,Si ¼ n21,S � S � V1 � V

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 � gp �m1

seðY�4�Y3=27�ðln SÞ2Þ � dxc ð7Þ

with

Y¼p � x2

1 � gkB � T

ð8Þ

where dxc represents the Kronecker-delta, n1,S denotes the mono-mer, respectively, silicon atom number concentration for a satu-rated vapour, m1 the mass and x1 the diameter of a silicon atom.

In the present model the growth of particles results from twodistinct mechanisms. On the one hand, a heterogeneous deposi-tion of monosilane on the surface of the SiNPs occurs as describedabove; furthermore, free monomers condense from the super-saturated vapour on the surface of existing particles.

The growth rate due to chemical surface reaction GSiH4with

respect to the diameter of a spherical particle

GSiH4¼ 2Rd ð9Þ

is obtained by adding twice the film growth rate stated in Eq. (5).Growth by surface reaction on existing particles is only relevantas long as the precursor is available.

The modelling of the condensation of free monomers from thegas phase on the surface of the existing particles follows anapproach of Panda and Pratsinis (1995) based on the kinetictheory of gases assuming also spherical particles

GSi ¼ 2 � n1,S � V1 � ðS�1Þ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikB � T

2 � p �m1

sð10Þ

The correction term for the extension of the equation into thecontinuum regime can be neglected due to the high Knudsennumbers (41000) in this work.

It has to be pointed out that the change of volume needs to beconsidered in the present setting due to the variable temperatureprofile. In consequence, for example Eq. (3) has to be adapted tothe present size of the control volume. For a complete descriptionof the reaction processes, the tracking of the remaining precursormass mSiH4

along with the available silicon mass mSi in the gasphase has to be set up. Considering the pyrolysis of monosilane(index r for gas phase reaction), of course a decrease (D) of theprecursor mass Dr,SiH4

is accompanied by an increase (I) of siliconmass Ir,Si in the vapour. As described above, new particlesoriginate through homogeneous nucleation (index n) from thesaturated vapour. To account for the corresponding reductionof available silicon mass in the balance equation, the rate ofdecrease is denoted by Dn,Si. Subsequently, the reduction rateof silane Dg,SiH4

due to the surface reaction (index g for growth) iscalculated by summing up the adsorbed mass on all particles.Analogously, the consumption of silicon from the gas phase Dg,Si

in the case of condensation can be derived.Thus, the complete model can be assembled from the equa-

tions for each individual mechanism. Following the previousfindings, it is has to be pointed out that this model is only valid


for reaction conditions with negligible agglomeration.

@

@tyðx,tÞ ¼�

@

@x½ðGSiH4

ðtÞþGSiðtÞÞyðx,tÞ�þBn,Siðx,tÞ

@mSiH4ðtÞ

@t¼�Dr,SiH4

ðtÞ�Dg,SiH4ðtÞ

@mSiðtÞ

@t¼ Ir,SiðtÞ�Dg,SiðtÞ�Dn,SiðtÞ, xAO�Rþ0 , tA ½tmin,tmax�

yðx,0Þ ¼ y0ðxÞ in O

yðx,tÞ ¼ 0 on @O� ½tmin,tmax�

mSiH4ð0Þ ¼m0

SiH4

mSið0Þ ¼m0Si ð11Þ

In summary, the described model comprises two ODEs reflect-ing the mass balances besides to the main PDE of the populationbalance equation. Coupled systems of this kind have rarely beeninvestigated by advanced optimisation techniques in order toidentify suitable processes conditions. Of particular importance isthe fact that the model agrees with the experimental data anddoes not employ any moment approximations.

2.3. Modelling results

The main objective of the performed simulations by Kormeret al. (2010b) was to understand the formation mechanism that isbehind the experimentally observed narrow PSDs which is veryuncommon in aerosol processing. A prospect of this formationmechanism was presented on the basis of the experimentalfindings in Kormer et al. (2010a). In the following the mainresults are stated for a brief review.

The process parameters of the simulations were chosen to matchthe experimental conditions in the best possible way. The precursorloss due to the deposition of silicon on the reactor walls isdetermined in the experiments and has subsequently been takeninto account for initialising the simulation (see Kormer et al.,2010b,p. 5). The temperature characteristics from the hot-wallreactor were transferred into time–temperature profiles by incor-porating the respective volume flow rates from the experiments.

For the remaining part of this contribution, the basic set ofprocess parameters refers to the configuration specified byT¼1100 1C, pSiH4

¼ 1 mbar, ptotal ¼ 25 mbar and a residence time of80 ms. The corresponding simulation result was validated againstthe experimental observations and yields a total particle numberconcentration of 2.4�1015 m�3 with a mean particle diameter of27 nm and a geometric standard deviation (GSD) of 1.1.

Furthermore, one key feature in the growth mechanism wasattributed to condensation. In the prior simulations withneglected condensation the final PSD exhibited much smallerparticle sizes and improper GSDs. Thus, any attempt to correlatesimilarities between the simulation and the experimentallyobtained PSDs fails. These findings discussed in Kormer et al.(2010b) confirm the significant impact of the condensationmechanism on the particle formation mechanism. Besides con-densation, different aspects concerning the influence of the mainprocess variables were highlighted. The importance of the totalpressure in the reactor was pointed out since it is only possible togenerate narrow size distributions at low pressures.

Following the experimental data, the maximum temperature of1100 1C was decreased in the simulations to 1000 1C and 900 1C.The decreased reaction kinetic and, consequently, the incompleteprecursor consumption reproduced the trend of a shift of the PSDto smaller sizes. Although the results show a narrow primary

particle size distribution, only a qualitative agreement could befound regarding the experimental measurements.

In the outline of the model, two additional parameters havebeen introduced in order to fit the model to the present reactionconditions. This adaption has become necessary since the globalreaction kinetics have been replaced by a parametrical model.Using two fitting parameters F1 and F2, the rate coefficient k1a (4)in the silane decomposition kinetic and the film growth rate Rd (5)have been adjusted empirically based on the case with standardprocess parameters. This choice will further be investigated in thepresent contribution. The optimisation approach presented in thenext section is expected to provide a better quantitative agree-ment with the experimental data.

Concerning the choice of process conditions, in Kormer et al.(2010a,b) a suitable regime for the production of narrow particlesize distributions has been identified. The results show that a lowstandard deviation can only be met in a certain temperaturerange as long as both the initial silane concentration and thereactor total pressure are low. Considering other combinations offeasible process conditions, even tighter particle size distributionsmay be achieved. However, changing parameters without beingconscious of their impact on the complex process is not bestpractice. Thus, tuning the process conditions within an optimisa-tion method provide new insight on how the reaction conditionscan be improved in the future.

3. Optimisation approach

For the simulations a finite volume solver based on theTVD-WAF (total variation diminishing weighted average flux)approach was chosen. Qamar and Warnecke (2007) alreadyconsidered growth and aggregation processes within the frame-work of the finite volume methods. In the simulations of theirmodel problem, they also incorporated nucleation as a boundaryterm. In contrast to their approach, the use of realistic reactionrates, a treatment of the nucleation as a source term, and thevalidation against experimental data posed additional challengesin the present case.

The contribution of the nucleation to the overall populationbalance equation is generally confined to a very narrow sizeinterval compared to the entire simulation range. Therefore, thecorresponding part needs to be treated with special care. Afterdiscretising only the spatial derivatives, referred to as the methodof lines, the resulting system of ODEs contains in particular thenucleation as a stiff source term besides the regular parts. Thetwo components were separated by an operator splitting techni-que of second order. The implementation guarantees an appro-priate treatment of the stiff term ensuring at the same time areasonable computational effort. Integrating in time, the stiffexpression undergoes an implicit treatment based on a trapezoi-dal rule and solved by a Newton-type iterative method; whereasthe explicit part is handled via a standard higher-order Runge–Kutta method.

The main goal is now to build up an inverse model capable offinding optimal process parameters. Moreover, the introduced fitparameters in the reaction kinetic as well as in the growth ratedue to chemical surface reaction (see Section 2) have to beascertained or recalibrated by a parameter estimation technique.By prescribing the total particle number, the mean particlediameter and the geometric standard deviation in a nonlinearleast squares optimisation problem, suitable process controlstrategies are finally obtained.

The described situation fits in the context of general nonlinearconstrained optimisation problems. Within such an optimisation,the simulation needs to be carried out repeatedly. Since no

Table 1Characteristic properties of the PSDs shown in Fig. 2a.

Data set T (1C) Ntotal (m�3) x (nm) sgeo

1 900 4.0�1015 22 1.08

2 1000 1.8�1015 29 1.08

3 1100 1.3�1015 32 1.07


derivatives are available, Trust-region methods constitute a wide-spread and popular approach due to their flexibility and efficiency(see, e.g., Conn et al., 2000). In the present case the time for asingle simulation run is significantly larger than the pure compu-tation time of the optimisation method. Therefore, a specialglobally convergent Trust-region based algorithm designed byThekale (2011) was applied in order to solve the consideredoptimisation problem.

The method is based on the idea of iteratively replacing thesimulation inside the optimisation by models with sufficientaccuracy. In addition, as much information as possible is pre-served from the original optimisation problem in order to keepthe number of simulation runs low. When minimising the modelfunction, the algorithm uses the derivative information availablefor the model to guarantee fast convergence. Therefore, themethod is of hybrid type using derivative-based techniques aswell as derivative-free ones. For the stated special class ofoptimisation problems, the proposed algorithm yields excellentconvergence behaviour using at the same time considerably fewerfunction evaluations.

The field of optimisation and control of systems governed bypartial differential equations is a very active area of research inapplied mathematics with a growing impact on engineeringapplications. Simulations provide in general a particular solutiony for a given set of process parameters u solving the correspond-ing PDE model. Nonlinear optimisation problems deal with theminimisation of a given objective function JðyðuÞÞwhich quantifiesin general the distance between the current solution and a giventarget

minuAL

JðyðuÞÞ ð12Þ

In particular, it becomes amply clear that a set of constraints Lhas to be taken into account since the process parameters areoften restricted to certain range limits.

In the simulation of the considered aerosol pyrolysis, theprimary objective is to realise a predefined particle size distribu-tion yn after the process. The solution of the simulation at the finaltime tmax is given by yðx,tmaxÞ. In many cases the properties of thefull PSD are not available, whereas the main characteristics maybe observed. Considering for example the total particle numberNtotal, the mean particle diameter x, and the geometric standarddeviation sgeo in a normalised observation function g(y)

gðyÞ ¼

1

NtotalðynÞNtotalðyÞ

1

xðynÞxðyÞ

1

sgeoðynÞsgeoðyÞ

0BBBBBBB@

1CCCCCCCA

ð13Þ

the general least squares objective function

JðuÞ ¼a2JgðynÞ�gðyðuÞÞJ2

2 ð14Þ

transforms to

JðuÞ ¼a2

1�1

NtotalðynÞNtotalðyðuÞÞ

1�1

xðynÞxðyðuÞÞ

1�1

sgeoðynÞsgeoðyðuÞÞ

0BBBBBBB@

1CCCCCCCA

��

��

2

2

: ð15Þ

As a consequence of the normalisation, all observables are equallyweighted balancing the different scales of the variables. Never-theless, a subsequent change of the individual weighting remainspossible.

The characteristic particle properties themselves are definedusing the principal moments of the distribution. Using themoment methodology a clear representation of the consideredquantities in the objective function at the final time tmax becomesfeasible. The total particle number concentration Ntotal for exam-ple is therefore stated by the zeroth moment and the meanparticle diameter can be calculated accordingly.

The geometric standard deviation sgeo was chosen since thePSDs in aerosol processes are generally represented by a lognor-mal distribution. However, there are no restrictions concerningthe shape of the distribution due to the used finite volumescheme.

4. Process control strategies

In the considered application of the silane pyrolysis, theidentification of process control strategies suitable to producenearly monodisperse PSDs is of great importance. Besides theresolution of potential model uncertainties, this section estab-lishes optimal process conditions in order to achieve the men-tioned target. For this purpose, the previously describedoptimisation method is used to solve the corresponding inverseproblem.

4.1. Model uncertainties

In Section 2.2, two parameters, F1 and F2, have been introducedin order to fit the described model to the present conditions in thereactor. This adaption has become necessary since the globalreaction kinetics have been replaced by a parametrical model asoutlined before. Up to now, it has been unclear whether this set ofparameters is unique or if there are other possibilities to repro-duce the previously published results. In the following, thisproposal will further be substantiated by a parameter estimationbased on several particular data sets available for a residence timeof 420 ms.

Table 1 summarises the characteristic properties of the experi-mental PSDs, obtained from measurements using the processparameters pSiH4

¼ 1:0 mbar, ptotal ¼ 25 mbar, tmax ¼ 420 ms andthe three varying temperature histories described in Section 2.3.

The values given in Table 1 have subsequently been employedin defining the objective function J of the corresponding optimi-sation problem. The value of J measures hereby the distance of thefinal properties between each simulation run and the experi-mental data. In the optimisation, the parameters F1 and F2 areadapted such that the value of the objective function decreases,i.e. that the simulations are approaching the experimental PSDs(see Table A1). The corresponding evolution of the numberdensity distributions in the optimisation process is depicted inFig. 1.

At first, the simulation has been adapted to a residence time of420 ms still using the parameter set F80 ¼ ½F1,F2� ¼ ½1:25�10�3,70� which was empirically determined at a residence timeof 80 ms. A first simulation run produces three almost indistin-guishable PSDs for the different temperature profiles (see Fig. 1a–c).Displaying the fifth iteration, Fig. 1d–f shows that the mean

Fig. 1. Evolution of the PSDs in the optimisation process varying the fit parameters F1 and F2 (pSiH4¼ 1:0 mbar, ptotal ¼ 25 mbar, tmax ¼ 420 ms, temperature profiles

according to Section 2.3). (a)–(c) iteration 0, (d)–(f) iteration 5, (g)–(i) iteration 14.


particle sizes of the final PSDs already differ significantly. Thischange is reflected by a drop in the value of the objective functionJ by three orders of magnitude (see Table A1). After 14 iterations(Fig. 1g–i), the optimisation routine obtains an optimised set of fitparameters F420 ¼ ½2:2� 10�4,37:4� for a residence time of420 ms. A much better correlation with the experimental data isachieved by using the newly found set of parameters (see Fig. 2for a comparison to Fig. 11 of Kormer et al., 2010b). Compared tothe initially almost identical PSDs, the mean particle diametersare now considerably more consistent with the predefined valuesfrom Table 1. Furthermore, it is important to note that themagnitude of the determined parameters remains in the rangeof the empirically found parameter set F80.

Fig. 3 shows the profiles of the present supersaturation againstthe relative residence times for a temperature of 1100 1C andresidence times of 80 ms and 420 ms, respectively. Thus, accord-ing to their position in the reactor, the particles encounter at thesame time the maximum degree of supersaturation. This sub-stantiates the reasoning stated in Kormer et al. (2010b), that theadopted parametrical representation constitutes an approvedmodel of the complex reaction kinetics in the gas phase. However,reviewing the comparison of the final PSDs resulting from the twodifferent residence times in Fig. 8 of Kormer et al. (2010a), a

longer exposure time causes an increased mean particle size and areduced total number of particles.

A detailed view on the different reaction kinetics reveals thateven though the supersaturations are comparable in their relativeprofiles, the growth and nucleation processes may differ to a greatextend. Fig. 4 shows the available silicon mass in the gas phase aswell as the corresponding precursor concentration on the top.Below, the homogeneous nucleation rate and the film growth rateare compared for the considered residence times. In the optimisa-tion, the fit parameters of the reaction kinetic were adapted tomatch the experimental data which also results in a reducedsilicon mass in the gas phase for the case of the longer exposuretime of 420 ms. Subsequently, the nucleation rate is diminishedand fewer particles are formed. Thus, a higher silane concentra-tion remains and the particle growth is sustained for a longertime span. Under these conditions the resulting particle sizedistribution is shifted to the right and contains fewer particlesdue to the mentioned reduced nucleation rate. However, Fig. 1shows that the new parameters do not only slow down thekinetics since the effect of the temperature variation is reflectedin a reasonable way. In consequence, the proposed model is ableto match three more data sets at a different residence time. Therobustness of the optimisation framework makes it reasonable to

15 20 25 30 35 40 450

0.05

0.1

0.15

0.2

0.25

0.3

particle diameter [nm]

q 0(x) [

nm−1

]

T=900 °Cfit 900 °C (GSD 1.08)T=1000 °Cfit 1000 °C (GSD 1.08) T=1100 °Cfit 1100 °C (GSD 1.07)

15 20 25 30 35 40 450

0.05

0.1

0.15

0.2

0.25

0.3


q 0(x) [

nm−1

]

T=900 °CT=1000 °CT=1100 °C

Fig. 2. Number density distributions q0 obtained from (a) measurements

taken from Kormer et al. (2010b) and (b) resulting after the optimisation,

i.e. with the determined parameter set F420 ¼ ½2:2� 10�4 ,37:4� (pSiH4¼ 1:0 mbar,

ptotal ¼ 25 mbar, tmax ¼ 420 ms).

0 0.05 0.1 0.150

2

4

6

8

10 x 104

rel. residence time

supe

rsat

urat

ion

[−]

residence time 0.08 sresidence time 0.42 s

Fig. 3. Supersaturation profiles for the obtained final values F420 ¼ ½2:20�

10�4 ,37:40� plotted up to 15% of the entire residence time.


assume that there are no other possible choices for the twoparameters. Since there is no possibility to match both cases withan identical set, the introduced fit parameters definitely requirefurther investigation to study their specific dependencies. Sincethe main process mechanisms have been analysed in detail inKormer et al. (2010a,b), the temperature distribution inside thereactor persists to be one of the possible remaining uncertainties.

4.2. Analysis of the temperature profile in the base case

Since the temperature profile has been identified as a possiblemodel uncertainty in the last section, the temperature curveapplied in the simulations requires a more detailed investigation.The simulation of the base case (see Section 2.3) yields a totalparticle number concentration of 2.4�1015 m�3 with a meanparticle diameter of 27 nm and a GSD of 1.1. The assumedtemperature history of the particles inside the reactor is depictedin Fig. 5. Now the question arises whether the experimentallyobtained PSD may also be attained by a different temperatureprofile in the simulations.

Starting from a completely different temperature history, anoptimisation run is supposed to determine the optimal profilewhich yields the same characteristic values of the final PSD. Thevariable temperature profile is built up using a five point splineinterpolation as shown in Fig. 5; whereas the initial precursormass m0

SiH4represents the sixth process parameter besides the five

nodes in the optimisation routine. The coarse interpolation waschosen since an increased number of nodes would allow forunphysical temperature oscillations which cannot be realised inthe reactor.

A first simulation runs with the initial values shown in Fig. 6aproduces a far too high number of particles combined with a verylow mean particle size (see Table A2 and Fig. 6b, c). Varying thetemperature profile (see Fig. 6d), the optimiser tries to match thefinal properties to the prescribed values. Thus, particle growth ispromoted in iteration 10. Now, however, a very low particlenumber density is attained due to the low initial precursor mass(see Fig. 6e, f). After 20 iterations Fig. 6g–i reveal a differentpicture. The temperature history has turned into a profile resem-bling quite more to the original profile. Although a first nucleationburst occurs around 10 ms, it is not visible in Fig. 6h, i since mostparticles are mainly formed after the first half of the process. Dueto the rapid increase of the temperature, the supersaturationdeclines for the next 10 ms. Then, after 40 ms, again a sufficientdegree of supersaturation is attained inducing a second nuclea-tion period. The number of nucleated particles exceeds thepreviously formed ones by two orders of magnitude since theavailable silicon mass in the gas phase is much higher this time. Inthe remaining iterations, the characteristic values are fine-tunedresulting in a final temperature profile and precursor mass(Fig. 6j) equivalent to the basic set of parameters. After 80iterations, the objective function has dropped to a value of9.9�10�1 indicating that a very good agreement with theprescribed data is attained (see Table A2). The correspondingevolution of the particle size distribution and the final PSD areshown in Fig. 6k, l.

Starting from a completely different trajectory, the optimisa-tion routine has finally determined a temperature profile which issimilar to the standard shape used in the simulations. The highlystable convergence indicates that the determined temperaturehistories are in good agreement with the effective thermalinfluences on the particles in the reactor. Therefore, the consid-ered model uncertainties on the process have been analysed

0 0.05 0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10−8

rel. residence time

silic

on m

ass

in th

e ga

s ph

ase

[kg] residence time 0.08 s

residence time 0.42 s

0 0.05 0.1 0.15 0.2 0.25 0.30

1

2

3

4

5

6

7x 10−5

rel. residence time

sila

n m

ass

[kg]


0 0.05 0.1 0.15 0.2 0.25 0.30

1

2

3

4

5

6

7x 1016

rel. residence time

nucl

eatio

n ra

te Bn,Si [

1/m

3 s]


0 0.05 0.1 0.15 0.2 0.25 0.30

0.5

1

1.5

2

2.5x 10−7

rel. residence time

grow

th ra

te GSiH

4 [m

/s]


Fig. 4. Process characteristics obtained for the two residence times using F80 and F420. (a) Silicon mass in the gas phase, (b) silane mass, (c) homogeneous nucleation rate,

and (d) growth rate due to surface reaction.

0 0.02 0.04 0.06 0.08

800

1000

1200

1400

residence time [s]

tem

pera

ture

[K]

profile for standard process conditionsvariable profile with five control nodes

Fig. 5. Temperature profile of the base case and spline interpolation of a variable

temperature profile using five nodes.


under the present assumptions and the findings of Kormer et al.(2010a,b) are consolidated as far as possible by the statedoptimisation results.

4.3. Control of the particle size distribution to predefined target

configurations

Besides the resolution of potential model uncertainties, thepossibility of controlling the simulation process to predefinedtarget configurations is of special interest for industrial applica-tions. In this section the inverse problem of finding the optimal

values of the process variables for a given PSD via the optimisa-tion routine is addressed.

After having identified process conditions which result inparticles with a mean diameter of 27 nm, the main challenge isto produce particle size distributions with a small predefinedstandard deviation and a specific mean diameter. Two distinctobjectives are therefore defined aiming at a PSD with a theoreticalGSD of 1.0 and a mean particle diameter of 10 nm and 60 nm,respectively. Subject to the optimisation, the correspondingoptimal process conditions are found by solving the inverseproblem. Additionally, in a next step the opposite direction ofproducing broad PSDs with a geometric standard deviation of2.0 is addressed. The related process conditions, obtained againby the optimisation method, help to analyse the underlyingprocess mechanisms. This work schedule not only demonstratesthe flexibility of the presented optimisation method, it also showsup new insights on how the reaction conditions can be improvedin the future.

Beginning with the small particles, the predefined character-istic properties ½xn

¼ 10 nm, Nn

total ¼ 2:4� 1015 m�3, sngeo ¼ 1:0� are

set in the objective function J. All target values are identified by anasterisk, specifying the prescribed properties of the final PSD inthe optimisation process. Again, the distance of the currentsolution to the specific target properties is to be minimised.Therefore, the underlying trust-region algorithm varies the tem-perature profile defined by the five nodes shown in Fig. 5 in a waywhich reduces the stated objective function.

An optimisation run attained in 99 iterations the minimal valueof the objective function for the narrow distribution corresponding

Fig. 6. Evolution of the number density distributions q0 in the optimisation process varying the temperature profile (left column) and the initial silane mass m0SiH4

(ptotal ¼ 25 mbar, tmax ¼ 80 ms). (a)–(c) Iteration: 0 m0SiH4¼ 5� 10�6 kg (d)–(f) iteration: 10 m0

SiH4¼ 1:51� 10�5 kg, (g)–(i) iteration: 20 m0

SiH4¼ 4:54� 10�5 kg,

(j)–(l) iteration: 80 m0SiH4¼ 5:97� 10�5 kg.


to the set of characteristic values ½x ¼ 10:5 nm, Ntotal ¼ 3:47�1015 m�3, sgeo ¼ 1:01�. The required mean particle diameter of10 nm and the low geometric standard deviation have apparentlybeen met; whereas the total number of particles exceeds theintended value. In the optimisation process, this deviation isusually an undesirable result; however, from a practical point ofview, an additional benefit with respect to the overall throughputis attained. The intended use of prescribing the total number of

particles though consists in avoiding the regime where particlesmay aggregate. For the present reaction conditions, a total numberof 5�1015 m�3 particles has been identified experimentally as anupper limit on the considered time scale.

Fig. 7 shows the temporal evolution of the PSD in the reactor.After 40 ms the final shape of the distribution is already attainedindicating that the nucleation has stopped and the particles willonly grow for the remaining time. Attention should be paid to the

0 2 4 6 8 10 12 140

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4


q 0(x) [

nm−1

]

t=0.04st=0.06st=0.08s

Fig. 7. Evolution of narrow number density distribution q0 in time corresponding

to the final values ½x ¼ 10:5 nm, Ntotal ¼ 3:47� 1015 m�3 , sgeo ¼ 1:01�.

0 20 40 60 80 100 12010−6

10−4

10−2

100

102

104

106

number of iterations

valu

e of

the

cost

func

tiona

l J

Fig. 8. Exemplary convergence behaviour of the optimisation method aiming for

the prescribed values ½xn¼ 10 nm, Nn

total ¼ 2:4� 1015 m�3 , sngeo ¼ 2:0�.


shift of the very sharp profile since the particle size distributionwould generally broaden due to a continuous numerical disper-sion. The used finite volume scheme was adapted to handle thiscase appropriately such that the peak of the distribution remainsalmost at the same height.

As outlined before, the next step consists in the aim to alterthe process conditions such that a broad particle size distributionwith an equal mean particle diameter of 10 nm is attained.Therefore, the objective function is changed in order to obtainthe following characteristic values ½xn

¼ 10 nm, Nn

total ¼ 2:4�1015 m�3, sn

geo ¼ 2:0�. An optimisation run attained after 115iterations almost perfectly the specified target. Using the identi-fied process conditions, the following set of characteristic values½x ¼ 10:0 nm, Ntotal ¼ 2:42� 1015 m�3, sgeo ¼ 2:00� was obtained.Remarkably, only 320 simulation runs have been performed inthe optimisation for this purpose which is less than threeevaluations per iteration. This number is notably low in order tobuild up a new model including the computation of a feasibletrust-region. Compared to other TR-methods, the algorithmpropagates thus in a very efficient way which saves in averagemore than 80% of computational time compared to standardMATLAB routines. Exemplary, Fig. 8 shows the convergencebehaviour for the previously mentioned optimisation run attain-ing a geometric standard deviation of 2.0.

Obviously, a suitable set of process conditions can be found forboth cases. This puts forward the question of what is thedifference between the two strategies. Fig. 9 shows the finalparticle number concentration along with the correspondingtemperature profiles obtained in the optimisation. For a detailedanalysis of the underlying process mechanisms, the nucleationrate as well as the growth rate due to surface reaction is depictedbelow. As it can be clearly seen, the main difference in producinga narrow size distribution instead of a broad one is due to asuccessful separation of the nucleation and growth process. Thispartition is achieved since the first part of the reactor is operatedat a low temperature in the first case. Thus, the nucleation processcan take place for a long time span in absence of growth,maintaining an extremely narrow PSD for even 50% of theresidence time. The final increase of the temperature sustainsthe growth of the particles until they reach their required meandiameter.

In the case of a high prescribed geometric standard deviationboth processes occur right at the beginning in parallel. Therefore,

the primarily formed particles grow while nucleation takes con-tinually place spreading the particle size distribution to a largersize range. When nucleation has ceased after about 30 ms, theavailable silane concentration has not been used up completely.Further growth was inhibited in the optimisation by a decreasein the temperature in order to comply with the requirementof a small mean diameter. It has to be mentioned that particlegrowth may not only be limited in practice by a decrease in thetemperature alone but also by a reduced initial precursor mass.

Conclusively, the main achievement of the optimisation con-sists in the ability of controlling the mechanisms in the reactorsuch that both cases are met. Starting from identical conditions,the optimiser identified two temperature profiles which result intwo PSDs with very different standard deviations. However, therestrictions of the present reactor set-up, i.e. the maximumtemperature of 1200 1C and the common shape of all temperatureprofiles, limit the possibilities of an experimental validation.

In a next step, the opposite test case is intended to produceparticles with a mean diameter of 60 nm specified in the objectivefunction by the target values ½xn

¼ 60 nm, Nn

total ¼ 2:4� 1015 m�3,sn

geo ¼ 1:0�. First optimisation runs revealed that the requestedfinal properties cannot be met under the prescribed conditions bysolely adjusting the temperature profile. Therefore, besides thefive parameters for the temperature spline, the precursor con-centration pSiH4

as well as the total system pressure ptotal havesubsequently been subjected to optimisation.

An optimisation run attained in 35 iterations the minimal valueof the objective function for the narrow distribution (see Fig. 10)corresponding to the set of characteristic values ½x ¼ 59:6 nm,Ntotal ¼ 5:90� 1014 m�3, sgeo ¼ 1:05� using pSiH4

¼ 2:6 mbar andptotal ¼ 15 mbar. While the mean diameter has been perfectlymet, the total number of particles as well as the geometricstandard deviation differs from the requested values. Therefore, itmust be suspected that the present means to influence the processare not sufficient to attain the prescribed PSD.

Fig. 10 reveals that after 20 ms the main mechanisms havealready ceased sustaining the particle formation process. Conse-quently, the remaining temperature profile has only little influ-ence upon the final properties of the distribution. The elevatedprecursor concentration enforces both, nucleation and growth,such that the slowly varying temperature profile in the first stageof the reactor is not capable of separating the processes in asatisfactory manner.

0 10 20 30 400

0.050.1

0.150.2

0.250.3

0.350.4


q 0(x) [

nm−1

]

0 10 20 30 400

0.01

0.02

0.03

0.04

0.05


q 0(x) [

nm−1

]0 0.02 0.04 0.06 0.08

1100115012001250130013501400145015001550

time [s]

tem

pera

ture

[K]

0 0.02 0.04 0.06 0.08700

800

900

1000

1100

1200

1300

1400

time [s]

tem

pera

ture

[K]

0 0.02 0.04 0.060

2

4

6x 1017

nucl

eatio

n ra

te Bn,Si [1

/m3 s

]

time [s]0.08

0

1

2

3x 10−7

grow

th ra

te GSiH

4 [m

/s]

0 0.02 0.04 0.06 0.080

1

2

3x 1017

nucl

eatio

n ra

te Bn,Si [1

/m3 s

]

time [s]

0

2

4

6x 10−7

grow

th ra

te GSiH

4 [m

/s]

Fig. 9. Comparison between narrow and broad size distributions for the case of small particles showing the final PSDs, the computed temperature profile as well as the

correlated nucleation and growth rate.


Nevertheless, the final process parameters yield a considerablylower standard deviation in comparison to the case where a broadparticle size distribution is objected. The latter was obtained withan objective function using the following target properties½xn¼ 60 nm, Nn

total ¼ 2:4� 1015 m�3, sngeo ¼ 2:0�. The correspond-

ing final PSD is shown on the top right of Fig. 10. In this casethe optimisation routine generated after 42 iterations the follow-ing set of characteristic values ½x ¼ 60:2 nm, Ntotal ¼ 2:82�1014 m�3, sgeo ¼ 1:15�. The identified process conditions consistin the temperature profile depicted in the subfigure belowtogether with the values pSiH4

¼ 1:3 mbar and ptotal ¼ 20 mbar.Recapitulating the analysis of the first test case, it has been

observed that a successful separation of the nucleation andgrowth processes allow for the production of a narrow sizedistribution instead of a broad one. In principle, the samebehaviour is encountered in the present case. The optimisationmethod consequently altered the corresponding process condi-tions in a manner that the broadening of the PSD is reduced.However, the lower temperature in the first part of the reactor iscapable of inhibiting particle growth only to a certain extend. This

limitation is due to the previously mentioned increase of theinitial silane concentration which is required for the production ofa PSD with a mean particle diameter of 60 nm. Therefore, acomplete detachment of both processes is not possible in thissetting. In consequence, a comparably narrow PSD as for the caseof the small particles may not be obtained under the consideredprocess conditions.

5. Conclusion

Conclusively, the discussed process control mechanisms pro-vide a deeper insight and understanding of the silane pyrolysis inthe framework of the considered hot-wall aerosol reactor.Although the computed process parameters still have to bevalidated against experiments, the principal statements open upnew approaches in choosing appropriate operating conditions forthe production of predefined particle size distributions. The appliednumerical scheme proved to be consistent with the experimentsand turned out to be very efficient and flexible. In combination

40 50 60 70 80 900

0.0020.0040.0060.008

0.010.0120.0140.016


q 0(x) [

nm−1

]

40 50 60 70 80 900

0.0020.0040.0060.008

0.010.0120.0140.016


q 0(x) [

nm−1

]0 0.01 0.02 0.03 0.04

700800900

1000110012001300140015001600

time [s]

tem

pera

ture

[K]

0 0.01 0.02 0.03 0.04700800900

1000110012001300140015001600

time [s]te

mpe

ratu

re [K

]

0 0.01 0.02 0.03 0.040

0.5

1

1.5

2

2.5x 1017

nucl

eatio

n ra

te Bn,Si [1

/m3 s

]

time [s]

0

0.2

0.4

0.6

0.8

1x 10−5

grow

th ra

te GSiH

4 [m

/s]

0 0.01 0.02 0.03 0.040

0.5

1

1.5

2

2.5 x 1017

nucl

eatio

n ra

te Bn,Si [

1/m

3 s]

time [s]

0

0.2

0.4

0.6

0.8

1x 10−5

grow

th ra

te GSiH

4 [m

/s]

Fig. 10. Comparison between narrow and broad size distributions for the second test case aiming at big particles (showing the final PSD, the computed temperature profile

as well as the correlated nucleation and growth rate).


with the chosen, problem-adapted optimisation routine, a versatilerange of product properties can be achieved under the determinedoptimal process conditions. Moreover, the prevailing model uncer-tainties concerning the parametrical reaction model and thetemperature profile inside the reactor are substantiated. Theachieved results presented in this contribution open up furtherresearch possibilities for the production of tailored particle sizedistributions which yield special product properties. As a next stepfurther investigation has already been set-up, optimising the sizedistribution of quantum dots where a direct relation betweenparticle sizes and the electronic band gap exists.

Nomenclature

B
general nucleation rate (s�1) Bn,Si homogeneous nucleation rate from a
supersaturated vapour (s�1)
Dg,Si decrease of silicon mass due to
condensational growth (kg s�1)
Dg,SiH4
decrease of silane mass due to surfacereaction (kg s�1)

Dn,Si
decrease of silicon mass due to nucleation(kg s�1)
Dr,SiH4
decrease of silane mass due to gas phasereaction (kg s�1)
F1
first fitting parameter used in Eq. (4) (–) F2 second fitting parameter used in Eq. (5) (–) F80 set of fit parameters [F1,F2] for a residence
time of 80 ms
F420 set of fit parameters [F1,F2] for a residence
time of 420 ms
G general growth rate (m s�1) GSi condensational growth rate (m s�1) GSiH4
growth rate due to chemical surface reaction(m s�1)

g
normalised observation function (–) Ir,Si increase of silicon mass due to gas phase
reaction (kg s�1)
J objective function (–) k1a rate coefficient of the silane decomposition
(m3 mol�1 s�1)
KSiH4
equilibrium adsorption constant of silane(m3 mol�1)

kB
Boltzmann constant (m2 kg s�1 K�1) kp heterogeneous rate coefficient (mol m�2 s�1) m1 mass of a silicon atom (kg) MSi molecular weight of silicon (kg mol�1)

Table A1Evolution of the fitting parameters F1 and F2 in the optimisation run together with the value of the objective function J and the corresponding characteristic values Ntotal, x ,

and sgeo (m�3)/(nm)/(�) of each data set.

Iteration Design variables Final characteristic properties

F1 (–) F2 (–) Data set 1 (900 1C) (m�3)/(nm)/(–) Data set 2 (1000 1C) (m�3)/(nm)/(–) Data set 3 (1100 1C) (m�3)/(nm)/(–) J (–)

0 1.3�10�3 70.0 N¼1.1�1016 N¼1.1�1016 N¼1.1�1016 3.0�106

x ¼ 16:2 x ¼ 16:2 x ¼ 16:3

s¼ 1:09 s¼ 1:09 s¼ 1:09

5 2.0�10�4 29.5 N¼3.4�1015 N¼2.2�1015 N¼1.0�1015 5.3�103

x ¼ 23:6 x ¼ 28:0 x ¼ 36:1

s¼ 1:06 s¼ 1:06 s¼ 1:05

14 2.2�10�4 37.4 N¼3.2�1015 N¼2.2�1015 N¼1.2�1015 4.7�103

x ¼ 24:7 x ¼ 28:1 x ¼ 34:5

s¼ 1:06 s¼ 1:06 s¼ 1:06

Table A2

Evolution of the process variables in the optimisation. The entries correspond to Fig. 6 showing the data points of spline interpolation T1�T5, the initial silane mass m0SiH4

,

the characteristic final values [Ntotal, x , sgeo] as well as the value of the objective function J.

Iteration Design variables Final characteristic properties

½T1 ,T2 ,T3 ,T4 ,T5� (K) m0SiH4

(kg) Ntotal (m�3) x (nm) sgeo (–) J (–)

0 [1350, 900, 950, 1050, 1200] 5.0�10�6 1.7�1016 1.3 1.22 3.8�106

10 [700, 896, 751, 1500, 1500] 1.5�10�5 9.9�1013 15.6 1.04 1.1�105

20 [749, 1401, 1465, 1394, 1034] 4.5�10�5 2.3�1015 22.3 1.14 3.5�103

40 [1069, 1356, 1395, 1341, 700] 4.8�10�5 2.3�1015 25.6 1.08 4.9�102

80 [1151, 1449, 1471, 1473, 761] 6.0�10�5 2.4�1015 27.2 1.10 9.9�10�1


MSiH4
molecular weight of silane (kg mol�1)
mSi
silicon mass in the gas phase (kg)
m0Si

initial silicon mass (kg)

mSiH4
precursor mass (kg)
m0SiH4

initial precursor mass (kg)

n1,S
silicon atom number concentration for asaturated vapour (m�3)
Ntotal
total particle number density (m�3)
Nn

total
predefined total particle number density(m�3)
pSi,sat
saturation vapour pressure of silicon(kg m�1 s�2)
pSi
silicon partial pressure (kg m�1 s�2) q0 number density distribution (m�3)
Rd
film growth rate (m s�1) S supersaturation (–) t residence time (s) ½tmin,tmax� temporal domain of computation (s)
T
current process temperature (K) ½T1,T2,T3,T4,T5� control nodes of temperature spline
interpolation (K)
u general set of design variables V current control volume (m3) V1 volume of a silicon atom (m3) x diameter of a particle (m) x mean particle diameter (m)
xn
predefined final mean particle diameter (m)
x1
diameter of a silicon atom (m) xc critical cluster diameter (m) y number distribution of particles (–) yn predefined final number distribution of
particles (–)
y0 initial seed particles (–)
dxc
Kronecker-delta(–)
g
temperature dependent surface energy(kg s�2)
L
general set of constraints on design variables O spatial domain of computation (m) rSi density of silicon atom (kg m�3)
sgeo
geometric standard deviation (–)
sngeo
predefined final geometric standard deviation
(–)

Acknowledgment

The authors gratefully acknowledge the funding of theDeutsche Forschungsgemeinschaft (DFG) through the Cluster ofExcellence ‘‘Engineering of Advanced Materials’’ and jointly by theGrant LE 595/23 within the priority program DFG-pp 1253‘‘Optimization with Partial Differential Equations’’.

Appendix A

The evolution of the fitting parameters in the optimization ofrate constants is shown in Table A1. Table A2 summarises theevolution of the process variables in the optimisation analysingthe temperature profile in the base case.

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process control strategies for the gas phase synthesis of silicon nanoparticles

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