Parameter estimation and statistical analysis of a new model for silicon nanoparticle synthesis

Transcription

1 Parameter estimation and statistical analysis of a new model for silicon nanoparticle synthesis Computational Modelling Group Department of Chemical Engineering and Biotechnology University of Cambridge 31 July 2013 GROUP 1

2 Contents 1 Introduction 2 Model development 3 Parameter estimation 4 Conclusions GROUP 2

3 Introduction: significance of Si nanoparticles Si nanoparticles are usually obtained from the thermal decomposition of silane SiH 4 Si + 2H 2 Are used for mainly research purposes Medical imaging (as quantum dots) Printable semiconductor inks Future touchscreen devices Properties of Si nanoparticles are strongly dependent on size and morphology May et al (2012) GROUP 3

4 Introduction: particle synthesis Particles grow through complex physiochemical pathways M gas phase clustering Sii H j H H Si i H j inception SiH 4 H 2 SiH 2 SiH 2 Si 2 H 4 H H gas phase decompostion H 2 H 2 Si i H j hydrogen release H 2 surface reaction sintering coagulation H 2 GROUP 4

5 Introduction: parameter estimation Particle synthesis models may have unknown/uncertain parameters Inverse problem: How do we choose these parameters? How do we solve this problem for a complex non-linear model? Which parameters are actually important? GROUP 5

6 Introduction: key aims 1 Present model for silicon nanoparticle synthesis Gas-phase kinetic model Particle population balance 2 Discuss parameter estimation efforts How do we estimate the parameters? Which parameters is the model most sensitive to? GROUP 6

7 Model: fully-coupled model The model consists of a gas-phase kinetic model fully-coupled with a particle population balance gas phase clustering (rev) A 2 H3SiSiH +SiH 2 H SiH 2 A 1 2 SiH 4 thermal decomposition A SiH 7 +H 2 2 +SiH +SiH A A A A SiH 5 3 +SiH 2 H 2 SiSiH 2 2 Si 2 H 6 + SiH Si 3 H 8 2 (rev) A 4 inception hydrogen condensation +SiH release 4 A H2 +SiH 2 A SR,SiH 4 H surface reaction 2 particle population balance coagulation sintering GROUP 7

8 Model: gas-phase The gas-phase model contains 8 reactions and 11 species This system of ODEs is solved with a conventional solver (rev) A 2 H3SiSiH +SiH 2 H SiH 2 2 +SiH +SiH A A A A SiH 5 3 +SiH 2 H 2 SiSiH 2 2 Si 2 H 6 + SiH Si 3 H 8 2 (rev) A 4 A 1 SiH 4 thermal decomposition A SiH 7 +H 2 2 GROUP 8

9 Model: particle type-space Each particle P q is represented as P q = P q ( p1,..., p nq, C ) Primaries are described by the number of silicon (η Si ) and hydrogen (η H ) units p x = p x (η Si, η H ) E.g., a particle with 4 primaries P q (p 1, p 2, p 3, p 4, C q ) p 1 p 3 p 2 C 12 p 3 (η Si, η H ) p 4 GROUP 9

10 Model: particle population balance The particle population balance includes terms for inception, heterogeneous growth, coagulation and sintering hydrogen release condensation +SiH 4 inception A H2 +SiH 2 A SR,SiH 4 H surface reaction 2 coagulation sintering GROUP 10

11 Model: type-space and coagulation The particle model is needed to capture process details For example, coagulation joins the particle tree structures P q (p 1, p 2, C q ) P r (p 3, p 4, C r ) P s (p 1, p 2, p 3, p 4, C s ) p 1 p 2 + p 4 p 3 p 1 p 2 p 4 p 3 GROUP 11

12 Model: population balance solver A stochastic solver is used Enables use of the detailed particle model Takes advantage of a number of developments 1 LPDA: accelerates surface reaction processes 2 Binary tree: reduces summation time over the ensemble 3 Majorant rates and fictitious jumps GROUP 12

13 Parameter estimation: the adjusted parameters Previous studies and preliminary work identified two areas with significant uncertainty gas-phase rates (A 1,LP, A 2,LP, A 3,LP, A 5,LP, A 8,rev ) heterogeneous growth rates (A SR,SiH4, A H2 ) Seven parameters were adjusted, giving parameter vector x x = (A 1,LP, A 2,LP, A 3,LP, A 5,LP, A 8,rev, A SR,SiH4, A H2 ) GROUP 13

14 Parameter estimation: case studies The model was tested against a range of experimental studies initial silane fraction, % Onischuk et al., 2000 Koermer et al., 2010 Frenklach et al., 1996 Wu et al., 1987 Flint + Haggerty, 1987 Nguyen and Flagan, temperature, o C GROUP 14

15 Parameter estimation: objective function The objective function for this system was defined as N exp ( Φ(x) = φ exp i i=1 φ sim i (x) ) 2 where φ i represents one of the N exp experimental datasets used for fitting We considered two forms of φ i : φ µ i : mean or mode of a particle size distribution : geometric standard deviation of a PSD φ σ i GROUP 15

16 Parameter estimation: initial optimisation 1 Locate parameter sets near the minimum of Φ(x) using low discrepancy (Sobol) sequences 2 Refine initial guess using SPSA algorithm 3 Select optimal parameter set x 4 Construct a response surface (surrogate model) around x 5 Conduct MCMC sampling to estimate the posterior distribution of parameters 6 Obtain confidence intervals for parameters 7 Evaluate model at mode of confidence interval GROUP 16

17 Parameter estimation: visualised A 1,LP 1. Low discrepancy sequence scan SPSA 3. x found A 2,LP 4. Make response surface sim φ A 2,LP x* A 1,LP 5. MCMC sampling of surface 6. Obtain parameter CIs probability density lb ub A 1,LP GROUP 17

18 Parameter estimation: model results 7. Evaluated model at mode of parameter confidence intervals Generally good agreement with experimental measurements kernel density, 1/nm spherical particles kernel density, 1000/nm aggregate particles exp. sim primary diameter, nm mobility diameter, nm GROUP 18

19 Parameter estimation: HDMR response surfaces The global sensitivities can be determined from a High Dimensional Model Representation (HDMR) Decomposes a response φ sim into a function of the parameters x i φ sim f (x) = f 0 + N param fi (x i ) }{{} first-order terms + N param i=1 We truncate f (x) to second-order terms N param j=i+1 f ij (x i, x j ) } {{ } second-order terms GROUP 19

20 Parameter estimation: HDMR sensitivities The first-order absolute sensitivity is given by the variance of the response function f (x i ) integrated over specified upperand lower-bounds S φ sim (x i ) = u.b. l.b. f 2 i (x i )dx i These tell us how much variance in output response φ sim can be achieved by adjusting the parameter x i GROUP 20

21 Parameter estimation: sensitivity analysis First-order sensitivities for the PSD mode objective function 40 S (Ai) from φ µ, nm Flint & Haggerty, 1986; φ µ 14 Körmer et al., 2010; φ µ 8 Nguyen & Flagan, 1991; φ µ 18 Frenklach et al., 1996; φ µ 10 Wu et al, 1987; φ µ 13 Onischuk et al., 2000; φ µ A 1,LP A 2,LP A 3,LP A 5,LP A 8,rev A SR,SiH4 A H2 parameter type GROUP 21

22 Parameter estimation: bubble plots (1) Global sensitivities of the initial decomposition step pre-exponential (A 1,LP ) to different experimental cases 10 1 S (A 1,LP ) from φ µ S (A 1,LP ) from φ σ initial silane pressure, kpa temperature, C Körmer et al., 2010 Frenklach et al., 1996 Wu et al, 1987 Flint & Haggerty, temperature, C Nguyen & Flagan, 1991 Onischuk et al., 2000 GROUP 22

23 Parameter estimation: bubble plots (2) Global sensitivities of the surface reaction pre-exponential (A SR,SiH4 ) to different experimental cases 10 1 S (A SR,SiH4 ) from φ µ S (A SR,SiH4 ) from φ σ initial silane pressure, kpa temperature, C Körmer et al., 2010 Frenklach et al., 1996 Wu et al, 1987 Flint & Haggerty, temperature, C Nguyen & Flagan, 1991 Onischuk et al., 2000 GROUP 23

24 Parameter estimation: next steps Bar chart indicated that A 1,LP, A 3,LP and A SR,SiH4 are the most important parameters Can we use fewer parameters for the parameter estimation? Bubble plots tell us which parameters influence which experiments Which experiments pull optimals in which directions? Which experiments (or their model representation) are poor? Answering these questions embarks us on a model and experimental discrimination pathway GROUP 24

25 Conclusions Parameter estimation for a detailed model is challenging! For models with many parameters, a systematic estimation procedure is necessary Global sensitivities from HDMR can potentially yield physical insight about the model These tools will be used for model and experimental discrimination GROUP 25

26 Acknowledgements The Computational Modelling Group, University of Cambridge GROUP Cambridge Australia Trust GROUP 26