In situ nanoparticle size measurements of gas-borne silicon nanoparticles by time-resolved...

In situ nanoparticle size measurements of gas-borne siliconnanoparticles by time-resolved laser-induced incandescence

T. A. Sipkens • R. Mansmann • K. J. Daun •

N. Petermann • J. T. Titantah • M. Karttunen •

H. Wiggers • T. Dreier • C. Schulz

Received: 13 September 2013 / Accepted: 8 December 2013 / Published online: 19 December 2013

� Springer-Verlag Berlin Heidelberg 2013

Abstract This paper describes the application of time-

resolved laser-induced incandescence (TiRe-LII), a com-

bustion diagnostic used mainly for measuring soot primary

particles, to size silicon nanoparticles formed within a

plasma reactor. Inferring nanoparticle sizes from TiRe-LII

data requires knowledge of the heat transfer through which

the laser-heated nanoparticles equilibrate with their sur-

roundings. Models of the free molecular conduction and

evaporation are derived, including a thermal accommoda-

tion coefficient found through molecular dynamics. The

model is used to analyze TiRe-LII measurements made on

silicon nanoparticles synthesized in a low-pressure plasma

reactor containing argon and hydrogen. Nanoparticle sizes

inferred from the TiRe-LII data agree with the results of a

Brunauer–Emmett–Teller analysis.

List of symbols

cg,t Thermal molecular speed of the gas at equilibrium

(m s-1)

co Speed of light in a vacuum (2.998 9 108 m s-1)

cp Specific heat of the nanoparticle (J kg-1 K-1)

cv,t Thermal speed of evaporating atoms (m s-1)

dp Nanoparticle diameter (nm)

E(mk) Complex absorption function

h Planck’s constant (6.626 9 10-34 J s)

DHv Heat of vaporization (J mol-1)

Ib,k Spectral blackbody intensity (W)

Jevap Evaporating mass flux (kg s-1)

Jk Spectral incandescence (a.u.)

kB Boltzmann constant

(1.38 9 10-23 J molecule-1 K-1)

mv Mass of vaporized atoms (kg)

mg Molecular mass of the gas (kg)

mk Complex index of refraction

Ng00 Incident number flux of gas molecules

Nv00 Number flux of vaporized atoms

ng Number density of gas molecules

nv Number density of evaporated vapor

P(dp) Probability density of particle diameters

pg Gas partial pressure (Pa)

pv Vapor pressure (Pa)

Qabs,k Spectral absorption efficiency

qcond Conduction heat transfer (W)

qevap Evaporation heat transfer (W)

qrad Radiation heat transfer (W)

R Universal gas constant (8.314 J mol-1 K-1)

Rs Specific gas constant (J kg-1 K-1)

Tcr Critical temperature of liquid silicon (K)

Teff Pyrometrically defined effective temperature (K)

Tg Gas temperature (K)

T. A. Sipkens � K. J. Daun (&)

Department of Mechanical and Mechatronics Engineering,

University of Waterloo, 200 University Ave West, Waterloo,

ON N2L 3G1, Canada

e-mail: [email protected]

R. Mansmann � N. Petermann � H. Wiggers � T. Dreier �C. Schulz

Institute for Combustion and Gas Dynamics – Reactive Fluids

(IVG), University of Duisburg-Essen, Duisburg, Germany

R. Mansmann � N. Petermann � H. Wiggers � T. Dreier �C. Schulz

Center for Nanointegration Duisburg-Essen (CENIDE),

University of Duisburg-Essen, Duisburg, Germany

J. T. Titantah � M. Karttunen

Department of Applied Mathematics, Western University,

London, ON, Canada

M. Karttunen

Department of Chemistry, University of Waterloo, Waterloo,

ON, Canada

123

Appl. Phys. B (2014) 116:623–636

DOI 10.1007/s00340-013-5745-2

ti Discrete time (ns)

Ti Initial temperature (K)

Tm Melting temperature of silicon (K)

Tp Nanoparticle temperature (K)

Ts Surface temperature (K)

Uij Interatomic potential between atoms i and j (eV)

v1 Incident gas velocity (m s-1)

v2 Scattering gas velocity (m s-1)

vxy Gas atom velocity parallel to surface (m s-1)

vz Gas atom velocity perpendicular to surface

(m s-1)

x Particle size parameter

X Uniformly distributed random number

a Thermal accommodation coefficient

d Tolman length (nm)

c Specific heat ratio

cs Surface tension of silicon (N m-1)

k Wavelength (nm)

l Ratio of gas atom mass to surface atom mass

q Nanoparticle density (kg m-3)

n Sticking coefficient

1 Introduction

The unique electromagnetic properties of silicon nanopar-

ticles have led to a multitude of existing and emerging

roles in diverse areas of science and engineering. In med-

icine, for example, silicon nanoparticles may be used for

biomedical diagnostics, targeted drug delivery, cancer

therapy, cell tracking and labeling, and tissue engineering

[1]. Photovoltaic device performance has undergone a

paradigm shift with the introduction of nanoscale films

containing silicon quantum dots, which can greatly

increase photoelectric conversion efficiency [2]. Silicon

nanoparticles also enhance the performance of other elec-

tronic equipment, including lithium-ion batteries [3], solid-

state devices, LEDs, and printable electronics [4]. Gas-

phase synthesis is the most economical route for mass

production of silicon nanoparticles, but since the electro-

magnetic properties of silicon nanoparticles depend

strongly on their size, these reactors must be designed and

operated to produce nanoparticles having a highly con-

trolled size distribution. Accordingly, there is a pressing

need for an instrument that can make temporally and spa-

tially resolved size measurements within the reactor to

elucidate nanoparticle formation and growth mechanisms,

pinpoint production problems, and eventually provide

feedback for closed-loop control.

Time-resolved laser-induced incandescence (TiRe-LII),

a combustion diagnostic normally used to measure the

volume fraction and size of soot primary particles, is a

promising candidate to fulfill this need. In this technique, a

laser pulse heats the nanoparticles within a sample volume

of aerosol to incandescent temperatures. Following the

laser pulse, the spectral incandescence is measured as the

nanoparticles equilibrate with their surroundings. Since

larger nanoparticles cool more slowly than smaller nano-

particles, in principle the average size, and to some extent

the size distribution, can be inferred from the observed

spectral incandescence decay.

While TiRe-LII was initially conceived to measure the

size and concentration of primary soot particles (e.g. [5–

8]), several studies have investigated the feasibility of

using this technique to size synthetic nanoparticles. Vander

Wal et al. [9] first showed this approach could be viable for

sizing metallic nanoparticles based on observed incandes-

cence decay data, but they did not use the technique to

recover particle sizes. Subsequent studies assessed the

ability to extend TiRe-LII to size metal aerosols containing

Ag [10], Fe [11–15], Mo [16, 17], and Ni [18] nanoparti-

cles and oxide aerosols containing MgO [19], TiO2 [20,

21], Fe2O3 [22], and SiO2 [23] with varying success. The

unique challenges associated with TiRe-LII measurements

on synthetic nanoaerosols are due to the following: lower

vaporization temperatures compared to carbonaceous

nanoparticles, resulting in comparably weak signals [15];

high-temperature chemistry that may change the chemical

composition of the nanoparticles [22]; and non-incandes-

cent laser-induced emission from excited fragments and

potential plasmas [9]. In this context, silicon is a promising

material due to its high boiling point and because elemental

silicon is chemically stable at high temperatures.

Despite the growing interest in gas-phase-synthesized

silicon nanoparticles with well-defined properties, there

have only been two prior attempts to size silicon nano-

particles using TiRe-LII [24, 25]. While strong TiRe-LII

signals were obtained from the laser-heated Si nanoparti-

cles in low-pressure plasmas, the subsequent analysis

neglected the 1/k dependence of emission and absorption

efficiency in the Rayleigh regime [26], as well as the

contribution of evaporation to nanoparticle cooling, which

has been shown to be very important in low-pressure

aerosols [27].

This paper presents pioneering experimental and theo-

retical work aimed at extending the capabilities of TiRe-LII

to silicon nanoparticles. The paper describes the procedure

and instrumentation used to collect the TiRe-LII data and

briefly introduces the gas-phase synthesis process. The

following section presents the heat transfer model required

to analyze the TiRe-LII data, including the thermal

accommodation coefficient, a, which is obtained by

molecular dynamics for Si/Ar and Si/He, starting from first

principles by using a combination of ab initio calculations

and atomistic MD simulations to realistically model the

624 T. A. Sipkens et al.

123

gas/surface scattering that underlies a. Nanoparticle sizes

inferred from the TiRe-LII data are found to be generally

consistent with those found using Brunauer–Emmett–

Teller (BET) analysis [28] and transmission electron

microscopy (TEM) [29] on material from the same syn-

thesis process.

2 Experimental procedure

Gas-borne non-agglomerated silicon nanoparticles are

produced from silane (SiH4) in a low-pressure microwave

plasma flow reactor shown schematically in Fig. 1. The

chamber is first evacuated and then purged with argon to

remove potential contaminants (e.g., O2) that could react

with the nanoparticles. Silane is premixed with dilution

gases H2 and Ar at a pressure of 12 kPa so that the volume

ratio of the constituents is approximately 1:12:60 for SiH4,

H2, and Ar, respectively. The SiH4:H2:Ar core flow of

3.7 slm is surrounded by a Ar/H2 coflow that stabilizes the

plasma. The microwave radiation of a 1,200-W magnetron

is focused in the center of a 7.7-cm diameter quartz tube,

producing a visible purple plasma in the lower region of the

reactor shown in Fig. 2. Due to unipolar particle charging,

plasma reactors form non-aggregated, electrostatically

confined nanoparticles with a narrow nanoparticle size

distribution; the microwave plasma reactor used here is

known to produce single crystalline silicon nanoparticles

with a geometric standard deviation of approximately

rg = 1.2 and nanoparticle sizes in the 5–50 nm range,

depending on pressure and precursor concentration [30].

Time-resolved laser-induced incandescence measure-

ments are carried out 20 cm downstream from the plasma

zone using the Artium 200M TiRe-LII system shown in

Fig. 2. The instrument consists of a transmitter module

containing a 1,064-nm Nd:YAG laser and optics, a receiver

module containing collection optics and two photomulti-

plier tubes, and a computer for instrument control and data

acquisition. Optical access to the aerosol is obtained

through three quartz windows in the reactor walls. Inert gas

flushing prevents particle deposition on the windows and

allows continuous operation of the reactor for several hours.

A laser pulse is shone across the reactor chamber through

two opposite windows. The laser was operated with a rep-

etition rate of 10 Hz. A nearly uniform ‘‘top-hat’’ beam

profile with a square 2.8 mm 9 2.8 mm cross section was

generated by relay imaging an aperture into the measure-

ment location where fluences were in the 0.12–0.16 J cm-2

range. The resulting incandescence signal of the laser-

heated nanoparticles is detected through the third quartz

window, perpendicular to the laser pulse; the probe volume

is defined by intersection of the laser beam and the detector

solid angle. The incandescence signal is split by a dichroic

mirror, passed through two band-pass filters centered at

442 and 716 nm (full width at half maximum of 50 nm),

and imaged onto the photomultiplier tubes. Further details

of this procedure are provided in Ref. [6].

Fig. 1 Schematic showing the experimental procedure: (1) mixing of

inlet gases; (2) electrical discharge dissociation of silane; (3)

nanoparticle formation; (4) in situ TiRe-LII nanoparticle sizing; and

(5) ex situ BET nanoparticle sizing

Fig. 2 The stream of glowing silicon nanoparticles within the low-

pressure plasma reactor (left), and the Artium 200M LII transmitter

and receiver units arranged around the plasma reactor (right)

Time-resolved laser-induced incandescence 625

123

The spectral incandescence from the laser-heated

nanoparticles can be modeled by integrating the incan-

descence emitted by all nanoparticle sizes

JkðtiÞ ¼ Ck

Z1

0

pd2p

4Qabs;kðdpÞPðdpÞIb;k½Tpðti; dpÞ�dðdpÞ ð1Þ

where Ib,k is the blackbody spectral intensity, Ck is a

constant that depends on the optical collection efficiency,

laser fluence, and nanoparticle volume fraction, Qabs,k is

the absorption efficiency of the nanoparticles, and P(dp) is

the probability density of nanoparticle diameters. The

nanoparticle diameters are expected to be much smaller

than the laser wavelength and the principal wavelengths of

emitted radiation, and consequently, the nanoparticles emit

and absorb in the Rayleigh limit:

Qabs;kðdpÞ � �4xEðmkÞ ¼ 4pdp

kIm

m2k � 1

m2k þ 2

� �ð2Þ

where x = pdp/k is the nanoparticle size parameter and

mk = nk ? i�jk is the complex index of refraction for

molten silicon [31]. The optical constants and E(mk) are

plotted versus wavelength in Fig. 3.

The two color LII data are used to calculate a pyro-

metrically defined effective temperature

TeffðtÞ ¼hco

kB

1

k2

� 1

k1

� �1

ln KoptJk1ðtÞ

Jk2ðtÞ

� �Emk2

Emk1

� �k1

k2

� �6� � ð3Þ

where Kopt accounts for the spectral variation of window

transmissivity; this parameter was found to be Kopt = 1.01

using a deuterium lamp over ultraviolet wavelengths and a

xenon lamp over visible wavelengths. Sample incandes-

cence data (averaged over 300 shots) and corresponding

effective temperatures found using Eq. (3) is shown in

Fig. 4.

The in situ size measurements of the silicon nanoparti-

cles by TiRe-LII are complemented with the measurement

of an average nanoparticle size calculated from their spe-

cific surface area as measured by BET analysis of silicon

powder collected via a filter behind the reactor [28]. BET

infers the specific surface area of nanoparticles from the

physisorption of N2 by a sample of nanoparticle powder,

which was kept at 150 �C and under vacuum overnight to

remove residual water. Assuming that the nanoparticles are

monodisperse spheres, the specific surface area can be

converted to a representative nanoparticle diameter (cor-

responding to the Sauter mean diameter for a polydisperse

powder) based on the knowledge of the sample mass and

density. This technique is often used to determine the size

of non-aggregated nanoparticles and has also been applied

to size silicon nanoparticles produced from the reactor in

previous studies [29]. Typical measurements carried out at

CENIDE show a repeatability with \1 % variation in

nanoparticle size, while a previous study that compared

BET measurements of reference powders made indepen-

dently by several laboratories showed a variation in the

results below 5 % [32]. Although the BET analysis is done

on the nanoparticles leaving the reactor, one would not

expect the nanoparticle morphology to differ considerably

between the TiRe-LII probe volume and the nanoparticle

exit due to the strong Columbic repulsive forces between

the Si nanospheres.

3 Heat transfer modeling

In TiRe-LII, the nanoparticle size distribution is inferred by

regressing modeled spectral incandescence data, derived

from a heat transfer model via Eq. (1), to experimental

measurements. The laser-heated nanoparticles equilibrate

with the surrounding bath gas according to

0.00

0.05

0.10

0.15

0.20

0

2

4

6

8

300 400 500 600 700 800 900

E(m

λ)

n λ, κ

λ

λ [nm]

nλ

E(mλ) κλ

Fig. 3 Radiative properties of silicon [31] and calculated E(mk) as a

function of wavelength

1900

2100

2300

2500

2700

2900

3100

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 1 2 3 4

Tef

f[K

]

J λλ

/J,m

ax

t [μs]

Teff

J442nm

J716nm

Fig. 4 TiRe-LII experimental data: scaled monochromatic incandes-

cence and pyrometrically defined effective temperature


123

qcp

pd3p

6

dTp

dt¼ �qcondðt; dpÞ � qevapðt; dpÞ � qradðt; dpÞ

ð4Þ

where q, cp, Tp, and dp are the nanoparticle density, specific

heat, temperature, and diameter. The bulk density and

specific heat of silicon are taken from Refs. [33, 34],

respectively.

Since the silicon nanoparticles formed within the reac-

tors are expected to be much smaller than the mean free

molecular path in the bath gas, heat conduction takes place

in the free molecular regime so that

qcondðt; dpÞ ¼ pd2pN 00g Eo � Eih i ð5Þ

where Ng00 = cg,t�ng/4 is the incident gas molecular number

flux, ng = pg/(kBTg) is the number density, cg,t = [8kBTg/

(pmg)]1/2 is the mean thermal speed of the gas molecules,

and hEo - Eii is the average energy transferred to a gas

molecule when a gas molecule scatters from the laser-

heated nanoparticle. (As noted above, Columbic forces

between the unipolar silicon nanoparticles result in

aggregates having an open structure [29], so aggregate

shielding is expected to be negligible). The energy transfer

term is defined using the thermal accommodation

coefficient

a ¼ Eo � Eih iEo � Eih imax

ð6Þ

where hEo - Eiimax = 2kB(Tp - Tg) for a monatomic gas.

Equation (5) can be rewritten as [35]

qcondðt; dpÞ ¼ pd2p

pgcg;t

2Tg

aðTp � TgÞ ð7Þ

The thermal accommodation coefficient is found using

molecular dynamics (MD) as described in the subsequent

section of this paper. Free molecular heat conduction with

argon and hydrogen molecules was considered simulta-

neously with two conduction terms identical to Eq. (7)

except that pg is the gas partial pressure and cg,t, and a are

specific to each gas species. Although the partial pressure

of H2 is much lower than that of Ar, and a should also be

much lower for H2 compared to Ar based on the relative

molecular weights, free molecular heat conduction due to

H2 molecules is still significant as its smaller mass trans-

lates into a higher cg,t and consequently a higher incident

number flux of H2 molecules.

Evaporation in the free molecular regime is given by

qevapðt; dpÞ ¼ DHvJevap ¼ DHvpd2p

nvcv;t

4f ð8Þ

where DHv is the enthalpy of vaporization per atom, Jevap is

the evaporating mass flux, nv = pv/(kBTp) is the vapor

number density, cv,t = [8kBTp/(pmv)]1/2 is the mean thermal

speed of the vapor, mv is the atomic mass of the vaporized

species, and f is the sticking coefficient, 0 B f B 1, which

accounts for the fraction of evaporated atoms that cannot

escape the potential well and adsorb back onto the surface.

In this work, we assume f = 1, since the high surface

energy of the nanoparticle relative to the gas–surface

potential well depth precludes reabsorption of evaporated Si

atoms [15, 17]. The vaporized species is assumed to consist

entirely of atomic silicon, which is supported by mass

spectrometry measurements carried out by Tomooka et al.

[36] over temperature ranges similar to those considered in

this study. The enthalpy of vaporization, DHv, was found

using Watson’s equation [37, 38]

DHv ¼ Kð1� TrÞ0:38 ð9Þ

where K is a material constant, Tr = Tp/Tcr is the reduced

temperature, and Tcr is the critical temperature of liquid

silicon [39]. The constant can be solved using a known

reference point on the vapor dome, most often the heat of

vaporization at the boiling temperature and atmospheric

pressure. Once DHv is known, the vapor pressure, pv, can

be determined by the Clausius–Clapeyron relation [13]

lnðpvÞ ¼ �DHv

R

1

Tp

þ C ð10Þ

where R is the universal gas constant and C is a material

constant. The vapor pressure is also corrected for the

increased surface energy due to nanoparticle curvature

using the Kelvin equation [40]

pv ¼ pv;o exp4cs

dpqRsTp

� �ð11Þ

where pv,o is the vapor pressure for a flat surface, cs is the

surface tension of the nanoparticle material, taken at the

melting point [33], and Rs is the specific gas constant.

Recent studies (e.g. [41]) suggest that using the bulk value

for cs significantly underestimates the surface energy of

nanoparticles. The influence of nanoparticle size on surface

tension can be incorporated into the model using the

Tolman equation [42]

cs ¼cs;o

1þ 4d�

dp

ð12Þ

where cs,o is the bulk surface tension taken to be that of

bulk silicon at 2,000 K [33] and d is the Tolman length

taken as the atomic diameter for d/D C 20. Figure 5 shows

that this effect is negligible except for small particle sizes

(\10 nm) and at lower nanoparticle temperatures. For

moderately sized nanoparticles (*30 nm) and tempera-

tures at which evaporation is important ([2,400 K), this

model provides vapor pressures consistent with values

measured above molten silicon as reported by Tomooka


123

et al. [36] and Desai [34]. Figure 5 also shows that heat

transfer due to evaporation corresponding to the vapor

pressure from Ref. [34] is also consistent with values

obtained through Eqs. (10–12).

The nanoparticle mass lost due to evaporation is found by

integrating Jevap over the temperature decay. Since laser

heating is excluded from the heat transfer model, mass loss was

estimated by assuming the peak temperature for prevaled for

10 ns of laser heating as a worst-case scenario. This calculation

showed that the nanoparticle mass would decrease less than

15 % (5 % reduction in dp) for all nanoparticles larger than

10 nm and only 4 % (2 % reduction in dp) for 30 nm nano-

particles. This is considered low enough that nanoparticle size

was assumed to be static during the cooling process.

Finally, thermal radiation from the nanoparticles is

given by [26]

qradðt; dpÞ ¼Z1

0

pd2pQabs;kðdpÞpIb;k½Tpðt; dpÞ�dk ð13Þ

Although thermal radiation from the laser-heated nano-

particles is the basis for TiRe-LII particle sizing, Fig. 5

shows that heat transfer due to thermal radiation is several

orders of magnitude smaller than conduction and evapo-

ration; consequently, radiation is neglected for the

remainder of the analysis. Figure 5 also shows that the heat

transfer due to evaporation decays very rapidly with

decreasing temperature and becomes negligible below

2,250 K for most nanoparticle size classes.

4 Predicting the thermal accommodation coefficient

using molecular dynamics

One of the main obstacles in extending TiRe-LII to new

aerosols is that the thermal accommodation coefficient is

unknown. Most published values for a (e.g. [43]) are found

under conditions different from those encountered in TiRe-

LII and thus cannot be applied directly to TiRe-LII ana-

lysis. Consequently, in most TiRe-LII experiments, a is

inferred by comparing incandescence decay data to nano-

particle sizes found using ex situ analysis, such as TEM

[11, 15, 44–46]. These accommodation coefficients are

then used to infer nanoparticle sizes from other TiRe-LII

data, a somewhat circular process.

It has been shown, however, that molecular dynamics

(MD) can be used to predict this parameter for various gas–

surface systems [47–49]. In the present work, this approach

is used to predict a for the Si/Ar system over a range of

surface and gas temperatures. We also carry out the same

procedure for Si/He to investigate how the reduced gas

molecular mass and potential well depth influences the

accommodation coefficient.

The first step of this procedure is to define the inter-

atomic potentials. The potential energy of silicon is defined

using the Stillinger–Weber potential [50], consisting of

two-body and three-body terms

UijðrijÞ ¼1

2

Xi

Xj [ i

/2ðrijÞ þX

i

Xj 6¼i

Xk [ j

/3ðrij; rik; hijkÞ

ð14Þ

The two-body term describes the Si–Si bonding within the

crystal

/2 ¼ eA Brrij

� �p

� rrij

� �q� �exp

rrij � ar

� �ð15Þ

where rij is the distance between the ith and jth atoms. The

three-body term promotes the bond angle, hijk, between

three silicon atoms

/3 ¼ ke cos hijk � cos h0

2exp

crrij � ar

þ crrik � ar

� �

ð16Þ

Fig. 5 Conduction, evaporation, and radiation as a function of surface temperature for silicon nanoparticles at 30 nm (left); and vapor pressure

as a function of particle size using the bulk value and those corrected using Kelvin and Tolman equations (right)


123

which keeps the silicon crystal in its diamond structure

below its melting temperature. The parameterization for

the Stillinger–Weber potential [50], summarized in

Table 1, has been shown to replicate the empirically

observed melting temperature and molten density of silicon

[51]. To verify the physicality of the Stillinger–Weber

potential for this application, the simulation density was

compared to the experimental density used in the heat

transfer model over a range of temperatures important to

TiRe-LII analysis [33]. Figure 6 confirms that the density

predicted by MD is within 10 % of the experimentally

derived value.

The gas atoms interact with the surface atoms through a

pairwise Morse potential

UijðrijÞ ¼ D e�2kðrij�reÞ � 2e�kðrij�reÞh i

ð17Þ

where D, k, and re are specific to the gas–surface molecular

pair. These parameters are found by fitting superimposed

pairwise potentials to the ground state energies derived

from density functional theory (DFT) for gas atoms at

various heights above a silicon surface. The silicon surface

is represented by a 2 9 2 9 2 supercell of 64 atoms with a

lattice parameter of 0.543 nm. The present work used the

WIEN2k code [52] with the generalized gradient approxi-

mation (GGA) parameterization of the exchange and cor-

relation functionals with RKM value of 5.0- and 12-k

points in the irreducible Brillouin zone. RKM is the

product of the largest plane-wave vector and the smallest

muffin-tin radius in the system. Muffin-tin radii of

0.117 nm (for Si) and 0.106 nm (for Ar and He) were used.

The DFT-derived parameterization of the Morse potential

for the gas–surface interaction is given in Table 2 and

plotted against the raw DFT data in Fig. 7. The binding

energies shown in Fig. 7 are comparable to the 8 meV

given by Bandler et al. [53] for helium in adsorption

experiments and 20 meV given by Lau et al. [54] for argon

in bombardment experiments.

The above parameterization forms the basis of MD

simulations of gas molecules scattering from the laser-

heated silicon nanoparticle. Simulations were carried out

using LAMMPS [55]. The silicon surface is modeled using

544 silicon atoms, initially arranged in a diamond crystal

lattice. Periodic boundary conditions are applied on the

lateral surfaces (perpendicular to the free surface) of the

computational domain. The silicon is initially brought to

the specified temperature by applying the Nose–Hoover

thermostat [56, 57] (NVT ensemble) with a damping

Table 1 Parameterization for the Stillinger–Weber potential for sil-

icon [50]

e 2.17 eV

r 2.01 A

a 1.80

k 21.0

c 1.20

cos(hb) -1/3

A 7.049556277

B 0.6022245584

p 4

q 0

0.9

0.95

1

1.05

1.1

1.15

0 1000 2000 3000

ρ MD/ρexp

Ts [K]

Fig. 6 MD-derived density of molten silicon averaged over the final

5,000 timesteps of the warming procedure, normalized by the

published density over a range of surface temperatures [33]. Error

bars denote two standard deviations of the mean

Table 2 Pairwise Morse potential parameters between the gas and

silicon atoms

Si/Ar Si/He

D 4.669 (meV) 1.130 (meV)

re 4.647 (A) 4.534 (A)

k 1.256 1.398

-30

-20

-10

0

10

20

300.3 0.4 0.5 0.6 0.7

Eto

t [m

eV]

z [nm]

USi/He

USi/Ar

Fig. 7 DFT-derived ground state energies for the Si/Ar and Si/He

systems (symbols) and fitted Morse potentials (dashed lines)


123

constant of 0.1 ps for 30 ps. The simulation is continued

for 5 ps under the NVE ensemble to ensure that the system

has reached equilibrium conditions at the desired temper-

ature, during which time the surface density is also tracked.

At the conclusion of this simulation, the silicon atom

positions and velocities are stored in a restart file.

The MD simulation forms the kernel of a Monte Carlo

integration over 1,500 incident gas molecular trajectories.

Incident gas velocities are sampled from a Maxwell–

Boltzmann distribution at the prescribed gas temperature

following [47], and the silicon atomic trajectories are ini-

tialized from the restart file. The atomic trajectories are then

traced until the gas atom exceeds its initial height above the

surface. The accommodation coefficient is then found by

a ¼mg

�2 v2

2 � v21

� �2kBðTp � TgÞ

ð18Þ

where mg is the gas molecular mass, and v1 and v2 are the

incident and scattered gas molecular velocities, respec-

tively. Figure 8a shows the progression of the initial

warming process from a perfect silicon crystal lattice to an

amorphous silicon molten surface at 2,500 K, while

Fig. 8b shows an argon molecule directly scattering from

the silicon surface.

For the MD study, a surface temperature of 2,500 K and

gas temperature of 1,300 K were chosen since they are

representative of TiRe-LII conditions, following previous

MD studies [46–48]. Under these conditions, the accom-

modation coefficients for Si/He and Si/Ar were found to be

0.11 ± 0.01 and 0.36 ± 0.02, respectively, using 1,500

Monte Carlo trials. (Uncertainties correspond to two

standard deviations of the mean.) These results follow the

same general trend seen in the thermal accommodation

coefficients versus potential well depth and reduced

molecular mass calculated between monatomic gases and

metals and graphite [49], as shown in Fig. 9.

The effect of gas temperature on the average energy

transferred to the gas molecule, DE = hEo - Eii, is plotted

in Fig. 10 with Ts held at 2,500 K. The change in transla-

tional normal and tangential energy components of the gas

molecule is also plotted. While the increase in tangential

translational energy is less than the normal component, it is

larger than that observed by Daun et al. [47] between the

graphite and both the monatomic gases and N2 [48]. This is

almost certainly due to the greater surface roughness of

(a)

t = 0 ps t = 0.2 ps t = 0.5 ps t = 1 ps t = 5 ps

(b)

t = 0 ps t = 5 ps t = 6.5 ps t = 8 ps t = 9.5 ps

Fig. 8 Visualization of the molecular dynamics simulation: a a

Nose–Hoover thermostat is used to transform the silicon surface from

its initial crystal configuration to amorphous molten silicon at

2,500 K, and b an argon molecule scatters directly from the silicon

surface. Illustrated particles sizes are 70 % of the Van der Waals

diameter

Fe/Ar

Fe/HeMo/Ar

Mo/He

Ni/Ar

Gr/He

Gr/Ne

Gr/Ar Gr/KrGr/Xe

Si/Ar

Si/He

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8

α

D/kBTs

Fig. 9 MD-derived values of a versus potential well depth. Data for

graphite are from Ref. [47], values for molybdenum, iron, and nickel

are reported in Ref. [58], and values for silicon are the present

study. Error bars denote two standard deviations of the mean


123

liquid silicon compared to solid graphite; in the latter case,

the thermal motion of carbon atoms is primarily normal to

the exposed surface, so surface energy is transferred pref-

erentially into the normal translational mode of the gas

molecule. In contrast, the motion of atoms in molten Si is

comparatively unconstrained, so energy is transferred into

the normal and tangential modes. Similar trends were

observed when comparing the MD-derived normal and

tangential modes of the accommodation coefficients for

molten Fe and Ni nanoparticles with those for Mo nano-

particles, which remain solid in TiRe-LII experiments [58].

Figure 10 also shows that the average energy increase is

zero when Ts = Tg, in accordance with the 2nd Law of

Thermodynamics. The individual normal and tangential

components of translational energy also appear to follow

the same rule, suggesting that the normal and tangential

modes of the gas molecule are uncoupled.

Figure 11 shows accommodation coefficients corre-

sponding to change in gas molecular energies shown in

Fig. 10. Because the denominator of Eq. (18) becomes very

small when Ts & Tg, a quadratic curve is fit to the points in

Fig. 10 and is forced to cross zero when Ts = Tg in

accordance with the 2nd Law. Substituting this expression

into Eq. (18) gives a linear relationship between a and Tg

that is plotted in Fig. 11. The fitted curves generally lie

within the error bounds (two standard deviations of the

mean of the Monte Carlo trials) in the entire range of

considered gas temperatures.

Figure 12 shows the simulated change in energy transfer

considering surface temperatures from 200 to 3,000 K for

Tg = 300 K. This plot reveals an inflection in the normal

and tangential components of the gas molecular energies

occurring when Ts & Tmelt, represented by the vertical

dashed line in Fig. 12c. This is expected, particularly for

the tangential component, due to increased mobility of the

surface atoms in the liquid state as described above. Fig-

ure 12 shows that the thermal accommodation coefficients

can be approximated by constant values above and below

the melting temperature

aSi=Ar ¼0:100; Ts\Tm

0:347; Ts [ Tm

aSi=He ¼0:078; Ts\Tm

0:107; Ts [ Tm

: ð19Þ

5 Analysis of TiRe-LII data

The MD-derived accommodation coefficients are then used

to interpret the experimental TiRe-LII data. Silicon nano-

particle diameters are initially found by nonlinear regres-

sion of the experimental pyrometrically defined effective

temperature, Teff, to the same effective temperature derived

-2000

-1500

-1000

-500

0

500

1000

15000 1000 2000 3000 4000

Δ

Δ ΔEn

[K]

Tg [K](a)

Si/He

Si/Ar

-2000

-1500

-1000

-500

0

500

1000

15000 1000 2000 3000 4000

Et[K

]

Tg [K]

Si/ArSi/He

(b)

-2000

-1500

-1000

-500

0

500

1000

15000 1000 2000 3000 4000

E [K

]

Tg [K]

Si/Ar

Si/He

(c)

Fig. 10 Variation of DE = hEo - Eii of Si/Ar and Si/He with Tg at Ts = 2,500 K decomposed into: a the normal component; b the tangential

component; and c the sum of both components. Error bars denote two standard deviations of the mean

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1000 2000 3000 4000

α

Tg [K]

Si/Ar

Si/He

Fig. 11 Variation of a for Si/Ar and Si/He with Tg at

Ts = 2,500 K. Error bars denote two standard deviations of the mean


123

from a simulated incandescence signal found by solving

Eq. (1). Since the procedure for inferring the nanoparticle

size distribution from TiRe-LII data requires an initial

condition of uniform nanoparticle temperatures (taken to

be the peak temperature in the case of a ‘‘top-hat’’ beam

profile), and because of a smoothing effect lasting several

nanoseconds around the peak temperature, it is necessary to

extrapolate a hypothetical peak nanoparticle temperature

that would be compatible with the conduction and evapo-

ration cooling models. Accordingly, instead of the experi-

mentally observed peak temperature of 3,075 K, a

somewhat higher initial temperature, Ti = 3,100 K, was

chosen as an initial condition to account for the smoothing

effect at the peak.

A thermocouple was inserted in the central gas flow

slightly above the TiRe-LII probe volume. After correcting

for radiation losses from the probe, it indicated a gas

temperature of 1,300 K, which is consistent with temper-

atures found through planar laser-induced florescence

measurements carried out in a similar reactor [59].

Accounting for free molecular heat conduction by the H2

molecules requires knowledge of the aH2, which was not

quantified using MD due to the complexity of deriving

ab initio potentials for a polyatomic molecule. Since previ-

ous work has shown that the mass ratio and the thermal

accommodation coefficient are closely related [47], H2 was

modeled as monatomic and aH2was assigned the value of

aHe = 0.11 due to similar mass of the two gases. Uncertainty

introduced by this assumption is addressed later in the paper.

Based on detailed TEM studies of nanoparticles

extracted from the same reactor (e.g., [32]), the nanopar-

ticle sizes are known to follow a lognormal distribution

PðxÞ ¼ 1

xffiffiffiffiffiffi2pp

ln rg

exp�ðln x� ln dp;gÞ2

2ðln rgÞ2

" #ð20Þ

defined by the geometric standard deviation, rg, and geo-

metric mean particle diameter, dp,g, which is also the

median diameter for a lognormal distribution. Figure 13

compares the modeled data corresponding to the maximum

likelihood estimates (MLEs) of the lognormal distribution

parameters, dp,g = 24.2 nm and rg = 1.43, reported in

Table 3. On the other hand, if the aerosols were mono-

disperse, the MLE nanoparticle diameter is 37.3 nm. Fig-

ure 13 also shows that the monodisperse model is unable to

predict the incandescence decay at measurement times

greater than 2 ls, which one would expect as the influence

of polydisperse nanoparticle sizes on the incandescence

decay becomes more pronounced at longer cooling times

[46].

-200

0

200

400

600

800

1000

1200

1400

0 2000 4000

Δ Δ ΔEn

[K]

Ts [K]

Si/He

(a)

Si/Ar

-200

0

200

400

600

800

1000

1200

1400

0 2000 4000

Et [

K]

Ts [K]

Si/Ar

Si/He

(b)

-500

0

500

1000

1500

2000

2500

0 500 1000 1500 2000 2500 3000 3500

E [K

]

Ts [K]

Si/ArSi/He

(c)

Tm

Fig. 12 Variation of DE = hEo - Eii of Si/Ar and Si/He with surface temperature at Tg = 300 K decomposed into: a the normal component;

b the tangential component; and c the sum of both components. Error bars correspond to two standard deviations of the mean

2000

2200

2400

2600

2800

3000

3200

0 1 2 3 4

Tef

f[K

]

t [μs]

Polydisperse E

ffects

Texp

Tlognornal

Tmonodisperse

Fig. 13 Experimentally observed pyrometric temperatures (solid

line), and modeled temperature decays corresponding to the most

probable monodisperse (long dash) and lognormal (short dash)

nanoparticle size distributions. The monodisperse assumption is

incapable of reproducing the observed pyrometric temperatures at

longer cooling times due to the polydispersity of nanoparticle sizes


123

Uncertainty in the distribution parameters caused by

noise in the monochromatic incandescence measurements

(due mainly to photomultiplier shot noise) was quantified

using robust Bayesian analysis similar to the procedure

described in Ref. [60]. In this approach, the posterior

probability, P(x|b), of the hypothesized set of distribution

parameters in x = [dp,g, rg]T is defined by

P xjbð Þ ¼ P bjxð ÞPprðxÞP bð Þ ð21Þ

where P(b|x) is the likelihood of the observed data in b

occurring for a hypothetical x, Ppr(x) is the probability of x

being correct based on prior knowledge of the distribution

parameters, and P(b) scales the posterior probability so that

the Law of Total Probability is satisfied. If the spectral

incandescence data are contaminated with independent,

normally distributed error, the likelihood is given by

P bjxð Þ ¼Y

j

1ffiffiffiffiffiffiffiffiffiffi2pr2

j

q exp �Texp;j � Tmod;j

2

2r2j

( )ð22Þ

where rj is the expected standard deviation of the measured

incandescence at the jth measurement time. The standard

deviation increases at longer cooling times, as the signal-

to-noise ratio in the incandescence traces drops with

decreasing signal intensity [61]. In order to account for this

fact, rj is modeled by a quadratic function fitted to the

standard deviations of the mean of 300 independent sets of

incandescence data evaluated at every measurement time.

The prior probability is defined as

PprðxÞ ¼1; if dp;g� 0 and rg� 1

0; otherwise

ð23Þ

since neither dp,g nor r = ln(rg) can hold non-positive

values. While Eq. (19) defines a two-dimensional plot of

the probability density of x, it is more convenient to

quantify the uncertainty of a distribution parameter of

interest with a credible interval over the marginalized

probability densities of each variable. Since the marginal-

ized posterior probabilities may be asymmetric, it is useful

to define the credible interval using the highest density

region (HDR) or highest density probability [62], which

can be quantified using the density quantile approach [63].

In this case, a set of 10,000 samples, X = {x1, x2, …, xn},

is generated using a Markov Chain Monte Carlo (MCMC)

algorithm [64]; these samples are then used to form

marginalized posterior distributions for dp,g and rg through

kernel density estimation [65].

In Fig. 14, the MCMC samples are plotted over contours

of the residual norm, showing the relationship between the

samples and the region of minimal residual. The 95 %

credible intervals and mean are given in Table 3. The

maximum a posteriori (MAP) estimate of rg = 1.43

obtained by minimizing the scaled residual is typical of a

self-preserving distribution for an aerosol in which nano-

particle growth mechanisms have stabilized [66]. On the

other hand, this value is larger than the rg = 1.2 typically

found from in situ particle mass spectrometry and ex situ

TEM analysis [67]; the narrower distribution is also what

one would expect based on the Coulomb repulsion of the

charged nanoparticles [30].

The large credible intervals associated with dp,g and rg

are typical of ill-posed inverse problems, since a wide

range of nanoparticle size distributions exist that explain

the experimental data within the standard deviation due to

signal noise. Figure 14 shows that these solutions lie along

a specified thin region of minimal residual. Daun et al. [68]

demonstrated that this family of solutions approximately

shares a narrow distribution of Sauter mean diameters,

defined for a lognormal distribution as

dp;32 ¼

R10

d3pPðdpÞddp

R10

d2pPðdpÞddp

¼ dp;g exp5

2lnðrgÞ2

� �ð24Þ

Credible intervals for dp,32 are also reported in Table 3. As

one would expect, the credible interval is considerably

smaller than that found for dp,g since the ill-posedness of

Table 3 Most probable nanoparticle size distribution parameters and

credible intervals

dp,g (nm) 24.2 (22.8, 25.7)

rg 1.43 (1.39, 1.46)

dp,32 (nm) 33.2 (32.6, 33.8)

Fig. 14 MCMC samples laid over the L2 norm of the residual

between experimental and modeled data. Histograms show the

posterior distribution resulting from MCMC sampling, while the

solid line corresponds to lognormal nanoparticle sizes that share a

Sauter mean of dp,32 = 33.2 nm


123

the problem is due to the narrow curvature of the residual

function along the locus of distributions that share a

common Sauter mean diameter.

We must also consider, separately, how model param-

eter uncertainty affects the recovered nanoparticle size

distribution parameters. As noted above, the gas tempera-

ture within the probe volume is difficult to measure pre-

cisely due to the limited access afforded by the reactor

geometry, but is approximately 1,300 K based on a ther-

mocouple measurement in near the probe volume. An

uncertainty of ±200 K is assigned as a conservative esti-

mate of this uncertainty, primarily due to uncertainty in

laser position with respect to the thermocouple location.

The extrapolated initial nanoparticle temperature used in

the sizing analysis is assigned an uncertainty of ±25 K,

based on the difference between the experimentally

observed peak temperature (3,075 K) and the assumed

value (3,100 K). The thermal accommodation coefficient

for H2 is assigned a conservative uncertainty of ±50 %.

Uncertainties in q, cp, pg, cs, Tcr, and the MD-derived

thermal accommodation coefficients are taken to be 10 %

of their nominal values.

The impact of these uncertainties on the inferred size

parameters is assessed through a perturbation analysis, in

which the relative sensitivity coefficients, i.e., the product

of the local sensitivity and the nominal model parameter

value, are estimated through a central finite difference

approximation. The error in an inferred parameter, y, due to

uncertainty in a model parameter, x, can then be found

using

e yjxð Þ ¼ PE(xÞ � RSC(x; yÞ ¼ PE(xÞ � xoy

ox

� �x0

ð25Þ

where e is the error in the units of the inferred parameter,

PE is the percentage error in a model parameter, and RSC

is the relative sensitivity coefficient. Table 4 shows that

these error bounds are comparable in magnitude to the

credible intervals associated with signal noise.

Ex situ measurements made by BET analysis on a

nanoparticle powder consisting of spherical, non-aggre-

gated particles loosely connected by point contacts give an

approximate diameter of 33.3 nm. This diameter can also

be interpreted as the Sauter mean diameter since the

nanoparticle size is derived from the ensemble volume of

nanoparticles (found from the mass of the sample and bulk

density of Si), divided by the specific surface area, which is

inferred from N2 adsorption. This value is in excellent

agreement with the TiRe-LII-derived Sauter mean of

33.2 nm and lies well within the error bars generated by

measurement noise and model parameter uncertainty.

6 Conclusions

This paper describes the application of TiRe-LII to size

silicon nanoparticles within a low-pressure microwave

plasma reactor. Inferring nanoparticle sizes from TiRe-LII

data requires a model of the free molecular heat conduction

and evaporation through which the laser-heated nanopar-

ticles equilibrate with the surrounding gas. Free molecular

heat conduction depends on the thermal accommodation

coefficient, which is calculated using molecular dynamics

with DFT-derived gas/surface potentials. A parametric

study elucidated how the thermal accommodation coeffi-

cient is expected to vary over a range of gas and particle

temperatures.

The nanoparticle diameters were then inferred from the

experimentally observed temperature decay, using a con-

duction model with separate terms for H2 and Ar and an

evaporation model using Watson’s equation for the heat of

vaporization. Maximum likelihood estimates of lognormal

distribution parameters were found through Bayesian ana-

lysis; the geometric mean (and median) diameter of

24.2 nm is generally consistent with expected nanoparticle

diameters, while the TiRe-LII-derived geometric standard

deviation of 1.43 is larger than what is typically observed

in TEM micrographs of silicon nanoparticle powders pro-

duced by the reactor. On the other hand, the Sauter mean

diameter inferred from the TiRe-LII data matches the

diameter found from BET analysis on a nanoparticle

powder.

Ongoing work is focused on improving the evaporation

model by including the influence of nanoparticle size on

the surface energy used in the Kelvin equation. Uncertainty

estimates will also be made more robust by combining

the model parameter uncertainty and signal noise within

an extended Bayesian framework. This lays the way for

a more rigorous statistical treatment of TiRe-LII data and

the possibility of obtaining more robust estimates of

Table 4 Relative sensitivity coefficients and estimated error for

inferred nanoparticle size distribution parameters due to model

parameter uncertainty

dp,g (nm) rg dp,32 (nm)

RSC e RSC e RSC e

Ti -148.0 ±1.2 3.90 ±0.03 -75.0 ;0.6

Tg 73.4 ±11.3 -2.38 ;0.37 8.7 ±1.3

pg -33.8 ;3.4 1.10 ±0.11 -1.7 ;0.2

cp -21.7 ;2.2 -0.05 - -31.9 ;3.2

q -24.3 ;2.4 - - -33.4 ;2.4

aAr -21.8 ;2.2 0.70 ±0.07 -1.1 ;0.1

aH2-11.7 ;5.9 0.37 ±0.19 -0.6 ;0.3

cs 2.6 ±0.3 -0.05 – 1.5 ±0.2

Tcr 47.9 ±4.8 -0.70 ;0.07 38.7 ±3.9


123

nanoparticle morphology by combining TiRe-LII data with

other techniques that provide complementary information.

Acknowledgments This research was supported by grants from the

Natural Science and Engineering Council of Canada (NSERC) and

the Deutsche Forschungsgemeinschaft (DFG). One of the authors (TA

Sipkens) was also supported by a scholarship from the Government of

Ontario. Compute Canada and SharcNet (www.sharcnet.ca) provided

the computational resources.

References

1. N. O’Farrell, A. Houlton, B.R. Horrocks, Int. J. Nanomed. 1, 451

(2006)

2. G. Konstantanos, E.H. Sargent, Nat. Nanotechnol. 5, 391 (2010)

3. M. Kummer, J.P. Badillo, A. Schmitz, H.-G. Bremes, M. Winter,

C. Schulz, H. Wiggers, J. Electrochem. Soc. 161, A40 (2014)

4. L. Pavesi, R. Turan, Silicon Nanocrystals: Fundamentals, Syn-

thesis, and Applications (Wiley, New York, 2010)

5. L.A. Melton, Appl. Opt. 23, 2201 (1984)

6. D.R. Snelling, G.J. Smallwood, F. Liu, O.L. Gulder, W.D. Bac-

halo, Appl. Opt. 44, 6773 (2005)

7. B.F. Kock, T. Eckhardt, P. Roth, Proc. Comb. Inst. 29, 2775

(2002)

8. S. Schraml, S. Will, A. Leipertz, SAE Technical Paper Series

1999-01-0146 (1999)

9. R.L. Vander Wal, T.M. Ticich, J.R. West, Appl. Opt. 38, 5867

(1999)

10. A.V. Filippov, M.W. Markus, P. Roth, J. Aerosol Sci. 30, 71

(1999)

11. R. Starke, B. Kock, P. Roth, Shock Waves 12, 351 (2003)

12. A.V. Eremin, E.V. Gurentsov, C. Schulz, J. Phys. D Appl. Phys.

41, 055203 (2008)

13. A.V. Eremin, E.V. Gurentsov, E. Popova, K. Priemchenko, Appl.

Phys. B 104, 289 (2011)

14. E.V. Gurentsov, A.V. Eremin, High Temp. 49, 667 (2011)

15. B.F. Kock, C. Kayan, J. Knipping, H.R. Orthner, P. Roth, Proc.

Comb. Inst. 30, 1689 (2005)

16. Y. Murakami, T. Sugatani, Y. Nosaka, J. Phys. Chem. A 109,

8994 (2005)

17. T.A. Sipkens, G. Joshi, K.J. Daun, Y. Murakami, J. Heat Transf.

135, 052401 (2013)

18. J. Reimann, H. Oltmann, S. Will, E.L. Bassano Carotenuto, S.

Losch, S. Gunther, in Laser Sintering of Nickel Aggregates

Produced from Inert Gas Condensation. Proceedings of the

World Congress on Particle Technology (Nuremburg, Germany,

2010)

19. T. Lehre, R. Suntz, H. Bockhorn, Proc. Combust. Inst. 30, 2585

(2005)

20. F. Cignoli, C. Bellomunno, S. Maffi, G. Zizak, Appl. Phys. B 96,

599 (2009)

21. S. Maffi, F. Cignoli, C. Bellomunno, S. De luliis, G. Zizak,

Spectrochim. Acta B 63, 202 (2008)

22. B. Tribalet, A. Faccinetto, T. Dreier, C. Schulz, in Evaluation of

Particle Sizes of Iron-Oxide Nano-Particles in a Low-Pressure

Flame-Synthesis Reactor by Simultaneous Application of TiRe-

LII and PMS. 5th International Workshop on Laser-Induced

Incandescence (Le Touquet, France, 2012)

23. I. Altman, D. Lee, J. Song, M. Choi, Phys. Rev. E 64, 052202

(2001)

24. G.S. Eom, S. Park, C.W. Park, Y.H. Shin, K.H. Chung, S. Park,

W. Choe, J.W. Hahn, Appl. Phys. Lett. 83, 1261 (2003)

25. G.S. Eom, S. Park, C.W. Park, W. Choe, Y. Shin, K.H. Chung,

J.W. Hahn, Jpn. J. Appl. Phys. 43, 6494 (2004)

26. M.F. Modest, Radiative Heat Transfer (Academic Press, San

Diego, 2013)

27. F. Liu, K.J. Daun, V. Beyer, G.J. Smallwood, D.A. Greenhalgh,

Appl. Phys. B 87, 179 (2007)

28. S. Brunauer, P.H. Emmett, E. Teller, JACS 60, 309 (1938)

29. N. Petermann, N. Stein, G. Schierning, R. Theissmann, B. Stoib,

C. Hecht, C. Schulz, H. Wiggers, J. Phys. D Appl. Phys. 44,

174034 (2011)

30. U. Kortshagen, L. Mangolini, A. Baost, J. Nanopart. Res. 9, 39

(2007)

31. M.S.K. Fuchs, J. Phys. Condens. Matter 12, 4341 (2000)

32. V.A. Hackley, A.B. Stefaniak, J. Nanopart. Res. 15, 1742 (2013)

33. W.K. Rhim, K. Ohsaka, J. Cryst. Growth 208, 313 (2000)

34. P.D. Desai, J. Phys. Chem. Ref. Data 15, 967 (1986)

35. F. Liu, K.J. Daun, D.R. Snelling, G.J. Smallwood, Appl. Phys. B

83, 355 (2006)

36. T. Tamooka, T. Shoji, T. Matsui, J. Mass Spectrom. Soc. Jpn. 47,

49 (1999)

37. K.M. Watson, Ind. Eng. Chem. 35, 398 (1943)

38. S. Velasco, F.L. Roman, J.A. White, A. Mulero, Fluid Phase

Equlib. 244, 11 (2006)

39. V. Svrcek, T. Sasaki, Y. Shimizu, N. Koshizaki, J. Appl. Phys.

103, 023101 (2008)

40. S.J. Gregg, K.S.W. Sing, Adsorption, Surface Area and Porosity

(Academic Press, London, 1982)

41. K.K. Nanda, A. Maisels, F.E. Kruis, H. Fissan, S. Stappert, Phys.

Rev. Lett. 91, 106102 (2003)

42. H.H. Lu, Q. Jiang, Langmuir 21, 779 (2005)

43. S.C. Saxena, R.K. Joshi, Thermal Accommodation Coefficients of

Gases (McGraw-Hill, New York, 1981)

44. A.V. Eremin, E.V. Gurentsov, M. Hofmann, B.F. Kock, C.

Schulz, Appl. Phys. B 83, 449 (2006)

45. E.V. Gurentsov, A.V. Eremin, C. Schulz, Kinet. Catal. 48, 194

(2007)

46. K.J. Daun, G.J. Smallwood, F. Liu, J. Heat Transf. 130, 121201

(2008)

47. K.J. Daun, G.J. Smallwood, F. Liu, Appl. Phys. B 94, 39 (2009)

48. K.J. Daun, Int. J. Heat Mass Transf. 52, 5081 (2009)

49. K.J. Daun, J.T. Titantah, M. Karttunen, Appl. Phys. B 107, 221

(2012)

50. F.H. Stillinger, T.A. Weber, Phys. Rev. B: Condens. Matter 31,

5263 (1985)

51. V.S. Dozhidikov, A.Y. Basharin, P.R. Levashov, J. Chem. Phys.

137, 054502 (2012)

52. P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, J. Luitz,

WIEN2k, An Augmented Plane Wave Plus Local Orbitals Pro-

gram for Calculating Crystal Properties (Vienna University of

Technology, Vienna, 2001)

53. S.R. Bandler, J.S. Adams, S.M.E.C. Brouer, R.E. Lanou, H.J.

Maris, T. More, F.S. Porter, G.M. Seidel, Nucl. Instrum. Methods

Phys. Res. A 370, 138 (1996)

54. W.M. Lau, I. Bello, L.J. Huang, M. Vos, I.V. Mitchell, J. Appl.

Phys. 74, 7101 (1993)

55. S. Plimpton, J. Comput. Phys. 117, 1 (1995)

56. S. Nose, Mol. Phys. 52, 255 (1984)

57. W.G. Hoover, Phys. Rev. A 31, 1695 (1985)

58. K.J. Daun, T.A. Sipkens, J.T. Titantah, M. Karttunen, Appl. Phys.

B. doi:10.1007/s00340-013-5508-0

59. C. Heicht, A. Abdali, T. Dreier, C. Schulz, Z. Phys. Chem. 225,

1225 (2011)

60. R. Charnigo, M. Francoeur, P. Kenkel, M.P. Menguc, B. Hall, C.

Srinivasan, JQSRT 113, 182 (2012)

61. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart, and

Winston, Inc., New York, 1971)

62. M. Chen, Q. Shao, J. Comput. Graph. Stat. 8, 69 (2012)

63. R.J. Hyndman, Am. Stat. 50, 120 (1996)


123

http://www.sharcnet.ca

http://dx.doi.org/10.1007/s00340-013-5508-0

64. W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery,

Numerical Recipes: The Art of Scientific Computing (Cambridge

University Press, Cambridge, 2007)

65. A.W. Bowman, A. Azzalini, Applied Smoothing Techniques for

Data Analysis (Oxford University Press, Oxford, 1997)

66. W.C. Hinds, Aerosol Technology: Properties, Behaviour, and

Measurement of Airborne Nanoparticles (John Wiley and Sons,

New York, 1982)

67. J. Knipping, H. Wiggers, B. Rellinghaus, P. Konjhodzic, C.

Meier, J. Nanosci. Nanotech. 4, 1039 (2004)

68. K.J. Daun, B.J. Stagg, F. Liu, G.J. Smallwood, D.R. Snelling,

Appl. Phys. B 87, 363 (2007)


123

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