8/12/2019 JVA061605 - Measurement of the Tangential Momentum Accommodation Coefficient of H2 on Stainless Steel Extre
1/8
Measurement of the tangential momentum accommodation coefficient ofH2 on stainless steel extreme ultraviolet-resist and polyimideJohannes F. M. Velthuisand Laurens van BokhovenCitation: J. Vac. Sci. Technol. A 31, 061605 (2013); doi: 10.1116/1.4816941View online: http://dx.doi.org/10.1116/1.4816941View Table of Contents: http://avspublications.org/resource/1/JVTAD6/v31/i6Published by theAVS: Science & Technology of Materials, Interfaces, and ProcessingRelated ArticlesPlasma etching: Yesterday, today, and tomorrowJ. Vac. Sci. Technol. A 31, 050825 (2013)Nucleation and growth of MgO atomic layer deposition: A real-time spectroscopic ellipsometry studyJ. Vac. Sci. Technol. A 31, 06F101 (2013)Enhanced response to molecular adsorption of structurally defective grapheneJ. Vac. Sci. Technol. B 31, 030602 (2013)Effects of Cs adsorption on the field emission characteristics of closed single-walled carbon nanotubesJ. Vac. Sci. Technol. B 31, 021802 (2013)Copper deposition on TiO2 from copper(II)hexafluoroacetylacetonateJ. Vac. Sci. Technol. A 31, 01A121 (2013)Additional information on J. Vac. Sci. Technol. AJournal Homepage: http://avspublications.org/jvstaJournal Information: http://avspublications.org/jvsta/about/about_the_journalTop downloads: http://avspublications.org/jvsta/top_20_most_downloadedInformation for Authors: http://avspublications.org/jvsta/authors/information_for_contributors
Downloaded 18 Sep 2013 to 139.63.40.192. Redistribution subject to AVS license or copyright; see http://avspublications.org/jvsta/about/rights_and_permissions
http://avspublications.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=JVTAD6&possible1=Johannes%20F.%20M.%20Velthuis&possible1zone=author&alias=&displayid=AVS&ver=pdfcovhttp://avspublications.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=JVTAD6&possible1=Laurens%20van%20Bokhoven&possible1zone=author&alias=&displayid=AVS&ver=pdfcovhttp://avspublications.org/jvsta?ver=pdfcovhttp://link.aip.org/link/doi/10.1116/1.4816941?ver=pdfcovhttp://avspublications.org/resource/1/JVTAD6/v31/i6?ver=pdfcovhttp://www.avs.org/?ver=pdfcovhttp://link.aip.org/link/doi/10.1116/1.4819316?ver=pdfcovhttp://link.aip.org/link/doi/10.1116/1.4816776?ver=pdfcovhttp://link.aip.org/link/doi/10.1116/1.4798649?ver=pdfcovhttp://link.aip.org/link/doi/10.1116/1.4790510?ver=pdfcovhttp://link.aip.org/link/doi/10.1116/1.4765644?ver=pdfcovhttp://avspublications.org/jvsta?ver=pdfcovhttp://avspublications.org/jvsta/about/about_the_journal?ver=pdfcovhttp://avspublications.org/jvsta/top_20_most_downloaded?ver=pdfcovhttp://avspublications.org/jvsta/authors/information_for_contributors?ver=pdfcovhttp://avspublications.org/jvsta/authors/information_for_contributors?ver=pdfcovhttp://avspublications.org/jvsta/top_20_most_downloaded?ver=pdfcovhttp://avspublications.org/jvsta/about/about_the_journal?ver=pdfcovhttp://avspublications.org/jvsta?ver=pdfcovhttp://link.aip.org/link/doi/10.1116/1.4765644?ver=pdfcovhttp://link.aip.org/link/doi/10.1116/1.4790510?ver=pdfcovhttp://link.aip.org/link/doi/10.1116/1.4798649?ver=pdfcovhttp://link.aip.org/link/doi/10.1116/1.4816776?ver=pdfcovhttp://link.aip.org/link/doi/10.1116/1.4819316?ver=pdfcovhttp://www.avs.org/?ver=pdfcovhttp://avspublications.org/resource/1/JVTAD6/v31/i6?ver=pdfcovhttp://link.aip.org/link/doi/10.1116/1.4816941?ver=pdfcovhttp://avspublications.org/jvsta?ver=pdfcovhttp://avspublications.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=JVTAD6&possible1=Laurens%20van%20Bokhoven&possible1zone=author&alias=&displayid=AVS&ver=pdfcovhttp://avspublications.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=JVTAD6&possible1=Johannes%20F.%20M.%20Velthuis&possible1zone=author&alias=&displayid=AVS&ver=pdfcovhttp://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/test.int.aip.org/adtest/L23/516020152/x01/AIP/Hiden_JVACovAd_1640x440Banner_09_10and09_17_2013/1640x440_-_23874-BANNER-AD-1640-x-440px_-_USA.jpg/7744715775302b784f4d774142526b39?xhttp://avspublications.org/jvsta?ver=pdfcov8/12/2019 JVA061605 - Measurement of the Tangential Momentum Accommodation Coefficient of H2 on Stainless Steel Extre
2/8
Measurement of the tangential momentum accommodation coefficientof H2on stainless steel, extreme ultraviolet-resist, and polyimide
Johannes F. M. Velthuisa)
TNO Science and Industry, P. O. Box 155, 2600AD Delft, The Netherlands
Laurens van Bokhovenb)
ASML Netherlands B.V., De Run 6501, 5504 DR Veldhoven, The Netherlands
(Received 12 March 2013; accepted 12 July 2013; published 9 September 2013)
The tangential momentum accommodation coefficient (TMAC) of H2 on electropolished stainless
steel 304, extreme ultraviolet-resist, and polyimide was determined by measuring the mass flow
through a macroscopic rectangular channel (100 cm 10cm 1 cm) as a function of the pressure
drop in the Knudsen range Knave 00.6. The TMAC and Knudsen number determine the amount
of velocity slip taking place at the wall at rarefied conditions. By comparing the measurements
with an analytical expression for the mass flow through the channel, including the 1st order
slip flow contribution to continuum, the only remaining unknown, that is, the TMAC, was extracted.
VC 2013 American Vacuum Society.[http://dx.doi.org/10.1116/1.4816941]
I. INTRODUCTION
The value of the tangential momentum accommodation coef-ficient (TMAC), also referred to as r, and the thermal accom-
modation coefficient (TAC), also referred to as a, is important
as boundary conditions for the accurate simulation of flow and
thermal effects in extreme ultraviolet (EUV)-wafer scanners.
These scanners operate in a hydrogen environment in the
slip flow regime (0.01 < Kn< 0.1) and the early transitionalflow regime (Kn 0.4). Simulations models that are being
used are CFD (computational fluid dynamics) and DSMC
(direct simulation Monte Carlo) amongst others. Continuum
flow (Kn < 0.01) is modeled by applying zero velocity slipat the wall. At rarefied conditions (small pressures and/or
small channel dimensions), both velocity and temperature
slip take place at the wall. The velocity slip (1st order slipflow expansion of continuum) is a function of Knudsen num-
ber Kn and TMAC. The Knudsen number is a measure of
rarefaction and hence of the amount of velocity slip. For
channels, the Knudsen number computed at the channel av-
erage pressure (of the channel inlet and outlet) is representa-
tive, denoted Knave. The TMAC is a measure of the
momentum exchange between the gas and the surface, which
is a combined property of the gas and the surface involved.
TMAC data of H2 on surfaces relevant for EUV-wafer
scanners could not be found in the literature; therefore, it
was decided to measure these parameters. The 1st order slip
flow expansion of continuum is also known to be valid in the
early transitional flow regime (Knave 0.4), the extent ofwhich is investigated by varying the Knudsen number over
the range Knave 00.6.
Several methods exist to measure TMAC, see, for exam-
ple, Agrawal and Prabhu.1 A popular method involves meas-
uring the flow rate over microchannels manufactured with
MEMS techniques usingthe rate of pressure rise (ROR) tech-
nique, see Arkilic et al.2 By comparing the measured mass
flow rate over the channel with the analytical expression for
the mass flow rate over the channel, including the 1st order
slip flow contribution to continuum, the remaining unknown,
that is, TMAC, can be determined.
We use a similar method but now applied to macroscopic
channels. Thus, the method is not limited to MEMS compati-
ble materials but can be applied to actual surfaces and condi-
tions, such as found in EUV-wafer scanners. Standard mass
flow sensors can be used, avoiding the difficulties associated
with the ROR technique. First, the TMAC of H2 in an elec-
tropolished stainless steel 304 (SS304) channel is measured.
Next, the channel is spray-coated to measure the TMAC on
EUV-resist (SPUR-V002). Finally, the channel is covered
with thin sheet material to measure the TMAC on polyimide
(Dupont Kapton type H-film).
II. THEORY
A. Channel mass flow rate
Figure1shows the channel with rectangular cross section.
Starting from the NavierStokes equations, a theoretical
expression for the laminar mass flow rate over the channel
can be derived, which includes a 1st order slip flow contribu-
tion to continuum as proposed by Maxwell in 1879.25 The
mass flow rate expression reads
_m
p2i p2o
A B 2
po pi
; (1)
where
A H3 W M
3
4 C1
24 l L R T ; (2)
B A C2
C1
2 r
r Kno po; (3)
C1 4
3 1
192
p5
H
W
X1
n1;3;5::
1
n5tanh
np
2
W
H
" #;
(4)a)
Electronic mail: [email protected])Electronic mail: [email protected]
061605-1 J. Vac. Sci. Technol. A 31(6), Nov/Dec 2013 0734-2101/2013/31(6)/061605/7/$30.00 VC 2013 American Vacuum Society 061605-1
Downloaded 18 Sep 2013 to 139.63.40.192. Redistribution subject to AVS license or copyright; see http://avspublications.org/jvsta/about/rights_and_permissions
http://dx.doi.org/10.1116/1.4816941http://dx.doi.org/10.1116/1.4816941http://dx.doi.org/10.1116/1.4816941http://dx.doi.org/10.1116/1.4816941http://dx.doi.org/10.1116/1.4816941http://dx.doi.org/10.1116/1.4816941mailto:[email protected]:[email protected]://crossmark.crossref.org/dialog/?doi=10.1116/1.4816941&domain=pdf&date_stamp=2013-09-09mailto:[email protected]:[email protected]://dx.doi.org/10.1116/1.4816941http://dx.doi.org/10.1116/1.4816941http://dx.doi.org/10.1116/1.4816941http://dx.doi.org/10.1116/1.48169418/12/2019 JVA061605 - Measurement of the Tangential Momentum Accommodation Coefficient of H2 on Stainless Steel Extre
3/8
C2 8 C1 256
p4 1
H
W
X1
n1;3;5::
1
n4tanh2
np
2
W
H
:
(5)
The mass flow rate _m (kg/s) is scaled with pi2po
2, where
p denotes pressure and subscript i and o refer to the
channel inlet and the channel outlet, respectively. The aver-
age pressure is defined as pave
(pi
po)/2. For an isother-mal channel of given dimension and given TMAC value r,
the terms A and B in Eq. (1) are constants, leading to a
linear relationship between the scaled mass flow rate and
inverse average pressure. The first term A represents the
scaled mass flow rate in the continuum limit, whose theoreti-
cal value is given by Eq. (2). The second term B accounts
for slip flow effects, whose theoretical value is given by
Eq.(3). Slip has the effect of enhancing the mass flow. The
slip term in Eq. (1) disappears when the absolute pressures
rises, despite B having a finite value. The Knudsen number
Kn is a measure for noncontinuum effects and is defined as
Kn k/H, where H is the channel height and k denotes
the mean free path of gas molecules between collisions
k l
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip M
2 R T
r : (6)
The density q of hydrogen is calculated with the ideal the
gas law, where R is the universal gas constant, T is the
temperature, and M is the molecular weight (2.016 g/mol).
The dynamic viscosity l of hydrogen is taken according to
TableI. For comparison, at isothermal conditions, the mean
free path of hydrogen is inversely proportional to pressure
and reads k 12 mm at 1 Pa and 293.5 K. We differentiate
between the outlet Knudsen number Kno (evaluated at the
channel outlet pressure po) and the average Knudsen num-ber Knave (evaluated at the average channel pressure pave).
As in our measurements, we always enforce p i to be greater
than or equal to two times po; this in practice means that
Knave is always less than 2/3Kno. Note that for isothermal
conditions, the product (Knp), for example, appearing in
Eq. (3), is constant. The coefficients C1 [Eq. (4)] and
C2 [Eq. (5)] account for the channel aspect ratio (H/W).
For a plan-parallel channel with infinitely small aspect ratio
(H/W 0), the coefficients read C1 4/3 and C2/C1 6.
For a channel with finite aspect ratio (H/W 0.1), as applies
for our experimental setup, the channel sides reduce the con-
tinuum mass flow rate by approximately 7%, when com-
pared to a channel with zero aspect ratio.
By linear curve-fitting the measured data for the scaled
mass flow rate versus inverse average pressure according to
Eq.(1), an experimental value is obtained for the line inter-
cept A and slope B. From these and the previous given theo-
retical expression for the slope [Eq. (3)], the unknown
TMAC valuer is extracted, according to
r 2
1B
A
1
Kno Po
C1
C2
: (7)
In this paper, we will determine the TMAC value, orr, in
the Knudsen range Kno 01.2, equivalent to Knave 00.6.
B. Assumptions
In the derivation of the scaled mass flow rate expression
[Eq.(1)], a number of simplifying assumptions are used. The
experimental conditions must meet these simplifying
assumptions (see Sec. IV), otherwise the extracted TMAC
values become ill-defined. The simplifying conditions are as
follows:
(1) Laminar flow. For this, the hydraulic Reynolds numbers
should stay below the laminar-turbulent flow transition,Re2H 2300.
(2) Negligible inertial effects. For this, the product of Mach
number at the channel inlet Mi, Mach number at the
channel outlet Mo, and heat capacity ratio c should be
smaller than unity (cMoMi 1), see Karniadakis and
Beskok.3
(3) Isothermal channel. For isothermal conditions, the varia-
tion in channel temperature in time and in place must be
limited, i.e., less than 3 K, which corresponds to 1% rela-
tive error in absolute ambient temperature.
(4) Slip flow (Knave< 0.1). Notwithstanding this condition, itis generally known that Eq. (1) with associated TMAC
(derived for the 1st order slip flow regime) also reprodu-
ces the mass flow rate in the early transitional flow regime
(Knave 0.4) quite well. When lumping higher order
Knudsen effects to Eq. (1) by simply modifying the
TMAC value, this per definition will result in a Knudsen
dependency of the TMAC. Plotting these lumped TMAC
values is of limited practical significance, other than as an
indicator for higher order Knudsen effects. As in our
experiments Knave does not exceed the value 0.6, we
expect that this Knudsen dependency is still limited.
(5) Developed flow. To say negligible entrance and exiteffects
on the channel pressure drop. Duan and Muzychka7 give
FIG. 1. Channel with rectangular cross section (L 100cm, W 10 cm,
H 1 cm).
TABLE I. Dynamic viscosity of H2[(reported (Ref.6) accuracy 60.5%].
T (K) l(lPas)
290 8.766
300 8.969
310 9.170
061605-2 J. F. M. Velthuis and L. van Bokhoven: Measurement of the TMAC 061605-2
J. Vac. Sci. Technol. A, Vol. 31, No. 6, Nov/Dec 2013
Downloaded 18 Sep 2013 to 139.63.40.192. Redistribution subject to AVS license or copyright; see http://avspublications.org/jvsta/about/rights_and_permissions
8/12/2019 JVA061605 - Measurement of the Tangential Momentum Accommodation Coefficient of H2 on Stainless Steel Extre
4/8
an expression for the apparent friction factor for flow in a
plane parallel channel including slip flow and entry
effects. Using this expression, it can be shown that the rel-
ative error in the scaled mass flow rate due to entry/exit
effects, relative to that of a channel with developed flow,
stays below 0.1% for Reynolds numbers Re2H< 25,where it is assumed that the exit effect is of the same
order of magnitude as the entrance effect.
III. EXPERIMENT
A. Vacuum system
The vacuum system consists of two vacuum vessels,
referred to as inlet vessel and outlet vessel, see Fig. 2.
Each vessel has a volume of1.4 m3. A stainless steel 304
channel connects the two vessels (see Sec.III Bfor details).
The inlet vessel is not pumped and the respective pump-
openings are closed. The outlet vessel is pumped by option-
ally four Turbo Molecular Pumps (TMP), type Shimadzu
TMP-H3153LMC. The pumps are water-cooled with achiller to lab temperature level (293 K). The individual
pump speed is 2600 l/s H2 at 24 000 rpm (100%) for abso-
lute pressures in the normal operating range (
8/12/2019 JVA061605 - Measurement of the Tangential Momentum Accommodation Coefficient of H2 on Stainless Steel Extre
5/8
C. Surface preparation
1. Stainless steel
The cleaning procedure of the electropolished stainless
steel 304 channel parts consisted of degreasing, rinsing in
demineralized water, alkaline cleaning, and repeated rinsing
in demineralized water of increasing purity, followed by dry-
ing in air. After mounting in the test setup, the channel
resided in vacuum (1 Pa) for a few days and was allowed
to outgas before the experiments started. This outgassing
procedure was followed for all materials studied.
2. EUV-resist
For the TMAC measurement on EUV-resist, the internal
wetted surface of the stainless steel channel was spray-
coated with several layers of EUV-resist (SPUR-V002, seeTableIV), with drying in between. The resist itself is almost
colorless in appearance. After applying a few layers, the
coating showed colorization, indicating a layer thickness of
typically 0.25 lm (1=4 wavelength of visible light). Between5 and 10 layers were applied in total until the colorization
disappeared. Subsequently, the resist was baked at 105 C,
using a lint heater wrapped around the channel.
3. Polyimide (Kapton)
For the TMAC measurement on polyimide, the internal
wetted surface of the stainless steel channel was coated with
a 25 lm thin polyimide sheet (Dupont Kapton type H film),
with negligible effects on channel height and the variation in
channel height.
D. Measurement procedure
A typical measurement series consists of tuning the outlet
channel pressure po to a constant value (within 1%) and
varying the inlet pressure to approximately multiples of the
outlet pressure (pi npo, n 2). Next, the scaled mass flow
rate is plotted versus inverse average pressure according to
Eq. (1), including error bars in both coordinate directions
due to sensor accuracies or resolution, whichever is larger.The error bars are determined using linear error propagation
techniques and assuming square-root weighted averaging of
the individual error contributions, assuming independent
measurement quantities. All uncertainties mentioned in this
paper are based on a 99.8% confidence level (3stdev/
mean). Next a linear curve fit of the plotted data is made
using the total least squares (TLS) technique.8 The TLS tech-
nique takes into account the errors bars present in the data,
in both coordinate directions. The curve fit procedure results
in a value for the line intercept A (scaled mass flow in the
continuum limit) and for the slope B (slip effect), together
with their associated uncertainties DA and DB. Next,
the TMAC value r is calculated according to Eq. (7). Theuncertainty Dr is determined again using linear error propa-
gation techniques, now with linear addition of the individual
error contributions. Thus, a certain measurement series
can be associated with a specific (constant) channel outlet
pressure po, a corresponding Knudsen number Kno, and a
TMAC value. By measuring multiple data series at varying
discrete channel outlet pressures, the TMAC values can be
plotted as function Kno. As already stated in Sec. II B, the
relevance of plotting TMAC beyond the early transition
range (Knave 0.4) is limited, other than as an indicator for
higher order Knudsen effects. Typical discrete outlet pres-
sures selected are in the range po 201 Pa. The highest
occurring channel outlet Knudsen numbers is Kno 1.2,
which is equivalent to a measurement series in which the
average Knudsen number is varied between Knave 0 and
0.6. Measurements at even higher Knudsen numbers (lower
pressures) were not conducted, as these where limited by the
resolution of the pressure sensors and accuracy of the mass
flow controllers.
To determine the TMAC valid for the slip flow regime
(and early transitional flow regime), all measurement series
were collected in one plot (scaled mass flow rate versus
inverse average pressure) and fitted to Eq. (1) in the range
Knave 0.4. The resulting scaled mass flow in the continuum
FIG. 3. (Color online) Stainless steel channel consisting of bottom part and lid.
TABLE III. Channel dimensions.
Item Descriptiona
Height H 10.16 6 0.17mm
Width W 100.0 6 0.1mm
Length L 1000.8 6 0.5mm
C1 1.248b
C2 7.588b
a99.8% confidence level (3stdev/mean).bAt nominal dimensions.
TABLE IV. EUV-resist (SPUR-V002). Source material safety data sheet.
Item Description
Product SPUR-V002
Supplier Shin-Etsu Chemical Co Ltd., Japan
Composition 50% cyclohexanone,
8/12/2019 JVA061605 - Measurement of the Tangential Momentum Accommodation Coefficient of H2 on Stainless Steel Extre
6/8
limit (intercept A) is compared with the theoretical value
Eq.(2), by inserting in this equation the channel dimensions,
temperature, and corresponding H2 dynamic viscosity. The
uncertainty in the theoretical value is determined based on
square-root weighted averaging of the individual error
contributions.
IV. RESULTS
Over the course of the electropolished stainless steel
experiments, the channel temperatures stayed within
293.8 6 1.5 K, indicating that the channel can be considered
isothermal during the experiments (Sec. II B). As already
mentioned in Sec. III D, uncertainties given in this paper
refer to a 99.8% confidence level (3stdev/mean). The tem-perature of the channel during all the EUV-resist experi-
ments stayed within 293.5 6 0.5 K. The temperature of the
channel during all the polyimide experiments stayed within
293.2 6 0.3K.
The Reynolds numbers (not shown) during all flow
experiments stayed below 10, well below the previously
determined criterion for the laminar to turbulent flow
transition of Re 2300 (Sec. II B). Entrance and exit
effects on the channel pressure drop can therefore be safely
neglected, and the flow can be considered developed
throughout the channel. Inertia effects are also considerednegligible, as the product of inlet Mach number, outlet
Mach number, and heat capacity ratio c during all experi-
ments stayed below 0.05, far less than the previously deter-
mined criterion of unity (Sec.II B).
Figure 4 shows a typical measured data series in which
the channel outlet pressure po is kept constant during
the experiments and the channel inlet pressure pi is varied.
For this particular example, po 10 Pa (equivalent to
Kno 0.12). The outlet pressure pi is varied between 20
and 100 Pa, which is equivalent to 1/pave 0.020.07 or
Knave 0.020.08. The mass flow rate is scaled with the the-
oretical continuum mass flow rate, which is equal to a con-
stant times (pi2po2). The inverse average pressure 1/pave isgiven along the x-axis. To convert the x-axis to Knave, a lin-
ear scaling factor of 1.2 must be used. The error bars in the
scaled mass flow rate become larger toward the right side of
the graph (lower pressure range), as the associated mass flow
rates fall into the lower, less accurate flow range of the
MFCs. The error bars in the x-direction are not visible on
this scale. As expected, the measured data fall on a straight
line conform Maxwells model. The associated TMAC for
this data series with Kno 0.12 is given in Fig. 5, together
FIG. 4. (Color online) Scaled mass flow rate vs inverse average pressure. To
convert to Knavemultiply the x-axis with 1.2. Shown is a typical data series
in which po is kept constant and pi is varied. Straight lines are Maxwells
model [Eq.(1)] with TMAC as fit parameter.
FIG. 5. (Color online) TMAC as function of Knoobtained by fitting a meas-
ured series of mass flow data to Maxwells model [Eq. (1)]. A specific value
of Knocorresponds to a data series in which pois kept constant and piis var-
ied (Knavevaries between roughly 0 and 2/3Kno).
FIG. 6. (Color online) Scaled mass flow rate vs inverse average pressure for
SS304. All data series collected. To convert to Knave multiply the x-axis
with 1.2. The straight line is Maxwells model [Eq. (1)] with the TMAC fit-
ted in the range Knave 0.4.
FIG. 7. (Color online) Similar as Fig.6but now for EUV-resist.
061605-5 J. F. M. Velthuis and L. van Bokhoven: Measurement of the TMAC 061605-5
JVST A - Vacuum, Surfaces, and Films
Downloaded 18 Sep 2013 to 139.63.40.192. Redistribution subject to AVS license or copyright; see http://avspublications.org/jvsta/about/rights_and_permissions
8/12/2019 JVA061605 - Measurement of the Tangential Momentum Accommodation Coefficient of H2 on Stainless Steel Extre
7/8
with the results of the other data series. The highest outlet
Knudsen number reads Kno 1.2, which corresponds to a
data series in which the average Knudsen number varies
between Knave 0 and 0.6. As remarked in Sec. II B4 plot-
ting TMAC values as a function of Kno based on measure-
ment data lying outside the early transition flow regime
(Knave 0.4) is of limited practical significance, other than
as an indicator of higher order Knudsen effects. The
Knudsen dependency of the TMAC shows a similar trend as
that observed in Refs. 2 and 3 for nitrogen and argon gas in
contact with a silicon microchannel in the range Kno< 0.4,namely, a small decrease in lumped TMAC value up to a
Kno 0.2, after which the TMAC becomes more or less
constant. This seems consistent with the finding that the 1st
order slip flow model still holds in the early transitional flow
regime (Knave 0.4) as the measurements stay close to this
range. The TMAC of SS304 at Kno 1.2 shows a deviation
and larger uncertainty than the EUV-resist and polyimide
results. This is most probably due to the limited variation ofthe channel inlet pressure used in the SS304 experiments. As
the SS304 channel at that time was already coated with
resist, this particular SS304 experiment could not be repaired.
This was rectified in later experiments for EUV-resist and pol-
yimide, by adding more pressure data points.
Figures 68 show all measured data series collected in
one graph per surface material. The mass flow rate is scaled
with the theoretical continuum mass flow rate, which is
equal to a constant time (pi2po
2). To convert the inverse
average pressure along the x-axis to Knave, a linear scaling
factor of 1.2 must be used. The straight lines are Maxwells 1st
order slip flow model [Eq.(1)], where the associated TMAC is
fitted on that subset of the data for which Knave 0.4, that
is, lying in the early transitional flow regime. These TMAC
values are summarized in Table V. This table also compares
the intercept of the fitted line with the theoretical value as
given by Eq.(2). These values are equal to each other within
the measurement uncertainty. The slope of the curve connect-
ing the measurement points increases only very gradually,
indicating the onset of higher order rarefaction effects at higher
Knudsen numbers. The data support the general finding that
the 1st order slip flow model can be used throughout the early
transitional flow regime (Knave 0.4).
Another way of representing the data is plotting the mass
flow rate versus the average Knudsen number, similar as in
given Ref.3 (Fig. 5.14). The mass flow rate is scaled with the
theoretical free molecular mass flow rate which is equal to aconstant times the pressure difference (pipo) over the channel.
The resulting graph is shown in Fig. 9, ignoring the exact value
of the constant. Maxwells model [Eq. (1)] with TMAC 0.83
is given as comparison. As the measurement range does not
exceed Knave 0.6, it is still too early to see a local minimum
in the scaled mass flow rate which in general is observed
around Knave 1, the so-called Knudsen minimum.3
TACs on the same prepared surfaces were separately
measured by Philips,9 using the heated parallel disk method
and by fitting the results with the ShermanLees10 relation-
ship for heat transfer between parallel gaps in the continuum
FIG. 8. (Color online) Similar as Fig.6but now for polyimide.
TABLE V. TMAC and TAC (Refs. 9and10) of H2on SS304, EUV-resists, and polyimide (Knave< 0.4). Confidence level 99.8% (3stdev/mean).
Surface
Theoretical
scaled mass
flow [Eq.(2)]
Measured scaled
mass flow in the
continuum limit [Eq.(1)]
Measured slope
[Eq.(1)]
Momentum accommodation
coefficient TMAC
Thermal accommodation
coefficient TAC (Refs.9and10)
Atheory Ameas Bmeas r a
[kg/(s Pa2
)] [kg/(s Pa2
)] [kg/(s Pa)] [] []
SS304 3.8 6 0.2 1010 3.64 6 0.02 1010 3.71 6 0.02 109 0.83 6 0.01 0.32 6 0.04
EUV-resist 3.8 6 0.2 1010 3.65 6 0.02 1010 3.70 6 0.02 109 0.84 6 0.01 0.39 6 0.04
Polyim ide 3.8 6 0.2 1010 3.67 6 0.02 10
10 3.78 6 0.02 109 0.83 6 0.01 0.35 6 0.04
FIG. 9. (Color online) Mass flow rate channel vs Knave. The mass flow rate is
scaled with the theoretical free molecular mass flow rate, which is propor-
tional to (pipo). The line is Maxwells model [Eq.(1)] with TMAC0.83
(fitted in the range Knave 0.4).
061605-6 J. F. M. Velthuis and L. van Bokhoven: Measurement of the TMAC 061605-6
J. Vac. Sci. Technol. A, Vol. 31, No. 6, Nov/Dec 2013
Downloaded 18 Sep 2013 to 139.63.40.192. Redistribution subject to AVS license or copyright; see http://avspublications.org/jvsta/about/rights_and_permissions
8/12/2019 JVA061605 - Measurement of the Tangential Momentum Accommodation Coefficient of H2 on Stainless Steel Extre
8/8
to transitional flow regime (Kn 01.2). These TAC values
can be found in TableV.
V. SUMMARY AND CONCLUSIONS
The tangential momentum accommodation coefficient
(TMAC) of H2on various surfaces was determined, by meas-
uring the mass flow through a macroscopic rectangular chan-
nel (100 cm 10cm 1 cm) as a function of pressure drop inthe Knudsen range Knave< 0.6. This works main advantage isthe possibility of using macroscopic channels (as compared to
frequently used MEMS microchannels and MEMS materials).
Therefore, this method can be easily applied to a number of
actual surfaces, for example, relevant for EUV-wafer scanners.
The results support the general finding that the 1st order slip
flow expansion can be used throughout the early transitional
flow regime (Knave 0.4). The TMAC value for H2 on stain-
less steel 304 and polyimide (r 0.83 6 0.01) was found to
be almost identical to that on EUV-resist (r 0.84 6 0.01).
ACKNOWLEDGMENT
This project is funded by ASML Netherlands B.V.
1A. Agrawal and S. V. Prabhu,J. Vac. Sci. Technol. A 26, 634 (2008).2
E. B. Arkilic, K. S. Breuer, and M. A. Schmidt, J. Fluid Mech. 437, 29
(2001).3G. E. Karniadakis and A. Beskok, Micro Flows(Springer, New York, 2002).4J. Jang and S. T. Wereley, J. Micromech. Microeng. 16, 493 (2006).5F. M. White,Viscous Fluid Flow(McGraw-Hill, New York, 2006).6
M. J. Assaei, S. Mixafendi, and W. A. Wakeham, J. Phys. Chem. Ref.
Data 15, 1315 (1986).7
Z. Duan and Y. S. Muzychka,J. Fluids Eng. 132, 011201(2010).8W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery,
Numerical Recipes in Fortran (Cambridge University Press, Cambridge,
1992).9S. Box and G. Hannen, Measurement of the thermal accommodation
coefficient for ASML, Philips Research, Report Ref: ATP591-11-0405/
01, 2011.10D. J. Rader, W. M. Trott, J. R. Torczynski, J. N. Casta~neda, and T. W.
Grasser, Measurements of thermal accommodation coefficients, Sandia
Report SAND2005-6084, 2005.
061605-7 J. F. M. Velthuis and L. van Bokhoven: Measurement of the TMAC 061605-7
JVST A - Vacuum, Surfaces, and Films
http://dx.doi.org/10.1116/1.2943641http://dx.doi.org/10.1017/S0022112001004128http://dx.doi.org/10.1088/0960-1317/16/3/004http://dx.doi.org/10.1063/1.555764http://dx.doi.org/10.1063/1.555764http://dx.doi.org/10.1115/1.4000692http://dx.doi.org/10.1115/1.4000692http://dx.doi.org/10.1063/1.555764http://dx.doi.org/10.1063/1.555764http://dx.doi.org/10.1088/0960-1317/16/3/004http://dx.doi.org/10.1017/S0022112001004128http://dx.doi.org/10.1116/1.2943641