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EXPERIMENTAL CONFIRMATION OF THE KNUDSEN EFFECT IN NANOPOROUS
INSULATION MATERIALS
L. GRASSBERGER1 and R. STREY1
1
Institute for Physical Chemistry, University of Cologne,
Luxemburger Str. 116, 50939 Cologne, Germany
Keywords: nanoporous polymer, PMMA, Knudsen effect, thermal insulation, nanoporous insulating
material.
INTRODUCTION
Nanoporous insulating materials (NIMs) promise enormous energy savings through improved thermal
insulation properties (Jelle et al. 2010). Unfortunately, no cost-efficient generation of such materials on
the large scale is in sight due to the necessity to apply high pressures of blowing agent or supercritical
drying. Grassberger et al. (2017) have developed an ambient pressure method for preparation of
nanoporous materials, which utilizes PMMA (polymethylmethacrylate) beads swollen in a mixture of two
miscible solvents (e.g. acetone and cyclohexane), one of which (acetone) is a swelling agent for the
polymer. When drying the swollen beads, sequential evaporation of the two liquids (acetone evaporates
more readily than cyclohexane and fixates thereby the polymer matrix) leaves behind nanoporous polymer
beads. The internal structure as visualized by SEM (c.f. Figure 1) shows a spongy morphology
(Grassberger 2016).
Figure 1. Nanoporous PMMA. By simple suspension polymerization with various amounts of crosslinker PMMA
beads of millimeter size were generated. After swelling the beads and sequential evaporation of the solvents
nanoporous PMMA material with mean pore sizes varying between 80 and 800 nm was obtained. Note the spongy,
open-celled nanostructure with a porosity of about P=0.7. For details refer to Grassberger et al. (2017).
For various average pore sizes we measured the gas thermal conductivity and found enormous reductions
as pressure or pore size decrease. Interestingly the individual gas thermal conductivities scale and collapse
onto a single curve as quantitatively predicted by the Knudsen effect. (c.f. Figure 2).
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THEORY
The thermal conductivity λ of an insulating nanoporous material consists of three contributions, the heat
transfer by (infrared) radiation λradiation, the heat conduction through the solid matrix λmatrix and heat
conduction by the cell gas λgas:   radiation  f (1  P)matrix  Pgas , where P is the porosity and f<1 is a
tortuosity factor taking into account the irregular nature of the matrix (Lu et al. 1995, Notario et al. 2015).
For ordinary insulation foams the heat conduction by the cell gas accounts for more than two thirds of the
thermal conductivity (Placido et al. 2005). Thus, taking advantage of the Knudsen effect the gaseous
thermal conductivity can be lowered enormously, if the pore size is made comparable to or smaller than
the mean free path length lmean of the enclosed gas molecules. Dpore is the pore diameter of the porous
material. Knudsen in 1909 introduced the ratio of the two numbers (hence the name Knudsen number)
l
(1)
Kn  mean ,
Dpore
where lmean of the gas molecules is given by
kBT
(2)
lmean 
2 d m2 p
with dm the molecular diameter (dm = 0.36 nm for air (Kennard 1938), p the pressure, T the temperature
and kB the Boltzmann constant. The Knudsen effect is characterized by (Lu et al. 1995, Jelle et al. 2010)
 gas 
 gas,0
(3)
1  2 Kn
where λgas,0 =0.0262 mWm-1K-1is the thermal conductivity of air (Lemmon and Jacobsen 2004), β a
parameter that takes the energy transfer between the gas molecules and the limiting solid structure into
account (β = 1.5 for air (Jelle et al. 2010)). Obviously, increasing the mean free path length of the gas by
reducing the pressure or decreasing the pore size increases the Knudsen number and in consequence λgas
decreases.
EXPERIMENT
In Figure 2, left, the gas thermal conductivities are measured as function of pressure. Note that with
decreasing pore size the gas thermal conductivity vanishes.
-1
-1
gas thermal conductivity,gas [mWm K ]
30
25
20
Dpore = 300 µm
Dpore = 200 µm
15
Dpore = 800 nm
Dpore = 450 nm
Dpore = 150 nm
Dpore = 80 nm
10
5
0
0,001
0,01
0,1
pressure, p [bar]
1
0,001
0,01
0,1
1
10
100
Knudsen number, Kn
Figure 2. Left: Measured gas thermal conductivities as function of pressure for porous materials. Right: Scaling of
gas thermal conductivities vs. Knudsen number; full line calculated according to theory, eq. 3 with =1.5
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We note that by plotting in Figure 2, right, the data versus the Knudsen number all data fall strikingly
(within experimental uncertainty) onto a single curve calculated from equation 3.
DISCUSSION
We notes earlier that the Knudsen effect in gas phase growth of water droplets provided an amazingly
accurate description of the transition regime (Fladerer et al. 2002, Miles et al. 2010, Seinfeld and Pandis
1998). Here we found further striking evidence of the validity of the Knudsen concept. A number of
interesting observations can be discussed. 1) The well-known theoretical independence of the gas thermal
conductivity on pressure is borne out by the measurements on the two macrofoams (Styropor® with 200
and Armaflex® with 300 cellsize, respectively). 2) The thermal conductivity decreases as the total
pressure is decreased as soon as the pore size drops below 1 micron. Therefore, if low pressure is an
option, as in vacuum isolation panels (VIPs), or is given, as in space applications, like for satellites in the
orbits around the earth, a mean pore size of 1 micron may suffice. This aspect is of interest for production
of nanoporous materials, as it is easier to reach low densities of the polymer matrix, if the requirement of
extremely small pore size is relaxed. 3) The scaled plot in Figure 2, right, shows that the Knudsen number
apparently is the only relevant tuning parameter, not the pressure or the pore size alone. Thus we provide
experimental gas thermal conductivities over more than six orders of magnitude of the corresponding
Knudsen numbers in conjunction with a perfect description by eq. 3 without any adjustable parameter
(=1.5 has been suggested by previous workers based on theoretical arguments). 4) As has been
mentioned previously (Grassberger et al. 2017) our results confirm and supplement Notario et al.’s (2015)
results on different nanoporous materials quantitatively. 5) Before application as low-cost insulation
materials the porosities of our nanoporous polymeric materials have to be increased (or the densities
reduced). This task in mind several thesis works (Müller 2013, Oberhoffer 2015 and Grassberger 2016))
were devoted to investigate the density - pore size trade off. To date the lowest pore sizes have the highest
densities and vice versa. Here the efforts are continuing. 6) Here we have demonstrated substantial
progress in reducing the gas thermal conductivity. For application, however, also contributions by
radiation and solid conduction to  will need to be considered (Placido et al. 2005, Notario et al. 2015)
CONCLUSIONS
The Knudsen effect on transport properties is well-known to atmospheric scientists and gas phase workers
when particle dimensions become comparable to the mean free path of the gas molecules (Seinfeld and
Pandis, 1998). In this work we explore the beneficial effect on the gas thermal conductivity that occurs in
nanoporous insulation materials (Jelle et al. 2010). The required high pressure of blowing agents or
supercritical extraction procedures is a major obstacle in industrial production processes of nanoporous
materials. Grassberger et al. (2017) demonstrated a new method for generating nanoporous polymer
materials without gaseous blowing agent at ambient conditions. By swelling crosslinked PMMA beads in
specially selected solvents leads to nanostructured PMMA materials with mean pore diameters as low as
Dpore = 80 nm (c.f. Figure 1). Varying the crosslink density nanoporous PMMA materials with different
pore sizes were generated. For these materials the gas thermal conductivity was measured as function of
pressure. In addition the gaseous thermal conductivity of conventional microfoams (Styropor® with 200
and Armaflex® with 300 cellsize, respectively) was measured for comparison (c.f. Figure 2, left). The
effect of the mean free path limitations in the nanoporous materials (the Knudsen effect) becomes obvious
considering Figure 2. By decreasing the pore size, the sigmoidal evolution of the gaseous contribution to
the thermal conductivity shifts to higher pressure. Remarkably, all measured data points collapse onto a
single curve that is quantitatively predicted by the Knudsen effect (c.f. Figure 2 right). Our best
nanoporous PMMA material has a three times better gaseous thermal insulation at ambient conditions than
e.g. Styropor® or Armaflex®.
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ACKNOWLEDGEMENTS
We gratefully acknowledge K. Koch’s input during her bachelor thesis work as well as ongoing
collaboration with Drs. A. Müller and R. Oberhoffer of Sumteq GmbH. RS is indebted to Prof. P. E.
Wagner, Vienna, for illuminating discussions on the Knudsen effect. We thank Prof. Meerholz of the
Institute for Physical Chemistry at the U. of C. for permission to use the SEM.
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