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Silver Sinter Heat Exchangers
Construction of a sinter press and a BET-system to measure
specific surface areas
Semester Project
Universität Basel
Supervised by
Prof. Dr. D. Zumbühl
K. Schwarzwälder
Tobias Bandi
July 7, 2008
Declaration
I have written this project thesis independently, solely based on the literature and
tools mentioned in the chapters and the appendix. This document – in the present
or a similar form – has not and will not be submitted to any other institution apart
from the University of Basel.
Bern, July 7, 2008
Tobias Bandi
Contents
1 Introduction
1
2 Theory
2.1 Dilution Refrigerators . . . . . . . . . .
2.1.1 Cooling cycle . . . . . . . . . .
2.2 Nuclear refrigeration . . . . . . . . . .
2.3 Kapitza resistance . . . . . . . . . . . .
2.4 The BET method . . . . . . . . . . . .
2.4.1 Adsorption . . . . . . . . . . .
2.4.2 Derivation of the BET equation
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3
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3 Materials and Methods
3.1 Sinter press . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 BET system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
14
16
4 Results and Discussion
4.1 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
22
23
Bibliography
24
List of Figures
27
A Appendix
28
B Appendix
29
C Appendix
31
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CHAPTER 1
Introduction
Many exciting physical phenomena like superconductivity or superfluidity can only
be observed at low temperatures near absolute zero. Once the physics is sufficiently
understood, applications of these effects, also at higher temperatures, can be designed.
In order to reach ultra low temperatures, complex experimental setups have to be
designed. Modern dilution refrigerators, which reach temperatures below 10mK,
rely on the ’evaporative cooling’ of mixtures of 3 He and 4 He. One difficulty limiting
the cooling power of such cryostats is the weak thermal coupling between liquid
helium and the surrounding matter. In order to minimize heat leaks, the incoming
Helium gas has to be cooled on its way to the mixing chamber. This is usually done
by transferring heat to the cold outgoing 3 He gas on its way to the still [1]. Also in
nuclear refrigeration stages, efficient heat exchangers are of special importance. As
the thermal boundary resistance, called Kapitza–resistance, is inversely proportional
to the interface area, heat exchangers with large surface areas are used for the heat
transfer. Usually sub–micron silver powder sinters are deployed for this purpose.
This project aimed at fabricating silver powder sinters and measuring their
surface area using the BET method. A press for compressing the silver powder was
constructed as well as an oven in which the sintering took place.
The BET method is based on adsorption of gas on a surface. Measuring the
amount of gas adsorbed at equilibrium pressure allows to deduct the surface area of
the sample. These measurements were done in a self–built system, optimized for
measuring surface areas in the expected range.
1
Chapter 1 Introduction
Tobias Bandi
This work is structured as follows: In the following chapter, a brief overview
over the theoretical background of dilution and nuclear refrigerators, the Kapitza–
resistance and the BET method is given. Chapter 3 covers the materials and
methods used to construct the press and the BET system. In chapter 4 the results
are presented and discussed. This final chapter also contains the conclusions and an
outlook.
2
CHAPTER 2
Theory
2.1 Dilution Refrigerators
Modern dilution refrigerators rely on
the ’evaporative cooling’ of mixtures
of 3 He and 4 He.
Below 0.85K, dilutions of 3 He and
4
He undergo a phase separation into
a 3 He–rich and a 3 He–poor phase.
Because of the higher zero–point energy of 3 He, the 3 He–rich phase is
less dense and floats above the 3 He–
poor phase. The amount of 3 He
in the 3 He-poor phase is about 6%
whereas the concentration of 4 He Figure 2.1: Schematic diagram of a dilution
refrigerator. From [1]
in the 3 He–rich phase is extremely
3
small. By pumping He–atoms from
the 3 He–poor phase, the equilibrium is disturbed and 3 He–atoms have to cross the
phase boundary to reestablish equilibrium. Energy is required to ’evaporate’ 3 He
from the 3 He–rich phase to the 3 He–poor phase, leading to cooling of the system [1].
Figure 2.1 shows a schematic setup of a dilution refrigerator.
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Chapter 2 Theory
Tobias Bandi
2.1.1 Cooling cycle
The 3 He is precooled in the helium bath and is then condensed at the 1K pot or
condenser. The 3 He at the 1K pot is connected to the 3 He-rich phase of the mixing
chamber through a flow impedance so that the pressure in the condenser is always
larger than the condensation pressure. This ensures that all 3 He is condensed,
minimizing heat transfer to the mixing chamber. The 3 He then flows into the mixing
chamber to the concentrated phase. On its way there, it is additionally cooled
by heat transfer to the outgoing gas in the 3 He-poor phase. Crossing the phase
boundary, 3 He–atoms provide cooling power. The phase boundary thus is the coldest
part of the system and the sample is located as near to it as possible. The 3 He atoms
then migrate through the 3 He–poor phase to the still where they evaporate due to
pumping and heating. The 4 He acts as a mechanical vacuum as there is virtually no
mutual friction between the two isotopes. This is why the dilution process is also
called ’upside–down evaporation’ [1, 2].
2.2 Nuclear refrigeration
Nuclear refrigeration is based on
the magnetocaloric effect. Exposing a material to a changing magnetic field causes a change in temperature. This phenomenon can be
used to reach extremely low temperatures down to a few microkelvin.
If no magnetic field is applied, the
nuclear spins in a paramagnet are
randomly ordered. Upon turning
Figure 2.2: Nuclear spins in an external magthe magnetic field on, 2I + 1 energy
netic field, [3]
levels with equidistant spacing are
generated where I is the nuclear spin. The spacing is given by
εm = −µn gn mB
(2.1)
4
Chapter 2 Theory
Tobias Bandi
where µn is the nuclear magneton, gn the Landé g-factor (for nuclei ∼
= 2), m the
magnetic quantum number and B the external magnetic field. If the system is
cooled while exposed to the magnetic field, all nuclear spins will eventually relax
into the ground state, see figure 2.2.
The entropy S of this system is
S=c
B2
T2
(2.2)
where c is a constant prefactor. From this relation follows, that if the magnetic
field is decreased adiabatically, e.i. without changing the entropy (S = const.), the
ratio B/T remains constant, and thus the temperature drops [1, 4, 5]. An overview
over a cooling cycle is shown in figure 2.3.
Figure 2.3: Schematic diagram of a magnetic refrigeration cycle. From [5]
An appropriate material for the magnetic refrigeration is copper, as it meets
the requirements of an ideal refrigerant better than most other materials [4]. The
demagnetization stage is connected to silver sinters placed in the mixing chamber.
After applying the magnetic field, the copper is cooled to the base temperature of
the dilution refrigerator by heat transfer to the helium mixture through the silver
heat exchangers. After that, the magnetic field is adiabatically decreased, which
leads to the magnetic cooling.
5
Chapter 2 Theory
Tobias Bandi
2.3 Kapitza resistance
A bottleneck in cooling the copper demagnetization stage is the heat transfer to the
liquid helium. At the boundary between liquid helium and the metal, a thin film
of 4 He is formed due to the lower zero-point energy of 4 He. The heat transfer thus
occurs in several steps, as schematically shown in figure 2.4. In the metallic bulk,
conducting electrons are mainly responsible for the heat transport. They couple to
phonons in the metal, which in turn couple to phonons in the 4 He–layer. From there
the energy is finally transferred to the 3 He–atoms.
Figure 2.4: Thermal coupling of liquid helium and metal [6]
The Kapitza conductance of heat over a phase boundary was first described by
the acoustic mismatch theory of Khalatnikov [7]. The theory bases on the analogy
of phonon coupling to boundary scattering in optics.
The efficiency of the crossing of phonons over the interface depends on the ratio
of the sound velocities of the two materials. The critical angle of incidence αlc at
which a phonon is transmitted is given by
αlc = arcsin(
vl
)
vs
(2.3)
where vl and vs are the sound velocities in the liquid and the solid respectively.
If the angle of incidence is larger than αlc , the phonon undergoes total reflection.
Figure 2.5 shows a schematic representation of the refraction/reflection principle.
The sound velocity of liquid helium is v4 He(l) = 200m/s and v3 He(l) = 183m/s. In
metals the sound velocity is much larger: vCu ∼
= 4700m/s and vAg ∼
= 3600m/s [8].
This huge discrepancy in the sound velocities results in a very small critical angle
(αlc < 3◦ ) and thus the probability for a phonon to cross the boundary is only about
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Chapter 2 Theory
Tobias Bandi
10−5 . The Kapitza–resistance for the heat flow over the interface is
RK =
c
∆T
=
AT 3
Q̇
(2.4)
Q̇ is the heat flow across the interface, ∆T is the temperature difference of the two
materials. c is a material–dependent constant and A is the interface area [1, 6, 7].
The acoustic mismatch theory is only
a crude approximation to the reality.
Nevertheless it is the best explanation
for the thermal boundary resistance
available today. The predicted Kapitza–
resistance is only about a factor 3 smaller
than experimentally determined values
[9]. This holds true for the low limit of
temperatures achievable with dilution
refrigerators. At higher temperatures
(T > 20mK) other effects like the thermal conductivity of the mixture in the
pores of the sinter and the thermal conductivity of the sinter become dominant Figure 2.5: Phonon reflection and refraction at a liquid–solid interface
and the temperature dependence of RK
[6]
goes down to ∼ T −1.5 [6, 9].
As the Kapitza–resistance is inversely proportional to the boundary area, it can
be minimized by increasing A. This is why powder sinters, which have large surface
areas, in the range of few square meters per gram, are good candidates for efficient
heat exchangers. In most cases, silver powder is used, as the sound velocity in silver
is small compared to other metals and because silver powders have a low sintering
temperature [10–12].
7
Chapter 2 Theory
Tobias Bandi
2.4 The BET method
The chemical, optical, mechanical and electrical properties of materials are largely
determined by their surfaces. Many important chemical reactions like the heterogeneous catalysis, sorption and stationary phases in chromatography occur at surfaces.
Porous materials, which have large surface areas, have a wide range of applications
like pharmaceutics, medical implants, cosmetics, paints and waste gas treatments.
In reactions which take place at surfaces, the
reaction rate depends heavily on the surface properties as well as on the surface area available for
the reaction. As an example, the Haber–Bosch
reaction for the synthesis of ammonium takes
place at the surface of appropriate catalysts. Figure 2.6 shows a cartoon and a chemical equation
of the reaction. Nitrogen gas adsorbs on the
surface and due to the interactions with the catalyst, the electronic orbitals are changed. In this
adsorbed state, the activation barrier for the ad- Figure 2.6: Haber–Bosch reaction at the surface
dition of hydrogen is lowered and the reaction
of a heterogeneous
can take place [13]. As the reaction is dependent
catalyst
on the surface, efforts were taken to maximize
the surface areas of the catalysts.
As the Haber–Bosch method is a perquisite for the synthesis of fertilizers and
many other reactions, the optimization of the reaction was very important. The
principle of the reaction was patented in 1910 by Carl Bosch, but it took till 1934,
that a method for measuring the surface area of porous materials was proposed by
Stephen Brunauer, Paul H. Emmett and Edward Teller, opening new possibilities
for the characterization of catalysts and other porous materials [14].
2.4.1 Adsorption
The method proposed by Brunauer, Emmett and Teller is based on the adsorption
of gas on a surface. Adsorption means the accumulation of molecules on a surface
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Chapter 2 Theory
Tobias Bandi
forming a film. The process is driven by the surface energy of atoms and molecules.
Dangling bonds at the surface can be saturated with adsorbed molecules by covalent,
ionic or van der Waals bonding. Many parameters influence the adsorption, from
which the gas pressure, the adsorption enthalpy, the surface area and the temperature
are the most important ones. The process is referred to as Physisorption if
the bonding is weak and mainly due to van der Waals forces. The enthalpy of
physisorption is less than 20kJ/mol (hydrogen bond ∼
= 21kJ/mol) and the sorption
is fast and reversible as there is no activation barrier. Physisorption can occur in
multiple layers. The BET method relies on this type of sorption. On the other hand,
if a stronger bonding occurs, e.g. covalent bonding, typically 50kJ/mol < ∆H <
400kJ/mol, the reaction is called Chemisorption. Only one layer can be formed
as a surface atom or molecule is one bonding partner.
For studying physisorption, it is convenient to adsorb gas on a surface at a constant
temperature (isotherm), which lies near the boiling temperature of the gas, and
to measure the amount adsorbed as a function of the gas pressure. Adsorption
isotherms contain informations about the strength of the interaction as well as the
surface area and the porosity of the samples. They are classified in six types by
IUPAC, see figure 2.7 [15].
Figure 2.7: Types of physisorption isotherms, [15]
The type I isotherm is characteristic for strong interactions between adsorbate
(the substrate) and adsorbent (the adsorbing substance). After the formation of one
monolayer no further adsorption occurs. Adsorption isotherms of the types III and
V reveal weak interactions, e.g. water on a hydrophobic surface.
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Chapter 2 Theory
Tobias Bandi
The most common situation is the type II isotherm. Initially, one complete
monolayer is formed (strong increase of the isotherm). After that, the adsorption
of multilayers takes place (linear region) and finally, at high relative pressures, the
condensation of gas on the surface is initiated. The relative pressure is the gas
pressure normalized with the condensation pressure of the adsorbent. Point B
indicates the pressure at which one complete monolayer is formed. The information
about the specific surface area lies in the low–pressure section of the isotherms.
Types IV and V I are special cases of the type II sorption isotherms. Stepwise
adsorption leads to type V I isotherms.
The hysteresis loops in the type IV and V
isotherms indicate the existence of pores. Capillary condensation leads to the formation of a
meniscus in the pores, as depicted in figure 2.8.
The process of formation of the meniscus is different from the process of desorption. This leads
Figure 2.8: Capillary condensato the hysteresis in the sorption isotherm and
tion in pores, [16]
allows to draw conclusions about the total pore
volume and the pore size distribution [17, 18].
All isotherms of the types II, IV and V I can be described by the BET theory.
2.4.2 Derivation of the BET equation
The BET theory is an extension to the Langmuir equation describing the monolayer
adsorption [19]. In this model, the fraction of the surface covered by adsorbate θ is
given by
θ=
Bp
1 + Bp
(2.5)
where B is a constant dependent on the adsorbate/adsorbent pair and p is the
relative gas pressure.
The first monolayer forms according to the Langmuir mechanism. On top of this
layer, multilayer–adsorption can take place. BET make five assumptions on which
their theory is based:
10
Chapter 2 Theory
Tobias Bandi
• The surface of the adsorbate is homogeneous and the adsorption potential is
equal at all points
• There is no lateral interaction between the layers
• Only the uppermost layer is in equilibrium with the gas phase
• There is a characteristic heat of adsorption for the first layer. The adsorption of
the second and upper layers underlies the heat of liquefaction. (Condensation
and evaporation of the gas above its liquid phase)
• At saturation pressure, the number of layers becomes infinite
At equilibrium, for the first layer, the rate of adsorption must be equal to the rate
of desorption:
a1 ps0 = b1 s1 e−EADS /RT .
(2.6)
p is the gas pressure, EADS is the heat of adsorption, T is the temperature, a1 and
b1 are are constants and s0 and s1 are the fractions of the surface area covered by 0
and 1 layers respectively. As every layer is only in equilibrium with the next–upper
layer, this can be extended to all layers so that for the i–th layer
ai psi−1 = bi si e−Ei /RT .
(2.7)
From the second assumption follows that for i = 1, E1 is equal to the heat of
adsorption and for the other layers Ei (i > 1) is the heat of liquefaction. As the
adsorption on the first and upper layers is equivalent to condensation, one can
assume that the ratio of ai /bi (i > 1) is constant and therefor can be rewritten as g.
The total surface area A of the sample and the total volume of gas adsorbed v
are given by
A=
∞
X
i=0
si and v = v0
∞
X
(2.8)
isi .
i=0
v0 is the volume required to form one layer of adsorbate on an area of 1cm2 . This
value depends on the crossectional area of one gas molecule. The fractions of the
11
Chapter 2 Theory
Tobias Bandi
surface covered by i layers, si , can now be expressed in the following way:
a1 EADS /RT
pe
b1
p
where x = peEL /RT
g
s1 = ys0 where y =
s2 = xs1 = yxs0
si = yxi−1 s0 = cxi s0
(2.9)
(2.10)
(2.11)
where c is
c=
a1 g EADS −EL /RT
y
=
e
x
b1
(2.12)
Thus the volume adsorbed v normalized with the volume required to form one
complete unimolecular layer vm gives
i
cs0 ∞
v
i=1 ix
=
P∞ i
vm
s0 [1 + c i=1 x ]
P
(2.13)
The sums in the numerator and the denominator are geometrical series and equation
2.13 can be expressed as
v
cx
=
vm
(1 − x)(1 − x + cx)
(2.14)
The fifth assumption states that the number of layers at the saturation pressure p0
goes to infinity. So if p = p0 then v = ∞ and x must be 1. Thus x is equal to p/p0 .
If x is substituted in equation 2.14, it can be rewritten as
cp
v
=
vm
(p0 − p)(1 + (c − 1)(p/p0 )
(2.15)
This equation connects the amount of gas adsorbed with the pressure [14, 19].
For the purpose of graphical illustration of the equation, it can be brought to the
following, linearized form:
p
1
c−1 p
=
+
v(p0 − p)
vm c
vm c p0
(2.16)
In this form, the equation corresponds to a straight line with 1/vm c being the intercept
with the y–axis and (c − 1)/vm c being the slope. In the range 0.05 < p/p0 < 0.3 the
12
Chapter 2 Theory
Tobias Bandi
linearity is usually best and a linear fit of the isotherm allows to determine c and vm .
Figure 2.9 shows a representation of equation 2.14, calculated with different values
of c.
Figure 2.9: Curves of eq. 2.15 (n being the volume in mols) for different values of
c: (A): c = 1; (B): c = 11; (C): c = 100; (D): c = 10000, From [19]
Equation 2.16 can explain experimental data extremely well and it has enjoyed
widespread use since its derivation by Brunauer, Emmett and Teller (some examples
can be found in [20–22]).
13
CHAPTER 3
Materials and Methods
In this section, the sinter press and the BET system that were built in this project
are presented. Step–by–step instructions, on how to use the two devices, are given
in the appendix.
3.1 Sinter press
The requirements for the sinter press were the following:
• The silver powder had to be pressed in an appropriate geometry under variable
pressure
• The sintering temperature had to be tunable
• The atmosphere under which the sintering was performed had to be controllable
The sinter was placed in a teflon box which was closed by a cap a with male
die part. The inner dimensions were 4mm × 4mm × 20mm which equals 0.32cm3 .
Teflon was chosen for its large expansion coefficient, compensating for the shrinking
of the powder during sintering [23].
The body of the press was made of brass and was placed on a copper piece in
which the heater (500W, 230V; Streuri GmbH, CH–9044 Wald) was placed. The
14
Chapter 3 Materials and Methods
Tobias Bandi
Figure 3.1: Section through a schematic picture of the sinter press
press was fixed to a cap of a cubic vacuum chamber and was thermally isolated from
the rest of the device, (Figure 3.1).
The temperature of the press was measured by a thermocouple on the copper
piece, and the heater was controlled with a commercial temperature–controller
(Panasonic AKT4111100J from Distrelec AG, CH–8606 Nänikon). The wires leading
to the heater and the thermocouple were transmitted through a feedthrough. The
pressure was measured by a manometer consisting of a pirani–gauge combined with
a capacitive membrane–sensor (THERMOVAC-Transmitter TTR 100; Oerlikon
Leybold Vacuum GmbH, D-50968 Köln).
The brass press body was screwed to the copper plate and thus was removable.
The silver powder was pressed under various pressures resulting in different packing
densities (pressing by hand or under a press at pressures up to 0.4 tons). Sintering
was performed under 1bar of helium gas for various temperatures and durations.
The silver powder used in all experiments was ’Silver Nanopowder’, 99.95% purity,
∼150 nm diameter (Inframat Advanced Materials LLC, 74 Batterson Park Road,
Farmington, CT 06032 USA). The sintering time was measured after reaching the
target temperature (which took about 7 min.).
15
Chapter 3 Materials and Methods
Tobias Bandi
3.2 BET system
The BET device for measuring surface areas was designed following previous researchers working with snow samples, which have specific surface areas similar to
silver sinters [24–26]. Figure 3.2 shows a representation of the setup. The device
consists of a dosing volume VD to which three valves are joined. The dosing volume
is a cross–shaped tube fitting with connections to the valves and the manometer.
The tube outer diameter is 6mm. All fittings and valves (except valve 4) were
obtained from Swagelok (Arbor Ventil & Fitting AG, CH–5443 Niederrohrdorf).
Valve 1 (quarter–turn plug) leads to a turbopump (Varian Turbo–V 81–M, Varian
Inc., D–64289 Darmstadt) and a rotary pump (Alcatel 2012 A). Valve 3 is a needle
valve that regulates the gas flow into the dosing volume. The connection to the
sample volume VS is made by valve 2 (quarter–turn plug). Right behind valve 2 the
tube outer diameter is reduced to 1/16” in order to minimize the dead volume of the
sample volume. Valve 4 is a three–way valve (High Pressure Equipment Company,
1222 Linden, Erie, Pennsylvania 16505, USA). In this valve, the connection from
valve 2 to the sample cell is always open, but there is another connection to the
pump which can be closed. This allows to pump out the sample cell more efficiently.
The system was checked for leaks with a leak detector (Smarttest HLT 560, Pfeiffer
Vacuum, 35608 Asslar, Germany). At a base leaking rate of 2.1*10−7 (mbar l)/s no
leaks were detected.
Figure 3.2: Schematic setup of the BET instrument
The volumina of the device were determined by expanding gas from another volume
to the system. During the measurement of the surface area, the dosing volume was
16
Chapter 3 Materials and Methods
Tobias Bandi
kept at room temperature and the sample was cooled with liquid nitrogen, allowing
adsorption. The temperature gradient had to be accounted for by introducing an
effective volume VS,ef f . The specific surface area was determined by adsorption of
N2 which is a standard adsorbent [19]. The volumina had to be designed according
to the requirements of the experiment. The BET theory predicts that the amount of
gas adsorbed depends on the pressure. If the dosing volume is too large, the pressure
difference after the expansion is too small to be detected. If it is too small, too
many points have to be recorded until higher relative pressures (p/p0 ) are reached.
The adsorption isotherm is recorded as follows: The sample is pumped out for 3
hours or over night. Then the sample cell is immersed in liquid nitrogen. A given
pressure of nitrogen gas is introduced in VD . From the ideal gas equation the exact
number of molecules can be determined. Now the gas is expanded to VS,ef f and the
pressure drops (1) because of the expansion to the dead volume of VS,ef f and (2)
because of adsorption of N2 on the sample. In order to minimize the first effect, the
sample volume is made as small as possible.
Figure 3.3: A typical example of an adsorption isotherm
The pressure at which the system settles, allows to determine the number of
molecules adsorbed. By repeating this procedure several times, points on the
adsorption isotherm are obtained. As the total amount adsorbed is the sum over
the adsorption of every step and every measurement has an uncertainty, the error
increases with every step. The error was estimated by error propagation calculations.
17
Chapter 3 Materials and Methods
Tobias Bandi
From a linear regression of the isotherm in the linear range the constant c and vm
can be extracted. Then the specific surface area is calculated, assuming a coverage
of 16.1 Å2 per nitrogen atom [11].
18
CHAPTER 4
Results and Discussion
Sinters have been fabricated with various packing densities, sintering temperatures
and sintering times. In the sintering process, the particles grow into each other by
material diffusion, without entering the liquid phase. However, below 250◦ C, only
partial sintering was observed and the results were very brittle samples that easily
broke apart. At 250◦ C the sinters showed a shiny surface which by eyesight looked
like solid silver. The microscopic structure was revealed in the Scanning Electron
Microscope (SEM).
Figure 4.1: Sinter 4; 200◦ C, 2h
19
Chapter 4 Results and Discussion
Tobias Bandi
Figure 4.1 shows a micrograph of a sample sintered at 200◦ C for two hours. Some
of the particles are already connected to their neighbors, but the majority of powder
particles are still not sintered. At 250◦ C the particles were much more sintered, as
can be seen in figures 4.2a and 4.2b. Figure 4.2a shows a picture of a fracture surface
of a sample sintered during one hour. The surface of the same sinter is depicted in
figure 4.2b. A sample sintered at 225◦ C (1h) had similar properties to the sinters
baked at 200◦ C.
(a) fracture surface
(b) surface
Figure 4.2: Sinter 3; 250◦ C, 1h
Adsorption isotherms were measured of several samples and repetitive measurement
was performed to estimate the reproducibility. A theoretical estimate of the surface
are of the powder, assuming 150nm particle diameter, is 4.01m2 /g. The specific
surface areas (SSA) measured, lied in the range of 1.5–2.3 m2 /g. The results of the
measurements are depicted in figure 4.3. A tabular form of these results are given
in the appendix, table A.1.
The sample sintered at 200◦ C had a packing density of 62% and an SSA of
2.69 ± 0.31m2 /g. This relatively high value is due to the partial sintering of the
powder. The samples sintered at 250◦ C had surface areas between 1.52 ± 0.32m2 /g
and 2.38 ± 0.33m2 /g. The error of ∼15% is mainly due to the error of the manometer
(15% up to 50mbar and 5% between 50mbar and 1000mbar). Compared to this,
other effects were small. However, the reproducibility of the results is better, ∼
2.7% which is an indication that the error is partially systematic and similar in all
measurements.
The specific surface area decreased with increasing packing density, see figure 4.4.
20
Chapter 4 Results and Discussion
Tobias Bandi
Figure 4.3: Specific surface area of sintered silver powder. Points with the same
color indicate the results of repeated measurement of a sample.
In order to connect the silver sinters with the demagnetization stage, copper wires
of 0.07” diameter (California Fine Wire, Grover Beach, CA 93483-0446, USA) were
placed in the silver powders and sintered to the powder. This was done by sawing
slits into the teflon boxes and the brass press body in which the wires were lain.
However, at 250◦ C sintering temperature, the wires were damaged during the baking
process and broke off when the sinter was taken out of the teflon box. Using thicker
wires could possibly solve this problem.
21
Chapter 4 Results and Discussion
Tobias Bandi
Figure 4.4: Specific surface area vs. packing density
4.1 Conclusions and Outlook
The present project aimed at fabricating silver sinter powders and measuring their
specific surface area. A press for sintering silver powder has been built and successfully tested. In the press, the packing density, the sintering temperature, the
atmosphere and the geometry of the sinters can be controlled. The sintering of the
powder was also affirmed by SEM micrographs.
The specific surface area of the samples was measured in a self–built BET system
using N2 –gas as adsorbent. The results are consistent with the expected values and
the literature [10–12]. The effect of several sintering parameters has been estimated.
The baking had to be performed at high enough temperatures in order to activate the
sintering process and the specific surface area was a function of the packing density.
However, for determining the optimal sintering parameters, further experiments are
22
Chapter 4 Results and Discussion
Tobias Bandi
required.
If the attachment of thicker copper wires to the sinters is successful, it would
be interesting to measure the electron temperature in a quantum dot coupled to
silver sinter heat exchangers. The latter could possibly be attached to the wires
connected to the experimental chip. In order to achieve this goal, a way to fit the
heat exchangers into the mixing chamber would have to be realized. Another aim is
the application of the silver heat exchangers in the demagnetization stage.
4.2 Acknowledgment
Special thanks goes to my tutor Kai Schwarzwälder who guided me through this
project for his great and very enthusiastic mentoring which allowed me to getting
an idea of some of the fascinations and challenges of ultra low temperature physics.
This project would have not been possible without the profound knowledge on the
BET method by Tony Clark. My thanks also go to Kristine Bedner for her kind
assistance at the SEM and all the members of the group of Prof. D. Zumbühl with
whom I had the pleasure to work together in these exciting months.
23
Bibliography
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In: JLTP 87 (1992), S. 595–633 (Cited on pages 1, 3, 4, 5, 7, and 27)
[2] Craig, N ; Lester, T: Hitchhikers Guide to the Dilution Refrigerator. http:
//marcuslab.harvard.edu/how\_to/Fridge.pdf, 2002 (Cited on page 4)
[3] University of Lancaster ULT Group. http://www.lancs.ac.uk/depts/
physics/research/condmatt/ult/demag.htm, . – visited on June 18th 2008
(Cited on pages 4 and 27)
[4] Pickett, G. R.: Microkelvin Physics. In: RPP 51 (1988), S. 1295–1340
(Cited on page 5)
[5] Brück, E: Developments in magnetocaloric refrigeration. In: JPD 38 (2005),
S. 381–391 (Cited on pages 5 and 27)
[6] deWaard, A.: PhD thesis. 2003. – Leiden University, Leiden, The Netherlands
(Cited on pages 6, 7, and 27)
[7] vanSciver, S. W.:
(Cited on pages 6 and 7)
Helium
Cryogenics.
Springer,
1986
[8] Lide, D. R.: Handbook of Chemistry and Physics. CRC Press, 1990-1991
(Cited on page 6)
[9] Cousins, D. J. ; Fisher, S. N. ; Guénault, A. M. ; Pickett, G. R. ; Smith,
E. N. ; Turner, R. P.: T −3 Temperature Dependence and a Length Scale for
the Thermal Boundary Resistance between Saturated Dilute 3 He–4 He Solution
and Sintered Silver. In: PRL 73 (1994), S. 2583–2586 (Cited on page 7)
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[10] Keith, V. ; Ward, M. G.: A Recipe for Sintering Submicron Silver Powders.
In: Cryogenics 24 (1984), S. 249–250 (Cited on pages 7 and 22)
[11] Itoh, W. ; Sawada, A. ; Shinozaki, A. ; Inada, Y.: New Silver Powders with
Large Surface Area as Heat Exchanger Materials. In: Cryogenics 31 (1991), S.
453–455 (Cited on page 18)
[12] Franco, H. ; Bossy, J. ; Godfrin, H.: Properties of Sintered Silver Powders
and their Application in Heat Exchangers at Millikelvin Temperatures. In:
Cryogenics 24 (1984), S. 477–483 (Cited on pages 7 and 22)
[13] Bozso, F. ; Ertl, G. ; Grunze, M. ; Weiss, M.: Interaction of Nitrogen
with Iron Surfaces. In: POC 49 (1977), S. 18–41 (Cited on page 8)
[14] Brunauer, S. ; Emmett, P. H. ; Teller, E.: Adsorption of Gases in Multimolecular Layers. In: JACS 60 (1938), S. 309–319 (Cited on pages 8 and 12)
[15] SING, K. S. W. ; EVERETT, D. H. ; HAUL, R. A. W. ; MOSCOU,
L. ; PIEROTTI, R. A. ; ROUQUEROL, J. ; SIEMIENIEWSKA, T.:
REPORTING PHYSISORPTION DATA FOR GAS/SOLID SYSTEMS with
Special Reference to the Determination of Surface Area and Porosity. In: PAC
57 (1985), S. 603–619 (Cited on pages 9 and 27)
[16] Fletcher, A. J.: Isotherms and Adsorption Theory. http://www.staff.
ncl.ac.uk/a.j.fletcher/isotherms.htm\#M, . – visited on June 21th 2008
(Cited on pages 10 and 27)
[17] Cohan, L. H.: Sorption Hysteresis and the Vapor Pressure of Concave Surfaces.
In: JACS 60 (1938), S. 433–436 (Cited on page 10)
[18] Barrett, E. P. ; Joyner, L. G. ; Halenda, P. P.: The Determination of
Pore Volume and Area Distributions in Porous Substances. I. Computations
from Nitrogen Isotherms. In: JACS 73 (1951), S. 373–380 (Cited on page 10)
[19] Gregg, S. J. ; Sing, K. S. W.:
and Porosity, 2nd edition.
London :
(Cited on pages 10, 12, 13, 17, and 27)
25
Adsorption, Surface Area
Academic Press, Inc., 1982
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[20] Peigney, A. ; Laurent, C. ; Flahaut, E. ; Bacsa, R. R. ; Rousset, A:
Specific Surface Area of Carbon Nanotubes and Bundles of Carbon Nanotubes.
In: Carbon 39 (2001), S. 507–514 (Cited on page 13)
[21] Hanot, L. ; Domine, F.: Evolution of the Surface Area of a Snow Layer. In:
EST 33 (1999), S. 4250–4255 (Cited on pages )
[22] Sing, K.: The use of Nitrogen Adsorption for the Characterisation of Porous
Materials. In: CSA 187 (2001), S. 3–9 (Cited on page 13)
[23] Kirby, R. K.: Thermal Expansion of Polytetrafluoroethylene (Teflon) From
−190◦ to +300◦ C. In: JRNBS 57 (1956), S. 91–94 (Cited on page 14)
[24] Bartels-Rausch, T. ; Ammann, M: A BET Instrument to Measure Very
Low Specific Surface Areas. http://lch.web.psi.ch/pdf/anrep03/17.pdf, .
– visited on June 21th 2008 (Cited on page 16)
[25] Domine, F. ; Cabanes, A. ; Taillandier, A.-S. ; Legagneux, L.: Specific
Surface Area of Snow Samples Determined by CH4 Adsorption at 77 K and
Estimated by Optical Microscopy and Scanning Electron Microscopy. In: EST
35 (2001), S. 771–780 (Cited on pages )
[26] Legagneux, L. ; Cabanes, A. ; Domine, F.: Measurement of the Specific
Surface Area of 176 Snow Samples Using Methane Adsorption at 77 K. In:
JGR 107 (2002), S. ACH5–1–15 (Cited on page 16)
26
List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
Schematic diagram of a dilution refrigerator. From [1] . . . . . . . .
Nuclear spins in an external magnetic field, [3] . . . . . . . . . . . .
Schematic diagram of a magnetic refrigeration cycle. From [5] . . .
Thermal coupling of liquid helium and metal [6] . . . . . . . . . . .
Phonon reflection and refraction at a liquid–solid interface [6] . . .
Haber–Bosch reaction at the surface of a heterogeneous catalyst . .
Types of physisorption isotherms, [15] . . . . . . . . . . . . . . . . .
Capillary condensation in pores, [16] . . . . . . . . . . . . . . . . .
Curves of eq. 2.15 (n being the volume in mols) for different values
of c: (A): c = 1; (B): c = 11; (C): c = 100; (D): c = 10000, From [19]
13
3.1
3.2
3.3
Section through a schematic picture of the sinter press . . . . . . .
Schematic setup of the BET instrument . . . . . . . . . . . . . . . .
A typical example of an adsorption isotherm . . . . . . . . . . . . .
15
16
17
4.1
4.2
4.3
Sinter 4; 200◦ C, 2h . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sinter 3; 250◦ C, 1h . . . . . . . . . . . . . . . . . . . . . . . . . . .
Specific surface area of sintered silver powder. Points with the same
color indicate the results of repeated measurement of a sample. . . .
Specific surface area vs. packing density . . . . . . . . . . . . . . .
19
20
B.1 Materials for pressing the silver powder . . . . . . . . . . . . . . . .
30
C.1 Schematic setup of the BET instrument . . . . . . . . . . . . . . . .
31
4.4
27
3
4
5
6
7
8
9
10
21
22
APPENDIX A
C
A [m2 /g]
Error in A
Sinter 2, 200 C, 1h
676 ± 4555
2.69 ± 0.307
11.44%
Sinter 3, 250◦ C, 1h,
58%, M1
1840 ± 5100948
1.79 ± 0.368
20.56%
Sinter 3, 250◦ C, 1h,
58%, M2
−124 ± 249
1.52 ± 0.323
21.28%
Sinter 3, 250◦ C, 1h,
58%, M3
2220 ± 47660
1.72 ± 0.314
18.22%
Sinter 6, 250◦ C, 1h,
53%, M1
4900 ± 248356
2.46 ± 0.340
13.80%
Sinter 6, 250◦ C, 1h,
53%, M2
−419 ± 2478
2.26 ± 0.356
15.78%
Sinter 6, 250◦ C, 1h,
53%, M3
−5860 ± 146272
2.32 ± 0.250
10.77%
Sinter 7, 250◦ C, 1h,
51%, M1
−1080 ± 7486
2.13 ± 0.23
11.01%
Sinter 7, 250◦ C, 1h,
51%, M2
−232 ± 435
2.15 ± 0.29
13.28%
Sinter 7, 250◦ C, 1h,
51%, M3
−599 ± 3869
2.38 ± 0.33
14.06%
Sinter-nr.
◦
Table A.1: Specific surface areas of sintered silver. Column 1 indicates sample
number, sintering temperature, sintering time, packing density and
number of measurement.
28
APPENDIX B
Howto
Fabrication of silver powder sinters
Caution! Silver powders are nanoparticles, invisible and can easily be inhaled.
When working with the powder, always use appropriate protection (gloves and
mask)
Pressing the Silver Powder
• Fill a the teflon box about half with silver powder. Distribute it evenly. Close
the teflon box with the cap with the long male die part.
• Put the box into the brass press body and press using cap C, see figure B.1.
Press either by hand or with the large press in the student practice, 3rd floor,
DoP.
• Repeat the procedure until the box is full of pressed silver powder. When the
box is nearly full, switch to the cap with the short male die part.
Sintering the pressed powder
• Put the teflon box into the press body and screw latter to the copper piece of
the heater
• Close the cap of the vacuum chamber and pump for ∼10 minutes with the
rotary pump. Flush the vacuum chamber with helium gas and pump again
29
Appendix B Appendix
Tobias Bandi
Figure B.1: Materials for pressing the silver powder
• Fill the vacuum chamber with the desired amount of gas
• Switch the Controller on. The upper red display shows the actual temperature
and the green number is the target temperature. By pressing ’MODE’ and
then ’↑’ or ’↓’ you can adjust the target temperature
• Plug in the cable of the heater. Now the heating starts
• After the desired time, unplug the heater and leave the system cool down (this
takes about 45 minutes)
• Remove the sinter from the teflon box and measure its weight to determine
the packing density
30
APPENDIX C
Howto
Measurement of the BET surface
Figure C.1: Schematic setup of the BET instrument
• Unscrew the sample cell and put the sinter into it. Screw it on firmly
• Flush the system several times with nitrogen gas
• Pump the whole system with the turbo pump for min. 3 hours or over night.
Heat out the system with the heating gun.
• Immerse the sample cell in liquid nitrogen up to a certain level. During the
whole experiment the level of liquid N2 has to be kept constant
• Close valves 1, 2 and 4. Add a certain pressure of N2 gas to the dosing volume.
Record this pressure
31
Appendix C Appendix
Tobias Bandi
• Open valve 2 and leave the system settle. Record the equilibrium pressure
• Close valve 2 and add another amount of gas to VD . Record the pressure
• Open valve 2, wait for equilibration and record the final pressure
• Repeat this procedure up to a pressure of ∼300mbar (p/p0 ∼
= 0.3). Record
∼10–15 points between (0.05 < p/p0 < 0.3)
Analysis of the Data
The amount of gas in the system, before the expansion, is the sum of the amounts
in the dosing volume and the sample volume:
V1 = VD,1 + VS,1 =
1
(pD VD + pS VS )
RT
(C.1)
1
peq (VD + VS )
RT
(C.2)
and after the expansion
V2 = VD,2 + VS,2 =
Thus the volume of gas adsorbed is given by:
∆VADS = V1 − V2 =
VD (pD − peq ) + VS (pS − peq )
RT
(C.3)
The effect of the temperature gradient and the volume of the sample have to be
accounted for in VS,ef f (effective volume). Therefor T in the above equations is
always equal to the room temperature. By summing up the adsorbed volume over
the steps, pairs of VADS and peq /p0 are obtained. They can then be plotted according
to equation 2.16, figure 3.3. The errors can be determined by an error propagation
calculation. A linear regression of the adsorption isotherm in the linear range leads
to a linear equation. The slope is equal to Vc−1
and the intercept with the y–axis is
mc
1/(Vm c). Thus
Vm =
1
1
and c =
slope + intercept
Vm ∗ intercept
(C.4)
The sizes of the dosing volume and the (empty) sample volume at room temperature were: VD = 9.76 ± 0.83 cm3 and VS = 5.43 ± 0.54 cm3 .
32