The Viscosity of Metal - Ammonia Solutions:
Are They Superfluids?
N.E. Shuttleworth
March 24th, 2006
Department of Physics and Astronomy, University College London
Email: nick.shuttleworth@gmail.com; Phone: 07834-365878;
WWW: www.ucl.ac.uk/~zcapxa5/Superfluids/
________________________________________
Abstract
The overall aim of this project was to attempt viscosity measurements of a range of
concentrations of lithium-ammonia solutions. Due to complications in the design of
the equipment required for the realisation of the above, this aim had to be scaled back
severely. Hence, this report details the theoretical basis, design, construction and basic
testing of a free oscillation rotational torsion pendulum viscometer designed for use
with low viscosity liquid samples, such as lithium-ammonia solution, and a Stirling
cooler. It contains detailed information on the induction sensing techniques that were
employed and discusses the types of signals produced and their analysis.
Keywords: Electromagnetic Induction; Induction Sensing; Lithium-Ammonia; Low
Temperature; Oscillating Rotational Viscometer; Stirling Cooler; Torsion Pendulum;
Viscometer; Viscosity.
1
Contents
1
Contents
2
2
Introduction
4
2.1
Viscosity
4
2.2
Alkali Metal-Ammonia Solutions
5
2.3
Viscometers and Viscosity Measurement
6
2.3.1
Rotational Viscometer Geometry
7
2.3.2 Free Oscillation Rotational Viscometers
8
2.4
Motion Sensing Techniques
10
2.5
Relevant Previous Work
12
2.6
Initial Project Aims and Objectives
13
2.6.1 Revised Project Aims and Objectives
13
3
Equipment Design
14
3.1
Overview
14
3.2
Torsion System
15
3.2.1 Torsion Bob
15
3.2.2 Torsion Wire
17
3.2.3 Motion Starting Equipment
18
3.2.4 Rotational Motion Lock
18
3.3
19
Vacuum Equipment
3.3.1 Lower Vacuum Sleeve
19
3.3.2 Test Chamber
19
3.3.3 Gas Connections
20
3.3.4 Wiring Connections
20
3.4
20
Cooling
3.4.1 Stirling Cooler Head
20
3.4.2 Sprung Thermal Contact
21
3.5
20
Induction Sensing
3.5.1 Magnets
21
3.5.2 Sensing Coils
21
3.5.3 Signal Collection
22
3.6
22
Incomplete Pieces
3.6.1 Torsion Bob
22
2
3.6.2 Motion Starting Equipment
23
3.6.3 Sprung Thermal Contact
23
3.6.4 Sensing Coils
24
4
Induction Signal
24
4.1
Signal Analysis
28
4.2
Signal Error Analysis
30
4.3
Signal Error Investigation
31
4.4
Error Correction
33
5
Testing
33
5.1
Torsion Wire Tests
33
5.2
Frequency Tests
34
6
Conclusions
36
6.1
Equipment Conclusions
36
6.2
Signal Conclusions
36
6.3
Testing Conclusions
36
6.4
General Conclusions
37
7
Future Work
37
[References
38]
3
2 Introduction
2.1 Viscosity
The viscosity of a liquid is a measure of its resistance to deformation under a shear
stress 1 . It characterises the degree of internal resistance in a liquid. This internal
friction, or viscous force, is associated with the resistance that two adjacent layers
have to moving relative to each other2. The more fluid the liquid, i.e. the more easily
it flows, the lower the viscosity value.
There are two types of viscosity, measured in different units. Dynamic viscosity is
measured in pascal seconds (Pa. s), although it is more commonly expressed as
centipoise (cP – 1cP ≡ 10-3 Pa.s). Kinematic viscosity values are as those of dynamic
viscosity but divided by the density. The SI unit of kinematic viscosity is m2 s-1. The
cgs unit is the stokes (cm2 s-1). This project will deal universally with dynamic
viscosities and use units of centipoise.
The viscosities of a variety of liquids and gases are included below (table 1).
Substance
Viscosity (cP)
Benzene (g)
0.0070
Hydrogen (g)
0.0084
Carbon Dioxide (g)
0.0137
Air (g)
0.0173
Neon (g)
0.0298
Methanol (l)
0.543
Benzene (l)
0.603
Water (l)
0.890
Mercury (l)
1.528
Castor Oil (l)
700
Table 1 - Viscosity values for various gases and liquids. Gas values are at 273K, liquid at 298K.3
The viscosities of liquids tend to be several orders of magnitude higher than those of
gases but not in all cases. A superfluid is a liquid with an exceptionally low viscosity
caused by all the atoms being in the same low quantum state. The only known
superfluids are phases of liquid 3He and 4He, which exist below 2.4mK and 2.2K
respectively. They have almost infinite thermal conductivity, low density and
effectively zero viscosity (4He viscosity ~ 10-8 cP). As a result, they exhibit a wide
4
range of unique phenomena such as thin film flow, the Meisner effect and quantised
vortices.
2.2 Alkali Metal-Ammonia Solutions
The alkali metals are lithium, sodium, potassium, rubidium, caesium and francium.
These metals dissolve readily in amines like ammonia without chemical reaction; the
bond formed is purely one of charge. Solvated electrons form during dissolution, their
numbers growing with increasing concentration to saturation at between 15.5
(sodium) and 21.1 (lithium) mole percent metal4 (MPM). At saturation the solutions
are very good electrical and thermal conductors with low densities and low
viscosities.
Figure 15 6, below, shows the phase diagram for lithium-ammonia (Li-NH3).
Figure 1 - Li-NH3 phase diagram. At low concentrations (I), below about 1MPM, the solutions have an
intense blue colour, due to the presence of the solvated electrons. Around 4MPM the solutions take on
a bronze colour and become metallic (II). Below the consolute temperature (T C) the metallic and nonmetallic liquids do not mix, and there is a phase separation in which the metal floats on the non-metal
(III). Regions IV and V are solid ammonia and solid lithium respectively. Note also the deep pseudoeutectic at saturation, which extends down to 88K to give the lowest temperature liquid metal.
5
Viscosity in metal-ammonia solutions decreases with increasing numbers of solvated
electrons, and the higher the metal concentration, the more electrons, the lower the
viscosity (see section 2.5). It is thought that the difference in viscosities between
lithium and the other alkali metals is due, at least in part, to the shape of the molecules
formed7. Lithium has the smallest radius and so the ammonia molecules pack together
most tightly around it, forming a spheroid. There are only very few ways of arranging
the ammonia molecules so all the Li-NH3 molecules are very similar. These
molecules are surrounded by a sea of the solvated electrons and move within it,
effectively lubricated by it. For the other metals, sodium, potassium etc, the molecules
formed with the ammonia are not as uniform simply because the metal atom is larger
so there are more ways in which the molecules can be configured. This lower
uniformity leads to more internal friction and reduced flow rates.
2.3 Viscometers and Viscosity Measurement
There are four main types of viscometer: capillary, falling ball, vibrating wire and
rotational. Capillary viscometers works on the principle that a pressure difference
across a capillary tube will cause material to flow through it. In most circumstances
the pressure difference along the capillary is measured as a function of the material
flow rate and viscosity is inferred from this data.8
Falling ball viscometers use the time taken for a smooth ball to fall a known distance
through a material to infer viscosity. The more viscous the material through which the
ball travels, the longer the time of travel. An air bubble viscometer works in a very
similar manner, but with a bubble passing upwards instead.
Vibrating wire viscometers measure the viscous damping of an oscillating electrically
conductive wire in a permanent magnetic field. The movement of the wire induces a
voltage and so the voltage is proportional to the velocity. Viscosity is found in one of
three ways: by measuring the power required to maintain the amplitude of oscillation;
by measuring the ring-down time of the oscillations (the ‘Q’ factor); by measuring the
frequency of the resonator as a function of phase angle between excitation and
response waveforms.9
There are many types of rotational viscometer, but all tend to run on one of two
principles: controlled stress or controlled strain. Controlled stress measurements use a
controlled harmonic torque applied to a boundary surface. For example, the torque
6
produced by a motor driving a bob immersed in a sample of material is kept constant
and the resulting speed attained measured. Controlled strain measurements use a
controlled angular displacement.10 For example, the torque required to rotate a torsion
bob at a constant angular speed within a sample of material is measured. These
principles are applied to a variety of geometries to create rotational and oscillating
viscometers, where the exact relationship between torque, angular speed and viscosity
depends on the particulars of the chosen geometry.
A free oscillation rotational type viscometer was chosen for this project. There
follows a description of the available geometries and a basic theoretical treatment of a
freely oscillating rotational viscometer.
2.3.1 Rotational Viscometer Geometry11
The main experimental arrangements for the measurement of viscosity in liquids are
shown below (figure 2).
a
b
c
Figure 2 - The main torsion bob geometries used for dynamic viscosity testing. a) Cone-plate; b)
Parallel plates; c) Coaxial cylinders
In the cone-plate geometry (figure 2a) the test material is held between a plate of
radius R and a cone of angle α. Therefore the shape factor, A, which mathematically
describes the geometry of the system, is:
A
2R 3
3
(eqn. 1)
This equation is valid only for small cone angles (1-10°) for laminar flow. This
geometry gives a constant shear rate at all points within the material under test.
In the parallel plate geometry (figure 2b) the test piece is a cylindrical rod of radius R
and thickness h:
A
R 4
2h
(eqn. 2)
7
In this geometry the shear rate is not uniform: it is maximum at the plate edge and
zero in the centre, and care must be taken to ensure the amplitude of oscillation is
small enough for the linearity hypothesis to be valid.
In the case of coaxial cylinders (figure 2c) the shearing occurs between an inner
cylinder of radius RI and length h and an outer cylinder of radius RO. The inner
cylinder is driven in oscillatory motion and the outer records the harmonic torque. In
the limit of small RO – RI there is a constant shear rate between the cylinders and the
shape factor is:
A  4h
2
RO RI
2
RO  RI
2
(eqn. 3)
2
2.3.2 Free Oscillation Rotational Viscometers12
Experiments involving free oscillations are a simpler alternative to forced oscillations
or constant angular speed experiments. The experimental setup is much simpler, and
is less flexible as a result, but yields viscosity directly from measurement of only
angular displacement. In most applications the basic experimental setup is that of a
torsion pendulum. An inertia member imposes a torque on the sample when given an
angular momentum. When the member is released it experiences free torsional
oscillations where the rate of damping depends on the sample viscosity.
In a free oscillation viscometer the member supporting the rotating mass undergoes
damped harmonic oscillations. In the viscoelastic domain, the angular displacement θ
is damped as:
 ( , t )   0 cost   e  dt
(eqn. 4)
The equation of motion for free oscillations is:
0I
d 2 
 AG   B  
dt 2


(eqn. 5)
where I is inertia
B is elastic constant
A is experimental shape factor
G is complex shear modulus
8
Assuming η’ and G’ = η’’/ω are slowly varying in the narrow frequency range
covered by the experiment:
  ( , t )   0 exp  j (t   )exp  dt 
(eqn. 6)
with
d
A '
2I
2 
(eqn. 7)
AG' B
d2
I
(eqn. 8)
To evaluate the above for frictional loses in the wire and other parts of the system,
solve for a purely viscous material:
0I
d 2 
d 



A


n
 B 
2
dt
dt
(eqn. 9)
where n is a term including the frictional forces.
Thus the equation of motion without sample is:
d 2 
d 
0I
 An
 B 
2
dt
dt
(eqn. 10)
which has the solution:
  ( , t )   0 exp  j ( 0 t )exp  d 0 t 
(eqn. 11)
with
d0 
An
2I
(eqn. 12)
2 
B
2
 d0
I
(eqn. 13)
where ω0 is the natural frequency of oscillation.
Thus, with sample:
  ( , t )   0 exp  j (t )exp  dt 
(eqn. 14)
with
d
A  n 
A
 d0 
2I
2I
2 
B A2   n 
nA2  2 A2
2





0
I
4I 2
2I 2
4I 2
(eqn. 15)
2
(eqn. 16)
where
ω is angular frequency
9
ω0 is natural frequency
η is sample viscosity
n is frictional term
A is shape factor
I is inertia
B is elastic constant
In words, the above equates to the following. The successive damping of the
oscillations of a torsion pendulum suspended in a sample occurs due to the frictional
forces exerted by the viscosity of the sample. Measurement of this decrease in
oscillation amplitude will, therefore, yield the value of the viscosity of the sample.
The motion shown below (figure 3) is expected. The frequency remains effectively
constant with time but the amplitude decreases exponentially.
Figure 3 - Damped harmonic motion
2.4 Motion Sensing Techniques
Motion sensing techniques for a rotational oscillating viscometer fall broadly into two
categories: optical and magnetic. The optical approach is quite basic and involves the
use of a laser-mirror system to detect the change in angular displacement of a bob
suspended from a torsion wire. The laser beam enters the test area, strikes a mirror
mounted on the top of the bob and reflects onto a detector (e.g. a screen or PSD). The
mirror oscillates with the same motion as the bob so the angle at which the laser beam
is reflected changes depending on the position of the bob. From this the motion of the
bob can be later reconstructed. This method has the advantage of giving a reading at
all points of the torsion bob’s motion. However, detection can be a problem; the use
of a screen requires manual readings to be taken, either concurrently or later from a
video playback, which limits the number of readings. Large arrays of PSDs could
10
solve this problem but are impractical due to their expense, so smaller arrays must be
used and placed closer to the oscillating mirror. This, however, significantly decreases
the positional resolution.
Magnetic motion sensing involves the application of the concept of electromagnetic
induction. The basic idea is that a magnet is mounted on the oscillating torsion bob
which, on passing an induction coil, induces a voltage proportional to the angular
velocity of the bob. Strictly, the voltage is proportional to the rate of change of flux
density, so that it increases with both increasing speed and increasing magnetic field
strength. This method has the advantage that the induced voltage can be read and
recorded automatically (using a suitable hardware connection, such as a LabJack) and
so a large number of readings can be taken. The main disadvantage of this method is
that is does not give a usable signal at all points in the bob’s motion. Because the
induced voltage is proportional to the angular velocity, and not the displacement,
there is little signal when the bob is travelling relatively slowly towards the extremes
of its motion and none at all when it is stationary at the extremes. A signal similar to
that shown below (figure 4) is expected.
Figure 4 - The expected signal shape. The minimum signal sections (marked ‘V=0’) occur when the
bob is near the extremes of its motion and it is travelling at its slowest. The signal maxima (marked
‘V=max’) occur as the bob passes the equilibrium position, when it has its greatest speed.
Despite the absence of a signal for periods of the torsion bob’s motion, this detection
method is sufficient for the purposes of this study. This is because measurement of the
exponential decrease in amplitude, used to calculate sample viscosity, is equivalent to
that of angular speed, due to the linear relationship between the two. Therefore, the
incremental decrease in the maximum signal strength can be used to find viscosity,
without the need for data from the ‘dead’ periods in the induced signal.
11
2.5 Relevant Previous Work
The viscosity of sodium-ammonia (Na-NH3) has been studied in the intermediate and
concentrated ranges between -30°C and +30°C by Kikuchi 13 . The viscosity was
observed to decrease with both increasing concentration and increasing temperature.
Potassium-ammonia (K-NH3) has been studied by O’Reilly and Meranda 14 and
Lobry 15 , who also studied Li-NH3. Together these studies yield the following
characteristics: metal-ammonia solutions become less viscous as concentration
increases and also as temperature increases. The viscosity begins to decrease more
rapidly between 1 and 2MPM. Around 9 to 10MPM the slope changes again with the
decrease becoming less rapid, and, for Na-NH3 and K-NH3, tends to a limit. This
change in slope is not seen in Li-NH3 until 18.2MPM (no measurements have been
made at higher concentrations). The following graph16 (figure 5) shows the results for
the viscosities of three alkali metal-ammonia solutions.
Figure 5 - Graph of viscosity vs. concentration for Li-NH3, Na-NH3 and K-NH3
Lobry’s work is the most recent study of the viscosity of Li-NH3 at midconcentrations (no specific reference could be found to another). He used an Ostwald
capillary viscometer to perform viscosity measurements over a concentration range
from 3 - 18.2MPM and a temperature range of -40°C to +10°C. The lowest viscosity
value he obtained was at the highest concentration solution reached: 0.135cP at
18.2MPM and -30°C 17. As has been stated above the viscosity-concentration curve
12
for Li-NH3 appears to continue to fall past this point so lower values are expected.
This continued fall in viscosity with concentration coupled with the already low
viscosity and other superfluid-like properties of high concentration Li-NH3 solution
(very good electrical and thermal conductivity, low density) makes it a good
candidate for being the first high temperature superfluid.
In 2004/5 a UCL student named Jo Bartlett made an attempt to fulfil a very similar set
of aims to those of this project (see section 2.6) 18 . She designed and built one
rotational and one capillary viscometer; the rotational viscometer was constructed
with controlled stress/strain measurements in mind. She encountered a number of
problems, mainly related to construction but also due to the original design ideas. The
process used to attempt viscosity measurements using the capillary was to freeze a
sample at the top of a tube, melt it and measure the time taken for it to pass through
the tube. The sample tended to travel so quickly down the tube that the time taken was
immeasurably small and so no usable figure for viscosity could be calculated. This,
and Lobry’s failure to measure above 18.2MPM with a capillary caused capillary
viscometers not to be included in this study.
2.6 Initial Project Aims and Objectives
1. To design, build and calibrate at least one rotational type viscometer.
2. To determine the viscosity of Li-NH3 solution over a range of concentrations
from 1MPM to 21MPM at ~240K.
3. To establish a relationship between absolute temperature and viscosity for LiNH3 solution at saturation.
2.6.1 Revised Project Aims and Objectives
1. To thoroughly research, carefully design and construct one rotational type
viscometer.
2. To design and build two torsion bobs.
3. To calibrate the torsion bobs and seek values for the viscosity of standards.
The revision of the project aims was caused by the realisation that the design process
would be much more complicated than originally thought. Little of the previous
practical work related to this project (see section 2.5) could be used due to the style of
the rotational viscometer chosen being different to that of the previous work and so
design had to start from a more fundamental stage than was expected.
13
3 Equipment Design
NB: Technical drawings and dimensions for all existing parts described below can be
found in the appendix.
3.1 Overview
Based on the research shown above an experimental setup was devised which would
be used to attempt measurements of the viscosity of Li-NH3 solution over a range of
temperatures and concentrations. The setup is shown below in figure 6.
Figure 6 - Overview/orientational view of the devised experimental setup.
14
The following sub-sections describe the important aspects of the design on a system
by system basis.
3.2 Torsion System
At its heart the principle underlying this equipment is that of the torsion pendulum,
with the torsion system consisting of the torsion bob, torsion wire, rotational motion
lock and motion starting equipment.
3.2.1 Torsion Bob
The torsion bob is the only member of the experimental setup which interacts directly
with the sample under test. As treated above (see section 2.3.2) the torsion bob
performs simple harmonic motion which is damped by a frictional force arising from
contact with the sample. The restoring force is due to the torsion wire, to which it is
attached, which is in turn attached to the rotating shaft at the top of the experiment.
The design of the torsion bob makes a large difference to both the types and
sensitivities of measurements that can be made using a given experimental setup.
Simple bob designs are shown above (figure 2). These simple designs became
impractical for a number of reasons. It was hoped that it would be possible to use a
parallel plate geometry (a flat bottomed torsion bob in a flat bottomed cup containing
the sample) for the torsion bob but this turned out to be impractical. This geometry
would have made for high quality surfaces between which the sample would be
tested, because of its simplicity; no manufacturing problems should occur with this
simple a setup. However, the spatial limitations imposed by the vacuum sleeves meant
that the plate radius would have to be so small that an effect would be unlikely to be
seen. This problem is common in the simple geometries: their simplicity makes them
easy to set up but also means that their range of applications is limited.
Because of the spatial limitations a more complex design was required in order to
effectively use the remaining space. The geometry best suited to the radially small but
axially unlimited space available in the test chamber is that of stacked plates (figures
6, 7). The theoretical basis of this geometry is complicated by the differences in the
environments of the plates. If close to the bottom of the cup containing the sample,
the bottom plate acts as in a parallel plate setup, although if moved away from the cup
it performs as a thin object moving through a fluid. The latter is also the case for all
the higher plates.
15
The stacked plates torsion bob has a potential characteristic which does not apply to
the simple parallel plate, involving the viscous penetration depth (δ) of the sample. If
the spacing between the stacked plates is sufficiently small, significantly less than δ,
viscous components interacting with both the plate above and below can become
trapped, rather than being accelerated by and oscillating with the torsion bob, as is
expected. This leads to a different effect than that desired being seen. An increase in
inertia, due to the trapped material, becomes apparent and the frequency of oscillation
falls. However, the damping effect is reduced, compared with the case of a wider
plate spacing, because the plates now move within a fluid of reduced viscous
elements. This phenomenon can be exploited to make viscosity measurements (see
E.L. Andronikashvili; J. Phys. USSR; 1946) but if attempting to calculate based on
the damping it will give erroneous results.
Despite the insurmountable problem (detailed above) with the simple parallel plate
geometry, it did prove useful as a working model with which aspects of the torsion
bob, other than the direct interaction of the bob and sample, could be developed.
Figure 7 shows some of these features, which are explained in detail below. The
features of the parallel plate torsion bob that appear in the diagram are carried over to
the more complex geometry and so appear in that image also.
a
b
Figure 7 - a) Simple parallel plate geometry torsion bob showing developed features. b) Stacked plate
torsion bob showing features carried over to more complex geometry.
1. Magnet Mounting Hole
This is a hole drilled through the centre of the torsion bob shaft in order to allow
for the mounting of a magnet for induction sensing. The round shaft around this
hole has been flattened on two opposite sides to also allow for the mounting of
16
low profile (less than 1mm thick) mirrors so that optical sensing methods are also
possible.
NB: This hole is designed specifically to fit the magnets detailed in section 3.5.1.
2. Screw Thread on Shaft
This allows for the addition of mass in order to tune the oscillating frequency of
the torsion bob. Circles of metal with holes in the centre can be slid down the
shaft to increase the mass and thereby decrease the frequency if required (see
section 5.2). These ‘doughnuts’ are firmly attached to the bob by a nut which is
screwed down the thread above them and locks then against the top side of the
parallel plate.
3. Balancing Screw Hole
Two holes with screw threads pass through the shaft parallel to the plate and at
right-angles to each other. These allow the insertion and retraction of balancing
screws which can be used to make fine corrections to the orientation of the
parallel plate below to ensure it is indeed parallel with the surface beneath it.
NB: See also section 3.6.
3.2.2 Torsion Wire
The torsion wire suspends the torsion bob from the shaft at the top. It is connected to
the rotational feedthrough at the top by a pin vice and to the torsion bob at the bottom
by a wire connector. The wire connector is a 10mm long, threaded piece of aluminium
into which the wire is secured, the whole piece screwing firmly into the top of the
torsion bob shaft.
There are strong limitations placed on the material from which the torsion wire can be
made. These come predominantly from the type of solution being tested. The gaseous
ammonia (which will be condensed onto an amount of lithium in the test chamber, in
order to form the Li-NH3 solutions) is highly reactive and will attack a wide variety of
torsion wire materials. Therefore the list of possible materials is very short:
practically, stainless steel and quartz, with a handful other much less practical
suggestions. The reason for the concern being so great about the torsion wire being
attacked is down to the sensitivity of measurement that will be required when
readings of the viscosity of low viscosity solutions are being taken. It is possible that
17
corrosive substances, such as gaseous and liquid ammonia and Li-NH3 solutions, will
progressively degrade the quality of the torsion wire. This will change the physical
characteristics of the wire and hence the manner in which it resists force, giving
different frequency and amplitude readings. Because the damping effect of low
viscosity samples will be very small it would be easy for it to become lost in
uncertainties arising from this.
It is important that the torsion wire used is straight. This is difficult with easily
commercially available wire, such as piano wire, because it is stored for extended
periods on rolls which causes bending. Bends in the wire amplify any axial
asymmetry and cause seemingly random variation in experimental voltage readings.
For example, if the rotational feedthrough, shaft, wire, torsion bob and cup are all
axially symmetrical and motion is begun from all components being absolutely
stationary then pure rotational motion occurs, with no linear motion at all. If,
however, the torsion wire is even slightly bent then, on rotating the shaft to begin
motion, the torsion bob will be swung out of line, along the preferential direction
created by the bend, and will start to oscillate linearly as well as rotationally. This
problem is discussed further in sections 4.2, 4.3, 4.4.
3.2.3
Motion Starting Equipment
Equipment is to be mounted to the outer end of the rotating shaft to allow for
computer control of the initial motion. This will consist of a motor and a brake. The
motor will be connected to the shaft with a series of gears. It will be started at a
reasonable speed (at which cumulative damage to the wire will not be caused by
repeated execution, affecting the elastic properties), allowed to rotate the wire for a
number of complete turns and turned off. The brake will then act to stop the rotations.
The rotational motion lock (see section 3.2.4) will then be applied to secure the wire.
Integer numbers of rotations are required in order to ensure that the induction sensing
coils sit at the equilibrium position of the motion once it begins. Computer control
aids reproducibility and comparability or results.
NB: See also section 3.6.
3.2.4
Rotational Motion Lock
The rotational motion lock consists of several pieces (figure 8).
18
Figure 8 - Schematic of rotational motion lock. The locking screws are relaxed to allow the centre
cylinder to rotate with the shaft. Once rotation is complete a brake slows motion and the screws are
retightened, stopping all motion.
The purpose of the lock is to ensure that none of the rotational energy of the torsion
system is lost through undesired movement of the shaft once oscillations are
underway. If energy were to be being lost in this way viscosity measurements would
appear larger than was actually the case. The effect would not be possible to adjust
measurements for and so the strength of the locking mechanism is great to ensure that
this does not occur at all.
3.3
Vacuum Equipment
3.3.1 Lower Vacuum Sleeve
The purpose of the lower vacuum sleeve is to keep a vacuum around the bottom end
of the test chamber and the Stirling cooler cold head. The reason for this is that
vacuum allows the heat transfer between the cold head and the test chamber to be
most efficient, thereby leaving the test chamber at the lowest possible temperature.
3.3.2 Test Chamber
The bottom of the test chamber is where the sample sits with the torsion bob
suspended in it. The entire chamber is removable to allow for the safe preparation of
hazardous samples in a fume cupboard. The bottom section, the test area, is
surrounded by vacuum to keep temperatures low. The surface inside is highly cleaned
and smoothed to ensure that the sample motion occurring during test is as regular and
predictable as possible.
19
3.3.3 Gas Connections
There are a total of three gas rig connections on the main vacuum sleeving and test
chambers. Two of these are for vacuum rig connections so that the test chamber and
inter-sleeve space can be evacuated. The third is a connection for gaseous samples to
be allowed into the test area. This allows for both gaseous addition with subsequent
cooling to the liquid phase in order to make liquid viscosity measurements, and the
formation of hazardous compounds in the test chamber, reducing the amount of
handling required. For example, Li-NH3 can be formed by condensing gaseous
ammonia onto small lithium pieces at the bottom of the test chamber removing the
need to handle the solutions.
3.3.4 Wiring Connections
Two electrical feedthroughs, one to the test chamber and one to the inter-sleeve space,
exist. The feedthrough on the main sleeve allows for the wires from the induction
sensing coils and the Stirling cooler temperature sensors to escape the system. The
feedthrough on the test chamber allows for direct measurement of the temperature of
the inside of the bottom of the test chamber for calibration against the readings from
the Stirling cooler’s sensors.
3.4
Cooling
3.4.1 Stirling Cooler Head
The Stirling cooler operates according to the principles of the Stirling cycle. This
cycle enables the cryogenerator to produce extremely low temperatures, and allows
virtually all gasses and liquids to be cooled. The Stirling cycle involves alternately
compressing and expanding a fixed quantity of an ideal gas in a closed cycle,
producing progressively lower temperatures in the process.19 This cooling power is
concentrated onto a piece of metal called a cold head with which the test chamber has
contact via a sprung thermal contact.
The cooler is computer controlled via temperature sensors on the head, calibrated
against probes in the test chamber. The wires from the sensors travel inside the intersleeve space and out of an electrical feedthrough. They do not leave the vacuum area
immediately because the thermal gradient between the cold head and outside
environment would be too great for successful operation. The temperature range of
the cooler extends down to 20K, which is more than adequate for use with Li-NH3
solutions, which freeze at 88K. The extra cooling power is however useful as it means
20
that the cooler will not need to be operating near the edge of its capability while
viscosity measurements are performed. It also allows the quality of the thermal
contacts to be less of an issue than they might otherwise be.
3.4.2 Sprung Thermal Contact
The sprung thermal contact is a self expanding piece which sits between the cold head
and test chamber. Its purpose is to effectively fill the gap between the two and in
doing so maximise the cooling power applied to the sample in the test chamber and
produce an absolute minimum of temperature gradients in the test area. It will use flat
springs of copper or aluminium and thermal contact grease sandwiched between
copper plates for best thermal transfer.
NB: See also section 3.6.
3.5
Induction Sensing
3.5.1 Magnets
The magnets are used in this setup to produce flux, the change of which induces a
voltage in sensing coils placed nearby. The magnets used are 6mm x 6mm (diameter x
length) nickel plated neodymium disc magnets with an in contact pull of 1kg (Eclipse
Magnetics N701-RB). The nickel plating ensures that the magnet material does not
react with and become corroded by the Li-NH3 test solution. For their size these
magnets are very strong. This is desirable because the strength on the magnet is
proportional to the magnitude of the voltage induced by their motion, so stronger
magnets give higher voltage readings which means higher resolution and smaller
errors.
3.5.2 Sensing Coils
The induction sensing coils are placed within 3-4mm of the oscillating magnets and
are mounted in the inter-sleeve space on perspex arms extending from the sprung
thermal couple. The changing flux produced by the moving magnets cuts through the
coils of thin wire inducing a voltage proportional to both the amount of flux (magnet
strength) and its rate of change (torsion bob speed). There are two sensing coils which
are placed opposite each other in a Helmholtz arrangement. In this arrangement the
magnet and induction coils have a common axis, so that the poles point directly at the
circular faces of the coils. This leads to the greatest change in field strength as the
magnet moves towards the coil, across the centre and away again on the other side.
21
The coils are wired together such that their signals add constructively; this connection
can be direct and equivalent if each coil sees the same type of magnet pole, but must
be reversed if the two poles of the same magnet are used. The oscillations of the
torsion bob are centred upon the two coils so that its maximum speed is reached as it
passes the coils, inducing the greatest voltage for a given amplitude.
The coils in the test setup (those for the main equipment were not built but have the
same electrical qualities of those of the test setup, except made from Tufnol and not
perspex, to cope with the extreme low temperatures) have 250 turns each of 0.081mm
(44 gauge) enamel-coated copper wire. At the typical maximum speeds of oscillations
of the magnets this setup produces a signal strength of 2-3mV. This can be increased
greatly through amplification as detailed below.
The induced voltage signal produced by the induction coils is treated in sections 4,
4.1.
NB: See also section 3.6.
3.5.3 Signal Collection
Before recording, the signal produced by the induction sensing coils is amplified
using an operational amplifier. This increases the peak voltage from 2-3mV to 2-3V.
This produces a signal which is easily automatically readable by a LabJack. The
LabJack U12 is a USB based measurement and automation device which can monitor
and record voltage inputs continuously at 1200Hz. It provides the data to a laptop as a
tab delineated file.
3.6 Incomplete Pieces
3.6.1 Torsion Bob
As mentioned in section 3.2.1 the complex stacked plate torsion bob was a late
addition to the experiment and, as such, has not been thoroughly thought through,
designed and constructed. It is likely to be of similar design to that shown in figure 7
although this is not certain. The only limitations on its design are that it must be a
torsion bob, fit inside the test chamber and produce sensitive enough data that low
viscosity samples can be studied accurately.
22
3.6.2 Motion Starting Equipment
As mentioned in section 3.2.3 the precise arrangement of equipment to initialise the
motion of the torsion bob has not been finalised. However, the general form of the
equipment will be as follows. A low voltage motor (less than 5V) will be attached to
the lid piece of the vacuum sleeving. It will be connected to the rotational shaft by
two cogs which will gear down the initially rapid motion of the motor to a lower,
more controllable pace which is less likely to damage the torsion wire. The motor will
be connected to the analog outputs of the LabJack for precise control of its motion.
This type of control will allow for the same start-up motion for each use of the
equipment aiding reproducibility. The braking mechanism to bring the shaft motion to
a halt is less certain but will likely be a LabJack controlled electromagnet.
3.6.3 Sprung Thermal Contact
As mentioned briefly in section 3.4.2 the sprung thermal contact design has not been
completed. The reason for this is a mechanical one; without the complete vacuum
sleeving, including the test chamber, it was not possible to measure accurately the size
of gap between the Stirling cooler cold head and the base of the test chamber. This
had to be a physical measurement, rather than a calculation, because the size of the
vacuum seal rings under pressure from the vacuum clamps was not known and this
affects the gap between the cold head and test chamber. Once this is known complete
design of the sprung thermal contact will be possible, although some things can be
said already. The vertical gap between the cold head and test chamber will be ~21mm
(±3). The top and bottom faces are likely to be copper, due to its very high thermal
conductivity. The springs are likely to be flat springs of a ‘Z’ or ‘J’ shape of either
copper or aluminium. Sprung guides will likely be required to ensure that the contact
travels correctly and to produce some of the buoyancy so that the flat springs do not
become fatigued and can be used predominantly for thermal transfer.
A possible design is included overleaf (figure 9).
23
Figure 9 - Possible sprung thermal contact design. There would also be a corresponding top piece
which would sit on top of that drawn above. It would have a male connector projecting from the centre
which would fit inside the female connector shown above.
3.6.4 Sensing Coils
As mentioned in section 3.5.2 all tests were carried out using a test set of induction
sensing coils made from perspex; the main experimental coils, to be made from
Tufnol, were not built. This is because the signal characteristics being tested would
not be affected at all by the difference in construction material, and sourcing the
Tufnol could have been a problem. The reason for construction from Tufnol and not a
more common material is due to the low temperatures involved in the viscosity
measurements; with the induction wire wound very tightly round the ‘former’ it is
important that the former not change shape or size. The thermal expansion of Tufnol
is low enough to ensure no significant contraction due to temperature in the working
range of the experiment (88K - 293K).
4 Induction Signal
NB: The discussion that follows concerns tests carried out on a freely oscillating
torsion bob (mass ~220g; diameter ~70mm; wire length ~70cm; wire thickness
0.064mm) operating in air outside of all experimental equipment mentioned above.
The test setup is similar enough to that of the main experimental setup that some
aspects are shared, enabling studies to be made which are, in part, applicable to the
24
main setup. However, the responses given and specific behaviour displayed should
not be assumed to be the same as those of the main experimental setup.
The induction sensing setup shown in the experimental overview (figure 6) is shown
schematically below (figure 10). The signal produced consists of an increase in
voltage as the magnet approaches the coil, a transition through zero voltage to a
negative peak as the magnet crosses the centre of the coil and a decrease as the
magnet moves away on the other side. The exact shape and importance of the three
described sections changes with time as the amplitude of the oscillations decreases.
Figure 10 - Schematic diagram showing the functional layout of the induction sensing components and
the wiring connections between them.
When studying the signal several factors must be considered: the overall signal
strength, the relative strength of different parts of a cycle, the overall signal shape and
the relative size of the noise in the signal. The signal produced by the induction
sensing setup evolves with time as the oscillations grow smaller (figure 11). As can be
seen in the figure the amplitude of the induced voltage decreases with time and the
curve becomes less punctuated and more smoothed in appearance. However, as the
induced voltage decreases the noise in the signal takes on a much more prominent
role.
25
Figure 11 - Graphs showing the evolution of the induced voltage signal with time. Top row left: 1-2
secs. Top row right: 51-52 secs. Second row left: 101-102 secs. Second row right: 151-152 secs. Third
row left: 201-202 secs. Third row right: 251-252 secs. Fourth row left: 301-302 secs. Fourth row right:
351-352 secs. Bottom row left: 401-402 secs. Bottom row right: 451-452 secs.
NB: The angular amplitude for the above graphs varies between ~30° (top left) and ~2° (bottom right).
26
At the beginnings of motion, when the maximum speed of the torsion bob is still
relatively rapid the signal produced shows a sharp positive and sharp negative peak in
quick succession followed by a period of noise until the next cycle starts (figure 12).
Figure 12 - a) Noise between voltage signals. The magnet is too far from the coil and travelling too
slowly to give any signal. b) Magnet approaching sensing coil. The induced voltage increase as the flux
passing through the coil increases. c) Magnet passing centre of coil. Swift change in polarity caused by
the rapid change from increasing to decreasing flux. d) Magnet leaving coil. The signal decreases
because the magnet is moving away and decelerating.
The signal shape produced is not that of a sinusoid, as might be expected. This is
because the range of motion of the magnet is too great, although sinusoidal variation
does occur at later times when the amplitude of the motion has decreased. Broadly
speaking, two regimes exist: sharp peaks and sinusoidal motion. The former is due to
the amplitude of the torsion bob’s oscillations being much larger than the diameter of
the sensing coil. This means that the induced voltage has a much stronger overall
dependence on position rather than speed, i.e. the sensing coil sees the magnet
approach from a distance but with apparent constant speed. There is a very large
signal as the magnet passes the coil but next to nothing for almost all of the rest of the
motion. The latter is due to the magnet always being within close enough range of the
coil to give a signal at all points when the torsion bob has motion. This means that
there is a dependence on both position and speed, and as both vary sinusoidally, so
does the induced signal. There is a gradual transition from one regime to the other as
the amplitude of the torsion bob’s oscillations decreases.
27
4.1 Signal Analysis
In order to calculate the viscosity of a sample interacting with the torsion bob the
exponential decrease in the induced voltage is required (see section 2.3.2). To find
this only the peak values of each induction cycle are needed and the rest of the data
can be discarded.
A typical raw dataset produced by the LabJack is shown below (figure 13).
Figure 13 - Left: Data collected over an ~8 minute period showing an exponential shaped decay in
amplitude. Right: The first 2 seconds of the same signal showing in detail the shape of the signal and
the distribution of the datapoints.
In order to render the data usable the signal produced is processed in three steps segmentation, thresholding and peak detection - each of which is described in the
following text and shown below (figure 14).
a
b
c
Figure 14 - a) The complete raw dataset.
All datapoints remain. b) The segmented
and thresholded dataset. All points
below the threshold values have been
removed. c) The peak values from the
dataset. The peaks visible in the
thresholded dataset have been selected
and all other points removed.
28
1. Segmentation
The complete dataset is cut into sections of several seconds (the size of the
sections is variable but must be constant over a dataset) and those sections
placed in an array. This has two purposes; firstly, it allows the entire analysis
to execute faster as, although the same calculations must be carried out at
subsequent points they are now performed on smaller arrays, so less data is
manipulated unnecessarily. Secondly, it is required for the best performance of
the thresholding procedures, as detailed below.
2. Thresholding
The thresholding procedures move through the data section by section setting
a threshold and removing all datapoints found below. The threshold level is
chosen based on the maximum voltage value within each section multiplied by
a factor between zero and one. The threshold acts to remove the noisy,
fluctuating data which arises when the magnet is far from the induction coils.
This aids the accuracy of the peak detection procedures (see below). It also
acts to separate the positive and negative exponential curves so that they can
be analysed separately.
In combination with the segmentation, the thresholding keeps the most
possible signal. Without segmentation the threshold value would be calculated
from the relatively large initial voltages but applied to the much smaller
voltages towards the end of the data run, resulting in lost datapoints amongst
the smaller peaks.
3. Peak Detection
The final step is to run through the data which remains after the thresholding
and search for peaks. Basically this can be done by comparing the height of
each point to those of the last and next. If a point is higher than those either
side it is recorded as a peak. In reality more stringent conditions must be set
(e.g. whether the surrounding 8-10 points are all also above the expected
background) in order to exclude erroneous peaks, although care must be taken
in their design to ensure that a minimum of legitimate peaks are excluded.
These conditions can be quite strong because of the sheer number of
datapoints present in a dataset.
29
Once the array of only peak values has been found the values can be plotted
and an exponential regression line fitted (figure 15).
Figure 15 - Plot of peak induction voltage values against time. The equation of the superimposed
exponential regression line is V = 0.6755 e-0.0047 t and has an R2 value of 0.9857.
The equation of the regression line in the above figure is:
V  0.6755e 0.0047t
(eqn. 17)
4.2 Signal Error Analysis
By comparison with the theoretical treatment (see section 2.3.2):
d
A  n 
A
 d0 
 0.0047
2I
2I
(eqn. 18)
and
d  4.7  0.5  10 3
(eqn. 19)
The error value quoted in the above equation is based on a slightly crude but useful
assessment of the errors apparent in the data. Contributions to the value are made by
the temporal and voltage resolutions and the spread in the voltage data. The temporal
resolution is extremely high, around a fraction of 10-5 of the timestamps on the
datapoints, and so was not included. The voltage resolution is also extremely high
~10-6 and so was also discarded as an error. However, there is a significant spread in
the voltage values of the datapoints (figures 14c, 15). This spread is calculated to be
around 10-11%. This leads directly to the uncertainty on the measured value for d and
is effectively the only contributing factor.20 Therefore, the error bars shown in figure
15 are based upon this.
30
4.3 Signal Error Investigation
In order to be able to make an assessment of the possibility of making measurements
of small viscosity effects several things must be known. First, the size of the error on
the calculated value for the systematic friction must be known (see section 4.2). Then,
an assessment must be made of the likely sources of the contributing errors and of
whether it would be possible to minimize these in practice to such a degree that small
viscosity measurements would be possible.
The most significant error in the value obtained above for the damping factor, d, is the
vertical spread of voltage results, as mentioned above. This kind of spread could
result from one or both of two sources: temporal resolution problems and nonrotational motion. These are treated in turn below.
Temporal resolution problems arise when the frequency of recording of a varying
quantity is too low, leading to features of the relationship being missed. In the context
of the induced voltage readings this would equate to the top values of the positive
induced peak being periodically missed. This would sometimes leave peaks appearing
lower than their true value. The figure below (figure 16) shows in detail the shapes of
the positive induced voltage peaks for the first seven cycles.
Figure 16 - Detailed view of the positive peaks of the first 7 cycles of the induced voltage. Top row:
cycles 1 - 4. Bottom Row: cycles 5 - 7. NB: The vertical scale and origin values are the same for all
images.
Investigation of the first seven positive induced voltage peaks shows no circumstance
in which it appears that increased temporal resolution would yield a datapoint
significantly (above 5%) higher than those already present. This shows that the cause
31
of the spread in peak induced voltage values is not due to the frequency of
measurement.
Non-rotational motion would arise, in this setup, from a component, or components,
not being sufficiently axially symmetric. A non-symmetric component would yield a
preferential direction in which linear motion could begin upon rotational motion being
initialised. This linear motion, in a simple form, would manifest itself as an envelope
applied to the desired induced voltage result. However, this is complicated by the
facts that the rotational and linear motions are not independent, and that other types of
non-rotational motion may also be present. Interconversion between the various types
of motion would degrade an initially clear linear envelope into seemingly random
fluctuations, rapidly making the source of the error unclear. The amplitudes of the
first seven positive peaks (figure 16) do appear show a sinusoidal envelope, as do the
negative peaks (figure 17). After these initial signals the variation in peak values
shown below deteriorates rapidly into seemingly random noise and the sinusoidal
envelope is no longer a good fit.
Figure 17 - Induced voltage signal data from t = 0 to t = 2.4s showing the first seven cycles. The
positive
and
negative
envelopes
shown
have
equations
0.048Cos8.7x  0.76  0.7
and
 0.03Cos8.7x  0.76  0.7 respectively.
Another sign that this variation in induced voltage peak height is due to non-rotational
oscillations is that the spread of the values becomes smaller with time (figures 14c,
15). This follows because it is reasonable to assume that any additional physical
oscillation would be damped in a similar way to the damping of the rotational
oscillatory motion, and so its amplitude would decrease with time.
32
4.4 Error Correction
Given that the source of the main error in the determination of the exponential
regression line, which yields values for both the systematic damping, d, and sample
viscosity, η, (eqn. 18) is the vertical spread in the datapoints, and that that is due to
superfluous non-rotational oscillations, it should be possible to reduce the uncertainty
in both d and η relatively easily. This is done by removing all non-purely rotational
oscillations from the system. This is achieved by ensuring that all components are
axially symmetric. In the context of the torsion wire this means that no bend exists so
there is no preferred direction in which unwanted oscillations can begin. If this error
can be removed to a sufficient degree there is very little limit to the potential accuracy
of the experiment, as then the only remaining errors are random errors associated with
the temporal and voltage measurements.
5 Testing
During the course of the project some testing was carried out into various aspects of
the main experimental setup. This was predominantly with the aim of investigating
the differences between the theoretically predicted and the practically observed
behaviour which would be expected in the main setup, e.g. the difference between the
purely theoretical and practically observed relationships between torsion bob mass
and oscillating frequency.
5.1 Torsion Wire Tests
Testing was performed on stainless steel torsion wires of various lengths and torsion
bob masses with the aim of obtaining a reasonable estimate of the magnitude of the
frictional damping inherent to the system. These tests were carried out using an
optical detection method (a laser beam reflected onto a detection screen from a mirror
attached to the shaft of the torsion bob), which, although crude, was accurate enough
to get a good idea of the types of behaviours occurring.
The results obtained are shown overleaf (figure 18).
33
Figure 18 - Top Left: Light torsion bob with long wire; d = 2.08x10 -3 Top Right: Light torsion bob with
short wire; d = 2.66x10-3 Bottom Left: Heavy torsion bob with long wire; d = 1.19x10 -3 Bottom Right:
Heavy torsion bob with short wire; d = 1.58x10 -3 - The ‘d’ value is the size of the exponential decrease
in an oscillator’s amplitude (eqn. 18).
Several things are clear from the above results. First, that the torsion frequency
decreases with increasing mass. Second, that the torsion frequency also decreases
with increasing wire length. Third, that the exponential damping factor, d, decreases
with increasing mass and increasing wire length and fourth, that d, is on the order of
10-3. This leads to a rough determination of the magnitude of the viscosity effect at
around d = 10-4 (calculations in appendix).
5.2 Frequency Tests
These tests were performed using the induction sensing setup so that they
approximate well to the main experimental setup. The purpose was to make frequency
determinations for a range of torsion bob masses to make a comparison between the
theoretical prediction and the practical observations, and to test frequency finding
computer code for accuracy. A further aim was to perform the same tests in vacuum
in a bell jar for comparison, although this was not achieved.
The graph overleaf shows the results (figure 19).
34
Figure 19 - Graph of oscillation frequency vs. torsion bob mass. Also shown is a regression line with
equation
y  24.094 x 0.6145 .
The expected relationship between oscillation frequency and torsion bob mass is
shown below (eqn. 20).

k
I
(eqn. 20)
where
ω is angular frequency
k is wire constant
I is moment of inertia
If this relationship (eqn. 20) were to hold in the circumstance of the test, the power to
which x is raised would be -0.5 and not -0.6. Although several approximations have
been made here, most notably that moment of inertia and mass are equivalent (this is
reasonable because the distribution of the additional mass was the same as that
already present), this is believed to be a genuine effect. It appears to arise from
changes occurring in the torsion wire due to heavier masses being applied. However,
the effect does appear to be irreversible, so that, as long as tests made using similar
setups record a frequency along with any other data, it should be possible to
compensate for this effect if required in subsequent calculations.
35
6 Conclusions
6.1
Equipment Conclusions
The experimental equipment that was developed is likely to be successful in making
measurements of low viscosity samples. This can be said despite the lack of specific
testing because of the degree of consideration put into every component and at every
stage. It is certain to say that the final setup will be functional, the only problems that
could arise would be due to a lack of sufficient sensitivity or incorrect oscillation
parameters. For every variable component it has been a very high priority to maintain
as much of that variability as possible into the final designs, e.g. the length, material
and thickness of the torsion wire are all variable, as are the mass, dimensions and
even geometry type of the torsion bob. This will ensure that no insurmountable
problems occur with the equipment.
6.2
Signal Conclusions
The signal production by a test setup approximate to the induction sensing setup in the
main equipment has been successful. The power of the signal over a wide range of
angular amplitudes has been more than sufficient to be able to reconstruct the relevant
motion effects. The signal is regular and although it does evolve with time, the
manner of this evolution will not be problematic to experimental readings (see section
6.3).
6.3
Testing Conclusions
Testing with the aim of illuminating potential problems within the main experimental
setup has been successful. Tests on the produced induction sensing signal showed
several aspects of the behaviour which needed to be known. Evolution in the signal
shape was identified and tests were performed to find out if its effect would be
problematic. It was found not to be. Testing also showed an erroneous signal
envelope. The source of this envelope was investigated and found to be superfluous
oscillations. This was further looked into and it was decided that this potential
problem would be unlikely to recur in the main experimental setup.
Testing of the effects of varying the torsion wire length were performed and showed
some effects. It lead to increased flexibility being incorporated into the main
equipment design to allow for the variation of results that it uncovered.
36
Frequency tests showed an interesting effect on the torsion constant of the wire which
could have caused potential problems when attempting viscosity readings. The testing
showed that a relatively simple measurement should be made in conjunction with any
other readings taken to ensure results are comparable and errors as small as possible.
6.4
General Conclusions
If comparison is made between the original aims for this project (see section 2.6) and
the final outcome, this project must be deemed a failure in that only part of one of the
main objectives has been completed, in that a rotational viscometer has been designed
and constructed. However, these original aims must also be examined and, with
hindsight, are found wanting. The complexity of the design process involved in
creating the main experimental setup has shown that the original aims were far too
optimistic and that the revised aims (see section 2.6.1) were much more accurate and
realistic. Against these rewritten aims the work performed is judged more favourably
as the first two aims were completed, although the third was not.
7 Future Aims
The future aims for this project fall broadly into two categories: things which were to
be attempted in this project but were not and points of interest which arose during the
project which deserve further attention. The items explained in section 3.6 fall into the
first of these categories. Any piece of the main experimental setup which is not
complete as of the end of this project must be continued in order that a fully
functional experimental apparatus is obtained and attempts to measure the viscosity of
Li-NH3 solutions are made. The items listed include the torsion bob, motion starting
equipment, sprung thermal contact and sensing coils. A connected area to study is the
torsion wire. One thing needed for further, more accurate, testing is straight wire and
the sourcing of this must be a priority.
The subject of the third revised aim (see section 2.6.1) should be investigated. The
study of the effect of different radii of torsion bobs is of interest on its own, but would
also be useful to any further study involving torsion bobs, as the effect of radius
increases is likely to be an important one.
In the category of points that have arisen but not been studied lies another aspect of
the torsion wire. The thermal contraction of the wire must be examined, as even the
most subtle changes in length will be very important if geometries like the parallel
37
plate are in use. In such circumstances the plate separation is very small and has a
significant effect on the outcome. The separation also must be known in order for
calculations of viscosity to be carried out and this will not be possible without a
thorough understanding of the thermal expansion characteristics of the torsion wire
and, to a lesser extent, other components.
Another point which deserves attention is the possibility of contamination of the
purely rotational oscillations of the torsion wire and bob by vibrations from the
Stirling cooler. The rapid pumping backwards and forwards of the cooler occurs in a
direction perpendicular to the torsion wire and is therefore likely to affect its motion.
It is feasible that the cooler could be used to reach the required temperature and then
switched off for the duration of the experiment proper, but this would only be possible
if the subsequent temperature rise was not significant. Otherwise some sort of
damping mechanism will have to be designed in order to reduce this effect to
acceptable levels.
1
Viscosity - Wikipedia; http://en.wikipedia.org/wiki/Viscosity; 22/02/06
R.A. Serway, R.J. Beichner; Physics For Scientists and Engineers 5 th ed.; Saunders College
Publishing, 2000; page 469
3
Kaye and Laby - Tables of Physical and Chemical Constants. National Physical Laboratory;
http://www.kayelaby.npl.co.uk/general_physics/2_2/2_2_3.html; 22/03/06
4
J.J. Lagowski, M.J. Sienko; Metal-Ammonia Solutions; London Butterworths, 1970; page 248
5
N.T. Skipper; Personal Correspondence; 07/03/06
6
Metal Ammonia; http://www.cmmp.ucl.ac.uk/~nts/metam.html; 07/03/06
7
N.T. Skipper; Personal Correspondence; 26/10/05
8
A.A Collyer, D.W. Clegg; Rheological Measurement 2 nd ed.; Chapman & Hall, 1998; page 167
9
Viscosity - Wikipedia; http://en.wikipedia.org/wiki/Viscometer; 22/02/06
10
A.A Collyer, D.W. Clegg; Rheological Measurement 2 nd ed.; Chapman & Hall, 1998; pages 10-11
11
A.A Collyer, D.W. Clegg; Rheological Measurement 2 nd ed.; Chapman & Hall, 1998; pages 11-13
12
A.A Collyer, D.W. Clegg; Rheological Measurement 2 nd ed.; Chapman & Hall, 1998; pages 23-25
13
S. Kikuchi; J. Soc Chem. Ind. Japan; 43; 1940; page 233
14
J.J. Lagowski, M.J. Sienko; Metal-Ammonia Solutions; London Butterworths, 1970; page 251
15
P. Lobry; These Doctorat; Universite de Lille, 1969
16
J.J. Lagowski, M.J. Sienko; Metal-Ammonia Solutions; London Butterworths, 1970; page 250
17
P. Lobry; These Doctorat; Universite de Lille, 1969
18
J. Bartlett; The Viscosity of Metal-Ammonia Solutions - Are They Superfluids?; (UCL
- unpublished) 2004
19
Thermodynamic Cycle - Stirling; http://www.Stirling.nl/Stirling/nl/page33.asp.htm; 20/03/06
20
M.J. Esten, A. Aruliah; Treatment of Experimental Data (Laboratory II Data Analysis Notes); (UCL
- unpublished) 2005
2
38