The Viscosity of Metal - Ammonia Solutions

The Viscosity of Metal - Ammonia Solutions:

Are They Superfluids?

N.E. Shuttleworth

March 24th, 2006

Department of Physics and Astronomy, University College London

Email: ; Phone: 07834-365878;

WWW:

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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.

1Contents

1Contents / 2
2Introduction / 4
2.1Viscosity / 4
2.2Alkali Metal-Ammonia Solutions / 5
2.3Viscometers and Viscosity Measurement / 6
2.3.1Rotational Viscometer Geometry / 7
2.3.2Free Oscillation Rotational Viscometers / 8
2.4Motion Sensing Techniques / 10
2.5Relevant Previous Work / 12
2.6Initial Project Aims and Objectives / 13
2.6.1Revised Project Aims and Objectives / 13
3Equipment Design / 14
3.1Overview / 14
3.2Torsion System / 15
3.2.1Torsion Bob / 15
3.2.2Torsion Wire / 17
3.2.3Motion Starting Equipment / 18
3.2.4Rotational Motion Lock / 18
3.3Vacuum Equipment / 19
3.3.1Lower Vacuum Sleeve / 19
3.3.2Test Chamber / 19
3.3.3Gas Connections / 20
3.3.4Wiring Connections / 20
3.4Cooling / 20
3.4.1Stirling Cooler Head / 20
3.4.2Sprung Thermal Contact / 21
3.5Induction Sensing / 20
3.5.1Magnets / 21
3.5.2Sensing Coils / 21
3.5.3Signal Collection / 22
3.6Incomplete Pieces / 22
3.6.1Torsion Bob / 22
3.6.2Motion Starting Equipment / 23
3.6.3Sprung Thermal Contact / 23
3.6.4Sensing Coils / 24
4Induction Signal / 24
4.1Signal Analysis / 28
4.2Signal Error Analysis / 30
4.3Signal Error Investigation / 31
4.4Error Correction / 33
5Testing / 33
5.1Torsion Wire Tests / 33
5.2Frequency Tests / 34
6Conclusions / 36
6.1Equipment Conclusions / 36
6.2Signal Conclusions / 36
6.3Testing Conclusions / 36
6.4General Conclusions / 37
7Future Work / 37
[References / 38]

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 other[2]. 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 inpascal 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 ism2 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 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 metal[4] (MPM). At saturation the solutions are very good electrical and thermal conductors with low densities and low viscosities.

Figure 1[5][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 (TC) the metallic and non-metallic 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 pseudo-eutectic at saturation, which extends down to 88K to give the lowest temperature liquid metal.

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 formed[7]. 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 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 Geometry[11]

The main experimental arrangements for the measurement of viscosity in liquids are shown below (figure 2).

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:

(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:

(eqn.2)

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:

(eqn.3)

2.3.2 Free Oscillation Rotational Viscometers[12]

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:

(eqn. 4)

The equation of motion for free oscillations is:

(eqn. 5)

where I is inertia

Bis elastic constant

Aisexperimental shape factor

Gis complex shear modulus

Assuming η’ and G’ = η’’/ω are slowly varying in the narrow frequency range covered by the experiment:

(eqn. 6)

with

(eqn. 7)

(eqn. 8)

To evaluate the above for frictional loses in the wire and other parts of the system, solve for a purely viscous material:

(eqn. 9)

where n is a term including the frictional forces.

Thus the equation of motion without sample is:

(eqn. 10)

which has the solution:

(eqn. 11)

with

(eqn. 12)

(eqn. 13)

where ω0 is the natural frequency of oscillation.

Thus, with sample:

(eqn. 14)

with

(eqn. 15)

(eqn. 16)

where

ω is angular frequency

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 thisdecrease 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 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.

2.5Relevant 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 graph[16](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 mid-concentrations (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 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.

2.6Initial 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 Li-NH3 solution at saturation.

2.6.1Revised 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.

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.