Development of a Novel Radial Heat Flow Apparatus for Measuring Solid-Liquid Interfacial Energy

178 Pages·2017·5.64 MB·English

Development of a Novel Radial Heat Flow
Apparatus for Measuring Solid-Liquid
Interfacial Energy

Thesis Submitted for the Degree of
Master of Philosophy
At the University of Leicester

by

SungWoo Son

Department of Engineering
University of Leicester

2016

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Abstract: Development of a Novel Radial Heat Flow Apparatus for measuring Solid-
Liquid interfacial energy by Sung Woo Son
The solid-liquid interfacial energy,𝜍 , plays an important role in all processes that involves
𝑆𝐿
nucleation and growth of solids from its liquid state. In nucleation, interfacial energy affects the
temperatures at which solids nucleate from its liquids during growth, the interfacial energy
affects the transformation rate. Solid-liquid interfacial energy also plays important role in
determining the growth morphology which may lead to solidification occurring in the preferred
crystallographic directions. The solid-liquid interfacial energy is also important in the
phenomena such as coarsening of dendrites, sintering and wetting.
It is thus very important to have quantitative values of the interfacial energy, as this will assist
in gaining fundamental knowledge into the structural nature and physics of interfaces.
To measure the 𝜍 , the grain boundary groove method is used. The grain boundary grooves
𝑆𝐿
are obtained in a stable temperature gradient and by using direct application of the Gibbs-
Thomson equation on the groove shape, the 𝜍 can be obtained.
𝑆𝐿
The main aim of this study is to design and construct aradial heat flow apparatus and then to
commission and test using a commercial Al alloy. The Radial heat flow apparatus provides ideal
set up to obtain grain boundary groove shape, as the central heating element in the centre of
the sample and the water cooling jacket around the sample provides the radial temperature
gradient to produce solid-liquid interface.
The equilibrated grain boundary groove shapes for an industrial alloy LM25 were obtained in
experiment at above eutectic temperature for 10 days with the temperature control to within
±0.05K. From the observed grain boundary groove shapes, the 𝜍 for LM25 has been
𝑆𝐿
determined to be 167.361± 8.312 mJ/m2.

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Publications
1. S. Son, H.B. Dong. “Measuring Solid-Liquid Interfacial Energy by Grain Boundary Groove
Profile Method” Materials Today: Proceedings, Volume 2, Supplement 2, 2015, Pages
S306-S313
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Acknowledgements
I would like to thank my supervisor, Professor Hongbiao Dong, who has supported me
throughout my study. He has guided me with encouragement and advice, and have given me
many opportunities to experience the life as a PhD student in sending me to many conferences
and training courses. He has given me endless support during my time at the university and I
am very grateful.
I would like to thank Professor Atkinson, who has also given me a lot of support. I am very
grateful to both Professor Dong and Professor Atkinson, and I hope that I can be in a position in
the future to repay the support and opportunities I have received from them, to someone else
in need.
I also thank all the technicians and colleagues at the Universityfor giving me a lot of support and
help in being able to complete an experiment, which I found very challenging. I could not have
done it without the help and support of the technicians and colleagues.
I thank my girlfriend Nahee for her endless prayer and support she has given me throughout my
PhD studyand her understanding in times of difficulty.
lthank all my families for their thoughts and prayers, and the love and support they have shown
me through both good and bad times.
But most of all I thank God for the opportunity that you have given me to study PhD. I thank
God for the guidance throughout the study.
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Table of Contents

1 Solidification Thermodynamics ............................................................................................................. 1
1.1 First Law of thermodynamics ........................................................................................................ 1
1.2 System work .................................................................................................................................. 2
1.3 Enthalpy change ............................................................................................................................ 3
1.4 Latent heat of pure material ......................................................................................................... 4
1.5 Entropy .......................................................................................................................................... 5
1.6 Second Law of thermodynamics ................................................................................................... 5
1.7 Gibbs free energy and phase transformation ............................................................................... 6
1.8 Driving Force ............................................................................................................................... 12
1.9 Determination of solid-liquid interfacial free energy ................................................................. 13
1.10 Solid liquid surface energy for pure materials ............................................................................ 16
1.11 Solid liquid surface energy for binary alloys ............................................................................... 17
2 Experimental Methods for Measuring Solid-Liquid Interfacial Energy ............................................... 32
2.1 The calculation of 𝜍𝑆𝐿 from the measurement of dihedral angle ............................................. 32
2.1.1 Equilibrium at three phase junction ................................................................................... 32
2.1.2 Sessile drop method............................................................................................................ 36
2.1.3 Dihedral angle method ....................................................................................................... 38
2.2 Direct application of the Gibbs Thomson equation .................................................................... 39
2.2.1 Skapski’s method ................................................................................................................ 41
2.2.2 The melting point of small crystals ..................................................................................... 42
2.2.3 The grain boundary groove shape in a temperature gradient ........................................... 43
2.3 Calculation of solid-liquid interfacial energy from classical nucleation theory .......................... 48
2.3.1 Calculating the solid-liquid interfacial energy in homogeneous nucleation ...................... 49
3 Design and Construction of a Radial Heat Flow Apparatus ................................................................ 53
3.1 Main crucible .............................................................................................................................. 55
3.1.1 The bottom crucible lid ....................................................................................................... 55
3.1.2 The top crucible lid .............................................................................................................. 57
3.1.3 Crucible container assembly ............................................................................................... 59
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3.2 Radial heat flow apparatus assembly ......................................................................................... 62
3.2.1 Top and bottom heater holder ........................................................................................... 62
3.2.2 Water cooling jacket ........................................................................................................... 63
3.2.3 An assembly with crucible container and water cooling jacket.......................................... 65
3.2.4 The apparatus casing with brass lids .................................................................................. 66
3.3 The complete assembly of the radial heat flow apparatus ........................................................ 68
3.4 Thermal analysis on the radial heat flow apparatus ................................................................... 69
3.5 Testing of the heating element ................................................................................................... 75
3.6 Parts list to build radial heat flow apparatus .............................................................................. 81
3.7 Temperature Control .................................................................................................................. 83
3.7.1 The proportional control ..................................................................................................... 83
3.7.2 The integral and derivative term ........................................................................................ 84
3.7.3 Temperature tuning ............................................................................................................ 85
3.8 Experimental procedure ............................................................................................................. 88
3.9 The final experimental setup ...................................................................................................... 90
4 Experimental Results and Discussion .................................................................................................. 97
4.1 The cooling curve of LM25 ........................................................................................................ 100
4.2 Obtaining the equilibrated grain boundary groove shape ....................................................... 102
4.3 Temperature gradient measurement ....................................................................................... 109
4.4 Thermal conductivity of the solid phase ................................................................................... 111
4.5 Numerical method used to evaluate the Gibbs Thomson coefficient ...................................... 118
4.5.1 Evaluation of the Gibbs Thomson coefficient ................................................................... 121
4.5.2 Taylor expansion ............................................................................................................... 123
4.5.3 Calculation of the undercooling ∆𝑇 .................................................................................. 126
4.6 The effective entropy change ................................................................................................... 128
4.7 The composition analysis .......................................................................................................... 130
4.8 Direct application of Gibbs Thomson equation on Grain boundary groove ............................. 135
4.9 The average value of Gibbs Thomson coefficient and solid-liquid interfacial energy .............. 143
4.10 Error analysis in measuring thermal conductivity value ........................................................... 148
4.11 The Error analysis in measuring temperature gradient ............................................................ 150
4.12 Fractional uncertainty in calculating the effective entropy change ......................................... 151
5 Conclusion and Further work ............................................................................................................ 151
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6 References ........................................................................................................................................ 154

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List of Tables
Table 3.1 A test result for the coiled tantalum heating element wire…………………….…….pg 83
Table 3.2 A test result for the tantalum wire with higher voltage………………………….…….pg 83
Table 3.3 A Parts list to build radial heat flow apparatus……………………………………….…….pg 85
Table3.4 Ziegler-Nichols coefficient to obtain PID values for input……………………………..pg 90
Table 4.1 Chemical composition of LM25………………………………………………………………..…..pg 96
Table 4.2 literature values for phase change temperatures and the value for present
work…………………………………………………………………………………………………………………….………pg 99
Table 4.3 Values used for calculating thermal conductivity……………………………………….…pg110
Table 4.4 Gibbs Thomson coefficient values for each analysis points on groove 1…….…pg 129
Table 4.5 Gibbs Thomson coefficient values for each analysis points on groove 2…….…pg 131
Table 4.6 Gibbs Thomson coefficient values for each analysis points on groove 3……….pg 132
Table 4.7 Gibbs Thomson coefficient values for each analysis points on groove 4……….pg 133
Table 4.8 Gibbs Thomson coefficient values for each analysis points on groove 5……….pg 135
Table 4.9 Gibbs Thomson coefficient values for each analysis points on groove 6……….pg 136
Table 4.10 Gibbs Thomson coefficient values obtained for the 6 grain boundary grooves
analysed………………………………………………………………………………………………………………………pg 137
Table 4.11 The variables used to calculate the effective entropy change to calculate interfacial
energy…………………………………………………………………………………………………………………………pg 139
Table 4.12 The calculation of solid-liquid interfacial energy by averaging the values obtained for
6 grain boundary grooves………………………………………………………………………………………..….pg 140
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Table 4.13 A comparison of the interfacial energy values from the literature survey on
aluminium based system………………………………………………………………...……………………………pg 141

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List of figures
Figure 1.1 System Work (a) Constant Pressure (b) Non Constant Pressure
Figure 1.2 Gibbs free energy from initial to final state
Figure 1.3Schematic representation of the Gibbs free energies of solid and liquid as a function
of temperature at constant pressure
Figure 1.4 Typical phase diagram of a binary alloy system where 𝒌 = 𝒎𝑳 and m and m are
S L
𝒎𝑺
gradient of solidus and liquidus lines, for this case, k<1.
Figure 1.5 Free energy verses composition for a binary alloy
Figure 1.6 Depression of equilibrium condition caused by the Gibbs Thomson effect on binary
alloy
Figure 1.7 Representation of non spherical surfaces used to calculate Gibbs Thomson equation
Figure 1.8Representation of diagrams for calculating crystallographic orientation effect
Figure 2.1. Showing surface tensions acting where the three interfaces between the phases 1,2
and 3 intersect
Figure 2.2. Schematic diagrams of three phase system for (a) solid-liquid system, (b) solid-
vapour system, (c) solid-liquid-vapour system
Figure 2.3. The left figure shows the schematic of the experiment set up and analysis, (a) The
liquid Ag-Bi droplet on the solid Ag substrate and (b) the cross-section after the experiment
Figure 2.4. Schematic of the forces acting in the groove at the intersection of a symmetrical tilt
boundary
Figure 2.5. The wedge shape set up used by Skapski
Figure 2.6. The tapered cylinder set up by Skapski
Figure 2.7 Schematic diagram of Bridgman type apparatus
Figure 2.8 an example of the cooling rate curve of liquid and solid phase of an alloy
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