liquid snowflake formation in superheated ice

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Liquid Snowflake Formation in

Superheated Ice

Matthew G. Hennessy

St Hughs College

University of Oxford

A thesis submitted for the degree of

Master of Science in Mathematical Modelling and Scientific

Computing

Trinity 2010


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This one is for my parents, Peter and Debra Hennessy.

Without their love and support,

I would not have made it to where I am today.


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Acknowledgements

First and foremost, I would like to extend my sincerest thanks to Dr

Stephen Peppin and Dr Richard Katz, who supervised this project. Their

genuine interest in all aspects of the project, in combination with their

endless enthusiasm, made carrying out this research the most enjoyable

experience, even when this involved a week of consecutive experimental

failures.

I am very thankful to Prof Grae Worster, as he not only hosted me during

my visit to the University of Cambridge, but he also provided invaluable

ideas with regards to the linear stability analysis. Furthermore, I would

like to acknowledge Colin Macdonald for his discussions and his advice

about the level set method. I am also grateful for the various discus-

sions Ive had about liquid snowflakes with Dr Rob Style and Prof John

Wettlaufer over the course of the summer.

Finally, I would like to thank my friend Iain Moyles, who was not only

brave enough to read the first draft of this thesis, but whose success on

the west coast is a constant source of motivation to always stay on top of

my academic game.

This publication was based on work supported in part by Award No KUK-

C1-013-04, made by King Abdullah University of Science and Technology

(KAUST), and with funding from the Natural Sciences and EngineeringResearch Council of Canada (NSERC).


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Abstract

Liquid snowflakes, also called Tyndall figures, are small volumes of water

that resemble the classical shape of a snowflake that is composed of ice.

They initially form as cylindrical discs of liquid in superheated crystals

of ice. Several experimental studies have shown that liquid snowflakes

acquire their shape through an instability that occurs in the circular in-terface.

This thesis aims to begin a novel mathematical investigation of the evolu-

tion of two dimensional, planar, liquid snowflakes. In particular, several

linear stability analyses are carried out in order to gain insight into the

physical mechanisms that drive the instability of the circular interface.

The nonlinear evolution of liquid snowflakes is studied via direct numeri-

cal simulation. The numerical methods are based on the level set method.

The results from the mathematical analysis show that the interfacial insta-

bility is driven by superheating in the solid and is inhibited by diffusion

of thermal energy in the liquid. Furthermore, the existence of a large

wavenumber mode that grows faster than all of the other modes is shown.

The wavenumber of this mode corresponds to the wavenumber of the inter-

face when it becomes unstable, and its theoretical value is shown to agree

with experimental measurements. Moreover, the functional relationship

between this wavenumber and the experimental parameters is found using

asymptotic techniques.

Linear stability theory predicts that the circular interface of a liquid

snowflake eventually restabilizes due to large temperature gradients in

the water. As this phenomenon has yet to be observed experimentally, a

hypothesis is formulated which suggests that growth in the vertical direc-

tion is required for keeping the temperature of the water small enough to

prevent thermal diffusion from stabilizing the interface. This hypothesis

is also the first of its kind to explain why liquid snowflakes remain as

circular cylinders if their vertical growth is inhibited.


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Contents

1 Introduction 1

1.1 Experimental Observations . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 A Similar Physical System . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Aims of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.1 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Mathematical Model 9

2.1 Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.1 Volumetric Heating due to Absorption of Radiation . . . . . . 11

2.2 Conditions at the Ice-Water Interface . . . . . . . . . . . . . . . . . . 12

2.2.1 The Gibbs-Thompson Condition . . . . . . . . . . . . . . . . . 132.2.1.1 Surface Energy Anisotropy . . . . . . . . . . . . . . 15

2.2.2 The Stefan Condition . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Non-dimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.1 Summary of Non-dimensional Equations . . . . . . . . . . . . 21

2.4 The Quasi-steady Approximation . . . . . . . . . . . . . . . . . . . . 21

3 Linear Stability Analysis 23

3.1 An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1 Stability of the Basic State . . . . . . . . . . . . . . . . . . . . 253.2 The Effects of Superheating due to Radiation Absorption . . . . . . . 29

3.2.1 The Basic State . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.2 Stability of the Basic State . . . . . . . . . . . . . . . . . . . . 31

3.2.3 Physical Interpretation of Results . . . . . . . . . . . . . . . . 38

3.3 Linear Stability of a Liquid Disc . . . . . . . . . . . . . . . . . . . . . 43

3.3.1 Stability of the Basic State . . . . . . . . . . . . . . . . . . . . 48

3.3.1.1 Asymptotic Analysis . . . . . . . . . . . . . . . . . . 51

3.3.1.2 Physical Interpretation of Results . . . . . . . . . . . 54

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3.3.2 Extensions to Three Spatial Dimensions . . . . . . . . . . . . 58

4 Numerical Simulations 594.1 A Note on Alternative Numerical Approaches . . . . . . . . . . . . . 60

4.2 The Level Set Method . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3 Implementation of the Numerical Scheme . . . . . . . . . . . . . . . . 63

4.3.1 Updating the Temperature Profiles . . . . . . . . . . . . . . . 64

4.3.1.1 Boundary Conditions on the Computational Domain 67

4.3.2 Extrapolating the Interface Velocity . . . . . . . . . . . . . . . 68

4.3.2.1 Additional Numerical Details . . . . . . . . . . . . . 70

4.3.3 Advancing the Level Set Function in Time . . . . . . . . . . . 71

4.3.4 Reinitializing the Level Set Function . . . . . . . . . . . . . . 72

4.3.5 Further Implementation Details . . . . . . . . . . . . . . . . . 74

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.4.1 Validation of Code . . . . . . . . . . . . . . . . . . . . . . . . 74

4.4.2 Long-term Stabilization of the Interface . . . . . . . . . . . . . 75

4.4.3 Effects of Radiation Absorption in the Liquid Phase . . . . . . 76

4.4.4 Breaking the Symmetry . . . . . . . . . . . . . . . . . . . . . 78

4.4.5 Surface Energy Anisotropy . . . . . . . . . . . . . . . . . . . . 78

5 Conclusion 81

5.1 Summary of Key Results . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.3 Final Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

A Experimental Details 85

A.1 Growing Large Crystals of Ice . . . . . . . . . . . . . . . . . . . . . . 85

A.2 The General Experimental Setup . . . . . . . . . . . . . . . . . . . . 88

A.2.1 Estimation of The Radiation Intensity . . . . . . . . . . . . . 89

B High Order WENO Discretizations 91

C Constructing the Initial Level Set Function 95

Bibliography 100

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List of Figures

1.1 A six-fold symmetric liquid snowflake that forms in superheated ice.

The dark spot is a vapour bubble. This figure was produced using the

experimental setup described in Appendix A. . . . . . . . . . . . . . . 2

1.2 The evolution of three liquid snowflakes, one of which shows a re-

markable hexagonal symmetry. The overlap occurs because each liquid

snowflake is at a different depth below the surface of the ice. These

images were produced using the experimental setup described in Ap-

pendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 The hexagonal structure of an ice crystal. The planes that are formed

by the hexagons are called basal planes and the axis that is normal

to the basal plane is called the c-axis. Circles denote the approximate

location of oxygen atoms. . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 A beam of light with area A passing through a slab of thickness z. . 12

2.2 The Gibbs-Thomson effect leads to the transfer of heat from liquid

fingers to solid fingers, which drives the interface to a planar state. . . 13

2.3 The surface energies associated with a liquid drop on a substrate that

is surrounded by a solid. . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 An interface moving through a solid with velocity vn. Also shown are

the diffusive fluxes, Jl

and Js, of the liquid and the solid, respectively. 17

3.1 A planar interface (dashed) growing into a solid with velocity v along

the z-axis. Linear stability theory will predict the growth of perturba-

tions to the interface, shown as a solid line. . . . . . . . . . . . . . . . 25

3.2 A sinusoidally perturbed interface requires superheating in the solid

and supercooling in the liquid in order to sustain the growth of both

solid fingers and liquid fingers. . . . . . . . . . . . . . . . . . . . . . . 28

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3.3 When the interface of a liquid snowflake goes unstable, liquid fingers

grow into the superheated ice. The initial circular interface can still

be seen as a faint line from which the fingers grow out of. The image

was obtained using the experimental setup described in Appendix A . 28

3.4 Schematic diagram of a solid-liquid system being heated by a light

source which travels with the interface. . . . . . . . . . . . . . . . . . 30

3.5 Temperature profiles of a solid being superheated by incident radiation

as viewed from a frame that moves with the planar interface. Param-

eter values for the non-dimensional numbers can be found in Table

2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.6 The growth rate of perturbations to a planar interface as a functionof the non-dimensional wavenumber (top) and the dimensional wave-

length (bottom). Also shown is the asymptotic growth rate for large

Stefan numbers (stars). Parameter values can be found in Table 2.1

and Table 2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.7 The critical wavenumber (top) and the critical absorption coefficient

(bottom) as functions of the critical beam radius. Also shown are the

regions of stability and instability. Parameter values can be found in

Table 2.1 and Table 2.2. . . . . . . . . . . . . . . . . . . . . . . . . . 383.8 Isotherms of the solid phase in front of two interfaces with different

curvature. The interface on the left has wavenumber a = 2, and the

interface on the right has wavenumber a = 6. The isotherms bunch

together near the liquid fingers in the right figure, indicating that dif-

fusion is enhanced near these regions. This figure was created using

solution in (3.12) with the parameters given in Table 2.2. . . . . . . . 39

3.9 The wavelength of the most unstable mode as a function of the beam

intensity at the surface of the ice. Parameter values are given in Table

2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.10 The critical radiation intensity as a function of the absorption coeffi-

cient for various depths under the ice surface. Also depicted are the

regions where a planar water-ice interface is expected to become unsta-

ble. Absorption coefficients that are less than 1 m1 correspond to the

visible spectrum, whereas the larger values correspond to the infrared

spectrum. Parameter values are given in Table 2.1. . . . . . . . . . . 42

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3.11 Evolution of the temperature profile associated with a growing liquid

disc. In each figure, the liquid region is given by r < R. Parameter

values can be found in Table 2.2. . . . . . . . . . . . . . . . . . . . . 46

3.12 The growth rate of a liquid disc surrounded by superheated solid. The

overdot denotes differentiation with respect to (dimensional) time. The

parameters used in this figure are those of Table 2.1 and they corre-

spond to an ice-water system. . . . . . . . . . . . . . . . . . . . . . . 47

3.13 The relative growth rates of perturbations with wavenumbers from

n = 2 (circles) to n = 35 (stars) as a function of the disc radius. The

non-dimensional parameters that were used to generate this figure are

given in Table 2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.14 Critical basic state radii as a function of the perturbation wavenumber.

The top figure shows the radii where perturbations first become un-

stable. The bottom figure shows the cut-off radii, where perturbations

begin to decay. Parameters can be found in Table 2.1. . . . . . . . . . 53

3.15 The top figure shows, for each wavenumber n, the basic state radius

where this mode achieves its maximum relative growth rate. The bot-

tom figure shows the maximum relative growth rate of each mode.

Parameters are from Table 2.1. . . . . . . . . . . . . . . . . . . . . . 543.16 Isotherms of the liquid region, which bunch together near solid fingers

that extend into the liquid. This leads to additional transfer of heat

which causes these fingers to melt and retreat. Figure was produced

using (3.33) and by setting R = 1, R = 0.1, and n = 6. Parameter

values can be found in Table 2.2. . . . . . . . . . . . . . . . . . . . . 55

3.17 The ratio of the perturbation amplitude to the basic state radius

as a function of time. Breakdown of the linearized equations is ex-

pected when this ratio becomes greater than one. The perturbation

has wavenumber n = 7 and the initial conditions were R(0) = 0.2 and

R(0) = 0.001. Parameter values can be found in Table 2.2. . . . . . . 56

4.1 The solid-liquid interface can be viewed as the zero level set of some

function . The signs of this function can also be associated with a

particular phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

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4.2 The two-stage extrapolation process that is used to define a velocity

for the level set equation. The first stage (left) involves computing

the temperature gradient of the liquid near the interface and then

propagating these values into the solid region. The second phase (right)

computes the temperature gradient of the solid near the interface and

then sends these values into the liquid region. . . . . . . . . . . . . . 69

4.3 Growth of a circular disc as predicted by the level set method. As the

number of grid points N increases, the curves converge to the exact

curve at a rate that is at least first order in h = 1/(N 1). Theabsolute error is the difference in the final radii. . . . . . . . . . . . . 75

4.4 The restabilization of the interface, which was initially assumed tobe a superposition of a circle and a wavenumber seven mode. The

simulation ran until t = 1, and the curves are 25 time steps apart.

Moreover, 250 grid points were used in each direction. . . . . . . . . . 76

4.5 The effect of increasing the relative absorption coefficient, which is

equivalent to decreasing the amount of radiation that is absorbed by

the liquid. The curves in the top three figures are 20 time steps apart.

The bottom figures show the corresponding temperature fields at the

final time-step. Each computation used 200 grid points in each direction. 774.6 The growth of a liquid snowflake that does not start from the centre

of the radiation beam. The figures on the left show the evolution of

the interface, whereas the figures on the right show the evolution of

the temperature profile. The simulation was carried out using 250 grid

points in each direction and the absorption coefficient in the liquid was

taken to be zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.7 The effects of six-fold anisotropy in the surface energy. The interfaces

are shown every 20 time steps and 400 grid points were used in each

direction. Furthermore, to reduce stabilization, the absorption of ra-

diation in the liquid was neglected. . . . . . . . . . . . . . . . . . . . 80

A.1 Liquid dendrites growing out of a grain boundary. . . . . . . . . . . . 87

A.2 Using polarizers, the individual crystals of ice can be seen as dark

spots. In this case, the ice block is composed of one large crystal, with

several smaller crystals scattered throughout. . . . . . . . . . . . . . . 88

A.3 Schematic diagram of the general experimental setup. . . . . . . . . . 90

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C.1 The parameter s is chosen to minimize the distance between the point

(x0, z0) that lies on the interface and the point (x, z). . . . . . . . . . 99

C.2 An example of the log of the residual error 0 1 when the initialinterface is assumed to be hexagonal. The exact parametrization of

the interface is given by x0 = cos(s)[1+(1/35) cos(6s)], z0 = sin(s)[1+

(1/35) cos(6s)], where 0 s < 2. A 100 100 grid was used in thecomputation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

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List of Tables

2.1 Typical parameter values for an ice-water system. A reference of [E]

denotes a value that was measured experimentally. . . . . . . . . . . . 18

2.2 Summary of the non-dimensional numbers and their typical values,

which are based on a temperature scale of T 2 K and a lengthscale of l 1 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

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Chapter 1

Introduction

In 1858, John Tyndall made a remarkable discovery when studying the effects of

sunlight on ice slabs from Norway and Wenham Lake [41]. Using a convex lens to

focus a concentrated beam of sunlight into the interior of the ice slab, Tyndall noticed

the appearance of a vast number of lustrous spots that resembled bubbles of air. Upon

further investigation, however, Tyndall found that these were not bubbles of air, but

were, in fact, volumes of water. Moreover, each volume of water had a remarkable

flower-like shape that exhibited a high degree of symmetry (see Figure 1.1). Tyndall

subsequently called these bright spots liquid flowers, but today they are commonly

referred to as Tyndall figures, internal melt figures, and liquid snowflakes [19, 29, 38].Despite being discovered over 150 years ago, the precise physical mechanisms that

govern the formation of liquid snowflakes are still unknown. However, it is known

that a small portion of the electromagnetic radiation that passes through the ice will

be absorbed, which, in turn, will raise the temperature of the ice. Although the exact

temperature rise will depend on the intensity and the wavelength of the incident light

[19], in many cases it is possible to bring the interior of ice to its equilibrium melting

temperature of approximately 273 K. If the block of ice is actually a single crystal

that is nearly free of defects and impurities, then mass heterogeneous nucleation ofwater will not occur when the ice reaches 273 K and the interior of the ice will not

melt. Instead, the ice will become superheated, that is, its temperature will be raised

above its equilibrium melting temperature. However, it is extremely rare for ice to be

completely free of defects, and therefore, some nucleation of water will occur. In fact,

it is this water which gives rise to liquid snowflakes. The liquid that forms absorbs

nearly 100 times more radiation than the surrounding ice [19, 44], and therefore, its

temperature is expected to rise rapidly. To moderate the temperature rise of the

liquid, heat exchange with the surrounding ice is expected to occur. This can cause

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Figure 1.1: A six-fold symmetric liquid snowflake that forms in superheated ice. The dark

spot is a vapour bubble. This figure was produced using the experimental setup describedin Appendix A.

the ice to melt, which effectively leads to a growth of the liquid snowflake. This is

merely a hypothesis, however, and it has yet to be validated.

1.1 Experimental Observations

There are several characteristics of liquid snowflakes that have been consistently ob-

served in experiments. While some of these can be explained using the physical and

molecular properties of water and ice, the cause of many of them remain unknown.

The purpose of this section is to give an overview of the common experimental obser-

vations, and to provide explanations for the ones that are well understood. Many of

these observations were made by the author using the experimental setup described

in Appendix A, and these observations agree with those which have been documented

in [29, 38].

One of the most prominent features of a liquid snowflake is the appearance of

a dark circle near its centre (see Figure 1.1). This circle is, in fact, a bubble ofwater vapour that forms because of the density difference between water and ice. In

particular, when a mass of ice melts, the resulting water has less volume than the

ice, and hence the bubble forms to fill the void. Originally, it was suspected that

this bubble was composed of air that had been trapped in the ice. However, Tyndall

disproved this hypothesis by placing a piece of ice that contained a large number of

liquid snowflakes in beaker of warm water [41]. Any air bubbles that were contained

in the liquid snowflakes would be liberated when the ice melted and they would then

rise to the surface of the water. No such bubbles were observed, however.

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A particularly interesting aspect of liquid snowflakes is the morphology of the

interface that forms between the ice and the water. Generally, liquid snowflakes begin

as circular discs that grow outwards in a radially symmetry manner. If the intensity

of the incident radiation is sufficiently strong, however, the circular interface becomes

unstable and small amplitude perturbations that resemble sine waves appear. These

perturbations can subsequently grow into vast dendritic structures, as Figure 1.2

shows. An interesting question arises about wavenumber selection at the interface

when it first becomes unstable. From Figure 1.2 it can be seen that corrugations

with a large wavenumber grow from the circular interface, which is surprising because

surface energy effects at the interface tend to inhibit the growth of such modes. Thus,

it seems likely that there is an alternative physical mechanism that is promoting thegrowth of high wavenumber modes, and that this mechanism is competing with the

dampening effects of surface energy.

Circular liquid snowflakes have been observed to grow into hexagons, and the

hexagonal interface can also become unstable. Hexagonal liquid snowflakes are par-

ticularly interesting because there appear to be two modes of instability that can

occur. The first mode is analogous to the circular liquid snowflake case where the

entire interface develops small, outward growing corrugations. This case can be seen

in Figure 1.2. The second mode of instability involves the vertices of the hexagongrowing into large lobes, so that the overall shape resembles a flower with six petals.

Figure 1.1 shows this case. In both of these cases, the six-fold symmetry is usually

preserved.

There are a number of open questions about the physical mechanisms that govern

the evolution of the interface. For example, the precise relationship between the

intensity of the radiation and the shape of the interface has yet to be determined.

Obtaining quantitative data about this relationship is not trivial, however, as it is

quite common to see liquid snowflakes that have highly different morphologies, but

which are in close proximity to each other.

When liquid snowflakes form in a single crystal of ice, they are always oriented in

the same direction [38, 41]. More specifically, if the liquid snowflakes are imagined

as planar objects with zero thickness, then their planes are parallel to one another.

This characteristic is intimately related to the underlying microscopic structure of

the ice crystal in which the liquid snowflakes are contained. In the absence of defects

and impurities, the crystalline lattice of ice at standard temperature and pressure is

a hexagonal prism [19], as Figure 1.3 shows. The planes that the hexagons lie in are

called basal planes, and the direction normal to basal plane is called the c-axis. The

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Figure 1.2: The evolution of three liquid snowflakes, one of which shows a remarkablehexagonal symmetry. The overlap occurs because each liquid snowflake is at a different

depth below the surface of the ice. These images were produced using the experimentalsetup described in Appendix A.

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c

Figure 1.3: The hexagonal structure of an ice crystal. The planes that are formed by thehexagons are called basal planes and the axis that is normal to the basal plane is called the

c-axis. Circles denote the approximate location of oxygen atoms.

anisotropy of an ice crystal causes melting to occur preferentially in the basal plane,

and therefore, liquid snowflakes grow into these planes. Since the basal planes in a

single crystal of ice are parallel, it follows that the liquid snowflakes will then have

the same orientation. Because of this fact, liquid snowflakes are often used to deduce

the orientation of the basal plane and the c-axis of an ice crystal [38].

Although most of the growth of a liquid snowflake occurs in the basal plane, there

is a small component of growth along the c-axis. Experiments have shown that growth

in this direction is, in fact, necessary for the ice-water interface to become unstable.

In particular, [25] found that if liquid snowflakes did not grow beyond a critical height

of 10 m, then no perturbations formed on the interface. Similar data is shown in

[38], however, no attempt is made to measure the critical height in this case. Neither

of these works were able to provide any explanation for why this critical height exists.

1.2 A Similar Physical System

The formation of liquid snowflakes in superheated ice is remarkably similar to ice

formation in supercooled water. In this latter system, highly purified water is lowered

to a temperature below 273 K and an initial mass of ice is nucleated by some form

of perturbation. Experiments typically show that the solid nucleus initially grows

as a circular disc. If the amount of supercooling is not too large, then most of the

growth will be in the macroscopic basal planes of the nucleus that are formed by

its circular cross-sections [35]. Eventually, the circular disc of ice becomes unstable,

resulting in a slightly corrugated interface [19]. If the water is sufficiently supercooled,

the corrugations develop into dendrites, similar to in the case of liquid snowflakes.

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Furthermore, [36] has shown that ice growth along the c-axis is particularly crucial for

instabilities to form. In fact, they deduce that morphological instability is controlled

by the thickness of the ice disc and not by its radius.

A particularly appealing feature of the ice-growth system is that it is relatively

simple and it doesnt require a constant source of external energy to supercool the

water. This is in contrast to liquid snowflake experiments, which require a source of

radiation to superheat the ice. The simplicity of the supercooled water system has

been exploited by numerous authors which have formulated mathematical models of

the freezing process. Many of these models aim to investigate the morphological in-

stabilities that appear. The first of these studies was carried out in 1963 by Mullins

and Sekerka [28], who investigated the stability of a solid sphere growing into super-cooled liquid. They found that the sphere must grow beyond a certain radius for any

instabilities to occur, and that this radius is related to the critical radius of nucleation

(see [10] for more information). Subsequent studies built upon the ideas of Mullins

and Sekerka using linear stability analysis and by investigating solids with more re-

alistic geometries. For example, [12] investigated the two dimensional evolution of a

cylindrical disc in a three dimensional temperature field. More specifically, growth

along the c-axis was neglected and the analysis assumed that the solid and the liquid

had the same thermal properties. This latter assumption, in particular, fails to holdfor an ice-water system. However, this work did produce growth rates for the disc

radius and showed how the growth rate of a perturbation depends on its wavenumber.

Unfortunately, a lack of experimental data prevented an in-depth discussion of these

theoretical results. The full three dimensional growth of a cylindrical disc of ice was

studied in [45], where it was found that symmetry breaking occurs when the cylinder

grows beyond a critical thickness and the circular faces acquire different radial growth

rates. This, in turn, leads to the formation of a tapered cylinder. The critical thick-

ness was also shown to be related to the morphological stability of the liquid-solid

interface. Moreover, the critical thickness was found to be inversely proportional to

the amount of supercooling, which compares favourably with experimental data. This

not overly surprising, however, because the growth along the c-axis was assumed to be

governed by slow interfacial kinetics and the analysis relied on an empirical formula

to relate the vertical growth rate to the supercooling.

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1.3 Aims of This Thesis

Liquid snowflakes have been used for decades to examine the mechanical and ther-modynamic properties of ice crystals [29, 38], yet no mathematical theory has been

developed for their formation and evolution. Perhaps this is due to a belief that the

growth of liquid snowflakes in superheated ice is sufficiently similar to the growth of

ice in supercooled water that the theoretical results for the latter system carry over

to the former system. However, [38] has shown that this is not the case, and that the

two systems can vary significantly. In particular, the dendritic growth rates of liquid

snowflakes were measured experimentally and then, using the theory developed in

[23], were compared to theoretical growth rates of ice dendrites in supercooled water.

The theoretical growth rates poorly matched those which were measured experimen-

tally. Moreover, experimental data from both systems was used to compare how the

tip radii of liquid and solid dendrites depended on the amount of superheating in

the ice and supercooling in the liquid, respectively. It was found that an order of

magnitude more supercooling was required to produce an ice dendrite with the same

tip radius as a liquid dendrite. Therefore, the two systems exhibit markedly different

quantitative behaviour.

The general goal of this thesis is to begin developing a theory for the formation

of liquid snowflakes. This will be done by first building a suitable mathematical

model that is based on well established physical principles and not on empirically

determined relationships. Using this model, a combination of analytical techniques

and numerical methods will be used to study the growth of liquid snowflakes and

their morphological instabilities. In particular, linear stability analysis will be used

to investigate the mechanisms that govern the transition from a circular interface to a

corrugated interface. Unfortunately, linear stability theory is expected to break down

once the corrugations become so large that the nonlinear terms in the equations can no

longer be neglected. To study the nonlinear evolution of the interface, which includesthe formation of dendrites, numerical simulations that are based on the level set

method will be used. Furthermore, this thesis will have an experimental component

to it, and this will allow data to be collected that can be used for comparison and

validation purposes. By combining analytical work with numerical simulations and

experimental observations, it is hoped that a deeper understanding of the physics

which govern the evolution of liquid snowflakes will be obtained and that this, in

turn, will provide new insight into the many unanswered equations of liquid snowflake

evolution.

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1.3.1 Outline of Thesis

The second chapter of this thesis will provide a detailed description of the mathemat-ical equations that are used to model the growth of liquid snowflakes. In particular,

the governing equations and their boundary conditions will be derived and put into

non-dimensional form. Chapter 3 will be dedicated to linear stability theory, and

several systems of increasing complexity and realism will be analyzed. Chapter 4 will

discuss the level set method and show the results of several numerical simulations.

Each of these three chapters will have a large focus on the physical interpretation of

the equations and the results that are obtained. Finally, the thesis will conclude in

Chapter 5, where the main results will be summarized and possible areas of future

work will be mentioned.

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Chapter 2

Mathematical Model

Obtaining a complete mathematical description of the formation and the evolution of

a liquid snowflake is a highly complex problem which would have to account for the

nucleation of the liquid disc and the vapour bubble, and then model their subsequent

growth. The nucleation dynamics are particularly complicated because they depend

on the exact nature of the microscopic defects in the ice crystal. However, for the

purposes of this study it will suffice to assume that nucleation has already occurred

and that the initial state of the system contains a liquid disc that is surrounded by

superheated ice. Further simplifications to a mathematical model can be obtained by

neglecting the appearance of the vapour bubble, which can be justified using the factthat the amount of thermal interaction between the water and the vapour is expected

to be small. In particular, by assuming the thermal properties of the vapour are

similar to air, owing to the fact that the vapour is expected to have a low density, it

can be shown that the thermal conductivity of water is approximately twenty times

larger than that of air [7]. This implies there will be little heat exchange between

these two regions. Moreover, the latent heat which is released upon vapourization of

liquid into gas can also be neglected because the temperature of the liquid is expected

to be small [5].By neglecting nucleation and the vapour bubble, a relatively simple mathematical

model that describes the evolution of liquid snowflakes can be developed. Such a

model will be based on two main parts; evolution equations for the liquid and the

solid, and governing equations for the dynamics at the water-ice interface. This

chapter will derive and discuss these key equations and put them into their most

simple, yet physically enlightening form, by performing a non-dimensionalization.

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2.1 Evolution Equations

The primary physical principle which governs the evolution of the solid phase andthe liquid phase is that of conservation of thermal energy. The mathematical mani-

festations of such conservation laws are well developed, and an equation representing

the conservation of thermal energy in a phase i can be written as [16]

t(icpiTi) + Ji = Qi, (2.1)

where i, cpi, and Ti denote the density, the specific heat capacity, and the temperature

of that phase, respectively, Ji is the flux of thermal energy, and Qi represents source

and sink terms. The quantity icpiTi is the thermal energy per unit volume.The main contribution to the thermal flux in both the solid phase and the liquid

phase is from diffusion, which arises from the tendency of heat to flow from regions of

high temperature to regions of low temperature. This behaviour is quantified using

Fouriers law of heat conduction [16],

Jd = kiTi, (2.2)

where Jd is the diffusive flux and ki is the thermal conductivity of phase i. A sec-

ondary contribution to the flux in the liquid phase arises from advection, which is

the transport of heat by the motion of the water within the liquid snowflake. The

velocity of the fluid is expected to be small, however, and therefore the advective flux

can be neglected from the model.

The temperature of each phase is expected to vary by at most a couple of degrees,

which allows the material properties of the system (heat capacity, density, thermal

conductivity) to be modelled as temperature independent. Moreover, the water and

the ice are assumed to be sufficiently homogeneous that these properties can, in fact,

be taken as constants. As the vapour bubble is also being neglected, the conservation

of mass requires the densities of water and ice to be equal. Using these assumptions

and by combining (2.1) and (2.2), the governing equations for the temperature profiles

of the two phases can be written as

cplTlt

= kl2Tl + Ql, (2.3a)

cpsTst

= ks2Ts + Qs, (2.3b)

where the subscripts l and s denote liquid and solid, respectively. The exact form

of the source terms Ql and Qs which, in this context, represent the absorption of

electromagnetic radiation, will be discussed below.

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The boundary conditions that are imposed away from the interface depend on the

particular problem that is being studied, but in general they will be of the standard

type (Dirichlet, boundedness, etc.).

2.1.1 Volumetric Heating due to Absorption of Radiation

When electromagnetic radiation is incident on water or ice, a portion of this radiation

is absorbed and converted into thermal energy, which then leads to an increase in the

temperature. The source of this radiation is usually from a focused beam of light, and

therefore, at the surface of the ice, the intensity profile I is approximately Gaussian

[26],

I = I0e(r/rb)2 ,

where r denotes the distance from the centre of the beam to where I is measured,

rb is a measure of the beam radius, and I0 denotes the peak intensity of the beam

(which occurs directly in the centre).

At the microscopic level, the absorption of radiation occurs when the electrons

that are bound to the constituent atoms of the medium enter an excited state by

absorbing a photon. A reduction of photons corresponds to a reduction in the intensity

of radiation. While absorption is a quantum mechanical phenomenon, it can be

formulated in a classical sense using wave theory, and it can be shown that the decay

rate of a beam as it travels through a medium is exponential [15]. This leads to an

intensity of the form

I(r, z) = I0 e(r/rb)

2z, (2.4)

where z is the distance travelled through the medium, and is the absorption coeffi-

cient of the medium. The inverse of the absorption coefficient represents the distance

that a beam of light can travel before its intensity is reduced by a factor of 1

e1.

Furthermore, the absorption coefficient is strongly dependent on the wavelength of

light and it may vary by several orders of magnitude over a small range of the elec-

tromagnetic spectrum.

To derive an expression for the volumetric heating that occurs as a medium absorbs

radiation, consider a beam of light that passes through a thin slab of material with

a thickness z, as shown in Figure 2.1. Recalling that intensity is a measure of the

power per unit area, the amount of power that is lost as the beam passes through

this material is

P = [I(r, z) I(r, z+ z)]A,

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z

A

I(r, z+ z)

I(r, z)

Figure 2.1: A beam of light with area A passing through a slab of thickness z.

where A is the area of the beam. The volumetric loss of power, which is equivalent to

the volumetric heating in the material (by conservation of energy), can be found by

expanding this expression about small z, keeping only the leading order term, and

then dividing by Az. The result is

Q = Iz

= I(r, z),

which is readily obtained using (2.4).

All of the problems that are investigated in this thesis are posed in a particular

two dimensional basal plane that is assumed to be a distance d below the surface

of the ice. Therefore, the radiation that is absorbed by the water and by the ice in

this plane has decayed by the same amount, in particular, by a factor of esd. By

defining a new incident intensity as I1 = I0esd, the volumetric heating terms that

appear in the evolution equations for the temperature (2.3) can be written as

Ql = l I1 e(r/rb)

2

, (2.5a)

Qs = s I1 e(r/rb)

2

. (2.5b)

2.2 Conditions at the Ice-Water Interface

Closing the mathematical model for liquid snowflake evolution requires additionalboundary conditions at the ice-water interface. The first condition is to impose con-

tinuity of the temperature field across the interface,

Ts = Tl = TI,

where TI denotes the interfacial temperature. This temperature can be calculated

from the Gibbs-Thomson condition, which uses thermodynamic arguments to estab-

lish the equilibrium melting temperature at a curved interface. The second and final

condition, called the Stefan condition, describes the velocity of the interface by con-

sidering local conservation of energy.

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Heat flow

TI > Tm

TI < Tm

Solid

Liquid

Figure 2.2: The Gibbs-Thomson effect leads to the transfer of heat from liquid fingers tosolid fingers, which drives the interface to a planar state.

2.2.1 The Gibbs-Thompson Condition

One of the most distinguishing features of a liquid snowflake is its shape, which is

the result of a highly curved interface that forms between the solid phase and theliquid phase. The intermolecular forces that act between the water and the ice at this

interface lead to a net surface energy, which nature tends to minimize. The physical

mechanism that drives this minimization is a shift in the phase equilibria at curved

interfaces that causes thermal energy to diffuse from liquid fingers to solid fingers [43].

This, in turn, causes the liquid fingers to freeze and the solid fingers to melt, thus

reducing the surface area of the interface and hence minimizing its surface energy.

Figure 2.2 shows a schematic of this effect.

This process can be formalized by first considering a two-dimensional liquid drop

that is placed on a substrate of length L. Furthermore, assume that the drop is

surrounded by a solid (see Figure 2.3) and let the surface energy per unit length

of the liquid-substrate interface, the solid-substrate interface, and the solid-liquid

interface be denoted by ls, ss, and , respectively. The net surface energy is then

given by

E = ls(b a) + ss(L + a b) +b

a

1 + h2x dx,

where a and b denote the contact points of the liquid on the substrate, and h denotes

the height of the drop as a function of distance along the substrate. Furthermore, thesubscript on h denotes differentiation with respect to x. The height of the drop will

be that which minimizes the surface energy E subject to a constraint which conserves

the initial area of the drop,

A =

ba

h(x) dx.

This minimization problem can be solved using the Euler-Lagrange equations [32],

and it can be shown that h will satisfy

= , (2.6)

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a b

Solid

Liquid

ss

ls

h(x)

x

Substrate

Figure 2.3: The surface energies associated with a liquid drop on a substrate that issurrounded by a solid.

where is the curvature of the interface given by

= ddx

hx

1 + h2x

and is a Lagrange multiplier that corresponds to the jump in pressure across the

interface [18]. This expression can be generalized to arbitrary geometries by writing

the curvature as the divergence of the normal vector n at the interface, = n.Moreover can be replaced by the surface energy per unit area if a two dimensional

interface is being considered.

The jump in the pressure at the interface is a result of a shift in the thermodynamic

equilibrium, which can be quantified using the Gibbs free energy . If Tm and pm

denote the equilibrium temperature and pressure of a planar solid-liquid interface1,

respectively, then at a curved interface, the Gibbs free energy looks approximately

like

(TI, pI) (Tm, pm) + p

(p pm) + T

(TI Tm), (2.7)

where pI is the pressure at the interface. In particular, pI = ps at the solid side of

the interface, and pI = pl at the liquid side, where pl and ps denote the interfacial

pressures of the solid and the liquid, respectively. The partial derivatives of the Gibbsfree energy are well known from classical thermodynamics (see [8], for example) and

are given by

p=

1

,

T= s,

where s is the entropy per unit mass. By demanding that the Gibbs free energy is

continuous across the interface,

(TI, pl) = (TI, ps),

1

For a water-ice interface, these values are well known as Tm = 273.15 K and pm = 101.3 kPa[3].

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and using the approximation in (2.7), the difference in pressure across the interface

is given by

ps pl = (sl ss)(TI Tm),where sl and ss denote the entropy per unit mass of the liquid phase and the solid

phase, respectively, and the assumption that ice and water have equal densities has

been used. The change in entropy across the interface is given by sl ss = L/Tm,where L is the latent heat of fusion. Therefore, the jump in interfacial pressure can

be written in terms of physical variables as

ps pl = LTIT

m

1 .Recalling that the Lagrange multiplier in (2.6) is exactly the jump in the pressure

across the interface, the interfacial temperature can be written as

TI = Tm

1

L

, (2.8)

which is known as the Gibbs-Thomson condition. For this equation to accurately

model the above physical process, the curvature of a liquid finger that extends into

the solid must be taken as negative.

2.2.1.1 Surface Energy Anisotropy

Surface energy effects arise because of the microscopic interactions between molecules

near the interface. The molecules in a solid, however, are typically arranged in a well

defined crystal lattice whose geometrical properties can give rise to strong anisotropies

in the surface energy. The crystalline structure of ice, for example, has six fold

symmetry [19], which is inherited in the shape of both conventional snowflakes and

liquid snowflakes (see Figure 1.1). This implies that the effects of surface energy

anisotropy, which are due to microscopic properties of the solid, are propagated to

the macroscale and should be incorporated into the mathematical model.

An extended Gibbs-Thomson condition can be derived by assuming the surface

energy per unit area is a function of the normal vector at the solid-liquid interface,

= (n), which models the fact that the surface energy will depend on the orientation

of the interface relative to the structure of the crystal. The resulting surface energy

minimization problem can solved with the Euler-Lagrange equations to obtain

+

d2

d2

= , (2.9)

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where denotes the angle that the normal vector makes with the horizontal axis, is

the curvature of the interface, and denotes the pressure jump across the interface.

Using this expression, the Gibbs-Thomson condition can be written as

TI = Tm

1 +

L

, (2.10)

where the subscripts with denote differentiation with respect to that variable.

The surface energy of a crystal with n fold symmetry is typically written as [10]

= 0

1 +

nn2 1 cos(n)

,

where n is a measure of the anisotropy. The case when |n| < 1 is considered weakanisotropy, as the expression + will have the same sign for all angles . On

the other hand, strong anisotropy occurs when |n| > 1. When the system is inthermodynamic equilibrium, (2.9) implies that ( + ) is constant. Thus, when

the anisotropy is weak, the interface will look approximately circular, as expected.

When the anisotropy is strong, however, the interface will be composed of straight

line segments and it will resemble a polygon, which can be deduced using the Wulff

construction (see [10, 33]).

Surface energy anisotropy is a particularly complicated phenomenon and it willonly be considered in the numerical simulations. Therefore, in the subsequent equa-

tions and analysis it can be assumed that the surface energy is isotropic, unless

otherwise stated.

2.2.2 The Stefan Condition

The Gibbs-Thomson condition provides an expression for the temperature at the

interface. However, the location of the interface is not known a priori, and it must be

solved as part of the problem. In this case, the velocity of the interface can be deducedfrom the conservation of energy. In order for a drop of water that is surrounded by

ice to grow, the neighbouring solid must absorb enough thermal energy to stretch

the interface and to overcome the latent heat of fusion. The thermal energy that is

required to do this is supplied by the net heat flux into that region.

To formalize this argument, consider an interface that is growing through the ice

with a normal velocity vn, as shown in Figure 2.4. The total mass of ice that melts

in a short interval of time t is given by m = A(t)vn t, where A is the area of

the interface at time t. The latent heat that is required to melt this mass of ice is

wl = LAvn t. Furthermore, if the area of the interface at time t + t is A(t + t),

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t

t+ t

Liquid

vn

A(t+ t)

A(t)

Jl

Js

n

Solid

Figure 2.4: An interface moving through a solid with velocity vn. Also shown are thediffusive fluxes, Jl and Js, of the liquid and the solid, respectively.

then the work that is required to stretch the interface is ws = [A(t + t) A(t)].Thus, the total energy that is required to advance the interface is

w = LA(t)vn t + [A(t + t) A(t)],

and this is supplied by thermal energy that flows into the ice via diffusion,

wd = (klTl + ksTs) nA(t) t,

where n denotes a unit vector that points into the solid. By conservation of energy,

these two expressions must be equal,

LA(t)vn t + [A(t + t) A(t)] = (klTl + ksTs) nA(t) t.

Dividing this equation by A(t)t and then taking the limit as t 0 yields

Lvn +

A

A

t= (klTl + ksTs) n.

From differential geometry it is known that (see [2], for example)

1

A

A

t= vn,

which allows the energy balance equation to be written as

(L + ) vn = klTl n + ksTs n. (2.11)

This expression for the velocity of the interface is known as the Stefan condition, and

it provides closure to the boundary value problem.

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Table 2.1: Typical parameter values for an ice-water system. A reference of [E] denotes avalue that was measured experimentally.

Parameter Description Value Unit Ref.

Density of water/ice 1000 kgm3 [3]cpl Heat capacity of water 4181 Jkg1K1 [7]cps Heat capacity of ice 2050 Jkg1K1 [7]kl Thermal conductivity of water 0.6 Wm1K1 [7]ks Thermal conductivity of ice 2.0 Wm1K1 [7]L Latent heat of fusion 3.33 105 Jkg1 [3]

TmMelting temperature of a

273 K [3]planar ice-water interface

Surface energy per unit area

0.033 Jm2

[19]of ice-water interfacel Absorption coefficient of water 1 104 m1 [44]s Absorption coefficient of ice 100 m

1 [19]d Distance below ice surface 0.01 m [E]rb Radius of light beam 0.01 m [E]I0 Intensity of incident radiation 300 Wm2 [E]

2.3 Non-dimensionalization

The governing differential equations and their associated boundary conditions dependon a large number of physical parameters. Moreover, the magnitudes of these param-

eters can vary significantly (see Table 2.1), and therefore, it is difficult to establish

the primary physics that drive the evolution of liquid snowflakes. To overcome this

difficulty, the equations can be written in terms of non-dimensional variables and

parameters that characterize the relative importance of each term in the equations.

To begin, a non-dimensional temperature u is defined for the solid and liquid

regions

ui =

Ti

Tm

T ,where i is either l or s, and T sets the temperature scale of the problem. A possible

choice for this scale is the amount of superheating that is measured in the ice, which is

usually in the range of 0.01 K to 0.1 K [19, 38]. However, this parameter will actually

be chosen to balance terms in the governing equations. The lengths are scaled by l so

that x = lx, where x is non-dimensional vector of the coordinate length variables

and l will be found later. Similarly, time is rescaled so that t = t, where is the time

scale and t is a unitless time. With these scalings, the equations for the temperature

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can be written in terms of characteristic time scales,

ult =

dl 2

ul +

hl e(r/)2

,

ust

=krcpr

dl2us + r

cpr

hle(r/)

2

,

where the primes have been dropped, = rb/l is a rescaled beam radius, and cpr =

cps/cpl, kr = ks/kl, and r = s/l are relative measures of the heat capacities,

thermal conductivities, and absorption coefficients, respectively. The time scale dl

measures the amount of time it takes heat to diffuse across a distance l in the liquid,

and hl is the time required for the radiation to raise the temperature of the liquid

by an amount T. Explicit expressions for these time scales are given by

dl =cpll

2

kl,

hl =cplT

lI1,

and the analogous time scales in the solid are ds = (cpr/kr)dl and hs = (cpr/r)hl.

Inserting typical parameter values shows that diffusion is approximately seven times

faster in ice compared to in water, and that water heats up fifty times quicker than

the surrounding ice. The time scales of diffusion can also be used to define the lengthscales of diffusion in the liquid and the solid, ldl and lds, respectively. Similarly, one

of the heating time scales can be used to choose a temperature scale.

The Gibbs-Thomson condition (2.8) can be written as

uI = ,

where uI is the non-dimensional interface temperature, is the non-dimensional cur-

vature of the interface, and is a non-dimensional number that characterizes the

relative effects of the surface energy,

=

Ll

TmT

.

Using this parameter, a capillary length scale lcap can be defined that represents the

smallest radius of curvature that an interface can obtain before surface energy begins

to flatten it out,

lcap =

L

TmT

.

Inserting parameter values shows that lcap

(2.7

105/T) mm

K, which implies

that for most temperature scales, the capillary length will be small. This most likely

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explains the appearance of a highly corrugated interface when a liquid snowflake first

becomes unstable.

The Stefan condition for the motion of the interface (2.11) can be put into non-

dimensional form by defining a non-dimensional velocity vn = vn/l. Upon dropping

the prime on the velocity, the Stefan condition becomes

m

(1 + ) vn = ul n + krus n, (2.12)

where is a non-dimensional number that represents the relative energy needed to

stretch the interface and m denotes the time scale of melting additional ice in front

of the interface. More specifically, = /(Ll), which is very small provided that

the length scale of the problem, l, is chosen to be larger than the capillary length.

The time scale of melting is

m =Ll2

T kl,

which is usually very large because of the high latent heat and the small temperature

scales.

The equations can be put into their final non-dimensional form by choosing scales

for the time, the length, and the temperature. These scales are often chosen to

balance the magnitudes of various terms in the governing equations, which ensures

that the key physics occur at leading order in the model. However, the physics that

drive the interfacial instability in liquid snowflakes are unknown, and when combined

with the fact that the physical processes occur on different time scales, ensuring that

the proper terms are in balance is a particularly nontrivial task to accomplish.

As previously mentioned, the time scale associated with melting is typically large,

and therefore it will not be balanced with the other time scales. Instead, the balancing

will occur between the diffusion and the absorption time scales of the liquid. This

leads to a relationship between the temperature scale and the length scale, T =

lI1l2/kl. The length scale is determined from the average size of a liquid snowflake,

which is on the order of millimeters. Hence, l 1 mm, which leads to a temperaturescale of T 2 K. The associated time scale of heating due to radiation absorption isroughly eight seconds, which is reasonable, as this implies that it takes eight seconds

for the water to heat up by two Kelvin. To simplify the equations, the time scale

is taken to be the time scale of thermal diffusion in the liquid.

With a time scale for the problem, the Stefan condition can be written as

S(1 + ) vn = ul n

+ krus n

,

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where S = L/(cplT) is often referred to as the Stefan number [11, 43]. From a

physical standpoint, this number measures the ratio of energy required to overcome

the latent heat of fusion to the energy required to raise the temperature of the liquid

by an amount T. For the ice-water system considered here, S 40, which isconsidered to be large. The interface stretching parameter is small, 107, andtherefore only a tiny percentage of the energy goes into stretching the interface. In

fact, this number is so small that it will be taken as zero, for simplicity.

2.3.1 Summary of Non-dimensional Equations

The full system of non-dimensional equations and their boundary conditions will now

be summarized. The evolution equations for the temperatures are given by

ult

= 2ul + e(r/)2, (2.13a)

cprust

= kr2us + re(r/)2. (2.13b)

The Gibbs-Thomson equation for the interface temperature reads

uI = (2.14)

and the Stefan condition for the velocity of the interface is

Svn = ul n + krus n. (2.15)

To close the problem, suitable boundary conditions away from the interface must

be imposed, and initial conditions for the temperature profiles and the interface are

required. These will be discussed in the individual problems that are considered. A

summary of the non-dimensional parameters and their typical values can be found in

Table 2.2.

2.4 The Quasi-steady Approximation

While the Stefan number represents a ratio of energies, it was obtained by dividing

the time scale of melting by the time scale of diffusion in the liquid, and hence it

also represents a ratio of these time scales. The Stefan number associated with liquid

snowflake experiments is large, which implies that diffusion of heat occurs much faster

than melting. Therefore, by the time the ice-water interface can advance, thermal

diffusion has nearly brought the temperature profiles to their steady state, which

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Table 2.2: Summary of the non-dimensional numbers and their typical values, which arebased on a temperature scale of T 2 K and a length scale of l 1 mm.

Parameter Definition Value

cpr cps/cpl 0.5kr ks/kl 3.3r s/l 0.01 rb/l 10

Ll

TmT

1.4 105

Ll1.0 107

SL

cplT

41

suggests that the transient dynamics of the temperatures are not significant and can

be neglected.

To make this statement rigorous, a new time t that is associated with the time

scale of melting can be defined as t = t/S, which leads to the evolution equations for

the temperatures becoming

1

S

ul

t

=

2ul + e

(r/)2,

cprS

ust

= kr2us + re(r/)2.

Thus, when the Stefan number is large, the terms involving the time derivatives can

be neglected and temperature profiles can be assumed to be in their steady state.

Note, however, that the temperature profiles will still be a function of time because

of the order one motions of interface which follow from the Stefan condition

vn = ul n + krus n, (2.16)

where vn = vn/S a rescaled velocity. Neglecting the transient dynamics of the tem-

peratures is commonly referred to as the quasi-steady approximation [11, 43], owing

to the fact that the temperature profiles continuously evolve despite being in their

steady state at any given moment in time.

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Chapter 3

Linear Stability Analysis

Experimental observations show that liquid snowflakes often begin as circular discs

which then grow outwards in a radially symmetric manner. However, after a certain

amount of time, the interface becomes unstable and small, sinusoidal perturbations

with a well defined wavenumber appear. Understanding this phenomenon from a

mathematical perspective is a problem that is well-suited for a linear stability analysis,

which investigates the growth of small amplitude perturbations to a known basic

state. In the context of liquid snowflakes, the basic state corresponds to the outward

growing liquid disc. The key assumption in linear stability theory is to assume that the

perturbations to the basic state are sufficiently small that their growth is dominatedby the linear terms in the governing equations. This allows the nonlinear terms to be

neglected which simplifies the analysis of the perturbations by a considerable amount.

In general, linear stability theory aims to predict the conditions and the physical

parameters that cause perturbations to grow in time. This corresponds to the basic

state becoming unstable, which is an experimentally observable phenomenon. Of

particular interest in this study is to determine how the wavenumber of the interface

depends on the intensity of the incident light, as this relationship could be determined

experimentally.This chapter will investigate the stability of several systems of increasing complex-

ity. By starting with the simplest models and then building upwards, the key physics

of the problem can be discovered and understood, which will aid the subsequent

interpretation of more realistic models.

3.1 An Illustrative Example

There are a number of physical mechanisms that could drive the interfacial instabil-

ity, for example, large temperature gradients in the water and superheating in the

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surrounding ice, both of which result from the absorption of radiation. Moreover, the

exchange of heat between these two regions is maximized by a highly curved interface,

which surface energy effects tend to minimize. Thus, there is competitive behaviour

which will play an important role in selecting the wavenumber of the interface. To

gain some insight about how these various physical mechanisms interact with each

other, the dynamics of a planar interface that has prescribed temperature gradients

in the solid and the liquid regions will first be studied.

As a basic state, consider a planar interface that is growing into the ice by travel-

ling along the z-axis with a velocity v, as shown in Figure 3.1. For convenience, the

system will be written in terms of a single coordinate that moves with the interface,

= z vt. Negative values of this coordinate will correspond to the liquid region,and positive values will correspond to the solid. Finally, it will be assumed that the

temperature profiles of the basic state are linear near the interface, so that

ul() = Gl, < 0,us() = Gs, > 0,

(3.1)

where the bars are used to denote variables that are associated with the basic state,

and Gl and Gs represent the temperature gradients in their respective regions. In

reality, these temperature gradients will depend on various physical parameters of

the system. The motivation for using such temperature profiles comes from the fact

that in a small enough region around the interface, any temperature profile will be

approximately linear. It should be noted that these profiles are consistent with the

Gibbs-Thomson interface condition (2.14), in particular, us = ul = 0 at = 0,

since the curvature of a planar interface is zero. Furthermore, a positive temperature

gradient in the solid implies that it is, in fact, superheated.

The interface velocity can be found by applying the Stefan condition (2.16), which,

in these moving coordinates, is given by

v = duld

+ krdusd

, = 0. (3.2)

Inserting the temperature profiles given in (3.1), the velocity of the planar interface

is given by

v = Gl + krGs. (3.3)

From this expression it can be seen that there are two mechanisms that lead to an

advance of the interface; superheating in the solid and a liquid with a temperature

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x

h(t) h(t) + h(x, t)

v

Liquid

Solidz

Figure 3.1: A planar interface (dashed) growing into a solid with velocity v along thez-axis. Linear stability theory will predict the growth of perturbations to the interface,shown as a solid line.

profile that decreases near the interface1. Both of these mechanisms make physical

sense, because they both lead to a transfer of thermal energy to the interface, which

can then be used to overcome the energy barriers that are associated with melting

additional ice. A particularly interesting feature of this result is that the effects of a

nonzero temperature profile in the solid are either amplified or quenched, depending

on the magnitude of kr, which denotes the ratio of thermal conductivities. For a

system composed of ice and water, this ratio is approximately three. Therefore, theeffects of superheating in the ice are magnified, which may have an important role in

the growth rate of a liquid disc.

3.1.1 Stability of the Basic State

The idea behind linear stability analysis is to determine the growth rate of pertur-

bations that are added to the basic state. Such perturbations will be denoted with

tildes, to make them distinct from the basic state variables. Like the temperature

profiles of the basic state, the form of the perturbations will also be prescribed,

ul(x,,t) = ule

t+a+iax, (3.4a)

us(x,,t) = use

ta+iax, (3.4b)

h(x, t) = het+iax, (3.4c)

where h denotes the perturbation to the planar interface, a represents the wavenumber

of the perturbations, and denotes the growth rate of the perturbations. The variable

1

A liquid with this temperature profile is really the typical case, since a liquid that has anincreasing temperature profile towards the interface is actually supercooled.

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t denotes a time that is independent of the time t that appears in the expression for

the moving coordinate . Furthermore, in these moving coordinates, the interface is

at = h(x, t). Therefore, the interfacial boundary conditions can be written as

ul(x, h, t) = us(x, h, t

) = ,which is the Gibbs-Thomson condition, and

v +h

t= ul n + krus n, = h(x, t)

is the Stefan condition. The normal vector and the curvature are given by

n =hxex + ez

1 + h2x, =

hxx

(1 + h2x)3/2, (3.5)

where the subscripts on h denote differentiation with respect to x, and ex and ez

represent unit vectors along the positive x and z axes, respectively. As it currently

stands, the above set of boundary conditions pose a nonlinear problem that must be

solved in order to obtain the growth rate of the perturbations. However, since it is

the growth of small amplitude perturbations that is of interest, these conditions can

be expanded about h = 0, which significantly simplifies the problem. Under such

assumptions, the expressions for the normal vector and the curvature simplify to

n ez, hxx.By writing the temperature fields as u = u + u, the interfacial boundary conditions

can be linearized to read

ul +duld

h + ul = hxx,

us +dusd

h + us = hxx,

v +h

t=

dul

d+

d2ul

d2h +

ul

+ kr dus

d+

d2us

d2h +

us

,

which hold at the linearized interface position = 0. Using the fact that the tem-

peratures of the planar basic state are zero at the interface and that the basic state

satisfies (3.2), these boundary conditions simplify to become

duld

h + ul = hxx, (3.6a)dusd

h + us = hxx, (3.6b)h

t = d2ul

d2 h +ul

+ kr

d2usd2 h +

us

, (3.6c)

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which are still valid at = 0. By inserting the expressions for the basic state (3.1) and

the perturbations (3.4) into these boundary conditions, a linear system of algebraic

equations for h, ul, and us emerges,

Glh + ul = a2h,Gsh + us = a

2h,

h = aul akrus.

These equations can be combined to yield an analytical expression for the growth of

the perturbations,

= a

krGs Gl (1 + kr)a2 . (3.7)This expression clearly shows the stabilizing effects of surface tension, which tend to

inhibit the growth of large wavenumbers. Moreover, this stabilization mechanism is

enhanced in a water-ice system because the relative thermal conductivity is greater

than one. Furthermore, it can be seen that superheating in the solid can drive the

growth of perturbations, whereas the same temperature gradient in the liquid that

acts to advance the planar interface, actually stabilizes the system. This compet-

itive behaviour can be understood by considering what is happening physically ata sinusoidally perturbed planar interface. Such an interface will have liquid fingers

that extend into the solid, and solid fingers that extend into the liquid, as Figure

3.2 shows. An unstable growth of the interface requires that both the solid fingers

and the liquid fingers will increase deeper into their respective regions. For a liquid

finger to continue its growth, the neighbouring solid must be superheated. Similarly,

supercooled liquid is required for the solid fingers to grow (see Figure 3.2). If, how-

ever, the temperature of the liquid decreases towards the interface (corresponding

to a positive Gl), the liquid is not supercooled and thus growth of the solid fingers

is inhibited. Therefore the system is, in some sense, stabilized, and this is what is

reflected in the linear stability analysis. It should be emphasized, however, that the

interface can still be unstable because the growth of liquid fingers will be sustained

by local superheating in the solid. The fact that linear stability theory tends to find

the average growth rate of the fingers is a severe shortcoming, and it remains unclear

how this will influence the predicted behaviour of more complex models.

As the water in a liquid snowflake is constantly absorbing incoming radiation, it is

not expected to be supercooled. The above arguments therefore predict that when the

interface becomes unstable, the only fingers that will grow are those extending from

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to growRequires superheating

Solid

Liquid

Requires supercoolingto grow

Figure 3.2: A sinusoidally perturbed interface requires superheating in the solid andsupercooling in the liquid in order to sustain the growth of both solid fingers and liquidfingers.

Figure 3.3: When the interface of a liquid snowflake goes unstable, liquid fingers grow intothe superheated ice. The initial circular interface can still be seen as a faint line from whichthe fingers grow out of. The image was obtained using the experimental setup described inAppendix A

the liquid region into the solid. This behaviour is, indeed, observed experimentally,

as Figure 3.3 illustrates. In particular, experiments consistently show the growthof liquid fingers into the superheated ice, a fact which can be deduced because the

circular shape of the interface can often still be seen even after it has become unstable.

More specifically, all of the fingers grow outwards from the circular interface, which

implies the fingers are composed of liquid. Furthermore, solid fingers would appear

as protrusions that extend into the centre of the liquid snowflake, and these have yet

to be observed experimentally.

The expression for the growth rate of the perturbations (3.7) can written in a

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simplified form as

= a(1 + kr) G

a2 ,

where G denotes a weighted average of the interfacial temperature gradients

G = ksGs klGlks + kl

.

For small wavenumbers, the surface energy effects are small and these perturbations

will grow provided that the average temperature gradient is positive. However, surface

energy tends to inhibit large wavenumber perturbations. In particular, perturbations

with wavenumbers greater than acut = (G/)1/2 are stabilized by surface energyeffects and hence they do not grow. The most unstable perturbation is that which hasa wavenumber that maximizes the growth rate, and this is readily computed as ad =

acut/

3. Knowing the wavenumber of the most unstable perturbation is particularly

useful because it is often this perturbation that is observed experimentally.

3.2 The Effects of Superheating due to Radiation

Absorption

The previous example showed that a local region of superheated ice near the interfacecan lead to the growth of perturbations. In the context of liquid snowflake formation,

this superheating is caused by the absorption of radiation, and therefore, it can be

expected that the physical characteristics of this radiation will largely influence the

stability of the system. To study the basic relationship between the properties of

the radiation and the evolution of the interface, another model involving a planar

interface will be used. In this case, however, the temperature of the ice will be solved

as part of the problem, and for simplicity, the temperature of the water will be taken

as constant.

3.2.1 The Basic State

Proceeding in similar manner to the previous example, the basic state is taken to be

a planar interface that is moving along the positive z axis with velocity v. Moreover,

it is assumed that there is a source of radiation that follows this interface, as shown

in Figure 3.4. The temperature of the liquid is assumed to be constant, and the

temperature of the solid will satisfy

cpr ust

= kr 2

usz2

+ re(zvt)2/2,

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= 0

SolidLiquid

v

x

v

Figure 3.4: Schematic diagram of a solid-liquid system being heated by a light sourcewhich travels with the interface.

together with the boundary conditions us = 0 at z = vt, which is the Gibbs-Thomson

condition for a planar interface, and the far field condition us 0 as z . Thisfar field condition is chosen to ensure that any superheating in the solid is solely due

to the absorption of radiation. The Stefan condition for the interface velocity is given

by

Sv = krusz

, z = vt.

To reduce the number of parameters in the problem, time can be scaled according to

t = (kr/cpr)t, which leads to the equation for the temperature becoming

ust

=2usz2

+ Ae(zvt)2/2,

where A = r/kr is a measure of the effective absorption coefficient of ice. The Stefan

condition becomes

Sv =usz

, z = vt,

where S = S/cpr is a modified Stefan number (which is still large), and v = (kr/cpr)v

is a rescaled velocity. The primes on these variables will be dropped in subsequent

computations for notational convenience.It is again useful to write these equations in terms of a variable = z vt that

travels with the interface. Under this coordinate transformation, the equation for the

temperature in the solid is given by

v dusd

=d2usd2

+ Ae(/)2

, (3.8)

which has boundary conditions us = 0 at = 0 and us 0 as . The Stefancondition reads

Sv =us , = 0. (3.9)

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The solution to this problem can be found analytically, and it can be written most

simply in terms of integrals involving the error function

us() =

2A e(v/2)

2

ev erf

v

2

d

ev

0

ev erf

v

2

d

, (3.10)

where is a dummy variable and the interface velocity is given by

v =

2

A

S.

To compute this velocity, the integral identity

ev erf

v2

d = v1ev erf

v2

+ v1e(v/2)2

1 erf

,

has been used, which is readily obtained via integration by parts.

The dependence of the temperature on the modified Stefan number is shown in

Figure 3.5. While the decay rate of these profiles varies significantly, their behaviour

near the interface is almost identical, which suggests that the Stefan number will

not play a significant role in the dynamics that are predicted by the linear stability

analysis. The relationship between the characteristics of the incident radiation and

the interface velocity is best seen by writing the velocity in dimensional form,

v =

2

sI1rbL

. (3.11)

This expression shows, rather unsurprisingly, that the interface velocity is determined

by the rate at which the radiation can supply the ice with enough energy to overcome

the latent heat. Since ice is a poor absorber of radiation, the velocity predicted by this

expression is orders of magnitude smaller than what is observed experimentally. This

may suggest that strong temperature gradients in the water play an important role in

driving the initial growth of a circular liquid snowflake. However, it is important to

keep in mind that (3.11) was the result of studying the evolution of a planar interface,

and this could be a poor representation of the circular interface of a liquid snowflake.

3.2.2 Stability of the Basic State

To investigate the growth of perturbations, the multidimensional analogues of the

basic state equations must be considered. In particular, the temperature of the solid

will satisfy

ust

= 2usx2

+ 2usz2

+ Ae(zvt)2/2 ,

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0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

/

Tus

[K]

S= 1

S= 10

S= 40

Figure 3.5: Temperature profiles of a solid being superheated by incident radiation asviewed from a frame that moves with the planar interface. Parameter values for the non-

dimensional numbers can be found in Table 2.2.

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with boundary conditions us = at z = h(x, t), where h denotes the location ofthe interface, and us

0 as z

. Moreover, the temperature is assumed to be

periodic in the x direction. The Stefan condition is given by

Sh

t= us n, z = h(x, t),

where, as mentioned above, S is a modified Stefan number. Expressions for the

curvature and the normal vector n are given in (3.5).

Before solving these equations, a change of variable is made so that the equations

can be written in terms of the usual coordinate = z vt that follows the planarinterface. Furthermore, a new time t = t/S is introduced that is independent from

the time in the moving coordinate . The derivatives of the old variables t and z are

related to the derivatives of the new variables and t via

t=

1

S

t v

,

z=

.

Therefore, the evolution equation for the non-dimensional temperature in these new

coordinates is given by

1

S

ust

v us

=2usx2

+2us2

+ Ae(/)2

,

and the interface is now at = h(x, t) Svt. The Gibbs-Thompson condition forthe interfacial temperature remains the same, and

liquid snowflake formation in superheated ice

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