surfaces in solid dynamics and fluid statics

Surfaces in Solid Dynamics and Fluid Statics

A thesis presented

by

Haoyu Henry Chen

to

The Department of Physics

in partial fulfillment of the requirements

for the degree of

Doctor of Philosophy

in the subject of

Physics

Harvard University

Cambridge, Massachusetts

January 2005

Thesis advisor Author

Michael P. Brenner Haoyu Henry Chen

Surfaces in Solid Dynamics and Fluid Statics

AbstractIn the first part of this work we formulate a nonperturbative, local description of the

dynamics of surfaces under ion bombardment. Our theory has the form of a nonlinear

advection equation with a fourth order diffusive term. This partial differential equa-

tion admits shock wave solutions with a selected velocity, which correspond to steep

propagating steps on the surface. This selection mechanism provides a precise test

of the transport and sputtering processes on the surface. Our equation also provides

an efficient means of simulating sputtering effects on any surface with regions of high

slope.

The second part presents a method of optimization for the problem of liquid

droplet reduction. We work in the adiabatic limit, wherein the droplet can be calcu-

lated by balancing pressure and surface tension. We show a bifurcation at a critical

pressure and find a relationship between this pressure and the shape of the nozzle

which forms the boundary of the droplet surface. Our formalism yields an algorithm

for changing the nozzle shape such that the droplet volume is minimized. A numerical

example demonstrates a reduction in volume of 20%.

Contents

Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivCitations to Previously Published Work . . . . . . . . . . . . . . . . . . . viAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

1 Introduction 1

2 Shocks on surfaces induced by ion sputtering 52.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Sigmund model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 The one dimensional description . . . . . . . . . . . . . . . . . 112.2.2 Planar limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 A local approximation to the Sigmund model . . . . . . . . . . . . . . 152.4 Surface diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5 Slope evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5.1 Comparison to Sigmund model . . . . . . . . . . . . . . . . . 242.5.2 Slope versus curvature . . . . . . . . . . . . . . . . . . . . . . 272.5.3 Step velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6 Shocks and undercompressive shocks . . . . . . . . . . . . . . . . . . 31

3 Reattachment kinetics 393.1 The closing hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 The structure of the boundary . . . . . . . . . . . . . . . . . . . . . . 40

4 Optimal design 444.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2 Keller’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.3 The optimal faucet . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.4 Pendant droplet instability . . . . . . . . . . . . . . . . . . . . . . . . 504.5 Optimization method . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

iv

Contents v

4.6 Coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.7 Explicit formula for δp. . . . . . . . . . . . . . . . . . . . . . . . . . . 554.8 The deformed faucet . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Bibliography 63

A Relation to previous theories of sputtering 67A.1 Limit of small slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67A.2 Rippling instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

B Numerical solution of surface evolution 72

C A comment on the numerical solution of the drop surface 76

Citations to Previously Published Work

Chapter 4.3 has appeared in the following paper:

“The optimal faucet”, H. H. Chen and M. P. Brenner, Phys. Rev. Lett.92, 166106 (2004), physics/0311046.

Electronic preprints (shown in typewriter font) are available on the Internet at thefollowing URL:

http://arXiv.org

Acknowledgments

I thank my advisor Michael Brenner. His creativity and optimism always con-

vinced me that a solution was forthcoming, even when it’s not. I hope to sustain this

most rational exuberance in my future pursuits.

Much appreciation to my committee members, Jene Golovchenko, Howard Stone,

and David Weitz. Howard made sure that nothing less than rigorous got through.

The sputtering experiments were carried out in the laboratory of Mike Aziz, who

has provided me an education in materials science. Stefan Ichim and Humberto

Rodriguez spent many hours in the basement collecting data.

In the early stages of research I benefitted from many discussions with Mike Burns.

My regards to Melissa Franklin, ”the only worthwhile human being on the physics

faculty” of 1999. (No disrespect to the other humans intended.)

The great raconteur John Barrett was the life of Lyman. His absence is still felt.

My deepest thanks to Sheila Ferguson and Mary Lampros. They made sure that

the place did not collapse.

I am grateful to David Norcross for improving the lot of all graduate students and

mine in particular.

Danny has been a model of decency. He remains the test audience for my most

politically incorrect statements.

Olivia is my devil’s advocate. She does not let me utter anything, no matter how

insignificant, without dissecting it until the logic is beyond reproach. She also has

the longest average conversational sentence length of anyone I know.

I wish Alexia could have been with us for the duration. I’m confident that her

future will be an upward trend.

Many compliments to Eric (he’s a good soul), Dominique, and young Alexis. They

are a persuasive argument for the propagation of the species in this age of so-called

“family values”.

Danae has made the recent years happy ones. She is a thaw in the winter of my

discontent.

Finally I want to salute the courage of my parents for leaving their homeland and

settling in a new country. Despite the recent events in American politics, I still think

they made the right decision.

For my grandmother. She would have enjoyed this.

Chapter 1

Introduction

In this work we address two problems of device design and manufacturing. The

first part is devoted to a study of the dynamics of a surface under erosion by energetic

ions, a process commonly referred to as ion sputtering. Ion sputtering is an important

tool in fabrication [36, 26, 37, 21] of nano- and micro-scale structures, but much of the

experimental phenomena is still poorly understood. In particular, much experimental

and theoretical activity has focused on the spontaneous formation of patterns on the

target substrate. These patterns come in a wide variety such as ripples, islands, or

highly disorganized structures [29, 30, 14, 8, 39]. Because pattern formation is a

widely occurring feature of ion sputtering, it is important for fabrication either to

prevent it or to exploit it. Perhaps it will become possible to expose a templated

surface to the ion source and allow the sputtering mechanism to generate the desired

structure.

The motivation for our present study of sputtering is an ongoing series of experi-

ments performed in the laboratory of Prof. Michael Aziz in the Division of Engineer-

ing and Applied Sciences at Harvard University. In these experiments, micrometer

scale pits are etched into a silicon substrate and then exposed to the ion beam. Dur-

ing sputtering, the larger pits expand, and the smaller ones contract. Figure 1.1 is

an example of a expanding pit. Remarkably, the pit edge remains sharp and steep as

it advances. In Chapter 2 we develop the theory to model this behavior and identify

the advancing pit edge as an example of an “undercompressive” shock [1]. Moreover,

1

Chapter 1: Introduction 2

Figure 10: SEM view at 52° from normal of the initial profile and final profile (after

Figure 1.1: Left: an initial pit in silicon with 3 µm diameter and 0.5 µm depth. Right:The enlarged pit after ∼ 15× 1018 ions/cm2 exposure. Courtesy of S. Ichim.

our theory predicts that the pit edge must maintain a particular slope and velocity

during its sputter-induced motion, independent of the initial conditions and material

parameters such as the adatom diffusion constant. This selectivity is not present in

conventional shocks and provides a very stringent test of the sputtering process.

In Chapter 3 we address the contracting pits. The phenomenon has some resem-

blance to previous work on the ion sculpting of nanometer sized holes [21, 34]. In

those experiments a hole is drilled through the target substrate and then exposed

to the ion beam. It was found that the hole can open or close, depending on the

temperature and on the time sequence of the ion beam (for a pulsed ion beam). We

will argue for a new diffusive process on the surface, different from Mullins-type sur-

face diffusion [25], that provides a length scale that distinguishes small contracting

pits from large expanding ones. However the details of the mechanism have not been

established, in part due to lack of experimental data. We offer a candidate model for

this new process in order to clarify the state of our ignorance, perhaps as a guide for

future experiments.

In Chapter 4 we turn to a problem of device design. The goal is the production

of small liquid droplets. Such technology has obvious application to inkjet printing,

and much work as been done in that industry. A less obvious application is in the

important technique of microarrays in biology [31]. Microarrays are chips on which

tiny samples are deposited, so many different specimens can be tested simultaneously.


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−1.5

−1

−0.5

0

0.5

1

1.5

2

Figure 1.2: The improved faucet.

One way of depositing the samples is by droplet ejection from an inkjet. Of course,

the smaller droplet one can make, the greater the throughput of the microarray.

Our approach to droplet reduction is to change the shape of the nozzle opening

from which the droplets are produced. We work under the assumption that the

droplet is adiabatically forced out of the nozzle by increasing the pressure. We show

that there is a critical pressure at which the droplet surface becomes unstable due to

a bifurcation, and derive a formula relating this critical pressure to the shape of the

nozzle. Our analysis makes clear that the usual circular nozzle is in fact the worst for

making small droplets! In addition, the formula provides an algorithm for decreasing

the droplet by changing the nozzle shape iteratively. Using this, we demonstrate that

a quasi-triangular nozzle (Figure 1.2) decreases the droplet by around 20% compared

to the circular nozzle which has the same critical pressure. The critical pressure is

the relevant experimental constraint due to material failure.


Our formalism is inspired by a classic work of Joseph Keller [19] which treats a

variation of the Euler buckling problem. This formalism is quite general and can be

used in other contexts. So we provide a brief summary of Keller’s problem by way of

introduction to our more complicated problem of droplet size optimization.

Chapter 2

Shocks on surfaces induced by ion

sputtering

2.1 Overview

In recent experiments, Aziz et al. used a focused ion beam (FIB) of gallium ions at

30 keV to sputter various target surfaces. Figure 2.1 is taken from an experiment done

on a magnesium target. Strikingly, pits form on the surface from void defects and

evolve into highly regular circular pits which subsequently enlarge under continued

ion exposure.

Another set of experiments was done on silicon surfaces with prefabricated pits.

Figure 2.2 is an example of the initial surface. The silicon target was exposed to the

ion beam over a region containing the pits, and Figure 1.1 shows the typical behavior

of a pit. Figure 2.3 gives the radius as a function of ion dose (which is proportional

to time) for a set of 0.5 µm deep pits with a range of initial radii. We see that, after

some initial transient behavior (up to ∼ 3× 1018 ions/cm2), the pits with initial radii

greater than 1 µm all expand at a fixed velocity. The transient behavior is due to the

implantation of ions into the pure silicon substrate, rendering it amorphous. As seen

in Figure 1.1, an expanding pit maintains a well defined edge with a steep wall slope.

This behavior is well outside the scope of existing theories [3, 23, 22] because they

assume small slopes. Another striking feature of the surface evolution seen in Figure

5

Chapter 2: Shocks on surfaces induced by ion sputtering 6

Figure 2.1: Formation of circular pits on magnesium under 30 keV gallium ion sput-tering. Courtesy of W. Zhou and M. J. Aziz.


Figure 2.2: Prefabricated circular pits on silicon. Depth of each pit is 1 µm. Diametersrange from 0.5 µm to 4 µm. The viewing angle is 52 from vertical. Courtesy of S.Ichim.

0.5 µm initial depth

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6 7 8

dose (1018

ions/cm2)

rad

ius (

µm

)

21.81.61.41.210.750.50.25

initial radius (µm):

diameter ambiguity

sample error bar

Figure 2.3: Radius versus dose for 0.5 µm deep circular pits. Figure taken from [17].


Figure 2.4: Threshold radius versus initial depth of pit. The solid line is a guide tothe eye. Figure taken from [17].

1.1 is that the surface remains smooth inside and outside the pit, whereas many

experiments have shown the development of surface instabilities on various targets

[5, 29, 30, 10, 11, 14, 35, 8, 39, 18]. Because the ion implantation renders silicon

amorphous, one can avoid complications due to crystallinity of the target material.

In the rest of this chapter we will develop the formalism to describe the expanding

pit behavior in silicon.

As for the smaller pits (initial radii below 1 µm) in Figure 2.3 that shrink and close,

a new mechanism other than sputtering is needed. Figure 2.4 shows the dependence

of the threshold radius (the initial radius below which the pit closes) on the initial

pit depth. This dependence is perplexing. We will address this in the next chapter

and offer some clues.

The rest of the chapter is organized as follows. We first introduce the classic

theory of sputtering due to Peter Sigmund [32, 33] which provides a simple geometric

picture for the sputtering of surfaces by ions. Then we show how Sigmund’s theory

may account for surfaces of large slope such as the expanding pit, and in doing so,


derive a local approximation to Sigmund’s model in the form of a partial differential

equation (PDE), valid for our experimental conditions. Finally, we show that our

PDE admits shock wave solutions which we identify as the edge of the expanding

pit. A central result of our analysis is that the shocks have the surprising feature of

selecting a particular slope of the pit wall corresponding to an angle of

θ ≈ 75 (2.1)

from the horizontal. A direct consequence of this selection mechanism is that the

ratio of the pit edge velocity to the sputter rate of the target surface is

VedgeVsurface

≈ 1.7. (2.2)

This number does not depend on experimental parameters such as temperature or sur-

face diffusion coefficient, but is a characteristic of the target material. Such predicted

dimensionless quantities constitute a severe experimental test of the mechanisms of

surface evolution.

2.2 Sigmund model

According to Sigmund [33], the probability of an atom being dislodged from the

target surface is simply proportional to the total energy that reaches the atom from

nearby ion collision processes. The key simplification is to suppose that each ion

penetrates a distance a below the surface as measured along its trajectory, upon

which the ion energy is released in a Gaussian distribution with widths σ along the

trajectory and µ perpendicular to it (Figure 2.5). This simple picture is based on

Sigmund’s earlier work on solutions to the Boltzmann transport equation [32] and is

the starting point for much of the theoretical work in sputtering.

In cartesian coordinates, the surface is conveniently described by a height function

z = h(x, y), with the ion trajectories along the z-axis. The distribution of energy due

to an ion at (x′, y,′ , z′) is

E(x, y, z; x′, y′, z′) =ε

(2π)3/2σµ2exp

(−(z′ − z)2

2σ2− (x′ − x)2 + (y′ − y)2

2µ2

), (2.3)


z

x

incident ion

Figure 2.5: Schematic picture of Sigmund model. The dashed curve indicates the lociof ion penetrations below the target surface (solid curve).


where ε is the total energy released. Since we are interested in the energy that reaches

the surface, let z = h(x, y) and z′ = h(x′, y′)− a, and the above expression becomes

E(x, y; x′, y′)

=ε

(2π)3/2σµ2exp

(−(h(x′, y′)− a− h(x, y))2

2σ2− (x′ − x)2 + (y′ − y)2

2µ2

).(2.4)

To obtain the power density at (x, y), we integrate Eq. 2.4. The velocity of a

surface element due to sputtering is then

v(x, y) = pJ∫

dx′dy′E(x, y; x′, y′), (2.5)

where J is the ion flux, and p is a proportionality constant relating sputtering rate

and power density. In principle the integration extends to the entire surface, but the

integrand is very much localized due to the Gaussian distribution. v is the velocity

normal to the surface, and in our coordinates is

v = − ∂h/∂t√1 + |∇h|2

. (2.6)

We now have a complete mathematical description of the surface evolution under ion

sputtering, as defined by Sigmund.

2.2.1 The one dimensional description

When the radius of a pit is large compared to the length parameters in the Sigmund

model (a, σ, µ), as is the case for the experiments we are studying, it suffices to treat

the edge as being straight and to consider its profile as a one dimensional step. Hence

we will work in the limit that the surface geometry is translationally invariant in the

y direction, unless stated otherwise. Then we can perform the y integration in Eq.

2.5 and arrive at

h√1 + h′(x)2

= − 1

2πTs

∫dx′ exp

[−(h(x′)− a− h(x))2

2σ2− (x′ − x)2

2µ2

], (2.7)

where h and h′ denote differentiation in time and space, respectively. We introduce

the time constant

Ts ≡ σµ

pJε. (2.8)


Figure 2.6: The ion is closer to the surface when the slope is nonzero (right) comparedto the zero slope case (left). However, the flux per surface area is reduced on the right.

2.2.2 Planar limit

To gain intuition into Eq. 2.7, we study the simplest case of a flat surface,

h(x) = bx + h0, (2.9)

with b constant. The integral in Eq. 2.7 is Gaussian and evaluates to

√2πµ√

1 + b2 µ2

σ2

exp

− a2/σ2

2(1 + b2 µ2

σ2 )

≡ I(b). (2.10)

This expression has a simple interpretation. In the exponent, the slope b effectively

reduces the penetration depth, because for off-normal incidence, the ion stops closer

to the surface. See Figure 2.6. Similarly, the b dependence in the prefactor is a

correction to the ion flux due to the surface inclination.

The sputter yield is defined to be the number of atoms sputtered per incident ion.

Since the ion flux is along the vertical direction, the sputter yield is proportional to

the vertical rate of surface erosion, h; hence we multiply Eq. 2.10 by√

1 + b2 and


0 10 20 30 40 50 60 70 80 900

2

4

6

8

10

12

14

Incidence angle (degrees)

Spu

tter

yiel

d (n

orm

aliz

ed)

experimentSigmundwith correction

Figure 2.7: Angular dependence of sputtering yield. The yield is normalized by thenormal incidence value. The parameters from fitting to the data are a/σ = 2.04,µ/σ = 0.658, and Σ = 0.0462. The data points are taken from Figure 2 in Vasile etal. [37], and fit to Eq. 2.12. The dashed curves is obtained by setting Σ = 0, leavingµ/σ and a/σ unchanged.

arrive at the sputter yield

Y (θ)

Y (0)=

√1 + b2

√1 + b2 µ2

σ2

exp

− a2/σ2

2(1 + b2 µ2

σ2 )+

a2/σ2

2

, (2.11)

where θ is the angle between the ion trajectory and the surface normal, and b = tan θ.

We have normalized this expression by the yield at normal incidence.

The dashed curve in Figure 2.7 shows the typical slope dependence of the sput-

tering yield. For not too oblique incidence angles, this is consistent with experi-

mental determinations [7, 37], although for large slopes the yield diminishes, so that

there is a peak at a high angle. Several authors [37, 12] have suggested the factor

exp(−Σ/ cos θ) = exp(−Σ√

1 + b2) to correct for large incidence angles, where Σ is a


fitting parameter. Then the normalized sputter yield becomes

Y (θ)

Y (0)=

√1 + b2

√1 + b2 µ2

σ2

exp

− a2/σ2

2(1 + b2 µ2

σ2 )+

a2/σ2

2− Σ(

√1 + b2 − 1)

. (2.12)

With three fitting parameters (a/σ, µ/σ, Σ), Eq. 2.12 is a reasonable analytic de-

scription of the experimental yield, as seen in Figure 2.7.

We note that there is a popular software package SRIM (Stopping and Range of

Ions in Matter [40]), freely obtainable from http://www.srim.org, which calculates

the parameters a, µ, σ for various ions and targets. For the case of 30 keV gallium

ions on silicon, SRIM gives a penetration depth of 270 A. However, the values it

gives for the energy distribution widths are inconsistent with our fitted ratios. The

discrepancy is perhaps related to limitations in SRIM recently studied [38]. We will

use the fitted ratios and use a = 270 A as a reference length when necessary.

The large angle correction should be taken as a phenomenological fitting parameter

and not the unique expression from some underlying theory. For a crystalline target,

it is expected that its orientation with respect to the ion trajectory would affect the

collision cascade, such as in surface channeling. But even for amorphous materials,

the probability that an ion would reflect off the surface rather than penetrate becomes

significant at glancing angles. Another source of this correction resides in the validity

of the Sigmund Model itself, since it is derived from Boltzmann’s transport equation

under the approximation of an infinite medium. Of course, at large angles the surface

becomes ever closer to the ion energy distribution, and there must be corrections to the

Gaussian distribution due to the presence of a boundary. To isolate the mechanism(s)

responsible in the case at hand is beyond the scope of our treatment. In any case,

as we shall see, the results contained in subsequent sections follow from the fitted

yield function itself and will not depend on the details of the Sigmund theory and its

possible extensions.

Having determined the form of the correction, we can now add it to Eq. 2.7,

h√1 + h′(x)2

= −e−Σ(√

1+h′(x)2−1)

2πTs

∫dx′ exp

[−(h(x′)− a− h(x))2

2σ2− (x′ − x)2

2µ2

],

(2.13)


with the understanding that the value of Σ is always determined by fitting the yield

versus slope, as we did above.

2.3 A local approximation to the Sigmund model

In order to study the dynamics of surfaces, we need to describe surfaces that are

not flat. Since we have seen that the Sigmund integral is Gaussian for a constant

slope, we are lead to consider a surface with slowly varying slope, in a sense to be

made precise shortly. This will yield a local description of the Sigmund model as

a partial differential equation (PDE). The striking feature of this PDE is that it

describes surfaces of arbitrarily large slopes, as in the case of the pit edge.

Let’s choose to calculate the surface velocity at x = 0, such that the surface has

the expansion

h(x) = h0 + bx +η

2x2 + ... , (2.14)

and has slope b at the origin and second derivative η. We wish to find the correction

to the flat surface evolution to leading order in η. Setting x = 0 in Eq. 2.7 and

dropping the prime in the dummy variable, the integral becomes∫

dx exp

[−(bx− a + ηx2/2)2

2σ2− x2

2µ2

], (2.15)

which to first order in η is∫

dx exp

[−(bx− a)2

2σ2− x2

2µ2

] (1− ηx2

2σ2(bx− a)

). (2.16)

This integral consists of a Gaussian multiplied by another function. If this function

varies slowly over the width of the Gaussian, then our approximation is good, and we

can neglect higher derivative terms in the Taylor expansion of h(x). After completing

the square in the exponent, the width ∆ of the Gaussian is found to be

1

∆2=

b2

2σ2+

1

2µ2. (2.17)

The η-independent term is already shown in Eq. 2.10, while the order η integral

evaluates to

aµ

√π

2

(µ2

σ2

) −2b4µ4/σ4 + (a2/σ2 − 1)b2µ2/σ2 + 1

(1 + b2µ2/σ2)7/2exp

[− a2/σ2

2(1 + b2µ2/σ2)

]η


≡ G(b) η. (2.18)

Using Eqs. 2.10 and 2.18, the surface evolution Eq. 2.13 becomes

h = −√

1 + b2

2πTs

e−Σ(√

1+b2−1)[I(b) + G(b)h′′(x)]

= −F (b) + D(b)h′′(x), (2.19)

where we have replaced η with h′′(x) and restored the x dependence. The slope is

now a local quantity: b(x) = h′(x). We see that the surface evolution consists of a

slope dependent erosion term and a diffusive term. (We call this diffusive because

of its formal resemblance to the diffusion equation and do not imply any diffusing

mechanism!) For clarity we have defined the flux

F (b) ≡√

1 + b2

2πTs

e−Σ(√

1+b2−1)I(b) (2.20)

and the diffusion coefficient

D(b) ≡ −√

1 + b2

2πTs

e−Σ(√

1+b2−1)G(b). (2.21)

Note that we have not made any restrictions on the magnitude of the slope. Eq. 2.19

is valid for regions of slowly varying, but arbitrarily high slope!

Though we have chosen normal incidence, our formalism actually contains all

incidence angles because any incidence angle can be brought to zero by suitably

tilting the surface, that is, shifting the slope by a constant. “Incidence angle” and

”angle of surface inclination” are equivalent. Hence all of our results will hold for

any incidence angle through a rotation or projection. Other authors have chosen to

work in a local coordinate system wherein the slope is locally zero by shifting the

incidence angle [3, 23, 22]. This needlessly complicates matters, and forces one to

take the small slope limit in order to connect regions of different slopes. (A locally

varying ”incidence angle” would be equivalent to our approach, but is an unnatural

concept.)

Of course, D(b) is constant, and therefore can be interpreted as a diffusion coef-

ficient, only for flat surfaces. Moreover it has the wrong sign for b = 0; D(b = 0)

is negative, and the surface is unstable to short wavelength perturbations. This is


a) b)

Figure 2.8: Sigmund sputtering of a valley (a) and a crest (b). As evident from thediagrams, the bottom of the valley is erodes faster than the top of the crest because theneighboring ion implantations are closer to the center in the case of the valley. Thisdifference in sputtering rates causes a ripple to be amplified under normal incidencesputtering. This observation is due to Bradley and Harper.

the celebrated Bradley-Harper instability [3]. Figure 2.8 illustrates this phenomenon.

However, as seen in Figure 2.9, D(b) changes sign at a higher slope, and this instability

disappears.

It should be noted that the story is more complicated for real surfaces because, in

addition to ripples, the instabilities can be intrinsically two dimensional. Figure 2.10

is a dramatic example of such a instability which we call “grass” or “hair”. Clearly

this cannot be described with a one dimensional model.

Aside from two dimensional instabilities, ripple instabilites are also trickier be-

cause rotational symmetry is broken. The ripples must pick an orientation. At

normal incidence, this can be done by, say, the raster pattern [39]. For off normal

incidence, ripples typically either form parallel to the ion trajectory (as seen from

above) or perpendicular to it (Figure 2.11). For very oblique incidence, the former

is the case. For close to normal incidence, we have the latter, which can be studied

in our one dimensional description. At intermediate incidence angles, the two ori-

entations compete [3], and the slope at with D(b) becomes positive marks the angle


0 10 20 30 40 50 60 70 80 90−1.5

−1

−0.5

0

0.5

1

1.5

Slope (Angle)

D(b

)/|D

(0)|

Figure 2.9: D(b) normalized by the zero slope value, as a function of slope angle.Note the sign change at 64.


Figure 2.10: Surface morphology of germanium after sputtering. Photo taken at 37

from normal. Courtesy of S. Ichim.


Figure 2.11: For off normal incidence angle, the ripples can align perpendicular tothe ion path (left) or parallel to it (right).

beyond which ripples will always be parallel.

2.4 Surface diffusion

Of course, at normal incidence, the instability cannot occur at arbitrarily small

wavelengths. Surface diffusion dominates at sufficiently small distances, and together

with the instability will select a finite wavelength. Surface diffusion is the result of

random walks of mobile adatoms, biased by the mean curvature.

According to the classic work of Herring and Mullins [16, 25], we can define a

chemical potential on the surface

µ = KγΩ, (2.22)

where K is the mean curvature, γ is the surface-free energy per area, and Ω is the

molecular volume. By the Nernst-Einstein relation, a gradient of the chemical poten-

tial gives rise to a surface current of adatoms, given by

J = −Dsν

kT∇µ = −DsγΩν

kT∇K, (2.23)

where Ds is the constant of surface diffusion, k is the Boltzmann constant, T is

temperature, and ν is the area density of adatoms. In almost all treatments, ν is


assumed to be a constant over the surface; that is, the mobile atoms that undergo

diffusion comprise a uniform outer layer. We will come back to this point later.

Mass conservation requires that the normal velocity of a surface element, vn, is

equal to the divergence of −J , with an additional factor of the molecular volume. So

the surface evolution is described by

vn =DsγΩ2ν

kT∇2K ≡ B∇2K. (2.24)

Note that in the context of surface transport the gradient operator∇ is that associated

with the surface. If our surface is described by a height function h(x), then the surface

gradient is

∇ =1√

1 + h2x

∂

∂x. (2.25)

Finally, the mean curvature is given by

K = − hxx

(1 + h2x)

3/2. (2.26)

Putting everything together, the evolution of the surface due to sputtering (Eq. 2.19)

and the Mullins mechanism is given by

∂h

∂t= −F (hx) + D(hx)hxx −B

∂

∂x

1√1 + h2

x

∂

∂x

hxx

(1 + h2x)

3/2. (2.27)

The constant B is defined in Eq. 2.24.

Consider the case of a level surface (b = 0). The dynamics consist of sputtering

and surface diffusion, which leads to the PDE

h = −F (0) + D(0)h′′ −Bh′′′′, (2.28)

where the coefficients, F and D, are evaluated at b = 0, and the Mullins term is

linearized. We assume that h is of the form

h = vt + ε exp(st + ikx). (2.29)

The first term is just a uniform shift from the sputtering, and the second term rep-

resents rippling of a fixed wavenumber k and growth rate s. Plugging this into the

PDE and choosing v to eliminate the F -term, we get

s = D(0)k2 −Bk4. (2.30)


From this we easily find that the wavenumber that maximizes the growth rate is

k2 =D(0)

2B. (2.31)

This is the predicted length scale of the instability, as derived by Bradley and Harper

[3].

In Appendix A we show the how our model reduces to previous models [3, 23, 22]

in the small slope limit. There we also give the condition under which such a limit is

justified.

2.5 Slope evolution

We now let the slope vary in space. Eq. 2.19 still applies, as long as the slope varies

slowly, i.e., ∆h′′ << 1 over the area of interest, where ∆ is given by Eq. 2.17. This

condition holds for the pit experiments we are considering. (It would presumably fail

in Figure 2.10). Since the rate of sputtering is unaffected by a uniform shift in surface

height, it is convenient to consider instead the evolution of the slope. Differentiating

Eq. 2.27 on both sides, we find

∂b(x, t)

∂t= − C(b(x, t))

∂b(x, t)

∂x+

∂

∂x

(D(b(x, t))

∂b(x, t)

∂x

)

− B∂2

∂x2

1√1 + b2

∂

∂x

bx

(1 + b2)3/2. (2.32)

In the above we have defined the nonlinear velocity

C(b) ≡ ∂

∂bF (b), (2.33)

and now D(b) is formally a diffusivity. Eq. 2.32 is a nonlinear advection-diffusion

equation for the slope, with velocity C(b) and diffusivity D(b) both functions of the

slope, with an additional surface diffusion term. We take this PDE to be our nonlinear

model of surface dynamics under ion sputtering.

Figure 2.12 shows the advective velocity C(b). For small slope, C(b) linearly

increases with the slope, hence the advective term will produce shockwaves, as in the

Burgers equation. An initial hump in the slope (corresponding to a step profile) will

“tip over” unless resolved by a diffusive term.


0 1 2 3 4 5 6−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

b (slope)

C(b

)

Figure 2.12: Velocity of advection as a function of the local slope.


2.5.1 Comparison to Sigmund model

We need to determine how effectively Eq. 2.32 describes the full Sigmund integral

theory under typical conditions in the experiments. The full Sigmund model is given

by

h = −√

1 + h2x

2πTs

e−Σ(√

1+h2x−1)

∫dx′ exp

[−(h(x′)− a− h(x))2

2σ2− (x′ − x)2

2µ2

]

− B∂

∂x

1√1 + h2

x

∂

∂x

hxx

(1 + h2x)

3/2. (2.34)

We simulate the above dynamics by performing the integration using Simpson’s

rule and solving the fourth order surface diffusion semi-implicitly. See Appendix

B for details. The results of the integration and the PDE model are presented in

Figure 2.13. In Figure 2.14 we show more clearly the advancement of the step. The

profile consists of a steep leading edge followed by a less steep tail. Eventually the

leading edge disappears, and the profile broadens completely. This broadening can

be understood from the advection itself. Consider a localized hump in b(x) - this

corresponds to a step in h(x), and the area under the hump must be conserved, since

it is just the step height. Because the velocity of advection is not constant and in fact

goes to zero for small slope, the base of the hump lags behind during advection, and

by area conservation the slope distribution must spread and decrease in magnitude.

The simulation shows an advancing step which corresponds to an expanding pit

in the experiments on silicon. Because the advection velocity is an odd function of

slope, a step will always move in the “ascending” direction, which means that a pit

always expands under sputtering alone. This suggests that another mechanism is

needed to cause a pit to close. We defer this discussion to the next chapter.

At present it is difficult to make detailed comparisons of the profile with exper-

iment due to lack of high resolution data. We will identify the leading edge as the

boundary of the ”pit”, so long as it’s well defined. Note that our theory neglects

secondary collisions from ions which glance off the steep region; this is a nonlocal

mechanism which could have a significant effect on the profile.


−50 0 50 100 150

−40

−20

0

20

40

60

80

x

h(x)

t = 0

t = 150

Figure 2.13: Surface evolution under the full Sigmund model (dotted) and the reducedPDE (solid). The agreement is excellent, and the discrepancy appears at later timesin the high slope regions. The profiles are separated by time intervals of ∆t = 10. Wechose the sputtering time constant Ts = 0.1 and the surface diffusion constant B = 20.The grid spacing is ∆x = 0.5 for both models. The initial profile is h = 50 erf(x/20).


−50 0 50 100 150

x

t = 0

t = 150

Figure 2.14: Profiles from the PDE model, aligned such that the sputtering of thehorizontal regions is eliminated for clarity.


−50 0 50 100 150

−40

−20

0

20

40

60

80h(

x)

x

t = 0

t = 150

Figure 2.15: Surface evolution from the Sigmund model (dotted) and the PDE withoutsecond order diffusion (solid). The parameters are the same as in Figure 2.13.

2.5.2 Slope versus curvature

Recall that our advection-diffusion PDE comes from a systematic expansion of

the Sigmund integral. To determine the effectiveness of this expansion, we eliminate

the second order (anti-)diffusive term (while retaining surface diffusion). Figure 2.15

shows the resulting profiles. We see that the advection alone is a reasonable descrip-

tion of sputtering. This is not surprising given that the profile is at a length scale

much larger than the sputtering parameters. Discrepancies appear at the top and

bottom of the leading edge, where curvature is most appreciable.

2.5.3 Step velocity

We’d like to predict the velocity of the leading edge as a function of the parameters.

Clearly this is given by the difference between the sputtering rate of the edge region


0 1 2 3 4 5 60

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Sho

ck V

eloc

ity (µ

/Ts)

Slope (b)

Figure 2.16: Step velocity (in units of µ/Ts) as a function of the step slope. Thecrosses are from simulations such as in Figure 2.13, for values of B = 10, 20, 30. (Wealso tried initial height profiles of various slopes and depths, and the slopes of theleading edges always evolve to be around 4.) Clearly a specific slope (and thereforevelocity) is dynamically selected.

and the horizontal rate, divided by the edge slope:

U =F (b)− F (0)

b, (2.35)

where b refers to the slope of the steep leading edge. Note that this is the Rankine-

Hugoniot condition familiar from shock theory, which we address in the next section.

Figure 2.16 shows this relation between the velocity and the slope of the edge region,

and where along this curve are our simulations.

The advection velocity C(b) tends to zero for decreasing slope, so that a hump

profile in the slope, corresponding to a step on the surface, will be ”fixed” at the base

during advection. As the hump travels, the bottom portion will drag behind, and

since the area under the hump (equal to the step height) is conserved, its amplitude


0 50 100 150 200 250

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x

Slo

pe

t = 0

t = 300

Figure 2.17: Evolution of step profiles in b(x, t). The amplitudes of the initial stepsare 3.5 (dashed) and 4.5 (solid). Both profiles develop a leading step with amplitude4.01. See Figure 2.18 for a close up of the leading region.

must diminish in time. This is known as rarefaction and has the effect of smearing

out the surface step, so that an advancing pit edge will eventually become less steep.

However, as seen in Figure 2.15, there is a window of time in which the upper portion

of the step remains steep. In actual experiments, the pits remain steep for the duration

of the experiment (as in Figure 1.1); this is possibly due to enhanced erosion at the

base of the step from reflected ions.

To avoid the rarefaction effect, let’s use a step instead of a hump as the initial

condition for the slope. Figure 2.17 shows the resulting evolution for two different

initial conditions. Both give rise to a leading step with the same amplitude. Figure

2.18 is a close up of this region. These simulations suggest not only the emergence

of shocks, but also a particular velocity of shock. In order to understand why this

velocity emerges in the simulations, we need to study shock wave solutions to the


160 170 180 190 200 210

0

0.5

1

1.5

2

2.5

3

3.5

4

x

Slo

pe

Figure 2.18: Close up of the leading step region of Figure 2.17.


a)

b)

Figure 2.19: Shock formation from a) a hump and b) a step. The dashed lines markthe location of the shock front when there is a diffusive mechanism that prevents thetipping over.

slope evolution, Eq. 2.32.

2.6 Shocks and undercompressive shocks

The prototype for all discussion of shocks is the Burgers Equation

bt = −bbx + εbxx, (2.36)

where b(x, t) is a quantity, say, density of a gas. This equation has the property

that the advection velocity is linear in the quantity itself, hence an an initial profile

such as a hump will tend to tip over, in which case b(x) is no longer single valued.

The diffusive term acts to prevent this from happening, and the propagating solution

instead has a front with thickness ∼ √ε. This is the simplest example of a shock. See

Figure 2.19.


Eq. 2.36 admits a steady state solution that is a shock with constant velocity U

with the boundary conditions

b|x→−∞ = b−, b|x→+∞ = b+, (2.37)

with b− > U > b+. In fact, it’s easily to verify that

U =b− + b+

2. (2.38)

Proceeding by analogy with our slope evolution, in which the surface remains

level ahead of the front, b+ is zero. The above expression gives a relation between

the shock velocity U and the boundary value b−. But any positive b− will give a

shock solution. Unlike our advancing pit edge, the Burgers Equation does not select

a particular value of b−.

To clarify the issue, let us suppose, for our slope advection equation, a traveling

solution

b(x, t) = b(

x− Ut

λ

), (2.39)

where λ is the width of the shock front. Recall that the second order term in Eq. 2.32

has a small effect on the evolution, we will neglect it in the following. We will also

pretend for the moment that the fourth order surface diffusion term is linear. This

will not affect the argument.

The advection velocity can be written as

C(b) =µ

Ts

∂f(b)

∂b, (2.40)

where f is the nondimensional flux

f(b) ≡ 1√2π

√1 + b2

√1 + b2 µ2

σ2

exp

− a2/σ2

2(1 + b2 µ2

σ2 )− Σ(

√1 + b2 − 1)

. (2.41)

Our equation is now

bt = − µ

Ts

f(b)x −Bbxxxx. (2.42)

Upon substituting the traveling solution we have

−U

λb′ = − µ

Ts

1

λf(b)′ − B

λ4b′′′′, (2.43)


where primes denote differentiation with respect to the dimensionless coordinate

z ≡ x− Ut

λ. (2.44)

We integrate the above equation from z to +∞ to get

−Ub = − µ

Ts

[f(b)− f(0)]− B

λ3b′′′, (2.45)

where the constant of integration comes from the condition that b → 0 as z → +∞.

After rearranging, we have

BTs

λ3µb′′′ =

UTs

µb− [f(b)− f(0)]. (2.46)

Choosing λ such that BTs

λ3µ= 1 and defining U ≡ UTs

µas the nondimensionalized shock

velocity, our shock equation is

b′′′ = Ub− [f(b)− f(0)]. (2.47)

On the left side of the shock, b = b− is constant, and we have

U =f(b−)− f(0)

b−(2.48)

which is again the Rankine-Hugoniot condition.

Figure 2.20 shows both the shock velocity U(b−) and the advection velocity C(b).

We see that shocks of amplitude less than a certain value (b− < 2.53) has a travel

velocity less than the advection velocity to the far left, i.e., U(b−) < C(b−). This is

the Lax condition which characterizes ordinary shocks such as in the Burgers case.

Intuitively, the shock is being “compressed” from both sides. The Lax condition is

violated when the shock amplitude is greater (b− > 2.53), for which U(b−) > C(b−).

In this case the shock is faster than the advection velocity to the far left, and for good

reason this kind of shock is known as “undercompressive” [1, 2].

Taking the variation of Eq. 2.47,

δb′′′ = Uδb− ∂f

∂bδb = (U − c(b))δb, (2.49)


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Slope (b)

Vel

ocity

(µ/T

s)

advectionshockselected

Figure 2.20: Comparison of the shock velocity U and the advection velocity C (inunits of µ/Ts) as functions of slope. Note that they cross at b = 2.53. Below this wehave ordinary Burgers-type shocks, and an undercompressive shock exists at b = 3.89,indicated by the dot. This is found from the solution of Equation 2.51.


where c(b) ≡ ∂f∂b

is the dimensionless advection velocity. For small deviations, this

equation has the homogeneous solutions

δb ∼ eaz, (2.50)

where a = 3√

U − c(b) has three values in the complex plane.

As z → +∞, U > c(0), hence there is one value of a with positive real part.

So one boundary condition is needed to eliminate this growing mode. On the other

side, z → −∞ and U > c(b−), giving two growing modes in the negative direction.

So a total of three boundary conditions is needed to eliminate exponential growth

in both directions. Eq. 2.47 is third order, so we have just enough boundary condi-

tions. However, the traveling solution b(z) needs to be translationally invariant, and

since we’ve exhausted the boundary conditions, this is not possible in general. The

remaining possibility is that for particular value(s) of U there exists a translationally

invariant solution. This is the selection mechanism for the shock velocity as evidenced

in the dynamical simulations. Note that the same argument can be applied to the

Lax shock, for which there is only one growing mode on the left, and counting shows

that we do not need to adjust U to have a solution. The undercompressive nature of

the shock and the diffusion being fourth order constitute necessary conditions for the

selection mechanism.

For the case of the nonlinear surface diffusion, our argument is unchanged because

it only assumes that the diffusive term is fourth order. Of course, there will be

additional terms on the right hand side of Eq. 2.47, and therefore the slope selection

will yield different values.

The boundary value problem is

(1√

1 + b2

(b′

(1 + b2)3/2

)′)′= Ub− [f(b)− f(0)] (2.51)

with the boundary conditions

b(z → −∞) = b−, b(z →∞) = 0 (2.52)

and

b′(z → −∞) = 0, b′(z →∞) = 0. (2.53)


−20 −15 −10 −5 0 5 10 15 20−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

z = (x − Ut)/λ

Slo

pe

Figure 2.21: Shock solution obtained from the boundary value problem (Eq. 2.51).b− is 3.89 and U is 0.57.

Here the unknown parameter b− is itself part of the solution to be determined. The

BVP4C solver in MATLAB r© allows for parameter determination, and Figure 2.21 is

the solution. The selected slope is b− = 3.89 which is close to the value of b− = 4.01

found in the simulation (Figure 2.17). The discrepancy is due to numerical diffusion

in the PDE simulation, which is in general proportional to the time step [15, 1], hence

the value obtained in the boundary value problem is the “correct” one. The width of

this solution scales like

λ =

(BTs

µ

)1/3

, (2.54)

which means that the strength of surface diffusion, B, sets the sharpness of the shock

front, but does not affect the slope selection. This is an enormous simplification

because the experimental value of B is a difficult quantity to measure.

We emphasize that this is the selected shock solution for b+ = 0. Indeed, we can


−15 −10 −5 0 5 10 151.5

2

2.5

3

3.5

4

4.5

z = (x − Ut)/λ

Slo

pe

Figure 2.22: Another shock solution, where b+ = 3.89, b− = 1.65, and U = 0.57.This solution can be ”glued” onto to the solution in Figure 2.21. Note that the twosolutions have the same velocity.

choose b+ = 3.89 and find another solution for some new b− to be determined. Figure

2.22 is the result. Because the right end of this solution has the same value as the

left side of the previous solution, they can be joined, and result is a two tiered shock!

This is characteristic of undercompressive shocks.

Since both the shock velocity and the etching rate of a flat surface scale as µ/Ts,

their ratio is a dimensionless number that depends only on the shape of the sputter

yield versus slope, i.e., the dimensionless parameters a/σ, µ/σ,and Σ. This is a

severe test of the theory because these parameters can be extracted by fitting. In our

particular case the ratio is

U

f(0)= U

√2π exp

a2/σ2

2= 1.7, (2.55)

where U = 0.086 is found by solving the boundary value problem, Eq. 2.51; hence


it depends implicitly on the sputter parameters which govern the yield as a function

of slope. Note that the surface diffusion constant B does not appear because it only

affects the sharpness of the shock wave, not its velocity. This is fortunate since B is

difficult to determine in the experiments.

Chapter 3

Reattachment kinetics

Surface diffusion as described by Mullins, Eq. 2.27, cannot account for interac-

tion between surface features separated by a flat region. We need a new non-local

mechanism of mass transport on the surface.

3.1 The closing hole

Mullins assumes that the adatoms reside in a uniform surface layer, i.e., that

the concentration of mobile adatoms is constant. This need not be the case, as

demonstrated by Li et al. [21]. In that experiment, a hole of diameter 60 nm was

drilled in a Si3N4 surface and exposed to an argon ion beam. Under certain conditions,

the hole closes. Li et al. modeled this behavior by supposing that the hole boundary

was a sink for adatoms which attach to the rim, thereby reducing the hole size. The

corresponding equation is

∂

∂tC(r, t) = FYa − C

τ+ D∇2C, (3.1)

where C(r, t) is the adatom concentration as a function of position and time. The

first term on the right is the source, proportional to the ion flux F through Ya. The

second term drains the adatoms through ion collision and trapping onto the surface,

with a combined rate 1/τ . The last term on the right is concentration diffusion with

coefficient D.

39

Chapter 3: Reattachment kinetics 40

The diffusion length in this model is lD =√

Dτ . Surface features separated by

a distance on the order of lD or smaller will be affected by each other. Similarly, a

feature of size lD or less will “self-interact”. In the case of a hole, Li et al. assumed that

the hole edge is a perfect sink, so C(r, t) satisfies the boundary condition C(Rh, t) = 0,

where Rh is the radius of the hole. In radial coordinates, the steady state diffusion

equation is

0 = FYa − C

τ+

D

r

∂

∂r

(r∂C

∂r

), (3.2)

with solution

C(r) =

FYaτ(1− I0(r/lD)

I0(Rh/lD)

)r < Rh

FYaτ(1− K0(r/lD)

K0(Rh/lD)

)r > Rh.

(3.3)

I0 and K0 are modified Bessel functions of the first and second kind, respectively.

The concentration far away is C∞ ≡ FYaτ . Of course, for the hole there is no inner

solution, but it does exist in the pit geometry if we can regard the pit edge as a sink

as well. Hence we provide the inner solution here for completeness. Several are shown

in Figure 3.1. Notice that the outer solution becomes steeper at the hole edge as the

hole radius becomes smaller. This means that a larger flux flows into the hole with

decreasing radius. In the model of Li et al., the velocity of radius decrease is taken

to be proportional to the flux at the edge, and the hole size as a function of time can

be readily obtained via numerical integration.

It is important to recognize that this dependence of the flux on the hole size has

no one dimensional analogue. Suppose we have a long narrow slit, the edges of which

act as sinks - the concentration on either side is unaffected by how narrow the slit is.

One side does not know about the other.

3.2 The structure of the boundary

The model of Li et al. has the desirable feature that it contains a length scale, the

diffusion length, that may mediate surface dynamics across relatively large distances.

However, the assumption that an edge acts as a sink for diffusing adatoms is ad hoc.

We would like to motivate the boundary condition - that the concentration vanishes


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

r/lD

C/C

∞

Figure 3.1: Steady state solutions to the diffusion equation for various hole radii. Thehorizontal and vertical axes are nondimensionalized by the diffusion length and thefar away concentration, respectively.


at the edge.

Clearly, the hole or pit edge is not a perfectly sharp object. It may be thought of

as a localized peak in the slope. In order for reattachment to discriminate the edge

region, the rate 1/τ should be a function of the slope b. For simplicity we neglect the

effect of adatom loss due to ion scattering. We propose the form

τ(b) =τ0

1 + b2, (3.4)

where τ0 is the value for zero slope. This gives a reattachment rate that is enhanced

at the pit edge which has a large slope. This form of τ is by no means unique; indeed,

it is unclear why the edge of the pit or hole should be a sink for adatoms. We use

Eq. 3.4 for the sake of illustration.

The steady state diffusion equation now becomes

0 = FYa − C

τ0

(1 + b2) +D

r

∂

∂r

(r∂C

∂r

). (3.5)

Solving for C, we have

C = [1 + b2 − l2D∇2r]−1FYaτ0, (3.6)

where lD =√

Dτ0 and the subscript on the Laplacian emphasizes that it is in the

radial form.

We assume that the reattachment contributes a term to the surface evolution that

is proportional toC

τ, (3.7)

where C is the equilibrium concentration Eq. 3.6. The modified equation for the

surface height evolution (Eq. 2.27) becomes

h = −F (b) + q(1 + b2)RF (b)−B∇4h, (3.8)

where R ≡ [1 + b2 − l2D∇2r]−1 is the inverse diffusion operator, and ∇4 is understood

to be the nonlinear surface diffusion operator as in Eq. 2.27. In this equation we

assume that the source of adatoms is proportional to the sputtered target atoms.

The dimensionless number q is a measure of the efficiency of the reattachment; q = 0

corresponds to no reattachment, and q = 1 means that all sputtered atoms reattach.


0 50 100 150

−60

−40

−20

0

20

40

Radius

Hei

ght

t = 0

t = 150

Figure 3.2: Simulation of reattachment Eq. 3.8 for q = 1 and lD = 50, with B = 20and Ts = 0.1. The sputter parameters are a = 2.04, µ = 0.658, σ = 1, and Σ = 0.0462.The axes are drawn to the same scale.

Note that we have neglected the D-term in Eq. 2.27 since its contribution is small.

We emphasize that the sputtering is still one dimensional as before, since only the

diffusion mechanism is affected by the new length scale lD.

Figure 3.2 is a simulation of Eq. 3.8.

We see that the bottom portion of the pit indeed seems to close. The flat outside

region does not change height because we have set q = 1. This shows that the slope

dependent reattachment mechanism can alter the behavior of a pit dramatically.

Of course, due to a lack of detailed knowledge about the reattachment process (at

the level of, say, Sigmund’s model of sputtering), we cannot single out the correct

form of the reattachment. We hope that this theoretical study is prelude to future

experimental research.

Chapter 4

Optimal design

4.1 Motivation

In many applications, such as inkjet printing or microarrays, it is important to

produce very small liquid droplets. The common method is to force the liquid out of a

small opening, and the opening size determines the volume of the droplets. However,

the opening size cannot be made arbitrarily small because the pressure needed to

eject the liquid will increase due to surface tension, resulting in material failure.

One strategy for creating smaller droplets without reducing the opening size is to

use a time varying forcing. This method has achieved an order of magnitude decrease

in droplet volume [6].

We study the problem of droplet minimization by considering the limit wherein the

forcing is slow, so that the droplet is quasi-static and does not require the numerical

solution of the Navier-Stokes equation with a free surface. Our strategy is to change,

instead of the size of the opening, its shape. In the following section we introduce the

optimization procedure with a classic optimization problem solved by Joseph Keller.

We then generalize the procedure to our droplet optimization. Finally, we provide

a numerical example giving an opening shape which reduces the droplet volume by

20%.

44

Chapter 4: Optimal design 45

4.2 Keller’s problem

The prototype for our class of optimal design is Keller’s variation on the Euler

buckling problem [19]. If a material column is compressed from the ends, it bends at

a critical load that is determined by the eigenvalue equation

(EIw(x)′′)′′ + Tw(x)′′ = 0, (4.1)

where E is Young’s modulus, I is the moment of inertia of the cross section, w is the

deflection, and the smallest T for which a solution exists is the critical compression

load [20]. The centerline of the column is along the x-axis. For simplicity, suppose

the ends of the column are hinged. Then

w = w′′ = 0 (4.2)

at both ends, and we can define y = w′′, which satisfies

EIy′′(x) + Ty(x) = 0. (4.3)

If the column is of length L, then the first nonvanishing solution is y ∝ sin(πx/L),

which gives a critical load

Tc =π2EI

L2. (4.4)

Keller posed the question: if the column is nonuniform, i.e., I is a function of x,

then what is the optimal I(x) such that the critical load is maximized, given a fixed

total column mass? Note that

I = αA2, (4.5)

where A is the cross-sectional area, and α depends only on the shape of the cross-

section, not its area. Then we can define λ = TEα

and rescale z = x/L, such that the

eigenvalue problem becomes

y′′(z) +λ

A2(z)y(z) = 0 (4.6)

y(0) = y(1) = 0. (4.7)


Under a variation in the profile A, the solution y and the eigenvalue λ must change

in response. Hence,

δy′′ +λ

A2δy = − δλ

A2y +

2δA

A3λy. (4.8)

We see that δy satisfies the an inhomogeneous version of Eq. 4.6, which implies that

the right side of Eq. 4.8 is orthogonal to y, i.e.,

∫ 1

0dz y

(− δλ

A2y +

2δA

A3λy

)= 0, (4.9)

since the operator ∂zz +λ/A2 is manifestly Hermitian. Rearranging the above expres-

sion,

δλ =2λ

∫dz y2

A3 δA∫

dz y2

A2

. (4.10)

Eq. 4.10 is the change in the critical load induced by a change in the column

profile. For any column, we can increase the load by choosing δA such that the

integral in the numerator is positive definite, e.g., by letting δA ∝ y2/A3. Of course,

this choice isn’t unique: we can multiply by another positive definite function!

We can narrow the choices as follows. First, let’s recall that we must impose the

fixed volume constraint ∫ 1

0dz δA = 0. (4.11)

If y2/A3 is a constant, we see from Eq. 4.10 that this constraint implies the change in

critical load is zero. In other words, for the column that maximizes the critical load

for a fixed volume, y2/A3 is a constant. Second, each time we change the profile by

δA, we must also rescale such that the volume is unchanged:

A → A + δA− εA, (4.12)

where

ε =

∫δA∫A

. (4.13)

Thus, if δA ∝ A then A remains unchanged and is the optimal profile. The two con-

ditions, y2/A3 constant and δA ∝ A, suggest the following prescription for changing

the profile:

δA ∝ y2

A2. (4.14)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

z

Are

a initial profilefirst iterationtenth iterationexact solution

Figure 4.1: Cross section profiles obtained by the iterative algorithm. The final profilematches the exact solution, shown by dots.

We implement Eq. 4.14 numerically with the normalization

δA =0.4

∫ 10 dz y2

y2

A2, (4.15)

and show the resulting sequence of profiles in Figure 4.1 and the corresponding critical

loads in Figure 4.2.

y at each step is obtained by solving Eq. 4.6 discretized on the interval [0, 1],

with a node spacing of 0.01, using the Matlab r© function eigs. We can compare the

numerical solution to the analytic result obtained by Keller. The exact optimal A(z)

can be expressed in the parametric form

A =4

3sin2 θ, z =

θ

π− 2θ

2π, (4.16)

where θ is between 0 and π. This cross-sectional profile has the critical load of 4π2/3,

an improvement of a third over the uniform column with the same volume, which


0 1 2 3 4 5 6 7 8 99.5

10

10.5

11

11.5

12

12.5

13

13.5

Iteration

Crit

ical

Loa

d

Figure 4.2: The sequence of critical loads obtained by iteration. The maximumcritical load is indicated by the horizontal line.


has critical load of π2. Figures 4.1 and 4.2 show excellent convergence to the analytic

solution.

4.3 The optimal faucet

A standard protocol for producing small droplets is as follows: a pipette, of circular

cross-section, is pressurized at one end, pushing out a small fluid droplet. If the

nozzle is sufficiently small, force balance requires that the droplet has constant mean

curvature. At a critical pressure, this equilibrium shape becomes unstable, ultimately

leading to the droplet detaching from the nozzle.

The volume of fluid entrained during this process is set by the total fluid volume

contained in the critical droplet. This volume scales like r3, where r is the nozzle

radius. On the other hand, the critical pressure for ejecting this droplet scales like

γ/r, where γ is the liquid surface tension. Thus, ejecting smaller droplets requires

higher pressures. The smallest size droplet that can be ejected is thus determined

by the highest pressure that can be reliably applied to the nozzle, without material

failure, etc.

However, typical nozzles use a circular cross section. It is not unreasonable to

imagine that changing the shape of the cross section to be some other shape may

decrease the ejected droplet volume, while maintaining the same applied pressure.

For example, imagine that we have a circular nozzle with a pendant droplet just

below the critical volume: by “squeezing” the shape of the nozzle cross section into

an elliptical shape, one might cause the droplet to detach at a lower volume.

In the following we address the question: what is the shape of a nozzle for which the

ejected droplet volume is minimized, for a given applied pressure? We demonstrate

that circular nozzles do not eject the smallest droplets; instead, the optimal nozzle

more closely resembles an equilateral triangle, albeit with “stretched” corners. The

best nozzle shape that we have found has an ejected droplet volume about twenty

percent smaller than the circular nozzle with the same critical pressure. Our method

is inspired by and extends J. Keller’s classic treatment of the Euler buckling problem

with a beam of nonuniform cross section [19]. Recently, the method has been applied


to the optimization of a bistable switch [4]. For a detailed mathematical treatment

of capillary surfaces in general, see [13].

The rest is organized as follows. We first explain the origin of the pendant droplet

instability. Then we describe our method for reducing droplet size. Lastly we provide

numerical calculations implementing the method, and present the candidate optimal

nozzle.

4.4 Pendant droplet instability

The instability of a droplet protruding from a nozzle is due to a bifurcation, most

easily seen in the case of a circular nozzle that is much smaller than the capillary

length, which allows us to neglect gravity. The shape of the droplet is then determined

by the Young-Laplace equation p = γK, where p is the pressure difference across the

liquid/air interface, γ is the surface tension, and K is the mean curvature of the

droplet surface. This equation describes a surface of constant mean curvature p/γ

with the nozzle edge as its boundary. If the boundary is a circle, then the solution

must be a section of the sphere with mean curvature p/γ. From the familiar relation

Ksphere =2

sphere radius(4.17)

we deduce that the radius of curvature of the droplet is 2γ/p. For small p, such that

the sphere radius is much greater than the nozzle radius, the solution is a shallow

spherical cap. But note that its complement, the rest of the sphere, is also a solution.

See Figure 4.3. As p is increased, these two solutions approach each other until both

become a hemisphere with the nozzle at the equator. The pressure at which the two

solutions meet is the critical pressure p∗, and the corresponding degenerate solution

is unstable. Note that the critical pressure is also the maximum pressure, for the

nozzle cannot support a sphere smaller than itself.

For a noncircular nozzle, we no longer have such a simple geometric picture, how-

ever key features remain. The unstable solution is still characterized by a bifurcation

at which two solutions meet, corresponding to the maximum pressure achievable for


a)

b) c)

Figure 4.3: a) At low pressure and curvature, there exist two solutions, the union ofwhich is a sphere. b) At a higher pressure, the two solutions become a hemisphere.c)Above the critical pressure, there is no pendant drop solution.


the given nozzle. The critical pressure for a general nozzle can be computed as fol-

lows: let the droplet surface be parameterized as a function ~R(u, v) over a domain D

in the uv-plane, which takes value in three dimensional physical space. The boundary

of the domain ∂D corresponds to a closed curve C which represents the nozzle. The

curvature is a nonlinear functional of the surface and its derivatives up to second

order, hence the equation for the droplet shape has the form

γK[~R, ~∇~R, ~∇~∇~R] = p, (4.18)

where ~∇ is the gradient operator in the uv-plane.

Upon increasing the pressure p → p + δp, the surface changes: ~R → ~R + δ ~R. Eq.

4.18 implies that the variation δ ~R and δp are related by

γLδ ~R = δp, (4.19)

where Lδ ~R is the change in mean curvature induced by the surface change. L is a

differential operator acting on δ ~R.

At the critical solution, the pressure is at a maximum; therefore, there must be a

solution w = δ ~R to Eq. 4.19 with δp = 0. The solution w satisfies

Lw = 0 (4.20)

with boundary condition w = 0 at ∂D. Note that the pressure dependence in this

formula arises because L = L[~R] depends implicitly on the pressure p through ~R.

Hence, the existence of a nonzero w is a diagnostic for finding the critical solution to

Eq. 4.18 and the corresponding critical pressure p∗.

4.5 Optimization method

Now, to find the optimal nozzle, we need to derive a relation between the change in

critical pressure and change in nozzle shape. Since pressure and volume are conjugate

variables, increasing critical pressure is tantamount to decreasing critical volume. By

iteratively changing the nozzle shape to increase critical pressure, we will thus arrive


at a nozzle which produces smaller droplets. We compare the critical volume of the

deformed nozzle with that of the circular nozzle that corresponds to the same critical

pressure, since pressure is the control variable in practical situations.

Suppose that a given nozzle shape C has a critical pressure p∗, a critical droplet

shape ~R∗, and a corresponding w. All of these quantities change when the nozzle

shape C → C + δC. The change in the droplet shape δ ~R is linearly related to the

pressure change δp by Eq. 4.19 with the boundary condition δ ~R = δC at ∂D. On

the other hand, since the critical solution maximizes the critical pressure, w does not

change to leading order in δC.

The change in critical pressure induced by δC can therefore be computed by taking

the inner product of both sides of Eq. 4.19 with w:

γ〈w, LδR〉 = γ〈δR, Lw〉+ γ∮

b(δR, w)

= 0 + γ∮

b(δC, w)

= 〈w, δp〉. (4.21)

Therefore

δp =γ

∮b(δC, w)

〈w, 1〉 . (4.22)

Here b(•, •) denotes the boundary integrand from integrating by parts. The deriva-

tion also uses the self adjointness of L, which is readily demonstrable by explicit

computation 1. Eq. 4.22 is an explicit relation between a change in the nozzle shape

(δC) and the resulting change in critical pressure.

4.6 Coordinate system

For calculation, we need to choose a suitable coordinate system for the drop

surface. The most obvious one would be polar coordinates, wherein the surface is

1In computing an inner product 〈f, Lg〉 ≡ ∫dudvfLg, where f and g are arbitrary functions of

u, v, and the integration is over D, we can undo the differentiation on g, and instead let the adjointoperator L† act on f . Self-adjointness (L = L†) can be verified explicitly. In the derivation, thisallows us to simply interchange f and g, while introducing the boundary terms from integration byparts.


given as the distance from the origin as a function of the azimuthal angle θ and polar

angle φ: R(θ, φ). However this is inconvenient due to the coordinate singularity at

the pole (θ = 0).

Alternatively, let’s adopt the coordinates u and v, such that

u = f(θ) cos φ, v = f(θ) sin φ, (4.23)

where f(θ) is to be determined. From this ansatz we have

du = f ′(θ) cos φ dθ − f(θ) sin φ dφ, dv = f ′(θ) sin φ dθ + f(θ) cos φ dφ (4.24)

and consequently

dθ =1

f ′(θ)(cos φ du + sin φ dv), dφ =

1

f(θ)(cos φ dv − sin φ du). (4.25)

The line element is

ds2 = dR2 + R2 dθ2 + R2 sin2 θ dφ2 (4.26)

and upon substitution becomes

ds2 = dR2 + R2

(cos2 φ

f ′2+

sin2 θ sin2 φ

f 2

)du2 + R2

(sin2 φ

f ′2+

sin2 θ cos2 φ

f 2

)dv2

+ 2R2 sin φ cos φ

(1

f ′2− sin2 θ

f 2

)du dv. (4.27)

We can eliminate the last cross term by requiring

f ′2 =f 2

sin2 θ, (4.28)

which has the solution

f(θ) = tanθ

2. (4.29)

Our new coordinates

u = tanθ

2cos φ, v = tan

θ

2sin φ (4.30)

map the unit hemisphere, parametrized by θ ∈ [0, π/2] and φ ∈ (0, 2π], to the unit

disk: u, v such that u2 + v2 ≤ 1. The line element in this new coordinate system is

ds2 = dR2 + 4R2 cos4 θ

2(du2 + dv2)

= dR2 +4R2

(1 + u2 + v2)2(du2 + dv2)

= dR2 + Γ(du2 + dv2), (4.31)


where Γ ≡ 4R2

(1+u2+v2)2. The metric is then

1 0 0

0 Γ 0

0 0 Γ

. (4.32)

As will soon be evident, the metric being diagonal will simply our equations im-

mensely.

The square root of the determinant of the metric is simply Γ, from which the

volume element is

dV = du dv dR4R2

(1 + u2 + v2)2= du dv

43R(u, v)3

(1 + u2 + v2)2. (4.33)

In the last expression we integrated R from the origin to the surface described by

R(u, v), so that the differential volume is that of an infinitesimal cone radiating from

the origin.

In order to find the surface area of the drop, we need the induced metric on the

surface. The total differential of R(u, v) is

dR = Ru du + Rv dv, (4.34)

where the subscripts denote differentiation. Substituting this into Eq. 4.31 we find

ds2 = (Γ + R2u) du2 + (Γ + Rv) dv2 + 2RuRv du dv (4.35)

which corresponds to the metric Γ + R2

u RuRv

RuRv Γ + Rv

. (4.36)

Taking the square root of the determinant, the surface area element is

dA = du dv√

Γ2 + Γ(R2u + R2

v). (4.37)

4.7 Explicit formula for δp.

We choose the nozzle C to lie in the xy-plane, enclosing the origin. Then the

droplet surface may be given by the distance from the origin (R) as a function of


u, v as defined above. Hence the surface is a scalar function R(u, v); its domain D is

the unit disk in the uv-plane. We retain φ to denote the polar angle in the uv-plane;

tan φ = v/u. Figure 4.4 illustrates our coordinate system.

The free energy has contributions from surface tension and pressure,

E =∫

(γdA− pdV ) =∫

du dv

(γ

√Γ2 + Γ(R2

u + R2v)− p

43R(u, v)3

(1 + u2 + v2)2

). (4.38)

The drop surface minimizes the free energy, therefore it satisfies the Euler-Lagrange

equation

~∇ · δE

δ~∇R− δE

δR= 0, (4.39)

which yields the Young-Laplace equation

−~∇ · (C ~∇R) + AR = F, (4.40)

where ~∇ is the usual gradient operator in the uv-plane,

~∇ = u∂

∂u+ v

∂

∂v. (4.41)

The coefficients are

C =1√

1 + (1+ρ2

2R)2(~∇R)2

,

A = C

(~∇R)2

R2+

8

(1 + ρ2)2

,

F = p4R2

(1 + ρ2)2, (4.42)

where ρ2 ≡ u2 +v2 is the radial coordinate in the uv-plane. ∂D corresponds to ρ = 1.

In our coordinate system, the pressure change is

δp =1

δwV

∮dφ δC

wρR(R2 + R2φ)

(R2 + R2ρ + R2

φ)3/2

, (4.43)

where δwV ≡ ∫d2ρ w 4R2

(1+ρ2)2. Here and in the following we use subscripts to denote

partial differentiation.

We can recast this expression into a form that is more geometric. First, the

contact angle α between the drop and the plane of the nozzle is given by cot α(φ) =


−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1

0.94

0.96

0.98

1

1.02

1.04

−1

−0.5

0

0.5

−1

−0.5

0

0.5

1

0

0.2

0.4

0.6

0.8

a)

b) c)

Figure 4.4: a) The transformation from polar coordinates to our u, v coordinates.b) A drop in physical space, with a slightly deformed boundary. c) The same droprepresented as R(u, v). Note the vertical scale has been exaggerated to emphasize thedeviation from a circular boundary.


Rρ/(R2 + R2

φ)1/2|ρ=1 where the right hand side is evaluated at the boundary. Second,

we define w⊥ ≡ wρ/(R2 + R2

φ)1/2|ρ=1 which can be understood as follows - note that

w is the difference between the outer and inner solutions as the pressure approaches

bifurcation. Using the contact angle given above, this expression is the difference

between the slopes (with respect to the vertical) of the outer and inner solutions at

the boundary. This is a coordinate independent quantity. Third, we observe that

dφ δc R =(dφ

√R2 + R2

φ

)δc

R√R2 + R2

φ

= dl δN, (4.44)

where dl is the line element, and δN is the change of the nozzle in the direction locally

normal to the nozzle. Lastly, the denominator δwV in (4.43) is just the change in

volume from changing the surface by w. Putting these facts together, the pressure

change is

δp =1

δwV

∮dl δN w⊥ sin3 α, (4.45)

which leads to the prescription for changing the nozzle

δN ∼ 1

δwVw⊥ sin3 α. (4.46)

Clearly, for the circular nozzle, symmetry implies that δN should be constant. But

this amounts to a mere reduction in the size of the nozzle; the shape remains a circle.

So the circular nozzle is at an extremum, in fact a minimum of critical pressure for

fixed nozzle area.

For a noncircular nozzle, the contact angle isn’t constant, and hence the change

according to the above formula cannot be constant. So one may change the critical

pressure while fixing the nozzle area. Moreover, since the circular nozzle is the only

one (except the infinite strip) with a constant contact angle, the process of deforma-

tion does not end.

4.8 The deformed faucet

We apply Eq. 4.46 iteratively to a perturbed circular nozzle to see how the shape

evolves away from the circle. Figure 4.5 shows the result of iterations starting with


a) b)

c) d)

Figure 4.5: Evolution of nozzle shape with threefold symmetry. a) Initial nozzle:V = 1.00; b) V = 0.97; c) V = 0.88; d) V = 0.82. V is a normalized volume given byEq. 4.47.


a circle deformed by a perturbation with a three-fold symmetry. The perturbation

grows with each iteration, and eventually the nozzle shape becomes concave. With

each iteration, we have applied a rescaling in order to maintain the nozzle area.

Without the area constraint, the nozzle would become arbitrarily small in accordance

with Eq. 4.46. We are interested in the shape of the nozzle, not its size. We also

apply the Savitzky-Golay filter [28] at each iteration to smooth out the mesh noise.

The solutions to the Young-Laplace equations are obtained using the nonlinear PDE

solver in the MATLAB r© PDE Toolbox, which implements the finite element method

for elliptic equations with variable coefficients, exactly of the form in Eq. 4.40. See

Appendix C for more details. For each nozzle shape, we start at a pressure below the

bifurcation and by choosing different trial solutions obtain both solutions. Then we

bring both solutions to just below the critical pressure by stepping up the pressure,

using the solution at each step as the trial solution for the next step. We then use the

average of the two solutions for our surface, and their difference for w. The validity

of this procedure can be rigorously shown for a circular nozzle, and we expect it

to remaind valid for noncircular nozzles as long as the pressure is brought close to

critical.

In order to compare and select among nozzle shapes, we need a measure of op-

timality independent of size. For every nozzle, we rescale its critical volume by the

critical volume corresponding to the circular nozzle with the same critical pressure.

This dimensionless volume is given by

V =v∗

2π3

(2p∗

)3 . (4.47)

Figure 4.6 shows a particular sequence of critical properties obtained through our

iteration procedure. We see that the critical pressure begins to increase rapidly about

the fifth iteration, after which the decrease in V slows down, and the nozzle shape

becomes stretched out (see Figure 4.5). This means that in order to decrease droplet

size at a given pressure, one should use a nozzle shape that is roughly triangular, per-

haps with somewhat stretched out corners; but further deformation does not lead to

significant improvement. Moreover, gravitational instabilities will inevitably become

relevant if the “arms” become too long [27, 24].


0 2 4 6 8 101

1.5

2cr

itica

l vol

ume

0 2 4 6 8 102

2.1

2.2

2.3

2.4

criti

cal p

ress

ure

0 2 4 6 8 100.8

0.85

0.9

0.95

1

iteration

norm

aliz

ed c

ritic

al v

olum

e

a)

b)

c)

d)

Figure 4.6: Sequence of iterations away from the circular nozzle with an initial three-fold perturbation. The normalized critical volume given by Eq. 4.47 is shown in thebottom graph. The arrows indicate the corresponding shapes in Figure 4.5.


It should be emphasized that we have shown a particular example of an improved

nozzle, generated by a choice of the initial perturbation. We have tried other pertur-

bations, leading to shapes with, say, four-fold symmetry or without any symmetry,

but the three-fold perturbation has yielded the biggest reduction in the normalized

critical volume.

So far we have ignored the effects of gravity, but our formalism applies just as

well to the problem with gravity. Including gravity means that the pressure would no

longer be constant throughout the drop surface, but rather a linear function of height:

p → p− ρmgh(u, v), where ρm is the mass density of the liquid, g is the gravitational

acceleration, h is the distance below the nozzle, and p now denotes the pressure at

the nozzle (h = 0). Although Eq. 4.40 acquires a new term as a result, this term

does not contain derivatives and thus does not contribute to the boundary integral.

So our formula for the pressure change remains the same in the presence of gravity.

To be sure, the nozzle evolution would differ because the contact angle and w⊥ will

be affected by gravity. Moreover, if the nozzle is too large relative to the capillary

length, then gravity destabilizes all solutions: it is not possible to suspend a water

drop from a meter wide faucet. It would be interesting to examine the case of the

intermediate sized nozzle, small enough to have stable solutions, yet large enough to

be affected by gravity.

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Appendix A

Relation to previous theories of

sputtering

A.1 Limit of small slope

The Bradley-Harper theory (Eq. 2.28) is linear with a finite wavelength rippling

instability as discussed in the previous section. By assumption, the amplitude of the

ripples grows exponentially in time. But as the ripples grow, we no longer have a low

slope surface, and the linear theory does not apply. It is useful to determine at what

ripple amplitude does the linear theory fail.

In order to model the long time behavior of the surface, Makeev et al.[9, 23]

improved upon the Bradley-Harper model by introducing the leading nonlinear terms

from the expansion of Sigmund’s integral. Their model (MCB), at normal incidence,

is partly given by

h = −[F (0) + Fb(0)b +1

2Fbb(0)b2]− [D(0) + Db(0)b]h′′ −Bh′′′′

= −[F (0) +1

2Fbb(0)h′2]−D(0)h′′ −Bh′′′′ (A.1)

in our notation. We see that this is the small slope limit of our model (Eq. 2.27).

Our model has the advantage of being nonperturbative, i.e., it holds to all orders

in the slope. The full MCB model also contains terms involving higher derivative

terms in h. We do not show those because our aim is to connect the MCB model

67

Appendix A: Relation to previous theories of sputtering 68

to our model, which contains only first and second derivatives in h. Of course, we

could have included higher derivative by extending the Taylor expansion (Eq. 2.14).

Subscripts on F and D denote differentiation. Note that both Fb(0) and Db(0) vanish

by symmetry. The Bradley-Harper and MCB models do not have the large angle

correction e−Σ(√

1+b2−1), but the effect of this is insignificant as we will soon see.

A.2 Rippling instabilities

An immediate issue is what effect the nonlinearities have on the growth of the

ripples predicted by Bradley and Harper. We choose the values of the sputtering

parameters

a = 2.04, µ = 0.658, σ = 1, (A.2)

and

B = 20, Ts = 0.01. (A.3)

The linear stability calculation gives us a fastest growing wavenumber of

k2 =1

4πTsB

√π

2

aµ3

σ2exp

(−a2/σ2

2

)= 0.0362 (A.4)

or

k = 0.19, (A.5)

giving a ripple wavelength of

λ =2π

k≈ 33. (A.6)

We simulate the ripple growth by choosing an initial sinusoidal perturbation with

k = 0.2. Figure A.1 shows that the growth of the ripple amplitude is indeed expo-

nential at early times, but saturates at a later time. Linear stability analysis predicts

the value of the growth rate to be

s = D(0)k2 −Bk4 ≈ 0.026 (A.7)

which agrees with the simulated value in the early exponential growth.

After the ripples saturate, the surface continues to be sputtered at a constant

rate, but the profile is fixed. Our simulation shows that the range of slopes that the


0 50 100 150 200 250 300 350 400

10−1

100

Time (Ts × 102)

Rip

ple

Am

plitu

de

Figure A.1: Ripples grow exponentially at outset, in accordance to a linear instability.Later the ripples reach saturation. The growth rate in the exponential regime is 0.026.


ripples attain is quite low, on the order of 0.1. This allows us to analyze the ripples

using the MCB model. Differentiating both sides of Eq. A.1 and assuming a steady

state solution, we have

b = −Fbb(0)bbx −D(0)bxx −Bbxxxx = 0. (A.8)

We can non-dimensionalize by using x ≡ kx as our spatial variable and seek a solution

with period 2π. After rearranging, the boundary value problem (BVP) is

bb′ + εb′′ + δb′′′′ = 0, b(x + 2π) = b(x), (A.9)

where primes denote differentiation respect to x, and the coefficients are given by

ε =D(0)k

Fbb(0)=≈ 0.037 (A.10)

and

δ =Bk3

Fbb(0)≈ 0.021. (A.11)

It is now clear why the slope remains low at saturation. Requiring the terms in Eq.

A.9 to balance, the magnitude of b must be on the order of ε and δ. This means that

the MCB model is adequate to describe the rippling behavior.

We note that for the fastest growing ripple,

k2 =D(0)

2B, (A.12)

which leads to

δ =ε

2. (A.13)

ε has the explicit form

ε =ka

2

µ2/σ2

µ2

σ2

(a2

σ2 − 1)

+ 1. (A.14)

We solve Eq. A.9, with the above coefficients and periodic boundary condition,

using MATLAB r©’s boundary value problem solver, BVP4C. Figure A.2 shows the

result, along with the solution obtained by the simulation of the fully nonlinear Eq.

2.28.


0 10 20 30 40 50 60−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

slop

e

x

BVPPDE

Figure A.2: Comparison of the saturated ripple profile obtained from Eq. 2.28 (PDE)and Eq. A.9 (BVP).

Appendix B

Numerical solution of surface

evolution

In the following we sketch the numerical procedure used in solving our partial

differential equations governing the surface dynamics. The method is adapted from

an earlier version written by Michael Brenner to solve the Navier-Stokes equation for

a slender jet. The method is semi-implicit with adaptive time step control. The novel

feature of our method is an algebraic method for calculating Jacobians which is much

more convenient than doing so explicitly.

We wish to solve a partial differential equation of the form

∂

∂tf(x, t) = W (f, f ′, ...), (B.1)

where W is a nonlinear function of f and its spatial derivatives. The discrete form of

this equation can be written as

fn+1i − fn

i

∆t= (1− θ)W (fn

i ) + θW (fn+1i ). (B.2)

Superscripts are time indices, and subscripts are space indices. We have suppressed

the derivatives of f for clarity. For θ = 0 the equation is explicit - fn+1i is determined

by a function of fni .

For θ = 1 the equation is fully implicit; in order to advance in time we need to

know fn+1i , the very quantity we are trying to obtain! (For accuracy one usually

72

Appendix B: Numerical solution of surface evolution 73

chooses θ to be a half or close to it [28].) Of course, this poses no difficulty if W (f)

is linear in f , in which case W (f) is a linear matrix operation W (f)i = Mijfj, and

we easily solve for fn+1i .

fn+1 = (1− θ∆tM)−1(1 + (1− θ)∆tM)fn. (B.3)

We use boldface f to emphasize that it is a vector with elements fi. 1 is the identity

matrix.

The story becomes complicated if W (f) is nonlinear. In that case we need to solve

for fn+1 iteratively with Newton’s method. We can regard our equation as finding the

zero of the residual function, which is defined as

r(fn+1) = fn+1 − fn −∆t[(1− θ)W (fn) + θW (fn+1)]. (B.4)

Suppose we have a trial solution f∗; then Newton’s method says that a better solution

is given by

f = f∗ − J−1r(f∗). (B.5)

J is the Jacobian matrix of the residual, defined as

Jij ≡ ∂r(fi)

∂fj

. (B.6)

In Eq. B.5 we evaluate J at the trial solution f∗. For our guess we simply use the

linear extrapolation

f∗ = fn + ∆t(fn − fn−1). (B.7)

In practice it suffices to apply Newton’s algorithm just once, provided we keep the

relative error small at each time step. This is accomplished by taking a single time

step ∆t and also two time steps of ∆t/2, and then comparing the relative error of

the two results. If the error is above our tolerance, we halve ∆t. If the error remains

below tolerance for a number of steps, we increase ∆t by some factor. This error

checking is far less costly than applying Newton iteration multiple times at each step,

for that involves large matrix inversions.

The most tedious part of implementing the above numerical method is evaluating

the Jacobian because the residual may be a complicated function, and hence its


derivative may be quite complicated indeed. Moreover, each spatial derivative in

W (f, f ′, ...) couples together adjacent points, and the resulting Jacobian in component

form soon becomes an unwieldy collection of indices.

Fortunately, we can recommend a trick which avoids this notational nightmare

altogether. For illustration, consider the term

W (f) =1

2∇2f 2. (B.8)

The one dimensional laplacian in discretized form becomes

∇2 → D

∆x2, (B.9)

where ∆x is the grid spacing, and D is the tridiagonal second derivative matrix

D ≡

−2 1 0 . . .

1 −2 1. . .

0 1 −2. . .

.... . . . . . . . .

(B.10)

with −2 on its main diagonal, 1 on the two adjacent diagonals, and 0 everywhere

else. Of course, since we always use finite matrices, boundary conditions need to be

imposed by suitably modifying the top and bottom rows.

Our expression is now

W =1

2∆x2Df2. (B.11)

f2 is a vector with components f 2i . Taking the variation with respect to f ,

δW =1

∆x2Dfδf , (B.12)

and fδf is a vector with components fiδfi. Its Jacobian, at least formally, is

J =δW

δf=

1

∆x2Df . (B.13)

This expression should be troubling - the Jacobian is a matrix, yet Df seems to be a

vector! But let’s be careful. Going back to component form, we find

δf 2i

δfj

= 2fiδfi

δfj

= 2fiδij. (B.14)


δij is the Kronecker delta, equal to one for i = j and zero otherwise. We see that the

right hand side corresponds to a matrix having the values fi on its main diagonal and

zero elsewhere. Back to matrix form, we have

δf2

δf= 2f , (B.15)

and f denotes the diagonal matrix with the vector f along its diagonal. Finally we

have

J =δW

δf=

1

∆x2Df , (B.16)

and it is manifestly a matrix. It is always clear from context that an expression is a

matrix, hence the caret may be dropped in practice. Confronted with a residual, it

is now merely an algebraic exercise to calculate its Jacobian! Aside from the consid-

erable improvement in speed, the absence of indices greatly reduces the likelihood of

mistakes.

Let’s go back to our residual

r(fn+1) = fn+1 − fn −∆t[(1− θ)W (fn) + θW (fn+1)]. (B.17)

Its Jacobian is simply

J(fn+1) =δr(fn+1)

δfn+1= 1− θ∆t

δW(fn+1)

δfn+1(B.18)

in our notation.

Appendix C

A comment on the numerical

solution of the drop surface

We solve the equation

−~∇ · (C ~∇u) + Au = F (C.1)

with the PDE Toolbox, an add-on package in MATLAB r©. It’s essentially a simpler

version of FEMLAB r©, and implements the finite element method for equations of the

form shown above. We adopt the convention that u is the function to be determined.

The function femdrop uses the function pdenonlin in the PDE Toolbox in order

to solve for drop surfaces of increasing pressures.

function u = femdrop(p,e,t,u0,pi,pf,dp)

% u0 is the initial drop surface with pressure pi.

% u is the final surface of pressure pf.

% dp is the increment in pressure.

% p, e, t are matrices of points, edges, and triangles, respectively.

% geometry

g=’circleg’; % domain is the unit circle

b=’circleb’; % boundary condition corresponding to the nozzle shape

% coefficients of the PDE of the form -grad(c grad u) + a u = f

c=’1./sqrt(1+(1+x.^2+y.^2).^2./(4*u.^2).*(ux.^2+uy.^2))’;

a=’1./sqrt(1+(1+x.^2+y.^2).^2./(4*u.^2).*(ux.^2+uy.^2))...

.*((ux.^2+uy.^2)./u.^2 + 8./(1+x.^2+y.^2).^2)’;

76

Appendix C: A comment on the numerical solution of the drop surface 77

rtol=1e-5; % Tolerance for nonlinear solver

%Solve the nonlinear problem

%generate initial solution

pressure=pi;

% f is the right hand side of the PDE

f=strcat(num2str(pressure),’*4*u.^2./(1+x.^2+y.^2).^2+0*4*u.^3...

.*(1-x.^2-y.^2)./(1+x.^2+y.^2).^3’);

% nonlinear PDE solver

u=pdenonlin(b,p,e,t,c,a,f,’tol’,rtol,’U0’,u0,’Jacobian’,’full’);

%change pressure

for pressure=pi+dp:dp:pf

f=strcat(num2str(pressure),’*4*u.^2./(1+x.^2+y.^2).^2’);

u=pdenonlin(b,p,e,t,c,a,f,’tol’,rtol,’U0’,u,’Jacobian’,’full’);

end

circleb is the function that expresses the nozzle shape as a Dirichlet boundary

condition. The boundary condition has the general form

hu = r, (C.2)

and we choose h = 1 and r to be the values of the edge points given by the vector

nozzle. One can similarly specify a Neumann boundary condition with the parame-

ters q and g, which we set to zero because our boundary condition is purely Dirichlet.

function [q,g,h,r] = circleb(p,e,u,time)

% Dirichlet boundary condition of the form h u = r

% (q and g are for Neumann boundary conditions)

global nozzle;

% nozzle is a function of the polar angle which defines

% the contour of the nozzle opening.

ne=size(e,2); %number of edge points

q = zeros(1,ne); % Neumann

Appendix C: A comment on the numerical solution of the drop surface 78

g = zeros(1,ne);

h = ones(1,2*ne); % Dirichlet

r = ones(1,2*ne);

r = nozzle; %u = nozzle at the boundary

surfaces in solid dynamics and fluid statics

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