The Effect of the Third Dimension on Rough Surfaces Formed by Sedimenting Particles in Quasi-Two-Dimensions

4 Pages·0.52 MB·English

by  K. V. McCloud

The Effect of the Third Dimension on Rough Surfaces Formed by Sedimenting
Particles in Quasi-Two-Dimensions
K. V. McCloud
Department of Physics and Engineering, Xavier University of Louisiana, New Orleans, LA 70125
M. L. Kurnaz
1 Department of Physics, Bogazici University,80815 Bebek Istanbul
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0 The roughness exponent of surfaces obtained by dispersing silica spheres into a quasi-two-
2 dimensional cell is examined. The cell consists of two glass plates separated by a gap, which
is comparable in size to the diameter of the beads. Previous work has shown that the quasi-one-
n
dimensionalsurfacesformedhavetwodistinctroughnessexponentsintwowell-definedlengthscales,
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which have a crossover length about 1cm. We have studied the effect of changing the gap between
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theplates to a limit of about twice thediameter of thebeads.
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PACSnumbers: PACSnumbers: 05.40.+j,47.15.Gf,47.53.+n,81.15.Lm

]
t
f I. INTRODUCTION widerangeofvolumefractionsandPecletnumbersinsed-
o
imentation problems. However so far there has been no
s
t. The formation of rough surfaces is a problem of theo- experimental or theoretical work to distinguish between
a retical and practical importance in many areas, ranging theeffects ofthe interactionbetweenthe particlesversus
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fromthestudyoffundamentalnonequilibriumstatistical the interactionsbetween the particles and the walls,and
- physicstovariousindustrialprocessessuchasthegrowth to correlate these interactions with the surface growth
d
of films by deposition [1, 2]. Rough surfaces formed by problem.
n
o thesedimentationofparticlesthroughaviscousfluidare In our previous work [22, 23] on the quasi-one-
c particularly interesting, since interactions between the dimensional surfaces formed by particles sedimenting
[ sedimenting particles, and between the particles and the throughaviscousfluidinaquasi-two-dimensionalcellwe
1 walls of the container, can be expected to have an effect havefoundthat the surfaces formedby sedimenting par-
v on the final interface. Surfaces formed by sedimentation ticles are rough on all length scales between the particle
5 areclosetotheoriginalproblemofsedimentationofpar- sizeandthecellsize. However,differentroughnessexpo-
2 ticles sedimenting along straight vertical lines first stud- nentswerefoundintwowell-definedlength-scaleregimes,
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iedbyEdwardsandWilkinson[3]. However,sincethehy- with a well-defined crossover length scale. These rough-
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drodynamicparticle/particleandparticle/wallforcesare ness exponents and the crossover length scale have been
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1 inprinciplelong-range,theroughsurfacesformedbypar- foundtobeindependentofthecellaspectratioorthevis-
0 ticlessedimentinginaviscousfluidareadifferentgrowth cosityofthefluidthroughwhichtheparticlessettle. The
/ situation from the simpler vertical deposition. The situ- exponent found at long length scales has been shown to
t
a ationis furthercomplicatedbythe presenceofbackflows depend on the rate at which particles are deposited into
m of fluid caused by the motion of significant numbers of thecell(hence tothestrengthoftheinteractionbetween
- particles [4, 5]. the particles) [24]. This leads to the conclusion that the
d
scaling exponent seen at long length scales depends on
n In this work we are primarily interested studying ex-
thedetailsofthehydrodynamicinteractionsbetweenthe
o perimentally the effect of the particle/wall interactions
particles, while the exponent seen at small length scales,
c on the roughness of the final interface. The motion of a
: which remained relatively unaffected by changes in the
v sphere parallel to a single wall is of interest as the lim-
deposition rate, may be due to more universal consider-
i iting case of motion of a small sphere in a cylindrical
X ations.
container when the sphere approaches the cylinder wall.
r This problem and the more generalone of the motion of In the present work, we are continuing to study the
a
asphereparalleltotwoexternalwallsweretreatedinthe rough surfaces formed by sedimenting particles, but we
1920s by Faxen [6]. Unfortunately, it is very difficult to are focusing specifically onthe effect of the particle/wall
commentontheexactnatureoftheinteractionsbetween interactions on the final interface while keeping the av-
the particlesinthe presenceofthe walls. Analyticalsed- erage volume fraction of the particles at the low value
imentation theory has succeeded only in analyzing the of φ = 0.01 to restrict the particle / particle interac-
effectivesettlingvelocityofparticlesinadiluteregimein tions to a minimum level allowed by the experimental
thepresenceofthewalls[7,8,9,10,11,12,13]andsome setup. All of our work to date has taken place in quasi-
featuresofmany-bodyinteractionsbetweenthe particles two-dimensionalsedimentationcells,whichconsistoftwo
[14, 15] when there are no walls [16]. Recent theoretical glass plates separated by a gap set by Teflon spacers,
[5, 17, 18] and experimental [19, 20, 21] work hold out through which glass beads are allowed to settle to form
somehopeofdeterminingparticleinteractionsthrougha a quasi-one dimensional rough surface at the bottom of
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the cell (Fig. 1). There are two types of cells, open and photographed during and at the end of the deposition
closed, which will be discussed in more detail below. In process and the photographs were digitized by a Nikon
each type of cell, the gap between the plates, although LS-2000film scanner. Individual particles were typically
variable, has been of the order of the diameter of the resolvable and thus the position of the particles on the
sedimenting particles. interface could be traced accurately. There is a limit to
the extent over which the gap can be widened without
changing the method of analysis, since at one point it
willnolongerbe possibletoanalyzethe roughsurfaceas
II. EXPERIMENTAL METHOD
a one-dimensional interface. We believe that we are al-
readypastthatlimitatR=2,buttogiveanestimateof
Two types of cells were used in the previous work, de-
the effects ofthe wallseparationto the interested reader
notedas”closed”and”open”cells. Closedcellswerecon-
we have included the data for R>2.
structed of 1/4 in. float glass,held 1mm apart by sealed
side frames of precision machined Plexiglas. Around
10,0000.6mm-diametermonodispersesilicasphereswere
placed in the cell, which was then filled with a viscous III. DISCUSSION
fluid (such as glycerin) and closed. Each cell could be
rotatedabout a horizontalaxis perpendicular to the gap
Asinthepreviouswork,wehaveanalyzedtheserough
direction. Whenthecellwasrotated,theparticleswhich
surfacesusingthescalingansatzproposedbyFamilyand
had been at rest on the bottom fell though the viscous
Vicsek [25]. In this ansatz, the rms thickness of the in-
fluid, slowly building up a new surface at the bottom of
terface is defined to be:
the cell. In the closed cells we only had a fixed gap size
between the cell walls. The open cell was constructed
of in float glass, separated by strips of Teflon of known
1
thickness. It had dimensions comparable to the closed 1 N 2
cell, but was open at the top, so that beads could be W(L,t)= h˜(xi,t)2 (1)
N
dispersedthroughafunnelwhichsteadilydroppedbeads " i=1 #
X
as it traveled back and forth across the top of the cell
(Fig. 2). In this way, the deposition rate of the beads where
into the cell could be controlled precisely by varying the
speedandthesizeofthefunnel. Thepresentexperiments h˜(x ,t)=h(x ,t)−h¯(t) (2)
i i
were designed to test the effect of changing the distance
between the walls on the roughness of the final interface
while keeping the interparticle distance constant. The
experiments took place in the open cell and we investi-
gated the effect of variability in the gap by setting the
gap at different values, and measuring the effect of the
walls on the roughnessof the final interface formed after
sedimentation. We investigated gaps ranging from 0.8
mm to 2.0 mm. The ratio of the gap thickness to the
bead diameter was defined as a dimensionless parameter
R, and our experiments spanned a gap/bead diameter
ratioofR=1.33toR=3.33. The previousexperiments
took place at either a gap of 1mm (R = 1.66) (closed
cell) or 0.8 mm (R = 1.33) (open cell), so this range of
gapvalueswentfarbeyondourearliermeasurements. In
all cases, the deposition rate of beads into the cell was
controlled at about 4 beads/sec so the average distance
between the particles was about 20R. The surface was
FIG. 2: The sedimentation cell consists of two glass plates
with a small gap between them, of the order of the diameter
of the glass beads. The cell is filled with a viscous fluid such
asoil,andafunnelsweepsacrossthetopofthecell,delivering
FIG.1: Portionofaroughsurfaceformedby0.6mmdiameter amixtureofoilandbeadstothecell. Thebeadssettletothe
glass beads sedimenting through heavy paraffin oil. bottom of thecell and build a rough surface.
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and deposition rate of one bead/sec to 0.5 at an average
deposition rate of 35 beads/sec.
N
1
h¯(t)= h(x ,t). (3)
N i We have investigated the effect of the interaction be-
i=1
X tweenthewallsofthecontainerandthesedimentingpar-
The scaling ansatz predicts that: ticlesontheroughnessexponentofthesurfaceformedby
this quasi-two-dimensional sedimentation. The rough-
ness exponent is found to be robust to the changes in
W(L,t)=Lαf(t/Lα/β) (4) the separation between the walls of the container. The
reasons for the slight increase in the of the roughness
where the exponents α and β are the static and dy- exponent at R = 2 is under further investigation using
namic scaling exponents. The function f(t/Lα/β) is ex- computationalmethodstosimulatetheeffectsofthewall
pected to have an asymptotic form such that separation, as well as direct experimental investigations
of correlations between particles in the fluid.
W(L,t)∼tβ for t≪Lα/β (5)
and
W(L,t)∼Lα fort≫Lα/β (6)
Fig. 3 shows an example of W(L,t) at a typical
gap/bead ratio. To minimize the wall effects at the hor- Acknowledgments
izontal edges, we have used only the middle 70% of each
interface for our analysis.
AtallvaluesofR(gap/beadratio)studied,westillsee
The authors wouldlike to acknowledgethe adviceand
two well defined roughness exponents. These roughness
loanofequipmentgraciouslyprovidedbyDr. JamesMa-
exponents have a crossover length scale at about 1 cm,
her at the University of Pittsburgh.
whichistypicalfromthepreviouswork. Ourearlierwork
correspondedto a gap width of 1.0 mm, and the present
workgivesthesameresultsatthisgapwidthasexpected.
As thevalue ofRisincreased(Fig.4),wedo notseeany
significantchangein the value ofeither exponent. While
there is a slightincrease in the scalingexponents around
R = 2, this slight increase is less significant than the
increaseintheroughnessexponentswhenweincreasethe
mean particle separation by increasing the feeding rate
of the particles in our previous experiments, where the
large length scale roughnessrose from 0.2 at an average
FIG.4: Averageroughnessexponentsfoundatdifferentbead
diameter/gapratios,R.Thecirclesdenoteal,thesquaresrep-
resentas. Thevaluesofeachaaretypicallyaveragedoverfour
experimental runs, each of which would have had relatively
FIG. 3: Roughnessfunction W(L,t) vs. L for a typical inter- small uncertainty in a. Thus, the stated uncertainties arise
face at a gap of 0.8mm from the reproducibility of roughness.
4
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