Self-Concentration of Bacteria

1
The Platonic Ideal of Stalactite Growth
Self-Concentration of Bacteria
R E Goldstein DAMTP Cambridge
2
What is this?
Hydrothermal vents? Tubeworms? Coral?
David A. Stone (U. Arizona)
3
Tubular Precipitation and Redox Gradients on a
Bubbling Template
5 mm
D A Stone and REG, Proc. Natl. Acad. Sci. (USA)
101, 11537 (2004).
4
Outline
Free-boundary theory
Tubes
Icicles
Fluid Jets
Verification
Speleothems
5
Tube Formation
ANODE ()
Curvature with magnetic field
CATHODE ()
6
Tubular Growth Templated by Bubbles (The Movie)
7
Electron Microscopy Reveals Redox Gradients
Tripartite layering green rust xtals
magnetite nodules magnetite xtals
5 mm
10 mm
100 mm
5 nm
inside
White rust Fe(II)(OH)2 Green rust
Fe(II)4Fe(III)2(OH)12SO4 Magnetite
Fe(II)Fe(III)2O4 Lepidocrocite
g-Fe(III)OOH
reduced
outside
oxidized
8
Tubular Precipitation Templated by Bubbles
NH3 as the local pH-changer
acidic
Redox gradients in the walls of hydrothermal
vents Tivey McDuff (1990), Tivey (1995)
basic
9
Liesegang Rings and Redox Gradients
Pourbaix diagram Génin et al. (Nancy)
10
Reversed-Phase Growth of a Tube
NH3 atmosphere
growing magnetite tube
pendant drop of FeSO4 solution
precipitate film
11
Soda Straws
12
Stalactite Growth
Stalactites have featured in written accounts
dating back thousands of years (Pliny, 1st
century A.D.). Only in the last century or so
has their chemical origin been understood. Yet,
their characteristic carrot-like shape has never
been explained.
M.B. Short, J.C. Baygents, J.W. Beck, D.A.
Stone, R.S. Toomey, III, and G., Phys. Rev.
Lett. 94, 018501 (2005). M.B. Short, J.C.
Baygents, and G., Phys. Fluids 17, 083101
(2005). See also New Scientist, May 2006
13
Inside Kartchner Caverns(Benson, Arizona)
REG
14
Draperies
REG
15
Draperies Viewed From Below
REG
16
Thin-Film Flow
y
air
fluid film
q
Q in cm3/hr R in cm
wrap around a cylinder
17
Separation of Time Scales

• In the century it takes to elongate a
• typical stalactite by 1 cm, with a
• drip rate of 40 cm3/hr
• 36,000 liters of water flow over
• the stalactite
• 5.4 kg of dissolved calcium was
• available for precipitation
• 200 g actually precipitated, a
• fractional depletion of 3-4
• Growth rate is about 2 A/minute!

• Length L 10-100 cm
• Radius r 1-10 cm
• Fluid flow Q 10-100 cm3/hr
• Fluid layer thickness h 10 mm
• Fluid velocity u0 0.1 cm/sec
• Reynolds number 0.01-0.1

Diffusion
Traversal
Growth
18
Reaction-Diffusion Dynamics

• Five equilibrium constants electroneutrality
  leaves one degree of freedom let it be CO2

Buhman and W. Dreybrodt Chemical Geology 48, 189
(1984) the slowest reactions are those
generating CO2

• Typically, k and k- have values 0.1-1 s-1
• Reaction timescale tR 1-10 s gt diffusional
  timescale 0.1 s
• Generation of CO2 is the rate limiting step for
  stalactite growth
• The growth rate is given directly by the
  outgassing of CO2

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Reaction-Diffusion Dynamics The Essentials
Henrys Law constant
chemistry
geometry
exterior diffusion
20
The Mathematics of Geometric Growth Laws
Dissipative Local (Lyapunov functional)
Curve-shortening equation
Dissipative Nonlocal (e.g Biot-Savart)
Lee, McCormick, Ouyang Swinney
Langmuir monolayers, type-I superconductors, react
ion-diffusion systems
Integrable (soliton hierarchies)
Modified KdV equation
21
The Local Geometric Growth Law
z
r(z)
t
?
n
Extreme enhancement near the tip due to thicker
fluid layer
22
Numerical Studies Reveal an Attractor
Flux conservation is key to the instability of a
surface A small downward bump leads to a
locally-thick film and a higher precipitation
rate, increasing the size of the bump, and so
on. What is this shape?
23
The Traveling Shape
A uniformly translating shape obeys
Rescale symmetrically
Consequences

• The rescaling was symmetric in z and r , and the
  equation describing the shape is parameter-free,
  hence
• all stalactites will tend toward the same
  universal shape
• The only difference between stalactites is the
  scaling factor a that magnifies the ideal shape
  onto the ones in nature

24
The Platonic Ideal of Stalactites
Convex shape from thin-film dynamics
25
Field Work in Kartchner Caverns
Martin Short
Jim Baygents
26
Image Analysis
posterize edge-detect
r
z
Nonlinear fit to obtain scale factor a
c2
a (mm)
27
Compilation of Data
Optimally-rescaled shapes are averaged and
compared to the Platonic ideal
28
Stalactites and Icicles
M.B. Short, J.C. Baygents, and G., Phys. Fluids
18, 083101 (2006).
29
(No Transcript)
30
Icicle Schematic
rising thermal boundary layer
Ti
Tm
Ta
Related work Makkonen, Szilder Lozowski
31
Shape of the Ideal Icicle
Thin-film law
Depletion
Integrate
Growth rate
Rescale, find ode
Integrate
Again!
32
Direct Comparison with Available Data (Images)
33
Hydothermal Vents
34
Setup for Controlled Tube Growth
Related previous experiments and
theory Hydrothermal vents Turner Campbell,
Earth Plan. Sci. Lett. 82, 36 (1987). Ice
stalactites Martin, J. Fluid Mech. 63, 51
(1974). Chemical gardens Thouvenel-Romans, van
Saarloos Steinbock,
Europhys. Lett. 67, 42 (2004).
35
Tubular Growth Up Close
t100 min
t315 min
t300 min
t200 min
White rust Fe(II)(OH)2
2 mm
t330 min
t400 min
t500 min
t345 min
Stone, Llewellyn, Baygents, and G, Langmuir 21,
10916 (2005).
36
Scaling the Growth Dynamics
5 ml/hr
1 ml/hr
Supersaturation model
Lateral diffusive depletion of local concentration
at tip
data collapse
37
Model Continued
a1.7
38
Future Directions (Ripples Nonaxisymmetry)
Icicle Ripples (Stephen Morris)
Stalactite Crenulations
Paleoclimate studies, isotope distribution,
aridity index?
39
Terraces and Domes at Hot Springs
Nigel Goldenfeld et al. (UIUC) guava.physics.uiuc.
edu
Yellowstone National Park