Dynamics of fronts in chemical and bacterial media: If you

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Dynamics of fronts in chemical and bacterial
mediaIf youve seen one front, youve seen them
all

• Paul Ronney
• Department of Aerospace Mechanical Engineering
  Univ. of Southern California, Los Angeles, CA,
  90089

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University of Southern California

• Established 125 years ago this week!
• jointly by a Catholic, a Protestant and a Jew -
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  in 2005
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Paul Ronney

• B.S. Mechanical Engineering, UC Berkeley
• M.S. Aeronautics, Caltech
• Ph.D. in Aeronautics Astronautics, MIT
• Postdocs NASA Glenn, Cleveland US Naval
  Research Lab, Washington DC
• Assistant Professor, Princeton University
• Associate/Full Professor, USC
• Research interests
• Microscale combustion and power generation
• (10/4, INER 10/5 NCKU)
• Microgravity combustion and fluid mechanics
  (10/4, NCU)
• Turbulent combustion (10/7, NTHU)
• Internal combustion engines
• Ignition, flammability, extinction limits of
  flames (10/3, NCU)
• Flame spread over solid fuel beds
• Biophysics and biofilms (10/6, NCKU)

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Paul Ronney
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Motivation

• Propagating fronts are ubiquitous in nature
• Flames
• (Fuel Oxidant) Heat ? More heat
• Solid rocket propellant fuels
• (Fuel Oxidant) Heat ? More heat
• Self-propagating high-temperature synthesis (SHS)
  - reaction of metal with metal oxide or nitride,
  e.g. Fe2O3(s) 2Al(s) ? Al2O3(s) 2Fe(l)
• (Fuel Oxidant) Heat ? More heat
• Frontal polymerization
• Monomer initiator heat ? polymer more heat
• Autocatalytic chemical reactions (non-thermal
  front)
• Reactants H ? Products more H
• Bacterial front (non-thermal front)
• Nutrient bugs ? more bugs
• All of these might be construed as
  reaction-diffusion systems
• Todays topic what is similar and what is
  different about these different types of fronts?

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Reaction-diffusion systems

• Two essential ingredients
• Reactive medium (e.g. fuel-air mixture)
• Autocatalyst - product of reaction that also
  accelerates the reaction (e.g. thermal energy)
• Self-propagation occurs when the autocatalyst
  diffuses into the reactive medium, initiating
  reaction and creating more autocatalyst, e.g. A
  nB ? (n1)B
• Enables reaction-diffusion fronts to propagate at
  steady rates far from any initiation site

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Premixed flame (SHS, solid propellant similar)
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Reaction-diffusion systems - characteristics

• After initial transient, fronts typically
  propagate at a steady rate
• Propagation speed (SL) (D?)1/2
• D diffusivity of autocatalyst or reactant
• ? characteristic reaction rate (reaction
  time)-1
• D depends on sound speed (c) mean free path
  (?)
• D c?
• Propagation rate generally faster in turbulent
  media due to wrinkling (increased surface area)
  of front
• Thermal fronts require high Zeldovich number (Ze)
  so that ?products gtgt ?reactants, otherwise
  reaction starts spontaneously!
• Flammability or extinction limits when loss rate
  of autocatalyst production rate of autocatalyst

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Instability mechanisms

• Instability mechanisms may preclude steady flat
  front
• Turing instability - when ratio of reactant to
  autocatalyst diffusivity differs significantly
  from 1
• Thermal fronts Dautocatalyst/Dreactant Lewis
  number
• Low Le additional thermal enthalpy loss in
  curved region is less than additional chemical
  enthalpy gain, thus local flame temperature in
  curved region is higher, thus reaction rate
  increases drastically, thus blip grows
• High Le pulsating or travelling wave
  instabilities
• Hydrodynamics - thermal expansion, buoyancy,
  Saffman-Taylor

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Polymerization fronts

• First demonstrated by Chechilo and Enikolopyan
  (1972) reviewed by Pojman et al. (1996), Epstein
  Pojman (1998)
• Decomposition of the initiator (I) to form free
  radicals (Ri)
• I ? R1 R2 - highest activation energy step
• e.g. (NH4)2S2O8 ? 2NH4SO4
• Followed by addition of a radical to a monomer
  (M)
• M Ri ? RiM - initiates polymer chain, grows
  by addition
• RiMn M ? RiMn1
• Most of heat release occurs through addition step
• Note not chain-branching like flames
• Chain growth eventually terminated by
  radical-radical reactions
• RiMn RjMm ? RiMnmRj
• Chain length can be controlled by chain transfer
  agents - affects properties of final product

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Polymerization front
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Polymerization fronts

• Potential applications
• Rapid curing of polymers without external heating
• Uniform curing of thick samples
• Solventless preparation of some polymers
• Filling/sealing of structures having cavities of
  arbitrary shape without having to heat the
  structure externally
• Limitations / unknowns
• Thermally driven system - need significant ?T
  between reactants and products to have??products
  gtgt ?reactants
• Previous studies use very high pressures or high
  boiling point solvent (e.g. DMSO) to avoid
  boiling since mixtures with Tad lt 100C wont
  propagate
• but water at ambient pressure is the solvent
  required for many practical applications
• Idea use a very reactive monomer (acrylic acid)
  highly diluted with water to minimize peak
  temperature, and control heat losses to avoid
  extinction
• but nothing is known about the extinction
  mechanisms!

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Polymerization fronts - approach

• Simple apparatus round tubes
• Need bubble-free model polymerization systems
• 2-hydroxyethyl methacrylate (HEMA) monomer in
  DMSO solvent
• Acrylic acid (AA) monomer in water solvent
• Both systems ammonium persulfate (AP) initiator,
  Cab-o-sil (fumed silica powder) viscosity
  enhancer
• Control thermal boundary conditions assess heat
  loss
• Varying tube diameter
• Water bath, ambient air or insulated tube to
  control external temperature

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Polymerization front

• Typical speeds 0.01 cm/s, SL (??)1/2 ? ?-1 14
  s
• From plot of ln(SL) vs. 1/Tad can infer E 13.5
  kcal/mole, Ze 20
• Extinction at Pe (0.004 cm/s)(1.6 cm)/(0.0014
  cm2/s) 4.6 - close to classical flame theory
  predictions
• Plot of SL vs. fuel concentration approaches
  vertical at extinction limit as theory predicts
• With insulation, limiting SL and AA much lower

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Polymerization fronts - thermal properties

• Far from limit
• Peak T same with or without insulation, speed and
  slope of T profile same, uninsulated case shows
  thermal decay in products
• Close to limit
• Uninsulated case shows substantial thermal decay
  in products ratio (peak slope)/(peak - slope)
  12
• Insulated case much slower, thicker flame, little
  or no thermal decay, limit not well defined

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Polymerization front

• High Lewis number - spiral travelling-wave
  instabilities like flames (middle and right
  videos, viscosity-enhancing agent added to
  suppress buoyant instabilities)

Movies courtesy Prof. J. Pojman, University of
Southern Mississippi
Lean C4H10-O2-He mixtures Pearlman and Ronney,
1994
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Autocatalytic aqueous reactions - motivation

• Models of premixed turbulent combustion dont
  agree with experiments nor each other!

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Modeling of premixed turbulent flames

• Most model employ assumptions not satisfied by
  real flames, e.g.
• Adiabatic (sometimes ok)
• Homogeneous, isotropic turbulence over many LI
  (never ok)
• Low Ka or high Da (thin fronts) (sometimes ok)
• Lewis number 1 (sometimes ok, e.g. CH4-air)
• Constant transport properties (never ok, 25x
  increase in ? and ? across front!)
• u doesnt change across front (never ok, thermal
  expansion across flame generates turbulence) (but
  viscosity increases across front, decreases
  turbulence, sometimes almost cancels out)
• Constant density (never ok!)

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Autocatalytic front (bacterial fronts similar)
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Liquid flame idea

• Use propagating acidity fronts in aqueous
  solution
• Studied by chemists for 100 years
• Recent book Epstein and Pojman, 1998
• Generic form
• A nB ? (n1)B - autocatalytic
• ??/? ltlt 1 - no self-generated turbulence
• ?T 3 K - no change in transport properties
• Zeldovich number ? 0.05 vs. 10 in gas flames
• Aqueous fronts not affected by heat loss!!!
• Large Schmidt number ?/D 500 (liquid flames)
  vs. 1 (gases) - front stays "thin even at
  high Re

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Approach - chemistry

• Iodate-hydrosulfite system
• IO3- 6 H 6e- ? I- 3 H2O
• S2O4-2 4 H2O ? 6 e- 8 H 2 SO4-2
• _________________________________________________
• IO3- S2O4-2 H2O ? I- 2 SO4-2 2 H
• Comparison with turbulent combustion model
  assumptions
• Adiabatic
• Homogeneous, isotropic turbulence over many LI
• Low Ka or high Da (thin fronts) due to high
  Schmidt
• Constant transport properties
• u doesnt change across front
• Constant density
• Conclusion liquid flames better for testing
  models!

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Gaseous vs. liquid flames

• Most model employ assumptions not satisfied by
  real flames
• Adiabatic (sometimes ok) (Liquid flames TRUE!)
• Homogeneous, isotropic turbulence over many LI
  (never ok) (Liquid flames can use different
  apparatuses where this is more nearly true)
• Low Ka or high Da (thin fronts) (sometimes ok)
  (Liquid flames more often true due to higher
  Sc)
• Lewis number 1 (sometimes ok, e.g. CH4-air)
  (Liquid flames irrelevant since heat transport
  not a factor in propagation)
• Constant transport properties (never ok, 25x
  increase in ? and ? across front!) (Liquid
  flames TRUE)
• u doesnt change across front (never ok, thermal
  expansion across flame generates turbulence) (but
  viscosity increases across front, decreases
  turbulence, sometimes almost cancels out) (Liquid
  flames TRUE)
• Constant density (never ok!) (Liquid flames
  true, although buoyancy effects still exist due
  to small density change)
• Conclusion liquid flames better for testing
  models!

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Taylor-Couette apparatus
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Capillary-wave apparatus
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Results - liquid flames
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Results

• Thin "sharp" fronts at low Ka (lt 5)
• Thick "fuzzy" fronts at high Ka (gt 10)
• No global quenching observed, even at Ka gt 2500
  !!!
• High Da - ST/SL in 4 different flows consistent
  with Yakhot model
• Low Da - ST/SL lower than at high Da - consistent
  with Damköhler model over 1000x range of Ka!
• Rising, buoyantly-unstable fronts in Hele-Shaw
  flow show unexpected wrinkling - subject of
  separate investigation

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Liquid flames - comparison to Yahkot (1988)
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Results - liquid flames - propagation rates

• Data on ST/SL in flamelet regime (low Ka)
  consistent with Yakhot model - no adjustable
  parameters
• Transition flamelet to distributed at Ka 5

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Results - liquid flames - propagation rates

• Data on ST/SL in distributed combustion regime
  (high Ka) consistent with Damköhlers model - no
  adjustable parameters

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Front propagation in one-scale flow

• Turbulent combustion models not valid when energy
  concentrated at one spatial/temporal scale
• Experiment - Taylor-Couette flow in Taylor
  vortex regime (one-scale)
• Result - ST/SL lower in TV flow than in turbulent
  flow but consistent with model for one-scale flow
  probably due to "island" formation reduction in
  flame surface (Joulin Sivashinsky, 1991)

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Fractal analysis in CW flow

• Fractal-like behavior exhibited
• D 1.35 (? 2.35 in 3-d) independent of u'/SL
• Same as gaseous flame front, passive scalar in CW
  flow
• Theory (Kerstein others)
• D 7/3 for 3-d Kolmogorov spectrum (not CW flow)
• Same as passive scalar (Sreenivasan et al, 1986)
• Problem - why is d seemingly independent of
• Propagating front vs. passively diffusing scalar
• Velocity spectrum
• Constant or varying density
• Constant or varying transport properties
• 2-d object or planar slice of 3-d object

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Fractal analysis in CW flow
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Bacterial fronts

• Many bacteria (e.g. E. coli) are motile - swim to
  find favorable environments - diffusion-like
  process - and multiply (react with nutrients)
• Two modes run (swim in straight line) tumble
  (change direction) - like random walk
• Longer run times if favorable nutrient gradient
• Suggests possiblity of flames

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Motile bacteria

• Bacteria swim by spinning flagella - drag on rod
  is about twice as large in crossflow compared to
  axial flow (G. I. Taylor showed this enables
  propulsion even though Re 10-4) (If you had
  flagella, you could swim in quicksand or
  molasses)
• Flagella rotate as a group to propel, spread out
  and rotate individually to tumble

http//www.rowland.org/bacteria/movies.html
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Analogy with flames
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Reaction-diffusion behavior of bacteria

• Bacterial strains E.coli K-12 strain W3110
  derivatives, either motile or non-motile
• Standard condition LB agar plates (agar
  concentration of 0.1 - 0.4)
• Variable nutrient condition Tryptone/NaCl plates
  (agar concentration of 0.1, 0.3)
• All experiments incubated at 37C

Fronts show a steady propagation rate after an
initial transient
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Propagation rates of motile bacteria fronts

• As agar concentration increases, motility of
  bacteria (in particular sound speed (c))
  decreases, decreases effective diffusivity (D)
  and thus propagation speed (s) decreases
  substantially
• No effect of depth of medium
• Above 0.4 agar, bacteria grow along the surface
  only
• Recently very similar results for Bacillius
  subtilis - very different organism - E. coli B.
  subtilis evolutionary paths separated 2 billion
  years ago

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Effect of nutrient concentration

• Increasing tryptone nutrient concentration
  increases propagation speed (either due to
  increased swimming speed or increased division
  rate) but slightly decreases propagation rate
  beyond a certain concentration - typically
  motility decreases for high nutrient
  concentrations (detectors saturated?)

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Quenching limit of bacteria fronts

• Quenching limit min. or max. value of some
  parameter (e.g. reactant concentration or channel
  width) for which steady front can exist
• Quenching channels made using filter paper
  infused with antibiotic - bacteria killed near
  the wall, mimics heat loss to a cold wall in
  flames
• Bacteria can propagate through a wide channel but
  not the narrow channel, indicating a quenching
  limit
• Quenching described in terms of a minimum Peclet
  Number
• Pe sw/D (w channel width)
• For the test case shown s 1.75 x 10-4 cm/s, D
  3.7 x 10-5 cm2/s, w at quenching limit 2.1 cm ?
  Pe 9.8 - similar to flames and polymer fronts

6 mm wide channel 35 mm wide channel E. coli,
0.1 agar, 100 µl of kanamycin per side, 6.5
hours after inoculation
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Comparison of fronts in Mot and Mot- bacteria

• Some mutated strains are non-motile but D due to
  Brownian motion  104 smaller
• Fronts of Mot- bacteria also propagate, but more
  slowly than Mot bacteria

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Quantitative analysis

• Bacteria D as estimated from measured front
  speeds
• SL for Mot 5.3 x 10-5 cm/s for 0.3 agar
• Reproduction time scale (?) of E.coli 20 min
• D s2? (5.3 x 10-5 cm/s)2(1200s) 3.3 x 10-6
  cm2/s
• Similarly, D 3.7 x 10-5 cm2/s in 0.1 agar
• Bacteria diffusivity estimated from molecular
  theory
• Mean free path (?) estimated as the sound
  speed (c) multiplied by the time (t) bacteria
  swim without changing direction
• c 21 µm/s, t  1.4 s
• ? 3.0 x 10-3 cm, D 6.3 x 10-6 cm2/s, similar
  to value inferred from propagation speed
• Diffusivity of Mot- E. coli due to Brownian
  motion (0.75 µm radius particles in water at
  37C) 2.9 x 10-9 cm2/s, 1700x smaller than
  Mot bacteria
• Fronts should be (1700)1/2 40x slower in Mot-
  bacteria
• Consistent with experiments (e.g. 8 mm/hr vs. 0.2
  mm/hr at 0.1 agar)

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Comparison of fronts in Mot and Mot- bacteria

• Dnutrient ( 10-5 cm2/s) close to Dbacteria, so
  Lewis number 1
• Do bacteria choose their run-tumble cycle time to
  produce D required for Le 1 and avoid
  instabilies???
• Switching from Mot to Mot- bacteria decreases
  the bacteria diffusivity (Dautocatalyst) by
  1700x but nutrient diffusivity (Dreactant) is
  unchanged - decreases the effective Lewis
  number
• Mot- fronts cellular but Mot fronts smooth -
  consistent with Lewis number analogy

Mot 5 hr 30 min after inoculation Mot- 50
hr after inoculation 0.1 Agar dyed with a 5
Xylene Cyanol solution (Petri dish 9 cm diameter)
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Biofilms

• Until recently, most studies of bacteria
  conducted in planktonic (free swimming) state,
  but most bacteria in nature occur in biofilms
  attached to surfaces
• Recently many studies of biofilms have been
  conducted, but the effects of flow of the
  nutrient media have not been systematically
  assessed
• No flow no replenishment of consumed nutrients
  - little or no growth
• Very fast flow attachment and upstream spread
  difficult
• Most flow studies have reported only volumetric
  flow rate or flow velocity - not a useful
  parameter - why should it matter what the flow is
  far from the surface when the biofilm is attached
  to the surface?
• Biofilms can spread upstream - is spread rate
  shear as with upstream flame spread on a solid
  fuel bed?
• Fluid mechanics tells us the shear rate at the
  surface is the key
• Our approach use flow in tubes (shear not
  separated from mean flow rate) and Taylor-Couette
  cells (shear and mean flow independently
  controlled)

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Biofilm experiments - laminar flow in tubes
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Biofilms - images
Experiments show an effect of flow velocity or
shear rate on growth rate and upstream spread
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Biofilms
Fixed flow / varying elapsed time more time,
more growth, but maybe some sloughing at long
times
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Biofilms
Fixed elapsed time / varying flow optimal
flow/shear rate that maximizes growth
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Biofilms - Taylor Couette cell concept
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Summary
Property Gaseous flame (e.g. CH4-air) SHS Aqueous autocatalytic front Polymer front Bacterial front
Reactant(s) Fuel, oxidant Reductant, oxidant Reactants (e.g. IO3- - S2O42-) Monomer initiator (e.g. IO3- - S2O42-) nutrient
Product(s) CO2, H2O, thermal energy Metal, metal oxide or nitride product polymer bacteria
Autocatalyst Thermal energy, free radicals Thermal energy H Thermal energy bacteria
Flame speed (S) 400 mm/s 10 mm/s 0.2 mm/s 0.1 mm/s 0.001 mm/s
Zeldovich (Ze) 10 - 20 10 - 20 0.05 20 0
Heat loss Important Important Unimportant (Zeltlt1) Important None
Prandtl (Pr) 1 8 7 1000 1000
Lewis (Le) 1 8 70 (but irrelevant) nearly infinite 1 (Mot) 10-4 (Mot-)
Density ratio ?????????a?/?? 6 (monotonic) 0.1 0.0006 (monotonic) -0.2 (non-monotonic) ?
Viscosity ratio ??a??8?/?? 25 8/8 0.1 1 - 8 1
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Conclusions

• Broad analogies can be drawn between different
  types of reaction-diffusion fronts in disparate
  types of physical / chemical / biological systems
• Steady propagation rates
• Effects of reactant and product diffusivities
• Instabilities (i.e. pattern formation)
• Quenching behavior
• Applications to
• Combustion engines
• Solid propellant rockets
• Synthesis of ceramics
• Polymer synthesis
• Assessment of turbulent combustion models
• Colonization of new environments by swarms of
  bacteria
• Biofilms - bacteria growing on surfaces - far
  more resistant to antibiotics other stresses
  than planktonic (free-swimming) bacteria

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Thanks to

• National Cheng-Kung University
• Prof. Y. C. Chao, Prof. Shenqyang Shy
• Combustion Institute (Bernard Lewis Lectureship)