Sidebranching in the Dendritic Crystal Growth of Ammonium Chloride
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Transcript of Sidebranching in the Dendritic Crystal Growth of Ammonium Chloride
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Andrew DoughertyFranklin Stinner (‘11)
Physics DepartmentLafayette College, Easton PA
http://sites.lafayette.edu/doughera
Sidebranching in theDendritic Crystal Growth of
Ammonium Chloride
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Experiments• NH4Cl growing in aqueous solution
• Growth cell: 40 x 10 x 2 mm3
• Obtain an approximately spherical seed.• Lower temperature T (~1oC) to initiate slow
growth.
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Apparatus
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Growth from a Nearly Spherical Seed
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Apparent tip oscillations – note the regular sidebranches close to the tip.However – such patterns are only rarely seen in this experiment, and we have not found any way to repeat them.
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Typical sidebranches—note the long smooth tip and the slightly irregular branches.
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Noise-induced Sidebranch Amplitude
2/1
2
3
0 *3)(
32exp)(
zzwSzA ave
202*
vDd
1/24
0 2 4
2 1 10( )
eqL
eq
C DSC v
wave(z) = average shape of the dendrite.
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Determining Materials Constants
d0: Capillary length: Measure the very slow growth and dissolution of an initially spherical seed.
v, and : Measure the tip of steady-state growing dendrites.
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Finding d0: Modeling the initial growth
TTdTd
Rd
RD
dtdR
eq
where2 0
Assume quasi-static, diffusion-limited, spherically-symmetric growth:
•Increasing supersaturation increases growth rate.•Growth rate proportional to local concentration gradient.•Surface tension limits sharpness•Unstable equilibrium at Rc, the critical radius for nucleation.
•2d0//R term is very small; need to optimize the experimentalprotocol to determine d0
02dRc
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Slow Growth of a Spherical Crystal
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Modeling the initial growth
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Fitting the Dendrite Tip
• First, model the tip, then look for sidebranches as deviations from the initially smooth tip.
• Approximate model for tip shape: (A4 -0.002)
• Measure tip position to determine v.
3
4
4
2)4cos(2
xAxz
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Preliminary Results for Materials Constants
d0 (2.2 + 0.1)x10-4 m
d/dT 0.0043 + 0.0001/oC
v2 12.1 + 0.1 m2/s
* 0.093 + 0.008
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Noise-induced Sidebranch Amplitude
2/1
2
3
0 *3)(
32exp)(
zzwSzA ave
202*
vDd
1/24
0 2 4
2 1 10( )
eqL
eq
C DSC v
wave(z) = average shape of the dendrite.
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Average Shape: No single simple shape –Different Scaling Regimes:
• Near tip, w ~ z1/2
• Very far back, w ~ z1
• Intermediate region: w ~ z3/5 ? Actual scaling varies more continuously.
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Average shape for 3 growth velocities. w ~ z0.6~0.8
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Modeling Initial Sidebranches
Approximate model for initial sidebranches (all distances are scaled by :
zzAzmzmw
zzwzwdev2sin)()(
2)()(
2210
5/2)/(0)( szeSzA
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Aggregated Fits for RMS Sidebranch Amplitude
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Aggregated Fits for RMS Sidebranch Amplitude
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Preliminary Fit Results forNoise Amplitude
S0 (expt) ~5 x 10-4
S0 (theory) ~1 x 10-4
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Conclusions:• No velocity oscillations were observed during
normal steady-state growth.• The functional form of the sidebranch amplitude
is reasonably-well described by the noise-driven scenario.
• The amplitude of the sidebranches is slightly larger, but of the same order of magnitude as predicted by the noise-driven scenario.
Limitations: The most important limitations are precise characterizations of both wave and actual sidebranch amplitudes.