andrew dougherty franklin stinner (‘11) physics department lafayette college, easton pa ...
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Andrew Dougherty
Franklin Stinner (‘11)
Physics DepartmentLafayette College, Easton PA
http://sites.lafayette.edu/doughera
Sidebranching in theDendritic Crystal Growth of
Ammonium Chloride
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.
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.
Noise-induced Sidebranch Amplitude
2/1
2
3
0 *3
)(
3
2exp)(
z
zwSzA ave
202
*
v
Dd
1/2
40 2 4
21 10
( )
eqL
eq
C DS
C v
wave(z) = average shape of the dendrite.
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.
Finding d0: Modeling the initial growth
TTdT
d
R
d
R
D
dt
dReq
where
2 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
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
x
Axz
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
Noise-induced Sidebranch Amplitude
2/1
2
3
0 *3
)(
3
2exp)(
z
zwSzA ave
202
*
v
Dd
1/2
40 2 4
21 10
( )
eqL
eq
C DS
C v
wave(z) = average shape of the dendrite.
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.
Modeling Initial Sidebranches
Approximate model for initial sidebranches (all distances are scaled by :
zzAzmzmw
zzwzwdev2
sin)()(
2)()(
2210
5/2)/(0)( szeSzA
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.