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.

Apparatus

Growth from a Nearly Spherical Seed

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.

Typical sidebranches—note the long smooth tip and the slightly irregular branches.

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

Slow Growth of a Spherical Crystal

Modeling the initial growth

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.

Average shape for 3 growth velocities. w ~ z0.6~0.8

Modeling Initial Sidebranches

Approximate model for initial sidebranches (all distances are scaled by :

zzAzmzmw

zzwzwdev2

sin)()(

2)()(

2210

5/2)/(0)( szeSzA

Aggregated Fits for RMS Sidebranch Amplitude

Aggregated Fits for RMS Sidebranch Amplitude

Preliminary Fit Results forNoise Amplitude

S0 (expt) ~5 x 10-4

S0 (theory) ~1 x 10-4

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.