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Understanding the movements of metal whiskers

Victor Karpov

University of Toledo

victor.karpov@utoledo.edu

Victor Karpov, Univ Toledo, April 2015 1

Outline • Motivation • Approach • Whisker mechanics in elastic beam model • 5 potential mechanisms of movements

– Air flow effects – Brownian movements – Mechanical vibrations – Garden hose instability – Electric instabilities: ionic diffusion and

nonequilibrium charges

• Conclusions

Victor Karpov, Univ Toledo, April 2015 2

Motivation: observations by many • Examples of movies

– https://nepp.nasa.gov/whisker/video/index.html

– https://www.youtube.com/watch?v=HbWRPAVzdmc

• Movies imply whisker movement due to minute air flows (expiration towards whiskers, etc.; ), thanks Jay B for explanations

• Anecdotal evidence by many: whiskers move spontaneously, without any intended stimuli (thanks to the discussions during one of the recent teleconferences).

• I saw spontaneous movements myself with a flashlight and a magnifying glass on huge zinc whiskers (thanks Jay B)

• But was it spontaneous? Could it be documented?

• Did others see spontaneous whisker movements?

• My contribution here: what is possible theoretically?

Victor Karpov, Univ Toledo, April 2015 3

https://nepp.nasa.gov/whisker/video/index.html

Note: these are high amplitude movements

o with characteristic times of 0.1 – 0.01 s o often, nearest neighbors move incoherently

Victor Karpov, Univ Toledo, April 2015 4

Proposed mechanisms of metal whisker movements

Air flow (NASA team) Brownian movements (G. Davy)

External mechanical vibrations (B. Rollins)

Garden hose instability (VK)

Random electric fields due to ionic diffusion (VK)

Random electric fields due to light induced recharging (VK)

Victor Karpov, Univ Toledo, April 2015 5

Announcing final results (in case you do not have time)

Air flow (NASA team) Brownian movements (G. Davy)

External mechanical vibrations (B. Rollins)

Garden hose instability (VK)

Random electric fields due to ionic diffusion (VK)

Random electric fields due to light induced recharging (VK)

Certainly possible

u n

lik e

ly

u n

lik e

ly

impossible

im p

o ss

ib le

plausible

Victor Karpov, Univ Toledo, April 2015 6

Elastic beam model for whiskers

u(x,t)

x l d

f

u – deflection, t – time, Y-Young’s moduli, I-area moment of inertia, r-material density, A-area, B-friction coefficient, f-force at point x on the beam

Movements reversible – elasticity Euler-Bernoulli equation

Can be solved exactly for cantilever beams under simple forces; otherwise -- useless

Victor Karpov, Univ Toledo, April 2015 7

Order-of-magnitude approximations • Phenomena are versatile and parameters unknown

approximations necessary, exact models excessive

• Neglect all numerical multipliers – Example: area of a circle ~ d2 instead of p(d/2)2

• Including even integrals and derivatives – Example:

• It works! Errors mostly cancel each other to the accuracy of insignificant multipliers similar to cancelations of random displacements in diffusion

• = ‘Errors propagate by diffusion’ (attributed to Einstein)

• It is efficient, fast, inexpensive... and almost forgotten

21

2 2

0

2 2

1 1 1 1 instead of

2 4 3

( ) instead of 2

x dx x x

d x x x x

dx x

Victor Karpov, Univ Toledo, April 2015 8

Basics of whisker bending

d a

R u(x,t)

x l d

f Torque sAd by stresses ~ Torque Fl by force F

• Major equation: deflection vs. force/geometry • Bending is easier than compressing • Very sensitive to whisker geometry

F

K is compressive spring constant,

Kb =K(d/l) 2

is bending spring constant

Victor Karpov, Univ Toledo, April 2015 9

Useful consequences

2 2

2

1

b b s

b

b b

K d Y d v

m l l

F m

r

Transversal vibrational frequency depends on whisker geometry and speed of sound in its material

Time of whisker bending is reciprocal of the above frequency

Typical times on the order of 1-100 ms consistent with observations

Frequency of harmonic oscillator

Newton’s second law

Victor Karpov, Univ Toledo, April 2015 10

Some other applications

d

d

x

Whisker with constriction (not sure how realistic)

Fraction of constricted regions

Significant lowering of vibrational frequency

Buckling instability

Critical compressive force

or critical whisker length

above which buckling begins Victor Karpov, Univ Toledo, April 2015 11

Viscous drag on a whisker Area A, Velocity v Force F

h Viscosity h

Av F

h h

h d

A ld

The only length across which the velocity field can change

Side surface of a cylinder

Recalling the concept of viscosity

aF v lh

Bottom at rest

Victor Karpov, Univ Toledo, April 2015 12

6 mechanisms of whisker movements

Relevant material parameters Sn (Zn)

Victor Karpov, Univ Toledo, April 2015 13

Structure of results

Quantity = estimated parameter that can be compared to observations Neglecting continuous (~lognormal distributions) of whisker lengths and diameters, four reference points are taken: Thin whiskers d=1 mm Thick whiskers d=10 mm Short whiskers l=1 mm Long whiskers l=10 mm

Victor Karpov, Univ Toledo, April 2015 14

1. Air flow

Victor Karpov, Univ Toledo, April 2015 15

Yes, air flow can be a cause Air velocity that produces deflections comparable to whisker lengths. Even thick whiskers can be moved under strong enough wind

Thin whiskers Thick whiskers

3 4

,

/

~ - citerion

aF v l

Fl Yd

l

h

Victor Karpov, Univ Toledo, April 2015 16

Brownian movements of ‘free’ particles ~ TT

mv v d

h

h

- coherency timeh

Typical Brownian displacements are small: ~ 10 mm per 1 s (It took microscope to discover Brownian movements).

Victor Karpov, Univ Toledo, April 2015 17

Modification: Brownian movements of a harmonic oscillator

The spring limits Brownian displacements to that of equipartition theorem

2

max

2 2

BKL k T

The ‘free’ particle Brownian model works for L

Brownian movements of elastic beams

l f

2

2 2

b BK k T

Limiting deflection is determined by bending elasticity Kb and thermal energy kBT

This theory is consistent with the known results for bending fluctuations of long molecules

Victor Karpov, Univ Toledo, April 2015 19

Brownian movements of metal whiskers is unlikely

Predicted angles are too small... maybe OK for severely constricted whiskers Even though predicted time scales can be realistic

Thin whiskers Thick whiskers

Victor Karpov, Univ Toledo, April 2015 20

Brownian movements vs. viscous drag: what is so different?

d

l0

l0 vT

vT vT

vT

v a (l

0 /d

)

v a (l

0 /d

)

Air flow velocity at distances of the mean free path Thermal velocities

Vertical air flow has a certain direction: momenta from both sides add

Brownian random momenta don’t have preferential direction and mutually cancel

0

2

0~ 1 (!),

atomic concentration

a T

l Nv N v

d

N d l n

n

Victor Karpov, Univ Toledo, April 2015 21

External vibrations

F F

w

3

4

Inertia force

deflection

truck on a rural road

standard lab 0.01

F mw

Fl

Yd

w g

w g

Victor Karpov, Univ Toledo, April 2015 22

External vibrations unlikely to move metal whiskers

Thin whiskers Thick whiskers

Displacements small, except maybe severely constricted whiskers

Victor Karpov, Univ Toledo, April 2015 23

Garden hose instability

l

d

F

Whisker growth velocity vw~ 1A/s is so much smaller than the sp

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