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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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