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• Some phenomena where short-circuits are important

Diffusion in noncrystalline materials

• Gasses and liquids

• Self-diffusion in metallic glasses

• Interstitials in metallic glasses

• Conformation of long-chain polymers

3.21 Spring 2001: Lecture 21

Diffusion in polymer melts

Motion of Dislocations (KPIM Chapter 11)

• Forces on dislocations

• Dislocation glide and climb

Dislocation motion

• Glide motion
  • plastic deformation of crystals at low temperature

  • movement of glissile interfaces at low temperature

  
  

• Climb motion
  • contributes to high temperature deformation

  • results from operation of sources and sinks of vacancies

  
  

Forces on Dislocations

• external stresses

• osmotic force $\vec{F} = \vec{F}_\sigma + \vec{F}_\mu + \vec{F}_\kappa$

• curvature force

• external stresses:
  • are given by the Peach-Koehler equation $\vec{F}_\sigma =(\vec{b} \cdot \sigma) \times \vec{\xi}$

• osmotic force:
  • arises if nonequilibrium concentrations of point defects are present:
    $\vec{F}_\mu = (\vec{\xi} \times \vec{b}) \frac{kT}{\Omega} \ln \frac{N_\nu}{N^{eq}}$

  
  

• curvature force:
  • line length can change if a curved dislocation moves $\vec{F}_\kappa \approx \frac{\mu b^2}{R} \hat{n} = \mu b^2 \frac{d \vec{\xi}}{ds}$

• Total force: $\vec{F} = \vec{F}_\sigma + \vec{F}_\mu + \vec{F}_\kappa$

Screw dislocation glide

• Dislocation velocity $v$ is limited by the sound velocity $c$ in the material--essentially a relativistic effect

• Drag forces arise from
  • Sound emission

  • Elastic dissipation

  • Phonon and electron scattering

  • Solute atom interactions

Figure 21-1:

Experimental studies of dislocation motion

• Extensive studies of LiF in 1960's by Gilman and Johnson

• Velocities increase rapidly with increasing stress and appear to level off as they approach sound velocity

Figure 21-2:

Dislocation climb

• Edge dislocations can absorb or emit vacancies by climbing

• Jogs are especially favorable sites for vacancy creation and destruction

  A

  vacancy diffuses from within bulk and annihilates at a jog

  B

  vacancy diffuses to dislocation core and attaches

  C

  vacancy diffuses along core

  D

  vacancy diffuses along core to jog and annihilates

Figure 21-3:

Vacancy condensation on screw dislocations

• Formation of helical edge segments

Figure 21-4: Left-hand screw dislocation in a primitive cubic crystal collects 1, 2, 3, and 4 vacancies.

Vacancy condensation kinetics on edge dislocations

• Supersaturations can be relieved by diffusion of vacancies to dislocation cores

  Figure 21-5:

  

• Model as diffusion within finite cylinder of radius $R$ where

  $$\begin{array}{lll} & R = \sqrt{\pi \rho} &\\ c_\nu = c_\nu^{... ...c{\partial c}{\partial r} = 0 & r = R & t \geq 0\\ \end{array}$$ (21-1)

  
  

• Approximate solution for fraction of excess vacancies remaining

  $$f(t) = \exp (-\alpha_1 D_\nu t)$$ (21-2)

  
  

Shrinkage of prismatic dislocation loops by vacancy condensation

• Driving force for loop shrinkage is $\Delta \mu$ between dislocation loop and free surface

  Figure 21-6:

  

• KPIM has a solution for the rate of decrease of loop radius adapted from electrostatics (Eq. 11.33)