Download Supplementary Material for manuscript “Viscoelasticity of stepped

Supplementary Material for manuscript
“Viscoelasticity of stepped interfaces”
by S. A. Skirlo and M. J. Demkowicz
Derivation of shear resistance equation
𝑑𝜀
A derivation of the expression relating steady state shear resistance 𝜎 and strain rate 𝑑𝑡
given in the paper may be found in references 8 or 19. We also give a brief derivation here.
Consider a two-state model describing atoms in the interface under the influence of stress. At
zero stress, the activation energy for transitions between these states is 𝐸𝑎 . Additionally, the
energy of each well is stress-dependent and equals −𝜎𝑉 and +𝜎𝑉. Thus, under non-zero stress,
the activation energy from transitions from the former state to the latter is 𝐸𝑎 + 𝜎𝑉 while the
reverse transition has activation energy 𝐸𝑎 − 𝜎𝑉.
Thus, at non-zero temperature, the probability of an atom transitioning between states
is proportional to 𝑒 −𝐸𝑎±𝜎𝑉 /𝑘𝑏𝑇 . To get the transition rate, we multiply this probability by an
attempt frequency 𝜈. From this we, find that the net transition rate of atoms from one state to
the other is given by 𝜈 (𝑒
−𝐸𝑎 +𝜎𝑉
𝑘𝑏 𝑇
−𝑒
−𝐸𝑎 −𝜎𝑉
𝑘𝑏 𝑇
) = 𝜈𝑒 −𝐸𝑎 /𝑘𝑏𝑇 sinh[𝜎𝑉 /𝑘𝑏 𝑇].
In steady state, the continuous net transition rate from one state to the other under a
constant applied stress 𝜎 causes a continuous accumulation of strain at a constant strain rate.
We associate each transition with an average atom displacement increment ∆𝑑. The strain rate
𝑑𝜀
𝑑𝑡
may then be expressed as
𝑑𝜀
𝑑𝑡
where 𝜀̇0 =
𝜈∆𝑑
𝜆
= 𝜀̇0 𝑒 −𝐸𝑎 /𝑘𝑏𝑇 sinh[𝜎𝑉 /𝑘𝑏 𝑇],
.
𝑑𝜀
This expression describes the average relationship between 𝜎 and 𝑑𝑡 and does not
account for fluctuations arising from individual transitions.
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