5.3 Viscosity: Hamiltonian methods
(Of dolls and practical jokes)
Subsections
Further Sections
Hoover-Cupido
[Phys.Rev.A 22(1980)1690]
Reverting to the $q,p$ notation we introduce an appropriate perturbative term in the Hamiltonian:
$
\Delta H = \sum_{i}
\left( \vec{q}_{i} \vec{p}_{i}\right):\left(
\vec{\nabla}\vec{u}\right)^{T}
$
where $\vec{u}(\vec{q})$ is the hydrodynamic velocity field, and
$
A: B^{T} \equiv
A_{\alpha \beta} \left( B^{T}\right)_{\beta \alpha}
$
(inner product.)
For reasons of his own, W. Hoover dubbed the construct
$(qp)$ Doll's tensor.
Written in dummy index notation we have (omitting the particle index $i$)
$
\Delta H = q_{\alpha} p_{\beta} \partial_{\alpha} u_{\beta}
$
and the resulting equations of motion are
$ \begin{eqnarray}
\dot{q}_{\gamma} &=& \frac{1}{m}p_{\gamma}
+q_{\alpha}\partial_{\alpha} u_{\gamma}\\
\dot{p}_{\gamma} &=& K_{\gamma}
-p_{\beta}\partial_{\gamma} u_{\beta}
\end{eqnarray} $
To avoid gradual heating of the sample, a Gaussian (or other) thermostat
may be added.
For the simple homogeneous shear with $\gamma \equiv \partial_{z} u_{x}$
as the only non-vanishing element of $\nabla \vec{u}$, the
above equations read
$ \begin{eqnarray}
\dot{x}&=&\frac{1}{m}p_{x}+\gamma z
\\
\dot{y}&=&\frac{1}{m}p_{y}
\\
\dot{z}&=&\frac{1}{m}p_{z}
\\
\dot{p}_{x}&=&K_{x}
\\
\dot{p}_{y}&=&K_{y}
\\
\dot{p}_{z}&=&K_{z}- \gamma p_{x}
\end{eqnarray} $
Shearing boundary conditions as introduced by Lees and Edwards
may be applied. But now the flow is driven by the additional terms
in the equations of motion.
Again, the stress tensor element $\sigma_{xz}$ is evaluated, and from
its average the viscosity is determined as
$\eta=-<\sigma_{xz}>/\gamma$.
The method may used with an oscillatory applied shear,
yielding the frequency-dependent viscosity. Also, a volume dilation may
be applied to determine the bulk visosity.
Evans-Morriss
[Computer Phys. Rep. 1/6(1984) 297]
These authors remarked that the in the stationary laminar shear case
the Hoover-(qp) algorithm does not produce the correct flow pattern.
As evident from equ. ... the shear force acts on the
"wrong" component of the momentum - $p_{z}$ instead of $p_{x}$.
This does not destroy the relation between $\sigma_{xz}$ and $\eta$,
but it results in an unphysical flow geometry.
Evans and Morris suggested a simple solution. Instead of
...-... they wrote
$ \begin{eqnarray}
\dot{q}_{\gamma} &=& \frac{1}{m}p_{\gamma}
+q_{\alpha}\partial_{\alpha} u_{\gamma}\\
\dot{p}_{\gamma} &=& K_{\gamma}
-p_{\alpha}\partial_{\alpha} u_{\gamma}
\end{eqnarray} $
For simple Couette flow we now have
$ \begin{eqnarray}
\dot{x}&=&\frac{1}{m}p_{x}+z\gamma
\\
\dot{y}&=&\frac{1}{m}p_{y}
\\
\dot{z}&=&\frac{1}{m}p_{z}
\\
\dot{p}_{x}&=&K_{x}- p_{z} \gamma \;\;\left[ -\frac{\lambda}{m}p_{x}
\right]\\
\dot{p}_{y}&=&K_{y}\;\;\;\;\; \left[-\frac{\lambda}{m}p_{y} \right]
\\
\dot{p}_{z}&=&K_{z}\;\;\;\;\;\left[ -\frac{\lambda}{m}p_{z}\right]
\end{eqnarray} $
where in $\left[ \dots \right]$
we have added optional Gaussian constraint forces to keep
$T=const$. It is clear that equ. ...
has now the right form to produce Couette flow. In addition, these authors
demonstrated that their method produces the correct viscosity up to
much higher (non-linear regime) shear values than the Doll's tensor
method.
Evans' and Morriss' equations of motion cannot be derived from a
perturbed Hamiltonian. However, they fulfill the necessary linear
response condition ....
The above equations of motion are known as S'LLOD equations,
as they make use of the tensor $(\vec{p}\vec{q})$, the transpose of
the DOLL'S tensor $(\vec{q}\vec{p})$.
5.4 Shear Thinning
Early NEMD simulations yielded an unexpected result: at high shear rates
($\gamma^{*} \approx 1$) the viscosity decreased strongly,
to increase again at even higher
$\gamma$. (See, e.g. [Heyes et al., Mol. Phys. 57(1986)1265].)
This "Non-Newtonian" behaviour may indeed be observed
experimentally. While a value of $\gamma^{*}=1$
is unrealistically high for gases, it
is in an accessible range for liquids composed of large molecules, i.e.
polymer melts:
$
\gamma^{*}= 1 \;\;\; \rightarrow \;\;\;
\gamma \equiv \frac{\textstyle \Delta u_{x}}{\textstyle \Delta z}
= \frac{\textstyle 1}{\textstyle t_{0}}
= \sqrt{\frac{\textstyle \epsilon}{\textstyle m \sigma^{2}}}
$
Argon:
$\gamma =1/t_{0}=0.3 \cdot 10^{12}s^{-1}$; with $\Delta z=10^{-4}m$
this would mean $\Delta u_{x}=0.3 \cdot 10^{8} ms^{-1}$
Colloid: Let $\sigma=10^{-6} m$ and $m=10^{-15} kg$; then
$t_{0}=1 s$ and with $\Delta z=10^{-3} m$ we have
$\Delta u_{x}=10^{-3} ms^{-1}$.
Explanation: When $u_{x}$
becomes comparable to the thermal speed,
the molecules spontaneously arrange in flow lines that glide along each
other without interdiffusion. A plane perpendicular to the lines will
be crossed at points that form a hexagonal lattice.
At even higher shear the formation of vortices destroys the flow line
pattern, and the viscosity increaes again.
Heyes' results were later challenged and suspected of being computational
artefacts after all, the argument being that the specific method of
thermostating the sample was the source of trouble. As with Spring 2002
the issue seems unsettled.
[Lit.??]
5.5 Diffusion by NEMD
To determine the diffusion constant $D$
Ciccotti et al. [J. Stat. Phys. 21(1979)1] suggest the following Hamiltonian
non-equilibrium method:
Let
$
H = \sum_{i}\frac{\textstyle p_{i}^{2}}{\textstyle 2m}
+\frac{1}{2}\sum_{i}\sum_{j}u_{ij}-\sum_{i}c_{i}x_{i}F
$
where $c_{i}=\pm 1$ for even/odd numbered particles is a
"color charge" coupling to the applied "color field"
$F$.
In other words, particles are drawn in opposite directions depending
on their number; otherwise, they are completely identical.
Keeping $F$ constant we have
$
A(t)= \sum_{i}c_{i}x_{i}(t); \;\;\;\;\;\;\;
\dot{A}(t)=V j_{c}(t) = \sum_{i}c_{i} \dot{x}_{i}
$
where $j_{c}$ is the "color current".
Again, the external field does work on the system, heating it:
$
\frac{\textstyle dH}{\textstyle dt}=-F V j_{c}
$
The heating may once more be avoided by Gaussian terms in the eqs. of
motion which now read, in 2 dimensions,
$ \begin{eqnarray}
\dot{x}_{i}&=&\frac{1}{m} p_{ix}
\\
\dot{y}_{i}&=&\frac{1}{m} p_{iy}
\\
\dot{p}_{ix}&=&K_{ix}+c_{i} F
\\
\dot{p}_{iy}&=&K_{iy} \;\;\;\;\;\;\;\;\;\;[- \frac{\lambda}{m}p_{iy}]
\end{eqnarray} $
In this way a "color conductivity"
$\sigma_{c} \equiv \lim_{F \rightarrow 0}
\frac{\textstyle \langle j_{c} \rangle }{\textstyle F}$
may be evaluated. Applying Linear Response Theory one may show that
$\sigma_{c}$ is closely related to the diffusion constant:
$
D=\frac{\textstyle (N-1)VkT}{\textstyle N^{2}} \sigma_{c}
$
vesely may-06