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remove \begin{equation} \end{equation} which are not needed and break epub

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Axel Kohlmeyer 2020-02-21 13:21:59 -05:00
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commit 7186b4795b
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7
doc/src/Howto_drude2.rst
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@ -9,10 +9,8 @@ USER-DRUDE package activated. Then, the data file and input scripts
have to be modified to include the Drude dipoles and how to handle
them.
----------
**Overview of Drude induced dipoles**
Polarizable atoms acquire an induced electric dipole moment under the
@ -35,7 +33,7 @@ polarizability :math:`\alpha` by
.. math::
\begin{equation} K_D = \frac 1 2\, \frac {q_D^2} \alpha\end{equation}
K_D = \frac 1 2\, \frac {q_D^2} \alpha
Ideally, the mass of the Drude particle should be small, and the
stiffness of the harmonic bond should be large, so that the Drude
@ -75,11 +73,8 @@ important features:
#. The possibility to thermostat the additional degrees of freedom associated with the induced dipoles at very low temperature, in terms of the reduced coordinates of the Drude particles with respect to their cores. This makes the trajectory close to that of relaxed induced dipoles.
#. The Drude dipoles on covalently bonded atoms interact too strongly due to the short distances, so an atom may capture the Drude particle (shell) of a neighbor, or the induced dipoles within the same molecule may align too much. To avoid this, damping at short of the interactions between the point charges composing the induced dipole can be done by :ref:`Thole <Thole2>` functions.
----------
**Preparation of the data file**
The data file is similar to a standard LAMMPS data file for

8
doc/src/compute_hma.rst
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@ -80,7 +80,7 @@ A detailed description of this method can be found in (:ref:`Moustafa <hma-Moust
.. math::
\begin{equation}\left< U\right>_{HMA} = \frac{d}{2} (N-1) k_B T + \left< U + \frac{1}{2} F\bullet\Delta r \right>\end{equation}
\left< U\right>_{HMA} = \frac{d}{2} (N-1) k_B T + \left< U + \frac{1}{2} F\bullet\Delta r \right>
where :math:`N` is the number of atoms in the system, :math:`k_B` is Boltzmann's
constant, :math:`T` is the temperature, :math:`d` is the
@ -93,7 +93,7 @@ The pressure is computed by the formula:
.. math::
\begin{equation}\left< P\right>_{HMA} = \Delta \hat P + \left< P_{vir} + \frac{\beta \Delta \hat P - \rho}{d(N-1)} F\bullet\Delta r \right>\end{equation}
\left< P\right>_{HMA} = \Delta \hat P + \left< P_{vir} + \frac{\beta \Delta \hat P - \rho}{d(N-1)} F\bullet\Delta r \right>
where :math:`\rho` is the number density of the system, :math:`\Delta \hat P` is the
difference between the harmonic and lattice pressure, :math:`P_{vir}` is
@ -108,9 +108,9 @@ pressure and harmonic pressure.
.. math::
\begin{equation}\left<C_V \right>_{HMA} = \frac{d}{2} (N-1) k_B + \frac{1}{k_B T^2} \left( \left<
\left<C_V \right>_{HMA} = \frac{d}{2} (N-1) k_B + \frac{1}{k_B T^2} \left( \left<
U_{HMA}^2 \right> - \left<U_{HMA}\right>^2 \right) + \frac{1}{4 T}
\left< F\bullet\Delta r + \Delta r \bullet \Phi \bullet \Delta r \right>\end{equation}
\left< F\bullet\Delta r + \Delta r \bullet \Phi \bullet \Delta r \right>
where :math:`\Phi` is the Hessian matrix. The compute hma command
computes the full expression for :math:`C_V` except for the

22
doc/src/fix_drude_transform.rst
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@ -54,58 +54,58 @@ Masses:
.. math::
\begin{equation} M' = M + m \end{equation}
M' = M + m
.. math::
\begin{equation} m' = \frac {M\, m } {M'} \end{equation}
m' = \frac {M\, m } {M'}
Positions:
.. math::
\begin{equation} X' = \frac {M\, X + m\, x} {M'}\end{equation}
X' = \frac {M\, X + m\, x} {M'}
.. math::
\begin{equation} x' = x - X \end{equation}
x' = x - X
Velocities:
.. math::
\begin{equation} V' = \frac {M\, V + m\, v} {M'}\end{equation}
V' = \frac {M\, V + m\, v} {M'}
.. math::
\begin{equation} v' = v - V \end{equation}
v' = v - V
Forces:
.. math::
\begin{equation} F' = F + f \end{equation}
F' = F + f
.. math::
\begin{equation} f' = \frac { M\, f - m\, F} {M'}\end{equation}
f' = \frac { M\, f - m\, F} {M'}
This transform conserves the total kinetic energy
.. math::
\begin{equation} \frac 1 2 \, (M\, V^2\ + m\, v^2)
= \frac 1 2 \, (M'\, V'^2\ + m'\, v'^2) \end{equation}
\frac 1 2 \, (M\, V^2\ + m\, v^2)
= \frac 1 2 \, (M'\, V'^2\ + m'\, v'^2)
and the virial defined with absolute positions
.. math::
\begin{equation} X\, F + x\, f = X'\, F' + x'\, f' \end{equation}
X\, F + x\, f = X'\, F' + x'\, f'
----------

2
doc/src/fix_electron_stopping.rst
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@ -55,7 +55,7 @@ to each atom as:
.. math::
\begin{equation}\vec{F}_i = \vec{F}^0_i - \frac{\vec{v}_i}{\|\vec{v}_i\|} \cdot S_e\end{equation}
\vec{F}_i = \vec{F}^0_i - \frac{\vec{v}_i}{\|\vec{v}_i\|} \cdot S_e
where :math:`\vec{F}_i` is the resulting total force on the atom.
:math:`\vec{F}^0_i` is the original force applied to the atom, :math:`\vec{v}_i` is

4
doc/src/fix_ffl.rst
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@ -49,12 +49,12 @@ by the equations
.. math::
\begin{equation} \frac {dq}{dt} = \frac{p}{m}, \end{equation}
\frac {dq}{dt} = \frac{p}{m},
.. math::
\begin{equation} \frac {dp}{dt} = -\gamma p + W + F, \end{equation}
\frac {dp}{dt} = -\gamma p + W + F,
where :math:`F` is the physical force, :math:`\gamma` is the friction coefficient, and :math:`W` is a
Gaussian random force.

16
doc/src/fix_langevin_drude.rst
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@ -60,34 +60,34 @@ Velocities:
.. math::
\begin{equation} V' = \frac {M\, V + m\, v} {M'} \end{equation}
V' = \frac {M\, V + m\, v} {M'}
.. math::
\begin{equation} v' = v - V \end{equation}
v' = v - V
Masses:
.. math::
\begin{equation} M' = M + m \end{equation}
M' = M + m
.. math::
\begin{equation} m' = \frac {M\, m } {M'} \end{equation}
m' = \frac {M\, m } {M'}
The Langevin forces are computed as
.. math::
\begin{equation} F' = - \frac {M'} {\mathtt{damp\_com}}\, V' + F_r' \end{equation}
F' = - \frac {M'} {\mathtt{damp\_com}}\, V' + F_r'
.. math::
\begin{equation} f' = - \frac {m'} {\mathtt{damp\_drude}}\, v' + f_r' \end{equation}
f' = - \frac {m'} {\mathtt{damp\_drude}}\, v' + f_r'
:math:`F_r'` is a random force proportional to
:math:`\sqrt { \frac {2\, k_B \mathtt{Tcom}\, m'} {\mathrm dt\, \mathtt{damp\_com} } }`.
@ -98,12 +98,12 @@ transform:
.. math::
\begin{equation} F = \frac M {M'}\, F' - f' \end{equation}
F = \frac M {M'}\, F' - f'
.. math::
\begin{equation} f = \frac m {M'}\, F' + f' \end{equation}
f = \frac m {M'}\, F' + f'
This fix also thermostats non-polarizable atoms in the group at
temperature *Tcom*\ , as if they had a massless Drude partner. The

14
doc/src/fix_phonon.rst
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@ -71,21 +71,21 @@ coefficients of the system in reciprocal space are given by
.. math::
\begin{equation}\mathbf{\Phi}_{k\alpha,k^\prime \beta}(\mathbf{q}) = k_B T \mathbf{G}^{-1}_{k\alpha,k^\prime \beta}(\mathbf{q})\end{equation}
\mathbf{\Phi}_{k\alpha,k^\prime \beta}(\mathbf{q}) = k_B T \mathbf{G}^{-1}_{k\alpha,k^\prime \beta}(\mathbf{q})
where :math:`\mathbf{G}` is the Green's functions coefficients given by
.. math::
\begin{equation}\mathbf{G}_{k\alpha,k^\prime \beta}(\mathbf{q}) = \left< \mathbf{u}_{k\alpha}(\mathbf{q}) \bullet \mathbf{u}_{k^\prime \beta}^*(\mathbf{q}) \right>\end{equation}
\mathbf{G}_{k\alpha,k^\prime \beta}(\mathbf{q}) = \left< \mathbf{u}_{k\alpha}(\mathbf{q}) \bullet \mathbf{u}_{k^\prime \beta}^*(\mathbf{q}) \right>
where :math:`\left< \ldots \right>` denotes the ensemble average, and
.. math::
\begin{equation}\mathbf{u}_{k\alpha}(\mathbf{q}) = \sum_l \mathbf{u}_{l k \alpha} \exp{(i\mathbf{qr}_l)}\end{equation}
\mathbf{u}_{k\alpha}(\mathbf{q}) = \sum_l \mathbf{u}_{l k \alpha} \exp{(i\mathbf{qr}_l)}
is the :math:`\alpha` component of the atomic displacement for the :math:`k`
th atom in the unit cell in reciprocal space at :math:`\mathbf{q}`. In
@ -95,9 +95,9 @@ according to the following formula,
.. math::
\begin{equation}\mathbf{G}_{k\alpha,k^\prime \beta}(\mathbf{q}) =
\mathbf{G}_{k\alpha,k^\prime \beta}(\mathbf{q}) =
\left< \mathbf{R}_{k \alpha}(\mathbf{q}) \bullet \mathbf{R}^*_{k^\prime \beta}(\mathbf{q}) \right>
- \left<\mathbf{R}\right>_{k \alpha}(\mathbf{q}) \bullet \left<\mathbf{R}\right>^*_{k^\prime \beta}(\mathbf{q})\end{equation}
- \left<\mathbf{R}\right>_{k \alpha}(\mathbf{q}) \bullet \left<\mathbf{R}\right>^*_{k^\prime \beta}(\mathbf{q})
where :math:`\mathbf{R}` is the instantaneous positions of atoms, and
:math:`\left<\mathbf{R}\right>` is the averaged atomic positions. It
@ -110,8 +110,8 @@ Once the force constant matrix is known, the dynamical matrix
.. math::
\begin{equation}\mathbf{D}_{k\alpha, k^\prime\beta}(\mathbf{q}) =
(m_k m_{k^\prime})^{-\frac{1}{2}} \mathbf{\Phi}_{k \alpha, k^\prime \beta}(\mathbf{q})\end{equation}
\mathbf{D}_{k\alpha, k^\prime\beta}(\mathbf{q}) =
(m_k m_{k^\prime})^{-\frac{1}{2}} \mathbf{\Phi}_{k \alpha, k^\prime \beta}(\mathbf{q})
whose eigenvalues are exactly the phonon frequencies at :math:`\mathbf{q}`.

88
doc/src/pair_granular.rst
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@ -98,7 +98,7 @@ on particle *i* due to contact with particle *j* is given by:
.. math::
\begin{equation}\mathbf{F}_{ne, Hooke} = k_N \delta_{ij} \mathbf{n}\end{equation}
\mathbf{F}_{ne, Hooke} = k_N \delta_{ij} \mathbf{n}
Where :math:`\delta_{ij} = R_i + R_j - \|\mathbf{r}_{ij}\|` is the particle
overlap, :math:`R_i, R_j` are the particle radii, :math:`\mathbf{r}_{ij} = \mathbf{r}_i - \mathbf{r}_j` is the vector separating the two
@ -112,7 +112,7 @@ For the *hertz* model, the normal component of force is given by:
.. math::
\begin{equation}\mathbf{F}_{ne, Hertz} = k_N R_{eff}^{1/2}\delta_{ij}^{3/2} \mathbf{n}\end{equation}
\mathbf{F}_{ne, Hertz} = k_N R_{eff}^{1/2}\delta_{ij}^{3/2} \mathbf{n}
Here, :math:`R_{eff} = \frac{R_i R_j}{R_i + R_j}` is the effective
radius, denoted for simplicity as *R* from here on. For *hertz*\ , the
@ -124,7 +124,7 @@ For the *hertz/material* model, the force is given by:
.. math::
\begin{equation}\mathbf{F}_{ne, Hertz/material} = \frac{4}{3} E_{eff} R_{eff}^{1/2}\delta_{ij}^{3/2} \mathbf{n}\end{equation}
\mathbf{F}_{ne, Hertz/material} = \frac{4}{3} E_{eff} R_{eff}^{1/2}\delta_{ij}^{3/2} \mathbf{n}
Here, :math:`E_{eff} = E = \left(\frac{1-\nu_i^2}{E_i} + \frac{1-\nu_j^2}{E_j}\right)^{-1}` is the effective Young's
modulus, with :math:`\nu_i, \nu_j` the Poisson ratios of the particles of
@ -139,7 +139,7 @@ is simply Hertz with an additional attractive cohesion term:
.. math::
\begin{equation}\mathbf{F}_{ne, dmt} = \left(\frac{4}{3} E R^{1/2}\delta_{ij}^{3/2} - 4\pi\gamma R\right)\mathbf{n}\end{equation}
\mathbf{F}_{ne, dmt} = \left(\frac{4}{3} E R^{1/2}\delta_{ij}^{3/2} - 4\pi\gamma R\right)\mathbf{n}
The *jkr* model is the :ref:`(Johnson-Kendall-Roberts) <JKR1971>` model,
where the force is computed as:
@ -147,8 +147,7 @@ where the force is computed as:
.. math::
\begin{equation}\label{eq:force_jkr}
\mathbf{F}_{ne, jkr} = \left(\frac{4Ea^3}{3R} - 2\pi a^2\sqrt{\frac{4\gamma E}{\pi a}}\right)\mathbf{n}\end{equation}
\mathbf{F}_{ne, jkr} = \left(\frac{4Ea^3}{3R} - 2\pi a^2\sqrt{\frac{4\gamma E}{\pi a}}\right)\mathbf{n}
Here, *a* is the radius of the contact zone, related to the overlap
:math:`\delta` according to:
@ -156,7 +155,7 @@ Here, *a* is the radius of the contact zone, related to the overlap
.. math::
\begin{equation}\delta = a^2/R - 2\sqrt{\pi \gamma a/E}\end{equation}
\delta = a^2/R - 2\sqrt{\pi \gamma a/E}
LAMMPS internally inverts the equation above to solve for *a* in terms
of :math:`\delta`, then solves for the force in the previous
@ -179,7 +178,7 @@ following general form:
.. math::
\begin{equation}\mathbf{F}_{n,damp} = -\eta_n \mathbf{v}_{n,rel}\end{equation}
\mathbf{F}_{n,damp} = -\eta_n \mathbf{v}_{n,rel}
Here, :math:`\mathbf{v}_{n,rel} = (\mathbf{v}_j - \mathbf{v}_i) \cdot \mathbf{n} \mathbf{n}` is the component of relative velocity along
:math:`\mathbf{n}`.
@ -208,7 +207,7 @@ user-specified damping coefficient in the *normal* model:
.. math::
\begin{equation}\eta_n = \eta_{n0}\end{equation}
\eta_n = \eta_{n0}
Here, :math:`\eta_{n0}` is the damping coefficient specified for the normal
contact model, in units of *mass*\ /\ *time*\ .
@ -218,7 +217,7 @@ For *damping mass\_velocity*, the normal damping is given by:
.. math::
\begin{equation}\eta_n = \eta_{n0} m_{eff}\end{equation}
\eta_n = \eta_{n0} m_{eff}
Here, :math:`\eta_{n0}` is the damping coefficient specified for the normal
contact model, in units of *mass*\ /\ *time* and
@ -233,7 +232,7 @@ damping is given by:
.. math::
\begin{equation}\eta_n = \eta_{n0}\ a m_{eff}\end{equation}
\eta_n = \eta_{n0}\ a m_{eff}
Here, *a* is the contact radius, given by :math:`a =\sqrt{R\delta}`
for all models except *jkr*\ , for which it is given implicitly according
@ -247,7 +246,7 @@ the normal model is interpreted as a restitution coefficient
.. math::
\begin{equation}\eta_n = \alpha (m_{eff}k_n)^{1/2}\end{equation}
\eta_n = \alpha (m_{eff}k_n)^{1/2}
For normal contact models based on material parameters, :math:`k_n = 4/3Ea`. The parameter :math:`\alpha` is related to the restitution
coefficient *e* according to:
@ -255,7 +254,7 @@ coefficient *e* according to:
.. math::
\begin{equation}\alpha = 1.2728-4.2783e+11.087e^2-22.348e^3+27.467e^4-18.022e^5+4.8218e^6\end{equation}
\alpha = 1.2728-4.2783e+11.087e^2-22.348e^3+27.467e^4-18.022e^5+4.8218e^6
The dimensionless coefficient of restitution :math:`e` specified as part
of the normal contact model parameters should be between 0 and 1, but
@ -267,7 +266,7 @@ damping components:
.. math::
\begin{equation}\mathbf{F}_n = \mathbf{F}_{ne} + \mathbf{F}_{n,damp}\end{equation}
\mathbf{F}_n = \mathbf{F}_{ne} + \mathbf{F}_{n,damp}
----------
@ -295,14 +294,14 @@ gran/hooke* style. The tangential force (\mathbf{F}\_t\) is given by:
.. math::
\begin{equation}\mathbf{F}_t = -min(\mu_t F_{n0}, \|\mathbf{F}_\mathrm{t,damp}\|) \mathbf{t}\end{equation}
\mathbf{F}_t = -min(\mu_t F_{n0}, \|\mathbf{F}_\mathrm{t,damp}\|) \mathbf{t}
The tangential damping force :math:`\mathbf{F}_\mathrm{t,damp}` is given by:
.. math::
\begin{equation}\mathbf{F}_\mathrm{t,damp} = -\eta_t \mathbf{v}_{t,rel}\end{equation}
\mathbf{F}_\mathrm{t,damp} = -\eta_t \mathbf{v}_{t,rel}
The tangential damping prefactor :math:`\eta_t` is calculated by scaling
the normal damping :math:`\eta_n` (see above):
@ -310,7 +309,7 @@ the normal damping :math:`\eta_n` (see above):
.. math::
\begin{equation}\eta_t = -x_{\gamma,t} \eta_n\end{equation}
\eta_t = -x_{\gamma,t} \eta_n
The normal damping prefactor :math:`\eta_n` is determined by the choice
of the *damping* keyword, as discussed above. Thus, the *damping*
@ -331,7 +330,7 @@ the normal force:
.. math::
\begin{equation}F_{n0} = \|\mathbf{F}_n\|\end{equation}
F_{n0} = \|\mathbf{F}_n\|
For cohesive models such as *jkr* and *dmt*\ , the critical force is
adjusted so that the critical tangential force approaches :math:`\mu_t F_{pulloff}`, see :ref:`Marshall <Marshall2009>`, equation 43, and
@ -341,7 +340,7 @@ form:
.. math::
\begin{equation}F_{n0} = \|\mathbf{F}_ne + 2 F_{pulloff}\|\end{equation}
F_{n0} = \|\mathbf{F}_ne + 2 F_{pulloff}\|
Where :math:`F_{pulloff} = 3\pi \gamma R` for *jkr*\ , and
:math:`F_{pulloff} = 4\pi \gamma R` for *dmt*\ .
@ -356,7 +355,7 @@ For *tangential linear\_history*, the tangential force is given by:
.. math::
\begin{equation}\mathbf{F}_t = -min(\mu_t F_{n0}, \|-k_t\mathbf{\xi} + \mathbf{F}_\mathrm{t,damp}\|) \mathbf{t}\end{equation}
\mathbf{F}_t = -min(\mu_t F_{n0}, \|-k_t\mathbf{\xi} + \mathbf{F}_\mathrm{t,damp}\|) \mathbf{t}
Here, :math:`\mathbf{\xi}` is the tangential displacement accumulated
during the entire duration of the contact:
@ -364,7 +363,7 @@ during the entire duration of the contact:
.. math::
\begin{equation}\mathbf{\xi} = \int_{t0}^t \mathbf{v}_{t,rel}(\tau) \mathrm{d}\tau\end{equation}
\mathbf{\xi} = \int_{t0}^t \mathbf{v}_{t,rel}(\tau) \mathrm{d}\tau
This accumulated tangential displacement must be adjusted to account
for changes in the frame of reference of the contacting pair of
@ -386,8 +385,7 @@ work:
.. math::
\begin{equation}\mathbf{\xi} = \left(\mathbf{\xi'} - (\mathbf{n} \cdot \mathbf{\xi'})\mathbf{n}\right) \frac{\|\mathbf{\xi'}\|}{\|\mathbf{\xi'}\| - \mathbf{n}\cdot\mathbf{\xi'}}
\label{eq:rotate_displacements}\end{equation}
\mathbf{\xi} = \left(\mathbf{\xi'} - (\mathbf{n} \cdot \mathbf{\xi'})\mathbf{n}\right) \frac{\|\mathbf{\xi'}\|}{\|\mathbf{\xi'}\| - \mathbf{n}\cdot\mathbf{\xi'}}
Here, :math:`\mathbf{\xi'}` is the accumulated displacement prior to the
current time step and :math:`\mathbf{\xi}` is the corrected
@ -404,7 +402,7 @@ discussion):
.. math::
\begin{equation}\mathbf{\xi} = -\frac{1}{k_t}\left(\mu_t F_{n0}\mathbf{t} + \mathbf{F}_{t,damp}\right)\end{equation}
\mathbf{\xi} = -\frac{1}{k_t}\left(\mu_t F_{n0}\mathbf{t} + \mathbf{F}_{t,damp}\right)
The tangential force is added to the total normal force (elastic plus
damping) to produce the total force on the particle. The tangential
@ -414,12 +412,12 @@ overlap region) to induce a torque on each particle according to:
.. math::
\begin{equation}\mathbf{\tau}_i = -(R_i - 0.5 \delta) \mathbf{n} \times \mathbf{F}_t\end{equation}
\mathbf{\tau}_i = -(R_i - 0.5 \delta) \mathbf{n} \times \mathbf{F}_t
.. math::
\begin{equation}\mathbf{\tau}_j = -(R_j - 0.5 \delta) \mathbf{n} \times \mathbf{F}_t\end{equation}
\mathbf{\tau}_j = -(R_j - 0.5 \delta) \mathbf{n} \times \mathbf{F}_t
For *tangential mindlin*\ , the :ref:`Mindlin <Mindlin1949>` no-slip solution is used, which differs from the *linear\_history*
option by an additional factor of *a*\ , the radius of the contact region. The tangential force is given by:
@ -427,7 +425,7 @@ option by an additional factor of *a*\ , the radius of the contact region. The t
.. math::
\begin{equation}\mathbf{F}_t = -min(\mu_t F_{n0}, \|-k_t a \mathbf{\xi} + \mathbf{F}_\mathrm{t,damp}\|) \mathbf{t}\end{equation}
\mathbf{F}_t = -min(\mu_t F_{n0}, \|-k_t a \mathbf{\xi} + \mathbf{F}_\mathrm{t,damp}\|) \mathbf{t}
Here, *a* is the radius of the contact region, given by :math:`a =\sqrt{R\delta}`
for all normal contact models, except for *jkr*\ , where it is given
@ -443,7 +441,7 @@ case, mixing of the shear modulus for different particle types *i* and
.. math::
\begin{equation}1/G = 2(2-\nu_i)(1+\nu_i)/E_i + 2(2-\nu_j)(1+\nu_j)/E_j\end{equation}
1/G = 2(2-\nu_i)(1+\nu_i)/E_i + 2(2-\nu_j)(1+\nu_j)/E_j
The *mindlin\_rescale* option uses the same form as *mindlin*\ , but the
magnitude of the tangential displacement is re-scaled as the contact
@ -452,7 +450,7 @@ unloads, i.e. if :math:`a < a_{t_{n-1}}`:
.. math::
\begin{equation}\mathbf{\xi} = \mathbf{\xi_{t_{n-1}}} \frac{a}{a_{t_{n-1}}}\end{equation}
\mathbf{\xi} = \mathbf{\xi_{t_{n-1}}} \frac{a}{a_{t_{n-1}}}
Here, :math:`t_{n-1}` indicates the value at the previous time
step. This rescaling accounts for the fact that a decrease in the
@ -485,7 +483,7 @@ the tangential force:
.. math::
\begin{equation}\mathbf{F}_{roll,0} = k_{roll} \mathbf{\xi}_{roll} - \gamma_{roll} \mathbf{v}_{roll}\end{equation}
\mathbf{F}_{roll,0} = k_{roll} \mathbf{\xi}_{roll} - \gamma_{roll} \mathbf{v}_{roll}
Here, :math:`\mathbf{v}_{roll} = -R(\mathbf{\Omega}_i - \mathbf{\Omega}_j) \times \mathbf{n}` is the relative rolling
velocity, as given in :ref:`Wang et al <Wang2015>` and
@ -494,7 +492,7 @@ velocity, as given in :ref:`Wang et al <Wang2015>` and
.. math::
\begin{equation}\mathbf{\xi}_{roll} = \int_{t_0}^t \mathbf{v}_{roll} (\tau) \mathrm{d} \tau\end{equation}
\mathbf{\xi}_{roll} = \int_{t_0}^t \mathbf{v}_{roll} (\tau) \mathrm{d} \tau
A Coulomb friction criterion truncates the rolling pseudo-force if it
exceeds a critical value:
@ -502,7 +500,7 @@ exceeds a critical value:
.. math::
\begin{equation}\mathbf{F}_{roll} = min(\mu_{roll} F_{n,0}, \|\mathbf{F}_{roll,0}\|)\mathbf{k}\end{equation}
\mathbf{F}_{roll} = min(\mu_{roll} F_{n,0}, \|\mathbf{F}_{roll,0}\|)\mathbf{k}
Here, :math:`\mathbf{k} = \mathbf{v}_{roll}/\|\mathbf{v}_{roll}\|` is the direction of
the pseudo-force. As with tangential displacement, the rolling
@ -519,12 +517,12 @@ opposite torque on each particle, according to:
.. math::
\begin{equation}\tau_{roll,i} = R_{eff} \mathbf{n} \times \mathbf{F}_{roll}\end{equation}
\tau_{roll,i} = R_{eff} \mathbf{n} \times \mathbf{F}_{roll}
.. math::
\begin{equation}\tau_{roll,j} = -\tau_{roll,i}\end{equation}
\tau_{roll,j} = -\tau_{roll,i}
----------
@ -551,7 +549,7 @@ the most straightforward treatment:
.. math::
\begin{equation}\tau_{twist,0} = -k_{twist}\xi_{twist} - \gamma_{twist}\Omega_{twist}\end{equation}
\tau_{twist,0} = -k_{twist}\xi_{twist} - \gamma_{twist}\Omega_{twist}
Here :math:`\xi_{twist} = \int_{t_0}^t \Omega_{twist} (\tau) \mathrm{d}\tau` is the twisting angular displacement, and
:math:`\Omega_{twist} = (\mathbf{\Omega}_i - \mathbf{\Omega}_j) \cdot \mathbf{n}` is the relative twisting angular velocity. The torque
@ -560,7 +558,7 @@ is then truncated according to:
.. math::
\begin{equation}\tau_{twist} = min(\mu_{twist} F_{n,0}, \tau_{twist,0})\end{equation}
\tau_{twist} = min(\mu_{twist} F_{n,0}, \tau_{twist,0})
Similar to the sliding and rolling displacement, the angular
displacement is rescaled so that it corresponds to the critical value
@ -569,7 +567,7 @@ if the twisting torque exceeds this critical value:
.. math::
\begin{equation}\xi_{twist} = \frac{1}{k_{twist}} (\mu_{twist} F_{n,0}sgn(\Omega_{twist}) - \gamma_{twist}\Omega_{twist})\end{equation}
\xi_{twist} = \frac{1}{k_{twist}} (\mu_{twist} F_{n,0}sgn(\Omega_{twist}) - \gamma_{twist}\Omega_{twist})
For *twisting sds*\ , the coefficients :math:`k_{twist}, \gamma_{twist}`
and :math:`\mu_{twist}` are simply the user input parameters that follow
@ -582,29 +580,29 @@ sliding friction coefficients, as discussed in
.. math::
\begin{equation}k_{twist} = 0.5k_ta^2\end{equation}
k_{twist} = 0.5k_ta^2
.. math::
\begin{equation}\eta_{twist} = 0.5\eta_ta^2\end{equation}
\eta_{twist} = 0.5\eta_ta^2
.. math::
\begin{equation}\mu_{twist} = \frac{2}{3}a\mu_t\end{equation}
\mu_{twist} = \frac{2}{3}a\mu_t
Finally, the twisting torque on each particle is given by:
.. math::
\begin{equation}\mathbf{\tau}_{twist,i} = \tau_{twist}\mathbf{n}\end{equation}
\mathbf{\tau}_{twist,i} = \tau_{twist}\mathbf{n}
.. math::
\begin{equation}\mathbf{\tau}_{twist,j} = -\mathbf{\tau}_{twist,i}\end{equation}
\mathbf{\tau}_{twist,j} = -\mathbf{\tau}_{twist,i}
----------
@ -690,7 +688,7 @@ models. In that case, the effective elastic modulus is computed as:
.. math::
\begin{equation}E_{eff,ij} = \left(\frac{1-\nu_i^2}{E_i} + \frac{1-\nu_j^2}{E_j}\right)^{-1}\end{equation}
E_{eff,ij} = \left(\frac{1-\nu_i^2}{E_i} + \frac{1-\nu_j^2}{E_j}\right)^{-1}
If the *i-j* coefficients :math:`E_{ij}` and :math:`\nu_{ij}` are
explicitly specified, the effective modulus is computed as:
@ -698,14 +696,14 @@ explicitly specified, the effective modulus is computed as:
.. math::
\begin{equation}E_{eff,ij} = \left(\frac{1-\nu_{ij}^2}{E_{ij}} + \frac{1-\nu_{ij}^2}{E_{ij}}\right)^{-1}\end{equation}
E_{eff,ij} = \left(\frac{1-\nu_{ij}^2}{E_{ij}} + \frac{1-\nu_{ij}^2}{E_{ij}}\right)^{-1}
or
.. math::
\begin{equation}E_{eff,ij} = \frac{E_{ij}}{2(1-\nu_{ij})}\end{equation}
E_{eff,ij} = \frac{E_{ij}}{2(1-\nu_{ij})}
These pair styles write their information to :doc:`binary restart files <restart>`, so a pair\_style command does not need to be
specified in an input script that reads a restart file.

12
doc/src/pair_thole.rst
View File

@ -73,9 +73,9 @@ short distances by a function
.. math::
\begin{equation} T_{ij}(r_{ij}) = 1 - \left( 1 +
T_{ij}(r_{ij}) = 1 - \left( 1 +
\frac{s_{ij} r_{ij} }{2} \right)
\exp \left( - s_{ij} r_{ij} \right) \end{equation}
\exp \left( - s_{ij} r_{ij} \right)
This function results from an adaptation to point charges
:ref:`(Noskov) <Noskov1>` of the dipole screening scheme originally proposed
@ -90,9 +90,9 @@ between the atom-specific values.
.. math::
\begin{equation} s_{ij} = \frac{ a_{ij} }{
s_{ij} = \frac{ a_{ij} }{
(\alpha_{ij})^{1/3} } = \frac{ (a_i + a_j)/2 }{
[(\alpha_i\alpha_j)^{1/2}]^{1/3} } \end{equation}
[(\alpha_i\alpha_j)^{1/2}]^{1/3} }
The damping function is only applied to the interactions between the
point charges representing the induced dipoles on polarizable sites,
@ -168,12 +168,12 @@ are defined using
.. math::
\begin{equation} \alpha_{ij} = \sqrt{\alpha_i\alpha_j}\end{equation}
\alpha_{ij} = \sqrt{\alpha_i\alpha_j}
.. math::
\begin{equation} a_{ij} = \frac 1 2 (a_i + a_j)\end{equation}
a_{ij} = \frac 1 2 (a_i + a_j)
Restrictions
""""""""""""