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small tweaks

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Axel Kohlmeyer 2020-02-25 00:42:57 -05:00
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21
doc/src/pair_colloid.rst
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@ -53,8 +53,7 @@ The colloid-colloid interaction energy is given by
\right] \\
& \\
U_R = & \frac{A_{cc}}{37800} \frac{\sigma^6}{r}
\left[ \frac{}{} \right. \\
&\qquad \frac{r^2-7r\left(a_1+a_2\right)+6\left(a_1^2+7a_1a_2+a_2^2\right)}
\biggl[ \frac{r^2-7r\left(a_1+a_2\right)+6\left(a_1^2+7a_1a_2+a_2^2\right)}
{\left(r-a_1-a_2\right)^7} \\
&\qquad +\frac{r^2+7r\left(a_1+a_2\right)+6\left(a_1^2+7a_1a_2+a_2^2\right)}
{\left(r+a_1+a_2\right)^7} \\
@ -66,8 +65,8 @@ The colloid-colloid interaction energy is given by
& \\
U = & U_A + U_R, \qquad r < r_c
where :math:`A_{cc}` is the Hamaker constant, a\_1 and a\_2 are the
radii of the two colloidal particles, and Rc is the cutoff. This
where :math:`A_{cc}` is the Hamaker constant, :math:`a_1` and :math:`a_2` are the
radii of the two colloidal particles, and :math:`r_c` is the cutoff. This
equation results from describing each colloidal particle as an
integrated collection of Lennard-Jones particles of size sigma and is
derived in :ref:`(Everaers) <Everaers1>`.
@ -76,12 +75,12 @@ The colloid-solvent interaction energy is given by
.. math::
U = & \frac{2 ~ a^3 ~ \sigma^3 ~ A_{cs}}{9 \left( a^2 - r^2 \right)^3}
U = \frac{2 ~ a^3 ~ \sigma^3 ~ A_{cs}}{9 \left( a^2 - r^2 \right)^3}
\left[ 1 - \frac{\left(5 ~ a^6+45~a^4~r^2+63~a^2~r^4+15~r^6\right) \sigma^6}
{15 \left(a-r\right)^6 \left( a+r \right)^6} \right], \quad r < r_c
where :math:A_{cs}` is the Hamaker constant, a is the radius of the colloidal
particle, and Rc is the cutoff. This formula is derived from the
where :math:A_{cs}` is the Hamaker constant, *a* is the radius of the colloidal
particle, and :math:`r_c` is the cutoff. This formula is derived from the
colloid-colloid interaction, letting one of the particle sizes go to
zero.
@ -90,16 +89,16 @@ Lennard-Jones formula
.. math::
U = & \frac{A_{ss}}{36} \left[ \left( \frac{\sigma}{r}
U = \frac{A_{ss}}{36} \left[ \left( \frac{\sigma}{r}
\right)^{12} - \left( \frac{ \sigma}{r} \right)^6 \right], \quad
r < r_c
with :math:`A_{ss}` set appropriately, which results from letting both
particle sizes go to zero.
When used in combination with :doc:`pair_style yukawa/colloid <pair_colloid>`, the two terms become the so-called
DLVO potential, which combines electrostatic repulsion and van der
Waals attraction.
When used in combination with :doc:`pair_style yukawa/colloid
<pair_colloid>`, the two terms become the so-called DLVO potential,
which combines electrostatic repulsion and van der Waals attraction.
The following coefficients must be defined for each pair of atoms
types via the :doc:`pair_coeff <pair_coeff>` command as in the examples