From 3f5bb96aed761621a4d08928396cbd76445e613e Mon Sep 17 00:00:00 2001 From: Axel Kohlmeyer Date: Sat, 15 Feb 2020 16:47:32 -0500 Subject: [PATCH] pair ufm to pair zbl --- doc/src/Eqs/pair_ufm.jpg | Bin 17830 -> 0 bytes doc/src/Eqs/pair_ufm.tex | 14 ------- doc/src/Eqs/pair_vashishta.jpg | Bin 73306 -> 0 bytes doc/src/Eqs/pair_vashishta.tex | 22 ---------- doc/src/Eqs/pair_yukawa.jpg | Bin 2189 -> 0 bytes doc/src/Eqs/pair_yukawa.tex | 9 ---- doc/src/Eqs/pair_yukawa_colloid.jpg | Bin 3067 -> 0 bytes doc/src/Eqs/pair_yukawa_colloid.tex | 9 ---- doc/src/Eqs/pair_zbl.jpg | Bin 32209 -> 0 bytes doc/src/Eqs/pair_zbl.tex | 11 ----- doc/src/pair_ufm.rst | 24 ++++++++--- doc/src/pair_vashishta.rst | 62 ++++++++++++++++------------ doc/src/pair_yukawa.rst | 8 ++-- doc/src/pair_yukawa_colloid.rst | 26 ++++++------ doc/src/pair_zbl.rst | 35 +++++++++------- 15 files changed, 93 insertions(+), 127 deletions(-) delete mode 100644 doc/src/Eqs/pair_ufm.jpg delete mode 100644 doc/src/Eqs/pair_ufm.tex 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\frac{\left[ \cos \theta_{ijk} - \cos \theta_{0ijk} \right]^2} - {1+C_{ijk}\left[ \cos \theta_{ijk} - \cos \theta_{0ijk} \right]^2} \times \\ -& & \exp \left( \frac{\gamma_{ij}}{r_{ij} - r_{0,ij}} \right) - \exp \left( \frac{\gamma_{ik}}{r_{ik} - r_{0,ik}} \right), r_{ij} < r_{0,ij}, r_{ik} < r_{0,ik} -\end{eqnarray*} - -\end{document} diff --git a/doc/src/Eqs/pair_yukawa.jpg b/doc/src/Eqs/pair_yukawa.jpg deleted file mode 100644 index 103edc604321d02e39545dd4e55acac3884edc8c..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 2189 zcmbW2cTm%b632gmgd!q@8XzG^QIR4=BM>o0x+o0bg(i3=QZz)R9#S3ZF;RHm= zA;l+03mv2e6X1aeNCGiPRRrX~NDB(&>3cIb_vf4U*_rP@`<>mL{bN@+DVzmlFWEZS z0w53wu>T%_kOx=+5K%A~EDHH95C}v}{IG=hcVwibBo9Mn z9fheP5J)6aURhIHQ(YUbj#L+R15y%zFGvIgf&#*q02}}TBBDQf`~^f@6f6c3k@%iH zCkueUA|Mg4sF)~N>WAZp8w`Pp$sHA!H^wR`x=1K#;rXHr%4nX8XD9jssFnlP)cvBXm85 zzc`6xwA&nUna@bx^^P!c?#9;z<4;^%ElW^v!z%rW-applyC?)6`moSR3!PiRRL{Qh 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zj)}TPSM?FGC|i#@|7~133!G!@zL340Zk-2=!>9>6PO@`zdiA(rT#2r2lM_*CJ}%cW z3WW5hf`%`;gk@L)&a@3;mqt^rpZ&!TZX&f7eVaKGgwi=tTH2;75JpZQu53vXRZU6uP diff --git a/doc/src/Eqs/pair_zbl.tex b/doc/src/Eqs/pair_zbl.tex deleted file mode 100644 index 3d68dbed23..0000000000 --- a/doc/src/Eqs/pair_zbl.tex +++ /dev/null @@ -1,11 +0,0 @@ -\documentclass[12pt]{article} - -\begin{document} - -\begin{eqnarray*} - E^{ZBL}_{ij} & = & \frac{1}{4\pi\epsilon_0} \frac{Z_i Z_j \,e^2}{r_{ij}} \phi(r_{ij}/a)+ S(r_{ij})\\ - a & = & \frac{0.46850}{Z_{i}^{0.23} + Z_{j}^{0.23}}\\ - \phi(x) & = & 0.18175e^{-3.19980x} + 0.50986e^{-0.94229x} + 0.28022e^{-0.40290x} + 0.02817e^{-0.20162x}\\ - \end{eqnarray*} - -\end{document} diff --git a/doc/src/pair_ufm.rst b/doc/src/pair_ufm.rst index 3bb4e83a0e..c8a7e42743 100644 --- a/doc/src/pair_ufm.rst +++ b/doc/src/pair_ufm.rst @@ -42,10 +42,16 @@ Description Style *ufm* computes pairwise interactions using the Uhlenbeck-Ford model (UFM) potential :ref:`(Paula Leite2016) ` which is given by -.. image:: Eqs/pair_ufm.jpg - :align: center +.. math:: -where rc is the cutoff, sigma is a distance-scale and epsilon is an energy-scale, i.e., a product of Boltzmann constant kB, temperature T and the Uhlenbeck-Ford p-parameter which is responsible + E & = -\varepsilon\, \ln{\left[1-\exp{\left(-r^{2}/\sigma^{2}\right)}\right]} \qquad r < r_c \\ + \varepsilon & = p\,k_B\,T + + +where :math:`r_c` is the cutoff, :math:`\sigma` is a distance-scale and +:math:`\epsilon` is an energy-scale, i.e., a product of Boltzmann constant +:math:`k_B`, temperature *T* and the Uhlenbeck-Ford p-parameter which +is responsible to control the softness of the interactions :ref:`(Paula Leite2017) `. This model is useful as a reference system for fluid-phase free-energy calculations :ref:`(Paula Leite2016) `. @@ -55,8 +61,8 @@ or in the data file or restart files read by the :doc:`read_data ` or :doc:`read_restart ` commands, or by mixing as described below: -* epsilon (energy units) -* sigma (distance units) +* :math:`\epsilon` (energy units) +* :math:`\sigma` (distance units) * cutoff (distance units) The last coefficient is optional. If not specified, the global *ufm* @@ -76,7 +82,13 @@ of a run: .. note:: - The thermodynamic integration procedure can be performed with this potential using :doc:`fix adapt `. This command will rescale the force on each atom by varying a scale variable, which always starts with value 1.0. The syntax is the same described above, however, changing epsilon to scale. A detailed explanation of how to use this command and perform nonequilibrium thermodynamic integration in LAMMPS is given in the paper by :ref:`(Freitas) `. + The thermodynamic integration procedure can be performed with this + potential using :doc:`fix adapt `. This command will + rescale the force on each atom by varying a scale variable, which + always starts with value 1.0. The syntax is the same described above, + however, changing epsilon to scale. A detailed explanation of how to + use this command and perform nonequilibrium thermodynamic integration + in LAMMPS is given in the paper by :ref:`(Freitas) `. ---------- diff --git a/doc/src/pair_vashishta.rst b/doc/src/pair_vashishta.rst index de29985f56..2dde0482be 100644 --- a/doc/src/pair_vashishta.rst +++ b/doc/src/pair_vashishta.rst @@ -63,16 +63,21 @@ including SiO2 :ref:`Vashishta1990 `, SiC The potential for the energy U of a system of atoms is -.. image:: Eqs/pair_vashishta.jpg - :align: center +.. math:: + + U & = \sum_i^N \sum_{j > i}^N U_{ij}^{(2)} (r_{ij}) + \sum_i^N \sum_{j \neq i}^N \sum_{k > j, k \neq i}^N U_{ijk}^{(3)} (r_{ij}, r_{ik}, \theta_{ijk}) \\ + U_{ij}^{(2)} (r) & = \frac{H_{ij}}{r^{\eta_{ij}}} + \frac{Z_i Z_j}{r}\exp(-r/\lambda_{1,ij}) - \frac{D_{ij}}{r^4}\exp(-r/\lambda_{4,ij}) - \frac{W_{ij}}{r^6}, r < r_{c,{ij}} \\ + U_{ijk}^{(3)}(r_{ij},r_{ik},\theta_{ijk}) & = B_{ijk} \frac{\left[ \cos \theta_{ijk} - \cos \theta_{0ijk} \right]^2} {1+C_{ijk}\left[ \cos \theta_{ijk} - \cos \theta_{0ijk} \right]^2} \times \\ + & \exp \left( \frac{\gamma_{ij}}{r_{ij} - r_{0,ij}} \right) \exp \left( \frac{\gamma_{ik}}{r_{ik} - r_{0,ik}} \right), r_{ij} < r_{0,ij}, r_{ik} < r_{0,ik} + where we follow the notation used in :ref:`Branicio2009 `. -U2 is a two-body term and U3 is a three-body term. The +:math:`U^2` is a two-body term and U3 is a three-body term. The summation over two-body terms is over all neighbors J within -a cutoff distance = *rc*\ . The twobody terms are shifted and +a cutoff distance = :math:`r_c`. The twobody terms are shifted and tilted by a linear function so that the energy and force are -both zero at *rc*\ . The summation over three-body terms -is over all neighbors J and K within a cut-off distance = *r0*\ , +both zero at :math:`r_c`. The summation over three-body terms +is over all neighbors *i* and *k* within a cut-off distance :math:`= r_0`, where the exponential screening function becomes zero. The *vashishta* style computes these formulas analytically. The @@ -126,20 +131,20 @@ and three-body coefficients in the formulae above: * element 1 (the center atom in a 3-body interaction) * element 2 * element 3 -* H (energy units) -* eta -* Zi (electron charge units) -* Zj (electron charge units) -* lambda1 (distance units) -* D (energy units) -* lambda4 (distance units) -* W (energy units) -* rc (distance units) -* B (energy units) -* gamma -* r0 (distance units) -* C -* costheta0 +* *H* (energy units) +* :math:`\eta` +* :math:`Z_i` (electron charge units) +* :math:`Z_j` (electron charge units) +* :math:`\lambda_1` (distance units) +* *D* (energy units) +* :math:`\lambda_4` (distance units) +* *W* (energy units) +* :math:`r_c` (distance units) +* *B* (energy units) +* :math:`\gamma` +* :math:`r_0` (distance units) +* *C* +* :math:`\cos\theta_0` The non-annotated parameters are unitless. The Vashishta potential file must contain entries for all the elements listed in the @@ -159,12 +164,14 @@ unambiguous, general, and simple to code, LAMMPS uses a slightly confusing method for specifying parameters. All parameters are divided into two classes: two-body and three-body. Two-body and three-body parameters are handled differently, as described below. -The two-body parameters are H, eta, lambda1, D, lambda4, W, rc, gamma, -and r0. They appear in the above formulae with two subscripts. The -parameters Zi and Zj are also classified as two-body parameters, even -though they only have 1 subscript. The three-body parameters are B, -C, costheta0. They appear in the above formulae with three -subscripts. Two-body and three-body parameters are handled +The two-body parameters are *H*\ , :math:`\eta`, :math:`\lambda_1`, +*D*\ , :math:`\lambda_4`, *W*, :math:`r_c`, :math:`\gamma`, +and :math:`r_0`. They appear in the above formulae with two subscripts. +The parameters :math:`Z_i` and :math:`Z_j` are also classified +as two-body parameters, even +though they only have 1 subscript. The three-body parameters are *B*\ , +*C*\ , :math:`\cos\theta_0`. They appear in the above formulae with +three subscripts. Two-body and three-body parameters are handled differently, as described below. The first element in each entry is the center atom in a three-body @@ -184,7 +191,8 @@ ensure that these values are equal. Two-body parameters appearing in entries where the 2nd and 3rd elements are different are stored but never used. It is good practice to enter zero for these values. Note that the three-body function U3 above contains the two-body parameters -gamma and r0. So U3 for a central C atom bonded to an Si atom and a +:math:`\gamma` and :math:`r_0`. So U3 for a central C atom bonded to +an Si atom and a second C atom will take three-body parameters from the CSiC entry, but two-body parameters from the CCC and CSiSi entries. diff --git a/doc/src/pair_yukawa.rst b/doc/src/pair_yukawa.rst index c3a8e9fc71..b7cb6818d2 100644 --- a/doc/src/pair_yukawa.rst +++ b/doc/src/pair_yukawa.rst @@ -38,10 +38,12 @@ Description Style *yukawa* computes pairwise interactions with the formula -.. image:: Eqs/pair_yukawa.jpg - :align: center +.. math:: -Rc is the cutoff. + E = A \frac{e^{- \kappa r}}{r} \qquad r < r_c + + +:math:`r_c` is the cutoff. The following coefficients must be defined for each pair of atoms types via the :doc:`pair_coeff ` command as in the examples diff --git a/doc/src/pair_yukawa_colloid.rst b/doc/src/pair_yukawa_colloid.rst index 76bf010fb9..2c337bdb94 100644 --- a/doc/src/pair_yukawa_colloid.rst +++ b/doc/src/pair_yukawa_colloid.rst @@ -35,11 +35,13 @@ Description Style *yukawa/colloid* computes pairwise interactions with the formula -.. image:: Eqs/pair_yukawa_colloid.jpg - :align: center +.. math:: -where Ri and Rj are the radii of the two particles and Rc is the -cutoff. + E = \frac{A}{\kappa} e^{- \kappa (r - (r_i + r_j))} \qquad r < r_c + + +where :math:`r_i` and :math:`r_j` are the radii of the two particles +and :math:`r_c` is the cutoff. In contrast to :doc:`pair_style yukawa `, this functional form arises from the Coulombic interaction between two colloid @@ -49,7 +51,7 @@ theory. :doc:`Pair\_style yukawa ` is a screened Coulombic potential between two point-charges and uses no such approximation. This potential applies to nearby particle pairs for which the Derjagin -approximation holds, meaning h << Ri + Rj, where h is the +approximation holds, meaning :math:`h << r_i + r_j`, where *h* is the surface-to-surface separation of the two particles. When used in combination with :doc:`pair_style colloid `, @@ -72,17 +74,17 @@ used in :doc:`pair_style yukawa `. For low surface potentials, i.e. less than about 25 mV, A can be written as: -.. parsed-literal:: +.. math:: - A = 2 \* PI \* R\*eps\*eps0 \* kappa \* psi\^2 + A = 2 \pi R\varepsilon\varepsilon_0 \kappa \psi^2 where -* R = colloid radius (distance units) -* eps0 = permittivity of free space (charge\^2/energy/distance units) -* eps = relative permittivity of fluid medium (dimensionless) -* kappa = inverse screening length (1/distance units) -* psi = surface potential (energy/charge units) +* *R* = colloid radius (distance units) +* :math:`\varepsilon_0` = permittivity of free space (charge\^2/energy/distance units) +* :math:`\varepsilon` = relative permittivity of fluid medium (dimensionless) +* :math:`\kappa` = inverse screening length (1/distance units) +* :math:`\psi` = surface potential (energy/charge units) The last coefficient is optional. If not specified, the global yukawa/colloid cutoff is used. diff --git a/doc/src/pair_zbl.rst b/doc/src/pair_zbl.rst index 8e34341172..bffcf9fbda 100644 --- a/doc/src/pair_zbl.rst +++ b/doc/src/pair_zbl.rst @@ -43,12 +43,16 @@ that ramps the energy, force, and curvature smoothly to zero between an inner and outer cutoff. The potential energy due to a pair of atoms at a distance r\_ij is given by: -.. image:: Eqs/pair_zbl.jpg - :align: center +.. math:: -where e is the electron charge, epsilon\_0 is the electrical -permittivity of vacuum, and Z\_i and Z\_j are the nuclear charges of the -two atoms. The switching function S(r) is identical to that used by + E^{ZBL}_{ij} & = \frac{1}{4\pi\epsilon_0} \frac{Z_i Z_j \,e^2}{r_{ij}} \phi(r_{ij}/a)+ S(r_{ij})\\ + a & = \frac{0.46850}{Z_{i}^{0.23} + Z_{j}^{0.23}}\\ + \phi(x) & = 0.18175e^{-3.19980x} + 0.50986e^{-0.94229x} + 0.28022e^{-0.40290x} + 0.02817e^{-0.20162x}\\ + +where *e* is the electron charge, :math:`\epsilon_0` is the electrical +permittivity of vacuum, and :math:`Z_i` and :math:`Z_j` are the nuclear +charges of the +two atoms. The switching function :math:`S(r)` is identical to that used by :doc:`pair_style lj/gromacs `. Here, the inner and outer cutoff are the same for all pairs of atom types. @@ -56,16 +60,17 @@ The following coefficients must be defined for each pair of atom types via the :doc:`pair_coeff ` command as in the examples above, or in the LAMMPS data file. -* Z\_i (atomic number for first atom type, e.g. 13.0 for aluminum) +* :math:`Z_i` (atomic number for first atom type, e.g. 13.0 for aluminum) -* Z\_j (ditto for second atom type) +* :math:`Z_j` (ditto for second atom type) -The values of Z\_i and Z\_j are normally equal to the atomic +The values of :math:`Z_i` and :math:`Z_j` are normally equal to the atomic numbers of the two atom types. Thus, the user may optionally -specify only the coefficients for each I==I pair, and rely +specify only the coefficients for each :math:`i==i` pair, and rely on the obvious mixing rule for cross interactions (see below). -Note that when I==I it is required that Z\_i == Z\_j. When used -with :doc:`hybrid/overlay ` and pairs are assigned +Note that when :math:`i==i` it is required that :math:`Z_i == Z_j`. +When used with :doc:`hybrid/overlay ` and pairs are +assigned to more than one sub-style, the mixing rule is not used and each pair of types interacting with the ZBL sub-style must be included in a pair\_coeff command. @@ -108,9 +113,11 @@ instructions on how to use the accelerated styles effectively. **Mixing, shift, table, tail correction, restart, rRESPA info**\ : -For atom type pairs I,J and I != J, the Z\_i and Z\_j coefficients -can be mixed by taking Z\_i and Z\_j from the values specified for -I == I and J == J cases. When used +For atom type pairs *i,j* and :math:`i \neq i`, the :math:`Z_i` and +:math:`Z_j` coefficients +can be mixed by taking :math:`Z_i` and :math:`Z_j` from the values +specified for +:math:`i == i` and :math:`j == j` cases. When used with :doc:`hybrid/overlay ` and pairs are assigned to more than one sub-style, the mixing rule is not used and each pair of types interacting with the ZBL sub-style