From 02e287bf51e8bf4116a218f2b91eed6e4d0002a1 Mon Sep 17 00:00:00 2001 From: Axel Kohlmeyer Date: Mon, 24 Feb 2020 15:29:57 -0500 Subject: [PATCH] convert pair styles dpd to exp6 --- doc/src/Eqs/e3b.jpg | Bin 28534 -> 0 bytes doc/src/Eqs/e3b.tex | 15 ---- doc/src/Eqs/eff_ECP1.jpg | Bin 5146 -> 0 bytes doc/src/Eqs/eff_ECP1.tex | 9 -- doc/src/Eqs/eff_ECP2.jpg | Bin 9744 -> 0 bytes doc/src/Eqs/eff_ECP2.tex | 8 -- doc/src/Eqs/eff_KE.jpg | Bin 3385 -> 0 bytes doc/src/Eqs/eff_KE.tex | 9 -- doc/src/Eqs/eff_NN.jpg | Bin 4153 -> 0 bytes doc/src/Eqs/eff_NN.tex | 9 -- doc/src/Eqs/eff_Ne.jpg | Bin 6425 -> 0 bytes doc/src/Eqs/eff_Ne.tex | 9 -- doc/src/Eqs/eff_Pauli.jpg | Bin 5676 -> 0 bytes doc/src/Eqs/eff_Pauli.tex | 9 -- doc/src/Eqs/eff_ee.jpg | Bin 6829 -> 0 bytes doc/src/Eqs/eff_ee.tex | 9 -- doc/src/Eqs/eff_energy_expression.jpg | Bin 9538 -> 0 bytes doc/src/Eqs/eff_energy_expression.tex | 9 -- doc/src/Eqs/pair_dpd.jpg | Bin 12376 -> 0 bytes doc/src/Eqs/pair_dpd.tex | 13 --- doc/src/Eqs/pair_dpd_energy.jpg | Bin 25636 -> 0 bytes doc/src/Eqs/pair_dpd_energy.tex | 12 --- doc/src/Eqs/pair_dpd_energy_terms.jpg | Bin 8883 -> 0 bytes doc/src/Eqs/pair_dpd_energy_terms.tex | 11 --- doc/src/Eqs/pair_dpd_omega.jpg | Bin 1689 -> 0 bytes doc/src/Eqs/pair_dpd_omega.tex | 9 -- doc/src/Eqs/pair_drip.jpg | Bin 58904 -> 0 bytes doc/src/Eqs/pair_drip.tex | 14 --- doc/src/Eqs/pair_eam.jpg | Bin 6693 -> 0 bytes doc/src/Eqs/pair_eam.tex | 10 --- doc/src/Eqs/pair_eam_fs.jpg | Bin 6547 -> 0 bytes doc/src/Eqs/pair_eam_fs.tex | 11 --- doc/src/Eqs/pair_edip.jpg | Bin 42939 -> 0 bytes doc/src/Eqs/pair_edip.tex | 22 ----- doc/src/Eqs/pair_eim1.jpg | Bin 6824 -> 0 bytes doc/src/Eqs/pair_eim1.tex | 9 -- doc/src/Eqs/pair_eim2.jpg | Bin 8494 -> 0 bytes doc/src/Eqs/pair_eim2.tex | 11 --- doc/src/Eqs/pair_eim3.jpg | Bin 40993 -> 0 bytes doc/src/Eqs/pair_eim3.tex | 25 ------ doc/src/Eqs/pair_exp6_rx.jpg | Bin 9372 -> 0 bytes doc/src/Eqs/pair_exp6_rx.tex | 9 -- doc/src/Eqs/pair_exp6_rx_oneFluid.jpg | Bin 16062 -> 0 bytes doc/src/Eqs/pair_exp6_rx_oneFluid.tex | 11 --- doc/src/Eqs/pair_exp6_rx_oneFluid2.jpg | Bin 7629 -> 0 bytes doc/src/Eqs/pair_exp6_rx_oneFluid2.tex | 11 --- doc/src/pair_dpd.rst | 31 ++++--- doc/src/pair_dpd_fdt.rst | 80 +++++++++++------- doc/src/pair_drip.rst | 7 +- doc/src/pair_e3b.rst | 17 +++- doc/src/pair_eam.rst | 15 +++- doc/src/pair_edip.rst | 28 ++++-- doc/src/pair_eff.rst | 32 +++---- doc/src/pair_eim.rst | 40 ++++++--- doc/src/pair_exp6_rx.rst | 46 ++++++---- examples/USER/{ => misc}/e3b/README | 0 examples/USER/{ => misc}/e3b/e3b_box.data | 0 examples/USER/{ => misc}/e3b/in.e3b-tip4p2005 | 0 .../e3b/log.29Mar2019.e3b-tip4p2005.g++.1 | 0 .../e3b/log.29Mar2019.e3b-tip4p2005.g++.4 | 0 60 files changed, 184 insertions(+), 376 deletions(-) delete mode 100644 doc/src/Eqs/e3b.jpg delete mode 100644 doc/src/Eqs/e3b.tex delete mode 100644 doc/src/Eqs/eff_ECP1.jpg delete mode 100644 doc/src/Eqs/eff_ECP1.tex delete mode 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doc/src/Eqs/pair_dpd_omega.tex delete mode 100644 doc/src/Eqs/pair_drip.jpg delete mode 100644 doc/src/Eqs/pair_drip.tex delete mode 100644 doc/src/Eqs/pair_eam.jpg delete mode 100644 doc/src/Eqs/pair_eam.tex delete mode 100644 doc/src/Eqs/pair_eam_fs.jpg delete mode 100644 doc/src/Eqs/pair_eam_fs.tex delete mode 100644 doc/src/Eqs/pair_edip.jpg delete mode 100644 doc/src/Eqs/pair_edip.tex delete mode 100644 doc/src/Eqs/pair_eim1.jpg delete mode 100644 doc/src/Eqs/pair_eim1.tex delete mode 100644 doc/src/Eqs/pair_eim2.jpg delete mode 100644 doc/src/Eqs/pair_eim2.tex delete mode 100644 doc/src/Eqs/pair_eim3.jpg delete mode 100644 doc/src/Eqs/pair_eim3.tex delete mode 100644 doc/src/Eqs/pair_exp6_rx.jpg delete mode 100644 doc/src/Eqs/pair_exp6_rx.tex delete mode 100644 doc/src/Eqs/pair_exp6_rx_oneFluid.jpg delete mode 100644 doc/src/Eqs/pair_exp6_rx_oneFluid.tex delete mode 100644 doc/src/Eqs/pair_exp6_rx_oneFluid2.jpg delete mode 100644 doc/src/Eqs/pair_exp6_rx_oneFluid2.tex rename 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\sum_{\substack{i,j,k,\ell \\\in \textrm{type B}}} f(r_{ij})f(r_{k\ell}) + E_C \sum_{\substack{i,j,k,\ell \\\in \textrm{type C}}} f(r_{ij})f(r_{k\ell}) \\ -f(r) =& e^{-k_3 r}s(r) \\ -s(r) =& \begin{cases} - 1 & rR_f\\ -\end{cases} -\end{align*} - -\end{document} diff --git a/doc/src/Eqs/eff_ECP1.jpg b/doc/src/Eqs/eff_ECP1.jpg deleted file mode 100644 index 3b8a3f8ff39ace5ad9c37d8ac6ada08077bf575d..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 5146 zcmbuDcQ72>yT^Bj1Es{EJKVr5kI^Z-?j5(%jAo(jCI7mV9t$gC+~OVu9=XW4mnCB_;nSFd|wfKH(hWG){y z$H1#5Gd>O7)am6mB1=R>pJxn{6MA5#PvkjZQyXYxzk`Zs)xx}#T>d_FzL`ZvT>y&F zdx1GuEoQ0C`X&*-c7nIsiY=t~@KC1dMC#hkpL2HgfW!8V4!z!34W~SW!lsEl>cjb> z7J{UOhOiv4Oh_XeKC0l6r#o8oi3J7 z?wfQ-Qh)l%b=UO1;NZ~2z~}nbc{c>&H`=~VQ!M)jHRZVPp_GSmBQ5&p2U`X-$)(st z-@D>&y-@8{gSi<$c0(3_lb9zzX`lMkVLjIhVR^BNwTx?(R)?y?X0kcmX7HI zg~+HsI+Y#Zz5rAn(v|X}Au+K*oIig**;BdzkV1^|g@u>9NcPqAxFYQ_oYj<|bAq+s 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e6813b125c..0000000000 --- a/doc/src/Eqs/eff_Ne.tex +++ /dev/null @@ -1,9 +0,0 @@ -\documentclass[12pt]{article} - -\begin{document} - -$$ -E_{Ne} = - \frac{1}{{4\pi \varepsilon _0 }}\sum\limits_{i,j} {\frac{{Z_i }}{{R_{ij} }}Erf\left( {\frac{{\sqrt 2 R_{ij} }}{{s_j }}} \right)} -$$ - -\end{document} diff --git a/doc/src/Eqs/eff_Pauli.jpg b/doc/src/Eqs/eff_Pauli.jpg deleted file mode 100644 index 61bb8652e390a0970aed8727753dd6a248ec9733..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 5676 zcmb`LcT5xPx5tZR3J3_YDM%SAONQ(zQmCj1$X-HOQYe&4%djG{w}LXtQig?>J<60N zOQ0ZowG5e}Yy@N!u5W%fH@Sb_M2!M);3SjUz z02d^HE`SC|O-&7?`FqgN(9qH`(bN5fg^`hgiH+szRW=qjHg-;aE_M#yYiw-X*SUEG z1Ox>I*|~&8gh3+wAVJVY8-S4>U`>@vMa2fV=mhWsr~sFM|7!dXG_*i!x=Zv_f3x>l z0aVmfz)RGBRp~D?{w?{pnwo}|?JC`Mc41i^4r51pIb_H+PVilM6Ti3bD{Dr%L_`&p zp7^GG95FSwaDus@U%zYrelY`}|5rl`a4N<2{|xH0n@QR7K}(Rq&~SiC6}Oo7>ENHn zFNzE}JJ$J}IVtMR4USCSez5_En9=EnJnuh{GhA=a$+t@~KlhZ8!qBB&(%vMsE}M1z 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-h(l,Z) & = & \lambda [(1-e^{-Q(Z)(l+\tau(Z))^2}) + \eta Q(Z)(l+\tau(Z))^2 ] \\ -Q(Z) & = & Q_0 e^{-\mu Z} \qquad \tau(Z) = u_1 + u_2 (u_3 e^{-u_4 Z} - e^{-2u_4 Z}) -\end{eqnarray*} - -\end{document} - diff --git a/doc/src/Eqs/pair_eim1.jpg b/doc/src/Eqs/pair_eim1.jpg deleted file mode 100644 index c243120b2a2076fa45697cbf10d05927295f6d98..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 6824 zcmb_gXIK+XlMezyK#-1tp+{- z5fcAp#Kgq6N$!x6{H43(P(n(!AHqNrrjR1=By*e~O+{Y&)Ez3jM++wF-U3JzEp-p_{_9bUtDI_i+acZy{}G+bXkXA+oFr%_BHtHh$-!A!(g2F z@LQk)yCX9hAW@}*HN(Eja=$)xv)WA`CrH~9`N!Bz6jSP#N1sO=)TCYbd*a{q$Ci8e zY4`}X(lF(>C5j8D(u7vGPZ?RA6ql=e%x4;w<}jMtn!cJc-A$>f1)WOMVT8LrL=Wgy zsTYg@9+L#PEEJi&Dtc*`KhIe#76J0xjmGtUa7pkn)cGy5X>+p>Fn>Lx}0!4y4nXDzBO8+p0v}cw`bWxmK_w zlU`gw+nJ$&migH0`!r?RO70I$_j`}`cq>=fC)v5M5VPn|#k$8*7y4iR#}&%BlY%yU z@6{VK^J&;;nxT$nZU8U7`ruO;v`;*X%C7tQf0gBi;#3);Adb8qK@z)apb<+%#%fT> zaKb@R)4w~E*>H~Keqce+p|9K(6g$o*tJyhnA}GA^RTN%>DG93xr}%?Glghnin~ISA 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ze8Q}!x%8Y`c0*+xvk}=2sQHUu;fhTr8w_@;JMIf}MwItr*6t^Tgr8Cal+;LSQfUu& zte-x|!jmu}DH_4` z$vIROL|lU`cwRNESFsRAS*EVHPav^&A@JpGzH6mIB!;bZv6pS3MyJnYhx z&9BsSr>-VNFj+1xzZq5aT>H)ST(h&JXLt)rL&04~)0SE?L7RF~N8pE3y*_Zr`d?Nw z;4t(PELL5Isom>7_UmL@`1mfqq}ye6E>d;UTo*bjY9sEO5eDmevQwm0Q>Iy;Vd1i6 z`xQ6VRM}>>jOo_b+g@ndA_OyH$Ey%_XQVqA&*~#hXeP*sZ1aH~Y&~eSZhsrn*MIST zgb$2Ch9h4)1LUK7J|8WKqwIrdcT5- r=(ULX?C{|5b9efW?~1BY!ib)+}Gp diff --git a/doc/src/Eqs/pair_exp6_rx_oneFluid2.tex b/doc/src/Eqs/pair_exp6_rx_oneFluid2.tex deleted file mode 100644 index 1d1214f229..0000000000 --- a/doc/src/Eqs/pair_exp6_rx_oneFluid2.tex +++ /dev/null @@ -1,11 +0,0 @@ -\documentstyle[12pt]{article} -\pagestyle{empty} -\begin{document} - -\begin{eqnarray*} - \epsilon_{ab} &=& \sqrt{\epsilon_{a}\epsilon_{b}} \\ - R_{m,ab} &=& \frac{R_{m,a}+R_{m,b}}{2} \\ - \alpha_{ab} &=& \sqrt{\alpha_{a}\alpha_{b}} \\ -\end{eqnarray*} - -\end{document} diff --git a/doc/src/pair_dpd.rst b/doc/src/pair_dpd.rst index fde0463f14..bf537a1006 100644 --- a/doc/src/pair_dpd.rst +++ b/doc/src/pair_dpd.rst @@ -67,17 +67,25 @@ pair interaction and the thermostat for each pair of particles. For style *dpd*\ , the force on atom I due to atom J is given as a sum of 3 terms -.. image:: Eqs/pair_dpd.jpg - :align: center +.. math:: -where Fc is a conservative force, Fd is a dissipative force, and Fr is -a random force. Rij is a unit vector in the direction Ri - Rj, Vij is -the vector difference in velocities of the two atoms = Vi - Vj, alpha -is a Gaussian random number with zero mean and unit variance, dt is -the timestep size, and w(r) is a weighting factor that varies between -0 and 1. Rc is the cutoff. Sigma is set equal to sqrt(2 Kb T gamma), -where Kb is the Boltzmann constant and T is the temperature parameter -in the pair\_style command. + \vec{f} = & (F^C + F^D + F^R) \hat{r_{ij}} \qquad \qquad r < r_c \\ + F^C = & A w(r) \\ + F^D = & - \gamma w^2(r) (\hat{r_{ij}} \bullet \vec{v_{ij}}) \\ + F^R = & \sigma w(r) \alpha (\Delta t)^{-1/2} \\ + w(r) = & 1 - r/r_c + + +where :math:`F^C` is a conservative force, :math:`F^D` is a dissipative +force, and :math:`F^R` is a random force. :math:`r_{ij}` is a unit +vector in the direction :math:`r_i - r_j`, :math:`V_{ij} is the vector +difference in velocities of the two atoms :math:`= \vec{v}_i - +\vec{v}_j, :math:`\alpha` is a Gaussian random number with zero mean and +unit variance, dt is the timestep size, and w(r) is a weighting factor +that varies between 0 and 1. :math:`r_c` is the cutoff. :math:`\sigma` +is set equal to :math:`\sqrt{2 k_B T \gamma}`, where :math:`k_B` is the +Boltzmann constant and T is the temperature parameter in the pair\_style +command. For style *dpd/tstat*\ , the force on atom I due to atom J is the same as the above equation, except that the conservative Fc term is @@ -97,7 +105,7 @@ the examples above, or in the data file or restart files read by the commands: * A (force units) -* gamma (force/velocity units) +* :math:`\gamma` (force/velocity units) * cutoff (distance units) The last coefficient is optional. If not specified, the global DPD @@ -125,7 +133,6 @@ the work of :ref:`(Afshar) ` and :ref:`(Phillips) `. includes all the components of force listed above, including the random force. - ---------- diff --git a/doc/src/pair_dpd_fdt.rst b/doc/src/pair_dpd_fdt.rst index c893f68a1c..dfde24a8e2 100644 --- a/doc/src/pair_dpd_fdt.rst +++ b/doc/src/pair_dpd_fdt.rst @@ -54,25 +54,34 @@ under isoenergetic and isoenthalpic conditions (see :ref:`(Lisal) `). For DPD simulations in general, the force on atom I due to atom J is given as a sum of 3 terms -.. image:: Eqs/pair_dpd.jpg - :align: center +.. math:: -where Fc is a conservative force, Fd is a dissipative force, and Fr is -a random force. Rij is a unit vector in the direction Ri - Rj, Vij is -the vector difference in velocities of the two atoms = Vi - Vj, alpha -is a Gaussian random number with zero mean and unit variance, dt is -the timestep size, and w(r) is a weighting factor that varies between -0 and 1. Rc is the cutoff. The weighting factor, omega\_ij, varies -between 0 and 1, and is chosen to have the following functional form: + \vec{f} = & (F^C + F^D + F^R) \hat{r_{ij}} \qquad \qquad r < r_c \\ + F^C = & A w(r) \\ + F^D = & - \gamma w^2(r) (\hat{r_{ij}} \bullet \vec{v_{ij}}) \\ + F^R = & \sigma w(r) \alpha (\Delta t)^{-1/2} \\ + w(r) = & 1 - r/r_c -.. image:: Eqs/pair_dpd_omega.jpg - :align: center + +where :math:`F^C` is a conservative force, :math:`F^D` is a dissipative +force, and :math:`F^R` is a random force. :math:`r_{ij}` is a unit +vector in the direction :math:`r_i - r_j`, :math:`V_{ij} is the vector +difference in velocities of the two atoms :math:`= \vec{v}_i - +\vec{v}_j, :math:`\alpha` is a Gaussian random number with zero mean and +unit variance, dt is the timestep size, and w(r) is a weighting factor +that varies between 0 and 1. Rc is the cutoff. The weighting factor, +:math:`\omega_{ij}`, varies between 0 and 1, and is chosen to have the +following functional form: + +.. math:: + + \omega_{ij} = 1 - \frac{r_{ij}}{r_{c}} Note that alternative definitions of the weighting function exist, but would have to be implemented as a separate pair style command. -For style *dpd/fdt*\ , the fluctuation-dissipation theorem defines gamma -to be set equal to sigma\*sigma/(2 T), where T is the set point +For style *dpd/fdt*\ , the fluctuation-dissipation theorem defines :math:`\gamma` +to be set equal to :math:`\sigma^2/(2 T)`, where T is the set point temperature specified as a pair style parameter in the above examples. The following coefficients must be defined for each pair of atoms types via the :doc:`pair_coeff ` command as in the examples above, @@ -80,33 +89,42 @@ or in the data file or restart files read by the :doc:`read_data ` or :doc:`read_restart ` commands: * A (force units) -* sigma (force\*time\^(1/2) units) +* :math:`\sigma` (force\*time\^(1/2) units) * cutoff (distance units) The last coefficient is optional. If not specified, the global DPD cutoff is used. -Style *dpd/fdt/energy* is used to perform DPD simulations -under isoenergetic and isoenthalpic conditions. The fluctuation-dissipation -theorem defines gamma to be set equal to sigma\*sigma/(2 dpdTheta), where -dpdTheta is the average internal temperature for the pair. The particle -internal temperature is related to the particle internal energy through -a mesoparticle equation of state (see :doc:`fix eos `). The -differential internal conductive and mechanical energies are computed -within style *dpd/fdt/energy* as: +Style *dpd/fdt/energy* is used to perform DPD simulations under +isoenergetic and isoenthalpic conditions. The fluctuation-dissipation +theorem defines :math:`\gamma` to be set equal to :math:`sigma^2/(2 +\theta)`, where :math:theta` is the average internal temperature for the +pair. The particle internal temperature is related to the particle +internal energy through a mesoparticle equation of state (see :doc:`fix +eos `). The differential internal conductive and mechanical +energies are computed within style *dpd/fdt/energy* as: + +.. math:: + + du_{i}^{cond} = & \kappa_{ij}(\frac{1}{\theta_{i}}-\frac{1}{\theta_{j}})\omega_{ij}^{2} + \alpha_{ij}\omega_{ij}\zeta_{ij}^{q}(\Delta{t})^{-1/2} \\ + du_{i}^{mech} = & -\frac{1}{2}\gamma_{ij}\omega_{ij}^{2}(\frac{\vec{r_{ij}}}{r_{ij}}\bullet\vec{v_{ij}})^{2} - + \frac{\sigma^{2}_{ij}}{4}(\frac{1}{m_{i}}+\frac{1}{m_{j}})\omega_{ij}^{2} - + \frac{1}{2}\sigma_{ij}\omega_{ij}(\frac{\vec{r_{ij}}}{r_{ij}}\bullet\vec{v_{ij}})\zeta_{ij}(\Delta{t})^{-1/2} -.. image:: Eqs/pair_dpd_energy.jpg - :align: center where -.. image:: Eqs/pair_dpd_energy_terms.jpg - :align: center +.. math:: -Zeta\_ij\^q is a second Gaussian random number with zero mean and unit + \alpha_{ij}^{2} = & 2k_{B}\kappa_{ij} \\ + \sigma^{2}_{ij} = & 2\gamma_{ij}k_{B}\Theta_{ij} \\ + \Theta_{ij}^{-1} = & \frac{1}{2}(\frac{1}{\theta_{i}}+\frac{1}{\theta_{j}}) + + +:math:`\zeta_ij^q` is a second Gaussian random number with zero mean and unit variance that is used to compute the internal conductive energy. The -fluctuation-dissipation theorem defines alpha\*alpha to be set -equal to 2\*kB\*kappa, where kappa is the mesoparticle thermal +fluctuation-dissipation theorem defines :math:`alpha^2` to be set +equal to :math:2k_B\kappa`, where :math:`\kappa` is the mesoparticle thermal conductivity parameter. The following coefficients must be defined for each pair of atoms types via the :doc:`pair_coeff ` command as in the examples above, or in the data file or restart files @@ -114,8 +132,8 @@ read by the :doc:`read_data ` or :doc:`read_restart ` commands: * A (force units) -* sigma (force\*time\^(1/2) units) -* kappa (energy\*temperature/time units) +* :math:`\sigma` (force\*time\^(1/2) units) +* :math:`\kappa` (energy\*temperature/time units) * cutoff (distance units) The last coefficient is optional. If not specified, the global DPD diff --git a/doc/src/pair_drip.rst b/doc/src/pair_drip.rst index ca04efcece..efff4ce25b 100644 --- a/doc/src/pair_drip.rst +++ b/doc/src/pair_drip.rst @@ -40,8 +40,11 @@ in :ref:`(Wen) `, which is based on the :ref:`(Kolmogorov) `. -.. image:: Eqs/e3b.jpg - :align: center +.. math:: + + E =& E_2 \sum_{i,j}e^{-k_2 r_{ij}} + E_A \sum_{\substack{i,j,k,\ell \\ + \in \textrm{type A}}} f(r_{ij})f(r_{k\ell}) + E_B \sum_{\substack{i,j,k,\ell \\ + \in \textrm{type B}}} f(r_{ij})f(r_{k\ell}) + E_C \sum_{\substack{i,j,k,\ell \\ + \in \textrm{type C}}} f(r_{ij})f(r_{k\ell}) \\ + f(r) =& e^{-k_3 r}s(r) \\ + s(r) =& \begin{cases} + 1 & rR_f\\ + \end{cases} + This potential was developed as a water model that includes the three-body cooperativity of hydrogen bonding explicitly. To use it in this way, it must be applied in conjunction with a conventional two-body water model, through *pair\_style hybrid/overlay*. @@ -103,7 +114,7 @@ If the neigh setting is too large, the pair style will use more memory than nece This pair style tallies a breakdown of the total E3B potential energy into sub-categories, which can be accessed via the :doc:`compute pair ` command as a vector of values of length 4. The 4 values correspond to the terms in the first equation above: the E2 term, the Ea term, the Eb term, and the Ec term. -See the examples/USER/e3b directory for a complete example script. +See the examples/USER/misc/e3b directory for a complete example script. ---------- diff --git a/doc/src/pair_eam.rst b/doc/src/pair_eam.rst index d3356007cd..575980eb39 100644 --- a/doc/src/pair_eam.rst +++ b/doc/src/pair_eam.rst @@ -102,8 +102,11 @@ Style *eam* computes pairwise interactions for metals and metal alloys using embedded-atom method (EAM) potentials :ref:`(Daw) `. The total energy Ei of an atom I is given by -.. image:: Eqs/pair_eam.jpg - :align: center +.. math:: + + E_i = F_\alpha \left(\sum_{j \neq i}\ \rho_\beta (r_{ij})\right) + + \frac{1}{2} \sum_{j \neq i} \phi_{\alpha\beta} (r_{ij}) + where F is the embedding energy which is a function of the atomic electron density rho, phi is a pair potential interaction, and alpha @@ -371,8 +374,12 @@ alloys using a generalized form of EAM potentials due to Finnis and Sinclair :ref:`(Finnis) `. The total energy Ei of an atom I is given by -.. image:: Eqs/pair_eam_fs.jpg - :align: center +.. math:: + + E_i = F_\alpha \left(\sum_{j \neq i}\ + \rho_{\alpha\beta} (r_{ij})\right) + + \frac{1}{2} \sum_{j \neq i} \phi_{\alpha\beta} (r_{ij}) + This has the same form as the EAM formula above, except that rho is now a functional specific to the atomic types of both atoms I and J, diff --git a/doc/src/pair_edip.rst b/doc/src/pair_edip.rst index 6cc388b3cc..7b7f5742ec 100644 --- a/doc/src/pair_edip.rst +++ b/doc/src/pair_edip.rst @@ -37,15 +37,27 @@ potentials, while *edip/multi* supports multi-element EDIP runs. In EDIP, the energy E of a system of atoms is -.. image:: Eqs/pair_edip.jpg - :align: center +.. math:: -where phi2 is a two-body term and phi3 is a three-body term. The -summations in the formula are over all neighbors J and K of atom I -within a cutoff distance = a. -Both terms depend on the local environment of atom I through its -effective coordination number defined by Z, which is unity for a -cutoff distance < c and gently goes to 0 at distance = a. + E = & \sum_{j \ne i} \phi_{2}(R_{ij}, Z_{i}) + \sum_{j \ne i} \sum_{k \ne i,k > j} \phi_{3}(R_{ij}, R_{ik}, Z_{i}) \\ + \phi_{2}(r, Z) = & A\left[\left(\frac{B}{r}\right)^{\rho} - e^{-\beta Z^2}\right]exp{\left(\frac{\sigma}{r-a}\right)} \\ + \phi_{3}(R_{ij}, R_{ik}, Z_i) = & exp{\left(\frac{\gamma}{R_{ij}-a}\right)}exp{\left(\frac{\gamma}{R_{ik}-a}\right)}h(cos\theta_{ijk},Z_i) \\ + Z_i = & \sum_{m \ne i} f(R_{im}) \qquad + f(r) = \begin{cases} + 1 & \quad ra + \end{cases} \\ + h(l,Z) = & \lambda [(1-e^{-Q(Z)(l+\tau(Z))^2}) + \eta Q(Z)(l+\tau(Z))^2 ] \\ + Q(Z) = & Q_0 e^{-\mu Z} \qquad \tau(Z) = u_1 + u_2 (u_3 e^{-u_4 Z} - e^{-2u_4 Z}) + + +where :math:`\phi_2` is a two-body term and :math:`\phi_3` is a +three-body term. The summations in the formula are over all neighbors J +and K of atom I within a cutoff distance = a. Both terms depend on the +local environment of atom I through its effective coordination number +defined by Z, which is unity for a cutoff distance < c and gently goes +to 0 at distance = a. Only a single pair\_coeff command is used with the *edip* style which specifies a EDIP potential file with parameters for all diff --git a/doc/src/pair_eff.rst b/doc/src/pair_eff.rst index fc4c5f0964..baf8e9bf6e 100644 --- a/doc/src/pair_eff.rst +++ b/doc/src/pair_eff.rst @@ -98,25 +98,19 @@ and the quantum-derived Pauli (E\_PR) and Kinetic energy interactions potentials between electrons (E\_KE) for a total energy expression given as, -.. image:: Eqs/eff_energy_expression.jpg - :align: center +.. math:: + +U\left(R,r,s\right) = E_{NN} \left( R \right) + E_{Ne} \left( {R,r,s} \right) + E_{ee} \left( {r,s} \right) + E_{KE} \left( {r,s} \right) + E_{PR} \left( { \uparrow \downarrow ,S} \right) The individual terms are defined as follows: -.. image:: Eqs/eff_KE.jpg - :align: center +.. math:: -.. image:: Eqs/eff_NN.jpg - :align: center - -.. image:: Eqs/eff_Ne.jpg - :align: center - -.. image:: Eqs/eff_ee.jpg - :align: center - -.. image:: Eqs/eff_Pauli.jpg - :align: center + E_{KE} = & \frac{\hbar^2 }{{m_{e} }}\sum\limits_i {\frac{3}{{2s_i^2 }}} \\ + E_{NN} = & \frac{1}{{4\pi \varepsilon _0 }}\sum\limits_{i < j} {\frac{{Z_i Z_j }}{{R_{ij} }}} \\ + E_{Ne} = & - \frac{1}{{4\pi \varepsilon _0 }}\sum\limits_{i,j} {\frac{{Z_i }}{{R_{ij} }}Erf\left( {\frac{{\sqrt 2 R_{ij} }}{{s_j }}} \right)} \\ + E_{ee} = & \frac{1}{{4\pi \varepsilon _0 }}\sum\limits_{i < j} {\frac{1}{{r_{ij} }}Erf\left( {\frac{{\sqrt 2 r_{ij} }}{{\sqrt {s_i^2 + s_j^2 } }}} \right)} \\ + E_{Pauli} = & \sum\limits_{\sigma _i = \sigma _j } {E\left( { \uparrow \uparrow } \right)_{ij}} + \sum\limits_{\sigma _i \ne \sigma _j } {E\left( { \uparrow \downarrow } \right)_{ij}} \\ where, s\_i correspond to the electron sizes, the sigmas i's to the fixed spins of the electrons, Z\_i to the charges on the nuclei, R\_ij @@ -229,11 +223,9 @@ representations, after the "ecp" keyword. Si. The ECP captures the orbital overlap between the core and valence electrons (i.e. Pauli repulsion) with one of the functional forms: -.. image:: Eqs/eff_ECP1.jpg - :align: center - -.. image:: Eqs/eff_ECP2.jpg - :align: center +.. math:: + E_{Pauli(ECP_s)} = & p_1\exp\left(-\frac{p_2r^2}{p_3+s^2} \right) \\ + E_{Pauli(ECP_p)} = & p_1\left( \frac{2}{p_2/s+s/p_2} \right)\left( r-p_3s\right)^2\exp \left[ -\frac{p_4\left( r-p_3s \right)^2}{p_5+s^2} \right] Where the 1st form correspond to core interactions with s-type valence electrons and the 2nd to core interactions with p-type valence diff --git a/doc/src/pair_eim.rst b/doc/src/pair_eim.rst index 133a81c0dc..7c1b4eca85 100644 --- a/doc/src/pair_eim.rst +++ b/doc/src/pair_eim.rst @@ -34,30 +34,42 @@ Style *eim* computes pairwise interactions for ionic compounds using embedded-ion method (EIM) potentials :ref:`(Zhou) `. The energy of the system E is given by -.. image:: Eqs/pair_eim1.jpg - :align: center +.. math:: + + E = \frac{1}{2} \sum_{i=1}^{N} \sum_{j=i_1}^{i_N} \phi_{ij} \left(r_{ij}\right) + \sum_{i=1}^{N}E_i\left(q_i,\sigma_i\right) The first term is a double pairwise sum over the J neighbors of all I -atoms, where phi\_ij is a pair potential. The second term sums over +atoms, where :math:`\phi_{ij}` is a pair potential. The second term sums over the embedding energy E\_i of atom I, which is a function of its charge -q\_i and the electrical potential sigma\_i at its location. E\_i, q\_i, -and sigma\_i are calculated as +q\_i and the electrical potential :math:`\sigma_i` at its location. E\_i, q\_i, +and :math:`sigma_i` are calculated as -.. image:: Eqs/pair_eim2.jpg - :align: center +.. math:: -where eta\_ji is a pairwise function describing electron flow from atom -I to atom J, and psi\_ij is another pairwise function. The multi-body + q_i = & \sum_{j=i_1}^{i_N} \eta_{ji}\left(r_{ij}\right) \\ + \sigma_i = & \sum_{j=i_1}^{i_N} q_j \cdot \psi_{ij} \left(r_{ij}\right) \\ + E_i\left(q_i,\sigma_i\right) = & \frac{1}{2} \cdot q_i \cdot \sigma_i + +where :math:`\eta_{ji} is a pairwise function describing electron flow from atom +I to atom J, and :math:`\psi_{ij}` is another pairwise function. The multi-body nature of the EIM potential is a result of the embedding energy term. A complete list of all the pair functions used in EIM is summarized below -.. image:: Eqs/pair_eim3.jpg - :align: center +.. math:: -Here E\_b, r\_e, r\_(c,phi), alpha, beta, A\_(psi), zeta, r\_(s,psi), -r\_(c,psi), A\_(eta), r\_(s,eta), r\_(c,eta), chi, and pair function type -p are parameters, with subscripts ij indicating the two species of + \phi_{ij}\left(r\right) = & \left\{ \begin{array}{lr} + \left[\frac{E_{b,ij}\beta_{ij}}{\beta_{ij}-\alpha_{ij}}\exp\left(-\alpha_{ij} \frac{r-r_{e,ij}}{r_{e,ij}}\right)-\frac{E_{b,ij}\alpha_{ij}}{\beta_{ij}-\alpha_{ij}}\exp\left(-\beta_{ij} \frac{r-r_{e,ij}}{r_{e,ij}}\right)\right]f_c\left(r,r_{e,ij},r_{c,\phi,ij}\right),& p_{ij}=1 \\ + \left[\frac{E_{b,ij}\beta_{ij}}{\beta_{ij}-\alpha_{ij}} \left(\frac{r_{e,ij}}{r}\right)^{\alpha_{ij}} -\frac{E_{b,ij}\alpha_{ij}}{\beta_{ij}-\alpha_{ij}} \left(\frac{r_{e,ij}}{r}\right)^{\beta_{ij}}\right]f_c\left(r,r_{e,ij},r_{c,\phi,ij}\right),& p_{ij}=2 + \end{array} + \right.\\ + \eta_{ji} = & A_{\eta,ij}\left(\chi_j-\chi_i\right)f_c\left(r,r_{s,\eta,ij},r_{c,\eta,ij}\right) \\ + \psi_{ij}\left(r\right) = & A_{\psi,ij}\exp\left(-\zeta_{ij}r\right)f_c\left(r,r_{s,\psi,ij},r_{c,\psi,ij}\right) \\ + f_{c}\left(r,r_p,r_c\right) = & 0.510204 \mathrm{erfc}\left[\frac{1.64498\left(2r-r_p-r_c\right)}{r_c-r_p}\right] - 0.010204 + +Here :math:`E_b, r_e, r_(c,\phi), \alpha, \beta, A_(\psi), \zeta, r_(s,\psi), +r_(c,\psi), A_(\eta), r_(s,\eta), r_(c,\eta), \chi,` and pair function type +*p* are parameters, with subscripts *ij* indicating the two species of atoms in the atomic pair. .. note:: diff --git a/doc/src/pair_exp6_rx.rst b/doc/src/pair_exp6_rx.rst index c44c87ae17..d5ec1e5dc1 100644 --- a/doc/src/pair_exp6_rx.rst +++ b/doc/src/pair_exp6_rx.rst @@ -43,17 +43,20 @@ one CG particle can interact with a species in a neighboring CG particle through a site-site interaction potential model. The *exp6/rx* style computes an exponential-6 potential given by -.. image:: Eqs/pair_exp6_rx.jpg - :align: center +.. math:: -where the *epsilon* parameter determines the depth of the potential -minimum located at *Rm*\ , and *alpha* determines the softness of the repulsion. + U_{ij}(r) = \frac{\epsilon}{\alpha-6}\{6\exp[\alpha(1-\frac{r_{ij}}{R_{m}})]-\alpha(\frac{R_{m}}{r_{ij}})^6\} + + +where the :math:`\epsilon` parameter determines the depth of the +potential minimum located at :math:`R_m`, and :math:`\alpha` determines +the softness of the repulsion. The coefficients must be defined for each species in a given particle type via the :doc:`pair_coeff ` command as in the examples above, where the first argument is the filename that includes the exponential-6 parameters for each species. The file includes the -species tag followed by the *alpha*\ , *epsilon* and *Rm* +species tag followed by the :math:`\alpha, \epsilon` and :math:`R_m` parameters. The format of the file is described below. The second and third arguments specify the site-site interaction @@ -74,22 +77,22 @@ to scale the EXP-6 parameters as reactions occur. Currently, there are three scaling options: *exponent*\ , *polynomial* and *none*\ . Exponent scaling requires two additional arguments for scaling -the *Rm* and *epsilon* parameters, respectively. The scaling factor +the :math:`R_m` and :math:`\epsilon` parameters, respectively. The scaling factor is computed by phi\^exponent, where phi is the number of molecules represented by the coarse-grain particle and exponent is specified -as a pair coefficient argument for *Rm* and *epsilon*\ , respectively. -The *Rm* and *epsilon* parameters are multiplied by the scaling +as a pair coefficient argument for :math:`R_m` and :math:`\epsilon`, respectively. +The :math:`R_m` and :math:`\epsilon` parameters are multiplied by the scaling factor to give the scaled interaction parameters for the CG particle. Polynomial scaling requires a filename to be specified as a pair coeff argument. The file contains the coefficients to a fifth order -polynomial for the *alpha*\ , *epsilon* and *Rm* parameters that depend +polynomial for the :math:`\alpha`, :math:`\epsilon` and :math:`R_m` parameters that depend upon phi (the number of molecules represented by the CG particle). The format of a polynomial file is provided below. The *none* option to the scaling does not have any additional pair coeff arguments. This is equivalent to specifying the *exponent* option with -*Rm* and *epsilon* exponents of 0.0 and 0.0, respectively. +:math:`R_m` and :math:`\epsilon` exponents of 0.0 and 0.0, respectively. The final argument specifies the interaction cutoff (optional). @@ -133,23 +136,30 @@ between sections. Following a blank line, the next N lines list the species and their corresponding parameters. The first argument is the species tag, the -second argument is the exp6 tag, the 3rd argument is the *alpha* -parameter (energy units), the 4th argument is the *epsilon* parameter -(energy-distance\^6 units), and the 5th argument is the *Rm* parameter +second argument is the exp6 tag, the 3rd argument is the :math:`\alpha` +parameter (energy units), the 4th argument is the :math:`\epsilon` parameter +(energy-distance\^6 units), and the 5th argument is the :math:`R_m` parameter (distance units). If a species tag of "1fluid" is listed as a pair coefficient, a one-fluid approximation is specified where a concentration-dependent combination of the parameters is computed through the following equations: -.. image:: Eqs/pair_exp6_rx_oneFluid.jpg - :align: center +.. math:: + + R_{m}^{3} = & \sum_{a}\sum_{b} x_{a}x_{b}R_{m,ab}^{3} \\ + \epsilon = & \frac{1}{R_{m}^{3}}\sum_{a}\sum_{b} x_{a}x_{b}\epsilon_{ab}R_{m,ab}^{3} \\ + \alpha = & \frac{1}{\epsilon R_{m}^{3}}\sum_{a}\sum_{b} x_{a}x_{b}\alpha_{ab}\epsilon_{ab}R_{m,ab}^{3} where -.. image:: Eqs/pair_exp6_rx_oneFluid2.jpg - :align: center +.. math:: -and xa and xb are the mole fractions of a and b, respectively, which + \epsilon_{ab} = & \sqrt{\epsilon_{a}\epsilon_{b}} \\ + R_{m,ab} = & \frac{R_{m,a}+R_{m,b}}{2} \\ + \alpha_{ab} = & \sqrt{\alpha_{a}\alpha_{b}} + + +and :math:`x_a` and :math:`x_b` are the mole fractions of a and b, respectively, which comprise the gas mixture. diff --git a/examples/USER/e3b/README b/examples/USER/misc/e3b/README similarity index 100% rename from examples/USER/e3b/README rename to examples/USER/misc/e3b/README diff --git a/examples/USER/e3b/e3b_box.data b/examples/USER/misc/e3b/e3b_box.data similarity index 100% rename from examples/USER/e3b/e3b_box.data rename to examples/USER/misc/e3b/e3b_box.data diff --git a/examples/USER/e3b/in.e3b-tip4p2005 b/examples/USER/misc/e3b/in.e3b-tip4p2005 similarity index 100% rename from examples/USER/e3b/in.e3b-tip4p2005 rename to examples/USER/misc/e3b/in.e3b-tip4p2005 diff --git a/examples/USER/e3b/log.29Mar2019.e3b-tip4p2005.g++.1 b/examples/USER/misc/e3b/log.29Mar2019.e3b-tip4p2005.g++.1 similarity index 100% rename from examples/USER/e3b/log.29Mar2019.e3b-tip4p2005.g++.1 rename to examples/USER/misc/e3b/log.29Mar2019.e3b-tip4p2005.g++.1 diff --git a/examples/USER/e3b/log.29Mar2019.e3b-tip4p2005.g++.4 b/examples/USER/misc/e3b/log.29Mar2019.e3b-tip4p2005.g++.4 similarity index 100% rename from examples/USER/e3b/log.29Mar2019.e3b-tip4p2005.g++.4 rename to examples/USER/misc/e3b/log.29Mar2019.e3b-tip4p2005.g++.4