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--- exercises:2015_ethz_mmm:monte_carlo_ice [2015/02/06 17:49] – external edit 127.0.0.1
+++ exercises:2015_ethz_mmm:monte_carlo_ice [2015/03/18 14:53] – sclelia
@@ Line 1: / Line 1: @@
 ====== Properties of Ice from Monte Carlo Simulations ======
-<note important>
-  - Add the line ''PRINT_COORDS .FALSE.'' in the ''TMC'' section. It disables the output of the coordinate trajectory and thus avoids the files-size problem.
-  - Add the line ''RND_DETERMINISTIC 42'' to the ''TMC'' section to choose a random number seed. You have to replace ''42'' with different values to obtain different samplings.
-</note>
-In this exercise we will use Monte Carlo sampling to calculate the [[wp>Relative_permittivity|dielectric constant]] and the [[wp>Thermal_expansion#Coefficient_of_thermal_expansion| thermal expansion coefficient]] of the disordered hexagonal phase (Ih) of water ice. In order to speed up the calculation we are going to use the simple SPC/E water model. For more information see: [[doi>10.1063/1.1568337]]. A DFT-version of the calculation can be found here: [[doi>10.1021/jp4103355]].
+In this exercise we will use Monte Carlo sampling to calculate the [[wp>Relative_permittivity|dielectric constant]] of the disordered hexagonal phase (Ih) of water ice. \\
-<note tip>
-You should run these calculations on 3 cores with ''bsub -n 3''.
-</note>
+In order to speed up the calculation, we are going to use the SPC/E water model.
+The model is a non polarizable forcefield model, with parameters for:
+  * Charge of the atomic species
+  * Harmonic O-H bond elongation $ U_{bond}= k_{bond}⋅(r-r_{eq})^2$
+  * Harmonic H-O-H angle bending $ U_{angle}= k_{angle}⋅(θ-θ_{eq})^2$
+  * Lennard Jones interaction between non bonded species (interaction between atoms belonging to different water molecules)
+ (A DFT-version of the calculation can be found here: [[doi>10.1021/jp4103355]]).
-===== Task 1: Calculate the dielectric constant =====
+===== Introduction =====
-The dielectric constant describes the response of a system to an external electric field. Within a linear response approximation the //Kubo Formula// is applicable. The Kubo Formula is an equation which expresses the linear response of an observable quantity due to a time-dependent perturbation. Hence, one can calculate the dielectric constant based on a sampling of the dipole moment.
-The advantage of Monte Carlo is that we can employ special hydrogen reordering moves, which lead to an effective sampling of the dipole.
+Water molecules are arranged according to the so called //"ice rules"//:
+  - Water molecules are present as neutral H2O
+  - Each molecule makes four hydrogen bonds with its four nearest neighbors, two as a hydrogen bond donor and two as an acceptor.
+All the possible proton arrangements in the ice strucrure are not energetically equivalent and we sample and weight them with a Monte Carlo approach. For this purpose, a specific algorithm was created (see: [[doi>10.1063/1.1568337]]), that allows to employ special proton reordering moves, which lead to an effective sampling of the ice dipole (N.B: different proton arrangement = different dipole). \\
-Run the input-file ''mc_exercise.inp''. It will create a file ''tmc_trajectory_T200.00.dip_cl'', which contains the dipole moment for each accepted step. Plot the histogram of the z-component of the dipole moment. The distribution should be symmetric, because the simulation cell is also symmetric in the z-direction.
+The dielectric constant of a system describes its response to an external electric field.
+If the dipole moment is properly sampled, one can compute the dielectric constant of ice, by applying the //Kubo Formula//. This is valid in the approximation that the response of the system to the time-dependent perturbation (the field) is linear. \\
-The dielectric constant can then calculated from the dipole moments via:
+The dielectric constant can then be calculated from the dipole moments via:
 \begin{equation}
 \epsilon = 1 + \left(\frac{4 \pi}{3 \epsilon_0 V  k_B T } \right ) \operatorname{Var}(M) \ ,
 \end{equation}
-where $M$ denotes the dipol moment of the entire simulation cell and $\operatorname{Var}(M)$ denotes the variance of the dipol moment of the sampling:
+where $M$ denotes the dipole moment of the entire simulation cell and $\operatorname{Var}(M)$ denotes the variance of the dipole moment of the sampling:
 \begin{equation}
 \operatorname{Var}(M) = (\langle M \cdot M\rangle - \langle M\rangle\langle M\rangle ) \ .
 \end{equation}
-To simplify this task you can use the following python-script:
+===== Task 1: Calculate the dielectric constant from one simulation =====
+      * As usual, download the 4 required files (at the end of this page: input, auxiliary input, ice coordinates and topology) in the same directory. \\
+      * Run the input-file ''mc_exercise.inp''
+⇒ It will create a file ''tmc_trajectory_T200.00.dip_cl'', which contains the dipole moment for each accepted step.
+<note tip>
+You should run these calculations on 3 cores with ''bsub -n 3''.
+</note>
+    * Plot the histogram of the z-component of the dipole moment. \\
+⇒ The distribution should be symmetric, because the simulation cell is also symmetric in the z-direction.
+  * Compute the dielectric constant of ice. To simplify this task you can use the following python-script:
 <code python calc_dielectric_constant.py>
@@ Line 92: / Line 102: @@
 You can also share trajectories with your colleges.
 </note>
-===== Task 3: Calculate the thermal expansion coefficient =====
-Run the MC sampling again at temperature of 150K . Plot the cell volume of each sampling. Read off the converged cell volume and plot it vs. the temperature. Determine the expansion coefficient via a linear fit. Do you trust your result? Why (not)?
 ===== Required Files =====