This tutorial uses an example study of square-well particles to introduce several topics:

This is a full study on the square-well fluid system.

## About the system

For the purpose of the tutorial, we'll want to simulate a fluid of square-well molecules. These are essentially hard-sphere particles with a short-range attractive well:

We will simulate this fluid at a reduced temperature of $k_B\,T=2$ and a reduced density of $\rho=0.1$. If you want to learn more about the square-well potential, its parameters, and how it corresponds to realistic intermolecular interactions please see the reference entry. If you're uncertain of the units please see the FAQ on units, but everything here is presented in reduced units.

We will again use periodic boundary conditions to allow us to simulate an infinite fluid without the effects of walls or other containers.

## The whole tutorial in brief

We're going to create a square-well fluid with $N=4000$ particles at a low density. We'll try to control the temperature using velocity rescaling, then using thermostats. Finally, we'll collect some measurements from the system. The commands we will use are

#Create the low-density square-well system
dynamod -m 1 -C 10 -d 0.1 --i1 0 -r 2.0 -o config.start.xml
#Run the system briefly to check the temperature
dynarun config.start.xml -c 1000000
#Add a thermostat, to allow us to control the temperature
dynamod config.out.xml.bz2 -T 2.0
#Run the system using the thermostat to set the temperature and let it equilibrate
dynarun config.out.xml.bz2 -c 1000000
#Run it some more to equilibrate it further
dynarun config.out.xml.bz2 -c 1000000
#Disable the thermostat and remove any momentum, so that we might collect accurate dynamic information
dynamod -T 0 -Z config.out.xml.bz2
#Run the simulation to collect data on the system
dynarun config.out.xml.bz2 -c 1000000 -o config.final.xml -L IntEnergyHist -L MSD
#Use dynatransport to analyse the transport coefficients
dynatransport output.xml -s 2 -c 10 -v


We'll now look in detail at each of these commands.

# Setting up the configuration file

When you first start using DynamO, it is not really practical to try to create a configuration file from scratch. The dynamod tool helps by providing many pre-designed systems to start your simulations from.

Following the same steps in tutorial 2, we again query the available options of the dynamod command using the --help option:

dynamod --help

We then look for the most useful mode and we see that square-well fluids can be made using dynamod's packing mode 1. We can get some more information on this mode by adding the --help option:

dynamod -m 1 --help

And a detailed description of the modes options will be outputted on screen:

...
Mode 1: Mono/Multi-component square wells
Options
-C [ --NCells ] arg (=7)    Set the default number of lattice unit-cells in each direction.
-x [ --xcell ] arg          Number of unit-cells in the x dimension.
-y [ --ycell ] arg          Number of unit-cells in the y dimension.
-z [ --zcell ] arg          Number of unit-cells in the z dimension.
--rectangular-box           Set the simulation box to be rectangular so that the x,y,z cells also specify the simulation aspect ratio.
-d [ --density ] arg (=0.5) System density.
--i1 arg (=FCC)             Lattice type (0=FCC, 1=BCC, 2=SC)
--f1 arg (=1.5)             Well width factor (also known as lambda)
--f2 arg (=1)               Well Depth (negative values create square shoulders)
--s1 arg (monocomponent)    Instead of f1 and f2, you can specify a multicomponent system using this option. You need to pass the the parameters for each species as follows --s1 "diameter(d),lambda(l),mass(m),welldepth(e),molefrac(x):d,l,m,e,x[:...]"
...

Here you can see many of the same options available for hard-sphere systems, as seen in tutorial 2. The only additions are the well width (--f1) and depth (--f2) options and the option for a multicomponent system (--s1).

Lets start by making a monocomponent mixture of square-wells using the following command:

dynamod -m 1 -C 10 -d 0.1 --i1 0 -r 2.0 -o config.start.xml

The options passed here are discussed in detail in tutorial 2. The only differences are that the number of particles has been increased to 4000 (-C 10), we're creating square-well molecules (-m 1) instead of hard spheres, and the density is lower (-d 0.1). An example of the configuration file is available below (it is a large XML file, so your browser may take some time to display it).

As we haven't specified the well depth and well width, they have been left at their default values of 1 and 1.5 respectively. Next, we're going to look at thermostatting the system.

# Rescaling velocities

When creating the configuration, we initially set the temperature through the rescale option -r. This option works by rescaling all of the velocities of the particles so that the instantaneous temperature is $k_B\,T=2$ (or whatever is passed as an argument to the option). For a system without rotational degrees of freedom, the temperature is given by $k_B\,T = \frac{1}{3\,N}\sum_i^N m_i\,v_i^2$ so it is clear that by scaling the velocities we can set the temperature to whatever we wish. However, rescaling the temperature only holds the temperature fixed (AKA thermostats) in "hard" systems such as the hard-sphere/parallel-cube/hard-lines systems. This is because these systems have no finite potential energy terms between the particles, therefore the temperature does not change with time (except if we perform work such as compression on the system).

For square-well systems, we can set the temperature at the start of the simulation, but it will change over time due to interactions converting energy between kinetic and potential modes. We can see this if we run a simulation on the starting configuration:

dynarun config.start.xml -c 1000000

Please note, we didn't set an output configuration file name using -o so the default config.out.xml.bz2 is used. Taking a look at the output, we can see the temperature (and excess internal energy $U$) is fluctuating over time:

...
ETA 16s, Events 100k, t 7.08388, <MFT> 0.141678, T 2.47483, U -0.71225
ETA 14s, Events 200k, t 13.6032, <MFT> 0.136032, T 2.48533, U -0.728
ETA 12s, Events 300k, t 20.1072, <MFT> 0.134048, T 2.48133, U -0.722
ETA 11s, Events 400k, t 26.6342, <MFT> 0.133171, T 2.48267, U -0.724
...


You should note that if the temperature fluctuates higher, the internal energy fluctuates lower as the total energy is constant. You can see this if you calculate the average energy per particle $\left\langle E\right\rangle=U + 3\,k_B\,T/2$ And see that for this system it remains constant at the starting value of $\approx3$. This is one of the nice properties of event-driven molecular dynamics, energy is exactly conserved. Unfortunately, we still need some way of setting the temperature. We could rescale again to take some energy out of the system to try to lower it to a temperature of $k_B\,T=2$, but this would need to be repeated over by hand until the temperature converged. Instead, we can use a thermostat to automatically add/remove energy from the system to reach a specified temperature.

# Adding a thermostat

To add an Andersen thermostat, again use the dynamod tool:

dynamod config.out.xml.bz2 -T 2.0 

Please note that this command loads the config.out.xml.bz2 file, adds an Andersen thermostat, and the result is saved into the default output file name, which is config.out.xml.bz2. This will overwrite the initial file, if you don't want to do this, specify a new file name with the -o option.

The dynamod command above will add an Andersen thermostat to the system with a target temperature of $k_B\,T=2$ (set by the -T argument). This thermostat will eventually bring the system to the specified temperature, even with changes in the configurational energy, by randomly reassigning particle velocities.

Note: If you wish to change the thermostat temperature at a later time, you can use the dynamod on the configuration again:

dynamod config.out.xml.bz2 -T 4.0

You can even use dynamod remove the thermostatt by using a temperature of zero (-T 0). Alternatively, you can open up the configuration file in a text editor, and edit or delete the Andersen type System event by hand:

<System Type="Andersen" Name="Thermostat" MFT="2.0" Temperature="1.0" SetPoint="0.05" SetFrequency="100">
<IDRange Type="All"/>
</System>


With the thermostat added and the temperature set to $k_B\,T=2$, we can see what the result is on the temperature of the system. Again, running the system

dynarun config.out.xml.bz2 -c 1000000

And the output should look like this:

...
ETA 16s, Events 100k, t 6.28632, <MFT> 0.129188, T 2.15641, U -0.75675
ETA 15s, Events 200k, t 12.6762, <MFT> 0.130097, T 2.05169, U -0.771
ETA 13s, Events 300k, t 19.1881, <MFT> 0.13125, T 2.03105, U -0.7735
ETA 11s, Events 400k, t 25.678, <MFT> 0.131705, T 2.01297, U -0.75525
ETA 9s, Events 500k, t 32.179, <MFT> 0.13203, T 2.06379, U -0.7915
ETA 7s, Events 600k, t 38.6795, <MFT> 0.132246, T 2.02205, U -0.77625
ETA 6s, Events 700k, t 45.1681, <MFT> 0.132363, T 2.01704, U -0.7615
ETA 4s, Events 800k, t 51.6511, <MFT> 0.132437, T 2.04454, U -0.78925
ETA 2s, Events 900k, t 58.1523, <MFT> 0.132537, T 2.01887, U -0.79125
ETA 0s, Events 1000k, t 64.6884, <MFT> 0.132689, T 1.9653, U -0.7795
...


We can see that the temperature approaches the required temperature at the end. Looking at the instantaneous $T$ and $U$ values it appears to have reached steady state after around 200k events. The average mean free time (MFT) is still changing but this is due to it accumulating samples during the equilibration. We can confirm this by running the configuration for another $10^6$ events.

dynarun config.out.xml.bz2 -c 10000000
...
ETA 16s, Events 100k, t 6.50405, <MFT> 0.13339, T 2.00641, U -0.7845
ETA 15s, Events 200k, t 13.0134, <MFT> 0.13345, T 1.98747, U -0.78325
ETA 13s, Events 300k, t 19.5232, <MFT> 0.133469, T 1.9498, U -0.7825
ETA 11s, Events 400k, t 26.1082, <MFT> 0.133857, T 1.97389, U -0.815
ETA 9s, Events 500k, t 32.6387, <MFT> 0.133873, T 2.01406, U -0.76675
ETA 7s, Events 600k, t 39.2339, <MFT> 0.134103, T 2.02729, U -0.79425
...


Here its easy to see that the mean free time is relatively stable as well. It is very difficult to conclusively prove that we're at steady state but previous experience with this system tells us that we're now ready to collect some data.

# Collecting Data

At this point we have a system which has been equilibrated with a thermostat. We want to collect some information on the properties of the system, namely the internal energy histograms, diffusion coefficients, viscosity, and thermal conductivity.

To find out what output plugins are available and how to load them please see the output plugin documentation. Most of what we want to collect is contained in the Misc plugin which is loaded by default, but we'll need to add the IntEnergyHist and MSD plugins to collect the energy histograms and diffusion data.

Unfortunately there is a problem with thermostats while collecting data which characterises the dynamics of the system, e.g. the transport coefficients. The Andersen thermostat changes the motion of the system when it randomly re-assigns the particle velocities. Thus, if we measure the properties of the system, they will be the those of the square-well fluid AND the thermostat, not the fluid alone. Also, if we take a look at the restrictions on using the thermal conductivity, we'll notice that it is restricted only to NVE/microcanonical simulations (systems without a thermostat).

We're going to have to disable the thermostat during data collection and hope (and check) that the system fluctuates close to the target temperature. We can use dynamod to disable the thermostat:

dynamod -T 0 -Z config.out.xml.bz2


Please note that we also zeroed the total momentum again using the -Z option as the Andersen thermostat causes the total momentum to fluctuate around zero. Now we're ready to collect some data! We just run dynarun while enabling the output plugins we wish to use:

dynarun config.out.xml.bz2 -c 1000000 -o config.final.xml -L IntEnergyHist


And we're now ready to process the results!

# Processing the results

## Thermodynamic properties

In the first instance, we can start processing the collected data in the same way tutorial 2 deals with processing collected data. Expanding the output file:

bunzip2 output.xml.bz2


We can then check the file to see how close the temperature is to $k_B\,T=2$ after we disabled the thermostat. We use the Temperature tag in the output file for this:

<Temperature Mean="1.9695813386316516" MeanSqr="3.8793330745797636" Current="1.9751905078351808" Min="1.942023841168514" Max="2.0076905078351808"/>


The average value has a deviation of $\approx2\%$ from the desired value, which can be expected with this system size. Larger systems (with longer equilibration times) will lower this value if needed as the fluctuations scale with $N^{-0.5}$. For publication results, I would recommend setting the temperature more accurately, perhaps using the calculated heat capacity to estimate the required energy to add or remove from the system to more accurately set the temperature.

It is interesting to note at this point that DynamO collects "exact" time averages wherever possible (see the FAQ on averages). By exact we mean that the mean values are not discretely sampled over the trajectory, but are true time averages integrated over the length of the simulation.

We can also get some scale for the fluctuation of the temperature by calculating its standard deviation. We can calculate this using the mean square value, e.g. $\sigma_T=\sqrt{\left\langle T^2\right\rangle - \left\langle T\right\rangle^2} \approx0.009079$ Again, this value is system size dependent. Interestingly, in NVE simulations this value is related the heat capacity of the system; however, we will calculate this property through the configurational internal energy below.

Taking a look at the UConfigurational tag, we have:

<UConfigurational Mean="-3161.3449847792272" MeanSqr="9997069.4161528479" Current="-3194.9999999999995" Min="-3389.9999999999995" Max="-2995.9999999999995"/>


where this is the total energy the system has through interactions (if you want the specific configurational internal energy you will need to divide by $N$). We could again calculate the standard deviation which is related to the residual heat capacity, $C_V^{ex.}$, by the following formula: $\frac{C_v^{ex.}}{k_B}=\frac{\left\langle U_{conf.}^2\right\rangle - \left\langle U_{conf.}\right\rangle^2}{k_B^2\,T^2}$ however, for convenience we can just use the ResidualHeatCapacity tag which calculates this for us:

<ResidualHeatCapacity Value="764.91663782252635"/>


Please note that, like the UConfigurational values, this value is extensive. We'll now take a look at processing collected data which is more complex, beginning with the internal energy histogram.

## Internal Energy Histogram

The internal energy histogram is extremely interesting as it allows us to begin to calculate key thermodynamic properties such as the density of states. This also allows us to use advanced techniques such as multicanonical simulations and histogram reweighting. We enabled the internal energy histogram plugin with the -L IntEnergyHist option to dynarun and its output is under the EnergyHist tag in the output file:

<EnergyHist BinWidth="1">
<HistogramWeighted TotalWeight="68.801254983242984" Dimension="1" BinWidth="1" AverageVal="-3161.3443994965492">
-3390 2.330168873287094e-06
-3389 4.9110740851148333e-06
-3388 3.4733255439184949e-06
...
</HistogramWeighted>
</EnergyHist>


For all the details on what the above attributes mean, please see the IntEnergyHist plugin documentation, but it is essentially a list of configurational internal energy values and the fraction of simulation time spent in each (again collected using exact time averaging and not periodic sampling. If we wanted to process or plot this data, we need to cut it out of the output.xml file. For a full description of how to handle XML files, please take a look at the reference. Here, we'll use the xmlstarlet tool to cut it out:

xmlstarlet sel -t -v '//EnergyHist/HistogramWeighted' output.xml > histogram.dat


Now we can plot the data and we should end up with a graph like the one on the right. It appears that this data is quite rough and longer simulations are needed to accurately obtain good averages, but this is a good initial estimate.

We've covered some basic properties and how to extract tabulated data, now we will take a look at the transport properties.

## Transport Properties

Transport properties, such as the viscosity, thermal conductivity, and diffusivity are difficult to measure in simulation. They require the use of correlators and need long simulation times to gain good averages. You also need to be careful of their definition, especially in multicomponent systems, and over what correlation times it is valid to collect data from. This is all documented in the reference entry for the thermal conductivity but there is no substitute for experience here. Please calculate known values from the literature to validate your understanding before attempting to measure new systems.

The easiest transport property to calculate is the self-diffusion coefficient, obtained from the MSD plugin:

<MSD>
<Species Name="Bulk" val="807.82538152569123" diffusionCoeff="1.9569056352302285"/>
</MSD>


Here, each Species and Topology will have a separate entry for the calculated diffusion coefficient. This value is just calculated from the total distanced travelled over the simulation by each particle, so there isn't a significant amount of work to do in processing it. We only need to be confident that the simulations have been run for sufficient time to reach the long-time behavior.

The other transport property data lies within several correlator tags. E.g.

<ThermalConductivity>
<Correlator>
0 0 0 0 0
0.016637884124075412 4135 0.00080420056234304738 0.00080194081779336385 0.00079775560698344759
0.033275768248150824 4134 0.0017904547411232278 0.0017381232563064169 0.0018186474468955217
...
</Correlator>
</ThermalConductivity>


For the thermal conductivity, the first column is the time, the second is the number of samples collected at that time, and the last three columns are the correlator values in the $x$, $y$, and $z$-directions. For more information please take a look at the reference entry for the thermal conductivity. As these are the Einstein correlators, the transport coefficients are the gradients of the correlation functions. If we cut the data out of output.xml using xmlstarlet or some other tool and plot each correlator we end up with the graph to the right.

Ideally, this plot should consist of points in a straight line which we can fit to extract the slope/thermal transport coefficient, $L_{\lambda\lambda}$. Unfortunately, at short times we have short-time effects from molecular processes which dominate the correlator. We are only interested in the behaviour at long times, where a "hydrodynamic" description applies, so we need to avoid these short-time effects. Unfortunately again, at long times we have poor statistics (see the low number of samples at long times) AND we have effects from the periodic boundary conditions.

We need to extract the correlator curves from a window which has good statistics, ignores short times and avoids long time effects and poor statistics. Once we have the window of time, we need to fit a linear function to the correlator, and to calculate the gradient/transport coefficient. Luckily, there is the dynatransport tool to help us do all of this.

### dynatransport

If we run the dynatransport tool on the output file, we can get an estimate of the transport coefficients.

dynatransport output.xml
ShearViscosityL_{\eta,\eta}= 0.0628751050012 +- 0.0 <R>^2= 0.386109814931
BulkViscosityL_{\kappa,\kappa}= 11.3811651077 +- 0.0 <R>^2= 0.567696677048
ThermalConductivityL_{\lambda,\lambda}= 0.178111094018 +- 0.0 <R>^2= 0.535953655856
ThermalDiffusionL_{\lambda,Bulk}= -3.03999278728e-18 +- 0.0 <R>^2= 0.736800451496
MutualDiffusionL_{Bulk,Bulk}= 1.04605741811e-34 +- 0.0 <R>^2= 0.893542145422


By default, dynatransport uses the full data set to calculate the correlators. You should be able to see that the $R^2$ values of the fits are significantly below $1$ which indicates that the correlators are not linear in the window selected. We can view the current fit by passing the -v option:

dynatransport output.xml -v


This will give plots, like the one presented to the right, for each transport property, including averaged correlator data, standard deviations between each dimension, and the regressed line used to calculate the coefficient.

Clearly, this linear fit is terrible and focuses too much on the long time section which has poor sampling. After examining the fit and testing a few different window sizes its clear that one suitable window appears to be $\Delta t \in [2,10]$. We can set this window for dynatransport using the start (-s) and cutoff (-c) options to give:

dynatransport output.xml -s 2 -c 10 -v
ShearViscosityL_{\eta,\eta}= 0.195306335342 +- 0.0 <R>^2= 0.999498595286
BulkViscosityL_{\kappa,\kappa}= 0.0707529418455 +- 0.0 <R>^2= 0.0110843745332
ThermalConductivityL_{\lambda,\lambda}= 0.508343229525 +- 0.0 <R>^2= 0.997347634149
ThermalDiffusionL_{\lambda,Bulk}= -4.22881210719e-19 +- 0.0 <R>^2= 0.866354789486
MutualDiffusionL_{Bulk,Bulk}= 2.2536681734e-35 +- 0.0 <R>^2= 0.968183740424


This fit is significantly better (see plot), although there is some strong variation between the dimensions it appears that the average is almost linear. We're only comparing the fit for the viscosity but all fits should be checked.

Using this window gives a much better fit for the thermal conductivity $L_{\lambda\,\lambda}\approx0.5083$ and viscosity $L_{\eta\,\eta}\approx0.1953$. The bulk viscosity is relatively hard to calculate and you should notice that the thermal diffusion, $L_{\lambda\,Bulk}$, and mutual diffusion, $L_{Bulk,Bulk}$, coefficients are close to zero. These coefficients are only non-zero for systems with multiple Species.

# Conclusions

We've simulated a square well system of $N=4000$ molecules/particles with a well-width of $\lambda=1.5$ at a reduced density of $\rho=0.1$ and temperature $k_B\,T=1.97\approx2$. Using dynarun we've equilibrated and run the simulation to collect data. We've then calculated the average configurational internal energy $U_{conf.}\approx-3161$ and internal energy histograms. Using dynatransport we've calculated the self-diffusion coefficient $D\approx1.957$, thermal conductivity $L_{\lambda\,\lambda}\approx0.5083$, and viscosity $L_{\eta\,\eta}\approx0.1953$.

This is our first proper study with DynamO. Now we will look at more complex systems and how to set them up.

Page last modified: Thursday 29th July 2021