Module: contour

Description	Usage
actions that take functions as grid as input and that find isocontours in these functions
Authors: Gareth Tribello

Details

The actions implemented in this module were inspired by concept of the Willard-Chandler surface that is described in the first two papers referenced below. The way that this idea and these implementations has been used for a range of applications is then discussed in the other paper referenced below.

Willard and Chander's method for finding the position of the water surface starts by introducing an instantaneous density field:

$$\rho(x,y,z) = \sum_{i=1}^N K\left(\frac{x-x_i}{\lambda}, \frac{y-y_i}{\lambda}, \frac{z-z_i}{\lambda}\right)$$

In this expression the sum runs over a collection of atoms. Each of these atoms is located at a position $(x_i,y_i,z_i)$ and the $K$ is a smooth, 3-dimensional Kernel function (e.g. a Gaussian) that integrates to one. Each atom thus contributes a function that is peaked at the atom's position and which decays over some characteristic length scale $\lambda$ to the final field $\rho(x,y,z)$. The value of $\rho(x,y,z)$ is thus large if this function is evaluated at a point that near to the specified atoms and small if it is evaluated at some position that is far from them. Consequently, we can find interfaces that separate the region where the atoms of interest are located and their surroundings by finding the 2-dimensional manifold containing points for which:

$$\varphi(x,y,z) - \varphi_0 = 0$$

In this expression $\varphi_0$ is a parameter and $\varphi(x,y,z)$ is calculated using the first equation above. This procedures thus converts a particle based representation of the atoms and their surroundings into a coarse grained representation that describes the extents region containing atoms and the extent of the region that does not contain atoms. In other words, if we have performed a simulation of coexisting liquid and gas phases we can use the expressions above to locate the interface between the two phases.

Similarly, if we search for a contour using the second expression above in a phase field that is defined as follows:

$$\rho'(x,y,z) = \frac{ \sum_{i=1}^N s_i K\left(\frac{x-x_i}{\lambda}, \frac{y-y_i}{\lambda}, \frac{z-z_i}{\lambda}\right) }{ \sum_{i=1}^N K\left(\frac{x-x_i}{\lambda}, \frac{y-y_i}{\lambda}, \frac{z-z_i}{\lambda}\right) }$$

where $s_i$ is the value of one of the symmetry functions described in the documentation for the symfunc that can be used to distinguish solid-like atoms from liquid-like atoms evaluated based on the environment around atom $i$, we can find solid-liquid interfaces. This works becuase $\rho'(x,y,z)$ is large in regions of the box where there are many solid-like atoms and small in regions of the simulation box that are filled with liquid-like atoms.

Actions

The following actions are part of this module

Tags

GRIDANALYSIS COLVAR

GRIDANALYSIS: actions that take a function computed on a grid in input COLVAR: a function of a set of atomic positions that returns a scalar

Name	Description	Tags
DISTANCE_FROM_CONTOUR	Calculate the perpendicular distance from a Willard-Chandler dividing surface.	COLVAR
DISTANCE_FROM_SPHERICAL_CONTOUR	Calculate the perpendicular distance from a Willard-Chandler dividing surface.	COLVAR
DUMPCONTOUR	Print the contour	GRIDANALYSIS
FIND_CONTOUR	Find an isocontour in a smooth function.	GRIDANALYSIS
FIND_CONTOUR_SURFACE	Find an isocontour by searching along either the x, y or z direction.	GRIDANALYSIS
FIND_SPHERICAL_CONTOUR	Find an isocontour in a three dimensional grid by searching over a Fibonacci sphere.	GRIDANALYSIS

References

More information about this module is available in the following articles:

• A. P. Willard, D. Chandler, Coarse-grained modeling of the interface between water and heterogeneous surfaces. Faraday Discuss. 141, 209–220 (2009)
• A. P. Willard, D. Chandler, Instantaneous Liquid Interfaces. The Journal of Physical Chemistry B. 114, 1954–1958 (2010)
• J. Klug, C. Triguero, M. G. Del Pópolo, G. A. Tribello, Using Intrinsic Surfaces To Calculate the Free-Energy Change When Nanoparticles Adsorb on Membranes. The Journal of Physical Chemistry B. 122, 6417–6422 (2018)
• E. Baldi, M. Ceriotti, G. A. Tribello, Extracting the interfacial free energy and anisotropy from a smooth fluctuating dividing surface. Journal of Physics: Condensed Matter. 29, 445001 (2017)
• B. Cheng, M. Ceriotti, G. A. Tribello, Classical nucleation theory predicts the shape of the nucleus in homogeneous solidification. The Journal of Chemical Physics. 152 (2020)