Chapter 10. Surface Tension of Liquid H2O and H2O on PTFE

10.1. Introduction

Determining the surface tension of bulk liquids is a very difficult problem with standard small-cluster or infinite periodic boundary condition simulation methods. Large-scale molecular mechanics offers the opportunity to use large clusters which more accurately simulate bulk properties and thus promises to enable the prediction of surface tensions.

10.2. Drop Procedure

Water cubes from Jorgensen et al. [1] were used to build large supercells. In particular, a 216 molecule cube was replicated 8 times in each Cartesian direction to form a 110,000 molecule box.

A sphere of radius 50 Å was then extracted from the center of this large box. The sphere contained 17,254 molecules or 51,762 atoms. The initial sphere density was 0.94 g/mm^3.

Microcanonical dynamics at 300 K was then performed on the system. A timestep of 1 fs was used, though longer timesteps should be feasible. An iterative rigid molecule procedure based on SHAKE [2] was used to constrain the bonds and angles in the water molecules. The temperature was rescaled periodically, every 10 fs. The CMM parameters used were a maximum level of 5 and farfield updates performed every timestep. The TIP3P forcefield parameters [3] were used to describe the water molecules.

Ideally, reflecting boundary conditions would be used to enable the formation of a vapor atmosphere around the drop. The performed simulations, however, were for relatively short times and hence had relatively little boil-off of surface molecules.

10.3. Drop Results

After 2450 timesteps, the system temperature was stable and the total and potential energies were also essentially constant, indicating that equilibration was achieved.

Visualization of the system shows that the water drop has lost the periodicity derived from the initial supercell; the molecular positions have been thoroughly randomized. Some shrinkage has occurred. The RMS radius from the center of mass decreased from 38.69 Å to 35.38 Å. Using R_sphere = sqrt(5/3) R_rms, the computed radius of the final sphere is 45.68 Å, a shrinkage of about 10%. The resulting density is 1.29 g/mm^3.

10.4. PTFE Simulation

Standard polymer simulations typically use one or a few chains or chain fragments, each composed of a handful of monomer units. These are then placed in a unit cell with periodic boundary conditions.

This approach has several significant limitations. Molecular weights are either very small or else infinite, both of which are unphysical. Interactions between a chain and the images of other chains, or even its own image, in a neighboring unit cell may lead to unphysical correlations.

Using large-scale molecular mechanics, a different model for bulk polymers may be generated that may better reproduce physical properties. Since thousands or millions of atoms may now be handled, molecular weights can be increased to the range of 10^5 or 10^6 typical of experimental values. In addition, the correlation problem may be avoided by using a very large, but finite system. This has the disadvantage of introducing edge effects, but hopefully the behavior in the center of the system will accurately reproduce the bulk. Perhaps better still is to use a very large unit cell with periodic boundary conditions, reducing the correlation problem to negligible levels.

10.5. Procedure

A chain of 143 monomer units (431 atoms) was built in two forms: a left-handed helix and a right-handed helix. The helix parameters were obtained from experiment [4].

Sets of helices were then packed together into hexagonal structures using rotations (about the helical axis) and translations (both in the X-Y plane to generate the hexagonal lattice and in the Z axis). The lattice parameters, helical rotations, and Z axis translations were determined from experimental values. In Figure 10-1, the left- and right-handed helices are depicted as circles with appropriate letters. The spacing between two helices of the same handedness in the vertical direction is 5.648 Å, while the spacing in the horizontal direction is 9.649 Å. If the left-handed helix at the bottom of the figure is assigned a Z coordinate of zero, then the right-handed helix above and to the left of it will be offset 0.15 Å in the positive Z direction (out of the plane of the paper). The Z offset between one helix and the next of the same handedness in the vertical direction, as indicated by the arrow, is 0.12 Å; the Z offset in the horizontal direction is 0.05 Å. These parameters are sufficient to allow a system composed of any number of helices to be built.

[IMAGE]
Figure 10-1. Experimental PTFE lattice geometry.

These hexagonal models expose three types of surfaces: one composed of the sides of helices having the same handedness (e.g. the right or left surfaces in Figure 10-1), one composed of the sides of helices having alternating handednesses (e.g. the angled surfaces in the figure), and one composed of the ends of the helices.

The sizes of the models used ranged from 7 chains (3000 atoms) to 2611 chains (1.1 million atoms).

Microcanonical dynamics was performed for 2.4 ps with a timestep of 2 fs. Since only heavy atoms are present, timesteps longer than the usual 1 fs are feasible for this system.

The forcefield used included accurate torsions with terms up to cos(12\phi) from quantum mechanical calculations on small model systems [5]. The charges used were also optimized based on simpler systems.

10.6. Results

The carbon backbones of the polymer chains were visualized to assess changes that occurred during the dynamics.

As expected, edge effects were seen, with the ends of surface chains and loops in the centers of surface chains moving into the vacuum. For the smaller systems, these effects dominated any bulk behavior.

[IMAGE]
Figure 10-2. End view of 2611 chain PTFE system after dynamics.

[IMAGE]
Figure 10-3. Enlargement of 2611 chain PTFE system after dynamics.

Figure 10-2 is a snapshot of the final state of the million-atom system. An enlargement of the central portion of this system is displayed in Figure 10-3.

Disorder of the helices occurred surprisingly readily, even in the central bulk portion of the system. With less accurate calculations, supercoiling of the helices was observed. It is possible that longer dynamics runs will produce the same effect in this calculation.

10.7. PTFE Surface Procedure

Two cubes of 216 water molecules were placed on the surface of a large, 1.1 million atom hexagon of PTFE. The polymer molecules were fixed in place in the experimental configuration, while the water was allowed to move. One cube was placed in the center of a surface (a YZ plane) composed of sides of helices, all with the same handedness, while the other was placed on a surface (an XY plane) composed of the ends of the helices. The side-surface drop is exposed to CF2 groups only, while the end-surface drop is primarily exposed to CF3 groups.

The positions of the cubes were adjusted so that the side of the cube was the TIP3P oxygen van der Waals radius away from the nearest PTFE atom.

Microcanonical dynamics at 300 K was then performed. CMM parameters were: level 6, farfield update every 5 steps. The temperature was rescaled to 300 K every 200 fs. The iterative SHAKE constraints were used. Timesteps of 2 fs were found to be feasible and were used.

The PTFE parameters from the previous section were used, along with TIP3P parameters for the water molecules.

10.8. PTFE Surface Results

After 24.4 ps, the water on PTFE had formed drops, showing no evidence of the original cubic shape. 7 molecules had boiled away from the drop on the CF3 end surface, while 10 had left the CF2 side surface drop. The drop shapes are shown in Figures 10-4 and 10-5. Qualitatively, we observe that the CF3 drop appears to extend farther in the direction perpendicular to the surface (the Z direction) and also seems to have a smaller contact area with the surface. This meshes with our expectation that the end surface should have lower surface tension, causing water to wet it less easily.

[IMAGE]
Figure 10-4. Water drop on CF3 surface of PTFE.

[IMAGE]
Figure 10-5. Water drop on CF2 surface of PTFE.

To quantitate these observations, the distances of the centers of mass of the drops from the surface were computed. The surface was defined as the minimum value of the X or Z coordinate over all the drop's molecules for the CF2 and CF3 surfaces, respectively. The center of mass for the CF2 drop was found to be 6.73 Å away from the surface, while that of the CF3 drop was found to be 9.78 Å distant. The CF3 drop is thus in fact elongated in the surface-normal direction compared with the CF2 drop. Since both drops have approximately the same number of molecules, the cross-sectional area of the CF3 drop must also be smaller than that of the CF2 drop.

We can compute moments of the drops along the three axes to further investigate the differences between them. We find that the root-mean-square values of the coordinates parallel to the surface are 9.35 by 8.07 Å for the CF3 drop and 9.01 by 10.88 Å for the CF2 drop, confirming the smaller surface area for the former.

Since the number of water molecules in each drop is relatively small, the drops are distorted from the ideal spherical sections. It is thus difficult to compute quantitative surface tension values from these results.

10.9. Conclusions

Proof-of-principle calculations were performed on a large-scale spherical drop of water and a million-atom finite PTFE system. These calculations demonstrated the ability of the code to handle systems much larger than could be used on ordinary workstations. The results of these simple calculations indicate possibly fruitful further directions for investigation.

For the water drop, future work with reflecting boundary conditions and quaternion rigid molecules should be able to make contact with experimental studies of drops on the scale of 0.1 um. For example, a drop containing 5 million atoms, the same size as previously-simulated argon clusters, would have a diameter of about 0.05 um.

The simulations of PTFE systems showed surprising amounts of disorder in the interior of the crystal. Further investigation of this disorder and possible supercoiling will be necessary.

The surface tension calculations used systems of greater than one million atoms. Even though most of those were fixed in place, their effects on the surface atoms were still computed rigorously. Use of the large, finite PTFE crystal allowed simulations to occur simultaneously on multiple crystal surfaces. The surfaces were large enough to avoid edge effects. The results show that drop formation can be simulated on this small scale, and they demonstrate the expected trends in drop shape, with the CF3 surface showing a significantly lower tendency to be wetted by the drop.

Future work in this area would include removing the constraint on the polymer, allowing it to respond to motions of the water on the surface, and use of larger drops with quaternion rigid body dynamics instead of the constraint-based technique used here.


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Kian-Tat Lim