Numerical investigation of microscale dynamic contact angles of the CO₂–water–silica system using coarse-grained molecular approach

Abstract

The dynamic contact angle of a gas–liquid–solid system depends on the contact line velocity and ignoring this effect could lead to inaccurate estimations of the capillary pressures in microporous media. While most existing coarse-grained molecular dynamics (CGMD) models use one particle to represent a few molecules, we present a novel CGMD framework to model microscale CO₂/water flows in silica with each particle representing hundreds of thousands of molecules. The framework can reproduce the densities and viscosities of water and CO₂, water–CO₂ interfacial tension, and static contact angle over a wide range of pressures. The validated framework is applied to study the velocity-dependency of contact angle of the microscale CO₂–water–silica system. The results indicate that the assumption in the molecular kinetic theory that liquid–solid interaction is similar to the reversible work of adhesion between liquid and solid may not hold for CO₂–water–silica systems.

Appendices

Appendices

1.1 Appendix 1: Calculation of density

The calculations of density at different pressures for MDPD and DPD particles follow a procedure similar to our previous work [37]. 8000 particles are created in a cubic simulation box with periodic boundary conditions in all three directions. The system is run in the NPT ensemble at a constant temperature T^* and pressure P^* for 100,000 time steps. The instantaneous density is calculated at every 20 time steps. The average density and the standard deviation are calculated from the instantaneous density obtained from the last 50,000 time steps. A time step Δt^* = 0.0025 is used.

1.2 Appendix 2: Calculation of viscosity

For the calculation of viscosity at a particular pressure, a system composed of 3000 particles (MDPD or DPD) is created in a periodic simulation box with a particle density calculated at the studied pressure. The aspect ratio of the simulation box is L^*_x :L^*_y :L^*_z  = 1:1:3. The system is first equilibrated with a Langevin thermostat for 100,000 time steps. The Müller-Plathe algorithm is then applied to calculate the viscosity [64]. A schematic diagram of the simulated system is shown in Fig. 11. The system is divided into 20 layers in the z-direction. At every 50 time steps, the particle in the first layer with the largest speed in the positive x-direction is selected. Similarly, the particle in the middle layer (11^th layer) with the largest speed in the negative x-direction is selected. The x-components of the momenta of the two particles are then exchanged. The exchange of momenta induces a shear velocity profile in the system with periodic boundary conditions. The simulations are run for 1,100,000 time steps at the NVT ensemble using the Nosé–Hoover chains thermostat. The first 100,000 time steps are considered as the equilibrium run, for the development of the velocity profile. The last 1,000,000 time steps are considered as the production run. The velocity profile is calculated at every 10,000 time steps by averaging the x-component of the particle velocities in each layer. The total momentum exchanged p^*_x is calculated at every 10,000 time steps. In the steady state, the momentum flux J^* can be calculated as:

$$ J^{*} = p_{x}^{*} /2t^{*} L_{x}^{*} L_{y}^{*} , $$

(22)

where t^* is the time duration of the production run; L^*_x and L^*_y are the length of the simulation box in the x- and y-directions, respectively. The factor of 2 is used here due to the periodic boundary conditions applied. Therefore, the viscosity η^* can be calculated by:

$$ J^{*} = - \eta^{*} \frac{{\partial v_{x}^{*} }}{{\partial z^{*} }}, $$

(23)

where ∂v^*_x /∂z^* is the gradient of the x-component of the fluid velocity with respect to the z-direction within the region between the first and middle layers. The averaged viscosity can be calculated using Eqs. (22) and (23) from the production run. The value of J^* is obtained at the end of the simulation. ∂v^*_x /∂z^* is calculated from the slope of the time-averaged velocity profile by linear regression at every 10,000 time steps. The average of ∂v^*_x /∂z^* is then calculated from the 100 samples of the slope. The simulations are repeated five times for each case, and the average and standard deviation of the viscosity are then calculated accordingly.

Fig. 11

A schematic diagram of the simulated system for the calculation of viscosity using the Müller–Plathe algorithm [64]. The x-component velocity profile is plotted in red colour. (Color figure online)

1.3 Appendix 3: Calculation of interfacial tension

The CO₂–water interface simulation is performed to calculate the interfacial tension of the CO₂–water system at a certain T^* and P^*. To construct the system, an 8000-water-particle cubic box, which is equilibrated at targeted T^* and P^* with periodic boundary conditions applied in all three dimensions, is prepared. The CO₂ box is also prepared with the same size as the water box. The particle numbers of CO₂ in the box is calculated from the equilibrium density at the targeted T^* and P^*. The two boxes are replicated twice in the z-direction and placed together in a periodic simulation box with a ratio of L^*_x :L^*_y :L^*_z  = 1:1:4. An energy minimization process is applied to the system at the beginning to avoid the particle overlapping at the two CO₂–water interfaces. The water phase is first relaxed by allowing the water particles to move at the Langevin thermostat for 10,000 time steps, while the CO₂ particles are fixed. The CO₂ phase is then relaxed at the Langevin thermostat with water particles being fixed for 10,000 time steps. Next, the whole system is relaxed together at the Langevin thermostat for another 10,000 time steps. An example of a relaxed system is shown in Fig. 12. The relaxed system is then further equilibrated under the NP_zT ensemble for 100,000 time steps to reach the targeted T^* and P^*. The system is then switched to the NVT ensemble and run for 1,000,000 time steps. The last 600,000 time steps are considered as the production run. The interfacial tension γ^* is calculated as:

$$ \gamma^{*} = \frac{1}{2}L_{z}^{*} \left\langle {P_{zz}^{ *} - \frac{{P_{xx}^{ *} + P_{yy}^{*} }}{2}} \right\rangle , $$

(24)

where P^*_xx , P^*_yy , and P^*_zz are the x-, y-, and z-components of the virial pressure of the system; L^*_z is the length of the simulation box in the z-direction; the time averages are performed inside of < > for the data recorded every 20 time steps. The statistical errors of the surface tension are estimated by calculating the statistical inefficiency [42, 65, 66].

Fig. 12

A snapshot of the system of the water–CO₂ interface simulation after being relaxed at the Langevin thermostat

1.4 Appendix 4: Water bridge simulation

A liquid bridge simulation is performed to ensure that the selected parameters of water–silica interactions would be able to generate a complete wetting of the silica surface. A periodic simulation box with L^*_x  = 50.67, L^*_y  = 10.48, and L^*_z  = 26.38 is created with water sandwiched between two silica slabs. The particle density of water is initially close to the bulk water density of the equilibrium water–vapour system. The water particles are equilibrated at the Langevin thermostat for 100,000 time steps, with the fixed solid particles. Next, the left and right portions of the water particles are deleted with around one-third of the water particles remaining in the middle between the two silica slabs. Figure 13 shows a snapshot of the initial state of the remained water particles between the two silica slabs. The system is further equilibrated under the NVT ensemble with a Nosé–Hoover chains thermostat until it reaches equilibrium. If the water particles spread over and cover all the silica surfaces inside, it indicates a complete wetting of the simulated silica surface.

Fig. 13

A snapshot of the initial state of the system for the liquid bridge simulation

1.5 Appendix 5: Contact angle measurements

There are different ways of measuring contact angles. In this work, the contact angle is measured by fitting a circle to the interface by assuming a circular shape of the interface. The simulation box is divided into small square bins with a size of 0.5r_c × 0.5r_c (equivalent to 0.01768 × 0.01768 μm²) in the x–z plane, and the time-averaged density in each bin is calculated at every 100,000 time steps throughout the simulations, with instantaneous density recorded every 20 time steps. For the simulations with the piston moving at a large velocity greater than 0.7 m/s, the 2D density profile is calculated at every 10,000 time steps and the movement of the piston in the x-axis is also checked at every 10,000 time steps. When the piston has moved one bin width forward (0.5r_c), the calculated density profile should be shifted one bin width backward in order to have a stable water–CO₂ interface at large piston velocity. In addition, the first bin at the x-direction should be moved to the end due to the periodic boundary condition applied along the x-axis. Hence, the shifted density profiles are averaged again to obtain the density profile at every 100,000 time steps, in order to calculate the contact angle. The 2D density contour is plotted using Matlab. The location of the interface between CO₂ and water can be calculated from the 2D density contour by calculating the density at the interface ρ_int = 0.5(ρ_water + ρ_CO2), where ρ_water and ρ_CO2 are the bulk densities of water and CO₂, respectively, which can be estimated from the bulk regions in water and CO₂ phases. Since the density perturbations can happen near the solid surface, only the interface data that are 2r_c away from the solid surface are used to fit the circle. An example of the circular fitting is shown in Fig. 14. The contact angle is calculated from the tangent of the circle where the circle and solid surfaces meet. The contact angle values are calculated by averaging all the top and bottom contact angles during the production run, and the standard deviations of the contact angles are calculated accordingly.

Fig. 14

The measurement of the contact angle by fitting a circle to the interface. The markers of small circles are the coordinates of the water–CO₂ interfaces. The solid blue lines indicate the location of the solid surface, and the interface data between the two blue dashed lines are used in the circle fitting