Supplemental Material for Molecular Simulations of Heterogeneous Ice
Total Page：16
File Type：pdf, Size：1020Kb
Supplemental Material for Molecular simulations of heterogeneous ice nucleation I: Controlling ice nucleation through surface hydrophilicity Stephen J. Cox,1, 2 Shawn M. Kathmann,3 Ben Slater,1 and Angelos Michaelides1, 2, a) 1)Thomas Young Centre and Department of Chemistry, University College London, 20 Gordon Street, London, WC1H 0AJ, U.K. 2)London Centre for Nanotechnology, 17{19 Gordon Street, London WC1H 0AH, U.K. 3)Physical Sciences Division, Pacific Northwest National Laboratory, Richland, Washington 99352, United States (Dated: 22 April 2015) This Supplemental Material contains further details of the simulation methods and the data fitting procedure used. A movie and still images of a nucleation event at the NP with Eads=∆Hvap ≈ 0:9 are also provided. a)Electronic mail: [email protected] 1 I. FURTHER SIMULATIONS DETAILS To construct the NP, the ASE package1 was used: 6 atomic layers were used in the f1,0,0g family of directions; 9 layers in the f1,1,0g family; and 5 layers in the f1,1,1g family, except along the (1¯; 1¯; 1)¯ direction where no layers were used. As the equations of motion for the atoms in the NP were not integrated (i.e. they were fixed) no interaction potential was defined between them, and the adsorption energy Eads of a water monomer to the NP was therefore simply defined as the total energy after geometry optimization of a single water molecule at the center of the (111) face. The velocity Verlet algorithm was used to propagate the equations of motion of the water molecules, using a 10 fs time step. Temperature and pressure were maintained using the Nos´e-Hoover thermostat and barostat (with a chain length of 10) with relaxation times of 1 ps and 2 ps respectively. For each NP with a given Eads, a 100 ns trajectory was first performed at 290 K and 1 bar from which initial configurations (at least 5 ns apart) were extracted for the nucleation simulations. The initial velocities of the nucleation simulations were drawn randomly from a Maxwell-Boltzmann distribution with a target temperature of 205 K. Simulations were stopped after 500 ns if nucleation did not occur. In the mW potential, the interaction energies become negligibly small at distances smaller than the theoretical cutoff distance. LAMMPS therefore introduces a virtual cutoff based on a user defined tol parameter. This parameter reduces the cutoff by a distance jσSW= log(tol)j. In Fig. S1, we present the radial distribution function g(r) for 1400 bulk mW molecules at 210 K (averaged over a 10 ns trajectory), as well as the mean squared displacement (MSD), for different values of tol (for tol = 0, the theoretical cutoff is used). With tol = 1 × 10−5, the g(r) is indistinguishable from the case with tol = 0. We can also see that the MSD is well converged: the extracted diffusion coefficients are D = 0:7402 ± 0:03 cm2/s and D = 0:7335 ± 0:02 cm2/s for tol = 0 and tol = 1 × 10−5, respectively. All of our simulations have therefore employed tol = 1 × 10−5 for interactions between mW molecules, with a reduction of approximately 30% in computational cost. 2 FIG. S1. Converging the tol parameter for the interactions between mW molecules. A value of tol = 0 corresponds to the theoretical cutoff. For tol = 1 × 10−5, both the g(r) and the MSD are well converged. This reduces the computational cost by approximately 30%. II. FITTING PROCEDURE For each system, 16 simulations were performed. As nucleation is a stochastic process, even in the same system, we had to wait for different times to see nucleation. We can determine the induction time to nucleation tind for each trajectory by monitoring the time evolution of the potential energy and fitting to the equation: ∆U U(t) = U0 + : (S1) 1 + exp(k(t − tind)) In Eq. S1, U0, ∆U, k and tind are all freely variable parameters. Equation S1 is analogous to the Fermi-Dirac distribution, from which is should be clear that the parameter tind can be used to fit the drop in potential energy. See Fig. S2 (a) for an example. From the distribution of induction times to nucleation, for each system we are then able to calculate the probability that a simulation will be in the liquid state at a time t after the simulation has started (at t = 0): Nsim 1 X (i) P (t) = 1 − Θ(t − t ); (S2) liq N ind sim i=1 (i) where Nsim = 16 is the total number of simulations performed for each system, tind is the induction time determined for the ith simulation, and Θ(x) is the heaviside step function. Typical Pliq(t) data are shown in Fig. S2 (b). From the shape of these curves, we can quantify 3 a nucleation rate R for each system by fitting: γ Pliq(t) = exp [−(Rt) ] ; (S3) where γ > 0 is also a fitting parameter. The reason for using this form of the fitting function rather than the regular exponential function (i.e. with γ = 1) is because the nucleation simulations start from a non-equilibrium distribution of initial phase space points. This means that the system needs time to relax towards its equilibrium state. When nucleation is fast, for example in the presence of the NP for Eads=∆Hvap ≈ 0:4, this relaxation time and the time required for nucleation become comparable, and non-exponential kinetics is observed. Examples are given in Fig. S3. In these cases, we actually find compressed exponential kinetics (γ > 1). Such behavior has also been reported in protein folding models.2 We note that we have used the same protocol for all simulations i.e. extracting a simulation from 290K for each system, and then giving the water molecules random momenta from the Maxwell-Boltzmann distribution at 205 K. This ensures that any dependence on the choice of initial conditions is equal for all systems investigated. Furthermore, we use the same simulation protocol in Ref. 3 to investigate ice nucleation at graphitic surfaces, and find good agreement with previous work4,5 that has employed non-equilibrium temperature ramps and isothermal simulations. Furthermore, for homogeneous ice nucleation we obtain −1 Rhom = 0:0210 ns , which appears to be in good agreement with the average time to crystallization τx ≈ 40−50 ns presented in Fig. 2(a) of Ref. 6. The actual fits were obtained using version 8.6 of the OriginPro software package from OriginLab, Northampton, MA. OriginPro uses the Levenberg-Marquardt algorithm to adjust the parameter values in an iterative procedure. It also reports standard errors SR and Sγ in the fitted parameters R and γ, respectively, by calculating the square root of the diagonal elements of the variance-covariance matrix. The error in the ratio is then calculated as: s 2 2 R SR SRhom SR=Rhom = + ; (S4) Rhom R Rhom and the error in the logarithm is computed as: S S = R=Rhom : (S5) log10 R ln(10) Rhom It is this value that is indicated by the error bars in Fig. 1 in the main article. 4 FIG. S2. In (a) we show an example of how the induction time to nucleation tind for a trajectory is determined. The black line shows the potential energy extracted from the simulation, and the red line shows a fit to these data using Eq. S1. For each system we obtain a distribution of tind, and by Eq. S2 we can calculate Pliq(t), shown in (b). Data for bulk homogeneous nucleation is shown by filled circles. The other data show typical Pliq(t) curves obtained in the presence of the NP (different symbols correspond to different values of Eads=∆Hvap). Specifically: olive triangles, 0.08; orange pentagons, 0.11; magenta stars, 0.14; cyan diamonds, 0.15; blue triangles, 0.30; red squares, 0.60; royal-blue crosses, 0.75; pink cross-hairs, 0.90; and brown triangles, 1.20. III. NUCLEATION EVENT AT NP WITH Eads=∆Hvap ≈ 0:9 Fig. S4 shows a nucleation event at the NP with Eads=∆Hvap ≈ 0:9. For nucleation to happen, a decreased coverage of water molecules at the surface must occur so that an hexagonal structure can form in the contact layer. A movie of this nucleation event is also supplied. REFERENCES 1S. R. Bahn and K. W. Jacobsen, Comput. Sci. Eng. 4, 56 (2002), ISSN 1521-9615. 2P. Hamm, J. Helbing, and J. Bredenbeck, Chem. Phys. 323, 54 (2006). 3S. J. Cox, S. M. Kathmann, B. Slater, and A. Michaelides, submitted -, (2014). 4L. Lupi, A. Hudait, and V. Molinero, J. Am. Chem. Soc. 136, 3156 (2014), http://pubs. acs.org/doi/abs/10.1021/ja411507a. 5 FIG. S3. In (a) we show Pliq(t) data for the NP with Eads=∆Hvap ≈ 0:15, 0:30 and 0:60, and their corresponding fits using Eq. S3. For these systems nucleation is fast and the Pliq(t) is clearly non- exponential, which is reflected by the obtained fitted values of γ = 4:79, 5:25 and 3:08, respectively. In (b), Pliq(t) is shown for Eads=∆Hvap = 0:08 and 0:11 and for the bulk homogeneous system. In these systems, nucleation is a much slower process than relaxation from the initial state, and exponential kinetics is observed. In these cases, γ = 0:96, 1:02 and 0:98 for the homogeneous and Eads=∆Hvap = 0:08 and 0:11 systems, respectively. 5L. Lupi and V. Molinero, J. Phys. Chem. A 118, 7330 (2014), http://pubs.acs.org/doi/ abs/10.1021/jp4118375. 6E. B.