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Uncertainty quantification in MD simulations of concentration driven ionic flow through a silica nanopore. I. Sensitivity to physical parameters of the pore
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10.1063/1.4804666
/content/aip/journal/jcp/138/19/10.1063/1.4804666
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/19/10.1063/1.4804666

Figures

Image of FIG. 1.
FIG. 1.

(a) Schematic of the dual-reservoir nanopore system. The silica is a rectangular block with a cylindrical pore of nominal radius . Only few surface layers of the silica block have dynamics (gray), shown in panel (b), while the rest comprises the “frozen” atoms (green). The volume of fluid is nominally .

Image of FIG. 2.
FIG. 2.

Silica pore model of dimensions (, , ) ≈ 54, ≈ 60, and Å obtained from a bulk crystal silica structure after carving a pore of nominal radius , ensuring stoichiometry, and hydroxylating the surface and pore defects. Color legend: silicon Si (ochre), bulk oxygens O (blue), hydroxide oxygens O (yellow), and hydroxide hydrogens (black).

Image of FIG. 3.
FIG. 3.

Snapshots of cross-sectional views of the systems for = 12.5 (a), 17 (b), 21 (c), and 27 Å (d), taken during the steady state of the CC stage. Color legend: Si (ochre), O (blue), O (yellow), (black), O (red), H (white), Na (purple), Cl (cyan). The silica is nearly transparent for visualization convenience.

Image of FIG. 4.
FIG. 4.

(a) Time/bin-averaged radial profile of the axial velocity , for Na and Cl computed for one replica of the case = 21 Å during the steady phase of the CC stage. (b) Time/bin-averaged axial velocity plotted as a function of the -coordinate computed for one replica of = 21 Å at the final step of the CC stage for water, Na and Cl. In (a), for the sake of visualization, the data obtained from the spatial binning are interpolated over a finer mesh. In (b) the filled circles are plotted at the location of the 24 bins.

Image of FIG. 5.
FIG. 5.

Time evolution of the running average conductance, (), for Na (a) and Cl (b) plotted for all 5 replicas and each diameter value showing the variation in steady state values and the timescale at which steady values are achieved.

Image of FIG. 6.
FIG. 6.

(a) Steady-state replica values (markers) of the Na (blue) and Cl (red) conductance as a function of the nominal pore diameter , with the superimposed lines outlining the replica-averaged values. (b) Coefficient of variation, σ/μ, i.e., the ratio of the standard deviation, σ, over the mean, μ, of the conductance data plotted as function of the pore diameter, , for both ions.

Image of FIG. 7.
FIG. 7.

Snapshots obtained for = 12.5 (a), 17 (b), 21 (c), and 27 Å (d), during the steady state of the CC stage, showing the distribution of water molecules around the ions passing through the pore: Na is color-coded purple, while Cl is color-coded cyan.

Image of FIG. 8.
FIG. 8.

Panel (a) Data set of the ionic conductance, , computed for all replicas as a function of the gating charge density for Na and Cl. The superimposed solid lines connect the replica-averaged values. Panel (b) The corresponding variances plotted as function of with a logarithmic scale on the -axis. Note that since = 0.265619083 C/m yields vanishing Na conductance for all three replicas, the corresponding variance is not shown in panel (b).

Image of FIG. 9.
FIG. 9.

(Top row) Scatter plots of 20 000 MCMC chain samples obtained for π( , ) for Na (a) and Cl (b), extracted from the original MCMC chain by removing the burn-in period comprising the first 15 000 samples. (Bottom row) Marginalized posterior, π( ), of each PC coefficient , = 0, …, 4, obtained via KDE for Na (c) and Cl (d). All panels show the results for a linear ( = 1), quadratic ( = 2), cubic ( = 3), and quartic ( = 4) expansion.

Image of FIG. 10.
FIG. 10.

Panel (a) shows the comparison between the data points of the Na (black circles) conductance and the corresponding predictions obtained from the MAP estimate of the PC regression function, (ξ), computed for a linear ( = 1), quadratic ( = 2), cubic ( = 3), and quartic ( = 4) expansion as a function of . The corresponding results for Cl are shown in panel (b).

Image of FIG. 11.
FIG. 11.

Panel (a) shows the data-based variance (black circles) of the Na conductance versus the corresponding predictions obtained from the MAP estimate of the noise PC coefficients { , }, as a function of the order, , of the regression function, (ξ). The corresponding results for Cl are shown in panel (b). Both plots are presented with a log scale on the -axis.

Image of FIG. 12.
FIG. 12.

Results showing the posterior predictive check samples (gray) obtained for the Na (a) and Cl (b) conductance using a third-order and fourth-order PC representation, respectively. Superimposed to the plots, we report the original data color-coded blue for Na and red for Cl, and the mean (black square) and error bars for , where is the standard deviation calculated from the original data set of conductances used in the inference.

Image of FIG. 13.
FIG. 13.

Results of the “model uncertainty” analysis obtained for Na (a) and Cl (b) showing how the posterior uncertainty in the inferred PC representations of the conductances is reflected in the corresponding predictions.

Tables

Generic image for table
Table I.

Lennard-Jones parameters (ɛ, σ) and Coulombic charge, in multiples of the electron charge ||, for each atom type present in the system: {H, O} are water hydrogen and oxygen atoms, respectively, {H, O} are hydrogen and oxygen atoms appearing in an hydroxide group, O are oxygen atoms in the bulk of the silica, Si are silicon atoms, and {Na, Cl} are the salt ions. The Lorentz-Berthelot mixing rules , and are used to define the interspecies Lennard-Jones interactions. For the remaining set of force-field parameters, refer to Ref. for , to Refs. for , and to Ref. for the .

Generic image for table
Table II.

Computed values of , obtained for all four different models and each ion Na and Cl. Note that due to the logarithmic scale adopted, for a given ion, the corresponding matrix of values is antisymmetric.

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/content/aip/journal/jcp/138/19/10.1063/1.4804666
2013-05-17
2014-04-24
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Uncertainty quantification in MD simulations of concentration driven ionic flow through a silica nanopore. I. Sensitivity to physical parameters of the pore
http://aip.metastore.ingenta.com/content/aip/journal/jcp/138/19/10.1063/1.4804666
10.1063/1.4804666
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