Insidious errors in dipole modeling due to shell model misspecification in a spherical model were examined analyzing multiple time-points using the constraints of a commonly-used DSL (Dipole Source Localization) method. The computer simulation examined the differences in the fit dipole parameters for the same generator under two circumstances: 1) when computed as a single dipole active alone, and 2) when computed as a member of a simultaneously-active dipole pair. The computations were done using a simplification by which the dipole parameters computed from multiple time-points can be correctly assessed by computing dipole parameters at only two virtual time-points. Using multiple time-points in the DSL generally resulted in less error than if only a single time-point was used. However, how much improvement cna be achieved by using multiple time-points, as compared with a single time-point, is a function of many factors, such as the location and orientation of the dipoles, and the relative magnitudes and overlap of the waveforms (i.e., time-varying magnitudes) of the dipoles, as well as the model used in the fitting. Further, it was shown that it is incorrect to assume that a multiple-time-point DSL will compute a zero magnitude for generators during quiescent intervals. Additionally, it was shown that a "correction" to reduce error for one pair of waveforms will not be applicable to other waveforms. Also, even if location errors are eliminated, magnitude and orientation errors can still be shown to be present. Finally, iterative reduction of the least-square error between the observed and predicted surface maps leads to increasing errors in dipole parameters. We conclude that a DSL with model misspecification can contain insidious (undetectable) errors.
Considerable
unique chemical properties of nanoparticles are related
to their surface thermodynamic properties which mainly depend on their
sizes. Herein, the theoretical relationships between partial molar
surface thermodynamic properties as well as molar surface thermodynamic
properties and size were derived, respectively. It was found that
the ratio of molar surface thermodynamic properties to the corresponding
partial molar surface thermodynamic properties for spherical nanoparticles
is 3:2 during the dissolution process. In experiment, the standard
dissolution equilibrium constants and the thermodynamic properties
of nanovaterite with different sizes were obtained by determining
the equilibrium conductivities in distilled water at different temperatures.
Then, the surface thermodynamic properties were calculated, and the
size dependence of the surface thermodynamic properties were discussed
eventually. The results show that particle size has remarkable influence
on the surface thermodynamic properties; the molar surface Gibbs energy,
the molar surface enthalpy, and the molar surface entropy increase
with decreasing size and are linearly related to the reciprocal of
particle radius, respectively, when the radius is close to or exceeds
10 nm. The relationships between the surface thermodynamic properties
and particle size together with the quantitative influence regularities
of particle size on these properties are valuable to understand some
unique chemical processes involving nanoparticles.
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