Output
Depending on the quantity, the requirements for calculations are
different. Some quantities require the exterior calculation, some the
interior, and some both. Additionally, some quantities are direct
outputs, while others are indirect—requiring only the direct outputs
coupled with algebraic expressions. Finally, some special quantities
require optional input such as the temperature. Each of these
dependencies are listed.
Capacitance
| Equation: |
\(C = tR\) |
| Calculation: |
Direct exterior |
| Explanation: |
\(t\) is the fraction of
random walks that hit the object
as opposed to go to infinity, and
\(R\) is the radius of the
launch sphere. |
| Uncertainty: |
Determined by directly estimating
the variance of \(t\) and
then applying propagation of
uncertainties. |
| Units: |
Length |
Electric polarizability tensor
| Equation: |
\(\mathbf{\alpha}\) is
expressed in
Ref. [9]. |
| Calculation: |
Direct exterior |
| Explanation: |
Several different counting
variables from the exterior
calculation are combined to
evaluate this quantity. |
| Uncertainty: |
Determined by directly estimating
the variances of \(t\),
\(u\), \(v\), and
\(w\) from
Ref. [9]
and then applying propagation of
uncertainties. |
| Units: |
Length cubed |
Eigenvalues of electric polarizability tensor
| Calculation: |
Indirect exterior |
| Explanation: |
The eigenvalues of the previously
computed electric polarizability
tensor \(\mathbf{\alpha}\)
are determined. |
| Uncertainty: |
Determined via propagation of
uncertainties. |
| Units: |
Length cubed |
Mean electric polarizability
| Equation: |
\(\langle \alpha \rangle =
\mathrm{Tr}(\mathbf{\alpha})/3\) |
| Calculation: |
Indirect exterior |
| Explanation: |
The trace of the previously
computed electric polarizability
tensor \(\mathbf{\alpha}\) is
computed and then divided by
three. |
| Uncertainty: |
Determined via propagation of
uncertainties. |
| Units: |
Length cubed |
Intrinsic conductivity
| Equation: |
\([\sigma]_\infty = \langle
\alpha \rangle/V\) |
| Explanation: |
\(\mathbf{\alpha}\) is the
electric polarizability tensor;
\(V\) is the volume. |
| Calculation: |
Indirect exterior and interior |
| Uncertainty: |
Determined via propagation of
uncertainties. |
| Units: |
None |
Volume
| Equation: |
\(V= p \frac{4}{3} \pi R^{3}\) |
| Explanation: |
\(p\) is the fraction of
points inside the object;
\(R\) is the radius of the
launch sphere. |
| Calculation: |
Direct interior |
| Uncertainty: |
Determined by directly estimating
the variance of \(p\) and
then applying propagation of
uncertainties. |
| Units: |
Length cubed |
Gyration tensor
| Equation: |
\(\mathbf{S}\) is expressed
in
Ref. [11] |
| Explanation: |
Several different counting
variables from the interior
calculation are combined to
evaluate this quantity. |
| Calculation: |
Direct interior |
| Uncertainty: |
Determined by directly estimating
the variance of sums and the sums
of products of interior sample
point coordinates, and then
applying propagation of
uncertainties. |
| Units: |
Length squared |
Eigenvalues of gyration tensor
| Calculation: |
Indirect interior |
| Explanation: |
The eigenvalues of the previously
computed gyration tensor
\(\mathbf{S}\) are
determined. |
| Uncertainty: |
Determined via propagation of
uncertainties. |
| Units: |
Length squared |
Capacitance of a sphere of the same volume
| Equation: |
\(C_0 = \left(3V/(4\pi)\right)^{1/3}\) |
| Calculation: |
Indirect interior |
| Explanation: |
\(V\) is the volume of the object. |
| Uncertainty: |
Determined via propagation of uncertainties. |
| Units: |
Length |
Hydrodynamic radius
| Equation: |
\(R_{h}=q_{R_{h}}C\) |
| Explanation: |
\(q_{R_{h}}\approx 1\), and
\(C\) is the capacitance. |
| Calculation: |
Indirect exterior |
| Uncertainty: |
Determined via propagation of
uncertainties assuming the
standard deviation of
\(q_{R_{h}}\) is
\(0.01\). |
| Units: |
Length |
Prefactor relating average polarizability to intrinsic viscosity
| Equation: |
\(q_\eta\) varies slowly with
shape and is expressed in
Ref. [8] |
| Calculation: |
Indirect exterior |
| Explanation: |
The electric polarizability
tensor plus a complicated Padé
approximate is used to determine
this quantity. |
| Uncertainty: |
\(0.015q_\eta\) |
| Units: |
None |
Viscometric radius
| Equation: |
\(R_{v}= (3 q_\eta \langle
\alpha \rangle/(10 \pi))^{1/3}\) |
| Explanation: |
\(q_\eta\) is the prefactor
for the intrinsic viscosity, and
\(\langle \alpha \rangle\) is
the mean polarizability. |
| Calculation: |
Indirect exterior |
| Uncertainty: |
Determined via propagation of
uncertainties. |
| Units: |
Length |
Intrinsic viscosity
| Equation: |
\([\eta]=q_\eta
[\sigma]_\infty\) |
| Explanation: |
\(q_\eta\) is a prefactor,
and \([\sigma]_\infty\) is
the intrinsic conductivity. |
| Calculation: |
Indirect exterior and interior |
| Uncertainty: |
Determined via propagation of
uncertainties. |
| Units: |
None |
Intrinsic viscosity with mass units
| Equation: |
\([\eta]_{m}=q_\eta \langle
\alpha\rangle/m\) |
| Explanation: |
\(q_\eta\) is the prefactor,
\(\alpha\) is the
polarizability tensor, and
\(m\) is the specified mass. |
| Calculation: |
Indirect exterior |
| Uncertainty: |
Determined via propagation of
uncertainties. |
| Units: |
Length cubed / mass |
| Requirements: |
Specified mass. |
Friction coefficient
| Equation: |
\(f = 6\pi\eta R_h\) |
| Explanation: |
\(\eta\) is the solvent
viscosity, and \(R_h\) is the
hydrodynamic radius. |
| Calculation: |
Indirect exterior |
| Uncertainty: |
Determined via propagation of
uncertainties. |
| Units: |
Mass / time |
| Requirements: |
Specified length scale and
solvent viscosity. |
Diffusion coefficient
| Equation: |
\(D = k_{B}T/f\) |
| Explanation: |
\(k_{B}\) is the Boltzmann
constant, \(T\) is the
temperature, and \(f\) is the
friction coefficient. |
| Calculation: |
Indirect exterior |
| Uncertainty: |
Determined via propagation of
uncertainties. |
| Units: |
Length squared / time |
| Requirements: |
Specified length scale, solvent
viscosity, and temperature. |
Sedimentation coefficient
| Equation: |
\(s = mb/f\) |
| Explanation: |
\(m\) is the mass, \(b\)
is the buoyancy factor, and
\(f\) is the friction
coefficient. |
| Calculation: |
Indirect exterior |
| Uncertainty: |
Determined via propagation of
uncertainties. |
| Units: |
Time |
| Requirements: |
Specified length scale, solvent
viscosity, mass, and buoyancy
factor. |