# 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. |