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

 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

 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.