Solution Tutorial — Spherical Diffusion

This tutorial describes the analytical solutions underlying the Spherical Diffusion applet. The model solves three scenarios of radial diffusion involving a sphere, all governed by the same radial diffusion equation (Section 5) but with different boundary and initial conditions.

For a guide to the applet controls and interface see the User Interface Tutorial.

1 Overview of Models

All three models use the analytical solutions of Crank (1975) for radial diffusion in a sphere. They differ in the initial and boundary conditions:

Model 1 — Diffusion into an initially empty sphere held at a constant outside concentration (uptake).
Model 2 — Diffusion out of a uniformly contaminated sphere into an infinite bath (release).
Model 3 — Two-phase process: the sphere is first loaded for time $t_0$ using diffusion coefficient $D_2$, creating a non-uniform profile; then mass diffuses out with coefficient $D_1$.

Each model plots (i) the concentration profile $C(r,t)$ at up to five snapshot times, and (ii) the fractional approach to equilibrium $f(t) = M_t / M_\infty$.

2 Model 1 — Diffusion into Empty Sphere from Infinite Bath

At time zero the sphere is empty ($C_1 = 0$). The concentration outside the sphere is held constant at $C_0$ for all $t > 0$. Mass diffuses inward.

Concentration Profile

$$\frac{C - C_1}{C_0 - C_1} = 1 + \frac{2a}{\pi r}\sum_{n=1}^{\infty} \frac{(-1)^n}{n}\sin\!\left(\frac{n\pi r}{a}\right) \exp\!\left(-\frac{Dn^2\pi^2 t}{a^2}\right)$$

(1)

Fractional Uptake

$$f = \frac{M_t}{M_\infty} = 1 - \frac{6}{\pi^2}\sum_{n=1}^{\infty} \frac{1}{n^2}\exp\!\left(-\frac{Dn^2\pi^2 t}{a^2}\right)$$

(2)

Figure 1. Concentration profiles — diffusion into sphere (Model 1).

Figure 2. Fractional approach to equilibrium (Models 1 & 2).

3 Model 2 — Diffusion Out of Uniformly Contaminated Sphere

At time zero the sphere is uniformly loaded at concentration $C_0$. The outside concentration is zero for all $t > 0$ (infinite bath). Mass diffuses outward.

Concentration Profile

$$\frac{C}{C_0} = \frac{2a}{\pi r}\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}\sin\!\left(\frac{n\pi r}{a}\right) \exp\!\left(-\frac{Dn^2\pi^2 t}{a^2}\right)$$

(3)

Fractional Release

$$f_\text{rel} = \frac{M_t}{M_0} = 1 - \frac{6}{\pi^2}\sum_{n=1}^{\infty} \frac{1}{n^2}\exp\!\left(-\frac{Dn^2\pi^2 t}{a^2}\right)$$

(4)

Note: Eq. 4 (fractional release) has the same mathematical form as Eq. 2 (fractional uptake), but $M_t/M_0$ now represents the fraction of the original mass remaining in the sphere.

4 Model 3 — Diffusion Out of Non-Uniformly Contaminated Sphere

The sphere is first loaded for time $t_0$ using diffusion coefficient $D_2$ (from an outside bath at concentration $C_0$), creating a non-uniform initial concentration profile. Then the outside concentration is set to zero and mass diffuses out with coefficient $D_1$ for additional time $t$.

Step 1 — Loaded Profile after Uptake

After loading for time $t_0$ with $D_2$ (using Model 1 equations), the initial condition for release is:

$$C_\text{load}(r,t_0) = C_0\!\left[1 + \frac{2a}{\pi r}\sum_{m=1}^{\infty} \frac{(-1)^m}{m}\sin\!\left(\frac{m\pi r}{a}\right) e^{-D_2 m^2\pi^2 t_0/a^2}\right]$$

(5)

Step 2 — Fourier Coefficients for Release

The loaded profile is decomposed into a Fourier sine series. Using the orthogonality of $\sin(n\pi r/a)$ on $[0,a]$, the Fourier coefficients are:

$$B_n = \frac{2C_0 a}{\pi n}(-1)^{n+1}\!\left[1 - e^{-D_2 n^2\pi^2 t_0/a^2}\right]$$

(6)

Step 3 — Release Profile and Fractional Fill

The concentration profile during release is:

$$C(r,t) = \frac{1}{r}\sum_{n=1}^{\infty} B_n \sin\!\left(\frac{n\pi r}{a}\right) e^{-D_1 n^2\pi^2 t/a^2}$$

(7)

And the fraction of sphere filled (mass remaining / maximum possible mass):

$$f_\text{fill}(t) = \frac{3}{\,C_0 a^3\,}\sum_{n=1}^{\infty} B_n \cdot \frac{a^2(-1)^{n+1}}{n\pi} \cdot e^{-D_1 n^2\pi^2 t/a^2}$$

(8)

5 Governing Equation & Boundary Conditions

All three models are governed by the radial diffusion equation in spherical coordinates:

$$\frac{\partial C}{\partial t} = D\!\left(\frac{\partial^2 C}{\partial r^2} + \frac{2}{r}\frac{\partial C}{\partial r}\right)$$

(9)

Symmetry at the sphere center requires:

$$\left.\frac{dC}{dr}\right|_{r=0} = 0$$

(10)

Analytical solution: The series solutions (Eqs. 1–8) are the exact analytical solutions to Eq. 9 given their respective initial and boundary conditions, as derived in Crank (1975). The infinite series converge rapidly for $t > 0$ and are truncated at $n = 150$ terms in the applet.

6 Symbol Definitions

Symbol	Definition
$C$	Concentration inside the sphere
$C_0$	Boundary concentration (outside sphere or initial inside, depending on model)
$C_1$	Initial concentration inside sphere (= 0 for Models 1 & 2)
$a$	Sphere radius
$r$	Radial distance from center (0 ≤ r ≤ a)
$t$	Time
$D$	Diffusion coefficient (Models 1 & 2)
$D_1$	Diffusion coefficient for release phase (Model 3)
$D_2$	Diffusion coefficient for uptake/loading phase (Model 3)
$t_0$	Loading time (Model 3)
$M_t$	Mass in sphere at time $t$
$M_\infty$	Equilibrium mass (fully loaded)
$f$	Fractional approach to equilibrium (uptake) or fractional release
$B_n$	Fourier sine series coefficients for Model 3 release (Eq. 6)

7 References

Crank, J. (1975). The Mathematics of Diffusion, 2nd ed. Oxford University Press, Oxford.