Diffusion Coefficient Calculator

The diffusion coefficient $D$ is the number that turns “this particle wanders around in the solvent” into something you can put into an equation. It has units of area per time, and its clearest reading is this: over a time $t$, a particle starting from one spot will typically be found a distance of about $\sqrt{2Dt}$ from where it began, in each direction it is free to move.

That single interpretation covers most of the reasons people look this number up. A 2 nm particle in water at room temperature has a diffusion coefficient near $1.2 \times 10^{-10}\ \mathrm{m^2/s}$, which means it spreads a couple of micrometres in the first second and about ten times that in the first hundred. A large protein under the same conditions can sit two orders of magnitude lower, and that contrast is exactly what makes the number useful for sizing things you cannot see.

This calculator works out $D$ from the absolute temperature, the solvent’s dynamic viscosity and the size and shape of the particle. It covers seven geometries, from a sphere through three ways of moving a disk to three ways of moving an ellipsoid, and either direction works: give it a friction coefficient and a size instead of a viscosity and the solvent viscosity is filled in for you, which is what turns a measured friction coefficient into a property of the fluid.

Every input except the shape is a property of the system, not of the particle alone. Work through them in this order and the answer follows immediately.

A worked example: a 2 nm sphere in water at 25 °C

This is the case most often quoted for the Einstein relation, and it is worth following once end to end. Choose Sphere, then enter $T = 25\ \degree\mathrm{C}$, $\eta = 0.00089\ \mathrm{Pa \cdot s}$ and $a = 2\ \mathrm{nm}$.

The calculator reports both intermediate numbers: a friction coefficient of $3.35522 \times 10^{-11}\ \mathrm{s/kg}$ and a diffusion coefficient of $1.22687 \times 10^{-10}\ \mathrm{m^2/s}$, the second displayed against the $\times 10^{-10}$ scale as $1.22687$.

The same calculation the other way round: 3 nm and 2 nm, tumbling, ξ = 4 s/kg

Suppose you have measured how strongly a fluid drags an ellipsoidal particle but you have no figure for the fluid’s viscosity. Choose Ellipsoid (tumbling), enter the two semi-axes $a = 3\ \mathrm{nm}$ and $b = 2\ \mathrm{nm}$, and enter $\xi = 4\ \mathrm{s/kg}$ where the viscosity would go.

At $T = 1\ \mathrm{K}$ the answer then follows from the Einstein relation: $D = k_{\mathrm{B}}T/\xi$ gives $3.451623 \times 10^{-24}\ \mathrm{m^2/s}$, and the viscosity box reports $77,710,923\ \mathrm{Pa \cdot s}$. That is a fluid roughly a hundred billion times thicker than water, which is the honest reading of a friction coefficient that large on a particle that small.

Only the friction coefficient changes between shapes, but it changes by a factor of several, and the answer moves with it. Two particles of exactly the same size can therefore have clearly different diffusion coefficients purely because of how they are shaped and how they tumble.

The chips give each shape factor at a glance with $\eta$ and $a$ set to one, so $6\pi$ is about 18.8 and the disks sit at 16, 12 and 10.7. Smaller means less drag, which means a larger diffusion coefficient for the same size.

The three ellipsoid entries are not three different particles: they are three answers to “which way is it going?” The sideways case costs exactly twice the lengthways case for the same axes, and the tumbling case sits between them once its smaller denominator is accounted for. When a measurement does not resolve orientation, the tumbling entry is the honest choice, because it is the average over orientations rather than a best case.

The calculator is two statements joined together. The Einstein relation turns thermal energy and friction into the diffusion coefficient, and a shape formula ties the friction coefficient to the solvent and the particle dimensions. Either statement can be read in either direction, which is what lets a friction coefficient stand in for a viscosity.

The two statements are solved together, so the boxes can be filled in any order: the shape factor $\sigma$ comes from the geometry card above, and whichever of the five quantities you leave out is the one the calculator works out.

How the two equations chain together

For an elongated particle the shape factor is not a constant, because the friction depends on how much of the particle is presented to the flow. Writing $a$ for the major semi-axis and $b$ for the minor one, the ratio $2a/b$ enters through a natural logarithm. Doubling the aspect ratio therefore does not double the friction; it adds a fixed amount to a denominator. The formulas used here describe a prolate ellipsoid, the elongated kind with $a > b$, which is why the minor axis has to stay strictly smaller than the major one.

Tracing the friction coefficient back to the solvent

The friction coefficient is where the two equations meet, and it is the one number a laboratory can measure without knowing anything about the particle. Give the calculator a friction coefficient and a particle size and the solvent viscosity box fills itself in, because$\eta = \xi / (\sigma a)$ is just the friction formula read the other way round. A friction coefficient of $3\ \mathrm{s/kg}$ on a 2 nm face-first disk therefore reports $9.375 \times 10^{7}\ \mathrm{Pa \cdot s}$, which is what a fluid thick enough to resist that particle that strongly would have to be.

The temperature and the particle size are the two quantities the calculator treats as measurements rather than results, so neither is ever overwritten: a derived answer is reached by adjusting the friction coefficient, the solvent viscosity or the answer itself, never the temperature or the geometry you typed. A set of values that already agree with each other moves nothing at all.

Almost every wild result from this calculation traces back to one of five inputs. These are the checks worth making before you trust a number.

The formulas behind this calculator are the classical low-Reynolds-number results. They are extremely accurate inside their range and quietly wrong outside it, and the range is worth knowing.

Two directions are worth exploring once you have a number. If the particle is settling or being dragged through the fluid rather than diffusing freely, the terminal velocity calculator uses the same balance of drag and driving force at higher speeds. If what you actually need is the mass of the particle rather than its size, the molar mass calculator converts between a formula and a molecular mass, which is often the quantity a size estimate gets compared against.