= Molecular diffusion
Molecular diffusion is the component of material transport caused by the random thermal motion of <molecules>, measured relative to bulk motion of their surroundings. A <diffusion coefficient> is a property of a particular substance in a particular medium under specified conditions, not a universal speed of spreading. Its dependence on the <solute>, <solvent> and <temperature> is precisely what makes a spreading experiment informative. A <random walk> idealizes this motion; the <Brownian motion> description provides a continuous approximation over sufficiently long times. https://ocw.mit.edu/courses/1-061-transport-processes-in-the-environment-fall-2008/8260e3e1577cd739ce85143510d9c165_lec_01.pdf[Nepf, MIT: molecular motion and diffusivity]{ref}
\b[A motivating experiment: watch dye enter a gel.] Samprovalaki, Robbins and Fryer studied rhodamine 6G and methylene blue entering agar gel from two side reservoirs. A covered slab, $10\times5\times1\,\mathrm{mm}$, sat on a controlled thermal stage; the dye was brought to the same <temperature>. They photographed the gel every ten minutes, averaged image intensity across a strip, and fitted the changing <concentration> to a <diffusion equation> model. The cover limited evaporation and stopped dye travelling over the gel surface. Runs at $30$, $50$ and $70\,\mathrm{{}^{\circ}C}$ were repeated in triplicate. The measured spatial colour change therefore supplied a quantitative transport test, rather than merely showing that dye eventually spreads. Changing lighting also changes image intensity, so the optical response had to be checked. Importantly for interpretation, their constant-<diffusion coefficient> fit was imperfect and the gel coefficient is an effective transport coefficient, not automatically the value in free <water>. https://doi.org/10.1016/j.jfoodeng.2012.03.024[Samprovalaki, Robbins and Fryer (2012), Investigation of the diffusion of dyes in agar gels, sections 2-4]{ref}
\b[A finite-pulse version to test width growth.] The following is a proposed teaching variant, not that paper's apparatus or a report of measurements. Make a long, narrow, transparent gel channel of uniform cross-section. Join a thin dyed gel band between two undyed lengths of the same gel, with good contact and no air gaps; the dye should initially fill the cross-section so the dominant <concentration> variation is along the channel coordinate $x$. After assembly, keep the covered channel horizontal at uniform <temperature>, away from vibration. The dye is a finite initial pulse: no reservoir supplies more dye. Choose a dye whose image response can be calibrated and whose interaction with the gel is small enough for an approximately constant <diffusion coefficient> to be plausible.
Photograph the whole pulse at recorded times with fixed lighting, focus and exposure, including a length scale. Use uniformly dyed standards of the same gel thickness and a blank gel to convert image intensity to <concentration> $C(x,t)$; do not assume raw brightness is proportional to <concentration>. Average across the channel width and examine several parallel strips to check the one-dimensional approximation. Measure the initial profile instead of treating the band as infinitely thin. For a finite pulse with negligible end effects, define its spatial integral $Q$, centre $\bar x$ and width $\sigma$ by
$$
Q=\int_{-\infty}^{\infty}C(x,t)\,dx,\qquad
\bar x(t)=\frac1Q\int_{-\infty}^{\infty}xC(x,t)\,dx,\qquad
\sigma^2(t)=\frac1Q\int_{-\infty}^{\infty}(x-\bar x(t))^2C(x,t)\,dx.
$$
Here $\sigma^2$ is the <variance> of the normalized profile $C/Q$, and $\sigma$ is its <standard deviation>. If $C$ is measured in $\mathrm{mol\,m^{-3}}$, then $Q$ has units $\mathrm{mol\,m^{-2}}$; multiplying by the channel cross-section gives the total <amount of substance>. For the unbounded one-dimensional <diffusion equation> with constant $D$, conserved $Q$, finite <variance> and sufficiently decaying tails,
$$
\sigma^2(t)=\sigma_0^2+2Dt,\qquad
D=\frac12\frac{d\sigma^2}{dt}.
$$
Thus plot width \i[squared] against time: its slope determines the <diffusion coefficient>, while its intercept includes the finite starting width. This moment law holds even for a non-<Gaussian distribution> initially; it follows by applying <integration by parts> twice to the <diffusion equation>. https://ocw.mit.edu/courses/1-061-transport-processes-in-the-environment-fall-2008/68b0a4a664246168ee639298b39384b1_lec_03.pdf[Nepf, MIT: profile moments and estimating diffusivity from variance growth]{ref}
\Image[/molecular-diffusion-width-experiment.png]
{title=Measuring <molecular diffusion> from the growth of width squared}
{description=Original schematic of the proposed finite-pulse gel experiment, with an illustrative calculation rather than measured data. A camera records the broadening band. The plotted points are exact samples of $\sigma^2=0.25+0.40t$ with $\sigma^2$ in $\mathrm{mm^2}$ and $t$ in hours; the slope gives the chosen $D=0.20\,\mathrm{mm^2\,h^{-1}}$. The schematic is not to scale and is not a reproduction of the published apparatus.}
For the particularly readable case of an initially <Gaussian distribution> centred at zero, uniform <advection> at speed $u$ and constant <diffusion coefficient> $D$ give the one-dimensional <concentration> profile
$$
C(x,t)=\frac{Q}{\sqrt{2\pi(\sigma_0^2+2Dt)}}\exp\!\left[-\frac{(x-ut)^2}{2(\sigma_0^2+2Dt)}\right].
$$
Its centre moves as $\bar x=ut$ and its <variance> grows as $\sigma_0^2+2Dt$. Our illustrative choice $\sigma_0=0.50\,\mathrm{mm}$ and $D=0.20\,\mathrm{mm^2\,h^{-1}}$ gives widths $0.50$, $0.81$ and $1.36\,\mathrm{mm}$ at zero, one and four hours. These are calculated values, not a measured dye coefficient. Once $2Dt$ dominates the initial <variance>, width grows as $\sqrt{2Dt}$: four times the elapsed time gives approximately twice the width. A rectangular starting band will not immediately have a <Gaussian distribution>, although the same width-squared law still applies under the stated assumptions.
\b[Controls and limits of the interpretation.] In the proposed experiment, check that $Q$ stays constant and that the centre remains stationary while the width increases. Include a blank and a uniformly dyed gel photographed over the same period: an apparent signal change there exposes lighting drift, dye fading or evaporation. Repeat with a lower starting dye <concentration> and a second initial width; agreement of the slopes is a check on constant $D$ and on preparation effects. Keep the pulse far from the channel ends and record its faint tails: truncation and background errors can strongly bias a <variance>. Poor contact, dye binding, <chemical reactions> or transport in a liquid film can invalidate the simple model. A gel suppresses bulk liquid motion, but its inferred <diffusivity> need not equal that in the pure <solvent>. A stationary centre alone does not exclude symmetric flows, and linear width-squared growth alone does not prove a molecular mechanism: <turbulence> can also produce an effective diffusive description. These are reasons to combine the profile test with the controlled setup.
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