Gold's Surface Share Rises 10x When You Shrink It 10x. Not Quite.

For a sphere, the surface-to-volume ratio is 3/r. Cut the diameter by 10 and the ratio rises 10 times. This is geometry, not physics. So when a paper says "small gold is more active" or "small gold melts lower", the first question is simple: does the measured change follow 3/r, or does it do something else?

I could not run code for this post. Every number I computed below is hand arithmetic, and you can repeat it with a calculator. I made no particle and I ran no experiment. The measured data come from papers I opened, and I say where I could only read an abstract.

The question

Take a gold sphere of diameter 3e-9 m (3 nm, about 30,000 times thinner than a human hair). Make a second sphere 10 times wider (3e-8 m), and a third 100 times wider (3e-7 m). The question: where does the source's own measured property depart from what 3/r predicts?

I changed my plan on one point while reading. I expected to show that melting points depart from the estimate below about 1e-8 m (10 nm). The one melting paper I could open says something different. Schlexer et al. imaged gold particles of 2 to 20 nm with high-resolution transmission electron microscopy (HRTEM, which resolves atom columns at about 1e-10 m) and ran molecular dynamics. Their abstract reports "a linear dependence of the melting temperature on the inverse particle size" [1]. Since 1/d is proportional to S/V, linear in 1/d means melting follows the surface-to-volume ratio over that range. The abstract prints no melting temperatures and no particle count, so I cannot give a slope. I treat it as a qualitative result.

Data and where it came from

I opened three sources.

1. Overbury et al., ORNL record (2006). Gold on titania (P25), made by deposition precipitation, at two loadings: 7.2 wt% and 4.5 wt%. The same catalyst was calcined in repeated steps so the mean gold size grew. Size was set by extended X-ray absorption fine structure (EXAFS), an ensemble method that infers a mean coordination number, not a picture of each particle. The range was about 2e-9 to 1e-8 m. The authors report turnover frequency (TOF, reaction events per site per second) at 298 K for CO oxidation, scaling as d^-1.7 (plus or minus 0.2) for the 7.2 wt% sample and d^-0.9 (plus or minus 0.2) for the 4.5 wt% sample [2]. They find activity falls past about 2 nm and see no maximum at small sizes [2].
2. Schlexer et al. (2019), described above. HRTEM plus simulation, 2 to 20 nm, melting depression "significant" below about 10 nm [1].
3. An in situ electron microscopy study (Nanotechnology, 2009). Gold on graphite at a substrate temperature of 1100 K. Particles went from solid to liquid as they shrank to about 5e-9 m [3]. No melting point and no particle count appear in the abstract I read.

I also found the classic reference, Buffat and Borel, Physical Review A 13, 2287 (1976), but I only saw its catalogue record [4]. A 2007 arXiv paper (which I could not read; the PDF was compressed) cites it [5]. A search summary says that paper reproduces the Buffat data and puts a 2e-9 m (20 Å) gold cluster near or below room temperature. I do not use that number, because I did not read it. A blog post says a Gibbs-Thomson model puts the room-temperature melting point near 1.4e-9 m [6]. That is a model estimate, not a measurement, and I did not check its inputs.

Count of sources that print usable numbers for the comparison: one (Overbury). That is thin, and I want it said before any result.

Method

I did two calculations by hand.

Calculation A: S/V at three sizes. For a sphere of radius r:

$$\frac{S}{V} = \frac{4\pi r^2}{(4/3)\pi r^3} = \frac{3}{r}$$

Calculation B: share of atoms in the outer shell. S/V is a ratio of areas to volumes. A chemist wants a share of atoms. If the outer layer has thickness t, the shell fraction is:

$$f = 1 - \left(1 - \frac{t}{r}\right)^3$$

I set t to 2.88e-10 m, the nearest-neighbour distance in gold. This is a standard crystallographic value that I took from memory, not from a source I opened. Treat it as an assumption. A shell of one atomic layer also ignores facets, edges and steps, which is exactly where Overbury says the activity sits [2].

Comparison with Overbury. If the quantity measured scales as S/V, the exponent of TOF on d is -1. I do not know from the abstract whether TOF was normalized per total gold atom or per surface atom [2]. That matters. If it was per total gold, -1 is the S/V prediction. If it was per surface atom, S/V predicts 0. I give the first reading, and flag it as unresolved.

Result

Geometry

Diameter (m)	Radius (m)	S/V = 3/r (m^-1)	Shell fraction f
3e-9	1.5e-9	2.0e9	0.473 (47%)
3e-8	1.5e-8	2.0e8	0.0565 (5.7%)
3e-7	1.5e-7	2.0e7	0.0058 (0.58%)

S/V falls exactly 10 times per step, by construction. The shell fraction does not. From 3e-8 m to 3e-9 m it rises 0.473 / 0.0565 = 8.4 times, not 10. From 3e-7 m to 3e-8 m it rises 9.8 times. The thin-shell approximation f ≈ 3t/r gives 58% at 3e-9 m, against the exact 47%. So the tenfold rule overstates the surface share at the small end by about a fifth, because the shell eats into itself. At 1e-8 m (10 nm), f is about 16%.

This is the first departure, and it is not a physics departure. It is the rule failing at its own edge. A reader who hears "10 times more surface" for a 3 nm particle is hearing a number that is 8.4 times in the exact count.

Against the measured exponents

Over the Overbury range (a factor of 5 in diameter, 2e-9 to 1e-8 m), an exponent of -1 gives 5 times more activity at the small end. The two measured exponents give:

Sample	Exponent	Activity gain over a factor of 5 in d	Range using plus or minus 0.2
4.5 wt%	-0.9 ± 0.2	4.3 times	3.1 to 5.9 times
S/V estimate	-1	5.0 times	none
7.2 wt%	-1.7 ± 0.2	15.4 times	11.2 to 21.3 times

The 4.5 wt% sample sits on the estimate. The 7.2 wt% sample is about 0.7 exponent units away, which is 3.5 times its own stated error of 0.2. If I ignore the unknown error on the -1 reference, that gap is well outside the error bar. I cannot say whether the authors' ± is a standard error or a different interval, since the abstract does not say.

Both samples came from one titania support, and each series used one parent batch, calcined in steps [2]. The sample count is therefore two batches, one per loading. That is a good design for isolating size, because it removes support and synthesis noise [2]. It is a poor design for judging whether another lab gets the same exponent. Two exponents from one lab differ by 0.8 between loadings. I would want to see a second lab get one number.

The authors' explanation is that activity tracks the population of low-coordinate sites, not total surface and not electronic structure [2]. That is consistent with the table above: edge and corner atoms are a smaller, faster-shrinking subset of the surface shell. I did not check this mechanism beyond the abstract.

Melting

Schlexer et al. give the linear 1/d law over 2 to 20 nm [1]. I read this as support for S/V as a coordinate and as no evidence of departure at 1e-8 m in that dataset. The 2009 electron microscopy paper reports a solid-to-liquid change near 5e-9 m on a graphite support, and the liquid particle bored a one-layer-deep hole into the carbon [3]. That means the support took part. A melting point measured on graphite is not a property of gold alone. I have no printed pair of (diameter, temperature) from either paper, so I cannot test the slope.

Sensitivity: which assumption moves the result most

Three inputs could move the result, and I rank them.

1. How the activity is normalized. This moves the target exponent from -1 to 0. It changes the whole reading. I could not resolve it from an abstract. This is the largest uncertainty.
2. The sizing method. EXAFS gives a mean size from an averaged coordination number [2]. For a broad size distribution, mean coordination and mean diameter are not the same. A 10% bias in d changes d^-1.7 by about 17%, because 1.1^1.7 = 1.18. That is smaller than the gap between the two loadings, so it will not rescue the S/V rule for the 7.2 wt% sample. I have no stated diameter uncertainty to use, so this is a what-if, not a result.
3. The shell thickness t. If I set t to 1.5 times 2.88e-10 m (a thicker active layer), f at 3e-9 m rises to about 0.64, and the ratio between 3e-9 m and 3e-8 m falls further. If t is half, f at 3e-9 m is about 0.27. The qualitative point, that the ratio is below 10 at the small end, holds for any t, since the exact count saturates at 1.

What I think

I will not claim a rule from a single paper. The honest summary: S/V is exact geometry, the shell share departs from it by about 16% at 3e-9 m, one gold catalyst series follows an exponent near -1, and another from the same lab departs from it by about 3.5 error bars. For melting, the one source I opened supports 1/d scaling and does not show a break at 1e-8 m.

Does it scale? For product claims, the answer depends on which of these is true for a given batch. A catalyst maker who sizes by EXAFS and promises a 15-fold gain is quoting the steep exponent. One who promises 5-fold is quoting S/V. Both can cite the same paper [2].

I would change my mind if a second lab, with printed diameters and activities, gets an exponent near -1.7 on a different support. I would also drop the word "departure" if the TOF turns out to be per surface atom, because then both samples would be far from the target. I plan to open the full text of the Overbury paper and compute the exponent from its table. I also still lack one melting paper that prints diameter and temperature pairs. This post is a first pass, not a verdict.