The Gibbs–Thomson Effect

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By Charles Xie ✉

The melting point of bulk gold is 1064 °C. The number is in every handbook, and it applies to any ordinary piece of gold. A gold nanoparticle melts well below that temperature. Gold nanoparticles 2 to 3 nm across have been measured to melt at roughly half the bulk temperature in kelvin. Pawlow showed in 1909, from thermodynamics, that a nanoparticle had to melt at a lower temperature than the bulk, decades before anyone could make nanoparticles of a chosen size. Takagi observed the effect in thin films of lead, tin and bismuth by electron diffraction in 1954. In 1976 Buffat and Borel put gold nanoparticles from 2 to 25 nm on the heated stage of an electron microscope and recorded, for each size, the temperature at which the diffraction rings disappeared. Their plot of melting point against diameter is the one that appears in nanoscience textbooks. The curve has a simple form: the drop below the bulk value is inversely proportional to the diameter. This is the Gibbs–Thomson effect.

The usual textbook treatment is a diagram of surface energies and the Buffat–Borel curve. In AIMS a student can run the experiment. They tell the AI agent what to cut and what to heat, and the agent cuts the nanoparticles out of a gold crystal, runs a molecular dynamics simulation of each one at a series of temperatures, and posts charts in the chat showing where each one melts. The melting points are computed from the motion of a few hundred atoms under a model of the forces between them. The example below has three turns: cut the nanoparticles, find the melting point of bulk gold in the model, then melt the nanoparticles and compare.

Why a Smaller Nanoparticle Has a Lower Melting Point

An atom inside a gold crystal has twelve nearest neighbors. An atom on the surface has eight or nine, and one on an edge or at a corner has fewer. Fewer nearest neighbors means fewer bonds holding the atom in place, so surface atoms start to move at a lower temperature than interior ones. In a bar of gold this hardly matters, because surface atoms are a negligible fraction of the total. In a nanoparticle a couple of nanometers across, they are the majority. The first turn of the example counts them. The agent finds the nearest neighbors of every atom. An atom with the full twelve is interior. An atom with fewer is on the surface, because each missing nearest neighbor leaves one side of the atom open to vacuum. The agent cuts a sphere 1.9 nm across (201 atoms, 61% of them on the surface), a sphere 1.5 nm across (87 atoms, 78%), and a 55-atom cuboctahedron about 1 nm across. The cuboctahedron is a closed-shell cluster, one atom surrounded by a shell of 12 and then a shell of 42, so its surface is a complete polyhedron with 12 atoms at corners, 24 on edges and 6 at the centers of its square faces. Thirteen atoms are inside, and 76% are on the surface.

Thermodynamics explains the size dependence. A surface costs energy: an atom at a surface has lost the bonds that would have reached into the missing half of its surroundings, so a surface holds extra energy in proportion to its area. This surface energy is measured in joules per square meter, and for a liquid it is the surface tension. When a nanoparticle melts, its solid surface is replaced by a liquid surface, and the surface energy of liquid gold is lower than that of solid gold. In the crystal the surface atoms are held on lattice sites, so every bond broken by the cut stays broken. In the liquid they are free to shift closer together and recover part of that bonding. The energy saved scales with the surface area. The latent heat needed to melt the interior scales with the volume. The ratio of area to volume is inversely proportional to the diameter d, so the depression of the melting point is inversely proportional to d as well. This is the Gibbs–Thomson equation, T_m(d) = T_bulk(1 − β/d), where β is a length set by the surface energies and the latent heat. For gold β is about 1 nm, which is why a 2 nm nanoparticle melts at about half the bulk temperature in kelvin. Plotted against 1/d, the equation is a straight line that starts at the bulk melting point. The third turn draws that line from simulated nanoparticles.

The Melting Point of Bulk Gold in the Model

The second turn asks for the melting point of bulk gold in the model. The handbook value is 1337 K. The gold in the lab is a model of gold. Its atoms interact through an embedded-atom potential fitted to properties of the bulk crystal, and the model reproduces some properties of the metal better than others. The agent finds the bulk melting point of the model by seeded growth. It builds a slab of 288 atoms, eight unit cells tall with a free top surface, melts the top seven cells while holding the bottom cell crystalline, then releases the seed and holds the whole slab at a chosen temperature. Below the melting point, crystal grows upward from the seed into the liquid. Above it, the seed melts. At 1100 K the crystal was still growing when the run ended. At 1200 K the seed melted. So bulk gold in the model melts between 1100 and 1200 K, or 1150 ± 50 K, which is about 190 K below the real 1337 K (1064 °C).

That difference is a property of the potential. It is the reason the bulk value has to be measured in the same model. Otherwise every depression would come out 190 K too large.

Three Sizes, Three Melting Points

In the third turn each nanoparticle is heated from 300 to 1500 K in steps of 50 K. At every step the nanoparticle is held for 24 ps to settle and then sampled for another 24 ps, which adds up to 1.2 ns per nanoparticle. Two things are measured at each step. The first is the heat capacity, the energy the nanoparticle absorbs per degree. A solid that is only vibrating absorbs about the same amount per atom at every temperature. While it is melting it absorbs more, because for a few steps the energy goes into breaking up the lattice, so melting appears as a peak, and the area under the peak is the latent heat. The second is the Lindemann index, which measures how much the distance between pairs of atoms fluctuates during the sampling. In a solid the distances fluctuate by a few percent. Once atoms start changing places, the index jumps. Where the heat capacity peaks and the Lindemann index jumps at the same temperature, the nanoparticle has melted.

The melting point drops as the nanoparticle shrinks, and the drop is inversely proportional to the diameter. The last chart of the example plots the four melting points against 1/diameter, with the bulk value at the origin, and fits the Gibbs–Thomson line through them. All four points fall within 50 K of the line, and the fitted β is 0.50 nm.

One detail of the plot needs explaining. Each nanoparticle is placed at its effective diameter, the diameter of the sphere its atoms would fill at the density of gold. The sphere cut to 1.9 nm ends up with 201 atoms, and 201 gold atoms fill a sphere 1.86 nm across. The 87-atom sphere fills 1.41 nm. The diameter of the cuboctahedron, 1.21 nm, comes from its atom count in the same way. The same thing came up in the quantum dot article, where a cube cut to 2.4 nm held fewer atoms than a 2.4 nm sphere and was plotted at a smaller effective diameter.

Small Nanoparticles Melt Differently

Besides melting at different temperatures, the three nanoparticles melt in different ways, and the heat-capacity chart shows it. The 201-atom sphere melts like a very small piece of bulk gold. Its heat capacity stays near the solid value up to 650 K, rises through 700 and 750, peaks at 850 K at almost three times the solid value, and has returned to the liquid value by 1000 K. The Lindemann index goes from 0.05 to 0.11 between 850 and 900 K. The latent heat under the peak is about 0.06 eV per atom, half of bulk gold's 0.13 eV. The surface accounts for the difference. Of the 201 atoms, 122 are on the surface and bound less tightly than the 79 inside, so less energy is needed to melt them.

The two smaller nanoparticles are different. With three quarters of their atoms on the surface, their Lindemann index starts climbing around 500 K, and the heat capacity rises with it. Much of each nanoparticle is rearranging well below the temperature at which the chart shows melting. The melting of the core, 13 atoms in the cuboctahedron and 19 in the sphere, appears as a modest hump in the heat capacity on top of that rise, with about 0.015 eV per atom under it. If the nanoparticle is made a little smaller, the hump disappears. A 43-atom sphere has one interior atom. It is a solid at room temperature, but neither an 8 nor a 24 ps hold produces a peak. It melts over a spread of temperatures.

The same facts explain why the Lindemann index of the 87-atom sphere has no jump like the one at 900 K for the 201-atom sphere, and why the cuboctahedron shows only a small one. The index is an average over every pair of atoms. In the 201-atom sphere the 79 interior atoms are a large share of the pairs, so when the core melts the average moves at once. In the 87-atom sphere only 19 atoms are interior, and their melting is a small change hidden in the steady rise of the surface atoms, which have been moving since 500 K. Its index rises steadily up to 850 K, and its melting point rests on the heat-capacity hump alone. The cuboctahedron, with 13 interior atoms, still shows a step, from 0.06 to 0.12 between 650 and 700 K, because its closed shell holds the 42 surface atoms in place until the whole cluster melts together. A sphere cut from the crystal at a chosen radius ends with a partial outer shell and some atoms with few nearest neighbors, which start moving early and smear the rise.

The closed shell is why the example uses a cuboctahedron for its smallest nanoparticle. On shorter holds of 8 ps per step, the 87-atom sphere shows no peak at all, but the cuboctahedron, with fewer atoms, still does. Fifty-five is one of the magic numbers for metal clusters, and this is why: a complete shell is harder to disturb than a partial one.

Running Experiments by Asking

The student asks for what they want to know in ordinary language, and the agent picks the tool, builds the structure, runs it, and posts the chart in the chat. Cutting a cuboctahedron, setting up seeded growth, and computing a Lindemann index are the agent's work.

The Model and Its Limits

The nanoparticles are fragments of the gold crystal, and the forces between their atoms come from the embedded-atom potential of Zhou, Johnson and Wadley (2004), the one AIMS uses for bulk gold. An embedded-atom potential adds to the usual pair interaction an energy that depends on how much electron density an atom receives at its site from the atoms around it, which is what lets one potential handle both an interior atom with twelve nearest neighbors and a surface atom with eight. The temperature is set with a Berendsen thermostat. The nanoparticle's overall translation and rotation are removed at every step, since otherwise rotation would be counted as thermal motion. The heat capacity on the chart is the slope of the potential energy against temperature, so a harmonic solid reads 1.5 k_B per atom. The peak of the 201-atom sphere reaches 5.2 k_B per atom.

The results are properties of this model and this protocol. The clearest sign of this is the slope of the line. β comes out at 0.50 nm where Buffat and Borel measured about 1 nm, so the model's nanoparticles have melting points only about half as far below bulk as real ones do. Several things contribute to the difference. A nanoparticle heated in 50 K steps can overshoot its melting point before the lattice has time to melt. The 1.9 nm sphere melted at about 900 K with 8 ps steps and at 870 K with 24 ps steps, so some superheating probably remains at 24 ps. The potential was fitted to properties of the bulk crystal, so its surface energies may differ from those of gold. The nanoparticles are cut from the face-centered cubic crystal, whereas real gold clusters of this size rearrange into icosahedra, which have a different surface. A single run of fifty or two hundred atoms is one sample. Repeating a run can shift its melting point by 50 to 100 K. Read 696 and 698 K in the table as about 700 K. The trend across the three sizes is more reliable than any single melting point.

Conclusion

This article uses the melting of gold nanoparticles to illustrate a new way to learn science: students converse with an AI agent that turns a plain-language question into a molecular dynamics experiment and hands back measured data. The Gibbs–Thomson effect, normally a formula in a nanoscience course, becomes something students measure for themselves. They cut nanoparticles of three sizes out of a gold crystal, find where each one melts, and discover that the melting points fall on a straight line against the inverse of the diameter. They also discover where the model disagrees with experiment, and why. By pairing a physically grounded simulation with an AI that can set it up and run it, AIMS lets students investigate a nanoscale phenomenon that they would otherwise only read about.

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