States of Matter Simulator

What happens to the particles in a solid, a liquid and a gas?

The same particles make all three, and what changes is how their motion compares with the pull between them. In a solid the attraction holds each atom in a fixed place in a regular pattern, where it vibrates; in a liquid the atoms are nearly as close but slide past each other; in a gas they have enough energy to escape each other and fill their container. This simulator shows that change happening. A box of argon-like atoms, 150 unless you change it, sits in a heat bath at the temperature you set, and every pair of atoms pulls and pushes on each other through one rule, the Lennard-Jones potential U(r) = 4ε[(σ/r)¹² − (σ/r)⁶]. Nothing tells the atoms which state to be in: the readouts work it out from where they are.

Raise the temperature and the crystal melts from its edges inwards, then the liquid evaporates until the box is full of gas. Lower it again and the gas condenses and freezes. The atoms are coloured by the state each one is in, and the two plots record the energy and the share of atoms in each state as it happens.

One rule for every pair of atoms

The (σ/r)⁶ term is the weak attraction between neutral atoms at a distance, the London dispersion force, and the (σ/r)¹² term is the steep repulsion when their electron clouds are pushed into each other. Between them the energy of a pair has a minimum of −ε at r = 2^(1/6) σ. With the values that fit argon, ε/k = 119.8 K and σ = 0.3405 nm (Rowley, Nicholson and Parsonage, 1975), the well is 1.654 × 10⁻²¹ J deep, or 0.996 kJ/mol for a mole of pairs, and its bottom lies 0.3822 nm apart, close to the spacing of the atoms in the crystal. Lennard-Jones proposed this 12-6 form in 1931.

Pairs more than 2.5σ, 0.85 nm, apart are ignored, a common shortcut: a pair that far apart has only 1.6 percent of the well depth left, and the potential is shifted by that much so that the energy does not jump when a pair crosses the line. The walls push back like stiff springs on any atom that touches them. Every atom then moves by Newton’s second law, stepped forward every 10.8 fs by the velocity Verlet method, and the heat bath gives each atom a fresh random velocity for the set temperature on average once every 10.8 ps, as if a molecule of something surrounding the box had struck it. Between those kicks the motion is Newton’s laws and nothing else.

Worked example: how fast are the atoms in the solid?

Temperature fixes the average kinetic energy of an atom, and in two dimensions that average is kT, half of kT for each direction it can move in. At 20 K it is 1.380649 × 10⁻²³ J/K × 20 K = 2.761 × 10⁻²² J. The mean speed in two dimensions is v̄ = √(πkT/2m), and an argon atom’s mass is 39.948 u, 6.634 × 10⁻²⁶ kg, so

v̄ = √(π × 2.761 × 10⁻²² / (2 × 6.634 × 10⁻²⁶)) = 80.9 m/s.

That is the crystal at 20 K, and the Mean speed readout sits close to 81 m/s there, about 290 km/h. The atoms of a solid are not still: the pull of their six neighbours turns each one back before it gets anywhere. At 120 K, where the box is all gas, the mean speed is √6, about 2.45, times larger, 198 m/s. Six times the temperature buys only two and a half times the speed.

The energy holding the crystal together is just as easy to estimate. Each bond sits near the bottom of the well, −0.996 kJ/mol, and each atom has six bonds, each shared with a neighbour, so a large flat crystal holds about 3 × −0.996 = −2.99 kJ/mol, or −3.15 kJ/mol counting the weaker pull of atoms further out. The 150-atom crystal starts at −2.82 kJ/mol, because 42 of its atoms sit at the edge with fewer neighbours to hold them. That is what the potential energy readout shows the moment the box is built, and it is the energy you would have to supply to pull every atom free. Warmed to 20 K, the crystal reads about −2.63 kJ/mol.

Watch a solid melt

Start from the crystal and raise the temperature a few kelvin at a time, giving it a moment at each setting. At the default size it is still about nine tenths solid-like at 30 K. Near 35 K the atoms at its edge start to come loose and turn liquid-like while the middle holds its pattern, which is why melting starts at a surface. By 40 K about half the atoms are solid-like, and a crystal this small does not settle there: the solid share rises and falls by tens of percent over a few hundred picoseconds as it freezes a little and melts again. By 45 K it is a drop of liquid with gas around it.

Watch the energy plot while it happens. Jump from 20 K to 45 K and the kinetic energy climbs most of the way within the first 50 ps or so, then lags while the crystal melts, because the heat bath’s energy is going into loosening the atoms rather than into speeding them up. The potential energy goes on rising for several hundred picoseconds. That energy, taken in with the temperature nearly steady, is the latent heat. The latent heat calculator works out how much a real sample needs, and the heating curve calculator adds it to the warming between the changes of state.

Why a sealed box has no single boiling point

Keep heating and the liquid evaporates, but gradually. At the default size about a fifth of the atoms are gas-like at 45 K, about half at 55 K and nine in ten by 70 K. A boiling point is the temperature at which a liquid’s vapour pressure reaches the pressure above it, 87.3 K for argon under one atmosphere, and in a sealed box of fixed size there is no fixed pressure above the liquid: the vapour builds up as the temperature rises, so the drop shrinks steadily instead of boiling away at one temperature.

There is a limit, too. Above the critical temperature a liquid and its vapour stop being different at all, and for this flat, cut-off version of the potential Smit and Frenkel (1991) put it at 0.459 ε/k, which is 55 K. Near it the drop becomes ragged and gas-like and liquid-like atoms are hard to tell apart; well above it, the box holds a gas and nothing else. Its atoms still attract, so it is not quite the ideal gas of the kinetic theory gas simulator, whose hard discs never do.

Cooling it down: condensing and freezing

Choose Scattered atoms and a low temperature and the reverse happens. The atoms stick where they meet and the clusters drift together, and at 25 K most of them are solid-like within about 300 ps. The solid that forms this way is usually not one crystal but several, joined along boundaries where the pattern breaks, and it holds more energy than the crystal the other start builds: at 25 K between −2.2 and −2.4 kJ/mol, against about −2.57 kJ/mol for the single crystal. That difference is energy locked into the boundaries, which a real solid gives up only if it is warmed and cooled slowly enough to reorganise. It is also why the simulator reads the state from the atoms instead of looking it up from the temperature: two boxes at the same temperature can hold different solids.

How the simulator decides which state an atom is in

Every 1.08 ps it counts each atom’s neighbours within 0.51 nm, one and a half atomic diameters, a distance that falls in the empty gap between the crystal’s first ring of neighbours and its second. It also records how the directions to those neighbours are arranged, as the bond order ψ₆, the average of e^(6iθ) over the bonds, which is 1 when the neighbours sit at the corners of a hexagon. Both are averaged over the last 10.8 ps. An atom with fewer than three neighbours on average is gas-like. One whose neighbours have also held a steady hexagonal pattern, an average bond order of at least 0.7, is solid-like. Any other is liquid-like: packed in, but rearranging.

The headline names every state at least a fifth of the atoms are in, and keeps naming it until fewer than 15 percent are, so it does not flicker when a share sits near the line. The thresholds are choices rather than laws, and an atom at a surface or a defect can fall either way, which is why the shares are shown as well as the headline.

Why the liquid is a round drop

There is no gravity in the box, so nothing pulls the liquid down into a puddle, and at this size it would make no difference. The strongest pull between two argon atoms is 1.16 × 10⁻¹¹ N, about 1.8 × 10¹³ times an atom’s weight. On Earth a drop of water is flattened by its own weight only once it is wider than about 2.7 mm, its capillary length, and a drop a few nanometres across is round.

What this model leaves out

• The third dimension. A flat crystal gives each atom six nearest neighbours instead of twelve, so these atoms melt and evaporate at lower temperatures than real argon, which melts at 83.8 K. The same potential in three dimensions melts at 0.69455 ε/k (Schultz and Kofke, 2018), 83.2 K. Melting in two dimensions can also pass through an in-between hexatic phase, though for these atoms only when they are hotter and squeezed far harder than anything in this box (Li and Pica Ciamarra, 2020).
• Size. A few hundred atoms at most, so every change of state is spread over several kelvin and the temperature itself wobbles by a few percent around the setting. A gram of argon holds 1.5 × 10²² atoms, and its changes of state are sharp.
• Pressure. The box never changes size, so you cannot squeeze the gas into a liquid, and a boiling point, which is defined at a fixed pressure, does not exist here.
• Quantum effects. The atoms follow classical mechanics, which gets the coldest behaviour of any real solid wrong.
• Anything but argon. Real argon also feels small three-atom forces, and molecules have shapes and bonds that a single pair potential cannot describe.

Common mistakes

• Thinking the particles of a solid are still. At 20 K these average 81 m/s. A solid’s atoms vibrate about fixed places; they do not stop.
• Thinking the particles themselves expand or melt. Every atom is the same in every state. Only their arrangement and their motion change.
• Thinking heat always raises the temperature. While the crystal melts, most of the energy goes into the potential energy, and the kinetic energy that the temperature measures barely moves.
• Picturing gas particles as bigger. They are the same size and much further apart, nearly a nanometre on average here against 0.38 nm in the crystal.
• Treating a melting point as fixed for any sample. Small crystals melt lower, because more of their atoms sit at a surface. Buffat and Borel measured it for small gold particles in 1976.
• Reading a sealed box as boiling at one temperature. With no fixed pressure above it, the liquid evaporates steadily as the temperature rises.