States of matter
Chips are made in vacuum chambers, and the reason is not cleanliness: it is that at 10⁻⁶ torr an evaporated atom travels fifty meters without colliding with anything.
01Gases, and why they are the easy ones
In a gas the molecules are so far apart that they hardly notice one another. That mutual indifference is what makes it possible to describe the gas with a single equation, the same for all gases:
The equation fails when the gas is compressed or cooled enough for the molecules to start noticing each other. Van der Waals corrects the two things the ideal model ignores —the volume of the molecules themselves and the attraction between them—:
In diffusion furnaces and deposition chambers, where the gas flow is measured in sccm —standard cubic centimeters per minute—, which is a unit of amount of substance disguised as a volume. Converting sccm to moles per second is applying \( PV = nRT \) with \( P \) and \( T \) fixed by convention.
02Liquids and solids: when the forces win
As the temperature drops, kinetic energy is no longer enough to overcome the intermolecular forces and matter condenses. In a liquid the molecules touch but slide past each other; in a solid they are fixed in definite positions.
| State | Volume and shape | Order | What dominates |
|---|---|---|---|
| Gas | Neither of its own | None | Thermal energy |
| Liquid | Volume yes, shape no | Short-range | A tie |
| Crystalline solid | Both | Long-range, periodic | The bonding forces |
| Amorphous solid | Both | Short-range, frozen | The viscosity on cooling |
The last row matters more than it seems. Glass and thermal silicon dioxide are amorphous: they have the local order of the crystal —each silicon surrounded by four oxygens— but no periodicity. And the silicon of wafers is the opposite, a single crystal in which the lattice repeats without interruption from one end to the other, because any grain boundary would be a leakage path for the carriers.
Ionic —NaCl, ceramic oxides—: hard, brittle, insulating as solids. Covalent network —Si, diamond, SiC—: the hardest, with a very high melting point, semiconductors or insulators. Metallic —Cu, Al—: ductile and conductive. Molecular —ice, paraffin, most plastics—: soft and with a low melting point, because what has to be overcome is not the bond but the force between molecules.
03The phase diagram
The state of a substance is not decided by temperature alone: temperature and pressure decide it together. The map of that decision is the phase diagram, and on it there are two points with names of their own: the triple point, where the three states coexist, and the critical point, beyond which the distinction between liquid and gas no longer makes sense.
Move the temperature and the pressure, and the point is located on the map. Look at silicon and water: their melting curves slope to the left, unlike all the others.
Almost all substances contract on solidifying, and that is why their melting curve slopes to the right: more pressure, harder to melt. Water and silicon expand on freezing —both form a tetrahedral network more open than the liquid—, so their curve slopes to the left. It is the same reason why ice floats and why a quartz crucible in which the silicon solidifies cracks.
04Vacuum, which is where chips are made
A good part of electronics manufacturing takes place in vacuum chambers: metal evaporation, sputtering, ion implantation. The reason is not cleanliness but geometry. For an evaporated atom to reach the wafer in a straight line, it must not collide with anything along the way, and how far it travels without colliding is the mean free path:
Those 50 meters are the number that justifies all of vacuum engineering: if the chamber measures half a meter, a mean free path of 50 m means that practically no atom collides before arriving. The lab above calculates \( \lambda \) for the chosen pressure.
| Regime | Pressure | \( \lambda \) | What for |
|---|---|---|---|
| Atmosphere | 760 torr | 65 nm | — |
| Medium vacuum | 10⁻³ torr | 5 cm | Sputtering, etching plasmas |
| High vacuum | 10⁻⁶ torr | 50 m | Metal evaporation |
| Ultra-high vacuum | 10⁻⁹ torr | 50 km | Molecular beam epitaxy |
05In the lab
With a capped 60 mL syringe and a pressure gauge, compress the air in steps and record \( P \) and \( V \). Plot \( P \) against \( 1/V \) and verify the linearity. Estimate the error introduced by the dead volume of the connection.
Observe solid CO₂ at atmospheric pressure: it sublimes without passing through a liquid. Explain with the phase diagram why, and calculate at what pressure it would have to be held to see it melt.
Melt solder tin in a crucible with a thermocouple inside and record the temperature as it cools. Identify the solidification plateau and compare it with the melting point of the alloy used.
06Common mistakes
- Using \( PV = nRT \) with the temperature in degrees Celsius. \( T \) always goes in kelvin.
- Believing that every solid is crystalline. Glass and thermal SiO₂ are amorphous, and that changes their properties.
- Thinking that more pressure always solidifies. In water and in silicon it is the other way around.
- Confusing evaporation with boiling. One occurs at the surface at any temperature; the other, throughout the volume, when the vapor pressure equals the external pressure.
- Assuming that vacuum means “having nothing.” At 10⁻⁶ torr about 10¹⁰ molecules remain per cubic centimeter; what changes is how far they travel without colliding.
07Self-assessment
What volume does one mole of gas occupy at 0 °C and 1 atm?
22.4 liters, and it does not depend on which gas it is.
What is the triple point?
The only combination of pressure and temperature at which solid, liquid and vapor coexist. For water, 0.006 atm and 0.01 °C.
Why does ice float?
Because water expands on solidifying: its tetrahedral network is more open than the liquid. Silicon does the same.
What distinguishes a crystalline solid from an amorphous one?
Long-range order. The amorphous one keeps the local order but not the periodicity.
What is high vacuum for in metal evaporation?
So that the mean free path greatly exceeds the size of the chamber and the atoms reach the wafer in a straight line.
What does the van der Waals equation correct?
The volume of the molecules themselves, with \( b \), and the attraction between them, with \( a \).
08Further reading
- Raymond Chang and Kenneth A. Goldsby. Chemistry. 12th ed. (Spanish edition, “Química”), McGraw-Hill, 2017. Chapters 5 and 11: ideal and real gases, intermolecular forces and phase diagrams.
- William D. Callister and David G. Rethwisch. Materials Science and Engineering: An Introduction. 9th ed. (Spanish edition, “Introducción a la ciencia e ingeniería de los materiales”), Reverté, 2019. Crystal structures, amorphous solids and equilibrium diagrams of alloys.
- Stephen A. Campbell. Fabrication Engineering at the Micro- and Nanoscale. 4th ed., Oxford University Press, 2013. The vacuum chapter: mean free path, pumps and the regimes in the table of this unit.