Lambda Ratio (Specific Film Thickness)

In 1902 Richard Stribeck ran a series of careful bearing tests and plotted friction against a duty parameter combining viscosity, speed and load. The curve he got is the most important picture in tribology: friction starts high, falls steeply to a minimum, then rises again. Three regimes, three different physical mechanisms, and almost everything you would want to know about a lubricated contact depends on which of them you are in. The lambda ratio is how you find out.

The definition is simply the minimum film thickness measured in units of the combined roughness of the two surfaces: $\Lambda = h_{\min}/\sqrt{R_{q1}^2 + R_{q2}^2}$. Both numbers are typically a fraction of a micrometre, and their ratio decides the regime.

$\Lambda < 1$: boundary lubrication. The film is thinner than the roughness, so the asperities are genuinely touching and carrying a large share of the load. What protects the surfaces here is not the oil's viscosity at all but its CHEMISTRY — anti-wear additives such as ZDDP, and extreme-pressure additives, reacting with the metal to form a soft sacrificial layer that shears in preference to the substrate. Friction is high and roughly constant, wear is continuous and real, and Archard's coefficient lives at the severe end of its range, $10^{-4}$ to $10^{-2}$. Every hydrodynamic bearing passes through this regime on every start and every stop, and that is where most of its lifetime wear actually occurs — which is why start-stop cycles matter more than running hours, and why big machines that start under load get hydrostatic jacking oil.

$1 < \Lambda < 3$: mixed lubrication. The film carries most of the load and the tallest asperities carry the rest. This is where a great deal of real machinery operates, and it is not a failure to be there — friction is near its minimum in this region, so a well-run mixed contact can be more efficient than a full-film one. Wear is present but modest, with $K$ around $10^{-6}$ to $10^{-4}$. It is a workable place to live and a nervous one, because the margin is small: a modest drop in speed or viscosity, or a modest rise in load, tips it into boundary.

$\Lambda > 3$: full film. The surfaces are completely separated and wear is essentially nil. A full-film contact does not wear out; it fails from contamination, from fatigue, or from the moments when the film is absent. Friction here comes entirely from shearing oil, so it RISES with viscosity and speed — the right-hand climb of the Stribeck curve — which means that past this point a thicker oil costs power and buys nothing. Wear coefficients run down at $10^{-8}$ and below.

This is why a friction or wear number quoted without its regime is close to useless. The same steel pair, same load, same counterface, can show $K$ spanning six orders of magnitude across these three regimes. When someone hands you a wear coefficient, the first question is not what the materials were — it is what $\Lambda$ was.

Two measurement points, and both bite. The ratio wants $R_q$, the root-mean-square roughness, and most surface testers report $R_a$, the arithmetic average. For a ground or turned surface $R_q$ runs about 1.1 to 1.3 times $R_a$, so quietly substituting $R_a$ makes $\Lambda$ look roughly 20 % better than it is — a difference that matters most exactly at the regime boundaries, where it is doing its work. And the two roughnesses combine in quadrature, not by adding, so the larger dominates: if one surface is 0.4 µm and the other 0.1 µm, polishing the smooth one to nothing barely moves the answer, while a modest improvement to the rough one moves it a great deal. Spend the money on the rougher surface.

Smoother is not unconditionally better, though, and the cylinder bore is the standing counterexample. A perfectly polished surface holds no oil. A plateau-honed finish — a smooth bearing plateau cut across a cross-hatched valley structure that retains lubricant and carries away debris — outperforms a mirror. Real surfaces also run themselves in: the roughness at ten hours is not the roughness on the drawing, and $\Lambda$ generally improves over the first hours of service. That is the entire justification for a run-in procedure, and the reason a machine run hard from cold on day one may never reach the film it was designed for.

One last point of scale. The films that separate these surfaces are a fraction of a micrometre. That is why a single hard particle a few microns across is a serious event rather than a nuisance, and why filtration is not housekeeping but part of the lubrication design.