# Deborah number

The Deborah number (De) is a dimensionless number, often used in rheology to characterize the fluidity of materials under specific flow conditions. It is based on the premise that given enough time even a solid-like material will flow. The flow characteristics are not inherent properties of the material alone, but a relative property which depends on two fundamentally different characteristic times.

## Definition

Formally, the Deborah number is defined as the ratio of the relaxation time characterizing the time it takes for a material to adjust to applied stresses or deformations, and the characteristic time scale of an experiment (or a computer simulation) probing the response of the material:

$De equals tc divided by tp$

where tc refers to the stress relaxation time (sometimes called the Maxwell relaxation time), and tp refers to the time scale of observation.

This incorporates both the elasticity and viscosity of the material. At lower Deborah numbers, the material behaves in a more fluidlike manner, with an associated Newtonian viscous flow. At higher Deborah numbers, the material behavior enters the non-Newtonian regime, increasingly dominated by elasticity and demonstrating solidlike behavior.[1][2]

## History

The Deborah Number was originally proposed by Markus Reiner, a professor at Technion in Israel, who chose the name inspired by a verse in the Bible, stating "The mountains flowed before the Lord" in a song by prophetess Deborah (Judges 5:5).

## Time-temperature superposition

The Deborah Number is particularly useful in conceptualizing the time–temperature superposition principle. Time-temperature superposition has to do with altering experimental time scales using reference temperatures to extrapolate temperature-dependent mechanical properties of polymers. A material at low temperature with a long experimental or relaxation time behaves like the same material at high temperature and short experimental or relaxation time if the Deborah number remains the same. This can be particularly useful when working with materials which relax on a long time scale under a certain temperature. The practical application of this idea arises in the Williams–Landel–Ferry equation. Time-temperature superposition avoids the inefficiency of measuring a polymer’s behavior over long periods of time at a specified temperature by utilizing the Deborah Number.[3]

## References

1. ^ Reiner, M. (1964), "The Deborah Number", Physics Today 17 (1): 62, Bibcode:1964PhT....17a..62R, doi:10.1063/1.3051374
2. ^ The Deborah Number
3. ^ Rudin, Alfred, and Phillip Choi. The Elements of Polymer Science and Engineering. 3rd. Oxford: Academic Press, 2013. Print. Page 221.