Deborah Number

The dimensionless Deborah number is one of the most fundamental numbers of rheology. It can be used to describe the viscoelastic behavior of any material. This number was first introduced by Marcus Reiner (1964). He defined the Deborah number as the ratio of the relaxation time of a material to the observation or experimental time¹:

De = time of relaxation / time of observation

or

De = λ(T) / t_obs

where t_obs is a characteristic time of the deformation process and λ(T) is the relaxation time. The viscoelastic response during the observation time is then proportional to its Deborah number, De; or to quote Reiner's original paper:² "The greater the Deborah number, the more solid the material; the smaller the the Deborah number, the more fluid it is."

To better understand the physical meaning of this number, we choose as an example the stress relaxation of a one-dimensional Maxwell element. In this experiment, a rapid strain rate is applied, so that the material undergoes an almost instantaneous deformation (strain), ε₀, which is kept constant through out the experiment. Then the strain rate, dε₀/dt, will vanish after a (short) time t₀, whereas the stress, σ(t), will relax slowly. The basic equation for strain rate in the Maxwell model is

dε / dt = 1/E · dσ / dt + σ / η = 0; (  t > t₀)

or

0 = λ(T) · dσ / dt + σ(t)

where λ(T) is the response or relaxation time of the one-dimensional Maxwell element which is represented by the ratio of viscosity η to Young's modulus E. Separation of the variables and integration yields

σ(t - t₀) = E λ₀ · exp[-(t - t₀) / λ(T)]

where E λ₀ = σ₀ is the integration constant, i.e. the stress at time t₀. The stress relaxation experiment is one of the best experiments to measure or predict the relaxation modulus:

E_R(t - t₀) =  σ(t - t₀) / λ₀ =  E · exp[-(t - t₀) / λ(T)]

If the time scale for the deformation process t_obs = t - t₀ is much shorter than the relaxation time λ(T), then the material behaves like a (rigid) solid, and E_R ≈ E. If, on the other hand, the temperature T or the observation time t_obs is increased, so that t_obs » t₀, then the relaxation modulus vanishes and E_R ≈ 0. In that case, the material exhibits liquid like behavior. Obviously, the Deborah number De = λ(T) / t_obs determines what behavior prevails. Or in other words, materials will show solid like bahavior for large Deborah numbers and liquid-like behavior for small Deborah numbers.

 

References and Notes

1. Reiner named this dimensionless number after the prophete Deborah who, in the Book of Judges, proclaimedThe mountains flowed before the lord.²

2. M Reiner, The Deborah Number, Phys. Today, 17, pp 62 (1964)

3. Another dimensionless number sometimes used in the study of viscoelastic flow is the Weissenberg number. It is defined as the ratio of viscous forces to elastic forces of a material which is equal to the relaxation time times the shear or strain rate: Wi = λdγ / dt or Wi = λ·dε / dt. The Weissenberg number can be used to describe the flow of a material with a constant stretch history like simple shear.