The Deborah number (De) is a
dimensionless number
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
, often used in
rheology
Rheology (; ) is the study of the flow of matter, primarily in a fluid ( liquid or gas) state, but also as "soft solids" or solids under conditions in which they respond with plastic flow rather than deforming elastically in response to an appl ...
to characterize the fluidity of materials under specific flow conditions. It quantifies the observation that given enough time even a solid-like material might flow, or a fluid-like material can act solid when it is deformed rapidly enough. Materials that have low relaxation times flow easily and as such show relatively rapid stress decay.
Definition
The Deborah number is the ratio of fundamentally different characteristic times. The Deborah number is defined as the ratio of 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:
:
where stands for the relaxation time and for the "time of observation", typically taken to be the time scale of the process.
The numerator,
relaxation time
In the physical sciences, relaxation usually means the return of a perturbed system into equilibrium.
Each relaxation process can be categorized by a relaxation time τ. The simplest theoretical description of relaxation as function of time ' ...
, is the time needed for a reference amount of deformation to occur under a suddenly applied reference load (a more fluid-like material will therefore require less time to flow, giving a lower Deborah number relative to a solid subjected to the same loading rate).
The denominator, material time,
is the amount of time required to reach a given reference strain (a faster loading rate will therefore reach the reference strain sooner, giving a higher Deborah number).
Equivalently, the relaxation time is the time required for the stress induced, by a suddenly applied reference strain, to reduce by a certain reference amount. The relaxation time is actually based on the rate of relaxation that exists at the moment of the suddenly applied load.
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.
For example, for a Hookean elastic solid, the relaxation time will be infinite and it will vanish for a Newtonian viscous fluid. For liquid water, is typically 10
−12 s, for lubricating oils passing through gear teeth at high pressure it is of the order of 10
−6 s and for polymers undergoing plastics processing, the relaxation time will be of the order of a few seconds. Therefore, depending on the situation, these liquids may exhibit elastic properties, departing from purely viscous behavior.
While is similar to the
Weissenberg number
The Weissenberg number (Wi) is a dimensionless number used in the study of viscoelastic flows. It is named after Karl Weissenberg. The dimensionless number compares the elastic forces to the viscous forces. It can be variously defined, but it is u ...
and is often confused with it in technical literature, they have different physical interpretations. The Weissenberg number indicates the degree of anisotropy or orientation generated by the deformation, and is appropriate to describe flows with a constant stretch history, such as simple shear. In contrast, the Deborah number should be used to describe flows with a non-constant stretch history, and physically represents the rate at which elastic energy is stored or released.
History
The Deborah number was originally proposed by
Markus Reiner
Markus Reiner ( he, מרכוס ריינר, born 5 January 1886, died 25 April 1976) was an Israeli scientist and a major figure in rheology.
Biography
Reiner was born in 1886 in Czernowitz, Bukovina, then part of Austria-Hungary, and obtained ...
, a professor at
Technion in
Israel
Israel (; he, יִשְׂרָאֵל, ; ar, إِسْرَائِيل, ), officially the State of Israel ( he, מְדִינַת יִשְׂרָאֵל, label=none, translit=Medīnat Yīsrāʾēl; ), is a country in Western Asia. It is situated ...
, who chose the name inspired by a verse in the
Bible
The Bible (from Koine Greek , , 'the books') is a collection of religious texts or scriptures that are held to be sacred in Christianity, Judaism, Samaritanism, and many other religions. The Bible is an anthologya compilation of texts of a ...
, stating "The mountains flowed before the Lord" in a song by the prophetess
Deborah
According to the Book of Judges, Deborah ( he, דְּבוֹרָה, ''Dəḇōrā'', " bee") was a prophetess of the God of the Israelites, the fourth Judge of pre-monarchic Israel and the only female judge mentioned in the Bible. Many scholars ...
in the
Book of Judges
The Book of Judges (, ') is the seventh book of the Hebrew Bible and the Christian Old Testament. In the narrative of the Hebrew Bible, it covers the time between the conquest described in the Book of Joshua and the establishment of a kingdom i ...
; הָרִ֥ים נָזְל֖וּ מִפְּנֵ֣י יְהוָ֑ה ''hā-rîm nāzəlū
mippənê Yahweh
Yahweh *''Yahwe'', was the national god of ancient Israel and Judah. The origins of his worship reach at least to the early Iron Age, and likely to the Late Bronze Age if not somewhat earlier, and in the oldest biblical literature he posse ...
'').
Time-temperature superposition
The Deborah number is particularly useful in conceptualizing the
time–temperature superposition
The time–temperature superposition principle is a concept in polymer physics and in the physics of glass-forming liquids.
This superposition principle is used to determine temperature-dependent mechanical properties of linear viscoelastic mat ...
principle. Time-temperature superposition has to do with altering experimental time scales using reference temperatures to extrapolate temperature-dependent mechanical properties of
polymers
A polymer (; Greek '' poly-'', "many" + ''-mer'', "part")
is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic an ...
. A material at low temperature with a long experimental or
relaxation time
In the physical sciences, relaxation usually means the return of a perturbed system into equilibrium.
Each relaxation process can be categorized by a relaxation time τ. The simplest theoretical description of relaxation as function of 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
The Williams– Landel– Ferry Equation (or WLF Equation) is an empirical equation associated with time–temperature superposition.
The WLF equation has the form
: \log(a_T) = \frac
where ''\log(a_T)'' is the decadic logarithm of the WLF shi ...
. 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.
[Rudin, Alfred, and Phillip Choi. The Elements of Polymer Science and Engineering. 3rd. Oxford: Academic Press, 2013. Print. Page 221.]
References
Further reading
* J.S. Vrentas, C.M. Jarzebski, J.L. Duda (1975
"A Deborah number for diffusion in polymer-solvent systems" AIChE
The American Institute of Chemical Engineers (AIChE) is a professional organization for chemical engineers. AIChE was actually established in 1908 to distinguish chemical engineers as a profession independent of chemists and mechanical engineers ...
Journal 21(5):894–901, weblink to Wiley Online Library.
{{NonDimFluMech
Dimensionless numbers of fluid mechanics
Fluid dynamics
Rheology