The Deborah Number

The relaxation of the stress resulting from a step strain can be observed experimentally and we can see that it is the result of diffusive motion of the microstructural elements. Although we can have a mechanistic picture, what does this mean in terms of our measurements We have the very striking result that our material classification must depend on the time t, i.e. the experimental or observation time. Hence, we can usefully classify material behaviour into three categories  

The most frequently quoted example to illustrate this behaviour is the children s toy Silly Putty , which is a poly(dimethyl siloxane) polymer. Pulled rapidly it shows brittle fracture like any solid but if pulled slowly it flows as a liquid. The relaxation time for this material is 1 s. After t = 5t the stress will have fallen to 0.7% of its initial value so the material will have effectively forgotten its original shape. That is, one could describe it as having a memory of around 5 s (about that of a mackerel ). Many other materials in common use have relaxation times within an order of magnitude or so of 1 s. Examples are thickened detergents, personal care products and latex paints. This is of course no coincidence, and this timescale is frequently deliberately chosen by formulation adjustments. The reason is that it is in the middle of our, 

It is diflicult to write a concise definition of a time constant that governs the rate at which stored elastic energy changes in a given deformation without reference to a specific flow, and we therefore give a definition of the Deborah number in general terms 

---

Stress relaxation time, obtained from rheograms based on viscometric flows, is used to define a dimensionless parameter called the Deborah number , which quantifies the elastic character of a fluid... 

A parameter indicating whether viscoelastic effects are important is the Deborah number, which is the ratio of the characteristic relaxation time of the fluid to the characteristic time scale of the flow. For small Deborah numbers, the relaxation is fast compared to the characteristic time of the flow, and the fluid behavior is purely viscous. For veiy large Deborah numbers, the behavior closely resembles that of an elastic solid. 

The Deborah number Db has been defined by Metzner, White and Denn 201 as ... 

This relative importance of relaxation and diffusion has been quantified with the Deborah number, De [119,130-132], De is defined as the ratio of a characteristic relaxation time A. to a characteristic diffusion time 0 (0 = L2/D, where D is the diffusion coefficient over the characteristic length L) De = X/Q. Thus rubbers will have values of De less than 1 and glasses will have values of De greater than 1. If the value of De is either much greater or much less than 1, swelling kinetics can usually be correlated by Fick s law with the appropriate initial and boundary conditions. Such transport is variously referred to as diffusion-controlled, Fickian, or case I sorption. In the case of rubbery polymers well above Tg (De < c 1), substantial swelling may occur and... 

Then calculate the Deborah number from Eq. (6-99), using k = 0.088 and k2 = 0.0431 from Table 6-2 ... 

It is helpful here to introduce the Deborah number De defined by... 

The fluid s relaxation time A is the characteristic time of the fluid and, for oscillatory shearing, cu 1 can be taken as a measure of the characteristic time of the flow process, so De = A to. Thus, viscous behaviour occurs when the Deborah number is low, reflecting the fact that the fluid is able to relax. When the Deborah number is high, elastic behaviour is observed because the fluid is unable to relax sufficiently quickly. 

Although a mechanism for stress relaxation was described in Section 1.3.2, the Deborah number is purely based on experimental measurements, i.e. an observation of a bulk material behaviour. The Peclet number, however, is determined by the diffusivity of the microstructural elements, and is the dimensionless group given by the timescale for diffusive motion relative to that for convective or flow. The diffusion coefficient, D, is given by the Stokes-Einstein equation ... 

The similarity to the Peclet number is obvious but we should also bear in mind the relationship to the Deborah number. This becomes clear when we consider the fact that the mechanism of stress relaxation is due to the... 

In order to observe linear viscoelasticity, structural relaxation by diffusion must occur on a timescale comparable to our measurement time. The ratio of these times is the Deborah number. When this is of the order of unity our experiment will follow the relaxation processes in the material and the material will appear to be viscoelastic ... 

We can get a first approximation of the physical nature of a material from its response time. For a Maxwell element, the relaxation time is the time required for the stress in a stress-strain experiment to decay to 1/e or 0.37 of its initial value. A material with a low relaxation time flows easily so it shows relatively rapid stress decay. Thus, whether a viscoelastic material behaves as a solid or fluid is indicated by its response time and the experimental timescale or observation time. This observation was first made by Marcus Reiner who defined the ratio of the material response time to the experimental timescale as the Deborah Number, Dn-Presumably the name was derived by Reiner from the Biblical quote in Judges 5, Song of Deborah, where it says The mountains flowed before the Lord. ... 

The dimensionless number in rheology that compares relative importance of the time scale of the deformation process tD over the observation time tQ is called the Deborah number (De) ... 

While the Deborah number is often used to compare the time for deformation with the time of observation in experiments, it also inspires us to identify and formulate other dimensionless groups that compare the various characteristic times and forces relevant in colloidal phenomena. We discuss some of the important ones. 

What is the Deborah number What is its physical significance What is the Peclet number Describe at least two ways of defining the Peclet number for flow of dispersions. 

With the average elongational strain rate of the flow field between the eddies and the relaxation time of the polymer molecules, one can define a dimensionless characteristic number, the Deborah number, which represents the ratio of a characteristic time of flow and a characteristic time of the polymer molecule, and thus one can transfer considerations in porous media flow to the turbulent flow region. 

The difference between solids and liquids is found in the magnitude of D. Liquids, which relax in small fractions of a second, have small D. Solids have a large D. A sufficient lime span can reduce the Deborah number of a solid to unity, and impact loading can increase D of a liquid. Viscoelastic materials are best characterized under conditions in which D lies within a few decades of unity. 

A useful parameter often used to estimate the elastic effects during flow is the Deborah2 number, De. The Deborah number is defined by... 

From the Song of Deborah, Judges 5 5 - "The mountains flowed before the Lord." M. Rainer is credited for naming the Deborah number Physics Today, 1, (1964). 

Note that since in this case, the Deborah number, De = Aw, the moduli and the loss tangent, G. G", tan S, are functions of the Deborah number. 

This upper limit decreases with increasing Mw, with increasing molecular weight distribution (MWD) at constant Mw, and with decreasing melt temperature. On physical grounds, it is considered to terminate roughly where the Deborah number reaches unity. 

---