Rheology

Rheology (; from Ancient Greek ῥέω (rhéō) 'flow' and -λoγία (-logía) 'study of') is the study of the flow of matter, primarily in a fluid (liquid or gas) state, as well as "soft solids", which experience conditions under which they respond with plastic flow rather than elastic deformation to forces applied to them. Rheology is the branch of physics that deals with the deformation and flow of materials, both solids and liquids.

The term rheology was coined by Eugene C. Bingham, a professor at Lafayette College, in 1920 from a suggestion by a colleague, Markus Reiner. The term was inspired by the aphorism of Heraclitus (often mistakenly attributed to Simplicius), panta rhei (πάντα ῥεῖ, 'everything flows') and was first used to describe the flow of liquids and the deformation of solids. It applies to substances that have a complex microstructure, such as muds, sludges, suspensions, and polymers and other glass formers (e.g., silicates), as well as many foods and additives, bodily fluids (e.g., blood) and other biological materials, and other materials that belong to the class of soft matter such as food.

Newtonian fluids can be characterized by a single coefficient of viscosity for a specific temperature. Although this viscosity will change with temperature, it does not change with the strain rate. Only a small group of fluids exhibit such constant viscosity. The large class of fluids whose viscosity changes with the strain rate (the relative flow velocity) are called non-Newtonian fluids.

Rheology generally accounts for the behavior of non-Newtonian fluids by characterizing the minimum number of functions that are needed to relate stresses with rate of change of strain or strain rates. For example, ketchup can have its viscosity reduced by shaking (or other forms of mechanical agitation, where the relative movement of different layers in the material actually causes the reduction in viscosity), but water cannot. Ketchup is a shear-thinning material, like yogurt and emulsion paint (US terminology latex paint or acrylic paint), exhibiting thixotropy, where an increase in relative flow velocity will cause a reduction in viscosity, for example, by stirring. Some other non-Newtonian materials show the opposite behavior, rheopecty (viscosity increasing with relative deformation), and are called shear-thickening or dilatant materials. Since Sir Isaac Newton originated the concept of viscosity, the study of liquids with strain-rate-dependent viscosity is also often called non-Newtonian fluid mechanics.

The experimental characterisation of a material's rheological behaviour is known as rheometry, although the term rheology is frequently used synonymously with rheometry, particularly by experimentalists. Theoretical aspects of rheology are the relation of the flow/deformation behaviour of material and its internal structure (e.g., the orientation and elongation of polymer molecules) and the flow/deformation behaviour of materials that cannot be described by classical fluid mechanics or elasticity.

Scope

In practice, rheology is principally concerned with extending continuum mechanics to characterize the flow of materials that exhibit a combination of elastic, viscous and plastic behavior by properly combining elasticity and (Newtonian) fluid mechanics. It is also concerned with predicting mechanical behavior (on the continuum mechanical scale) based on the micro- or nanostructure of the material, e.g. the molecular size and architecture of polymers in solution or the particle size distribution in a solid suspension. Materials with the characteristics of a fluid will flow when subjected to a stress, which is defined as the force per area. There are different sorts of stress (e.g. shear, torsional, etc.), and materials can respond differently under different stresses. Much of theoretical rheology is concerned with associating external forces and torques with internal stresses, internal strain gradients, and flow velocities.

Rheology unites the seemingly unrelated fields of plasticity and non-Newtonian fluid dynamics by recognizing that materials undergoing these types of deformation are unable to support a stress (particularly a shear stress, since it is easier to analyze shear deformation) in static equilibrium. In this sense, a solid undergoing plastic deformation is a fluid, although no viscosity coefficient is associated with this flow. Granular rheology refers to the continuum mechanical description of granular materials.

One of the major tasks of rheology is to establish by measurement the relationships between strains (or rates of strain) and stresses, although a number of theoretical developments (such as assuring frame invariants) are also required before using the empirical data. These experimental techniques are known as rheometry and are concerned with the determination of well-defined rheological material functions. Such relationships are then amenable to mathematical treatment by the established methods of continuum mechanics.

The characterization of flow or deformation originating from a simple shear stress field is called shear rheometry (or shear rheology). The study of extensional flows is called extensional rheology. Shear flows are much easier to study and thus much more experimental data are available for shear flows than for extensional flows.

Viscoelasticity

• Fluid and solid character are relevant at long times:
  We consider the application of a constant stress (a so-called creep experiment):
  • if the material, after some deformation, eventually resists further deformation, it is considered a solid
  • if, by contrast, the material flows indefinitely, it is considered a fluid
• By contrast, elastic and viscous (or intermediate, viscoelastic) behaviour is relevant at short times (transient behaviour):
  We again consider the application of a constant stress:
  • if the material deformation strain increases linearly with increasing applied stress, then the material is linear elastic within the range it shows recoverable strains. Elasticity is essentially a time independent processes, as the strains appear the moment the stress is applied, without any time delay.
  • if the material deformation strain rate increases linearly with increasing applied stress, then the material is viscous in the Newtonian sense. These materials are characterized due to the time delay between the applied constant stress and the maximum strain.
  • if the materials behaves as a combination of viscous and elastic components, then the material is viscoelastic. Theoretically such materials can show both instantaneous deformation as elastic material and a delayed time dependent deformation as in fluids.
• Plasticity is the behavior observed after the material is subjected to a yield stress:
  A material that behaves as a solid under low applied stresses may start to flow above a certain level of stress, called the yield stress of the material. The term plastic solid is often used when this plasticity threshold is rather high, while yield stress fluid is used when the threshold stress is rather low. However, there is no fundamental difference between the two concepts.

Dimensionless numbers

Deborah number

On one end of the spectrum we have an inviscid or a simple Newtonian fluid and on the other end, a rigid solid; thus the behavior of all materials fall somewhere in between these two ends. The difference in material behavior is characterized by the level and nature of elasticity present in the material when it deforms, which takes the material behavior to the non-Newtonian regime. The non-dimensional Deborah number is designed to account for the degree of non-Newtonian behavior in a flow. The Deborah number is defined as the ratio of the characteristic time of relaxation (which purely depends on the material and other conditions like the temperature) to the characteristic time of experiment or observation. Small Deborah numbers represent Newtonian flow, while non-Newtonian (with both viscous and elastic effects present) behavior occurs for intermediate range Deborah numbers, and high Deborah numbers indicate an elastic/rigid solid. Since Deborah number is a relative quantity, the numerator or the denominator can alter the number. A very small Deborah number can be obtained for a fluid with extremely small relaxation time or a very large experimental time, for example.