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Rheology (from Greek ῥέω rhéō, "flow" and -λoγία, -logia, "study of") is the study of the flow of matter, primarily in a liquid 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 applied force. Rheology is the science of deformation and flow within a material. It is a branch of physics which 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 Simplicius (often attributed to Heraclitus), panta rhei, "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, 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 flow of materials, that exhibits a combination of elastic, viscous and plastic behavior by properly combining elasticity and (Newtonian) fluid mechanics. It is also concerned with establishing predictions for 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 and 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 empirically establish the relationships between strains (or rates of strain) and stresses, by adequate measurements, 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 with 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
We consider the application of a constant stress (a so-called creep experiment): We again consider the application of a constant 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.
 * Fluid and solid character are relevant at long times:
 * 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):
 * 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:

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 behaviour of all materials fall somewhere in between these two ends. The difference in material behaviour is characterized by the level and nature of elasticity present in the material when it deforms, which takes the material behaviour to the non-Newtonian regime. The non-dimensional Deborah number is designed to account for the degree of non-Newtonian behaviour 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) behaviour 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.

Reynolds number
In fluid mechanics, the Reynolds number is a measure of the ratio of inertial forces (vsρ) to viscous forces ($μ⁄L$) and consequently it quantifies the relative importance of these two types of effect for given flow conditions. Under low Reynolds numbers viscous effects dominate and the flow is laminar, whereas at high Reynolds numbers inertia predominates and the flow may be turbulent. However, since rheology is concerned with fluids which do not have a fixed viscosity, but one which can vary with flow and time, calculation of the Reynolds number can be complicated.

It is one of the most important dimensionless numbers in fluid dynamics and is used, usually along with other dimensionless numbers, to provide a criterion for determining dynamic similitude. When two geometrically similar flow patterns, in perhaps different fluids with possibly different flow rates, have the same values for the relevant dimensionless numbers, they are said to be dynamically similar.

Typically it is given as follows:


 * $$ \mathrm{Re} = \frac{\rho \frac{u_{s}^2}{L} }{ \mu \frac{u_{s}}{L^2}} = \frac{\rho u_{s} L}{ \mu} = \frac{u_{s} L}{ \nu} $$

where:
 * us – mean flow velocity, [m s−1]
 * L – characteristic length, [m]
 * μ – (absolute) dynamic fluid viscosity, [N s m−2] or [Pa s]
 * ν – kinematic fluid viscosity: ν = $μ⁄ρ$, [m2 s−1]
 * ρ – fluid density, [kg m−3].

Measurement
Rheometers are instruments used to characterize the rheological properties of materials, typically fluids that are melts or solution. These instruments impose a specific stress field or deformation to the fluid, and monitor the resultant deformation or stress. Instruments can be run in steady flow or oscillatory flow, in both shear and extension. Different measuring techniques are used depending on the size of the system of interest. More recent advances in these techniques have allowed for the measurement of rheological properties of materials only nanometers (10−9 meters) in length. Such measurements are considered under the heading of microrheology.

Measuring viscoelasticity
One way of probing the viscoelastic properties of the material is to expose it to a small oscillatory stress or strain. For example, a material placed in a concentric cylinder (couette) rheometer undergoes controlled shear strain as one of the cylinders is rotated. As long as the oscillation amplitude is small, the material will experience an oscillatory shear stress at the same frequency as the applied strain, but shifted by a phase offset. Mathematically, the stress and strain in such experiment are given by:

$$\gamma(t) = \gamma_o\sin(\omega t)$$

$$\sigma(t) = \sigma_o\sin(\omega t + \delta) = \gamma_o(G^\prime\sin(\omega t) + G^{\prime\prime}\cos(\omega t))$$

where:


 * $\gamma(t)$ - time-dependent applied strain, [unit-less]
 * $\sigma(t)$ - time-dependent response stress, [Pa] or [N m−2]
 * $\gamma_o$ - maximum strain amplitude, [unit-less]
 * $\sigma_o$ - maximum stress amplitude, [Pa] or [N m−2]
 * $\delta$ - phase offset between the strain and stress, [rad]
 * $\omega$ - angular oscillation frequency, [Rad s−1]
 * $G^{\prime}$ - elastic storage shear modulus, [Pa] or [N m−2]
 * $G^{\prime\prime}$ - loss shear modulus, [Pa] or [N m−2].

The moduli $G^\prime$ and $G^{\prime\prime}$  are related to the complex dynamic modulus via $G^* = G^{\prime} + iG^{\prime\prime}$  (here $$i$$ is the imaginary unit given by $i^2 = -1$ ), and will be frequency dependent for most viscoelastic materials. In the limiting case of an ideal elastic material, these moduli are given by $ G^\prime = G$ and $G^{\prime\prime} = 0$  where $$G$$ is the shear modulus of the material. In the other limiting case of an ideal Newtonian fluid, the two moduli are given by $ G^\prime = 0$ and $G^{\prime\prime} = \omega\mu$. $G^\prime$ is called the elastic storage shear modulus, because ideal elastic material store the energy used when a stress is applied to them. The energy applied to an ideal Newtonian fluid, however, is completely lost. Thus, $G^{\prime\prime}$ is the loss shear modulus. The viscoelastic behavior of a material resulting from an axial stress and strain can be described similarly. In this case, the stress is written:

$$\epsilon(t) = \gamma_o(E^\prime\sin(\omega t) + E^{\prime\prime}\cos(\omega t))$$

where $E^{\prime}$ and $E^{\prime\prime}$  are the Young storage and loss moduli respectively.

A frequency sweep experiment measures the values of $G^\prime$ and $G^{\prime\prime}$  over a range of oscillation frequencies, $ \omega$. Such experiments are useful in determining under what conditions the material is predominantly elastic, $ G^\prime > G^{\prime\prime}$, or predominantly viscous, $ G^\prime < G^{\prime\prime}$.

Applications
Rheology has applications in materials science, engineering, geophysics, physiology, human biology and pharmaceutics. Materials science is utilized in the production of many industrially important substances, such as cement, paint, and chocolate, which have complex flow characteristics. In addition, plasticity theory has been similarly important for the design of metal forming processes. The science of rheology and the characterization of viscoelastic properties in the production and use of polymeric materials has been critical for the production of many products for use in both the industrial and military sectors. Study of flow properties of liquids is important for pharmacists working in the manufacture of several dosage forms, such as simple liquids, ointments, creams, pastes etc. The flow behavior of liquids under applied stress is of great relevance in the field of pharmacy. Flow properties are used as important quality control tools to maintain the superiority of the product and reduce batch to batch variations.

Polymers
Examples may be given to illustrate the potential applications of these principles to practical problems in the processing and use of rubbers, plastics, and fibers. Polymers constitute the basic materials of the rubber and plastic industries and are of vital importance to the textiles.i also deals with the ghoo industry as well.

In viscoelastic materials, such as most polymers and plastics, the presence of liquid-like behaviour depends on the properties of and so varies with rate of applied load, i.e., how quickly a force is applied. The silicone toy 'Silly Putty' behaves quite differently depending on the time rate of applying a force. Pull on it slowly and it exhibits continuous flow, similar to that evidenced in a highly viscous liquid. Alternatively, when hit hard and directly, it shatters like a silicate glass.

In addition, conventional rubber undergoes a glass transition (often called a rubber-glass transition). E.g. The Space Shuttle Challenger disaster was caused by rubber O-rings that were being used well below their glass transition temperature on an unusually cold Florida morning, and thus could not flex adequately to form proper seals between sections of the two solid-fuel rocket boosters.

Sol-gel


With the viscosity of a sol adjusted into a proper range, both optical quality glass fiber and refractory ceramic fiber can be drawn which are used for fiber optic sensors and thermal insulation, respectively. The mechanisms of hydrolysis and condensation, and the rheological factors that bias the structure toward linear or branched structures are the most critical issues of sol-gel science and technology.

Geophysics
The scientific discipline of geophysics includes study of the flow of molten lava and study of debris flows (fluid mudslides). This disciplinary branch also deals with solid Earth materials which only exhibit flow over extended time-scales. Those that display viscous behaviour are known as rheids. For example, granite can flow plastically with a negligible yield stress at room temperatures (i.e. a viscous flow). Long-term creep experiments (~10 years) indicate that the viscosity of granite and glass under ambient conditions are on the order of 1020 poises.

Physiology
Physiology includes the study of many bodily fluids that have complex structure and composition, and thus exhibit a wide range of viscoelastic flow characteristics. In particular there is a specialist study of blood flow called hemorheology. This is the study of flow properties of blood and its elements (plasma and formed elements, including red blood cells, white blood cells and platelets). Blood viscosity is determined by plasma viscosity, hematocrit (volume fraction of red blood cell, which constitute 99.9% of the cellular elements) and mechanical behaviour of red blood cells. Therefore, red blood cell mechanics is the major determinant of flow properties of blood.

Food rheology
Food rheology is important in the manufacture and processing of food products, such as cheese and gelato.

Thickening agents, or thickeners, are substances which, when added to an aqueous mixture, increase its viscosity without substantially modifying its other properties, such as taste. They provide body, increase stability, and improve suspension of added ingredients. Thickening agents are often used as food additives and in cosmetics and personal hygiene products. Some thickening agents are gelling agents, forming a gel. The agents are materials used to thicken and stabilize liquid solutions, emulsions, and suspensions. They dissolve in the liquid phase as a colloid mixture that forms a weakly cohesive internal structure. Food thickeners frequently are based on either polysaccharides (starches, vegetable gums, and pectin), or proteins.

Concrete rheology
Concrete's and mortar's workability is related to the rheological properties of the fresh cement paste. The mechanical properties of hardened concrete increase if less water is used in the concrete mix design, however reducing the water-to-cement ratio may decrease the ease of mixing and application. To avoid these undesired effects, superplasticizers are typically added to decrease the apparent yield stress and the viscosity of the fresh paste. Their addition highly improves concrete and mortar properties.

Rheology in biology
Rheology has important applications to the field of biology as it helps one understand the ways cells and molecules interact with their surroundings. It has been shown that certain cells respond as strongly to physical stimuli as they do to chemical agonists, and that these stimuli impact cellular development, shape, and size.

One of the most useful ways to describe biological materials is as a power-law fluid. This behavior can be understood in terms of the material's response to a so call creep experiment. Here, a constant force, $F$, is applied to the material, and the resulting deformation $d(t)$  is recorded as a function of time. One can define a creep function, $J(t) = d(t)/F$, which has time dependence according to some power, $J(t)\propto t^\beta$ , for $\beta$ between zero and one. The case of $\beta = 0$ corresponds to an ideal elastic material, and $\beta = 1$  to an ideal Newtonian fluid. This description has been found in a large number of systems and using a variety of measurement techniques. Other characteristic properties of biological materials are an increase in stiffness in response to applied stress as well as becoming more fluid like under stress.

Biopolymers
The study of biopolymers, or polymers produced by living organisms, can include the use of rheological techniques to learn how they align and link in networks. Biopolymers make up the cytoskeleton found in eukaryotic cells. The three most important biopolymers present in the cytoskeleton are the actin-myosin network, the microtubule network, and the intermediate filament network.

Solutions of biopolymers behave differently when the concentration of bioploymers is high. For some, entangled networks form due to the essentially random alignment of the individual monomers. In other stiff or ordered biopolymers, structured fluids or gels are formed. These two cases can be distinguished using a frequency sweep experiment described above. In the case of an entangled network, $G^{\prime\prime}$ will dominate at low angular frequencies and $G^{\prime}$ will dominate at high ones. In gels and structured fluids, $G^{\prime}$ will be larger than $G^{\prime\prime}$, and both will be relatively independent of angular frequency.

Filled polymer rheology
The incorporation of various types of fillers into polymers is a common means of reducing cost and to impart certain desirable mechanical, thermal, electrical and magnetic properties to the resulting material. The advantages that filled polymer systems have to offer come with an increased complexity in the rheological behavior.

Usually when the use of fillers is considered, a compromise has to be made between the improved mechanical properties in the solid state on one side and the increased difficulty in melt processing, the problem of achieving uniform dispersion of the filler in the polymer matrix and the economics of the process due to the added step of compounding on the other. The rheological properties of filled polymers are determined not only by the type and amount of filler, but also by the shape, size and size distribution of its particles. The viscosity of filled systems generally increases with increasing filler fraction. This can be partially ameliorated via broad particle size distributions via the Farris effect. An additional factor is the stress transfer at the filler-polymer interface. The interfacial adhesion can be substantially enhanced via a coupling agent that adheres well to both the polymer and the filler particles. The type and amount of surface treatment on the filler are thus additional parameters affecting the rheological and material properties of filled polymeric systems.

It is important to take into consideration wall slip when performing the rheological characterization of highly filled materials, as there can be a large difference between the actual strain and the measured strain.

Rheologist
A rheologist is an interdisciplinary scientist or engineer who studies the flow of complex liquids or the deformation of soft solids. It is not a primary degree subject; there is no qualification of rheologist as such. Most rheologists have a qualification in mathematics, the physical sciences (e.g. chemistry, physics, geology, biology), engineering (e.g. mechanical, chemical, materials science, plastics engineering and engineering or civil engineering), medicine, or certain technologies, notably materials or food. Typically, a small amount of rheology may be studied when obtaining a degree, but a person working in rheology will extend this knowledge during postgraduate research or by attending short courses and by joining a professional association (see below).