# Rheology

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Rheology () is the study of the flow of matter, primarily in the 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.[1] It applies to substances which have a complex molecular structure, 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.

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 flow rate or strain rate. Only a small group of fluids exhibit such constant viscosity, and they are known as Newtonian fluids. But for a large class of fluids, the viscosity change with the strain rate (or relative velocity of flow) and are called non-Newtonian fluids. Rheology generally accounts for the behaviour of non-Newtonian fluids, by characterizing the minimum number of functions that are needed to relate stresses with rate of change of strains 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, as an increase in relative velocity caused a reduction in viscosity, while some other non-Newtonian materials show the opposite behaviour: viscosity going up with relative deformation, which 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.[1]

The term rheology was coined by Eugene C. Bingham, a professor at Lafayette College, in 1920, from a suggestion by a colleague, Markus Reiner.[2][3] The term was inspired by the aphorism of Simplicius (often misattributed to Heraclitus), panta rei, "everything flows"[4]

The experimental characterization 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 behaviour by properly combining elasticity and (Newtonian) fluid mechanics. It is also concerned with establishing predictions for mechanical behaviour (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 for different stresses. Much of theoretical rheology is concerned with associating external forces and torques with internal stresses and internal strain gradients and velocities.[1][5][6][7]

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, solids 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 deformations and stresses, respectively their derivatives 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.

## Rheologist

A rheologist is an interdisciplinary scientist/engineer who studies the flow of complex liquids or the deformation of soft solids. It is not taken as a primary degree subject, and there is no general qualification. He or she will usually have a primary qualification in one of several fields: mathematics, the physical sciences (e.g. chemistry, physics, biology), engineering (e.g. mechanical, chemical, Materials Science & 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 the professional will extend this knowledge during postgraduate research or by attending short courses and by joining one of the professional associations (see below).

## Viscoelasticity

The classical theory of elasticity deals with the behaviour of elastic solids under small deformations, for which,(1) according to Hooke's Law, stress is directly proportional to the strain — but independent of the rate of strain, or how fast the deformation was applied, and (2) the strains are completely recoverable once the stress is removed. Materials that can be characterized by classical theory of elasticity is known as linear elastic materials, even for such materials the linear relationship between stress and strain may be valid only for a certain range of strains. A large number of solids show non-linear relationship between stress and strain even for small stresses (such as rubber), but if the strains are still recoverable they are known as non-linear elastic materials. The classical theory of fluid mechanics, governed by the Navier-Stokes equation, deals with the behaviour of viscous fluids, for which, according to Newton's Law, the stress is directly proportional to the rate of strain, but independent of the strain itself. These behaviour are, of course, generally observed for ideal materials under ideal conditions, although the behaviour of many solids approaches Hooke's law for infinitesimal strains, and that of many fluids approaches Newton's law for infinitesimal rates of strain. Two types of deviations from linearity may be considered here.

1. When finite strains (larger strains, as opposed to infinitesimal strains) are applied to solid bodies, the stress-strain relationships are often much more complicated (i.e. Non-Hookean). Similarly, in steady flow with finite strain rates, many fluids exhibit marked deviations in stress-strain rate proportionality from Newtons law.
2. Even if both strain and rate of strain are infinitesimal, a system may exhibit both liquid-like and solid-like characteristics. A good example of this is when a body which is not quite an elastic solid (i.e. an inelastic solid) does not maintain a constant deformation under constant stress, but rather continues to deform with time – or "creeps" under the same stress at constant temperature. When such a body is constrained at constant deformation, the stress required to hold it at that stretch level gradually diminishes—or "relaxes" with time.

Similarly, a fluid while flowing under constant stress may show some elastic properties as well, such as storing some of the energy input instead of dissipating it all as heat and random thermal motion of its molecular constituents or having some recovery of strains after stresses are removed, although it may never recover all of its deformation upon removal of the initial applied stress. When such bodies are subjected to a sinusoidally oscillating stress, the strain is neither exactly in phase with the stress (as it would be for a perfectly elastic solid) nor 90 degrees out of phase (as it would be for a perfectly viscous liquid) but rather exhibits a strain that lags the stress at a value between zero and 90 degrees: i.e., Some of the energy is stored and recovered in each cycle, and some is dissipated as heat. These are viscoelastic materials.

Thus, fluids are generally associated with viscous behaviour (a thick oil is a viscous liquid) and solids with elastic behaviour (an elastic string is an elastic solid). A more general point of view is to consider the material behaviour at short times (relative to the duration of the experiment/application of interest) and at long times.

• Fluid and solid character are relevant at long times:</br>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):</br>We again consider the application of a constant stress[8]:
• 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 stains appear the moment the stress is appliled, without any time delay.
• if the material deformation 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 behaviour observed after the material is subjected to a yield stress:</br>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.

## 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 concrete, 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 behaviour 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.

### Materials science

#### Polymers

The viscoelastic properties of polymers are determined by the effects of the many variables, including temperature, pressure, and time. Other important variables include chemical composition, molecular weight and weight distribution, degree of branching and crystallinity, types of functionality, component concentration, dilution with solvents or plasticizers, and mixture with other materials to form composite systems. With guidance by molecular theory, the dependence of viscoelastic properties on these variables can be simplified by introducing additional concepts such as the free volume, the monomeric friction coefficient, and the spacing between entanglement loci, to provide a qualitative understanding and in many cases a quantitative prediction of how to achieve desired physical and chemical properties and ultimate microstructure.

Viscoelastic behaviour reflects the combined viscous and elastic responses, under mechanical stress, of materials which are intermediate between liquids and solids in character. Fundamentally, the viscoelasticity can be related to the motions of flexible polymer molecules and their entanglements and network junctions—the molecular basis of viscoelasticity. Thus, rearrangements on a local scale (kinks) are relatively rapid, while on a long-range scale (convolutions) very slow. In addition, a new assortment of configurations is obtained under stress. The response to the local aspects of the new distribution is rapid, while the response to the long-range aspects is slow. Thus there is very wide and continuous range of timescales covering the response of such a system to externally applied stress. From measurements of the viscoelastic properties of polymers, information can be obtained about the nature and the rates of change of the configurational rearrangements, and the nature of the (macro)molecular interactions over a range of time scales.

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 textile, petroleum, automobile, paper, and pharmaceutical industries. Their viscoelastic properties determine the mechanical performance of the final products of these industries, and also the success of processing methods at intermediate stages of production.

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.

#### Biopolymers

Linear structure of cellulose -- the most common component of all organic plant life on Earth. * Note the evidence of hydrogen bonding which increases the viscosity at any temperature and pressure. This is an effect similar to that of polymer crosslinking, but less pronounced.

A major but defining difference between polymers and biopolymers can be found in their structures. Polymers, including biopolymers, are made of repetitive units called monomers. While polymers are often randomly constructed with massive entanglement, biopolymers often have a well defined structure. In the case of proteins, the exact chemical composition and the sequence in which these units are arranged is called the primary structure.

Many proteins spontaneously fold into characteristic compact shapes—which determine their biological functions and depend in a complicated way on their primary structures. Structural biology is the study of the structural properties of the biopolymers, much of which can be determined by their viscoelastic response to a wide range of loading conditions.

#### Sol-gel

Polymerization process of tetraethylorthosilicate (TEOS) and water to form amorphous hydrated silica particles (Si-OH) can be monitored rheologically by a number of different methods.

Sol-gel science (aka chemical solution deposition) is a wet-chemical technique widely used in the fields of materials science, glass production and ceramic engineering. Such methods are used primarily for the fabrication of materials (typically a metal oxide) starting from a chemical solution which acts as the precursor for an integrated network (or gel) of either discrete nanoparticles or network polymers. Typical precursors are metal alkoxides and metal chlorides, which undergo various forms of hydrolysis and polycondensation reactions in order to form a viscoelastic network (or solid).

One of the largest application areas is thin films and coatings, which can be produced on a piece of substrate by spin coating or dip coating. Other methods include spraying, electrophoresis, inkjet printing or roll coating. Optical coatings, protective and decorative coatings, and electro-optic components can be applied to glass, metal and other types of substrates with these methods. 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

Geophysics includes the flow of molten lava and debris flows (fluid mudslides). Also included in this disciplinary branch are solid Earth materials which only exhibit flow over extended time scales. Those that display viscous behaviour are known as rheids. E.G. granite can flow plastically with a vanishingly small yield stress at room temperatures, (i.e. a viscous flow). Long term creep experiments (~ 10 years) indicate that the viscosity of granite under ambient conditions is on the order of 1020 poises.[9]

Piledriving for a bridge in Napa, California.

Deep foundations are used for structures or heavy loads when shallow foundations cannot provide sufficient capacity. They may also be used to transfer building loads past weak or compressible soil layers. While shallow foundations rely solely on the bearing capacity of the soil beneath them, deep foundations can rely on end bearing resistance, frictional resistance along their length, or both in developing the required capacity. Geotechnical engineers use specialized tools, such as the cone penetration test, to estimate the amount of skin and end bearing resistance available in the subsurface.

In addition, pile driving is often used to check for stability in varying soil types such as clay, sand, gravels, fractured shale, etc. Geotechnical engineering (or 'soil engineering') often utilizes soil logs or bore logs to show what may be evidenced while driving piles through given stratum and soil lenses. Wave equations must often be employed when using vibratory or mechanical impact hammers. The harmonics set up by vibratory or impact hammers drastically change the ability of given soils to create wall friction on a given pile type, as well as the elastic alteration or resistance to penetration in a normal state.

Dynamic testing of soils may involve the attachment of transducers to pilings while they are being driven. In addition, theoretical bearing calculations using a nuclear densometer may be carried out in the field. In the end, a fairly simple linear equation may suffice to give a good approximation of the bearing capacity of the soil.

### 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.[10]

### Food rheology

Food rheology is important in the manufacture and processing of food products, e.g. cheese.[11] Food rheology is the study of the rheological properties of food, that is, the consistency and flow of food under tightly specified conditions. The consistency, degree of fluidity, and other mechanical properties are important in understanding how long food can be stored, how stable it will remain, and in determining food texture. The acceptability of food products to the consumer is often determined by food texture, such as how spreadable and creamy a food product is. Food rheology is important in quality control during food manufacture and processing.

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.[12]

### 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 are better if less water is used in the preparation of concrete paste, 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.[13]

## 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.

## 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 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.[14][15] 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:

$\mathit{Re} = {\rho v_{s}^2/L \over \mu v_{s}/L^2} = {\rho v_{s} L\over \mu} = {v_{s} L\over \nu}$

where:

• vs - mean fluid 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].

## References

1. W. R. Schowalter (1978) Mechanics of Non-Newtonian Fluids Pergamon ISBN 0-08-021778-8
2. J. F. Steffe (1996) Rheological Methods in Food Process Engineering 2nd ed ISBN 0-9632036-1-4 page 1
3. The Deborah Number
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5. R. B. Bird, W. E. Stewart, E. N. Lightfoot (1960), Transport Phenomena, John Wiley & Sons, ISBN 0-471-07392-X
6. R. Byrin Bird, Charles F. Curtiss, Robert C. Armstrong (1989), Dynamics of Polymeric Liquids, Vol 1 &2 , Wiley Interscience, ISBN 0-471-51844-1 and 978-0471518440
7. Faith A. Morrison (2001), Understanding Rheology, Oxford University Press, ISBN 0-19-514166-0 and 978-0195141665
8. William N. Findley, James S. Lai, Kasif Onaran (1989), Creep and Relaxation of Nonlinear Viscoelastic Materials, Dover Publications
9. Kumagai, N., Sasajima, S., Ito, H., Long-term Creep of Rocks, J. Soc. Mat. Sci. (Japan), Vol. 27, p. 157 (1978) Online
10. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
11. S. Gunasekaran, M. Mehmet (2003), Cheese rheology and texture, CRC Press, ISBN 1-58716-021-8
12. {{#invoke:citation/CS1|citation |CitationClass=book }}
13. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
14. M. Reiner (1964) Physics Today volume 17 no 1 page 62 The Deborah Number
15. The Deborah Number