The Rheology Handbook

Thomas G. Mezger

The Rheology Handbook

For users of rotational and oscillatory rheometers

5th Revised Edition

Thomas G. Mezger

The Rheology Handbook

For users of rotational and oscillatory rheometers

5th Revised Edition

# Foreword

People working in industry are often confronted with the effects of rheology, the science of deformation and flow behavior. When looking for appropriate literature, they find either short brochures which give only a few details and contain little useful information, or highly specialized books overcharged of physical formulas and mathematical theories. There is a lack of literature between these two extremes which reduces the discussion of theoretical principles to the necessary topics, providing useful instructions for experiments on material characterization. This book is intended to fill that gap.

The practical use of rheology is presented in the following areas: quality control (QC), production and application, chemical and mechanical engineering, industrial research and development, and materials science. Emphasis is placed on current testing methods related to daily working practice. After reading this book, the reader should be able to perform useful tests with rotational and oscillatory rheometers, and to interpret the achieved results correctly.

### How did this book come into existence?

The first computer-controlled rheometers came into use in industrial laboratories in the mid-1980s. Ever since then, test methods as well as control and analysis options have improved with breath-taking speed. In order to organize and clarify the growing mountain of information, company Anton Paar Germany – and previously Physica Messtechnik – has offered basic seminars on rheology already since 1988, focused on branch-specific industrial application. During the “European Coatings Show” in Nuremberg in April 1999, the organizer and publishing director Dr Lothar Vincentz suggested expanding these seminar notes into a comprehensive book about applied rheology.

### What is the target audience for this book? For which industrial branches will it be most interesting?

The Rheology Handbook is written for everyone approaching rheology without any prior knowledge but is also useful to people wishing to update their expertise with information about recent developments. The reader can use the book as a course book and read from beginning to end or as a reference book for selected chapters. The numerous cross-references make connections clear and the detailed index helps when searching. If required, the book can be used as the first step on the ladder towards theory-orientated rheology books at university level. In order to break up the text, there are as well many figures and tables, illustrative examples and small practical experiments, as well as several exercises for calculations. The following list reflects how the contents of the book are of interest to rheology users in many industrial branches.

• Polymers: Solutions, melts, solids; film emulsions, cellulose solutions, latex emulsions, solid films, sheetings (uni-laminar, multi-laminar), laminates; natural resins, epoxies, casting resins; silicones, caoutchouc, gums, soft and hard rubbers; thermoplastics, elastomers, thermosets, blends, foamed materials; uncrosslinked and cross-linked polymers containing or without fillers or fibers; polymeric compounds and composites; solid bars of glass-fiber, carbon-fiber and synthetic-fiber reinforced polymers (GFRP, CFRP, SFRP); polymerization, cross-linking, curing, vulcanization, melting and hardening processes; powder rheology, resin powders, granulates
• Adhesives and sealants: Glues, single and multi-component adhesives, pressure sensitive adhesives (PSA), UV curing adhesives, hotmelts, plastisol pastes (e. g. for automotive underseals and seam sealings), construction adhesives, putties; uncured and cured adhesives; curing process; tack, stringiness
• Coatings, paints, lacquers: Spray, brush, dip coatings; solvent-borne, water-based coatings; metallic effect, textured, low solids, high solids, photo-resists, UV (ultra violet) radiation curing, powder coatings; glazes and stains for wood; coil coatings; reactive fire-protection coatings; solid coating films; powder rheology, powder coatings, colored powders (e. g. titanium dioxide, soot), e. g. for additive manufacturing (AM)
• Printing inks and varnishes: Gravure, letterpress, flexographic, planographic, offset, screen printing inks, UV (ultra violet) radiation curing inks; ink-jet printer inks; writing inks for pens; mill-base premix, color pastes, “thixo-pastes”; liquid and pasty pigment dispersions; printing process; misting; tack; powder rheology: materials for additive manufacturing (AM)
• Paper coatings: Primers and topcoats; immobilization process
• Foodstuffs: Water, vegetable oils, aroma solvents, fruit juices, baby food, liquid nutrition, liqueurs, syrups, purees, thickeners as stabilizing agents, gels, pudding, jellies, ketchup, mayonnaise, mustard, dairy products (such as yogurt, cream cheese, cheese spread, soft and hard cheese, curds, butter), emulsions, chocolate (melt), soft sweets, ice cream, chewing gum, dough, whisked egg, cappuccino foam, sausage meat, sauces containing meat chunks, jam containing fruit pieces, animal feed; bio-technological fluids; gel formation of hydrocolloids (e. g. of corn starch and gelatin); interfacial rheology (e. g. for emulsions, foams); rheology of powders and granulates: milk powder, cocoa powder, coffee powder, coffee whitener, flour, starch powder (e. g. as a binder), powdered sugar, granulated sugar, spices, animal feed (as granulates, pellets), grain, corn, rice, spray-dried products; influence of humidity (e. g. biscuits, cookies, crackers); food tribology (e. g. for creaminess); tack
• Cosmetics, beauty care products: Perfume oils, emulsions (e. g. skin care, hair-dye), lotions, nail polish, roll-on fluids (deodorants), shampoo, shower gels, skin creams, abrasive peeling creams, hair gels, styling waxes, shaving creams, tooth-gels, toothpastes, makeup dispersions, lipstick, mascara, medical adhesives (e. g. for diapers), super-absorbers; hairs, sponges; interfacial rheology (e. g. for emulsions, foams); powder rheology: make-up powders, rouge, deodorant powders, dry shampoo, baby powders, hygienics powders
• Pharmaceuticals, medicaments, bio-tech products, health and personal care products: Cough mixtures, wetting agents, nose sprays, vaccines, blood (hemo-rheology), blood-plasma substitutes, emulsions, saliva, mucus, hydrogels, skin creams, synovia fluid (e. g. for joints), hyaluronan acid (HA), ointments, vaseline, natural and synthetic membranes, silicone pads and cushions, dental molding materials, tooth filling, sponges, contact lenses, medical adhesives (e. g. for skin plasters, dental prothesis), denture fixative creams, hair, bone cement, implants, organic-inorganic compounds (hybrids); “biologically active” suspensions and gels (e. g. microalgae, bacteria); tribolgy: bacterial bio-films, biological cells, tissue engineered medical products (TEMPs), cartilage, catheters; interfacial rheology (e. g. emulsions, foams); powder rheology: tablets, disinfection powders
• Agrochemicals: Plant or crop protection agents, solutions and dispersions of insecticides and pesticides, herbicides and fungicides
• Detergents, home care products: Household cleaning agents, liquid soap, disinfectants, surfactant solutions, dispersions containing viscoelastic surfactants (VES), washing-up liquids, dish washing agents, laundry, fabric conditioners, washing powder concentrate, fat removers; interfacial rheology: emulsions, foams; powder rheology: superabsorbers (e. g. for diapers)
• Surface technology: Polishing and abrasive suspensions; cooling emulsions; powder rheology, tribology: polishing powders, abrasive suspensions
• Electrical engineering, electronics industry: Thick film pastes, conductive, resistance, insulating, glass paste, soft solder and screen-printing pastes; SMD adhesives (for surface mounted devices), insulating and protective coatings, de-greasing agents, battery fluids and pastes, coatings for electrodes
• Petrochemicals: Crude oils, petroleum, solvents, de-icing agents, fuels, mineral oils, light and heavy oils, lubricating greases, paraffines, waxes, petrolatum, vaseline, natural and polymer-modified bitumen (PmB), asphalt binders, distillation residues; from coal and wood: tar and pitch; interfacial rheology (e. g. for emulsions); tribology: lubricating behavior
• Ceramics and glass: Casting slips, kaolin and porcelain suspensions, glass powder and enamel pastes, glazes, plastically deformable ceramic pastes, glass melts, aero-gels, xero-gels, sol/gel materials, composites, organo-silanes (hybrids), basalt melts; powder rheology: ceramic powders (e. g. for additive manufacturing, AM), clay, loam
• Construction materials: Self-levelling cast floors, plasters, mortar, cement suspensions, tile adhesives, dispersion paints, sealants, floor sheeting, natural and polymer-modified bitumen (PmB), and GTR (ground tire rubber) modified asphalt binder (for road pavement); bulk and powder rheology: sand, lime, chalk, gypsum
• Metals: Melts of magnesium, aluminum, steel, alloys, slags; molding process in a semi-solid state (“thixo-forming”, “thixo-casting”, “thixo-forging”), compounds: ceramic fiber reinforced light-weight metals; powder rheology: metal powders (e. g. for additive manufacturing, AM)
• Waste industry: Waste water, sewage sludges, animal excrements (e. g. of fishes, poultry, cats, dogs, pigs, cattle), residues from refuse incineration plants; powder rheology: sludges, filter cakes
• Geology, soil mechanics, mining industry: Sludges from coal, peat, soil, drilling muds; river and lake sediment masses; soil deformation (e. g. due to mining operations, earthwork, canal and drain constructions, operations of vehicles in agriculture); drilling fluids, fracturing fluids (e. g. containing “flow improvers”); melts of volcanic stones (e. g. basalt), lava, magma, salt melts; powder rheology: coal powder, briquet manufacturing
• Disaster control: Foam for fire extinguishers, deformation behavior of burning materials, soil deformation due to floods and earthquakes
• Materials for special functions (e. g. as “smart fluids”): Magneto-rheological fluids (MRF), electro-rheological fluids (ERF), di-electric (DE) materials, self-repairing coatings, materials showing self-organizing superstructures (e. g. surfactants), dilatant fabrics (shock-absorbing, “shot-proof”), mesogenic fluids (MF), liquid crystals (LC), ionic fluids, micro-capsule paraffin wax (e. g. as “phase-change material” PCM), shape-memory materials (SPM); tribology: haptic sensation (when prooving the shape of the whole sample) or tactile sensation (when touching or scanning the surface); systems reacting by a change in shape due to an external excitation (e. g. temperature, light, pressure); powder rheology: materials used for additive manufacturing (AM)

It is pleasing that the first four editions of The Rheology Handbook, published in 2002, 2006, 2011 and 2014 sold out so unexpectedly quickly. It was positive to hear that the books met with approval, not only from laboratory technicians and practically oriented engineers, but also from teachers and professors of schools and colleges of applied sciences. Even at universities, The Rheology Handbook is meanwhile taken as an introductory teaching material for explaining the basics of rheology in lectures and practical courses, and as a consequence, many students worldwide are using it when writing their final paper or thesis. This textbook is also available in German language, and between 2000 and 2016 also here, five editions were published meanwhile (title: Das Rheo­logie Handbuch).

New in this fifth edition is Chapter 13 (shear tests with powders and bulk solids). Further present-day examples have been added resulting as well from contacts to industrial users as well as from corporation with several working groups, e. g. for developing modern standardizing measuring methods for diverse industrial branches. The references and standards have been updated (e. g. in Chapter 15).

I hope that The Rheology Handbook will prove itself a useful source of information for characterizing the above mentioned products in an application-oriented way, assuring their quality and helping to improve them wherever possible.

Stuttgart, June 2020

Thomas G. Mezger

# 1Introduction

## 1.1Rheology, rheometry and viscoelasticity

##### a) Rheology

Rheology is the science of deformation and flow. It is a branch of physics and physical chemistry since the most important variables come from the field of mechanics: forces, deflections and velocities. The term rheology originates from the Greek: rhei or rheo meaning to flow [1.1]. Thus, rheology is literally flow science. However, rheological experiments do not merely reveal information about flow behavior of liquids but also about deformation behavior of solids. The connection here is that a large deformation produced by shear forces causes many materials to flow.

All kinds of shear behavior, which can be described rheologically in a scientific way, can be viewed as being in between two extremes: flow of ideal-viscous liquids on the one hand and deformation of ideal-elastic solids on the other. Illustrative examples coming close to these two extremes of ideal behaviors are a low-viscosity mineral oil and a rigid steel ball. Viscosity and flow behavior of fluids are explained in Chapter 2. Elasticity and deformation behavior of solids are described in Chapter 4.

Behavior of all real materials is based on the combination of both a viscous and an elastic portion and therefore, it is called viscoelastic. Wallpaper paste is a viscoelastic liquid, for example, and a gum eraser is a viscoelastic solid. Information on viscoelastic behavior can be found in Chapter 5. Complex and extraordinary rheological behavior is presented in Chapter 9 using the example of surfactant systems.

Table 1.1 shows the most important terms, all of which will be covered in this book. This chart can also be found at the beginning of Chapters 2 to 8, with those terms given in bold print being discussed in the chapter in hand.

 Table 1.1: Overview on different kinds of rheological behavior Liquids Solids (ideal-) viscousflow behaviorviscosity law viscoelasticflow behaviorMaxwell model viscoelasticdeformation behaviorKelvin/Voigt model (ideal-) elasticdeformation behaviorelasticity law (according to Newton) (according to Hooke) flow/viscosity curves creep tests, relaxation tests, oscillatory tests

Rheology was first seen as a science in its own, right not before the beginning of the 20th century. However, scientists and practical users have long before been interested in the behavior of liquids and solids, although some of their methods have not always been very scientific. A list of important facts of the historical development in rheology is given in Chapter 14. Of special interest are here the various attempts to classify all kinds of different rheological behavior, such as the classification of Markus Reiner in 1931 and 1960, and of George W. Scott Blair in 1942; see also [1.2]. The aim of the rheologists’ is to measure deformation and flow behavior of a great variety of matters, to present the obtained results clearly and to explain it.

##### b) Rheometry

Rheometry is the measuring technology used to determine rheological data. The emphasis here is on measuring systems, instruments, and methods for testing and analysis. Both liquids and solids, but also powders, can be investigated using rotational and oscillatory rheometers. Rotational tests which are performed to characterize viscous behavior are presented in Chapter 3. In order to evaluate viscoelastic behavior, creep tests (Chapter 6), relaxation tests (Chapter 7) and oscillatory tests (Chapter 8) are performed. Chapter 10 contains information on measuring systems (e. g. measuring geometries) and special measuring devices, and Chapter 11 gives an overview on diverse instruments used. Shear experiments on slightly compressed powders and on strongly compressed bulk materials are explained in Chapter 13.

Analog programmers and on-line recorders for plotting flow curves have been on the market since around 1970. Around 1980, digitally controlled instruments appeared which made it possible to store measuring data and to use a variety of analysis methods, including also complex ones. Developments in measuring technology are constantly pushing back the limits. At the same time, thanks to standardized measuring systems (geometries) and procedures, measuring results can be compared world-wide today. Meanwhile, several rheometer manufacturers can offer test conditions to customers in many industrial branches which come very close to simulate even complex process conditions in practice.

A short guideline for rheological measurements is presented in Chapter 12 in order to facilitate the daily laboratory work for practical users.

##### c) Appendix

Chapter 15 (Appendix) shows all the used signs, symbols and abbreviations with their units. The Greek alphabet and a conversion table for units (SI and CGS system) can also be found there.

More than 500 standards are listed in Chapter 16 (ISO, ASTM, EN and DIN). The refer
ences, publications and books are specified at the end of the respective chapter. They can be identified by the number in brackets (e. g. [12.34] as reference 34 in Chapter 12).

##### d) Information for “Mr. and Ms. Cleverly”

Throughout this textbook, the reader will find sections for “Mr. and Ms. Cleverly” which are marked with a symbol showing glasses:

These sections are written for those readers who wish to go deeper into the theoretical side and who are not afraid of a little extra mathematics and fundamentals in physics. However, these “Cleverly” explanations are not required to understand the information given in the normal text of later chapters, since this textbook is also written for beginners in the field of rheology. Therefore, for those readers who are above all interested in the practical side of rheology, the “Cleverly” sections can simply be ignored.

## 1.2Deformation and flow behavior

We are confronted with rheological phenomena every single day. Some experiments are listed below to demonstrate this point. The examples given will be discussed in detail in the chapters mentioned in brackets.

##### Experiment 1: Behavior of mineral oil, plasticine, and steel

Completely different types of behavior can be seen when the following three subjects hit the floor (see Figure 1.1):

1. The mineral oil is flowing and spreading until it shows a very thin layer finally (ideal-viscous flow behavior: see Chapter 2.3.1)
2. The plasticine will be deformed when it hits the floor, and afterwards, it remains deformed permanently (inhomogeneous plastic behavior outside the linear viscoelastic deformation range: see Chapter 3.3.4.2c)
3. The steel ball bounces back, and exhibits afterwards no deformation at all (ideal-elastic behavior: see Chapter 4.3.1)

Figure 1.1: Deformation behavior after hitting the floor:
a) mineral oil, b) plasticine, c) steel ball

##### Experiment 2: Playing with “bouncing putty” (some call it “Silly Putty”)

The silicone polymer (uncrosslinked PDMS) displays different rheological behaviors depending on the period of time under stress (viscoelastic behavior of polymers: see Chapter 8.4, frequency sweep):

1. When stressed briefly and quickly, the putty behaves like a rigid and elastic solid: If you mold a piece of it to the shape of a ball and throw it on the floor, it is bouncing back.
2. When stressed slowly at a constantly low force over a longer period of time, the putty shows the behavior of a highly viscous, yielding and creeping liquid: If it is in the state of rest, thus, if you leave it untouched for a certain period of time, it is spreading very slowly under its own weight due to gravity to show an even layer with a homogeneous thickness finally.
##### Experiment 3: Do the rods remain in the position standing up straight?

Three wooden rods are put into three glasses containing different materials and left for gravity to do its work.

1. In the glass of water , the rod changes its position immediately and falls to the side of the glass (ideal-viscous flow behavior: see Chapter 2.3.1).
• Additional observation: All the air bubbles which were brought into the water when immersing the rod are rising quickly within seconds.
1. In the glass containing a silicone polymer (uncrosslinked PDMS), the rod moves very, very slowly, reaching the side of the glass after around 10 minutes (polymers showing zero-
shear viscosity: see Chapters 3.3.2.1a).
• Additional observation concerning the air bubbles which were brought into the polymer sample by the rod: Large bubbles are rising within a few minutes, but the smaller ones seem to remain suspended without visible motion. However, after several hours even the smallest bubble has reached the surface. Therefore, indeed long-term but complete de-aeration of the silicone occurs finally.
1. In the glass containing a hand cream , the rod still remains standing straight in the initial position even after some hours (yield point and flow point: see Chapters 3.3.4, 4.4 and 8.3.4).
• Additional observation concerning the air bubbles: All bubbles, independent of their size, remain suspended, and therefore here, no de-aeration takes place at all.
##### Summary

Rheological behavior depends on many external influences. Above all, the following test conditions are important:

• Type of loading (preset of deformation, velocity or force; or shear strain, shear rate or shear stress, respectively)
• Temperature (see Chapters 3.5 and 8.6)

Further important parameters are, for example:

• Concentration (e. g. of solid particles in a suspension: see Chapter 3.3.3; of polymer molecules in a solution: see Chapter 3.3.2.1a; of surfactants in a dispersion: see Chapter 9). Using an immobilization cell, the amount of liquid can be reduced under controlled conditions (e. g. when testing dispersions such as paper coatings: see Chapter 10.8.1.3).
• Ambient pressure (see Chapter 3.6)
• pH value (e. g. with surfactant systems: see Chapter 9)
• Strength of a magnetic or an electric field when investigating magneto-rheological fluids or electro-rheological fluids (MRF, ERF), respectively (see Chapters 10.8.1.1 and 2).
• UV radiation curing (e. g. of resins, adhesives and inks: see Chapter 10.8.1.4).
• Air humidity (see Chapter 10.8.1.5)
• Amount of air, flowing through a fluidized mixture of powder and air (see Chapter 13.3)
• Degree of solidification in a powder or compressed bulk material (e. g. granulate; see Chapter 13.2)

## 1.3References

[1.1]Beris, A. N., Giacomin, A. J., Panta rhei – everthing flows, J. Appl. Rheol. 24 (2014) 52918

[1.2] McKinley, G., A hitchhikers guide to complex fluids, Rheol. Bull., 84(1), (2015)

# 2Flow behavior and viscosity

In this chapter are explained the following terms given in bold:

 Liquids Solids (ideal-) viscous viscoelasticflow behaviorMaxwell model viscoelasticdeformation behaviorKelvin/Voigt model (ideal-) elasticdeformation behaviorelasticity law flow behavior (according to Hooke) viscosity law (according to Newton) flow/viscosity curves creep tests, relaxation tests, oscillatory tests

## 2.1Introduction

Before 1980 in industrial practice, rheological experiments on pure liquids and dispersions were carried out almost exclusively in the form of rotational tests which enabled the characterization of flow behavior at medium and high flow velocities. Meanwhile since measurement technology has developed, many users have expanded their investigations on deformation and flow behavior performing measurements which cover also the low-shear range.

## 2.2Definition of terms

Figure 2.1: The Two-Plates model for shear tests to illustrate the velocity distribution of a flowing fluid in the shear gap

Figure 2.2: Laminar flow in the form of planar fluid layers

The Two-Plates model is used to define fundamental rheological parameters (see Figure 2.1). The upper plate with the (shear) area A is set in motion by the (shear) force F and the resulting velocity v is measured. The lower plate is fixed (v = 0). Between the plates there is the distance h, and the sample is sheared in this shear gap. It is assumed that the following shear conditions are occurring:

1. The sample shows adhesion to both plates without any wall-slip effects.
2. There are laminar flow conditions, i. e. flow can be imagined in the form of layers. Therefore, there is no turbulent flow, i. e. no vortices are appearing.

Accurate calculation of the rheological parameters is only possible if both conditions are met.

##### Experiment 1: The stack of beer mats

Each one of the individual beer mats represents an individual flowing layer. The beer mats are showing a laminar shape, and therefore, they are able to move in the form of layers along one another (see Figure 2.2). Of course, this process takes place without vortices, thus without showing any turbulent behavior.

The real geometric conditions in rheometer measuring systems (or measuring geometries) are not as simple as in the Two-Plates model. However, if a shear gap is narrow enough, the necessary requirements are largely met and the definitions of the following rheological parameters can be used.

### 2.2.1Shear stress

Definition of the shear stress:

Equation 2.1

τ = F/A

τ (pronounced: tou); with the shear force F [N] and the shear area (or shearing surface area) A [m2], see Figure 2.1. The following holds: 1 N = 1 kg · m/s2

The unit of the shear stress is [Pa], (pascal).

Blaise Pascal (1623 to 1662 [2.1]) was a mathematician, physicist, and philosopher.

For conversions: 1 Pa = 1 N/m2 = 1 kg/m · s2

A previously used unit was [dyne/cm2]; with: 1 dyne/cm2 = 0.1 Pa

Note: [Pa] is also the unit of pressure

100 Pa = 1 hPa (= 1 mbar); or 100,000 Pa = 105 Pa = 0.1 MPa (= 1 bar)

Example: In a weather forecast, the air pressure is given as 1070 hPa (hecto-pascal; = 107 kPa).

Some authors take the symbol σ for the shear stress (pronounced: sigma) [2.2] [2.3]. However, this symbol is usually used for the tensile stress (see Chapters 4.2.2, 10.8.4.1 and 11.2.14). To avoid confusion and in agreement with the majority of current specialized literature and standards, here, the symbol τ will be used to represent the shear stress (see e. g. ISO 3219-1, ASTM D4092 and DIN 1342-1).

### 2.2.2Shear rate

Definition of the shear rate:

Equation 2.2

$Formel$ = v/h

$Formel$ (pronounced: gamma-dot); with the velocity v [m/s] and the distance h [m] between the plates, see Figure 2.1.

The unit of the shear rate is [1/s] or [s -1 ], called “reciprocal seconds”.

Sometimes, the following terms are used as synonyms: strain rate , rate of deformation, shear gradient , velocity gradient .

Previously, the symbol D was often taken instead of $Formel$. Nowadays, almost all current standards are recommending the use of $Formel$ (see e. g. ISO 3219-1, ASTM D4092). Table 2.1 presents typical shear rate values occurring in industrial practice.

For “Mr. and Ms. Cleverly”

##### a) Definition of the shear rate using differential variables

Equation 2.3

$Formel$ = dv/dh

flowing layers, and the “infinitely” (differentially) small thickness dh of a single flowing layer (see Figure 2.2).

 Table 2.1: Typical shear rates of technical processes Process Shear rates $Formel$ (s-1) Practical examples physical aging, long-term creep within days and up to several years 10-8 ... 10-5 solid polymers, asphalt cold flow 10-8 ... 0.01 rubber mixtures, elastomers sedimentation of particles ≤ 0.001 ... 0.01 emulsion paints, ceramic suspensions, fruit juices surface leveling of coatings 0.01 ... 0.1 coatings, paints, printing inks sagging of coatings, dripping, flow under gravity 0.01 ... 1 emulsion paints, plasters, chocolate melt (couverture) self-leveling at low-shear conditions in the range of the zero-shear viscosity ≤ 0.1 silicones (PDMS) mouth sensation 1 ... 10 food dip coating 1 ... 100 dip coatings, candy masses applicator roller, at the coating head 1 ... 100 paper coatings thermoforming 1 ... 100 polymers mixing, kneading 1 ... 100 rubbers, elastomers chewing, swallowing 10 ... 100 jelly babies, yogurt, cheese spreading 10 ... 1000 butter, spreadcheese extrusion 10 ... 1000 polymer melts, dough,ceramic pastes, tooth paste pipe flow, capillary flow 10 ... 104 crude oils, paints, juices, blood mixing, stirring 10 ... 104 emulsions, plastisols,polymer blends injection molding 100 ... 104 polymer melts, ceramic suspensions coating, painting, brushing, rolling, blade coating (manually) 100 ... 104 brush coatings, emulsion paints, wall paper paste, plasters spraying 1000 ... 104 spray coatings, fuels, nose spray aerosols, adhesives impact-like loading 1000 ... 105 solid polymers milling pigments in fluid bases 1000 ... 105 pigment pastes for paints and printing inks rubbing 1000 ... 105 skin creams, lotions, ointments spinning process 1000 ... 105 polymer melts, polymer fibers blade coating (by machine), high-speed coating 1000 ... 107 paper coatings, adhesive dispersions lubrication of engine parts 1000 ... 107 mineral oils, lubricating greases

There is a linear velocity distribution between the plates, since the velocity v decreases linearly in the shear gap. Thus, for laminar and ideal-viscous flow, the velocity difference between all neighboring layers are showing the same value: dv = const. All the layers are assumed to have the same thickness: dh = const. Therefore, the shear rate is showing a constant value everywhere between the plates of the Two-Plates model since

$Formel$ = dv/dh = const/const = const (see Figure 2.3).

Figure 2.3: Velocity distribution and shear rate in the shear gap of the Two-Plates model

Both $Formel$ and v provide information about the velocity of a flowing fluid. The advantage of selecting the shear rate is that it shows a constant value throughout the whole shear gap. Therefore, the shear rate is independent of the position of any flowing layer in the shear gap. Of course, this applies only if the shear conditions are met as mentioned in the beginning of Chapter 2.2. However, this does not apply to the velocity v which decreases from the maximum value vmax on the upper, movable plate to the minimum value vmin = 0 on the lower, immovable plate. Therefore, when testing pure liquids, sometimes as a synonym for shear rate the term velocity gradient is used (e. g. in ASTM D4092).

##### b) Calculation of shear rates occurring in technical processes

The shear rate values which are given below are calculated using the mentioned formulas and should only be seen as rough estimations. The main aim of these calculations is to get merely an idea of the dimension of the relevant shear rate range.

##### 1) Coating processes: painting, brushing, rolling or blade-coating

$Formel$ = v/h, with the coating velocity v [m/s] and the wet layer thickness h [m]

##### 1a) Painting with a brush:

With v = 0.1 m/s and h = 100 µm = 0.1 mm = 10-4 m; result: $Formel$ = 1000 s-1

With v = 0.1 m/s and h = 1 mm = 10-3 m; result: $Formel$ = 100 s-1

##### 1c) Applying emulsion paint with a roller

With v = 0.2 m/s (or 5 s per m), and h = 100 µm = 0.1 mm = 10-4 m; result: $Formel$ = 2000 s-1

With the application rate AR (i. e. mass per coating area) m/A [g/m2]; for the coating volume V [m3] applies, with the mass m [kg] and the density ρ [1 g/cm3 = 1000 kg/m3]: V = m/ρ

Calculation: h = V/A = (m/ρ)/A = (m/A)/ρ = AR/ρ; with AR = 1 g/m2 = 10-3 kg/m2 holds:
h =10-6 m = 1 µm; and then: $Formel$ = v/h. See Table 2.2 for shear rates occurring in various kinds of blade-coating processes [2.4] [2.5].

 Table 2.2: Shear rates of various kinds of blade-coating processes for adhesive emulsions Coating process Application rateAR [g/m2] Coating velocityv [m/min] Coating velocityv [m/s] Layer thicknessh [µm] Approx. shear rate range $Formel$ [s-1] metering blade 2 to 50 up to 250 up to 4.2 2 to 50 80,000 to 2 mio. roller blade 15 to 100 up to 100 up to 1.7 15 to 100 10,000 to 100,000 lip-type blade 20 to 100 20 to 50 0.33 to 0.83 20 to 100 3000 to 50,000 present maximum 2 to 100 700 12 2 to 100 120,000 to 6 mio. future plans up to 1500 up to 25 250,000 to 12.5 mio.

##### 2) Flow in pipelines, tubes and capillaries

Assumptions: horizontal pipe, steady-state and laminar flow conditions (for information on laminar and turbulent flow see Chapter 3.3.3), ideal-viscous flow, incompressible liquid. According to the Hagen/Poiseuille relation , the following holds for the maximum shear stress τw and the maximum shear rate $Formel$w in a pipeline (index w for “at the wall”):

Equation 2.4

τw = (R ⋅ Δp) / (2 ⋅ L)

Equation 2.5

$Formel$w = (4 ⋅ $Formel$) / (π ⋅ R3)

With the pipe radius R [m]; the pressure difference Δp [Pa] between inlet and outlet of the pipe or along the length L [m] of the measuring section, respectively (Δp must be compensated by the pump pressure); and the volume flow rate $Formel$ [m3/s]. This relation was named in honor to Gotthilf H. L. Hagen (1797 to 1848) [2.6] and Jean L. M. Poiseuille (1799 to 1869) [2.7].

##### 2a) Pipeline transport of automotive coatings [2.8] [2.9]

For a closed circular pipeline with the diameter DN 26 (approx. R = 13 mm = 1.3 ⋅ 10-2 m), and the volume flow rate $Formel$ = 1.5 to 12 L/min = 2.51 ⋅ 10-5 to 2.00 ⋅ 10-4 m3/s; results: $Formel$w = 14.6 to 116 s-1 = approx. 15 to 120 s-1. For a pipeline branch with DN 8 (approx. R = 4 mm = 4 ⋅ 10-3 m), and $Formel$ = 0.03 to 1 L/min = 5.06 ⋅ 10-7 to 1.67 ⋅ 10-5 m3/s; results: $Formel$w = 10.1 to 332 s-1 = approx. 10 to 350 s-1

##### 2b) Drinking water supply, transport in pipelines [2.10]

For a pipeline with the diameter DN 1300 (approx. R = 650 mm = 0.65 m), and a volume flow rate of max. $Formel$ = 3300 L/s = 3.30 m3/s; and for a second pipeline with DN 1600 (approx.
R = 800 mm = 0.80 m) with max. $Formel$ = 4700 L/s = 4.70 m3/s; results: max. $Formel$w = 15.3 and 11.7 s-1, respectively.

##### 2c) Filling bottles using a filling machine (e. g. drinks in food industry):

Filling volume per bottle: V = 1 L = 0.001 m3; filling time per bottle: t = 5 s, then:

$Formel$ = V/t = 2 ⋅ 10-4 m3/s; diameter of the circular geometry of the injection nozzle: d = 2R = 10 mm; result: $Formel$w = 2037 s-1 = approx. 2000 s-1

##### 2d) Squeezing an ointment out of a tube (e. g. pharmaceuticals):

Pressed out volume: V = 1cm3 = 10-6 m3; time to squeeze out: t = 1 s; then: $Formel$ = V/t = 10 -6 m3/s; diameter of the tube nozzle: d = 2R = 6 mm; result: $Formel$w = 47.2 s-1 = approx. 50 s-1

##### 2e) Filling ointment into tubes using a filling machine (e. g. medicine):

Filling volume per tube: V = 100 ml = 10-4 m3; filling time per tube (at 80 work-cycles per minute, where 50 % is filling time): t = (60 s/2)/80 = 0.375 s; then: $Formel$ = V/t = 2.67 ⋅ 10-4 m3/s, using an injection nozzle with an annular geometry and a cross-sectional area of A = 24 ⋅ 10-6 m2, which for a rough estimation, corresponds to a circular area showing R = 2.76 ⋅ 10-3 m (since A = π ⋅ R2); result: $Formel$w = 16,200 s-1

##### 2f) Transport process of a stucco gypsum suspension during production of architectural plates [2.11]

Size of the plates to be produced, made of stucco gypsum: Thickness h = 1.2 cm = 0.012 m und width b = 1.20 m; production speed v = L/t = 60 m/min = 1 m/s, with the length L of the plates; thus: necessary volume flow rate $Formel$1 = V/t = (h · b · L)/t = 0.0144 m3/s; for a mixer with three outlet pipes, each of them with a diameter of d = 2R = 75 mm; thus, for each single pipe counts: $Formel$ = (14.4 · 10-3 m3/s) / 3 = 4.80 · 10-3 m3/s, resulting in: $Formel$w = 116 s-1

##### 3) Sedimentation of particles in suspensions

Assumptions: fluid in a state-at-rest; the particles are almost suspended and therefore they are sinking very, very slowly in a steady-state process (laminar flow, at a Reynolds number Re ≤ 1; more about Re numbers: see Chapter 10.2.2.4b); spherical particles; the values of the weight force FG [N] and the flow resistance force FR [N] of a particle are approximately equal in size.

According to Stokes’ law (Georges G. Stokes, 1819 to 1903 [2.12]):

Equation 2.6

FG = Δm ⋅ g = FR = 3 ⋅ π ⋅ dp ⋅ η ⋅ v

with the mass difference Δm [kg] between a particle and the surrounding fluid, the gravitation constant g = 9.81 m/s2, the mean particle diameter dp [m], the shear viscosity of the dispersion fluid η [Pas], and the particles’ settling velocity v [m/s].

The following applies: Δm = Vp ⋅ Δρ, with the volume Vp [m3] of a particle, and the density difference Δρ [kg/m3] = (ρp - ρfl) between the particles and the dispersion fluid; particle density ρp [kg/m3] and fluid density ρfl [kg/m3].

The following applies for spheres: Vp = (π ⋅ dp 3) / 6; and therefore, for the settling velocity

Equation 2.7

Assumption for the shear rate: $Formel$ = v/h

with the thickness h of the boundary layer on a particle surface, which is sheared when in motion against the surrounding liquid (the shear rate occurs on both sides of the particle). This equation is valid only if there are neither interactions between the particles, nor between the particles and the surrounding dispersion fluid.

Assuming simply, that h = 0.1 ⋅ d, then: $Formel$ = (10 ⋅ v)/d

##### 3a) Sedimentation of sand particles in water

With dp = 10 µm = 10-5 m, η = 1mPas = 10-3 Pas, and ρp = 2.5 g/cm3 = 2500 kg/m3 (e. g. quartz silica sand), and ρfl = 1 g/cm3 = 1000 kg/m3 (pure water); results: v = approx. 8.2 ⋅ 10-5 m/s

Such a particle is sinking a maximum path of approx. 30 cm in 1 h (or approx. 7 m per day).
With h = 1 µm results: $Formel$ = v/h = approx. 80 s-1

##### 3b) Sedimentation of sand particles in water containing a thickener

With dp = 1 µm = 10-6 m, and η = 100 mPas (e. g. water containing a thickener, measured at
$Formel$ = 0.01 s-1), and with the same values for ρp and ρfl as above in Example (3a), results: v = approx. 8.2 ⋅ 10-9 m/s (or v = 0.7 mm per day). With h = 0.1 µm results: $Formel$ = 0.08 s-1 approximately.

Note 1: Calculation of a too high settling velocity if interactions are ignored

Stokes’ sedimentation formula only considers a single particle sinking, undisturbed on a straight path. Therefore, relatively high shear rate values are calculated. These values do not mirror the real behavior of most dispersions, since usually interactions are occurring. The layer thickness h is hardly determinable. We know from colloid science: It depends on the strength of the ionic charge on the particle surface, and on the ionic concentration of the dispersion fluid (interaction potential, zeta-potential) [2.28] [2.29]. Due to ionic adsorption, a diffuse double layer of ions can be found on the particle surface. For this reason, in reality the result is usually a considerably lower settling velocity. Therefore, and since the shear rate within the sheared layer is not constant: It is difficult to estimate the corresponding shear rate values occurring with sedimentation processes.

Note 2: Particle size of colloid dispersions, and nano-particles

In literature, as medium diameters of colloid particles are mentioned different specifications: between 10-9 m and 10-6 m (or 1 nm to 1 µm) [2.14] [2.25], or between 10-9 m and 10-7 m (or 1 nm to 100 nm) [2.13], or between 10-8 m and 10-6 m (or 10 nm to 1 µm) [2.26]. In ISO 80004-1 of 2015 is stated: Nano-scaled particles are in the range of approximately 1 nm to 100 nm [2.27]. Due to Brownian motion, the nano-particles usually are remaining in a suspended state and do not tend to sedimentation. Above all, the limiting value of the settling particle size depends on the density difference of particles and dispersing fluid.

End of the Cleverly section

### 2.2.3Viscosity

For all flowing fluids, the molecules are showing relative motion between one another, and this process is always combined with internal frictional forces. Therefore, for all fluids in motion, a certain flow resistance occurs which may be determined in terms of the viscosity. All materials which clearly show flow behavior are referred to as fluids (thus: liquids and gases ).

##### a) Shear viscosity

For ideal-viscous fluids measured at a constant temperature, the value of the ratio of shear stress τ and corresponding shear rate $Formel$ is a material constant. Definition of the shear viscosity, in most cases just called “viscosity“:

Equation 2.8

η = τ/$Formel$

η (eta, pronounced: etah or atah), the unit of the shear viscosity is [Pas], (pascal-seconds).

The following holds: 1Pas = 1N ⋅ s/m2 = 1 kg/s ⋅ m

For low-viscosity liquids, the following unit is usually used:

1 mPas (milli-pascal-seconds) = 10-3 Pas

Sometimes, for highly viscous samples the following units are used:

1 kPas (kilo-pascal-seconds) = 1000 Pas = 103 Pas, or even

1 MPas (mega-pascal-seconds) = 1000 kPas = 1,000,000 Pas = 106 Pas

A previously used unit was [P], (“poise”; at best pronounced in French); and: 1 P = 100 cP; however, this is not an SI unit [2.15]. This unit was named in honor to the doctor and physicist Jean L. M. Poiseuille (1799 to 1869) [2.7].

The following holds: 1 cP (centi-poise) = 1 mPas, and 1 P = 0.1 Pas = 100 mPas

Sometimes, the term dynamic viscosity is used for η (as in DIN 1342-1). However, some people use the same term to describe either the complex viscosity determined by oscillatory tests, or to mean just the real part of the complex viscosity (the two terms are explained in Chapter 8.2.4b). To avoid confusion and in agreement with the majority of current international authors, here, the terms viscosity or shear viscosity will be used for η. Table 2.3 lists viscosity values of various materials.

The inverse value of viscosity is referred to as fluidity Φ (phi, pronounced: fee or fi) [2.17]. However today, this parameter is rarely used. The following holds:

Equation 2.9

Φ = 1/η with the unit [1/Pas] = [Pas-1]

 Table 2.3: Viscosity values, at T = +20 °C when without further specification; own data and from [2.2] [2.3] [2.16] Material Viscosity η [mPas] gases/air 0.01 to 0.02 / 0.018 pentane/acetone/gasoline, petrol (octane)/ethanol 0.230 / 0.316 / 0.538 / 1.20 water at 0 / +10 / +20 / +30 / +40 / +50 / +60 / + 70 / +80 / +90 / +100 °C 1.79 / 1.31 / 1.00 / 0.798 / 0.653 / 0.547 / 0.467 / 0.404 / 0.354 / 0.315 / 0.282 mercury 1.55 blood plasma at +20 / +37 °C 1.7 / 1.2 wine, fruit juices (undiluted) 2 to 5 milk, coffee cream 2 to 10 blood (from a healthy body) at +20 / +37 °C 5 to 120 / 4 to 15 (at $Formel$ = 0.01 to 1000 s-1) light oils 10 glycol 20 sulphuric acid 25 sugar solutions (60 %) 57 motor oilsSAE 10W-30, at +23 / +50 / +100 °C 50 to 1000175 / 52 / 20 olive oils approx. 100 gear oils 300 to 800 glycerine 1480 honey, concentrated syrups approx. 10 Pas polymer melts (at processing conditions, e. g. between T = +150 and +300 °C, and at $Formel$ = 10 to 1000 s-1) 10 to 10,000 Pas polymer melts: zero-shear viscosity at $Formel$ ≤ 0.1 s-1 and at T = +150 to +300 °C 1 kPas to 1 MPas silicone (PDMS, uncrosslinked, zero-shear viscosity) 10 to 100 kPas hotmelts (maximum processing viscosity for melt extruders) 100 kPas bitumen (example): at T = +80 / +60 / +40 / +20 °C and at T = 0 °C 200 Pas / 1 kPas / 20 kPas / 0.5 MPas and 1 MPas, i. e., then almost like a viscoelastic solid

For “Mr. and Ms. Cleverly”

Note 1: Usually, samples with high viscosity values are viscoelastic

Many rheological investigations showed that at values of η > 10 kPas, the elastic portion should no longer be ignored. These kinds of samples should no longer be considered simply viscous only, but visco-elastic (see also Chapter 5).

Note 2: Shear viscosity η and extensional viscosity ηE

For ideal-viscous fluids under uniaxial tension the following applies for the values of the extensional viscosity (in Pas) and shear viscosity η (also in Pas): ηE($Formel$) = 3 ⋅ η($Formel$), if the values of the extensional strain rate $Formel$ [s-1] and shear rate $Formel$ [s-1] are equal in size (see also Chapter 10.8.4.1: Trouton relation).

End of the Cleverly section

##### b) Kinematic viscosity

Definition of the kinematic viscosity:

Equation 2.10

ν = η/ρ

ν (ny, pronounced: nu or new), with the density ρ [kg/m3], (rho, pronounced: ro).

For the unit of density holds: 1 g/cm3 = 1000 kg/m3

The unit of kinematic viscosity is [mm2/s]; and: 1 mm2/s = 10-6 m2/s

A previously used unit was [St] (stokes); with: 1 St = 100 cSt. This unit was named in honor to the mathematician and physicist George G. Stokes (1819 to 1903) [2.12].

The following holds: 1 cSt (centistokes) = 1 mm2/s.

##### Conversion of the values of kinematic viscosity and shear viscosity

Preset: A liquid shows ν = 60 mm2/s = 60 ⋅ 10-6 m2/s, and ρ = 1.1 g/cm3 = 1100 kg/m3

Calculation: η = ν ⋅ ρ = 60 ⋅ 10-6 ⋅ 1100 (m2/s) ⋅ (kg/m3) = 66 ⋅ 10-3 kg/(s ⋅ m) = 66 mPas

Usually, kinematic viscosity values are measured by use of flow cups, capillary viscometers, falling-­ball viscometers or Stabinger viscometers (see Chapters 11.3 to 11.6).