[
    {
        "id": 1,
        "chunk": "Stuart G. Croll  \n\nNorth Dakota State University Coatings and Polymeric Materials  \n\nDiagrams by Dr. Olena Shavranska  \n\nCopyright S. G. Croll, NDSU  \n\nSummary  \n\n• Background and Definitions   \nPaint Properties   \n• Viscometers   \n• Solution rheology   \n• Suspension rheology  \n\nRheology  \n\n• Rheology $\\mathbf{\\tau}=$ science of flow and deformation ( materials characteristics)  \n\n• Fluid Mechanics $\\mathbf{\\tau}=$ science of where fluids flow to in given processes.",
        "category": " Introduction"
    },
    {
        "id": 2,
        "chunk": "# Importance of Rheology  \n\n• Most convenient state to apply coatings is as a liquid (can also be done as a powder or gas)  \n\n– Brush – Roll – Spray, etc  \n\n• Therefore paint must be made into a liquid form  \n\n– Solution properties – Suspension properties – Mixing – Drying and Curing  \n\nCopyright S. G. Croll, NDSU  \n\nPaint, Inks, Sealants, Caulks, Cosmetics, and Packaged Foods etc.  \n\n• Consist of:  \n\n– Solutions: binder polymers, dispersants, thickeners, cross-linkers  \n\n– Suspensions: latex, pigments, extenders, non-aqueous dispersions, emulsions, defoamers, surfactant micelles  \n\nHow do we define flow properties?  \n\n• Materials flow when pushed (forced) – Stress is important  \n\n• How much they flow is important (changes of shape)  \n\n– Simple liquids require no force to retain their shape  \n\n– How fast they are strained is the important factor that determines how much force is needed",
        "category": " Introduction"
    },
    {
        "id": 3,
        "chunk": "# Viscosity Definition  \n\n• Liquids’ flow is usually determined by the shear stress imposed, or the change of flow rate through the liquid, Newton’s law:  \n\n$$\n{\\boldsymbol{\\tau}}=\\eta{\\dot{\\boldsymbol{\\gamma}}}\n$$  \n\nWhere  \n\nThis is Newtonian behavior $\\mathbf{\\tau}=\\mathbf{\\tau}$ linear, no time dependence   \nShear rate is the time differential of the shear strain and is given, in shear, by the velocity, $\\nu$ , gradient across the flow:  \n\n$$\n\\dot{\\gamma}=\\frac{\\delta\\nu}{\\delta y}\n$$  \n\nCopyright S. G. Croll, NDSU",
        "category": " Materials and methods"
    },
    {
        "id": 4,
        "chunk": "# Viscosity  \n\nViscosity $\\mathbf{\\tau}=\\mathbf{\\tau}$ resistance to flow  \n\n$\\mathbf{\\Sigma}=$ resistance to movement of molecules; solvent, solute and suspended matter through space.   \n\\~ inverse of diffusion coefficient of molecules or particles through the medium  \n\nNewtonian flow is for liquids what Hooke’s law is for solids (elastic solids)  \n\nHowever, since liquids flow to accommodate stress, the complementary variables are stress and strain-rate (in shear usually).",
        "category": " Introduction"
    },
    {
        "id": 5,
        "chunk": "# Shear Deformation  \n\n• Shear Deformation can be visualized like a pack of cards   \n• A force $F$ is applied to the uppermost volume element (thickness $d y)$ , the material will deform by the displacement $d x$ of adjacent elements.  \n\n• Shear rate $\\mathbf{\\tau}=\\mathbf{\\tau}$ rate of change of the shear strain $\\mathbf{\\Sigma}=$ change in the velocity across the thickness direction (y here) of the element $\\mathbf{\\Sigma}=\\mathbf{\\Sigma}$ [distance/time $\\div$ distance $\\mathbf{\\tau}=\\mathbf{\\tau}/\\mathrm{tim}\\mathbf{e}]=\\mathbf{s}^{-1}$  \n\n• Shear Stress $\\mathbf{\\tau}=\\mathbf{\\tau}$ Shear force, $\\mathrm{~F~}\\div$ Area, A  \n\n![](images/b8bd3f6c1ba4e80a6797182e0188f68d3f21e30b65376001ec67ff19adfdc9e8.jpg)  \nShear Deformation  \n\n<html><body><table><tr><td>Shear Strain</td></tr><tr><td>In shear, strain is not a relative increase in length, area or volume, but a change in shape (angle) given by: 4x</td></tr><tr><td>4y And in the infinitesmal limit by: dx 2</td></tr></table></body></html>  \n\n<html><body><table><tr><td>Units and Conversions</td></tr></table></body></html>  \n\nCopyright S. G. Croll, NDSU   \n\n\n<html><body><table><tr><td></td><td>CGS</td><td>MKS</td><td>SI</td></tr><tr><td>Strain</td><td>dimensionless</td><td>dimensionless</td><td>dimensionless</td></tr><tr><td>Strain rate</td><td>s-1</td><td>s-1</td><td>s-1</td></tr><tr><td>Stress</td><td>dyne/cm²</td><td>Newton/m²</td><td>Pascal (Pa) (=1 Newton/m2)</td></tr><tr><td>Viscosity (liquids)</td><td>Poise (P) (=1 dyne-s/cm²) centipoise (cP) (=0.01 P = 1 m Pa-s)</td><td>Newton-s/m²</td><td>Pa-s (=10 P) mPa-s (= 10-3 Pa-s = 1 cP)</td></tr><tr><td>Modulus (solids)</td><td>dyne/cm²</td><td>Newton/m2</td><td>Pa</td></tr></table></body></html>  \n\n<html><body><table><tr><td>Note: Kinematic Viscosity</td></tr><tr><td>Occasionally used = viscosity/density</td></tr><tr><td>Units: “Stokes\" in c.g.s. m²/s in S.I. or M.K.S.</td></tr><tr><td>Copyright S.G.Croll,NDSU 13</td></tr></table></body></html>  \n\nExamples: orders of magnitude",
        "category": " Materials and methods"
    },
    {
        "id": 6,
        "chunk": "# Viscosity Application Shear Rates  \n\nAir,  a Gas $10^{-5}$ Pa.s Water 10-3 Glycerine 1 Syrup $10^{2}$ Glass, a solid $10^{21}$  \n\nBrushing 4000-10000 s-1 Brush Pick-up $5{\\mathrm{~}}{\\mathrm{{s}}}^{-1}$ Spraying $10^{3}-10^{6}$ Settling $\\sim10^{-3}$ Sagging $10^{-2}\\ –\\ 10^{-1}$ Leveling $10^{-1}$ Coil Coating $10^{4}$ Hand Rolling $\\mathord{\\sim}500$  \n\nCopyright S. G. Croll, NDSU",
        "category": " Materials and methods"
    },
    {
        "id": 7,
        "chunk": "# Example: Balance of Properties in HousePaint  \n\nDifferent applications have different requirements on viscosity:  \n\nSagging needs high viscosity to counter it   \nLeveling needs low viscosity so brushmarks etc. disappear (note the problem vs. sag)   \nApplication by brush and roll needs a low viscosity so that it is easy (but the coating must level and not sag)   \nIf the brushing or rolling is done at too low a viscosity – the coating is too thin and may bead up   \nThe viscosity has to be high enough that the brush picks up enough paint.   \nSpraying viscosity must be low (for pumping and atomization)  \n\nThe time-dependence and non-linear behavior of typical paint gives us ways to achieve all these properties.",
        "category": " Results and discussion"
    },
    {
        "id": 8,
        "chunk": "# Leveling Geometry  \n\n![](images/806a020b288aaf3b2fc02e7a33db675ff0f01beefe374faf122af51d65d2620a.jpg)  \n\n<html><body><table><tr><td>Example: Leveling in Newtonian Paint</td></tr><tr><td>Leveling expresses how fast the brushmarks or roller spatter disappear. To a first approximation they disappear exponentially in time: Amplitude(t)= Ampt=o.exp(-t/t.)</td></tr><tr><td>3L'n (2π)4oh3 Where the time constant, t is given by: t, = -</td></tr><tr><td>L = wavelength of the brushmark, longer goes away more slowly, o= surface tension of the fluid, high values help leveling</td></tr><tr><td>speed. h = thickness, so leveling is very sensitive to thickness, thicker coatings level quicker</td></tr></table></body></html>  \n\nExample: Sagging in Newtonian Paint  \n\nThere is a balance of forces between the viscous drag within a paint film and the force of gravity making the paint run down the wall.",
        "category": " Results and discussion"
    },
    {
        "id": 9,
        "chunk": "# Coating  \n\n$$\n\\scriptstyle{\\dot{\\gamma}}\\left(x\\right)={\\frac{\\rho g\\left(h-x\\right)}{\\eta}}\n$$  \n\n$g=$ acceleration due to gravity $\\rho=$ density of the paint $h=$ thickness of the paint At $x=0$ (outer surface of paint), sag movement is greatest At wall, $x=h$ , there is no movement  \n\n![](images/53bf1fb5bf2d0475a9cdc7ac72d919701332a079e2ba5cde7ba17212be42a1f0.jpg)  \n\nHigh viscosity helps stop movement",
        "category": " Results and discussion"
    },
    {
        "id": 10,
        "chunk": "# Real Materials (inc. paints) are NonNewtonian  \n\n• Newtonian liquids are “ideal” – Liquid does not change under flow so viscosity does not change Real Liquids are not ideal – They change so their properties depend on the previous motion They have memory, i.e. $\\mathbf{\\Sigma}=\\mathbf{\\Sigma}$ time dependent – Molecular and particle interactions depend on rate of deformation They are non-linear – They may also be reacting and drying as well",
        "category": " Introduction"
    },
    {
        "id": 11,
        "chunk": "# High vs. Low shear rate  \n\n• High Shear rates  \n\n– All specific interactions are overcome by high energy of flow   \n– Viscosity depends mainly on viscosity of solvent and the volume concentration of dissolved or dispersed phases  \n\n• Low Shear rates  \n\nSpecific interactions  remain between components Viscosity is often orders of magnitude higher than at high shear rates because of these interactions  \n\n<html><body><table><tr><td>Time Dependent Behavior = Solid-like behavior (in liquids)</td></tr><tr><td>If liquids have memory, then they have some elements of solid-like behavior and we can use solid-like concepts to describe these attributes:</td></tr><tr><td>Elasticity</td></tr><tr><td>Yield Stress Normal Forces</td></tr><tr><td>Extensional Viscosity The overall combination of viscous and elastic behavior is</td></tr><tr><td>termed “viscoelastic\",as in solids Time-dependent solids, e.g. solid polymers, are also called</td></tr><tr><td>viscoelastic</td></tr><tr><td>PAINTSARE VISCOELASTIC Copyright S.G.Croll,NDSU 21</td></tr></table></body></html>  \n\n<html><body><table><tr><td>Elasticity - a form of time-dependence in liquids</td></tr><tr><td>Liquids may recoil when shearing stops, i.e. solid-like - hence“elasticity” The response to a changing stress or shear rate may not be completely in phase with it - i.e. time-dependent properties (relaxation) - for liquids, the in-phase part of the response is the viscosity and the out-of-phase component is the shear modulus (solid-like part of the response)</td></tr></table></body></html>  \n\n![](images/5e3d8b92d4583f0ff48a87eb35911365f5b4efd891819fcfb994dee32abe6ff9.jpg)  \nCopyright S. G. Croll, NDSU  \n\nWhat if the material does not respond proportionally?  \n\n• Non-linear fluids exhibit a great variety of behaviors.  \n\n– Usually paints, solutions and dispersions are “shear-thinning”   \n– They may exhibit a “yield stress” (plastic behavior)   \n– Sometimes they “shear-thicken”   \n– Occasionally they conform to other jargon  \n\n“Newtonian” liquids are linear and are not timedependent.",
        "category": " Results and discussion"
    },
    {
        "id": 12,
        "chunk": "# Shear-Thinning Fluids  \n\nViscosity decreases with increasing shear-rate or increasing shear stress,  another term for this is “pseudoplastic” The term does not  imply time-dependence necessarily (but most paints are as well)  \n\nDispersions and solutions are usually shear-thinning above a dilute concentration.  \n\nThe two main reasons for shear-thinning in suspensions:  \n\nBreakdown of flocculation, i.e. releasing more liquid from within the flocc. to lubricate the particles and decrease the effective volume solids.   \nIn non-flocculated systems, often there are inter-particle correlating forces or arrangements that are overcome at higher shear stresses when the system becomes randomized or the particles may order themselves along the direction of flow.  \n\nSolutions shear-thin for similar reasons; entanglements breakdown and molecules elongate along the flow direction",
        "category": " Results and discussion"
    },
    {
        "id": 13,
        "chunk": "# As particles align along the flow , the viscosity diminishes.  \n\nLining-up occurs more at both higher deformation rates and longer times. Produces non-linear and time-dependent elements of the viscosity response  \n\n![](images/f0f8612f32859bef43442f2ed26a9abbc74628743f12c0a50f8d3a6a142fa25a.jpg)  \n\nLining-up occurs in spatially confined flows – most coatings applications, and in shear-thinning fluids – most of coatings, see: S. V. Loon, J. Fransaer, C. Clasen, J. Vermant, “String Formation in sheared suspensions in rheologically complex media: The essential role of shear thinning,” J. Rheol., 58(1), 237 – 254, 2014  \n\n![](images/8b47d2b526efb261b48bb52ff6f65873a11a7aa0700f50bd7a506c6537998849.jpg)  \n\n![](images/b5424a79ec76ab1ef961948c8e50e8552b15e3307e459e196e99fbd1eafacf37.jpg)  \nPaints are Shear-Thinning and Time-Dependent, example:  \n\n![](images/a06ff19bbbbeca68082caf08fdd8cb0e043a8b5c37d9497f7ce884b206b12cf1.jpg)  \nCopyright S. G. Croll, NDSU  \n\n![](images/9c45802bcca86e50b95fc161471e43835ee1c50e1c1be9771cc8b702135edcb9.jpg)  \nCopyright S. G. Croll, NDSU",
        "category": " Results and discussion"
    },
    {
        "id": 14,
        "chunk": "# “Yield Stress”  \n\n• Some fluids seem not to flow at very low stresses, but only flow above a certain critical stress referred to as the “yield” stress. – This is called “plastic” behavior by analogy to solids.   \n• Below this yield, there is no flow so the viscosity is infinite(?)   \nThere are those who believe that yield cannot really exist (we just need more sensitive rheometers) – but it can be a useful description of fluid behavior.  \n\n![](images/00bc2289b4e20787279dc778feccbef1d7cfc7cd31d18c06e9b2d7ec6ae52e82.jpg)  \n\n![](images/555f3c6e323bfd3b18d81f4503ff8ad44a3d331c5f51c8fc0925d0b02a4cd5a6.jpg)  \nCopyright S. G. Croll, NDSU  \n\n33",
        "category": " Introduction"
    },
    {
        "id": 15,
        "chunk": "# Shear-Thickening  \n\nUsually happens at high shear-rates and in systems that are more concentrated. May happen because particles get jammed together and do not have enough room to move around each other quickly enough Disperse phase effectively increases in concentration $\\mathbf{\\sigma}=\\mathbf{\\sigma}$ “dilatant” behavior May also happen because strings of particles tumble at high flow rates. Flocculation can be induced at the high collision rates imposed by high shearrates.  \n\nSee also: E. Brown, H. M Jaeger, “Shear thickening in concentrated suspensions: phenomenology, mechanisms and relations to jamming,” Rep. Prog. Phys. 77 (2014) 046602 (23pp)  \n\n![](images/abc5c8ec24d01496ef791f8214214d491c7f1ecf6ae9b01b1f2c792d5e91a315.jpg)  \nN. J. Wagner, J. F. Brady, \"Shear Thickening in colloidal dispersions, Physics Today, 62(10), 27 - 32 (2009)",
        "category": " Results and discussion"
    },
    {
        "id": 16,
        "chunk": "# More about Time-Dependence  \n\n“Thixotropic” behavior $\\mathbf{\\tau}=\\mathbf{\\tau}$ viscosity diminishes with time under shear – Very typical behavior in paints, food etc. – Intermolecular or inter-particle interactions breakdown with time under stress • Flocculation, hydrophobic association or polar interactions – Detected in “hysteresis” loop experiments  \n\n• “Rheopexy” is the opposite behavior and unusual  \n\n<html><body><table><tr><td colspan=\"4\">Thixotropic Behavior, some examples (time-dependent) >  between j, and 0</td></tr><tr><td colspan=\"4\">=0 >0</td></tr></table></body></html>  \n\n![](images/86a9c84faf6d4a74ac3a7645dee3bed389e207d14386c527e01b4f65c411ee4c.jpg)",
        "category": " Results and discussion"
    },
    {
        "id": 17,
        "chunk": "# Normal Forces  \n\n• In Newtonian liquids, the shear in one direction does not change the interaction or shape of molecules or particles, so everything remains balanced.  \n\nIn a real liquid, shear changes interactions (depends on their closeness) or causes anisotropy, e.g. orientation of molecules, etc.  \n\n– System is no longer in the initial random configuration   \n– So the stresses in the three cartesian directions (normal directions) may no longer be the same $\\tau_{11}\\neq$ $\\tau_{22}\\neq\\tau_{33}$   \nWe measure “normal forces” as $\\Nu_{1}=\\tau_{11}-\\tau_{22}$ , really the first normal stress difference • $\\mathrm{{N}}_{1}$ usually exerts an outward (positive) force as measured in cone and plate rheometer Can make a difference to brush drag and other properties.",
        "category": " Results and discussion"
    },
    {
        "id": 18,
        "chunk": "# Extensional viscosity  \n\n• Unusually high extensional viscosity shows up in systems that contain high molecular weight, flexible polymers in solution.  \n\n– causes roller spatter and poor atomization – Stabilizes thick and long strings of paint behind roller, therefore big spatter drops  \n\n• This is an extension property - not a shear property Elongational viscosity $\\mathbf{\\Psi}=\\mathbf{\\Psi}$ stress difference / elongation rate  \n\n• Even Newtonian materials have viscosity in extensional flows $\\mathbf{\\tau}=3\\mathbf{x}$ shear viscosity  \n\nCopyright S. G. Croll, NDSU",
        "category": " Results and discussion"
    },
    {
        "id": 19,
        "chunk": "# Dr. Glass’ work (Union Carbide and NDSU)  \n\n![](images/ba29c30b0ce1a935bfdca0905bc1a0c2cf8993a8f1e843c98fa9e79456cc694f.jpg)  \nFigure3.17.Fiber development inrollcoating a high extensional viscosity paint.(From Ref.[29], withpermission.)  \n\nHigh extensional viscosity means that the fibers become large before they break into large spatter droplets, and leave greater roller stipple on the painted surface.",
        "category": " Results and discussion"
    },
    {
        "id": 20,
        "chunk": "# Rheometers $\\mathbf{\\tau}=\\mathbf{\\tau}$ Viscometers",
        "category": " Materials and methods"
    },
    {
        "id": 21,
        "chunk": "# Brookfield: rotating disk or spindle  \n\nimposes a shear-rate field (rate varies across the disk)   \nmeasures the resistance, gives shear stress   \nCalibrated to give viscosity – Needs periodic calibration – Fluid container must be bigger than disk or spindle and provide good clearance underneath   \nQuite useful and accurate but cannot impose much shear stress or rate - limited in application  \n\n![](images/1dc69267f7439c0135823a2a2ac2130ff11251b35dc62dbf8cad2b53a618865b.jpg)  \nCopyright S. G. Croll, NDSU",
        "category": " Materials and methods"
    },
    {
        "id": 22,
        "chunk": "# Stormer Viscometer  \n\n![](images/18c0df542e60e170f32c6e25a2cfa03faf72a18b0f916747e0b39b980d10ee1b.jpg)  \n\nImposes a shear stress (very approximately)   \nusing vanes that move through the fluid at   \n200 r.p.m.   \nRotation rate usually monitored by stroboscope   \nMeasure the weight that maintains this rotation rate   \nResult is given in Krebs units (KU)ASTM   \nD562 - difficult to relate to any other,   \nscientific unit for viscosity   \nOriginally intended to emulate stirring   \nOften 90-100 KU is held to provide good brush pick-up performance and is used as a target for the lower shear-rate performance of an architectural paint  \n\nCopyright S. G. Croll, NDSU  \n\n![](images/ea3be5f8f8322de1ca0534caf72dd522af53ef92796ecf600660411d795b9d1c.jpg)  \nCopyright S. G. Croll, NDSU",
        "category": " Materials and methods"
    },
    {
        "id": 23,
        "chunk": "# Efflux Cups  \n\n• Measure the time taken for a standard quantity of fluid to pour out of the bottom. • Result is usually given in seconds • Used in Quality Control more than anything else, for low viscosity, sprayed paints! – Common type is the Ford No. 4 cup.",
        "category": " Materials and methods"
    },
    {
        "id": 24,
        "chunk": "# Schematic of Ford Cup, number 4  \n\n![](images/89f2e620c254fddc64be9d7caf6415d9419ae4ca39ccd3fcdb820d54a33714f6.jpg)  \nCopyright S. G. Croll, NDSU",
        "category": " Materials and methods"
    },
    {
        "id": 25,
        "chunk": "# Scientific Viscometers  \n\nThese impose rigorous shear stresses or shear-rates, usually in a rotation sense.  Possibilities:  \n\n$\\succ$ Steady ramps   \n$\\blacktriangleright$ Constant levels   \n$\\succ$ Sinusoidal stresses or shear-rates   \n$\\succ$ Or combinations  \n\nUsually, a wide range of stress or shear-rate is available with very sensitive transducers for measuring the response.  \n\nTwo geometries are common (I) cone and plate (best defined strain-rate and stress) (II) parallel plate (can achieve higher shear-rates)  \n\nTwo types are used:  \n\n(I) controlled stress is the input, resultant shear-rate is measured - most common   \n(II) controlled shear-rate is the input, stress is measured. - requires bigger motors, more expensive",
        "category": " Materials and methods"
    },
    {
        "id": 26,
        "chunk": "# Cone and Plate Geometry  \n\n![](images/799c394dbafb8dd0d1125e5c9eca7703821912868d911eb50288b53cb346d509.jpg)  \nCopyright S. G. Croll, NDSU",
        "category": " Materials and methods"
    },
    {
        "id": 27,
        "chunk": "# Cone and Plate Geometry  \n\n• Shear Stress is given by:  \n\n$$\n\\tau=\\frac{3T}{2\\pi r^{3}}\n$$  \n\n$\\mathrm{{T}=}$ torque $r=$ radius of cone  \n\nShear-rate is given by: γ = $\\dot{\\gamma}=\\frac{\\omega}{a}$  \n\nWell defined while the angle is less $\\sim4^{\\circ}$ $\\boldsymbol{\\upomega}\\ =$ rotational speed, radians/second ${\\mathfrak{a}}=$ cone angle, radians (see previous slide)",
        "category": " Materials and methods"
    },
    {
        "id": 28,
        "chunk": "# “I.C.I” Viscometers  \n\nDeveloped by I.C.I Ltd (Imperial Chemical Industries, UK) (paints division now part of AkzoNobel and PPG).  \n\n![](images/8c8e26c29fefa9363ee2e345f07de8e245d004f9fef8cd4f739982f5311fc75b.jpg)  \n\n• Cone and plate viscometers with single or defined range of shear rates  \n\n• Most common form is the model that operates at a shear rate of $10,000\\ \\mathbf{s}^{-1}$ Gauges performance in application process, e.g. brushing etc.  \n\nSupplied by instrument makers.  \n\nhttp://www.researchequipment.com/researchequipment.html",
        "category": " Materials and methods"
    },
    {
        "id": 29,
        "chunk": "# Bubble Viscometers  \n\nBubble rises at a speed that can be   \nused to calculate a viscosity Usually determined by comparison to a standard  \n\n![](images/6b5690d6d3c73ec3d72bc6e56f63f822660e3c6311ccbd728cf1fb22b828a019.jpg)  \nCole-Parmer  \n\nQuick and simple but useful only for clear solutions, resins, varnishes  \n\n• Usually reported by the letter grade of the standard closest in bubble speed, A5 through Z10, (0.005 to 1,000 Stokes) ASTM D1131, D1545, D1725 Tubes do have marks, so a timing can be done and a result calculated in standard viscosity units Sets of sealed standards are available, open tubes for your sample, and a holder so that sample tubes are inverted at same time as the standard Accurate provided that temperature does not vary too much and the bubble shape remains stable.  \n\nCopyright S. G. Croll, NDSU",
        "category": " Materials and methods"
    },
    {
        "id": 30,
        "chunk": "# Solution Viscosity  \n\n• Overall, solution viscosity depends on:  \n\n– Temperature   \n– Molecular weight   \n– Molecular weight distribution   \n– Solvent viscosity   \n– Polymer-solvent interactions   \n– Concentration",
        "category": " Results and discussion"
    },
    {
        "id": 31,
        "chunk": "# Resin Solutions  \n\nHigher molecular weight resins within about $100^{\\circ}\\mathrm{C}$ of their $T_{g}$ seem to follow a WLF (Williams-Landel-Ferry) type of dependence, as do lower molecular weight resins at all temperatures:  \n\n$$\nl n\\ \\eta=\\eta_{r}-{\\frac{c_{I}\\left(T-T_{r}\\right)}{c_{2}+{\\left(T-T_{r}\\right)}}}=27.6-{\\frac{A{\\left(T-T_{g}\\right)}}{B+{\\left(T-T_{g}\\right)}}}\n$$  \n\n• Subscript $^{\\ast}\\mathrm{r}^{\\ast}$ means some curve-fitted reference value. There are “universal” values if one uses the $T_{g}$ of the polymer as the reference. $\\mathrm{A}{\\sim}17.44$ and $\\mathrm{B}{\\sim}51.6$ Important variable is the difference between the actual temperature and $\\mathrm{T_{g}}$ .",
        "category": " Results and discussion"
    },
    {
        "id": 32,
        "chunk": "# Resin Solutions  \n\n• Lower $\\mathrm{T_{g}}$ leads to lower viscosity – Polymer is more flexible and thus poses less resistance to flow.  \n\n• At even higher temperatures and molecular weights the temperature dependence is closer to Arrhenuis:  \n\n$$\nl n\\eta{=}l n A^{\\prime}+\\frac{E_{\\nu}}{R T}\n$$  \n\n![](images/82a4e067b65a7d26a940d151ecf3009cb1797453ff51d929467344edd1a35268.jpg)  \nFigure 3.12. Viscosity dependence of standard liquid BPA epoxy resin on temperature. (From Ref.",
        "category": " Results and discussion"
    },
    {
        "id": 33,
        "chunk": "# Dilute Solutions  \n\n• If a solution is dilute, i.e. the polymer molecules act independently, the solution viscosity can be expressed as:  \n\n$$\nl n~\\eta_{\\boldsymbol{r}}=\\left[\\pmb{\\eta}\\right]c+\\left[\\pmb{\\eta}\\right]^{2}c^{2}\n$$  \n\n$\\eta_{r}$ is the solution viscosity/solvent viscosity $\\mathbf{\\Sigma}=$ relative viscosity  \n\nThe intrinsic viscosity [] depends on the temperature (naturally) and the hydrodynamic volume swept out by the polymer molecule - which in turn depends on the molecular weight:  \n\nMark-Houwink-Skarada equation. $a$ goes from 0.3 to ${\\sim}0.8$  \n\n$$\n\\scriptstyle{[\\pmb{\\eta}]=\\pmb{K}\\pmb{M}^{a}}\n$$",
        "category": " Materials and methods"
    },
    {
        "id": 34,
        "chunk": "# Characterizing Polymers in Dilute Solutions  \n\n• The Intrinsic Viscosity [] is defined by:  \n\n$$\n\\eta_{r e l}=\\frac{\\eta}{\\eta_{s}}=1+\\big[\\eta\\big]c+k c^{2}+\\dots\n$$  \n\nan alternative to equation on previous slide.  \n\n• It is measured by viscosity measurements taken on a succession of dilute concentration solutions in capillary viscometers (very accurate) – usually in a tightly controlled temperature bath.  \n\nCopyright S. G. Croll, NDSU  \n\n![](images/b581cfa24460d2d599a90972b7680feaa1c73298da262e86c165e5c411924c48.jpg)",
        "category": " Materials and methods"
    },
    {
        "id": 35,
        "chunk": "# Concentrated Solutions  \n\n• There is not so much simplicity on the dependence of the viscosity on concentration:  \n\n$$\nl n\\eta_{r}=\\frac{w_{r}}{k_{I}-k_{2}w_{r}+k_{3}w_{r}^{2}}\n$$  \n\n• This equation works fairly well, $w_{r}=$ weight fraction of the polymer; sometimes simpler equations are fine.   \n• Polymer molecules extend in good solvents and thereby increase viscosity - choice of solvent is crucial. Poor solvents cause polymer molecules to coil upon themselves, and if the solvent is poor enough - the polymer comes out of solution.",
        "category": " Results and discussion"
    },
    {
        "id": 36,
        "chunk": "# Dispersion Viscosity  \n\nDilute suspensions adhere well to Einstein’s equation: $\\eta_{r}=1+2.5\\phi$ Where $\\phi=$ volume fraction of the suspended particles $\\mathrm{\\sim}<0.1$ The factor of 2.5 assumes that the particles are spheres.  \n\nFor more concentrated suspensions other equations have been proposed. The most common are, (i) the Mooney equation:  \n\n$$\n\\eta_{\\mathrm{~r~}}=e x p\\left[\\frac{2.5\\phi}{I\\mathrm{~-~}\\displaystyle\\frac{\\phi}{\\phi_{\\mathrm{~m~}}}}\\right]\n$$  \n\nAnd, the most successful is (ii) the Krieger-Dougherty equation, but see next slide:  \n\n$$\n\\eta_{r}=\\left(I-\\frac{\\phi}{\\phi_{m}}\\right)^{-2.5\\varphi_{m}}\n$$  \n\nCopyright S. G. Croll, NDSU  \n\n![](images/27a62319727b0e8f91ec5067173c3131e5fa3c453bf9f023fbac351922e417ea.jpg)  \nCopyright S. G. Croll, NDSU",
        "category": " Results and discussion"
    },
    {
        "id": 37,
        "chunk": "# How the behavior of a suspension will change with concentration  \n\n![](images/dd2c8c2718e78fae55dfa2b1fdd3e9ba75686eb2ee54eab1d23f3825927faeff.jpg)  \nJ. M. Brader, “Nonlinear rheology of colloidal dispersions,” J. Phys.: Condens. Matter 22 (2010) 363101   \nFigure 3. A schematic illustration of the phase diagram of hard-spheres as a function of volume fraction. Monodisperse systems undergo a freezing transition to an FCC crystal with coexisting densities $\\phi=0.494$ and 0.545. Polydispersity suppresses the freezing transition resulting in a glass transition at $\\phi\\sim0.58$ ,which lies below the random-close-packing value of $\\phi\\sim0.64$",
        "category": " Results and discussion"
    },
    {
        "id": 38,
        "chunk": "# Dispersion Viscosity  \n\n• Both the Mooney equation and the Krieger-Dougherty equations exhibit infinite viscosity at the maximum packing fraction, $\\Phi_{\\mathrm{m}}$ . The maximum packing fraction is a geometric constraint on the number of particles that can be accommodated in a given arrangement.   \n0.63 – dense random packing of spheres (good 1st.  choice for rheology)   \n0.59 – loose random packing of spheres (good second choice)   \n0.52 – simple cubic packing of spheres   \n0.74 – hexagonal close packing of spheres   \nFlakes, discs and rods may tumble and get in each others way so $\\Phi_{\\mathrm{m}}$ can be as low as 0.1.  \n\nFlocculation usually incorporates solvent into the aggregates and so effectively increases the volume fraction of the solids and so the viscosity becomes much higher.  \n\nCopyright S. G. Croll, NDSU",
        "category": " Results and discussion"
    },
    {
        "id": 39,
        "chunk": "# Possible Advance in Rheology of Dispersions (and Granular Matter)  \n\n• F. Boyer, E. Guazzelli, O. Pouliquen, “Unifying Suspension and Granular Rheology,” PRL 107, 188301 (2011) Different, but successful form of the equations for the relative viscosity of dispersions. Draws common ground between suspension and granular flow with viscous and collision contributions.  \n\n![](images/41d72ad8bd4ed88029373717031143642f31700a3b40ca093d4d03023fac6170.jpg)  \n$\\eta_{n}$ correlations of Eilers (red dashed line) and Krieger-Dougherty $\\eta_{s}$ $\\eta_{\\ast}$",
        "category": " References"
    },
    {
        "id": 40,
        "chunk": "# Analytical Use of Rheology  \n\n• QC and Control – is it the same as the control material? – Best done at low shear rates (see below)  \n\n• High Shear Rate viscosity – Will it flow/atomize well in the equipment and with the power available?  \n\n• Low Shear Rate viscosity  \n\n– Low shear rates do not break down flocculation or other interactions, so it is good at detecting when materials or their behavior are different   \n– Very sensitive to material differences but only diagnostic by comparison with known control materials  \n\nCopyright S. G. Croll, NDSU",
        "category": " Materials and methods"
    }
]