# Notes on Polymer Rheology Outline

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1 1 Why is rheology important? Examples of its importance Summary of important variables Description of the flow equations Flow regimes - laminar vs. turbulent - Reynolds number - definition of viscosity Characterizing Viscosity Notes on Polymer Rheology Outline Viscosity Models - effects of temperature pressure molecular weight structure How do we measure rheological properties? - melt index measurements - capillary rheometry - cone and plate rheometers Solution of flow equations for simple geometries

2 2 Why is rheology important? In the making of plastic articles, the manufacturer often starts with pellets of the plastic and ends up with the finished article -- often both the starting point and the final article being in a 'solid' state. To make the finished article, we need to take the polymer into an intermediate state. This intermediate step in the forming or shaping of plastic articles involves the passage of the polymer through either the viscous flow region or the rubbery region. In these regions the material can be deformed using relatively small forces. Of course, both these regions are above the glass transition temperature. After shaping and forming, the polymer is quenched to a temperature below the glass transition temperature where it will keep its new shape. Of course, there are a number of articles that are in the rubbery state after being formed but they still tend to be 'solid' in that they keep their shape. You should all be familiar with the effect of temperature. That is, as temperature increases, the properties of a material will change. Sometimes the changes are small, sometimes they are large. For polymers, if we look at the mechanical properties as a function of temperature, we see a behavior similar to that on this graph. 11 glassy 10 log (Modulus) rubbery melt 6 5 T g Temperature Rheology is the study of flow and deformation of materials and how that flow is affected by stresses, strains and time.

3 3 Examples of it's importance - extrusion (e.g. extrusion blow molding) & die design to analyze, we may need to take into account high viscosities odd shaped flow channels other effects include sag, swell, and stretching We want to be able to : 1) design a die head a) get correct thickness distribution b) get the correct diameter c) insure that shear stresses are not too large 2) predict pressure drops 3) multilayer dies -- need to study flow patterns in the die to insure that layers are of the correct thickness and evenly distributed. - injection molding high injection pressures are due to : high viscosities long flow channels thin parts desire for tight tolerances We may want to be able to predict : 1) pressures needed 2) fill time of part 3) temperature of the melt versus time a) mold cooling time b) shear heating c) melt degradation - calendering - quality control (calculation of MIV) Properties of a plastic article are dependent on the molecular weight of the polymer. It's rigidity, ability to resist fracture, and chemical resistance, just to name a few properties, can all be affected by changes in molecular weight. The polymer chains can 'break down' during processing resulting in a degraded polymer with a reduced molecular weight. As we will see later, the viscosity of a melt is a very sensitive indicator of the molecular weight of a polymer. A very simple test for viscosity is the melt flow indexer. Thus rheology (particularly the study of the viscous flow of polymers) is very important in processing and proves useful as a quality control tool. Viscous properties determine processing issues such as form of the velocity profile magnitude of pressure drops or amount of heat generation

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7 7 Summary of important variables viscosity density heat capacity, thermal conductivity Description of the flow equations If we wish to predict or analyze pressure drops or flow rates or velocity profiles we need to solve the flow equations for the system If we wish to compute temperature profiles or heat generated or transferred we need to solve the thermal energy equations The basic equations for flow usually include : equation of continuity momentum balance - pressure forces - viscous forces - inertial forces - gravitational forces We also need a relationship to describe the behavior of viscosity as a function of - stresses - strain rates - temperature - pressure

8 8 Flow regimes - laminar flow - turbulent flow the transition between these regimes is characterized by a quantity called the Reynolds Number N Re = ρ V D µ Characterizing Viscosity ρ density µ viscosity V characteristic velocity D characteristic length Often the definition of viscosity is explained by using a simple flow situation such as drag flow. No pressure gradients are present. Movement of the fluid contained between two plates is sustained by the motion of the top plate while the bottom plate is held stationary. A force F is necessary to maintain the motion where the top plate moves at a velocity V. dx F V dy drag flow The shear stress τ is given by : τ = F A We can also show that the strain rate is proportional to the velocity gradient. γ = dv dy For many fluids, we assume that the shear stress is proportional to the strain rate -- so : τ = η γ This equation defines the viscosity (even when the viscosity is not constant).

9 9 Units of viscosity shear rate sec -1 or 1/sec shear stress Newtons / meter 2 (N / m 2) Pascals viscosity Pa-sec in CGS viscosity is given in terms of poise 1 poise = 1 dyne / cm 2 - sec = 1 gm/(cm-sec) = 1 gm cm -1 sec -1 Note also that : 10 poise = 1 Pa-sec = 1 kg/(m-sec) = 1 N-sec/m 2 1 poise = 2.09 * 10-3 lbf/(ft 2 -sec) Conversion factors Pascal (Pa) = Newton(N)/m 2 Joule (J) = N m N = kilogram - m/sec 2 = kg-m/sec 2 dyne = 10-5 N 1 MPa = 10 6 Pa = 10 6 N m -2 = 10.2 Kg cm -2 = 145 psi Shear Rates of Different Processes process shear rate (sec -1 ) compression molding 1-10 calendering extrusion injection molding Viscosities of Various Materials substance viscosity (Pa-sec) air 10-5 gas water 10-3 liquid paint 10-2 to 10-1 glycerol 10 syrup 10 2 liquid polyurethane 10 2 to 10 3 polymer melts 10 2 to 10 6 viscous liquid pitch 10 9 flowing solid glass rigid solid

10 10 Limitations of flow At high enough stress levels or strain rates, the flow of polymer melts becomes unstable. Thus, turbulent flow is not generally possible. These instabilities limit flow rates.

11 11 Viscosity models - ideal fluid the viscosity is zero, this is used for the analysis of potential flow problems such as found in aerodynamics - Newtonian fluid a good viscosity model for gases and simple liquids such as water - non Newtonian fluid the viscosity of the fluid changes with shear rate. For some fluids, it increases with shear rate (dilatant fluids) while for others it decreases with shear rate (pseudoplastic behavior). This pseudoplastic (or shear thinning) behavior is what we observe for polymers. The most common model for shear thinning is the power-law model. This states that the viscosity is proportional to the shear rate raised to a power. τ = η γ For a power-law fluid, where η = m γ n 1 or m γ n 1 n is called the power-law index This model will give a straight-line curve on a log-log plot. log(η) = log(m) + (n 1)*log( γ) One can plot either shear stress versus shear rate or viscosity versus shear rate on the log-log plots. Factors affecting viscosity - pressure - molecular weight - temperature - shear rate or shear stress Any factor which increases entanglements between chains increases the viscosity. - branching - molecular weight - side groups - pressure These factors affect the ability of the chains to slide past each other.

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14 14 Polymer melts are : non-newtonian usually shear thinning (decrease in viscosity with increasing shear rate) decrease in viscosity with increasing temperature increase in viscosity with increasing pressure increase in viscosity with increasing molecular weight

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16 16 Temperature Effects WLF Equation (Williams-Landel-Ferry) This represents the temperature dependency of a response log η η Tg = a T T g b + (T T g ) for T > T g where the universal constants are a = ; b = 51.6 It can also be shown that the free volume at the glass transition temperature is about 2.5% of the total free volume. Also, viscosity at the glass transition temperature ( η Tg ) is about poise for most plastics. Prediction of viscosity at any temperature log η = T T g (T T g ) Prediction of injection molding temperature a) need to estimate required viscosity at molding conditions e.g. assume η = 7000 poise b) need to determine T g e.g. for polystyrene T g = 100 C Then log 7000 = T T g (T T g ) or T = 157 C = 430 K

17 17 Solution of flow equations for simple geometries The viscosity we are most familiar with is the shear viscosity. Sometimes we need to be concerned with other material quantities Bulk modulus (which relates a volume change to applied pressure) K 10 9 Pa K = P V/V Elongational viscosity which is important in - spinning of fibers - calendering How do we measure rheological properties? - melt index measurements (for quality control) - cone and plate rheometers - capillary rheometry For the capillary tube, the momentum equation becomes a balance between the pressure forces and the viscous forces. If we know the viscosity function, we can then compute the velocity profile and from this get the shear rates.

18 18 Melt Flow Index (MFI) To compute the melt flow index (MFI) of a polymer, the polymer is brought up to a specified melt temperature in a cylindrical chamber. A piston on top of the melt is loaded with a given weight, applying a specified pressure to the melt which is extruded through a given die. Sometimes this is also referred to as the melt index (MI). For example, for polyethylene (PE) the melt flow index is equivalent to the grams of material extruded from the die in 10 minutes with the melt at 190 C. Weight applied to the piston equates to 43.5 psi (or 300 kpa). For polypropylene, the melt temperature is 230 C. This test is just meant as a comparison between similar materials -- it does not compute a material property. It is mainly useful for quality control -- to see if the material has changed from batch to batch, or to compare the relative flow behavior of different materials. Both the load and melt temperature may be different for different materials. More information can be obtained by either taking the Polymer Lab course or by referring to ASTM procedure D 1238.

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20 20 Capillary Rheometer A force balance between the shear stress and the pressure drop results in τ w = R ( P) 2 L The apparent strain rate is given by ϕ = γ a = 4 Q π R 3 * Note that often the minus sign is dropped but remember that for flow through a tube, strain rates are either zero or negative. This is related to the actual strain rate by γ w = 3 n' n' ϕ Thus for capillary flow we can compute the shear stress and the actual strain rate and from these compute the viscosity. η = τ w γw The variable n is computed from a plot of the log(shear stress) versus log(strain rate). This index is the slope of the curve log( τ ) w slope = n log( φ)

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22 22 Equations for pressure flow of a power-law fluid in a tube

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