Viscosity and Polymer Melt Flow Rheology-Processing / Chapter 2 1
Viscosity: a fluid property resistance to flow (a more technical definition resistance to shearing) Remember that: τ μ du dy shear stress F/A shear rate: U/H viscosity Rheology-Processing / Chapter 2 2
POLYMER MELTS THE VISCOSITY DECREASES AS THE SHEAR RATE (du/dy) INCREASES (due to molecular alignments and disentanglements) A characteristic value is the limiting viscosity for zero shear rate (frequently referred to as zero shear viscosity) usually denoted with η o. Rheology-Processing / Chapter 2 3
ZERO SHEAR VISCOSITY is a function of POLYMER MOLECULAR WEIGHT Critical molecular weight M c... entanglements to become effective. for M c critical molecular weight for entanglements. M c 4,000 for PE, M c 36,000 for PS Most (NOT all) commercial polymers have M w =50,000 to 500,000 Rheology-Processing / Chapter 2 4
In characterization of polymers, the solution viscosity is often used. For a polymer dissolved in a solvent: viscosity as concentration (solution viscosity definitions in pages 2.4 and 2.5) For instance, the specific viscosity: or the intrinsic viscosity: η solution solvent solvent The intrinsic viscosity is found by plotting η sp /C against C and extrapolating to zero concentration sp η η ηsp [ η] C η C0 Rheology-Processing / Chapter 2 5
The so called viscosity average molecular weight, introduced in chapter 1, can be found from the Mark-Houwink equation: [ η] ΚΜ α [η]: intrinsic viscosity M: (viscosity) average molecular weight K, a: experimentally determined constants (depend on polymer solvent system) Rheology-Processing / Chapter 2 6
SHEAR-THINNING BEHAVIOR OF POLYMERS From Newton s law of viscosity μ τ du dy shear stress shear rate Pa s Pa 1 s Viscosity is constant for Newtonian fluids BUT it DECREASES as SHEAR RATE INCREASES for polymer solutions and melts (non-newtonian fluids) This behavior is called: SHEAR-THINNING (due to molecular alignments and disentanglements ) Rheology-Processing / Chapter 2 7
We cannot talk about a constant viscosity μ in polymers, but rather about vis cosity 1 shear stress shear rate Pa s We usually symbolize it with the Greek letter η Rheology-Processing / Chapter 2 8
Most popular model to express the shear-thinning behavior: POWER-LAW τ mγ (shear stress) = m (shear stress) n n or τ ηγ or η mγ n1 viscosity is a function of shear rate m is called consistency index (the larger the m the more viscous the melt) n indicates the degree of Non-Newtonian behavior n=1 means Newtonian n<1 for SHEAR THINNING POLYMERS Rheology-Processing / Chapter 2 9
The power-law relation gives log η log m n 1log γ Note that consistency index m is equal to viscosity η at γ 1s 1 On a log-log paper η vs γ is a straight line and the slope is equal to n-1 The power-law fit is OK for high shear rates, but for low shear rates polymers exhibit a Newtonian plateau (e.g. at values say γ 3s 1 ) The power-law model is good for most polymer processing operations because the shear rate is between 100 s -1 and 5000 s -1 Rheology-Processing / Chapter 2 10
WHAT ABOUT FITTING THE WHOLE VISCOSITY CURVE? THE MOST POPULAR MODELS: Carreau Yasuda: 1 n1 a a where η o is the viscosity at zero shear and λ, α and n are fitted parameters Cross model: 1 1n UNFORTUNATELY NO ANALYTICAL SOLUTIONS ARE POSSIBLE WITH THESE VISCOSITY MODELS USED IN COMPUTER SIMULATIONS FOR DETAILED LOCAL FLOW ANALYSIS Rheology-Processing / Chapter 2 11
The consistency index m is sensitive to temperature. A common representation is: m m o e b T T o where m o the consistency index at a reference temperature T o Rheology-Processing / Chapter 2 12
Typical values: For polymer melts at usual processing conditions m=1000 100000 Pa s n n=0.2 0.8 b=0.01 0.1 o C -1 For example, for a commercially available polystyrene (PS) the following parameters were obtained by curve fitting of viscosity data m o =10800 Pa s n, n=0.36, T o =200 o C, b=0.022 o C -1 The above value b corresponds to a viscosity reduction ~20% for 10 o C temperature rise. For isothermal flows we can use the power-law viscosity model to solve problems of practical importance using analytical methods of solution. Rheology-Processing / Chapter 2 13
To solve general flow problems we must set a momentum balance. It turns out that the momentum balance can be written verbally Rheology-Processing / Chapter 2 14
Polymers are characterized by extremely high viscosities (about a million times more viscous than water) in molten state. Rheology-Processing / Chapter 2 15
Therefore: 0 p τ Pressure p is a scalar Velocity is vector V i.e. Vx, Vy, Vz having components in the x, y and z directions Stress is defined as the ratio Force/Area and can be normal or tangential Stress is a tensor having nine components: (τ xx, τ yy, τ zz ) normal stresses The rest components shear stresses Rheology-Processing / Chapter 2 16
For planar unidirectional flows the equation 0 p τ is simplified to 0 p x τ yx y 0 A good way to remember it, is symbolically: p ( shear stress) direction of flow normal to flow Rheology-Processing / Chapter 2 17
PRESSURE DRIVEN FLOW OF A POWER-LAW FLUID BETWEEN TWO FLAT PLATES (book page 2.15) The absolute value is needed because sometimes V x y is negative and n-1<0 for polymers. Rheology-Processing / Chapter 2 18
boundary condition Delicate point: the right-hand side is negative and thus V x y must be negative. Rheology-Processing / Chapter 2 19
Apply NO-SLIP boundary condition: V x =0 at y=b This is the velocity profile! Rheology-Processing / Chapter 2 20
The maximum velocity V max is at y=0 The average velocity V avg is calculated by Rheology-Processing / Chapter 2 21
The volume flow rate per unit width W (plate width) is given by The pressure drop may then be easily calculated Rheology-Processing / Chapter 2 22
From a previous relation we saw that which means that the stress varies linearly in the gap. The maximum value is at the wall (i.e. at y=b) the w subscript refers to. wall The negative sign simply indicates that when this quantity is multiplied by the area, it gives a force (i.e. F w =τ w A) that is exerted by the plate on the fluid which is, of course, in the negative x direction. The force exerted by the fluid on the wetted plate should therefore be positive. Rheology-Processing / Chapter 2 23
Another important quantity that we need to calculate is the shear rate We may then calculate the shear rate at the wall (absolute value) The maximum shear stress at the wall can also be expressed as Rheology-Processing / Chapter 2 24
PRESSURE DRIVEN FLOW OF A POWER-LAW FLUID IN A TUBE (book page 2.19) governing equation Rheology-Processing / Chapter 2 25
Boundary conditions The solution gives the following velocity profile maximum velocity average velocity volume flow rate pressure drop Rheology-Processing / Chapter 2 26
In the previous equations if we set n=1 we end up with the well-known Hagen- Poiseuille formula for Newtonian fluids. Pressure driven flows are also referred to as Poiseuille flows. For SHEAR-THINNING fluids, the velocity profiles are more flat than the parabolic profiles of Newtonian fluids. Rheology-Processing / Chapter 2 27
From the previous equations it is easy to see that the stress varies linearly and the maximum value is at the wall (i.e. at r=r) The linearity of the stress is often expressed as τ w is taken as positive Rheology-Processing / Chapter 2 28
The shear rate is: And the shear rate at the wall: The maximum shear stress at the wall can be then calculated from Rheology-Processing / Chapter 2 29
CAPILLARY VISCOMETER ANALYSIS (book page 2.23) For Newtonian fluids, the Hagen-Poiseuille formula can be used for direct determination of viscosity μ, from the measurement of pressure drop ΔP at flow rate Q, through a tube of length L and radius R. For non-newtonian fluids, the viscosity η is a function of the shear rate (i.e. η γ ) and special treatment is necessary. We will determine the viscosity from its basic definition γ The wall shear stress can be calculated from η τ γ γ Rheology-Processing / Chapter 2 30
The shear rate requires special manipulations. Again for Newtonian fluids (n=1) the shear rate can be obtained by differentiating the velocity profile and using the Hagen-Poiseuille formula we get For non-newtonian fluids we will develop a general expression for the shear rate at the wall by starting from the definition of the volume rate of flow through a tube (see subsequent steps followed in pages 2.24 and 2.25. Next slide goes directly to the result) Rheology-Processing / Chapter 2 31
This equation is usually referred to as the Rabinowitsch equation. It gives the shear rate in terms of Q, R and τ w. The term in the parentheses may be considered as a correction to the Newtonian expression which is simply 4Q/πR 3. To obtain γ w we must plot Q versus τ w on logarithmic coordinates to evaluate the derivative dlnq/dlnτ w for each point of the curve. Rheology-Processing / Chapter 2 32
The previous method can be simplified if POWER-LAW fluid is assumed n τ mγ We may then write an empirical expression 4Q τ m 3 πr in which n is the slope of the logτ w versus log(4q/πr 3 ) plot, that is d log τw n 4Q d log 3 πr It turns out that for the derivative dlnq/dlnτ w we have dlnq/dlnτ w =1/n, therefore γ 4Q 3 1 4Q 3n 1 w 3 3 πr 4 4n πr 4 n This means that the relation between the apparent m and the true m is 4n m m 3n 1 n n Rheology-Processing / Chapter 2 33
STEPS FOR DETERMINATION OF m and n 1. Determine τ w from pressure drop Δp data 2. Determine γ app 4Q πr 3 3. Plot logτ w versus log(4q/πr 3 ) to get n (slope) and m (intercept at γ 1s ) 1 app 4. Correct m using m m 4n 3n 1 n Rheology-Processing / Chapter 2 34