VI CURSO DE INTRODUCCION A LA REOLOGÍA Madrid, 8-9 julio 2013 NON-LINEAR VISCOELASTICITY Prof. Dr. Críspulo Gallegos Dpto. Ingeniería Química. Universidad de Huelva & Institute of Non-Newtonian Fluid Mechanics (UK)
OUTLINE 1. Introduction 2. Continuous Mechanics description of non-linear phenomena 3. Normal Stresses 4. Relationships between viscometric functions and linear viscoelastic functions 5. Extensional flow
1. INTRODUCTION Linear viscoelastic behavior is exhibited by a material that is subjected to a deformation that is either very small or very slow. Linear properties are of interest, because they are closely related to molecular structure. On the other hand, the industrial processing of viscoelastic materials always involves large, rapid deformations in which the behavior is nonlinear.
1. INTRODUCTION For linear behavior, the Boltzmann superposition principle describes the response to any deformation, as long as it is very small or very slow. When a viscoelastic material is subjected to a deformation that is neither very small nor very slow, its behaviour is no longer linear, and there is no universal rheological constitutive equation that can predict the response of the material to such a deformation. One approach to describing nonlinear behaviour is based on continuum mechanics principles and attempts to establish a rheological constitutive equation to replace the Boltzmann principle.
1. INTRODUCTION While continuum mechanics models sometimes contain elements inspired by molecular or thermodynamic concepts, they are basically empirical. This means that their applicability outside of the conditions under which their predictions can be tested by experiment is unreliable. Another approach is to build up a model of flow behavior starting from a picture of the material at the molecular level. This is very complex, and some degree of success has been achieved only when the problem is drastically simplified: attention is focused on one molecule, with the influence of the surrounding molecules modeled by representing them as a tube or a series of slip-links that severely restrict the motion of the molecule of interest.
2. CONTINUOUS MECHANICS DESCRIPTION Wagner (1979) proposed the introduction of a nonlinear memory function to correct some of the deficiencies of the rubberlike liquid model. Taking into account that the relaxation of stress following a large step strain can often be separated into time-dependent and straindependent factors, Wagner proposed the use of a memory function defined as the product of the linear memory function and the damping function. This function h(i 1,I 2 ) depends on the Finger strain tensor: ij t ( ) m( t t) h( I1, I 2 t ) B ( t, t) dt In this model, the damping function is an empirical function whose parameters are determined by fitting experimental data. This model predicts material shear-thinning characteristics. ij
2. CONTINUOUS MECHANICS DESCRIPTION SIMPLE SHEAR t ( t) m( t t') h( ) ( t, t') d t' h( ) G(, t t' ) G( t t' )
h () 2. CONTINUOUS MECHANICS DESCRIPTION Simple exponential (Wagner): Sum of exponentials (Osaki): h( ) exp Sigmoidal-type equations (Soskey-Winter, Papanastasiou et al, ): k 1 1 2 h( ) a exp k ( a) exp k h( ) h( ) 1 1 a 2 2 1,0 0,8 0,6 egg yolk sucrose ester 0 % 8 % 1 % 7 % 2 % 6 % 4 % 4 % 6 % 2 % 7 % 1 % 8 % 0 % Soskey-Winter's model h( ) 1 1 a b 0,4 0,2 0,0 0,0 0,5 1,0 1,5 2,0
2. CONTINUOUS MECHANICS DESCRIPTION SHEAR HISTORY t t' h t, t ' ' dg t t, dt dt' LINEAR RELAXATION MODULUS Transient viscosity: G ( t) H ( ) e t / d (ln( )) 1 t, t H ( ) e tt' / h( ) t, t' d ln dt'
2. CONTINUOUS MECHANICS DESCRIPTION t t for t t ' ' WAGNER DAMPING FUNCTION ln 1 2 d k H e h k SOSKEY-WINTER DAMPING FUNCTION b a h 1 1 ' ln ' ' 1 ) ( / ' dt d t t t t a e H b t t SHEAR HISTORY FOR STEADY-STATE CONDITIONS
2. CONTINUOUS MECHANICS DESCRIPTION 10 7 10 5 B 10 6 10 5 10 4 (Pa.s) 10 4 (Pa.s) 10 3 10 3 10 2 10 1 Experim ental data Sos key-winter function Wagner function Carreau A model 10 2 10 1 5 s -1 0.5 s -1 0.05 s -1 Soskey-Winter f unction 10 0 10-7 10-6 10-5 10-4 10-3 10-2 10-1 10 0 10 1 10 2 10 3 Shear rate (s -1 ) 10 0 10-2 10-1 10 0 10 1 10 2 t (s) Experimental and predicted values of the steady-state viscosity for a lubricating grease (T = 25ºC) Experimental and predicted values of the transient flow viscosity for a lubricating grease (T = 25ºC).
2. CONTINUOUS MECHANICS DESCRIPTION n g ( t t')/ ( t) i e i e k ( t, t') ( t t') dt' i 1 i ( ) n i 1 g i i 1 ki 2 (Pa.s) 10 5 10 4 Experimental flow curve Fit from continuous spectrum Fit from discrete spectrum 10 3 Experimental and predicted values of the steady-state viscosity for a food emulsion showing wall-slip phenomena (T = 25ºC) 10 2 10 1 10 0 10-1 10-2 10-3 10-2 10-1 10 0 10 1 10 2 10 3 10 4 (1/s)
3. NORMAL STRESSES In large deformations, normal stress differences in shear flow may appear due to non-linear viscoelasticity effects. In this sense, unless the shear rate is very low, nonlinearity manifests itself by a dependency of viscosity on shear rate and the appearance of differences between the normal stresses in three orthogonal directions. Simple shear and other flows that are rheologically equivalent to it are called viscometric flows. The components of the shear rate tensor are shown below: ij 0 0 0 0 0 0 0
3. NORMAL STRESSES Three material functions completely describe the nonlinear behavior of a fluid in a viscometric flow, the viscosity and the first and second normal stress differences. These are called the viscometric functions: ( ) / N ( 1 ) 11 22 N ( 2 ) 22 33
3. NORMAL STRESSES The first and second normal stress difference coefficients are defined as follows: 2 ( ) N ( ) / 1 1 2 ( ) N ( ) / 2 2 Newtonian fluid Non-Newtonian fluid Weissenberg effect N 1 N2 0 Interesting effects: Barus effect (die swell)
3. NORMAL STRESSES N 1 m 1 m 2 In a wide range of shear rates
3. NORMAL STRESSES A plot of N1 vs. σ at various temperatures for the polymer solution D2, which is a 10% w/v solution of polyisobutylene (Oppanol B50) in dekalin N 1 a a is approximately 2
3. NORMAL STRESSES Viscometric data for a Boger fluid: 0.184% polyisobutylene in a mixture of kerosene and polybutene. 25ºC.
4. RELATIONSHIPS BETWEEN VISCOMETRIC FUNCTIONS AND LINEAR VISCOELASTIC FUNCTIONS Since the departure from a Newtonian response in the viscometric functions and the dynamic functions can be ascribed to viscoelasticity, it is not surprising to find that there are relationships between the various rheometrical functions. It is not difficult to deduce the exact relationships in the lower limits of frequency and shear rate:
4. RELATIONSHIPS BETWEEN VISCOMETRIC FUNCTIONS AND LINEAR VISCOELASTIC FUNCTIONS Various attempts have been made to develop empirical relationships between η and η' at other than the lower limits of shear rate and frequency. The most popular, and most successful in this respect, certainly for polymeric liquids, is the so-called Cox-Merz rule, which proposes that η should be the same function of the shear rate as η* is of ω.
4. RELATIONSHIPS BETWEEN VISCOMETRIC FUNCTIONS AND LINEAR VISCOELASTIC FUNCTIONS The Cox-Merz rule applied to the polymer solution D1, which is a 2% w/v polyisobutylene solution in dekalin. 25ºC.
4. RELATIONSHIPS BETWEEN VISCOMETRIC FUNCTIONS AND LINEAR VISCOELASTIC FUNCTIONS A relationship analogous to the Cox-Merz rule could be expected between G' and N1: Asymptotic approach of oscillatory and steady shear parameters. Steady shear and dynamic data for the polymer solution D3, which is a 1.5% w/v polyisobutylene solution in dekalin. 20ºC.
5. EXTENSIONAL FLOW
5. EXTENSIONAL FLOW Extensional flows are of particular importance in the study of nonlinear viscoelasticity. In this type of flow, material elements are stretched very rapidly along streamlines. Uniaxial (tensile), equibiaxial (usually called biaxial), and planar extension have all been used, but the response to uniaxial extension is the easiest to generate, and the response to this deformation has been found to be quite sensitive to certain aspects of polymer molecular structure.
5. EXTENSIONAL FLOW A good example is found in the flow of a particle in and out of a short tube, which is the kind of flow experienced when liquids such as ketchup, washing-up liquid and skin lotion are squeezed from plastic bottles, or when toothpastes, meat pastes and processed cheese are squeezed from tubes.
5. EXTENSIONAL FLOW UNIAXIAL EXTENSION
5. EXTENSIONAL FLOW Effect of open siphon
5. EXTENSIONAL FLOW
5. EXTENSIONAL FLOW
5. EXTENSIONAL FLOW
5. EXTENSIONAL FLOW
5. EXTENSIONAL FLOW A fluid for which the extensional viscosity increases with increasing strain rate is said to be 'tension-thickening', whilst, if decreases with increasing strain rate, it is said to be 'tension-thinning'. Experimentally, it is often not possible to reach the steady state. Under these circumstances, it is convenient to define a transient extensional viscosity, which is clearly a function of t as well as. ( t, ) ( t, ) / E At longer times, it is generally assumed that the stress will approach a limiting constant value E lim (, ) E t t ( ) E
5. EXTENSIONAL FLOW For the special case of linear viscoelastic behaviour, the tensile stress growth coefficient reduces to just a function of time, and becomes equal to 3 times the shear stress growth coefficient: ) ( 3 ) ( ), ( lim 0 t t t E E
5. EXTENSIONAL FLOW Extensional viscosity growth as a function of time t for a low-density polyethylene melt. 423 K
5. EXTENSIONAL FLOW The shear (dotted line) and extensional (solid line) viscosities of a dilute fibre suspension, at comparable deformation rates.
5. EXTENSIONAL FLOW The shear (dotted line) and extensional (solid line) viscosities of a dilute solution of linear polymer.
5. EXTENSIONAL FLOW Viscometric data for aqueous solutions of polyacrylamide (1175 grade)
5. EXTENSIONAL FLOW
5. EXTENSIONAL FLOW Extensional viscosity and shear viscosity as functions of stress for the low-density polyethylene designated IUPAC A. 423 K
5. EXTENSIONAL FLOW The effect of branching on the extensional viscosity of polymer melts. The shear and extensional viscosities of two polymer melts and their 50/50 blend.
5. EXTENSIONAL FLOW The effect of temperature and molecular weight on the shear (dotted lines) and extensional (solid lines) viscosity of a polymer melt. The effect of branching and molecular weight distribution on extensional viscosity of polymer melts.
5. EXTENSIONAL FLOW