Report

Polymer Rheology (高分子流變學) Instructor: Prof. Chi-Chung Hua (華繼中 教授) Complex Fluids & Molecular Rheology Laboratory, National Chung Cheng University, Chia-Yi 621, Taiwan, R.O.C. 國立中正大學 複雜流體暨分子流變實驗室 Homepage: http://www.che.ccu.edu.tw/~rheology/ Textbook R. B. Bird, R. C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids. Vol I: Fluid Mechanics, 2nd edition, Wiley-Interscience (1987). References 1. R. G. Larson, The Structure and Rheology of Complex Fluids, Oxford University Press (1998). 2. M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford Science: New York (1986). 3. C. W. Macosko, Rheology-Principles, Measurements, and Applications, WileyVCH (1994). 4. G. G. Fuller, Optical Rheometry of Complex Fluids, Oxford University Press (1995). Scope and Goal Rheology is a science that concerns the mechanical stresses arising during processing of complex fluids, as well as the microstructures that develop in responses to the external flow. This course focuses on the phenomenology, general concepts, analytical tools, and applications that are central to the interest of researchers and engineers in related fields. Rheology will not necessarily become your expertise after this course; rather, you might find yourself indulged in a fantastic world rich in the physics of a broad diversity of fluids. Course Outline Non-Newtonian Flows: Phenomenology (2 weeks) Mechanical Characterizations: Measurements and Material Functions (3 weeks) Optical Characterizations: Flow Birefringence/Dichroism and Light Scattering (3 weeks) General Analyses: Scaling Laws, Time-Temperature Superposition, Solvent Quality, and Fundamental Material Constants (3 weeks) Constitutive Equations and Modeling of Complex Flow Processing (2 weeks) Colloidal Rheology (1 week) Student Presentations on Ongoing Researches and Future Perspectives (2 weeks) Chapter Zero Introduction of Rheology Terminology What is Rheology? It normally refers to the flow and deformation of “non-classical materials” or the so-called non-Newtonian Fluids. What are “non-classical materials” ? They include rubber, molten plastics, polymer solutions, slurries & pastes, electrorheological fluids, blood, muscle, composites, soils, paints etc. [Excerpt from the website of the Institute of Non-Newtonian Fluid Mechanics (INNFM), http://innfm.swan.ac.uk/innfm_mms/index.html] Rheological Properties—from Microscopy to Macroscopy Kinetic Theory Fluid Mechanics Rheological parameters acting as a “link” between monomer structure and final properties of a polymer. [Reproduce from M. Gahleitner, “Melt rheology of polyolefins”, Prog. Polym. Sci., 26, 895 (2001).] Rheological Circle [Reproduced from C. Clasen and W. M. Kulicke, “Determination of viscoelastic and rheo-optical material functions of water-soluble cellulose derivatives”, Prog. Polym. Sci., 26, 1839 (2001).] Chapter I Non-Newtonian Flows: Phenomenology “The mountains flowed before the Lord” [From Deborah’s Song, Judges, 5:5] Contents of Chapter I Viscosity Thinning/Thickening (pp. 60-61) Normal Stress Differences and Elasticity (pp. 62-69, 72-83) Thixotropy The Deborah/Weissenberg numbers (pp. 92-95) Flow Regimes of Typical Processing Secondary Flows and Instabilities (pp.69-72) Length/Time scales & Probing Techniques 什 麼 是 流 變 (Rheology)? Rheology is the science of fluids. More specifically, the study of Non-Newtonian Fluids Y 流體 牛頓流體 - 水、有機小分子溶劑等 V yx V V Y 黏度η為定值 非牛頓流體 - 高分子溶液、膠體等 Small molecule Macromolecule V ● Newton’s law of viscosity Deformable 黏度不為定值 (尤其在快速流場下) I.1 Shear Thinning/Thickening Dilatants (Shear thickening) Dilatants (Shear thickening) Newtonian Fluids 0 pleatau Newtonian Fluids Pseudoplastics (Shear thinning) Pseudoplastics (Shear thinning) (a) Shear stress vs shear rate and (b) log viscosity vs log shear rate for Dilatants, Newtonian fluids and Pseudoplastics. For very high shear rates the pseudoplastic material reaches a second Newtonian pleatau. [Reproduced from G. M. Kavanagh and S. B. Ross-Murphy, “Rheological characterisation of polymer gels”, Prog. Polym. Sci., 23, 533 (1998).] I.1 Shear Thinning/Thickening (cont.) Tube flow and “shear thinning”. In each part, the Newtonina behavior is shown on the left (N); the behavior of a polymer on the right (P). (a) A tiny sphere falls at the same rate through each; (b) the polymer flows out faster than the Newtonian fluid. [Reproduced from R. B. Bird, R. C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids. Vol I: Fluid Mechanics, 2nd edition, Wiley-Interscience (1987), p. 61.] [Retrieved from the video of Non-Newtonian Fluid Mechanics (University of Wales Institute of Non-Newtonian Fluid Mechanics, 2000)] I.2 Normal Stress Difference and Elasticity Rod-Climbing Fixed cylinder with rotating rod. (N) The Newtonian liquid, glycerin, shows a vortex; (P) the polymer solution, polyacrylamide in glycerin, climbs the rod. [Reproduced from R. B. Bird, R. C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids. Vol I: Fluid Mechanics, 2nd edition, Wiley-Interscience (1987), p. 63.] [Retrieved from the video of Non-Newtonian Fluid Mechanics (University of Wales Institute of Non-Newtonian Fluid Mechanics, 2000)] I.2 Normal Stress Difference and Elasticity (cont.) Extrudate Swell (also called “die swell”) Behavior of fluid issuing from orifices. A stream of Newtonian fluid (N, silicone fluid) shows no diameter increase upon emergence from the capillary tube; a solution of 2.44 g of polymethylmethacrylate (Mn = 106 g/mol) in 100 cm3 of dimethylphthalate (P) shows an increase by a factor in diameter as it flows downward out of the tube. [Reproduced from A. S. Lodge, Elastic Liquids, Academic Press, New York (1964), p. 242.] [Retrieved from the video of Non-Newtonian Fluid Mechanics (University of Wales Institute of Non-Newtonian Fluid Mechanics, 2000)] I.2 Normal Stress Difference and Elasticity (cont.) Tubeless Siphon When the siphon tube is lifted out of the fluid, the Newtonian liquid (N) stops flowing; the macromolecular fluid (P) continues to be siphoned. [Reproduced from R. B. Bird, R. C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids. Vol I: Fluid Mechanics, 2nd edition, WileyInterscience (1987), p. 74.] [Retrieved from the video of Non-Newtonian Fluid Mechanics (University of Wales Institute of Non-Newtonian Fluid Mechanics, 2000)] I.2 Normal Stress Difference and Elasticity (cont.) Elastic Recoil An aluminum soap solution, made of aluminum dilaurate in decalin and m-cresol, is (a) poured from a beaker and (b) cut in midstream. In (c), note that the liquid above the cut springs back to the breaker and only the fluid below the cut falls to the container. [Reproduced from A. S. Lodge, Elastic Liquids, Academic Press, New York (1964), p. 238.] A solution of 2% carboxymethylcellulose (CMC 70H) in water is made to flow under a pressure gradient that is turned off just before frame 5. [Reprodeced from A. G. Fredrickson, Principles and Applications of Rheology, © Prentice-Hall, Englewood cliffs, NJ (1964), p. 120.] I.3 Time-dependent effect_Thixotropy Thixotropy behavior Anti-thixotropy behavior A decrease (thixotropy) and increase (anti-thixotropy) of the apparent viscosity with time at a constant rate of shear, followed by a gradual recovery when the motion is stopped The distinction between a thixotropic fluid and a shear thinning fluid: A thixotropic fluid displays a decrease in viscosity over time at a constant shear rate. A shear thinning fluid displays decreasing viscosity with increasing shear rate. I.3 The Deborah/Weissenberg Number Dimensionless groups in Non-Newtonian fluid mechanics: the Deborah number (De) De / t flow : the characteristic time of the fluid, tflow: the characteristic time of the flow system the Weissenberg number (We) We : the characteristic strain rate in the flow Dimensionless groups in Newtonian fluid mechanics: the Reynolds number (Re) Re LV / L: the characteristic length; V, and are the velocity, the density and the viscosity of fluid I.3 The Deborah/Weissenberg Number (cont.) Streak photograph showing the streamlines for the flow downward through an axisymmetric sudden contraction with contraction ratio 7.675 to 1 as a function of De. (a) De = 0 for a Newtonian glucose syrup. (b-e) De = 0.2, 1, 3 and 8 respectively for a 0.057 % polyacrylamide glucose solution. [Reproduced from D. B. Boger and H. Nguyen, Polym. Eng. Sci., 18, 1038 (1978).] I.4 Flow Regimes of Typical Processing Typical viscosity curve of a polyolefin- PP homopolymer, melt flow rate (230 C/2.16 Kg) of 8 g/10 minat 230 C with indication of the shear rate regions of different conversion techniques. [Reproduced from M. Gahleitner, “Melt rheology of polyolefins”, Prog. Polym. Sci., 26, 895 (2001).] I.5 Secondary Flows and Instabilities Secondary flow Newtonian Fluids Non-Newtonian Fluids Secondary Flow Primary Flow Secondary flow around a rotating sphere in a polyacrylamide solution. [Reporduce from H. Giesekus in E. H. Lee, ed., Proceedings of the Fourth International Congress on Rheology, Wiley-Interscience, New York (1965), Part 1, pp. 249-266] Secondary Flow Primary Flow I.5 Secondary Flows and Instabilities (cont.) Secondary flow Newtonian Fluids Non-Newtonian Fluids Secondary Flow Secondary Flow Primary Flow Primary Flow Newtonian fluid (N): water-glycerin Non-Newtonian fluid (P): 100 ppm polyacrylamide in water-glycerin Steady streaming motion produced by a long cylinder oscillating normal to its axis. The cylinder is viewed on end and the direction of oscillation is shown by the double arrow. The photographs do not show streamlines but mean particles pathlines made visible by illuminating tiny Spheres with a stroboscope synchronized with the cylinder frequency. [Reproduced from C. T. Chang and W. R. Schowalter, Nature, 252, 686 (1974).] I.5 Secondary Flows and Instabilities (cont.) Melt instability Sharkskin Melt fracture Photographs of LLDPE melt pass through a capillary tube under various shear rates. The shear rates are 37, 112, 750 and 2250 s-1, respectively. [Reproduced from R. H. Moynihan, “The Flow at Polymer and Metal Interfaces”, Ph.D. Thesis, Department of Chemical Engineering, Virginia Tech., Blackburg, VA, 1990.] [Retrieved from the video of Non-Newtonian Fluid Mechanics (University of Wales Institute of Non-Newtonian Fluid Mechanics, 2000)] I.5 Secondary Flows and Instabilities (cont.) Taylor-Couette flow Taylor vortex R1 R2 [S. J. Muller, E. S. G. Shaqfeh and R. G. Larson, “Experimental studies of the onset of oscillatory instability in viscoelastic Taylor-Couette flow”, J. Non-Newtonian Fluid Mech., 46, 315 (1993).] Flow visualization of the elastic Taylor-Couette instability in Boger fluids. [http://www.cchem.berkeley.edu/sjmgrp/] I.6 Length/Time Scales & Probing Techniques [Reproduced from G. M. Kavanagh and S. B. Ross-Murphy, “Rheological characterisation of polymer gels”, Prog. Polym. Sci., 23, 533 (1998).] 流體加工性質 macrorheology 基本流變性質 the De, Wi numbers 機械量測 microrheology microscopy/spectroscopy birefringence/dichroism light/ neutron scatterings particle tracking 光學量測 G 0 N molecular orientation / alignment particle size distribution/ diffusivity micro/mesoscopic structures 本質方程式 分子動力理論 flow pattern 模流分析 Traditional route Modern (predictive) route monomer mobility, elastic modulus etc. 量子、原子、多尺度計算 物質特性 (化學合成) Chapter II Mechanical Characterizations *Most of the figures appearing in this file are taken from the textbook “Dynamics of Polymeric Liquids” (Vol. 1) by R. B. Bird et al. For more details, you are referred to the textbooks and references cited therein. Topics in Each Section §2-1 Rheometry Shear and Shearfree Flows Flow Geometries & Viscometric Functions §2-2 Basic Vector/Tensor Manipulations Vector Operation Tensor Operation §2-3 Material Functions in Simple Shear Flows Steady Flows Unsteady Flows §2-4 Material Functions in Elongational Flows 2.1. Rheometry Two standard kinds of flows, shear and shearfree, are frequently used to characterize polymeric liquids (b) Shearfree (a) Shear vx y FIG. 3.1-1. Steady simple shear flow v x yx y ; v y 0; vz 0 Elongation rate vx Shear rate vy FIG. 3.1-2. Streamlines for elongational flow (b=0) x 2 2 vz z y The Stress Tensor y x z Shear Flow Total stress tensor* Elongational Flow Stress tensor p xx p yx 0 yx 0 p yy 0 0 p zz p xx p 0 0 0 0 p yy 0 0 p zz Hydrostatic pressure forces S h e a r S tre ss : yx First N o rm a l S tre ss D iffe re n ce : xx yy S e co n d N o rm a l S tre ss D iffe re n ce : yy zz Tensile Stress : zz xx *See §2.2 Classification of Flow Geometries (a) Shear Pressure Flow: Capillary Drag Flows: Concentric Cylinder (b) Elongation Cone-andPlate U n ia x ia l E lo n g a tio n ( b 0 , 0 ) : Moving Parallel Plates Typical Shear/Elongation Rate Range & Concentration Regimes for Each Geometries (a) Shear Concentrated Regime Homogeneous deformation:* Cone-andPlate Nonhomogeneous deformation: 10 (b) Elongation Dilute Regime Concentric Cylinder Parallel Plates 3 10 2 10 1 Capillary 10 0 10 1 10 2 10 3 10 4 -1 γ (s ) 10 5 Moving clamps (s ) -1 For Melts & High-Viscosity Solutions *Stress and strain are independent of position throughout the sample Viscometric Functions & Assumptions Example: Concentric Cylinder W1 R1 A ssu m p tio n s : (1 ) S te a dy, la m in a r, iso th e rm a l flo w (2 ) v R1W 1 o n ly a n d v r v z 0 R2 (3 ) N e gligible gra vity a n d e n d e ffe cts H (4 ) S ym m e try in , 0 (From p.188 of ref 3) FIG. Concentric cylinder viscometer S h e a r ra te : S h e a r - ra te d e p e n d e n t v isc o sity ( ) : W 1 R1 R 2 R1 ( ) (homogeneous) T 2 R1 H 2 w h e re th e to rqu e a ctin g o n th e su rfa ce o f th e in n e r cylin de r T is : T r r R1 (2 R1 H ) R1 R1 , R 2 : R a dii of in n e r a n d ou te r cylin de rs W 1 : A n gu la r ve lo city o f in n e r cyl in de r H : H e igh t of cylin de rs T : To rqu e o n in n e r cyli n de r Flow Instability in a Concentric Cylinder Viscometer for a Newtonian Liquid Tay lo r n u m b e r (T a) C e n trifu g al fo rce V isco u s fo rce 41.3 Laminar Secondary Turbulent Onset of Secondary Flow T a 4 1.3; R e 9 4 T a 141; R e 322 Ta (or Re) plays the central role! Taylor vortices Turbulent T a 387; R e 868 T a 1, 715; R e 3, 960 Rod Climbing is not a subtle effect, as demonstrated on th cover by Ph.D. student Sylvana Garcia-Rodri from Columbia. Ms. Garcia-Rodrigues is stud rheology in the Mechanical Engineering Department at U. of Wisconsin-Madison, US The Apparatus shown was created by UWMadison Professors Emeriti John L. Schra and Arthur S. Lodge. The fluid shown is a 2% aqueous polyacrylamide solution, and the rotational speed is nominally 0.5 Hz. Photo b Carlos Arango Sabogal (2006) Example 2.3-1: Interpretation of Free Surface Shapes in the Rod-Climbing Experiment (N) (P) I. Phenomenological Interpretation: (F): For the Newtonian fluids the surface near the rod is slightly depressed and acts a sensitive manometer for the smaller pressure near the rod generated by centrifugal fo (P): The Polymeric fluids exhibit an extra tension along the streamlines, that is, in the direction. In terms of chemical structure, this extra tension arises from the stretching alignment of the polymer molecules along the streamlines. The thermal motions make the polymer molecules act as small “rubber bands” wanting to snap back ‧P z r :1 r :2 (N) z :3 (P) II. Use of Equations of Change to Analyze the Distribution of the Normal Pressure The resultant formula derived in this example is: d ( 33 p ) d ln r 2 21 d d 21 ( 22 33 ) ( 11 22 ) v1 2 (2.3 5) (N ) : Fo r N e w to n ia n flu id s th e first a n d se c o n d n o rm a l stre ss d iffe re n c e s, 1 1 2 2 a n d 2 2 3 3 , a re b o th z e ro in sh e a r flo w , a n d e q . 2 .3 - 5 sh o w s th a t th e n o rm a l p re ssu re e x e rte d o n th e l u b ric a te d li d in c reases w ith th e ra d iu s (P) : Fo r po lym e r ic flu ids w e h a ve a lre a dy in dica te d th a t 11 22 is pra tica lly a lw a ys n e ga tive w ith a n u m e rica l va lu e m u ch la rge r th a n th a t o f 22 33 . W e se e th a t th e n o rm a l stre sse s m a y ca u s e th e to ta l n o rm a l pre ssu re to decrease in th e ra dia l dire ctio n . Examples 1.3-4 & 10.2-1: Cone-and-Plate Instrument A s s u m p tio n s : (1 ) S te a d y , la m in a r, iso th e rm a l flo w (2 ) v ( r , ) o n ly ; v r v 0 (3 ) 0 0.1 rad ( 6 ) (4 ) N e g lig ib le b o d y fo rc e s (5 ) S p h e ric a l liq u id b o u n d a ry (From p.205 of ref 3) S h e a r ra te : S h e a r - ra te de pe n de n t The first norm al stress visco sity ( ) : difference coefficient 1 ( ) : W0 0 FIG. 1.3-4. Cone-and-plate geometry ( ) (homogeneous) 3T 0 2 R W 0 3 2F 1 ( ) T 2 R 2 R r 2 W 0 : A n gu la r ve lo city o f co n e T : T o rqu e o n pla te 0 : C o n e a n gle F : Fo rce re qu ire d to ke e p tip o f co n e R : R a diu s o f circu la r pla te 0 in co n ta ct w ith c ircu la r pla te 0 2 2 dr d Example: p.530 Uniaxial Elongational Flow H e n c k y stra in : max t max ln ( Lmax L0 ) L 0 : In itia l sa m p le le n g th L m ax : M a x im u m sm a p le le n g th The Norm al Stress Difference : zz rr F ( t ) A ( t ) F ( t ): To ta l fo rce pe r u n it a re a e xe rte d by th e lo a d ce ll A ( t ): In sta n ta n e o u s co rss - se ctio n a l a re a o f th e sa m ple The Transient Elongational Viscosity z r : F (t ) ( zz rr ) 0 A0 e 0t 0 0 : Elongation rate A0 : Initial cross - sectional area of the sam p le FIG. 10.3-1a. Device used to genera uniaxial elongational flows by separati Clamped ends of the sample Exptl. data see §2.4 Supplementary Examples Capillary: o Example 10.2-3: Obtaining the Non-Newtonian Viscosity from the Capillary Concentric Cylinders o Problem 10B.5: Viscous Heating in a Concentric Cylinder Viscometer Parallel Plates: o Example 10.2-2: Measurement of the Viscometric Functions in the ParallelDisk Instrument o Problem 1B.5: Parallel-Disk Viscometer o Problem 1D.2: Viscous Heating in Oscillatory Flow 2.2. Basic Vector/Tensor Manipulations Vector Operations (Gibbs Notation) Vector: u u i i 3 x 1 y 2 z 3 u1 1 u 2 2 u 3 3 i u 3 1 Th e K ron ecker delta ij 1 ij 1, ij 0 , Dot product: u v u i i i Le t j i i (A .2 1, 2) if i j u j j j ( u v ) i if i j i ii uv i i i (u v i ij j )( i j ) 2 2 Th e perm u tation sym bol ijk ijk 1, ijk 1, ijk 0 , if ijk 123 , 231, o r 312 if ijk 321, 132 , o r 213 (A .2 3, 4, 5) if a n y tw o in dice s a re a like Cross product: u v u i i i (u v i j u j j j (u v i j )( i j ) ij )( ijk k ) ijk 3 U sin g i j k 1 ijk k (A .2 15) 1 2 3 u v u1 u2 u3 v1 v2 v3 Tensor Operations 2 Tensor: i i 2 ( 21 , 22 , 23 ) ij j 1 ( 11 , 12 , 13 ) j Stresses acting on plane 1 1 1 11 1 2 12 1 3 13 1 2 1 21 2 2 22 2 3 23 3 1 31 3 2 32 3 3 33 3 3 ( 31 , 32 , 33 ) FIG. The stress tensor The total momentum flux tensor for an incompressible fluid is: Normal stresses p 11 21 31 12 p 22 32 13 23 p p 33 Hydrostatic pressure forces Stress tensor or Momentum flux tensor Example: T h e to ta l m o m e n tu m flu x th ro u g h a su rfa c e o f o rie n ta t io n n is : n i i j ij n k k j k ij n k ( i j k ) ijk Let k j ij ij n j i Some Definitions & Frequently Used Operations: † † T ra n sp o se : : i j ij ij G ra d ie n t : : i Cartesian coordinate i j L a p la cia n : : Cartesian coordinate i 2 x i v 2 i 1 U n it te n so r : : i j ij 0 = ij 0 0 1 0 0 0 1 j xi ij xi 2 ij i v ji v j ik k xi ik ( ij v k )( i ij v jl )( i l ) l ) ijkl v ( ijk v ijk k ijk v jkl ij ij ji v j xi 2.3. Material Functions in Simple Shear Flows Preface: A variety of experiments performed on a polymeric liquid will yield a host of material functions that depend on shear rate, frequency, time, and so on. The fluid behavior can be better understood by means of representative rheological data. Descriptions of the nature and diversity of material responses to simple shearing or shearfree flow are given. FIG. 3.4-1. Various types of simple shear experiments used in rheology I. Steady Shear Flow Material Functions Exp a: Steady Shear Flow FIG. 3.3-1. Non-Newtonian viscosity η of a low-density polyethylene at several Different temperatures The shear-rate dependent viscosity η The first and second normal stress is defined as: coefficients are defined as follows: xx yy 1 ( ) yx 2 yx ( ) yx yy zz 2 ( ) yx 2 Relative Viscosity: rel s : S olu tion viscosity s : S olven t viscosity FIG. 3.3-2. Master curves for the viscosity and first normal stress difference coefficient as functions of shear rate for the low-density polyethylene melt shown in previous figure s Intrinsic Viscosity: [ ] lim c 0 c s c : M a s s c o n c e n tra tio n FIG. 3.3-4. Intrinsic viscosity of polystyrene Solutions, With various solvents, as a function of reduced shear rate β II. Unsteady Shear Flow Material Functions Exp b: Small-Amplitude Oscillatory Shear Flow T h e sh e a r s tre ss o sc illa te s w ith fre q u e n c y , b u t is n o t in p h a se w ith e ith e r th e sh e a r s tra i n o r sh e a r ra te S h e a r S tre ss : yx A ( ) sin ( t ) 0 S h e a r ra te : S h e a r s tra in : FIG. 3.4-2. Oscillatory shear strain, shear rate, shear stress, and first normal stress difference in small-amplitude oscillatory shear flow ( t ) co s t 0 yx ( t ) sin t 0 yx It is customary to rewrite the above equations to display the in-phase and out-of-phase parts of the shear stress Storage modulus yx G ( ) sin t G ( ) co s t 0 0 Loss modulus FIG. 3.4-3. Storage and loss moduli, G’ and G”, as functions of frequency ω at a reference temperature of T0=423 K for the low-density polyethylene melt shown in Fig. 3.3-1. The solid curves are calculated from the generalized Maxwell model, Eqs. 5.2-13 through 15 Exp c: Stress Growth upon Inception of Steady Shear Flow Transient Shear Stress: yx 0 Fo r la rg e sh e a r ra te s + d e p a rts fro m th e lin e a r visco e la stic e n ve lo p e , g o e s th ro u g h a m a x im u m , a n d th e n a p p ro a ch e s th e ste a d y - sta te va lu e . + FIG . 3 .4 - 7 . Shear stress grow th function ( t , 0 ) data for a low - density polyethylene m elt. The solid curve is calculated from Eq. 5 .3 - 25 w ith the spectrum in Table 5.3 - 2 F IG . 3 .4 - 8 . S h e a r stre ss gro w th fu n ctio n ( t , 0 ) ( 0 ) fo r tw o po lym e r so lu tio n s (a a n d b) a n d a n a lu m in u m so a p so lu tio n (c) FIG . 3 .4 - 9 . First norm al stress grow th function 1 ( t , 0 ) for the low - density polyethylene m elt Exp e: Stress Relaxation after a Sudden Shearing Displacement (Step-Strain Stress Relaxation) Relaxation Modulus:* G (t , 0 ) yx 0 T h e sh e a r stra in 0 c a n b e in d u c e d b y a p p ly in g a la rg e , c o n sta n t sh e a r ra te 0 fo r a sh o rt tim e in te rv a l t , so th a t 0 t 0 For small shear strains lim G ( t , 0 ) G ( t ) 00 In th is lim it, th e s h e a r s tre s s is lin e a r in s tra in The Lodge-Meissner Rule: G (t , 0 ) G 1 (t , 0 ) 1 F IG . 3 .4 - 1 5 . T h e re la x a tio n m o du lu s G ( t , 0 ) (o pe n sym bo ls) a n d n o rm a l stre ss re la x a tio n fu n ctio n G 1 ( t , 0 ) (so lid sym bo ls) fo r a lo w - de n sity po lye th yle n e m e lt *Example 5.3-2: Stress Relaxation after a Sudden Shearing Displacement F IG . 3 .4 - 1 6 . T h e stre ss re la xa tio n m o du lu s G ( t , 0 ) fo r 20 % po lystyre n e ( M w 1.8 10 ) in A ro clo r. 6 Pa rt (a ) sh o w s h o w G ( t , 0 ) va rie s w ith sh e a r stra in . In (b) th e da ta a re su pe rpo se d by ve rtica l sh if tin g to sh o w th e sim ila rity in G ( t , 0 ) a t la rge tim e s re ga rdle ss o f th e im po se d sh e a r sta in 2.4. Material Functions in Elongational Flows Shearfree Flow Material Functions Zero - elon gation - rate Fo r U n ia x ia l E lo n g a tio n a l Flo w ( b 0 , 0 ) : elon gation al viscosity 0 zz xx ( ) : E lo n gatio n al visco sity : E lo n gatio n rate Zero - shear - rate viscosity 0 F IG . 3 .5 - 1 . E lo n ga tio n visco sity a n d visco sity fo r a po lystyre n e m e lt a s fu n ctio n s o f e lo n ga tio n ra te a n d sh e a r ra te , re spe ctive ly Elongational Stress G row th Function T h e a b ru p t u p tu rn , o r " stra in h a rd e n in g , " o c c u rs a t a ro u g h ly c o n sta n t v a lu e o f H e n c k y st r a in (0, t ) 0 t The number average and weight average molecular weights of the samples: Monodisperse, but with a tail in high M.W. (GPC results) FIG . 3 .5 - 2 . Tim e dependence of the elongational stress grow th viscosity + for four polystyrene m elts Chapter III Optical Characterizations Topics in Each Section §3-1 Introduction to Rheo-Optics Method 3-1.1. Introduction & Review of Optical Phenomena 3-1.2. Characteristic Dimension & Optical Range §3-2 Typical Experimental Set-ups 3-2.1 Flow Dichroism and Birefringence Measurements [for Case Study 12] 3-2.2 Combined Rheo-Optcial Measurements (including Rheo-SALS) [for Case Study 3] §3-3 Information Retrieval in Individual Measurements Case Study 1: Flow Dichroism and Birefringence of Polymers Case Study 2: Dynamics of Multicomponent Polymer Melts Case Study 3: Combined Rheo-Optcial Measurements References 3-1. Introduction to Rheo-Optics Method 3-1.1 Introduction & Review of Optical Phenomena A rheological measurement entails the measurement of: (a) (b) Force (related to the stress) Displacement (related to the strain) In a rheo-optical experiment, both the force and optical properties of the sample are measured Table: A comparison of some important features in optical and mechanical measurements Optical Mechanical Measured Quantity Molecular orientation and shape Dissipation and/or storage of energy Polymer Contribution Dominates the measured signal Relatively small for dilute solutions Spatial Resolution Possible Impossible Molecular Labeling Possible Impossible Sensitivity Good precision - Time Scale Shorter Longer When incident electromagnetic radiation interacts with matter, three broad classes of phenomena are of interest: I. Transmission of Light • • II. The light can propagate through the material with no change in direction or energy, but with a change in its state of polarization Birefringence; “Dichroism”; Turbidity Scattering Radiation • • III. The radiation can be scattered (change in direction) with either no change in energy (elastic) or a measruable change in energy (inelastic) Static Light, X-Ray, and Neutron Scattering; Dynamic Light Scattering Absorption and Emission Spectroscopies • • Energy can be absorbed with the possible subsequent emission of some or all of the energy Fluorescence; Phosphorescence 3-1.2 Characteristic Dimension & Optical Range Typical levels of structures in polymeric systems are listed below Structural length scales probed by various techniques 3-2. Typical Experimental Set-ups 3-2.1. Flow Dichroism and Birefringence of Polymers in Shear Flows Basic Concepts: Turbidity The Lambert-Beer’s Law: I : In ten sity o f th e tra n sm itted ligh t I I0 exp( l ) I 0 : In ten sity o f th e in ciden t ligh t : Tu rbidity l : Pa th len gth W h e n lig h t p a sse s th ro u g h a m a te ria l th e in te n sity o f th e tra n sm itte d lig h t I w ill b e sm a lle r th a n th e in te n sity I 0 o f th e in cid e n t lig h t b e a m T h e lo ss o f in te n sity is d u e to a b so rp tio n a n d /o r sca tt e rin g o f th e lig h t In m o st m a te ria ls is in d e p e n d e n t o f th e p o la risa tio n sta te o f th e l ig h t Dichriosm Fo r dich ro ic m a te ria ls, ligh t is a tte n u a te d diffe re n tly w ith diffe re n t po la riza tio n sta te s, th a t is, i s po la riza tio n sta te de pe n de n t EX 1: Polariod Sun Glasses (A daily Eexample of dichrism resulting from absorption) FIG. Representation of a Polaroid sheet. Light with a polarization direction parallel to the aligned polymers is absorbed more strongly as compared to light with a polarization direction perpendicular to the polymers EX 2: Colloids under Shear Flow The total amount of scattered light depends on the polarisation direction due to the anisotropic nature of the microstructure under shear flow Birefringence A m a te ria l is ca lle d bire frin ge n ce , w h e n th e re fra ctive in de x is po la risa tio n sta te de pe n de n t R e su ltin g in a ph a se diffe re n ce be tw e e n ligh t w ith diffe re n t po la risa tio n sta t e s a fte r h a vin g pa sse d th e bire frin ge n ce m a te ria l FIG. Linear polarised light is generally transmitted as elliptically polarized light through a birefringent material More specifically, the polarisation direction of the light can be decomposed into a component parallel to the direction where the refractive index is large and a component parallel to the direction where the refractive index is small After having traversed the sample, there is a relative phase shift of the two field components and the sum of the two fields is generally elliptically polarised Phase-Modulated Flow Birefringence Design: M 9 ( m , 45 ) M S ( , ) M 11 (90 ) S0 Polarizer Light source S1 PEM S2 M 11 ( 4 5 ) S3 Analyzer S4 Detector Sample FIG. Phase-Modulated Flow Birefringence schematic • In transmission exps, one is normally concerned with the measurement of light polarization The Stokes Vector Entering the Detector is: S 4 M 11 ( 45 ) M S ( , ) M 9 ( m , 45 ) M 11 (90 ) S 0 The specific Mueller matrix components (optical properties) of the sample can be identified [Frattini, P. L. and G. G. Fuller, J. Rheol. 28, 61-70 (1984) ] Typical Arrangement for Flow Birefringence and Dichroism Measurements: FIG. Schematic of the experimental set-up for dichroism and birefringence measurements A somewhat simpler set-up can be sued: (1) Dichroism only: removal of R2, P2, and D2 (2) Turbidity only: removal of R1, R2, P2, and D2 BS’,BS: Beam splitter D1-D3: Detectors P1,P2: Polarizers R1,R2: Rotating quarter wave platelets Optical Train Mounted on a Rheometer FIG Experimental apparatus for determination of flow birefringence and flow dichroism FIG A combination of rheomechanical and rheo-optical measurements http://www.chemie.uni-hamburg.de/tmc/kulicke/rheology/rheo2.htm Flow Birefringence Setup [Frattini, P. L. and G. G. Fuller, J. Rheol. 28, 61-70 (1984) ] oscilloscope DAQ card Polarizer Lock in amplifier PEM Head PEM controller PMT Analyzer Reflecting mirror PEM：The PEM100 photoelastic modulator is an instrument used for modulating or varying (at a fixed frequency) the polarization of a beam of light.(δ=A sinωt) Lock in amplifier： Experimentally Idc, Iω, and I2ω are determined with lock in amplifier. The Measured Intensity I for a Sample with Coaxial Birefringence and Dichroism oriented at an Angle θ is: I I 0 1 cos 2 sin sin m cos 2 sin 2 (1 cos ) cos m 2 S4 I : In ten sity o f th e tran sm itted lig h t I 0 : In ten sity o f th e in cid en t lig h t o n th e P E M : O rien tatio n an g le o f th e sam p le : R etard an ce o f th e sam p le m : R etard an ce o f th e P E M g iven b y m A sin t I I0 2 1 cos 2 sin sin m cos 2 sin 2 (1 cos ) cos m [Frattini, P. L. and G. G. Fuller, J. Rheol. 28, 61-70 (1984) ] Fourier series expansion of cos δm and sin δm , and the intensity at the detector, I I dc I sin t I 2 cos 2 t ........ where Idc = I0 /2 Iω =2cos2θsinδJ1(A) Idc I2ω =2cos2θsin 2θ(1-cosδ)J2(A) Idc The birefringence Δn of the sample is defined from the retardance by n 2 d [Frattini, P. L. and G. G. Fuller, J. Rheol. 28, 61-70 (1984) ] Solid State Birefringence Measurement Analyer(45°) PEM PMT laser Quarter wave(sample1/4λ) Polarizer(+45 °) [Hinds Instruments, Inc., PEM-100 technical note] A=0.3125 λ I I0 1 2 1 cos A sin( t ) λ/4 0.3125λ 50kHz 1 .0 I/I 0 0 .5 0 .0 0 .0 0 0 0 0 .0 0 0 1 0 .0 0 0 2 0 .0 0 0 3 0 .0 0 0 4 tim e (s ) Fig. Waveforms obtained from calculation 0 .0 0 0 5 Fig. Waveforms obtained from oscilloscope 0 .1 5 0 .1 0 I/I0 0 .0 5 0 .0 0 0 2 e -5 4 e -5 6 e -5 tim e (s ) Fig. Waveforms obtained from DAQ card 8 e -5 A= 0.5 λ I I0 1 2 1 cos A sin( t ) λ/4 0.5λ 50kHz 1 .0 I/I 0 0 .5 0 .0 0 .0 0 0 0 0 .0 0 0 1 0 .0 0 0 2 0 .0 0 0 3 0 .0 0 0 4 tim e (s ) Fig. Waveforms obtained from calculation 0 .0 0 0 5 Fig. Waveforms obtained from oscilloscope 0 .1 0 I/I0 0 .0 5 0 .0 0 0 2 e -5 4 e -5 6 e -5 tim e (s ) Fig. Waveforms obtained from DAQ card 8 e -5 3-2.2. Combined Rheo-Optical Measurements Optical Setup for Shear-Small-Angle-Light-Scattering (SALS) Mesurements [Kume et al. (1997)] FIG. Schematic diagram of the experimental setup for shear-light scattering: one-dimensional detector (photodiode array), cone-and-plate type shear cell to generate Couette flow, and coordinate system used in this study Small-Angle Light Scattering (SALS) A monochromatic beam of laser light impinges on a sample and is scattered into a 2D detector 1 2 3 4 Hammouda, B., Probing Nanoscale Structures: The SANS Toolbox (unpublished book) Brief Introduction to SALS Description A monochromatic beam of laser light impinges on a sample and is scattered into a 2D detector; the interference of the scattered light is of interest in scattering experiments Particle 1 or light Phase difference : Particle 2 2 (Q P O R ) 2 2 s r , w here s Geometry of the path length difference Probed Length Scale Application S S0 C onventionally, the quantitiy q ( 2 s is defined as the scattering vec tor Angular Range (S 0 r S r ) 4 n sin ) 2 By the law of cosines θ =1o to 10o ~500 nm to ~5 μ m Larger systems, such as polymer solutions, gels, colloids, micelles, etc, contain structures that fall into the mesoscopic region (100 nm to 2 μ m) II. Why Large (Small) Structural Length Can be Probed at Small (Large) Angles? Focus on the “red” only (λ=633 nm)! Large structural length slits are closer Small structural length FIG. The diffraction pattern illustrated in Fig. (a) was captured by a 40x objective imaging of the lower portion of the line grating in Fig 2(b), where the slits are closer together. In Fig. (c), the objective is focused on the upper portion of the line grating, and more spectra are captured by the objective. Large structural length Small structural length Double-slit Fraunhofer pattern Schematic Drawing of SALS Apparatus 1 Monochromatic Light 2 Collimation Polarier Objective lens 3 Scattering Lens 4 Detection 1 to 10 Sample Analyzer Mirror Pinhole Iris CCD Iris Spatial filter & Beam expander Iris Lens 1 Spatial filter Beam expansion Spatial filter Mirror Lens 2 Beam expansion SALS Apparatus 2 Collimation 3 Scattering 4 Detection 1 Monochromatic Light Sample stage Laser Polarizer 1,2 Laser pointer Lens 1,2 CCD Mirror Spatial filter Mirror Iris 1,2,3 Calibration: Diffraction Pattern of a Pinhole Using a 50 μm pinhole as a sample, whose diffraction pattern is known (airy func.) Polarier Objective lens Lens Analyzer Pinhole Iris Iris Pinhole Iris Mirror Lens 1 Spatial filter & Beam expander Lens 2 1 M ea su red d iffra ctio n p a ttern A iry fu n ctio n 0 .1 I( ) / I(0 ) Mirror CCD 0 .0 1 0 .0 0 1 0 .0 0 0 1 0 2 4 6 k a sin 8 10 Calibration: Scattering of a PS Colloidal Dispersion 100-nm-diameter PS colloidal dispersion Polarier Objective lens Lens Analyzer Mirror Pinhole Iris CCD Iris Sample Mirror Lens 1 Spatial filter & Beam expander 10 S catterin g in ten sity (a.u .) The Rayleigh-Gans-Debye theory predicts that the scattering profile of the measured sample is of no angular dependence, as was confirmed experimentally Iris Lens 2 6 105 104 103 100 200 300 P ix el 400 500 Versatile Optical Rheometry Lens Iris PEM Iris Objective lens Polarier Pinhole Spatial filter & Beam expander Flow-LS (large-angle detection) Couette cell Rheology CCD Analyzer Rheo-SALS Lens Screen with aperture (from PEM) 1f 2f Photodiode Rheo-Birefringence Lock-in amplifiers Rheo-SALS (under construction) Flow cell 12.67cm 10cm Rheometer Lens 1 f=10 cm 37.5cm Lens 2 f=12.5 cm 18.75cm Lens 3 f=6 cm 9 ± 0.15 cm 18 ± 0.62 cm Butterfly Pattern (abnormal type, in this example) Uniaxially stretched with “butterfly” scattering patterns Isointensity curves for the uniaxially stretched sample (calculated) This anisotropy is the source of unusual “butterfly” scattering patterns: density fluctuation are the largest along the stretching direction Bastide et al., “Scattering by deformed swollen gels: butterfly isointensity patterns,” Macromolecules 23, 1821 (1990) Complementary to SALS: Multi-Angle Dynamic/Static Light Scattering 30 to 150 Temperature Controller 10oC to 70oC Polarizer 1 Sample cell Polarizer 2 Photomultiplier tube 30 to 150 Circulating water Detection arm 3-3. Information Retrieval in Individual Measurements CASE STUDY 1: Flow Dichroism and Birefringence of Polymers in Shear Flows A Rheo-Optical Study of Shearing Thickening and Structure Formation in Polymer Solutions [Kishbaugh and Muhugh (1993); Figs. Reproduced from Sondergaard and Lyngaae-Jorgensen (1995)] FIG. Schematic of photoelastic modulation rheo-optical device. Optical elements in the alignment configuration K ish ba u gh a n d M cH u gh stu die d m o n o dispe rs e po lystyre n e s disso lve d in de ca lin . In m o st ca se s, so lu tio n s w e re in th e dilu te to se m idilu te tra n sitio n re gio n , i .e ., c c 1 . In th e h igh sh e a r ra te ra n ge w h e re th e reversible shear thickening o cc u rre d (i.e ., 500 s -1 10, 000 s ) -1 Note that only data for the case of Mw=1.54 x 106 is shown in the following 3 pages One-to-one correlation between the onset of shear thickening and the occurrence of a maximum in the dichroism Dichroism Viscosity • The viscosity and dichroism patterns for the lowest concentration are similar to those exhibited by a lower molecular weight sample (Mw=4.3 x 105). Namely, the dichroism rises to a plateau, while viscosity undergoes a monotonic drop with shear rate to an eventual Newtonian plateau • At higher concentrations, a dramatic and distinctive pattern emerges. One sees a shaper rise in the dichroism to an eventual maximum, while the viscosity simultaneously drops to a minimum. This is followed by a region of shear thickening in which the viscosity continuously rises, while the dichroism decreases and eventually turns negative • This figure shows that, in this range, the orientation angle dropped to a constant near-alignment with the flow axis • Throughout the entire flow curve, the birefringence exhibits a steady monotonic increase with shear rate • These data offer strong evidence that the overall orientation of the chain segments is independent of the structuring processes, which may take place as indicated in the dichroism CASE STUDY 2: Dynamics of Multicomponent Polymer Melts Infrared Dichroism Measurements of Molecular Relaxation in Binary Blend Melt Rheology [Kornfield et al. (1989)] 1. Chains are identical in chemical composition, but differ in M.W.. Isotopic labeling with deuterium (D) can be used to distinguish one M.W. component from another 2. At 2,180 cm-1 the C-D bond absorbs but the C-H bond does not 3. The most interesting result is that the longest relaxation time of the the shorter chains is a strongly increasing function of the volume fraction of longer chains. This contrasts with the predictions of the basic reptation model CASE STUDY 3: Combined Rheo-Optical Measurements Rheo-Optical Studies of Shear-Induced Structures in Semidilute Polystyrene Solutions [Kume et al. (1997)] 1. Shear-induced structure formation in semidilute solutions of high molecular weight polystyrene was investigated using a wide range of rheo-optical techniques 2. The effects of shear on the semidilute polymer solutions could be classified into some regimes w.r.t. shear rate c : O n se t o f th e sh e a r - e n h a n ce d co n ce n tra tio n flu ctu a tio n s a : O n se t o f th e a n o m a lie s in th e rh e o lo gica l a n d sca tte rin g be h a vio r s FIG. A complete picture of the shear-induced phase separation and structure formation from a wide range of techniques on the same polymer solutions Continued Homogeneous solution Strong butterfly-type LS pattern Streaklike LS pattern Oblate-ellipsoidal structures Long stringlike structures Shear-microscopy results Change of the sign Due to the stringlike structures oriented parallel to the flow dir. Chains weakly orient along the flow dir. c : O n se t o f th e sh e a r - e n h a n ce d co n ce n tra tio n flu ctu a tio n s Chains in the strings with their end-to-end vectors parallel to the flow dir. a : O n set o f th e an o m alies in th e rh eo lo gical an d scatterin g beh avio r s Continued Comparisons with Mechanical Characterizations: 6.0 w t% P S /D O P so lu tio n ( c c 30 ) M w 3.84 10 ; M 6 Mechanical w M n 1.06 FIG Th e plots of sh e a r viscosity ( ), bire frin ge n ce ( n ), a n d dich roism ( n ) of th e solu tion a s a fu n ction of sh e a r ra te ( ) Notice that the behavior of the shear viscosity is also classified into three regimes References (1 ) B a a ije n s, J. P . W ., E valuation of C onstitutive E quations for P olym er M elts and Solutions in C om plex F low s , E in d h o v e n U n iv e rsity o f T e ch n o lo g y , D e p a rtm e n t o f M e ch a n ica l E n g in e e rin g , E in d h o v e n , T h e N e th e rla n d s (1 9 9 4 ). (2 ) C o lly e r, A . A . a n d L . A . U tra ck i, P olym er R heology and P rocessing , E ls e v ie r S cie n ce P u b lish e rs L td , L o n d o n (1 9 9 0 ). (3 ) Fu lle r, G . G ., O ptical R heom etry of C om plex F luids , O x fo rd U n iv e rsity P re ss, N e w Y o rk (1 9 9 5 ). (4 ) K ish b a u g h , A . J. a n d A . J. M u H u g h , "A R h e o - O p tica l S tu d y o f S h e a r - T h ick e n in g a n d S tru ctu re Fo rm a tio n in P o ly m e r S o lu tio n s. P a rt I. E x p e r im e n ta l , " R heo A cta 3 2 , 9 - 2 4 (1 9 9 3 ). ( 5 ) K o rn fie ld , J. A ., G . G . Fu lle r, a n d D . S . P e a rso n , "In fra re d D ich ro ism M e a su re m e n ts o f M o le cu la r R e la x a tio n in B in a ry B le n d M e lt R h e o lo g y , " M acrom olecules 2 2 , 1 3 3 4 - 1 3 4 5 (1 9 8 9 ). (6 ) K u m e , T ., T . H a sh im o to , T . T a k a h a sh i, a n d G . G . Fu lle r, "R h e o - O p tica l S tu d ie s o f S h e a r - In d u ce d S tru ctu re s in S e m id ilu te P o ly sty re n e S o lu tio n s , " M acrom olecules 3 0 , 7 2 3 2 - 7 2 3 6 (1 9 9 7 ) . (7 ) L e n stra , T . A . J., C o llo id s n e a r p h a se tra n sitio n lin e s u n d e r sh e a r, P h .D . th e s is, U n iv e rsity o f U tre ch t, N e th e rla n d s (2 0 0 1 ). (h ttp : //ig itu r - a rch iv e .lib ra ry .u u .n l/d isse rta tio n s/1 9 5 2 3 9 4 / in h o u d .h tm ) (8 ) M a co sk o , C . W ., R heology : P rinciples, M easurem ents, and A pplications , W ile y - V C H , N e w Y o rk ( 1 9 9 4 ). (9 ) S o n d e rg a a rd , K . a n d J. L y n g a a e - Jo rg e n se n , R heo - P hysics of M ultiphase P olym er Syste m s : C haracterization by R heo - O ptical T echniques , T e ch n o m ic P u b l. C o ., L a n ca ste r, P A (1 9 9 4 ). (1 0 ) T a p a d ia , P ., S . R a v in d ra n a th , a n d S . - Q . W a n g , "B a n d in g in E n ta n g le d P o ly m e r Flu id s in O scilla to ry S h e a r in g , " P hys R ev L ett 9 6 , 1 9 6 0 0 1 (2 0 0 6 ). Chapter IV General Analyses: Scaling Laws, Time-Temperature Superposition, Solvent Quality, and Fundamental Material Constants Fig 3.3-1 (p 105) in the textbook Content of Chapter IV Effects of Solvent Quality (pp. 139-143) Molecular-Weight Scaling Laws (pp.143-150) Retrieval of Fundamental Material Constants Time-Temperature Superposition: Application and Failure (pp. 105-108, 139-143) The ability to measure viscoelasticity of low viscosity fluids without TTS data shifting Case Study IV.1 Effects of Solvent Quality 6 6 P o lystyren e, M w = 7 .1 4 x 1 0 g /m o l P o lystyren e, M w = 7 .1 4 x 1 0 g /m o l 1 .2 1000 [ ] / [ ] 0 [ ] (m l/g ) 1 .0 0 .8 o b en zen e(3 0 C ) o 1 -ch lo ro b u tan e(3 8 C ) o tra n s-d ecalin (2 3 .8 C ) o benzene(30 C ) 0 .6 o 1-chlorobutane(38 C ) o trans-decalin(23.8 C ) 100 0 .0 0 0 1 0 .0 0 1 0 .0 1 0 .1 1 W eissen b erg N u m b er Magnitude of intrinsic viscosity -temperature & Solvent Flow curve 10 0 .0 0 0 1 0 .0 0 1 0 .0 1 0 .1 1 10 W eissen b erg n u m b er Fig 3.3-4 (p 107) in the textbook, or T. Kotaka et al., J. Chem. Phys. 45, 2770-2773 (1966). IV.1 Effects of Solvent Quality The solvent quality is an index describing the strength of polymer-solvent interactions. This interaction strength is a function of chemical species of polymer & solvent molecules, temperature, and pressure. Scaling law of polymer size and molecular weight (<R2>end-to-end 1/2 ~ Mw). Root mean square end-to-end distance Solvent condition Good <R2>end-to-end 1/2 Bad Temperature T Index T> T= T< 3/5 1/2 1/3 IV.1 Effects of Solvent Quality P M S in cyclohexane 1 .0 poor good Sample Mw g/mol Mw / Mn (SEC) Poly(methylstyrene) 1.14×106 1.11 [ ] (m l/g) 0 .9 0 .8 0 .7 0 .6 -temperature 0 .5 0 .4 15 20 25 30 35 o T ( C) 40 45 50 Advantages of PMS: 1. High plasticized speed 2. Good temperature tolerance 3. Contamination resistance 4. Compatibility with other additives 5. Environment friendly N. Hadjichristidis et al., Macromolecules 24, 6725-6729 (1991). IV.1 Effects of Solvent Quality The (temperature, weight fraction) phase diagram for the polystyrene-cyclohexane system for samples of Indicated molecular weight. S. Saeki et al, Macromolecules 6, 246-250(1973). TU: upper critical solution temperature TL: lower critical solution temperature IV.1 Effects of Solvent Quality Poly(N-isopropylacrylamide) in water Mw = 4.45x105 g/mol, c = 6.65x10-4 g/ml Mw = 1.00x107 g/mol, c = 2.50x10-5 g/ml coil globule coil globule x X. Wang et al., Macromolecules 31, 2972-2976 (1998). PN IPA M /w ater, heating cooling PN IPA M /SD S/w ater, cooling H. Yang et al., Polymer 44, 7175-7180 (2003). IV.1 Effects of Solvent Quality on intrinsic viscosity (i) The Rouse model: N N b 2 A Ms 2 N A : Avogadro constant 36 M : Molecular (ii) The Zimm model for Θ solvent: 0 . 425 NA ( N b) N : number 3 M (iii) The Zimm model for good solvent: N A N 3 b 3 weight of segments per polymer b : effective bond length : friction constant η s : solvent viscosity ν is equal to the α M We write the molecular weight dependence of [η] as 1 0 .5 3 1 0 . 8 M Rouse model (Θ solvent) Zimm model (Θ solvent) Zimm model (good solvent) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics. P113~114 IV.2.1 Molecular-Weight Dependence For linear polymer melts Molecular weight, Mw Zero-shear Relaxation Diffusivity, viscosity, DG time, 0 < Mc ~ Mw ~ Mw2 ~ 1/Mw > Mc ~ Mw3.4 ~ Mw3 ~ 1/Mw2 Mc (=2Me): critical molecular weight Me: entangled molecular weight Plot of constant + log 0 vs. constant + log M for nine different polymers. The two constants are different for each of the polymers, and the one appearing in the abscissa is proportional to concentration, which is constant for a given undiluted polymer. For each polymer the slopes of the left and right straight line regions are 1.0 and 3.4, respectively. [G. C. Berry and T. G. Fox, Adv. Polym. Sci. 5, 261-357 (1968).] IV.2.2 Concentration Effect Relative r viscosity solution solvent Specific sp : 1 c k c viscosity 2 : solution solvent Intrinsic solvent viscosity sp c c 0 2 r 1 : [cf. p109] An example of viscosity versus concentration plots for polystyrene (Mw=7.14106 g/mol) in benzene at 30 C. White circles: plot of sp / c vs. c; black circles: plot of (lnr)/c vs. c. (1) Zimm-Crothers viscometer (3.710-3 ~7.610-2 dyn/cm2); (2)Ubbelohde viscometer (8.67 dyn/cm2); (3)Ubbelohde viscometer (12.2 dyn/cm2). T. Kotaka et al., J. Chem. Phys. 45, 2770-2773 (1966). IV.2.2 Concentration Effect for semi-dilute solution PAM copolymer with hydrophobic blocks Pure polyacrylamides • The viscosity of polymer solutions increases steeply ( roughly in proportion to C 4~5 ) above the overlap concentration. • Combining the Mw dependence and concentration effect , the zero-shear viscosity can be estimated by [H] : Hydrophobe content in the monomer feed 0 C 4~5 M 3 .4 NH : Number of hydrophobe per micelle w Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics.P157 Enrique J. R. et al., Macromolecules 32,8580- 8588 (1999) IV.2.3 Impact of Molecular Weight Distribution H. Munstedt, J. Rheol. 24, 847-867 (1980) IV.2.4 Molecular Architecture Linear Polymer Star Polymer Pom-Pom Polymer polybutadiene Polyisoprene Polyisoprene IV.3 Retrieval of Fundamental Material Constants Newtonian Power law Zero-shear viscosity, 0 0 Relaxation time, 1 / critical Fig 3.3-1 (p 105) in the textbook IV.3 Retrieval of Fundamental Material Constants G 0 J e 2 2 0 1 , 0 6 2 0 5G N 0 2 0 0 G N cRT / M e 12 0 d 2 0 0 GN d e ~ M0 d ~ M3 Storage modulus vs. frequency for narrow distribution polystyrene melts. Molecular weight ranges from Mw = 8.9x103 r/mol (L9) to Mw = 5.8x105 g/mol (L18). Theoretical results of (a) G(t) and (b) G’() for polymer melts. M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford Science: New York (1986), pp 229-230. IV.4 Time-Temperature Superposition Time-temperature superposition holds for many polymer melts and solutions, as long as there are no phase transitions or other temperature-dependent structural changes in the liquid. Time-temperature shifting is extremely useful in practical applications, allowing one to make prediction of timedependent material response. WLF (Williams - Landel - Ferry) equation c1 T T 0 0 log a T c T T0 0 2 c1 T T 0 0 T T : IV.4 Time-Temperature Superposition WLF temperature shift parameters WLF (Williams - Landel - Ferry) equation c1 T T 0 0 log a T c 2 T T0 0 : c1 T T 0 0 T T J. D. Ferry, Viscoelastic Properties of Polymers, 3rd ed., Wiley: New York (1980). IV.4 Time-Temperature Superposition Non-Newtonian viscosity of a low-density polyethylene melt at several different temperatures. Master curves for the viscosity and first normal Stress coefficient as functions of shear rate for a low-density polyethylene melt Fig 3.3-1 and 3.3-2 (pp105-106) in the textbook. IV.4 Time-Temperature Superposition A master curve of polystyrene-n-butyl benzene solutions. Molecular weights varied from 1.6x10 5 to 2.4x106 g/mol, concentration from 0.255 to 0.55 g/cm3, and temperature from 303 to 333 K. Fig 3.6-5 (p 146) in the textbook. The ability to measure viscoelasticity of lowviscosity fluids without TTS data shifting The relaxation times for low-viscosity fluids are usually quite short and fall in the time domain of milliseconds or below. G’ and G” measurements must cover an enormously wide scale of times or frequencies in order to capture the relaxation process of these fluids. Conventional rheometers are usually limited to frequencies ≦100 Hz due to inertial effects. This range of frequencies is insufficient to reach the true high-frequency, limiting behavior of these fluids. High-frequency rheometry such as piezoelastic axial vibrator (PAV) or torsion resonator(TR) provides a way to characterize the dynamic properties of these low-viscosity fluids. The ability to measure viscoelasticity of lowviscosity fluids without TTS data shifting PAV gives reliable mechanical spectra for frequencies between 1 and 4000 Hz The TR can be used only at given (high) frequencies The ability to measure viscoelasticity of lowviscosity fluids without TTS data shifting FIG. 1. Fluid: DEP-10 wt% of monodisperse PS Mw=210000. (■) η*, (●) G”, and (▲) G’. FIG. 2. Fluid: DEP-2.5 wt% of monodisperse PS Mw=110000. (■) η*, (●) G”, and (▲) G’. Figure1 shows that the combination of mechanical rheometer and PAV give a reasonable match in the overlapping region. Figure 2 shows that the LVE data for both PAV and TR. Overlapping data are not possible using these two rheometers. However, consistency between the two data sets appears reasonable. Reference 1. Vadillo, D.C., T.R. Tuladhar, A.C. Mulji, and M.R. Mackley, “The rheological characterization of linear viscoelasticity for ink jet fluids using piezo axial vibrator and torsion resonator theometers,” J. Rheol. 54, 781795(2010) 2. Crassous, J., R. Regisser, M. Ballauff, and N. Willenbacher, “Characterization of he viscoelastic behavior of complex fluids using the piezoelastic axial vibrator,” J. Rheol. 49, 851-863 (2005) 3. Fritz, G., W. Pechhold, N. Willenbacher, and N. J. Wagner, “Characterization complex fluids with high frequency rheology using torsional resonators at multiple frequencies,” J. Rheol. 47, 303-319 (2003) Chapter V Constitutive Equations and Modeling of Complex Flow Processing Content of Chapter V Models for Generalized Newtonian Fluids Constitutive Equations for Generalized Linear Viscoelasticity Objective Differential/Integral Constitutive Equations Simulations of complex Flow Processing Case Study V.1 Models for Generalized Newtonian Fluids In many industrial problems the most important feature of polymeric liquids is that their viscosities decrease markedly as the shear rate increases. The generalized Newtonian model incorporates the idea of a shear-rate-dependent viscosity into the Newton’s consitutive equation. The generalized Newtonian model cannot, however, describe normal stress effects or time-dependent elastic effects. Incom pressible N ew tonian fluids: τ - γ , f tem perature, pressure, com position Incom pressible generalized N ew tonian flu ids: τ - γ , f scalar invariants of γ , .... V.1 Models for Generalized Newtonian Fluids The Carreau-Yasuda model 0 1 a n 1 / a : viscosity , 0 : zero - shear visc osity, : infinite : relaxation time, n : power - law exponent, - shear visc osity, : shear rate a : dimensionl The power-law model m n 1 n < 1, shear-thinning (pesudoplastic) fluids n = 1 and m = , Newtonian fluids n > 1, shear-thickening (dilatant) fluids ess parameter V.1 Models for Generalized Newtonian Fluids The Eyring model arcsinh 0 T he E yring equation w as the first expre ssion obtained by a m olecular theory . The Bingham model 0 0 / 0 : yield stress , 0 0 τ : τ / 2 Other empirical functions in the generalized Newtonian fluid model (see Table 4.5-1, p 228 in the textbook) V.2 Constitutive Equations for Generalized Linear Viscoelasticity Goal: To introduce an equation that can describe some of the timedependent motions of fluids under a flow with very small displacement gradients Why do we concern the linear viscoelasticity (LVE) of fluids? (1) To interrelate molecular structure with the linear mechanical responses (2) To proceed to the subject of nonlinear viscoelasticity How to combine the idea of viscosity and elasticity into a single constitutive equation that describes various rheological features? A natural combination of the Newton’s law for Newtonian fluids & the Hookean law for perfect elastic solids. V.2 Constitutive Equations for Generalized Linear Viscoelasticity The Maxwell model (for melts or concentrated solutions) shear stress for a Newtonian yx yx a. the differential form : 1 t shear stress for a Hookean solid 0 The nature of flow τ yx G b. the integral form : t t 0 / 1 e t t / 1 fluid t dt Relaxation modulus, G(t): The nature of fluid τ yx yx G t u x y yx replace by 0 and μ / G by 1 V.2 Constitutive Equations for Generalized Linear Viscoelasticity The Jeffreys model (for dilution solutions) a. the differential form : 1 0 2 t t b. the integral form : 0 t 1 t 02 2 t t / 1 2 t t t dt 1 e 1 1 Relaxation modulus, G(t) (contribution of both polymer and solvent) V.3 Objective Differential/Integral Constitutive Equations Quasi-linear model is obtained by reformulating the linear viscoelastic model. The convected Jeffreys model or Oldroyd’s fluid B τ 1 τ 1 0 γ 1 2 γ 2 Convected time derivative γ 1 γ γ n 1 τ 1 D Dt D Dt γ n v γ n γ n v T τ v τ τ v T The convected Jeffreys model is derived from the kinetic theory for dilute solutions of elastic Hookean dumbbell. If 2 = 0, the model reduces to the convected Maxwell model. V.3 Objective Differential/Integral Constitutive Equations Nonlinear differential model The Giesekus model: τ τs τp τ s s γ τ p 1 τ p 1 1 p τ p τ p p γ The model contains four parameters: a relaxation time, 1; the zero-shearrate viscosities (s and p) of solvent and polymer; and the dimensionless “mobility factor”, . is associated with anisotropic Brownian motion and/or anisotropic hydrodynamic drag on the polymer molecules. V.3 Objective Differential/Integral Constitutive Equations Nonlinear integral models The factorized K-BKZ model: τ t W I1 , I 2 W I1 , I 2 M t t 0 I1 I 2 t 0 The factorized Rivlin-Sawyers model: τ t t M t t 1 I 1 , I 2 0 2 I 1 , I 2 0 d t M t t : time - dependent factor W I 1 , I 2 or i I 1 , I 2 : strain dependent factor dt V.3 Objective Differential/Integral Constitutive Equations Advantages of nonlinear integral models: (1) they include the general linear viscoelastic fluids (2) they provide a framework of constitutive equations with molecular and empirical origins (3) it is possible to use these constitutive equations to interrelate various material functions Disadvantages of nonlinear integral models: (1) the models generally predict too much recoil in elastic recoil experiments (2) these models have been omitted for the cases of memory-strain coupling V.4 Simulations of Complex Flow Processing relaxation section Stretching section Polymer properties Governing equations (balance equations of mass, momentum and energy) Power-law constitutive equation Finite element method A. Makradi et al, J. Appl. Polym. Sci. 100, 2259-2266 (2006). V.4 Simulations of Complex Flow Processing 1D Post Draw model for IPP Spinning Roller 2 Roller 3 Polymer properties Density 0.85 (Kg/m3) Glass transition temperature 253 (K) Surface tension 35 (dyn/cm) Melt shear modulus 9x108 (Pa) Maximum crystallization rate 0.55 (1/s) Maximum rate temperature 65 (K) Crystallization half width temperature 60 (K) Avrami index 3 Maximum percent crystallinity 70 (%) Roller 1 CAEFF (Center for Advanced Engineering Fibers and Films) software V.4 Simulations of Complex Flow Processing Model properties Heat capacity parameters Orientation hardening parameter 9 Cs1 0.25 (cal/g/C) Frictional coefficient 0.6 Cs2 7.0x10-4 (cal/g/C2) Room temperature 298 (K) Cs3 0 Initial tensile stress 5x105 (Pa) Cl1 0.32 (cal/g/C) Initial percent crystallinity 45 (%) Cl2 5.7x10-4 (cal/g/C2) Crystallization rate 0.1 Cl3 0 Amorphous shear stress 8.5x106 (Pa) Hf 30 (cal/g) Poisson ratio 4.3x105 (Pa) Rubber elasticity 1.5x105 (Pa) Roller parameters activation energy 10800 Temperature 35 (C) Pre-exponential shear strain rate 2.3x107 Radius 8 (cm) Activation volume 4.7x10-29 Activation parameter 3.65 Mass flow rate 1.3x10-6 (Kg/s) Roller 1 Velocity of Roller 2 Roller 3 80 (m/s, conter-clockwise) 80, 160 (m/s, clockwise) 160 (m/s, conter-clockwise) V.4 Simulations of Complex Flow Processing Velocity of Roller 2 = 160 m/s Velocity of Roller 2 = 80 m/s Chapter VI Shear Thickening in Colloidal Dispersions Content of chapter VI Introduction to shear thickening fluids Onset of shear thickening : the Péclet number Lubrication hydrodynamics and hydroclusters Controlling shear thickening fluids: to modify colloidal surface VI.1 Introduction to the shear thickening fluids The unique material properties of increased energy dissipation combined with increased elastic modulus make shear thickening fluids ideal for damping and shock-absorption applications. Example: The different velocity at which a quarter –inch steel ball required to penetrate various layers For single layer of Kevlar is measured at about 100 m/s For Kevlar formulated with polymeric colloids is about 150 m/s For Kevlar formulated with silica colloids is about 250 m/s VI.1 Introduction to shear thickening fluids Right video : two layers containing shear thickening fluids and Nylon. The popular interest in cornstarch and water mixers known as “oobleck” is due to their transition from fluid-like to solid-like behavior when stressed. Left video : three layers containing neat Nylon. VI.1 Introduction to shear thickening fluids Beyond the critical stress, the fluid’s viscosity decreases (shear thinning). At high shear stress, its viscosity increases (shear thickening) The viscosity of colloidal latex dispersions, as a function of applied shear stress. The actual nature of the shear thickening will depend on the parameters of the suspended phase: phase volume, particle size (distribution), particle shape, as well as those of the suspending phase (viscosity and the details of the deformation.) VI.2 Onset of shear thickening : the Péclet number Fluid drag on the particle leads to the Stokes-Einstein relationship: D k BT 6 a a : particle's hydrodynam ic radius Dt The mean square of the particle’s displacement is Accordingly, the diffusivity sets the characteristic time scale for the 2 particle’s Brownian motion. a x t particle 2 D A dimensionless number known as Péclet number, Pe Pe a D 2 a 3 k BT VI.2 Onset of shear thickening : Péclet number Low shear rate ( Pe <<1 , t particle ) is close to equilibrium that Brownian motion can largely restore equilibrium microstructure on the time scale of slow shear flow. Pe ～1, shear thinning is evident around that regime. At high shear rates or stress (Pe >>1), deformation of colloidal microstructure by the flows occurs faster than Brownian motion can restore it. Accordingly, the High Pe triggers the onset of shear thickening. VI.3 Lubrication hydrodynamics and hydroclusters Pe~1 Pe<<1 The flow-induced density fluctuations are known as hydroclusters which lead to an increase in viscosity. The formation of hydroclusters is reversible, so reducing the shear rate returns the suspensions to a stable fluid Pe>>1 At (Pe<<1) regime, random collisions among particles make them naturally resistant to flow. As the shear rate increase (Pe~1), particles become organized in the flow, which lowers their viscosity. At (Pe>>1) regime, the strong hydrodynamic coupling between particles leads to the formation of hydroclusters (red particles) which cause an increase in viscosity. Normalized lubrication force VI.3 Lubrication hydrodynamics and hydroclusters The force required to drive two particles together is lubrication force, as well as the force is the same one required to separate two particles. Distance between particle surfaces In simple shear flow, particle trajectories are strongly coupled by the hydrodynamic interaction if the particle are close together. When two particles approach each other, rising hydrodynamic pressure between them squeezes fluid from the gap. VI.3 Lubrication hydrodynamics and hydroclusters Red region indicates the most probable particle position as nearest neighbors At low Pe number (0.1), the distribution of neighboring particles is isotropic. At Pe =0.1, shear distortion appears in neighbor distribution, such that particles are convected together along the compression axes At high Pe regime, particles aggregate into closely connected clusters, which manifest as yet greater anisotropy in the micro structure. Particles are more closely packed and occupy a narrow region (red) , indicative of being trapped by the lubrication forces. Viscosity (Pa*s) VI.3 Lubrication hydrodynamics and hydroclusters Pe>>1 Pe~1 ■ :the viscosity of concentrated colloidal suspension ● : stochastic motion of particles component ▲ : hydrodynamic interaction component Pe~1 Pe<<1 Pe>>1 Shear stress (Pa) The equilibrium microstructure is set by balance of stochastic and interparticle force, including electrostatic and van der Waals force, but is not affected by hydrodynamic interactions. The low shear (Pe<<1) viscosity has two components, one due to interparticle force, and the other due to hydrodynamic interactions. At Pe~1 regime, the stochastic motion dominates the flow behavior At high shear rates (Pe>>1), hydrodynamic interactions between particles dominate over stochastic ones. VI.4 Controlling shear thickening fluids: to modify colloidal surface The addition of a polymer “brush” grafted or absorbed onto the particles’ surface can prevent particles from getting close together. The figure shows that shear thickening is suppressed by imposing a purely repulsive force field. With the right selection of grafted density, molecular weight, and solvent , the onset of shear thickening moves out of the desired processing regime References N. J. Wagner, J. F. Brady, Physical today, October 2009 B. J. Maranzano, N. J. Wagner, J. Chem. Phys. 114, 10514 (2001)