R heological Behavior and Polymer Properties G. C.

R heological Behavior and Polymer Properties G. C.

R heological Behavior and Polymer Properties G. C. Berry Department of Chemistry Carnegie Mellon University Colloids, Polymers and Surfaces e-mail: [email protected] web site: http://www.chem.cmu.edu/berry Carnegie Mellon 1 Introduction 3 (12)

Rheological methods 16 (19) Linear elastic parameters 26 (5) Linear viscoelastic functions 33 (12) Several viscoelastic experiments

44 (16) Relations among linear viscoelastic functions 62 (10) Examples of linear viscoelastic functions 73 (9) Time-temperature equivalence 83 (9)

The glass transition temperature 93 (13) The viscosity 107 (26) Effects of polydispersity 134 (4) Network formation 139

(13) Isochronal Behavior 153 (6) Examples from the literature 160 (45) Branched and linear metallocene polyolefins 161 (10) Colloidal dispersions 172 (9)

Wormlike Micelles 182 (4) Deformation of rigid materials 187 (4) Nonlinear shear behavior 192 (16) 209 (6) Carnegie Mellon Linear and nonlinear bulk properties 2 Carnegie Mellon

Introduction Rheological methods Linear elastic parameters Linear viscoelastic functions Several viscoelastic experiments Relations among linear viscoelastic functions Examples of linear viscoelastic functions

Time-temperature equivalence (Thermo-rheological simplicity) The glass transition temperature The viscosity Effects of polydispersity Network formation Isochronal Behavior Examples from the literature 3 POLYMERS

NATURAL SYNTHETIC PROTEINS GUMS POLYNUCLEOTIDES RESINS THERMOPLASTIC THERMOSETTING POLYSACCHARIDES ELASTOMERS Carnegie Mellon 4 Some Common Elastomers, Plastics and Fibers ELASTOMERS PLASTICS

Polyisoprene polyethylene polyisobutylene polytetrafluoroethylene poybutadiene polystyrene FIBERS poly(methyl methacrylate) Phenol-formaldehyde Urea-formaldehyde Melamine-formaldehyde Carnegie Mellon Poly(vinylchloride)) Polyure)thane)s

Polysiloxane)s Polyami de) Polye)ste)r Polypropyle)ne) 5 Mn Mw Fraction of Molecules With

Molecular Weight M Mz Molecular Weight M A Schematic Illustration of a Typical Distribution of Molecular Weights, showing Mn, Mw, and Mz Carnegie Mellon 6 A generalized Average of molecular weights: w is the weight fraction of polymer with molecular weight M: M(a) = a1/a w M pe) cial C ase) s: Num be)r ave)rage) : =

1 1 M(0)/M(1) = 1/ wM Mw = M(1)/M(0) Mz = M(2)/M(1) Mn We ight av era ge: = wM z-averag e : 2

2 = wM/ wM G. C. Berry "Molecular Weight Distribution" Encyclopedia of Materials Science and Engineering, ed. M. B. Bever, Pergamon Press, Oxford, 3759-68 (1986) Carnegie Mellon 7 Specific Volume Tm Temperature A schematic v-T diagram for a typical nonpolymeric material. Carnegie Mellon 8 Specific Volume Tg

Tm Temperature A schematic v-T diagram for a typical semi-crystalline polymeric material. Carnegie Mellon 9 Specific Volume Tg Temperature A schematic v-T diagram for a typical noncrystalline polymeric material. Carnegie Mellon 10 Rigid Plastic Stress Flexible Plastic

Elastomer Strain Typical Stress-Strain Behavior for Plastics and Elastomers Carnegie Mellon 11 F. W. Billmeyer Jr. (1976): J. Polym. Sci.: Symp. (1976) 55: 1-10 "characterization of polymers is inherently more difficult than that of other materials. Polymers are roughly equivalent in complexity to, if not more complex than, other materials, at every physical level of organization from microscopic to macroscopic" "We would wish, ideally, to characterize all aspects of a polymer structure in enough detail to predict its performance from first principles. I seriously doubt that this will ever be possible, and I am sure that even if it were, it would never be economically feasible." Carnegie Mellon 12 Microscopic Characterization Needed at Many Resolutions:

A B A A B B B A A B A B A A B A

B B C C C Carnegie Mellon C C 13 2-D projection of a random arrangement of a chain with 1000 non-overlapping bonds, each step otherwise randomly selected Carnegie Mellon 14 Mean chain dimensions: For a linear chain with contour length L (without excluded volume effects): Mean square-end-to-end dimension: RL2 = 2L is the persistence length (2 is the Kuhn length)

for a flexible chain, << L. Mean square-radius of gyration: RG2 = RL2 /6 = L/3 Carnegie Mellon 15 Carnegie Mellon Introduction Rheological methods Linear elastic parameters Linear viscoelastic functions Several viscoelastic experiments

Relations among linear viscoelastic functions Examples of linear viscoelastic functions Time-temperature equivalence (Thermo-rheological simplicity) The glass transition temperature The viscosity Effects of polydispersity Network formation

Isochronal Behavior Examples from the literature 16 Schematic of Rheometer System Computer System for Data Acquisition and Instrument Control Shear Stress vs Torque Transducer

Time (Frequency) Shear Strain Force vs Transducer Time (Frequency) Position Normal Force Transducer vs Time (Frequency) Shape Transducer Temperature vs Temperature Time

Transducer Carnegie Mellon Rheometer Output Interfaces 17 CONTROLLED STRESS IN TENSION "Frictionless" Bearing Position Transducer Sample Removable Tare Weight

Device Carnegie Mellon Output Removable Weight Input Controlled weight PositionTransduc er Measure of shaft position Voltage(current) Controlledforce 18 CONTROLLED DEFORMATION IN TENSION Drive Screws Crosshead

Position Transducer Sample Device Carnegie Mellon Output Crosshead Drive Input Controlled Drive PositionTransduc er Measure of shaft position Voltage(current) Controlledforce 19

CONTROLLED STRESS RHEOMETER Controlled Torque Drive Angle Position Transducer Shaft "Frictionless mount" Sample Fixtures Fixed Shaft (Alternate: controlled rotation) Carnegie Mellon Device

Input Output Controlled Torque Drive Controlled voltage Controlled torque Angle Position Transducer Measure of shaft angle Vo ltage (current) 20 CONTROLLED DEFORMATION RHEOMETER Controlled Rotation Drive Angle

Position Transducer Shaft "Frictionless mount" Sample Fixtures Torque Transducer (Force Transducer) Carnegie Mellon Device Input Output Controlled Deformation Drive Controlled voltage Controlled shaft rotation

21 Electromagnetic Coils I : A-F II E : a-f F d e , c f D G Iron Core

C H b g a h B A Aluminum Cylinder Attached to Rotor Carnegie Mellon Phasing of the currentsin Coils I and II can produce a timedependent o t rque: Constanttorque amplitude

Sinusoidal torque amplitude Torque amplitude may readily be varied over a factorof 106. 22 Paralle)l Plate)s Sample Fixture)s Height h 2R Cone) & Plate) ample) Fixture)s Angle) a

2R Conce)ntric Cylinde)rs ample) Fixture)s h 2R Carnegie Mellon R 23 Geometric Factors in Rheometry Geometry Measured Translational geometries

Parallel Plate Force: a Calculated F Stress: s = F/wb width,w; bre)adth b; se)paration h isplace)m e)nt: Force): F Conce)ntric Cylinde)rs inne)r radiusR ; gap; he)ight h isplace)m e)nt: train: g = /h tre)ss: train: s = F/2pR h g = /R ln(1 +/R )

s = (2r/R )M/pR 3 Ro tationalge)om e)trie)s Parall e)lPlate) Torque): M tre)ss: Rotation: train: Cone) & Plate) oute)r radiusR ; cone) angle) p - a Torque): Rotation: M tre)ss: train:

s = (3/2)M/pR 3 g = (1/a) Conce)ntric Cylinde)rs inne)r radiusR ; gap; he)ight h Torque): Rotation: M tre)ss: train: s =(R /2h)M/pR 3 g(r) =(R /R ) f(R ,r) oute)r radiusR ; se)paration h g(r) = (r/h) 2 1 + /R f(R ,r) = (R/ r) 1 + /2R a s and g are) the) she)a

r stre)ss and tr sain, re)spe)ctive)ly Carnegie Mellon 24 Functions and Parameters Used Function/Parameter Symbol Units Time t w T Frequency train Com pone)nt Elongational tr sain e) ---

he)ar strain R ate) of she)ar g g, e) --T-1 tre)ss Co m pone)nt ij ML-1T-2 he)ar stre)ss s ML-1T-2 Modulus Com pliance) G, K, E J, B, h

ML-1T-2 M-1LT2 V iscosity Carnegie Mellon e)ij T-1 --- ML-1T-1 25 Carnegie Mellon Introduction Rheological methods Linear elastic parameters

Linear viscoelastic functions Several viscoelastic experiments Relations among linear viscoelastic functions Examples of linear viscoelastic functions Time-temperature equivalence (Thermo-rheological simplicity) The glass transition temperature The viscosity

Effects of polydispersity Network formation Isochronal Behavior Examples from the literature 26 Linear elastic phenomenology Shear stress s he)ar straing g = Js =(1/G )s Elongational tr se)ss sT Elongational tr saine) e) = sT =(1/E)sT Pre)ssure) P V olume) change) V V /V = B P =(1/K) P

Carnegie Mellon 27 Linear Elastic Functions Shear Compliance J Shear Modulus G Bulk Compliance B Bulk Modulus K Tensile Compliance D = J/3 + B/9 Tensile Modulus

1/E = 1/3G + 1/9K Carnegie Mellon 28 Linear elastic phenomenology e)ij uj 1 ui = 2 x + x ; j i u is the displacement vector 2ij = J [Sij 1 3 Sij = 2G [ij ij S ] + (2/9) ij B S 1

3 ij ] + ij K ij = 1 if i = j, and ij = 1 if i j In this notation, Shear stress = S12 Shear strain = 212 Carnegie Mellon 29 Relations Among Linear Elastic Constants K, G E, G K, E K, n E, n G, n K

K EG 3[3G E] K K E 3[1 n2] 2G[1 + n] 3[1 n2] E 9KG 3K + G E E 3K(1 2n) E 2G(1 +n)

G G G 3KE 9K E 3K[1 n2] 2[1 + n] E 2[1 + n] G n 3K G 2 6K + G 2 E 2G 1 3K E

6K n n n Carnegie Mellon J = 1/G, B = K, 1/ =1/E 30 Carnegie Mellon 31 Carnegie Mellon Introduction Rheological methods

Linear elastic parameters Linear viscoelastic functions Several viscoelastic experiments Relations among linear viscoelastic functions Examples of linear viscoelastic functions Time-temperature equivalence (Thermo-rheological simplicity) The glass transition temperature

The viscosity Effects of polydispersity Network formation Isochronal Behavior Examples from the literature 32 Linear Elastic Functions Shear Compliance J Shear Modulus G

Bulk Compliance B Bulk Modulus K Tensile Compliance D = J/3 + B/9 Tensile Modulus 1/E = 1/3G + 1/9K Carnegie Mellon 33 Linear Viscoelastic Functions Shear Compliance J(t) Shear Modulus

G(t) Bulk Compliance B(t) Bulk Modulus K(t) Tensile Compliance D(t) = J(t)/3 + B(t)/9 Tensile Modulusa 1/E(s) = 1/3G(s) + 1/9K(s) a. The superscript "" denotes aLaplace transform. Carnegie Mellon 34 Linear viscoelastic phenomenology Stress Controlled g(t)

J(t ti) i = = i=1 t -duJ(t u) J(t u) (u) u = (t) J(u) = Jo(t) + 0 du (t u) u

2 1 (t) (t) t Carnegie Mellon (t) d[(u)] 0 (t) (t) t

1 t 2 35 Linear viscoelastic phenomenology Strain Controlled = s(t) G(t ti) i i=1 (t) d[(u)]G(t 0 (t) =

(u) u) u (t) = Go(t) + 0 du (t u) t duG(t - = G(u) u

(t) (t) 2 (t) Carnegie Mellon u) t 1 t 1 t 2

36 Linear elastic phenomenology e)ij uj 1 ui = 2 x + x ; j i u is the displacement vector 2ij = J [Sij 1 3 Sij = 2G [ij ij S ] + (2/9) ij B S 1 3 ij ] + ij K

ij = 1 if i = j, and ij = 1 if i j In this notation, Shear stress = S12 Shear strain = 212 Carnegie Mellon 37 Linear viscoelastic phenomenology uj 1 ui e)ij = 2 x + x ; j i u is the displacement vector Sij(s) 2ij(t) = -ds{J(t s)[ s t 1 3 S (s) ij s ]

S (s) + (2/9) ij B(t s) s } Sij(t) = ij(s) {2G(t s)[ s t ds - (s) s ] 1 3 ij (s) + ij K(t s) s } In this notation, Shear stress (t) = S12 (t) Shear strain (t) = 212(t) Carnegie Mellon 38

Relation between G(t) and J(t) 1 t t 0du G(t u) J(u) = 1 with Laplace transform: 2 s G(s)J(s) Carnegie Mellon = 1 39 Shear Compliance J(t) and Recoverable Shear Compliance R(t) R(t) = J(t) t/h = J [J ]a(t)

oJ a(t): Re)tardation Function he)ar ModulusG(t) G(t) = G e) + [G o G e)]f(t) f(t): Relaxation Function the (linear) viscosity, with1/ =0 for a solid, Ge the equ ilibriummodulu s, withGe =0 for a fluid, Go the "instantan eous" modulu s, withJoGo =1, and J the limit of R(t) for larget: Solid: J =Je =1/Ge; equilibrium compliance Fluid: J =Js; steady -staterecoverable compliance Carnegie Mellon 40 Creep Shear Compliance J(t)

R(t) = J(t) t/h = J [J Jo]a(t) he)ar Modulus G(t) G(t) = G e) + [G o G e)]f(t) Linear elastic solid: 1/ = 0, J = Je = 1/Ge, (t) = (t) = 0 Linear viscousfl uid: 1/ > 0, Go = 0,

(t) = (t) = (t) Linear viscoelastic soli d: 1/ = 0, J = Je = 1/Ge, 0 < (t) < (t) 1 Linear viscoelastic fluid: 1/ > 0, J = Js (= Joe), 0 < (t) < (t) 1 Bulk Compli ance Carnegie Mellon Bulk Modulus B(t)

= Be [Be Bo](t) K(t) = K e + [K o K e](t) 41 Retardation function a(t) R (t) = (t) J t/h = J [J Jo]a(t) Re) laxationfunction f(t) G(t) = G e) + [G o G e)]f(t) 1 Viscoelastic Fluid or Solid f

(t) (t) Linear Viscous Fluid Linear Elastic Solid 0 Time Carnegie Mellon 42 Simple example of the relation between G(t) and J(t) Maxwell fluid: G(t) = Goexp(- t/t); J(t) = Js + t/h; t = h/G o Js = Jo = 1/G o R(t) = Js Note: (t) = exp(-t/) and (t) = 0 for this model.

Carnegie Mellon 43 Often used relations for f(t) and a(t) A we) ight se) t of e) xpon e) ntials with N e)rlaxation tim e) s: N-1 a(t) = i exp(t/ i) m N (t) = i exp(t/ i) 1 = 1 d(ln ) L()exp(t/) J Jo - 1

= G G - d(ln ) H()exp(t/) o e Notes: i = i = 1, and m is equal to 0 or 1 for a solid and fluid, resp. (1 m) 0 > 1 > 1 > > i > i > i+1 > > N-1 > N (The contribution 0 is absent for a fluid) Carnegie Mellon 44 Carnegie Mellon Introduction Rheological methods Linear elastic parameters

Linear viscoelastic functions Several viscoelastic experiments Relations among linear viscoelastic functions Examples of linear viscoelastic functions Time-temperature equivalence (Thermo-rheological simplicity) The glass transition temperature The viscosity Effects of polydispersity

Network formation Isochronal Behavior Examples from the literature 45 Carnegie Mellon 46 Creep and recovery with a step shear stress Stress history: s(t) = 0 t< 0 s(t) = so 0 t Te)

s(t) = 0 t > Te) (t) = o Stress 0 (t) = a + bt R ( )=

(t = T e )- (t) Strain (t) t=T e 0 q =t-T e

t Time Carnegie Mellon 47 The strain in creep for t Te: g(t) t = so0du J(t u) d(u - 0) = soJ(t) =so[R (t) + t/h] Carnegie Mellon 48 The strain in creep for t Te: g(t) t = so0du J(t u) d(u - 0) = soJ(t) =so[R (t) + t/h] The) strain for J =t Te) > 0 in re)cove)ry:

Te) t g(t) = so0 du J(t u) d(u - 0) soT du J(t u) d(u - Te)) e) g(J) = so[J(J + Te)) J(J )] =so[R (J + Te)) R (J) + Te)/h] Carnegie Mellon 49 The strain in creep for t Te: g(t) t = so0du J(t u) d(u - 0) = soJ(t) =so[R (t) + t/h] The) strain for J =t Te) > 0 in re)cove)ry: T t g(t) = so0e)du J(t u) d(u - 0) soT du J(t u) d(u - Te)) e)

g(J) = so[J(J + Te)) J(J )] =so[R (J + Te)) R (J) + Te)/h] The) re)cove)rabl e) strain gR (J) =g(Te)) g(t) forJ > 0: gR (J) = so{ J(Te)) [J(J + Te)) J(J )]} Carnegie Mellon = so{ R (J) + R (Te)) R (J + Te))} 50 Carnegie Mellon 51 The stress response for t > 0: t s(t) = go0du G(t u)d(u - 0) =goG(t) = go{G e) + (G o G e))f(t)} s() = goG e) Carnegie Mellon

52 Carnegie Mellon 53 The stress response for t Te : t t s(t) = g 0 du G(t u) =g [G e)t + (G o G e))0dsf(s)] For afluid in ste)ady-state) de)formation, s =hg, or h = s()/g = G o 0 dsf(s) The) strain for t > Te): Te) 0 s(t) = 0 = g du G(t u) + t du G(t Te) g(u)

u) u Forlarge) Te) and t, (fullre)coil afte)rste)adyflow) it can be) shown that for a fluidthis give) s: tc =hJs = Carnegie Mellon 0 dssf(s)/0 dsf(s) 54 Carnegie Mellon 55 The strain response for t > 0: t g(t) = wso0du J(t )cos( u wu) In the) te)ady-s s tate) im l it with large) : t

g(t) = so{J'(w)sin(wt) J''(w)cos(wt)} In-phase) (or re)alor tor s age)) dynam ic compliance): J'(w) = J w[J oJ] 0 ds a(s)sin(ws) Out-of-phase) (orimaginary orloss) dyn ami c com pliance) J"(w) = (1/wh) +w[J oJ] 0 ds a(s)cos(ws) Carnegie Mellon 56 Alternatively g(t) = so |J*(w)|sin [wt d(w)] "ynamic compliance)": |J*(w)|2 = [J'(w)] 2 + [J"( w)] 2 Phase) angle)d(w): tan d(w) = J"( w)/J'(w) For sm allw: J'(w) J, Carnegie Mellon

J"(w) 1/wh, and J"(w) 1/wh w 57 Oscillation with a sinusoid shear strain Strain history: g(t) = 0 t < 0 g(t) = gosin(wt) t 0 The) stre)ssre)sponse) fort > 0 is giv e)n by t s(t) = wgo 0du G (t u)cos(wu) In the)ste)ady-stat e) lim it withlarge) t, s(t) = go{G'(w)sin(wt) + G''(w)cos(wt)} In-phase) (or re)alor tor s age)) dynam ic compliance): G'(w) = G e) + w[G o G e)] 0 ds f(s)sin(ws) Out-of-phase) (orimaginary orloss) dyn ami c com pliance)

Carnegie Mellon G'' (w) = w[G o G e)] 0 ds f(s)cos(ws) 58 Alternatively s(t) = go |G *(w)|sin [wt + d(w)] "ynamic compliance)": 2 2 |G*(w)| = [G'(w)] + [G"(w)] 2 Phase) angle)d(w): tan d(w) = G"(w)/G'(w) For sm allw: 2 2 fluid G'(w) G e) + w [G o Ge)] 0 dssf(s) (wh) Js

flu id G''(w) = w[G o G e)] 0 ds f(s) Carnegie Mellon wh 59 Exact relations among the dynamic moduli and compliances: |G*(w)||J*(w)| = 1 2 J'(w) = G'(w)/|G*(w)| J"( w) = G"(w)/|G*(w)|2 2 G'(w) = J'(w)/|J*(w)| G"(w) = J"( w)/|J*(w)|2 tan d(w) = J"( w)/J'(w) = G"(w)/G'(w) Carnegie Mellon

60 The dynamic viscosity: In-phase with the strain rate: h'(w) = G"(w)/w Out-of-phase) with the)train s rate): h"(w) = G'(w)/w For sm allw: id h'(w) = [G o G e)] 0 ds f(s) flu h h''( w) G e)/w + w[G o Ge)] 0 dssf(s) flu id wh Js Carnegie Mellon 2 61 Carnegie Mellon

Introduction Rheological methods Linear elastic parameters Linear viscoelastic functions Several viscoelastic experiments Relations among linear viscoelastic functions Examples of linear viscoelastic functions

Time-temperature equivalence (Thermo-rheological simplicity) The glass transition temperature The viscosity Effects of polydispersity Network formation Isochronal Behavior Examples from the literature 62 Linear Viscoelastic Functions

Shear Compliance J(t) Shear Modulus G(t) Bulk Compliance B(t) Bulk Modulus K(t) Tensile Compliance D(t) = J(t)/3 + B(t)/9 Tensile Modulusa 1/E(s) = 1/3G(s) + 1/9K(s) a. The superscript "" denotes aLaplace transform. Carnegie Mellon 63

Relation between G(t) and J(t) 1 t t 0du G(t u) J(u) = 1 s2G(s)J(s) = 1 with Laplace transform: Carnegie Mellon 64 Carnegie Mellon 65 Carnegie Mellon 66

Carnegie Mellon 67 An often used relation between G(t) and J(t) A weight set of exponentials with N relaxation times: N-1 a(t) = m 1 exp(t/ ) = d(ln ) L()exp(t/) i i J Jo - N (t) =

exp(t/ ) 1 i i = 1 d(ln ) H()exp(t/) Go Ge - Notes: i = i = 1, and m is equal to 0 or 1 for a solid and fluid 0 > 1 > 1 > > i > i > i+1 > > N-1 > N ( 0 absent for a fluid) Carnegie Mellon 68 Determination of L(l) (or the) ai-li se)t) from J(t) (im ilar con sid e)ration s apply to the) de)te) rm ination of H((t) (or the) f i-ti se)t) from G (t)) e) rivative) m e) thod s for L(l): 1st Approx.:

L(l) M(m) [R(t)/ln t]t = l M(m) = lnL(l)/ln l (inte) rative) ) 2nd Approx.: Carnegie Mellon L(l) 2 2 [J(t)/ln t J(t)/(ln t) ] t = 2l 69 Carnegie Mellon 70 Determination of L(l) (or the) ai-li se)t) from J(t) (im ilar con sid e)ration s apply to the) de)te) rm in ation of H((t) (or the) f i-ti se)t) from G (t))

In ve)rse) transform m e) thod s for ai-li: The) inve)rse) transform is "ill -pose) d ", a n d a sta ble) solutions re) quire) s constrain ts (e) .g., ai 0) In an ofte) n use)d strate)gy, ase) t of logarithm icall y space) d li are) chose) n such that the) span in 1/lI d oe) s no t e) xce) e)d the) tim e) span in the) e) xpe) rim e) ntald ata. A onstra c ine) d nonline) ar e)last squ are)s analysis is the) n use) d to d e)te) rm in e) the) ai. Com m e)rcialpackage) s ar e) available) for this transform . Carnegie Mellon 71 Carnegie Mellon 72 Carnegie Mellon

Introduction Rheological methods Linear elastic parameters Linear viscoelastic functions Several viscoelastic experiments Relations among linear viscoelastic functions Examples of linear viscoelastic functions Time-temperature equivalence (Thermo-rheological simplicity)

The glass transition temperature The viscosity Effects of polydispersity Network formation Isochronal Behavior Examples from the literature 73 Low Molecular Weight Glass Former slope H(

L( tslope) h J(t) G(t) l J o G log h'( an og ltw) )d(w) lw t== or =1/3 G''( 1log w) t t/h Carnegie Mellon 74 Low Molecular

Polyme)ric Fluid Weight with Glass M

11/3 -1/2 log w) t t/h Carnegie Mellon 75 Polymeric Low Polyme)ric Mole)cular Fluid Fluidwith e)ight with M >> MGlass M

J(t) G(t) l J o G log h'( an og ltw) )d(w) lw t== or =1/3 G''( 11/3 -1/2 log w) t t/h /h N N Carnegie Mellon 76

Carnegie Mellon 77 Peak I with L(l) line)arin l1/3 be)fore) the)pe)akde)cre)ase)s sharplyto ze)ro. The behavior ascribed to peakI, first reported by Andrade, is seenin a variety of materials, suchas metals, ceramics, crystall ine and glassy polymers and small organic molecules; the decrease of L() to zero being evident in examples of the latter. The area under peak I provides the contribution JA Jo to the total recoverable compliance Js. It seems lik ely that the mechanism giving rise to peak I may be distinctly different from the largely entropic origins of peaks II and III described inthe following. Carnegie Mellon 78 Carnegie Mellon 79 Peak II that increases in peak area with increasing M until reaching a certain level, beyond which the peak is invariant with increasing M, both in area and position in l

Peak II is ascribed to Rouse-like modes of motion, either fl uidlike for low molecular weight in therangefor which the area increases with M, or pseudo-solid like (on the relevant time scale) in the range of M after peak II I develops. For low molecular weight, the Rouse model gives the area of peak II as Js (JA + Jo) = (2M/5RT). For the pseudo-solid like behavior, obtaining when peakIII has developed, refl ecting the eff ects of intermolecular entanglement, the area of peak II becomes invariant with M and given by JN (JA + Jo) = (M e/RT). Carnegie Mellon 80 Carnegie Mellon 81 Peak III that develops as peak II area ceases to increase with increasing fM, with pe)ak III de)ve)loping anre)a a invariant with fM, and a m axim um at lMAX that mov e)s to rla ge)r l aslMAX (fM/Mc)3.4 for fM >Mc The area under peak III, also invariant with M, ascribed to the effects of chain entanglements is given by

2+s Js (JN + JA + Jo) = (kM e/ RT), where k is in therange6-8 in most cases, and s 2( 1)/(3 2) 0 to 1/4with = ln RG2 /ln M Overall, Js (JA + Jo) = (2M/5RT)[1 + ( 1+s 1/ M/kM c) ] 2.2 2.0 1.8 2 Log (J f )

+ Cst. 1.6 1.4 1.2 1.0 Carnegie Mellon 1.0 1.5 2.0 2.5 ~ Log (X) 3.0 3.5 82

Carnegie Mellon Introduction Rheological methods Linear elastic parameters Linear viscoelastic functions Several viscoelastic experiments Relations among linear viscoelastic functions

Examples of linear viscoelastic functions Time-temperature equivalence (Thermo-rheological simplicity) The glass transition temperature The viscosity Effects of polydispersity Network formation Isochronal Behavior Examples from the literature

83 Consider the following reduced expressions: [J(t/tc) o]/J J s = [R (t/tc) o]/ J Js + /h t Js [J(t/tc) o]/ J Js = [R (t/tc) o]/ J Js + /tt c tc = Jsh'(0) (= Jsh) The) "tim e)te)mp e)rature) e)quiva le)nce)"approxim ation: [J(t/tc) oJ ]/Js is a singl e)-valu e)d fun ction fot/tc ov e)ra range) of te)mpe) rature). Although rare)ly, fie)ve)r,truly accurate) for l ale)mpe)

t rature), itis ne)ve)r-the)-le)ss a e)fu us land wide)ly us e)d app roximationfor s ue) with m ate)rialse)xhibiting no phase) transition ove)r th e) te)mpe) rature) range) ofinte)re)st. Carnegie Mellon 84 Since tc may notbe) known ov e)r the) range) of te)mpe) rature) of inte)re)st, it is o fte)n use)fulto "r e)duce)"data to a com mon re)fe)re)nce) te)mpe) rature) TR EF. Fo rmally , this m aybe) accomplishe)d with [J(t) Jo]/bTJs(TR EF) = [R (t) Jo]/bTJs(TR EF) + t/hbTJs(TR EF) [J(t) Jo]/bTJs(TR EF) = [R (t) Jo]/bTJs(TR EF) + t/hTbTtc(TR EF) [J(t/aT) Jo]/bT = [R (t/aT) Jo]/bT + t/

hTbTh(TR EF) [J(t/aT) Jo]/bT = [R (t/aT) Jo]/bT + t/ aTh(TR EF) bT = b(T, TR EF) =Js(T)/Js(TR EF) hT = h(T, TR EF) =h'(0)[T]/h'(0)[TR EF] { =h(T)/h(TR EF)} aT = bT hT Carnegie Mellon 85 Carnegie Mellon 86 Carnegie Mellon 87 Carnegie Mellon

88 Carnegie Mellon 89 Carnegie Mellon 90 Carnegie Mellon 91 Carnegie Mellon 92 Carnegie Mellon Introduction Rheological methods

Linear elastic parameters Linear viscoelastic functions Several viscoelastic experiments Relations among linear viscoelastic functions Examples of linear viscoelastic functions Time-temperature equivalence (Thermo-rheological simplicity) The glass transition temperature

The viscosity Effects of polydispersity Network formation Isochronal Behavior Examples from the literature 93 Specific Volume Tg Temperature A schematic v-T diagram for a typical noncrystalline polymeric material.

Carnegie Mellon 94 A Free Volume Model: (vf)i = (v vo)i at a certain position ri, v = (specific) volume vf = free volume vo = occupied volume Carnegie Mellon 95 Carnegie Mellon 96 The glass transition temperature Tg Tg depends on both intramolecular conformation and intermolecular interactions. Various Models/Treatments: Iso Free Volume: f(Tg) = constant

Iso Viscous: (Tg) = constant Iso Entropic: S(Tg) = constant None of theseare fully satisfactory are free of arbitrary assumptions, andall contain parameters that can not be independently evaluated. The free volume andentropic models provide similar expectations re the dependence of T g on chain length and diluent. Carnegie Mellon 97 120 PMMA

100 T (C) 80 g 60 40 0 0.2 0.4 0.6 0.8 1 Syndiotactic fraction Carnegie Mellon 98 Estimation of Tg and Tm via Group Contributions Tg

M-1Yg,i Tm M-1 Ym ,i The Yx,i represent molar group contributions to the relevant property Higher order approximations are available for both cases D. W. van Krevelen, Properties of polymers : their correlation with chemical structure, their numerical estimation and prediction from additive group contributions, 3rd Ed., Elsevier; Amsterdam ; New York, 1990. Carnegie Mellon 99 Group contributions: Y g (Kg/mol) Group Yg,i

Group Yg,i Group Yg,i 1. Polyisobutylene only Carnegie Mellon 100 Group contributions: Ym Group Carnegie Mellon Ym,i Group (Kg/mol) Ym,i Group

Y m,i 101 Carnegie Mellon 102 2.4 2.2 boyer krevelen avg 2.0 krevelen calc g T m/T1.8 1.6 1.4 1.2 200 300

400 500 600 700 Tm /K D.W. Van Krevelen, op cit R. F. Boyer, Rubber Reviews 36:1303-421 Carnegie Mellon 103 Both free volume and entropic models give results that may be cast in the forms: Tg(M) Tg (){1 + kM/Mn} w 1 1 w w + R1 1( - w)

= + R Tg(w, M) Tg;IL Tg(Mn) BothKM and R are) mode) lspe)cific param e)te)rs, be)st e)valua te)d e)xpe)rim e)ntally. For e)xample), in the) fre)e) volume) m ode)l, KM and R arise) fr om the) e)xtra fre)e) volume) provide)d bychaine)ndsand dilue)nt, re)spe)ctive)ly : typically, Ris in the) range) 0.5 to 1.5. Note), that fiTg;IL > Tg(Mn), the)n Tg(w, Mn) is incre)ase)d bythe) dilue)nt. [G. C. Be)rry J. Phys . Che)m. 70:1194-8 (1966)] Carnegie Mellon 104

120 100 Free Radical p(Syndio) ~ 0.76 T (C) g 80 p(Syndio) ~ 0.50 60 0 1 2 3 4 5 104 /Mn Carnegie Mellon

105 Carnegie Mellon 106 Carnegie Mellon Introduction Rheological methods Linear elastic parameters Linear viscoelastic functions Several viscoelastic experiments

Relations among linear viscoelastic functions Examples of linear viscoelastic functions Time-temperature equivalence (Thermo-rheological simplicity) The glass transition temperature The viscosity Effects of polydispersity Network formation

Isochronal Behavior Examples from the literature 107 h(T) hLOC(T) F(largescale structure, T) LOC(T) F(largescale structure) "Arrheniu s" form: LOC(T) exp[W/T] if T > (1.5-2)Tg For melts of crystalline polym ers, Tm > (1.5-2)Tg, permittinguse of this simpleform. "Vogel-Fulcher"form: For amorphouspolymers with 0 (T Tg)/K < 200: LOC(T) exp[C/(T To)] Carnegie Mellon if T < (1.5-2)Tg 108

Carnegie Mellon 109 The temperature dependence of the viscosity: h(T) hLOC(T) F(largescale structure, T) LOC(T) F(largescale structure) For amorphouspolymers with 0 (T Tg)/K < 200: LOC(T) exp[C/(T To)] if T < (1.5-2)Tg "WLF form": LOC(T)/LOC(TREF) = exp[C/(T To) C/(TREF To)] C(T TREF) = exp REF(T TREF +REF) withC andTo being con stants, and REF =TREF To. Carnegie Mellon 110

If TREF = Tg then K (T Tg) hLOC(T)/hLOC(Tg) = e)xp T Tg + where =Tg To andK = C/. For many polym ers: K = 2300 K and = 57.5 K These parameters may be in terpreted in erm t s of the "free-volum e" model Carnegie Mellon 111 Carnegie Mellon 112

Viscosity of Polymers and Their Solutions h(M, c, T) hLOC(T) F(M, c,T) Dilute solutions LOC(T) Solvent(T) F(M, c,T) 1 +[]c + [] = NNAKRG2RH/M G. C. Berry J. Rheology 40:1129-54 (1996) Carnegie Mellon 113 F(M, c, T) 1 + [h]c + [h] = pNAKR G2 R H(/M phe)ricalParticle)s R = R H( = (5/3)1/2R G ; K = 50/9 [h]c = (5/2)f Carnegie Mellon 114

F(M, c, T) 1 + [h]c + [h] = pNAKR G2 R H(/M Fle)xible) Chainine)ar L Polyme)rs R G2 = (L/3)a2; a the) cha in e)xpansion factor the) pe) rsiste)nce) le)ngth Carnegie Mellon 115 F(M, c, T) 1 + [h]c + [h] = pNAKR G2 R H(/M Fle)xible) ChainLine)ar P olyme)rs R G2 = (L/3)a2; a th e) chain e)xpansionfactor th e) pe)rsiste)nce) l e)ngth H(igh M: 3R H(/2 R G L1/2;

K 10/3 ML[h] = pNA(20/9)(/3)3/2a3L1/2 = F'(/3)3/2a3L1/2 Carnegie Mellon 116 F(M, c, T) 1 + [h]c + [h] = pNAKR G2 R H(/M Fle)xible) ChainLine)ar P olyme)rs R G2 = (L/3)a2; a th e) chain e)xpansionfactor th e) pe)rsiste)nce) l e)ngth H(igh M: 3R H(/2 R G L1/2; K 10/3 ML[h] = pNA(20/9)(/3)3/2a3L1/2 = F'(/3)3/2a3L1/2 Low M:

R H( L; ML[h] = pNA(/3)L Carnegie Mellon K 1 ( e)bye)) 117 Flexible Chain Branched Polymers ML[h] = pNAKR G2 R H(/L g =R G2 /(R G2 )LIN; calculate)d a =1 H(igh M: h = R H(/(R H()LIN; h g1/2 K KLINf(g, shape)) [h] = f(g, shape))g3/2[h] LIN Carnegie Mellon 118 Flexible Chain Branched Polymers ML[h] = pNAKR G2 R H(/L g =R G2 /(R G2 )LIN; calculate)d a =1

H(igh M: h = R H(/(R H()LIN; h g1/2 K KLINf(g, shape)) [h] = f(g, shape))g3/2[h] LIN tar: [h] = g1/2[h] LIN Comb: [h] = g3/2[h] LIN Random : [h] = g[h] LIN Carnegie Mellon 119 Flexible Chain Branched Polymers ML[h] = pNAKR G2 R H(/L g =R G2 /(R G2 )LIN; calculate)d a = 1 H(igh M:

h = R H(/(R H()LIN; h g1/2 K KLINf(g, shape)) [h] = f(g, shape))g3/2[h] LIN tar: [h] = g1/2[h] LIN Com b: [h] = g3/2[h] LIN R andom: [h] = g[h] LIN Low M: [h] = pNAKR G2 R H(/LML R H( L; Carnegie Mellon K 1 120 [h] = g[h] LIN

Carnegie Mellon 121 Viscosity of Polymers and Their Solutions h(M, c, T) hLOC(T) F(M, c, T) Concentrated solutions and undiluted linear flexible chain polymers LOC(T) LOC(Tg)exp{ K(T Tg)/(T Tg +)} F(M, c, T) 1 + [] (c)c Low M (Rouse behavior; = 1): ~ ~ F(M, c, T) 1 + X X ~ X = [] (c)c; range) Carnegie Mellon M L[] (c) = a modified Fox parameter NA (/3)L;

([] (c) independent of c in this 122 Carnegie Mellon 123 High M (Entanglement regime) ~ ~~ ~ ~~ F(M, c,T) 1 +XE(X/X c ) XE(X/X c ) ~~ ~~ E(X/X c ) = {1 + (X/X c )4.8}1/2 ~ X c = NNA(/3)rMc 100 for many polym e)rs ~ Mc = X c/pNA(/3)r 100/pNA(/3)r Carnegie Mellon 124 Carnegie Mellon 125

The dependence of Tg on the diluent concentration must be considered for polymer solutions: K (T Tg) hLOC(T)/hLOC(Tg) = e)xp T T + g where =Tg To andK =C/. For manypolymers: K =2300K and =57.5 K is approximately independen t of thepolym erconcent ration Carnegie Mellon 126 400 Polystyrene/Dibenzyl ether 300 Tg

Temperature/K 200 To 100 Tg To 0 0 0.2 0.4 0.6 0.8 1 Volume Fraction Polymer G. C. Berry and T. G Fox Adv. Polym. Sci. 5:261-357 (1968) Carnegie Mellon 127

1.0 5 0.75 4 0.50 3 2 log( h 0.25 /Pas)s) 1 0.125 0 -1

-2 3 4 5 log( f 6 M ) w Carnegie Mellon 128 Viscosity of Polymers and Their Solutions h(M, c, T) LOC(T) F(M, c, T) Bra nc hed C hain Po lymer s (Concen tra te d or und iluted) LOC(T) [ LOC(T)]LIN; Rare exceptions to this known F(M, c, T) 1 + [](c)c ML[](c) =

NA(/3) gL ~ ~ ~ F(M, c, T) 1 + XE(X/Xc ); ~ X = [](c)c ~ ~ ~ ~ E(X/Xc ) = {1 + B(g, MBR/Mc)(X/Xc )4.8}1/2 B(g, MBR/Mc) 1 unless the branch molecular MBR > Mc ~ Xc = NA(/3)Mc 100 for many polymers Carnegie Mellon 129 Carnegie Mellon 130 Carnegie Mellon 131 Moderately Concentrated Solutions

Carnegie Mellon 132 Viscosity of Polymers and Their Solutions h(M, c, T) LOC(T) F(M, c, T) Modera te ly Con ce ntra te d S olutions LOC(T) [ LOC(T)]1c -=0 [ LOC(T)]c = ; = c/ F(M, c, T) 1 + [](c)c ML[](c) = NA(/3) (c)2(RH(c)/L)L ~ ~ ~ F(M, c, T) 1 + H(c)XE(X/Xc ); ~ X = [](c)c ~ ~ ~ ~ E(X/Xc ) = {1 + (X/Xc )4.8}1/2 ~ Xc = NA(/3)Mc 100 for many polymers [G. C. Berry J. Rheology 40:1129-54 (1996)] Carnegie Mellon

133 Carnegie Mellon Introduction Rheological methods Linear elastic parameters Linear viscoelastic functions Several viscoelastic experiments Relations among linear viscoelastic functions

Examples of linear viscoelastic functions Time-temperature equivalence (Thermo-rheological simplicity) The glass transition temperature The viscosity Effects of polydispersity Network formation Isochronal Behavior

Examples from the literature 134 Molecular Weight Polydispersity LOC(T) scales with Mn through Tg / LOC(T) scales with Mw, except perhaps for unusual distributions Peak I in L() is essentially unaffected by molecular weight dispersion Peak II in L() may comprise two pieces: i) an area proportional to LMzMz+1/Mw, with the averages calculated for chains with M < Me at volume fraction L, and ii) an area proportional to (1 L)Me for chains with M > Mc at volume fraction 1 L Peak III in L() has an area proportional to (1 L)-2(Mz/Mw)2.5 The maxima for peaks II and III separate in as (1 L)Mw Carnegie Mellon 135 Theoretical treatments are usually cast in terms of G(t), often in the form: G(t) = { w iG i(t)n} 1/n i G i(t) = she)ar modulus or f chains with Mi

at we)ight fra ction wi For e)xample): n =1 in the)"r e)ptation mode)l n =1/2 in the) "double)-re)ptation"mode) l Carnegie Mellon 136 Carnegie Mellon 137 Theoretical treatments are usually cast in terms of G(t), often in the form: n 1/n G(t) = { w iG i(t) } i G i(t) = she)ar modulus or f chains with Mi at we)ight fra ction wi For e)xample): n =1 in the)"r e)ptation mode)l

n =1/2 in the) "double)-re)ptation"mode) l The) e)ffe)cts of incre)ase)d dispe)rsityof m ole)cula r spe)cie)s si usually mo st prom ine)nt inPe)ak III in L(l), followe)d bye)ffe)cts in e)ak P II in L(l). This is se)e)n in L(l) for a polyme)r unde)rgoing crosslinking to form a branche)dpoly m e)r, le)ading to a ne)twork poly m e)r Carnegie Mellon 138 Carnegie Mellon Introduction Rheological methods Linear elastic parameters

Linear viscoelastic functions Several viscoelastic experiments Relations among linear viscoelastic functions Examples of linear viscoelastic functions Time-temperature equivalence (Thermo-rheological simplicity) The glass transition temperature The viscosity

Effects of polydispersity Network formation Isochronal Behavior Examples from the literature 139 Carnegie Mellon 140 Carnegie Mellon 141 Carnegie Mellon 142

Carnegie Mellon 143 Carnegie Mellon 144 Carnegie Mellon 145 Power-law behavior G(t) = [Go Ge]f(t) +G e) J(t) = Jo + y(t) +t/h y(t) = (Js Jo)[1 a(t)] Suppose that for all t (note, this involves permissible, but peculiar behavior for large t): (t) =(t/) Withthisexpression,and 1/ = 0: [J'() Jo]/Jo =()cos(/2)()- J"()/Jo = ()sin(/2) ()- Use of the con volution integral relating J(t) and G(t) gives (t) =E(-k(t/)) withGe = 0 and1/ = 0,where k = () and

Carnegie Mellon E(x) = (nxn+ 1) : n=0 The Mittag-Leffler function 146 For small , G(t) Go{1 + (t/l)} 1 For any , forarg l e) t/l G(t) G osin(p)/p(t/l) G(t)J(t) sin(p)/p < 1 (G'(w) G e))/(G o G e)) G(1-m) sin[(1-m)p/2] (wl)m G"(w)/(G o G e)) G(1-m) cos[(1-m)p/2] (wl)m [J'(w) Jo]/Jo = G()cos(p/2) (wl)- J"(w)/Jo = G()sin(p/2) (wl)- Carnegie Mellon 147 Bounded power-law behavior for f(t) might be) obt aine)d in th

e) form f(t) = 1; for ttN = (tN/t)m; for tN < t t1, with0 < < 1 = (tq/t)m ; for t >t1, withm > 1 whe)re) tq =t1(tN/t1)m/m .The)n, G '(w) G e) w2 and G "(w) w forw << 1/t1; G '(w) = G o and G "(w) = 0 fo rw >> 1/tN; (G'(w) G e))/(G o G e)) G(1-m) sin[(1-m)p/2] (wtN)m G"(w)/(G o G e)) G(1-m) cos[(1-m)p/2] (wtN)m for th e) inte)rval 1/ t1 < w < 1/tN. Carnegie Mellon 148 An alternative relation that also exhibits partial power-law behavior is given by: f(t)

N N = (t1/ti) n/m e)xp(t/ti)/ (t1/ti) n/m i =1 i = 1 whe)re) ti =t1/im ; m =2 and n=0 in the) R ouse) mod e)l. For th e) inte)rm e)diate) inte)rval /1t1 < w < 1/tN, (G'(w) G e))/(G o G e)) { p/2m sin[(1-m)p/2]} (wt1)m G"(w)/(G o G e)) { p/2m sin[(2-m)p/2]} (wt1)m whe)r e) Mellon =(1 + n)/m Carnegie ( = 1/2, fo r th e) R ouse) mod e)l).

149 Carnegie Mellon 150 Carnegie Mellon 151 Carnegie Mellon 152 Carnegie Mellon Introduction Rheological methods Linear elastic parameters

Linear viscoelastic functions Several viscoelastic experiments Relations among linear viscoelastic functions Examples of linear viscoelastic functions Time-temperature equivalence (Thermo-rheological simplicity) The glass transition temperature The viscosity

Effects of polydispersity Network formation Isochronal Behavior Examples from the literature 153 ISOCHRONAL BEHAVIOR In some cases, the temperature is scanned while the dynamic properties are determined at fixed frequency; such experiments might typically be reported as G'(w;T) and tan d(w; T) or '(;T) versus T, depending on the application. Ins ofar a s G'( c (T)) a nd G "( c (T)) a s func tions of c (T) a re inde pen dent of T, the isoc hrona l plots a re s e en t o be mapp ings in which c (T) inc reases with dec reas ing tem pera ture with:

K c (T) ex p T - (Tg - ) For a reference temperature equal to the glass temperature Tg, so that a = c(T)/c(Tg): K T - Tg ln a = ln 2 1 +(T - Tg) k1 + k2(T - Tg) + with the linear approximation valid for (T - Tg) <<; k1 = ln and Carnegie Mellon k2 =K /2. 154 1

Log G' /Go T - T g = 0 0 Log tan d -1 -2 -3 -4 Log G' /G o and

-2 0 2 4 d Log tan log (a w ) T 1 -1 w

= 1 s 0 Log tan d -1 Log G' /Go -2 -3 Carnegie Mellon -10 10 0 T - T g 20

155 Carnegie Mellon 156 Carnegie Mellon 157 Carnegie Mellon 158 Carnegie Mellon 159 Carnegie Mellon Introduction Rheological methods

Linear elastic parameters Linear viscoelastic functions Several viscoelastic experiments Relations among linear viscoelastic functions Examples of linear viscoelastic functions Time-temperature equivalence (Thermo-rheological simplicity) The glass transition temperature

The viscosity Effects of polydispersity Network formation Isochronal Behavior Examples from the literature 160 Examples from the literature Carnegie Mellon Branched and linear metallocene polyolefins

Colloidal dispersions Wormlike Micelles Deformation of rigid materials Nonlinear shear behavior Linear and nonlinear bulk properties 161 0120og G''( l-1 5 4 3 2 1 log

Unmodified G'( )) Linear w Metallocene polyethylenes Claus Gabriel and Helmut Mnstedt Rheol. Acta 38: 393-403 (1999) Carnegie Mellon 162 1og G''( 2 1 0 -1 lUnmodified 5 4 3 2 log Modified G'( )) Branched Linear 1 0

w Carnegie Mellon Metallocene polyethylenes Claus Gabriel and Helmut Mnstedt Rheol. Acta 38: 393-403 (1999) 163 01log -1 M 2345l log Modifie)d og odifie)d G'( w) w) Branche)d 12 wG''( Carnegie Mellon Metallocene polyethylenes Claus Gabriel and Helmut Mnstedt Rheol. Acta 38: 393-403 (1999) 164

Unmodifie)d Modifie)d M Line)ar 12 -4odifie)d 2 1 0 -1 -2 log -5 log 4 3 w h'( J'( w)Branche)d Carnegie Mellon Metallocene polyethylenes Claus Gabriel and Helmut Mnstedt Rheol. Acta 38: 393-403 (1999) 165 Unmodifie)d Modifie)d

Branche)d Line)ar 21 -4og J'(w) 6 5 4 3 2 log -5 log l 0 -1 h'(w)/h'( w h'(0) 0) Carnegie Mellon Metallocene polyethylenes Claus Gabriel and Helmut Mnstedt Rheol. Acta 38: 393-403 (1999) 166 Unmodifie)d Branche)d

Line)ar 21M1 23456-log -5 log 0Modifie)d -1 4 h'( wJ'( w)/b w)/h'( h'(0) bU 0) M2 log h'(0) 3.28 3.68 4.00 log b -0.7 0 Carnegie Mellon 0

167 Carnegie Mellon 168 From creep/recovery Carnegie Mellon 169 Carnegie Mellon 170 Carnegie Mellon 171 Examples from the literature Carnegie Mellon Branched and linear metallocene polyolefins

Colloidal dispersions Wormlike Micelles Deformation of rigid materials Nonlinear shear behavior Linear and nonlinear bulk properties 172 Colloidal dispersions: Linear and nonlinear viscoelastic behavior. Dilute dispersion of spheres interacting via a hard-core potential: h = hLOC{1 + (5/2)f + k'(5/2) f + }

2 f (5/2)f 2 = volum e) fraction = c/r = [h]c hLOC hsolv. k' 1.0 Carnegie Mellon 173 Concentrated dispersion of hard-core spheres: Empirical relations: h hLOC{1 f/n } 5n /2 h

hLOC{1 (5/2)f[1 f/n ] 5n k'/2} 1 1 2 2 de)signe)d to force) agre)e)m e)nt with the) v iriale)xpansion at le)ast to orde)r f andf 2, re)spe)ctive)ly, n1 = 5/8 to give) k'1.0 n1 = f m ax 0.64

The)ore)ticalre)lations: h = hLOC{ 1 + (5/2)f + k'[y1(f) +y2(f)](5/2)2f 2} y1(f): hyd rodynami cs y2(f): the)rmodynami cs y1(0) +y2(0) =1 U: y1(f) (4/5)(1 f/f m ax) 2 Carnegie Mellon y2(f) (1/5)(1 f/f m ax) (se)mi- e)m pircial) 174 Carnegie Mellon 175 Concentrated dispersion of hard-core spheres: Linear Viscoelastic Response:

'() = '(0) for small , as expected, but also show a plateau'() '(L ) for a regime at an intermediate rangeof L , before decreasing to zero with increasing . '(L ) is estimated with 2() = 0, reflecting the suppression of thermodynamic interactionsat high G1R3/kT 2 0() for spheres of radius R G'(L ) G1; 0() 0.78('(L )/ solv)g(2, ) g(2, ) is the radial distribution at the contact conditi on r/R = 2 Theory: g(2, ) = (1 /2)2/(1 )3 for < 0.5 and g(2, ) = (6/5)(1 / max) for 0.5 Carnegie Mellon 176 Carnegie Mellon

177 Concentrated dispersion of hard-core spheres: Linear Viscoelastic Response: Theory: '(0) = LOC{ 1 + (5/2) + k'[1() + 2()](5/2)2 2} '(L ) = LOC{ 1 + (5/2) + k'[1()](5/2)2 2} J'EFF() -1/2 for a range of < L

J'EFF(L ) 1/G'(L ) 1/G1 R3/kT 20() Carnegie Mellon 178 Carnegie Mellon 179 Concentrated dispersion of interacting spheres: Van der Waals interactions Electrostatic interactions among charged spheres Interactions among spheres and a dissolved polymer True or apparent yield behavior may obtain Carnegie Mellon 180 Carnegie Mellon

170 nm beads (0.05 to 0.2 volume fraction), in 15% polystyrene solution D. Meitz, L. Yen, G. C. Berry and H. Markovitz J. Rheol. 32:309-51 (1988) 181 Examples from the literature Carnegie Mellon Branched and linear metallocene polyolefins Colloidal dispersions Wormlike Micelles Deformation of rigid materials Nonlinear shear behavior

Linear and nonlinear bulk properties 182 Wormlike micelles Certain amphillic molecules organize to form curvilinear cylinders, or wormlike micelles. For example, in an aqueous medium, the amphiphile might organize with its hydrophobic parts aggregated in the interior of the cylinder, and its hydrophopic pieces arranged on the "surface" of the cylinder The micelle structure will exhibit a lifetime trupture) for rupture) of its com pone) nts If trupture) is e)lss h t an alonge)st rhe)ologicaltim e) constant trhe)ol the) intact wormlike) mi ce)ll e) would e)xh ibit,the)n the) rupture) dynamics maydomi nate) the) obse) rve) d rhe)o logicalbe)havior, The) chain m ay re)spond to a de)form ation by mi ce) ll ar dynamics similar to those) for a structure) without rupture), abe)tte)d by the) rupture) proce) ss. ith one) mode) l,this approxim ate) s Maxwe) llbe) havior with a tim e) constant 1/2 te)ffe)ctive) (trupture) trupture) ) Carnegie Mellon 183

Cetyl triethylammonium tosylate - CTA+ T hydrophobic + hydrophilic + + + + + ++ - - + + + + + + +

+ + + + + + + micelles grow 10 nm micellar network Schematic courtesy Dr. Lynn M. Walker Carnegie Mellon 184 In an extreme case, the system might approximate behavior for the Maxwell model, with a single relaxation time te)ffe)ctive) so that J(t) = Js + t/h; with Jsh = te)ffe)ctive) G( t) = (1/Js)e) xp(-t/te)ffe)ctive) ) ith this simple) m ode) l,

J'(w) = Js 2 h'(w) = (1/Js)/[1 + (wte)ffe)ctive) ) ] Carnegie Mellon 185 0 10% -1 h '(w) /h p J'( w )/J p T (C)C) 30 These data reveal several deviations

from simple Maxwell behavior, including: 35 -2 The rate of decrease of h'(w) with increasing w for larger w, to the extent of an increase in h'(w) with increasing w for the data on the less concentrated Sample 40 -3 p 3 Calculate)d 2 h '(w) /h or log J'(

w )/J The increase of J'(w) above the imputed Js for smaller w for the data on the more concentrated sample p 1 0 20% -2 s log J /Pa -1 -3 4.5 -2

than the experimental range, and that J p solve)nt 4.0 log It may be likely that these samples exhibit solid-like behavior with a Je at smaller w is truly Js 10% h/h 3.5 20% -3 30 35 40 Temperature (C) -4 -3

-2 0 -1 log w h J p Carnegie Mellon p 1 2 The relatively constant J'(w) is expected with the Maxwell model, but this may be fortuitous J. F. A. Soltero and J. E. Puig Langmuir 12: 141-8 (1996) 186

Examples from the literature Carnegie Mellon Branched and linear metallocene polyolefins Colloidal dispersions Wormlike Micelles Deformation of rigid materials Nonlinear shear behavior Linear and nonlinear bulk properties

187 Deformation of Rigid Materials Creep and Recovery in Tension Creep for 0 t Te e)(t) = so(t) = so[ R (t) + NR (t)] Re) cove)ry for J =t Te) > 0 e)(J, Te)) = so[ R (J + Te)) R (J) + NR (Te))] e)R (J, Te)) = e)(Te)) e)(J, Te)) = so{ R (Te)) R (J + Te)) + R (J)} Carnegie Mellon 188 G. C. Berry J. Polym. Sci.: Polym. Phys. Ed. 14:451-78 (1976) Carnegie Mellon 189 Carnegie Mellon 190 Andrade Creep (with DNR(t) 0) A nonrecoverable logarithmic creep is frequently observed under larger stress:

DNR(t) DL ln(1 + t/DL) (a) t/DL <<1 t (b) 3 -1 D(t)/MPa 2 1 0 0 5 10 1/3

(t/sec) Carnegie Mellon 15 20 0 5 10 15 20 25 1/3 /sec) 1/3 (/sec)1/3 or [ + T )/sec] ( 191 Examples from the literature

Carnegie Mellon Branched and linear metallocene polyolefins Colloidal dispersions Wormlike Micelles Deformation of rigid materials Nonlinear shear behavior Linear and nonlinear bulk properties 192 An "Incompressible" Isotropic Elastic Material

Suppose K >> G, then for infinitisimal strains d Sij = 2 G {e)ij 3ije)aa} dijP More) ge)ne)rall y, for finite) strains: -1 ij = 1 Bij + 2Bij dijP i = i (I B;1, IB;2) IB;i For simp le) e)xte)nsion: f/ A 2(l2 l-1)( 1 + 2/ l) For simp le) she)ar: 12 = 2( 1 + 2) g =G g Carnegie Mellon 11 33 = 2 1 g2 ; 22 33 = 2 2 g2 193 An expansion of the strain energy function gives the MooneyRivlin Equation for small deformations:

W C1 (IB;1 3) +C2 (IB;2 3) W1 = C1 and W2 = C2 For the original Kinetic Theory of Rubber Elasticity the contributions to C1 are entropic in origin, and.: 2C1 = nEkT = rR T/ MXL 2C2 = 0 nE = Numb e)r of chains unde)r stre)ss M XL = Mole)cular w e)ight of chains be)tw e)e)n crosslinks The) pre)ce)di ng e)stim ate)s for C1 and C2 are) not accurate), and have) be)e) n m odifie)d in m ore) m ode) rn Carnegie Mellon the)orie)s, e).g., the)se) give) C 2 > 0. 194 An "Incompressible" Viscoelastic Material Suppose K(t) >> G (t), then for infinitisimal strains t Sij(t) =

2 G (t e)ij(s) s) s dije)aa(s) 3 s ds dijP - e)ve)ralre)lations are) propose)d for inite) f strains, includ ing that d ue) toBe)rnste)in, Ke)ars le)y and Zapas:: t ij(t) = U 2 I

B;1 U -1 (t)ij(s) (t) B IB;2 B ij(s) ds dijP - Carnegie Mellon 195 An "Incompressible" Viscoelastic Material Suppose K(t) >> G (t), then for infinitisimal strains t Sij(t) = 2 G (t

e)ij(s) s) s dije)aa(s) 3 s ds dijP - e)ve)ralre)lations are) propose)d for inite) f strains, includ ing that d ue) toBe)rnste)in, Ke)ars le)y and Zapas:: t ij(t) = U 2 I B;1

U -1 (t)ij(s) (t) B IB;2 B ij(s) ds dijP - Carnegie Mellon 196 Nonlinear Responsein SimpleShear fora Fluid (In the appro ximation with t >>tR ) he)ar t re)sss(t) = 12(t): G(u) s(t) = [(t,u)] F1[(t,u)] u du 0

t (u) (t) = G(t u) u M1[(t,u)] du - (t,u) = (t) (u) Carnegie Mellon F1() M1[(t,u)] = l n F1() = F1()1 + l n 197

Nonlinear Responsein SimpleShear fora Fluid (In the appro ximation with t >>tR ) FirstNormal Stress Difference (1)(t) = 11(t) 22(t) : G(u) (1)(t) = [(t,u)]2 F1[(t,u)] u du 0 t (u) (1) (t) = G(t u) u M2[(t,u)] du - 2 F1() M2[(t,u)] = Carnegie Mellon

l n F1() = F1()2 + l n 198 Carnegie Mellon 199 Carnegie Mellon 200 Response to a Ramp Deformation t g(t) = t >0 Stress Growth:

(t) (1)(t) = = 2 t t 0 G(s) s) l n F1( ds F1( s) 1 + s

ln 0 sG(s) s)2 + F1( Carnegie Mellon l n F1(s) ds s l n 201 Steady-State Flow Viscosity s)

lim s(t) = s(g) t >> tc s) = s(g)/ s) gs) h(g) s) = h(0) = h lim h(g) g=0 s) = h(0) H((tcgs)/g'') h(g) G(u)M [ g u]du 1 0 s) s)/g'') = H((tcg Carnegie Mellon

G(u)du 0 202 Steady-StateFlow First-Normal St ress Difference s) lim n(1)(t) = n(1) ( g) t >> tc s) = n(1)(g) s) /2{s (g)} s) 2 N(1)(g) s) = Js lim N(1)(g) g=0 s) = Js (tcg/g'' s) )

N(1)(g) N uG(u)M [ gu]du 2 0 s) s) ) = (tcg/g'' N Carnegie Mellon uG (u)du 0 -2 s)

0 G (u)M 1[ gu]du G(u)du 0 203 Steady-StateFlow Stead y-State Re coverable C ompliance s) lim gR (t,q) = gR (g) t;q >>tc s) = g (g)/s s) (g) s) R (g) R s) = Js lim R (g) g=0

s) = JsR (tcg/g'' s) ) R (g) s) ) = R e)sultof an ite)rative) calculation R (tcg/g'' involving G(t) and F1(g) Carnegie Mellon 204 Suppose G(t) = Gof ie)xp(t/ti); f i = 1 The)n, withthe) app roxim ate)F1(g) give)n above) h(gs) ) =G of i ti H((gs)ti/g '') H((gs)ti/g '') 1 ; s) [1 + (bgti/g '')e)] 2/e)

e) 6/5, b 1 By comparison, 1 h'(w) =G of i ti [1 + (wti)2] In both case)s, the) factors f i ti in the) te)rm s in the) su m mation are) we)ighte)d byfunctions th at de)cre)ase) te)rmbyte)rm withincre)asing gs) or w. Conse)que)ntly, th e)se) e)xpre)ssions e)xhibit the) Cox-Me)rz approximation: Carnegie Mellon h(gs)) h'(w = gs)) 205 Carnegie Mellon 206 0 log[ h(g )/

h(0 )] s) -1 0 0 s s s) log[J s) log[ (1) (g )/J

] s (g )/J ] -1 -1 -3 -2 -1 1 0 2 3

s) log( t g ) c Polyethylene K. Nakamura, C.-P. Wong and G. C. Berry J. Polym. Sci: Polym. Phys. Ed. 22:1119-48 (1984) Carnegie Mellon 207 0 0 -1 -1 log[ h

'( w )/ h(0 )] log[ h s) ( g )/ h(0 )] -2 -2 s s

0 0 s) log[J'( w )/J ] log[ (g )/J ] (1) -1 -1

-2 -1 1 0 2 3 s) log( t g ) c Linear and nonlinear behavior for a polymer with a relatively narrow MWD Carnegie Mellon 208

Examples from the literature Carnegie Mellon Branched and linear metallocene polyolefins Colloidal dispersions Wormlike Micelles Deformation of rigid materials Nonlinear shear behavior Linear and nonlinear bulk properties

209 Carnegie Mellon 210 Carnegie Mellon D. J. Plazek and G. C. Berry , in Glass: Science and Technology, Vol. 3 Viscosity and Relaxation, ed. D. R. Uhlmann and N.J. Kreidl, Academic Press (1986). 211 An Inherent Nonlinearity in Response B(t) = B(0) +B b(t) b(t) = ^ b(t/ tk) But tk = tk(V ,T) An atte)mp t to account for thise)ffe)ct make)s use) foan ma te)rialtime) constant ave)rage)d ove)r the) tim e) inte)rval of inte)re)st: 1 tk1(t2 ,t1) = (t - t ) 2 1 V( t) V( 0)

= V( 0) t - t2 t1 tk1(u) du P(s) B[(t s) tk1(t ,s)] s d s Fre)qu e)ntly, Carnegie Mellon B(t) = BA{1 + (t/ A)1/3}; t < 212 D. J. Plazek and G. C. Berry , in Glass: Science and Technology, Vol. 3 Viscosity and Relaxation, ed. D. R. Uhlmann and N.J. Kreidl, Academic Press

(1986). Carnegie Mellon 213 Carnegie Mellon D. J. Plazek and G. C. Berry , in Glass: Science and Technology, Vol. 3 Viscosity and Relaxation, ed. D. R. Uhlmann and N.J. Kreidl, Academic Press (1986). 214 Carnegie Mellon 215

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