Polymer Physics
90 CHAPTER 5 POLYMER PHYSICS Here T'I is the relaxation time of the elastic d umbbell The graphs of equations
5.34 and 5.36 are displayed in figure 5 . 1 2. Because G raessley's theory is based on entanglements of the polymer coils, it is a more realistic model than Williams' for semi-dilute polymer solutions. Therefore, we will use Graessley 's theory for working out the entanglement formation time from the measured power law index.
0.0 -0.2 � � Q -0.4 - C) � � -0.6 � Q - -0.8 - 1 .0 -0.5 log 111110 (Williams) log n (Willi am s) log 111110 (Graessley) log n 0.0 (Graessley) 0.5 log
(r
1:11 ) 1 .0 1 .5Figure 5.12: The viscosity is plotted vs. shear rate using equations 5.34 and 5.36. The shear rate dependence of the power law exponent using equation 5.37 is also shown for both theories.
A generalised , shear-rate dependent power law index can be defined as
. d log 7]
n(t) = 1 + -d l . ' (5.37) og t
Using equation 5.36, an analytical expression for n ( -y) can be obtained:
. 1
n (t) = 1 + (-YT'I) 2 . (5 .38)
This curve is also shown in figure 5 . 12 .
In chapter 7 w e will use equation 5 . 3 7 along with Graessley's finding for 7]( -YT'I) in equation 5.34 to obtain the temperature dependence of T'I from the measured power law indices.
Adam and Delsanti measured the longest viscoelastic relaxation time for PS in cyclohexane in the Newtonian regime [1 15] . They found that this parameter scales as M�pY , where x and y are numerical constants with the values of x :::::; 3.8
and y :::::; 2.8. The reptation model predicts x = y = 3.0 for theta solutions [61] .
5.3. R HEOLOG Y OF POLl?vIER SOL UTIONS
5 . 3 . 3 The Glass Transition
9 1
When a polymeric m aterial i s cooled from the liq uid o r r u bbery state, it becomes m uch stiffer as i t goes through a certain tern peratu re range. This stiffening is the result of one of two possible events: crystallisation or glass t ransition . For crystallisation to occur , the polymer molecules must be sufficiently regular along their length to allow formation of crystallisation lattices, and the cooling rate m ust be slow enough. When a polymer fails to crystallise, the amorphous, liquidlike struct u re of the polymer is retained , but the molec ular motion frozen , and the m aterial t u rns i nto a glass. Such a glass transition occu rs over a finite tem peratu re interval , but is still realised abruptly enough to merit the term transition. I t can be recognised by the change in many properties of the m aterial , such as an i ncrease i n viscosity or a change of the specific heat and thermal expansion .
The val ues o f the glass transition tem perature
Tg
reported for a certain poly mer often d iffer by as m uch as 1 0 20° C, because the glass transition occurs over a temperat u re range rather than at a single, sharply defined temperatu re, and be cause the observedTg
varies somewhat depending on the method of meas u rement used and the thermal h istory of the sample. For polystyrene,1�
h as a literatu re value of :373 K [68J . I f the polymer is i n solution ,Tg
can be much lower. T h is decrease can be attributed to the introd uction of add itional free vol u me with the solvent [ 1 1 6J . For example, for polystyrene i n cyclohexane a Tg of around 280 K h as been reported [ 1 1 7] .Near the glass transition , the self-diffusion coefficient Ds (T) can be described by a law named after Williams, Landel and Ferry (WLF) [ 1 1 8, 1 1 9, 1 20J :
Ds (T) = Doe ( .5 .39)
The ratio (tT (T) of mechanical and electrical relaxation times at temperatu re T
to t heir values at a reference temperatu re
Ts
can be expressed by the equation [ 1 18, 1 20]T - T
log aT(T) = - C l r S T
C2 + T - s
where Cl and 1'2 are ('onst a n ts which deppnd on the choice of Ts .
( 5 .40)
In chapter G we will measure the temperat u re dependence of Ds of PSjCYH solutions near i he de-mixing transition . \\'(' will show that it can be described by equation 5 .39. I n chapter I we will DW<lS U re the pO\ver law exponent n (T ) in order to obtain TT) ( T) for the same sam pIes. \Ve will show that the behaviou r of T,lT) Ileal' the de-mixi n g transition ca.n be described by equation 5 .40.
92 CHA P TER 5. POLYMER PI-f'lSICS
5 . 4 Flow-Induced Structures in PolYluer S olutions
W hen a polymer sol ution is subjected to shear, the free energy of mixing given by equation .5 . 12 m ust be adjusted by an add itional term w h ich acco unts for t he entropy change due to flow. A large n umber of d ifferent p henomena h as been observed , rangin g from macroscopic phase separations through mesoscopic concentration fl uctuations to microscopic alignment of molecules. Reviews on this topic have been pu blished by Rangel�Nafaile et al. [121J and Larson [122] . A recently p u blished book by Nakatani [12:3] contains several detailed articles with a large n umber of references. I n this section we will give a short review on t his topic.
5 . 4 . 1 S he ar-Ind uced Phase Transitions
The fi rst systematic i nvestigations on shear�induced phase transitions iNere per formed by K u h n and Silberberg i n 1 9.52 (124) . T hey observed that a solu tion of polystyrene and ethyl cellulose i n benzene, w hich was p h ase separated at room temperat u re, resulted i n a homogenous one�phase sol u tion when a velocity gra d ient was applied . After the velocity grad ient was t urned off, t he sol u tion p h ase separated again . The shift in the phase transition temperature was fou n d to be dependent on the shear rate.
Vel' Strate and P hilippoff reported in 191·1 [ 12.5] on experiments where a c lear polymer sol ution , PS in dioctyl phthalate (DOP) , was p u m ped from a reservoir through a capillary into a second reservoir. A bove a certain shear rate which was fou n d to depend on polymer, solvent, temperature and concentration, the sol u tion t urned cloudy u pon passing i nto the capillary and clear when leaving it. T his effect was i nterpreted as a phase separation induced by a shear rate dependent change i n free energy. T his phase separation was found to be reversible.
An irreversi ble shear�ind Hced change in the refractive index h as been observed in PSjCYH solutions by Debeauvais et al. and h as been i nterpreted as degradation of the polymer coils [ 126] . Because this was fou n d to happen at shear rates higher than we used in the experiments of chapter I, this degradation ,vas not i mportant for our experiments.
Rangel�� afaile et al. obser\'ed a shear�dependent incr ease in t.he cloud�poin t t e m perat u re of PSjDOP solutions. They found a. shift o f u p t o 28° C i n t h e phase t ransition temperat u re at a shear rate of 220 S- 1 [ 1 2 1] . In a review they also s u m marised a whole variety of shear-induced changes in the degree of mixing of polymer solu tions. These effects include shear�ind uced cloudiness, precipitation of gel�like particles and flow-ind uced crystallisation.
5 . e1 . FLO W-IND UCED STR UCT URES [V POLYJ'vIER SOL UTIONS 9:3 Barham and I\eller reported Oil the existence of d i fferent disti nct types of rheological behaviou r in high molecular weigh t P?v['ylA sol utions, depending on the shear rate. These types were attributed to the formation of layers formed by the m u tu al entanglement of m olecules in the flowing solu tion and those adsorbed along the rheometer su rfaces. They can be much t hicker than the rad i us of gyration and are called adsorption-entanglement layers. With increasing shear rate, t hese layers grow, and wild fl uctuations in shear stress are observed . At h igher shear rates, sponta.neous local phase separations occ u r , and fi nally t hese phase separations grow u ntil a stable two-phase system h as been established [ 1 27] . Link and Springer also reported the o nset of shear-induced phase separations in very dilute PS/DOP sol utions [ 128] .
A theoretical model for these shear -induced shifts in t he phase transition tem peratu re has been s uggested independently by Rangeh\;afaile et al. [ 1 2 1J and \Volf ( 1 29] . These authors add a shear -dependent term to the free energy of mix ing Q..Gm in eq u ation .5. 1 2 which corresponds to the elastic free energy stored i n the system d ue to chain deformation and i s proportional t o t h e shear stress tensor T .
5 . 4 . 2 Enhanced Concentration Fl uctuations
The thermodyn amic approach by Rangel-Nafaile et al. has been criticised by H elfand and Fred rickson [ 1:30] , and by Onuki [ 1 3 1 ) ' who h ave considered t h is problem from the poin t of view of the growth of concentration fl uctuations u po n t h e application of shear. I n their t reatment, t h e F lofY interaction parameter X is assumed to be shear-dependent. lvIi/ner extended the Helfand-Fredrickson model to the case of entangled polymers [ 1 32] . The shear-dependence of X has been measu red by Ha.mmouda et al. [ 1:33J for PS/DOP solutions. A steep increa.se of \: was found when the shear rate l' exceeded a critical value which was below the spinodal value where phase separatioll occ u r red .
Hashimoto et al. performed light scattering experiments on a binary polymer solution u nder shear [ 1 3-1J . With increasing shear rate. they report on five d ifferent regimes of concentration fl uctuations, including the formation of polymer-rich d roplets which elongate with i nc reasing shear rate.
\Vu e t al. have studied the orientation dependence of the enhanced concentra tion fl uct uations i n t he Couette geometry [ 1 35] . They showed that the structu res were a ligned i n t h e plane d efi ned by t he d irection of flow an d t h e direction of shear. ;\s the shear rate \vas increased . the structures rotated by 90° in this plane. Thei r length was fou nd to be o f order 1000 A . For high shear rates, Yanase et al. fou n d t h e dichroism � n/l t o b e time- dependent [ 1 :36J . This effect i s believed t o b e caused
94
by a shear-induced phase t ransitio n .
CHA.P TEH 5. POL 'y'.\IEH PH'{SICS
The d irect visualisation of enhanced concentration fluctuations i n semi-dilu te polymer sol utions h as been reported by Moses et a!. [ 137] . With a microscope, periodic structu res with a length scale of approximately 10 {lm could be see n . T hese structu res are in qualitative agreement with t h e ones fou nd from t he c har acteristic butterfly patterns i n light scattering [ l34] and neutron scattering [ 138] experiments . .1 i and Helfand calculated the polymer structure factor using a phe nomenological approach to write down a coupled set of Langevi n equations for polymer concentratio n , velocity and strain [139] . The calculated butterfly pat terns were fou nd to be i n q ualitative agreement with the experiments by \Vu et al. [135] .
5 .4 . 3 S hear-Induced Ordering
So far , we h ave only considered shear effects on scales much larger than the size of the molecules. However, for randomly oriented chain molecules, one expects t hat an i n homogeneous fl ow field somehow alters the configuration of the molecu les.
N akatani et al. [ 140] performed NMR experiments on sheared polymer melts. I n the next section we will describe these experiments in more detai l .
L i n k and Springer and M u ller et al. measured t h e shear dependence of all t h ree radii of gyration of PS coils i n DOP via light scatterin g [ 128] and of polymer melts via neutron scattering [ 1 4 1] . Both groups fou n d t h at t he coils elongate i n the plane defined by the flow and shear directions, i n q uali tative agreement with the t heory.
5 . 5 Rheo-NMR
In chapter 2 vve showed that NMR spectroscopy is a technique to meas u re interac tions on a molecular level. For exam ple, the d ipolar and quadrupolar i nteractions are both dependent on the orientation of the molecule i n the magnetic field, w hi le the relaxation times 1'1 and T2 revea.l information abo u t the dynamics of the m olecule u nder inws tigation . 0i�IR is t herefore an ideal t ech nique t o monitor changes on a molecular level due to macroscopic shear. Such a combination of rheological experiments with was fi rst demonstrated by Nakatani et al. and termed rheo-0i;\[ R [ 1 ,1OJ . These a.uthors designed a. cone and plate rheometer to operate i nside a standard probe. This apparatus was used to monitor the proton linewidt h of polymer melts u nder shear. An i ncrease i n the l inewidth was fonnd with i ncreasing shear which h as been attributed to a redistribution of the i ni tially isotropic dipolar cou pling.
5. 6. RHEO;\[ETEnS
G rabowski and Schmidt describe a cone and plate viscometer which attaches t o a standard N M R probe. With this setu p it is possible to monitor the depen dence of viscosity and order parameter defined in equ ation 2 .57 on the shear rate sim u ltaneously [ 1 "12J .
All t hese methods make some ass u m ptions about the velocity profile in the rheometer. With dyn amic NMR imaging it is possible to meas u re the flow proft le directly. This approach h as been used by Callaghan et al. in different experiments. Velocity profiles of non-Newtonian fl u ids have been monito red both in capillary flow [H3] and in the Couette geometry [55] . In principle, it is possible to meas u re the How field and 0JMR specifi c parameters at different flow rates. From the experimental flow profiles the local shear rate can be obtained and therefore the shear dependence of NMR parameters. No assumptions about the flow field need to be m ade.
Hopkins et al. reported on the N]\IR measurement of flow p rofiles in a Cou ette cell using oscillating magnetic field gradients [144] . This method, h owever, assu mes cylindrical sym metry of the sam ple which is not the case if it is not centred accurately.
5 . 6 RheOlneters
The s hear stress vs. shear rate relationship is measured in a rheometer. One somehmv h as to measu re the torque req u i red to shear the fl u id at a certain speed . F ig ure 5 . 1 3 shows two common r heometers. The plate in figure 5 . 1 3 ( a) or the o u ter cylinder i n figure .5. 1 3 (b) is spun , and the torque exerted by the liquid on the cone (a) or i n ner cylinder ( b) is m on itored . I f this is performed for different rotation frequencies, a graph like in fig u re .5 . 1 0 can be obtained .
I n the Couette geometry it is possible to obtain an analytical expression for the velocity profile of a power law fl u id across the gap [.5S] . If the i n ner cylinder has rad ius l'i and spins with the angular velocity Wo , the angular fl uid velocity at radius I ' is given by
1'(1') (5A l )
1'0 i s the radius o f the outer cylinder.
rn ch apter I' we will use a red uced spatial coordinate l's which we defi ne as:
96 CHA PTER 5. POLYMER PHYSICS