PARTE I: ESTUDIO PREVIO
Capítulo 2: Proceso de fabricación de culata y bloque motor
to Ds 'X <[>-3.0 with increasing concentration . However. this transition is verv s mooth over a wide concentration range. Wesson et al. [ 1 0 1] and K i m et al. [ 104J m easu red t h e polymer and tracer diffusion in good solvents via forced Rayleigh scattering. T hey also fou nd a smooth transition from Ds ex <[>- 1 .75 to Ds ex p-3.0 behaviour. A mis et al. [99] and Wheeler et al. [105] measured the self-diffusion coefficients of PS in theta solutions via light scattering. These authors also re port a smooth transition between the two scaling regimes. Nemoto et al. (1 03) re-analysed the Ds data reported in [ l O lJ using a concentration and temperat u re dependent local viscosity correction . With this correction term the d ata reported in [ 1 0 1] fit better into the scaling predictions of the Doi-Edwards modeL Fleischer
e t al. [ 106] performed PG S E NMR experiments on PS i n deuterated benzene and
in deuterated chloroform in sem i-d il u te concentration . The scalin g l aw Ds ex 1\11;; 2 could be confirmed , but no linear region with a concentration exponent of - 1 .7.5 could be fou n d . The concentration exponent of the m aster curve decreased fro m about - 1 .8 t o - :3 .0 a t t h e e n d o f t h e sem i-dilute region . and then exhi bited a further d ramatic decrease.
Because t here seems to be no sharp transitions between scaling in good and t h et a solvents, the question arises w hether t here is a u niversal scaling law for
Ds (p) . Phillies suggests in a series of papers [ 108)-[1 1 1) that rept ation is not an i mporta.nt process in polymer sol utions. Instea.d he suggests a u niversal scaling l aw for the concentration dependence of Ds :
(5. 28)
w here Do, Q and ;\ a.re adjustable parameters. \Vith t his empirical equ ation, he can obtain a good fi t to the experimental data mentioned above. A m ajor criticism of t h is description is t hat the apparent success of equ ation 5 . 28 reflects more the flexibility of the stretched exponential form than the universali ty of d i ffusion mechanisms i n semi-dilute polymer sol u tions [ 1 0.'5] .
Tb oll r knowledge, there have so far been no studies of the temperature de pendence of Ds near the de-mixing transitions of polymer solu tions. I n chapter 6 we report on PGSE :'-i:\;IR measu rements of the tem perature dependence of Ds for polymer solutions at different chain lengths and d i fferent concentrations. \Ve will show then that t here exists a single temperatu re scaling dependence.
5.3. R HEOL OG Y OF POLY.HER SOU)TIONS
5 . 3 Rheology of PolYlner Solutions
5 . 3 . 1 Introduction an d Definitions
R h eology is t he interdisciplinary science of deformation and flow of condensed matter. The term \vas suggested by E. C. Bingham who also founded t he Society and the JOllrnal of Rheology in 1 929. Rheologists study the behaviou r of m aterials which , because of thei r nature or because they are su bject to large deformations. do not obey eit her Hooke's law of elasticity or Newto n 's law of viscosity.
Mac roscopic flow is described by constitutive equ ations and parameters w hich relate observed stresses and rates of straining to m aterial properties. In princi ple. k nowing all relevant molecular properties. the parameters of t he conti n uu m 's mechanical eqllations of motion should be obtainable.
,
dv
+
2 1v + dv V
Figure 5 .9 : A velocity profile of fluid flowing a long a boundary
The laminar flow of a real fluid along Cl solid boundary is shown i n fig u re 5 . 9 . T h e velocity increases w i t h the distance from t h e bou ndary. Two volu m e elements 1 and :2 starting in adjacent layers on the same vertical line as shown in figure 5 . 9
move d i fferent distances cl l :::::: vdl and d2 :::::: (v + dv)dt in an infinitesim al t i m e elt. One defines the s hear strain Ai and shear rate '1' as
d2 - d l
dy ( 5 .:29 )
clu
AI -
dy ( .').:30)
The shear stress T is defi ned as the shearing force between the fl uid l ayers per u nit of contact area. For laminar motion . T is dependent on "':(. The shear viscosity Tl
is den !led as:
T I) = -;- .
88 CHA PTER ·5. POL '(:\fER P H YSICS Fluids for which 1) is constant are k nown as Newtonian fluids. Other fluids are classed as non-Newtonian fl u ids.
.
r
Figure .5 . 1 0 : A shear stress vs. shear rate d i agram for N ewtonian a n d non-Newto n i a n fl uids.
Some examples for non-Newtonian fl u ids are shovvn i n fi gu re 5 . 1 0 . A Bingh am
fl u id is characterised by the yield stress '1 which is the minimum stress requ ired for the fl uid to fl ow. The fl uid exhibits solid-like properties up to the stress '1 su fficient to cause a transition to flow. The shear stress vs. shear rate relationship is given by
( 5 . 32 ) Toothpaste is a classical example of a Bingham flu id .
For many polymer melts and sol utions i t is fou n d t hat the viscosity defined in equation 5 . 3 1 is dependent on the shear rate. A typical graph of viscosity vs.
shear rate looks l ike the one shown in figure 5 . 1 1 . I n region A, the viscosity can be approximated by a power l aw
17 ex 1,,- 1 . ( 5 .:33)
n is known as the power law index. For n = 1 the viscosity is independent on the shear rate and the fluid is �ewtoniall . If n < 1 . the viscosity decreases with increasing shear rate. Such cL flu id is called shear thinning. while a fluid with n > 1 is shear thickening.
5 . 3 . 2 Dependence o f Non-Newtonian Viscosity on Shear Rate
For polymer sol u tions with d ifferent concentrations it has been found that \vith decreasing concentration , the onset of non-Newtonian behaviou r shifts to higher
5.3. nHEOL OG Y OF POLYMEn SOL UTIONS
A log y.
89
Figure 5 . 1 1 : A viscosity vs . shear rate d ia gra m for a non-Newtonia n fluid . I n the reg',on A the viscosity c a n be a pproximated by the power law of equ a tion 5.33.
shear rates [ 1 1 2] . At high shear rates, the plot of log 1,1 vs. log "':! becomes linear. but the slope becomes less negative with decreasing concentration . G raessley showed that all t.hese curves can be plotted on one m aster curve by plotting log vs. log )'T71 , where 770 is the viscosity at zero shear rate and T,/ the entanglement for mation time w hich is characteristic for each solution ( 1 1 3) . He gives an i m plicit equation for this m aster curve [ 1 1 2] :
where 77 170
(
2)
5/2
[
-1 () ]3/2 [ -1 (}(1 _ (}2)]
;;: cote
+ 1 +e2
cot(}
+ (1 +(}2)2 '
(}
= !l "':(TI) . 1,10 2 (5.34) (5.35) The 17 vs. l' relationship obtained from equation 5 .34 is shown in figure 5 . 1 2 . The n u merically obtained power law index is shown i n the same figure.The concept underlying G raessley's theory is that the viscosity is governed by entanglements of the polY'mer coils. If the inverse of the shear rate is smaller than the time to form entanglements, the viscosity will decrease with increasing shear rate. It is therefore not surprising that the time T7) is found to bt� of the same order as Tei , w here Td is the tube disengagement time from the Doi-Edward model a nd is defined in equation 5.n.
Equation 5 .:}"! is difficult to handle for analytical calcul ations. 'vVilliams cal culated 17 b) by modelling the poly'mer molecules as elastic d u mb bells [ 1 1 4] . He obtai ned a monotonically decreasing fu nction of 1 which is algebraically com pli cated , but roughly obeys the form
I)
170
(
1 +)
1/2
90 CHAPTER 5. POLYMER PHYSICS