Polymer Physics
80 CHAPTER 5 POLY.\IER PHYSICS
5.2. D YNAMICS OF POLYMER CHAINS IN SOL UTION
(a)
(b)
8 1
Figure 5 . 5 : (a) Free energy c urves for d iffe rent tem peratures. (b) Schematic represen tation of a p h a se diagra m .
val ues o f X t h e spinodals and binodals can b e calculated . Einaga et al. s uggested an e mpirical correction for X [88] . In this approach X does not depend only o n T , b u t also on c{! and :r . O t h er corrections u s e a. virial t:�xpansion of t h e chemical potentia.1 [89] . The so� called apparent second viria.l coefficient r is assu m ed to depend o n T, c{! and x as wel l . These m odels describe the experi m ental d ata [85} m uch better, b u t are stil l not satisfactory. A review on t his topic is given i n [63} .
5 . 2 . 5 Concentration Regimes
Sufficiently d i l u te polymer sol utions m ay be viewed as systems i n w h ich "islands" of polymer coils are sca.ttered around i n a "sea" of sol vent. The spatial distribution of chai n molecu les is t herefore q u ite i n homogeneous and u ndergoes considerable fl uctuations. As the polymer concentration increases, the distribution of polymer coi ls becomes more homogenous a,nd local fl uctu ations get m ore and more s u p� pr�;ssed . Collisions m ay cause the chains to overlap and entangle i n a com plex fashion. The con centration w here this transition occu rs depends on the size of t h e
82
Ca)
Cb)
11 C l 1 0 .1 .2CHA P TER 5. POL'r?vIER PH}rSICS
Figure 5.6: (a) Measu red cloud-point curves for various sol utions of polystyrene i n cyclohex a ne. 'P i s t h e volume fraction of the solute. The molar m asses are given by: 10-4 Mw :::: 4.36 (A), 8.9 (B), 25 (C), a n d 127 (D). (b) Dependence of the critical temperatu re on the m o l a r m a s s . Both d iagrams ta ken from [85].
polymer chai n . It is com mon to define the overlap concentration c* as
(5 . 2 1 )
w here NA is Avogadro's constant .
I t i s i m portant to note that c * i s not a. critical concentration . No sharp changes in the concentration dependence of physical properties of the solu tion take place at t his concentration . However, the macroscopic d istribution of chai n segments a.cross the solu tion becomes homogenous when the polymer concentration is i ncreased beyond c*. \Vhen vie\ved microscopically, even sol u tions with concentrations above c* are not u ni form . The segment density in each volume element fl uctuates abou t
5.2. D Y.\'.·L \lICS OF POLY.\IER CHAINS IN SOL UTION 83 a mean value c. Because of the chain con nectivity, these fl uctuations can not take place independen tly. They have been observed by light scatterin g
[90. 91]
and can be enhanced by external shear and lead to shear-indu ced phase transitions. vVe will come back to this effect later in section .5.4.From eq u ation 5.2 1 it follows that if Alw is suffi ciently hig h , there is a rather wide range where the concentration is higher than c* , but its absol ute value is still low. S uch solu tions are dilu te in the sense that the concentration of polymer is small, but concentrated enough for coils to overlap. They are called sem i-dilute.
As the concentration increases, the polymer solution enters a regime where the chains overlap extensively. Density fl uctuations are m ainly screened. The concen t ration where the t ransition from semi-dilute to concentrated solu tions occu rs is termed eX" and is independent of Alw '
Equ ation
.5.21
allows us to estimate the dependence of c" on A[w ' Because of relationship ·5.6 Ry scales as A[:'u. Therefore"
C . (.5.22)
Adam and Delsanti [92] measured Ry of PS in a t heta solvent at dif1'erent m olar m asses and obtained the phenomenological relationshi p
" - �1
c
-
b . w w hich is i n agreement with relationship .5.22 for v .5.23 is shown in fig u re .5.7.5 . 2 .6 The Reptation Model
(.5 .23) 1/2. A graph of equ ation
I f the polymer concentration is high enough for entanglements to take place, the motion of each polymer chai n is restricted by neighbouring chains. This effect d u e to chain u llcrossability is called topological constraint and shown in figure .5.8 (a) . B rochard and de Gen nes
[9:3]
s uggested replacing the chain by a wire t rapped i n a curvilinear t u be fixed i n space ( figure.5.8
( b) ) . It can only move along its own path. Figure ;') .8 (c) shows the primitive path which is given by the shortest pat.h between the ends of the polymer and has the same topology relative to the obstacles a s the chai n . A p rimitive chain can then be used to describe the sim plified motion of the polymer moving along the primitive path by neglecting small scale fl uctuations across the path . Doi and Echva rds [72] extended this idea by a pproximating the chain of discrete molecu le's by a continuolls chain. The�r calcu lated different time scale regimes of polymer reptation [G l . 72] . If the time over \vhich d i ffusion is measu red is longer than the longest relaxation time Td of the Doi-Edwards model , the diffusion is a long range centre-of-mass diffusion;.:q -0.8 - 1 .0 -1.2 * (..) eJl - l A .$ - 1 .6 - 1 .8 -2.0 5 .0 5 . 5
CHA PTER. 5. POLYMER. P H YSICS
6.0 6.5
log Mw
7 . 0 7 . 5
Figu re .5 .7: A gra p h of c ' vs. M," for P S i n a t heta solvent using equ a tion 5.23. T h e d ata
poin ts i n dicate the Mw v a lues of the s a m p les from table 5.1, and the d ashed line is a t c* .5%.
and can be described by a time i ndependent B rown ian self-di ffusion coefficient
Ds.
T he so-called tube d isengagement or reptation time Td is a relaxation time characterising the time it takes for a primitive chain to disengage from the t u be it was conflIled to initially and is given by( 5 . 24)
w here the radius of gyration of the polymer coil which was defined in equation
5 .5 . to Doi and Edwards, Ds scales as N - 2 and
R;
roughly as iV, and t h us the reptation time scales as Tei ':xDs
also depends on the polymer concentration <1>. The Doi-Edwards theory predicts t hatDs
scales for semi-dilute polymer solutions as(5 .25)
For a gooe! solvent lJ = O.G, and v = 0.:3 for a t heta solvent. Therefo re we obtain two different scaling regimes:
D s x
Ds :X
;\" -2([>-3.0for (L good solH'nt for a theta sol vent
(� ')(') 0) . - ) (.5 . 27) The scaling of
Ds
\\"it h ;\" - 2 hils been investigated experimentally by different techn iques. :\ review is gi\Oen in [G3] . However. t he Ds lX S -2 dependence is not5 . 2. D ) 't\:\MICS OF POLL\IE'R CHA ISS I:V SOL UTIOIV
Ca)
(b)
• - - - -... •Cc)
• - - - -.... .... - - - -." • • • • • • • - , • i- - - -_ _ _ - " . I • 85Figure 5.8: ( a ) E n ta ngled polymer chains w i t h one polymer h ighlighted . ( b ) Schematic picture of the h ighlighted polymer in (a) placed on a plane with dots s howing t h e other polymer c h a i ns crossing this plane. T he Doi-Edwards tube is shown as well . (c) The primitive c h a i n in the Doi-Edwards tube.
only chara.cteristic for reptation [9el, 95. 96] . Furthermore, other scali n g exponents for reptation m ay be possible, for exam ple Ds 0( N-2A, given the k nown scaling of zero shear viscosity as ')0 0(
1V3.1
[97] .The concentration dependence of Ds i n good and t heta solvents h as been ex a m i ned by several gro u ps with different methods [98]-[1 06J . The Ds ':x: N - 2 depen dence could be confirmed for concentrations above the overlap concentratio n . The first experimental results in semi -dilute solutions were reported by Hervet et al. in l ()i�) [DS. 1 0i l T hese authors perfo rmed forced Rayleigh scattering st.udies on PS ch ains i n benzene solut.ions. T hey report a scaling of Ds 0( A/-2p-L75 . Callaghan a nd Finder [ lOO] and :"leenvall et al. [ 1 02J performed PGSE NMR experiments on PS in a good solvent. Both grou ps found a transition from Ds 0( p- 1 .75