Polymer Physics
76 CHAPTER 5 POLYMER PHYSICS
(a)
(b)
Origin
Figure 5 . 2 : (a) A polymer molecule ca n be modelled a s a succession of N l i n ks . (b) T h e l i n ks c a n not i n terpenetrate each other. This is modelled b y excluding some volume a ro u n d the l i n ks from the configuration space.
The value 1/ == 1 /2 corresponds to a chain with i ndependent links and 1/ = 1 to a rigid chai n . The effect of the excluded volume is therefore a swelling of the chai n and i ts behaviour i s betvleen t hat o f a r andom chain and a completely stretched chain or rod. The value of 1/ depends on the theoretical model and usual ly has a value of 1/ � 0 . 5 0 .6. I nstead of choosing R to characterise the mean size of the polymer i n solution, it is common to use the radi us of gyration Rg , because t h is can be m easured d irectly. It is defi ned as
(5 .5)
Rg scales in the same way with the chai n length 1\/ as R. T herefore we can w ri te
(.5. 6 )
The viscosity of a polymer solution will depend on the mean size of the polymer u nder investigat ion. The im portant parameter here is not R:J 1 but the hyd rody n amic radius Rh . It is defined as
5 . 2 . 2 Thermodynamics of Polymer So lutions
(5.7)
I n order to calculate the thermodynamic behaviou r of polymer solutions, the mean-field lattice approximation of FIory [69] and H uggins [70] h as been very suc-
5.2. D LYAJIICS OF POL Y.HER CHAINS IN SOL UTIO;\'
cessfu l . I n this model. one considers the polymer solu tion on a th ree�di mensional lattice, w here each solvE'nt molecule a.nd ea.ch monomer unit occ u py one lattice site. This is shown in fig u re .5.3. Because the monomer u nits are con nected , there are less configurations available t han in an ideal solutio n . This reduces the entropy of mixing srn .
(a)
0 0
•0 0
•0
•0 0 0 0
0 0 0 0 0
•0
•0
•0 0
0 0 0 0 0 0
• •0 0 0 0 0
• •0 0
•0 0
0 0 0
• •0
0 0 0 0 0 0
0 0
0
•0 0
0 0
•0
0 0
0 0
0 0
0
•0 0
•0
0 0
0
•0 0
•0
0 0
0 0
(b)
0 00 0 0 0 0 0 0 0
0 00
0 0
0 00 0 0 0 0 0
0 0 0
0 0
0 0 0 0
0 0
Figure 5.3: ( a ) I n a n idea l solution the solvent ( 0 ) a nd sol u te (.) molecules are randomly
distrib u ted . ( b ) I n a polymer solution the segments must be l inked together.
If t here are Ns solvent molecules and Np polymer molecules each with x monomer u n its, the entropy of mixing can be approximated by [69]
[
Ns N1
6Srn = -kB Ns log N TV + Np log r.y' p TV .
s + :1:1 p i 's + Xl p ( 5 .8 ) I t i s more convenient to use volume fractions instead o f t h e n u m ber of molecules. The volume fractions for solvent (4;05) and polymer (q)p) are defi ned as:
Equation 5.8 becomes t hen
Ns + �:Np x
( 5 . 9 ) ( .5 . 1 0 )
( 5. 1 1 ) By introducing t he Flory interaction parameter \: [69. 7:)] which characterises the interaction bet\veen polymer and solvent molecules and decreases with increasing tem perature . one can write the free energy of mixing as
78 CHA PTER 5. POLYi\I[ER PHYSICS
5 . 2 . 3 Osmotic P ressure a n d t h e Flory Temp erature
From equation 5 . 1 2 it is possible to obtain the osmotic pressure JI of the polymer solution [7:3J
.. kG T
[([>D
1 2]
n :::: --::::-- -' + (;- - \)t:P + . . . .L's .1: 2 p ( 5 . 1 3 )
T h is i s a virial expansion o f van 't Hoff's law [74] . T h e term
(�
- .\) i s k nown as the second virial coefficient. If this term is zero, the polymer solu tion behaves like an ideal solutio n . Because of the tem perature dependence of x, t h is occurs at a particular tem perature which is called the F!ory, or theta, temperature e. A solvent is called a t heta solvent i f X ::::�.
At the theta temperat u re, the polymer coil takes on its " natu ral size" , and excluded volume effects are cancelled out. Above the t heta temperat ure, X <�
and polymer-solvent contacts are favou red. The coil expands i n this region. Below the theta tem perature, X >�
and the coil collapses into a globule. This coil-to-globule t ransition has been a topic of extensive research in recent years [75]-[82] .T heta temperatures can vary widely. depend i ng on the particular polymer and solvent . For example. for polystyrene i n cyclohexane, it is :35° C [83] , \vhile for polystyrene i n benzene it is 50° C [68] and 22° C for polystyrene i n d ioctyl phthalate (DOP) [8-t] .
The interaction paxameter X d oes not depend on the tem perature alone, but also on other parameters such as <Pp and x . Strictly speaking, the t heta tem perat u re is defined for an infinitely long chain in a solution with infini tesim al concentration .
5 . 2 . 4 P hase Equilibria
For the formation of a single phase solution 6Cm of equation .5. 1 2 m ust always be negative. However, this is not the only condition , as is illustrated in figure 5.4. The cu rve i n figure 5 A (a) is always concave u pwards, and this means that any poin t Q on the curve has a lower free energy than a two phase syste m of the same overall composition. Howewr. if the free energy cu rve has a shape such as that shown in figure ;) .-1 ( b) . the free energy is smaller if a two-phase system is formed . i f the polymer concentration i s between ifJp1 and t:Pp2 ' T h e points PI and P2 can be connected by a. line whic h is a tangent at both points. T h is line represents the free energy of a phase ::;eparatpc! mixture. and each position on the line represents d i fferent proportions of t he two phase separated mixtures. Any hypothetical single phase mixture between these com positions h as a larger free energy than the phase separated mixture. so that the system is i m m iscible over this composition range. The composition range w here phase separation occurs is defined by the points of
.5.2. D YNAMICS OF POLL\[ER CHATvS IN SOL U TION
Ca)
Cb)
7D
Figu re 5.4: Schemati c diagrams of the free energies of mixi ng as a fu nction of polymer concentration . Polymer and solvent a lways form a single phase in (a), while in (b) p hase separation can occu r . The straight line represen ts the free energy for a phase-separated m ixture.
contact of the dou ble tangent . This gives the relationshi p
( 5 . 1 4)
T h e derivative is the chemical potential �tp of the polymer. The meaning of ("q u ation ,') . 1 ·1 is t h at t h e c h e m i cal potentials jlp ! at point P! a nd /'p2 at poi n t
P::. must be equal:
( 5 . 1 5) Tht' p h ase separation points Pt and P2 C<1.n be calculated from equa.tion S . U .
T h e points F\ and Pz represent t h e com positions o f the two phases t h at would be present i n equilibri u m . HO\w�ver , the shape of the free t'nergy curve arou nd