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					            Thermodynamic properties of a phase
Connection of macroscopic heat capacity
with microscopic motion
                                                                                 Glass
At Constant Pressure             At Constant Volume           CP
                  T                                T                              and
  H  H         C p dT        U  U          C V dT                      Crystal
        T   Cp                               CV
 S    
                                         T
                 dT               S             dT              Glass
                                                                                  TS
       T                                 T                    H
 G  H  TS                       F  U  TS
                                                              & Crystal
                                                              G                   TS
C P  C V  TV              
                         2

: expansivity
             ;         : compressibility
                                                                   Glass
                                                              S
Enthalpy H                   Hliquid > Hsolid
                                                                   Crystal
Entropy S                    Sliquid > Ssolid
                                                                             T
 Possible phase transitions
                    in one component system

               Solid          Intermediate        Liquid

Increasing    Crystal
  order
                                   Td               Tm
                         Tg                  T
             Mesophase                        i
                              Mesophase           Melt
              Glass

                                                     Tg
Disorder       Glass


           Immobile                       Increasingly mobile
            Equilibrium between Two Phases

                                     Ehrenfest’s Definition
     '
    G1    G2
            '       '
                ( dG1        '
                            dG2 )
                                     (Proc. Acad. Sci., Amsterdam, 36, 153, 1933)
G’: Gibbs free energy per mole of each phase
                                                                First order   Second order

 Phase Rule:       p+F=C+2                                           0           =0
 p: number of phases
 F: degree of freedom                                                0           =0
 C: number of components
                                                2 DG    DC
                                                        P        0           0
 At the transition:           DG=0              T 2  P   T


                                                 2 DG 
                                                P 2   VD
                                                       
                                                                     0           0
                                                       T
                     First-order transitions

    DGtran. = DHtran. - Ttran. DStran. = 0    G/ T = -S

    Ttran. = DHtran./ DStran.                 2G/ T2 = -Cp/T

     Melt




                                              , Cp, 
                         V, H, S
G




     Crystal




            Ttran.                 Ttrans.               Ttrans.
    One example of a first-order
               transition in polymer
The melting of polyethylene extended chain crystals
                             1.30
       1/density (cm/g3)-1



                             1.25

                             1.20

                             1.15

                             1.10

                             1.05

                             1.00
                                 80   90   100 110   120 130 140 150 160
                                               Temperature (°C)
              Enantiotropic and monotropic
                     phase behavior

Enantiotropic Phase Behavior   Monotropic Phase Behavior

    Melt                       G   Melt
G
    L. C.                          L. C.
    Crystal                        Crystal




               Td   Ti                       Ti   Tm
One example of monotropic
           behavior in polymer

         67 C




                           C




    67           87             107
                      C
                 Glass transitions
The Liquid (melt) has always the low Gibbs free energy (G).
•
Kinetic reasons cause the change to the higher G of the
glass at Tg on cooling.


    Glass
                          V, H, S




                                          ,Cp , 
G




    Melt




            Tg                      Tg               Tg
On example of polymer glass transition
                 for poly(vinyl acetate)
                     85.5

                                    Equilibrium values
                                    Values after quenching
                     85.0
    × 100 (cm3/g)




                     84.5



                     84.0



                     83.5
                                                    Tg Tg

                     83.0
                        -20   -10     0   10   20     30    40   50   60
                                       Temperature (°C)
         Phase transitions
                      in two-component system
Miscible Mixture of A and B Phase separated mixture of A and B
DG                                    DG
 0                                       0

                  Q*
                             B2
        B1                                              S1
                                                             S2

                                                   B1
                   Q                                              B2

          XB1             XB2                       XB1           XB2
     Concave upwards of a free energy curve with a minimum: miscible
                  Phase transitions
                           in two-component system
The slope of B1B2 line is defined by If in a polymer solution system and the
                                                 Flory-Huggins equation for the Gibbs free
      DG                   DG                  energy is used
                          
      X B   X B  X B1
                            X B    X B  X B2
                                                  s  
or in terms of numbers of moles
                                                        s
                                                           ln1   P   1    1
                                                                                  M     P  2 
                                                                                               P
                                                    RT
      DG                  DG                                  s  1  P
                         
      n B n               n B n
               B  nB1              B  nB2
                                                 µs°: the chemical potential of pure solvent
Therefore,                                       M: the number of segments in each polymer
             1   2
              B     B
                                                 chain

                                       Remember: DGm/RT = nslns + nplnp + nsp
µ: chemical potential
             Phase transitions
                      in two-component system
                             2 DG      Single phase UCST phase diagram
At the inflection point            0
                             X B2
                                                     Binodal
                      2G                            Spinodal
In B1S1 and B2S2                   0
                            X B
                               2

                                                                      Metastable
                                           TC                         Region
                       G
                       2
                                   0
while in S1S2               X B
                               2




For a stable single phase mixture, it                  Two Phase
requires the Gibbs free energy should be
negative. The second derivative of the
Gibbs free energy with respect to                            B C
composition should be positive.
                                                                 3DG
                                                At (B)C              0
                                                                 X B
                                                                    3
             Phase transitions
                      in two-component system
If the Flory-Hugginsequation is again used     At thecritical point
in the volume fraction formula                                         ½
                                                        A C    ½ MB
DGm  A                                                          M A  MB
                                                                         ½
        ln  A  B ln  B   A  B 
 RT   MA         MB
                                               and

                                                          C  1  1½ 
                                                                 
                                                               2 M
                                                                           1 
                                                                          MB 
And then,                                                                   ½
                                                                    A

 2 DGm RT 
               1  1  2  0                 For a polymer solution             MB = M
       2      A M A B MB
                                                          C  1        
         A
                                                                       1 ½
3 DGm RT                                                    2 1     M
               21  21  0
     3
       A         A M A B MB                  M  ,   0.5
                                             For a polymer blend, MA ~ MB  
                                                               C  0
       Liquid-liquid demixing
                in a two component system
                                           Single Phase
              Single Phase

T




     Phase Separation


                                  Ideal equilibrium   In practical case
    Concentration of Polymer   two macroscopic phases dispersion phase
                                                     (Metastable phase)
   Liquid-Solid Crystallization
Single Component                    Two Component (Solution)

               Melt                                                Tm
                                                Solution
      Tm
                                    T
                                                Crystal
              Crystal                              +
                                                Solvent

                                        Concentration of Polymer
     In most cases,         Single Lamellar Crystals      Spherulites
 spherulites or axialites                Dentrites and Others?



           Ideal Equilibrium Extended Chain Crystals
       Liquid–solid vitrification
                      in two-component system

                               Tg   For example:
                                    • Kelley–Bueche Equation
T

                                            A1Tg1  2Tg 2
                                     Tg 
                                              A1  2



    Concentration of Polymer
     Liquid-liquid phase separation
                             with vitrification

                                                                       Tg
Dispersed Glass Particles        T
   in Rubbery Matrix


                                                           BP
   Interpenetrating
       Network

                                        Concentration of Polymer
                                       •
                              BP: Berghmans Point
  Solution Droplets                    •
                              At BP, phase at concentrated side of the line
   in Glass Matrix            vitrifies.
                                       •
                        Tg becomes independent of concentration,
                            and phase morphology is frozen in.
                            One example of liquid-liquid phase
                                       separation with vitrification
                    a-PS (MW = 2.75×106 ) in cyclohexanol solution
                   83                                                                                120

                                                             metastable                                   90
                                                             nucleation




                                                                                Tg and Cloud Point (°C)
                   82
Cloud Point (°C)




                                                                                                          60
                                    fully stable
                   81                 spinodal                                                            30


                            K                                                                              0
                   80
                                                                                                          -30

                   79                                                                                     -60
                        0   2   4       6    8     10   12    14   16     18   20                               0   20      40       60        80   100
                                Polymer Concentration (%w/w)                                                        Polymer Concentration (%w/w)
        Phase separation morphologies
                              with vitrification

                                                              1
                         1   2     3
                                                         Tg
                                                               dispersed glass
   dispersed glass
                                                              in rubber matrix


                                                  BP
                                                              2

 interpenetrating
                                                               bicontinuous IPN
     network      T
                      cpol

                                                             3
                                        droplets of solution
                                        glassy matrix
                                                     inversely dispersed rubber
dispersed solution glassy matrix                          in glassy matrix
          Liquid–liquid phase separation
                            with crystallization

                                           T                   Crystallization T
Phase morphology of M1:                                                         m



                                                    M2
Crystal morphology of M2:
                                                    A           B
                                                         M1              M4
                                               M3
                                                                                    C*
Phase + crystal morphology of M3:                   A*        B* Invariant Line

                                                                              Tg
Phase Morphology of M4: Spherulites,                                          (?)
                        axialites, and           Concentration of Polymer
                        single lamellar crystals
    One example of liquid-liquid phase
                   separation with crystallization

Two mixed LLDPE fractions (H. Wang, C. C. Han et al. Macromolecules)

				
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posted:10/1/2011
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