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									London Metropolitan Polymer Centre




Determination of volume
fraction of rubber in a swollen
vulcanisate
Dr Ahmed S Farid


January 2006
London Metropolitan Polymer Centre


      VOLUME FRACTION OF RUBBER IN A
       SWOLLEN RUBBER VULCANISATE

Introduction
The crosslink density of rubber can be obtained from the Flory-Huggins equation.
However, this equation contains a parameter known as “the volume fraction of
rubber in a swollen gel or rubber vulcanisate”; the symbol used to denote this
parameter is usually  2 or  r . We shall use the symbol  2 . The present
document explains the way in which an expression for  2 is derived.

Theory

The definition of  2 is given by

                             Volume of rubber hydrocarbon
       2                                                                      (1)
               Volume of rubber hydrocarbon    Volume of swelling liquid
Let the mass of a test-piece cut from a sheet of vulcanised rubber be mu g.
Suppose the mass of none-rubber ingredients in the test-piece is mn . If the mass
of the rubber content in the test-piece is mr then

                                    mu  mn  mr                                (2)

If F represents the mass fraction of added none-rubber ingredients then

                                           mn
                                    F                                          (3)
                                         mr  mm

In determining 2 , rubber test-pieces are acetone-extracted and then dried. The
purpose of this is to remove residual substances that are present in the raw
rubber. If the dry mass of the test-piece after acetone-extraction is Me then the
mass of residual material extracted by acetone is

                                       mu  Me                                  (4)

If the mass fraction of acetone-extractable materials is E then




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London Metropolitan Polymer Centre


                                            mu  Me
                                       E                    (5)
                                              mu

Simplifying (5)

                                      Me  mu 1  E        (6)

Putting (2) into (6)

                                Me   mn  mr 1  E      (7)

Simplifying (7)

                                            Me
                                  mr             m         (8)
                                          1  E  n
From (3)

                                       1 mr  mn
                                                            (9)
                                       F   mn

Further simplifications of (9) give

                                        1 mr
                                            1
                                        F mn

                                        1    m
                                          1 r
                                        F    mn

                                                 mr
                                       mn                  (10)
                                              1     
                                               F  1
                                                    

Putting (10) into (8)

                                          Me        mr
                                mr                        (11)
                                        1  E   1  1
                                                  F    
                                                       

Further simplifications of (11) give




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London Metropolitan Polymer Centre


                                     Me           1
                             1              
                                  mr 1  E   1 
                                                F  1
                                                     

                                     1              Me
                             1              
                                  1            mr 1  E 
                                   F  1
                                        

                          1 
                           F  1
                                   1
                                            
                                              Me
                           1   1  mr 1  E 
                           F  1  F  1
                                       

                              1 
                               F  1  1
                                         
                                                Me
                                1          mr 1  E 
                                 F  1
                                      

                                  1
                                  F       Me
                                        
                                1  mr 1  E 
                                F  1
                                     

                                   1        Me
                                        
                                1  F  mr 1  E 
Hence

                                          Me 1  F 
                                  mr                                         (12)
                                             1  E 
Upon swelling, the rubber network that is not crosslinked can be extracted to
some extent by the swelling liquid. This extractable material that does not form
part of the crosslinked structure is known as sol. The amount of sol can be
determined by first swelling the test-piece to the point of equilibrium and
subsequently drying the test-piece to constant mass. Thus, if the mass of the
test-piece after drying once it had been swollen to an equilibrium state is Md then

                         Mass of sol extracted  Me  Md                      (13)




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London Metropolitan Polymer Centre


Note the implication of equation (13): swelling is always carried out on test-pieces
that have been acetone-extracted.

If the mass of crosslinked rubber hydrocarbon is mrh then

                           mrh  mr  Mass of sol extracted

Hence, noting (13)

                                    mrh  mr   Me  Md                      (14)

Simplifying (14)

                                     mrh  mr  Md  Me                        (15)

Putting (12) into (15)

                                          Me 1  F 
                                  mrh                   Md  Me              (16)
                                            1  E 
Further simplifications to (16) give

                                  Me 1  F   D 1  E   T 1  E 
                          mrh 
                                                  1  E 
                                 Me  Me F  Md  Md E  M e  M e E
                         mrh 
                                              1  E 
                                         Md  Md E  M e F  M e E
                             mrh 
                                                 1  E 

                                         Md 1  E   Me  F  E 
                                 mrh 
                                                  1  E 

                             mrh  Md  Me 1  E             F  E 
                                                          1
                                                                               (17)

If the density of the rubber hydrocarbon is rh ; then the volume of rubber
hydrocarbon is

                             Md  Me 1  E 1  F  E  rh1
                                                             
                                                                               (18)
                                                         


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London Metropolitan Polymer Centre


Let the mass of the test-piece at the point of equilibrium swelling be Ms . Hence,
the volume of liquid that has entered the test-piece under equilibrium conditions
is

                                       Ms  Md  s1                         (19)

The parameter  s represents the density of the swelling liquid.

Putting (18) and (19) into (1) gives

                             Md  Me 1  E 1  F  E   rh1

                 2                                         
                      Md  Me 1  E 1  F  E   rh1  Ms  Md  s1
                                                                              (20)
                                                   

Recall that in equation (20)

                                 mu  Md                   mn
                            E                ; F                             (21)
                                   mu                    mr  mn




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