Transcript Polymers PART.2 - Tunghai University

Polymers PART.2
Soft Condensed Matter Physics
Dept. Phys., Tunghai Univ.
C. T. Shih
Random Walks and the
Dimensions of Polymer Chains
 Goal of physics: to find the universal behavior
of matters
 Polymers: although there are a lot of varieties
of polymers, can we find their universal
behavior?
 The simplest example: the overall dimensions
of the chain
 Approach: random walk, short-range
correlation, excluded volume (self-avoiding
walk)
Freely Jointed Chain (1/2)
 There are N links (i.e., N+1 monomers) in
the polymeric chain
 The orientations of the links are
independent
 The end-to-end vector is simply (|a| is the
length of the links):
N 

r   ai
i 1
Freely Jointed Chain (2/2)
 The mean end-to-end distance is:
 
r r 
 Na 
2

 
i 1

ai 
i j
 
ai  a j

N
N
j 1

aj

 For the freely jointed (uncorrelated) chain, the
second (cross) term of the equation is 0. Thus
<r2>=Na2, or Dr～N1/2 (|r|=0)
 The overall size of a random walk is
proportional to the square root of the number of
steps
Distribution of the End-to-End
Distance － Gaussian
 The probability density distribution
function is given by:
 2Na

P(r , N )  
 3
2




3 / 2

3r
exp  
2
 2 Na
2




Proof of the Gaussian
Distribution (1/4)
 Consider a walk in one dimension first: ax
is the step length, N+(N-) is the forward
(backward) steps, and total steps
N=N++N After N steps, the end-to-end distance
Rx=(N+-N-)ax
 The probability of this Rx is given by:
x 
N!
N ! N !
Proof of the Gaussian
Distribution (2/4)
 Using the Stirling’s approximation for very large
N: lnx! ~ Nlnx-x and define f=N+/N we get
ln  x ~ N ln N  N  ln N   N  ln N 
  N  f ln f  (1  f ) ln( 1  f )
 This function is sharply maximized at f=1/2.
That is, the probability far away from this f is
much smaller
Proof of the Gaussian
Distribution (3/4)
 Use the Taylor expansion near f=1/2:
1
 d ln  x
ln  x ( f ) ~ ln  x  f  1 / 2     f 
2
 df
2
2

d
ln  x
11

   f  
2
22
  df



 f 1/ 2


 f 1/ 2
Proof of the Gaussian
Distribution (4/4)
 At f=1/2, the first derivative equals to 0
and the second derivative equals to -4N:
1

ln  x ( f ) ~ N ln 2  2 N   f 
2

2

Rx
  x  exp  
2
2
Na
x

 For 3D,
 3R 2
  exp  
2
 2 Na








2
Configurational Entropy
 Since the entropy is proportional to the log of the
number of the microscopic states (→ the probability),
the entropy comes from the number of possible
configuration is:
2
3k B r

S (r )  
 const .
2
2 Na
 The free energy is thus increased
2
3k BTr

F (r )  
 const.
2
2 Na
 Thus a polymer chain behaves like a spring
 The restoring force comes from the entropy rather than
the internal energy.
Real Polymer Chain － Short
Range Correlation (1/4)
 The freely jointed chain model is unphysical
 For example, the successive bonds in a
polymer chain are not free to rotate, the bond
angles have definite values
 A model more realistic: the bonds are free to
rotate, but have fixed bond angles q
q
Real Polymer Chain － Short
Range Correlation (2/4)
 Now the cross term becomes nonzero:
 
2
m
ai  ai m  a cos q
 Since |cosq| ≦ 1, the correlation decays
exponentially
 <ai‧ai-m> can be neglected if m is large enough,
say m ≧ g
 Let g monomers as a new unit of the polymer,
the arguments for the uncorrelated polymers
are still valid
Real Polymer Chain － Short
Range Correlation (3/4)
 Let ci denotes the end-to-end vector of the i-th
subunit
 Now there is N/g subunits of the polymer
 From the free jointed chain model we get:
r
2

N
c
2
 Nb
g
2
 Here b is an effective monomer size, or the
statistical step length
 The effect of the correlation can be
characterized by the “characteristic ratio”:
C  b / a
2
2
Real Polymer Chain － Short
Range Correlation (4/4)
 From the discussions above we see
 The long-range structure (the scaling of the
chain dimension with the square root of the
degree of N) is given by statistics
 This behavior is universal – independent of
the chemical details of the polymer
 All the effects of the details go into one
parameter – the effective bond length
 This parameter can be calculated from
theory or extracted from experiments
Real Polymer Chain –
Excluded Volume
 In the previous discussions, interactions
between distant monomers are neglected
 The simplest interaction: hard core
repulsion – no two monomers can
occupy the same space at the same time
 This is a long-range interaction which
may causes long-range correlation of the
shape of the chain
Real Polymer Chain –
Excluded Volume
 There are N monomers in the space with
volume V=r3
 The concentration of the monomers c ~
N/r3
 If the volume of the monomer is v, the
total accessible volume becomes V-Nv
Entropy Change from the
Excluded Volume
 Entropy for ideal gas Sideal  k B ln( aV / N )
 Due to the volume of the monomers v, the
number of possible microscopic states is
reduced
Nv 
 a (V  Nv) 

S  k B ln 
  S ideal  k B ln 1 

N
V 



~ Sideal  k B ( Nv / V )
Free Energy Change of the Polymer
Chain with Excluded Volume
 Thus the free energy will be raised (per
particle):
F  Fideal  k BT ( Nv / V )
 Elastic free energy contributed from the
configurational entropy:

Fel (r ) 
3k BTr
2 Na
2
2
 The total free energy is the summation of these
two terms
 Minimizing the total free energy we get r  N 3 / 5
 The experimental value of the exponent is ~
0.588