1. Polymers PART.2 Soft Condensed Matter Physics Dept. Phys., Tunghai Univ. C. T. Shih

2. 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)

3. 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):

4. Freely Jointed Chain (2/2)  The mean end-to-end distance is:  For the freely jointed (uncorrelated) chain, the second (cross) term of the equation is 0. Thus =Na 2, or  r ～ N 1/2 (|r|=0)  The overall size of a random walk is proportional to the square root of the number of steps

5. Distribution of the End-to-End Distance － Gaussian  The probability density distribution function is given by:

6. Proof of the Gaussian Distribution (1/4)  Consider a walk in one dimension first: a x 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 R x =(N + -N - )a x  The probability of this R x is given by:

7. 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  This function is sharply maximized at f=1/2. That is, the probability far away from this f is much smaller

8. Proof of the Gaussian Distribution (3/4)  Use the Taylor expansion near f=1/2:

9. Proof of the Gaussian Distribution (4/4)  At f=1/2, the first derivative equals to 0 and the second derivative equals to -4N:  For 3D,

10. 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:  The free energy is thus increased  Thus a polymer chain behaves like a spring  The restoring force comes from the entropy rather than the internal energy.

11. 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  

12. Real Polymer Chain － Short Range Correlation (2/4)  Now the cross term becomes nonzero:  Since |cos  | ≦ 1, the correlation decays exponentially  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

13. Real Polymer Chain － Short Range Correlation (3/4)  Let c i 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:  Here b is an effective monomer size, or the statistical step length  The effect of the correlation can be characterized by the “characteristic ratio”:

14. 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

15. 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

16. Real Polymer Chain – Excluded Volume  There are N monomers in the space with volume V=r 3  The concentration of the monomers c ~ N/r 3  If the volume of the monomer is v, the total accessible volume becomes V-Nv

17. Entropy Change from the Excluded Volume  Entropy for ideal gas  Due to the volume of the monomers v, the number of possible microscopic states is reduced

18. Free Energy Change of the Polymer Chain with Excluded Volume  Thus the free energy will be raised (per particle):  Elastic free energy contributed from the configurational entropy:  The total free energy is the summation of these two terms  Minimizing the total free energy we get  The experimental value of the exponent is ~ 0.588