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Polymers PART.2

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

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Polymers PART.2

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  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 <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

  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: 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:

  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 q q

  12. Real Polymer Chain － Short Range Correlation (2/4) • Now the cross term becomes nonzero: • 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

  13. 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: • 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=r3 • The concentration of the monomers c ~ N/r3 • 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

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