1 / 18

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

phipp
Download Presentation

Polymers PART.2

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related