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Polymers PART.2 Soft Condensed Matter Physics Dept. Phys., Tunghai Univ. C. T. Shih
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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)
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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):
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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
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Distribution of the End-to-End Distance - Gaussian The probability density distribution function is given by:
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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:
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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
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Proof of the Gaussian Distribution (3/4) Use the Taylor expansion near f=1/2:
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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,
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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.
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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
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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
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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”:
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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
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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
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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
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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
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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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