Polymers PART.2

1
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., N1 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
  ltr2gtNa2, or DrN1/2 (r0)
• 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 ax is the
  step length, N(N-) is the forward (backward)
  steps, and total steps NNN-
• After N steps, the end-to-end distance
  Rx(N-N-)ax
• The probability of this Rx is given by

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Proof of the Gaussian Distribution (2/4)

• Using the Stirlings approximation for very large
  N lnx! Nlnx-x and define fN/N we get
• This function is sharply maximized at f1/2. 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 f1/2

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Proof of the Gaussian Distribution (4/4)

• At f1/2, 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 q

q
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Real Polymer Chain - Short Range Correlation (2/4)

• Now the cross term becomes nonzero
• Since cosq ? 1, the correlation decays
  exponentially
• ltai?ai-mgt 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 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

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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
  Vr3
• The concentration of the monomers c N/r3
• 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