Module 7 Single chain conformation

How to characterize the size of a polymer chain? How does the size change with increasing degree of polymerization?

2 Module objectives

  • Derive the correlation between the root mean square (RMS) end-to-end distance and the degree of polymerization (\(DP\)) for three chain models: freely hinged, freely rotation, and hindered rotation.
  • Compare the RMS end-to-end distances of polymers with the same \(DP\) across the three different models.
  • Understand the universal scaling rule between chain size (RMS end-to-end distance and radius of gyration) and \(DP\)$, irrespective of the chain model.

3. Classroom implementation ideas

Experiments:

Set the chain type as freely hinged. Perform a series of experiments for different degree of polymerization (\(DP\)) and record the RMS end-to-end distance (\(h\)) and radius of gyration (\(R_g\)) in the table:

\(DP\) \(h\) \(R_g\)
50
100
200
400
800

Set the bond angle as 109.5 \(^\circ\) and keep it constant. Repeat the process as described above for freely rotation and hindered rotation models.

Questions:

  1. In general, does the RMS end-to-end distance or the RMS radius of gyration have narrower distribution? Explain why.

  2. Plot \(\ln{h}\) vs \(\ln(DP)\) and \(\ln(R_g)\) vs \(\ln(DP)\), find out how do \(h\) and \(R_g\) change with increasing \(DP\). Is the trend always the same across different models?

  3. Compare the \(h\) and \(R_g\) values for different models with the same \(DP\). What trend do you find?

4. Example practice questions

  1. The sizes (the RMS end-to-end distance or the RMS radius of gyration) of a linear polystyrene sample with the average \(DP\) 10000 should be roughly \(\underline{\qquad}\) times larger than a sample with the average \(DP\) 100 based on the freely hinged model. Based on freely rotation and hindered rotation model will the answer be the same or different?

  2. Why the RMS radius of gyration usually has narrower distribution than the RMS end-to-end distance?

  3. Which statement is true?

  1. The end-to-end distance of a freely rotation chain must be greater than that of a freely hinged chain if both chains have the same \(DP\).
  2. The radius of gyration of a freely rotation chain must be greater than that of a freely hinged chain if both chains have the same \(DP\).
  3. If \(DP\) increase by a factor of 3, the RMS end-to-end distance increase by a factor of 6.
  4. If the RMS end-to-end distance of freely rotation polymer chains with \(DP\) 50 is \(c\) times larger than that of freely hinged chians with the same \(DP\), the RMS end-to-end distance of freely rotation polymer chains with \(DP\) 500 should also be \(c\) times larger than that of freely hinged chains with the same \(DP\).