Solutions are due at the start of class on Thur 2/12

1. [15 pts] This question involves use of MATLAB to generate trajectories for random walks in three-dimensions.
   

   Generate ensembles of at least 1000 trajectories for an ideal random walk and a self-avoiding random walk (monomers cannot revisit an occupied point in space) on a cubic lattice. Make calculations for N=100, 500, 1000, 5000 and 10000 monomers or steps. The chain is allowed to take steps of unit length in any of the x, y or z directions.

   1. Provide a plot and show how < R² > scales with N in each case. Recall that

   2. What is the probability distribution function P(R) for N=10000?
   3. Calculate the radius of gyration, < R_g² > and show how it varies with N. How does < R² > ∕ < R_g² > vary with N? Recall that

   4. Subdivide your ideal random walk N = 10000 chain by considering only every n^th monomer where n = 2,4,8,32 and 128. Are the reduced trajectories still Gaussian in their statistics? At what n do they deviate?
2. [20 pts]
   

   Consider the pairwise interaction between uncharged colloidal particles, mediated by a polymer chain attached at their surfaces along a line joining their centers. We examine a linear arrangement of three such particles, as shown in Figure 1.

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   Figure 1: Colloidal particles joined by polymer chains

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   The attractive interaction between spherical particles due to Van der Waals forces is written as

   

   where s is the separation between the surfaces of the spheres (t - 2r in Figure 1), R is the radius of the spheres and A is the Hamaker constant. A good value for the Hamaker constant is 5E - 20 J or about 10 kT at room temperature. For R ≫ s, this expression reduces to

   
   1. Make a plot of the inter-particle potential (in units of kT) between the middle and either end particle due solely to the Van der Waals interaction, out to a center-center distance of 10r.
   2. Consider the displacement of the center particle away from equilibrium (x = 0). The free energy change associated with the change in the end-end dimensions of the left and right polymers modifies the interaction potential between the particles. Let’s examine polymers with R_g = r,r∕2 and r∕100, separated by t = 4r,8r and 16r (9 cases).
      1. Construct and plot the modified potentials (i.e. add the contribution from the polymer chains to the potential from part I above) under the assumption that the chains are Gaussian. That is, they follow force-displacement relationships of the form

         where < R₀ > is the unperturbed end-end distance of the chain.

      2. Construct and plot the modified potentials under the assumption that the chains are worm-like. That is, they follow force-displacement relationships of the form

         given R_max = 50b. In the WLC model, b = 2l_p and the radius of gyration is given by

         

   3. What other factors could be considered in this picture to make it more realistic?
3. [10 pts] Problem 2.12 from the text
   

   Consider a polymer containing N Kuhn monomers (of length b) in a dilute solution where ideal chain statistics apply. The molar mass of the polymer is M.

   1. What is the mean-square end-to-end distance R₀² of the polymer?
   2. What is the fully extended length R_max?
   3. What is the mean-square radius of gyration R_g² of this polymer?
   4. Estimate the overlap concentration c^* for this polymer, assuming that the pervaded volume of the chain is a sphere of radius R_g
   5. How does this overlap concentration depend on the degree of polymerization
   6. What is the ratio of its fully extended length to the average root mean square end-to-end distance R_max∕R₀?
   7. Consider an example of a polymer with molar mass M = 10⁴ g/mol. consisting of N = 100 Kuhn monomers (of length b = 10Å and determine R₀, R_g, R_max, c^* and R_max∕R₀.
4. [5 pts]
   

   The concept of the tension blob was introduced to make a simple scaling argument for the force-extension response of a Gaussian chain. Describe the concept and its utility in your own words.