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END TO END DISTANCE [Sperling-173] [Flory-399] [Williams-234] Flory defines two cases of end to end distance: Displacement Length- distance between one end of the polymer chain and the other for a coiled polymer Contour Length- distance between one end of the polymer chain and the other when the polymer is stretched out (maximum possible displacement length) In the above figure both polymer chains have a degree of polymerization of 12. There are two types of calculations: The first is the random walk, which uses the correct bond length for the monomer, but ignores the angle between the one monomer line and the next, as shown in the figure below: [Flory-414] and [Hiemenz-52] show which Flory refers to as a 'freely jointed chain.' [Williams-230] shows which he refers to as a 'freely orienting chain model.' h (or r) is the end-to-end distance, or the displacement length. n (or sigma) is the number of monomers in the polymer chain l is the length of the monomer The second builds in the correct bond angles. Notice that in the pictures above for displacement length and contour length that each angle is 120 degrees. There are two angles to consider; in our two dimensional representation you can only see one. If there were just θ then we could orient the first monomer in the plane of the screen, and the polymer would stay in the plane of the screen. This angle is defined by three atoms. The angle φ is called a dihedral or torsional angle, and it gives the polymer the three dimensional character. This angle is defined by four atoms. (Sperling-74) (Flory-418) (Hiementz-58)