Radii of Gyration for Polyethylene

The root-mean-square radius of gyration can be estimated by a variety of methods which make different assumptions about the behavior of the chain.

The extremes are defined by the first and last model summarized below.

Only the fourth and fifth examples below require specification of the temperatures.

Using a polyethylene chain with M = 100,000, the results with several approximations are:

• Minimum radius of gyration: 3.3 nm, calculated using a partial specific volume of 1.43 cm³/g.
• Freely jointed chain: 5.3 nm, calculated using a bond length of 0.154 nm and a molecular weight per bond of 14.
• Freely rotating chain: 7.5 nm, calculated using a bond length of 0.154 nm, a molecular weight per bond of 14, and a tetrahedral bond angle.
• Chain with symmetric hindered rotation about independent bonds: 10 nm, calculated using a bond length of 0.154 nm, a molecular weight per bond of 14, a tetrahedral bond angle, three states with torsion angles of 180, 60, and -60 deg, an energy of 2 kJ/mol for the two gauche states, and T = 140 deg C.
• Chain with pairwise interdependent bonds, via the rotational ismeric state model: 14 nm, calculated using a bond length of 0.154 nm, a molecular weight per bond of 14, a tetrahedral bond angle, three states with torsion angles of 180, 60, and -60 deg, an energy of 2 kJ/mol for the two gauche states, an energy of 8 kJ/mol for the pentane effect, and T = 140 deg C. This result is closest to the result that would be obtained for the chain in dilute solution in a Theta solvent. In a good solvent, the dimensions would be larger.
• Approximated as a fully extended chain: 183 nm, using a bond length of 0.154 nm, a molecular weight per bond of 14, and a tetrahedral bond angle.

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