Molecular geometry and chain entanglement: parameters for the tube model

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Abstract
The tube diameter in the reptation model is the distance between a given chain segment and its nearest segment in adjacent chains. This dimention is thus related to the cross‐sectional area of polymer chains and the nearest approach among chains, without effects of thermal fluctuation and steric repulsion. Prior calculated tube diameters are much larger, about 5 times, than the actual chain cross‐sectional areas. This is ascribed to the local freedom required for mutual rearrangement among neighboring chain segments. This tube diameter concept seems to us to infer a relationship to the corresponding entanglement spacing. Indeed, we report here that the critical molecular weight, Mc, for the onset of entanglements is found to be Mc = 28 A/(〈R2〉0/M), where A is the chain cross‐sectional area and 〈R2〉0 the mean‐square end‐to‐end distance of a freely jointed chain of molecular weight M. The new, computed relationship between the critical number of backbone atoms for entanglement and the chain cross‐sectional area of polymers, Nc = A0,44, is concordant with the cross‐sectional area of polymer chains being the parameter controlling the critical entanglement number of backbone atoms of flexible polymers.