A chain of N monomers is attached to a small colloidal particle, and is pulled (at a velocity V) inside a polymer melt (chemically identical, with P monomers per chain). The main parameter for this problem is the number X(V) of P chains entangled with the N chain. Earlier estimates of X are criticised in this appendix, which is based on work by A. Ajdari, F. Brochard-Wyart, C. Gay, and J. L. Viovy (1995), and a new form is proposed: at large, we are led to a ‘Stokes’ regime, X = N½, while at smaller, we find a ‘Rouse’ regime, X = N/Ne (where Ne is the number of monomers per entanglement).

The motion of a long tethered chain (N monomers) inside a polymer melt (P) is special: the N chain cannot reptate inside the P matrix. This occurs in star polymers, and also in two recent experimental situations (figure 2):

1. (a) The N chain is grafted to a colloidal particle (of size smaller than the coil radius RN of the N chain). The particle can be driven by sedimentation or by optical tweezers.

2. (b) The N chain is grafted on a flat wall, and the P melt flows tangentially to the wall (figure 15). (In all that follows, we assume that the grafting density is very small: no coupling between different Nchains.)

Problem (b) was first considered theoretically (for the low V limit) in reference [30]. The starting point is that a certain number X(V) of P chains are entangled with the N chain.