Completing the dynamical regime map for slippage of entangled polymer melts on grafted-chain surfaces: short mobile chains ($P<N$), dense brushes, and ultra-long grafted chains ($N>N_{\rm e}^{2}$)

An open problem in the interfacial rheology of polymer melts: completing the quantitative dynamical regime map for the slippage of an entangled melt (mobile chains of length $P$) over a surface carrying end-grafted chains (length $N$, areal density $\nu$). The scaling theory of Brochard-Wyart, Gay and de Gennes (1996) predicted, for $P>N$, sub-brush grafting and $N<N_{\rm e}^{2}$, a cascade of regimes — linear friction, cooperative saturation of trapped chains, a marginal coil–stretch state at $V^{*}(\nu)\propto\nu$, and decoupling at $V^{**}$ — and explicitly left untreated the high-velocity regime above $V^{**}$, the $P<N$ regime, the dense-brush regime, the ultra-long-graft regime $N>N_{\rm e}^{2}$ where single-graft friction becomes Stokes-like, and the quantitative comparison between binary and collective entanglement pictures. Later work confirmed the linear $V^{*}(\nu)$ law and the saturation of the weak-slip length, established the qualitative character of the $V>V^{**}$ regime (a steady high-slip state produced by shear-induced disentanglement of brush and melt chains, without graft loss), and reproduced this molecular scenario in simulation; however, no treatment was found for $P<N$, for $N>N_{\rm e}^{2}$, for the dense-brush crossovers (including the experimentally observed non-monotonic marginal shear rate at high grafting density), for the entanglement-picture comparison, or for a quantitative theory of the decoupled state. The problem is to construct these missing parts of the map — friction laws, slip lengths and regime boundaries expressed in $N$, $P$, $N_{\rm e}$, $\nu$ and the melt viscosity — matching the already-established limits.

Background

A polymer melt is a dense liquid of long, flexible chain molecules. Long-chain melts are entangled: the chains cannot cross one another, so each chain is effectively confined to a tube-like region and relaxes primarily by reptation, a snake-like diffusion along its own contour, on a time scale $T_{\rm rep}$. The mean number of monomers between successive topological constraints along a chain is denoted $N_{\rm e}$. When such a melt flows along a solid wall, the tangential fluid velocity at the wall need not vanish: the Navier boundary condition relates the slip velocity $V$ to the shear stress $\sigma$ through an interfacial friction coefficient, $\sigma=kV$, or equivalently through the extrapolation length $b=\eta_{\rm p}/k$ (with $\eta_{\rm p}$ the melt viscosity), the distance at which the velocity profile extrapolates to zero. Entangled melts on ideal non-adsorbing walls slip strongly, but grafting polymer chains to the wall suppresses slip. For grafted chains of length $N$ (coil size $R_N=N^{1/2}a$, with $a$ the monomer size) at areal density $\nu$, two regimes are distinguished: the mushroom regime, where grafts are spaced farther apart than their coil size ($\nu<\nu_{\rm b}\sim 1/R_N^{2}$), and the brush regime above this overlap threshold. Ajdari, Brochard-Wyart, de Gennes, Leibler, Viovy and Rubinstein (1994) computed the friction of an entangled melt on dilute, independent grafted mushrooms, assuming mobile chains longer than the grafts ($P>N$). Brochard-Wyart, Gay and de Gennes (1996) extended the theory to cooperative grafted layers: a mobile chain within the interfacial skin of thickness $R_p=P^{1/2}a$ can be entangled with several grafted chains, so the areal density of such trapped mobile chains saturates at $\sim a^{-2}P^{-1/2}$; beyond a critical grafting density $\nu_{\rm c}$ the low-velocity friction per unit area stops growing with $\nu$ and the slip length saturates at $b\sim R_p$. They predicted a cascade of characteristic velocities: $V_1$ (onset of deformation of the grafted coils), $V^{*}(\nu)$ (the marginal, coil–stretch state, in which each grafted chain is drawn into a trumpet-shaped sequence of blobs of diameter equal to the tube diameter $D^{*}=aN_{\rm e}^{1/2}$, with a velocity-independent stress), growing linearly with $\nu$ in the cooperative regime, and $V^{**}=D^{*}/T_{\rm rep}(P)$, above which trapped mobile chains can no longer re-entangle with the grafts. The number $X$ of mobile chains entangled with one graft was obtained mainly in the binary-entanglement picture (pairwise topological constraints, $X=N/N_{\rm e}$ for $N<N_{\rm e}^{2}$), with the collective picture of Kavassalis and Noolandi — in which an entanglement involves a fixed number of chains sharing a pervaded volume — as the alternative. The theory assumed $P>N$, grafting below the brush threshold, and $N<N_{\rm e}^{2}$, where single-graft friction is Rouse-like (proportional to chain length) rather than Stokes-like (hydrodynamic, set by the coil size). The authors explicitly left open the high-velocity regime above $V^{**}$, the quantitative comparison of the two entanglement pictures, the regime of mobile chains shorter than the grafts ($P<N$), the dense-brush regime, and the ultra-long-graft regime $N>N_{\rm e}^{2}$. A subsequent study by Gay (1999) reformulated the interfacial friction using the bulk tube-renewal mechanism (the renewal of a chain's topological constraints by the motion of the surrounding chains), which justified the earlier friction results more precisely, predicted a progressive thinning of the grafted layer above a threshold melt velocity, and gave only a qualitative outline of the chain behaviour expected at very high grafting densities, with topological effects akin to combing; it did not supply the missing quantitative regime map.

Problem Statement

The objective is to complete the quantitative dynamical regime map of interfacial slippage for an entangled polymer melt (mobile chains of length $P$, viscosity $\eta_{\rm p}$) sheared against a plane carrying end-grafted chains (length $N$, areal density $\nu$), within the scaling framework of Brochard-Wyart, Gay and de Gennes (1996): for each dynamical regime, the friction law $\sigma(V,\nu)$ and slip length $b$, together with the crossover boundaries between regimes expressed in terms of $N$, $P$, $N_{\rm e}$, $\nu$, $\eta_{\rm p}$ and the monomer size $a$. The regimes to be established are precisely those the 1996 theory left untreated or that later experiments left unexplained: (a) mobile chains shorter than the grafted chains, $P<N$, including the crossover to the established $P>N$ laws; (b) grafting densities beyond the cooperative-saturation threshold $\nu_{\rm c}$, through the mushroom–brush overlap threshold $\nu_{\rm b}\sim 1/R_N^{2}$ and into the dense-brush regime, including the non-monotonic dependence of the marginal shear rate on $\nu$ observed at high grafting density and a quantitative treatment of the combing-like topological effects at very high grafting densities that Gay (1999) could only outline; (c) ultra-long grafted chains, $N>N_{\rm e}^{2}$, where the number of trapped chains per graft crosses from $X=N/N_{\rm e}$ to $X\sim N^{1/2}$ and single-graft friction crosses from Rouse-like to Stokes-like; (d) the dependence of these predictions on the microscopic entanglement picture — binary two-chain constraints versus the collective Kavassalis–Noolandi packing picture — that is, how the entanglement count $X$ and the resulting crossover laws differ between the two frameworks; and (e) a quantitative theory of the high-velocity ($V>V^{**}$) decoupled state, whose qualitative character — a steady high-slip state produced by shear-induced disentanglement of brush and melt chains, without graft tearing or detachment — has been established experimentally and in simulation, while a progressive thinning of the grafted layer above a threshold velocity has been predicted theoretically (Gay 1999), but whose out-of-equilibrium grafted-chain conformations, re-entanglement kinetics and friction law are not predicted. The completed map must reduce, in the limits already treated ($P>N$, $\nu<\nu_{\rm c}$, $N<N_{\rm e}^{2}$, $V<V^{**}$), to the established predictions.

References

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