Closed‐ F orm Expressions for the Form Factor of a Polymer under Strong Confinement

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A recent model for unentangled polymer chains in confinement [M. Dolgushev and M. Krutyeva, Macromol. Theory Simul. 21, 565 (2012)] is scrutinized by small-angle neutron scattering (SANS) with respect to its static prediction, the single-chain structure factor. We find a remarkable agreement although the model simplifies the effect of the confinement to a harmonic potential. The effective confinement size from fits of SANS data with the model agrees well with the actual pore size. Starting from this result we discuss the possibility of an experiment on the dynamic structure factor predicted by the model. It turns out that such an experiment would need a large ratio polymer dimension/pore size which is difficult but not impossible to achieve.

“…A comparison of the mean square distance of the monomers from the axis of the channel, which is ρ 2 = R 2 cyl /2 in the case of a cylinder, with the value calculated in Ref. 5…”

Section: Recap Of Current Theoriesmentioning

confidence: 88%

Section: Leads Tomentioning

confidence: 97%

Description of poly(ethylenepropylene) confined in nanopores by a modified Rouse model

Muthmann

Krutyeva

Willner

et al. 2017

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Structure and dynamics of confined flexible and unentangled polymer melts in highly adsorbing cylindrical pores

Carrillo

Sumpter

2014

Coarse-grained molecular dynamics simulations are used to probe the dynamic phenomena of polymer melts confined in nanopores. The simulation results show excellent agreement in the values obtained for the normalized coherent single chain dynamic structure factor, S(Q,Δt)/S(Q,0). In the bulk configuration, both simulations and experiments confirm that the polymer chains follow Rouse dynamics. However, under confinement, the Rouse modes are suppressed. The mean-square radius of gyration ⟨R(g)(2)⟩ and the average relative shape anisotropy ⟨κ(2)⟩ of the conformation of the polymer chains indicate a pancake-like conformation near the surface and a bulk-like conformation near the center of the confining cylinder. This was confirmed by direct visualization of the polymer chains. Despite the presence of these different conformations, the average form factor of the confined chains still follows the Debye function which describes linear ideal chains, which is in agreement with small angle neutron scattering experiments (SANS). The experimentally inaccessible mean-square displacement (MSD) of the confined monomers, calculated as a function of radial distance from the pore surface, was obtained in the simulations. The simulations show a gradual increase of the MSD from the adsorbed, but mobile layer, to that similar to the bulk far away from the surface.

Structure factor of a Gaussian chain confined between two parallel plates

Liao

Miao

2015

We study the structure factor of a single Gaussian chain confined between two macroscopic parallel plates theoretically. The chain propagator is constructed in terms of the eigen-spectrum of the Laplace operator under the Dirichlet boundary condition enforced at the two plates, by which the confinement effect enters the treatment through size-dependent eigen-spectrum. In terms of the series expansion solution for the chain propagator, we first calculate the confinement free energy and the confinement force for an arbitrary confinement strength. It is found that the confinement force scales to the distance between the two confining surfaces with a power of -3 for strong confinements and of -2 for weak confinements. Based on the ground state dominance approximation for strong confinements and the Euler-Maclaurin formula for weak confinements, we develop approximation theories for the two limit situations, which agree with the numerical results well. We further calculate the structure factor of the confined Gaussian chain in this slit geometry. While the scattering function of the transverse chain fluctuations perpendicular to the confinement direction is still a Debye function form, the structure factor for the longitudinal fluctuations along the confinement dimension starts with the monotonic Debye function behavior for weak confinements and develops a decaying oscillation behavior with the increase of confinements. The numerical results for the structure factor are also interpreted by developing approximation theories in different confinement regimes. Finally, the orientational average of the anisotropic structure factor is performed and an analytic expression for the averaged structure factor is derived under the ground state dominance approximation for strong confinements.