## 1 Introduction
For example, poly(ethylene oxide) (PEO), having a molecular weight (_Mw_) of 5,000,000, entangled with proteins, contributing to their partitioning [34]. The entanglement features of polymers determine peptide partitioning as well as polymer-polymer interaction [35]. The entanglement of polymers or polymer-peptide originates from the fact that polymers with long-chain structures cannot cross over each other. In the field of polymer chemistry and physics, the entanglement of polymers has been accordingly characterized by the entanglement molecular weight (\(M_{\text{e}}\)), which corresponds to the _Mw_ between entanglements. The \(M_{\text{e}}\) value can be estimated from the viscoelastic modulus [11; 36] and \({}^{1}\)H-NMR [37]. The physicochemical properties of polymers at the interface are generally different from those of the bulk polymer phase [38]. This is (i) because polymers in the random coiled state collapse, and thus, charge a higher elastic energy, and (ii) because of the constraint of self-diffusion of polymers due to entanglement [39; 40]. This property is prominent in polymer membranes prepared by the layer-by-layer method [41]. If this effect would result in an interfacial environment that is advantageous for the binding of amphiphilic molecules, the induction of various morphologies of A\(\beta\) fibrils [29] might be explained. However, the influence of the entanglement between polymers or between polymers and peptides on peptide binding is still unknown. It is anticipated that the influence of the entanglement of polymers at the interface would provide helpful insights into the development of biocompatible polymers for medical devices, such as artificial organs or biosensors, with low protein adsorption. In this study, we selected various polymers, viz., PVP, PLLA, PMMA, and PSf, as shown in Figure 1. These linear polymers were selected to discuss the effect of their \(M_{\text{e}}\)[5; 38; 39; 42] (Table 1) and hydration structure [16; 23; 43; 44; 45]. PLLA with different \(Mw\) (5000 to 50,000) was used to examine the effect of the entanglement of polymers while avoiding the influence of its chemical structure. PVP and PSf were selected as hydrophilic and hydrophobic standard polymers, respectively. Syndiotactic-PMMA (\(syn\)-PMMA) and isotactic-PMMA (\(iso\)-PMMA) were used to investigate the influence of the tacticity of the polymers. Phospholipids, viz., 1,2-dioleoyl-phosphatidylcholine (DOPC), 1,2-dipalmitoyl-phosphatidylcholine (DPPC), and phosphoethanolamine-\(N\)-[Methoxy(Polyethylene Glycol)] (PEG-PE) were also used for comparison with polymer membranes, because the planar phospholipid membranes have been regarded as model biomembranes. In the first series of experiments, the fibril formation behavior of A\(\beta\) on various planar polymer membranes was investigated in terms of the morphology of fibrils and kinetics according to previous reports [28] using total internal fluorescence microscopy (TIRFM) [29]. Kinetic analysis of fibril formation was performed according to a previous report [28] to obtain the kinetic parameters, such as the nucleation rate. The nucleation rate was compared with A\(\beta\) accumulation on the planar membrane. Here, A\(\beta\) accumulation was estimated using a quartz crystal microbalance (QCM) following immobilization of planar membranes [23]. Finally, we discussed the interaction of A\(\beta\) with planar polymer membranes in terms of entanglement and hydration structure. Figure 1: Chemical structure of phospholipids and polymers used herein.