# Hybrid Colloids Made with Polymers
## 8 Polymer Wrapping  
The [wrapping/protruding] ratio is >0; the partition implies that the two states differ in energy. Surface-bound segments are termed \(\alpha\) (\(\leq\)1), the protruding ones (trains) \(\varepsilon\). \(X_{b}\) is the amount of bound polymer, which moves as a whole kinetic entity with _NPs_. Multilayer adsorption is possible. Wrapping can be analyzed by Langmuir-like isotherms [59]. It applies to _SiO_, _TiO_2, carbon nanotubes, _CNTs_, graphenes, latexes, droplets, vesicles, etc. Wrapping holds, therefore, in both hard- and soft-matter matrices. It is assumed that:  
1. The polymer length is \(\leq\)_NPs_' diameter, and wrapping length is lower than the extended polymer one. Different parts of the same polymer chain may wrap;
2. Polymers and _NPs_ are size mono-disperse;
3. Polymer partition between bulk and surface states is possible, and can be substantial. The amount of surface-bound polymer is lower than \(X_{tot}\), its overall mole fraction in the medium. This implies some partition with the bulk. Polymer chains consist in loops, \(\alpha\), whose physicochemical properties differ from those of the protruding ones, \(\varepsilon\); T is the number of segments in a single polymer chain. The interaction energy of _NP_-polymer systems differs from those of polymer-polymer and polymer-solvent ones. The solvent is ubiquitous in the coronas around _NPs_. From the amount of free and bound polymer, one obtains the equilibrium between \(\alpha\) and \(\varepsilon\) states. Polymer affinity towards _NPs_ is related to the forces acting among protruding chains. The statistical model for homopolymers wrapping considers the chain as a ribbon of constant width. Several equivalent polymer units adsorb on the particle, and form domains containing \(l_{i}\) subunits (with \(l_{i}\) > 1). The adsorption energy obtained by the model, \(E_{ad}\), is proportional to the number of wrapped segments. The weight of each state is calculated from the distribution function. For the most probable number of \(l_{i}\) values, a maximum in the probability occurs (Figure 6). Wrapping can be cooperative, as in _ss-DNA_ wrapping onto _CNTs_[62], or when poly-electrolytes adsorb on oppositely charged _NPs_[63; 64]. These processes occur because attractive terms among adsorbed chains are much higher than repulsive ones, and depend, thus, on the polymer affinity towards _NPs_. Different block co-polymers are used in wrapping. Pluronics _F127_, with its peculiar _PEO_-_PPO_-_PEO_ sequence [65], is a good example. _F127_ has different blocks in separate parts of the chain; its sequence-sensitive _HLB_ (hydrophilic-lipophilic balance) depends on the molecular details. In this system, wrapping is related to _HLB_, \(T\), and to the _PPO/PEO_ number ratios. Poly-(L-lysine)-_PEO_ block copolymers show a similar behavior [66]; there, wrapping depends also on the length and charge of the polypeptide moiety. The _PEO-SiO\({}_{2}\)_ system is an excellent model to quantify wrapping efficiency. This is because both components are finely mono-dispersed. In the following, we consider _SiO\({}_{2}\)_ sizes of 200 nm, and 50-60 _EO_ units in the range; the polymer length is nearly equivalent to the _NPs_ diameter. The conformational energy is hardly quantified. The hydration one equals the transfer energy from water to the _NP_ surface, and is about 5.5 kJ per _EO_ unit [67]. Wrapping can be continuous (the _EO_ units adsorb with continuity) or random. The polymer releases water upon adsorption; the process is similar to micelle formation. The reason is that wrapping is a phase transition from a hydrated to a less hydrated regime. The number of surface bound units (usually 8-10 _EO_ groups) comes from the maximum of the distribution function in Figure 6. For the _PEO-SiO\({}_{2}\)_ system, the number of wrapping units gives an interaction energy, \(W\), in the range of 45 \(\pm\) 5.0 kJ mol\({}^{-1}\). Polymers increase the medium osmotic pressure, modulated by interactions with _NPs_.