Adsorption of a single chain

For the final example comparing the properties of ideal and real chains, consider a polymer in dilute solution near a weakly adsorbing surface. Let 

The thickness ads of the adsorbed layer defines the adsorption blob size (see Fig. 3.12), This adsorption blob size is the length scale on which the cumulative interaction energy of a small section of the chain with the sur-face is of the order of the thermal energy kT. On smaller length scales, the 

In order to calculate the size of the adsorption blob ads we need to calculate the number of monomers in contact with the surface for a chain section of size ads- The average volume fraction in a chain section of size Cads containing gads monomers is 0  

The number of monomers in each adsorption blob that are in direct contact with the surface (within a layer of thickness b from it) is estimated as the product of the mean-field number density of monomers in the blob 0/6 and the volume of this layer within distance b of the surface, Cads  

The energy gain per monomer in contact with the surface is 6kT. Therefore, the energy gain per adsorption blob is 

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This chapter is concerned with the application of liquid state methods to the behavior of polymers at surfaces. The focus is on computer simulation and liquid state theories for the structure of continuous-space or off-lattice models of polymers near surfaces. The first computer simulations of off-lattice models of polymers at surfaces appeared in the late 1980s, and the first theory was reported in 1991. Since then there have been many theoretical and simulation studies on a number of polymer models using a variety of techniques. This chapter does not address or discuss the considerable body of literature on the adsorption of a single chain to a surface, the scaling behavior of polymers confined to narrow spaces, or self-consistent field theories and simulations of lattice models of polymers. The interested reader is instead guided to review articles [9-11] and books [12-15] that cover these topics. 

At low concentrations, adsorption is a single-chain phenomenon. The adsorption takes place when the enthalpy gain by the monomer-surface contact with respect to the monomer-solvent contact surpasses the loss of the conformational entropy. In a good solvent the adsorption is not likely unless there is a specific interaction between monomers and the surface. At high concentrations, however, interactions between monomers dominate the free energy of the solution. The adsorption takes place when the enthalpy gain by the mono-... 

Real polymer processes involved in polymer crystallization are those at the crystal-melt or crystal-solution interfaces and inevitably 3D in nature. Before attacking our final target, the simulation of polymer crystallization from the melt, we studied crystallization of a single chain in a vacuum adsorption and folding at the growth front. The polymer molecule we considered was the same as described above a completely flexible chain composed of 500 or 1000 CH2 beads. We consider crystallization in a vacuum or in an extremely poor solvent condition. Here we took the detailed interaction between the chain molecule and the substrate atoms through Eqs. 8-10. 

For the polymer to be effective, it must adsorb to the interface and maintain a certain configuration. Thus the following discussion describes various experimental techniques used for the study of adsorption density and configuration of polymer at the interface. After adsorption occurs, the main mechanisms of flocculation are due to the adsorption of a single polymer molecule on separate particles, interaction through the interpenetration of adsorbed polymer, and interactions due to the loss of freedom of movement of the polymer chains. 

Using this C picture, the elastic force produced by the stretching of a single chain is /ei — FvTb/, the total number of chains per unit area is wo = and among these chains, only a fraction Tb/( tb + ff) are in an adsorbing state. The frictional stress due to the adsorption of polymer chains is given as [48, 65] ... 

Reinikainen, T. et al.. Comparison of the adsorption properties of a single-chain antibody fragment fused to a fungal or bacterial cellulose-binding domain. Enzyme Microb. Technol., 20, 143,1997. 

This slight hysteresis of traces one, three, four and six, may be due to backfolding of the chain on itself, interaction of a second chain picked up at low extension or possibly even nonspecific adsorption of the single chain to surface at low extension and at a position along the chain of within 100 nm of extended length from the attachment point. Regardless of the source of the barely detectable non-overlap of several of the traces, ideal (entropic) elasticity has been... 

Milchev et al. [74, 75] found that it makes a big difference whether one studies the case in which in equilibrium a chain is not yet absorbed on the tram side (and still in a mushroom state when the chain gets through the pore) or whether adsorption occurs. In the first case, the problem is similar to unbiased translocation (which occurs by thermal fluctuations only [81, 82]), i.e., for any finite fraction of monomers that have already passed to the tram side there is still a non-zero probability that the whole chain returns to the cis side (and diffuses away). In this case, the translocation time is found to scale as t oc N N -, i.e., the time is simply of the order of the Rouse time of a single chain in a good solvent (note that the Monte Carlo modeling of Milchev et al. [74,75] uses implicit solvent... 

After having discussed the adsorption behavior of a single chain, a word of caution is in order. Experimentally, one never looks at single chains adsorbed to a siuface. First, this is because one always works with polymer solutions, where there is a large number of polymer chains contained in the bulk reservoir, even when the bulk monomer (or polymer) concentration is quite low. Second, even if the bulk polymer concentration is very low, and in fact so low that polymers in solution rarely interact with each other, the siuface concentration of polymer is enhanced relative to that in the bulk. Therefore, adsorbed polymers at the smface usually do interact with neighboring chains, owing to the higher polymer concentration at the siuface [34]. 

Nevertheless, the adsorption behavior of a single chain serves as a basis and guideline for the more complieated adsorption scenarios involving many-chain effects. It will turn out that the scaling of the adsorption layer thickness D and the proximal volume fraction profile, Eqs (11) and (12), are not affected by the... 

Even with all the simplifying assumptions, the emerging physical picture is quite rich and robust. Adsorption of polymers from dilute solutions can be understood in terms of a single-chain adsorption on the substrate. Mean-field theory is quite successful, but in some cases fluctations in the local monomer concentration play an important role. Adsorption from more concentrated solutions offers rather complex and rich density profiles, with several regimes (proximal, central, distal). 

Then we extended the 2D-model to a 3D one [21]. We considered crystallization of a single polymer chain C500 from a vapor phase onto a solid substrate, taking into account detailed interactions between the chain and the substrate. Though the polymer molecule in a vacuum was collapsed, like in a very poor solvent, under the influence of bare van der Waals interactions between atoms, the molecule was found to show quick adsorption and crystallization into a rather neat chain folded lamella. 

These assumptions are partially different from those introduced in our previous model.10 In that work, in fact, in order to simplify the kinetic description, we assumed that all the steps involved in the formation of both the chain growth monomer CH2 and water (i.e., Equations 16.3 and 16.4a to 16.4e) were a series of irreversible and consecutive steps. Under this assumption, it was possible to describe the rate of the overall CO conversion process by means of a single rate equation. Nevertheless, from a physical point of view, this hypothesis implies that the surface concentration of the molecular adsorbed CO is nil, with the rate of formation of this species equal to the rate of consumption. However, recent in situ Fourier transform infrared (FT-IR) studies carried out on the same catalyst adopted in this work, at the typical reaction temperature and in an atmosphere composed by H2 and CO, revealed the presence of a significant amount of molecular CO adsorbed on the catalysts surface.17 For these reasons, in the present work, the hypothesis of the irreversible molecular CO adsorption has been removed. 

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