Brush thickness

If the turbulent flame is ever proven to have asymptotically a constant flame brush thickness and constant speed in constant, i.e., nondecaying, turbulence, then the aforementioned turbulent flame speed and the flame brush thickness (5 give a well-defined sufficient characterization of the flame in its asymptotic behavior. However, it is not proven up to now that the studied experimental devices have been large enough to ensure that this asymptotic state can be reached. Besides, the correct definitions for the turbulent flame speed or flame brush thickness, as given above, are far from... 

Temporal evolution of the flame brush thickness for the previously described mixtures of hydrogen, methane, and propane with air. (Reproduced from Renou, B. and Boukhalfa, M., Combust. Set. Technol., 162, 342 2001. With permission. Figure 2, p. 353, copyright Gordon Breach Science Publishers (Taylor and Francis editions).)... 

The brush thickness and the axial distance per side chain are interrelated due to the condition of fixed monomer density r in the collapsed brush as R = Vn/ 11 h. The axial tension is given by... 

Recently, Quirk and Mathers [264] performed LASIP of isoprene on silicon wafers. A chlorodimethylsilane-functionalized diphenylefhene (DPE) was coupled onto the surface and lithiated with n-BuLi to form the initiating species. The living poly(isoprene) (PI) was end- functionalized with ethylene oxide. A brush thickness of 5 nm after two days of polymerization (9.5 nm after four days) was obtained in contrast to a polymer layer thickness of 1.9 nm by the grafting onto method using a telechelic silane functionahzed PI. 

Fig. 19 Dependence of the brush thickness reduced by the number of polymer repeat units for monovalent co-ions, H/AT, on the concentration of the external salt, ( s> for strong (solid line) and weak (dashed line) polyelectrolyte brushes in neutral brush (NB), salted brush (SB), and osmotic brush (OB) regimes, a and ao denote the bulk and internal (for weak polyelectrolyte brushes only) degree of dissociation, respectively (Reproduced with permission from [89])...

Poly(methyl methacrylate) with a variable degree of polymerization anchored to silica surfaces was synthesized following the room temperature ATRP polymerization scheme described earlier [45,46]. In the main part of Fig. 25 we plot the variation of the PMMA brush thickness after drying (measured by SE) as a function of the position on the substrate. Thickness increases continuously from one end of the substrate to the other. Since the density of polymerization initiators is (estimated to be 0.5 chains/nm ) uniform on the substrate, we ascribe the observed change in thickness to different lengths of polymer chains grown at various positions. 

Fig. 26 Dry thickness of poly(acryl amide) as a function of the position on the silica substrate prepared by slow ( ) and fast ( ) removal of the polymerization solution by utilizing the method depicted in Fig. 24. The inset shows the dry poly(acryl amide) thickness as a function of the polymerization time. Note that both data sets collapse on a single curve at short polymerization times. Regardless of the drain speed, the brush thickness increases linearly at short polymerization times and levels off at longer polymerization times. The latter behavior is associated with premature termination of the growing polymers...

These results illustrate the intimate interplay between the particle size and the brush thickness that determines the spatial distribution of the particles inside the polymer brushes. 

Correlation of Brush Thickness with Molecular Weight. 135... 

Table 2 Average AFM domain diameter vs. diblock brush thickness [30]...

Filled Brush thickness after both blocks Open Brush thickness after first block... 

Alexander s treatment, the grafted brush is assumed to have a uniform segment density (step function distribution), and each chain to consist of connected semidilute blobs. The osmotic repulsion between blobs tends to stretch the chains, while the elastic free energy of the chains has the opposite effect. For a single brush, the minimization of the overall free energy with respect to the brush thickness yields the equilibrium brush thickness L0, given by... 

However, as shown in Figure la, the approximation of the thickness of the brush by the length of the most probable configuration is in many cases arbitrary, since configurations that end up at different distances can contribute notably to the brush thickness. In addition, the derivation of eq 14 implies that the random walk can be described by a Gaussian extended from z = 0 to z = oo,... 

The present model is based on several assumptions (i) the possible configurations of the grafted chain are described by a random walk (ii) their free energy densities are expressed as functions of the local monomer volume fraction alone (iii) the configurations of minimum energy dominate the partition function of the system (iv) only the configurations with monomers distributed between the surface and the position of the last monomer of the chain, assumed to be the farthest one, are taken into account. The latter assumption basically implies that the probability that the most distant monomer from the surface reaches the distance z is equal to the probability that the last monomer of the chain reaches this distance this approximation clearly fails when z is in the vicinity of the surface. However, in swollen brushes the behavior of the monomers in the vicinity of the surface is less important than the behavior of the distant monomers, which are primarily responsible for the brush thickness and for the interactions between brushes. 

Using a self-consistent field theory, Misra et al. [10] examined the effect of the brush charge, electrolyte concentration, and surface charge density on the brush thickness. They extended the self-consistent field polymer brush theory suggested by Milner et al. [11] to the case of a polyelectrolyte brush. The theory involves a parabolic monomer concentration profile rather than the step-function suggested by Alexander [12] and de Gennes [13,14], The repulsion force... 

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