Scaled density profiles

Single chains confined between two parallel purely repulsive walls with = 0 show in the simulations the crossover from three- to two-dimensional behavior more clearly than in the case of adsorption (Sec. Ill), where we saw that the scaling exponents for the diffusion constant and the relaxation time slightly exceeded their theoretical values of 1 and 2.5, respectively. In sufficiently narrow slits, D density profile in the perpendicular direction (z) across the film that the monomers are localized in the mid-plane z = Djl so that a two-dimensional SAW, cf. Eq. (24), is easily established [15] i.e., the scaling of the longitudinal component of the mean gyration radius and also the relaxation times exhibit nicely the 2 /-exponent = 3/4 (Fig. 13). 

Theoretical predictions [73,74] concern also the monomer density profile p(z), which is predicted to scale as... 

The resolution of infra-red densitometry (IR-D) is on the other hand more in the region of some micrometers even with the use of IR-microscopes. The interface is also viewed from the side (Fig. 4d) and the density profile is obtained mostly between deuterated and protonated polymers. The strength of specific IR-bands is monitored during a scan across the interface to yield a concentration profile of species. While in the initial experiments on polyethylene diffusion the resolution was of the order of 60 pm [69] it has been improved e.g. in polystyrene diffusion experiments [70] to 10 pm by the application of a Fourier transform-IR-microscope. This technique is nicely suited to measure profiles on a micrometer scale as well as interdiffusion coefficients of polymers but it is far from reaching molecular resolution. 

The computational efficiency of the dynamics in Step 2 on the 2nnd lattice permits the study of the cohesion (and complete mixing) of two thin films [173]. Time scales for equilibration of the density profile, redistribution of chain ends, and complete intermixing of the chains span several orders of magnitude of time, expressed in MC steps, and are all accessible via the 2nnd simulation [173],... 

Computer simulations of confined polymers have been popular for several reasons. For one, they provide exact results for the given model. In addition, computer simulations provide molecular information that is not available from either theory or experiment. Finally, advances in computers and simulation algorithms have made reasonably large-scale simulations of polymers possible in the last decade. In this section I describe computer simulations of polymers at surfaces with an emphasis on the density profiles and conformational properties of polymers at single flat surfaces. 

First of all it is seen that the SCF results are free of any noise, whereas there is plenty of noise in the MD profiles (note, however, that the density profiles on both halves of the bilayer are in this case not averaged the close resemblance between the profiles on both halves thus indicates that the membranes are well equilibrated). Apart from this, inspection of Figure 18 shows a remarkable resemblance between the two set of predictions. Many details are in semi-quantitative agreement. Moreover, many of the features of membranes composed of SOPC resemble those of DMPC discussed above. For example, the width of the membrane-water interface is about 1 nm, i.e. the size of just two to three water molecules. This width is consistent with the scaling arguments mentioned at the beginning of this chapter. A more accurate comparison... 

The distribution of the center-to-end distance, F(R(,), in a star can also be predicted from scaling theory. For EV chains, it is expected to be close to Gaussian [26], except for small R. Applying scaling arguments and RG theory, Ohno and Binder [27] obtained a power-law behavior for small R, F(Rj,)=(Rj,/) with the exponent value 0(f)=l/2 for high f. They also considered the case of a star center adsorbed on a planar surface, evaluating the bead density profiles and the distribution of center-to-end distance in the directions perpendicular and parallel to the surface in terms of similar power-laws. 

Fig. 2. Bead density profiles. Solid line Brushes, mean-field and scaling theory (step function) dashed-dotted line generalization of the Milner et al. theory for brushes in the theta state dashed-double dotted line Milner et al. theory for brushes (EV chains) dashed line EV stars dotted line EV combs. Variable r is scaled to give zero bead density for the smooth curves of brushes at r=l. The brush curves are normalized to show equal areas (same number of units). The comb and star densities are arbitrarily normalized to show similar bead density per volume unit as the step function and EV curves for brushes at the value ol r where these curves intercept...

The internal distribution of beads in a uniform star predicted by the Daoud and Cotton scaling theory [11] can be tested by computing bead density profiles. These profiles can be compared with the scaling predictions for the density within EV blobs, Eq. (16), that can be explicitly written for EV stars as... 

According to molecular dynamics studies on microscopic structures of a flat oil/water interface, the interface shows a relatively smooth density profile in the long-term average, while that in the time scale of tens of picoseconds fluctuates thermally and is highly irregular owing to capillary distortions, which induce solute transfer between the two phases [36,87],... 

PS P4VP Toluene (selective for PS) Determination of scattering density profile and comparison to scaling theory SANS Forster et al. (1996)... 

A simple scaling model of block copolymer micelles was derived by de Gennes (1978). He obtained scaling relations assuming uniformly stretched chains for the core radius, RB, of micelles with association number p.This model can be viewed as a development of the Alexander de Gennes theory (Alexander 1977 de Gennes 1976,1980) for polymer brushes at a planar interface, where the density profile normal to the interface is a step function. In the limit of short coronal (A) chains (crew-cut micelles) de Gennes (1978) predicted... 

The scaling theory for spherical polymer brushes due to Daoud and Cotton (1982) (Section 3.4.1) has been applied to analyse the coronal density profile of block copolymer micelles by Forster et al. (1996). If the density profile is of the hyperbolic form r as found by FOrster et al. (1996) for the coronal layer of block copolymer micelles, the brush height scales as... 

However, the surface coverage is the same for both copolymers when weakly adsorbed to the surface. Surface density profiles were also compared. Finally, scaling relationships for triblock copolymer adsorption under weak adsorption conditions were derived (Haliloglu et al. 1997). In a related paper (Nguyen-Misra et al. 1996), adsorption and bridging of triblock copolymers in an athermal solvent and confined between two parallel flat surfaces were studied, and the dynamic response of the system to sinusoidal and step shear was examined. 

Numerical simulations indicate that virialized DM halos on the scales of galaxies and galaxy clusters show a cuspy density profile p(r) = pog(r) where... 

Observations of galaxy duster do not help in this respect since the available data stop our understanding of the DM density profile at distances r 10 kpc from their centres (see Fig.5.1). Down to these scales the NFW is still allowed and at smaller scales the inner slope remains quite uncertain even when the combined analysis of X-ray, gravitational lensing and galaxy dynamics data are taken into account (Dalai Keeton 2003). 

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