Perfection, defects, dispersity

Most studies performed partly on molecular models [33] but also on real POPAM and PAMAM dendrimers support the latter model concept [34]. Careful studies on the three-dimensional structure of flexible dendrimers in solution were performed by Ballauff et al. by means of SANS (Small Angle Neutron Scattering) [35] (see Section 7.6). 

The degree of branching DB defined by Frechet et al. provides a criterion for the classification of dendritic molecules with regard to their perfection [36], 

IT number of terminal monomer units ID number of dendritic monomer units IL number of linear monomer units 

In contrast to perfect dendrimers with a degree of branching of 100%, hyper-branched polymers (Sections 2.7 and 4.1.5.4) will have degrees of branching between 50 and 85%, depending upon the monomer - for example, whether AB2 or ABg monomers were used as starting materials [37]. 

Area per end group = (dendrimer surface area)/(number of end groups) 

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Dendrimers and hyperbranched polymers are two groups of materials resembling each other. The architectural difference is that dendrimers are perfectly branched structures, while hyperbranched polymers contain defects. Dendrimers are mono-dispersed while hyperbranched polymers are more dispersed which can be an advantage in some applications. 

Real substances often deviate from the idealized models employed in simulation studies. For instance, many complex fluids, whether natural or synthetic in origin, comprise mixtures of similar rather than identical constituents. Similarly, crystalline phases usually exhibit a finite concentration of defects that disturb the otherwise perfect crystalline order. The presence of imperfections can significantly alter phase behavior with respect to the idealized case. If one is to realize the goal of obtaining quantitatively accurate simulation data for real substances, the effects of imperfections must be incorporated. In this section we consider the state-of-the-art in dealing with two kinds of imperfection, poly-dispersity and point defects in crystals. 

In contrast, we can determine the absolute entropy of a substance. To do so requires application of the third law of thermodynamics, which states that a perfect crystal has zero entropy at a temperature of absolute zero S ys = 0 at 0 K. Perfect means that all the particles are aligned flawlessly in the crystal structure, with no defects of any kind. At absolute zero, all particles in the crystal have the minimum energy, and there is only one way it can be dispersed thus, in Equation 20.1, W = 1, so 5 = k In 1 =0. When we warm the crystal, its total energy increases, so the particles energy can be dispersed over more microstates (Figure 20.3). Thus, W > 1, In W > 0, and S > 0. 

A systematic SANS study of the PS-PA block copolymer has been achieved. The PS-PA co-polymers were obtained through thermal annealing of the diblock precursor PS-PPVS (polyphenylsulfoxide), a detailed description of the synthesis and preparation of the PA sequence free from defect can be found in references [54-55], To get a perfectly dispersed micelle structure in solution, the PA must be short and so the study was devoted to the diblock PS64-PA1 [56-58], The contrast factor of the PS unit is about 95 x 10 cm , while the contrast of the PA chain is negligible. As given in Section 1.2.2 the structure factor of a particle varies as a function of its geometrical shape. For instance, a... 

Compared with the results from the dynamics of simple impure 3D lattices, in the case of ID lattices (polymers) localized modes (or quasi-localized modes) can sometimes also be generated if the frequency of the defect unit occurs within the frequencies spanned by a dispersion branch of the perfect host lattice. As the... 

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