Chains in a Plane

Folding as discussed here is not just a property of structures embedded in aplane (such as were the forms of long chains superimposed on the graphite lattice considered here). The approach is general and extends to curves in 3D space 

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We emphasize at the outset that this article deals with flexible linear chains only, neither branched polymers [54,159] nor the packing of stiff chains near surfaces [52,53] will find much attention. However, we also shall not cover films formed by end-grafted chains ( polymer brushes [160-172]), although in brushes formed from two different types of chains A,B interesting phase separation behavior can occur [165,166] that is related to the phase separation in non-grafted films as treated here. Also films formed from strictly two-dimensional chains in a plane [173-175] are outside of our attention.,... 

The free energy of a Ginzburg-Landau field describing a system of weakly coupled chains in a plane is identified with the ground-state energy of a linear array of quantum anharmonic oscillators. The equivalent Hamiltonian is simplified for both the real and complex fields using a truncated basis of states. Results for both the real and complex fields will be discussed. In addition, the behavior of the specific heat and inverse correlation length for finite numbers of weakly coupled chains will be discussed. 

AWi is the energy needed to elongate a rod A/, and AW is the energy needed per bond to open the bond angle a Aa. If 4 is the effective cross-sectional area of the chain (in a plane perpendicular to F) show that the Young s modulus of the chain is. 

Figure 2.21 Projections of sheets of hydrogen bonded chains in a plane parallel to chain axes of even nylons (a, b) and even-even nylons (c) in the a form with chains in a zigzag planar conformation [122] (a, c) and in the y form with chains in a twisted less extended conformation [122] (b). The cases of a and /forms of nylon 6 are shown in (a) and (b) respectively, whereas the case of the a form of nylon 66 is shown in (c). The unit cell is monoclinic for nylon 6 with a = 9.56 A, b (chain axis) = 17.24 A, c = 8.01 A, and p = 67.5° for the a form [123] and a = 9.33 A, b (chain axis) = 16.88 A, c = 4.78 nm, and P = 121° for the Y form [123] and triclinic for the a form of nylon 66 with a = 4.9 A, = 5.4 A, c (chain axis) = 17.2 A, a = 48.5°, and p = 77°, /= 63.5° [118]. The distance of chain axes in the sheets (a - c) is =4.8 A regardless of crystalhne polymorph (a or /forms) and the numbers n and/ or m of carbon atoms per constitutional unit in the chains of nylon n and nylon n,m. The directionality of the chains of nylon 6 in (a) and (b) is indicated by arrows pointing up or down according to the direction of N-C(=0) bonds along the chains. Adjacent chains are anticlined in (a) and isoclined in (b). (See color insert.)...

Figure 2.21 Projections of sheets of hydrogen bonded chains in a plane parallel to chain axes of even nylons (a, b) and even-even nylons (c) in the ot form with chains in a zigzag planar conformation [122] (a, c) and in the 7 form with chains in a twisted less extended conformation [122] (b). (See text for full caption.)...

Figure 1.2 shows one way of dividing a polypeptide chain, the biochemist s way. There is, however, a different way to divide the main chain into repeating units that is preferable when we want to describe the structural properties of proteins. For this purpose it is more useful to divide the polypeptide chain into peptide units that go from one Ca atom to the next Ca atom (see Figure 1.5). Each C atom, except the first and the last, thus belongs to two such units. The reason for dividing the chain in this way is that all the atoms in such a unit are fixed in a plane with the bond lengths and bond angles very nearly the same in all units in all proteins. Note that the peptide units of the main chain do not involve the different side chains (Figure 1.5). We will use both of these alternative descriptions of polypeptide chains—the biochemical and the structural—and discuss proteins in terms of the sequence of different amino acids and the sequence of planar peptide units.

Simulations of monolayers have focused on internal phase transitions, e.g., between the expanded phase and the condensed phases, between different tilted phases, etc. These phenomena cannot be reproduced by models with purely repulsive interactions. Therefore, Haas et al. [148,149] represent the amphiphiles as stiff Lennard-Jones chains, with one end (the head bead) confined to move in a plane. In later versions of the model [150-152], the head bead interactions differ from those of the tail beads they are taken to be purely repulsive, and the head size is variable. 

If the chain is fully extended so that all of the chain atoms lie in a plane (planar zigzag form) as in formula X... 

Figure 2.24 Models of packing of chains in a-form of sPS according to space groups (a) / 3cl52 and (b) P3150. In (a) dotted lines indicate crystallographic glide planes coincident with local glide planes of chains. In (b) triplets of chains are rotated around threefold axes and crystallographic glide planes are lost.

Fig. 1. a Star polymer, b Comb polymer, c Brush, d Miktoarm star copolymer, e Star copolymer, f Star chain center-adsorbed in a plane, g Dendrimer... 

Natural biological membranes consist of lipid bilayers, which typically comprise a complex mixture of phospholipids and sterol, along with embedded or surface associated proteins. The sterol cholesterol is an important component of animal cell membranes, which may consist of up to 50 mol% cholesterol. As cholesterol can significantly modify the bilayer physical properties, such as acyl-chain orientational order, model membranes containing cholesterol have been studied extensively. Spectroscopic and diffraction experiments reveal that cholesterol in a lipid-crystalline bilayer increases the orientational order of the lipid acyl-chains without substantially restricting the mobility of the lipid molecules. Cholesterol thickens a liquid-crystalline bilayer and increases the packing density of lipid acyl-chains in the plane of the bilayer in a way that has been referred to as a condensing effect. 

In his early papers Natta referred to the state of the art of that time along the lines discussed by Floty (28). Until the early 1950s, the lack of crystallinity in vinyl polymers was attributed to the configurational disorder present in such compounds, considered as copolymers of d and I stmctural units. These units were defined with reference to a macromolecule having the chain in a zigzag planar disposition, the substituents of one type being above the plane... 

Linear macromolecules having a constitutional repeating unit such as -CH2-CHX- (X H) show two further stereoisomerisms, i.e., optical isomerism and tacticity. The stereoisomerism named tacticity has its origin in the different spatial arrangements of the substituents X. When we arrange the carbon atoms of the polymer main chain in a planar zigzag conformation in the paper plane. 

Furthermore from the computed area of the cross-section of the interface occupied by one soap molecule it is clear that the molecules of the soap are relatively close together and orientated in a plane at right angles to the interface. As has already been noted in the case of the air-water interface the fatty acids are orientated with their polar carboxyl groups in the water phase we would consequently anticipate that in the oil-water interface the same orientation would occur, the hydrocarbon chain being immersed in the paraffin phase and the polar —OOONa or —COOK group in the aqueous phase. Such orientation is an important factor in the... 

RiS theory is applied to investigate chain configuration of POLA. Independent conformations for each repeat monomer unit of the chain are assumed in the calculations of the unperturbed dimensions. Rotations about the oxygen-phenytene-carbon bonds are considered to be free with twofold symmetric potentials. The trans and cis conformations of the carbonyl-phenylene-carbon and the indan-carbonyl residues are assumed to have equal probability. The bond vectors for this model lie in a plane because every torsion angle is 0D or 180°. 

Figure 8-6 (A) Molecular packing of 2,3-dimyristoyl-D-glycero-l-phosphocholine dihydrate. The two molecules in the asymmetric unit are labeled 1 and 2. The position of the water molecules is indicated either by W1-W4 or by small open circles. Hydrogen bonds are represented by dotted lines. From Pascher et al.66 (B) Two-dimensional "orthorhombic" arrangement of hydrocarbon chains in a crystalline alkane. The a-b plane corresponds to the plane of the bilayer surface the long axes of the acyl chains project from the page. From Cameron et al.67...

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