The Stretch Energy

Estr is the energy function for stretching a bond between two atom types A and B. In its simplest form, it is written as a Taylor expansion around a namral , or equilibrium , bond length Rq- Tenninating the expansion at second order gives the expression 

The derivatives are evaluated at R = Rq and the (0) term is normally set to zero this is just the zero point for the energy scale. The second term is zero as the expansion is around the equilibrium value. In its simplest form the stretch energy can thus be written 

The harmonic form is the simplest possible, and is in fact sufficient for determination of most equilibrium geometries. There are certain strained and crowded systems where the results from a harmonic approximation are significantly different from experimental 

This of course has a price more parameters have to be assigned. 

The a constant is the same as that appearing in the Morse function, but is usually taken as a fitting parameter. 

Here is the force constant for the A—B bond. This is the form of a harmonic oscillator, with the potential being quadratic in the displacement from the minimum. 

The harmonic form is the simplest possible, and sufficient for determining most equilibrium geometries. There are certain strained and crowded systems where the results from a harmonic approximation are significantly different from experimental values, and if the force field should be able to reproduce features such as vibrational frequencies, the functional form for Es,r must be improved. The straightforward approach is to include more terms in the Taylor expansion. 

For each bond type, i.e. a bond between two atom types A and B, there are at least two parameters to be determined, and The higher-order expansions and the Morse potential have one additional parameter a or U) to be determined. 

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The Alexander model allows a simple approach to this problem. Within this model, each tethered chain is, in effect, confined within a cylindrical capillary of diameter d. Combining Eq. 5 and 7, we can express the stretching energy as ... 

The stretching energy is given by a sum of quadratic (harmonic) and cubic teims ... 

The brush thickness is determined by the balance between the stretching energy of the chains and the osmotic Interactions. The probability distribution of a Gauss... 

The stretching energy is of order kT per monomer when either chain is nearly fully stretched (i / Nb) resulting in/kTjb. The force required to stretch the real chain increases more rapidly with Rf, but is always smaller than the force required to stretch the ideal chain to the same end-to-end distance Rf, as shown in Fig. 3.9. Both chains have fewer possible conformations when they are stretched, but the real chain has fewer possible conformations to lose, resulting in a smaller stretching force. 

The height /f increases linearly with the number of monomers N per chain at constant grafting density. The stretching energy per chain chain in the grafting layer is A T times the number of correlation blobs per chain ... 

The stretching energy per unit volume in the brush is balanced by the interchain repulsive energy per unit volume, which is proportional to the osmotic pressure in the layer ... 

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