Elastic harmonic chain

The Alexander-de Gennes approach unrealistically assumes the free ends all to be at the midplane. In order to realize this physically, a fixed force would have to be applied to the chain ends only. In reality, a block is exposed to a force that originates from the thermal motion of all the surrounding blocks. This problem was first analyzed by Semenov who showed that this force originates from a molecular field that compensates the elastic harmonic potential of ideal entropic springs. This field can be calculated self-consistently and is given by... 

For coarse-grained models of linear biopolymers—such as DNA or chromatin— two types of interactions play a role. The connectivity of the chain implies stretching, bending, and torsional potentials, which exist only between directly adjacent subunits and are harmonic for small deviations from equilibrium. As mentioned above, these potentials can be directly derived from the experimentally known persistence length or by directly measuring bulk elastic properties of the chain. 

Another simple model that includes elasticity is the Frenkel Kontorova (FK) model [98], which plays an important role in understanding various aspects of solid friction. The simplest form of the FK model [98] consists of a onedimensional chain of N harmonically coupled atoms that interact with a periodic substrate potential (see Fig. 7). The potential energy is... 

As discussed in Chapter 1, a Gaussian chain is physically equivalent to a string of beads connected by harmonic springs with the elastic constant ikT/lP (Eq. (1.47) with 6 given by Eq. (1.44)). Here each bead is regarded as a Brownian particle in modeling the chain d3mamics. Such a model was first proposed by Rouse and has been the basis of molecular theories for the dynamics of polymeric liquids. 34... 

We consider a long Gaussian chain made up of n + 1 beads connected by n harmonic springs, each of elastic constant k = 3 (this is actually K = iksT/f, but takes the simpler form because of our choice of the energy and length units). Its Hamiltonian is ... 

As discussed in previous chapters, the choice of the bond / o rf( r/ ) and short-range —r l) potential varies from simulation to simulation. Off-lattice models, for example, have used the harmonic-spring potential, the FENE (finitely extendable, nonlinear elastic) potential, the rigid bond with fixed valence angles, and the freely-jointed chain model to represent the bonding interaction between adjacent monomers. For the short-range... 

Compared to the rigid polymer method, the average fluctuating polymer method improves the treatment of how polymer chains move during the penetrant diffusion process. Rather than remaining fixed in place, polymer chains execute harmonic vibrations about their equilibrium positions. Penetrant jumps are then coupled to elastic fluctuations of the polymer matrix and are independent of structural relaxation of the polymer chains [24,97]. After a penetrant jump completes, chains near the final sorption state will likely show slight elastic deviations as they swell to accommodate the penetrant molecule. Since no chain conformation relaxations are allowed, other polymer chains will essentially retain their initial conformation. The penetrant jump rate then depends only on the local, quasiharmonic fluctuations in the sorption state and the transition state [24,97]. 

The highest value of tan 8 is attained when the frequency of molecular relaxation harmonizes with the experimental frequency. As the frequency is progressively raised, it becomes difficult for the polymer to respond to oscillating load, and it behaves more elastically than it really is. This fact is evidenced by the decrease of tan 8 peak height with increasing frequency because the increase of E modulus takes place at the expense of the E" modulus (Figure 10.4). When other phenomena that confer additional mobility to polymer chains overlap a-relaxation, an increase of tan 5 peak with increasing frequency will take place (Cristea et al. 2009). 

In the following chapters of this book, we will discuss fluorescence measurements performed mainly on PE solutions in thermodynamically poor solvents. Therefore, the behavior of PEs in poor solvents is our main sphere of interest, but we will first mention the classical Kuhn treatment of polyelectrolytes in )9-solvents [53]. The potential energy of a polyelectrolyte chain in a given conformation (described by a set of r, position vectors of segments) can be written within the framework of the mean-field Debye-Hiickel (DH) theory [54] as a sum of three contributions the energy corresponding to (i) the entropic elasticity of harmonic bonds, Ui, with bond lengths I, which connect the monomers in the polymer chain. This contribution depends on the set of all position vectors, r, ... 

To model the block copolymers, the total force can also have an elastic contribution, which is derived from the harmonic force used to connect two consecutive particles in the chains of polymer [10]. This contribution is expressed as... 

---