Chains ideal

In solution, linear polymers are not stiff rods, but the chains are usually flexible. The groups along the chain can rotate around their bonds and in this way change the direction. If we start at a certain monomer, we may be able to predict the direction of the next monomer, but after a certain distance the orientation of the following monomers is not correlated anymore with the orientation of our starting monomer. This characteristic decay length for orientation correlation depends on the specific polymer we are dealing with and on the solvent. 

In many applications, we do not need to consider the detailed molecular structure of the polymer including bond lengths, bond angles, rotation energy, and so on. 

Typical variations from one solvent to another are 10% for Coo. Values of I, were calculated by Vangehs Harmandaris by applying energy minimization in trimers or tetramers to avoid end effects and by defining 1, as the distance between the center of mass of two consecutive monomers. Depending on the specific force field, they vary between 5 and 10%. 

In fact, Eq. (11.2) holds for any ideal chain, irrespective of the specific way the segments are linked. To determine the mean end-to-end distance, we could either average over many polymers at a given moment or observe one polymer chain and average its conformation over time. Due to thermal fluctuations, the chain will assume many configurations in a short time. 

The size of a polymer can be measured by light or neutron scattering. In light scattering, the radius of gyration is determined. It is related to the size of a polymer by 

Let us now at first treat in more detail the ideal chains, on the basis of the introduced freely jointed segments model. We choose numbers, from 0 to iVg, 

As explained above, p R) is identical with the distribution function for the displacement of a Brownian particle, after Ns uncorrelated steps. The latter is derived in many of the textbooks of statistical physics It equals a Gaussian function with the form 

As is important to notice, Eq. (2.10) includes one parameter only, namely the mean squared end-to-end distance related to v R) by 

When dealing with polymer chains, one requires a parameter for estimating the size of the volume occupied by a polymer chain in the fluid phase. A suitable measure is provided by the quantity Rq, defined as 

Equivalent equations hold for the distribution functions of the internal distance vectors rij between two junction points in the chain, which follow as 

1 Random Walk A linear flexible polymer chain can be modeled as a random walk. The concept of the random walk gives a fundamental frame for the conformation of a polymer chain. If visiting the same site is allowed, the trajectory of the random walker is a model for an ideal chain. If not allowed, the trajectory resembles a real chain. In this section, we learn about the ideal chains in three dimensions. To familiarize ourselves with the concept, we first look at an ideal random walker in one dimension. 

The probability distribution is called a binomial distribution, because P is equal to the nth term in the expansion of 

Using the identity, the mean (expectation) of n is calculated as follows  

On the average, the random walker moves half of the steps to the right. Likewise, the average of is calculated as 

Its square root, (An ) called the standard deviation, is a measure for the broadness of the distribution. Note that both (n) and (An ) increase linearly with N. Therefore, the relative broadness, (An )i /(n), decreases with increasing N. 

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In summary, we see now how tire change from tire expanded chains in dilute solutions to tire ideal chains in a melt is accomplished. Witli increasing polymer concentration, tire chain overlap increases and tire lengtli scale over... 

S. Livne, H. Meirovitch. Computer simulation of long polymers adsorbed on a surface I. Corrections to scaling in an ideal chain. J Chem Phys . 4498-4506, 1988. 

As has probably become obvious already, the study of micelles has been one of the big topics in simulations of systems with surfactants. We have cited many of the related publications in the previous sections. Here, we shall discuss some special aspects of micelle simulations in order to illustrate the use of idealized chain models for this type of problem. 

Detailed discussion of polymer tacticity can be found in texts by Randall,2 Bovey,1-3 Koenig,4-5 Tonelli6 and Hatada.7 In order to understand stereoisomerism in polymer chains formed from mono- or 1,1-disubstiluted monomers, consider four idealized chain structures ... 

The quantity b has the dimension of a volume and is known as the excluded volume or the binary cluster integral. The mean force potential is a function of temperature (principally as a result of the soft interactions). For a given solvent or mixture of solvents, there exists a temperature (called the 0-temperature or Te) where the solvent is just poor enough so that the polymer feels an effective repulsion toward the solvent molecules and yet, good enough to balance the expansion of the coil caused by the excluded volume of the polymer chain. Under this condition of perfect balance, all the binary cluster integrals are equal to zero and the chain behaves like an ideal chain. 

In good solvents, the mean force is of the repulsive type when the two polymer segments come to a close distance and the excluded volume is positive this tends to swell the polymer coil which deviates from the ideal chain behavior described previously by Eq. (1). Once the excluded volume effect is introduced into the model of a real polymer chain, an exact calculation becomes impossible and various schemes of simplification have been proposed. The excluded volume effect, first discussed by Kuhn [25], was calculated by Flory [24] and further refined by many different authors over the years [27]. The rigorous treatment, however, was only recently achieved, with the application of renormalization group theory. The renormalization group techniques have been developed to solve many-body problems in physics and chemistry. De Gennes was the first to point out that the same approach could be used to calculate the MW dependence of global properties... 

In addition to the use of cyclohexane, dimethylformamide (DMF) was used as a good solvent. So as to pick up the thiol-modified terminal, gold-coated cantilevers were used. The nominal values of their spring constant ki were 30 or 110 pN nm . A typical force-extension curve measured in cyclohexane is shown in Figure 21.4. The solvent temperature was kept at about 35°C, which corresponded to its temperature for PS chains. Thus, a chain should behave as an ideal chain. The slope at the lowest extension limit (dashed line in Figure 21.4) was 1.20 X lO" N m . ... 

The scaling exponent a can be related to the particle shape. One finds a = 2,0, 0.5, and 0.8 for a thin rod, solid sphere, ideal chain, and swollen chain, respectively. For most polymers K and a have been tabulated [23]. For a monodisperse sample Equation (36) can be used for a crude determination of the molar mass ... 

Figure 1.2. A number molecular mass distribution N (M) of an ideal chain polymer. N (M) is defined for integer multiples of Mm, the monomer mass. The integer factor, P, is called the degree of polymerization... 

To see that this is exact, consider an ideal chain fluid in a external field [Pg.122]

Nj,=N/f is the number of beads per branch or arm). For larger chains, however, the solvent can penetrate in outer regions of the star and the situation within these regions is more Hke a concentrated solution or a semi-dilute solution. These portions of the arms constitute a series of blobs, whose sizes increase in the direction of the arm end. The surface of a sphere of radius r from the star center is occupied by f blobs. Then the blob size is proportional to rf. Most internal blobs are placed in conditions similar to concentrated solutions and, consequently, their squared size is proportional to the number of polymer units inside them as in an ideal chain. This permits one to obtain the density of units inside the blob, as a function of r ... 

The introduction of branching in the Kirkwood formula and the KR calculations can be accomplished in a relatively easy way if Gaussian statistics corresponding to ideal chains are maintained. This description cannot, however, be very accurate in molecules with centers of high functionality because of the presence of cores with a high density of polymer units, which profoundly perturbs the internal distribution of distances. Stockmayer and Fixman [81 ] employed the Kirwood formula and Gaussian statistics to calculate h in the case of uniform stars, obtaining an analytical formula. They also performed a KR evaluation of the viscosity and proposed that g could be evaluated from the approximation... 

There is no distinction between Xg and Xp in the Rouse description of linear chains. A third relaxation mechanism contemplates the disentangling of two or more intertwined arms. This relaxation is considerably slower and strongly dependent on f. Obviously, this feature cannot be described by the ideal chain model. 

According to the results, it is determined that the asphericities can be described in terms of polynomials in Forni et al. [140] also used an off-lattice model and an MC Pivot algorithm to determine the star asphericity for ideal, theta, and EV 12-arm star chains. They also found that the EV stars chains are more spherical than the ideal and theta star chains. In these simulations the theta chains exhibit a remarkable variation of shape with arm length, so that short chains (where core effects are dominant for all chains with intramolecular interactions) have asphericities closer to those to those found with EV, while longer chains asymptotically approach the ideal chain value(see Fig. 10). 

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