Random-flight chains

Another simplified model is the freely jointed or random flight chain model. It assumes all bond and conformation angles can have any value with no energy penalty, and gives a simplified statistical description of elasticity and average end-to-end distance. 

The same effect happens inside a random flight chain where the close proximity of the polymer segments offers mutual screening from the bulk flow field. The idea of a chain being non-drained was first considered by Debye Bueche who introduced the concept of a shielding length defined as [46] ... 

The dashed line represents values evaluated by Casassa (10) for random-flight chains with infinitely many bonds of infinitesimal length. The dot-dashed line shows results of Giddings et al. (11) for rigid rods. 

Here x=qNb/6=qR where Ru is the radius of an undeformed Gaussian random flight chain. I deriving Eq. 10, the sum in Eq. 7 is replaced by an integral. The effect of the free chain segments, exclusive of the position of the junctions appears in the first two terms in the exponential. The deformations of the chains depend on the constraints on the junctions. Results are immediately derivable from Eq. 10. 

Figure 3 for four chain lengths, expressed as the number of segments per chain, r. As expected, in dilute solutions A is independent of In this dilute regime, A is approximately proportional to the square root of the chain length for not too short chains A/l = 0.56 (Sr - 2), which is of the order of rg/1. Note that for a random flight chain rg/1 = /r/6 = 0.41 Sr, and a lattice chain is expected to be slightly more expanded. 

If interactions between parts of the molecule separated by many links (the excluded volume effect ) is absent, so that the chains obey random-flight statistics, takes its unperturbed value, (s ). Theoretical calculations of the dimensions of branched molecules usually assume random flight chains, and values of the mean-square radius so obtained are estimates of . 

The unperturbed mean-square radius is calculable for polymers of known structure, on the assumption of random-flight chains (Section 3). It has usually been assumed that random-flight conformations are adopted at the temperature Al at which A2 is zero, according to Flory s theory (18). Light... 

If we suppose a very dilute gas of random flight chains. As a result of thermal rotation of chain segments each chain will take up a great number of conformations, with a very short interval of time being spent for the passage from one conformation to another. In so doing, it automatically avoids taking those conformations in which... 

Figure 5.2. The chain conformation of fig. 5.1a, but now modelled as a random-flight chain of JV bonds of length /. The (fluctuating) distance between the end points is r.

Note that. In a sense, the polyelectrolyte behaves now as If it were an "ideal" (Gaussian) chain with a relatively small number of Kuhn lengths. This coil size Is determined by the local stiffness, but not by long-range excluded volume. Should q Increase even furher (approaching L), then the wormlike chain would be better described by a slightly curved rod (as expressed by (5.2.21)) than by a random-flight chain. 

The simplest possible polymer model for a flexible polymer chain is the freely jointed (or random flight) chain. In this model, it is assumed that each bond angle and each rotational angle may take with equal probability any value in the range 0-360°. The conformational energies are the same irrespective of conformation. The mean-squared, end-to-end distance of a chain in this approximation is given by... 

Mean Square End-to-End Distance of a Chain on Hypercubic Lattices. Let us consider a random flight chain on hypercubic lattices. Let rn be a vector from a given site to the nth site. Then we have... 

For flexible linear polymers the energy barriers associated with rotation around the bonds are small with respect to the thermal motion. Such molecules have a randomly fluctuating three-dimensional tertiary structure that is referred to as the random coil (as illustrated in Figure 4.1). The chain conformation is described as a random flight chain of N bonds of length 1. The fluctuating distance between... 

One parameter that is commonly used to specify the dimension of a linear polymer molecule is the root-mean-square (rms) end-to-end length. The simplest, and also the most primitive, model for a polymer molecule is the random flight chain, also termed the freely jointed chain. In this model, the bonds are represented by volumeless lines in space, and there are no restrictions on the valency angles or on the rotations about the bonds. The rms end-to-end length, can be... 

The random flight chain has the simplest mathematical properties, but, unfortunately, also has the smallest degree of strnctnral similarity to real polymers. A more complex model considering the fixed valency angle (x) and the restrictions of free rotation has been proposed ... 

The characteristic ratio of a real chain can be considered crudely in terms of a time averaged spherical model (polymers adopt at any instant a shape that is more sausage like than spherical but undergo rotational random motion). Accordingly, C is the ratio of the average cross-section of the real chain compared with that of the corresponding random flight chain. Some values of Coo for common polymers are shown in Table 4.1. 

The average span <5 c> of a random flight chain is given by... 

A simpler approach to steric and depletion stabilization is to use the predictions of the random flight chain at the same reduced distance, say, HoKr y. The chain dimensions are used here as an arbitrary reduction parameter. The validity of this simple procedure can only be assessed after an exact theory has been elaborated. Unfortunately, such an... 

Theories of elastic steric stabilization Dolan and Edwards (1974) have calculated the elastic free energy of repulsion for isolated polymer chains attached terminally to parallel flat plates. It was shown in Section 11.4.1.1. that probability distribution function for a random flight chain obeys the diffusion equation... 

Plots of the normalized segment volume fraction (vj/vj ) as a function of the reduced distance dl<,r > (where d= distance normal to the surface and i = free solution rms end-to-end distance of the chains) are presented in Fig. 17.5. Results for both a short (27 bonds) and a longer (272 bonds) poly(oxyethylene) chain are given, together with data for a random flight chain of 4 (XX) bonds. 

Fig. 17.5. The distance dependence of the relative segment density for tails (1) poly(oxyethylene) of molecular weight 400 poly(oxyethylene) of molecular weight 4 000 3, random flight chain of 4 000 bonds (after Feigin and Napper, 1980b).

Also apparent from Fig. 17.5 is the close agreement between the 272 bond poly(oxyethylene) and the 4 000 bond random flight chains when plotted on the reduced distance scale. As the characteristic ratios C of these two chains are quite different (4 vy 1, respectively), the results suggest that the molecular structure of the chain does not drastically influence the segment density vs distance curves plotted in this reduced fashion. This conclusion requires further confirmation. 

Scheutjens and Fleer (1982) have developed a theory for depletion stabilization and depletion flocculation based upon their statistical thermodynamic approach to polymer adsorption and steric stabilization. This theory is cast in terms of the most primitive model for a polymer molecule, the random flight chain. This weakens the theory in so far as providing quaintitative predictions at the fundamental level for real systems is concerned. The theory does, however, offer qualitative results over a wide range of conditions, being especially powerful in establishing the various trends involved. 

Richmond and Lai (1974) considered the pressure exerted on two parallel flat plates by a random flight chain confined between them. Adsorption on the plates was not allowed. Not surprisingly, the loss of configurational entropy of the chains on close approach resulted in the generation of a repulsion. This situation is quite unphysical for colloidal systems since the polymer chains in free solution can escape from the compressional zone. Accordingly, the predictions of Richmond and Lai do not relate to depletion stabilization. 

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