ρ(r)dr = N, 1 N
ρ(r)r2dr =S2.
This allows us to approximate the function ρ(r), for example, by a two-parameter exponential function
ρ(r) =
3
2πS2
3/2
N exp
−3 2
r2
S2
. (1.24)
1.5 Excluded-Volume Effects
One says that the above results are valid for a chain with non-interacting particles. However, the monomers in a real macromolecule interact with each another, and this ensures, above all, that parts of the molecule cannot occupy the place already occupied by other parts; i.e. the probabilities of successive steps are no longer statistically independent, as was assumed in the derivation of the above probability distribution functions and mean end-to-end distance (Flory 1953). So, considering the coarse-grained model, one has to introduce lateral forces of attractive and repulsive interactions. The potential energy of lateral interactions U depends on the differences of the position vectors of all particles of the chain and, in the simplest case, can be written as a sum of pair interactions
U =1 2
N ν=0
N γ = 0 γ= ν
u (|rγ− rν|) . (1.25)
The presentation of potential in this form can be apparently justified only for large numbers N of subchains.
The effective potential u(r) between two fictious particles of the chain can be chosen in a convenient form. For analytical calculations, the potential function is approximated (Doi and Edwards 1986) by the delta function,
u(r) = vT δ(r). (1.26)
The parameter v has the dimension of volume and is called the excluded volume parameter. The above approximation of repulsive force can be appar-ently valid for a long macromolecule, when a very large number of subchains
1.5 Excluded-Volume Effects 9
N can be introduced. For a finite number of subchains, the potential can be approximated ( ¨Ottinger 1995) by a Gaussian function
u(r) = vT
where the parameter of interaction σ depends on the number of subchains of a macromolecule in such a way, that, at N → ∞, σ → 0, and repulsive potential turns into function (1.26). The dependence ought to be chosen in such a way, that properties of macromolecular coil do not depend on number of division of macromolecule into subchains.2
For the subchain model under consideration, an equilibrium distribution function that includes the particle interaction potential, takes the form
W = C exp where C is the normalisation constant. The definition of the quantity μ in (1.27) does not coincide with expression (1.7), so as the internal interac-tions are taken into account, but nevertheless the quantity can be expressed, on the basis of scaling speculations, through the mean end-to-end distance of a subchain as
μ∼ b−2.
The free energy of a macromolecule, instead of (1.9), is given by
F (r0, r1, . . . , rN) = μT Aαγrαrγ+ U (r0, r1, . . . , rN). (1.28) However, if one is not interested in observing the variables r0, r1, . . . , rN at all, the independent on these parameters free energy can be defined. This quantity can be calculated, starting from expression (1.25) and (1.27), so that it depends on the parameters T, N, b, v, whereby the arbitrary quantity N cannot influence the free energy of the macromolecular coil and the explicit
2 The problem of how to chose the effective potential for simulation purposes was recently discussed by M¨uller-Plathe (2002). At least, one parameter σ with dimension of length is usually included also in the function u(r). The magnitude of interaction decreases when N increases, so that, for long chains, the potential can be presented in universal form as
u(r) = T Nηvr
σ
.
The universality also assumes that σ∼ b. The index η can be estimated, when one calculates free energy of the coil. Specifically, the Lennard-Jones potential
u(r) = 4
is often used (Kremer and Grest 1990; Ahlrichs and D¨unweg 1999; Paul and Smith 2004) to describe interaction between particles.
10 1 Introduction: Macromolecular Systems in Equilibrium
dependence on arbitrary parameter N has to be excluded. So, after dimen-sional considerations has been taken into account, one has to write free energy of the coil as a function of the only parameter
F (T ) = T g
v b3
. (1.29)
A relation between the mean end-to-end distance of the entire chainR2 and the mean end-to-end distance of a subchain b can be found from simple speculation. This relation includes temperature T , mean distance b between the nearest along chain particles, excluded volume parameter v and the num-ber of particles on the chain N . When dimensional considerations are taken into account, the relation can be written in the form
R2 = b2f
N, v
b3
. (1.30)
Of course, the end-to-end distance of the entire macromolecule R2 does not depend on the arbitrary number of subchains N at N → ∞, when the ratio v/b3 is constant. This means that the relation betweenR2 and a finite number of subchains should be written in a way, which keeps the form of the relation under repeating divisions of the macromolecule, so that the mean square end-to-end distance of the macromolecule has to be written as a power function
R2 ∼ N2νb2.
It is easy to see that this relation is valid for an arbitrary number of divisions.
Thus, general consideration leads to the power dependence of the end-to-end distance of the macromolecule on its length
R2 ∼ M2ν. (1.31)
We can guess that the dimensions of a macromolecular coil with the excluded-volume effect are larger than those of the ideal coil, so that ν≥ 1/2.
However, it is necessary to fulfil a number of special and sophisticated cal-culations to find a specific value of power 2ν in expression (1.31) (Alkhi-mov 1991). The first estimates of the index (Flory 1953; Edwards 1965) were done by simple self-consistent methods. Then the mean end-to-end distance was calculated by a perturbation method, while the chain in a imaginable 4-dimensional space is considered to be non-perturbed, and deviation of di-mensionality of the imaginable space from the real physical space is believed to be the small parameter of expansion. The first-order term gives (Gabay and Garel 1978) the following value of index
2ν = 9 8.
The answer is known to many decimal places (Alkhimov 1991).