This section studies the effect of long-range interaction. Molecular interaction through van der Waals forces, hydrogen bonding, electrostatic forces, and hydrophobic forces, all fall into this category.
We first consider the van der Waals force. The total interaction energy is given by the sumof the pairwise potential
V =
i<j
u(ri,j), (1.64)
whereu(r) is the effective interaction potential between the monomer i and j in the solvent. It is assumed to have a hard core repulsive part and a long-range attractive part (Figure1.6(a)).
The partition function (1.8) for a given end-to-end vector R is
Z(R,T ) =
Following Mayer’s perturbation theory in the classical statistical mechanics of interacting particles, let us try to expand it in powers of the strength of the interaction [15].
The interaction potential u(r) goes to infinity when the two monomers come into contact, and hence a simple expansion in powers of it leads to the divergence of each
termin the series. To avoid this problem, we introduce the Mayer function, defined by
χ(r) ≡ e−βu(r)−1, (1.66)
and expand the interaction part in the partition function in powers of this function as
We then carry out the integration term by term. This is the method referred to as cluster expansion, which was developed in the theory of condensation of interacting gases [15].
For a polymer chain, the condition of linear connectivity is added.
To calculate termby termin the power expansion, let us introduce a further approx-imation. The Mayer function takes the form shown in Figure1.6(b). It can be roughly replaced by
χ(r) v(T )δ(r) (1.68)
in order to study the chain properties in scales larger than the size of monomers, where the excluded volume of a monomerv(T ) is defined by the integral
v(T ) ≡ −
χ(r)dr. (1.69)
Carrying out this integration separately fromthe repulsive force inside the diameterσ of the hard core and fromthe repulsive force outside of it, we find
v(T ) =4π 3 σ3+β
∞
σ u(r)4πr2dr, (1.70)
where the attractive part is expanded in powers of u(r) because it is finite. The first termv0≡ 4πσ3/3 gives the volume of the space region to which the monomers cannot enter due to the existence of other monomers. This is the origin of the term “excluded volume.” The second term takes a negative value due to the attractive nature ofu(r).
When summed, they are combined as
v(T ) = v0(1−θ/T ). (1.71)
The parameterθ is defined by
θ ≡ − 3 kBσ3
∞
σ u(r)r2dr, (1.72)
which is a positive number with the dimension as temperature. This gives the reference temperature for the study of the polymer chain, and called the theta temperature.1In
1 The theta temperature of polymer solutions is conventionally defined by the temperature at which the second virial coefficient of the osmotic pressure vanishes. In this book, we write8 for this theta
Table 1.2 Coefficients in the expansion factors
Expansion factor Mean end-to-end distance Radius of gyration
C1 4/3= 1.333 134/105= 1.276
C2 2.075 2.082
C3 6.297 –
the following, we measure the temperature in terms of the dimensionless temperature deviation
τ ≡ 1−θ/T , (1.73)
fromthe reference theta temperature.
In the high-temperature regionτ > 0, the repulsive force is dominant. The monomers mutually repel each other, and as a result the chain swells from the ideal state. The solvent serves as a good solvent. On the contrary, in the low-temperature regionτ < 0, the chain contracts due to the attractive interaction among the monomers. The solvent serving such an environment is a poor solvent. At the theta temperature lying in between them, these two effects exactly cancel each other, so that the chain remains ideal.
If the integrals are carried out termby termin the power series of the cluster expansion for the BS model chain (1.4), the series turns out to be in powers of the parameter
z ≡
3
2πa2
3/2
v(T )√ n τ√
n. (1.74)
This parameter is called the excluded-volume parameter.
On the basis of this perturbational method, we can calculate the statistical averages of various physical quantities [17,16]. For instance, the mean square end-to-end distance is expressed in the asymptotic series
αR2≡ R2 / R2 0= 1+C1z−C2z2+C3z3−··· (1.75) which has alternating coefficients. The series does not converge, but the absolute value gradually approaches the exact value as higher terms are included. The ratioαR2defined by the left of this equation is called the expansion factor of the mean end-to-end distance.
The expansion factorαSof the radius of gyration is similarly defined. The results obtained so far are summarized in Table1.2. We can see clearly that the chain expands or shrinks, depending on the temperature.
The perturbational analysis is theoretically clear in principle, but has weak points such as (1) calculation of the higher-order terms is seriously difficult, (2) it is not a convergent
temperature of the solutions to distinguish from the single-chainθ at which the attractive and repulsive interactions balance and the total excluded volume vanishes. The relation between the two is studied in the following chapters.
series, and (3) the series has physical meaning only in the vicinity of the theta temperature because the excluded-volume parameter is proportional to √
n. Therefore, we need a more efficient method to understand the statistical properties of the polymer chain over a range temperature wide.
In fact, experiments report that the average end-to-end distance obeys the power law R2 1/2 anν, ν = 3/5. (1.76)
The exponent ofn changes from1/2 to this Flory’s 3/5 law at high temperatures [18]. The exponent 3/5 of the polymer dimensions is called the Flory exponent. We cannot reach this result by continuing the calculation of higher-order terms in the cluster expansion.