2.2 Bond vector correlation function
2.2.5 Semiflexible chains
In this section we will discuss the long-range correlation effect in semiflexible chains with per- sistence length lplarger than bond length a.lp≪L. So far we considered flexible polymers with lp.a and Gaussian probability of contact between all pairs of monomers. However, the probability of monomeric contacts for semiflexible chains with lp&1 is strongly reduced at monomeric separations shorter than ∼5lp[27–29]. The reduction of contact probability sup- presses the effect of the long-range correlations, and short chain segments behave as elastic rods with exponential decay of correlations (Eq (2.1)). To account for this fact we need to in- clude the small s contact probability in the effective Mayer f -function of connected monomers
Eq (2.20):
fp(r,s) =ζ(s,r)f(r,s), (2.27) whereζ(s,r)describes the deviation of contact probability between i and j monomers from the Gaussian approximation. The asymptotic limits ofζ(s,r=0)are evidently lim
s→0ζ(s) =0 and lim
s→∞ζ(s) =1. For small r/s→0 we can expand the new monomer contact probability function
and expect that h(s)≈ζ(s)h0(s), where h0(s)denotes the bond vector correlation function of flexible chains, as given by Eq (2.22).
The complete bond-vector correlation function for semiflexible chains can be represented as the sum of short-range and long-range contributions:
hcosφ(s)i=e−s/lp+c
φζ(s)h0(s) (2.28) where cφ is the normalization coefficient.
The approximate form of the ”stiffness function” ζ(s)can be obtained by considering the formation of a loop in a semiflexible chain of length s. The bending energy in this case is [30]
E(s) =kBT lp 2 s Z 0 κ(s′)2ds′. (2.29)
whereκ(s)is the curvature (inverse of the radius of the circle that describes the bend).
In order to test our idea about the nature of the crossover (Eq (2.28)) from exponential to power law decay of the bond vector correlation function, we define ζ(s)as the Boltzmann weight of the loop of length s:
ζ(s) =exp −E(s) kBT =exp −kllp s (2.30)
where kl is the numerical constant to be found from our simulation results∗.
The chain stiffness in our simulations is introduced through the bending potential
U(φ) =uφ(1−cosφ) (2.31)
whereφ is the angle between the neighboring bonds i and i+1, and uφ is the stiffness of the bending potential. The persistence length of such chains depends on uφand for large uφ≫kBT becomes proportional to it: lim
uφ→∞lp=σuφ/kBT . 10-4 10-3 10-2 10-1 100 5 10 15 20 25 30 〈 cos φ〉 s/lp lp = 0.64 lp = 1.04 lp = 1.33 lp = 1.67 lp = 2.05 lp = 2.47
Figure 2.3: Bond vector correlation function of polymer chains with different stiffness. Points represent the simulation results for N=199 and T =3.1ε/kB. Horizontal axis is scaled by the corresponding persistence length lp, dotted line shows the exponential function e−s/lp.
The magnitude of the long-range correlation effect decays with increasing stiffness. This can be explained by the decrease of the probability to form a loop between the interacting monomers. The growing value of the persistence length lp leads to an overall decrease of polymer concentration c∗ inside the volume hr2i3/2 ∼(slp)3/2 occupied by the segment of length s, and the probability of pair contacts decays correspondingly c∗∼ d3
hR2i3/2 ∼
d3
(slp)3/2.
The magnitude of the long-range correlations effect is proportional to the probability of contact between the two monomers (estimated as the ratio of the monomer interaction volume d3 to the pervaded volume of the segment(slp)3/2).
10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 1 10 lp 3 /d 3 [ 〈 cos φ〉 - exp(-s/l p )] s / lp flexible, lp=0.9 lp=0.9 lp=1.043 lp=1.330 lp=1.666 lp=2.050 charged, c=0.05, lp=4.34 charged, c=0.10, lp=3.11 charged, c=0.15, lp=3.03 theory
Figure 2.4: The residual non-exponential decay of the bond vector correlation function. The dashed and solid lines mark the exponential and non-exponential components of the bond vec- tor correlation function (the first and the second terms in Eq (2.28)), respectively. The coeffi- cients cφ·A=1.84±0.04 and kl =4.3±0.07 were determined by fitting simulation data to theoretical prediction Eq (2.32). Persistence length lpis given in units ofσ, and concentration is units ofσ−3.
Fig. 2.3 shows the decay of the bond vector correlation function for semiflexible chains with different lp. The coefficient of the Lennard-Jones potential (Eq (2.23)) was set to ε=
kBT/3.1, because from the linear fits in Fig.2.2 we have found that the effective second virial coefficient is the smallest |B| ≈0 for ε=kBT/3.1 (the solid line in Fig.2.2 corresponding to ε=kBT/3.1 has the smallest slope). The persistence length lp was determined from the exponentially decaying section of the hcosφ(s)i at a ≤s≤lp. Using these values of lp we extract the long-range component of the hcosφ(s)i and plot it using the reduced coordinate axes in Fig. 2.4. The horizontal axis is scaled by persistence length lp, and vertical axis is scaled by l3p/d3. The rescaling of the vertical axis reflects the expected dependence of the
monomer contact probability on the persistence length d3/(slp)3/2= (d/lp)3(lp/s)3/2. Thus for a given value of s/lpthe factor(lp/d)3is expected to collapse the data. For the short-ranged LJ potential the interaction radius is about the size of the monomer d/σ∼1. The extracted exponential decay component of thehcosφi(s)is shown in (Fig. 2.4) by the dashed line e−s/lp,
and the theoretical expectation for the long-range decay Eq (2.28)
h(s)−exp(s/lp) =cφζ(s)h0(s) =cφe− kllp s 3 2π 3/2 ·52 A b7/2s3/2 (2.32) is shown by the solid line. The coefficients obtained by fitting the numerical data are cφ·A=
1.84±0.04 and kl =4.3±0.07. The data for the flexible chain are shown on the same plot for the guidance purposes, to illustrate the universal nature of the long-range correlations in polymers. Since the regular definition of the persistence length (see Eq (2.1)) is inapplicable for flexible chains, the numerical data were scaled using the empirical value of lp =0.9σ required to collapse them with the results for the semiflexible chains.
The long range intramolecular correlations in a dilute solution of charged polymers are the natural consequence of the long range electrostatic interactions. In semidilute polyelec- trolyte solutions these interactions are screened by charges of neighboring chains and by coun- terions. To compare the behavior of charged polymers with our previous findings we dis- cuss the coarse-grained model, in which the semidilute polyelectrolyte solution is consid- ered as the melt of chains, consisting of correlation volumes (blobs). Each such blob of size
ξ≈ f−1/3b−1/3c−1/2lB−1/6is neutral and consists of g≈ f−1b−1lB−1/2c−1/2monomers [31, 32] (where f is the fraction of charged monomers, lB is the Bjerrum length and solvent is θ-like for uncharged backbone). The chain inside the correlation volume is a stretched array of elec- trostatic blobs. Charged polymers in semidilute solutions become flexible only at the scales larger than correlation length ξ, and have the persistence length of order of correlation length l ∼ξ[16].
The results of the molecular dynamics simulation∗for the bond vector correlation function of polyelectrolytes chains in semidilute solutions are shown in Fig. 2.4 with solid symbols. This function also exhibits long-range correlation. The corresponding curves can be collapsed with our results for neutral semiflexible chains by associating the range d of interactions with correlation lengthξ leading to the scaling factor l3p/d3≈l3p/ξ3. For the solutions of concen- trations c=0.05σ−3, 0.10σ−3and 0.15σ−3we have found the values of the cube of the ratio of persistence length to correlation length l3p/ξ3 to be 1.1±0.1, 1.2±0.1 and 1.3±0.1, cor- respondingly. The ratio l3p/ξ3 increases with concentration, which is in agreement with the predictions made in ref. [16].
Significant deviation from the exponential decay of the bond vector correlation function begins at monomer separations of s≈4÷5lp, as can be seen in the Fig. 2.4. The magnitude of the correlations falls as ∼0.01l3p at this crossover point. This decreasing magnitude of the effect makes observations of the non-exponential decay of the bond vector correlation function in polymers with large persistence length, such as polyelectrolytes increasingly more difficult.