Reinhard
Sigel
Visiting Scientist, Max-Planck-Institut für Eisenforschung GmbH, 40237 Düsseldorf, Germany. E-mail: res314159@aol.de
First published on 12th December 2017
The statistical presence of kinks which form defects in semi-flexible polymer chains leads to a polydispersity in the effective persistence length. The form factor of a distorted semi-flexible polymer results as an average over this persistence polydispersity. It turns out that the scattering behavior of an ensemble of distorted semi-flexible polymer chains is quite well approximated by a form factor of an undistorted chain with a R_{g}-equivalent persistence length. An apparent length polydispersity is observed for short distorted chains. The R_{g}-equivalent persistence length is significantly smaller than the innate persistence length multiplied by the fraction of regular monomers. The results are compared to related work on DNA from the literature.
Fig. 1 Comparison of different form factors S(q) for flexible and semi-flexible polymer chains multiplied by the number of monomers N. The Debye form factor (grey solid line) describes a flexible polymer in a theta solvent, while semi-flexible polymers are represented here by the form factor of Pedersen and Schurtenberger^{2} with excluded volume interactions, method 3 (olive solid line) and without excluded volume interactions, method 2 (red solid line) as well as the form factor of Kholodenko^{3} (blue dashed line). The polymers here consist of N = 4000 monomers of length b_{0}, and their Kuhn segment length b is equal to 20b_{0}. For the semi-flexible polymers, b constitutes the persistence length . A locally cylindrical shape with a radius equal to b_{0}/2 is considered for all polymers. |
Scattering experiments are indirect methods. The extraction of the structural parameters from experimental data requires the fitting of a theoretical FF as a model function. Only if a suitable fitting model is used, one gets reasonable values for the fit parameters. An example is the polydispersity. Besides a polydispersity in length, there might be in addition other types of polydispersity in a sample, e.g. chain branching and imperfect monomer structure, or in random copolymers the distribution of monomer types.^{1} When the fitting model considers only the length polydispersity, the resulting fit parameter is affected by the other types of polydispersity. Further calculations which combine the extracted fitting parameters can then lead to inconsistent results. Usually, it is considered advisable to combine scattering experiments with complementary microscopy investigations. In microscopy, a small number of molecules adsorbed to a surface is imaged in detail, so it is possible to cross-check if the assumptions of a FF model are fulfilled. Scattering experiments give access to the polymer structure in bulk. They involve a large number of sample molecules, so good statistics is achieved for quantitative results for the structural parameters. The FFs for different architectures of macromolecules are available in the literature.
The motivation for this work stems from the investigation of dendronized polymer molecules^{20,21} by a combination of light and neutron scattering experiments. The presence of bulky dendritic structures attached to a polymer backbone leads to a stiffening of the polymer molecule. It turned out, however, that data analysis based on the model of a SFPC with a polydispersity of the overall length led to results for which a consistent interpretation has not yet been reached so far. It remains unclear whether the experimental persistence lengths represent molecular properties. On the other hand, microscopy data might hint at the occurrence of right angle kinks in the molecules, besides regularly persistent strands. It should be noted, however, that the apparent kinks might be an artefact in the picture due to the involved projection, or due to the adsorption process of the SFPCs to the substrate.^{22} So, the often claimed complementarity of the scattering experiments and microscopy might not work in this practical example, unfortunately. In any case, a real occurrence of kinks would not fit the assumed model of SFPCs without distortions. The present work investigates the effect of distortions in SFPCs on scattering experiments.
A geometrical approach to a SFPC is the freely rotating chain model.^{23} In this model, a polymer chain consists of N segments of length b_{0}. Two consecutive segments have a fixed angle θ. The rotations of segment (n + 1) around the direction of segment n with fixed θ provides flexibility to the chain. The persistence length l_{p} and the Kuhn segment length b of this model result as:^{23}
(1) |
The values in Fig. 1 correspond to θ = 25.2°, while Fig. 2 is based on the Kholodenko FF^{3} and θ = 10°, leading to l_{p} ≈ 65b_{0}. The notion of a fixed bond length b_{0} and a discrete number of monomers forms the basis of the discussion below. In particular, b_{0} enters the discussion as the minimum distance between two kinks. Furthermore, a chain with N segments of length b_{0} and mass M has the total mass NM and the total length Nb_{0}. The mass per length which determines the q^{−1} behavior of the scattering intensity at high q results as M/b_{0}. As M is usually a known quantity, the value of b_{0} of a SFPC can thus be determined in a scattering experiment.
The mathematical treatment is built up on FFs for undistorted SFPCs from the literature. Based on empirical fitting functions for computer simulation data, Pedersen and Schurtenberger (PS) describe 3 methods to calculate the FF for SFPCs without EVIs and two methods to calculate the FF for SFPCs with EVIs.^{2} A completely different access is offered by Kholodenko, who used an analogy between Dirac's fermions and SFPCs to derive a semi-analytical result for SFPCs without EVIs.^{3} The good matching in Fig. 1 of the PS FF without EVIs and the FF of Kholodenko despite quite different derivations and calculations is remarkable. For the case of dilute solutions with no overlap of the polymer chains considered here, the scattering contributions of different molecules are uncorrelated and the cross terms average to zero. The self-interferences of scattering contributions from each molecule build up the FF. The absence of cross terms permits the weighted averaging process of FFs with different effective persistence lengths.
Any difference in the scattering of a kink compared to a regular monomer constitutes an incoherent contribution not connected to the chain FF. The concept of coherent and incoherent scattering contributions is common in small angle neutron scattering (SANS) due to the different possible spin orientations of neutrons and sample atoms in the scattering process.^{1} Coherent and incoherent scattering contributions have also been discussed^{24} in connection with ellipsometric light scattering.^{25} In general, the coherent scattering contributions represent the average scattering properties of the molecules, and the incoherent contributions originate from deviations in individual molecules from the average scattering properties. As the kinks occur statistically in the chains, they cause deviations from the average scattering properties in individual molecules, which form the incoherent contribution. This contribution has a radius of gyration comparable to the size of a defect, and so the size of a monomer. It thus stands clearly apart from the chain FF in the coherent contribution, which is connected to a radius of gyration comparable to the size of the whole chain. For simplicity we neglect here any difference in the scattering contrast of kinks and regular monomers and focus on the coherent contribution only. A more detailed discussion of the incoherent contribution where the approach of Endo and Shibayama for flexible chains^{26} is generalized to SFPCs is postponed to a future extension of the concept of this work to the case of random copolymers, where a contrast difference between monomers of different types can be substantial.
In the first step in Section 2.1, the distribution in the number of kinks in a chain is calculated from a given probability for a monomer to be a kink. For chains with a fixed number of kinks, the length distribution of undistorted strands described in Section 2.2 is used in Section 2.3 in the derivation of the effective persistence length of the chains. After a short discussion of the problems to define the persistence length of chains in a good solvent in Section 2.4, the weighted averaging of FFs to build up the FF for an ensemble of distorted chains is described in Section 2.5. The implementation in Section 2.6 is used for an example plot in Section 2.7. It turns out, that the resulting FF of a distorted SFPC can be quite well approximated by a FF of an undistorted SFPC with a suitable equivalent persistence length, as discussed in Section 2.8. A still better approximation in Section 2.9 is based on a fit of a FF of a distorted SFPC by a FF of an undistorted SFPC with a polydispersity in length. Following an important hint from the peer review process of this publication, the results are compared to related work on DNA in Section 2.10.
(2) |
The weighting in a scattering experiment follows the weight distribution of the polymer chains. Since kinks are based on chemical defects in the chain, the molecular weight M + ΔM of a kink monomer might deviate from the molecular weight M of a regular monomer. A mass decrease in a kink is described by the value ΔM < 0. The weight distribution corresponding to (2) reads
(3) |
(4) |
The numerator of the fraction in (3) is the molecular weight of a polymer with D defects, and the denominator is calculated from the normalization condition for the weight distribution; since Nϕ is the average number of defects per chain, the denominator is the average molecular weight of the chains. For a small relative mass deviation δ = ΔM/M ≪ 1, (4) can be approximated (see Appendix) by a binomial distribution
(5) |
(6) |
The increase of ϕ_{w} relative to ϕ for positive δ reflects the higher weighting of the heavier defective chains in the weight average. ΔM and δ take negative values in cases where the defects have a lower molecular weight than regular monomers; in this case, distorted polymer chains are less weighted in the weight average. With the mean value of defects per chain 〈D〉 = Nϕ_{w} and the variance σ^{2} = Nϕ_{w}(1 − ϕ_{w}), (6) can be approximated as a Gaussian distribution. The Laplace rule of thumb for the validity of such an approximation reads σ^{2} > 9. In almost perfect chains with (1 − ϕ_{w}) ≈ 1, on average 9 or more defects per chain are needed to justify the Gaussian approximation. In cases where this criterium is not fulfilled, the calculations need to stick to (3) or (5).
(7) |
The strand lengths l in this chain with D kinks show an exponential length distribution
(8) |
The minimum distance between two kinks is the monomer bond length b_{0} = L/N, and so P_{l}(l|D) in (8) is defined for l ≥ b_{0}. The exponential length distribution can be derived based on the conditional probability ϕ_{D} that in a chain with D defects a certain monomer is a kink. For a series of monomers in this polymer chain, a conditional length distribution P_{n}(n|D) of kink-free chain strands results as
P_{n}(n|D) = (1 − ϕ_{D})^{(n−1)}ϕ_{D}. | (9) |
So, a strand consists of (n − 1) regular persistent units and one kink which ends the otherwise undistorted series. The exponential distributions (8) and (9) have identical mean and are thus equivalent for
(10) |
The result is unexpected, since a simple approach would assign the value D/N to the probability that a specific monomer in a chain of length N with D defects is a kink. The occurrence of [D + 1] can be rationalized, when the two chain ends are considered as one additional kink. When N monomers are addressed as a sequence with periodic boundary conditions, and [D + 1] defects 0, 1, 2,…,D are distributed among them, the periodic sequence can be opened at the defect 0. This procedure creates a linear chain with D defects among [D + 1] undistorted strands. In other words: in (9), the undistorted strands are ended by a distortion. The last strand ends at the chain end, and for this reason the chain end has to be counted as an additional kink.
K_{or}(Δl) = 〈(l)·(l + Δl)〉 | (11) |
(12) |
(13) |
Here, l_{0} = 0 and l_{D+1} = L are used for the start and the end point of a chain, respectively. The mean squared end-to-end vector results as
(14) |
Due to the loss of directional correlation at the kinks, the average scalar product of the unit vectors of two different strands i≠j is zero. With (12), (14) is written as^{23}
(15) |
Performing the double integration yields
(16) |
The contour lengths (l_{i+1} − l_{i}) are averaged with the exponential distribution (8). With (7), the result reads
(17) |
(18) |
The second form results from the Taylor expansion exp[−b_{0}/l_{p}] ≈ 1 − b_{0}/l_{p}, which is justified for a SFPC with l_{p} ≫ b_{0}. An effective persistence length _{p}(D) of a chain with D kinks is introduced based on the same functional form as (18) for the kink free case D = 0, where (7) yields l_{d}(0) = L:
(19) |
Now, (19) with 〈_{D}^{2}〉 from (18) inserted can be solved for _{p}(D). The result can be written in the intuitive form
(20) |
The physics behind this equation becomes clear, when it is multiplied by −Δl and exponential functions of the left and right hand side are taken, similar to (12). With the orientation correlation functions K^{(0)}_{or}(Δl) = exp[−Δl/l_{p}] and K^{(d)}_{or}(Δl) = exp[−Δl/_{p}(D)] for the undistorted and the distorted SFPCs, respectively, the result reads
(21) |
There are two mechanisms in (21) which lead to a decay of K^{(d)}_{or}(Δl): there is a continuous loss of orientation correlation similar to K^{(0)}_{or}(Δl) in the first factor and a probability that the correlation gets completely lost by a kink in the second factor. For the kink-free case D = 0 with l_{d}(0) = L, the second and third terms on the right side of (20) cancel out, and the second factor in (21) becomes unity; the third term in (20) is thus a correction for the finite chain length. The subtraction l_{d}(D) − b_{0} of one monomer length in (20) and (21) reflects (9), where an undistorted strand consists of (n − 1) stiff monomers and 1 non-stiff kink.
The transition of K_{or}(Δl) to a power law for a good solvent and large Δl can be understood by an intuitive reasoning. As the EVIs are repulsive, they cause an expansion of the polymer coil and a local stretching of the chains. This point of view is the basis of Flory's mean field derivation of ν, where the pressure of an ideal gas of chain segments is compensated by the elastic energy of a stretched coil.^{29} For a chain in a good solvent with EVIs, it appears thus more appropriate to address chain segments as extended by a local stretching force. The deviation of K_{or}(Δl) from an exponential at large Δl can be attributed to the local stretching force. As the stretching force at a specific chain location is built up by the EVIs of the chain sections at both sides of this location, it is clear that the chain stretching and the local persistence length decrease towards the chain ends. Based on such a mechanical approach, it might be worth investigating SFPCs in a good solvent by depolarized light scattering, as a local stress is transferred to a birefringence via the polymer's stress-optical coefficient.^{30,31}
The EVIs between the monomers extend the chain, and this effect has a longer range than the chain persistence. The EVIs are basically not affected by chain distortions, and so the large scale behavior of distorted and undistorted SFPCs is very similar. The persistence length thus affects only the Kuhn segment lengths, or the chain orientation correlation at short length scales, where K_{or}(Δl) is well described by an exponential decay. The calculated _{p}(D) is the appropriate short length scale parameter for distorted SFPCs. In the procedure below, the FF of distorted SFPCs is built up by FFs of undistorted chains with different effective persistence lengths. The power-law behavior of K_{or}(Δl) at large Δl is already built in the employed FF of a semi-flexible excluded volume chain. The superposition of such FFs of different short scale behaviors yields automatically the appropriate long scale algebraic decay for the orientation correlation of the chains. The model for the distorted SFPCs is as good or as bad as the employed FF for undistorted SFPCs.
The distorted SFPC involves the probability of defects ϕ and the relative monomer mass difference of defects δ as additional model parameters. In cases where these parameters are known from independent experiments, e.g. NMR measurements, the FF Ŝ(q) with polydispersity of the persistence length can be written based on the distribution (4) as
(22) |
Without previous knowledge of δ and ϕ, it is more suitable to describe the chain distortions by the modified probability (6) with a single additional fitting parameter based on the distribution (5)
(23) |
(24) |
The second form (24) is the Gaussian approximation, valid when the Laplace criterion Nϕ_{w}(1 − ϕ_{w}) ≥ 9 is fulfilled. Like for a polydispersity in length, there is no closed equation for the FF, and the averaging has to be done numerically. The bond length enters (22)–(24) as the minimum distance between two kinks.
The cross sectional profile of a SFPC is usually considered as a product of a FF S_{chain}(q) of a chain with a negligible cross section and a cross section FF S_{cs}(q):
S(q) = S_{chain}(q)S_{cs}(q). | (25) |
The latter is an integration
(26) |
S_{cs} = |2J_{1}(qR_{cs})/(qR_{cs})|^{2}. | (27) |
Here, J_{0}(·) and J_{1}(·) are Bessel functions of the 0th and 1st order, respectively. The product approximation (25) survives the averaging process in (22)–(24). The chain cross thus can either be considered on the level of S(q) before averaging, or equivalently as a multiplication of an average of infinite thin chain FFs S_{chain}(q) with S_{cs}(q). Similarly, the high-q rod-behavior S_{chain}(q) ∼ M/Lq^{−1} does not depend on _{p} and is thus fulfilled for all FFs. At high q, the averaging process (22)–(24) recovers the q^{−1} rod behavior with the unchanged prefactor M/L.
For the case of SFPCs without EVIs the available FFs offer different advantages, which are illustrated in Fig. 3. The analytical form of the PS FF^{2} method 2 is very fast in the simulations, while the numerical integration required for the Kholodenko FF takes significantly more time. In particular for the data fitting where numerical parameter derivatives have been used, a high accuracy of this integration is required. On the other hand, for short chains with a length comparable to or smaller than l_{p}, the PS FF in Fig. 3 shows a small bump at intermediate q values which appears to be an artefact. As we scrutinize the calculated FFs of distorted SFPCs for differences to FFs of undistorted SFPCs especially at a short chain length, the smoother Kholodenko FF^{3} has been used in the simulations. For efficient fitting, the PS FF method 2 was employed as a fitting function in a first step for the case of no EVIs, and the obtained parameter values are inserted as start parameters for the fit with the Kholodenko FF. For the case of SFPCs with EVIs, the corresponding PS FF, method 3 was used directly. The polydispersity in the fit was described as a log-normal length distribution. In all examples below, an innate persistence l_{p} ≈ 65b_{0} corresponding to an angle θ = 10° in (1) was used.
Fig. 3 Comparison of the form factors S(q) of Kholodenko^{3} (grey solid lines) and of Pederson and Schurtenberger,^{2} method 2 for SFPCs without EVIs (orange dashed lines). The numbers indicate the ratio n = L/l_{p}, and N is chosen as n^{2}. |
The FF of a chain with the same length N = 100 and a higher fraction of distortions ϕ = 0.15 is shown in blue in Fig. 4. The behavior of l_{d}(D) and _{p}(D) is identical to the previous case, but the distribution P_{w}(D) now emphasizes higher values of D in the averaging process (22)–(24), with smaller values of _{p}(D). As a result, the radius of gyration decreases compared to the previous case, as can be seen in the Guinier range of the FFs in Fig. 4c: the initial down-bending of the FF is shifted to higher q values. A further effect occurs in the resulting persistence length. Although the q^{−2} range in the FFs without EVIs and the range in the FFs with EVIs are not well built up for the considered short chains, the transition to the q^{−1} behavior at higher q is well visible in Fig. 4c. The transition at higher q values for ϕ = 0.15 compared to the location of the transition for is indicative of a shorter effective persistence length.
The third FF in Fig. 4 in grey describes a longer distorted SFPC with N = 400 and the same fraction of distortions ϕ = 0.15 as in the second case. The increased chain length results in a larger radius of gyration, while the resulting persistence lengths are comparable for the two cases. The higher N leads to an increase in the average number of chain defects 〈D〉 = Nϕ_{w} and the standard deviation of P_{w}(D). The relative standard deviation decreases with . This decrease shows up in Fig. 4a, where P_{w}(D) is plotted versus D/N. The decrease of the relative standard deviation of the binomial distribution (5) with increasing N is an inherent property of the statistics of distorted polymer chains in Section 2.1. This constitutes a second reason why the effects of persistence polydispersity are less prominent for longer chain lengths.
(28) |
The small q behavior of all PS FFs is traced back to the Debye FF,^{2} so the latter can be used here. Its squared radius of gyration
(29) |
(30) |
As (29) and (30) increase monotonically with b in the physical range L, b > 0, there is an unambiguous inversion b(R_{g}^{2}). With this inversion and (28), the R_{g}-equivalent persistence length results as
(31) |
Depending on the solvent quality, the calculation of R_{g}^{2} and the inversion b(R_{g}^{2}) in (31) are based on (29) or (30). For simplicity, _{p} is calculated numerically.
The approximations S(q,_{p}) by the undistorted FF and the R_{g}-equivalent persistence length _{p} are included in Fig. 4. They match closely the corresponding FFs Ŝ(q) of distorted SFPCs. The resulting values for _{p}/b_{0} are listed in Table 1. There is a strong decrease of _{p} compared to the innate persistence l_{p} ≈ 65b_{0}. The larger values for the case N = 100, reflect the higher weighting of small D values by P_{w}(D) in Fig. 4b. The values of _{p}/b_{0} with and without EVIs are almost identical, so the solvent quality has basically no effect. Any effect of chain stretching in a good solvent in the R_{g}^{2} calculation in (31) is compensated by an opposite effect for the inversion b(R_{g}^{2}). Also included in Table 1 are results for δ = −0.5. Even for this large value of the relative mass defect where a kink has only half the mass of a regular monomer, there is basically no change compared to the result of a distorted SFPC without a mass defect.
N | ϕ | δ | No EVIs | With EVIs | ||||
---|---|---|---|---|---|---|---|---|
_{g} ^{2} | Fit | _{g} ^{2} | Fit | |||||
_{p}/b_{0} | Ñ _{n} | _{p}/b_{0} | _{p}/b_{0} | Ñ _{n} | _{p}/b_{0} | |||
400 | 0.15 | 0 | 5.26 | 396.0 | 5.7 | 5.25 | 389.6 | 5.7 |
100 | 0.15 | 0 | 5.36 | 96.9 | 5.8 | 5.34 | 95.9 | 5.8 |
100 | 1/30 | 0 | 18.46 | 92.9 | 19.4 | 18.41 | 97.0 | 20.0 |
400 | 0.15 | −0.5 | 5.26 | 394.2 | 5.7 | 5.25 | 397.5 | 5.7 |
100 | 0.15 | −0.5 | 5.39 | 98.2 | 5.8 | 5.37 | 97.7 | 5.9 |
100 | 1/30 | −0.5 | 18.46 | 93.2 | 19.5 | 18.41 | 97.0 | 20.1 |
The strong decrease of _{p} compared to the innate persistence l_{p} indicates a synergetic effect of the bonds to achieve a semi-flexibility of the chain. It can be stated, that a chain with a high perfection of 95% of regular monomers and only 5% of defects does not show a R_{g}-equivalent persistence length of 95% of the innate persistence length, but a much lower value. An interruption of the exponential decay of K_{or}(Δl) in (12) at a kink has a strong effect on the overall chain persistence. For long chains N ≫ 1, the distribution P_{w}(D) is peaked at the average number 〈D〉 = Nϕ_{w} of distortions in a chain, as the relative standard deviation decreases with . In this limit, _{p} can be approximated by the effective persistence length _{p}(〈D〉) of a chain with the average number of defects. A combination of (20) and (7) yields a formula for _{p}(〈D〉). For large N, it simplifies to
(32) |
The numerator describes the fraction (1 − ϕ_{w}) of the innate persistence length l_{p}, which corresponds to the fraction of regular monomers. The additional decrease due to the denominator accounts for the distortion of the synergy of the semi-flexible monomers by the defects.
An unexpected outcome of the comparison of Ŝ(q) with the approximation S(q,_{p}) is the very good matching around ql_{p} ≈ 1, and hence the region sensitive to the persistence length. In Fig. 2, FFs of quite different effective persistence lengths _{p}(D) contribute to the averaging process to build up Ŝ(q). The expectation was to find some kind of smearing or broadening of the transition of the q^{−2} behavior for no EVIs or the behavior for SFPCs with EVIs to the rod limit q^{−1} behavior. Such a smearing could be a starting point for an approach to work out the innate persistence length from data of distorted SFPCs. However, basically no smearing is found here after the averaging process. The scattering behavior of undistorted and distorted SFPCs is thus very similar. As a consequence it is not possible to detect chain distortions from scattering experiments.
In a mathematical treatment by Caravenna et al.,^{33} it is shown that in the limit of infinite chain length the large scale appearance of semi-flexible statistical hetero-polymers without excluded volume interactions looks like a homo-polymer with a suitable persistence length.^{33} So for a marginal solvent, the form factor of such a monodisperse heteropolymer starts at small q like in Fig. 1 with a behavior described by , which bends over to a q^{−2} decay. The results here go one step further. Also around ql_{p} ≈ 1, which corresponds to a short scale appearance on a monomer level, a suitably chosen average persistence length captures the behavior of an ensemble of semi-flexible polymer chains, where the heterogenity in a chain results from the statistical occurrence of kinks. The result is valid without and with excluded volume interactions.
The finding that the persistence polydispersity has no effect on the FF can be rationalized with arguments similar to the calculation of _{g} by (28) based on an averaging process of Taylor expansions. In Section 2.4, l_{p}^{−1} was discussed as a short range property of K_{or}(Δl), so within a first order Taylor expansion to describe K_{or}(Δl) for small Δl. When K_{or}(Δl) is averaged over chains with different _{p}(D), the result has the same functional form as a first order Taylor expansion with a zero order term equal to 1 and a linear term with a slope which defines the resulting inverse persistence length. This form can in turn be considered as the short range behavior of an exponential which is connected to a not smeared transition of Ŝ(q) around ql_{p} ≈1. Deviations are of higher order. The calculation of the statistical weights for the averaging process is not as straightforward as for the small q behavior in the determination of _{g}^{2}. However, the formal argument applies for any normalized distribution used in the averaging process, especially as _{p}(D) > 0, so an average of value 0 for the resulting inverse persistence length can be excluded. This averaging property of the short term behavior of K_{or}(Δl) is furthermore relevant for the discussion of the orientation correlation function for the good solvent case in Section 2.4. It elucidates why it is possible to assign an average persistence length to SFPC in a good solvent with EVIs, although the local decay of orientation correlation depends on the position within the chain.
In cases where the fraction ϕ_{w} of chain distortions is known from other experiments, scattering data can be used for an estimation of the innate persistence length. For long chains N ≫ 1, an inversion of (32) reads:
(33) |
In cases where the denominator in (33) becomes very small or even negative, the finding might be interpreted as that the effect of chain distortions is so strong that it is not possible to estimate l_{p}. A test with the values of Table 1 yields rough estimates for l_{p}, with the closest value l_{p} ≈ 73b_{0} for the longest chain N = 400. The estimates for l_{p} based on a blowing up of _{p} by a small denominator in (33) should generally be used rather carefully. Less critical is an estimation of ϕ_{w} based on (33), in cases where l_{p} and _{p} are known.
The results for Ñ_{n} are slightly smaller than Ñ_{w} = N, so there is a small apparent length polydispersity in the distorted SFPCs. The value of u decreases with increasing N, as expected from the arguments in Section 2.7 ((1) The value of l_{d}(D)^{−1} for fixed D is smaller for higher N, so l_{p}^{−1} in (20) dominates more and there is less effect of D on _{p}(D)^{−1}. (2) The relative standard deviation σ/〈D〉 of the distribution of the number of defects in the chains of an ensemble becomes smaller with ). The smallness of the apparent length polydispersity can be rationalized as follows: a polymer chain in a dilute solution is a soft and highly fluctuating object. Its FF is an average over all possible conformations, which show already a broad distribution in size. For a distorted SFPC, there is the additional distribution due to the persistence polydispersity. In this case, the total size distribution for all chain conformations and all possible locations of kinks corresponds to a convolution of the conformation distribution and the kink distribution. Obviously, the variance of the conformation distribution of an undistorted chain, which is equal to R_{g}^{2}, exceeds the additional width contribution due to the kink distribution. So the variance of the convoluted distribution is dominated by the variance of the conformation distribution.
The fit result _{p} for the persistence length is comparable to the R_{g}-equivalent persistence length _{p}, with a slightly larger value. The results for δ = 0 and δ = −0.5 are basically the same, so a mass defect of kinks has no substantial effect. It was expected that values δ < 0 would increase the apparent length polydispersity, as chains with more kinks and thus a higher total mass defect would contribute with a smaller R_{g} value. This argument goes in the same direction as the mechanism in conventional length polydispersity, where shorter chains have a smaller R_{g} and less weight in scattering due to their smaller mass. Such an effect of δ, however, turned out to be below the resolution of the fitting procedure, even for the higher data quality of simulated data compared to true experimental data.
While in the classical WLC model the bending energy is quadratic in chain curvature,^{4} the theoretical description of DNA by Wiggins and coauthors introduces a saturation of the quadratic bending energy at a high chain curvature.^{37,38} In this way, the conventional WLC behavior at large length scales is combined with the occurrence of local sharp kinks.
There are several similarities and a number of clear differences between the work of Wiggins et al. and the approach described here. In a distinction used by Caravenna et al.,^{33} the localization of kinks in the description of DNA by Wiggins et al. is ergodic, and hence roughly speaking mobile within the chain. An averaging procedure over the conformations of a single chain is sufficient. The work here classifies as quenched within the Caravenna distinction. The locations of the kinks and their number are fixed and inherent in the chains, and after the averaging over conformations a second averaging step over an ensemble of chains is required. The distribution of the number of kinks in the chains and polydispersity questions become an issue. The mathematical approach of Wiggins et al. is based on quantum mechanical propagators, path integration, Fourier Laplace transformations and the Faltungs theorem, similar to Kholodenko's FF.^{3} It is far more advanced than the down-to-earth approach discussed here, which is an extension of the freely rotating chain model^{23} and uses basic geometry and probability theory only. The derivation of Wiggins et al. jumps back and forth between discrete and continuous descriptions of a chain. In this process the notion of a minimum distance between kinks as described here and finite chain length corrections like in (20) and (21) get lost. The access to partition functions in the work of Wiggins et al. allows the description of several experiments, while here the focus is on the FF only. On the other hand, the approach of Wiggins et al. is very specific to chains without EVIs, and hence the marginal solvent case. Here, FFs with and without EVIs are discussed on the same level, and the method can further be applied to distorted macromolecules with a morphology different from a linear chain arrangement of monomers, as long as a FF for non-distorted macromolecules of such a morphology is available. It appears, that EVIs are mostly not considered in the DNA community, apart from rare exceptions.^{39,40} From a polymer physicists viewpoint, the discarding of EVIs might limit the validity of theoretical descriptions of DNA in solution.
For scattering experiments, Wiggins et al. find no effect of kinks. In this comparison, the parameters of the saturated bending energy are selected in such a way as to obtain the same long range persistence length as in the reference WLC model.^{38} This finding corresponds to the result of the present work. The apparent length polydispersity as discussed in Section 2.9 has not been addressed by Wiggins et al. For a detection of the saturation of the bending energy, Wiggins et al. discuss experiments which involve a strong bending:^{38} fluorescence resonance energy transfer (FRET)^{41} and the formation of circles in short DNA segments (cyclization).
In summary of Section 2.10, the previous work of Wiggins et al.^{37,38} addresses a similar problem with rather different methods. Their approach and the work here have complementary advantages and disadvantages.
These conclusions have been achieved by a detailed description of the persistence polydispersity caused by statistically distributed kink distortions in the chains. For the dilute systems considered here, inter-chain interferences in scattering experiments average to zero, and the description of intra-chain interferences can be traced back to existing form factors from the literature. Depending on the solvent quality, form factors with and without excluded volume interactions have been considered. The statistical occurrence of kinks leads to a binomial distribution in the number of kinks in a chain. For a chain with a fixed number of kinks, the inverse effective persistence length is derived as the sum of the inverse innate persistence length, the inverse average length of an undistorted chain segment, and a correction term for the finite chain length. The corresponding orientation correlation function is a product of the innate orientation correlation function and the probability of complete correlation loss by a kink. The form factor of an ensemble of distorted chains results as an average based on the distribution in the number of kinks per chain and form factors where the effective persistence lengths are inserted. Three examples of distorted SFPCs calculated with an implementation of this approach are discussed. For a detailed look at the effect of persistence polydispersity, the example form factors of distorted SFPCs are compared to form factors of undistorted SFPCs with the same radius of gyration. Besides the strong decrease of the related R_{g}-equivalent persistence length compared to the innate persistence length, there is a slight change in curvature at small q, which in a conventional interpretation of experimental data would be assigned to a length polydispersity. A fit of a distorted SFPC form factor by an undistorted SFPC form factor with a length polydispersity provides a very good description. There is basically no smearing in the region ql_{p} ≈ 1.
The procedure developed here can be adapted for the description of semi-flexible random copolymers, as a generalization of the discussion of flexible random copolymers by Endo and Shibayama.^{26} The occurrence of different monomers introduces different contrast factors with further complications. The variation of scattering contrast might be an interesting tool in scattering experiments on semi-flexible random copolymers. Similar to the distorted SFPCs discussed here, it is expected that there are only minor deviations for semi-flexible random copolymers compared to the scattering behavior of homo-polymers, apart from a strong effect on the R_{g}-equivalent persistence length. Such a result, however, would be a valuable corner stone for the investigation of semi-flexible random copolymers by scattering experiments, as it could justify data fitting and interpretation by models and methods developed for homo-polymers. Furthermore, the averaging over the persistence polydispersity of SFPCs introduced here might become useful for more advanced scattering experiments. For colloidal particles, experimental light scattering techniques are available to distinguish coherent and incoherent contributions^{24} and excited and relaxed fluctuations of a soft colloidal shell.^{42,43} A theoretical description of other advanced light scattering techniques adapted to polymers to be developed in the future with similar abilities might be based on the methods introduced here.
(34) |
For δ ≪ 1, the approximation ln(1 + x) ≈ x can be used. The difference in the numerator of the right side of (34) leads to a product of two exponentials. The factors (N − D) and D are extracted as external exponents for the exponentials:
(35) |
First order Taylor expansions of the exponentials in δ yield
(36) |
(37) |
The modified probability ϕ_{w} in (6) is extracted from the term of power D in (37). The term of power (N − D) in (37) is equal to (1 − ϕ_{w}), and so (37) is really a probability distribution with correct normalization.
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