3.1

Einstein-Stokes relation

Brownian motion of a particle of mass m is described by the Newton’s equation

where ξ(t) denotes the white noise: < ξ(t) >= 0, < ξ(t)ξ(t′) >∝ δ(t – t′) and the brackets < … > denote the average over the realizations of ξ(t).^3,4 μ is the mobility, its inverse is called a friction constant ζ. The solution of Eq.(4) is the Ornstein-Uhlenbeck process.³ In equilibrium the Einstein relation holds:

where the diffusion constant reads (via Fokker-Planck equation³) D = (𝑔/m)². This formula relates the thermal fluctuations (D) to the response of the system (ζ) and is a specific version of the fluctuation-dissipation theorem.⁵

In an important case of a spherical particle of radius a immersed in a solvent of viscosity η Stokes formula holds: . The Einstein formula takes the form of Einstein-Stokes formula:

3.2

Zimm dynamics

Zimm model describes the dynamics of a single chain in a solvent. It is a non-draining model which assumes that the motion of the solvent close to the chain is strongly suppressed. The hydrodynamic effects are accounted for by the Navier-Stokes equation and lead to the long-range hydrodynamic interactions characterized by the Oseen tensor.² Each of the beads undergoes a Brownian motion in a harmonic potential due to the harmonic forces between the neighbouring beads. In the equilibrium the dynamics of the beads is described by the following equation,⁶ where describes the flow of the solvent close to the bead , and k₀ stands for the spring constant:

The excluded volume interaction resulting from the fact that two elements of a chain cannot occupy the same volume element in the space at the same time can be also taken into the account. Calculations² show that the diffusion coefficient D^(Zimm) of the center of the mass of the chain is proportional to (l_c)^–1/2. This is in a fair agreement with experimental result:⁷

Another seminal model of the dynamics of polymer chains - the Rouse model - predicts that D^(Rouse) ∝ (l_c)^–1. Because of this (and other, see Ref²) failures, the Rouse model is regarded as inappropriate for the description of the hydrodynamic-driven dynamics of diluted polymer systems.

The final expression for diffusion coefficient in Zimm dynamics reads²

Equation (9) has the form of Einstein-Stokes equation (6) and thus sets one of the milestones for an experimental analysis of the sizes of polymer chains. Namely, it states that the polymer diffusion in the dilute solution is similar to that of a dilute system of spheres with radius R = r_𝑔. In other words, as far as the hydrodynamic diffusion is taken into account, polymer chains can be treated as spheres. Once the diffusion coefficient is measured (see Section (3.3)), Eq. (9) yields the giration radius of the chain. Comparison with an experiment shows that Zimm model is accurate enough to approximate the diffusion of DNA when R ≈ r_𝑔/3.⁷ Then, the above equation reads

3.3

Dynamic Light Scattering

The Brownian motion of polymers in the solvent can be studied using Dynamic Light Scattering (DLS) technique which yields the time correlations of the intensity of the scattered light. The dynamical structure factor (the correlation function) reads:

In the regime kr_𝑔 ≪ 1 only the overall translatory motion of the polymer can be observed, represented by the translatory movement of the center of mass of the chain. In the long time limit²

where

Formulas (12) and (13) illustrate the principle of DLS measurements of the characteristic size of a polymer chain - radius of giration - from DLS measurements: it is inferred from the decay constant of the correlation function 𝑔(,t). Let us point again that this result is valid only for a single chain or for a strongly diluted system.

5.1

Dilute and semidilute solutions

Let us discuss the spatial organization of the solution. To start with, let us divide polymer solutions in good solvents into three groups, following Ref.:² dilute, semidilute and concentrated. In a dilute solution the DNA chains are away from each other and their interaction can be neglected. In this regime the coil size is independent on the concentration of the solution. For the dilute solution the interactions play a minor role and various physical quantities attain the dependence on the density through virial corrections which, as a rule, are small. As the density increases, the polymer chains come closer and start to overlap. The concentration ρ* at which the overlap starts is estimated as follows²

with M the molecular mass and N_A the Avogadro constant. This condition simply states that in the volume with radius equal to the giration radius there is, on the average, a single coil. Since ρ* ∝ M^–4/5, for large molecular weight ρ* can become quite small. For example, for a polystyrene of M ≈ 10⁶, ρ* is about 5 g/L (about 0.5% in weight). For M = 8 · 10⁶ an extra factor 8 in M decreases ρ⁸ by 8^4/5 ≈ 5 and yields ρ* about 1 g/L. Assuming that this argumentation is weakly dependent on the nature of the polymer chain we tentatively estimate the concentration for which the M = 8 · 10⁶ DNA coils starts to overlap as

This analysis shows that for large molecular weight one can have a solution where the molecules are overlapped but occupy a small volume fraction - by the definition, it is a semidilute solution.

5.2.

Mean chain to chain distance: an example

To illustrate the transition form dilute to semidilute solutions let us consider the case of M = 8 · 10⁶ Da double stranded DNA-CTMA-DR1 (dsDNA) solution in butanol, studied recently using DLS technique; the concentrations varied form 0.01 g/L to 10 g/L.⁸ The mass of DNA backbone is

Consider the case of the concentration . The number N of DNA chains in V = 1 L of the solvent is equal to

Thus, the average volume V_ch per chain is:

If every chain occupies a box, then its side is equal to

This result means that mean distance between adjacent DNA chains is approximately equal to 0.11 μm. On the other hand, experimental DLS data⁸ show that the typical coil size of DNA-CTMA biopolymer at this concentration is larger than 0.5 μm. Those estimations clearly show that the DNA chains overlap and the system is not dilute.

For an arbitrary concentration c the box side a reads:

The plot of function a(c) is shown in Fig. 2. We conclude that even for the lowest concentrations used in experiment (0.01 g/L), the distance between centers of chains is comparable with the characteristic coil size. This result is supported by the Monte Carlo simulations using bond fluctuating method.⁹ Fig. 3 shows an instantaneous configuration of polymer chains consisting of 20 beads for the concentration of chains comparable to the concentration 0.01 g/L in experiment. An eye inspection shows that some of the chains overlap.

Figure 2.

Distance between centers of mass of DNA-CTMA chains against concentration

Figure 3.

Instantaneous configuration of a model polymer system with low concentration (see text for the details).