Soft, coarse-grained polymer model

I start the discussion with models that describe the molecular conformation of macro-molecules with statistical methods. Advantageous for these models, is their treatment of

1The sketch for continuum models is actually a simulation snapshot from an SCMF simulation, but these length scales are typically only accessible with continuum models.

polymers as long chains with many repeat units. Instead of describing all details of the atomistic conformation, each state of each unit gets a probability assigned. Hence, an ensemble of chain realizations can be investigated. In the limit of many chains, such a description becomes accurate as each of the possible states is populated according to the assigned probability. Like in statistical mechanics studying the ensemble characteristics can give insights into the physics of these systems.

One such model has been previously mentioned to introduce a general description of polymers: the freely rotating chain (FRC), see the previoussection 2.1.1 andFigure 2.1.

A key feature of the freely rotating chain (FRC) is the correlation between adjacent bonds, the bond angle ψ, and the fix bond length b₀. Each bond has a fixed length b²_i =b²₀ and the angle formed with the previous bond is also fixedbibi−1=b²₀cosψ. The latter condition introduces a direct correlation of bondiwith its predecessor i−1. This correlation propagates along the backbone and can be quantified with the dot product of two bond vectors bibj. For uncorrelated bonds, the ensemble average of this dot product is equal to zero as the two vectors point in random directions. For the FRC, the ensemble average of the dot product decays exponentially with the distance between the monomers along the backbone [35]

⟨b_ib_j⟩=b²₀cos(ψ)^|i−j|=b²₀exp (

−|i−j|b₀ L_p

)

(2.3) with the correlation lengthLp :=b0/|log(|cosψ|)|. Consequently, forψ̸= 0, π one can assume monomers that are further apart thanLp as uncorrelated. This specific result of the FRC model is furthermore transferable to many real polymer species. Especially, I discuss long and flexible chains with a much longer contour lengthLc:= (N−1)b0 ≫Lp

than persistence length.

During the coarse-graining process, many monomers are described by a single in-teraction center. If each inin-teraction center contains more monomers than required by the persistence length, a different statistical model, the freely jointed chain (FJC), is sufficient. The only restriction of the FJC is that all bond vectors have a constant bond length, b0. There is no correlation between the bond vectors in this model. Ef-fectively, the polymer backbone is now a random walk withN −1 steps of length b₀. The analogy to the random walk reveals another critical property of polymer confor-mations: they are fractal [35]. The self-similarity on different length scales promotes the decoupling of molecular weight and discretization, which is discussed in the next step of the coarse-graining. Furthermore, the fractal dimension of random walks and polymer conformations is known to bed_f = 2, such that polymers do not densely fill the three-dimensional space. Because space is completely covered with polymers in a melt, many chains overlapping each others’ extension are present. As a result, the number of interacting chains, ¯N, is usually high. A high invariant degree of polymerization, ¯N, is important for many calculations as the HamiltonianHof a polymer system usually scales linearly with√

N¯, compare with Equation 2.29. In a canonical ensemble, the probability of a given state is Boltzmann distributedp∝exp⁽−_k^H

BT

). A typical distribution has a single maximum at the most probable state and decays continuously for all other states.

As the Hamiltonian H scales with ¯N, this decay becomes exponentially steeper with increasing ¯N. The mean-field approximation (MFA) approximates this distribution

2.1. Molecular conformations of polymers

as a delta distribution centered around the most probable state. The approximation becomes more accurate the narrower the distributionp∝exp⁽−_k^H

BT

)is. This justifies calculations in the MFA, which turn exact for ¯N → ∞, like in SCFT.

The average squared end-to-end distance of this model is found to be⟨R²_e0⟩=b²₀(N−1).

In addition, the full distribution of the end-to-end vector can be calculated by various methods [35,40, 45]. The central limit theorem (CLT) is the underlying reason why the distribution converges for long chains towards the Gaussian distribution.

p(Re) =⁽ 3 The full distribution of the end-to-end vector in equilibrium helps with the next step of coarse-graining.

Combining, again, many of these quasi-monomers to quasi-quasi-monomers leads to the Gaussian chain model. Every bond Gaussian chain has a fluctuating length, in contrast, to the fixed bond length b₀ of the FJC model. From now on, unless explicitly otherwise stated, the terms bead and monomer refer to this coarse-grained beads. Consequently, the statistics of a single bond in this model is again Gaussian (Equation 2.4).

The harmonic spring constant k = 3kBT /⟨b²⟩ relates to the average bond elonga-tion ^√⟨b²⟩. In the next section 2.1.2a the harmonic bond potential fulfilling this distribution is discussed. Note that this distribution is for the equilibrium scenario. For nonequilibrium conditions, variations of this potential are discussed as well.

This coarse-graining procedure can also be obtained by formally coarse-graining polymers via the renormalization group theory. The Gaussian chain is the fixed point in the renormalization to which all coarse-grained chains converge. A detailed discussion about this concept can be found in Refs. [46,47].

A major advantage of coarse-graining a polymer chain before simulations is the massive reduction in particles to simulate. Each simulated bead in the coarse-grained model represents many atomic repeat units,N. This reduces, on the one hand, the degrees of freedom, making computer simulations of many polymer chains tractable. On the other hand, the relaxation time of polymers scales with powers of the number of repeat units. In the Rouse model, the longest relaxation timescales with the square,N², and in the tube model with the power threeN³ (section 2.4.1). Thus reducing the number of simulated repeat units reduces the number of time steps to simulate a single relaxation time for polymers dramatically. Additionally, the resulting softer pairwise interactions, seesection 2.2, allow a faster time stepping for computer simulations.

Despite the mentioned technical advantages of coarse-graining, there are also physical advantages of investigating coarse-grained polymers. The most important one is uni-versality. The aforementioned similarity to random walks determines the universality class. In the motivation of the Gaussian chain, it becomes clear that any chemical species of a long flexible polymer can be represented by the Gaussian chain model.

This increases the versatility of the developed model. It also highlights that the model captures the universal properties of polymers instead of focusing on less important details. Additionally, the model enables the decoupling of molecular weightMw and discretizationN. For the coarse-graining, it is not required to specify exactly the number of atomistic repeat units. Thus, one simulation with a single discretization, N, can represent chains of different molecular weightMw and the other way around. Thereby, neither the discretization N nor the molecular weight is a good measure to characterize polymers, instead, the invariant degree of polymerization ¯N is used. Combining both characteristics, universality, and decoupling means that a single simulation can be interpreted for a whole range of polymer species and molecular weights. The chemical details of the polymers, like bond angles or specific side groups, are not important to build a coarse-grained model. Instead, the generic model can be used with the universal parametersχN, ¯N andReo. More details on the nonbonded interactions parameterχN are discussed insection 2.2.

In the next sections, the specific potentials employed for this work are discussed and motivated.