Freely Rotating ChainEdit
Freely rotating chain is a foundational model in polymer physics that helps scientists understand how local geometry shapes the large-scale structure of long chain molecules. It assumes fixed bond lengths and a fixed bond angle between successive bonds, while allowing the dihedral angle to rotate freely. In effect, each bond can swivel around the preceding one, subject to a constant angle constraint. This creates a chain that is stiffer than a freely jointed chain but far more flexible than a rigid rod, capturing a middle ground that is analytically tractable yet physically meaningful for many synthetic polymers and coarse-grained descriptions of biopolymers. See also polymer and end-to-end distance for related concepts.
The freely rotating chain sits between two classic limiting models. On one side is the freely jointed chain (FJC), where bond directions are uncorrelated from one segment to the next, producing simple random-walk statistics. On the other side is the worm-like chain (WLC) model, which treats the polymer as a continuous filament with bending stiffness that suppresses sharp turns. The freely rotating chain incorporates a fixed geometric constraint—the bond angle—while still allowing rotational freedom around bonds, thereby introducing finite persistence of direction without the full stiffness of a WLC. See also freely jointed chain and worm-like chain.
Overview
In the freely rotating chain, N bonds of fixed length l are connected end to end. Each bond i is represented by a unit vector u_i, and the chain direction is given by the sum of these vectors:
R = ∑_{i=1}^N l u_i
The angle between successive bonds is fixed to θ, so the dot product between consecutive bond vectors satisfies u_i · u_{i+1} = l^2 cos θ. Because the dihedral angle is free to rotate, there is no restriction on the relative orientation of non-adjacent bonds beyond the fixed bond-angle constraint. This structure yields a characteristic correlation in bond directions: the correlation between bond i and bond i+n is ⟨u_i · u_{i+n}⟩ = l^2 (cos θ)^n. The mean-square end-to-end distance is therefore
⟨R^2⟩ = ∑{i=1}^N ⟨(l u_i)·(l u_i)⟩ + 2 ∑{i<j} ⟨(l u_i)·(l u_j)⟩ = N l^2 + 2 l^2 ∑_{k=1}^{N-1} (N - k) (cos θ)^k.
This expression makes explicit how local geometry (the fixed bond angle θ) propagates into global chain size and shape. For θ near 90 degrees (cos θ ≈ 0), successive bonds are essentially uncorrelated and ⟨R^2⟩ ≈ N l^2, recovering the familiar freely jointed-chain result. For small θ (bond angles close to straight, cos θ ≈ 1), correlations persist over many bonds, and the chain behaves as a much stiffer object.
The degree of direction persistence in the freely rotating chain can be captured by a discrete persistence length. The tangent-tangent correlation decays approximately as (cos θ)^n with the number of bonds n between segments, so the characteristic decay length in bonds is ξ = -1 / ln(cos θ). In physical units, this translates into a persistence length p ≈ l / [ -ln(cos θ) ], which reduces to intuitive limits: as θ → 0 (straightened chain), p diverges and the chain behaves more like a rod; as θ → 90° (cos θ → 0), p becomes small and the chain behaves more like a flexible coil.
Mathematical formulation
- Bond geometry: fixed bond length l and fixed bond angle θ between consecutive bonds.
- Bond direction correlations: ⟨u_i · u_{i+n}⟩ = (cos θ)^n.
- End-to-end statistics: ⟨R^2⟩ = N l^2 + 2 l^2 ∑_{k=1}^{N-1} (N - k) (cos θ)^k.
- Persistence of direction: correlation decays as (cos θ)^n; characteristic length in bonds ξ = -1 / ln(cos θ); physical persistence length p ≈ l / [ -ln(cos θ) ].
Related concepts include Kuhn length, which for many chain models serves as the length of an effective freely jointed segment that reproduces the same large-scale statistics, and persistence length, a broader measure of stiffness used across polymer physics. For readers who want to connect to continuous models, the freely rotating chain provides a discrete-step analogue to the more widely used worm-like chain model, with a clear geometric interpretation of stiffness in terms of θ.
Relations to other models and limits
- Freely rotating chain vs freely jointed chain: The FRC reduces to the FJC when θ = 90°, since then consecutive bond directions are uncorrelated. In that limit, ⟨R^2⟩ = N l^2, the textbook result for an ideal random walk.
- Freely rotating chain vs worm-like chain: The WLC treats the polymer as a continuous, differentiable curve with a bending energy. The FRC retains a discrete geometry but captures a finite stiffness through a fixed θ. In many regimes, the FRC provides intuitive analytic insight that complements numerical simulations based on the WLC.
- Excluded volume and solvent quality: The expressions above assume ideal chain statistics without self-avoidance. In good solvents or at high concentrations, excluded volume interactions modify ⟨R^2⟩ and related properties, often requiring corrections or alternative models. See excluded volume and theta solvent for related discussion.
- Temperature and chemistry: In real systems, θ, l, and θ-dependent stiffness can shift with temperature, solvent, and chemical structure. While the FRC is a coarse-grained description, it remains a useful starting point for understanding how local geometries influence macroscopic chain dimensions.
Applications and critiques
The freely rotating chain is a workhorse in teaching and in coarse-grained modeling because it yields closed-form expressions for key quantities while staying closer to physical chain geometry than the simplest random-walk models. It is widely used to interpret experimental measurements of polymer dimensions, to benchmark numerical simulations, and to provide intuition about how local bond angles propagate into global conformations. See also polymer physics.
Critiques of the freely rotating chain emphasize its idealizations. Real polymers exhibit variations in bond lengths and bond angles along the chain, distributions of torsional angles rather than a single fixed angle, and substantial excluded-volume effects that produce deviations from Gaussian statistics in good solvents. While the FRC can be extended (for example, by allowing distributions of θ or by incorporating self-avoidance) to better mimic real systems, some researchers argue that other models—such as the worm-like chain for semi-flexible polymers or more detailed coarse-grained representations—are preferable for quantitatively accurate descriptions. See also Gaussian chain for the idealized limit of many polymer models and excluded volume for the impact of self-avoidance.
In educational contexts and initial design work, proponents stress the practical benefits of the FRC: its analytic tractability, its clear link between microscopic geometry and macroscopic statistics, and its role as a bridge between simpler models and more comprehensive simulations. Critics, while acknowledging these strengths, caution against overreliance on any single idealization when guiding real-world material design or interpreting intricate experimental data.