Полімерна ланцюг: Чому розмір росте з квадратом кореня довжини

The Random Walk Model

The random walk model describes a polymer as a collection of points moving randomly through space. Each point represents a segment of the chain, and its movement is governed by probabilities. This simple approach captures the essence of how polymers explore their environment, driven by thermal fluctuations.

Root-mean-square end-to-end distance

Because each bond vector is independent and random, the cross terms in the end-to-end vector average to zero, exactly as they do for ordinary Brownian motion. What survives is a simple sum of squared bond lengths:

R = ∑ bⁱ (end-to-end vector, sum of N bond vectors) ⟨R²⟩ = N·bⁱ² (freely-jointed chain, ideal/theta conditions) R_rms = b·√N (grows as the SQUARE ROOT of chain length) The √N scaling is the headline result: double the number of monomers and the coil's typical size grows by only √2 ≈ 1.41×, not 2×. A polymer coil is a fractal, sparse object — most of the volume it spans is solvent, not chain.

R = ∑ bⁱ                (end-to-end vector, sum of N bond vectors)
⟨R²⟩ = N·b²             (freely-jointed chain, ideal/theta conditions)
R_rms = b·√N              (grows as the SQUARE ROOT of chain length)

Radius of gyration and what an experiment actually measures

You cannot directly see the two chain ends in a real sample, so experiments (light scattering, SANS, viscometry) report the radius of gyation R_g, the root-mean-square distance of every monomer from the chain's centre of mass. For an ideal chain it is related to R² by a clean numerical factor:

R_g^2 = R^2 / 6 (ideal chain) R_g ~ sqrt(N) (real chain, nu the Flory exponent) That second line generalises the result to chains where the freely-jointed assumption breaks down.

⟨R_g²⟩ = ⟨R²⟩ / 6 = N·b² / 6      (ideal chain)
R_g ~ N⊃ν              (real chain, ν the Flory exponent)

Excluded volume and the Flory exponent

Real monomers cannot occupy the same point in space – a chain cannot cross itself. In a good solvent (one that prefers contact with the polymer over contact with itself) this excluded-volume effect systematically swells the coil beyond the ideal √N prediction. Paul Flory's mean-field argument balances two competing free energies: the entropic cost of stretching the chain against the repulsive energy of overlapping monomers, and minimising their sum gives a single exponent that has proven remarkably accurate:

R ~ N^β ν = 3/5 good solvent (Flory's mean-field value; renormalisation group: ν ≈ 0.588) ν = 1/2 theta solvent (ideal chain — excluded volume exactly cancelled by solvent) ν = 1/3 poor solvent (chain collapses into a dense globule) The theta point is the special temperature at which monomer-monomer and monomer-solvent interactions balance so perfectly that the chain behaves as if excluded volume did not exist, recovering the ideal ν = 1/2 scaling even though individual monomers still physically exclude each other – a remarkable cancellation, not an approximation.

R ~ Nν
ν = 3/5        good solvent  (Flory's mean-field value; renormalisation group: ν ≈ 0.588)
ν = 1/2        theta solvent (ideal chain — excluded volume exactly cancelled by solvent)
ν = 1/3        poor solvent  (chain collapses into a dense globule)

Why the scaling law matters

The exponent ν controls nearly every macroscopic property of a polymer solution or melt: how viscosity scales with molecular weight, how quickly a polymer diffuses, how densely chains can pack, and how a gel swells. Measuring R_g as a function of molecular weight by light scattering is, in effect, a direct measurement of ν, and hence of how good a solvent is for that particular polymer — the same reasoning that tells a chemist which solvent to use to process a plastic or which conditions will make a protein's unfolded state expand or collapse.

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