A dilute solution contains ideal (Gaussian) polymer chains with Kuhn segments of…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Condensed Matter, Materials & Soft Matter · Fourier analysis and integral transforms

Soft Matter at the Thermal Scale, Ch The size and shape of a polymer chain

Problem

A dilute solution contains ideal (Gaussian) polymer chains with N=2500N=2500 Kuhn segments of length b=0.70 nmb=0.70\,\mathrm{nm} each. ((a)) The scattering amplitude of NN segments at positions rj\mathbf r_j is A(q)=∑j=1Neiq⋅rjA(\mathbf q)=\sum_{j=1}^N e^{i\mathbf q\cdot\mathbf r_j}, so that I(q)=⟨∣A(q)∣2⟩=∑j,k⟨eiq⋅rjk⟩I(\mathbf q)=\langle|A(\mathbf q)|^2\rangle=\sum_{j,k}\langle e^{i\mathbf q\cdot\mathbf r_{jk}}\rangle with rjk=rj−rk\mathbf r_{jk}=\mathbf r_j-\mathbf r_k. For a solution of randomly oriented chains, the orientational average of a plane wave over a fixed separation r=∣rjk∣r=|\mathbf r_{jk}| is ⟨eiq⋅r⟩Ω=sin⁡(qr)/(qr)\langle e^{i\mathbf q\cdot\mathbf r}\rangle_\Omega=\sin(qr)/(qr). Expand this to order q2q^2 and hence derive the Guinier law I(q)I(0)=1−q2Rg23+O(q4),Rg2≡12N2∑j,k⟨rjk2⟩.\frac{I(q)}{I(0)} = 1-\frac{q^2R_g^2}{3}+O(q^4), \qquad R_g^2\equiv\frac{1}{2N^2}\sum_{j,k}\langle r_{jk}^2\rangle. ((b)) Using the result Rg2=Nb2/6R_g^2=Nb^2/6 for an ideal chain, evaluate RgR_g for the chains described above. ((c)) Small-angle neutron scattering from this solution is measured at q=0.028 nm−1q=0.028\,\mathrm{nm^{-1}}. Compute qRgqR_g, state whether this point lies inside the Guinier range (qRg≲1qR_g\lesssim1), and give the predicted value of I(q)/I(0)I(q)/I(0) to the order derived in part (a).

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