Light-scattering measurements of during the early

PHYSICS · Subjective · Fermi University Physics · Thermal, Statistical & Complex Systems · Condensed Matter, Materials & Soft Matter · Non-equilibrium statistical mechanics and stochastic thermodynamics

Soft Matter at the Thermal Scale, Ch Spinodal decomposition and coarsening

Problem

Light-scattering measurements of R(q)/q2R(q)/q^2 during the early, linear stage of decomposition of a quenched blend give q2=1.0 μm−2,R/q2=1.0×10−2 s−1 μm2;q2=5.0 μm−2,R/q2=0.6×10−2 s−1 μm2.\begin{aligned} q^2 &= 1.0\,\mathrm{\mu m^{-2}}, & R/q^2 &= 1.0\times10^{-2}\,\mathrm{s^{-1}\,\mu m^2}; \\ q^2 &= 5.0\,\mathrm{\mu m^{-2}}, & R/q^2 &= 0.6\times10^{-2}\,\mathrm{s^{-1}\,\mu m^2}. \end{aligned} Someone fits a straight line through the two points and writes: "slope =1.0×10−2−0.6×10−25.0−1.0=1.0×10−3 μm4 s−1= \dfrac{1.0\times10^{-2} - 0.6\times10^{-2}}{5.0 - 1.0} = 1.0\times10^{-3}\,\mathrm{\mu m^4\,s^{-1}}, a positive number. Since R(q)/q2=−Mf′′−Mκgq2R(q)/q^2 = -Mf'' - M\kappa_gq^2, a positive slope means MκgM\kappa_g is negative, so the gradient coefficient κg\kappa_g must be negative for this sample." Find the error in this working, give the correct slope and intercept, and state the resulting Mf′′Mf'' and MκgM\kappa_g with their signs.

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