Scaling of spin-echo amplitudes with frequency, diffusion coefficient, pore size, and susceptibility difference for the NMR of fluids in porous media and biological tissues.

Abstract

Both Carr-Purcell-Meiboom-Gill (CPMG) measurements and single-spin-echo measurements have been made at frequencies of \ensuremath{\nu}=10, 20, and 50 MHz for two relatively homogeneous porous porcelain materials with different pore sizes, both saturated separately with three liquids of different diffusion coefficients. The CPMG transverse relaxation rate is increased by an amount R by diffusion in the inhomogeneous fields caused by susceptibility differences \ensuremath{\chi}; R shows the dependence on \ensuremath{\tau} (half the echo spacing) given by the model of Brown and Fantazzini [Phys. Rev. B 47, 14 823 (1993)] if relaxation is slow enough that there are several CPMG echoes in a transverse relaxation time. For \ensuremath{\tau} values over a range of a factor of about 40, the increase of R with \ensuremath{\tau} is nearly linear, with a slope that is independent of pore dimension a and diffusion coefficient D. For this nearly linear region and a short initial region quadratic in \ensuremath{\tau}, we find R\ensuremath{\propto}(\ensuremath{\chi}\ensuremath{\nu}${)}^{2}$. In these regions we can scale and compare measurements of R taken for different values of \ensuremath{\chi} \ensuremath{\nu}, a, and D by plotting RD/(1/3\ensuremath{\chi}\ensuremath{\nu}a${)}^{2}$ vs D\ensuremath{\tau}/${\mathit{a}}^{2}$. The asymptotic values of R for large \ensuremath{\tau} for CPMG data can be inferred from the asymptotic slope, -${\mathit{R}}_{\mathit{s}}$, of lnM (magnetization) for single spin echoes as a function of echo time t=2\ensuremath{\tau}.It is shown from the Bloch-Torrey equations for NMR with diffusion that, for any combination of parameters \ensuremath{\chi}, \ensuremath{\nu}, a, or D, the magnetization M is a function of both a dimensionless time (either ${\mathit{t}}_{\mathit{u}}$=Dt/${\mathit{a}}^{2}$ or ${\mathit{t}}_{\mathit{v}}$=1/3\ensuremath{\chi}\ensuremath{\nu}t) and a dimensionless parameter \ensuremath{\xi}=1/3\ensuremath{\chi}\ensuremath{\nu}${\mathit{a}}^{2}$/D. If \ensuremath{\xi}2 (for our particular porous media and definition of the distance a), the asymptotic slope of -lnM is approximately ${\mathit{R}}_{\mathit{s}}$=1/3\ensuremath{\chi}\ensuremath{\nu}, that is, it is proportional to only the first power of \ensuremath{\chi}\ensuremath{\nu} and does not depend on either a or D. These results are compatible with the existence of a long-tailed distribution of phases, such as a truncated Cauchy distribution, at echo time. Diffusion does not lead to a reduction of ${\mathit{R}}_{\mathit{s}}$ because averages of choices from a Cauchy distribution give the same distribution rather than a narrower one as for the Gaussian distribution. For larger \ensuremath{\xi} the decay of lnM decreases and no longer approaches a linear asymptote during measurement times. A semiempirical expression for the large-\ensuremath{\xi} case is given. These scaling laws should help in predicting the effects of changes in frequency and of susceptibility contrast as well as of changes in temperature, fluid, or range of pore sizes or other characteristic dimensions.