Using NMR to Measure Fractal Dimensions
D. Candela
fracom.tex
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\title{Using NMR to Measure Fractal Dimensions}
\author{D. Candela}
\author{Po-zen Wong}

\affiliation{Physics Department, University of Massachusetts, Amherst MA
01003}
\begin{abstract}
\end{abstract}
\pacs{61.43.Hv, 61.43.Gt,91.65.-n}
\date{19 April 2002 v3}

\maketitle


\begin{figure}
\includegraphics[scale=0.8]{Fig1.eps}
\caption{\label{D0D}Time-dependent diffusion data for water-saturated samples of Indiana limestone (squares) and unconsolidated 15~$\mu$m dia. polystyrene beads (circles).
	The lines show least-squares fits to Eq. \ref{Dfrac2}.
	For the dashed line $D_s$ was fixed at 2.
	For the solid lines $D_s$ was allowed to vary, yielding $D_s =  2.58 \pm 0.14$ for Indiana limestone and $D_s = 2.2 \pm 0.4$ for polystyrene beads.
	To accurately determine these power laws it was necessary to separately measure $D_0$ on a bulk water sample in the same apparatus, at the same temperature.
}
\end{figure}

	In a recent Letter \cite{stallmach02}, Stallmach et al. reported pulsed
field gradient (PFG) NMR measurements of the time-dependent diffusion constant $D(t)$ in packings of water saturated sands.
	According to theory \cite{mitra93}, $D(t)$ decreases from the bulk water value $D_0$ with increasing observation time $t$ due to restrictions imposed by the pore surface:
 	\begin{equation}\label{Dsmooth}
\frac{D(t)}{D_{0}}=1-\frac{4S}{9\sqrt{\pi }V}\sqrt{D_{0}t}+\text{higher
order terms.}\end{equation}
Here $S/V$ is the surface-to-volume ratio of the pore space and $r = \sqrt{D_0 t}$ is the diffusion length.
	Stallmach et al. studied samples with different grain diameters $d_{g}$ and found that $S/V\varpropto d_{g}^{-0.7}$, which they interpreted with a fractal picture.
	If $d_{g}$ is identified as the upper cut-off scale $L$ for a fractal surface of dimension $D_s$, one expects $S/V\varpropto d_{g}^{D_{S}-3}$.
	In this Comment, we argue that the analysis of Ref. \onlinecite{stallmach02} is flawed and we propose an alternative.

	The key point is that Eq. \ref{Dsmooth} was derived for a \emph{nonfractal} surface such that $S/V$ is constant as the length scale $r$ varies.
	In Ref. \onlinecite{stallmach02}, Eq. \ref{Dsmooth} is used for $2 \ \mu\text{m} < r < 10 \ \mu\text{m}$ while $d_g$ is in the range 100-1000 $\mu$m.
	For the analysis to be valid, the surface would have to be smooth below 10 $\mu$m and abruptly turn into a fractal above this scale.
	This is implausible as sands and rocks are known to have fractal surfaces below 1 $\mu$m \cite{wong88a}.
	For example, BET measurements on rocks (for which the measurement scale is $r \approx 0.4$ nm) yield $S/V$ values one to two orders of magnitude greater than PFG NMR results \cite{hurlimann94}.

	To show how Eq. \ref{Dsmooth} is modified for fractal surfaces, we note that the term $4Sr/9\sqrt{\pi}V$ arises because molecules within a layer of volume $V_B \approx Sr$ can on average reach the pore surface within time $t$.
	Following Ref. \onlinecite{wong88} and allowing for $d_{g}>L$, it is easy to see that $V_{B}\varpropto (d_{g}/L)^{2}(L/r)^{D_s}r^{3}/(3-D_s)$.
	Hence Eq. \ref{Dsmooth} becomes
\begin{equation}\label{Dfrac}
\frac{D(t)}{D_{0}}=1-\frac{A}{3-D_s} \left( \frac{L}{d_g}\right)^{D_s -2}
\left( \frac{d_g}{r}\right)^{D_s} \left( \frac{r}{d_g} \right)^3+\text{...}
\end{equation}
where $A$ is a constant.
	This expression makes it clear that the analysis of Ref. \onlinecite{stallmach02} requires $L \varpropto d_g$, but also $r$ to be independent of $d_{g}$. 
	For a given sample with fixed $d_{g}$ and $L$, the limiting form of $D(t)$ at short times is
\begin{equation}\label{Dfrac2}
1 - D(t)/D_0 \varpropto r^{3-D_s} 
\varpropto t^{(3-D_{s})/2}.
\end{equation}

	To illustrate this method, we show some preliminary PFG NMR data on Indiana limestone and a packing of plastic beads; experimental details are given elsewhere \cite{ding96}.
	Figure \ref{D0D} shows a log-log plot of $1-D(t)/D_0$ vs. $r$.
	For a fractal surface, the data should fall on a straight line with slope $3-D_s$.
	As seen in previous studies, the data for smooth plastic beads are consistent with $D_s=2$.
	However, the limestone data fall on a line with distinctly smaller slope, giving  $D_s=2.58 \pm 0.14$.
	Optical micrographs of this limestone reveal that the pore surface has a wide range of features on scales above 1 $\mu$m.
	Although these data span less than a decade of length scales, the difference in slope between limestone and plastic beads is unmistakable.
	If the grain surfaces studied in Ref. \onlinecite{stallmach02} were truly fractal, they would show $D(t)$ time dependence similar to the limestone data in Fig. \ref{D0D}.

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\bibitem[{\citenamefont{Stallmach et~al.}(2002)\citenamefont{Stallmach, Vogt,
  K{\"{a}}rger, Helbig, and Jacobs}}]{stallmach02}
\bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Stallmach}},
  \bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Vogt}},
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{K{\"{a}}rger}},
  \bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{Helbig}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Jacobs}},
  \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{88}},
  \bibinfo{pages}{105505} (\bibinfo{year}{2002}).

\bibitem[{\citenamefont{Mitra et~al.}(1992)\citenamefont{Mitra, Sen, and
  Schwartz}}]{mitra93}
\bibinfo{author}{\bibfnamefont{P.~P.} \bibnamefont{Mitra}},
  \bibinfo{author}{\bibfnamefont{P.~N.} \bibnamefont{Sen}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{L.~M.} \bibnamefont{Schwartz}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{47}},
  \bibinfo{pages}{8565} (\bibinfo{year}{1992}).

\bibitem[{\citenamefont{Wong}(1988)}]{wong88a}
\bibinfo{author}{\bibfnamefont{P.-z.} \bibnamefont{Wong}},
  \bibinfo{journal}{Physics Today} \textbf{\bibinfo{volume}{41}},
  \bibinfo{pages}{24} (\bibinfo{year}{1988}).

\bibitem[{\citenamefont{H{\"{u}}rlimann
  et~al.}(1994)\citenamefont{H{\"{u}}rlimann, Helmer, Latour, and
  Sotak}}]{hurlimann94}
\bibinfo{author}{\bibfnamefont{M.~D.} \bibnamefont{H{\"{u}}rlimann}},
  \bibinfo{author}{\bibfnamefont{K.~G.} \bibnamefont{Helmer}},
  \bibinfo{author}{\bibfnamefont{L.~L.} \bibnamefont{Latour}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{C.~H.} \bibnamefont{Sotak}},
  \bibinfo{journal}{J. Magn. Reson. A} \textbf{\bibinfo{volume}{111}},
  \bibinfo{pages}{169} (\bibinfo{year}{1994}).

\bibitem[{\citenamefont{Wong and Bray}(1988)}]{wong88}
\bibinfo{author}{\bibfnamefont{P.-z.} \bibnamefont{Wong}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{A.~J.} \bibnamefont{Bray}},
  \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{60}},
  \bibinfo{pages}{1344} (\bibinfo{year}{1988}).

\bibitem[{\citenamefont{Ding and Candela}(1996)}]{ding96}
\bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Ding}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Candela}},
  \bibinfo{journal}{Phys. Rev. E.} \textbf{\bibinfo{volume}{54}},
  \bibinfo{pages}{656} (\bibinfo{year}{1996}).

\end{thebibliography}
