26 June 2015

The structure factor of a liquid - part V

Finite-size effects

In previous posts, we have always considered that the liquid system was infinite and homogeneous. This may no longer be the case if:
  • the particles are confined in relatively small spaces or
  • their attractive interaction leads to the formation of dense aggregates, separated by more dilute regions.
Although physically very different, those two situations have similar effects on the structure and we will treat them together.

 This finite-size effect (as we will call it) is illustrated by the Figure below:


As we recall from Eq. 2 in post II, the structure factor \(S(q)\) is 1 plus the Fourier transform of the radial distribution function \(g(r)\) and  has two components: a "well-behaved" one, \(S'(q)\), which is the Fourier transform of \(g(r)-1\), and a Dirac delta peak at the origin, which is simply the Fourier transform of \(1\). This result holds in an infinite system (top row), where the singular contribution can be safely ignored, as it is hidden in the experimentally inaccessible area "behind the beamstop" (grayed out in the Figure).

 What happens if the particles are confined within spherical regions (middle row), far enough from each other so we can neglect the interaction between regions? For a reference particle at the center of the sphere, the radial distribution function is the same as for the bulk liquid until the edge of the sphere, where it falls abruptly to zero. Mathematically, the finite-size radial distribution function \(g(r)_{\text{finite}}\) is related to the bulk one by multiplication with a "top-hat function" \(\Pi(r)\) (shown in the Figure underneath the system), which is constant for \(0 \leq r < R\) (radius of the confining sphere) and zero beyond this value:
\(g(r)_{\text{finite}} = g(r) \Pi(r)\).

The Fourier transform of a product being the convolution of the Fourier transforms of the two factors, we have:
\begin{equation}
\label{eq:Sfinite}
S(q)_{\text{finite}} = S(q) * \mathcal{F}\left [ \Pi(r)\right ](q)
\end{equation}
The Fourier transform of \(\Pi(r)\) is a narrow peak (width of the order \(1/R\)) centered at the origin1. Convolution with this function does not change by much the regular part of the bulk structure factor, but it does enlarge the singular contribution at the origin, which now has the same width, \(\sim 1/R\) and becomes detectable around the beamstop. At the origin, the peak height \(S(0)_{\text{finite}}\) is proportional to the number of particles present in the region of confinement.

Finally, the scattering signal at low \(q\) values (on length scales larger than the size of the individual particles) is the same as for an object of the size of the region of confinement and with constant density (bottom row in the Figure).

1. In fact, we must convolve \(S(q)\) with the squared absolute value of the Fourier transform of \(\Pi(r)\). This is because the correct —but more complicated— mathematical procedure is to truncate the density profile \(\rho(r)\), and \(g(r)\) is the density-density correlation function.






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