## 6 March 2013

### de Gennes narrowing

In colloidal solutions, a widely-used relation connects the scale-dependent collective diffusion constant and the structure factor:

D_c(q) =\frac{D_0}{S(q)}
\label{eq:dGn}
and is generally known as de Gennes narrowing since its use by de Gennes in the context of quasi-elastic neutron scattering from liquids [1].

Intuitively, equation \eqref{eq:dGn} makes sense: if $$S(q)$$ is high for a certain value of $$q$$ then fluctuations with that particular $$q$$ are frequent, meaning that their energetic cost is low and that they will decay slowly. However, I have not yet found in the literature a simple yet rigorous derivation. This is what I will attempt below.

Consider a system characterized by a conserved scalar parameter $$\phi(\mathbf{r})$$ (for instance, the local particle concentration in a suspension). The free energy only depends on this field: $$\mathcal{F}[\phi(\mathbf{r})]$$ or, equvalently, on its Fourier components: $$\mathcal{F}[\phi(\mathbf{q})]$$.

For an isotropic system in the absence of applied fields, all fluctuations $$\phi(\mathbf{q})$$ with $$\left |\mathbf{q} \right | > 0$$ will eventually decay to zero. To fix the ideas, we will consider an overdamped relaxation (model B in the Hohenberg-Halperin classification [2], but of course far from criticality).

Let us write Fick's laws, with $$\mathbf{j}$$ and $$\mu$$ the current and chemical potential associated to $$\phi$$ (this is similar to the presentation in [2], Eqs. (2.2)-(2.9)):
$\frac{\partial \phi(\mathbf{r},t)}{\partial t} = - \nabla \mathbf{j} \, ; \quad \mathbf{j} = - \lambda \nabla \mu \, ; \quad \mu = \frac{\delta \mathcal{F}}{\delta \phi(\mathbf{r},t)} \, \Rightarrow \, \frac{\partial \phi(\mathbf{r},t)}{\partial t} = \lambda \nabla ^2 \frac{\delta \mathcal{F}}{\delta \phi(\mathbf{r},t)}$ where $$\delta$$ denotes the functional derivative. Introducing the Fourier components yields:
\label{eq:relax}
\frac{\partial \phi(\mathbf{r},t)}{\partial t}= \lambda \nabla ^2 \frac{\delta \mathcal{F}}{\delta \phi(\mathbf{r},t)}  \Rightarrow \frac{\partial \phi(\mathbf{q},t)}{\partial t} = - \lambda q^2 \frac{\text{d} \mathcal{F}}{\text{d}\phi(\mathbf{q},t)}
If the different Fourier modes are uncoupled, we can write the equipartition relation:

\label{eq:equi}
\mathcal{F} = \sum_{\mathbf{q}}  \frac{A(\mathbf{q})}{2} \phi(\mathbf{q}) ^2 \Rightarrow \frac{\text{d} \mathcal{F}}{\text{d}\phi(\mathbf{q},t) } = A(\mathbf{q}) \phi(\mathbf{q})
From \eqref{eq:relax} and \eqref{eq:equi}, the evolution of mode $$\mathbf{q}$$ is given by:
\label{eq:evol}
\frac{\partial \phi(\mathbf{q},t)}{\partial t}=- \lambda q^2 A(\mathbf{q}) \phi(\mathbf{q,t}) \Rightarrow \frac{\phi(q,t)}{\phi(\mathbf{q},0)} = \exp \left \lbrace - D_c(q) q^2 t \right \rbrace
where we invoked the isotropy of the system. The collective diffusion coefficient is given by:
\label{eq:Dc}
D_c(q) = \lambda A(q)
On the other hand,equipartition also implies:
S(\mathbf{q}) = \left \langle \phi(\mathbf{q}) \phi(\mathbf{-q}) \right \rangle = \frac{k_B T}{A(\mathbf{q})}
\label{eq:Sq}
with $$\left \langle \cdot \right \rangle$$ the ensemble average. From \eqref{eq:Dc} and \eqref{eq:Sq} we finally obtain:

\label{eq:final}
D_c(q) = \frac{\lambda k_B T}{S(q)}

For simplicity's sake I glossed over some technical details, such as the proper use of $$\phi(\mathbf{q})$$ and $$\phi(\mathbf{-q})$$ and the rigorous form of the time evolution in Eq. \eqref{eq:evol}.

[1] P. G. de Gennes, Liquid dynamics and inelastic scattering of neutrons, Physica A 25, 825-839 (1959).
[2] P. C. Hohenberg and B. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys. 49, 435-479 (1977).