content_image/351227.jpg content_image/351225.jpg content_image/351226.jpg content_image/351224.jpg content_image/351223.jpg
where ˜ v $^{∗}$( k ) = ˜ v $^{T}$( − k ), and 

<img src='content_image/351227.jpg'>

The correlation functions for the fluctuations of the solvent composition ϑ and the density of ions η in the Fourier representation are given by the matrix ˜ G = ˜ C − $^{1}$. Each component of this matrix is inversely proportional to det( J − $^{1}$C0 + k $^{2}$I ) = ( λ 2 1 + k $^{2}$)( λ 2 + k $^{2}$), where λ 2 1 and λ 2 are the two eigenvalues of the matrix 

and are given by 

<img src='content_image/351225.jpg'>

<img src='content_image/351226.jpg'>

The correlations are thus given by a linear combination of the two terms, ( λ 2 1 + k $^{2}$) − 1 and ( λ 2 + k $^{2}$) − $^{1}$. The asymptotic decay of correlations in real space is described by the inverse correlation length ξ − 1 b = min( λ$_{1}$, λ$_{2}$ ). 

The uniform phase is stable in the range of parameters T, ¯ s and ¯ ρ$_{c}$ such that det ˜ C ( k ) > 0 for all k < k$_{max}$ , where k$_{max}$ is an upper cutoff on wave numbers | k | ≤ k$_{max}$ = π/a . a is usually identified with some appropriate microscopic length, e.g., the lattice spacing or the molecular diameter. Because det ˜ C ( k ) = J det ( M + k $^{2}$I ) = J ( λ 2 1 + k $^{2}$)( λ 2 + k $^{2}$) where 

<img src='content_image/351224.jpg'>

for the instability analysis we have to distinguish cases of positive and negative J . 

If J > 0, from the condition det ˜ C (0) ≥ 0 (one-phase stability) we obtain after some algebra that T r M ≥ 0 and λ 2 $_{1}$, λ 2 > 0. In that case the instability of the uniform phase occurs at k = 0 when 

<img src='content_image/351223.jpg'>

The condition ˜ C$_{ss}$ = 0 determines the instability for the case of the vanishing field η (the same concentration of ions in the phase-separated solvent). From (34) it follows that λ 2 1 → 0 + so that the asymptotic inverse decay length is 1 /ξ$_{b}$ = λ$_{1}$ . 

For J < 0, on the other hand, the stability condition of the uniform phase, det ˜ C ( k ) > 0, requires det ( M + k $^{2}$I ) = ( λ 2 1 + k $^{2}$)( λ 2 + k $^{2}$) < 0 for all k < k$_{max}$ . In particular, the 

