Signature of a critical point

As the potential between inverse droplets becomes strongly attractive, a critical behavior is observed. The curves of reduced compressibility versus the micellar volume fraction for three microemulsions formed with water, dodecane, and SDS (sodium dodecylsulfate) and heptanol or hexanol or pentanol, illustrate this effect [22,23]. The heptanol microemulsion is close to a hard-sphere system. In the case of pentanol, both dnld(f) and nfdcfr are close to zero, which is the signature of a critical point. Thus for this system it appears that attractions between droplets are strong enough to induce a phase separation into a micelle-rich phase and a micelle-poor phase. 

The nature of a critical point is denoted as (rank, signature). For example, a minimum in p is designated by (3, + 3) and a maximum by (3,-3). The two remaining types of critical points, (3,-1) and (3, +1) are saddle points, called bond critical point and ring critical point, respectively. 

Each maximum, minimum or saddle point occurs at a so-called critical point Tc, where the gradient vanishes. The nature of the critical point is determined by the eigenvalues of the Hessian. All the eigenvalues are real at the critical point, but some of them may be zero. The rank co of the critical point is defined to be the number of non-zero eigenvalues. The signature o is the sum of the signs of the eigenvalues, and critical points are discussed in terms of the pair of numbers (w, o). 

The Hessian matrix H(r) is defined as the symmetric matrix of the nine second derivatives 82p/8xt dxj. The eigenvectors of H(r), obtained by diagonalization of the matrix, are the principal axes of the curvature at r. The rank w of the curvature at a critical point is equal to the number of nonzero eigenvalues the signature o is the algebraic sum of the signs of the eigenvalues. The critical point is classified as (w, cr). There are four possible types of critical points in a three-dimensional scalar distribution ... 

Then, the critical points are characterized by two numbers, co and a, where to is the number of nonzero eigenvalues of H at the critical point (rank of the critical point) and a (signature) is the algebraic sum of the signs of the eigenvalues. Generally for molecules, the critical points are all of rank 3 then, four possible critical points may exist ... 

This discussion has shown that the principal topological features of a charge distribution can be summarized using the rank and signature classification scheme of its critical points. It has further demonstrated the existence... 

An eigenvalue and its associated eigenvector of the Hessian of p (a principal curvature and its associated axis) at a critical point define a onedimensional system. If the eigenvalue or curvature is negative, then p is a maximum at the critical point on this axis and a gradient vector will approach and terminate at this point from both its left- and right-hand side as illustrated in Fig. 2.6 for the case (1, — 1), a system of rank 1 and signature... 

The high sensitivity of the topography of II(p) to computational details has been examined. A critical point is characterized by the rank r and signature s of the Hessian... 

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