Yukawa particles phase diagram

Charged particles in polar solvents have soft-repulsive interactions (see section C2.6.4). Just as hard spheres, such particles also undergo an ordering transition. Important differences, however, are that tire transition takes place at (much) lower particle volume fractions, and at low ionic strengtli (low k) tire solid phase may be body centred cubic (bee), ratlier tlian tire more compact fee stmcture (see [69, 73, 84]). For tire interactions, a Yukawa potential (equation (C2.6.11)1 is often used. The phase diagram for the Yukawa potential was calculated using computer simulations by Robbins et al [851. 

We will focus on one experimental study here. Monovoukas and Cast studied polystyrene particles witli a = 61 nm in potassium chloride solutions [86]. They obtained a very good agreement between tlieir observations and tire predicted Yukawa phase diagram (see figure C2.6.9). In order to make tire comparison tliey rescaled the particle charges according to Alexander et al [43] (see also [82]). At high electrolyte concentrations, tire particle interactions tend to hard-sphere behaviour (see section C2.6.4) and tire phase transition shifts to volume fractions around 0.5 [88]. 

The phase behavior of the hard-core Yukawa potential has been studied experimentally and by numerical simulation, see e.g. Ref. [65, 66, 67]. The computed phase diagram of Ref. [67] shows a fluid-solid (bcc/fcc) and a solid-solid (bcc-fcc) coexistence line and it exhibits two fluid-bcc-fcc triple points, (see Fig. 19). The main difference between the phase diagram of the hard-core Yukawa model and that of the pure (i.e. point-particle) Yukawa potential [68] is the presence of the second triple point. This triple point sets a lower limit for the strength of the Yukawa interaction for which a bcc phase exists. 

Subsequently, we employ the Gibbs-Duhem method as first proposed by Kofke [31,32] to determine the phase coexistence lines in the (rj, kct) plane for a fixed Pe. Using these methods, we study the phase behavior of hard-core Yukawa particles, whose interactions are described by the pair potential given by Equation 8.8. The phase diagrams are calculated for fixed contact values Pe and they are given in the (ri, l/xa) representation. We calculate the phase diagram for four contact... 

FIGURE 8.3 Phase diagram of hard-core Yukawa particles with pe = 20 presented in the (q, 1/ko) plane. The symbols and lines are the same as in Fignre 8.2. Note the difference in the q scale compared to Fignres 8.2 throngh 8.5. (From Hynninen AP and Dijkstra M. 2003. Physical Review E 68 021407. With permission.)... 

In conclusion, we have shown that the phase diagram of charged colloids, where the interactions are given within the DLVO theory by hard-core repnlsive Yukawa pair potential, can be obtained for any snfficiently high contact valne Pe by mapping the well-known phase boundaries of the point Yukawa particles onto those of the hard-core repulsive Yukawa system and bearing in mind that the stable bcc region is bounded by a bcc-fcc coexistence at ti 0.5. 

FIGU RE 8.19 Phase diagram of dipolar soft-sphere particles with Yukawa parameters kdipole moment strength y, packing fraction T]) representation. The circles denote points where the phase boundary was determined and the gray area denotes the coexistence region (where tie lines are vertical). The upper horizontal axis gives an estimate of the root-mean-square external electric Field (see text for details). (From Hynninen AP and Dijkstra M. 2005. Physical Review Letters 94 138303. With permission.)... 

Hynninen AP and Dijkstra M. 2003. Phase diagrams of hard-core repulsive Yukawa particles. Physical Review E 68 021407. 

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