Micelles simulated

As has probably become obvious already, the study of micelles has been one of the big topics in simulations of systems with surfactants. We have cited many of the related publications in the previous sections. Here, we shall discuss some special aspects of micelle simulations in order to illustrate the use of idealized chain models for this type of problem. 

Table 1 Final micelle simulation parameters. In all these simulations /J-i = 1.18, v = -2.0and8=1.0...

Key words Association - kinetics -Monte Carlo - micellization -simulation - surfactant... 

Figure C2.3.6. Illustration of micelle stmcture obtained by Monte Carlo simulations of model octanoate amphiphiles. There are 30 molecules simulated in this cluster. The shaded spheres represent headgroups. Reproduced by pennission from figure 2 of [35].

Figure C2.3.7. Snapshot of micelle of sodium octanoate obtained during molecular dynamics simulation. The darkest shading is for sodium counter-ions, the lightest shading is for oxygens and the medium shading is for carbon atoms. Reproduced by pennission from figure 2 of [36].

Mesoscale simulations model a material as a collection of units, called beads. Each bead might represent a substructure, molecule, monomer, micelle, micro-crystalline domain, solid particle, or an arbitrary region of a fluid. Multiple beads might be connected, typically by a harmonic potential, in order to model a polymer. A simulation is then conducted in which there is an interaction potential between beads and sometimes dynamical equations of motion. This is very hard to do with extremely large molecular dynamics calculations because they would have to be very accurate to correctly reflect the small free energy differences between microstates. There are algorithms for determining an appropriate bead size from molecular dynamics and Monte Carlo simulations. 

Computer Simulations of Living Polymers and Giant Micelles... 

In contrast to statics, the relaxational kinetics of living polymers and of giant wormlike micelles is unique (and different in both cases). It is entirely determined by the processes of scission/recombination and results in a nonlinear approach to equilibrium. A comparison of simulational results and laboratory observations in this respect is still missing and would be highly desirable. 

M. Kroger, R. Makhloufi. Wormlike micelles under shear flow A microscopic model studied by nonequihbrium molecular dynamics computer simulations. Phys Rev E 55 2531-2536, 1996. 

Lattice models have the advantage that a number of very clever Monte Carlo moves have been developed for lattice polymers, which do not always carry over to continuum models very easily. For example, Nelson et al. use an algorithm which attempts to move vacancies rather than monomers [120], and thus allows one to simulate the dense cores of micelles very efficiently. This concept cannot be applied to off-lattice models in a straightforward way. On the other hand, a number of problems cannot be treated adequately on a lattice, especially those related to molecular orientations and nematic order. For this reason, chain models in continuous space are attracting growing interest. 

Beyond the CMC, surfactants which are added to the solution thus form micelles which are in equilibrium with the free surfactants. This explains why Xi and level off at that concentration. Note that even though it is called critical, the CMC is not related to a phase transition. Therefore, it is not defined unambiguously. In the simulations, some authors identify it with the concentration where more than half of the surfactants are assembled into aggregates [114] others determine the intersection point of linear fits to the low concentration and the high concentration regime, either plotting the free surfactant concentration vs the total surfactant concentration [115], or plotting the surfactant chemical potential vs ln( ) [119]. 

Hence the sizes of spherical micelles are distributed around a most probable aggregation number M, which depends only on molecular details of the surfactants in this simplest approximation. Indeed, micelle size distributions at concentrations beyond the CMC have shown a marked peak at a given aggregation number in many simulations [37,111,112,117,119,138,144,154,157]. 

Systematic variation in the water temperature, (WW), will produce a profile reflecting this influence. Vary the / b(WW) and J(WW) values in Example 5.3 to simulate different water temperatures. Run the dynamics for these different water temperatures to observe its influence. Note whether this is a linear or nonlinear effect on the cluster size. The structures formed may be quantified by recording the average micelle cluster size. The typical pattern looks like the examples in Figure 5.5. 

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