Chaotic mixtures

As far as chemistry and life sciences are concerned, there are for me and many others two main reasons for this fascination, summarized in Figure 9.4 firstly, above a certain critical concentration, structural order is achieved starting from the chaotic mixture of disordered surfactant molecules. As discussed earlier, this increase of order is attended by an increase of entropy and a decrease of free energy. 

Figure 9.4 Two good reasons for fascination in the field of surfactants (a) spontaneous order out of a chaotic mixture of monomers and (b) the emergence of compartments (microheterogeneous reactions...). (Vesicle and micelle are not to scale.)...

I (and most scientists) would answer, By accident. But what an absolutely unlikely accident it must have been The earth on which life first appeared - prebiotic earth - was most inhospitable a violent place, wracked by storms and volcanoes, wrenched by the pull of a moon that was much closer than the one we know now, still battered by cosmic impacts. On its surface and in its oceans were myriads of organic compounds, some formed in processes occurring on earth, some imported by infalls from space. Out of this universe of tumult and molecules, somehow a small subset of chemical processes emerged and accidentally replicated, thus stumbling toward what became the first cells. How could such a chaotic mixture of molecules have generated cells Order usually decays toward disorder Why do the tracks that led to life point in the opposite direction ... 

The bottleneck in the origin of life is the formation of the functional biopolymers— enzymes and nucleic acids. The answer cannot be the random polycondensation from a chaotic mixture of the monomers, as this process would afford an astronomic number of different chains—ca. 10 for chains with a polymerization degree of 60. Given that, the probability that the same chain is produced more than once by a random polymerization process is in first approximation equal to zero the single active individual macromolecule, even if formed, would decompose before it could be made again by another chance event. How then can active macromolecules be formed ... 

During the MD simulations of chaotic mixtures of water, CO2, TBP, U02(N03)2 and 30 TBP (D) or 60 TBP (E), demixing occurred, leading to separated aqueous and CO2 phases and (in relation to the imposed 3D periodicity) to two interfaces (Figure 4). In both systems, the bulk aqueous phase finally contains some CO2 molecules, but neither TBP nor uranyl species. 

Figure 8.16 Periodic-chaotic sequence in the BZ reaction, (a) Bifurcation diagram as a function of flow rate, (b) simple periodic state L, (c) mixed-mode periodic state l, (d) mixed-mode periodic state 1, (e) chaotic mixture of 1 and 1 patterns. (Adapted from Swinney, 1983.)...

The amount of energy made available in tliis way can indeed be very large, more than enough to break any bond. One might therefore expect a chaotic mixture of products to be formed by random bond cleavage. If this... 

Figure 5.1.7a shows a side view of a lean propane flame, 10 cm in diameter, propagating downward in a top-hat flow. The flame speed is 9cm/s, below the stability threshold, and the flame is stable at all wavelengths. Figure 5.1.7b shows a near stoichiometric flame in the same burner. The flame is seen at an angle from underneath. The mixture is diluted with nitrogen gas to reduce to flame speed to the instability threshold (10.1 cm/s), so that the cells are linear in nature. The cell size here is 1.9 cm. Figure 5.1.7c shows a flame far above the instability threshold, the cell shape becomes cusped, and the cells move chaotically. 

In equilibrium, impurities or vacancies wiU be distributed uniformly. Similarly, in the case of two gases, as above, once a thorough mixture has been formed on both sides of the partition, the diffusion process is complete. Also at that stage, the entropy of the system has reached its maximum value because the information regarding the whereabouts of the two gases has been minimized. In general, it should be remembered that entropy of a system is a measure of the information available about that system. Thus, the constant increase of entropy in the universe, it is argued, should lead eventually to an absolutely chaotic state in which absolutely no information is available. 

Ya.B. s more recent papers have been devoted to the study of nonlinear problems. In 1966 Ya.B. turned his attention to the stabilizing effect of accelerated motion through a hot mixture of a boundary of intersection of two flame fronts, convex in the direction of propagation, and proposed an approximate model of a steady cellular flame. G. I. Sivashinsky, on the basis of this work, proposed a nonlinear model equation of thermodiffusional instability which describes the development of perturbations of a bent flame in time and, together with J. M. Michelson, studied its solution near the stability boundary Le = Lecrit. It was shown numerically that the flat flame is transformed into a three-dimensional cellular one with a non-steady, chaotically pulsating structure. The formation of a two-dimensional cellular structure was also the subject of a numerical investigation by A. P. Aldushin, S. G. Kasparyan and K. G. Shkadinskii, who obtained steady flames in a wider parameter interval. 

A microliter-level chaotic mixer - used to combine reagent solutions from the rotary mixer with microlitre quantities of enzyme solution in buffer to give a homogeneous reaction mixture (CA in PBS, pH 7.4) ... 

The Second Law is sometimes stated as the Entropy Law. Entropy is a measure of randomness or disorder in a system. Systems that are more randomized, chaotic, or evenly mixed have more entropy. The Second Law states the entropy of the universe is constantly increasing. One clear implication of the Second Law is that the universe never, and a system almost never, spontaneously becomes more organized. So, hot molecules will not spontaneously separate themselves from cold molecules. Mixtures of oxygen and nitrous oxide will not spontaneously separate and send the oxygen to the patient separately from the nitrous oxide. IV fluids will mix evenly throughout the circulatory system, and not congregate in just the left arm. 

Texture is important in polymer processing because (a) laminar and even chaotic distributive mixing inevitably lead to it, (b) many products are visually examined for lack of texture or for a certain desired texture, and (c) mechanical properties of blends depend on the texture of the mixture. 

Well-defined products from the chaotic turmoil, which is a chemical reaction, result from a balance between external thermodynamic factors and the internal molecular parameters of chemical potential, electron density and angular momentum. Each of the molecular products, finally separated from the reaction mixture, is a new equilibrium system that balances these internal factors. The composition depends on the chemical potential, the connectivity is determined by electron-density distribution and the shape depends on the alignment of vectors that quenches the orbital angular momentum. The chemical, or quantum, potential at an equilibrium level over the entire molecule, is a measure of the electronegativity of the molecule. This is the parameter that contributes to the activation barrier, should this molecule engage in further chemical activity. Molecular cohesion is a holistic function of the molecular quantum potential that involves all sub-molecular constituents on an equal basis. The practically useful concept of a chemical bond is undefined in such a holistic molecule. 

We have investigated the transitions among the types of oscillations which occur with the Belousov-Zhabotinskii reaction in a CSTR. There is a sequence of well-defined, reproducible oscillatory states with variations of the residence time [5]. Similar transitions can also occur with variation of some other parameter such as temperature or feed concentration. Most of the oscillations are periodic but chaotic behavior has been observed in three reproducible bands. The chaos is an irregular mixture of the periodic oscillations which bound it e.g., between periodic two peak oscillations and periodic three peak oscillations, chaotic behavior can occur which is an irregular mixture of two and three peaks. More recently Roux, Turner et. al. 

Schulman and Hughes1 find that, in a mixture of 3 molecules of oleic acid to 1 of hexadecyl alcohol (a substance giving a condensed film ordinarily), the area is decidedly larger than the sum of the areas of the constituents measured separately, as if the chaotic oscillations of the oleic acid chains were communicated to the hexadecyl alcohol chains. 

When flow rates are high, a litjuid will not flow in a laminar fashion, but will become turbulent. The litjuid will start to flow in a chaotic way, forming large and small swirls and eddies. The flow rate at which this happens depends on the geometry of the machine, on the flow rate applied, and on the overall viscosity of the mixture. The transition from laminar to turbulent flow is characterized by the Reynolds number the critical Reynolds number depends on the geometry and the product properties (see Equations (15.9)-(15.10)). 

Besides the one little regular island at 0 and — 2 there are undoubtedly more regular islands in the phase space of the double pendulum at E = 2. We missed them by our rather coarse choice of initial conditions. As indicated in Fig. 3.3(b), their total area in phase space is probably very small. Nevertheless, Fig. 3.3(b) illustrates an important feature of the phase space of most physical systems the phase space contains an intricate mixture of regular and chaotic regions. The system is said to exhibit a mixed phase space. 

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