Disordered state

Equation (A3.3.57) must be supplied with appropriate initial conditions describing the system prior to the onset of phase separation. The initial post-quench state is characterized by the order parameter fluctuations characteristic of the pre-quench initial temperature T.. The role of these fluctuations has been described in detail m [23]. Flowever, again using the renomialization group arguments, any initial short-range correlations should be irrelevant, and one can take the initial conditions to represent a completely disordered state at J = xj. For example, one can choose the white noise fomi (i /(,t,0)v (,t, 0)) = q8(.t -. ), where ( ) represents an... 

Element-selective phase identification and quandfica-tion, structural characterization of disordered states... 

In another study Milehev and Landau [27] investigated in detail the transition from a disordered state of a polydisperse polymer melt to an ordered (liquid erystalline) state, whieh oeeurs in systems of GM when the ehains are eonsidered as semiflexible. It turns out that in two dimensions this order-disorder transition is a eontinuous seeond-order transformation whereas in 3d the simulational results show a diseontinuous first-order transformation. Comprehensive finite-size analysis [27] has established... 

The above result show that the concentration dependaiice of the Intensity maps is purely a statistical mechanics effect. In order to illustrate this important conclusion, we calculate transition temperature, we Indeed observe the experimentally observed splitting of the diffuse intensity maxima, with a saddle point at (100). 

Figure 5 Free energy surface at l l(Fig. 5a) [22, 24, 28] and 1 3 (Fig. 5b) [23, 24, 33] stoichiometries in the vicinity of disordered state ( f=0.0) at T—. 7Q and 1.6, respectively. The solid line in left-hand (right-hand) figure indicates the kinetic path evolving towards the L q LI2 ordered phase when the system is quenched from T—2.5 (3.0) down to 1.70 (1.60), while the broken lines are devolving towards disordered phase. The open arrows on the contour surface designate the direction of the decrease of free energy, and the arrows on the kinetic path indicate the direction of time evolution or devolution.

That the first stage of ordering (resistivity decrease) is correlated with excess vacancies not being in thermal equilibrium can be seen from measurement during isochronally lowering the temperature from the disordered state (0), which shows that atomic mobility is frozen below 280°C. 

Figure 2 Relative change of resistivity during isochronal annealing (AT=10K, At=10min) of deformed samples Deformed in disordered state 40% (A) and 80% reduction ( ) deformed in ordered state 30% reduction ( )...

The samples deformed in the disordered state show a behaviour different for the two degrees of deformation The sample cold-rolled to 40% at 260°C starts to decrease continuously to the completely recrystallized value, whereas the more highly deformed sample (80%) increases slightly (18%) until 390°C where a drastic decrease in hardness starts. 

From a comparison of the evolution of hardness of all samples during isochronal armealing it can be concluded that for high deformation in the disordered state and deformation in the ordered state, recovery and recrystallization is prevented up to T, in the sample deformed to 40% reduction in the disordered state recovery and recrystallization processes seem to start as soon as atomic mobility is enabled (260°C). 

A typical evolution of a disordered state under a radius r = 3 parity-rule FA, showing a dissolution into, and subsequent soliton-like interaction between, several particles with different velocities, is shown in figure 3.37. 

Fig. 3.37 Typical soliton-like evolution of the range r = 3 parity-rule filter automata, starting from an initially disordered state.

A remarkable, but (at first sight, at least) naively unimpressive, feature of this rule is its class c4-like ability to give rise to complex ordered patterns out of an initially disordered state, or primordial soup. In figure 3.65, for example, which provides a few snapshot views of the evolution of four different random initial states taken during the first 50 iterations, we see evidence of the same typically class c4-like behavior that we have already seen so much of in one-dimensional systems. What distinguishes this system from all of the previous ones that we have studied, however, and makes this rule truly remarkable, is that Life has been proven to be capable of universal computation. 

Microbial polysaccharides in solution lose their ordered conformation on heating. The temperature at which the polymer melts to a disordered state is known as the melting temperature (Tm) and is determined by a variety of factors ... 

Microdomain stmcture is a consequence of microphase separation. It is associated with processability and performance of block copolymer as TPE, pressure sensitive adhesive, etc. The size of the domain decreases as temperature increases [184,185]. At processing temperature they are in a disordered state, melt viscosity becomes low with great advantage in processability. At service temperamre, they are in ordered state and the dispersed domain of plastic blocks acts as reinforcing filler for the matrix polymer [186]. This transition is a thermodynamic transition and is controlled by counterbalanced physical factors, e.g., energetics and entropy. 

Liquid crystalline elastomers (LCEs) are composite systems where side chains of a crystalline polymer are cross-linked. Their mesogenic domains can be ordered nematically and undergo a phase transition to a disordered state at a temperature well above the glass-transition temperamre (Tg) of the polymer. Although the phase transition is thermally driven, LCEs demonstrate electrical conductivity and thus can be electrically stimulated." Ratna" has reported contractions of nearly 30% due to the phase transition of acrylate-based LCEs. 

Self-organization seems to be counterintuitive, since the order that is generated challenges the paradigm of increasing disorder based on the second law of thermodynamics. In statistical thermodynamics, entropy is the number of possible microstates for a macroscopic state. Since, in an ordered state, the number of possible microstates is smaller than for a more disordered state, it follows that a self-organized system has a lower entropy. However, the two need not contradict each other it is possible to reduce the entropy in a part of a system while it increases in another. A few of the system s macroscopic degrees of freedom can become more ordered at the expense of microscopic disorder. This is valid even for isolated, closed systems. Eurthermore, in an open system, the entropy production can be transferred to the environment, so that here even the overall entropy in the entire system can be reduced. 

That the most likely coarse velocity is equal to the most likely terminal velocity can only be true in two circumstances either the system began in the steady state and the most likely instantaneous velocity was constant throughout the interval, or else the system was initially in a dynamically disordered state, and x was large enough that the initial inertial regime was relatively negligible. These equations are evidently untrue for x —> 0, since in this limit the most... 

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