Line, equilibrium spinodal

Figure 5.13 Predicted phase diagrams for physical gels made from low-molecular-weight molecules with junctions of unrestricted functionality 4> is the total volume fraction of polymer, and Tr is here the reduced distance from the theta temperature, Tr = — Q/T. The parameter Aq controls the equilibrium constant among aggregates of various sizes. The outer solid lines are binodals, the inner solid lines are spinodals, and the dashed lines are gelation transitions. CP is a critical solution point, CEP is a critical end point, and TCP is a tricriti-cal point. (Reprinted with permission from Tanaka and Stockmayer, Macromolecules 27 3943. Copyright 1994 American Chemical Society.)...

The adsorption and desorption isotherms have been calculated for the Nj sorption at 77K in cylindrical pores of MCM-41 materials in the range 1-12 nm. The points of spinodal and equilibrium transitions are plotted in Fig. 2. There are several features worth noticing. As the pore size increases, the line of spinodal desorption saturates at the value corresponding to the spinodal decomposition of the bulk liquid. The line of equilibrium capillary condensation asymptotically approaches the Kelvin equation for the spherical meniscus and the line of spontaneous capillary condensation asymptotically approaches the Kelvin equation for the cylindrical meniscus. This asymptotic behavior is in agreement with the classical scenario of capillary hysteresis [12] capillary condensation occurs spontaneously after the formation of the cylindrical adsorption film on the pore walls while evaporation occurs after the formation of the equilibrium meniscus at the pore end. Most interestingly, the NLDFT predictions of equilibrium and spontaneous capillary condensation transitions for pores wider than 6 nm are approximated by the semi-empirical equations of the Deijaguin-Broekhoff-de Boer theory [13]. 

Fig. 9 Phase diagram (demixing into two liquid phases) of the system CH/PS for the indicated molar masses of the polymer (kg/ mol). Cloud points (open symbols) and critical points (stars) are taken from the literature [45], The data for the high temperatures refer to the equilibrium vapor pressure of the solvent. Binodals (solid lines) and spinodals (dotted lines) were calculated as described in the text by means of the temperature dependent parameters [46] C and v...

Figure A3.3.5 Tliemiodynamic force as a fiuictioii of the order parameter. Three equilibrium isodiemis (fiill curves) are shown according to a mean field description. For T < J., the isothemi has a van der Waals loop, from which the use of the Maxwell equal area constmction leads to the horizontal dashed line for the equilibrium isothemi. Associated coexistence curve (dotted curve) and spinodal curve (dashed line) are also shown. The spinodal curve is the locus of extrema of the various van der Waals loops for T < T. The states within the spinodal curve are themiodynaniically unstable, and those between the spinodal and coexistence...

Fig. 1. Equilibrium phase diagram T, c)=iT/Tc,c) for the alloy model used in Ref.. Solid lines boundaries of the disordered (a) and homogeneously ordered (6) fields areas c, d and e corre.spond to the two-phase region. Dashed line i.s the ordering spinodal separating the metastable disordered area c from the. spinodal decompo.sition area d. Dot-dashed line is the conditional spinodal that separate.s the area d from the ordered metastable area e.

If we imagine a line drawn on the primitive surface dividing all parts of the surface which are convex downwards in all directions from those which are concave downwards in one or both directions of principal curvature, this curve will have the equation (26), and is known as the spinodal carve. It divides the surface into two parts, which represent respectively states of stable and unstable equilibrium. For on one side A is positive, and on the other it is negative. If we assume that the tie-line of corresponding points on the connodal curve is ultimately tangent to that the direction of equations ... 

Fig. 8 Phase diagram for PI-fc-PEO system. Only equilibrium phases are shown, which are obtained on cooling from high temperatures. ODT and OOT temperatures were identified by SAXS and rheology. Values of /AT were obtained using /AT = 65/T + 0.125. Dashed line spinodal line in mean-field prediction. Note the pronounced asymmetry of phase diagram with ordered phases shifted parallel to composition axis. Asymmetric appearance can be accounted for by conformational asymmetry of segments. Adopted from [53]...

In Fig. 1.14, the dotted lines for each curve show the activity of the coexisting phases at chemical equilibrium. Similarly in Fig. 1.16 the dotted line BDF shows the activity of the coexisting phases (5 = 0.185 and 0.815). The coexisting phases, which have the same structure, differ in the concentration of vacancies. This phenomenon is generally called phase separation or spinodal decomposition (it is observed not only in the solid phases but also in the liquid phases), and originates from the sign of the interaction energy... 

Figure 2. Capillary hysteresis of nitrogen in cylindrical pores at 77 K. Equilibrium desorption (black squares) and spinodal condensation (open squares) pressures predicted by the NLDFT in comparison with the results of Cohan s equation (the BJH method) for spherical (crosses and line) and cylindrical (line) meniscus.

Figure 3.4a shows a normal crystallization curve with the spinodal (supersaturation limit) curve (CD) and equilibrium curve (AB). At point P neither nuclei nor crystal growth will occur since the solution is superheated by the amount RP. Once the saturation line (AB) is crossed, either through cooling or increase in concentration, nuclei and crystals may or may not form in the metastable region. Metastable point Q is shown between point R and the crosshatched line CD. 

Figure 1. Nitrogen in a 10 (Di teniai=3.3 nm) cylindrical pore of MCM-41 at 77.4 K. The chemical potential of equilibrium transition BF, j,e- io=-1.42 kT, is obtained from the Maxwell s rule and also corresponds to the intersection point of the Grand Potential (solid line). Lines CG and EA, which bound the hysteresis loop, correspond to the spinodal condensation and desorption, respectively.

Figure 2. The pore size dependence of the relative pressure of equilibrium condensation-evaporation (black squares), spinodal condensation (open squares), and spinodal desorption (open triangles) of nitrogen at 77K in cylindrical pores of MCM-41 materials. The Broekhoff-de Boer approximation [13] for condensation (solid line) and desorption (crosses) in cylindrical pores is plotted for comparison.

Herein, we expand on the discussion of our recently observed isothermal amorphous-amorphous-amorphous transition sequence. We achieved to compress LDA in an isothermal, dilatometric experiment at 125 K in a stepwise fashion via HDA to VHDA. However, we can not distinguish if this stepwise process is a kinetically controlled continuous process or if both steps are true phase transitions (of first or higher order). We want to emphasize that the main focus here is to investigate transitions between different amorphous states at elevated pressures rather than the annealing effects observed at 1 bar. The vast majority of computational studies shows qualitatively similar features in the metastable phase diagram of amorphous water (cf. e.g. Fig.l in ref. 39) at elevated pressures the thermodynamic equilibrium line between HDA and LDA can be reversibly crossed, whereas by heating at 1 bar the spinodal is irreversibly crossed. These two fundamentally different mechanisms need to be scrutinized separately. 

For binary mixtures, the binodal line is also the coexistence curve, defined by the common tangent line to the composition dependence of the free energy of mixing curve, and gives the equilibrium compositions of the two phases obtained when the overall composition is inside the miscibility gap. The spinodal curve, determined by the inflection points of the composition dependence of the free energy of mixing curve, separates unstable and metastable regions within the miscibility gap. 

On the basis of standard criteria for equilibrium, stability limits, and criticality yielding coexistence curve (binodal), spinodal line, and critical point, the phase behavior maybe predicted using Eq. (1) ... 

At the critical point, (0c, Tc), the coexistence curve (or binodal) and spinodal curve meet. It is important to note that the spinodal is never physically reached in thermal equilibrium (except at the critical point itself) since it is precluded by the first-order phase separation described in the (0, T) plane by the coexistence curve. While the coexistence curve separates the single phase and two-phase regions in thermal equilibrium, kinetic effects can allow the existence of metastable states — Le., one can have the system below the binodal curve and still observe only a single phase for a rather long time. In contrast, the region below the spinodal curve is unstable and even small, thermal fluctuations will drive the system toward equilibrium. The spinodal line represents a line of large fluctuations in the concentration 0, since the free energy cost for fluctuations of 0 away from its minimal values, as determined from Eq. (1.72), is proportional to 9 //90 when this quantity is small, the... 

When the thickness of a liquid film on a non-wettable substrate is thinner than that in equilibrium (usually nun), the film ruptures with cylindrical holes and shrinks from the con line, on which friree pha (liquid, substrate and air) meet. For relatively thicker films, initial holes are nucleated by dust particle and defects on die substrates. In the case of sub-micron thick films, another origin of spinodal decomposition dominates for the formation of initial holes, and hence the density distribution of the holes depends on the initial film thickness (/). 

For the calculation of the stability of a multicomponent polymer solution (spinodal line and critical solution points), the stability theory can be applied [42]. One possible consequence of the poiydispersity, especially if the distribution function is bimodal, is the appearance of tri-critical solution points [2]. Suggestions for the phase equilibrium calculations of such systems can be found in the literature [2, 39, 43). 

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