Isothermal compressibility divergence

The asymptotic relations (Equations 7.9 and 7.10) are exact for any one-component liquid with spherical interactions, except when the Taylor expansion (Equation 7.7) fails to hold, which is near a critical point (Fisher 1964). At the critical point, the DCF becomes a nonanalytic function, and the correlations develop a nonexponential algebraic decay in l/r +i where ii = 0.041 is a critical exponent (Fisher 1964). In all that follows, we will stay away from criticality, hence the OZ forms. Equations 7.9 and 7.10, are essentially exact. When the critical point is approached, the isothermal compressibility diverges, and the analysis above shows that the correlation length diverges as the square root of the compressibility. The divergence of leads to a Coulomb decay of the pair correlation in Equation 7.10, but the correct exponent is slightly faster than pure Coulomb decay. 

At temperatures much higher than Za, the polymer fluid formally undergoes a fluid-gas transition where the isothermal compressibility Kr diverges, but this high temperature regime is normally inaccessible in polymer systems because of thermal decomposition. 

So, c(q = 0) is a finite quantity even when the isothermal compressibility %T tends to diverge in the critical region of any fluid. As will be seen, the former quantity is useful, because of being accessible from direct experiment [12] thanks to recent SANS measurements (see Section V). At variance from g(r), the direct correlation function c(r) in not zero inside the core region (r < a) and its knowledge is crucial in this range of distances, since it is directly related to y T so that... 

Apparent hydration numbers have been derived from experimental measurements assuming the formation of a hydration complex studied as a chemical reaction. xhe change of volume for the reaction is calculated from an equation of state which includes variation of the dielectric constant based on the solvent isothermal compressibility, while the bare ion and the complex are assumed spherical with crystallographic and Stokes-Einstein radii respectively. The latter radius is obtained from conductance measurements. Due to these assumptions, the apparent hydration numbers increase when temperature increases and diverge near the critical point due to the divergence of the solvent compressibility. Furthermore, negative values are obtained when the Stokes-Einstein radius for the complex is smaller than the crystallographic radius. 

The possible existence of an endpoint for the supercooled liquid locus is particularly Interesting in view of the experiments of Angell and coworkers (7,8,9,10). They find that pure water at ordinary pressures (even very finely dispersed) cannot apparently be supercooled below about —40 "C, and that virtually all physical properties manifest an impending "lambda anomaly at T, = —45 . The most striking features of this anomaly are the apparent divergences to infinity of isothermal compressibility, constant-pressure heat capacity, thermal expansion, and viscosity. We now seem to have in hand a qualitative basis for explaining these observations. 

The supercooled liquid catastrophe, if it exists, would necessarily be associated with diverging fluctuations in the structural order parameter F. This stems from the fact that the Y surface develops a vanishing curvature in the F direction as this endpoint is approached. Because the bicyclic octamer elements are bulky, fluctuations in their coiKentration amount to density fluctuations. Diverging density fluctuations then imply diverging isothermal compressibility. Furthermore the infinite slope of the metastable liquid locus at its endpoint implies the divergence of thermal expansion. Potential energy fluctuations remain essentially normal, so constant-volume heat capacity remains small. But the volumetric divergence creates an unbounded constant-pressure heat capacity. 

Now, we may let the volume fraction ip vary on the isotherm r = 0 then the osmotic compressibility diverges when cp - cpc and the variation law is... 

A third exponent y, usually called the susceptibility exponent from its application to the magnetic susceptibility x in magnetic systems, governs what in pure-fluid systems is the isothermal compressibility Kj, and what in mixtures is the osmotic compressibility, and determines how fast these quantities diverge as the critical point is approached (i.e. as 7. —> 1). 

There are several circumstances in which the quadratic fluctuation theory presented here breaks down. When derivatives of any of the intensive parameters with respect to the extensive parameters are very small, the corresponding fluctuations are very large. The Taylor expansion of SE in the fluctuations are then very large and the Taylor expansion of SE in the fluctuations cannot be truncated at the second-order term. For example, the mean-square density fluctuation is given by Eq. (10.C.28), where the isothermal compressibility and correspondingly (Sp2y diverges when (dP/d V)t - 0. This happens at the gas liquid critical point. Likewise at the critical consolute point... 

Figure 2.3. Temperature dependence of isothermal compressibility (kj) in liquid water. The dashed line represents the behavior of typical liquids. Note the turnaround and divergence-like behavior for water. The figure is reproduced fiom the thesis of Dr. Pradeep Kumar, http //polymer.bu.edu/ hes/water/thesis-kumar.pdf...

Fluctuations of Pi,(r) can be caused by density or concentration fluctuations. Real density fluctuations in simple (one-component) fluids occur near the critical point, where the isothermal compressibility of the liquid diverges. In multicomponent fluids, the concentration fluctuations can be observed when (at least one of) the components want to demix. Naturally, in more complex samples, like colloids in some solvents, fluctuations are present because, on the atomic length scale point of view, there are several (at least two) different phases present. SANS (and also, its X-ray counterpart SAXS) can be applied with success in these latter instances. 

Figure 1 shows that one must distinguish first-order phase transitions [where first derivatives of the appropriate thermodynamic potential F, such as the enthalpy U = [9(y3F)/9/3]p, where p = I/HbT, Hb being Boltzmann s constant, or volume V = dF/dp)T exhibit a jump] and second-order transitions, where U,V are continuons, bnt second derivatives are singular (1,2,7). The classical example for the latter case is the gas-liquid critical point, where the specific heat Cp = dU/dT)p or the isothermal compressibility kt = - VV) dVldp)T diverge. 

Meso-scale heterogeneities can be probed by the intensity of electromagnetic or neutron scattering at a selected wave number q, the instrumental scale. A good example of the scale-dependent meso-thermodynamic property is the isothermal compressibility of fluids or osmotic susceptibility of binary liquids near the critical point of phase separation. " In the limit of zero wave number and/or when the correlation length is small (c g 1) the intensity becomes the thermodynamic susceptibility, which diverges at the critical point as... 

The problem of temperature and pressure control, temperature stability, and temperature homogeneity becomes particularly important in studies close to the liquid-vapor critical point. The difficulties are directly related to the strong critical divergences of the isothermal compressibility, xt = = —V dVldp)j and the isobaric expan-... 

Water can be supercooled to -39.5 °C at a pressure of one atmosphere, if ice nucleation and impurities are prevented. Water can also remain liquid if its pressure is lowered below its freezing point, at fixed temperature this is called stretched water. Compared with other supercooled and stretched materials, water has two anomalous properties the heat capacity and the isothermal compressibility grow large as the temperature approaches -45 °C (see Figure 29.18). Such divergences often indicate a phase transition. The microscopic origins of this behavior are not yet understood 8J. 

Figure 29.18 (a) The isothermal compressibility k and (b) the heat capacity C, of water diverge upon supercooling to -45 °C. Source PC DeBenedetti, Metastable Liquids, Concepts and Principles, Princeton University Press, Princeton, 1966. 

When the gas can be liquefied, it is easy to see that the isothermal compressibility Kj = — (l/V)(0y/0p)j diverges during the transition (the flat part of the p-Visotherm). Above the critical temperature, since there is no transition to liquid, there is no divergence. According to classical theory, as the critical temperature is approached from above, the divergence of Kj should be given by... 

At a eritieal point, the phase transition under consideration is the second-order transition. For the uniform system, in the critical region the correlation length of statistical fluctuations, the isothermal compressibility and the heat capacity diverge to the infinity, according to the well-known power laws [337]. Moreover, the order parameter, which, for the gas-Uquid transition, is defined as the difference between the densities of both coexisting phases, approaches zero at the critical temperature. 

Figure4.10 shows the volume dependence of the isothermal compressibility, kt, below, at, and above the critical temperature. The dashed lines are the continuation of Kt into the metastable region. Notice that kt diverges when dp/dv r = 0. This condition defines the stability limit, kt > 0, i.e. inside the gap between the dashed lines the van der Waals equation gives negative kt. Notice that kt also diverges at the critical temperature. Above the critical temperature kt exhibits a maximum near...

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