Hysteresis

Hysteresis is another cycle process. In contrast to all other cycles discussed hitherto, in this case no work is generated, but entropy. 

Hysteresis is not a problem of kinetics (which is treated in Chapter 8). Normally, the energy dissipated in an electrochemical cell varies as the square of the current, so the corresponding overpotential t] is proportional 

Hysteresis is characterized by the fact that the right hand side / does not only depend on the solution values at the actual time but also on the history of the motion, i.e. on solution values at previous time points. 

Hysteresis occurs, e.g., in the modeling of elasto-plastic behavior, e.g. in crash-test simulation. 

We introduce a hysteresis parameter a in order to describe these phenomena  

G summarizes all information from previous time points which are necessary to compute the right hand side at the actual time. Fig. 6.32 shows a hysteresis curve 

The realization in a program can be done with the help of an additional memo-vector, which corresponds to a. In the right hand side it must be read-only . It is allowed to be changed only by the switching algorithm after a root of of change has been localized. 

Quantitatively, hysteresis is represented by the free energy difference  

The first desorption curve shows two kinds of hysteresis. From P/Pq 1.0 to 0.3 the observed values are 

FIGURE 11.27 Adsorption of water on silica gel [31], showing hysteresis. The vertical axis shows gadsorbed/100 gadsorbem. rather than the conventional mmol/g. The three curves correspond to the first adsorption of water, the first desorption and then the second adsorption. 

For the second adsorption the curve lies somewhat above the first adsorption curve and practically retraces the lower part of the desorption curve to P/Pq 0.3, but then does not foUow it upward, but rather practically parallels the first adsorption curve. 

Such curves in which the values going up and down are not the same are called hysteresis cmves. Their explanation is far from agreed upon among experts, but the following, advanced in [31] is widely accepted. On the first desorption, hquid adsorbed into small pores is held in place by surface tension and does not evaporate at the same external pressure as it would if it were a flat sheet. This is called capillary condensation, discussed in Chapter 14. As the pressure falls, more and more of the liquid bound that way desorbs, and at about P/Pq 0.3 those pores are empty, and the behavior parallels that of the first adsorption curve. On the second adsorption those pores are filling again but capillary condensation does not affect their adsorption behavior. 

The first desorption and second absorption curves are reported as equilibrium curves, meaning that one could go up and down on them, finding the same values. The offset of the second adsorption curve from the first is explained by the authors as an irreversible type of hysteresis possibly due to imperfect rigidity in the gel strucmre.  

In Fig. 7.32, a stress-strain response, appearing as hysteresis loops, is shown under the indicated conditions, namely for the steady states of an homogeneous 

In zirconia, two kinds of hysteresis loops develop. As mentioned previously, a tetragonal-to-monoclinic transformation takes place under the influence of stress, 

When a material is subjected to cyclic loading, its stress-strain response may change with the number of applied cycles. If the maximum stress increases with the number of cycles, the material is said to cyclically harden . If maximum stress decreases over the number of cycles, the material is said to cyclically soften . If the maximum-stress level does not change, the material is said to be cyclically stable . As seen in Fig. 7.33, the nature of these transformation-induced hysteresis loops is cyclically stable when the stress level is considered. However, the strain of these cycles upon unloading and under compression are different, possibly due to the asymmetric stress characteristic of phase transformation (the peak strain at compression point E is less than that at tension point B). 

Microcracking-induced hysteresis may be seen in Fig. 7.34, where surface-crack friction and sliding occur. A comparison is made between such a case and one without crack friction and sliding. 

A tendency to strain softening with increased strain range has been observed and is illustrated in Fig. 7.36 by the fact that the hysteresis loop tilts progressively toward the strain axis as cycling proceeds. Also, the increase in the width of the loop is a sign of softening in the case of Mg-PSZ, a relatively small loop width increase may be observed. 

If a drug was to be administered orally, the following graph may be obtained. 

Plasma After drawing and labelling the axes, plot the concentration versus time curve for an orally administered drug. Label this curve plasma to show how the concentration rises and falls with time following an oral dose. 

Effector site Now draw a second, similar curve to the right of the first. This shows the concentration of the drug at its site of action. The degree of displacement to the right of the first curve is determined by the factors mentioned above. 

Key points When both curves are drawn, mark a fixed concentration point on the y axis and label it C. Demonstrate that the plasma concentration curve crosses this value twice, at times tx and t2. At time f, the concentration in the plasma is rising and at t2 it is falling. The crucial point now that enables you to define hysteresis is to demonstrate that the effector site concentration is different at these two times depending on whether the plasma concentration is rising (giving concentration Ej) or falling (giving concentration E2). 

In essence, the magnetic state of a ferromagnetic solid will depend on its history, a phenomenon 

In order to produce a stable poling direction in the first place, the cyclic process that is most widely used is the Bauer process. This room-temperature method, called the corona poling method, involves applying a large electric field up to 8kV. 

FIGURE 16.4 Hysteresis loop for PDVF at room temperature (20°C) at 1 MHz. (Experimental data from Takase, Y., J. W. Lee, J. 1. Scheinbeim, and B. A. Newman, Macromolecules, 24(25), 6644-6652,1991.) 

Piezoelectric properties between samples can be reproduced within 2% with this process. Bauer and Bauer describe the process in detail. 

In order to analyze the hysteresis, it is useful to use complex algebra. Define a complex piezoelectric coefficient as 

Then by analogy with Equation 2.35, the electric displacement D will lag the electric field and is given by 

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Lin et al. [70, 71] have modeled the effect of surface roughness on the dependence of contact angles on drop size. Using two geometric models, concentric rings of cones and concentric conical crevices, they find that the effects of roughness may obscure the influence of line tension on the drop size variation of contact angle. Conversely, the presence of line tension may account for some of the drop size dependence of measured hysteresis. 

Surfactant-coated surfaces may also rearrange on contact with a liquid as shown by Israelachvili and co-workers [77]. This mechanism helps to explain hysteresis occurring on otherwise smooth and homogeneous surfaces. 

It is clear from our discussion of contact angle hysteresis that there is some degree of variability in reported contact angle values. The data collected in Table X-2, therefore, are intended mainly as a guide to the type of behavior to be expected. The older data comprise mainly results for refractory and relatively polar solids, while newer data are for polymeric surfaces. 

There is appreciable contact angle hysteresis for many of the systems reported in Table X-2 the customary practice of reporting advancing angles has been followed. 

The situation is complicated, however, because some of the drag on a skidding tire is due to the elastic hysteresis effect discussed in Section XII-2E. That is, asperities in the road surface produce a traveling depression in the tire with energy loss due to imperfect elasticity of the tire material. In fact, tires made of high-elastic hysteresis material will tend to show superior skid resistance and coefficient of friction. 

Cationic surfactants may be used [94] and the effect of salinity and valence of electrolyte on charged systems has been investigated [95-98]. The phospholipid lecithin can also produce microemulsions when combined with an alcohol cosolvent [99]. Microemulsions formed with a double-tailed surfactant such as Aerosol OT (AOT) do not require a cosurfactant for stability (see, for instance. Refs. 100, 101). Morphological hysteresis has been observed in the inversion process and the formation of stable mixtures of microemulsion indicated [102]. 

Thus D(r) is given by the slope of the V versus P plot. The same distribution function can be calculated from an analysis of vapor adsorption data showing hysteresis due to capillary condensation (see Section XVII-16). Joyner and co-woikers [38] found that the two methods gave very similar results in the case of charcoal, as illustrated in Fig. XVI-2. See Refs. 36 and 39 for more recent such comparisons. There can be some question as to what the local contact angle is [31,40] an error here would shift the distribution curve. 

As also noted in the preceding chapter, it is customary to divide adsorption into two broad classes, namely, physical adsorption and chemisorption. Physical adsorption equilibrium is very rapid in attainment (except when limited by mass transport rates in the gas phase or within a porous adsorbent) and is reversible, the adsorbate being removable without change by lowering the pressure (there may be hysteresis in the case of a porous solid). It is supposed that this type of adsorption occurs as a result of the same type of relatively nonspecific intermolecular forces that are responsible for the condensation of a vapor to a liquid, and in physical adsorption the heat of adsorption should be in the range of heats of condensation. Physical adsorption is usually important only for gases below their critical temperature, that is, for vapors. 

Fig. XVll-19. Adsorption of CH4 on MgO(lOO) at 77.35 K. The vertical line locates each vertical step corresponds to the condensation of a monolayer. There was no hysteresis. Desorption points are shown as . (From Ref. 110.)...

The question is not trivial such agreement is not assured in the case of systems showing hysteresis (see Section XVII-16), and it has been difficult to affirm it on rigorous thermodynamic grounds in the case of a heterogeneous surface. 

Below the critical temperature of the adsorbate, adsorption is generally multilayer in type, and the presence of pores may have the effect not only of limiting the possible number of layers of adsorbate (see Eq. XVII-65) but also of introducing capillary condensation phenomena. A wide range of porous adsorbents is now involved and usually having a broad distribution of pore sizes and shapes, unlike the zeolites. The most general characteristic of such adsorption systems is that of hysteresis as illustrated in Fig. XVII-27 and, more gener-... 

The section cd can be regarded as due to relatively large cone-shaped pores that would fill and empty without hysteresis. At the end of section cd, then, all pores should be filled, and the adsorbent should hold the same volume of any adsorbate. See Ref. 200 for a discussion of this conclusion, sometimes known as the Gurvitsch rule. 

Adsorbents such as some silica gels and types of carbons and zeolites have pores of the order of molecular dimensions, that is, from several up to 10-15 A in diameter. Adsorption in such pores is not readily treated as a capillary condensation phenomenon—in fact, there is typically no hysteresis loop. What happens physically is that as multilayer adsorption develops, the pore becomes filled by a meeting of the adsorbed films from opposing walls. Pores showing this type of adsorption behavior have come to be called micropores—a conventional definition is that micropore diameters are of width not exceeding 20 A (larger pores are called mesopores), see Ref. 221a. 

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