The Kelvin Solid

To evaluate the constant of integration C again engineering reasoning is used. At time i = 0, the dashpot locks up the Kelvin element, because it takes time for the dashpot to move. Thus the initial condition is = 0 at t = 0. This relation yields 

This equation is plotted in Eigure 3.6a. The Kelvin solid has no initial elastic response (Eq = 0), but has delayed elasticity with an asymptotic modulus, namely as 

From the creep solution Equation 3.21, the time factor qlE in the exponential is called the retardation time t, such that 

FIGURE 3.6 Creep and relaxation of the Kelvin solid, (a) Creep curve for the Kelvin solid and (b) relaxation of the Kelvin solid. 

Problem Determine material specifications for a design of a shock absorber for a design that requires 63% of the elastic strain to be delayed (retarded) 2.0 s. Assume a Kelvin solid-type material behavior and assume the maximum elastic strain is limited to 0.01 when a constant stress of 1000 psi is applied. 

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Pharmaceutical materials are rarely described by simple mechanical equivalents such as the Kelvin (solid-like behavior) or the Maxwell (liquid-like behavior) model. 

The Kelvin model is also frequently used to describe the phenomena of creep. Recall the Kelvin solid from Fig. 3.21. 

In order to derive the differential equation describing the stress (strain) behavior of the Kelvin solid, the same principles of mechanics are applied, except that now account must be made of the parallel configuration of the spring and dashpot element shown in Figure 3.2b. From a force balance,... 

Thus, the Kelvin solid relaxes immediately to a constant stress, Oi < Oo- This behavior is characteristic of a solid—that as f —> o , o > 0 (i.e., the stress does not fully relax). 

Instead of solving this differential eqnation, the creep solution is more easily constructed from physical considerations by combining the solutions already available for the linear elastic solid, the Kelvin solid, and the linear viscous fluid. Thus,... 

Thus, the Kelvin solid approaches the Euler load as t which can be seen from Equation 4.9 and as shown in Figure 4.5. However, for a Maxwell fluid. 

While Eq. III-18 has been verified for small droplets, attempts to do so for liquids in capillaries (where Rm is negative and there should be a pressure reduction) have led to startling discrepancies. Potential problems include the presence of impurities leached from the capillary walls and allowance for the film of adsorbed vapor that should be present (see Chapter X). There is room for another real effect arising from structural peiturbations in the liquid induced by the vicinity of the solid capillary wall (see Chapter VI). Fisher and Israelachvili [19] review much of the literature on the verification of the Kelvin equation and report confirmatory measurements for liquid bridges between crossed mica cylinders. The situation is similar to that of the meniscus in a capillary since Rm is negative some of their results are shown in Fig. III-3. Studies in capillaries have been reviewed by Melrose [20] who concludes that the Kelvin equation is obeyed for radii at least down to 1 fim. 

As already indicated in Section 3.1, the study of mesoporous solids is closely bound up with the concept of capillary condensation and its quantitative expression in the Kelvin equation. This equation is, indeed, the basis of virtually all the various procedures for the calculation of pore size... 

Now, in principle, the angle of contact between a liquid and a solid surface can have a value anywhere between 0° and 180°, the actual value depending on the particular system. In practice 6 is very difficult to determine with accuracy even for a macroscopic system such as a liquid droplet resting on a plate, and for a liquid present in a pore having dimensions in the mesopore range is virtually impossible of direct measurement. In applications of the Kelvin equation, therefore, it is almost invariably assumed, mainly on grounds of simplicity, that 0 = 0 (cos 6 = 1). In view of the arbitrary nature of this assumption it is not surprising that the subject has attracted attention from theoreticians. 

The formation of a liquid phase from the vapour at any pressure below saturation cannot occur in the absence of a solid surface which serves to nucleate the process. Within a pore, the adsorbed film acts as a nucleus upon which condensation can take place when the relative pressure reaches the figure given by the Kelvin equation. In the converse process of evaporation, the problem of nucleation does not arise the liquid phase is already present and evaporation can occur spontaneously from the meniscus as soon as the pressure is low enough. It is because the processes of condensation and evaporation do not necessarily take place as exact reverses of each other that hysteresis can arise. 

These models, though necessarily idealized, are sufficiently close to the actual systems found in practice to enable useful conclusions to be drawn from a given Type IV isotherm as to the pore structure of a solid adsorbent. To facilitate the discussion, it is convenient to simplify the Kelvin equation by putting yVJRT = K, and on occasion to use the exponential form ... 

Effect of an error of 0-05 K in temperature of the solid on the value of r calculated from the Kelvin equation... 

A vast amount of research has been undertaken on adsorption phenomena and the nature of solid surfaces over the fifteen years since the first edition was published, but for the most part this work has resulted in the refinement of existing theoretical principles and experimental procedures rather than in the formulation of entirely new concepts. In spite of the acknowledged weakness of its theoretical foundations, the Brunauer-Emmett-Teller (BET) method still remains the most widely used procedure for the determination of surface area similarly, methods based on the Kelvin equation are still generally applied for the computation of mesopore size distribution from gas adsorption data. However, the more recent studies, especially those carried out on well defined surfaces, have led to a clearer understanding of the scope and limitations of these methods furthermore, the growing awareness of the importance of molecular sieve carbons and zeolites has generated considerable interest in the properties of microporous solids and the mechanism of micropore filling. 

In writing the present book our aim has been to give a critical exposition of the use of adsorption data for the evaluation of the surface area and the pore size distribution of finely divided and porous solids. The major part of the book is devoted to the Brunauer-Emmett-Teller (BET) method for the determination of specific surface, and the use of the Kelvin equation for the calculation of pore size distribution but due attention has also been given to other well known methods for the estimation of surface area from adsorption measurements, viz. those based on adsorption from solution, on heat of immersion, on chemisorption, and on the application of the Gibbs adsorption equation to gaseous adsorption. 

It would be difficult to over-estimate the extent to which the BET method has contributed to the development of those branches of physical chemistry such as heterogeneous catalysis, adsorption or particle size estimation, which involve finely divided or porous solids in all of these fields the BET surface area is a household phrase. But it is perhaps the very breadth of its scope which has led to a somewhat uncritical application of the method as a kind of infallible yardstick, and to a lack of appreciation of the nature of its basic assumptions or of the circumstances under which it may, or may not, be expected to yield a reliable result. This is particularly true of those solids which contain very fine pores and give rise to Langmuir-type isotherms, for the BET procedure may then give quite erroneous values for the surface area. If the pores are rather larger—tens to hundreds of Angstroms in width—the pore size distribution may be calculated from the adsorption isotherm of a vapour with the aid of the Kelvin equation, and within recent years a number of detailed procedures for carrying out the calculation have been put forward but all too often the limitations on the validity of the results, and the difficulty of interpretation in terms of the actual solid, tend to be insufficiently stressed or even entirely overlooked. And in the time-honoured method for the estimation of surface area from measurements of adsorption from solution, the complications introduced by... 

One of the first attempts to calculate the thermodynamic properties of an atomic solid assumed that the solid consists of an array of spheres occupying the lattice points in the crystal. Each atom is rattling around in a hole at the lattice site. Adding energy (usually as heat) increases the motion of the atom, giving it more kinetic energy. The heat capacity, which we know is a measure of the ability of the solid to absorb this heat, varies with temperature and with the substance.8 Figure 10.11, for example, shows how the heat capacity Cy.m for the atomic solids Ag and C(diamond) vary with temperature.dd ee The heat capacity starts at a value of zero at zero Kelvin, then increases rapidly with temperature, and levels out at a value of 3R (24.94 J-K -mol-1). The... 

By comparing Figure 11.9 and the characteristic Po2(Uwr) rate breaks of the inset of Fig. 11.9 one can assign to each support an equivalent potential Uwr value (Fig. 11.10). These values are plotted in Figure 11.11 vs the actual work function G>° measured via the Kelvin probe technique for the supports at po2-l atm and T=400°C. The measuring principle utilizing a Kelvin probe and the pinning of the Fermi levels of the support and of metal electrodes in contact with it has been discussed already in Chapter 7 in conjunction with the absolute potential scale of solid state electrochemistry.37... 

A triple point is a point where three phase boundaries meet on a phase diagram. For water, the triple point for the solid, liquid, and vapor phases lies at 4.6 Torr and 0.01°C (see Fig. 8.6). At this triple point, all three phases (ice, liquid, and vapor) coexist in mutual dynamic equilibrium solid is in equilibrium with liquid, liquid with vapor, and vapor with solid. The location of a triple point of a substance is a fixed property of that substance and cannot be changed by changing the conditions. The triple point of water is used to define the size of the kelvin by definition, there are exactly 273.16 kelvins between absolute zero and the triple point of water. Because the normal freezing point of water is found to lie 0.01 K below the triple point, 0°C corresponds to 273.15 K. 

Vapor sorption onto porous solids differs from vapor uptake onto the surfaces of flat materials in that a vapor (in the case of interest, water) will condense to a liquid in a pore structure at a vapor pressure, Pt, below the vapor pressure, P°, where condensation occurs on flat surfaces. This is generally attributed to the increased attractive forces between adsorbate molecules that occur as surfaces become highly curved, such as in a pore or capillary. This phenomenon is referred to as capillary condensation and is described by the Kelvin equation [19] ... 

In a similar way, the Systeme Internationale has defined other common physicochemical variables. The SI unit of temperature T is the kelvin. We define the kelvin as 1/273.16th part of the thermodynamic temperature difference between absolute zero (see Section 1.4) and the triple point of water, i.e. the temperature at which liquid water is at equilibrium with solid water (ice) and gaseous water (steam) provided that the pressure is 610 Pa. 

As with the elastic solid we can see that as the stress is applied the strain increases up to a time t = t. Once the stress is removed we see partial recovery of the strain. Some of the strain has been dissipated in viscous flow. Laboratory measurements often show a high frequency oscillation at short times after a stress is applied or removed just as is observed with the stress relaxation experiment. We can replace a Kelvin model by a distribution of retardation times ... 

Numerous instructions on determining 7S or the specific energy 7sl of the melt -solid interface use the equation first derived by W. Thomson (Lord Kelvin) in 1871. One of its derivations follows. 

Figure 3.10 Basic mechanical elements for solids and fluids a) dash pot for a viscous response, b) spring for an elastic response, c) Voigt or Kelvin solid, d) Maxwell fluid, and e) the four-parameter viscoelastic fluid...

When dash pot and spring elements are connected in parallel they simulate the simplest mechanical representation of a viscoelastic solid. The element is referred to as a Voigt or Kelvin solid, and it is shown in Fig. 3.10(c). The strain as a function of time for an applied force for this element is shown in Fig. 3.11. After a force (or stress) elongates or compresses a Voigt solid, releasing the force causes a delay in the recovery due to the viscous drag represented by the dash pot. Due to this time-dependent response the Voigt model is often used to model recoverable creep in solid polymers. Creep is a constant stress phenomenon where the strain is monitored as a function of time. The function that is usually calculated is the creep compliance/(f) /(f) is the instantaneous time-dependent strain e(t) divided by the initial and constant stress o. ... 

Analyses of the results obtained depend on the shape of the specimen, whether or not the distribution of mass in the specimen is accounted for and the assumed model used to represent the linear viscoelastic properties of the material. The following terms relate to analyses which generally assume small deformations, specimens of uniform cross-section, non-distributed mass and a Voigt-Kelvin solid. These are the conventional assumptions. 

Note 4 Damping curves are conventionally analysed in terms of the Voigt-Kelvin solid giving a decaying amplitude and a single frequency. 

Note 5 Given the properties of a Voigt-Kelvin solid, a damping curve is described by the equation... 

Note 2 The logarithmic decrement can be used to evaluate the decay constant, p. From the equation for the damping curve of a Voigt-Kelvin solid. 

Note 2 For a Voigt-Kelvin solid, with P(D)=1 and Q y)=a+pD, where a is the spring constant and P the dashpot constant, the equation describing the deformation becomes... 

Note 7 Notes 2 and 5 show that application of a sinusoidal uniaxial force to a Voigt-Kelvin solid of negligible mass, with or without added mass, results in an out-of-phase sinusoidal uniaxial extensional oscillation of the same frequency. 

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