Size, critical

The resistance to nucleation is associated with the surface energy of forming small clusters. Once beyond a critical size, the growth proceeds with the considerable driving force due to the supersaturation or subcooling. It is the definition of this critical nucleus size that has consumed much theoretical and experimental research. We present a brief description of the classic nucleation theory along with some examples of crystal nucleation and growth studies. 

The dynamic picture of a vapor at a pressure near is then somewhat as follows. If P is less than P , then AG for a cluster increases steadily with size, and although in principle all sizes would exist, all but the smallest would be very rare, and their numbers would be subject to random fluctuations. Similarly, there will be fluctuations in the number of embryonic nuclei of size less than rc, in the case of P greater than P . Once a nucleus reaches the critical dimension, however, a favorable fluctuation will cause it to grow indefinitely. The experimental maximum supersaturation pressure is such that a large traffic of nuclei moving past the critical size develops with the result that a fog of liquid droplets is produced. 

A homogeneous metastable phase is always stable with respect to the fonnation of infinitesimal droplets, provided the surface tension a is positive. Between this extreme and the other thennodynamic equilibrium state, which is inhomogeneous and consists of two coexisting phases, a critical size droplet state exists, which is in unstable equilibrium. In the classical theory, one makes the capillarity approxunation the critical droplet is assumed homogeneous up to the boundary separating it from the metastable background and is assumed to be the same as the new phase in the bulk. Then the work of fonnation W R) of such a droplet of arbitrary radius R is the sum of the... 

AS )) the function to be minimized is exp (-AS p/R)/ [36]. A quantitative expression for AS can be found by noting that the A monomers in an unstrained loop (N > 4) have essentially two possible confonnations, pointing either inwards or outwards. For loops smaller than a critical size the inward ones are in an apolar environment, since the enclosed water no longer has bulk properties, and the outward ones are in polar bulk water hence the electrostatic charges on... 

The collapse of the free volume below a critical size for molecular motion is... 

Eor an impact strength of 34 J (25 ft-lbf) the equivalent fracture toughness (150) is approximately 120 MPay. The fracture toughness dictates the critical size of crack above which fast fracture intervenes, so the smaller its value the smaller the critical crack and hence the greater significance of the transverse impact requirement specified by Manning. 

Once the precipitates grow beyond a critical size they lose coherency and then, in order for deformation to continue, dislocations must avoid the particles by a process known as Orowan bowing(23). This mechanism appHes also to alloys strengthened by inert dispersoids. In this case a dislocation bends between adjacent particles until the loop becomes unstable, at which point it is released for further plastic deformation, leaving a portion behind, looped around the particles. The smaller the interparticle spacing, the greater the strengthening. 

Below a critical size the particle becomes superparamagnetic in other words the thermal activation energy kTexceeds the particle anisotropy energy barrier. A typical length of such a particle is smaller than 10 nm and is of course strongly dependent on the material and its shape. The reversal of the magnetization in this type of particle is the result of thermal motion. 

The desked balance of ductility and strength can be obtained in age-hard-enable alloys, such as beryllium copper, by controlling the amount of precipitate. For higher strength, aging is conducted to provide a critical size dispersion. Greater amounts of precipitate are obtained by increasing the beryllium content of the alloy. 

When the nucleus is formed on a solid substrate by heterogeneous nucleation the above equations must be modified because of the nucleus-substrate interactions. These are reflected in the balance of the interfacial energies between the substrate and the environment, usually a vacuum, and the nucleus-vacuum and the nucleus-substrate interface energies. The effect of these terms is usually to reduce the critical size of the nucleus, to an extent dependent on... 

The critical size of the stable nucleus at any degree of under cooling can be calculated widr an equation derived similarly to that obtained earlier for the concentration of defects in a solid. The configurational entropy of a mixture of nuclei containing n atoms widr o atoms of the liquid per unit volume, is given by the Boltzmann equation... 

Using the Stokes-Einstein equation for the viscosity, which is unexpectedly useful for a range of liquids as an approximate relation between diffusion and viscosity, explains a resulting empirical expression for the rate of formation of nuclei of the critical size for metals... 

The left-hand side of our equation says that fast fracture will occur when, in a material subjected to a stress a, a crack reaches some critical size a or, alternatively, when material containing cracks of size a is subjected to some critical stress cr. The right-hand side of our result depends on material properties only E is obviously a material constant, and G, the energy required to generate unit area of crack, again must depend only on the basic properties of our material. Thus, the important point about the equation is that the critical combination of stress and crack length at which fast fracture commences is a material constant. 

First, the pressure vessel must be safe from plastic collapse that is, the stresses must everywhere be below general yield. Second, it must not fail by fast fracture if the largest cracks it could contain have length 2a (Fig. 16.4), then the stress intensity K CTV must everywhere be less than K. Finally, it must not fail by fatigue the slow growth of a crack to the critical size at which it runs. 

If the critical flaw size for fast fracture is less than the wall thickness (t) of the vessel, then fast fracture can occur with no warning. But suppose the critical size (2a jt) is... 

Pure titanium is cooled from a temperature at which the b.c.c. phase is stable to a temperature at which the c.p.h. phase is stable. As a result, lens-shaped nuclei of the c.p.h. phase form at the grain boundaries. Estimate the number of atoms needed to make a critical-sized nucleus given the following data AH = 3.48 kJ moT atomic weight = 47.90 - T = 30 K = 882°C y= 0.1 ]ra density of the c.p.h. 

During fatigue the stress amplitude usually remains constant and brittle failure occurs as a result of crack growth from a sub-critical to a critical size. Clearly the rate at which these cracks grow is the determining factor in the life of the component. It has been shown quite conclusively for many polymeric materials that the rate at which cracks grow is related to the stress intensity factor by a relation of the form... 

Assuming that the geometry function, Y, does not change as the crack grows then this equation may be integrated to give the number of cycles, N/, which are necessary for the crack to grow from its initial size (2n,) to its critical size at fracture (lOc). 

During cyclic loading, any cracks in the material will propagate until they reach this critical size. If the article is to have an endurance of at least 10 cycles then equation (2.119) may be used to determine the size of the smallest flaw which can be present in the material before cycling commences. 

While ideally structures should be designed and fabricated so that environment-sensitive cracking is avoided, in practice it is sometimes necessary to live with the problem. This implies an ability to detect and measure the size of cracks before they reach the critical size that may result in catastrophic failure. Such inspection has important implications for plant design, which should be such as to allow inspection at relevant locations. The latter are regions of high residual stress (welded, bolted or riveted joints) and regions of geometrical discontinuity (notches, crevices, etc.) where stress or environment concentration may occur. 

---