Hard phase volume fraction

Equation (35-9) describes the experimental data of a large number of systems quite well (Figure 35-11). Low values of the parameter, n, lead to high continuity factors that only vary slightly with the volume fraction of the hard phase (Table 35-6). High values of n, on the other hand, give continuity factors that decrease sharply with diminishing hard-phase volume fraction. 

Lagues et al. [17] found that the percolation theory for hard spheres could be used to describe dramatic increases in electrical conductivity in reverse microemulsions as the volume fraction of water was increased. They also showed how certain scaling theoretical tools were applicable to the analysis of such percolation phenomena. Cazabat et al. [18] also examined percolation in reverse microemulsions with increasing disperse phase volume fraction. They reasoned the percolation came about as a result of formation of clusters of reverse microemulsion droplets. They envisioned increased transport as arising from a transformation of linear droplet clusters to tubular microstructures, to form wormlike reverse microemulsion tubules. 

The continuity factor/must vary with the hard- and soft-phase volume fractions. The greatest change is expected when about the same fractions of hard and soft phases are present, since then, a phase reversal occurs. The following can be given for this change ... 

Fig. 4.26 Contours of tensile modulus (MPa) for a PU with hard segment volume fraction tpH = 0.354 and Poisson s ratio V = 0.5, according to the empirical log law for shear modulus of a two-phase composite material (equation (4.7)) [135]...

In summary one finds, that theoretical investigations of welding induced phase transformations concentrate, in most cases, on the resulting phase volume fractions. It is possible to estimate the distribution of hardness as well as the mechanical material behaviour. 

Based on various experimental studies, one can schematically represent the morphology of segmented polyurethane (elastomer or flexible foam polymer) on the nano- and micro-scale as shown in Figm-e 2.1. For the range of hard segments volume fraction less than 50%, much of the space is occupied by the soft phase matrix. Microphase-separated nano-domains of the hard phase are dispersed in this matrix they can be individual islands or can form percolated networks. Finally, there could also be some larger (micron-sized) macrophase-separated domains of hard phase, where hard phase domains are ordered at the macro-scale (this is especially true in the case of flexible foams). The relative amounts of all these elements depend on the formulation and processing history. 

While the existing approaches (such as the model of Qi and Boyce) often provide a good description of polyurethane tensile curves, th typically treat hard and soft phase volume fractions as adjustable (fitting) parameters. In a fully predictive theory, one needs to combine the Qi-Boyce or similar framework with a thermodynamic model to predict hard and soft phase volume fractions, as we discussed in the previous section. Below, we illustrate how one can build such a theory and obtain a qualitative, if not quantitative, agreement with experiment. We start from a micromechanical model of Figure 2.7. The initial value of Vfj (volume fi action of the elastically active regions of the percolated hard phase) is determined on the basis of thermodynamic considerations and the percolation model, as described in the previous section. We assume that each elastically active region of the har d phase can be described as an elasto-plastic material ... 

Figure B3.3.9. Phase diagram for polydisperse hard spheres, in the volume fraction ((]))-polydispersity (s) plane. Some tie-lines are shown connecting coexistmg fluid and solid phases. Thanks are due to D A Kofke and P G Bolhuis for this figure. For frirther details see [181. 182].

Altliough tire behaviour of colloidal suspensions does in general depend on temperature, a more important control parameter in practice tends to be tire particle concentration, often expressed as tire volume fraction ((). In fact, for hard- sphere suspensions tire phase behaviour is detennined by ( ) only. For spherical particles... 

Experimentally, tire hard-sphere phase transition was observed using non-aqueous polymer lattices [79, 80]. Samples are prepared, brought into the fluid state by tumbling and tlien left to stand. Depending on particle size and concentration, colloidal crystals tlien fonn on a time scale from minutes to days. Experimentally, tliere is always some uncertainty in the actual volume fraction. Often tire concentrations are tlierefore rescaled so freezing occurs at ( )p = 0.49. The widtli of tire coexistence region agrees well witli simulations [Jd, 80]. 

Charged particles in polar solvents have soft-repulsive interactions (see section C2.6.4). Just as hard spheres, such particles also undergo an ordering transition. Important differences, however, are that tire transition takes place at (much) lower particle volume fractions, and at low ionic strengtli (low k) tire solid phase may be body centred cubic (bee), ratlier tlian tire more compact fee stmcture (see [69, 73, 84]). For tire interactions, a Yukawa potential (equation (C2.6.11)1 is often used. The phase diagram for the Yukawa potential was calculated using computer simulations by Robbins et al [851. 

We will focus on one experimental study here. Monovoukas and Cast studied polystyrene particles witli a = 61 nm in potassium chloride solutions [86]. They obtained a very good agreement between tlieir observations and tire predicted Yukawa phase diagram (see figure C2.6.9). In order to make tire comparison tliey rescaled the particle charges according to Alexander et al [43] (see also [82]). At high electrolyte concentrations, tire particle interactions tend to hard-sphere behaviour (see section C2.6.4) and tire phase transition shifts to volume fractions around 0.5 [88]. 

Low—medium alloy steels contain elements such as Mo and Cr for hardenabiHty, and W and Mo for wear resistance (Table 4) (7,16,17) (see Steel). These alloy steels, however, lose their hardness rapidly when heated above 150—340°C (see Fig. 3). Furthermore, because of the low volume fraction of hard, refractory carbide phase present in these alloys, their abrasion resistance is limited. Hence, low—medium alloy steels are used in relatively inexpensive tools for certain low speed cutting appHcations where the heat generated is not high enough to reduce their hardness significantly. 

Abrasive wear is encountered when hard particles, or hard projections on a counter-face, are forced against and moved relative to a surface. In aUoys such as the cobalt-base wear aUoys which contain a hard phase, the abrasion resistance generaUy increases as the volume fraction of the hard phase increases. Abrasion resistance is, however, strongly influenced by the size and shape of the hard-phase precipitates within the microstmcture, and the size and shape of the abrading species (see Abrasives). 

The abrasion resistance of cobalt-base alloys generally depends on the hardness of the carbide phases and/or the metal matrix. For the complex mechanisms of soHd-particle and slurry erosion, however, generalizations cannot be made, although for the soHd-particle erosion, ductihty may be a factor. For hquid-droplet or cavitation erosion the performance of a material is largely dependent on abiUty to absorb the shock (stress) waves without microscopic fracture occurring. In cobalt-base wear alloys, it has been found that carbide volume fraction, hence, bulk hardness, has Httie effect on resistance to Hquid-droplet and cavitation erosion (32). Much more important are the properties of the matrix. 

Colloidal crystals . At the end of Section 2.1.4, there is a brief account of regular, crystal-like structures formed spontaneously by two differently sized populations of hard (polymeric) spheres, typically near 0.5 nm in diameter, depositing out of a colloidal solution. Binary superlattices of composition AB2 and ABn are found. Experiment has allowed phase diagrams to be constructed, showing the crystal structures formed for a fixed radius ratio of the two populations but for variable volume fractions in solution of the two populations, and a computer simulation (Eldridge et al. 1995) has been used to examine how nearly theory and experiment match up. The agreement is not bad, but there are some unexpected differences from which lessons were learned. 

The crucial question is at what value of <)> is the attraction high enough to induce phase separation De Hek and Vrij (6) assume that the critical flocculation concentration is equivalent to the phase separation condition defined by the spinodal point. From the pair potential between two hard spheres in a polymer solution they calculate the second virial coefficient B2 for the particles, and derive from the spinodal condition that if B2 = 1/2 (where is the volume fraction of particles in the dispersion) phase separation occurs. For a system in thermodynamic equilibrium, two phases coexist if the chemical potential of the hard spheres is the same in the dispersion and in the floe phase (i.e., the binodal condition). 

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