Fraction volume

Kunz and coworkers [161,162] found for CTBN- and amine-terminated copolymer of butadiene and acrylonitrile (ATBN)-modified epoxies that, once a volume fraction of about 0.1 had been achieved in the polymer, further increase in volume fraction resulted in only minor increase in the value of fracture energy. The apparent discrepancy can be attributed to the use of matrices with different inherent ductility, curing conditions and test conditions employed [163,164]. Fracture energy in rubber-toughened epoxy is dependent on the test temperature and test rate [164—166]. The fracture energy increases with increase in the test temperature and decreases in test rate below the T.  

Kinloch and Hunston [163, 167] and others used the concept of time-temperature superpositioning, and separated the effect of changing the volume fraction and the properties of the matrix phase. No definite relationship between fracture energy and volume fraction of rubber was observed. The maximal value of volume fraction that can be achieved is about 0.2-0.3. Attempts to produce higher volume fraction result in phase inversion and loss of mechanical fracture properties. Though in most of the studies [168, 169], phase inversion was reported at a volume fraction 20%, in some recent studies inversion has been reported [170,171] at a very low volume fraction ( 2%) of rubber. 

---

V, = molar volume of component i Xj, = volume fraction of component i Xj = mole fraction of component i l = conductivity of the component i in the liquid state A, = conductivity of the mixture in the liquid state... 

The flash curve of a petroleum cut is defined as the curve that represents the temperature as a function of the volume fraction of vaporised liquid, the residual liquid being in equilibrium with the total vapor, at constant pressure. 

The flash curve at atmospheric pressure can be estimated using the results of the ASTM D 86 distillation by a correlation proposed by the API. For the same volume fraction distilled one has the following relation ... 

Ecole Nationale Superieure du Petrole et des Moteurs Formation Industrie end point (or FBP - final boiling point) electrostatic precipitation ethyl tertiary butyl ether European Union extra-urban driving cycle volume fraction distilled at 70-100-180-210°C Fachausschuss Mineralol-und-Brennstoff-Normung fluid catalytic cracking Food and Drug Administration front end octane number fluorescent indicator adsorption flame ionization detector... 

The pore system is described by the volume fraction of pore space (the fractional porosity) and the shape of the pore space which is represented by m , known as the cementation exponent. The cementation exponent describes the complexity of the pore system i.e. how difficult it is for an electric current to find a path through the reservoir. 

The volume fraction of water (S J and the saturation exponent n can be considered as expressing the increased difficulty experienced by an electrical current passing through a partially oil filled sample. (Note is only a special case of C, when a reservoir... 

Compositional data is expressed in two main ways components are shown as a volume fraction or as weight fraction of the total. 

The volume fraction would typically be used to represent the make up of a gas at a particular stage in a process line and describes gas composition e.g. 70% methane and 30% Ethane (also known as mol fractions) at a particular temperature and pressure. Gas composition may also be expressed in mass terms by multiplying the fractions by the corresponding molecular weight. 

The actual flowrate of each component of the gas (in for example cubic mefres), would be determined by multiplying the volume fraction of that component by the total flowrate. 

Piezocomposite ceramic can be tailored to our needs in contrast to conventional piezoceramic. The first parameter we can modify is the ceramic volume fraction. Fig. 2 indicates that the thickness coupling factor of the 1-3 composite is higher over a wide range of ceramic volume fraction between 15% and 95% than the coupling factor for PZT of about 0.52. Between 25% and 70% of volume fraction it is nearly constant at a high level of approximately 0.65. 

Fig. 2 Coupling factor as function of ceramic volume fraction...

Within this range we can modify the volume fraction without losses in the coupling factor and thereby can adjust the acoustic impedance to our demands. 

By variation of ceramic volume fraction and selection of the best fitting PZT material we can as well adjust the dielectric constant of the piezocomposite within a wide range. Therefore, we can choose the best piezocomposite material for each probe type to get optimum pulse form and amplitude. 

Marlow and Rowell discuss the deviation from Eq. V-47 when electrostatic and hydrodynamic interactions between the particles must be considered [78]. In a suspension of glass spheres, beyond a volume fraction of 0.018, these interparticle forces cause nonlinearities in Eq. V-47, diminishing the induced potential E. 

An equation algebraically equivalent to Eq. XI-4 results if instead of site adsorption the surface region is regarded as an interfacial solution phase, much as in the treatment in Section III-7C. The condition is now that the (constant) volume of the interfacial solution is i = V + JV2V2, where V and Vi are the molar volumes of the solvent and solute, respectively. If the activities of the two components in the interfacial phase are replaced by the volume fractions, the result is... 

Fig. XI-6. Polymer segment volume fraction profiles for N = 10, = 0-5, and Xi = 1, on a semilogarithinic plot against distance from the surface scaled on the polymer radius of gyration showing contributions from loops and tails. The inset shows the overall profile on a linear scale, from Ref. 65.

The polymer concentration profile has been measured by small-angle neutron scattering from polymers adsorbed onto colloidal particles [70,71] or porous media [72] and from flat surfaces with neutron reflectivity [73] and optical reflectometry [74]. The fraction of segments bound to the solid surface is nicely revealed in NMR studies [75], infrared spectroscopy [76], and electron spin resonance [77]. An example of the concentration profile obtained by inverting neutron scattering measurements appears in Fig. XI-7, showing a typical surface volume fraction of 0.25 and layer thickness of 10-15 nm. The profile decays rapidly and monotonically but does not exhibit power-law scaling [70]. 

Fig. XI-7. Volume fraction profile of 280,000-molecular-weight poly(ethylene oxide) adsorbed onto deuterated polystyrene latex at a surface density of 1.21 mg/m and suspended in D2O, from Ref. 70.

Apart from chemical composition, an important variable in the description of emulsions is the volume fraction, outer phase. For spherical droplets, of radius a, the volume fraction is given by the number density, n, times the spherical volume, 0 = Ava nl2>. It is easy to show that the maximum packing fraction of spheres is 0 = 0.74 (see Problem XIV-2). Many physical properties of emulsions can be characterized by their volume fraction. The viscosity of a dilute suspension of rigid spheres is an example where the Einstein limiting law is [2]... 

The HLB system has made it possible to organize a great deal of rather messy information and to plan fairly efficient systematic approaches to the optimiza-tion of emulsion preparation. If pursued too far, however, the system tends to lose itself in complexities [74]. It is not surprising that HLB numbers are not really additive their effective value depends on what particular oil phase is involved and the emulsion depends on volume fraction. Finally, the host of physical characteristics needed to describe an emulsion cannot be encapsulated by a single HLB number (note Ref. 75). 

Let us first consider two identical polymers, one deuterated and the other not, in a melt or a glassy state. The two polymers (degree of polymerization d) differ from each other only by scadermg lengths and b. If the total number of molecules is N, x is the volume fraction of the deuterated species x = N- / N, with Aq -t = A). According to equation (B1.9.116), we obtain... 

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].

Figure C2.1.10. (a) Gibbs energy of mixing as a function of the volume fraction of polymer A for a symmetric binary polymer mixture = Ag = N. The curves are obtained from equation (C2.1.9 ). (b) Phase diagram of a symmetric polymer mixture = Ag = A. The full curve is the binodal and delimits the homogeneous region from that of the two-phase stmcture. The broken curve is the spinodal.

Figure C2.1.11. Morjrhologies of a microphase-separated di-block copolymer as function of tire volume fraction of one component. The values here refer to a polystyrene-polyisoprene di-block copolymer and ( )pg is tire volume fraction of the polystyrene blocks. OBDD denotes tire ordered bicontinuous double diamond stmcture. (Figure from [78], reprinted by pemrission of Annual Reviews.)...

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... 

---