Mass distribution function

Before we go back to Eq. (14.30), we shall evaluate the mass distribution function for the particles whose size distribution is of the form (14.33), i.e., normal probability size distribution. 

Leadbetter AJ, Norris EK (1979) Molec Phys 38 669. There are different contributions which give rise to a broadening a of the molecular centre of mass distribution function f(z). The most important are the long-wave layer displacement thermal fluctuations and the individual motions of molecules having a random diffusive nature. The layer displacement amplitude depends on the magnitude of the elastic constants of smectics ... 

The program then requests specification as to type of polymer (line 320). If the "polymer" component is a collection of oligomers, the number of unique species is sought (line 360). The values for the mole (or weight) fraction, functionality and molecular weight of each species is then entered (lines 380-650). The number, site, and mass expectation values of the functionality and molecular weight (lines 650-810) are computed. The necessary site and mass distribution functions are also computed (lines 820-850). 

These parameters are used calculate the site and mass distribution functions assuming a Schulz-Zimm molecular weight distribution. The Schulz-Zimm parameters are calculated in lines 930-950. The weight fraction of diluent (as a fraction of the amount of polymer) is then sought. If there is no diluent enter 0. If there is a diluent, the functionality and molecular weight of the diluent is requested (line 1040). The necessary expectation values are computed (lines 1060-1150). 

Continuous distribution with the differential mass-distribution function of the form... 

Discrete distribution with the differential mass-distribution function of the form / (x) = a x(l -x)""". ... 

In some cases the molecular-weight distribution can be determined by turbi-dimetric titration, a technique which is based on the fractional precipitation. A precipitant is added to a very dilute solution of the polymer, and the resulting turbidity is measured as a function of the amount of added precipitant the preparative separation of the fractions is thereby avoided. If the polymer is chemically homogeneous, the mass distribution function can then be calculated. Tur-bidimetric titration is also suitable as a means for establishing the best fractionation conditions (e.g., choice of solvent/precipitant combination, size of fractions, etc.), before carrying out a full-scale fractionation by precipitation. 

Once the amounts and molecular weights of the fractions have been determined, the molecular-weight distribution of a polydisperse material can be expressed graphically in the form of a distribution curve. The mass distribution function is written as ... 

A polymer is the more uniform with respect to the molecular weight the steeper the integral distribution curve is. The differential mass distribution function... 

Fig, 2.17. Integral (1) and differential (2) mass distribution function of a polystyrene sample... 

Using Jl = dt)M + m again, the mass distribution function can be expressed in terms of the invariant radial displacement as... 

With the data given in Prob. 1, plot the number distribution function An/ (nT A log d) and the mass distribution function Am/(mT A log d) as a function of the logarithm of the particle diameter. Assume all particles within a size interval are spheres having a diameter equal to the midpoint of the size interval. The density of the particles equals 1 g/cm3. 

A distribution in more than one variable may be illustrated by the function describing the distribution of matter in a sphere with its center at the origin of a cartesian system. If the radius is c and density p, then the distribution of matter Pix y z) is a function of three variables with the definition that P Xjy,z) dx dy dz is the amount of matter in the element of volume bounded by the set of six cartesian planes through the points x,y,z) and (x + dx, y + dy, z + dz). The mass-distribution function P x,yyZ) is then equal to the density p, for all points satisfying the condition x + / + 2 c. If we are interested in the distribution of matter in only two dimensions, we can obtain a function P(x,y) from P(x,yjz) by integrating over all z ... 

Consider a physical property (such as the total Gibbs free energy G) of a continuous mixture, the value of which depends on the composition of the mixture. Because the latter is a function of, say, the mole distribution n(x), one has a mapping from a function to (in this case) a scalar quantity G, which is expressed by saying that G is given by afunctional of n(x). [One could equally well consider the mass distribution function m(x), and consequently one would have partial mass properties rather than partial molar ones.] We use z for the label x when in-... 

The light scattering methods provide statistically averaged quantities when applied to polydisperse samples (e.g., micellar or polymer solutions). The case of independent scatterers can be rigorously treated 2 by using the mass distribution function of the particles, f M). By definition, dm =f(M)dM is the mass of particles in the range between M and (M + dM), scaled by the total particle mass. As shown by Zimm, the scattering law in such a system can be presented similarly to the case of monodisperse particles (see Equation 5.405) ... 

In a real polydisperse system the particle radii, r, are distributed within some size range between rmin and rmax, and the fractional composition can be described by an appropriate mass distribution function,/(F) ... 

It is frequently useful to express bexl in terms of the aerosol mass distribution function... 

Liquid-liquid demixing in solutions of polymers in low molar mass solvents is not a rare phenomenon. Dembcing depends on concentration, temperature, pressure, molar mass and molar mass distribution function of the polymer, chain branching and end groups of the polymer, the chemical nature of the solvent, isotope substitution in solvents or polymers, chemical composition of copolymers and its distributions, and other variables. Phase diagrams of polymer solutions can therefore show a quite complicated behavior when they have to be considered in detail (see Ref la). 

In the foregoing, we have considered the heterodisperse system as composed of a number of distinct homodisperse fractions. However, as a rule, the distributions are continuous. This is approximated by taking - Mi(=dM) infinitesimally small n(M) is the number of particles having a molar mass between M and M + dM. The function n M) is called the (differential) number distribution function. Note that n(M) dM is a number and because M is expressed in kg mol", n M) has the dimension mol kg". In a similar way, w(M) dM is the mass of the particles having a molar mass between M and M + dM w(M) is the (differential) mass distribution function, having the dimension mole. 

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