Joseph T. Bailey, W. M. Beattie, ond C. ~ 0 0 t h Shell Chemical Company Torrance, California
I
I
Average Quantities in Colloid Science
Macromolecules differ from the substances more familiar t o the chemist in their large and greatly varied sizes. For instance, the molecular weights in a sample of polymer may range from lo2 to lo6. The problem of characterizing molecules of large and various sizes has led to the development of appropriate experimental methods, many of them peculiar to this field. The need to express experimental results concisely, meaningfully, and clearly, and to reconcile data obtained by experimental techniques based on different physical principles, has led t o the common use of average quantities. The two developments have been complementary, and an appreciation of the nature of the averages involved is as important t o understanding t,he experimental results as the specialized experimental technique is to obtaining them. A similar state of affairs is found in the field of colloid chemistry, where, for instance, the diameters of the particles in a sample of pigment may range from 0.1 to 10 microns. The problems of size and inhomogeneity with respect to size are evident here also, and the methods of experimentation and data evaluation are often very similar to those used in the case of macromolecules. The purpose of this paper is t o describe some of the averages and the averaging processes which are in common use in the field of colloids and macromolernles. We first show how the averages can be defined in a consistent and precise manner. Second, we show how the experimental method used t o measure a quantity often determines the nature of the average obtained. Last, we point out certain ambiguities which exist in the use of average quantities. ~
~~
Deflnition of Averages
I n general, an average may be defined in terms of a weighting factor, p, and the quantity being averaged, Q, by the expression L' PI&
b, > D, > DL> b , . This result is illustrative of a general relationship which is true whatever the quantity being averaged. The differences between the vaiious averages are greater for more heterogeneous or broader distributions and it is a common device to measure two averages, for instance, the weight- and number-averages, and by comparing the values gain some insight into the polydispersity of a sample. The Measurement of Average Molecular Weights a n d Particle Sizes
The simple process of summation illustrated in the preceding section is not usually the may in which an average is determined. Quite often the experimental procedure used in measuring a quantity involves an averaging process. We can, for convenience, distinguish two distinct processes by which averages are measured: (1) those averages which are determined entirely by the experiment, i.e., physical averaging processes; (2) those averages which are determined by the summation of data obtained on fractions of the sample which have been physically separated or individually classified on the basis of the quantity being averaged. In the following paragraphs severalmethods of determination, of both types, are illustrated. Physical Averaging Processes
the averages were obtained by making the substitution w , = n,DISand are given in Table 5 with the resulting Table 5.
The Values of the Average Diameters of the Spheres of Toble 4
Average
Working definition
Value
Number (D.)
Z n 9 i l z ni
2.48
Linear (&) (diameter)
F
niDia/z n P 3
3.13
Surface (D.)
niDiP/Z niDi2
3.85
Weight (Dm)
niDi4/z n 9 i a
4.50
values of the averages. Figure 1 illustrates the position of the averages with relation to the distribution. It is
M, by Omnometry. When a polymer solution is separated from pure solvent by a semipermeable membrane (i.e., a membrane permeable to solvent but not to polymer) there is a flow of solvent through the membrane in the direction necessary to dilute the solution. The reverse pressure which will stop this flow completely can be measured and is the osmotic pressure, II, of the solution. The relation between II and the number of moles in solution, n, was demonstrated by van't Hoff and expressed by the equation nV = nRT (1) where V is the volume of the solution, R the gas constant, and T the absolute temperature. If the total weight of polymer in solution is w , then the concentration of polymer, c, is equal to w/V and equation (1) can be rewritten
For a polydisperse polymer the equalities n w
= 2
= 2
n , and
w , are obvious, and we have already noted that
w, = n,M,; thus, the quantity w / n is recognized as the number-average molecular weight 2 n,M J 2 n,. This simple treatment assumes that the solution is ideal, as can be seen from the similarity of equation (1) to the equation of state of a perfect gas. In fact, polymer solutions are never ideal and equation (2) must be evaluated a t infinite dilution. In practice the value of (II/c) a t infinite dilution, ( I I / C ) ~is, obtained by extrapolat.ion from higher concentrations 'ud the equation used to calculate AT, is DIAMETER
Figure 1. Average diameters and number dirtribvtion of spheres of varying diameter.
1
nsi. = RT/(ll/c)o AT, by Light Scattering. When light is shone upon a Volume 39, Number 4, April 1962
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197
polymer sohtion (or a swpension of particles) it is scattered and the intensity of t,his scattered light can he measured. Since the refractive index of the polymer molecule is close to that of the surrounding medium, the theory of Rayleigh, Gans, and Debye can be applied. This t,heory predicts that the intensity of light,, Is, scattered at an angle 8 to the continuing incident beam is related to the molecular weight, M, of the polymer by the equation ID = KcM
(3)
where c is the concentration of polymer and K is a constant depending upon the refractive indices of the solvent and the polymer, the wavelength of the light, and the angle 8. (For large molecules this eq~at~ion is only correct a t 6 = 0, i.e., for light scattered in the direction of the continuing incident beam. The intensity I8is found by measuring l a at several angles and extrapolating these values to zero angle.) For a polydisperse:polymer, equation (3) becomes
In practice, corrections must he made for the effect of p d p w vmwmtn~rion (srdirnenti~tion rnt*nsnrrmenti :Ire nut t~xtrmolntrdto infinite tlilutiolnl. , For a monodisperse polymer the same value of M will be obtained by applying equation (6) or (7) at any level in the cell. When a polydisperse polymer is centrifuged, each individual species forms an independent equilibrium concentration distribution for which equation (5) will be true. In this case the total polymer concentration distribution is such that different values of M are obtained by applying equations (6) or (7) at different levels in the cell. According to the method used to analyze the data, different types of average molecular weight can be obtained. For example, when the concentration gradient a t equilihrium is measured a t the ends of the cell (r = a and r = b) by determination of the refractive index (n) gradient, an average molecnlar weight is given by: ~
~~
~
Z ciMi
Ie = K Z ciMi
=
Kc
' 24
or, since ciis:proportional to w,,
in which we recognize the weight-average molecular weight, Mu. In practice equation (4) is used in the form where (Kc/Ia) is evaluated at infinite dilution (c.f. osmometry) and zero angle (for large molecnle~only). I& by Sedimentation Equilibrium. When a polymer dissolved in a solvent of different density is rotated in an ultracentrifuge cell, the polymer molecules move toward one end of the cell. At sedimentation equilibrium, the tendency of the polymer to migrate is exactly balanced by the tendency of the polymer to remix with the solvent by diffusion. The relationship between the molecnlar weight of the polymer and the equilibrium concentration gradient is given by the equation of Goldberg as
(where T is distance from the center of rotation, c is the concentration, and K is a constant peculiar to the system). The concentration gradient (dc/dr) can be measured by a refractometric optical method, and the concentration a t any point in the cell can be calculated from dc/dr and the original concentration of polymer in solution. The molecular weight can be calculated from the integrated form of equation (5), i.e.,
or, since
and it is not difficult to show that this average is, in fact, Ms though the formal algebra is cumbersome and it is not reproduced here (1). The name 2-average was first used to describe this average because the displacement of a scale viewed through the ultracentrifuge cell was used to measure dn/dr, and this scale displacement was denoted by 1. b, by Sorption. The surface area of colloidal particles may be obtained from measurements of the sorption of certain snbetances on the surface of the particles. The most commonly wed substances are radioactive isotopes, fatty acids, dyes, and gases. The total surface of the particles in a sample, s, determined in this way can be divided by the total volume (v) of particles in the sample (the total weight divided by the density) to obtain a quantity known as the surface-to-volume ratio: .=-
u
nisi Zn;Vt
In the case of spherical particles
(D,being the diameter of the particle) and the surfaceaverage particle diameter is obtained from the surface to volume ratio as follows:
Particles of other shapes yield more complex averages. by Ultramicroscopy. Particles which are too small to he resolved mith a conventional optical microscope can be seen with an ultramicroscope as bright spots in a dark field. One can count the number of spots in a given volume but one cannot measure the size of the particles. The concentration (weight) of particles in the sample is known and the average particle weight is calculated from the ratio:
is the number-average ljarticle weight. 198
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Journal of Chemical Education
Pwby Turbidimetry. For small particles (diameter less than '/lo of the wavelength of the incident light) of any refractive index, the turbidity, T , of a dilute suspension due to light scattering is related to t,he particle volume, V , by the Rayleigh equation: where K is a constant depending upon refractive index and wavelength, and n is the number of particles per unit volume. For a polydisperse polymer this equation becomes 7
=
K Z niV?
For particles of uniform density, ( c = g/ml) is given by
p,
the concentration
and an average particle volume can be calculated from the relation
Since w, is proportional to n,V,, the average is the weight-average particle volume, i.e.
In order to determine a molecular weight of a polymer from the intrinsic viscosity, a prior calibration is required. This will normally consist of preparation of a number of linear homologs of narrow molecular weight distribution and determination of their intrinsic viscosities, Irl], and their molecular weights, M, by an absolute method. On plotting log [ q ] against log M a straight line is obtained whose intercept and slope respectively yield the constants K and a in the empirical relationship: [nI = K M a If this relationship is used to calculate a viscosityaverage molecular weight, ATo, from the intrinsic viscosity of a polydisperse polymer, the nature of M , can be seen hy substituting for [ q ] in equation (8), giving:
-
(MsP =
Z w;Mi' Z "A
For most polymer solutions the exponent a lies in the range 0.5 to 0.8. Thus, 2"< < AZw and M , lies closer to am than to
a*.
Fractionation Processes
a.by V i s c m e t r y .
The greatly increased viscosity, v, resulting from the dissolution of even low concentrations of polymer in a liquid, forms the basis for a very widely used method for determining molecular weight. However, the method is not absolute and requires calibration against one of the absolute methods described above. At vanishingly small concentrations each polymer species in a polymer solution will make an independent contribution to the increase in viscosity and the overall increase in viscosity will he the sum of these contributions, i.e.,
and by the concentration, c, Dividing by q,,,., and stating the limiting concentration condition we obtain :
n [%I, , -1
=
[Z[(nr)r-lI c
The intrinsic viscosity,
[?I,
where , ,=
Ic+o
n8dven~
of the polymer is defined by:
which upon substitution in the previous equation gives: f: ci[nli
[?I
=
The fractionation methods yield data which describe the distribution of the measured quantity among the species. In this article we are concerned only with the calculation of average quantities from the data and clearly, since we know something about the distribution, more than one average can be computed. However, one particular average, which we shall call the primary average, can be calculated directly from the experimental data and so obtained far more accurately than the others. This one average is emphasized in the descriptions which follow. b. by Microscopy. The sample is examined under the optical or elect,ron microscope and the particles are classified according to their projected diameters (or projected areas). The data obtained are the number of particles in each class (n,)and the mid-point diameter of each class interval (Do),. The average diameter is calculated from the equation
7
and since c
