Methods for Measuring Molecular Weights of Polymers

To determine the intrinsic viscosity, both inherent and reduced viscosities are plotted against concentration (C) on the same graph paper and extrapolated to zero. If the intercepts coincide then this is taken

Fig. 2.17 Cannon–Fenske capillary viscometer as the intrinsic viscosity. If they do not, then the two intercepts are averaged. The relationship of intrinsic viscosity to molecular weight is expressed by the Mark–Houwink–Sakurada equation [66]:

where K and a are constants. Various capillary viscometers are available commercially. Figure 2.17 illustrates a typical capillary viscometer. The logarithms of intrinsic viscosities of fractionated samples are plotted against log Mw or log Mn. The constants a and K of the Mark–Houwink–Sakurada equation are the intercept and the slope, respectively, of that plot. Except for the lower molecular weight samples, the plots are linear for linear polymers. Many values of K and a for different linear polymers can be found in the literature [66]. Actually, the Mark–Houwink–Sakurada equation applies only to narrow molecular weight distribution polymers. For low molecular weight polydisperse polymers this equation is useful, because the deviations due to chain entangle ment are still negligible. On the o ther hand, chain entanglement in high molecular weight polydisperse polymers affects viscosity and this equation does not really apply. The determinations of molecular weights of polymers rely, in most cases, upon physical methods. In some special ones, however, when the molecular weights are fairly low, chemical techniques can be used. Such determinations are limited to only those macromolecules that possess only one functional group that is located at the end of the chain ends. In place of the functional group, there may be a heteroatom. In that case, an elemental analysis might be sufficient to determine the molecular weight. If there is a functional group, however, a reaction of that group allows calculating the molecular weight. Molecular weights above 25,000 make chemical approaches impractical. In chemical determination each molecule contributes equally to the total. This determination, therefore, yields a number average molecular weight. With the development of gel permeation chromatography (discussed below), this method is hardly ever used today. There are various physical methods available. The more prominent ones are ebullioscopy, cryoscopy, osmotic pressure measurements, light scattering, ultracentrifugation, and gel permeation chromatography (also called size exclusion chromatography). All these determinations are carried out on solutions of the polymers. Also, all, except gel permeation chromatography, require that the results

Fig. 2.18 Membrane osmometer

of the measurements be extrapolated to zero concentrations to fulfill the requirements of theory. The laws that govern the various relationships in these determinations apply only to ideal solutions. Only when there is a complete absence of chain entanglement and no interaction between solute and solvent is the ideality of such solutions approached. A brief discussion of some techniques used for molecular weight determination follows Ebullioscopy, or boiling point elevation ,as well as cryoscopy ,or freezing point depression, are well known methods of organic chemistry They are the same as those used in determining molecular weights of small molecules.

The limitation to using both of these methods to determine the molecular weight of macromolecules is that ∆Tb and ∆Tf become increasingly smaller as the molecular sizes increase. The methods are limited, therefore, to the capabilities of the temperature sensing devices to detect very small differences in temperature. This places the upper limits for such determinations to somewhere between 40,000 and 50,000. The thermodynamic relationships for these determinations are:

The above two determinations, because each molecule contributes equally to the properties of the solutions ,yield number average molecular weights .How much this technique is used today is hard to tell. A method that can be used for higher molecular weight polymers is based on osmotic pressure measurements. It can be applied to polymers that range in molecular weights from 20,000 to 500,000 (some claim 1,000,000 and higher). The method is based on van’t Hoff’s law. When a pure solvent is placed on one side of a semi-permeable membrane and asolution on the other, pressure develops from the pure solvent side. This pressure is due toa tendency by the liquids to equilibrate the concentrations. It is inversely proportional to the size of the solute molecules. The relationship is as follow:

where p is the osmotic pressure, C is the concentration, T is temperature, and R is the gas constant, A2 is a measure of interaction between the solvent and the polymer (second viral coefficient). Astatic capillary osmometer is illustrated in Fig. 2.18. Rather than rely on the liquid to rise in the capillary on the side of the solution in response to osmotic pressure, as is done in the static method, a dynamic equilibrium method can be used. Here a counter pressure is applied to maintain equal levels of the liquid in both capillaries and prevents flow of the solvent. Different types of dynamic membrane osmometers are available commercially. The principle is illustrated in Fig. 2.19.

Fig. 2.19 Schematic of a high speed osmometer (from ref [68])

Fig. 2.20 Extrapolation to zero concentration , The results obtained from either method must still be extrapolated to zero concentration for van’t Hoff’s law to apply. Such extrapolation is illustrated in Fig. 2.20. Because all molecules contribute equally to the total pressure ,osmotic pressure measurements yield a number average molecular weight.

Light scattering measurement is a technique for determining the weight average molecular weight. When light passes through a solvent, a part of the energy of that light is lost due to absorption, conversion to heat, and scattering. The scattering in pure liquids is attributable to differences in densities that result from finite nonhomogenuities in the distribution of molecules within adjacent areas. Additional scattering results from a presence of a solute in the liquid. The intensity or amplitude of that additional scattering depends upon concentration, the size, and the polarizability of the solute plus some other factors. The refractive index of pure solvent and a solution is also dependent upon the amplitude of vibration. The turbidity that arises from scattering is related to concentration:

where n_o is the refractive index of the solvent, n is the refractive index of the solution, l is the wavelength of the incident light, No is Avogadro’s number, and c is the concentration. The dn/dc relationship is obtained by measuring the slope of the refractive index as a function of concentration. It is constant for a given polymer, solvent, and temperature and is called the specific refractive increment.

Fig. 2.21 A typical Zimm plot

Because scattering varies with different angles from the main beam of light, the results must be extrapolated to zero concentration and zero angle of scattering. This is done simultaneously by a method developed by Zimm. A typical Zimm plot is illustrated in Fig. 2.21. Avery popular technique for determining molecular weights and molecular weight distributions is gel permeation chromatography. It is also called size exclusion chromatography [69, 70]. The proce dure allows one to determine Mw and Mn, and the molecular weight distribution in one operation.

The procedure resembles HPLC. It separates molecules according to their hydrodynamic volumes or their effective sizes in solutions. The separation takes place on one or more columns packed with small porous particles. As the solution travels down the columns, there is retention of the polymer molecules by the pores of the packing .It was postulated in the past that the separation that takes place by molecular sizes is due to smaller molecules diffusing into all the pores while the larger ones only into some of the pores. Thelargestmoleculeswerethoughttodiffuseintononeoftheporesandpassonlythroughtheinterstitial volumes. As a result, polymer molecules of different sizes travel different distances down the column. Thismeansthatthemoleculesofthelargestsize(highestmolecularweight)areelutedfirstbecausethey f it into the least number of pores. The smallest molecules, on the other hand, are eluted last because they enter the greatest number of pores and travel the longest path .The rest fall in between .The process, however, is more complex than the above postulated picture. It has not yet been fully explained. It was found, for in stance ,that different gels display an almost identical course in the relation of dependence of VR (retention volume) to the molecular weight. Yet study of the pores of different gels show varying VR (retention volume) to the molecular weight. Yet study of the pores of different gels show varying cumulative distributions of the inner volumes. This means that there is no simple function correlating e volume and/or the size of the separated molecules with the size and distribution of the pores [69]. Also ,the shape of the pores that can be in ferred from the ratio of the area and volume of the inner pores is very important[70].Different models were proposed to explain these parathion phenomenon .These were reviewed thoroughly in the literature. They are beyond the scope of this book.

As indicated above, the volume of the liquid that corresponds to a solute eluting from the columns is called the retention volume or elution volume (VR). It is related to the physical parameters of the column as follows:

VR =Vo + KV1

where, Vo = the interstitial volume of the column(s) K = the distribution coefficient V1 = the internal solvent volume inside the pores The total volume of the columns is VT that is equal to the sum of VO and V1. The retention volume can then be expressed as follows:

VR =VO(1+K) +KV_T

Fig. 2.22 Molecular weight calibration curve for gel permeation chromatography

From the earlier statement it should be clear that polymer fractionation by gel permeation chroma tography depends upon the spaces the polymer molecules occupy in solution. By measuring, experi mentally, the molecular weights of polymer molecules as they are being eluted one obtains the molecular weight distribution. To accomplishthis, however, onemusthaveachromatographequipped with dual detectors. One must detect the presence of polymer molecules in the effluent. The other one must measure their molecular weights. Such detectors might be, for instance, a refractive index monitor and a low angle laser light scattering photogoniometer to find the absolute value of M. Manymolecular weight measurements, however, are done on chromatographs equipped with only one detector that monitors the presence of the solute in the effluent. The equipment must, therefore, be calibrated prior to use. The relationship of the ordinate of the chromatogram, commonly represented by F(V), must be related to the molecular weight. This relationship varies with the polymer type and structure. There are three methods for calibrating the chromatograph. The first, and most popular one, makes use of narrow molecular weight distribution reference standards. The second one is based upon a polydisperse reference material. The third one assumes that the separation is determined by molecular size. All three methods require that an experimentally established calibration curve of the relationship between the molecular size of the polymer in solution and the molecular weight be developed. A chromatogram is obtained first from every standard sample. A plot is then prepared from the logarithms of the average weights against the peak retention volumes (VR). The values of VR are measured from the points of injection to the appearances of the maximum values of the chromatograms. Above M1and below M4there is no effective fractionation because of total exclusion in the first place and total permeation in the second case. These are the limits of separation by the packing material.

To date the standard samples of narrow molecular weight distribution polymers that are available commercially are mainly polystyrenes. These samples have polydispersity indexes that are close to unity and are available over a wide range of molecular weights. For determining molecular weights of polymers other than polystyrene, however, the molecular weights obtained from these samples would be only approximations. Sometimes they could be in error. To overcome this difficulty a universal calibration method is used. The basis for universal calibration is the observation [51] that the multiplication products of intrinsic viscosities and molecular weights are independent of the polymer types. Thus, [n]M is the universal calibration parameter. As a result, a plot of log ([n]M) vs. elution volume yields a curve that is applicable for many polymers. The log ([n]M) for a given column (or columns), temperature, and elution volume is assumed to be a constant for all polymers. This is illustrated in Fig. 2.22.

Fig. 2.23 A typical digitized gel permeation chromatogram

Fig. 2.24 Schematic illustration of gel permeation equipment (the illustration only shows one sample column. Several sample columns are often used) (from ref [68])

Numerous materials have been used for packing the columns. Semi rigid crosslinked polystyrene beads are available commercially. They are used quite frequently. Porous beads of glass or silica are also available. In addition, commercial gel permeation equipment is usually provided with automatic sample injection and fraction collection features. The favorite detectors are refractive index and ultraviolet light spectroscopic detectors. Some infrared spectroscopic detectors are also in use. Commercially available instruments also contain pumps for high-pressure rapid flow and are usually equipped with a microcomputer to assist in data treatment. Also, they come with a plotter in the equipment to plot the detector response as the samples are eluted through the column or columns. Atypical chromatogram is illustrated in Fig. 2.23 and a schematic for the basic equipment is shown in Fig. 2.24. When polydisperse samples are analyzed, quantitative procedures consist of digitized chromatograms with indication of equally spaced retention volumes. These can be every 2.5 or 5.0 mL of volumes. The resultant artificial fractions are characterized by their heights hi, their solute concentrations Ci, and by the area they occupy within the curve Ai. The cumulative polymer weight values is calculated according to:

After conversion of the retention volumes Vi into molecular weights (using the calibration curve), the molecular weights, Mw, Mn, and Mz can be calculated:

If the chromatogram is not equipped with a microcomputer for data treatment, one can easily determine results on any available PC. Programs for data treatment have been written in various computer languages. They are available from many sources. Recently, there were several reports in the literature on combining size exclusion with high pressure liquid chromatography for more comprehensive characterization of polymers. Thus, Gray et al. reported that a combination of high pressure liquid chromatography with size exclusion chromatography allows comprehensive structural characterization of macromolecules [71]. On the other hand, Chang et al. reported on using a modified form of high pressure liquid chromatography analysis, referred to as interaction chromatography for polymer characterization. The process utilizes enthalpic interaction of polymeric solutes with the stationary phase. Such interaction depends on both, the chemical composition and on molecular weight. It is claimed to be less sensitive to chain architecture and to offer superior resolution to SEC. The typical HPLC instrument is modified to precisely control the temperature of the column. The temperature of the column and the mobile phase is controlled by circulating a fluid through the column jacket from a programmable bath/circulator. The mobile and stationary phases require careful choices to adjust the interaction strength of the polymer solutes with the stationary phase so that the polymer solutes elute out in a reasonable elution time. The process depends upon variations of the column temperature for precise control of the solute retention in the isocratic elution mode. Mixed solvent system of a polar and a less polar solvent are often employed to adjust the interaction strength [72].

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مواضيع ذات صلة

Radius of Gyration

Solutions of Polymers

Orientation of Polymers

The Mesomorphic State, Liquid Crystal Polymers

Kinetics of Crystallization

Thermodynamics of Crystallization

Autoacceleration

Ceiling Temperature

Effect of Reaction Medium

Steric, Polar, and Resonance Effects in the Propagation Reaction

Propagation

Capture of Free Radicals by Monomers