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Molecular Weight of Polymers

Module by: Sehmus Ozden, Andrew R. Barron. E-mail the authorsEdited By: Andrew R. Barron

Introduction

Knowledge of the molecular weight of polymers is very important because the physical properties of macromolecules are affected by their molecular weight. For example, shown in Figure 1 the interrelation between molecular weight and strength for a typical polymer.

Figure 1: Dependence of mechanical strength on polymer molecular weight. Adapted from G. Odian, Principles of Polymerization, 4th edition, Willey-Interscience, New York (2004).
Figure 1 (Fig1.jpg)

The melting point of polymers are also slightly depend on their molecular weight. (Figure 2) shows relationship between molecular weight and melting temperatures of polyethylene (Figure 3). Most linear polyethylenes have melting temperatures near 140 °C. The approach to the theoretical asymptote, that is a line whose distance to a given curve tends to zero, indicative that a theoretical polyethylene of infinite molecular weight (i.e., M = ∞) would have a melting point of 145 °C.

Figure 2: The molecular weight-melting temperature relationship for the alkane series. Adapted from L. H. Sperling, Introduction to physical polymer science, 4th edition, Wiley-Interscience, New York (2005).
Figure 2 (Fig2.jpg)
Figure 3: Structure of polyethylene.
Figure 3 (Fig3.jpg)

There are several ways to calculate molecular weight of polymers like number average of molecular weight, weight average of molecular weight, Z-average molecular weight, viscosity average molecular weight, and distribution of molecular weight.

Molecular weight of polymers

Number average of molecular weight (Mn)

Number average of molecular weight is measured to determine number of particles. Number average of molecular weight is the total weight of polymer, Equation 1, divided by the number of polymer molecules, Equation 2. The number average molecular weight (Mn) is given by Equation 3, where Mi is molecular weight of a molecule of oligomer n, and Ni is number of molecules of that molecular weight.

Eq14.jpg
(1)
Eq13.jpg
(2)
Eq12.jpg
(3)

Example 1

Consider a polymer sample comprising of 5 moles of polymer molecules having molecular weight of 40.000 g/mol and 15 moles of polymer molecules having molecular weight of 30.000 g/mol.

Eq19.jpg
(4)
Eq16.jpg
(5)
Eq20.jpg
(6)

Weight average of molecular weight (Mw)

Weight average of molecular weight (Mw) is measured to determine the mass of particles. Mw defined as Equation 7, where Mi is molecular weight of a molecule of oligomer n, and Ni is number of molecules of that molecular weight.

Eq21.jpg
(7)

Example 2

Consider the polymer described in Example 1.

Eq22.jpg
(8)
Eq25.jpg
(9)

Exercise 1

Calculate the MW for a polymer sample comprising of 9 moles of polymer molecules having molecular weight of 30.000 g/mol and 5 moles of polymer molecules having molecular weight of 50.000 g/mol.

Solution

Eq27.jpg

Z-average molecular weight (Mz)

The Z-average molecular weight (Mz) is measured in some sedimentation equilibrium experiments. Mz isn’t common technique for molecular weight of polymers. The molar mass depends on size and mass of the molecules. The ultra centrifugation techniques employ to determine Mz. Mz emphasizes large particles and it defines the EQ, where Mi is molecular weight and Ni is number of molecules.

Eq28.jpg
(10)

Example 3

Consider the polymer described in Example 1.

Eq29.jpg
(11)
Eq32.jpg
(12)
Eq31.jpg
(13)

Exercise 2

Calculate the MZ for a polymer sample comprising of 10 moles of polymer molecules having molecular weight of 20.000 g/mol and 2 moles of polymer molecules having molecular weight of 25,000 g/mol.

Solution

Eq33.jpg

Viscosity average molecular weight (Mv)

One of the ways to measure the average molecular weight of polymers is viscosity of solution. Viscosity of a polymer depend on concentration and molecular weight of polymers. Viscosity techniques is common since it is experimentally simple. Viscosity average molecular weight defines as Equation 14, where Mi is molecular weight and Ni is number of molecules, a is a constant which depend on the polymer-solvent in the viscosity experiments. When a is equal 1, Mv is equal to the weight average molecular weight, if it isn’t equal 1 it is between weight average molecular weight and the number average molecular weight.

Eq34.jpg
(14)

Distribution of molecular weight

Molecular weight distribution is one of the important characteristic of polymer because it affects polymer properties. A typical molecular distribution of polymers show in Figure 4.. There are various molecular weight in the range of curve. The distribution of sizes in a polymer sample isn’t totally defined by its central tendency. The width and shape of distribution must be known. It is always true that the various range molecular weight is Equation 15. The equality is occurring when all polymer in the sample have the same molecular weight.

Eq35.jpg
(15)
Figure 4: A schematic plot of a distribution of molecular weights along with the rankings of the various average molecular weights. Adapted from J. A. Nairn, Oregon State University (2003).
Figure 4 (graphics17.jpg)

Molecular weight analysis of polymers

Gel permeation chromatography (GPC)

Gel permeation chromatography is also called size exclusion chromatography. It is widely used method to determine high molecular weight distribution. In this technique, substances separate according to their molecule size. Firstly, large molecules begin to elute then smaller molecules are eluted Figure 5. The sample is injected into the mobile phase then the mobile phase enters into the columns. Retention time is the length of time that a particular fraction remains in the column. As shown in Figure 5, while the mobile phase passes through the porous particles, the area between large molecules and small molecules is getting increase. GPC gives a full molecular distribution, but its cost is high.

Figure 5: Solvent flow through column. Adapted from A. M. Striegel, W. W. Yau, J. J. Kirkland, and D. D. Bly. Modern Size-Exclusion Liquid Chromatography- Practice of Gel Permeation and Gel Filtration Chromatography, 2nd Edition. Hoboken. N.J. (2009).

According to basic theory of GPC, the basic quantity measured in chromatography is the retention volume, Equation 16, where V0 is mobile phase volume and Vp is the volume of a stationary phase. K is a distribution coefficient related to the size and types of the molecules.

Eq36.jpg
(16)

The essential features of gel permeation chromatography are shown in Figure 6. Solvent leaves the solvent supply, then solvent is pumped through a filter. The desired amount of flow through the sample column is adjusted by sample control valves and the reference flow is adjusted that the flow through the reference and flow through the sample column reach the dedector in common front. The reference column is used to remove any slight impurities in the solvent. In order to determine the amount of sample, a detector is located at the end of the column. Also, detectors may be used to continuously verify the molecular weight of species eluting from the column. The flow of solvent volume is as well monitored to provide a means of characterizing the molecular size of the eluting species.

Figure 6: Schematic of gel permeation chromatography system.
Figure 6 (Picture 2.jpg)

As an example, consider the block copolymer of ethylene glycol (PEG, Figure 10) and poly(lactide) (PLA, Figure 8), i.e., Figure 9. In the first step starting with a sample of PEG with a Mn of 5,700 g/mol. After polymerization, the molecular weight increased because of the progress of lactide polymerization initiated from end of PEG chain. Varying composition of PEG-PLA shown in Table 1 can be detected by GPC (Figure 10).

Figure 7: The structure of polyethyleneglycol (PEG).
Figure 7 (Fig4.jpg)
Figure 8: The ring-opening polymerization of lactide to polylactide.
Figure 8 (Fig5.jpg)
Figure 9: The structure of PEG-PLA block copolymer.
Figure 9 (Fig6.jpg)
Figure 10: Gel permeation chromatogram of (a) PEG (MW = 5,700 g/mol) and (b) PEG-PLA block copolymer (MW = 11,000 g/mol). Adapted from K. Yasugi, Y. Nagasaki, M. Kato, K. Kataoka, J. Control. Release, 1999, 62, 89.
Figure 10 (Fig7.jpg)
Table 1: Characteristics of PEG-PLA block copolymer with varying composition. Adapted from K. Yasugi, Y. Nagasaki, M. Kato, and K. Kataoka, J. Control Release, 1999, 62, 89.
Polymer Mn of PEG Mw/Mn of PEG Mn of PLA Mw /Mn of block copolymer Weight ratio of PLA to PEG
PEG-PLA(41-12) 4100 1.05 1200 1.05 0.29
PEG-PLA(60-30) 6000 1.03 3000 1.08 0.50
PEG-PLA(57-54) 5700 1.03 5400 1.08 0.95
PEG-PLA(61-78) 6100 1.03 7800 1.11 1.28

Light-scattering

One of the most used methods to characterize the molecular weight is light scattering method. When polarizable particles are placed in the oscillating electric field of a beam of light, the light scattering occurs. Light scattering method depends on the light, when the light is passing through polymer solution, it is measure by loses energy because of absorption, conversion to heat and scattering. The intensity of scattered light relies on the concentration, size and polarizability that is proportionality constant which depends on the molecular weight. Figure 11 shows light scattering off a particle in solution.

Figure 11: Modes of scattering of light in solution.
Figure 11 (Picture 3.jpg)

A schematic laser light-scattering is shown in Figure 12. A major problem of light scattering is to prepare perfectly clear solutions. This problem is usually accomplished by ultra-centrifugation. A solution should be as possible as clear and dust free to determine absolute molecular weight of polymer. The advantages of this method, it doesn’t need calibration to obtain absolute molecular weight and it can give information about shape and Mw information. Also, it can be performed rapidly with less amount of sample and absolute determinations of the molecular weight can be measured. The weaknesses of the method is high price and most times it requires difficult clarification of the solutions.

Figure 12: Schematic representation of light scattering. Adapted from J. A. Nairn, polymer characterization, Material science and engineering 5473, spring 2003.
Figure 12 (Picture 5.jpg)

The weight average molecular weight value of scattering polymers in solution related to their light scattering properties that define by Equation 17, where K is the wave vector, that defined by Equation 18. C is solution concentration, R(θ) is the reduced Rayleigh ratio, P(θ) the particle scattering function, θ is the scattering angle, A is the osmotic virial coefficients, where n0 solvent refractive index, λ the light wavelength and Na Avagadro’s number. The particle scattering function is given by Equation 19, where Rz is the radius of gyration.

Eq37.jpg
(17)
Eq38.jpg
(18)
Eq39.jpg
(19)

Weight average molecular weight of a polymer is found from extrapolation of data in the form of a Zimm plot (Figure 13). Experiments are performed at several angles and at least at 4 different concentrations. The straight line extrapolations provides Mw.

Figure 13: A typical Zimm plot of light scattering data. Adapted from M. P. Stevens, Polymer Chemistry an Introduction, 3rd edition, Oxford University Press, Oxford (1999).
Figure 13 (Fig8.jpg)

X-ray scattering

X-rays are a form of electromagnetic wave with wavelengths between 0.001 nm and 0.2 nm. X-ray scattering is particularly used for semicrystalline polymers which includes thermoplastics, thermoplastic elastomers, and liquid crystalline polymers. Two types of X-ray scattering are used for polymer studies:

  1. Wide-angle X-ray scattering (WAXS) which is used to study orientation of the crystals and the packing of the chains.
  2. Small-angle X-ray scattering (SAXS) which is used to study the electron density fluctuations that occur over larger distances as a result of structural inhomogeneities.

Schematic representation of X-ray scattering shows in Figure 14.

Figure 14: Schematic diagram of X-ray scattering. Adapted from B. Chu, and B. S. Hsiao, Chem. Rev. 2001,101, 1727.
Figure 14 (Picture 6.jpg)

At least two SAXS curves are required to determine the molecular weight of a polymer. The SAXS procedure to determine the molecular weight of polymer sample in monomeric or multimeric state solution requires the following conditions.

  1. The system should be monodispersed.
  2. The solution should be dilute enough to avoid spatial correlation effects.
  3. The solution should be isotropic.
  4. The polymer should be homogeneous.

Osmometer

Osmometry is applied to determine number average of molecular weight (Mn). There are two types of osmometer:

  1. Vapor pressure osmometry (VPO).
  2. Membrane osmometry.

Vapor pressure osmometry measures vapor pressure indirectly by measuring the change in temperature of a polymer solution on dilution by solvent vapor and is generally useful for polymers with Mn below 10,000–40,000 g/mol. When molecular weight is more than that limit, the quantity being measured becomes very small to detect. A typical vapor osmometry shows in the Figure 15. Vapor pressure is very sensitive because of this reason it is measured indirectly by using thermistors to measure voltage changes caused by changes in termperature.

Figure 15: Schematic vapor pressure osmometry. Adapted from http://www.gallay.com.au/node/186
Figure 15 (Picture 13.jpg)

Membrane osmometry is absolute technique to determine Mn (Figure 16). The solvent is separated from the polymer solution with semipermeable membrane that is strongly held between the two chambers. One chamber is sealed by a valve with a transducer attached to a thin stainless steel diaphragm which permits the measurement of pressure in the chamber continuously. Membrane osmometry is useful to determine Mn about 20,000-30,000 g/mol and less than 500,000 g/mol. When Mn of polymer sample more than 500,000 g/mol, the osmotic pressure of polymer solution becomes very small to measure absolute number average of molecular weight. In this technique, there are problems with membrane leakage and symmetry. The advantages of this technique is that it doesn’t require calibration and it gives an absolute value of Mn for polymer samples.

Figure 16: Schematic representative of membrane osmometry. Adapted from http://www.flickr.com/photos/mitopencourseware/3327963527/.
Figure 16 (Picture 14.jpg)

Summary

Properties of polymers depend on their molecular weight. There are different kind of molecular weight and each can be measured by different techniques. The summary of these techniques and molecular weight is shown in the Table 2.

Table 2: Summary of techniques of molecular weight of polymers.
Method Type of molecular weight Range of application
Light scattering Mw
Membrane osmometry Mn 104-106
Vapor phase osmometry Mn 40,000
X-ray scattering Mw, n, z 102 to

Bibliography

  • G. Odian, Principle of Polymerization, 4th edition, Willey Intersicence, New York (2004).
  • M. P. Stevens, Polymer Chemistry an Introduction, 3rd Edition, Oxford University Press, Oxford (1998).
  • T. Tanaka, Experimental Methods in Polymer Science, Academic Press, New York (1999).
  • L. H. Sperling, Introduction to physical polymer science, 4th edition, Wiley-Interscience, New York (2006).
  • R. W. Nunes, J. R. Martin, and J. F. Johnson, Poly. Eng. Sci., 1982, 22, 4.
  • A. Horta and M. A. Pastoriza, J. Chem. Educ., 2007, 84, 7.
  • K. Yasugi, Y. Nagasaki, M. Kato, and K. Kataoka, J. Control. Release, 1999, 62, 89.
  • B. Chu and B. S. Hsiao, Chem. Rev., 2001, 101, 1727.
  • V. E. Eskin, Sov. Phys. Ups., 1964, 7, 2.
  • B. H. Bersted, J. Appl. Polym. Sci., 1973, 17, 1415.
  • A. H. Compton, Physics, 1925, 11, 303
  • C. H. Hsu, P. M. Peacock, R. B. Flippen, S. K. Manohar, and A. G. MacDiarmid, Synth. Met., 1993, 60, 233.
  • http://www.gallay.com.au/node/186
  • http://www.flickr.com/photos/mitopencourseware/3327963527/

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