The Molecular Weight Distribution of Polymer Samples - Journal of

Research: Science and Education

The Molecular Weight Distribution of Polymer Samples Arturo Horta* and M. Alejandra Pastoriza Departamento de Fisicoquímica (CTFQ), Facultad de Ciencias, Universidad a Distancia (UNED), 28040 Madrid, Spain; *[email protected]

In linear polymerization reactions, monomer molecules are bonded together forming macromolecular chains. In each of these chain macromolecules, the degree of polymerization is the number of monomers that are bonded in the chain, and the molecular weight, M, is equal to the degree of polymerization times the molecular weight of one monomer unit.1 The chains formed in a polymerization reaction are of different degrees of polymerization, such that the resulting polymer is a mixture of chains of different molecular weights, covering a range of M. The so-called “molecular weight” of a polymer sample is then an average over the values of M covered, each value being weighted according to its abundance in the sample. Such abundance, as function of M, constitutes the molecular weight distribution (MWD) of the polymer sample. The molecular weight averages and distribution that can be attained in a polymerization reaction depend on the characteristics of the reaction itself. This connection between the reaction and the product is one of the most interesting topics discussed in textbooks (1–11) and other references (12). It appears when studying step-growth polymerization, where monomers with two reacting groups condense together. The average molecular weight obtained in such reaction is related to the degree of conversion of the monomers and to the stoichiometric imbalance between the two reacting groups; see, for example, the generalized Carothers’ equation (3, p 25). The student learns that forcing high degrees of conversion is needed to obtain molecular weights in the polymer range, and that, to prevent too high molecular weights that may hinder processability, one tunes the stoichiometric imbalance. The connection between the reaction mechanism and the average molecular weight also appears when studying the chain reaction of addition polymerization. Here, the transfer of reactivity from a growing chain to other species in the medium limits the length of the final chains. To detect the presence of a transfer agent and to measure its efficiency (tranfer constant) one studies the decrease of average degree of polymerization on increasing the molar concentration of the tranfer agent: see, for example, the Mayo–Walling equation (3, p 53). Another interesting connection between mechanism of polymerization and average molecular weight is found in the free radical addition polymerization. In this case, the termination step of the chain reaction can occur through two different mechanisms: combination of two growing macroradicals or disproportionation of a single macroradical. The distribution of molecular weights of the resulting polymer is different depending on the prevailing mechanism, the MWD being narrower when termination is by combination than when it is by disproportionation (3, p 59). These topics are standard in general polymer courses. One makes use of the average molecular weight and the breadth of the distribution as a source of information about the reaction and its mechanism. These source data refer to the polymer as obtained in the polymerization. However, to www.JCE.DivCHED.org

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determine molecular weights, the polymer has to be isolated from the reaction medium. After the polymerization, the polymer is converted into a “sample”; that is, it is isolated and purified. The student learns about these steps in the laboratory, where the polymerization is followed by precipitation of the polymer, then drying, and, probably, re-dissolution and precipitation again. The isolation of the polymer from the reacting medium by precipitation is a necessary step to proceed further with the characterization of the sample. But, is the precipitated polymer identical to the polymer resulting from the polymerization? When studying thermodynamics of polymer solutions, the student learns that the solubility of the macromolecular chains composing a polymer decreases exponentially as the length of the chain grows (3, p 209). Thus, it is likely that the shorter chains in the polymer are, in fact, soluble enough that they do not precipitate with the other chains. Then, the isolated polymer sample lacks some of the shorter chains. Is this a problem? It depends. If the yield in polymer is close to 100% of the expected value, the quantity lost is certainly of little importance. But seldom is a 100% yield attained, more often, the yield is several percent units below the theoretical 100%. Low polymer yields are usually due to reasons other than poor precipitation: for example, low conversion of the monomer or deficient manipulation of the obtained polymer (in the transfer from vessels, removal from filters, etc.). These can be the main reason for the overall loss of polymer, but, in general, they do not alter the molecular weight distribution as differential precipitation does. Again, can a few percent loss of short chains affect the molecular weight averages and breadth of the distribution? As shown by the number distribution of Figure 1, the short chains are the more abundant in the distribution. Hence, these short chains not recovered may be a small fraction of the weight, but correspond to a much larger fraction in the number of chains, because, by solubility, the weight lost corresponds to the more numerous shorter chains. This problem is mentioned in Billmeyer (1, p 69), which is probably the most influential textbook written in English. Here, we propose a simple numerical exercise by which the students can find answers to the following questions: (i) Can we calculate the error in molecular weight averages committed when a fraction of the shorter chains, present in the original polymer coming from the polymerization but lost during the isolation steps, are no longer present in the polymer sample that is subject to characterization? (ii) Can this error vitiate the connection between molecular weight and polymerization mechanism that is so neatly established in textbooks and polymer courses? The task we pursue here is to develop simple answers that can be formulated with the material that the students learn in general polymer courses. First, we review the usual definitions of molecular weight averages and distribution, then we give a method to calculate the error, and, finally, we discuss its consequences.

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Definitions of Molecular Weight Averages and Distribution The most common graph for a molecular weight distribution is in terms of the weight fraction, W(M), as shown in Figure 1. The meaning of such weight distribution is that W(M )dM gives the fraction of mass of the total polymer sample whose molecular weight value, M, lies in the range from M to M + dM. Another way of expressing the molecular weight distribution of a polymer is through the number fraction, N(M ). Here, N(M)dM gives the number of molecules (polymer chains) whose molecular weight value is in the range from M to M + dM, expressing that number as fraction of the total number of molecules that compose the polymer sample. The characteristics of the molecular weight distribution of a polymer sample are unique, and thus, these two functions are two ways of expressing the same reality and are not independent, but related through M N (M ) (1) Mn – where Mn is the number average molecular weight of the polymer sample, namely, the mean M of all the macromolecules in the sample. This averaging among macromolecules is done by multiplying each M value by the number of macromolecules having it—actually, the fraction of total number, N(M )dM—and summing to all M values possible W (M ) =

∞

M N (M ) dM

Mn = 0

probability that a macroradical adds a newer monomer (propagation) instead of losing activity (termination), and is equated to the rate of propagation (vp) divided by the rate of propagation plus termination (vt), vp兾(vp + vt). Then, the change of variable y = ᎑ln p, and the approximations ᎑ln p ≅ 1 − p and X − 1 ≅ X (valid for large enough X ), convert the 2 equation for the most probable distribution into – W(M) = y M according to exp(᎑yM ) and, using this form to calculate M n – eq 2, identifies y as y = 1兾Mn, so that, the most probable distribution of molecular weights is W (M ) =

1 Mn2

M exp −

M Mn

(4)

This form of the most probable distribution W(M ), and the corresponding one for N(M ), deduced from it by applying eq 1 (which is a general relationship, valid for any form of the distribution), are the ones shown on Figure 1. According to the mechanisms of radical addition polymerization and of step-growth polymerization, the polymer should ideally obey this most probable distribution. Deviations from ideal polymerizations, because of transfer, competing terminations, branching, monofunctional agents, and so forth give distributions that cannot be represented by a single simple function. However, in most cases, the form of Figure 1 is a good representation of the weight distributions found, and the comparison– of experimental values of the – molecular weight averages, Mn and Mw, with those predicted theoretically from a kinetic mechanism is taught as a way of discerning the mechanism of the polymerization.

The function W(M ) is used more often than N(M ), because mass is more practical than number –of molecules; hence, the number average molecular weight, Mn, is obtained also from W(M), as ∞

Mn = 0

W (M ) dM M

−1

(2)

which is deduced from eq 1, by dividing both sides through M, and integrating. Similarly, the weight average molecular – weight, Mw, is the mean M per unit mass in the sample, which is obtained by multiplying M times the fraction of mass having that value, W(M )dM, and summing: ∞

M W (M ) dM

Mw =

(3)

0

The student is taught that the mechanisms of free radical polymerization and of step-growth polymerization both give the same mathematical form for the MWD, the one called most probable distribution (3, p 27). The function of this distribution is usually deduced from the mechanism of the polymerization with the general result W = (1 − p)2XpX−1 where X is the degree of polymerization X = M兾M0 (M0 is the monomer unit molecular weight) and p is a probability. In step-growth polymerization, p is the probability that a given reactive group has already reacted, which is equated to the degree of advancement or monomer conversion of the polymerization, α. In free radical polymerization, p is the 1218


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Figure 1. Typical form of the molecular weight distribution (MWD) of a polymer. N(M) is the number distribution (or number of polymer chains having molecular weight value M) and W(M) is the weight distribution (or mass of the polymer chains whose molecu– – lar weight is M). Mn and M–w are the average values – of molecular weight for the distribution (Mn, number average; Mw, weight average). The figure is drawn for the most probable distribution, as an example. This function provides an adequate mathematical representation of the molecular weight distribution over a wide range of M values, although it has the limitation that N(M) fails to predict a vanishing probability for the value M = 0 (as should be in real polymers), owing to assumptions in its derivation which are not valid at very low M.

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Weight Distribution of a Polymer “Sample” This form of W(M) is adequate for our purpose of teaching how to calculate the difference between the polymer as polymerized and the polymer as a sample. The most probable distribution has several advantages for this purpose: (i) it is derived in polymer textbooks; (ii) it is applicable in the most common types of polymerization (especially those most easily done in the teaching laboratory); and (iii) its mathematical handling is simple and accessible to the student (e.g., the calculations require only simple integrals of the functions e᎑kx, xe᎑kx, and x2e᎑kx as shown in textbooks). Once we have the form of the distribution, we use it twice: first, for the original polymer and, then, for the fraction of short chains that is lost when the polymer sample is isolated and purified. The distribution of the original polymer we call W(M ), and the distribution of the lost soluble fraction we call F(M–). The form of W(M) is the function given in eq 4, with Mn the value of the number average molecular weight corresponding to the original polymer. The form of F(M) can be found with the following strategy. Since F(M ) represents the shorter chains fraction of W(M ), it can have the same form of W(M ) but decaying to zero at much shorter values of M than W(M ). Hence, for the fraction,–we write the decaying exponential of eq 4 as exp (᎑f M兾Mn), where f is a multiplying factor, that is much larger than unity, in order to make F(M ) a small fraction containing just the shorter chains present in W(M). The smaller the fraction, the larger f has to be to ensure a faster decay of the distribution F(M ).

The precise value of this factor can be easily calculated from the size of the fraction, S. This is because the function F(M ) should integrate to the mass of lost polymer, not recovered in the sample, relative to the mass of original polymer. Let us call S the fraction of the original mass that is lost when isolating the sample. Then F(M ) integrates to the value S, namely, ∞

F (M ) d M = S 0

1 Mn

=

2 ∞

M exp 0

−f M dM = Mn

1 f

2

which allows us to obtain f in terms of S. The form of the MWD of the fraction is then F (M ) =

1 M 2 M exp − Mn Mn S1 / 2

(5)

The original distribution, W(M ), and the distribution of the fraction, F(M ), that corresponds to a given value of S are shown in Figure 2. The distribution of the sample that has lost the fraction S of the original polymer is W ⬘(M ) and is equal to the distribution of the original polymer, W(M ), less the distribution of the fraction, F(M ): W ′ (M ) =

W (M ) − F (M ) 1− S

(6)

The denominator arises because the mass of polymer in the sample is only (1 − S ) times the mass of the original polymer, and so, we have to divide by 1 − S to make the distribution W ⬘(M ) a normalized function, the same as is W(M ). The distribution of the polymer sample, W ⬘(M ) can be compared with that of the original polymer, W(M), as shown in Figure 2. We can see that the distribution of the polymer sample is displaced with its maximum moved to a higher value of molecular weight. The fraction of short chains not recovered in the polymer sample represents 5% of the original polymer. The displacement of the maximum increases with the quantity of lost fraction, S. Molecular Weight Averages of a Polymer “Sample”

Figure 2. Fraction of the original distribution that is lost when preparing the polymer sample and the resulting distribution after losing such a fraction. W(M) is the original polymer, F(M) the fraction that is lost (total weight S ), and W’ (M) the distribution of the resulting sample. Both W and W‘ are normalized to unity, while F – integrates to S. The number average molecular weight is Mn in the – original polymer and raises to Mn‘ in the sample that has lost the low molecular weight fraction S. The figure is drawn for the most probable distribution with a 5% loss (S = 0.05).

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We –now proceed to calculate the molecular weight av– erages, Mn⬘ and Mw⬘, that correspond to the polymer sample and to compare these averages of the – sample – with the same averages of the original polymer, Mn and M w. To this end, we use the molecular weight distribution for the sample, W ⬘(M), and substitute it into the standard definitions of the number and weight averages of molecular weight (eqs 2 and 3). The results of these integrations, using for W ⬘(M ) the function given in eq 6, are

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Mn ′ = Mn 1 + S 1 / 2

(7)

1 − S 3/ 2 1− S

(8)

Mw ′ = Mw

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The averages with primes correspond to the polymer sample and those without primes –to the original polymer. In Figure – 2 the difference between Mn⬘ and Mn is shown for a value of S. The differences of the polymer sample with respect to the original polymer obviously grow with the quantity of short chains lost, that is, with the value of S. The variations with S are shown in Figure 3, where the deviations in molecular weight averages of the polymer sample with respect to the original polymer, are plotted as function of the fraction lost, S. The number average molecular weight and the polydispersity – – index – are – shown in terms of the relative changes, Mn⬘兾Mn, Mw⬘兾Mw. The– results are clear. The number average molecular –weight, Mn⬘, increases, and the – polydispersity index, Mw⬘兾Mn⬘, decreases, as the size of lost fraction, S, grows. Both quantities are sensitive to the loss of low molecular weight chains, even if the loss represents – only a tiny fraction of the whole polymer. A growth in Mn⬘ if low molecular weights are lost was to be expected without any detailed calculations, – but the numbers in Figure 3 show how the deviations in Mn⬘ value start with a steep slope (parabola), such that, in the range where – S is only a small percent of the polymer, the changes in Mn⬘ are the more pronounced. For– a fraction representing 1% of the polymer, the increase in M n⬘ is 10%, for a 2% fraction the increase is 14%,– and for a fraction that is 5% of the polymer,–the increase in Mn⬘ reaches – 22%. The polydispersity index, Mw⬘兾Mn⬘, is also affected in greater measure in the range of small fractions, S. For a fraction representing only 1% of the polymer, the polydispersity index decreases by 8%, for a 2% fraction the decrease is 11%, and for a 5% fraction it is 15%. These numbers can be obtained by the students as a practical calculation exercise. The results should be discussed in class to evaluate whether the deviations are of enough magnitude to affect the discussion of some polymerization mechanisms. For example, how these deviations in average molecular weight between the polymer as originally synthesized and the sample isolated for characterization may – affect the followup of step-growth polymerization, where M n is re– lated to the degree of conversion, α, through Mn ∼ (1 − α)᎑1 (3, p 23) or may affect the evaluation of transferences in ad– dition polymerization, where differences in Mn with molar concentration of possible transference agents are used to determine the corresponding transference constants (3, p 53). Also discussion can focus on whether these deviations in polydispersity index affect the mechanistic interpretation of free radical polymerization where the distinction between termination by recombination or by disproportionation represents – – only a 25% difference in the Mw兾Mn value of the resulting polymer (3兾2 versus 2) (3, p 59). Number Distribution of a Polymer “Sample” The shape of the weight distribution, W(M ), is intuitive and usually causes no surprise to the student. It seems reasonable that, – – if the polymer has molecular weight averages Mn and Mw, most of the mass of the polymer lies in the proximity of the mean values and that the fraction of polymer mass should progressively decrease to finally vanish for M values much lower or much higher than the mean. It is a statistical curve that is easily understood. However, the problem arises when the graph for the distribution of molecular 1220


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– Figure 3. Deviation in the number molecular weight, Mn, – average – and in the polydispersity index, Mw/Mn, between the original polymer and the polymer sample that has lost a fraction of low molecular weight chains, S. Without primes, the original polymer; with primes, the polymer sample that has lost the low molecular weight fraction, S (for each S, the polymer and fraction are represented as in Figure 2).

weights is the number fraction, N(M ). The number of macromolecules that have a given value of M is not maximum in the proximity of the averages. And the number of macromolecules that have a given value of M does not decrease progressively as M gets lower than the averages; on the contrary, there are more and more macromolecules as the molecular weight gets smaller and smaller deviating from the averages. In other words, the shorter the polymer chain, the more abundant it is in the sample (Figure 1). This behavior appears striking to most students when they first see the number distribution, but they have to accept it because this form derives directly from the mechanism of the polymerization. The calculations presented here have the advantage that they can reconcile the rigorous mechanistic derivation with the naive intuition that opposes a function without maximum. This is so because such function without maximum corresponds only to the polymer obtained in the reaction, not to the polymer sample purified and isolated after the reaction. To show this, the student can obtain the number distribution of the polymer sample, which we call N⬘(M ), and compare it with the N(M ) of the original polymer as it is obtained in the polymerization. The number distribution of the polymer sample, N⬘(M), can be obtained from the weight distribution W ⬘(M), using the same relationship of eq 1, but now applied –with the number average molecular weight of the sample, Mn⬘. In Figure 4, N⬘(M) goes to zero as M decreases, contrary to the endless growth of the distribution in the original polymer. The vanishing of the number distribution for the sample, as M approaches zero, agrees with the first intuitive idea of students. With this calculation, they can also be assured that the samples they obtain in the teaching laboratory truly have the distribution that they first expected.

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Acknowledgment Financial support received from Ministerio de Educación y Ciencia, Spain, under project CTQ2004-05706. Note 1. Keeping with the most popular use in the polymer field, we use the symbol M for molecular weight instead of Mr, which is recommended by IUPAC.

Literature Cited

Figure 4. Comparison between the molecular weight distribution (MWD) of the original polymer and of the polymer sample that has lost a fraction of low molecular weight chains. The polymer sample is denoted with primes, the original polymer without primes. The difference is most notable in the number distribution, where N‘ starts at zero and passes through a maximum, while N starts at its highest value and has no maximum. The figure is drawn for the most probable distribution with a 5% loss (S = 0.05).

Concluding Remarks The results of this calculation help in understanding the real meaning of molecular weight averages and distribution determined on purified samples. The calculation can be a good numerical exercise to practice on molecular weight distributions and fractions of a given polymer. All the knowledge needed is contained in the material covered by a standard introductory course on polymers.

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1. Billmeyer, F. W., Jr. Textbook of Polymer Science, 3rd ed.; John Wiley and Sons: Chichester, U.K., 1984. 2. Cowie, J. M. G. Polymers: Chemistry and Physics of Modern Materials, 2nd ed.; Blackie and Son Limited: Glasgow, 1991. 3. Young, R. J.; Lovell, P. A. Introduction to Polymers, 2nd ed.; Chapman and Hall: London, 1991. 4. Painter, P. C.; Coleman, M. M. Fundamentals of Polymer Science: An Introductory Textbook, 2nd ed.; Technomic: Lancaster, PA, 1997. 5. Rempp, P.; Merrill, E. W. Polymer Synthesis, 2nd ed.; Hüthing and Wepf: Basel, 1991. 6. Munk, P. Introduction to Macromolecular Science; John Wiley and Sons: Chichester, U.K., 1989. 7. Allcock, H. R.; Lampe, F. W. Contemporary Polymer Chemistry, 2nd ed.; Prentice-Hall: Englewood Cliffs, NJ, 1990. 8. Rodríguez, F. Principles of Polymer Systems, 4th ed.; Taylor and Francis: New York, 1996. 9. Elias, H. G. Macromolecules, 2nd ed.; Plenum Press: New York, 1984; translated from German by J. W. Stafford. 10. Champetier, G.; Monnerie, L. Introduction a la Chimie Macromoléculaire; Masson et Cie: Paris, 1969. 11. Horta, A. Macromoléculas; UNED: Madrid, 1982 and 1991. 12. Horta, A. J. Chem. Educ. 1985, 62, 286–292.

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