- Email: [email protected]
Fitting molecular weight distributions using a log-normal distribution model
European Polymer Journal xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
European Polymer Journal journal homepage: www.elsevier.com/locate/europolj
Fitting molecular weight distributions using a log-normal distribution model Michael J. Monteiro Australian Institute for Bioengineering and Nanotechnology, The University of Queensland, Brisbane, QLD 4072 Australia
a r t i c l e
i n f o
Article history: Received 11 December 2014 Received in revised form 26 December 2014 Accepted 11 January 2015 Available online xxxx Keywords: Size exclusion chromatography ‘Living’ radical polymerization Molecular weight distribution Log-normal distribution
a b s t r a c t ‘Living’ radical polymerization has opened the way to producing complex polymer architectures, including cyclic polymers, dendrimers, stars and many other structures. A major issue in the field of polymer science is the characterization of these polymer products, especially when made to high molecular weights, well beyond the accuracy of NMR or MALDI. Questions of purity arise due to the amount of side products (e.g. dead polymer through termination or cyclic purity). Here, we use the log-normal distribution model to fit distributions of all products to the overall experimental distribution. The fits in all cases were excellent. The log-normal distribution model was simple to implement as it only relies on a two-parameter fit, in which these two parameters, the standard deviation and median, were obtained from the number-average molecular weight and the polydispersity of the molecular weight distribution. This type of analysis can play a role in elucidating the side product distributions, leading to new or modified mechanistic insights. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction ‘Living’ radical polymerization (LRP) has provided unprecedented access to polymers of a wide range of chemical compositions and molecular weight distributions (MWDs) [1–5]. Control of the MWD in, for example, reversible addition-fragmentation chain transfer (RAFT) polymerization is through the reactivity of the RAFT chain transfer agent (CTA) [6]. A CTA with a high chain transfer constant (>10) will produce a linear increase in the number-average molecular weight (Mn) with conversion and a narrow MWD as found from the low values of the polydispersity index (PDI) below 1.1. On the other hand, a CTA with a low chain transfer constant (61) will result in a relatively constant Mn with conversion, in which Mn is close to that determined from theory at 100% conversion. The MWD was broad with a PDI close to 2. The PDI can be controlled to any desired value between 1.1 and 2 by using a E-mail address: [email protected]
single difunctional and highly reactive RAFT agent with varying amounts of initiator [7]. The modeling of the MWD when using a MacroCTA to produce diblock copolymers can provide insight into the mechanism and the percentage of MacroCTA remaining [8], and it would be useful to determine this information from the experimental MWD. The introduction of ‘click’ reactions into polymer chemistry [9] coupled with the high chain-end functionality of polymers made by LRP has provided a method to create a wide range of polymer architectures, including welldefined stars [10], dendrimers [11–14], hyperbranched polymers [15], multiblock copolymers [16], and even cyclic [17–20] structures. Characterization of these polymers becomes increasingly difficult with an increase in architectural complexity. This leaves size exclusion chromatography (SEC) as the only recognized and accessible method of characterization. The ring-closure method is one of the most utilized method to produce cyclic polymers from a difunctional linear analog [17–21]. All publications claimed
http://dx.doi.org/10.1016/j.eurpolymj.2015.01.009 0014-3057/Ó 2015 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Monteiro MJ. Fitting molecular weight distributions using a log-normal distribution model. Eur Polym J (2015), http://dx.doi.org/10.1016/j.eurpolymj.2015.01.009
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M.J. Monteiro / European Polymer Journal xxx (2015) xxx–xxx
the production of cyclic polymers by the ring-closure procedure based on the reduction of the hydrodynamic volume in the MWD determined by refractive index SEC. It would be useful to determine the percentage of other species (i.e. uncyclized starting linear or multiblock products). The purity of cyclic plays an important role in determining the effects of cyclic on the physical properties of the polymer [22]. In our previous paper [23], we derived the MWDs from SEC, and provided a method of analysis to gain greater insight into the mechanisms of LRP. The aim of this paper is to provide a quantitative method to determine the MWD from SEC that consists of the addition of many narrow MWDs. With this knowledge, one can use this method to determine, for example, the amount of dead polymer in an LRP, the ratio of linear to cyclic species in a ring-closure method, or the quantity of number of arms attached to a dendritic structure. The method utilizes a two-parameter equation to produce a log-normal distribution (LND) that describes the MWD of a polymer product. The two parameters can be readily determined from the SEC Mn and PDI values. Since this is only a two parameter fit, the fitting to real experimental data is limited to a MWD that consists of the addition of Gaussian MWDs. Thus the LND method allows quantification of structures produced from the combination of LRP and ‘click’ reactions. 2. Results and discussion
2
expððM MÞ =2r2 0:5
ð2pr2 Þ
wðln MÞ ¼
2 expð ln M ln M =2r2 ð2pr2 Þ
0:5
ð2Þ
or
wðMÞ ¼
2 expð ln M ln M =2r2 Mð2pr2 Þ
0:5
ð3Þ
The median value, M, is related to the Mn and the weightaverage molecular weight (Mw) by the following equations.
Mn ¼ Mexpðr2 =2Þ
ð4Þ
Mw ¼ Mexpðr2 =2Þ
ð5Þ
PDI ¼ expðr2 Þ
ð6Þ
and, from Eqs. (4) and (5), we obtain
The most commonly used distribution functions to describe a polymer made by LRP are the Poisson and Gaussian distributions [24]. The Poisson distribution is a one-parameter equation, and as such the breath of the distribution is predetermined to be quite narrow. This assumption becomes valid when the experimental MWD has a PDI less than 1.05 (or even lower depending on the molecular weight). However, there are many cases where the MWD can be greater than 1.1 and even increase to 2 for a LRP [6,25]. The Gaussian (or normal) distribution is a two-parameter equation that represents a symmetrical (bell-shaped) distribution about the median. The two parameter are the median and the standard deviation, and are used in the following equation to obtain a number distribution [24]:
nðMÞ ¼
the Gaussian (or normal) distribution (i.e. Eq. (1)) is that it cannot describe a broad distribution [24]. The reason is that as the MWD becomes broader through an increase in r, negative M values (in Eq. (1)) will be found. Obviously, negative values of molecular weight do not exist. Converting to the log-normal distribution overcomes this problem and still allows use of the same two parameters as in Eq. (1). This is accomplished by assuming that the weight distribution, w(M), is symmetrical about the median on a ln M abscissa (i.e. x-axis). Thus Eq. (1) can be transformed as follows:
ð1Þ
where M is the molecular weight, M is the median molecular weight of the distribution, r is the standard deviation. The standard deviation is defined as half width of the distribution at half height. The same nomenclature will be used as found in our pervious paper [23]. In polymer science there is the misconception that the PDI value represents the breath of the distribution [26]. What the PDI value tells us is the deviation of a polymer distribution from a distribution in which all the chains are of equal length (i.e. a PDI = 1). As the MWD broadens, the PDI increases to values greater than 1. As shown below there is an equation based on the log-normal distribution equation that equates r to the PDI. The main problem with
M ¼ ðM n M w Þ0:5
ð7Þ
The maximum (i.e. peak maximum) of the distributions are located at
Mp;w ¼ M n3=2 M 1=2 w
ð8Þ
Mp;n ¼ Mn5=2 M 3=2 w
ð9Þ
where Mp,w and Mp,n are the peak maximums for the weight and number distributions, respectively. To produce a weight distribution, the Mn and Mw (or PDI) values are substituted into Eqs. (6) and (7) to obtain the two parameters M and r required for Eq. (3). The area under the distribution from using Eq. (3) should equal 1. We can convert the weight distribution to either a number (n(M)) or log-weight distribution (x(M)) using the transformations given in our previous paper [23]. It should be noted that when a polymer is fractionated, the MWD becomes skewed and thus may no longer represent a lognormal distribution. 2.1. Case study 1: fitting a polymer made by ATRP using the log-normal model The atom radical transfer polymerization (ATRP) of styrene usually produces polystrene (PSTY) with a narrow MWD. We made PSTY by ATRP with an Mn of 3490 and a PDI of 1.088, and its MWD from SEC was given by curve a in Fig. 1. The weight (w(M)) was used as the ordinate and log M as the abscissa. It can be see that curve a was relatively symmetrical about the peak maximum (or median) of the distribution, suggesting that the log-normal
Please cite this article in press as: Monteiro MJ. Fitting molecular weight distributions using a log-normal distribution model. Eur Polym J (2015), http://dx.doi.org/10.1016/j.eurpolymj.2015.01.009
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M.J. Monteiro / European Polymer Journal xxx (2015) xxx–xxx
0.0005
(a) Expt 0.00045
(c)
0.0004 0.00035
w (M)
0.0003 0.00025 0.0002 0.00015
(b) 0.0001 0.00005 0 3
3.2
3.4
3.6
3.8
4
4.2
log M Fig. 1. Curve a (blue line): SEC data for the ATRP of styrene (Mn = 3490, PDI = 1.088). Curve b (red line): log-normal distribution (Eq. (3)) using Mn = 3490, PDI = 1.088. Curve c (green line): log-normal distribution (Eq. (3)) using two distributions (Dist 1: Mn = 3490, PDI = 1.062, and Dist 2: Mn = 6980, PDI = 1.062). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
distribution model can be used. Curve b utilized the Mn and PDI above to determine the two parameters M and r required to generate a MWD from Eq. (3). It can be seen that the log-normal fit gave quite a poor fit. The broader MWD from the log-normal distribution was a direct result from a small population of a higher molecular weight distribution to that of the main ATRP polymer. This higher molecular weight distribution formed through radical– radical bimolecular termination. Curve c represents the combination of two distributions; one from the main population but with a lower PDI value of 1.062, and the second with double the Mn and a PDI of 1.062. The final fitted distribution was calculated using Eqs. (3) and (10).
wðMÞfinal ¼
X wp;i wðMÞi
ð10Þ
i
where wp,i is the weight fraction of distribution i and w(M)i is the weight at molecular weight M. The fit is excellent and allows us to quantify the amount of dead polymer formed during the polymerization, which is 5.7 wt%. Two pieces of information can be gained from this analysis: (i) the polymer with halide end-groups has a very narrow MWD, and (ii) the amount of dead polymer can be accurately determined. It should be noted that a good choice of the baseline using the SEC software is essential to obtain good fits. 2.2. Case study 2: fitting the log-normal distribution for a polymer after cyclization Cyclization of linear polymers with two different functional chain-ends (e.g. –PSTYn–N3) that can undergo a coupling reaction via the ring-closure method should be carried out under dilute conditions [18,27]. The reason for this is that there is competition of the chemical
end-groups not only with that of the same chain but also to that from other chains (making multiblocks). It is quite difficult from SEC to determine whether or not 100% cyclic polymer has formed. Using RI–SEC to obtain MWDs before and after cyclization shows that there is a distinct shift in hydrodynamic volume (Fig. 2A). This shift for PSTY has been shown to be approximately 0.75 due to cyclic’s more compact structure and fewer degrees of freedom [22,28]. Fig. 2A shows the linear difunctional polymer (–PSTYn–N3, Mn = 3490, PDI = 1.088, curve a) that after cyclization produces a cyclic c-PSTY (curve b, Mn = 2650, PDI = 1.305). The linear PSTY is the same as that used in Fig. 1. It is found that the c-PSTY has a much greater PDI compared to linear PSTY because of the terminated product and unreacted linear PSTY. Using the same PDI value as the linear PSTY (Fig. 1, curve c) found from the log-normal model, the fit using a cyclic distribution with Mn = 2650, PDI = 1.062 (i.e. 94.3 wt%), starting linear PSTY (close to 0 wt%) and termination product (5.7 wt%) gave an excellent fit to the experimental data (curve c in Fig. 2A). To get close to pure cyclic, the MWD can easily be fractionated by using a preparative SEC system. The hydrodynamic volume (HDV) shift based on the log-normal fitted Mn’s was 0.76, which is very close to theory. There are many examples in the literature where the shift in hydrodynamic volume is close to 0.9, and the authors claim to have produced a high percentage of cyclic polymer. In Fig. 2B, we show theoretically using the lognormal model the cyclic purity at different hydrodynamic volume shifts, assuming that a shift of 0.75 is that of pure cyclic product. The PSTYnN3 distribution (Mn = 3490, PDI = 1.088, curve a) is added to the pure c-PSTY distribution (Mn = 2620, PDI = 1.062) at different weight ratios to obtain a final distribution in which its Mp,w (final)/Mp,w (linear) equals the desired HDV. The final distribution can
Please cite this article in press as: Monteiro MJ. Fitting molecular weight distributions using a log-normal distribution model. Eur Polym J (2015), http://dx.doi.org/10.1016/j.eurpolymj.2015.01.009
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M.J. Monteiro / European Polymer Journal xxx (2015) xxx–xxx
(A) 0.0007 0.0006
(b) 0.0005
(a)
w (M)
0.0004 0.0003
(c) 0.0002 0.0001 0 3
3.2
3.4
3.6
3.8
4
4.2
log M
(B)
1.2
(d) (c) (b) 1
(a) starting linear
w (M)
0.8
(e) Pure cyclic
0.6 0.4 0.2 0 3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
log M Fig. 2. (A) Cylcization of –PSTYn–N3 to c-PSTYn. curve a (blue line): experimental SEC distribution for –PSTYn–N3 (Mn = 3490, PDI = 1.088); curve b (green line): experimental SEC distribution after cyclization (c-PSTY, Mn = 2650, PDI = 1.305); curve c (red line): log-normal model fit to c-PSTY using 3 distributions (Dist 1: c-PSTY, Mn = 2650, PDI = 1.062, 94.3 wt%; Dist 2: Mn = 3490, PDI = 1.062, 0 wt%; Dist 3: linear terminated product, Mn = 6980, PDI = 1.062, 5.7 wt%). (B) Height normalized log-normal distributions with varying shifts in hydrodynamic volumes (HDV). Curve a: HDV = 0, curve b: HDV = 0.95, curve c: HDV = 0.85, curve d: HDV = 0.80, curve e: HDV = 0.75. The –PSTYn–N3 distribution (Mn = 3490, PDI = 1.088, curve a) is added to the pure c-PSTY distribution (Mn = 2620, PDI = 1.062) to produce the final distribution. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
be determined using Eqs. (3) and (10). Curve a represents the linear polymer before cyclization (i.e. 100 wt% linear) with HDV equal to 1. Curve b represents the final distribution to give a hydrodynamic shift of 0.95, which was fit using a 80:20 wt% ratio of linear to pure cyclic. Decreasing the hydrodynamic volume to 0.85 (curve c) was fit using a 50:50 wt% ratio of linear to pure cyclic. A further decrease in the hydrodynamic volume to 0.8 was fit using a 25:75 wt% ratio of linear to pure cyclic. This analysis shows that even with a large shift in hydrodynamic volume to 0.8 there still is a large fraction of non-cyclic (i.e. linear) species in the polymer sample. Therefore, it is recommended that the above analysis be carried out for all cyclic formed through the ring-closure method. 3. Conclusion In conclusion, the log-normal distribution can be used to fit MWDs to obtain characterization information for a multitude of cases involving polymers. The log-normal
distribution model provides a two parameter fit if Mn and the PDI are known. The log-normal distribution can be used effectively to deconvolute a broad MWD provided that the individual populations consist of log-normal distributions. We have demonstrated that the log-normal distribution model can be used to quantify the amount of dead polymer formed through radical–radical termination, and also obtain the true (or close to the true) PDI for the polymer formed primarily through LRP. It should be noted that the SEC will never give a PDI value of 1 due to column broadening. The log-normal distribution model can also be used to quantify the purity of cyclic polymer formed through the ring-closure method, and it was shown that only when the hydrodynamic volume decreased to close to 0.75 did the percentage of cyclic increase to greater than 95%. The method of analysis given in this paper can be used to extract information about a mechanism of polymerization, side products and amounts of product distributions in a broad molecular weight distribution.
Please cite this article in press as: Monteiro MJ. Fitting molecular weight distributions using a log-normal distribution model. Eur Polym J (2015), http://dx.doi.org/10.1016/j.eurpolymj.2015.01.009
M.J. Monteiro / European Polymer Journal xxx (2015) xxx–xxx
Acknowledgment M.J.M acknowledges financial support from the ARC Discovery Grant (DP140103497). References [1] Solomon DH, Rizzardo E, Cacioli P. US Patent 1986;4(581):429. [2] Kato M, Kamigaito M, Sawamoto M, Higashimura T. Macromolecules 1995;28(5):1721–3. [3] Wang JS, Matyjaszewski K. J Am Chem Soc 1995;117(20):5614–5. [4] Le TP, Moad G, Rizzardo E, Thang SH. WO 98/01478 Chem Abstr 1998;128:115390. [5] Percec V, Guliashvili T, Ladislaw JS, Wistrand A, Stjerndahl A, Sienkowska MJ, et al. J Am Chem Soc 2006;128(43):14156–65. [6] Monteiro MJ. J Polym Sci Part A: Polym Chem 2005;43(15): 3189–204. [7] Goh YK, Monteiro MJ. Macromolecules 2006;39(15):4966–74. [8] Monteiro MJ. J Polym Sci Part A: Polym Chem 2005;43(22):5643–51. [9] Iha RK, Wooley KL, Nystrom AM, Burke DJ, Kade MJ, Hawker CJ. Chem Rev 2009;109(11):5620–86. [10] Lammens M, Fournier D, Fijten MWM, Hoogenboom R, Du Prez F. Macromol Rapid Commun 2009;30(23):2049–55. [11] Whittaker MR, Urbani CN, Monteiro MJ. J Am Chem Soc 2006;128(35):11360–1. [12] Urbani CN, Bell CA, Lonsdale DE, Whittaker MR, Monteiro MJ. Macromolecules 2007;40(19):7056–9.
5
[13] Urbani CN, Bell CA, Whittaker MR, Monteiro MJ. Macromolecules 2008;41(4):1057–60. [14] Urbani CN, Lonsdale DE, Bell CA, Whittaker MR, Monteiro MJ. J Polym Sci Part A: Polym Chem 2008;46(5):1533–47. [15] Konkolewicz D, Monteiro MJ, Petrie S. Macromolecules 2011;44(18):7067–87. [16] Hu D, Zheng SX. Eur Polym J 2009;45(12):3326–38. [17] Laurent BA, Grayson SM. J Am Chem Soc 2006;128(13):4238–9. [18] Lonsdale DE, Bell CA, Monteiro MJ. Macromolecules 2010;43(7): 3331–9. [19] Lonsdale DE, Monteiro MJ. Chem Commun 2010;46(42):7945–7. [20] Hossain MD, Jia ZF, Monteiro MJ. Macromolecules 2014;47(15): 4955–70. [21] Hossain MD, Valade D, Jia ZF, Monteiro MJ. Polym Chem 2012;3(10): 2986–95. [22] Hossain MD, Lu D, Jia Z, Monteiro MJ. ACS Macro Lett 2014;3(12): 1254–7. [23] Gavrilov M, Monteiro MJ. Eur Polym J. doi:http://dx.doi.org/10.1016/ j.eurpolymj.2014.11.018. [24] Peebles LH. Molecular weight distributions in polymers. New York: Wiley-Interscience; 1971. [25] Johnston-Hall G, Monteiro MJ. Macromol Theory Simul 2010;19(7):387–93. [26] Rudin A, Choi P. Elements of polymer science and engineering. 3rd ed. London: Academic Press; 2013. p. Xvii–Xviii. [27] Lonsdale DE, Monteiro MJ. J Polym Sci Part A: Polym Chem 2010;48(20):4496–503. [28] Roovers J, Toporowski PM. Macromolecules 1983;16(6):843–9.
Please cite this article in press as: Monteiro MJ. Fitting molecular weight distributions using a log-normal distribution model. Eur Polym J (2015), http://dx.doi.org/10.1016/j.eurpolymj.2015.01.009
Contents lists available at ScienceDirect
European Polymer Journal journal homepage: www.elsevier.com/locate/europolj
Fitting molecular weight distributions using a log-normal distribution model Michael J. Monteiro Australian Institute for Bioengineering and Nanotechnology, The University of Queensland, Brisbane, QLD 4072 Australia
a r t i c l e
i n f o
Article history: Received 11 December 2014 Received in revised form 26 December 2014 Accepted 11 January 2015 Available online xxxx Keywords: Size exclusion chromatography ‘Living’ radical polymerization Molecular weight distribution Log-normal distribution
a b s t r a c t ‘Living’ radical polymerization has opened the way to producing complex polymer architectures, including cyclic polymers, dendrimers, stars and many other structures. A major issue in the field of polymer science is the characterization of these polymer products, especially when made to high molecular weights, well beyond the accuracy of NMR or MALDI. Questions of purity arise due to the amount of side products (e.g. dead polymer through termination or cyclic purity). Here, we use the log-normal distribution model to fit distributions of all products to the overall experimental distribution. The fits in all cases were excellent. The log-normal distribution model was simple to implement as it only relies on a two-parameter fit, in which these two parameters, the standard deviation and median, were obtained from the number-average molecular weight and the polydispersity of the molecular weight distribution. This type of analysis can play a role in elucidating the side product distributions, leading to new or modified mechanistic insights. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction ‘Living’ radical polymerization (LRP) has provided unprecedented access to polymers of a wide range of chemical compositions and molecular weight distributions (MWDs) [1–5]. Control of the MWD in, for example, reversible addition-fragmentation chain transfer (RAFT) polymerization is through the reactivity of the RAFT chain transfer agent (CTA) [6]. A CTA with a high chain transfer constant (>10) will produce a linear increase in the number-average molecular weight (Mn) with conversion and a narrow MWD as found from the low values of the polydispersity index (PDI) below 1.1. On the other hand, a CTA with a low chain transfer constant (61) will result in a relatively constant Mn with conversion, in which Mn is close to that determined from theory at 100% conversion. The MWD was broad with a PDI close to 2. The PDI can be controlled to any desired value between 1.1 and 2 by using a E-mail address: [email protected]
single difunctional and highly reactive RAFT agent with varying amounts of initiator [7]. The modeling of the MWD when using a MacroCTA to produce diblock copolymers can provide insight into the mechanism and the percentage of MacroCTA remaining [8], and it would be useful to determine this information from the experimental MWD. The introduction of ‘click’ reactions into polymer chemistry [9] coupled with the high chain-end functionality of polymers made by LRP has provided a method to create a wide range of polymer architectures, including welldefined stars [10], dendrimers [11–14], hyperbranched polymers [15], multiblock copolymers [16], and even cyclic [17–20] structures. Characterization of these polymers becomes increasingly difficult with an increase in architectural complexity. This leaves size exclusion chromatography (SEC) as the only recognized and accessible method of characterization. The ring-closure method is one of the most utilized method to produce cyclic polymers from a difunctional linear analog [17–21]. All publications claimed
http://dx.doi.org/10.1016/j.eurpolymj.2015.01.009 0014-3057/Ó 2015 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Monteiro MJ. Fitting molecular weight distributions using a log-normal distribution model. Eur Polym J (2015), http://dx.doi.org/10.1016/j.eurpolymj.2015.01.009
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M.J. Monteiro / European Polymer Journal xxx (2015) xxx–xxx
the production of cyclic polymers by the ring-closure procedure based on the reduction of the hydrodynamic volume in the MWD determined by refractive index SEC. It would be useful to determine the percentage of other species (i.e. uncyclized starting linear or multiblock products). The purity of cyclic plays an important role in determining the effects of cyclic on the physical properties of the polymer [22]. In our previous paper [23], we derived the MWDs from SEC, and provided a method of analysis to gain greater insight into the mechanisms of LRP. The aim of this paper is to provide a quantitative method to determine the MWD from SEC that consists of the addition of many narrow MWDs. With this knowledge, one can use this method to determine, for example, the amount of dead polymer in an LRP, the ratio of linear to cyclic species in a ring-closure method, or the quantity of number of arms attached to a dendritic structure. The method utilizes a two-parameter equation to produce a log-normal distribution (LND) that describes the MWD of a polymer product. The two parameters can be readily determined from the SEC Mn and PDI values. Since this is only a two parameter fit, the fitting to real experimental data is limited to a MWD that consists of the addition of Gaussian MWDs. Thus the LND method allows quantification of structures produced from the combination of LRP and ‘click’ reactions. 2. Results and discussion
2
expððM MÞ =2r2 0:5
ð2pr2 Þ
wðln MÞ ¼
2 expð ln M ln M =2r2 ð2pr2 Þ
0:5
ð2Þ
or
wðMÞ ¼
2 expð ln M ln M =2r2 Mð2pr2 Þ
0:5
ð3Þ
The median value, M, is related to the Mn and the weightaverage molecular weight (Mw) by the following equations.
Mn ¼ Mexpðr2 =2Þ
ð4Þ
Mw ¼ Mexpðr2 =2Þ
ð5Þ
PDI ¼ expðr2 Þ
ð6Þ
and, from Eqs. (4) and (5), we obtain
The most commonly used distribution functions to describe a polymer made by LRP are the Poisson and Gaussian distributions [24]. The Poisson distribution is a one-parameter equation, and as such the breath of the distribution is predetermined to be quite narrow. This assumption becomes valid when the experimental MWD has a PDI less than 1.05 (or even lower depending on the molecular weight). However, there are many cases where the MWD can be greater than 1.1 and even increase to 2 for a LRP [6,25]. The Gaussian (or normal) distribution is a two-parameter equation that represents a symmetrical (bell-shaped) distribution about the median. The two parameter are the median and the standard deviation, and are used in the following equation to obtain a number distribution [24]:
nðMÞ ¼
the Gaussian (or normal) distribution (i.e. Eq. (1)) is that it cannot describe a broad distribution [24]. The reason is that as the MWD becomes broader through an increase in r, negative M values (in Eq. (1)) will be found. Obviously, negative values of molecular weight do not exist. Converting to the log-normal distribution overcomes this problem and still allows use of the same two parameters as in Eq. (1). This is accomplished by assuming that the weight distribution, w(M), is symmetrical about the median on a ln M abscissa (i.e. x-axis). Thus Eq. (1) can be transformed as follows:
ð1Þ
where M is the molecular weight, M is the median molecular weight of the distribution, r is the standard deviation. The standard deviation is defined as half width of the distribution at half height. The same nomenclature will be used as found in our pervious paper [23]. In polymer science there is the misconception that the PDI value represents the breath of the distribution [26]. What the PDI value tells us is the deviation of a polymer distribution from a distribution in which all the chains are of equal length (i.e. a PDI = 1). As the MWD broadens, the PDI increases to values greater than 1. As shown below there is an equation based on the log-normal distribution equation that equates r to the PDI. The main problem with
M ¼ ðM n M w Þ0:5
ð7Þ
The maximum (i.e. peak maximum) of the distributions are located at
Mp;w ¼ M n3=2 M 1=2 w
ð8Þ
Mp;n ¼ Mn5=2 M 3=2 w
ð9Þ
where Mp,w and Mp,n are the peak maximums for the weight and number distributions, respectively. To produce a weight distribution, the Mn and Mw (or PDI) values are substituted into Eqs. (6) and (7) to obtain the two parameters M and r required for Eq. (3). The area under the distribution from using Eq. (3) should equal 1. We can convert the weight distribution to either a number (n(M)) or log-weight distribution (x(M)) using the transformations given in our previous paper [23]. It should be noted that when a polymer is fractionated, the MWD becomes skewed and thus may no longer represent a lognormal distribution. 2.1. Case study 1: fitting a polymer made by ATRP using the log-normal model The atom radical transfer polymerization (ATRP) of styrene usually produces polystrene (PSTY) with a narrow MWD. We made PSTY by ATRP with an Mn of 3490 and a PDI of 1.088, and its MWD from SEC was given by curve a in Fig. 1. The weight (w(M)) was used as the ordinate and log M as the abscissa. It can be see that curve a was relatively symmetrical about the peak maximum (or median) of the distribution, suggesting that the log-normal
Please cite this article in press as: Monteiro MJ. Fitting molecular weight distributions using a log-normal distribution model. Eur Polym J (2015), http://dx.doi.org/10.1016/j.eurpolymj.2015.01.009
3
M.J. Monteiro / European Polymer Journal xxx (2015) xxx–xxx
0.0005
(a) Expt 0.00045
(c)
0.0004 0.00035
w (M)
0.0003 0.00025 0.0002 0.00015
(b) 0.0001 0.00005 0 3
3.2
3.4
3.6
3.8
4
4.2
log M Fig. 1. Curve a (blue line): SEC data for the ATRP of styrene (Mn = 3490, PDI = 1.088). Curve b (red line): log-normal distribution (Eq. (3)) using Mn = 3490, PDI = 1.088. Curve c (green line): log-normal distribution (Eq. (3)) using two distributions (Dist 1: Mn = 3490, PDI = 1.062, and Dist 2: Mn = 6980, PDI = 1.062). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
distribution model can be used. Curve b utilized the Mn and PDI above to determine the two parameters M and r required to generate a MWD from Eq. (3). It can be seen that the log-normal fit gave quite a poor fit. The broader MWD from the log-normal distribution was a direct result from a small population of a higher molecular weight distribution to that of the main ATRP polymer. This higher molecular weight distribution formed through radical– radical bimolecular termination. Curve c represents the combination of two distributions; one from the main population but with a lower PDI value of 1.062, and the second with double the Mn and a PDI of 1.062. The final fitted distribution was calculated using Eqs. (3) and (10).
wðMÞfinal ¼
X wp;i wðMÞi
ð10Þ
i
where wp,i is the weight fraction of distribution i and w(M)i is the weight at molecular weight M. The fit is excellent and allows us to quantify the amount of dead polymer formed during the polymerization, which is 5.7 wt%. Two pieces of information can be gained from this analysis: (i) the polymer with halide end-groups has a very narrow MWD, and (ii) the amount of dead polymer can be accurately determined. It should be noted that a good choice of the baseline using the SEC software is essential to obtain good fits. 2.2. Case study 2: fitting the log-normal distribution for a polymer after cyclization Cyclization of linear polymers with two different functional chain-ends (e.g. –PSTYn–N3) that can undergo a coupling reaction via the ring-closure method should be carried out under dilute conditions [18,27]. The reason for this is that there is competition of the chemical
end-groups not only with that of the same chain but also to that from other chains (making multiblocks). It is quite difficult from SEC to determine whether or not 100% cyclic polymer has formed. Using RI–SEC to obtain MWDs before and after cyclization shows that there is a distinct shift in hydrodynamic volume (Fig. 2A). This shift for PSTY has been shown to be approximately 0.75 due to cyclic’s more compact structure and fewer degrees of freedom [22,28]. Fig. 2A shows the linear difunctional polymer (–PSTYn–N3, Mn = 3490, PDI = 1.088, curve a) that after cyclization produces a cyclic c-PSTY (curve b, Mn = 2650, PDI = 1.305). The linear PSTY is the same as that used in Fig. 1. It is found that the c-PSTY has a much greater PDI compared to linear PSTY because of the terminated product and unreacted linear PSTY. Using the same PDI value as the linear PSTY (Fig. 1, curve c) found from the log-normal model, the fit using a cyclic distribution with Mn = 2650, PDI = 1.062 (i.e. 94.3 wt%), starting linear PSTY (close to 0 wt%) and termination product (5.7 wt%) gave an excellent fit to the experimental data (curve c in Fig. 2A). To get close to pure cyclic, the MWD can easily be fractionated by using a preparative SEC system. The hydrodynamic volume (HDV) shift based on the log-normal fitted Mn’s was 0.76, which is very close to theory. There are many examples in the literature where the shift in hydrodynamic volume is close to 0.9, and the authors claim to have produced a high percentage of cyclic polymer. In Fig. 2B, we show theoretically using the lognormal model the cyclic purity at different hydrodynamic volume shifts, assuming that a shift of 0.75 is that of pure cyclic product. The PSTYnN3 distribution (Mn = 3490, PDI = 1.088, curve a) is added to the pure c-PSTY distribution (Mn = 2620, PDI = 1.062) at different weight ratios to obtain a final distribution in which its Mp,w (final)/Mp,w (linear) equals the desired HDV. The final distribution can
Please cite this article in press as: Monteiro MJ. Fitting molecular weight distributions using a log-normal distribution model. Eur Polym J (2015), http://dx.doi.org/10.1016/j.eurpolymj.2015.01.009
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M.J. Monteiro / European Polymer Journal xxx (2015) xxx–xxx
(A) 0.0007 0.0006
(b) 0.0005
(a)
w (M)
0.0004 0.0003
(c) 0.0002 0.0001 0 3
3.2
3.4
3.6
3.8
4
4.2
log M
(B)
1.2
(d) (c) (b) 1
(a) starting linear
w (M)
0.8
(e) Pure cyclic
0.6 0.4 0.2 0 3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
log M Fig. 2. (A) Cylcization of –PSTYn–N3 to c-PSTYn. curve a (blue line): experimental SEC distribution for –PSTYn–N3 (Mn = 3490, PDI = 1.088); curve b (green line): experimental SEC distribution after cyclization (c-PSTY, Mn = 2650, PDI = 1.305); curve c (red line): log-normal model fit to c-PSTY using 3 distributions (Dist 1: c-PSTY, Mn = 2650, PDI = 1.062, 94.3 wt%; Dist 2: Mn = 3490, PDI = 1.062, 0 wt%; Dist 3: linear terminated product, Mn = 6980, PDI = 1.062, 5.7 wt%). (B) Height normalized log-normal distributions with varying shifts in hydrodynamic volumes (HDV). Curve a: HDV = 0, curve b: HDV = 0.95, curve c: HDV = 0.85, curve d: HDV = 0.80, curve e: HDV = 0.75. The –PSTYn–N3 distribution (Mn = 3490, PDI = 1.088, curve a) is added to the pure c-PSTY distribution (Mn = 2620, PDI = 1.062) to produce the final distribution. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
be determined using Eqs. (3) and (10). Curve a represents the linear polymer before cyclization (i.e. 100 wt% linear) with HDV equal to 1. Curve b represents the final distribution to give a hydrodynamic shift of 0.95, which was fit using a 80:20 wt% ratio of linear to pure cyclic. Decreasing the hydrodynamic volume to 0.85 (curve c) was fit using a 50:50 wt% ratio of linear to pure cyclic. A further decrease in the hydrodynamic volume to 0.8 was fit using a 25:75 wt% ratio of linear to pure cyclic. This analysis shows that even with a large shift in hydrodynamic volume to 0.8 there still is a large fraction of non-cyclic (i.e. linear) species in the polymer sample. Therefore, it is recommended that the above analysis be carried out for all cyclic formed through the ring-closure method. 3. Conclusion In conclusion, the log-normal distribution can be used to fit MWDs to obtain characterization information for a multitude of cases involving polymers. The log-normal
distribution model provides a two parameter fit if Mn and the PDI are known. The log-normal distribution can be used effectively to deconvolute a broad MWD provided that the individual populations consist of log-normal distributions. We have demonstrated that the log-normal distribution model can be used to quantify the amount of dead polymer formed through radical–radical termination, and also obtain the true (or close to the true) PDI for the polymer formed primarily through LRP. It should be noted that the SEC will never give a PDI value of 1 due to column broadening. The log-normal distribution model can also be used to quantify the purity of cyclic polymer formed through the ring-closure method, and it was shown that only when the hydrodynamic volume decreased to close to 0.75 did the percentage of cyclic increase to greater than 95%. The method of analysis given in this paper can be used to extract information about a mechanism of polymerization, side products and amounts of product distributions in a broad molecular weight distribution.
Please cite this article in press as: Monteiro MJ. Fitting molecular weight distributions using a log-normal distribution model. Eur Polym J (2015), http://dx.doi.org/10.1016/j.eurpolymj.2015.01.009
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Please cite this article in press as: Monteiro MJ. Fitting molecular weight distributions using a log-normal distribution model. Eur Polym J (2015), http://dx.doi.org/10.1016/j.eurpolymj.2015.01.009