Influence of polymer architecture on the averaging effects in PGSE NMR attenuations for bimodal solutions of linear and star poly(vinyl acetates)

Journal of Molecular Liquids 167 (2012) 110–114

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Influence of polymer architecture on the averaging effects in PGSE NMR attenuations for bimodal solutions of linear and star poly(vinyl acetates) Scott A. Willis, William S. Price, I. Kristina Eriksson-Scott, Gang Zheng, Gary R. Dennis ⁎ Nanoscale Organisation and Dynamics Group, University of Western Sydney, Locked Bag 1797, Penrith, NSW 2751, Australia

a r t i c l e

i n f o

Article history: Received 14 December 2011 Received in revised form 3 January 2012 Accepted 3 January 2012 Available online 15 January 2012 Keywords: Bimodal solutions PGSE NMR PVAc Self-diffusion Star polymer

a b s t r a c t Pulsed gradient spin-echo (PGSE) NMR is a non-invasive technique that is useful for studying the self-diffusion of molecules in solution. However, even when the solutions are dilute the data analysis can be complicated by the effects of macroscopic and microscopic averaging. Here the averaging effects were measured for dilute bimodal solutions of linear poly(vinyl acetate) (PVAc) or 4-arm star PVAc. Star polymers are useful for biomedical applications due to their compactness and high functionality. Since synthetic polymers are polydisperse, the diffusive averaging effects, which have not yet been investigated in star polymer solutions, need to be considered as the magnitude of this may be important when designing star polymers for these applications. PGSE NMR attenuations were compared to predictions from two limiting case simulations. The results provide information on the magnitudes of the averaging effects for bimodal solutions of polymers with different polymer architectures and suggested that the averaging effects are not strongly dependent on polymer architecture. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Self-diffusion is the most fundamental form of transport and provides information about molecules in solution. Diffusion is related to molecular size through the Stokes–Einstein–Sutherland equation [1] and is sensitive to molecular interactions [2,3]. Pulsed gradient spin-echo (PGSE) NMR is a powerful non-invasive technique that can be used to measure self-diffusion [2]. The analysis of an NMR diffusion measurement of a freely diffusing monodisperse polymer in dilute solution is a monoexponential echo decay since there is only one diffusion coefficient for the polymer. However, in a polydisperse polymer there must be a range of diffusion coefficients and a multiexponential decay which complicates the analysis. Averaging of the self-diffusion coefficients through macroscopic and microscopic averaging has been observed for polydisperse macromolecular systems [4] but not for star polymers. Other phenomena, such as entanglement, also influence the diffusion coefficients in concentrated solutions of polymers [5–13]. The macroscopic average may be used to describe the PGSE NMR attenuation by averaging each of the component diffusion coefficients scaled by their concentrations in the polydisperse system [4]. This is what is expected for polydisperse solutions [4,14] but deviations have been observed [15,16]. This deviation is known as microscopic averaging or ensemble averaging [2,4,15,17]. The distribution of

⁎ Corresponding author. Tel.: + 61 2 9685 9939; fax: + 61 2 9685 9915. E-mail address: [email protected] (G.R. Dennis). 0167-7322/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.molliq.2012.01.006

diffusion coefficients measured in the polydisperse sample was narrower than expected [4,15,18]. Bimodal mixtures of polymers are often used to study the averaging process and in some cases deuterated polymers have been used so that one component is “NMR invisible” [4,19,20]. However, previous studies have only examined mixtures of linear polymers in dilute solution and so a comparison of the averaging effects in dilute solutions of polymers with other architectures (i.e., star polymers) is of interest. Star polymers have biomedical applications (e.g., drug delivery) [21,22] and have characteristics that are different from their linear equivalents (e.g., solution viscosity) [22–24]. Understanding the diffusive behaviour of molecules of different architectures may allow for better design of drug delivery molecules [21]. As synthetic polymers are polydisperse, the effects of diffusive averaging are present and understanding this for systems of different architectures is important particularly when the applications involve diffusion processes. Investigating the averaging effects of polydispersity is of great significance for NMR diffusion measurements and their application to synthetic and natural polymer characterisation and applications. In previous work [25], the averaging effects were measured in dilute bimodal solutions of linear poly(methyl methacrylate) (PMMA) and the effects were shown to be dependent on the molecular weight of the components in the bimodal solution. In this work, the diffusive averaging was measured for dilute bimodal solutions of mixtures of either linear poly(vinyl acetate) or 4-arm star PVAc in deuterated chloroform at 298 K and compared to simulations using the diffusion coefficients in monomodal solutions. This provided a comparison of the diffusive averaging effects in the PGSE NMR attenuations for

S.A. Willis et al. / Journal of Molecular Liquids 167 (2012) 110–114

bimodal solutions of star polymers compared to the equivalent solutions of linear polymers. These results provide useful insights into the diffusive averaging processes and represent, to these authors knowledge, the first reports of the independence of diffusive averaging effects on polymer architecture (linear and 4-arm star) for dilute bimodal polymer solutions. 2. Experimental 2.1. Synthesis and characterisation of PVAc The linear and 4-arm star PVAc were synthesised using methods described previously [26]. The peak molecular weights (Mp) and polydispersity index (PDI) of the polymers were measured by GPC analysis using a HP 1100 instrument and ChemStation GPC data analysis software (Agilent Technologies) with three individual pore size columns (Polymer Laboratories; 103, 104 and 105 Å), and polymers were eluted with tetrahydrofuran (THF) at a flow rate of 1 mL min− 1 with a toluene flow marker. GPC calibration was determined using 10 narrow molecular weight polystyrenes (PS) standards and universal calibration was used to convert the PS equivalent molecular weights of the PVAc to the corrected PVAc (Mark–Houwink–Sakurada equation [11,27,28] correction factors were K = 16 × 10− 3 mL g− 1 and α = 0.70 (for PVAc) and K = 14 × 10− 3 mL g− 1 and α = 0.70 (for PS)) [27] and these values are given in Table 1. 2.2. PGSE NMR 1 H NMR diffusion experiments were performed at 298 K with a Bruker Avance 400 MHz with a 5 mm broadband probe equipped with a z-axis gradient with maximum gradient strength of 0.521 T m − 1. Software for data fitting for NMR calibration (temperature and gradient calibrations) and diffusion measurement analysis were described previously [25,26]. Polymer diffusion measurements were performed using the double stimulated echo (DSTE) pulse sequence [29] with trapezoidal gradient pulses. This pulse sequence was chosen as it is designed to minimise the problems arising from convection [29]. Actually to investigate if convection was present the diffusion coefficient for residual CHCl3 in 400 μL of CDCl3 was measured with the stimulated echo (STE) pulse sequence [30] with rectangular gradient pulses and the chosen pulse sequence with diffusion times of 0.07 s and 0.1 s. The diffusion coefficients measured with the STE pulse sequence were found to increase (although only slightly) with diffusion time while those measured with the DSTE were constant, and so the DSTE pulse sequence was used for the diffusion measurements. The attenuation of the DSTE pulse sequence with trapezoidal gradient pulses is given by [25],

  

  2δ E ¼ exp − γ2 g 2 δ2 Δ− þ Btrapezoid D ¼ expð−bDÞ 3

ð1Þ

where Btrapezoid ¼

2

−2δεΔ þ 2δ ε þ

! 16ε3 35δε2 2 − þε Δ ; 15 15

ð2Þ

and δ is the gradient pulse duration in seconds, γ is the gyromagnetic ratio of the nucleus studied in rad s − 1 T − 1, Δ is the diffusion time in seconds, g is the gradient strength in T m − 1, ε is the rise/fall time of the trapezoidal gradient pulses in seconds, and D is the self-diffusion coefficient in m 2 s − 1. As shown in Eq. (1), all of the magnetic gradient parameters can be grouped into the b coefficient. The data was normalised to the signal at 0.011 T m − 1 gradient strength since the signal at 0 T m − 1 gradient strength of stimulated echo type sequences can be affected by cosine modulation dependent on chemical shift [31,32]. 2.2.1. Diffusion in monomodal solutions The diffusion coefficients were measured for each of the component polymers which were to be used to make the subsequent bimodal solutions. The monomodal samples (linear PVAc, star PVAc) were prepared at three concentrations (approximately 0.31, 0.75 and 1.5% w/v) and diffusion measurements and analysis of diffusion coefficients at infinite dilution, D0, were performed as described elsewhere [26]. The diffusion coefficients were found from non-linear regression of Eq. (1) onto the PGSE NMR data to avoid unequal weighting to noise (as would be the case if the logarithmic form of Eq. (1) was used). 2.2.2. Relaxation in monomodal solutions Longitudinal and transverse relaxation was measured for the monomodal linear and star PVAc solutions for the CH peak (4.6–5.4 ppm). This allowed the dependences of T1 and T2 on the molecular weight and/or concentration to be found so that they could considered in the analysis of the results for bimodal polymer solutions. The spin relaxation term, R(A), is given by [29], 2 RðAÞ ¼ 4−

Linear PVAc

i¼1

PDI

Mp × 10− 3 (g mol− 1)

PDI

5.1 11.0 26.5 58.7 115.5

1.20 1.26 1.38 1.46 1.63

6.5 17.3 58.2 89.4 –

1.17 1.22 1.30 1.33 –

ð4Þ

ð expðRi Þ expð−bDi ÞÞ 2 P i¼1

Mp × 10− 3 (g mol− 1)

T 2 ðAÞ

 3 Δ−2δ−2τ g þ t e 5; − T 1 ðAÞ

2.2.3. Diffusion in bimodal solutions The PGSE NMR attenuation plots and diffusion coefficients were measured in bimodal (50:50 by weight) polymer solutions with total polymer concentration of ~ 1.5% w/v. The bimodal linear PVAc solutions were 5100 g mol − 1/115 500 g mol − 1 (Mp ratio = 22.7), 11 000 g mol − 1/58 700 g mol − 1 (Mp ratio = 5.3) and 26 500 g mol − 1/58 700 g mol − 1 (Mp ratio = 2.2). The bimodal star PVAc solutions were 6500 g mol − 1/89 400 g mol − 1 (Mp −1 ratio = 13.8), 17 300 g mol /89 400 g mol − 1 (Mp ratio = 5.2) and 17 300 g mol − 1/58 200 g mol − 1 (Mp ratio = 3.4). Duplicate samples were made for each mixture and the PGSE NMR attenuations were recorded twice for each sample. The theory of PGSE NMR attenuations for discrete and continuous mixtures can be found in the literature [2,4,14,15,19,25]. In the simple case of a discrete bimodal solution with equal weight fractions the PGSE NMR attenuation is given by,

E¼

Star PVAc

 4 δ þ τg

where τg and te are other delays in the pulse sequence (in seconds). R(A), T1(A) and T2(A) indicate that the relaxation may be dependent on the variable, A, which may be the molecular weight and/or concentration for this work.

2 P

Table 1 Peak molecular weights and polydispersities of the PVAc.

111

;

ð5Þ

ð expðRi ÞÞ

where Ri and Di are the spin relaxation term and self-diffusion coefficient of component i, respectively. If the relaxation terms are independent of the molecular weight then R1 = R2 and so, E ¼ 0:5 expð−bD1 Þ þ 0:5 expð−bD2 Þ:

ð6Þ

112

S.A. Willis et al. / Journal of Molecular Liquids 167 (2012) 110–114

Two simulations were made to predict the PGSE signal attenuations measured for bimodal PVAc solutions: Simulation A — the diffusion coefficients used were those measured for the component polymer in a monomodal solution of the same total polymer concentration as in the bimodal solution (i.e., ~1.5% w/v); Simulation B — the diffusion coefficients used were those of the component polymer in a monomodal solution of the same component concentration as in the bimodal solution (i.e., ~0.75% w/v). Relaxation terms were included for completeness in the simulations and analysis (i.e., Eq. (5)), although they are often assumed to be identical for all resonances irrespective of M for simplicity to allow for normalisation [4,14,15]. Importantly, studying bimodal polymer solutions of equal weight fractions allows the molecular weight dependence to be removed from the analysis (see Eq. (5); i.e., only R1 or 2 and D1 or 2 are important). While the dependence of R1 or 2 on Mp may be different depending on the value of Mp (since it may be different depending on the measurement technique; i.e., GPC, NMR, DLS etc.), the value of R1 or 2 is the same despite the different functional dependence on Mp if a different molecular weight measurement technique was used. That is, it is the value of R1 or 2 not its functional dependence on Mp that is used in the simulations. 3. Results and discussion 3.1. Relaxation in monomodal solutions The longitudinal relaxation for linear and star PVAc was found to be independent of molecular weight and concentration in these experiments. For linear PVAc, T1 = 0.68 s, and for star PVAc, T1 = 0.69 s. The concentration dependence of the transverse relaxation of linear and star PVAc was negligible and was ignored; the molecular weight (i.e., Mp from GPC) dependence, however, could be described by a single exponential function. For linear PVAc this was given by,
 −Mp ; T 2 ðsÞ ¼ ð0:144  0:004Þ þ ð0:036  0:007Þ exp ð23400  12300Þ ð7Þ and for star PVAc this was given by,
 −M p : T 2 ðsÞ ¼ ð0:156  0:001Þ þ ð0:051  0:013Þ exp ð8960  2760Þ

ð8Þ

The relaxation times (T1 and T2) and DSTE parameters were used to calculate the relaxation terms (i.e., Eq. (4)) for use with Eq. (5) for the simulations of the attenuations for the bimodal solutions, and these are given in Table 2. Recall that Eqs. (7) and (8) are the dependence of T2 on the Mp measured by GPC, and while the functional dependence would change if another M was used; the T2 values would be the same. While the analysis of the relaxation terms showed that the relaxation terms diverge only for the highest Mp ratio bimodal solutions, the relaxation terms were included in all simulations and data analysis. Table 2 The exponential values of the relaxation terms (Eq. (4)) used in the analysis of the results for the bimodal solutions. Polymer

Component 1 Mp × 10− 3 (g mol− 1)

Component 2 Mp × 10− 3 (g mol− 1)

exp (R1)

exp (R2)

Linear PVAc

5.1 11.0 26.5 6.5 17.3 17.3

115.5 58.7 58.7 89.4 89.4 58.2

0.758 0.757 0.751 0.735 0.726 0.722

0.737 0.746 0.746 0.722 0.722 0.718

Star PVAc

3.2. Diffusion in bimodal polymer solutions While biexponential analysis could be performed to extract the component diffusion coefficients (i.e., using Eq. (5) with the relaxation terms fixed to the values given in Table 2); a simple biexponential fit can be affected by the signal-to-noise ratio, instrumental errors, the ratio of the contributing signals and the ratios of the two diffusion coefficients [33]. Hence, comparison of the simulated PGSE NMR attenuations using the results from monomodal solutions to the experimental PGSE NMR attenuations provides useful information about the bimodal systems without the need for biexponential analysis. The experimental and simulated attenuations are shown in Fig. 1, and the logarithmic form is shown to allow better visual comparison of the data and simulations. Mp ratios and D 0 ratios (or equivalently the ratio of RH) for the three bimodal solutions for this work and the linear PMMA of studied previously [25] are given in Table 3. Markedly non-single exponential (i.e., apparently ‘non-linear’) behaviour was observed for the samples with the highest Mp ratios (R 2 = 0.909 ± 0.002 (Fig. 1a) and 0.938 ± 0.006 (Fig. 1d)). For the other ratios, R 2 = 0.983 ± 0.002 (Fig. 1b), 0.996 ± 0.001 (Fig. 1c), 0.976 ± 0.003 (Fig. 1e) and 0.986 ± 0.001 (Fig. 1f). The PGSE NMR attenuations of the bimodal solutions of the lowest ratios are described best by Simulation A. As the ratios increased the PGSE NMR attenuations approached Simulation B. While for the linear PMMA the PGSE NMR attenuation for the bimodal solution of the highest ratio was best described by Simulation B [25], the results for the linear and star PVAc are not fully described by Simulation B (see Fig. 1). This is likely to be due to the ratio for the linear and star PVAc being less than the corresponding ratio for the linear PMMA (see Table 3). It is speculated that since the two lower ratios for linear and star PVAc give similar behaviours in the observed attenuations that the trend is to approach Simulation B when the ratios increase (i.e., compare the result for star PVAc to the result for the linear PVAc with the highest ratios). The results suggest that when the ratios are high the PGSE NMR attenuations are described mostly by Simulation B, though not fully reached due to the limitations of the available molecular weights, and when the ratios are low the PGSE NMR attenuations are described mostly by Simulation A. One possible explanation for why the solutions of the lowest ratios are better described by Simulation A is that when the two component molecules approach similar sizes (and also similar number of molecules in solution because of the use of equal weight fractions) the solution becomes more like that of a monomodal solution of the same total polymer concentration (i.e., the same obstruction effects). When the bimodal solution is one where the ratio of molecular weights is high (and overall better described by Simulation B) then the large molecule is simply in a solution of much smaller molecules and so behaves as if it was in a monomodal solution of the same component concentration (i.e., less obstruction effects than in a solution of the same total polymer concentration). As for the small molecule it is probably affected by the increased polymer concentration, and it must diffuse around the larger molecules, which is the likely reason why for low b values – the faster diffusing component – seems to be better fit with Simulation A (see the insets of Fig. 1a and d; and the following paragraph) but overall the attenuation is better described by Simulation B. However, for the linear and star PVAc of this work there is a notable difference in the behaviour for the Mp ratio ~ 5.3 (and D 0 ratio ~ 2.6) compared to the linear PMMA. These results suggests that there is a difference in the averaging behaviour due to polymer type and not architecture (i.e., linear and star). As mentioned, although the attenuation appeared to be better described by Simulation B for the attenuation for the highest ratio of linear PVAc, the data at low b values was better described by Simulation A. A simulation was also considered that was a combination of

S.A. Willis et al. / Journal of Molecular Liquids 167 (2012) 110–114

113

Fig. 1. PGSE NMR signal attenuations of bimodal linear PVAc solutions (approx. Mp ratios: a. 22.7, b. 5.3 and c. 2.2) and bimodal star PVAc solutions (approx. Mp ratios: d. 13.8, e. 5.2 and f. 3.4) with component polymer concentrations of ~ 0.75% w/v, giving a total polymer concentration of ~ 1.5% w/v. Shown are the average experimental data (■) and simulated data using Simulation A (—) and Simulation B (− − −). The error bars of the experimental data points are the standard deviation of the four experimental attenuations measured for each bimodal solution. The insets are included to better visualise the fits at low b values.

Simulation A and Simulation B. However such a simulation only improved the suitability to the data for the low b values and overall was similar to Simulation B.

Table 3 Mp ratios and D0 ratios (RH ratio) for the bimodal solutions. Polymer

Bimodal solution 1

Linear PVAc Star PVAc Linear PMMAa a

Bimodal solution 3

D0 ratio (RH ratio)

Mp ratio

D0 ratio (RH ratio)

Mp ratio

D0 ratio (RH ratio)

22.7 13.8 30.4

5.3 4.6 7.4

5.3 5.2 5.4

2.5 2.7 2.6

2.2 3.4 3.4

1.6 2.1 1.8

4. Conclusions Microscopic averaging is a complicated phenomenon and in this work it has been observed in bimodal solutions of either linear or 4-arm star PVAc polymers. The diffusive averaging effects were

Bimodal solution 2

Mp ratio

These bimodal solutions were studied in previous work [25].

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S.A. Willis et al. / Journal of Molecular Liquids 167 (2012) 110–114

found to be independent of the polymer architecture. This was demonstrated by comparing the experimental PGSE NMR attenuations to two simple simulated attenuations for bimodal solutions. The bimodal solutions consisted of polymers of the same type of polymer architecture. Both linear and 4-arm star PVAc showed similar diffusive averaging behaviours in bimodal solutions with respect to the simulated attenuations using the diffusion coefficients and relaxation parameters measured in monomodal solutions. A comparison with previous work [25] on the averaging effects in linear PMMA for the same temperature and solvent demonstrates that the magnitudes of the effects of microscopic averaging are dependent on the polymer type (i.e., PVAc or PMMA) rather than architecture. Acknowledgements This research was supported by a University of Western Sydney Honours Scholarship (S.A.W.) and a N.S.W. BioFirst Award from the N.S.W. Ministry for Science & Medical Research (W.S.P.). References [1] W. Sutherland, Philosophical Magazine 9 (1905) 781. [2] W.S. Price, NMR Studies of Translational Motion, Cambridge University Press, New York, 2009. [3] H. Therien-Aubin, X.X. Zhu, C.N. Moorefield, K. Kotta, G.R. Newkome, Macromolecules 40 (2007) 3644. [4] P.T. Callaghan, D.N. Pinder, Macromolecules 18 (1985) 373. [5] W. Brown, P. Zhou, Polymer 31 (1990) 772. [6] W. Brown, K. Mortensen, Macromolecules 21 (1988) 420. [7] W. Hess, Macromolecules 20 (1987) 2587. [8] W. Hess, Macromolecules 19 (1986) 1395. [9] P.G. De Gennes, Macromolecules 9 (1976) 587.

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