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Light Scattering – The Application of Static Light Scattering: Zimm Plot
Light Scattering – The Application of Static Light Scattering: Zimm Plot H Wu, ETH Zurich, Zurich, Switzerland M Lattuada, University of Fribourg, Fribourg, Switzerland ã 2013 Elsevier Inc. All rights reserved.
Introduction Theoretical Background of Zimm Plot Applications of the Zimm Plot Determination of the Rayleigh Ratio, Rex A Case Study on Applications of Zimm Plot to Investigate Stability of Bovine Serum Albumin Clusters Step 1: Treat the intensity curve based on Zimm plot II to get the zero-angle intensity and the radius of gyration Step 2: Perform Zimm plot I to obtain the average molecular weight and the second virial coefficient Conclusions References
1 1 4 4 5 6 6 8 9
Introduction Light scattering has been one of the most commonly used tools to investigate liquid dispersions, such as suspensions of nano- and colloidal particles, macromolecules or polymers or protein solutions, emulsions, and so forth. The so-called particles cover a variety of materials including organic and inorganic particles, droplets, micelles, liposomes, oligomers, fractal objects, and even gels. Thus, light scattering techniques have been widely applied in the fields of physics, chemistry, biology, material science, pharmaceutics, environmental science, and food science. Classical light scattering measures the time-averaged intensity of scattered light as a function of the scattering angle, which is typically referred to as static light scattering (SLS). Well-known applications of SLS include measurement of average molecular weight and root mean square radius (also called radius of gyration) of macromolecules, polymers, proteins, oligomers, micelles and clusters in disperse media, pair-interaction potential between two particles through measurement of the second virial coefficient, form and size of particles, structure of clusters, gels, and concentrated particulate systems, among others.1 Since coherent laser sources had begun to be available, new scattering methods were developed during the sixties of the last century, leading to the possibility of recording the temporal variations of scattered light. The temporal variation of scattered light is related to the dynamic properties of scatters in the medium, such as motion of the scatters, that is, diffusion (Brownian motion) and flow. Thus, such scattering experiments are referred to as dynamic light scattering (DLS), or sometimes called quasi-elastic light scattering. The most successful application of DLS occurred after the development of a digital autocorrelator,2 giving rise to the creation of techniques for fast characterization of particle diffusion coefficients and sizes based on DLS.3 Further extension of DLS also led to other applications, such as diffusing-wave spectroscopy for monitoring the dynamic properties of concentrated dispersions,4 and laser Doppler velocimetry or laser light velocimetry for determining particle mobility,5 etc. This contribution focusses on the application of SLS for the determination of average molecular weight, radius of gyration, and second virial coefficient of particles in disperse medium, that is, the well-known Zimm plot method.6 Here the so-called particles can be macromolecules, polymers, proteins, oligomers, micelles, or clusters. After a short introduction to the theoretical background of SLS and the Zimm plot, how to apply a Zimm plot to obtain the three quantities mentioned earlier is described in detail and through examples. The emphasis is on the practical problems typically faced during the applications and on how they can be solved so as to effectively perform the experiments and treat the obtained data. This article is written in such a way that all the material can be well understood by (upper) undergraduate students. The readers who would benefit from the content of this article are the active researchers in both academia and industry. The included material can be used as a reference manual for researchers to refer to during scattering experiments and data treatment.
Theoretical Background of Zimm Plot When light is shone on a liquid sample, part of the incident radiation is scattered anytime it encounters along its path a change in optical properties.1a,3a Such changes can be the result of the presence of particles, droplets, bubbles, proteins, or any object suspended in the medium,7 or even to the presence of density fluctuations in the medium itself.8 Here the latter case is not considered, and the focus is on the situation where the sample is a suspension, containing particles, droplets, proteins, or macromolecules dispersed in a liquid. For the sake of simplicity, the suspended objects will henceforth be referred to generically as particles. The optical properties of the suspension are usually described in terms of the difference in the refractive index between
Reference Module in Chemistry, Molecular Sciences and Chemical Engineering
http://dx.doi.org/10.1016/B978-0-12-409547-2.05431-7
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Light Scattering – The Application of Static Light Scattering: Zimm Plot
the suspended particles and the surrounding medium. The following quantitative treatment closely follows the notation of Pusey.9 A very general expression for the intensity (I) of the light scattered by a collection of N particles is given by the following expression: * + N X N IðqÞ 1 X ¼ 2 bi ðqÞbj ðqÞ exp q Ri Rj [1] I0 r i¼1 j¼1 where I0 is the intensity of the incident light; r, the sample distance from the detector; N, the number of suspended particles; Ri, the position vector of the ith particle; and q, the scattering wave vector, with a magnitude given by 4pn0 y q ¼ jqj ¼ sin [2] l 2 with y being the scattering angle and l the wavelength of the radiation in vacuum. The quantities indicated by bi(q) are the so-called scattering lengths, and are expressed as ! 2 2 ð pn2 np nl bi ðqÞ ¼ 20 exp ðqri Þd3 ri [3] n20 l Vi where np, nl, and n0 are the refractive indexes of the particles, the liquid, and the suspension, respectively, and the integral is carried out over the volume of the ith particle, Vi. Equation [1] can be rearranged and simplified in several ways. In the case where the dispersed phase is made of very small, identical particles at sufficiently low concentrations such that the correlation among them can be neglected, the intensity of the scattered light takes the following form9: 2 IðqÞ p2 ¼ 4 2 n2p n2l NVp PðqÞ I0 l r
[4]
where P(q) is the particle form factor, accounting for the particle shape, and Vp the particle volume. With this notation, P(q) ! 1 when q ! 0. In the case where the correlation among the identical particles cannot be neglected, eqn [1] becomes 2 IðqÞ p2 ¼ 4 2 n2p n2l NVp PðqÞSðqÞ I0 l r
[5]
where the scattering structure factor, S(q), has been introduced, which is normalized in such a way that S(q) ! N when q ! 0. It is often useful to reformulate the aforementioned equations for the scattered intensity in terms of the so-called Rayleigh ratio, R, defined as1b,9 2 IðqÞr 2 p2 2 ¼ 4 np n2l CVp P ðqÞSðqÞ R¼ [6] I0 V l where V is the volume of the sample, and C the number concentration of the particles. Very often, this last formulation is further modified in order to better highlight the role of the refractive index mismatch between solvent and particles. Assuming first that the particle concentration is low enough for the scattering structure factor to be neglected, taking the limit q ! 0 and considering that the difference in the refractive index between particles and solvent is not too large, eqn [6] can be rewritten as9: 2 4p2 n20 4p2 n20 dn0 2 M n n CV c ¼ KMc [7] R p p l dc NA l4 l4 with K¼
4p2 n20 dn0 2 l4 NA dc
[8]
where c is the molar concentration of particles; M, their molecular weight; and NA, the Avogadro number. The appearance of the gradient in the refractive index of the solution with the concentration of particles is very handy, because this quantity can be further manipulated using results from classical thermodynamics. In practical applications, it is more convenient to define an excess Rayleigh ratio, Rex, where, instead of the scattered intensity, the difference between the scattered intensity at a given particle concentration and the intensity at zero particle concentration (i.e., the solvent is used): Rex ¼
½Ic ðqÞ Ic¼0 ðqÞr 2 I0 V
[9]
In this manner, one can eliminate the contribution of the scattered light from the solvent. Hereafter, this definition of the Rayleigh ratio will be used. In the case of polydisperse samples, eqn [7] is modified as follows: Rex ¼ KMw c
[10]
where Mw is the weight-average molecular weight of the particles. This indicates that for measuring molecular weight, the scattering technique will enhance the contribution of particles with higher masses.
Light Scattering – The Application of Static Light Scattering: Zimm Plot
3
In order to see how particle size affects eqn [10], it is sufficient to expand in Taylor series the form factor (Guinier regime)1a,1b,7: 16p2 2 q2 2 y R sin [11] þ ¼ 1 R2g þ PðqÞ ¼ 1 g 2 3 2 3l where Rg is the radius of gyration, in order to recover the following famous expression1b,10:
Kc 1 q2 ¼ 1 þ R2g þ 3 Rex Mw
[12]
which is one of the basic equations of the Zimm plot.10 Whenever the concentration of particles is not negligible, the effect of the scattering structure factor needs to be taken into account. An instructive derivation of the effect of the concentration can be obtained from the definition of structure factor in terms of pair-correlation function g(r) 10: 1 ð sin ðqr Þ 2 SðqÞ ¼ 1 þ 4pC ½gðr Þ 1 r dr qr
[13]
0
and expanding it in power series of q: SðqÞ ¼ 1 2CA2 þ where A2 is the second virial coefficient, defined as
[14]
9
1 ð
1 ð
A2 ¼ 2p ½gðr Þ 1r dr 2p 2
0
0
U ðr Þ 1 r 2 dr exp kB T
[15]
with U(r) being the pair-interaction potential between two particles. The above expansion is valid at not too high concentrations. In this way, a Zimm plot at a relatively high particle concentration can also be used to gain information about interparticle interactions.9 The general equation of the Zimm plot in this case can be obtained by substituting eqn [14] into eqn [6] to replace S(q) so as to have, after rearrangement, Kc 1 1 ¼ þ 2B2 c [16] Rex PðqÞ Mw Note that B2 is also called the second virial coefficient, which is basically equal to A2/Mw. It is worth noting the three limiting cases of eqn [16]: I. For y ! 0, since P(q) ! 1, we have Kc Kc 1 ¼ ¼ þ 2B2 c Rex Rex ð0Þ Mw
[17]
Kc 1 1 q2 1 q2 ¼ , i:e:, ¼ 1 þ R2g ¼ 1 þ R2g 3 3 Rex Mw P ðqÞ Mw P ðqÞ
[18]
lim y!0
II. For c ! 0, eqn [12] is recovered: lim c!0
III. For both y ! 0 and c ! 0, eqn [10] is recovered: Kc 1 lim ¼ R M ex w c!0 y!0
[19]
Therefore, with the measured values of the Rayleigh ratio, Rex, for different concentrations and for different angles, these three equations constitute the basis for the SLS (Zimm plot) to determine the molecular (mass) weight, Mw, radius of gyration, Rg, and second virial coefficient, B2, of macromolecules (e.g., polymers and proteins) in disperse media.10 In addition to the derivation presented earlier, other theoretical approaches have been developed to obtain information about interparticle interactions from a Zimm plot for multicomponent systems. An interesting example is to start from the following definition of the Rayleigh ratio (measured at a scattering angle of 90 )11 Rex, 90 ¼
4p2 2 4 2 ðDnÞ C l n0
[20]
and expand the ensemble-average refractive index mismatch using the relations derived from multicomponent thermodynamics. More explicitly, since
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Light Scattering – The Application of Static Light Scattering: Zimm Plot
Dn dn ¼
X @n @Nj
j
dNj
[21]
T,P,Nk6¼j
substituting it in eqn [20] leads to:
ðDnÞ
2
12 2 @n A @ ¼ Ni hNi i2 þ @N j j T,P,Nk6¼j 0 1 0 1 2 0 13 X @n N @n @ A @ A 4 Nj @dij þ j Gij A5 2 V @N @N j i j
0
[22]
where the last term Gij is defined in terms of the pair-correlation function between species i and j 11: 1 ð
Gij ¼ 4p
gij ðr Þ 1 r 2 dr
[23]
0
which clearly depends on the level of correlation among the species (i and j) included in the pair-correlation function gij(r), and thus on their pair-interaction potentials. In the following section, the theoretical background illustrated earlier is made use of, particularly focusing on the applications of the Zimm plot.
Applications of the Zimm Plot Determination of the Rayleigh Ratio, Rex From eqns [16] to [19], it is clear that in all Zimm plot applications, we need to determine the absolute value of the Rayleigh ratio, Rex, which from eqn [9] requires the setup details of the instrument (e.g., intensity of the incident light, I0, sample distance from the detector, r, and sample volume, V). Precise determination of these operation parameters is feasible, but it is rather cumbersome. To avoid this, one typically uses an indirect approach, that is, one calibrates the Rex value using a known Rayleigh ratio (Rref) of a reference (pure) liquid such as toluene or benzene from which the Rex value can be computed based on the following relation1a: 2 Iex n0 Rref [24] Rex ¼ Iref nref where nref is the refractive index of the reference liquid, and Iref is the scattered intensity of the reference liquid under the same operation conditions. Thus, if one measures the Iex value and at the same time the Iref value, with the known Rref value, the Rex value can be easily obtained from eqn [24]. In the cases of toluene and benzene, the value of the Rayleigh ratio has been repeatedly measured by different researchers for several wavelengths such as l ¼ 488, 514.5, 546, and 633 nm. However, in recent years, diode-pumped solid-state lasers with various wavelengths (e.g., l ¼ 457, 473, 523, 532, 660, and 671 nm) have been introduced in applications of light scattering, for which reliable Rayleigh ratio values of toluene or benzene have not been reported in the literature. Therefore, to avoid the measurements of the Rayleigh ratios of toluene and benzene at these wavelengths, Wu12 recently collected from the literature as many values as possible of the Rayleigh ratio at different wavelengths for toluene and benzene and derived a reliable power-law correlation that can be used to predict any values of the Rayleigh ratio for toluene and benzene, in the range of the wavelengths between 420 and 700 nm, without further experimental measurements. The results are reported in Figure 1, where the average values of the Rayleigh ratio for toluene and benzene from the literature at T ¼ 25 C and y ¼ 90 are compared with the predictions of the following correlations for toluene, Ru,T cm1 ¼ 4:90 106 lðnmÞ4:17 [25] and benzene, Ru,B cm1 ¼ 1:59 107 lðnmÞ4:38
[26]
The following points need to be noted for the applications of the two correlations: (a) Ru is the Rayleigh ratio corresponding to an unpolarized incident light and a detector without a polarizer. For a vertically polarized incident light and a detector without a polarizer, the Rayleigh ratio, Rv, can be easily computed from Ru by13 Rv ¼
2Ru 1 þ ru
[27]
Light Scattering – The Application of Static Light Scattering: Zimm Plot
5
Figure 1 The average values (symbols) of the Rayleigh ratio of toluene and benzene with an unpolarized incident light and a detector without a polarizer, Ru,ref from the literature, as a function of the wavelength, l, which are compared with the predictions of the correlations. T ¼ 25 C; scattering angle, y ¼ 90 .
where ru is the depolarization factor for unpolarized incident light, whose value from analysis of the literature data should be 0.492 and 0.424, respectively, for toluene and benzene12. (b) In the case of a vertically polarized incident light and a detector with a vertical polarizer, the Rayleigh ratio, Rvv, can also be computed from Ru: Rvv ¼
Ru ð2 ru Þ 1 þ ru
[28]
(c) Based on the Rayleigh scattering law, the Rayleigh ratio should scale with the wavelength Ru ∝ l4. The exponent of eqn [25] for toluene is 4.17, which is very close to 4.0. In fact, the small difference can be explained by the effect of the wavelength on the refractive index.12 For benzene, however, the deviation from the Rayleigh law in eqn [26] is more significant, which must be related to additional unknown factors. Therefore, it is strongly suggested that toluene, instead of benzene, should be generally used as the reference liquid in Zimm plot applications.
A Case Study on Applications of Zimm Plot to Investigate Stability of Bovine Serum Albumin Clusters Let us now work with examples to understand how one works with the Zimm plots to obtain the three quantities, the weightaverage molecular weight, Mw, the radius of gyration, Rg, and the second virial coefficient, B2. To this aim, the bovine serum albumin (BSA) (nano-) clusters dispersed in water (pH 7), which were prepared ad hoc by thermal denaturation of native BSA, are taken for consideration.14 Since the isoelectric point of BSA is 4.7, at pH 7 the stability of the BSA clusters is related to negative charges on their surface. Conventionally, to perform the Zimm plot, one needs to prepare samples (at least) at four different particle concentrations and measure their scattered light intensity at a few selected angles. Then, a double extrapolation (i.e., c ! 0 and y ! 0) leads to the estimate of Mw, Rg, and B2. The typical scheme is shown in Figure 2, where the I, II, and III arrows correspond to the three limiting cases given by eqns [17]–[19]. In fact, suppliers of light scattering instruments often offer preinstallation of Zimm plot programs based on the scheme in Figure 2. In practical applications, however, it was found that the feasibility of such preinstalled Zimm plot programs is rather limited. The main problems may be summarized as follows. First, the selected angles for the intensity measurement are typically too few ( 10) to complete all the intensity measurements within a minimum time so as to avoid the effect of the time fluctuation of the intensity of the incident light. The consequence is that, because of the large experimental errors in the intensity, particularly when the particles are small and their concentrations are low, the results from the Zimm plot scheme are often irreproducible. Second, for relatively large particles, the scattered intensity is strongly angle-dependent on account of the form factor, P(q). In this case, the experimental data in Figure 2 do not follow the (approximate) straight line, and are substantially curved. This leads to difficulty in performing the extrapolation. Third, even though the operation time of the proposed approach for the Zimm plot is considered short, in reality it is not so short. The key point is that the operator cannot leave the instrument to do anything else, because he or she needs to periodically change the samples. Owing to the above limitations, although the Zimm plot program is preinstalled in the authors’ two light scattering instruments (Brookhaven and Malvern), it is seldom used by the researchers. Therefore, based on their research experience, the authors propose a more flexible approach to perform the Zimm plot analysis.
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Light Scattering – The Application of Static Light Scattering: Zimm Plot
Figure 2 Conventional scheme of Zimm plot, where I, II, and III arrows correspond to the three limiting cases given by eqns [17]–[19].
The proposed approach is also based on Zimm plots I and II, that is, eqns [17] and [18]: Zimm plot I, Kc 1 ¼ þ 2B2 c Rex ð0Þ Mw
[17]
1 q2 ¼ 1 þ R2g 3 PðqÞ
[18]
Zimm plot II,
However, we do not work according to the scheme mode, as shown in Figure 2. Instead, for each sample at a given particle concentration, we measure the scattered intensity in the entire angle range of the applied instrument at least two or three times so as to obtain a scattered intensity curve with good statistics. A typical intensity curve, Iex(q), of the BSA clusters at c ¼ 0.30 g l1 measured in this way is shown in Figure 3(a) (diamonds). Note that after the measurements for each sample, the intensity of pure toluene (here we use toluene as the reference liquid) is immediately measured as well. After an intensity curve is ready, one can start to treat the data, and meanwhile leave the instrument to measure the intensity curve for another particle concentration.
Step 1: Treat the intensity curve based on Zimm plot II to get the zero-angle intensity and the radius of gyration For a given intensity curve as shown in Figure 3(a), the data is plotted in the form of 1/P(q) versus q2/3, that is, based on the above Zimm plot II, eqn [18], where the form factor is given by P ð qÞ ¼
Iex ðqÞ Iex ð0Þ
[29]
Note that initially the Iex(0) value is not precisely known, and the location of the 1/P(q) versus q2/3 plot in Figure 3(b) is rather random. Therefore, the data points (toward low q2/3 values) that exhibit the best straight line are selected, as those given by the triangles in Figure 3(b). Note that the number of points that follow a straight line decreases and moves toward smaller q2/3 values as the particle size increases. For the selected data points, a linear trendline simulation is performed using Iex(0) as the fitting parameter, and the objective is to have the intercept of the straight line at q2/3 ¼ 0 equals 1, as given by the straight line in Figure 3(b). In this way, the Iex(0) value, as well as the radius of gyration, Rg, is obtained from the slope of the line. Continuing to work in the same way for the other intensity curves at different particles concentrations, we obtain the Iex(0) and Rg values as a function of the particle concentration, as reported in Rows 2 and 3 of Table 1 where the measured intensity of toluene at each corresponding Iex(0) value is also reported (Row 4). Note that the obtained Rg value should be independent of the particle concentration. The reported Rg values in Table 1 decline slightly with c, which may indicate some restriction on the extension of the external random chains of the BSA clusters as their concentration increases.14
Step 2: Perform Zimm plot I to obtain the average molecular weight and the second virial coefficient From eqn [17], to perform Zimm plot I, computing K and Rex(0) is required. To estimate the constant, K, defined by eqn [8], the only missing parameter is dn0/dc, which can be measured by means of a refractive index detector. For known materials, it can easily be found in the literature. In this case of BSA, the dn0/dc value has been well determined in the literature,14,15 as being equal to 0.186 cm3 g1. To compute Rex(0) from Iex(0) through eqn [24], the refractive index and Rayleigh ratio of the reference liquid (toluene), nT (¼1.496) and RT, respectively, are required. Since the wavelength of the laser beam for the applied scattering instrument is l ¼ 532 nm, from eqn [25] or Figure 1, we obtain Ru,T ¼ 2.10 105 cm1. Considering the vertically polarized
Light Scattering – The Application of Static Light Scattering: Zimm Plot
7
Figure 3 A working example to show how one can obtain the values of the zero-angle intensity, Iex(0), and the radius of gyration, Rg, from the source intensity curve, Iex(q), through Zimm plot II.
Table 1 Values of the zero-angle intensity (Iex(0)) and radius of gyration (Rg) obtained from Zimm plot II, eqn [18], at various concentrations of the BSA clusters (c), where IT is the measured scattered intensity of toluene and Kc/Rex(0) is the quantity for Zimm plot I, eqn [17] c, g ml1 Iex(0) Rg, nm IT Kc/Rex(0)
5.00 105 3.68 103 51.9 4.51 102 9.12 108
1.00 104 7.11 103 51.3 4.51 102 9.43 108
2.00 104 1.00 104 51.0 3.28 102 9.76 108
3.00 104 1.99 104 49.8 4.51 102 1.01 107
4.00 104 2.54 104 48.9 4.51 102 1.06 107
5.00 104 3.36 104 48.7 4.98 102 1.10 107
incident light and detector with a vertical polarizer, with Ru,T and eqn [28], we obtain RT ¼ 2.13 105 cm1. It is now possible to compute Rex(0) from Iex(0), with which Kc/Rex(0) can be computed as a function of c. The results are reported in Table 1 and in the form of Zimm plot I in Figure 4. All the data points follow a straight line, and from its intercept and its slope, the average molecular weight and the second virial coefficient of the BSA clusters, Mw ¼ 1.12 107 g mol1 and B2 ¼ 2.03 105 cm3 mol g2, respectively, are obtained. The positive B2 value indicates that the BSA cluster dispersion is stable. Since the molecular weight of single BSA proteins is MBSA ¼ 6.6 104 g mol1, with the obtained Mw value of the BSA clusters, the average aggregation number in a BSA cluster, Nagg ¼ Mw/MBSA ¼ 169, can be obtained. This approach has been applied to investigate the colloidal stability (B2) of the BSA clusters mentioned earlier as a function of the CaCl2 concentration, Cs, and the results have been reproduced from the original work14 in Figure 5 (open circles). A particularly interesting finding is that at low CaCl2 concentration, the B2 value decreases as Cs increases, that is, as expected, the stability of the BSA clusters decreases as the CaCl2 concentration increases. However, when the B2 value reaches a local minimum, it changes trend and increases with Cs, that is, the stability improves with further addition of salt, CaCl2, which is an unexpected phenomenon. The most important result is that such stability behavior observed from the B2 values has
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Light Scattering – The Application of Static Light Scattering: Zimm Plot
Figure 4 An example of Zimm plot I, eqn [17], to estimate the average molar mass (Mw) and second virial coefficient (B2) of bovine serum albumin (BSA) clusters, as described in the literature. Reproduced from Wu, H.; Arosio, P.; Podolskaya, O. G.; Wei, D.; Morbidelli, M. Phys. Chem. Chem. Phys. 2012, 14(14), 4906–4916.
Figure 5 The second virial coefficient (B2) values obtained from Zimm plot I, eqn [17], at various CaCl2 concentrations (Cs), compared with the experimentally measured Fuchs stability ratio (W). Reproduced from Wu, H.; Arosio, P.; Podolskaya, O. G.; Wei, D.; Morbidelli, M. Phys. Chem. Chem. Phys. 2012, 14(14), 4906–4916.
been confirmed by independent experiments investigating the stability of the same system through determination of the Fuchs stability ratio, W, defined as 1 ð
W¼2
exp ½U ðr Þ=kB T dl Gl2
[30]
2
where l ¼ r/a, a is the radius of the particles, and G is a hydrodynamic function. Comparing eqn [30] with eqn [15], one can see that both the Fuchs stability ratio and the second virial coefficient are directly related to the pair-interaction potential, U(r), thus reflecting the stability of the particles. In fact, the measured W values for the same system, as shown in Figure 5, follow exactly the same trend as the B2 values do. These results suggest the possibility of interrogating the stability of colloidal suspensions through measurement of the second virial coefficient using Zimm plot analysis. The advantage of this approach is that it is suitable for investigating stable suspensions, while the more traditional method based on measuring aggregation rate constants is more suitable for investigating less stable suspensions; thus, they complement each other.
Conclusions Since the seminal publication by Zimm almost 70 years ago, the Zimm plot method has been routinely used to measure the molecular weight of polymers in solutions. In this work, this method has been critically reviewed, and it has been shown that a Zimm plot can be equally well applied to general colloidal suspensions. In the latter cases, care has to be taken to properly account
Light Scattering – The Application of Static Light Scattering: Zimm Plot
9
for the finite size of particles (proteins, clusters, etc.), which is included in the form factor and can lead to deviations from the traditional straight lines encountered in traditional Zimm plots. The added value of applying a Zimm plot to colloidal suspensions is the possibility of measuring not only the aggregation number, but also the second virial coefficient. An example has been discussed, showing how the trend of the measured second virial coefficient versus the ionic strength closely matches that of the Fuchs stability ratio for suspensions of protein clusters. This opens up new avenues in the understanding of colloidal interactions, which are crucial in determining the colloidal stability of particulate suspensions.
References 1. (a) Kerker, M. The Scattering of Light and Other Electromagnetic Radiation. Academic Press: New York, 1969; (b) Johnson, C. S.; Gabriel, D. A. Laser Light Scattering. Dover Publications: New York, 1995; (c) Brown, W. Light Scatterings – Principles and Development. Clarendon Press: Oxford, 1996; (d) Chu, B. Laser Light Scattering: Basic Principles and Practice. Academic Press: Boston, MA, 1991. 2. Cummins, H. Z.; Pike, E. R. Photon Correlation and Light Beating Spectroscopy. Plenum: New York and London, 1973. 3. (a) Berne, B. J.; Pecora, R. Dynamic Light Scattering. Wiley: New York, 1976; (b) Finsy, R. Adv. Colloid Interface Sci. 1994, 52, 79–143; (c) Schmitz, K. S. An Introduction to Dynamic Light Scattering by Macromolecules. Academic Press: Boston, MA, 1990. 4. Pine, D. J.; Weitz, D. A.; Chaikin, P.M.; Herbolzheimer, E. Phys. Rev. Lett. 1988, 60 (12), 1134–1137. 5. Durst, F.; Melling, A.; Whitelaw, J. H. Principles and Practice of Laser Doppler Anemometry. Academic Press: London, 1976. 6. Zimm, B. H. J. Chem. Phys. 1948, 16 (12), 1099–1116. 7. Bohren, C. F.; Huffman, D. R. Absorption and Scattering of Light by Small Particles. Wiley-VCH: New York, 1998. 8. Einstein, A. Annalen Der Physik 1910, 33 (16), 1275–1298. 9. Lindner, P.; Zemb, T. Neutron, X-Rays and Light. Scattering Methods Applied to Soft Condensed Matter. North Holland: Amsterdam, 2002. 10. Zimm, B. H. J. Chem. Phys. 1948, 16 (12), 1093–1099. 11. Blanco, M.A.; Sahin, E.; Li, Y.; Roberts, C. J. J. Chem. Phys. 2011, 134 (22), 225103–225112. 12. Wu, H. Chem. Phys. 2010, 367 (1), 44–47. 13. Moreels, E.; De Ceuninck, W.; Finsy, R. J. Chem. Phys. 1987, 86 (2), 618–623. 14. Wu, H.; Arosio, P.; Podolskaya, O. G.; Wei, D.; Morbidelli, M. Phys. Chem. Chem. Phys. 2012, 14 (14), 4906–4916. 15. (a) Huglin, M. B. Specific Refractive Index Increments of Polymers in Dilute Solution. In Polymer Handbook; Brandrup, J.; Immergut, E. H., Eds.; Wiley: New York, 1991; pp VII/ 409–VII/471; (b) Tumolo, T.; Angnes, L.; Baptista, M. S. Anal. Biochem. 2004, 333 (2), 273–279.
Introduction Theoretical Background of Zimm Plot Applications of the Zimm Plot Determination of the Rayleigh Ratio, Rex A Case Study on Applications of Zimm Plot to Investigate Stability of Bovine Serum Albumin Clusters Step 1: Treat the intensity curve based on Zimm plot II to get the zero-angle intensity and the radius of gyration Step 2: Perform Zimm plot I to obtain the average molecular weight and the second virial coefficient Conclusions References
1 1 4 4 5 6 6 8 9
Introduction Light scattering has been one of the most commonly used tools to investigate liquid dispersions, such as suspensions of nano- and colloidal particles, macromolecules or polymers or protein solutions, emulsions, and so forth. The so-called particles cover a variety of materials including organic and inorganic particles, droplets, micelles, liposomes, oligomers, fractal objects, and even gels. Thus, light scattering techniques have been widely applied in the fields of physics, chemistry, biology, material science, pharmaceutics, environmental science, and food science. Classical light scattering measures the time-averaged intensity of scattered light as a function of the scattering angle, which is typically referred to as static light scattering (SLS). Well-known applications of SLS include measurement of average molecular weight and root mean square radius (also called radius of gyration) of macromolecules, polymers, proteins, oligomers, micelles and clusters in disperse media, pair-interaction potential between two particles through measurement of the second virial coefficient, form and size of particles, structure of clusters, gels, and concentrated particulate systems, among others.1 Since coherent laser sources had begun to be available, new scattering methods were developed during the sixties of the last century, leading to the possibility of recording the temporal variations of scattered light. The temporal variation of scattered light is related to the dynamic properties of scatters in the medium, such as motion of the scatters, that is, diffusion (Brownian motion) and flow. Thus, such scattering experiments are referred to as dynamic light scattering (DLS), or sometimes called quasi-elastic light scattering. The most successful application of DLS occurred after the development of a digital autocorrelator,2 giving rise to the creation of techniques for fast characterization of particle diffusion coefficients and sizes based on DLS.3 Further extension of DLS also led to other applications, such as diffusing-wave spectroscopy for monitoring the dynamic properties of concentrated dispersions,4 and laser Doppler velocimetry or laser light velocimetry for determining particle mobility,5 etc. This contribution focusses on the application of SLS for the determination of average molecular weight, radius of gyration, and second virial coefficient of particles in disperse medium, that is, the well-known Zimm plot method.6 Here the so-called particles can be macromolecules, polymers, proteins, oligomers, micelles, or clusters. After a short introduction to the theoretical background of SLS and the Zimm plot, how to apply a Zimm plot to obtain the three quantities mentioned earlier is described in detail and through examples. The emphasis is on the practical problems typically faced during the applications and on how they can be solved so as to effectively perform the experiments and treat the obtained data. This article is written in such a way that all the material can be well understood by (upper) undergraduate students. The readers who would benefit from the content of this article are the active researchers in both academia and industry. The included material can be used as a reference manual for researchers to refer to during scattering experiments and data treatment.
Theoretical Background of Zimm Plot When light is shone on a liquid sample, part of the incident radiation is scattered anytime it encounters along its path a change in optical properties.1a,3a Such changes can be the result of the presence of particles, droplets, bubbles, proteins, or any object suspended in the medium,7 or even to the presence of density fluctuations in the medium itself.8 Here the latter case is not considered, and the focus is on the situation where the sample is a suspension, containing particles, droplets, proteins, or macromolecules dispersed in a liquid. For the sake of simplicity, the suspended objects will henceforth be referred to generically as particles. The optical properties of the suspension are usually described in terms of the difference in the refractive index between
Reference Module in Chemistry, Molecular Sciences and Chemical Engineering
http://dx.doi.org/10.1016/B978-0-12-409547-2.05431-7
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Light Scattering – The Application of Static Light Scattering: Zimm Plot
the suspended particles and the surrounding medium. The following quantitative treatment closely follows the notation of Pusey.9 A very general expression for the intensity (I) of the light scattered by a collection of N particles is given by the following expression: * + N X N IðqÞ 1 X ¼ 2 bi ðqÞbj ðqÞ exp q Ri Rj [1] I0 r i¼1 j¼1 where I0 is the intensity of the incident light; r, the sample distance from the detector; N, the number of suspended particles; Ri, the position vector of the ith particle; and q, the scattering wave vector, with a magnitude given by 4pn0 y q ¼ jqj ¼ sin [2] l 2 with y being the scattering angle and l the wavelength of the radiation in vacuum. The quantities indicated by bi(q) are the so-called scattering lengths, and are expressed as ! 2 2 ð pn2 np nl bi ðqÞ ¼ 20 exp ðqri Þd3 ri [3] n20 l Vi where np, nl, and n0 are the refractive indexes of the particles, the liquid, and the suspension, respectively, and the integral is carried out over the volume of the ith particle, Vi. Equation [1] can be rearranged and simplified in several ways. In the case where the dispersed phase is made of very small, identical particles at sufficiently low concentrations such that the correlation among them can be neglected, the intensity of the scattered light takes the following form9: 2 IðqÞ p2 ¼ 4 2 n2p n2l NVp PðqÞ I0 l r
[4]
where P(q) is the particle form factor, accounting for the particle shape, and Vp the particle volume. With this notation, P(q) ! 1 when q ! 0. In the case where the correlation among the identical particles cannot be neglected, eqn [1] becomes 2 IðqÞ p2 ¼ 4 2 n2p n2l NVp PðqÞSðqÞ I0 l r
[5]
where the scattering structure factor, S(q), has been introduced, which is normalized in such a way that S(q) ! N when q ! 0. It is often useful to reformulate the aforementioned equations for the scattered intensity in terms of the so-called Rayleigh ratio, R, defined as1b,9 2 IðqÞr 2 p2 2 ¼ 4 np n2l CVp P ðqÞSðqÞ R¼ [6] I0 V l where V is the volume of the sample, and C the number concentration of the particles. Very often, this last formulation is further modified in order to better highlight the role of the refractive index mismatch between solvent and particles. Assuming first that the particle concentration is low enough for the scattering structure factor to be neglected, taking the limit q ! 0 and considering that the difference in the refractive index between particles and solvent is not too large, eqn [6] can be rewritten as9: 2 4p2 n20 4p2 n20 dn0 2 M n n CV c ¼ KMc [7] R p p l dc NA l4 l4 with K¼
4p2 n20 dn0 2 l4 NA dc
[8]
where c is the molar concentration of particles; M, their molecular weight; and NA, the Avogadro number. The appearance of the gradient in the refractive index of the solution with the concentration of particles is very handy, because this quantity can be further manipulated using results from classical thermodynamics. In practical applications, it is more convenient to define an excess Rayleigh ratio, Rex, where, instead of the scattered intensity, the difference between the scattered intensity at a given particle concentration and the intensity at zero particle concentration (i.e., the solvent is used): Rex ¼
½Ic ðqÞ Ic¼0 ðqÞr 2 I0 V
[9]
In this manner, one can eliminate the contribution of the scattered light from the solvent. Hereafter, this definition of the Rayleigh ratio will be used. In the case of polydisperse samples, eqn [7] is modified as follows: Rex ¼ KMw c
[10]
where Mw is the weight-average molecular weight of the particles. This indicates that for measuring molecular weight, the scattering technique will enhance the contribution of particles with higher masses.
Light Scattering – The Application of Static Light Scattering: Zimm Plot
3
In order to see how particle size affects eqn [10], it is sufficient to expand in Taylor series the form factor (Guinier regime)1a,1b,7: 16p2 2 q2 2 y R sin [11] þ ¼ 1 R2g þ PðqÞ ¼ 1 g 2 3 2 3l where Rg is the radius of gyration, in order to recover the following famous expression1b,10:
Kc 1 q2 ¼ 1 þ R2g þ 3 Rex Mw
[12]
which is one of the basic equations of the Zimm plot.10 Whenever the concentration of particles is not negligible, the effect of the scattering structure factor needs to be taken into account. An instructive derivation of the effect of the concentration can be obtained from the definition of structure factor in terms of pair-correlation function g(r) 10: 1 ð sin ðqr Þ 2 SðqÞ ¼ 1 þ 4pC ½gðr Þ 1 r dr qr
[13]
0
and expanding it in power series of q: SðqÞ ¼ 1 2CA2 þ where A2 is the second virial coefficient, defined as
[14]
9
1 ð
1 ð
A2 ¼ 2p ½gðr Þ 1r dr 2p 2
0
0
U ðr Þ 1 r 2 dr exp kB T
[15]
with U(r) being the pair-interaction potential between two particles. The above expansion is valid at not too high concentrations. In this way, a Zimm plot at a relatively high particle concentration can also be used to gain information about interparticle interactions.9 The general equation of the Zimm plot in this case can be obtained by substituting eqn [14] into eqn [6] to replace S(q) so as to have, after rearrangement, Kc 1 1 ¼ þ 2B2 c [16] Rex PðqÞ Mw Note that B2 is also called the second virial coefficient, which is basically equal to A2/Mw. It is worth noting the three limiting cases of eqn [16]: I. For y ! 0, since P(q) ! 1, we have Kc Kc 1 ¼ ¼ þ 2B2 c Rex Rex ð0Þ Mw
[17]
Kc 1 1 q2 1 q2 ¼ , i:e:, ¼ 1 þ R2g ¼ 1 þ R2g 3 3 Rex Mw P ðqÞ Mw P ðqÞ
[18]
lim y!0
II. For c ! 0, eqn [12] is recovered: lim c!0
III. For both y ! 0 and c ! 0, eqn [10] is recovered: Kc 1 lim ¼ R M ex w c!0 y!0
[19]
Therefore, with the measured values of the Rayleigh ratio, Rex, for different concentrations and for different angles, these three equations constitute the basis for the SLS (Zimm plot) to determine the molecular (mass) weight, Mw, radius of gyration, Rg, and second virial coefficient, B2, of macromolecules (e.g., polymers and proteins) in disperse media.10 In addition to the derivation presented earlier, other theoretical approaches have been developed to obtain information about interparticle interactions from a Zimm plot for multicomponent systems. An interesting example is to start from the following definition of the Rayleigh ratio (measured at a scattering angle of 90 )11 Rex, 90 ¼
4p2 2 4 2 ðDnÞ C l n0
[20]
and expand the ensemble-average refractive index mismatch using the relations derived from multicomponent thermodynamics. More explicitly, since
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Light Scattering – The Application of Static Light Scattering: Zimm Plot
Dn dn ¼
X @n @Nj
j
dNj
[21]
T,P,Nk6¼j
substituting it in eqn [20] leads to:
ðDnÞ
2
12 2 @n A @ ¼ Ni hNi i2 þ @N j j T,P,Nk6¼j 0 1 0 1 2 0 13 X @n N @n @ A @ A 4 Nj @dij þ j Gij A5 2 V @N @N j i j
0
[22]
where the last term Gij is defined in terms of the pair-correlation function between species i and j 11: 1 ð
Gij ¼ 4p
gij ðr Þ 1 r 2 dr
[23]
0
which clearly depends on the level of correlation among the species (i and j) included in the pair-correlation function gij(r), and thus on their pair-interaction potentials. In the following section, the theoretical background illustrated earlier is made use of, particularly focusing on the applications of the Zimm plot.
Applications of the Zimm Plot Determination of the Rayleigh Ratio, Rex From eqns [16] to [19], it is clear that in all Zimm plot applications, we need to determine the absolute value of the Rayleigh ratio, Rex, which from eqn [9] requires the setup details of the instrument (e.g., intensity of the incident light, I0, sample distance from the detector, r, and sample volume, V). Precise determination of these operation parameters is feasible, but it is rather cumbersome. To avoid this, one typically uses an indirect approach, that is, one calibrates the Rex value using a known Rayleigh ratio (Rref) of a reference (pure) liquid such as toluene or benzene from which the Rex value can be computed based on the following relation1a: 2 Iex n0 Rref [24] Rex ¼ Iref nref where nref is the refractive index of the reference liquid, and Iref is the scattered intensity of the reference liquid under the same operation conditions. Thus, if one measures the Iex value and at the same time the Iref value, with the known Rref value, the Rex value can be easily obtained from eqn [24]. In the cases of toluene and benzene, the value of the Rayleigh ratio has been repeatedly measured by different researchers for several wavelengths such as l ¼ 488, 514.5, 546, and 633 nm. However, in recent years, diode-pumped solid-state lasers with various wavelengths (e.g., l ¼ 457, 473, 523, 532, 660, and 671 nm) have been introduced in applications of light scattering, for which reliable Rayleigh ratio values of toluene or benzene have not been reported in the literature. Therefore, to avoid the measurements of the Rayleigh ratios of toluene and benzene at these wavelengths, Wu12 recently collected from the literature as many values as possible of the Rayleigh ratio at different wavelengths for toluene and benzene and derived a reliable power-law correlation that can be used to predict any values of the Rayleigh ratio for toluene and benzene, in the range of the wavelengths between 420 and 700 nm, without further experimental measurements. The results are reported in Figure 1, where the average values of the Rayleigh ratio for toluene and benzene from the literature at T ¼ 25 C and y ¼ 90 are compared with the predictions of the following correlations for toluene, Ru,T cm1 ¼ 4:90 106 lðnmÞ4:17 [25] and benzene, Ru,B cm1 ¼ 1:59 107 lðnmÞ4:38
[26]
The following points need to be noted for the applications of the two correlations: (a) Ru is the Rayleigh ratio corresponding to an unpolarized incident light and a detector without a polarizer. For a vertically polarized incident light and a detector without a polarizer, the Rayleigh ratio, Rv, can be easily computed from Ru by13 Rv ¼
2Ru 1 þ ru
[27]
Light Scattering – The Application of Static Light Scattering: Zimm Plot
5
Figure 1 The average values (symbols) of the Rayleigh ratio of toluene and benzene with an unpolarized incident light and a detector without a polarizer, Ru,ref from the literature, as a function of the wavelength, l, which are compared with the predictions of the correlations. T ¼ 25 C; scattering angle, y ¼ 90 .
where ru is the depolarization factor for unpolarized incident light, whose value from analysis of the literature data should be 0.492 and 0.424, respectively, for toluene and benzene12. (b) In the case of a vertically polarized incident light and a detector with a vertical polarizer, the Rayleigh ratio, Rvv, can also be computed from Ru: Rvv ¼
Ru ð2 ru Þ 1 þ ru
[28]
(c) Based on the Rayleigh scattering law, the Rayleigh ratio should scale with the wavelength Ru ∝ l4. The exponent of eqn [25] for toluene is 4.17, which is very close to 4.0. In fact, the small difference can be explained by the effect of the wavelength on the refractive index.12 For benzene, however, the deviation from the Rayleigh law in eqn [26] is more significant, which must be related to additional unknown factors. Therefore, it is strongly suggested that toluene, instead of benzene, should be generally used as the reference liquid in Zimm plot applications.
A Case Study on Applications of Zimm Plot to Investigate Stability of Bovine Serum Albumin Clusters Let us now work with examples to understand how one works with the Zimm plots to obtain the three quantities, the weightaverage molecular weight, Mw, the radius of gyration, Rg, and the second virial coefficient, B2. To this aim, the bovine serum albumin (BSA) (nano-) clusters dispersed in water (pH 7), which were prepared ad hoc by thermal denaturation of native BSA, are taken for consideration.14 Since the isoelectric point of BSA is 4.7, at pH 7 the stability of the BSA clusters is related to negative charges on their surface. Conventionally, to perform the Zimm plot, one needs to prepare samples (at least) at four different particle concentrations and measure their scattered light intensity at a few selected angles. Then, a double extrapolation (i.e., c ! 0 and y ! 0) leads to the estimate of Mw, Rg, and B2. The typical scheme is shown in Figure 2, where the I, II, and III arrows correspond to the three limiting cases given by eqns [17]–[19]. In fact, suppliers of light scattering instruments often offer preinstallation of Zimm plot programs based on the scheme in Figure 2. In practical applications, however, it was found that the feasibility of such preinstalled Zimm plot programs is rather limited. The main problems may be summarized as follows. First, the selected angles for the intensity measurement are typically too few ( 10) to complete all the intensity measurements within a minimum time so as to avoid the effect of the time fluctuation of the intensity of the incident light. The consequence is that, because of the large experimental errors in the intensity, particularly when the particles are small and their concentrations are low, the results from the Zimm plot scheme are often irreproducible. Second, for relatively large particles, the scattered intensity is strongly angle-dependent on account of the form factor, P(q). In this case, the experimental data in Figure 2 do not follow the (approximate) straight line, and are substantially curved. This leads to difficulty in performing the extrapolation. Third, even though the operation time of the proposed approach for the Zimm plot is considered short, in reality it is not so short. The key point is that the operator cannot leave the instrument to do anything else, because he or she needs to periodically change the samples. Owing to the above limitations, although the Zimm plot program is preinstalled in the authors’ two light scattering instruments (Brookhaven and Malvern), it is seldom used by the researchers. Therefore, based on their research experience, the authors propose a more flexible approach to perform the Zimm plot analysis.
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Light Scattering – The Application of Static Light Scattering: Zimm Plot
Figure 2 Conventional scheme of Zimm plot, where I, II, and III arrows correspond to the three limiting cases given by eqns [17]–[19].
The proposed approach is also based on Zimm plots I and II, that is, eqns [17] and [18]: Zimm plot I, Kc 1 ¼ þ 2B2 c Rex ð0Þ Mw
[17]
1 q2 ¼ 1 þ R2g 3 PðqÞ
[18]
Zimm plot II,
However, we do not work according to the scheme mode, as shown in Figure 2. Instead, for each sample at a given particle concentration, we measure the scattered intensity in the entire angle range of the applied instrument at least two or three times so as to obtain a scattered intensity curve with good statistics. A typical intensity curve, Iex(q), of the BSA clusters at c ¼ 0.30 g l1 measured in this way is shown in Figure 3(a) (diamonds). Note that after the measurements for each sample, the intensity of pure toluene (here we use toluene as the reference liquid) is immediately measured as well. After an intensity curve is ready, one can start to treat the data, and meanwhile leave the instrument to measure the intensity curve for another particle concentration.
Step 1: Treat the intensity curve based on Zimm plot II to get the zero-angle intensity and the radius of gyration For a given intensity curve as shown in Figure 3(a), the data is plotted in the form of 1/P(q) versus q2/3, that is, based on the above Zimm plot II, eqn [18], where the form factor is given by P ð qÞ ¼
Iex ðqÞ Iex ð0Þ
[29]
Note that initially the Iex(0) value is not precisely known, and the location of the 1/P(q) versus q2/3 plot in Figure 3(b) is rather random. Therefore, the data points (toward low q2/3 values) that exhibit the best straight line are selected, as those given by the triangles in Figure 3(b). Note that the number of points that follow a straight line decreases and moves toward smaller q2/3 values as the particle size increases. For the selected data points, a linear trendline simulation is performed using Iex(0) as the fitting parameter, and the objective is to have the intercept of the straight line at q2/3 ¼ 0 equals 1, as given by the straight line in Figure 3(b). In this way, the Iex(0) value, as well as the radius of gyration, Rg, is obtained from the slope of the line. Continuing to work in the same way for the other intensity curves at different particles concentrations, we obtain the Iex(0) and Rg values as a function of the particle concentration, as reported in Rows 2 and 3 of Table 1 where the measured intensity of toluene at each corresponding Iex(0) value is also reported (Row 4). Note that the obtained Rg value should be independent of the particle concentration. The reported Rg values in Table 1 decline slightly with c, which may indicate some restriction on the extension of the external random chains of the BSA clusters as their concentration increases.14
Step 2: Perform Zimm plot I to obtain the average molecular weight and the second virial coefficient From eqn [17], to perform Zimm plot I, computing K and Rex(0) is required. To estimate the constant, K, defined by eqn [8], the only missing parameter is dn0/dc, which can be measured by means of a refractive index detector. For known materials, it can easily be found in the literature. In this case of BSA, the dn0/dc value has been well determined in the literature,14,15 as being equal to 0.186 cm3 g1. To compute Rex(0) from Iex(0) through eqn [24], the refractive index and Rayleigh ratio of the reference liquid (toluene), nT (¼1.496) and RT, respectively, are required. Since the wavelength of the laser beam for the applied scattering instrument is l ¼ 532 nm, from eqn [25] or Figure 1, we obtain Ru,T ¼ 2.10 105 cm1. Considering the vertically polarized
Light Scattering – The Application of Static Light Scattering: Zimm Plot
7
Figure 3 A working example to show how one can obtain the values of the zero-angle intensity, Iex(0), and the radius of gyration, Rg, from the source intensity curve, Iex(q), through Zimm plot II.
Table 1 Values of the zero-angle intensity (Iex(0)) and radius of gyration (Rg) obtained from Zimm plot II, eqn [18], at various concentrations of the BSA clusters (c), where IT is the measured scattered intensity of toluene and Kc/Rex(0) is the quantity for Zimm plot I, eqn [17] c, g ml1 Iex(0) Rg, nm IT Kc/Rex(0)
5.00 105 3.68 103 51.9 4.51 102 9.12 108
1.00 104 7.11 103 51.3 4.51 102 9.43 108
2.00 104 1.00 104 51.0 3.28 102 9.76 108
3.00 104 1.99 104 49.8 4.51 102 1.01 107
4.00 104 2.54 104 48.9 4.51 102 1.06 107
5.00 104 3.36 104 48.7 4.98 102 1.10 107
incident light and detector with a vertical polarizer, with Ru,T and eqn [28], we obtain RT ¼ 2.13 105 cm1. It is now possible to compute Rex(0) from Iex(0), with which Kc/Rex(0) can be computed as a function of c. The results are reported in Table 1 and in the form of Zimm plot I in Figure 4. All the data points follow a straight line, and from its intercept and its slope, the average molecular weight and the second virial coefficient of the BSA clusters, Mw ¼ 1.12 107 g mol1 and B2 ¼ 2.03 105 cm3 mol g2, respectively, are obtained. The positive B2 value indicates that the BSA cluster dispersion is stable. Since the molecular weight of single BSA proteins is MBSA ¼ 6.6 104 g mol1, with the obtained Mw value of the BSA clusters, the average aggregation number in a BSA cluster, Nagg ¼ Mw/MBSA ¼ 169, can be obtained. This approach has been applied to investigate the colloidal stability (B2) of the BSA clusters mentioned earlier as a function of the CaCl2 concentration, Cs, and the results have been reproduced from the original work14 in Figure 5 (open circles). A particularly interesting finding is that at low CaCl2 concentration, the B2 value decreases as Cs increases, that is, as expected, the stability of the BSA clusters decreases as the CaCl2 concentration increases. However, when the B2 value reaches a local minimum, it changes trend and increases with Cs, that is, the stability improves with further addition of salt, CaCl2, which is an unexpected phenomenon. The most important result is that such stability behavior observed from the B2 values has
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Light Scattering – The Application of Static Light Scattering: Zimm Plot
Figure 4 An example of Zimm plot I, eqn [17], to estimate the average molar mass (Mw) and second virial coefficient (B2) of bovine serum albumin (BSA) clusters, as described in the literature. Reproduced from Wu, H.; Arosio, P.; Podolskaya, O. G.; Wei, D.; Morbidelli, M. Phys. Chem. Chem. Phys. 2012, 14(14), 4906–4916.
Figure 5 The second virial coefficient (B2) values obtained from Zimm plot I, eqn [17], at various CaCl2 concentrations (Cs), compared with the experimentally measured Fuchs stability ratio (W). Reproduced from Wu, H.; Arosio, P.; Podolskaya, O. G.; Wei, D.; Morbidelli, M. Phys. Chem. Chem. Phys. 2012, 14(14), 4906–4916.
been confirmed by independent experiments investigating the stability of the same system through determination of the Fuchs stability ratio, W, defined as 1 ð
W¼2
exp ½U ðr Þ=kB T dl Gl2
[30]
2
where l ¼ r/a, a is the radius of the particles, and G is a hydrodynamic function. Comparing eqn [30] with eqn [15], one can see that both the Fuchs stability ratio and the second virial coefficient are directly related to the pair-interaction potential, U(r), thus reflecting the stability of the particles. In fact, the measured W values for the same system, as shown in Figure 5, follow exactly the same trend as the B2 values do. These results suggest the possibility of interrogating the stability of colloidal suspensions through measurement of the second virial coefficient using Zimm plot analysis. The advantage of this approach is that it is suitable for investigating stable suspensions, while the more traditional method based on measuring aggregation rate constants is more suitable for investigating less stable suspensions; thus, they complement each other.
Conclusions Since the seminal publication by Zimm almost 70 years ago, the Zimm plot method has been routinely used to measure the molecular weight of polymers in solutions. In this work, this method has been critically reviewed, and it has been shown that a Zimm plot can be equally well applied to general colloidal suspensions. In the latter cases, care has to be taken to properly account
Light Scattering – The Application of Static Light Scattering: Zimm Plot
9
for the finite size of particles (proteins, clusters, etc.), which is included in the form factor and can lead to deviations from the traditional straight lines encountered in traditional Zimm plots. The added value of applying a Zimm plot to colloidal suspensions is the possibility of measuring not only the aggregation number, but also the second virial coefficient. An example has been discussed, showing how the trend of the measured second virial coefficient versus the ionic strength closely matches that of the Fuchs stability ratio for suspensions of protein clusters. This opens up new avenues in the understanding of colloidal interactions, which are crucial in determining the colloidal stability of particulate suspensions.
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