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Analysis of macromolecular polydispersity by dynamic light scattering and thermodiffusion
Volume 23, number 2
OVFICS COMMU ICATIONS
November 1977
ANALYSIS OF MACROMOLECULAR POLYDISPERSITY BY DYNAMIC LIGHT SCATTERING AND THERMODIFFUSION ~
M. CORTI, V. DEGIORGIO, M. GIGLIO and A. VENDRAMINI
CI.S.L, Segrate {Milano),Italy Received 16 August 1977
We show that tile ratio between the average diffusion coefficient DLS determined by dynamic light scattering and the average diffusion coefficient DTD deternfined by dynamic thermodiffusion can be used to characterize sample polydispersity. As an example, we present data on solutions of polystyrene in cyclohexane.
As it has been recently shown [1], thermodiffusion processes can be exploited, by using a simple optical method, to accurately determine the diffusion coefficient D for various macromolecules in solution. This method can be an useful alternative to standard techniques such as free diffusion or dynamic light-scattering. Whereas all methods give tile same information in the case of monodisperse samples, the interpretation of the experiment and the comparison among the results obtained with different techniques become more complex if the macromolecular solution is polydisperse. Through an appropriate data analysis procedure an average diffusion coefficient can be obtained from both the dynamic light-scattering and the dynamic thermodiffusion method. We show in this paper that the two values can be appreciably different, since they represent two distinct averages over the size probability distribution of the sample. The situation is analogous to that encountered in molecular weight determinations by static light scattering (or centrifuging) and osmometry, by which the so-called weight-average M w and number-average M n molecular weight are derived respectively. We discuss here how joint measurements of D by light scattering and thermodiffusion can be used to obtain information on polydispersity, and we apply our considerations to experimental results obtained on polymers solution. Work supported by Consiglio Nazionale delle Ricerche/
CISE contract no. 76.00976.02. 282
The diffusion coefficient determination by intensity correlation-spectroscopy is based on the measurement of the autocorrelation function G20- ) of the l i ~ t intensity scattered from the solution. From G ()(z) the electric field normalized correlation Igtl)(r)l is easily derived [2]. For a polydisperse system Ig(1)(z)l consists of a sum of exponentials [g(1)O)l = ~
Gs(Di) exp(-Di k2 7),
(1)
where k = (4nn/X) sin 0/2 is the modulus of the scattering vector, 0 is the scattering angle, X is the wavelength of laser light, n is the index of refraction o f the solution, and Gs(Di) represents the fraction of the total scattered light which is due to macromolecules having diffusion coefficient D i. Straightforward analysis of the experimental correlation function G(2)(r) yields the average diffusion coefficient DLS DLS = ~DiGs(Di).
(2)
In the dynamic thermodiffusion method described in ref. [1] one utilizes the time evolution of the concentration gradient Vc at midheight of a Soret cell after suddenly applying a temperature difference between the two plates. The concentration gradient attains asymptotically a steady-state value (Vc)s = -(KT/T)VT, where VT is the applied temperature gradient, and K T is the thermal diffusion ratio. The quantity A(t) = (VC)s - Vc(t) can be written, for a polydisperse solution, as
Volume 23, number 2
OPTICS COMMUNICATIONS
November 1977
Its relation with the more familiar polydispersity
(3)
A(t) = ~ GT(Di) exp(-Dik'2t),
where k' = rr/a is the cell thickness, and GT(Di) is the fraction of the total steady-state concentration gradient which is due to macromolecules having diffusion coefficient D i. Eq. (3) has the same structure as eq. (1). From k(t) one derives the average diffusion coefficient DTD given by DTD = ~DiGT(Di).
(4)
The typical situation we consider here is that of a sample which may contain a mixture of linear polymer molecules with different numbers of monomer units per molecule. If the macromolecules under study are noninteracting and small compared to 1/K, the average intensity of light scattered by N macromolecules of molecular weight M is proportional to NM 2. If we further assume that tile thermal diffusion ratio K T is proportional to NM l+t3, and that the diffusion coefficient D is proportional to M - a , where the positive numbers a and/3 depend on both the solute and the solvent, it is easy to show that DLS and DTD can be written as
DES = A
~Ni M2-a " • ~._~N.tM2 '
DN;V]+a DTD = A
~ N i M ] +¢s
(5) where A is a proportionality constant. The exponent a is known for many polymer solutions [3]. The values range from 0.33 to 0.80. Less well known is the behavior of K T. The available experimental data suggest that the product KTD is proportional to the concentration and independent of the molecular weight for polymer solutions [4]. This result applies also to micellar solutions, according to unpublished data obtained in our laboratory. Incidentally, we recall that measurement_s along the critical isochore of a binary mixture near the consolution point have also shown the constancy of the product KTD [5]. If, consistently with all these results,/3 is taken coincident with c~, eq. (5) indicate that DTD is larger than DLS for polydisperse solutions, and equal to DLS when the solution is monodisperse. The ratio r = DTD/DLs can be used as an indication of the degree of polydispersity of macromolecular solutions.
indexMw/M n depends in general on the form of the molecular weight probability distribution p(M). If p(M) is sufficiently narrow, the fractional moments appearing in eqs. (5) can be expressed, by truncated Taylor series, as functions of the first moment M n and of the variance of p(M), so that DLS and DTD can be approximately written as DLS = A Mn c~{1
-
~-a(3
-
a)(SM2)/M2n) (6)
DTD = A M n c~{ l - ½c¢(1 + c0 (SM 2)/M 2 } and the ratio r becomes
r = 1 + c~l - ~)(SM2)/M 2.
(7)
The light scattering apparatus consists of a scattering photometer with temperature-controlled water bath and an argon laser operating at 5145 A, and a digital correlator built in our laboratory [6]. Measurements were made at a scattering angle of 90 °. The same samples used in the light scattering apparatus are successively used to fill the Soret cell for the thermodiffusion measurement. The cell has a thickness of 1 mm. The typical temperature difference between the plates was 0.7°C. The concentration gradient at midheight of the cell is measured by determining the corresponding index-of-refraction gradient by me ans of a laser-beam-deflection technique [ 1 ]. We have studied three samples of polystyrene in cyclohexane at 35°C, the theta point for this system. The first two samples are two narrow fraction of polystyrene supplied by the Pressure Chemical Company. The third sample is a mixture of the two narrow fractions. We report in table 1 the diffusion coefficients DLS and DTD measured with the three sample at two distinct concentrations 5 and 7.5 mg/cm a . It can be noticed that DTD is rather close to DLS for samples 1 and 2, but it is considerably larger than DES for the polydisperse sample 3. The values ofK T and DLS (or DTD) obtained with the samples 1 and 2 are consistent with the hypothesis that the product KTD is proportional to the concentration and independent of the molecular weight. This means that the exponent 13coincides with c~. The diffusion coefficients are in agreement with literature data [7,8]. In particular ref. [8] shows that ct = 0.5 for polystyrene in cyclohexane at the theta tempera283
Volume 23, mm~ber 2
OPTICS COMMUNICATIONS
November 1977
"Fable 1 Diffi~sion coefficient of three samples of polystyrene in cyclohexane at 35°(7 measured by light scattering and thermal diffusion at two distinct concentrations Sample
M w X 10 -4
Mw/Mn
1. Monodisperse
3.7 a)
1.06 a)
2. Monodispersc
11.0 a~
1.06 a)
3. Mixture of 1 and 2 705~ wt 1 30% wt 2
5.89 b)
1.281 b)
c mg/cm a
DLS X 10 7 cm 2/s
DTD X 10 7 cm 2/s
5 7.5 5 7.5 5 7.5
6.76 6.60 3.73 3.60 497 4.88
6.99 6.90 3.83 3.74 5.59 5.53
+- 0.10 -+ 0.10 -+ 0.06 +- 0.06 ,+ 0.08 -+ 0.08
± 0.10 +-0.10 ,+ 0.06 _+0.06 -+0.08 -+0.08
a) Manufiacmrer's data. b) ('alculated.
ture and that the effect o f interparticle interactions leads to a n o n negligible d e p e n d e n c e o f the diffusion coefficient on p o l y m e r concentration. Our calculations of tile ratio r for the m e a s u r e m e n t s reported in table 1 include the small correction due to the concentration d e p e n d e n c e o f D. We obtain r' = 1.035 -+ 0.020 as the average value relative to samples 1 and 2. and r 3 = 1.119 -+ 0.020 for sample 3. The fact that the difference DTD - DLS is larger than the e x p e r i m e n t a l u n c e r t a i n t y for the narrow fractions 1 and 2 can be due partially to the nonnegligible polydispersity o f the two samples and partially to calibration errors. According to the m a n u f a c t u r e r the ratio Mw/M n is ~<1.06 for b o t h samples. If we take Mw/M n = 1.06, the ratio r, as calculated f r o m eq. (7), is 1.015. Calibration errors are c o n n e c t e d with the fact that in order to obtain the diffusion coefficient f r o m the experimental t i m e constants b o t h k (see eq. (1)) and k ' (see eq. (3)) have to be k n o w n w i t h good accuracy. We estimate our calibration error to be certainly smaller than 2%. In any ease, we can correct r 3 for the c o m b i n e d effects of miscalibration and nonnegligible polydispersity o f samples 1 and 2 simply by dividing it by r'. Note that ttris procedure is rigorous i n d e p e n d e n t l y f r o m the k n o w l e d g e o f the origin of the departure f r o m 1 o f r ' . We obtain r/r' = 1.081 which is in excellent agreement w i t h the theoretical value 1.076 calculated from eq. (5). A l t h o u g h the ratio DTD/DLs is less sensitive to polydispersity than Mw/Mn, our m e t h o d can be in some cases m o r e c o n v e n i e n t than the molecular weight d e t e r m i n a t i o n s by light scattering (or centrifuging) and o s m o m e t r y , because o f the higher accuracy o f the diffusion coefficient measurements. The so-called 284
c u m u l a n t fit [7,9] of the intensity correlation function, which determines the deviation f r o m a single e x p o n e n t i a l shape, is potentially more attractive for two reasons: it requires only the dynamic-lightscattering apparatus and m a y provide, in principle, several m o m e n t s o f the size probability distribution. In practice, however, only the first m o m e n t and the variance can be extracted with some confidence from the c u m u l a n t fit [2]. It is interesting to note that, when c~ = 0.5, the relative variance u obtained by the c u m u l a n t fit coincides with r - 1. With spherical particles (c~ = 1/3), r - 1 = 2u. For this latter case, therefore, our m e t h o d is twice as sensitive as the cumulant fit.
References [1] M. Giglio and A. Vendramini, Phys. Rev. Letters 38 (1977) 26. [2] H.Z. Cummins and P.N. Pusey, in Photon correlation spectroscopy and velocimetry, eds. ft.Z. Cummins and E.R. Pike (Plenum Press, New York, 1977)p. 164. [3] C. Tanford, Physical chemistry of macromolecules (Wiley, New York, 1961). [4 ] G. Meyerhoff and K. Nachtigall, J. Polymer Sci. 57 (1962) 227. [5] M. Giglio and A. Vendramini, Phys. Rev. Letters 34 (1975) 561. [6] M. Corti and V. Degiorgio, J. Phys. C 8 (1975) 953; V. Degiorgio, in Photon correlation spectroscopy and velocimetry, eds. H.Z. Cummins and E.R. Pike (Plenum Press, New York, 1977) p. 142. [7] J.C. Brown, P.N. Pusey and R. Dietz, J. Chem. Phys. 62 (1975) 1136. [8] T.A. King, A. Knox, W.1. Lee, and J.D.G. MeAdam, Polymer 14 (1973) 151. 19] D.E. Koppel, J. Chem. Phys. 57 (1972) 4814.
OVFICS COMMU ICATIONS
November 1977
ANALYSIS OF MACROMOLECULAR POLYDISPERSITY BY DYNAMIC LIGHT SCATTERING AND THERMODIFFUSION ~
M. CORTI, V. DEGIORGIO, M. GIGLIO and A. VENDRAMINI
CI.S.L, Segrate {Milano),Italy Received 16 August 1977
We show that tile ratio between the average diffusion coefficient DLS determined by dynamic light scattering and the average diffusion coefficient DTD deternfined by dynamic thermodiffusion can be used to characterize sample polydispersity. As an example, we present data on solutions of polystyrene in cyclohexane.
As it has been recently shown [1], thermodiffusion processes can be exploited, by using a simple optical method, to accurately determine the diffusion coefficient D for various macromolecules in solution. This method can be an useful alternative to standard techniques such as free diffusion or dynamic light-scattering. Whereas all methods give tile same information in the case of monodisperse samples, the interpretation of the experiment and the comparison among the results obtained with different techniques become more complex if the macromolecular solution is polydisperse. Through an appropriate data analysis procedure an average diffusion coefficient can be obtained from both the dynamic light-scattering and the dynamic thermodiffusion method. We show in this paper that the two values can be appreciably different, since they represent two distinct averages over the size probability distribution of the sample. The situation is analogous to that encountered in molecular weight determinations by static light scattering (or centrifuging) and osmometry, by which the so-called weight-average M w and number-average M n molecular weight are derived respectively. We discuss here how joint measurements of D by light scattering and thermodiffusion can be used to obtain information on polydispersity, and we apply our considerations to experimental results obtained on polymers solution. Work supported by Consiglio Nazionale delle Ricerche/
CISE contract no. 76.00976.02. 282
The diffusion coefficient determination by intensity correlation-spectroscopy is based on the measurement of the autocorrelation function G20- ) of the l i ~ t intensity scattered from the solution. From G ()(z) the electric field normalized correlation Igtl)(r)l is easily derived [2]. For a polydisperse system Ig(1)(z)l consists of a sum of exponentials [g(1)O)l = ~
Gs(Di) exp(-Di k2 7),
(1)
where k = (4nn/X) sin 0/2 is the modulus of the scattering vector, 0 is the scattering angle, X is the wavelength of laser light, n is the index of refraction o f the solution, and Gs(Di) represents the fraction of the total scattered light which is due to macromolecules having diffusion coefficient D i. Straightforward analysis of the experimental correlation function G(2)(r) yields the average diffusion coefficient DLS DLS = ~DiGs(Di).
(2)
In the dynamic thermodiffusion method described in ref. [1] one utilizes the time evolution of the concentration gradient Vc at midheight of a Soret cell after suddenly applying a temperature difference between the two plates. The concentration gradient attains asymptotically a steady-state value (Vc)s = -(KT/T)VT, where VT is the applied temperature gradient, and K T is the thermal diffusion ratio. The quantity A(t) = (VC)s - Vc(t) can be written, for a polydisperse solution, as
Volume 23, number 2
OPTICS COMMUNICATIONS
November 1977
Its relation with the more familiar polydispersity
(3)
A(t) = ~ GT(Di) exp(-Dik'2t),
where k' = rr/a is the cell thickness, and GT(Di) is the fraction of the total steady-state concentration gradient which is due to macromolecules having diffusion coefficient D i. Eq. (3) has the same structure as eq. (1). From k(t) one derives the average diffusion coefficient DTD given by DTD = ~DiGT(Di).
(4)
The typical situation we consider here is that of a sample which may contain a mixture of linear polymer molecules with different numbers of monomer units per molecule. If the macromolecules under study are noninteracting and small compared to 1/K, the average intensity of light scattered by N macromolecules of molecular weight M is proportional to NM 2. If we further assume that tile thermal diffusion ratio K T is proportional to NM l+t3, and that the diffusion coefficient D is proportional to M - a , where the positive numbers a and/3 depend on both the solute and the solvent, it is easy to show that DLS and DTD can be written as
DES = A
~Ni M2-a " • ~._~N.tM2 '
DN;V]+a DTD = A
~ N i M ] +¢s
(5) where A is a proportionality constant. The exponent a is known for many polymer solutions [3]. The values range from 0.33 to 0.80. Less well known is the behavior of K T. The available experimental data suggest that the product KTD is proportional to the concentration and independent of the molecular weight for polymer solutions [4]. This result applies also to micellar solutions, according to unpublished data obtained in our laboratory. Incidentally, we recall that measurement_s along the critical isochore of a binary mixture near the consolution point have also shown the constancy of the product KTD [5]. If, consistently with all these results,/3 is taken coincident with c~, eq. (5) indicate that DTD is larger than DLS for polydisperse solutions, and equal to DLS when the solution is monodisperse. The ratio r = DTD/DLs can be used as an indication of the degree of polydispersity of macromolecular solutions.
indexMw/M n depends in general on the form of the molecular weight probability distribution p(M). If p(M) is sufficiently narrow, the fractional moments appearing in eqs. (5) can be expressed, by truncated Taylor series, as functions of the first moment M n and of the variance of p(M), so that DLS and DTD can be approximately written as DLS = A Mn c~{1
-
~-a(3
-
a)(SM2)/M2n) (6)
DTD = A M n c~{ l - ½c¢(1 + c0 (SM 2)/M 2 } and the ratio r becomes
r = 1 + c~l - ~)(SM2)/M 2.
(7)
The light scattering apparatus consists of a scattering photometer with temperature-controlled water bath and an argon laser operating at 5145 A, and a digital correlator built in our laboratory [6]. Measurements were made at a scattering angle of 90 °. The same samples used in the light scattering apparatus are successively used to fill the Soret cell for the thermodiffusion measurement. The cell has a thickness of 1 mm. The typical temperature difference between the plates was 0.7°C. The concentration gradient at midheight of the cell is measured by determining the corresponding index-of-refraction gradient by me ans of a laser-beam-deflection technique [ 1 ]. We have studied three samples of polystyrene in cyclohexane at 35°C, the theta point for this system. The first two samples are two narrow fraction of polystyrene supplied by the Pressure Chemical Company. The third sample is a mixture of the two narrow fractions. We report in table 1 the diffusion coefficients DLS and DTD measured with the three sample at two distinct concentrations 5 and 7.5 mg/cm a . It can be noticed that DTD is rather close to DLS for samples 1 and 2, but it is considerably larger than DES for the polydisperse sample 3. The values ofK T and DLS (or DTD) obtained with the samples 1 and 2 are consistent with the hypothesis that the product KTD is proportional to the concentration and independent of the molecular weight. This means that the exponent 13coincides with c~. The diffusion coefficients are in agreement with literature data [7,8]. In particular ref. [8] shows that ct = 0.5 for polystyrene in cyclohexane at the theta tempera283
Volume 23, mm~ber 2
OPTICS COMMUNICATIONS
November 1977
"Fable 1 Diffi~sion coefficient of three samples of polystyrene in cyclohexane at 35°(7 measured by light scattering and thermal diffusion at two distinct concentrations Sample
M w X 10 -4
Mw/Mn
1. Monodisperse
3.7 a)
1.06 a)
2. Monodispersc
11.0 a~
1.06 a)
3. Mixture of 1 and 2 705~ wt 1 30% wt 2
5.89 b)
1.281 b)
c mg/cm a
DLS X 10 7 cm 2/s
DTD X 10 7 cm 2/s
5 7.5 5 7.5 5 7.5
6.76 6.60 3.73 3.60 497 4.88
6.99 6.90 3.83 3.74 5.59 5.53
+- 0.10 -+ 0.10 -+ 0.06 +- 0.06 ,+ 0.08 -+ 0.08
± 0.10 +-0.10 ,+ 0.06 _+0.06 -+0.08 -+0.08
a) Manufiacmrer's data. b) ('alculated.
ture and that the effect o f interparticle interactions leads to a n o n negligible d e p e n d e n c e o f the diffusion coefficient on p o l y m e r concentration. Our calculations of tile ratio r for the m e a s u r e m e n t s reported in table 1 include the small correction due to the concentration d e p e n d e n c e o f D. We obtain r' = 1.035 -+ 0.020 as the average value relative to samples 1 and 2. and r 3 = 1.119 -+ 0.020 for sample 3. The fact that the difference DTD - DLS is larger than the e x p e r i m e n t a l u n c e r t a i n t y for the narrow fractions 1 and 2 can be due partially to the nonnegligible polydispersity o f the two samples and partially to calibration errors. According to the m a n u f a c t u r e r the ratio Mw/M n is ~<1.06 for b o t h samples. If we take Mw/M n = 1.06, the ratio r, as calculated f r o m eq. (7), is 1.015. Calibration errors are c o n n e c t e d with the fact that in order to obtain the diffusion coefficient f r o m the experimental t i m e constants b o t h k (see eq. (1)) and k ' (see eq. (3)) have to be k n o w n w i t h good accuracy. We estimate our calibration error to be certainly smaller than 2%. In any ease, we can correct r 3 for the c o m b i n e d effects of miscalibration and nonnegligible polydispersity o f samples 1 and 2 simply by dividing it by r'. Note that ttris procedure is rigorous i n d e p e n d e n t l y f r o m the k n o w l e d g e o f the origin of the departure f r o m 1 o f r ' . We obtain r/r' = 1.081 which is in excellent agreement w i t h the theoretical value 1.076 calculated from eq. (5). A l t h o u g h the ratio DTD/DLs is less sensitive to polydispersity than Mw/Mn, our m e t h o d can be in some cases m o r e c o n v e n i e n t than the molecular weight d e t e r m i n a t i o n s by light scattering (or centrifuging) and o s m o m e t r y , because o f the higher accuracy o f the diffusion coefficient measurements. The so-called 284
c u m u l a n t fit [7,9] of the intensity correlation function, which determines the deviation f r o m a single e x p o n e n t i a l shape, is potentially more attractive for two reasons: it requires only the dynamic-lightscattering apparatus and m a y provide, in principle, several m o m e n t s o f the size probability distribution. In practice, however, only the first m o m e n t and the variance can be extracted with some confidence from the c u m u l a n t fit [2]. It is interesting to note that, when c~ = 0.5, the relative variance u obtained by the c u m u l a n t fit coincides with r - 1. With spherical particles (c~ = 1/3), r - 1 = 2u. For this latter case, therefore, our m e t h o d is twice as sensitive as the cumulant fit.
References [1] M. Giglio and A. Vendramini, Phys. Rev. Letters 38 (1977) 26. [2] H.Z. Cummins and P.N. Pusey, in Photon correlation spectroscopy and velocimetry, eds. ft.Z. Cummins and E.R. Pike (Plenum Press, New York, 1977)p. 164. [3] C. Tanford, Physical chemistry of macromolecules (Wiley, New York, 1961). [4 ] G. Meyerhoff and K. Nachtigall, J. Polymer Sci. 57 (1962) 227. [5] M. Giglio and A. Vendramini, Phys. Rev. Letters 34 (1975) 561. [6] M. Corti and V. Degiorgio, J. Phys. C 8 (1975) 953; V. Degiorgio, in Photon correlation spectroscopy and velocimetry, eds. H.Z. Cummins and E.R. Pike (Plenum Press, New York, 1977) p. 142. [7] J.C. Brown, P.N. Pusey and R. Dietz, J. Chem. Phys. 62 (1975) 1136. [8] T.A. King, A. Knox, W.1. Lee, and J.D.G. MeAdam, Polymer 14 (1973) 151. 19] D.E. Koppel, J. Chem. Phys. 57 (1972) 4814.