pubs.acs.org/Langmuir © 2009 American Chemical Society
Passive Microrheology of Solvent-Induced Fibrillar Protein Networks† Adam M. Corrigan and Athene M. Donald* Sector of Biological & Soft Systems, Department of Physics, University of Cambridge, Cambridge, U.K. Received December 21, 2008. Revised Manuscript Received February 23, 2009 Particle tracking microrheology (PTM) has been used to study the sol-gel transition in solvent-induced fibrillar β-lactoglobulin gels at room temperature and pH 7. The passive nature of microrheology allowed measurements to be made around and below the critical gelation concentration. The method of superposition introduced by Larsen and Furst (Larsen, T. H.; Furst, E. M. Phys. Rev. Lett. 2008, 100, 146001) was applied to the one-particle mean square displacement (MSD), yielding a critical relaxation exponent of n = 0.58 at concentrations close to the measured critical concentration of 4% (w/v). At a higher concentration of 12% (w/v), n was observed to decrease. The pregel and gel master curves were used to find the viscoelastic moduli over 8 decades of frequency. Combined with the measured shift factors, this allowed cure curves at 1 Hz to be constructed for direct comparison with results from bulk rheology. Timeindependent modulus superposition was found for all concentrations. Good agreement for concentration scaling was found between the traditional methods for characterizing gels and the recently described microrheological determination of the gel time and critical behavior.
Introduction A large amount of research has been dedicated to the study of gelation of the globular protein β-lactoglobulin ( βlg) because of a wide range of realized and potential applications from those in the food industry2 to the possibility of using milk proteins in cellular scaffolds.3 Depending on the conditions that the protein is exposed to, the networks formed can have very diverse morphologies. Under conditions where the native structure of the protein is only partially unfolded, fibrillar aggregates are slowly formed. One well-known form of such aggregates is the amyloid fibril, which forms typically for βlg at low pH and temperatures of up to 80 C4 and has been shown to form gels at concentrations as low as 2% (w/w) and below,5 depending on the ionic strength of the solvent. Fibrils have also been shown to form at room temperature in a mixture of pH 7 buffer and trifluoroethanol (TFE), among other similar organic solvents.6 These aggregates tend to be more wormlike and have not been specifically classified as amyloid. Differences in the fibril morphology imply differences in the reaction mechanism and kinetics for the solventinduced aggregation as compared to those for the heat-set amyloid gels. The time evolution of TFE-induced fibrillar gels has previously been studied using oscillatory shear rheometry,7 where measurements were made down to a concentration of 7% (w/v), still a distance away from the calculated critical concentration of ∼3%.7 Gel times showed a high power law (∼4) in concentration, tentatively explained in terms of the cooperative nucleation and † Part of the Gels and Fibrillar Networks: Molecular and Polymer Gels and Materials with Self-Assembled Fibrillar Networks special issue. *Corresponding author. E-mail:
[email protected].
(1) Larsen, T. H.; Furst, E. M. Phys. Rev. Lett. 2008, 100, 146001. (2) Kavanagh, G. M.; Clark, A. H.; Ross-Murphy, S. B. Int. J. Biol. Macromol. 2000, 28, 41–50. (3) Gravelend-Bikker, J.; de Kruif, C. Trends Food Sci. Technol. 2006, 17, 196– 203. (4) Bromley, E. H.; Krebs, M. R.; Donald, A. M. Faraday Discuss. 2005, 128, 13–27. (5) Veerman, C.; Ruis, H.; Sagis, L. M.; van der Linden, E. Biomacromolecules 2002, 3, 869–73. (6) Gosal, W. S.; Clark, A. H.; Ross-Murphy, S. B. Biomacromolecules 2004, 5, 2408–19. (7) Gosal, W. S.; Clark, A. H.; Ross-Murphy, S. B. Biomacromolecules 2004, 5, 2430–8.
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growth of oligomers, which then aggregated to form fibrils. Cure curves at different concentrations were superposed with limited success, possibly because of a systematic discrepancy in the gel time, which was defined as the point of logarithmic discontinuity in the elastic modulus. Microrheology is a recently developed collection of techniques that infer the local viscoelastic properties of a medium from the motion of embedded micrometer-sized tracer particles.8,9 The mean squared displacement (MSD) is calculated as a function of lag time, τ, for the tracers. For a simple viscous fluid, the MSD is diffusive with ÆΔr2(τ)æ ∼ τ, where the constant of proportionality is inversely proportional to the viscosity. In viscoelastic systems, the MSD becomes subdiffusive; that is, exponent R in ÆΔr2(τ)æ ∼ τR becomes less than unity for some range of τ. In this case, the MSD can be related to the viscoelastic modulus using a generalized Stokes-Einstein relation (GSER)10 G~ðsÞ ¼
dkB T 3πap sÆΔ~r2 ðsÞæ
ð1Þ
In the GSER, ÆΔ~r2(s)æ is the Laplace transform of the MSD, d is the number of dimensions in which measurements are made, ap is the particle radius, and s is the Laplace frequency. Calculating the viscoelastic moduli allows direct comparison with bulk rheological data; however, converting between time and frequency can introduce significant truncation errors near the extrema of frequency.11 A method introduced by Larsen and Furst1 avoids these errors by superposing the MSDs throughout the sol-gel transition to form a master curve; the divergence of the MSD shift factors identifies the gel point. Particle tracking microrheology (PTM) directly shows the tracers, typically using optical microscopy. This allows spatial information on the viscoelasticity to be found, meaning that the degree of heterogeneity can be quantified.12 (8) Mason, T. G.; Weitz, D. A. Phys. Rev. Lett. 1995, 74, 1250–1253. (9) Cicuta, P.; Donald, A. M. Soft Matter 2007, 3, 1449–1455. (10) Mason, T. Rheol. Acta 2000, 39, 371–8. (11) Dasgupta, B. R.; Tee, S. Y.; Crocker, J. C.; Frisken, B. J.; Weitz, D. A. Phys. Rev. E 2002, 65, 051505. (12) Savin, T.; Doyle, P. S. Phys. Rev. E 2007, 76, 021501.
Published on Web 04/03/2009
DOI: 10.1021/la804208q
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In this article, one-particle13 PTM is used to study the sol-gel transition of solvent-induced fibrillar β-lactoglobulin networks. The noninvasive nature of passive microrheology allows measurements at concentrations at and below the critical gelation concentration, c*, without disrupting the evolution of the system and provides reliable results before, after, and during the sol-gel transition. The method of Larsen and Furst is used to measure the critical exponents of the system, and from the resulting master curves, the GSER is used to extract the viscoelastic moduli, G0 and G00 , without truncation errors, thus allowing a direct comparison with previous bulk rheology in the literature.
Experimental Section Bovine β-lactoglobulin ( βlg, 90% pure) was purchased from Sigma-Aldrich in lyophilized powder form and used without further purification. Solutions of βlg in 50:50 TFE/pH 7 phosphate buffer were prepared as follows: βlg powder was allowed to dissolve fully in 0.5 mL of pH 7 phosphate buffer. Sulfatemodified latex polystyrene tracer particles (Sigma-Aldrich) of diameter 0.5 μm were added to give a final volume fraction of 0.01% before 0.5 mL of TFE was added to start the agregation process. A 40 μL aliquot was immediately taken and placed into an airtight cell14 of depth 1200 μm. In all cases, the incubation time was measured from when the sample was placed in the microrheology cell. The sulfate modification of the probe particles had a negative surface charge, minimizing the attraction of the negatively charged protein molecules. The samples were observed using a Zeiss LSM500 confocal microscope operated in bright-field mode. The field of view was fixed in the center of the cell to minimize edge effects. Videos were captured throughout the incubation of samples using an AVT Pike CCD camera at a frame rate of 60 Hz and a shutter time of 500 μs (sufficiently low to ensure that the dynamic error15 was negligible) and analyzed using in-house particle tracking software with a Laplacian-of-Gaussian (LoG) filter to determine particle centers to subpixel accuracy. Video lengths were limited so that the sample did not evolve during a single video. From particle positions, the ensemble-averaged mean squared displacement (MSD) was calculated, and from this, the complex modulus, G* (ω) = G0 (ω) + iG00 (ω) was extracted by finding the first and second logarithmic lag time derivatives of the MSD, analytically calculating the Laplace transform, and applying the GSER, following the method of Dasgupta et al.11
Results and Discussion The time evolution of β-lactoglobulin network formation in 50:50 TFE/pH 7 phosphate buffer was observed using 0.5-μmdiameter sulfate-modified latex probe particles for concentrations in the range of 4-12% w/v. Figure 1 shows the ensemble-averaged MSDs throughout the sol-gel transition for 6% (w/v) protein concentration. Each curve corresponds to a quasistatic time point during the incubation; the uppermost curve is the earliest incubation time (10 min), and as the incubation time increases, the magnitude of the MSD decreases. The collection of MSD curves is typical for all concentrations studied. At short incubation times, t, the system is purely viscous over the range of lag times, τ, accessible (1/60-2 s), giving a MSD that scales linearly with lag time: ÆΔr2(τ)æ = Aτ. As the incubation progresses, the value of coefficient A decreases, corresponding to an increase in the viscosity of the system. After a period of time, the MSD becomes subdiffusive at the shortest values of τ while remaining diffusive at longer lag times. (13) Gardel, M. L.; Valentine, M. T.; Crocker, J. C.; Bausch, A. R.; Weitz, D. A. Phys. Rev. Lett. 2003, 91, 158302. (14) Corrigan, A. M.; Donald, , A. M. Eur. Phys. J. E, DOI: 10.1140/epje/i200810439-7. (15) Savin, T.; Doyle, P. Biophys. J. 2005, 88, 623–638.
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Figure 1. Mean-squared displacement versus lag time for 0.5 μm latex probe particles in 6% (w/v) β-lactoglobulin in 50:50 TFE/ pH 7 phosphate buffer. Each curve corresponds to a single video under quasistatic conditions recorded at a given point during incubation. The dashed line, with a logarithmic slope of 0.59, marks the gel point, separating the pregel curves with upward curvature from the postgel curves with downward curvature.
(The upward curvature in the MSD curve at this incubation time is more clearly seen in Figure S1 in the Supporting Information.) The sample is now viscoelastic: there is a high-frequency elastic component in the system, which quickly relaxes and is dominated by the viscous response at low frequencies. As the incubation time increases still further, the MSD remains subdiffusive to longer and longer lag times until a point where the MSD is a subdiffusive power law at all frequencies; this is the gel point,16 shown as the dashed line with a power law exponent of 0.59 in Figure 1. The postgel MSDs at longer incubation times curve toward plateaus instead of recovering viscous behavior at large τ; now the probe particles are “trapped”, indicating that a viscoelastic solid has formed. The set of MSDs were found to superpose onto a master curve by application of horizontal and vertical logarithmic shifts. Timecure superposition is a long-established phenomenon17 arising because in many gelling polymer systems the structure of the system is not changing; only the characteristic length and time scales are shifting as polymer clusters grow over time. In practice, the shift factors are found as the multiplicative factors required to match the value and logarithmic time derivative of adjacent MSD curves. The MSDs of Figure 1 have been superposed to form two master curves, pregel and postgel, in Figure 2. Each curve has had the horizontal component, the lag time, multiplied by an independent shift factor, a, and the vertical component, the MSD, multiplied by a shift factor, b. Shift factors a and b were manually chosen to match the MSD gradient and curvature, giving the best superposition. At early incubation times, the system is purely viscous and the MSDs are linear over the range of lag times accessible (1/60-2 s), and the choice of shift factors is largely arbitrary to obtain satisfactory superposition. As the reaction proceeds, the fibrils grow and entangle to form regions of connectivity. These “clusters” have a spectrum of Rouse-like relaxation modes, each with a characteristic relaxation time. As fibrils grow longer, the connectivity length increases, as does the longest relaxation time. When the longest (16) Chambon, F.; Winter, H. H. J. Rheol. 1987, 31, 683–697. (17) Adolf, D.; Martin, J. E. Macromolecules 1990, 23, 3700–3704.
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Figure 2. Master curves produced by superposing MSD data before (upper) and after the gel point for 6% β-lactoglobulin in 50:50 TFE/pH 7 phosphate buffer. For clarity, the gel curve has been given an arbitrary shift downward to separate it from the pregel curve. (Inset) Shift factors, a (circles) and b (squares), as functions of incubation time.
relaxation time, τz, enters the observable range (i.e., becomes greater than 1/60 s), subdiffusive behavior is observed in the MSD. This relaxation time is illustrated by the lag time above which the MSD becomes diffusive. As the incubation time increases, τz continues to increase, reflecting the increase in the correlation length (length scale of connectivity) in the system until a samplespanning “infinite cluster” is formed. At this point, referred to as the gel point, the sample can support shear stress, so the longest relaxation time, τz, and the steady-state creep compliance, J0, diverge to infinity.1 The self-similar fractal nature of the network at this point and the presence of relaxation mechanisms over all time scales lead to critical power law scaling of the MSD, ÆΔr2(τ)æ ∼ τ n, over all lag times. In this system, the critical exponent, n, was measured as 0.59 ( 0.01 (Figure 2). The fibrils continue to grow after the gel point, and the mesh size of the network decreases from infinity. The sample is now a viscoelastic solid, meaning that the MSD tends to a plateau for long τ as the probe particles become trapped. This change in curvature from the pregel state means that the gel MSDs map onto a second master curve (Figure 2). As the mesh size decreases, the longest relaxation time also decreases, and the elastic plateau moves within the experimentally observed range of lag times to shorter times. The method of superposition is strictly valid only around the gel point.17 Here, the self-similar nature of the system means that changing the extent of reaction changes only the characteristic length scales (and therefore time scales) and not the structure of the network. By analogy to time-temperature superposition, measuring the system at one (pregel) stage in the reaction at a certain lag time is the same as measuring the system at a later stage and a longer lag time. The system may not be self-similar at very short or very long incubation times, and this leads to problems assigning vertical shift factors at early times and horizontal shift factors at long times. Nevertheless, for time points where unique superposition is possible, MSD data from other time points can be used to infer information about the MSD at different frequencies, allowing the construction of MSD curves that span many decades in frequency. The connection between the shift factors and the physical scales in the system can be seen as follows: the shift factors effectively map the curves so that the MSDs all display the same τz (the longest relaxation time) and the same J0 (steady-state creep Langmuir 2009, 25(15), 8599–8605
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compliance). Therefore, a MSD curve with a horizontal shift factor, a, of 0.1, for example, has a τz that is 10 times larger than the unshifted curve, so a ∼ τ-1 z . A similar argument applies for the creep compliance, giving b ∼ J0-1. Thus, the divergence of τz and J0 around the gel point1 is reflected by the critical behavior of the shift factors (inset in Figure 2): a εy and b εz, where ε = |pg - p|/pg is the distance from the gel point in terms of the bond probability, p, y, and z are the dynamic scaling exponents; and pg is the bond probability at the gel point. Because in practice p is rarely known, it is commonly assumed that it scales linearly with incubation time for cure experiments: |p -pg|/pg = |t - tg |/tg where tg is the gel time, measured as the incubation time for which the MSD(τ) is a constant power law and the shift factors diverge. This power law behavior is plotted in Figure 3, where the critical time exponent y is calculated to be 3.92 ( 0.09 and the critical modulus exponent z is calculated to be 2.43 ( 0.07. The extent of the critical region has been empirically defined as 10-2 < ε < 10-1,18 so the degree to which the exponents measured here, or indeed by Larsen and Furst, measure critical behavior is debatable.19 The relationship between the shift factors20 can be used to calculate the critical exponent, n. Larsen and Furst use the wellknown17 relationship n = z/y, giving n = 0.62 ( 0.02. However, if reliable measurements have not or cannot be made very close to the gel point, then the gel time, tg, may not be known precisely, giving systematic errors in y and z. As an alternative, we plot b against a (Figure 4). Because a εy and b εz, b az/y an, yielding n = 0.61 ( 0.02. Determining the critical relaxation exponent in this way has the advantage that the accuracy of the exponent does not rely on a precise choice of the gel point, though the calculation of dynamic scaling exponents still requires the gel time to be determined accurately. Consistency between our two values of n verifies that the gel time has been accurately determined. The critical gelation concentration, c*, was found experimentally to lie close to 4%, this being a concentration for which sometimes the gel point was reached, whereas other aliquots remained in a pregel state for more than 1 week. Samples with concentrations below 4% consistently failed to reach the microrheologically determined gel point. This value of c* is higher than the estimate by Gosal et al.,7 found by fitting bulk rheological data using the cascade theory of gelation. Increasing the β-lactoglobulin concentration increases the rate of gelation; at up to 12% (w/v), the time evolution was sufficiently slow to allow quasi-static measurements. Master curves were constructed independently for each concentration. The shape of the curves and the value of the exponent at the gel point are largely independent of concentration (Supporting Information Figure S2); however, deviation is seen at 12% w/v, especially near the gel point, which is found to have a smaller power law exponent of 0.45 ( 0.02. The ability to superpose the curves sheds light on a common underlying reaction mechanism and its concentration dependence. For concentrations between 4.5 and 7%, the critical exponent was found to be approximately constant within an experimental error of 0.58 ( 0.01 (Figure 5), which corresponds well with the MSD power law exponent at the gel point. This value is also in reasonable agreement with that obtained by Larsen and Furst for a β-hairpin peptide aggregate, 0.60 ( 0.02.1 There is more scatter in the values of the dynamic exponents (Supporting Information (18) Stauffer, D.; Coniglio, A.; Adam, M. Adv. Polym. Sci. 1982, 44, 103–158. (19) Ross-Murphy, S. B. Polym. Bull. 2007, 58, 119–126. (20) Cicuta, P.; Stancik, E. J.; Fuller, G. G. Phys. Rev. Lett. 2003, 90, 236101.
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Figure 3. Critical scaling of horizontal (left) and vertical (right) shift factors for a 6% (w/v) concentration. The dotted lines are fits to power law relationships, measuring the dynamic scaling exponents. (See the text.)
Figure 4. Relationship between shift factors b versus a for a 6% (w/v) concentration. The exponent of the power-law relationship is the critical exponent, n.
Figure 6. Elastic and loss moduli for the pregel (lower two curves) and postgel (upper two curves) systems, calculated from the MSD master curve. For clarity, the postgel curves have been shifted upward by a factor of 102, and only every fourth data point is plotted for each curve.
Figure 5. Critical exponent, n, as a function of concentration. Note that n is determined from the power law relationship between a and b, rather than z/y.
Figure S3); however, mean values were found for concentrations between 4.5 and 7% to be y = 3.9 ( 0.2 and z = 2.31 ( 0.08. These values are close to those expected from Rouse theory17 ( y = 4, z = 8/3, n = 2/3) and similar to those obtained by Larsen and Furst, especially in the deviation of the critical exponent, n, below the predicted Rouse exponent. The 12% data is different in the value of the critical exponent, 0.42 ( 0.01 8602 DOI: 10.1021/la804208q
(though again in agreement with the critical MSD power law for this concentration). As discussed previously, taking the Laplace transform or Fourier transform of the MSD in order to find the viscoelastic moduli can introduce truncation errors due to the finite nature of the MSD data. The size of the errors varies depending on the method of transform used,11 but in general becomes significant only near the extrema of frequency. Particularly in the case of the fast-evolving high concentrations, the length of the videos were limited such that the truncation errors significantly affect the value of the moduli obtained for a frequency of 1 Hz. However the method of superposition of the MSDs creates master curves for the pregel and gel states that are reliable over ∼8 orders of magnitude (Figure 6). This allows the moduli to be found over a similarly wide frequency range for all incubation times; only the scaling of the modulus and frequency changes as the gelation proceeds. The GSER is applied by calculating the first and second logarithmic time derivatives of the MSD, using the method of Dasgupta.11 This method was chosen because it minimizes the truncation errors, particularly for MSD data that is a slowly varying power law. The moduli master curves are shown in Figure 6 for a concentration of 6%. Langmuir 2009, 25(15), 8599–8605
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Figure 7. Extracted elastic modulus (+) and loss modulus (o) at 1 Hz for 6% w/v β-lactoglobulin in 50:50 TFE/pH 7 phosphate buffer. The MSD master curves were used to avoid truncation errors when applying the GSER. The dashed line indicates the gel point as determined from the MSD shift factors.
The pregel curve (lower two curves in Figure 6) shows viscous behavior at low frequencies, with a linear loss modulus given by η0ω and a negligibly small elastic modulus. Zero-shear viscosity η0 is given by the linear gradient of G00 as the frequency tends to zero and scales proportionally to the ratio of shift factors, b/a, because G00 ∼ b and ω ∼ a. Thus, effectively by definition, the zero-shear viscosity diverges, η0 ∼ ε-k, as expected17 at the gel point with viscosity exponent k = y - z. At high frequencies, the moduli approach critical behavior, with both G0 and G00 described by the same power law in frequency. As expected, the power law exponent in this region is 0.59, the same as the critical MSD exponent. As the incubation time increases, the experimentally accessible region of the frequency spectrum shifts toward higher frequencies, as the value of horizontal shift factor a decreases and the longest relaxation time in the sample increases. For an arbitrary incubation time after the gel point (upper two curves in Figure 6), a plateau is seen in G0 /b at low frequencies. Thus, the value of the plateau modulus, G0 , scales directly with b as the incubation time changes, increasing with time. As the incubation time increases, the horizontal shift factor decreases, so the plateau extends to higher and higher frequencies. At high frequency, there is a crossover in the two moduli, with G0 becoming greater than G00 . This occurs because the critical exponent for the 6% system is greater than 0.5, meaning that at the critical point the loss modulus exceeds the elastic modulus. At the gel point, there is no curvature in the MSD, which is a constant power law: ÆΔr2æ ∼ τn. Analytically applying the GSER10 yields G0 , G00 ∼ ωn. Hence, at the gel point the elastic and loss moduli are both described by the same power law in frequency, so the Winter-Chambon criterion16 is satisfied at the gel point with the same critical exponent, n, as found for the MSD. The moduli master curves can be used with the shift factors to evaluate G0 and G00 for all incubation times at 1 Hz, allowing a direct comparison with cure curves from the literature obtained using bulk rheology.7 The cure data derived from the 6% master curves is shown in Figure 7. The curve is very similar to those found for βlg using bulk rheometry,7 with the initial loss modulus significantly greater than the corresponding elastic modulus. After an early lag period, both moduli begin to increase rapidly, and G0 crosses G00 . This agreement with bulk rheology verifies that the probe particles are not affecting the gelation mechanism and Langmuir 2009, 25(15), 8599–8605
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also that (at the higher concentrations previously studied in the literature) the oscillatory shear rheometry did not disrupt the network formation. At long incubation times, G0 appears to plateau. This, however, is an artifact caused by the limited resolution of PTM; for modestly high concentrations, the true infinite-time modulus is beyond the resolution of 1-P microrheology. When the tracer movement becomes very small (∼5 nm), the apparent MSD is strongly affected by the static tracking error,15 placing an upper limit of ∼10 Pa on G0 for the 50 objective lens used. A distinct peak is visible in G00 shortly after the gel point, and similar findings in previous work7 suggest that this is a real feature and not an artifact. Previously, the maximum in G00 was tentatively identified as the point where the system has the largest fraction of dangling ends,21 corresponding to the greatest viscous dissipation. The identification of the gel time from cure curves is a widely debated issue in rheology;22,23 not having the frequency dependence of the moduli means that approximations must be used. Those most commonly employed are the point where G0 becomes greater than the corresponding G00 and the point in time where the logarithmic gradient of G0 jumps to a finite value.22 In this work, the accuracy of these approximations is investigated and compared with the true gel point where the MSD is a constant power law in lag time. To examine the concentration dependence of the gel properties, the change in MSD shift factors, a and b, at constant ε could be used. However, in practice the degree of scatter, combined with the arbitrary nature of the shift factors at short and long incubation times means that it is more reliable to use the extracted moduli shown in Figure 7. This has the added advantage of being able to compare the concentration dependence directly with bulk cure data. It has become common practice to attempt the superposition of the cure data for different concentrations using horizontal and vertical shifts.7,24,25 The ability to superpose the curves shows that varying the concentration does not change the mechanism of gelation, only the scaling of the viscoelasticity. Cure curve superposition was attempted (Supporting Information Figure S4) by normalizing the incubation time by the gel time and the moduli by the value of the modulus at the gel point. This differs from the approach taken by Gosal et al.,7 who used the elastic modulus extrapolated to infinite time. The curves superpose with limited success. Consistent with the MSD master curve interpretation, the 12% curves have a markedly different shape from the other concentrations; this difference is discussed later. Between 5 and 7% reasonable superposition is found, and the 4.5% curves tend asymptotically toward a constant value as the supply of monomer becomes exhausted before the elastic modulus increases above 10 Pa, the upper limit for 1-P microrheology. This shows, then, that the modulus scaling is different at the gel point than at long times because normalizing the vertical axis with the infinite-time modulus, as in Gosal et al.,7 does not give agreement at the gel point and vice versa. Therefore, good superposition will never be obtained for different concentrations over all incubation times. Instead, we propose the introduction of the loss angle, δ = arctan(G00 /G0 ), to remove the time dependence from the cure data. A loss angle of 90 corresponds to a purely viscous sample (21) Bibbo, M. A.; Valles, E. M. Macromolecules 1984, 17, 360–365. (22) Gosal, W. S.; Clark, A. H.; Ross-Murphy, S. B. Biomacromolecules 2004, 5, 2420–2429. (23) Kavanagh, G. M.; Ross-Murphy, S. B. Prog. Polym. Sci. 1998, 23, 533–562. (24) Meunier, V.; Nicolai, T.; Durand, D.; Parker, A. Macromolecules 1999, 32, 2610–2616. (25) Normand, V.; Muller, S.; Ravey, J. C.; Parker, A. Macromolecules 2000, 33, 1063–1071.
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Figure 8. Superposed master curve of the time-independent sample evolution, plotting |G| at 1 Hz as a function of loss angle, δ. Each curve is normalized by dividing by the gel-point modulus, G (tg). (Inset) Scaling of G(tg) with concentration.
whereas 45 is the point where G0 = G00 . The loss angle at the gel point is related to the critical exponent by δg = 90n,26 giving a critical loss angle of 52.2 at the gel point. This allows the evolution of the sample to be shown in a time-independent fashion by plotting the modulus against the balance between viscous and elastic behavior, given by δ. The shape of the timeindependent curves is very similar for all concentrations (Supporting Information Figure S5), including the highest and lowest concentrations, for which the time scaling broke down. The curves can be excellently superposed by normalizing the moduli by the value of the modulus at the gel point (Figure 8). The superposition holds before the gel point (δ > 52) and after (δ < 52) using only one shift factor, suggesting that the pregel viscosity and the postgel elasticity have the same dependence on concentration. There is significant scatter in the scaling of the modulus at the gel point with concentration (inset in Figure 8), reflecting the random nature of the aggregation particularly for small samples. Nevertheless, a power law dependence of the gel modulus on concentration is identified, with an exponent of 1.85 ( 0.10. This is similar to the value of 2 found and predicted for several polymer systems26 but cannot be compared directly with the exponent of 2.3 found by Gosal et al. for the infinite-time modulus. Figure 8 illustrates the concentration-independent evolution of the system beginning in the lower right corner of the figure, that is, the region that is purely viscous with a relatively low viscosity. As the aggregation process begins, the viscosity increases and the system moves upward along the curve before a network begins to form through physical cross-linking of the growing fibrils, giving rise to elasticity (δ < 90). The system evolves along the curve to the left as G0 begins to increase rapidly and the behavior of the system shifts from being mainly viscous to being dominated by the network elasticity. The evolution stops at the point where the monomer becomes exhausted. It is interesting that whereas the 12% data differed from the other concentrations in the MSD master curve, the value of the critical exponent, and the cure curve, the time-independent modulus superposes well in Figure 8, perhaps suggesting that any change in the gelation behavior affects the rates of reaction rather than the type of structures formed. Concentrations below c* move along the same curve, but (26) Nijenhuis, K. T. Adv. Polym. Sci. 1989, 130, 411–414.
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the supply of monomer becomes exhausted before the critical loss angle of δg = 52 is reached. The MSD gel point for low concentrations measured a critical exponent of 0.58, roughly independent of concentration near c*, giving a shorter gel time than the G0 - G00 crossover approximation and a longer gel time than the logarithmic derivative method. Despite these differing definitions of the gel point, the concentration dependence of all of the gel times are consistent (Supporting Information Figure S6), with concentration exponents of 3.2 ( 0.2 for the MSD gel time, 3.3 ( 0.3 for the G0 - G00 crossover, and 3.4 ( 0.2 for the logarithmic divergence in G0 . This perhaps implies that, at least around the gel point, all reaction kinetics have the same concentration dependence. The value of the gel-time exponent is consistent with results from bulk rheology measured further from the critical concentration.7 Similar and higher exponents have been found for other biopolymer gels. The cascade model in its simplest form predicts a gel-time exponent of -1, with a rapid divergent increase around c*27 that is not consistent with our data. A simple interpretation of the gel-time exponent might suggest that the rate-limiting step in the gelation involves three or four monomers. However, very high exponents (∼13) have been obtained for other biopolymer networks, suggesting that the exponent does not directly reflect the reaction order28 but rather that a critical divergence is being measured in the vicinity of the critical concentration. The kinetics of the gelation appeared to change at the higher concentration of 12%. This can be explained by an increase in the heterogenity, observed through an increase in the opacity of the sample, as well the excess kurtosis29 of the van Hove correlation function (results not shown). However, a further effect of the increased heterogenity is a greater amount of light scattering by the sample and increased sample opacity, leading to a larger static tracking error 30 Such an effect would cause an artifactual deviation in the master curve of the form seen for the 12% concentration (Figure S1 in Supporting Information), giving a lower apparent critical exponent. This cannot be ruled out as a possible cause of the difference.
Conclusions In summary, master curve behavior has been found for solventinduced fibrillar β-lactoglobulin gels using particle tracking microrheology, with viscoelastic properties showing the superposition of incubation time and concentration. Microrheology allows gelation to be monitored passively even at concentrations around c*, found experimentally to be ∼4% (w/v). The analysis introduced by Larsen and Furst yields a critical relaxation exponent of 0.58 for concentrations of up to 7% (w/v) and allows the gel time to be found with greater precision than from cure data. Care must be taken to control the magnitude of the static tracking error when finding critical exponents. The loss angle was introduced as a parameter to remove the time dependence from the data. Strong superposition for the timeindependent complex modulus at all concentrations suggested a common mechanism of aggregation independent of concentration. The agreement of this microrheological data with bulk rheological measurements from the literature suggests that, at least at the higher concentrations previously studied, the shear rheometry did not significantly disrupt the network formation. (27) Clark, A. H. Polym. Gels Networks 1993, 1, 139–158. (28) Ross-Murphy, S. B. Carbohydr. Polym. 1991, 14, 281 – 294. (29) Houghton, H.; Hasnain, I.; Donald, A. Eur. Phys. J. E 2008, 25, 119–127. (30) Papagiannopoulos, A.; Waigh, T. A.; Hardingham, T. E. Faraday Discuss. 2008, 139, 337–357.
Langmuir 2009, 25(15), 8599–8605
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The shift factor for incubation time had a concentration dependence proportional to c-3.2, consistent with previous bulk data and in common with other fibrillar gels. Acknowledgment. We thank Pietro Cicuta for helpful discussions. This work was funded by the EPSRC.
Langmuir 2009, 25(15), 8599–8605
Article
Supporting Information Available: Detailed pregel MSD curve, MSD master curves at all concentrations, dynamic exponents measured for each concentration, attempted superposition of cure data, time-independent representation of cure data, and concentration dependence of different geltime estimates. This material is available free of charge via the Internet at http://pubs.acs.org.
DOI: 10.1021/la804208q
8605
