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Langmuir 2007, 23, 776-784
Chain Dynamics in Microgels: Poly(N-vinylcaprolactam-co-N-vinylpyrrolidone) Microgels as Examples Volodymyr Boyko,† Sven Richter,*,† Walther Burchard,‡ and Karl-Friedrich Arndt† Physical Chemistry of Polymers, Dresden UniVersity of Technology, Mommsenstrasse 13, D-01062 Dresden, and Institute of Macromolecular Chemistry, Albert-Ludwigs-UniVersity of Freiburg, Stefan-Meier-Strasse 31, D-79104 Freiburg, Germany ReceiVed July 25, 2006. In Final Form: September 21, 2006 Microgels are highly swollen colloids built up of flexible cross-linked chains. We studied the static and dynamic light scattering (LS) behavior of thermosensitive microgels based on N-vinylcaprolactam and N-vinylpyrrolidone prepared by precipitation copolymerization in H2O (CP-1) and D2O (CP-2). Striking differences in behavior were observed in the two solvents. In both cases the angular dependence of static LS could reasonably well be described by a soft sphere model (J. Polym. Sci., Polym. Phys. Ed. 1982, 20, 157) with small deviations at large qRg. At temperatures larger than the collapse temperatures, the CP-1 sample in water started to aggregate whereas the CP-2 sample in D2O showed no association and developed the expected change toward hard sphere behavior. Dynamic LS permitted the determination of internal or segmental mobility. A remarkable shift toward large qRg was found for CP-1 compared to the behavior of linear chains. The dynamic behavior is clearly displayed in a plot of Γ*(q) ) (Γ1(q)/ q3)(η0/kT), with Γ1(q) the first cumulant of the field time correlation function and the common meaning of the other parameters. A long range of hard sphere behavior indicated the suppression of internal modes, but at large qRg the swollen microgel CP-1 in water displayed internal motions with a spectrum similar to that of Zimm relaxations. No internal mobility could be detected with the CP-2 sample in D2O. The behavior is in agreement with observations in the literature. The differences in the two similar solvents were attributed to the poorer solvent quality of D2O.
Introduction For a long time the interest of polymer scientists was concerned with macroscopic gels, i.e., polymer networks which are highly swollen in a solvent. Recently more attention has been paid to water-soluble polymers and the corresponding hydrogels. Fascinating properties were obtained with hydrophobic networks in water exhibiting a lower critical solution temperature or with ionically charged networks under the influence of pH and salt.1 Combined with phase separation, a pronounced shrinking of the volume is observed, unfortunately only at a rather low rate. Soon, this recognition was responded by the idea to reduce the size of the gels, which immediately led to the preparation of microgels. Water-swollen microgels of nanosize became of interest to pharmacists. Such nanogels, injected into the blood stream, would be capable of circulating unrestricted through capillary vessels. This capability comprises the possibility of using the nanogels as carriers for drugs which would be released in a controlled manner on surfaces of special cells. Certainly, the rate of collapse and drug release are related to the internal or segmental mobility. Contrary to that in macrogels, the mobility of chains in microgels cannot be characterized by common rheology. Here dynamic light scattering offers a new possibility with future aspects. Theories and experiments on large linear chains revealed that internal modes of motion are probed at sufficiently large values of qRg or qRh, where Rg is the radius of gyration and Rh the hydrodynamic radius, defined via the translational diffusion coefficient. The parameter q ) (4πn0/λ0) * To whom correspondence should be addressed. E-mail:
[email protected]. Phone: ++49-351-463-32492. Fax: ++49-351-463-37122. † Dresden University of Technology. ‡ Albert-Ludwigs-University of Freiburg. (1) Shibayama, M.; Tanaka, T. AdV. Polym. Sci. 1993, 109, 1.
sin(θ/2) is related to the scattering angle θ, the wavelength of the light λ0 in a vacuum, and the refractive index n0 of the solvent. The q parameter has a dimension of 1/r such that qr is a dimensionless parameter. Thus, small q values correspond to large distances in space and large q values to short chain sections. Let us assume values of qRg > 4 could be realized in light scattering. Then qRg > 4 indicates length scales being probed more than 4 times shorter than the radius of gyration. In static light scattering (SLS) the information of internal structure can be discriminated from the global structures or shape of the particles. Similarly, in dynamic light scattering (DLS) the translational diffusion coefficient is measured at low qRg, but at high qRg the motion of the internal structures can be detected. Most of the recent investigations were concerned with crosslinked poly(NIPAM) chains. An overview on thermally sensitive examples is given by Pelton.2 Our interest was directed to a generalization of the collapse behavior on heating, and for this reason we were looking for other polymers of similar behavior which so far have not been considered. We chose N-vinylcaprolactam (VCL) as the major chain component. Many years ago one of the present authors performed a systematic study3a,b of the solution properties of three linear poly(N-vinyllactam)s, i.e., poly(N-vinylpyrrolidone), poly(N-vinylpiperidine), and poly(N-vinylcaprolactam).3b A systematic decrease of the lower critical solution temperature was observed of about 30 °C per -CH2group added in the lactam ring. A comment appears to be necessary on how microgels can be prepared. One possibility consists of the cross-linking emulsion polymerization within uniformly sized latex particles.4 Soluble particles were obtained after removal of the surrounding (2) Pelton, R. AdV. Colloid Interface Sci. 2000, 85, 1. (3) (a) Eisele, M.; Burchard, W. Makromol. Chem. 1990, 191, 169. (b) Eisele, M. Ph.D. Thesis, University of Freiburg, 1990. (c) Burchard, W. Unpublished data, 1965.
10.1021/la062181+ CCC: $37.00 © 2007 American Chemical Society Published on Web 11/18/2006
Chain Dynamics in Microgels
surfactant. These particles showed properties very similar to the properties of common macrogels, with the exception of the critical behavior, which significantly deviated from that of macrogels.5-7 Another possibility goes back to Frisch8 and Kilb.9 If the overall concentration of monomers is much too low to form a connective network that extends from wall to wall in the reaction vessel, the growth of the clusters becomes limited to a finite size. Still present nonreacted cross-linking agent causes excessive ring formation, and a micronetwork is obtained. Such microgels are unstable colloids and cannot fully be redispersed after precipitation. Fortunately, most water-soluble radical initiators carry ionic charges. Once such a radical has initiated polymerization of a hydrophobic chain, it remains attached to the end group and forms in a way a charged polymer-surfactant. This structure stabilizes the microgel via Coulomb repulsive interaction among the particles if the microgel has reached a sufficiently large size.2 A particular situation occurs if at high temperature the monomers are soluble but the polymer chain changes the hydrophilicity/ hydrophobicity balance to hydrophobicity and precipitates. This procedure is sometimes called precipitating microgel polymerization.2 In the present study we applied the latter method. General characteristics of this technique were given by Pelton2 and Gilbert.10 The paper is organized as follows. We first give an outline of the theoretical background of DLS. Although DLS is now a rather commonly applied technique, we give a rather detailed outline, because in the literature the effect of segmental motions is often not taken into account. No special theoretical outline on SLS is given. We presume this technique is well-known. It is now described in all modern textbooks of polymer and colloidal science. After a short Experimental Section, the experimental results are given. In the ensuing section we discuss at greater detail the observations made by static and dynamic light scattering. Finally, the main findings are emphasized in the Conclusions.
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are made at any angle. These particles are essentially rigid, and only the translational motion of the center of mass can be observed. A weak angular dependence arises from a size distribution of the particles. In contrast, macromolecules are composed of mostly flexible linear, branched, or cross-linked chains. Now, segmental modes of motion are superimposed upon the translational motion.11a,b The technique and possibilities of DLS are well explored. In the following we still repeat some relevant relationships to avoid misunderstanding in the notation. The theory of Gaussian chain dynamics was developed by Rouse14 under the neglect of hydrodynamic interactions (HIs) and was later extended by Zimm15 to linear chains with hydrodynamic interactions. The corresponding relaxation modes were found to scale asymptotically as shown in eqs 1 and 2.
(4) (a) Nerger, D. Ph.D. Thesis, University of Freiburg, 1978. (b) Schmidt, M.; Burchard, W.; Nerger, D. Polymer 1979, 20, 582. (5) Burchard, W. Polym. Bull. 2007, 58, 3. (6) Flory, P. J. Principles of Polymer Chemistry: Cornell University Press: Ithaca, NY, 1953. (7) Stepto, R. F. T., Ed. Polymer Networks. Principles of their Formation, Structure and Properties; Blackie Academic & Professional: London, 1998. (8) Frisch, H. Proceedings of Polymer DiVision Meeting; American Chemical Society: Washington, DC, 1955; Vol. 128. (9) Kilb, R. W. J. Phys. Chem. 1958, 62, 969. (10) Gilbert, R. Emulsion Polymerization: A Mechanistic Approach; Academic Press: London, 1995. (11) (a) Berne, B. J.; Pecora, R. Dynamic Light Scattering; Wiley: New York, 1976. (b) Lo¨we, H.; Mu¨ller, P.; Zippelius A. J. Phys.: Condens. Matter 2005, 17, S1659. (12) Benmouna, M., Akcasu, A. Z. Macromolecules 1980, 13, 409. (13) Han, C. C.; Akcasu, A. Z. Macromolecules 1981, 14, 1080.
(1)
τj(Zimm) ) τ1/j3
(2)
Often the notation of free draining and nonfree draining is used. This notation corresponds to limiting situations which are described by the Rouse or Zimm relaxation models.11b In dilute solutions the hydrodynamic interaction is dominant, but at high concentrations, when random coils strongly overlap, HI is largely screened, and Rouse relaxation starts to dominate.16 First calculations of the time correlation function (TCF) in dynamic light scattering were made by de Gennes et al.17-19 for infinitely long and flexible chains. Pecora11a and later Perico et al.20 and Akcasu et al.21 treated finite chains. The dynamic structure factor of flexible chains has a complex form11a,22 and may be written as ∞
S(q,t) ) exp[-Dq2t]
Theoretical Background of Dynamic Light Scattering DLS is mostly used for a quick determination of translational diffusion coefficients. The procedure is simple for spherical particles or for irregularly shaped particles which are small compared to the wavelength of the used light. Measurements at only one scattering angle are sufficient in these cases, but a wrong answer is obtained with large linear or branched chain molecules. The translational diffusion coefficient is a macroscopic transport coefficient, but DLS is actually a technique that records all types of motions excited by thermal fluctuations. If the measurements are made in the limit of small q values such that qRg < 1, the whole particle is seen.11a-13 On the other hand, scattering measurements made under conditions of qRg . 2 probe distances in a particle which are much smaller than the particle diameter. For most colloidal particles there will be no difference in the result for the diffusion coefficient when the experiments
τj(Rouse) ) τ1/j2
∑Sn(q,t) n)0
(3)
with
Sn(q,t) ) Pn(q) exp(-t/τn)
(4)
where D is the translational diffusion coefficient, τn is the nth relaxation mode in the particle, and Pn(q) are the weight factors for the nth mode and depend on the scattering angle. For the first five modes analytic equations were derived.11a,20-22 Experimentally there will scarcely be a chance to separate the discrete contribution of higher than five modes, because in real systems the various relaxation modes have a certain time distribution. This can arise from chemical and size heterogeneities. The TCF merges into a function decaying continuously with qn, as was already found by de Gennes et al.17-19 Akcasu et al.21 completed the calculations for finite chain lengths and came to a number of important conclusions: First, the asymptotic behavior of the TCF was confirmed, which may be written in the logarithmic form (14) Rouse, P. E., Jr. J. Chem. Phys. 1953, 21, 1272. (15) Zimm, B. H. J. Chem. Phys. 1956, 24, 269. (16) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, 1986. (17) de Gennes, P. G. Physics 1967, 3, 37. (18) Dubois-Violette E.; de Gennes, P. G. Physics 1967, 3, 181. (19) Jannink, G.; de Gennes, P. G. J. Chem. Phys. 1968, 48, 2260. (20) Perico, A.; Piaggio, P.; Cuniberti, C. J. Chem. Phys. 1975, 62, 2690. (21) Akcasu, A. Z.; Benmouna, M.; Han, C. C. Polymer 1980, 21, 866. (22) Sorlie, S. S.; Pecora, R. Macromolecules 1988, 21, 1437.
778 Langmuir, Vol. 23, No. 2, 2007
ln g1(q,t) f ln a - b(t/t*)β
Boyko et al.
(5)
with β ) 1/2 for Rouse and β ) 2/3 for Zimm relaxations under hydrodynamic interactions. The arrow in eq 5 indicates asymptotic behavior at large q. Second, the scaling time t* is well represented by the first cumulant Γ1(q) ) 1/t* in the cumulant expansion of the TCF23
ln g1(q,t) ≈ Γ0 - Γ1(q)t +
Γ2(q) 2 Γ3(q) 3 t t + higher terms 2! 3! (6)
The first cumulant can be written as
Γ1(q) ) q2Dapp(q)
(7)
where formally Dapp(q) is a diffusion coefficient, but it is only an apparent one, because it contains a q dependence which arises from the internal modes of motion. Only after extrapolation to zero scattering angle (q f 0) is the translational diffusion coefficient obtained (Γ1(q f 0) ) q2Dtrans). Applying the StokesEinstein relationship, the hydrodynamic radius Rh is obtained. At large qRg and for chains with HIs, the asymptotes are
Γ1(q) f 0.0789(kT/η0)q3 (in a good solvent)
(8a)
Γ1(q) f 0.0625(kT/η0)q3 (in a Θ solvent)
(8b)
and
which were obtained without using the common hydrodynamic preaveraging.12 It is convenient to define a reduced first cumulant
Γ*(q) ≡
η0 Γ1(q) kT q3
(9)
which is a dimensionless quantity. It is worth emphasizing that in the asymptotic regime Γ*(∞) is a constant and has universal character. It remains valid for any type of flexible chain. Another way of checking the theoretical results consists of the consideration of Dapp(q)/Dz as a function of qRh, which, again, for linear chains should be a universal function. Benmouna and Akcasu12 considered in detail the two limiting regions of very small qRh and of qRh > 4. In the low regime the translational coefficient is obtained, but beyond qRh ≈ 1 a transition to a linear increase should be observed. These conclusions from the dynamic theory were very satisfactorily confirmed by experiments.13 An overview is given in ref 24. All these equations have been derived for linear chains. Similar behavior was expected also for branched macromolecules and swollen microgels. This expectation was based on the fact that at large qRg essentially flexible linear chain sections, connecting two points of branching points, are probed in scattering experiments. These chains should obey random coil behavior, perturbed only by excluded volume interactions. On the other hand, for hard spheres Dapp(q)/Dz is independent of q, because there is no internal mobility. This makes clear that the Dapp(q)/Dz curve cannot be a universal function for any polymer architecture.26,27 Indeed, a shift toward larger values than qRg . 1 for the onset of a linear increase was observed for samples of intermediate rigidity. Examples are shown in Figure 4 and are (23) Koppel, D. E. J. Chem. Phys. 1972, 57, 4814. (24) Trappe, V.; Burchard, W. Local dynamics in branched polymers. In Light Scattering and Photon Correlation Spectroscopy; Pike, E. R., Abbiss, J. B., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1997; pp 142-160.
Figure 1. Chemical structure of statistical VCL/VP copolymer, the cross-linker MBA, and the water-soluble initiator AMPA. A random composition of the monomer units along the chain length was assumed. Table 1. Composition of the Monomers VCL and VP, the Cross-Linker MBA, and the Initiator AMPA in the Reaction Volume for the Preparation of the Microgels CP-1 and CP-2 sample
solvent
VCL mass/g
VP mass/g
MBA mass/g
initiator vol/mL
vol/ mL
CP-1 CP-2
H2O D2O
1.67 0.835
0.38 0.19
0.06 0.066
5 5
150 100
discussed later in this paper. In this context measurements by Wu et al.26,27 with linear poly(NIPAM) chains and with poly(NIPAM) microgels are of interest. The authors could separate at low q values up to five modes of motion. They observed that the onset of the lowest relaxation was shifted for the microgel by a factor of 2.4 to larger q values compared to that of linear chains. The observed Zimm relaxations times τn in eq 4 were quantitatively assigned by some authors.20,22,26-28 Experimental Section In this work, the dynamics of a thermosensitive microgel based on a cross-linked copolymer of VCL and VP was studied by static and dynamic LS over a wide range of scattering angles. Components. Figure 1 shows the chemical structures of the two monomers, the cross-linker, and the water-soluble initiator. Synthesis. We followed a recipe that was suggested by Pelton and Chibante29 and applied it to the cross-linking copolymerization of N-vinylcaprolactam/N-vinylpyrrolidone comonomers. The moieties of the various components are given in Table 1 for the two samples CP-1 in water and CP-2 in heavy water. For example, sample CP-1 was obtained with 1.67 g of VCL, 0.38 g of VP, and 0.06 g of the cross-linker N,N′-methylenebisacrylamide (MBA) dissolved in 145 mL of deionized water. The components were filled in a double-wall glass reactor, equipped with a stirrer and reflux condenser. The solution of the monomers was placed into the reactor and stirred at 70 °C for 1 h while being purged with nitrogen. No turbidity of the reaction solution was observed at this point. After this time, 5 mL of an aqueous solution of 2,2′-azobis(2-methylpropionamidine) dihydrochloride (AMPA) (5 mg/mL) was added under continuous stirring to initiate the radical polymerization. Turbidity soon developed, indicating phase separation to a stabilized suspension of collapsed microgels. The time of reaction was 8 h in both cases. The polymer dispersion was freed from monomer and un-cross-linked polymers (the sol fraction) by dialysis for 24 h against water, using a Biomax 100 (Millipore) membrane. In both reactions (25) Trappe, V.; Bauer, J.; Weissmu¨ller, M.; Burchard, W. Macromolecules 1997, 30, 2365. (26) Wu, C.; Zhou, S. Macromolecules 1996, 29, 1574. (27) Wu, C.; Chan, K. K.; Xia, K.-Q. Macromolecules 1995, 28, 1032. (28) Chu, B.; Wang, Z.; Yu, J. Macromolecules 1991, 24, 6832. (29) Pelton, R.; Chibante, P. Colloids Surf. 1986, 20, 247.
Chain Dynamics in Microgels
Figure 2. Guinier plot (logarithmically modified Zimm plot) from the CP-1 microgel in H2O at 15 °C. Measurements were made in the concentration range of 0.01-015 mg/mL. The molecular parameters of these two samples at 15 °C are collected in Table 2. the conditions were almost identical, with the exception of the amount of cross-linker, which was 2 times larger in the D2O solution, and the overall concentration in D2O was more diluted. See Table 1. All chemicals were products of Aldrich. The monomers were purified by distillation under vacuum prior to use. Static Light Scattering. SLS experiments were performed with a computer-driven FICA 50 photogoniometer. The modification of the original FICA system was made by “SLS Systemtechnik” G. Baur, Denzlingen, Germany. A He-Ne laser (λ0 ) 632.8 nm) served as the light source. The measurements were made in the scattering angle range between 15° and 145°, in steps of 5°, and in a temperature range of 15-50 °C. The refractive index increment was measured separately in a differential refractometer (SLS Systemtechnik G. Baur) with the light of a He-Ne laser. The values were dn/dc ) 0.200 cm3/g for CP-1 in water and dn/dc ) 0.223 cm3/g for CP-2 in D2O at 15 °C. Dynamic Light Scattering. An ALV DLS/SLS-5000 light scattering system was used equipped with an ALV-5000/EPP multiple digital time correlator. A Uniphase 1145P 22 mW He-Ne laser (λ ) 632.8 nm) served as the light source. The DLS experiments were carried out in the scattering angle range between 30° and 110° in steps of 10°. The temperature was varied between 15 and 50 °C ((0.1 °C) by thermostating the toluene refractive index matching bath. The viscosities of water and heavy water, needed for the calculation of the hydrodynamic radius Rh, were taken from refs 30 and 31.
Results A preparation of stabilized microgels, based on N-vinylcaprolactam alone, was not possible without using surfactants or other additives. The copolymerization of VCL with VP, above the lower critical solution temperature (LCST) of poly(Nvinylcaprolactam) (PVCL) (>32 °C), in the presence of MBA as cross-linker was more successful. The hydrophilic PVP chain segments in the copolymer are placed near the surface of the particles. Apparently this fact supported the stabilization of microgels in addition to the effect of the ionic groups of the initiator, attached at the end of the chain.32 To improve the accuracy of the zero angle extrapolation, the data from CP-1 at 15 °C are displayed in Figure 2 as a Guinier33 plot, a logarithmic modification of the more common Zimm34 plot. PVCL has an LCST of about 32 °C.3c The introduction of VP units to the copolymer network is expected to increase this (30) Swindells, J. F. CRC Handbook of Chemistry and Physics; The Chemical Rubber Co.: Clevelend, OH, 1970. (31) ηD2O: NIST Chemistry WebBook. http://webbook.nist.gov/chemistry/. (32) Siu, M.; Zhang, G.; Wu, C. Macromolecules 2002, 35, 2723. (33) Guinier, A.; Fournet, G. Small-Angle Scattering of X-Rays; Wiley: New York, 1955.
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Figure 3. Change of the radii of gyration and hydrodynamic radii with temperature and the corresponding F ) Rg/Rh values of the two types of microgels (CP-1 in H2O and CP-2 in D2O). The measurements of CP-1 in water displayed a weak aggregation and were not further investigated. Table 2. Weight Average Molar Mass Mw, Radius of Gyration Rg, Hydrodynamic Radius Rh, Second Virial Coefficient A2, and the Ratio G of the Two Samples at 15 °C, Determined by SLS and DLS sample
Mw × 106/ g mol-1
Rg/ nm
Rh/ nm
F ) Rg/Rh
A2/ mol cm3 g-2
CP-1 (H2O) CP-2 (D2O)
721 1330
236 189
371 365
0.63 0.52
6.05 × 10-7 1.77 × 10-7
temperature, because its LCST was found at 103 °C. We measured the influence of the temperature on the size of the microgel particles by combined static and dynamic LS. The ratio F ) Rg/Rh is a conformation-sensitive parameter and gives valuable information on the segmental density in the particle.35 Figure 3 shows the change of Rg and Rh and their ratio F for both samples, prepared in water and heavy water. The radii Rg and Rh continuously decreased with heating for both samples. Around 38 °C aggregation occurred in the CP-1 sample, prepared in water. Measurements from higher temperatures were not evaluated. The aggregation indicated insufficient sterical stabilization by the hydrophilic PVP chain segments and the ionic charges from the initiator. On the contrary, the CP-2 sample, prepared in heavy water, was stable in the whole temperature range of investigation. A broad transition in a range from 20 to 40 °C was observed for both samples. Table 2 gives a list of molecular parameters measured at 15 °C. The initial slope in the field TCF is the first cumulant Γ1(q), and the ratio of Γ1(q)/q2 ≡ Dapp(q) gives an apparent diffusion coefficient that has an initial dependence of36
Dapp(q)/Dz ) 1 + Cq2Rg2 - ...
(10)
where Dz is the z-average translational diffusion coefficient, obtained at q ) 0. Figure 4 shows the experimental curves from CP-1 and CP-2 at some selected temperatures. In theoretical derivations qRg was used as an independent parameter. However, since the hydrodynamic radius is measured at q ) 0, qRh appears to be the more sensible parameter. Indeed, qRh was found the better adjusted parameter in other examples.24,25,32 (34) Zimm, B. H. J. Chem. Phys. 1948, 16, 1099. (35) Burchard, W.; Schmidt, M.; Stockmayer, W. H. Macromolecules 1980, 13, 1265. (36) Stockmayer, W. H.; Burchard, W. J. Chem. Phys. 1979, 70, 3138.
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Boyko et al.
Figure 5. Log-log plot of 1/P(u) against u2 ) (qRg)2. Approximating the asymptotic regime by a straight line, the slope is half of the fractal dimension df/2 and would for self-similar objects correspond to mass-fractal dimensions of df ) 5.6 for CP-1 and 4.08 for CP-2.
Figure 4. Normalized diffusion coefficient Dapp/Dz as a function of qRh (a), top, and qRg (b), bottom, on the log-log scale for the CP-1 and CP-2 microgels at different temperatures. (s) Theoretical curve for random coils of linear chains according to a theory by Akcasu et al.21
Discussion The experimental data show a number of features which are not observed with common flexible linear chains and not with hard spheres. Evidently, the microgels are subjects from a large group of special colloid particles built up from cross-linked flexible chains. We start the discussion with the global parameters, i.e., the molar mass Mw, the radii Rg and Rh, and the corresponding F ) Rg/Rh parameter. G Parameter (G ) Rg/Rh). A continuous decrease of the particle radii with increasing temperature was observed and an inflection point, but no sharp transition to a collapsed particle (Figure 3). This inflection point coincides for CP-2 at about 35 °C with the half-point of the transition of the F parameter. The transition was rather broad and did not allow us to estimate the correct LCST. Apparently the chemical heterogeneity in the copolymer masks the expected increase in the LCST. For CP-1 we could not trace the whole transition curve toward the microgel collapse, since at 38 °C aggregation among the microgels was observed. This aggregation was weak but still disturbed the correct determination of the radii Rg and Rh of the nonaggregated particles at the higher temperatures. Nonetheless, a few relevant conclusions can be drawn by comparing the absolute values Fexp from the present samples with those from other microgels and the theoretical ones derived for special models.35 For linear flexible chains in good and Θ solvents F values between 1.86 and 1.50 were predicted for chains without application of hydrodynamic preaveraging.12,13 Branching reduces the value,35 but it never became smaller than F ) 0.775 for hard spheres. Experimentally always somewhat smaller values were found, e.g., for linear polystyrene38 and star-branched macromolecules.39 Later these observations were confirmed by Freire40 on the basis of Brownian simulations. The deviations
from Akcasu’s result12,13 are based on the somewhat incorrect Kirkwood approximation for the hydrodynamic interaction (see ref 41). A drastic decrease of the F parameter was observed for poly(vinyl acetate) microgels from a value of F ) 1.8 in the pregel state to 0.5 for the finally formed microgel.4b Even lower values were found in other cases. The findings with the present samples at low temperatures confirm the behavior of swollen microgels. The gradual increase of the F values with increasing temperature indicates deswelling and finally collapse to a stable globule. For CP-1 in water the F parameter approaches at 35 °C the value for hard spheres, just before aggregation of the microgels occurred. For CP-2, prepared and measured in D2O, the transition did not reach the value for a hard sphere but kept the typical characteristics of a still swollen microgel. This aroused the suspicion that the result F ≈ 0.7 for CP-1 may be a result of two effects (see below). The observed F parameter for microgels lower than F < 0.775 has not yet found a theoretical explanation. Intuitively a strong binding of the solvent in the outskirts of the microgel by dangling chains is speculated. An extended solvent layer would drastically increase the hydrodynamic radius, but not the radius of gyration. Angular Dependence of the Static Particle Scattering Factor P(qRg). In Figure 2 we used the logarithmic modification of a Zimm plot, which gave the best estimation for the radius of gyration and the molar mass of the microgels. For comparison with results from other laboratories, and those obtained by small angle neutron (SANS) or X-ray (SAXS) scattering, it is appropriate to choose a universal plot, which is the particle scattering factor P(u) ≡ i(θ)/i(θ)0) as a function of u ) qRg (note that both P(u) and u are dimensionless quantities). As a function of u2 the particle scattering factor has a universal slope of -1/3 for any structure, but it shows structurally sensitive behavior at u2 > 4. In a plot of P(u) against u the scattering curve for globular particles has decayed already at u > 2 to very small values, and this would make discrimination between different structures very inaccurate. In Figure 5 1/P(u) as a function u2 is used, but here on a log-log scale, and in Figure 6 the Kratky42 version, i.e., u2P(u) as a function of u. The latter has the advantage that the scattering function in the large u regime is amplified by u2. This makes differences between various structures better discernible. (37) Galinsky, G.; Burchard, W. Macromolecules 1997, 30, 6966. (38) Schmidt, M.; Burchard, W. Macromolecules 1981, 14, 370. (39) Huber, K.; Burchard, W.; Fetters L. J. Macromolecules 1984, 17, 541. (40) Freire, J. J. AdV. Polym. Sci. 1994, 143, 35. (41) Yamakawa, H. Modern Theory of Polymer Solution; Harper & Row: New York, 1976. (42) Kratky, O.; Porod, G. J. Colloid Sci. 1949, 4, 35.
Chain Dynamics in Microgels
Figure 6. (Left) Kratky plot of the static scattering data from CP-1 in water and CP-2 in D2O at 15 °C. The dashed line represents the Guinier approximation,33 and the dotted line represents the DebyeBueche43 scattering function. (Right) A dendrimer model of four generations, redrawn after ref 47.
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Figure 7. Kratky plots of the scattering data from CP-1 in H2O and CP-2 in D2O at temperatures for the “collapsed” state of the microgels. Compared to the data at low temperatures, the curves are shifted toward the Guinier curve for CP-2 at 50 °C, but for CP-1 a slight shift away from the Guinier curve is observed. The Guinier curve corresponds to a rigid spherical particle with a Gaussian density distribution that has its highest value at the center of the particle and decays exponentially with increasing distance from the center.50
The log-log plot is commonly used in SANS. For particles of mass-fractal behavior the asymptotic slope presents half the mass-fractal dimension df/2. It is tempting to draw a straight line through the data at large u2. The slope would give fractal dimensions of df ) 5.6 and df ) 4.08 for the microgels in H2O and D2O, respectively. However, it remains doubtful that the asymptote was already reached. Indeed, the Kratky plot of Figure 6 shows that fractal behavior is not observed, but the experimental data lie between those of a structure that follows the DebyeBueche43 space correlation function with a fractal dimension of df ) 4 [P(u)Debye-Bueche ) 1/(1 + u2/6)2] and those of the Guinier approximation33 [P(u)Guinier ) exp(-u2/3)], which displays no power law behavior. In a previous paper,44 dealing with SANS measurements from CP-2 in D2O, an attempt of fit was made with a hard sphere model, introducing a certain size distribution and adding a Ornstein-Zernicke function,45,46 to take into account a certain internal flexibility. We wondered whether a reasonably good fit would be possible with a monodisperse soft sphere model in which the segments obey Gaussian behavior. This led one of us47 to the derivation of a generalized dendrimer in which the various branching generations were linked by monodisperse linear spacers. The full lines in Figure 6 represent the theoretical curves for such dendrimers with five to eight generations. Up to u ) 3 the experimental data from the CP-2 microgel were well described by this model with five generations, and for the CP-1 microgel by seven generations. At u > 3 significant deviations to higher generation numbers occurred. In fact, full agreement was not expected. The assumption of uniform size of the microgels is experimentally well confirmed by Pelton2 with similar microgels, but the uniform length of the spacer chains is certainly an oversimplification. Nonetheless, the experimental curves are over a large initial q domain well described by the soft sphere model, without terms being artificially added to take into account internal density fluctuations. (The equation of the generalized dendrimer is given in the Supporting Information). Similar Kratky plots were obtained also for CP-1 at 35 °C and CP-2 at 50 °C, at which the collapse of the thermosensitive microgels should have occurred. Figure 7 disclosed a weak shift of the CP-1 (water) away from the Guinier curve (in the soft
sphere model characterized by a lower number of generations). Contrary to that, the CP-2 (in heavy water) showed a marked shift toward the Guinier curve (i.e., to larger generation numbers). The Guinier approximation is often considered as a good approximation to spherical particles.33,48 Van de Hulst50 gave a more precise interpretation and showed that, for a spherical particle with a radial density distribution of d ) d0 exp(-(r/b)2), but no density fluctuations, the Guinier function P(u)Guinier ) exp(u/3) is obtained with Rg2 ) 12b2. Thus, the slight shift for CP-1 at 35 °C away from the Guinier curve indicates gain of internal mobility but the shift for CP-2 at 50 °C toward Guinier a remarkable loss of segmental mobility. Still, a transition to hard spheres was not obtained. The opposite behavior of the two samples is strange. At 50 °C for CP-2 and 35 °C for CP-1 a collapse of the microgels was expected, but only the CP-2 in D2O showed the expected behavior of hardening the soft sphere (see also the discussion of the F parameters). The conclusions from static LS measurements are confirmed by dynamic LS. This will be shown in the next section. The soft sphere model was also successfully applied to the characterization of poly(vinyl acetate) microgels swollen in methanol47 and for poly(N-vinylimidazole) aggregates in ethanol,51,52 yet we have to bear in mind that the regularly branched soft sphere model is an oversimplified model, as already outlined above. A full description by the regular soft sphere was not expected. Angular Dependence of the Dynamic Light Scattering Data from CP Microgels. In a careful study Sorlie and Pecora22 showed that a thoughtless application of first cumulant analysis leads to systematical deviations from the results of multiexponential relaxation modes, as determined by the CONTIN inversion of the time correlation function. The reason for this behavior arises from the curvature of ln(g1(t)) as a function of time. Figure 4 shows the angular dependence of the normalized diffusion coefficient Dapp/Dz for the measurements in the range of θ ) 30-98°. At low angles (qRg < 2.0), within experimental errors, a constant value of the diffusion coefficient was observed. A
(43) Debye, P.; Bueche, A. M. J. Appl. Phys. 1949, 20, 518. (44) Boyko, V.; Richter, S.; Grillo, I.; Geissler, E. Macromolecules 2005, 38, 5266. (45) Ornstein, L. S.; Zernicke F. Proc. Acad. Sci. Amsterdam 1914, 17, 793. See also ref 46. (46) Stanley, H. E. Introduction to Phase Transitions and Critical Phenomena; Clarendon Press: Oxford, 1971; Chapter 7.4. (47) Burchard, W.; Kajiwara, K.; Nerger, D. J. Polym. Sci., Polym. Phys. Ed. 1982, 20, 157.
(48) Glatter, O., Kratky, O., Eds. Small Angle X-Ray Scattering; Academic Press: London, 1982. (49) Higgins, J. S.; Benoit, H. C. Polymers and Neutron Scattering; Clarendon Press: Oxford, 1994; Chapter 6.5.1. (50) van de Hulst. Light Scattering by Small Particles; Dover Publications: New York, 1981; Chapter 7.23. (51) Savin, G.; Burchard, W.; Luca, C.; Beldie, C. Macromolecules 2004, 37, 6565. (52) Savin, G.; Burchard, W. Macromolecules 2004, 37, 3005.
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similar long range was also observed with poly(NIPAM) microgels by Wu and Zhou.26 The authors found in the CONTIN analysis over a broad range of qRg a constant diffusion coefficient, but after that an exponential increase of Dapp/Dz with qRg (in the range of 2.4 < qRg < 4.2). For the apparent diffusion coefficient Dapp(q) ) Γ1(q)/q2 at large qRg asymptotic power law behavior, Dapp(q)/Dz ∝ qn, was predicted,17,18 with an exponent of n ) 1 for chains with hydrodynamic interaction (Zimm limit). A linear dependence, Dapp(q)/Dz ∝ q1(0.05, was often found with linear chains,54 and also for end-linked polystyrene star molecules25 approximately the same dependence was obtained. However, for a copolymer of styrene and divinylbenzene clusters,55 densely branched polycyanurates,25 highly branched polyester clusters in the pregel state, and the branched clusters from the sol fraction of the gel,25 smaller exponents of n ≈ 0.8 were found. Similar exponents (n e 0.8) were observed with linear stiff chains, i.e., DNA fragment,22 actin filaments,56 and two bacterial polysaccharides,57,58 which all formed double helices. In recent theories59,60 the lower exponents for the stiff chains were shown to be the result of bending motions, which become dominant for chains with less than 10 Kuhn segments per contour length. In the present study a nearly linear asymptotic line was reached, but only at qRg > 4.2, which is considerably larger than qRg ≈ 1.7 observed for linear chains. The exponent n ) 0.96 ( 0.05 appeared to be in good agreement with the predicted value for chains in the Zimm limit. However, all curves in Figure 4 are shifted by a factor of 2.4 to larger q values, compared to those of linear flexible chains. We got the impression that in the microgels the segment mobility is suppressed, until chain sections are probed which by a factor of 2.4 are shorter than those of freely moving linear chains. This conclusion is confirmed by the q dependence of the reduced first cumulant Γ*(q) ≡ [Γ1(q)/q3](η0/kT) as a function of qRh. For hard spheres the simple equation Γ*(q)hardsphere ) 1/(6πqRh) is obtained, but for linear chains, under the influence of hydrodynamic interaction (Zimm limit), Γ*(q) should reach a constant value of Γ*(∞) ) 0.0798 for a good solvent and 0.0625 for a Θ solvent.12,13 Figure 8 shows the plots for CP-1 at three selected temperatures and for CP-2 in D2O at 15 °C. No change in behavior was found for CP-2 at temperatures of 15-50 °C. In all cases the experimental data initially follow the curve of hard spheres. Only at very low Γ*(q) a deviation from the straight line to a possibly constant value of Γ*(∞) ) 0.01 was found for CP-1 in H2O. For CP-2, in the whole qRh range, no deviation from the hard sphere behavior was found. The results may be compared with the findings with poly(NIPAM) microgels. Wu and Zhou26 found a similar value of Γ*(∞) ) 0.013, approximately 4.8-5.0 times smaller than for freely moving linear chains. (The experimental data for linear flexible chains were close to 0.062.) A collection of measurements from various laboratories was given by Trappe and Burchard.24 The observed behavior is caused by a loss in internal flexibility. In their study on branched amylopectin fragments Galinsky and Burchard37 concluded that branching (53) Nordmeier, E. J. Phys. Chem. 1993, 97, 5770. (54) See ref 24, which gives a list from various laboratories. (55) Delsanti, M.; Munch, J. P. J. Phys. II 1994, 4, 265. (56) (a) Piekenbrock, Th.; Sackmann, E. Biopolymers 1992, 32, 1471. (b) Fujime. S. AdV. Biophys. 1972, 3, 1. Schmidt, C. F.; Ba¨rmann, M.; Isenberg, G.; Sackmann, E. Macromolecules 1989, 22, 3638, (57) Dentini, M.; Coviello, T.; Burchard W.; Crescenzi, V. Macromolecules 1988, 21, 3312. (58) Coviello, T.; Burchard, W.; Dentini, M.; Crescenzi, V.; Morris, V. J. J. Polym. Sci., Part B: Polym. Phys. 1995, 33, 1833. (59) Go¨tter, R.; Kroy, K.; Frey, E.; Ba¨rmann, M.; Sackmann, E. Macromolecules 1996, 29, 30. (60) Harnau, L.; Winkler, R. G.; Reineker, P. J. Chem. Phys. 1996, 104, 6355.
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Figure 8. Reduced cumulant Γ*(q) versus qRh for the CP microgel dispersion at different temperatures. (s) Theoretically predicted dependence of the reduced cumulant for hard spheres (StokesEinstein).
introduces strong perturbations to the simple spring-bead model of linear chains. Instead of being connected by only two springs to neighbors in the linear chains, the ends of segments in the branched structure are actually connected by three or four springs. These additional springs may strongly impede the internal mobility. It was conjectured that branching or cross-linking may alter the Zimm relaxation spectrum. For the microgels the impediment appeared to be much more drastic. The experiments indicate that all internal motions were completely suppressed until the length of the chains connected by two branching points was probed. For long segments between branching points the impediment is small, but evidently it becomes significant when the branching density is high. The dynamic behavior of CP-1 shows no difference in the reduced first cumulant plot for low temperatures (Figure 8), but at 35 °C, just before a macroscopic phase transition occurs, the particles appear to be more flexible. This is contrary to expectation. A loss of mobility could be concluded from the decrease in microgel dimensions near the collapse or, in other words, a change toward hard sphere behavior. At present we have no well-founded explanation for the behavior of CP-1 near the collapse and can only make a guess. The collapse of a fraction of microgels may indeed have taken place, but this phenomenon could be superimposed by a weak association among dangling chains from two or three other microgels. The flexibility of the noncollapsed chains will contribute to segmental dynamics, while the fraction of collapsed microgels will give no contribution to flexibility. No current theory accounts for this result. Before moving to a discussion of the time dependence of the TCF, we demonstrate that the here reported dynamic behavior is not limited to microgels but is observed with many other branched examples. The result from some selected examples24,37,61 is shown in Figure 9. Time Dependence of the Time Correlation Function g1(q,t). The field time correlation function g1(q,t) is a complex function since the delay time also contains a q dependence. For hard spheres, where only the translational diffusion coefficient defines the TCF, a generalized scaling is obtained by the scaled time t* ) q2t. Han and Akcasu et al.13,21 noticed that for flexible chains the scaling can be generalized to t* ) Γ1(q)t, with Γ1(q) being the first cumulant of the TCF. This scaling of the delay (61) (a) Adam, M.; Delsanti, M. Macromolecules 1977, 10, 1229. (b) Han, C. C.; Akcasu, A. Macromolecules 1981, 14, 1080. (c) Tsunashima, Y.; Nemoto, N.; Kurata, M. Macromolecules 1983, 16, 1184. (d) Nemoto, N.; Makita, Y.; Tsunashima, Y.; Kurata, M. Macromolecules 1984, 17, 425. (e) Tsunashima, Y.; Hirata, M.; Nemoto, N.; Kurata, M. Macromolecules 1987, 20, 1992. (f) Tsunashima, Y.; Hirata, M.; Nemoto, N.; Kajiwara, K.; Kurata, M. Macromolecules 1987, 20, 2862. (g) Wiltzius, P.; Cannell, D. S. Phys. ReV. Lett. 1986, 56, 61. (h) Bhatt, M.; Jamieson, A. M. Macromolecules 1988, 21, 3015. (i) Bhatt, M.; Jamieson, A. M.; Petschek, R. G. Macromolecules 1989, 22, 2724.
Chain Dynamics in Microgels
Figure 9. Reduced first cumulant Γ*(q) ≡ (Γ1(q)/q3)(η0/kT) for a number of branched macromolecules compared with linear flexible chains (lin PS, linear polystyrene; PI, polyisoprene)61 and the present microgel (CP-1). Key: PS, b, end-linked three-arm PS stars;62 PE, b pre and PE, b post, cross-linked poly(ester chain) clusters from the pregel and postgel state;24,25 starch fragments,37 fragments obtained from amylopectin by controlled degradation (cleaving only the C1-C6 branching points); CP-1, CP-2, microgels in water and heavy water. The branching density increases for the curves from top to bottom. Note that the 35 °C curve deviates toward higher Γ*(∞), although a significant shrinking of the microgel was observed (see the text).
Figure 10. Semilogarithmic plot of the field correlation function g1(q,t) versus the scaled time Γ(q)t at angles of observation of (0) 30° and (b) 98° for a CP-1 microgel: solid lines, shape functions predicted for linear chains in the asymptotic range in the Zimm and in the Rouse limits according to Akcasu et al.;21 dashed line, singleexponential limit.
time includes two limits, diffusion at small q and asymptotic segment relaxation at large q. According to theory, the asymptotic time dependence is given by a stretched exponential g1(q,t) f exp[-(t/t*)β], with β ) 2/3 for chains with hydrodynamic interaction (Zimm limit) and β ) 1/2 for Rouse relaxation without hydrodynamic interaction. For a high segment concentration the hydrodynamic interaction is assumed to become screened,16 and a change to Rouse relaxation is expected. However, no hydrodynamic screening could be observed despite the high segment concentration in the microgel, in agreement with other examples.25,27,63,64 Figure 10 shows two selected curves for CP-1 at θ ) 30° and 98° and the theoretical curves for pure diffusion and Zimm and Rouse relaxations. At low scattering angles solely the expected translational diffusion was observed. At large scattering angles the curve showed approximately the correct asymptotic slope of β ) 2/3, but with a lower amplitude. The experimental curve displays bimodal behavior, in which the translational diffusion (62) (a) Weissmu¨ller, M. Burchard, W. Acta Polym. 1997, 48, 571. (b) Weissmu¨ller, M. Ph.D. Thesis, University of Freiburg, 1996. (63) Higgins, J. S.; Allan, G.; Gosh, R. E.; Howells, W. S.; Farnoux, B. Chem. Phys. Lett. 1977, 49, 197. (64) Nicholson, L. K.; Higgins, J. S.; Hayter, J. B. Macromolecules 1981, 14, 836.
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Figure 11. Relaxation time distributions H(τ) obtained for the CP-1 microgel (C ) 0.185 g/L) at 15 °C at scattering angles of (a) 30° and (b) 106°. At the low scattering angle only the peak of the diffusion coefficient was observed, whereas at the large scattering angle two faster modes of motion appeared which are assigned to the slowest relaxation modes in the microgel.
still contributes slightly to the time dependence. Only the second contribution to the TCF is determined by the Zimm limit, but with a lower amount. We checked this conclusion by CONTIN inversion of the TCF at 30° and 106°. The corresponding curves as a function of the delay time are shown in Figure 11. At 30° only one peak was obtained, which is assigned to the translational diffusion, but at 106° two faster relaxation peaks were found in addition to the diffusion peak. Similar results were observed by Chu et al.28
Conclusions The measured F parameter F ) Rg/Rh clearly indicated the presence of highly swollen microgels. The observed change toward hard spheres at elevated temperatures gave evidence of a drastic volume shrinking at a critical transition temperature. The angular dependence of the static LS data could satisfactorily be described by a soft sphere model that represents a dendrimer in which the various generations are connected via long linear chain spacers. The weak but systematic deviations of the experimental data from this model are probably caused by an oversimplification. The soft sphere model takes into account the uniform size of the microgel and the spacer chain segmental flexibility, but the uniform length of the spacer chains is an unrealistic feature that in further theories has to be avoided. Despite this deficiency, the soft sphere model correctly demonstrates for CP-2 in D2O a structure change toward hard sphere behavior, when the collapse temperature is exceeded. For the CP-1 sample in water a commencing association at the collapse temperature prevented a clear interpretation. The dynamic LS study revealed internal relaxation modes in the CP-1 microgel, but these were noticeable only at qRg > 2.4 instead of qRg > 1 for linear flexible chains. At values of qRg < 2.4 only the translational motion of the center of mass was observed for the microgel in water. The effect of internal mobility was clearly displayed in the plot of Γ*(q) and showed a similarity to the Zimm relaxation spectrum of linear chains. Comparison with other examples made clear that for increasing branching
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density the onset of internal motions is systematically shifted to shorter chain sections. This length may tentatively be taken as the length of chains between two branching points. Two well-separated peaks of internal modes of motion and one translation diffusion could be distinguished at qRg > 4. These modes are represented by the relaxation times τ1, τ2, and τ0, respectively, where τ0 represents the diffusive motion of the center of mass and τ1 and τ2 represent the slowest relaxation modes in the microgel. The relaxation times could not be assigned to special modes because of the lack of a theory for complicated structures. No internal modes could be recorded for CP-2 in D2O in the accessible qRh domain. The difference from CP-1 in water
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is contributed to differences in the solvent quality. No heterodyne or nonergodic behavior was found. Acknowledgment. We thank the European Graduate School “Advanced Polymer Materials“ (EGK 720-1, DFG) for financial support of the present work. Supporting Information Available: Equations of the used soft sphere model. This material is available free of charge via the Internet at http://pubs.acs.org. LA062181+
