pubs.acs.org/Langmuir © 2009 American Chemical Society
Confinement-induced fluid-gel transition in polymeric solutions Catalina Haro-Perez, Andres Garcı´ a-Castillo, and Jose Luis Arauz-Lara* Instituto de Fı´sica “Manuel Sandoval Vallarta”, Universidad Aut onoma de San Luis Potosı´, Alvaro Obreg on 64, 78000 San Luis Potosı´, S.L.P., Mexico Received June 9, 2009. Revised Manuscript Received July 3, 2009 The viscoelastic properties of confined polymer solutions are probed by particle-tracking microrheology. The mean squared displacement of spherical probe particles embedded in the solution and the storage and loss moduli of the system are measured as the level of confinement is increased. It is found that those quantities change continuously as the confinement increases, and, at severe conditions, when the constrain reaches the size of the polymer molecule, the system undergoes a transition from a viscoelastic fluid to a gel.
Natural soft matter systems are usually found to occur under some form of confinement. Some examples are cellular cytoplasm,1 blood in capillaries,2 and crude oil in porous or fractured rocks.3 Thus, it is of great interest to gain an understanding of the effects of confinement. Recent reports have shown that the physical properties of different confined systems may differ considerably from their bulk values. For instance, studies of the mechanical properties of confined molecular fluids,4 polymer melts,5,6 and colloidal fluids7 report dramatic changes of those quantities and, more interestingly, that those systems can undergo a fluid-solid phase transition when they are constrained in geometries with one dimension being comparable to the size of the molecular species. The description of such phenomena in terms of basic quantities is a rather complex matter and is not completely developed yet. Nevertheless, phenomenological interpretations of the main subject of attention, namely, the elucidation of the nature of the phase transition, have been advanced,7-9 but their conclusions are controversial. For instance, some experimental studies on molecular liquids report an abrupt firstorder transition,9 but other studies on similar systems find a continuous second-order transition.8 Here we are concerned with the study of the effects of confinement on the properties of a particular kind of soft materials, namely, viscoelastic polymer solutions, which are of great interest in different fields of science and technology. Previous experimental,1,5,6 theoretical,10 and simulation11,12 studies have shown that the mechanical and structural properties of polymer liquids change when they are confined in a narrow region between surfaces. Thus, the question here is whether this kind of system also undergoes a phase transition and under what conditions. *Corresponding author. E-mail:
[email protected]. (1) Claessens, M. M. A. E.; Tharmann, R.; Kroy, K.; Bausch, A. R. Nat. Phys. 2006, 2, 186. (2) Pries, A. R.; Neuhaus, D.; Gaehtgens, P. Am. J. Physiol. 1992, 263, H1770. (3) Chauveteau, G.; Tirrell, M.; Omari, A. J. Colloid Interface Sci. 1984, 100, 41. (4) Heuberger, M.; Z€ach, M.; Spencer, N. D. Science 2001, 292, 905. (5) Hu, H. W.; Granick, S. Science 1992, 258, 1339. (6) Luengo, G.; Schimitt, F. J.; Hill, R.; Israelachvili, J. Macromolecules 1997, 30, 2482. (7) Nugent, C. R.; Edmond, K. V.; Patel, H. N.; Week, E. R. Phys. Rev. Lett. 2007, 99, 025702. (8) Demirel, A. L.; Granick, S. Phys. Rev. Lett. 1996, 77, 2261. (9) Klein, J.; Kumacheva, E. Science 1995, 269, 816. (10) Subbotin, A.; Semenov, A.; Doi, M. Phys. Rev. E 1997, 56, 623. (11) Pakula, T. J. Chem. Phys. 1991, 95, 4685. (12) Romiszowski, P.; Sikorski, A. J. Chem. Phys. 2002, 116, 1731.
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The present work is aimed to address that question by investigating the effects of confinement on the mechanical properties of well-characterized polymer solutions. In the present work, we study the viscoelastic response of semidilute solutions of water-soluble polymers of high molecular weight, confined between two parallel glass plates as the distance between the walls is reduced. The polymer used here is polyethylene oxide (PEO), of two different molecular weights: Mw=8 106 (Polysciences, Inc.) and 9 105 Da (Scientific Polymer Products, Inc.), at polymer mass fractions of 0.15% and 0.5% (w/w), respectively. The choice of this system is based on its wide-spread use in microrheology experiments as a nonadsorbing water-soluble linear polymer. There are several works on this issue.13,14 These authors studied the bulk viscoelastic properties of PEO solutions in the semidilute regime, using polystyrene spheres as probes, and varied both the sizes and surface chemistry of the particles. They found an excellent comparison of microrheology with mechanical measurements, confirming that adsorption plays no role in determining the viscoelastic properties by microrheology. The present study is performed using particle-tracking microrheology, i.e., we follow the motion of probe particles dispersed in the confined sample by means of optical microscopy. From the mean squared displacement of the probes, we determine the rheological moduli of the suspension as the degree of confinement is varied. We show that confinement induces drastic changes in those properties and that, at severe conditions, when the confinement reaches the size of the polymer molecule, the system undergos a fluid-gel transition. This transition can be described as a gellification process where the gelling parameter is the free space available to the polymer molecules normalized by the size of the polymer. Thus, the presence of constraints can produce strong effects, which should have to be considered when dealing with macromolecular fluids under confinement, a frequent situation for biological fluids in vivo. The samples are prepared as follows: Stock solutions are prepared by mixing PEO powder in deionized water (resistivity of 18 MΩ cm, Barnstead) that has been filtered using 0.2 μm poresize filters. The solutions are incubated for several days at 40 C prior to use to ensure that the polymer is completely dissolved. PEO solutions have been reported to degrade when they are (13) Gisler, T.; Weitz, D. A. Curr. Opin. Colloid Interface Sci. 1998, 3, 586. (14) Dasgupta, B. R.; Tee, S.; Crocker, J. C.; Frisken, B. J.; Weitz, D. A. Phys. Rev. E 2002, 65, 051505.
Published on Web 07/13/2009
DOI: 10.1021/la902070q
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stored for periods longer than 30 days.15 Thus, we only use samples less than 15 days old. The radius of gyration of PEO ( 0.031 ˚ 16 A, in water can be estimated from Rg = 0.215 M0.583 w the overlap polymer concentration is obtained from c* = Mw/ (4/3NAπRg3), where NA is the Avogadro’s number,17 and the average mesh size of the network is ξ=Rg(c*/c)0.75 with c being the polymer mass concentration.17 For the polymers used here we have Rg=225 and 64 nm, c*=0.03% and 0.15% (w/w), and ξ= 68 and 26 nm, for the higher and lower Mw, respectively. Thus, in both cases the polymer concentration is above the corresponding c*, i.e., both solutions are in the semidilute viscoelastic regime. We add to the solutions a small amount of polystyrene spheres of diameter d (the probe particles), carrying ionizable sulfate end groups on the surface, which become negatively charged when immersed in polar solvents (Duke Sci.). We also add a small amount of larger polystyrene spheres of diameter h>d (the spacers). Let us note here that d . ξ, i.e., the solution appears as a continuous media in the scale of d. The system is then confined following the procedure described in ref 18. Briefly, a small volume of the sample is located between two carefully cleaned glass plates (a slide and a coverslip), which are uniformly pressed one against the other until the separation between the plates coincides with the size h of the larger particles, serving here as fixed interplates spacers. The system is then sealed with epoxy resin, and the particles of diameter d are allowed to equilibrate in this confined geometry for at least 2 h at room temperature (24 C) prior to the experiments. The final concentration of mobile particles is kept very low to avoid interparticle interactions. The system is observed using an inverted optical microscope, Olympus IX71 with a 100 objective. The motion of the particles is recorded using a charge-coupled device (CCD) camera with a time resolution of 30 frames/s. The position (x,y), along the plane parallel to the plates, of every particle in the field of view is determined from digitized images using the method devised by Crocker and Grier,19 which allows us to locate the sphere centers with a precision of 1/5 pixel. From the particles’ positions in consecutive frames, their trajectories are reconstructed. We then quantify the particle motion by calculating the two-dimensional mean square displacement (MSD) ÆΔr2(t)æ=ÆΔx2 þ Δy2æ= Æ[xi(τ þ t) - xi(τ)]2 þ yi(τ þ t) - yi(τ)]2æ, where the average is taken over all particles i and all initial times τ. We studied samples with different degrees of confinement, covering the range from bulk to the size of the polymer molecule. In Table 1 we show the confinement conditions of the PEO solutions of Mw=8000 kDa. Under severe conditions of confinement, like those of our experiment, the actual confinement is provided by one plate and the probe particle’s surface, whose average distance is (h - d)/2. Thus, the parameter representing the degree of confinement is the inverse of that distance, normalized with the polymer size 2Rg, i.e., χ=4Rg/(h - d). Figure 1 shows the MSDs of the probes moving in the systems of Table 1 versus lag time t. The MSD is scaled with the diameter d of the probes in order to eliminate tracer size effects. It is well established that the motion of particles diffusing in a pure viscous fluid confined between two walls is slowed as a result of particle-wall hydrodynamic interactions. Thus, in order to account for this effect, the MSDs are normalized by the ratio D/D0, where D0 and D are the (15) Kuzma, M. R.; Wedler, W.; Saupe, A.; Shin, S.; Kumar, S. Phys. Rev. Lett. 1992, 68, 3436. (16) Devanand, K.; Selser, J. C. Macromolecules 1991, 24, 5943. (17) Cooper, E. C.; Johnson, P.; Donald, A. M. Polymer 1991, 32, 2815. (18) Ramı´ rez-Saito, A.; Chavez-Paez, M.; Santana-Solano, J.; Arauz-Lara, J. L. Phys. Rev. E 2003, 67, 050403(R). (19) Crocker, J. C.; Grier, D. G. J. Colloid Interface Sci. 1996, 179, 298.
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Haro-P erez et al. Table 1. Confining Conditions of the PEO Solutions of Mw = 8000 kDa and Polymer Concentration 0.15% (w/w)a χ 3 10-3 0.26 0.56 0.90 1.20 1.70 2.0 h ( μm) 300 5.0 3.15 2.0 1.53 1.53 2.0 d ( μm) 1.55 1.55 1.55 1.0 0.78 1.0 1.55 h/d 194 3.23 2.03 2.0 1.96 1.53 1.29 1.0 0.70 0.57 0.56 0.55 0.48 0.45 D/D0 a χ = 4Rg/(h - d) is the parameter characterizing the degree of confinement, realized by combining different values of the probe’s diameter d and plates separation h. The ratio D/D0, between the probe’s diffusion coefficient in pure solvent, under confinement D and in the bulk D0, is taken from20.
Figure 1. Normalized MSD of probe particles in samples of increasing confinement χ. Inset: MSD in the bulk obtained from DLS (open squares) and DVM (closed squares).
values of the diffusion coefficient of the probe particle in the pure solvent, in the bulk D0, and confined between two walls D. The values for D/D0 (quoted in Table 1) are taken from refs 20 and 21, assuming the particles move mainly in the middle between the two walls. Filled squares correspond to the MSD of probe particles for χ=310-3. In this sample, the interplate distance is h=300 μm (achieved by using glass spacers of that thickness between the plates; the other values of h are achieved as described above). Here, we analyze the motion of particles moving at the middle of the sample to avoid particle-wall interactions. Then, in this case, the MSD corresponds to that in the bulk. As one can see here, the MSD is linear with slope 1 (dashed line), i.e., the particle motion is diffusive. Thus, although c > c*, the viscoelastic response of the system is not captured by the time resolution of standard digital video microscopy (DVM), but only the long-time regime where the network relaxes, allowing the particles to diffuse as in an effective viscous medium. Diffusion in the bulk at a shorter time scale is measured by dynamic light scattering (DLS) using a standard set up.22 The results from both measurements are shown in the inset of Figure 1 where one can see the characteristic behavior of the MSD in a viscoelastic fluid, i.e., an initial lower slope and then an increase of it at larger times.23 Figure 1 also shows the MSD of probe particles moving in confined samples (symbols). As one can observe, moderate confinement (χ=0.26, closed circles) already has a notable effect on the motion of the probes, and it becomes sufficiently slow that DVM can capture the viscoelastic nature of the polymeric matrix. Further increasing of confinement further constrains the particle motion, the long-time slope of the MSD decreases, and, at severe conditions (χ=1.7 and 2.0, left triangles and hexagons, respectively), the particles are (20) Dufresne, E. R.; Altman, D.; Grier, D. G. Eur. Phys. Lett. 2001, 53, 264. (21) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Kluwer: Dordrecht, The Netherlands, 1991. (22) Haro-Perez, C.; Andablo-Reyes, E.; Dı´ az-Leyva, P.; Arauz-Lara, J. L. Phys. Rev. E 2007, 75, 041505. (23) Mason, T. G.; Ganesan, K.; van Zanten, J. H.; Wirtz, D.; Kuo, S. C. Phys. Rev. Lett. 1997, 79, 3282.
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dynamically arrested, i.e., the MSD reaches a plateau. This behavior is pretty similar to that observed during the gelation process in physically and chemically cross-linked polymeric gels.24 Thus, it suggests a confinement-induced gelation transition in the system studied here. Indeed, as we show below, further analysis of our data using different criteria confirms the existence of such a transition, located at χ ∼ 1, i.e., when the separation between confining surfaces is about the size of the individual polymer molecule. Measurements of Rg for c > c* show that the radius of gyration of polymers in good solvents decreases with c following a scaling law.25 Adapting that scaling law for the polymers used here, we obtained the corrected values of Rg, at the polymer concentrations of our experiment. Those values are smaller than their free values, but only for less than 19%. With this correction, the values of χ change, but the transition still occurs at χ around 1. Let us note too that such scaling law was obtained for polymers in the bulk, where they are surrounded by other polymers. For the conditions of our experiment, particularly when the confinement reaches the size of the polymer, the polymers are not completely surrounded by other polymers, the screening effect leading to the reduction of Rg should be much less than that in the bulk, and Rg should be closer to its free value. Thus, we use this value here for the determination of χ. In order to corroborate that the observed arrest of the probes is due to actual changes in the mechanical properties of the polymer solution and not due to the confinement itself (h f d), we studied solutions of PEO of lower Mw (900 kDa) but at higher polymer mass concentration to have a similar viscoelastic response in the bulk. Here too, the long time slope of the MSD decreases as χ increases, but, although the same range of h/d was covered, no dynamic arrest of the probe particles is observed (data not shown). Then, confinement itself can be ruled out as the cause of particle arrest in our system. This polymer of lower Mw has a smaller Rg. Thus, the parameter χ varies in a narrower range. In fact, in this case, χ e 0.57. This is in agreement with our suggestion that particle arrest should not be observed for χ < 1. The gel point in our system is located using different methods. In the first one presented here, we extend a procedure used to locate the gel point of either chemically or physically gelling polymer solutions,24 which is itself based on the scaling principles of time-cure superposition.26 According to this method, it should be possible to construct two master curves by superposing the MSD of the probes. The MSDs in pregel states would produce a master curve, and those corresponding to postgel states would produce a different master curve of lower slope. Such superposition is achieved by scaling both time and MSD, by factors a and b, respectively. Moreover, the behavior of a and b, as functions of the gelling parameter, provides a different and more precise criterium to locate the gel point. Both parameters should decrease as the system approaches the transition from the pregel state, and then they should increase in the postgel state as the system becomes a stronger gel.27 The scaling procedure carried out for the systems of Table 1 indeed produce two master curves, as shown in Figure 2. On the upper curve fall the MSDs corresponding to the less confined samples (χ < 1), and on the lower curve fall those corresponding to the more confined samples (χ > 1). In the inset of Figure 2 are shown the corresponding (24) Larsen, T. H.; Furst, E. M. Phys. Rev. Lett. 2008, 100, 146001. (25) Cheng, G.; Graessley, W. W.; Melnichenko, Y. B. Phys. Rev. Lett. 2009, 102, 157801. (26) Adolf, D.; Martin, J. E. Macromolecules 1990, 23, 3700. (27) Chambon, F.; Winter, H. H. J. Rheol. 1987, 59, 683.
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Figure 2. Scaled MSDs from Figure 1. Inset: Scaling parameters a and b (pregel), and a0 and b0 (postgel).
scaling factors a and b for the pregel states, and a0 and b0 for the postgel states. As one can see here, both parameters first decrease with χ and then increase with the turning point located around χ=1. Thus, according the criterium described above, the process of confining a polymer solution drives it trough a fluid-solid transition with the gel point located at χ ∼ 1. On the other hand, the scaling procedure of the MSDs applied to the solution of lower Mw leads only to a single master curve, similar to the upper curve in Figure 2, and the corresponding scaling parameters, a and b, decrease as χ increases, implying that all the samples are in the pregel state. This is in agreement with the fact that, in this system, χ < 1 for all the samples. A complementary study of the mechanical properties of the system studied here, as it is constrained, can be carried out in terms of G0 (ω) and G00 (ω), the elastic and viscous components of the complex shear modulus, respectively. The response of gelling solutions to a shear strain of frequency ω is generally more viscous-like (G00 (ω)>G0 (ω)) in pregel states and more elastic-like in postgel states when ω is below the critical frequency. Thus, the gel point is the state of the solution at which both moduli coincide. Thus, at a fixed frequency below the critical value, a plot of G0 (ω) and G00 (ω) versus the gelling parameter (time, temperature, chemical composition, χ in our case) will show a crossover of both curves at the gel point. Furthermore, near the transition, the viscoelastic moduli are expected to have a power-law dependence on frequency, i.e., G0 ∼ G00 ∼ ωn, with n being the critical viscoelastic exponent given by G00 (ω)/G0 (ω) = tan(nπ/2).27 Figure 3a,b shows the viscoelastic moduli for the samples in Table 1, obtained from the MSDs of the probe particles following standard procedures28 where the storage and the loss moduli are obtained as follows: πRðωÞ G0 ðωÞ ¼ GðωÞ cos½ 2
ð1Þ
πRðωÞ G00 ðωÞ ¼ GðωÞ sin½ 2
ð2Þ
where GðωÞ ¼
kB T πaÆΔr2 ðωÞæΓ½1 þ RðωÞ
ð3Þ
Here, ÆΔr2(ω)æ denotes the magnitude of ÆΔr2(t)æ evaluated at t= 1/ω, and R(ω) is the first order logarithmic time derivative of the MSD, i.e., R(ω)=[∂ lnÆΔr2(t)æ/∂ ln t]t=1/ω. Thus, we found that, in the range of ω accessible in our experiment, the samples with χ 1 are more elastic, whereas, (28) Mason, T. G. Rheol. Acta 2000, 39, 371.
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Figure 3. (a) Elastic (lines) and viscous (symbols) moduli for samples with χ = 0.56, 0.9, and 2.0, from bottom to top, respectively. (b) Elastic (solid circles) and viscous (open circles) moduli at ω = 0.5 rad/s as a function of χ. Inset: Elastic (dashed line) and viscous (symbols) moduli for the system with χ= 0.9, indicated by an arrow in the main figure. Lines are guides to the eye.
in the sample where χ ≈ 1, it is satisfied that G00 (ω) ≈ G0 (ω). This can be seen in Figure 3a, where (for clarity) the elastic (lines) and the viscous (symbols) moduli are shown only for the samples with χ=0.56, 0.9, and 2.0, from bottom to top, respectively. Figure 3b shows in more detail G0 (ω) and G00 (ω) at ω=0.5 rad/s versus χ. Here, one can clearly see the crossover of both curves near to χ=1. Finally, the inset of Figure 3b shows that, in the sample where χ ≈ 1, both rheological moduli follow a power law of ω, with very similar exponents 0.41 and 0.44. These values are very close to the value of the critical viscoelastic exponent n=0.42, obtained from
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the experimental value for tan(nπ/2)=G00 (ω)/G0 (ω)=0.79. Let us mention that similar results are obtained for a wide range of values of ω. Thus, the different analysis of the viscoelastic moduli of the polymer solution studied here all concur with the existence of a fluid-gel transition located around χ=1. In this work we investigate the effect of confinement on the mechanical properties of viscoelastic fluids. As it is shown here, that effect can be very strong when the solution is confined in geometries whose dimensions are on the order of a few times the size of the polymer molecule. Moreover, we provide consistent evidence of a continuous confinement-induced fluid-gel transition in the system studied in the present work, with the gel point occurring when the confinement reaches the effective size of the polymer. This scenario can be simply summarized in terms of the parameter χ, which measures the geometrical constraint in units of the polymer size, i.e., samples with χ < 1 are in a pregel state, whereas samples with χ>1 are in a postgel state, with the gel point located around χ=1. A similar transition, in this case to a glassy state, has been observed confining simple fluids.8 In spite of the very different nature of our system and the latter, these two transitions, the fluid-gel and the fluid-glass transitions, can be described by the same time-cure superposition principle. This common description could imply that both transitions have a common origin, i.e., dynamical instead of thermodynamical. Additionally, a continuous transition from the liquid to the solid state has also been observed in colloidal fluids,7 and in some polymer melts,5,6 as the confinement is increased in contraposition to the first-order phase transition observed by others.9 Acknowledgment. We acknowledge the assistance of A. Ramı´ rez-Saito with sample preparation, financial support from COPOCYT and CONACYT, Mexico (Grant 84076), and Ministerio de Ciencia e Innovacion, Spain.
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