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Scaling of activation energy for macroscopic flow in poly(ethylene glycol) solutions: Entangled – Non-entangled crossover
Polymer 55 (2014) 4651e4657
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Scaling of activation energy for macroscopic flow in poly(ethylene glycol) solutions: Entangled e Non-entangled crossover ski a, Tomasz Kalwarczyk a, Agnieszka Wisniewska a, Krzysztof Sozan lik a, Christoph Pieper b, Stefan A. Wieczorek a, Sławomir Jakieła a, Karolina Ke˛ dra-Kro b €rg Enderlein , Robert Hołyst a, * Jo a b
Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland €ttingen, Friedrich-Hund-Platz 1, D-37077 Go €ttingen, Germany III. Institute of Physics, Georg August University Go
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 June 2014 Received in revised form 16 July 2014 Accepted 18 July 2014 Available online 25 July 2014
We postulate an empirical scaling equation, which accurately describes flow of polymer solutions, complimenting the paradigm of length-scale-dependent viscosity. We investigated poly(ethylene glycol) aqueous solutions and observed an exponential dependence of viscosity on the hydrodynamic radius of a single coil Rh divided by the correlation length x. Properties of the system changed abruptly with the onset of chain entanglement at concentration corresponding to x ¼ Rh. We propose a single equation valid for all the investigated systems, analyze the physical meaning of parameters appearing therein and discuss the impact of chain entanglement. Viscous flow is treated as an activated process, following the Eyring rate theory. We show that the difference of activation energy for flow between pure solvent and polymer solution, DEa, is a function of concentration, whose derivative has a discontinuity at the crossover concentration. For dilute PEG solutions DEa takes values of up to several kJ/mol and is proportional to the intrinsic viscosity. We successfully apply the scaling approach to the diffusive motion of a protein (aldolase) in solutions of 25 kg/mol PEO (concentrations of 2e20%), investigated by fluorescence correlation spectroscopy (FCS). A significant difference in the influence of crowding on translational and rotational motion of the protein is revealed. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Viscosity scaling Polymer solutions PEG
1. Introduction Transport properties of complex liquids are often determined not only by the chemical composition and molecular characteristics of their constituents, but also by the characteristic length scales of the system [1,2]. In polymer solutions, the key ones are the hydrodynamic and gyration radii of single coils (Rh and Rg, respectively) and the correlation length x. These parameters are the basis for estimation of the effective viscosity experienced by diffusing probes, which may be lower than the bulk viscosity by orders of magnitude [3,4]. The intrinsic microstructure of the solution depends heavily on the molecular weight and concentration of macromolecules: increasing congestion of coils leads to qualitative changes in the structure of the liquid, shifting the balance of predominant interactions and severely affecting the dynamics [5e9]. In this paper, we demonstrate a scaling approach allowing for
* Corresponding author. Tel.: þ48 22 343 3123. E-mail address: [email protected] (R. Hołyst). http://dx.doi.org/10.1016/j.polymer.2014.07.029 0032-3861/© 2014 Elsevier Ltd. All rights reserved.
unified description of viscosity of polymer solutions at different concentration regimes, fully consistent with the paradigm of length-scale-dependent viscosity. We show that the intrinsic viscosity is a function of activation energy for viscous flow. We provide an explicit definition of this activation energy and quantitatively describe its changes upon the onset of entanglement of polymer chains, which marks the crossover between different concentration ranges. Polymer handbooks define three main concentration regimes: dilute, semi-dilute, and concentrated [5,10]. In the dilute regime polymer coils are separated and behave similarly to hard spheres with respect to each other [11], while the dynamics of the system are well approximated by the Zimm model and predominated by the hydrodynamic interactions [2,12]. Various subdivisions of the dilute regime have been suggested, based on changes in the interactions between coils [13e16]. Although the implications of the existence of the extremely dilute regime are potentially important, this concentration range is in fact approaching the infinite dilution limit and is not accessible in most experimental setups nor practically exploited.
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In the semi-dilute regime polymer coils start to overlap, gradually loosing their individuality to constitute a uniform mesh e a toy-model depicting the shifts occurring in the solution is presented in Fig. 1a. With further increase of the volume fraction of macromolecules, the solution approaches the polymer melt conditions, where chains are assumed to be ideal. This situation corresponds to the concentrated regime. The critical concentration separating dilute and semi-dilute regimes, c*, is usually defined as the concentration at which the whole volume is occupied by polymer coils, so that they touch, but not interpenetrate each other significantly [5,10,17,18]. Therefore, it can be simply defined as
Mw
c ¼ . ; 4 3pR3g NA
The dynamic, internal structure of complex liquids is of key importance for description of probe diffusion therein. Numerous experimental observations of unexpectedly high mobility of small probes in such systems [3,27e31] suggested a breakdown of the Stokes-Sutherland-Einstein relation (D ¼ kBT/6phrp, where: D e diffusion coefficient, h e viscosity, rp e probe radius) for rp < Rg [31,32]. In our recent works on semi-dilute systems [3,4], we demonstrated that the application of de Gennes' scaling framework [5] to the description of size-dependent viscosity [33] allows to resolve this issue. In this approach, the notion of effective radius Reff is introduced, where 2 2 R2 eff ¼ Rh þ rp :
(1)
where Mw is the polymer molecular weight, Rg is the polymer gyration radius and NA is the Avogadro number. We shall use this definition further on, although it must be noted that it does not constitute a sharp boundary [6,19,20]. Several definitions of c* other than Equation (1) have also been proposed [10,19,21] that give slightly different values. Moreover, even if we assume the polymer coils at c* to be impenetrable spheres of radius Rg, including different theoretically possible sphere packing arrangements (Fig. 1) could shift the predicted overlap concentration value even by a factor of 2 from the value given by Equation (1). Another uncertainty to this definition comes from Refs. Rg, which usually is determined for very dilute solutions and may shift upon concentration changes [10,13,22]. The theoretical and experimental difficulties concerning the intermediate concentration range may conceal important changes occurring in the system. Graessley et al.19 proposed to divide the semi-dilute region into distinct subregimes: entangled and non-entangled [2]. It has been shown that the onset of entanglement between chains crucially influences the properties of the system e e.g. its rheology and elasticity [23], grafting efficiency [24], or fiber formation [25,26]. We shall use this notion further on to explain the changes in transport properties of polymer solutions.
(2)
Viscosity experienced by the probe is then given by
" a # R h ¼ h0 exp b eff ; x
(3)
where: h0 e solvent viscosity, a and b e parameters of the order of unity. x is the correlation length of local monomer concentration fluctuations, interpreted also as the mesh or blob size in entangled systems [5,10,18]. It is estimated as
x ¼ Rg
c b c
;
(4)
where b is a scaling exponent given by Ref. b ¼ n(1 3n)1 [5,34,35]. The n parameter relates the coil size to the Mw, accounting for repulsive excluded volume interactions within the mean-field approach. According to Flory [11], n should be equal to 3/5 for a three-dimensional polymer coil in a good solvent, which gives b ¼ 3/4. Equation (3) is in line with the well established paradigm of stretch exponential dependence of effective solution viscosity on the polymer concentration [36e39]. It has been proven functional for different complex liquids, including systems other than flexible polymer solutions, e.g. suspensions of rigid micelles [4] or bacterial cytoplasm [40]. Recently [41], it was merged with Eyring's rate theory [42,43] to describe mass transport in complex systems as an activated process [44e50]. It was found that parameter b from Equation (3) is temperature-dependent, and the viscosity scaling can be written in terms of the Arrhenius equation as [41]
DEa ; h ¼ h0 exp RT
(5)
where
a R DEa ¼ g eff ; x
Fig. 1. a) Schematic changes of the internal structure of a polymer solution with increasing concentration. c*, overlap concentration, is usually treated as the threshold separating dilute and semi-dilute regimes. b) Different possible packing models of polymer chains treated as impenetrable hard spheres at c*. Real polymer coils are not that well defined in terms of shape, and the onset of overlap as well as entanglement is gradual. Therefore, strict determination of critical crossover concentration is somewhat dubious.
(6)
which is the excess activation energy for viscous flow over the one observed for pure solvent. g is a system-dependent parameter expressed in terms of energy and equal 4.0 ± 0.4 kJ/mol for PEG (PEO) aqueous solutions [41]. It should be noted that in the large probe limit (rp [ Rh) the effective radius reduces to the polymer hydrodynamic radius and therefore both Equations (3) and (5) apply also to the macroscopic viscosity and viscous flow of the polymer solution. However, most of the aforementioned scaling considerations refer to the case of heavily entangled systems. In this paper we attempt to extend this reasoning to dilute systems, investigating the experimentally observed changes of scaling parameters. We provide a simple, qualitative physical explanation of the observed changes and crossover between concentration regimes, along with
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a semi-empirical quantitative formulation based on extensive rheological measurements and literature data. We prove that intrinsic viscosity is directly proportional to DEa and can be used to estimate the latter. Finally, we apply the obtained formulas to translational and rotational diffusion of probes in dilute polymer solutions, demonstrating the universality of the viscosity scaling approach and its relevance to both probe mobility as well as bulk flow in complex systems. 2. Materials and methods 2.1. Polymer systems The presented research utilized a common model polymer system e aqueous solutions of poly(ethylene glycol) e PEG and poly(ethylene oxide) e PEO.1 Due to very good solubility of the polymers in water, we were able to perform measurements in a broad range of molecular masses (1 e 8000 kg/mol), concentrations (0.1 e 50% w/w) and temperatures (278 e 323 K), while the good solvent conditions were satisfied in all cases. As the investigated polymer system is well-described, we could estimate hydrodynamic and gyration radii of the coils (denoted as Rh and Rg, respectively) form accurate, empirical formulas available in the literature: [51,52] 0:583 Rg ¼ 0:0215Mw
0:571 Rh ¼ 0:0145Mw :
(7)
PEG (PEO) aqueous solutions are broadly used in polymer research as convenient model systems, while due to their availability and biocompatibility they are of much interest for the industry, e.g. in cosmetics or pharmaceutics [53,54]. We used both molecular weight standards of low polydispersity, obtained from Polymer Standard Service GmbH, Mainz, Germany (molecular weights: 1, 3, 6, 12, 18, 500, and 1000 kg/mol e see Supporting Information for exact Mw and PDI values), as well as more polydisperse polymers obtained from Fluka and SigmaeAldrich (molecular weights: 6, 8, 12, 20, 600, 2000, and 8000 kg/ mol). All solutions were prepared in deionized water and stirred for at least 24 h to ensure uniformity of the samples. The overall range of viscosities covered in the experiments was ca. 1 e 1000 mPa s. 2.2. Rheometry and fluorescence correlation spectroscopy Viscosity measurements at 298 K for high-PDI samples were performed using an Anton Paar rotational viscometer with a coneplate geometry. For low-PDI samples, Anton Paar AMVn rolling ball/ falling ball microviscometer was used to decrease the total solution volume necessary for each measurement. Densities of solutions, required in the latter method, were obtained using an Anton Paar DSA-48 vibrating tube densimeter. The temperature dependence of viscosity was investigated with a Malvern Kinexus rotational rheometer in the range of 278 e 323 K. Temperature was controlled within ±0.1 K. In case of dilute polymer solutions, the measured viscosities were close to the solvent viscosity. To provide additional, even more accurate data in this region, we performed experiments based on estimating the flow rate in a capillary. Custom-assembled equipment, controlled by an application written in LabVIEW, consisted of a microfluidic pump, a loosely coiled stainless steel capillary (2 m long, inner radius of 2 mm) and a receiver beaker
1 PEG and PEO are commonly used, historically justified names for poly(oxyethylene), the only difference being the Mw: customarily, PEG is used for polymers of Mw < 20 kg/mol, while PEO e for longer chains.
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placed on an electronic balance. The whole setup was enclosed in a thick-walled box made of expanded polystyrene, wherein the temperature was controlled within ±0.1 K. The viscosity values were calculated from the flow rates using the Poiseuille equation. It should be noted that a variety of methods was used for measurements of macroscopic viscosity in this study and all the results were fully consistent. We applied fluorescence correlation spectroscopy (FCS) to validate the applicability of obtained scaling to molecular mobility. We measured coefficients of translational and rotational diffusion (denoted as Dt and Drot, respectively) of fluorescently labeled aldolase in dilute and semi-dilute polymer solutions. Highprecision FCS measurements of lateral diffusion where conducted with dual-focus FCS; see Refs. [55,56] for the technical details. Rotational diffusion measurements where also performed with fluorescence correlation spectroscopy as described in Ref. [57]. Aldolase was labeled with bis-functional Cy5 dye which binds at both ends to lysin residues on the protein surface and thus ensures co-rotation of the dye with the protein. The results could be easily compared with the macroviscosity data, since in the paradigm of size-dependent viscosity the inverse proportionality of D and h is conserved across all length scales and therefore [4]
D h ¼ 0: D0 h
(8)
This relation holds for both translation and rotations of the protein, on condition that Dt is only referred to D0 measured for translation and Drot is referred to D0 measured for rotation (since coefficients of translational and rotational diffusion in pure solvent are obviously different). This also means that effective viscosity experienced by the protein may be different for translational and rotational motion. 3. Results and discussion 3.1. Measurements at 298 K First, we performed extensive viscometric experiments at a constant temperature of 298 K for samples falling in both dilute and semi-dilute concentration regimes. According to the de Gennes' general scaling idea, we plotted all the results against c/c* (Fig. 2a), where both c and c* were put in terms of mass of polymer per volume unit of solvent. As could be expected, a clear dependence was observed. We applied an extension of the viscosity scaling paradigm (Equation (3)), originally developed for semi-dilute systems [3,4]. The result is shown in Fig. 2. For solutions of relatively high concentration the relationship is fully satisfied. However, for low-viscosity samples a significant, systematic deviation is observed. Although monotonicity and general trend is conserved, which suggests validity of the basic c/c* scaling, some rapid change in the particular form of the equation seems obvious. Interestingly, the crossover does not occur at c ¼ c* (as defined in Equation (1)), but at higher concentration, namely when (Rh/x) ¼ 1. If we put Rh/ Rg ¼ 0.6, which is roughly accurate across a broad Mw spectrum, and a, b parameters established for semi-dilute PEG solutions, we obtain a shift of the observed crossover from the c ¼ c* value by a factor of 2. Parameter b ¼ 3/4 is based on Flory's calculation [11] of the exponent n and is essential for estimation of the correlation length x. Although the blob size in the case of isolated coils is equal to Rg, it should not be still identified with the correlation length, as it is in the entangled regime. x does not describe the range of correlation of fluctuations of local monomer density any more: with decreasing concentration, polymer coils becomes hard-sphere-like, while in
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description of a is also valid for other polymer systems e for example, thus calculated value of a for solution of polystyrene in acetophenone (theta solvent) [62] reproduces the results of viscosity data fitting within a 7% error [4]. In the non-entangled regime, polymerepolymer interactions become less pronounced and the mesh structure recedes. The polymer solution approaches then a model of freely diffusing separate hard spheres. Since in this approximation the chains do not interpenetrate each other and are simply treated as spheres of radius of Rh, the swelling-related factor 1 RhR1 g can be dropped in this situation, leaving a ¼ b . For dilute polymer solutions in good solvent this produces a ¼ 4/3, which is in good agreement with the fitted value of 1.29. Although such interpretation of the scaling parameters would require a thorough validation in other polymer systems to be considered as exact and universal, it may for now serve as a useful approximation. In Fig. 3 we present data from rheological measurements at 298 K plotted according to the two-regime scaling. For fitting of the parameters we used only the results obtained for molecular mass standards, while data for polymers of broad Mw distribution were overlayed upon the fitted curve. b was treated as a free fitting parameter, common for both regimes. A value of 1.615 was obtained. b is related to the activation energy for the viscous flow and is discussed in detail in the next section of this manuscript. 3.2. Activation energy for viscous flow
Fig. 2. Results of viscosity measurements for aqueous polymer solutions: a) Relative viscosity plotted against concentration reduced by overlap concentration; b) Same data, plotted according to Equation (3), with one set of fitted parameters across the whole concentration range (a ¼ 0.75, b ¼ 1.75). All data concern macroscopic viscosity (limit of infinitely large probe), therefore Rh is used instead of Reff (according to Equation (2)). Inset shows the non-entangled regime in detail, where a significant, systematic offset appears. Apparent low quality of the fit suggests that a more sophisticated approach is needed. Legend applies to both panels; asterisk denotes polymers of broad Mw distribution.
the limit of infinite dilution they can be treated as material points. Therefore, a definition of correlation length useful for dilute solutions should account for the mean distance between centers of mass of neighboring coils. Such approach can be generalized by looking at x as the mean distance from a given monomer, at which a monomer belonging to a different coil can be found, which is in line with the definition postulated by Rubinstein [58] and retains validity across all concentration ranges, reproducing the blob size in semi-dilute solutions. Parameter a is related to the internal structure of the complex liquid, while its value is characteristic for a given polymer/solvent system [4]. Therefore, it is natural to expect a change of a upon crossover from non-entangled to entangled solutions [5,19,23,59,60]. The systematic deviation from Equation (3) observed for dilute solutions (Fig. 2b) confirms that it is actually the case. We analyzed the data, taking into account the effects of the qualitative change of the structure of the complex liquid occurring at x ¼ Rh. From fitting, we obtained values of a equal 0.78 for entangled PEG systems (which is congruent with our previous results) [41] and 1.29 for dilute solutions. The latter value reproduces the result obtained for solutions of hard spheres [61], which represent a good model for dilute polymer solutions. On the basis of fitting of data presented here and previously [4], 1 we propose a possible interpretation of a ¼ RhR1 for entangled g b solutions, which matches the obtained value within a 3% error. Such
Following the recently proposed [41] application of Eyring's rate theory [42,43] to size-dependent transport properties of complex liquids, we performed viscosity measurements in a broad temperature range (278 e 323 K). The objective of these experiments was to extend the applicability of the proposed scaling of activation energy for viscous flow to dilute polymer solutions and include it in the interpretation of intrinsic viscosity of such systems. We found that parameter b appearing in Equation (3) is temperaturedependent, which was in line with the previously described results [41]. We therefore implemented Equation (5) to the description of viscosity of dilute polymer solutions. In the discussed limiting case of macroscopic viscous flow (which corresponds to an infinitely large probe), effective radius Reff is reduced to the polymer hydrodynamic radius Rh. By analogy to Equation (6), the excess of activation energy for flow of solution over the activation energy for flow of pure solvent can therefore be described as
Fig. 3. Results of viscosity measurements at 298 K plotted according to Equation (3), including the shift of parameters at the non-entangled e entangled crossover (marked with a vertical dashed line). Asterisk denotes polymers of broad Mw distribution.
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DEa ¼ g
Rh x
a :
(9)
Following the above definition, we analyzed both new and literature [63,64] data on the temperature dependence of viscosity of PEG (PEO) solutions. The results are presented in Fig. 4. Parameter g is kept constant and equal to the value obtained previously [41] for aqueous PEG (PEO) solutions, i.e. 4.0 kJ/mol g is an empirical parameter stemming from polymerepolymer and polymer-solvent interactions. It is therefore expected to change from system to system, depending on the chemical composition. The values of a and b parameters applied here are same as those established in experiments at 298 K. All the numeric parameters are compiled in Table 1. DEa directly reflects the contribution of the polymer to the overall resistance of solution against flow. For a given polymersolvent system, activation energy for viscous flow is a continuous
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Table 1 Compilation of parameters appearing in the scaling equations. b is the de Gennes' exponent used for estimation of correlation length, calculated as n(1 3n)1 [5,34,35].a concerns the structure of the liquid and can be approximated as b1 in 1 non-entangled solutions and RhR1 in the entangled ones. g, expressed in energy g b units, is related to the polymerepolymer and polymer-solvent interactions.
Entangled Non-entangled
a
b
g [kJ/mol]
0.78 1.29
0.75 0.75
4.0 4.0
function of concentration e an exemplary curve for PEG (18 kg/ mol)-water solutions is plotted in Fig. 5. First derivative of DEa with respect to c presents a discontinuity at the point of crossover between non-entangled and entangled regimes, when Rh ¼ x and the a parameter changes. If we apply the interpretation of a ¼ b1 suggested for nonentangled solutions, we may write Equation (9) as
a g Rh c : h ¼ h0 exp RT Rg c
(10)
If we now approximate the above formula by the first term of the expansion of the exponential function (which is valid at low concentration), we obtain
g Rh a c : h ¼ h0 1 þ RT Rg c
(11)
This form is equivalent to the basic equation for viscosity of polymer solutions, h ¼ h0(1 þ c[h]), with intrinsic viscosity [h] given as
½h ¼
g Rh a 1 : RT Rg c
(12)
It is generally accepted [66e68] that, according to the MarkeHouwink equation, intrinsic viscosity is inversely proportional to the overlap concentration: [h] ¼ a(c*)1 a is often assumed to be 1 [10], although other values have also been suggested a [68,69]. From Equation (12), we get a ¼ g(RT)1(RhR1 g ) , which for dilute aqueous PEG solutions at 298 K gives a value of a ¼ 0.85. Equation (12) allows for easy estimation of the intrinsic viscosity of dilute polymer solutions. An even more accurate formula could be obtained if we added the next term of the exponential expansion to Equation (11), i.e. [h]2c2/2. We would then obtain an analog of the frequently used form of the Huggins equation [70]. However, the quality of currently available data is not sufficient to validate such minor corrections and we shall confine ourselves to the more coarse estimation of [h] given by Equation (12). Following this simplification, a direct relationship between DEa and [h] can be postulated for non-entangled polymer solutions:
DEa ¼ ½hcRT:
(13)
3.3. Applicability to probe diffusion
Fig. 4. Viscosity scaling including the temperature dependence. Panel b) shows the data for the non-entangled regime in more detail. The notion of activation energy for viscous flow is introduced through Equation (6). Single asterisk denotes polymers of broad Mw distribution. Data marked in a legend with y, z, and x are adapted from Refs. [63e65], respectively.
In our earlier work we have stated that the length-scaledependent viscosity approach can be used for uniform description of both probe diffusion and macroscopic viscous flow of complex liquids [3,4,41]. To test the validity of this statement in reference to the hereby postulated formulas for non-entangled solutions we performed fluorescence correlation spectroscopy measurements. We used a labeled protein, aldolase (rp ¼ 4.5 nm), as a probe freely diffusing in solutions of 25 kg/mol PEO. We covered both the non-entangled and entangled concentration regimes. We measured translational and rotational diffusion coefficients,
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phase boundary, originating from an entropic background (conformational freedom of a chain near an interface is lower than in bulk solution. This effect diminishes its entropy and makes such location of a chain unfavorable). The DL typically spans up to a few nm away from the interface and may appear e.g. at the surface of diffusing microspheres or nanoparticles [71]. It has been shown that the DL may have a significant influence on both rotational and translational diffusion in crowded environment [72,73]. However, probe rotation is usually facilitated by the DL much more than translation [73]. Such theoretical predictions are strongly supported by the experimental results presented hereby. 4. Conclusions
Fig. 5. Excess of activation energy for flow of polymer solution, DEa (which is the difference between Ea for solution and for pure solvent) plotted against polymer concentration. Curve drawn for aqueous solution of PEG, Mw ¼ 18 kg/mol, at 298 K. The plot is continuous across the whole concentration range; an inflection is observed at the entangledenon-entangled crossover.
translating them into effective viscosity using Equation (8). Effective radius was estimated according to Equation (2). In case of probe diffusion, parameter g may differ from the one observed for macroscopic flow (roughly reproducing it in the large probe limit) [41]. It is justified by the influence of probe-polymer interactions on the average probe velocity. On the basis of simple fitting of viscosity data, g for aldolase diffusion in PEG is estimated to be 3.7 kJ/mol. All other parameters are kept the same as for macroscopic flow. The results are plotted in Fig. 6. Solid line refers to the scaling function given by Equation (9), where effective radius Reff is used instead of Rh [4]. Good agreement between Equation (9) and the data for translational diffusion can be observed. On the other hand, rotational viscosity experienced by the probes is significantly lower than the translational across the whole concentration range. Such observation is an experimental confirmation that the depletion layer (DL) effect occurs not only in entangled, but also in highly diluted polymer solutions. The DL is a region of decreased polymer concentration in the vicinity of a
Fig. 6. Effective translational and rotational viscosities experienced by a labeled protein, aldolase (rp ¼ 4.5 nm), diffusing in 25 kg/mol PEO aqueous solutions. Polymer concentration ranged from 2 to 20% w/w. Values for translation match the hereby presented scaling (solid line) well. Rotational diffusion is significantly facilitated due to spontaneous formation of a depletion layer around an object immersed in a polymer solution.
We have demonstrated that viscosity scaling for PEG (PEO) aqueous solutions, previously reported for entangled systems, can be extended to the dilute regime. The clearly observed crossover between the regimes occurs at (Rh/x)a ¼ 1, i.e. at polymer concentration about twice higher that c*. This finding may be interpreted as indication of an intermediate concentration regime between dilute and semi-dilute: semi-dilute non-entangled. However, no change of scaling parameters is observed at the c* threshold, which would be the dilute e semi-dilute non-entangled border. We may therefore interpret the onset of entanglement as the determining factor inducing changes in system properties instead of coil overlap. This is in line with two earlier observations: that the onset of chain entanglement happens at concentration significantly higher than c* [19,60,74], and that properties of polymer systems heavily depend on the presence of chain entanglements [9,24,75]. What influences the rheology of complex systems is not just the volume fraction occupied by macromolecules, but also the topology of the dynamic supramolecular structures they form. It should also be noted here that there is no general agreement on the exact definition of c* and the observed entanglement-related crossover may in fact be close to the dilute e semi-dilute shift, if not even equivalent. In Equation (9) we explicitly formulate an applicable description of viscosity of polymer solutions across a broad concentration range. The proposed approach is based on notions commonly used for characterization of polymer systems: hydrodynamic and gyration radii, correlation length and the Flory exponent. We offer an interpretation of the parameters appearing in the equations and analyze their shifts at the crossover between different concentration regimes. Expansion of the exponential formula (Equation (11)) allows to estimate the intrinsic viscosity of polymer solutions in the dilute regime (Equation (12)), providing a reference to Einstein's simplistic approach to hard sphere solutions [76]. The notion of activation energy for viscous flow can be successfully introduced into the scaling formula, allowing to consider viscous flow in terms of an activated process (Equation (9)). This may open new ways to explore the interactions in polymer solutions and understand the principals of transport properties of such systems. Moreover, also intrinsic viscosity of dilute solutions can be easily described in terms of the activation energy (Equation (13)). The presented conclusions are based on precise viscosity measurements performed using various methods for a model good solvent system: PEG and PEO aqueous solutions. We have already shown previously that scaling equations of this kind can be easily adapted to different complex liquids (including even biological systems) [4,40]. Validity of the hereby proposed physical description is supported by literature data, which fall on the curves obtained from our experimental results. Furthermore, we prove the applicability of the scaling approach to description of probe diffusion. It is another important step towards a uniform and universal description of transport of mass in complex systems. We also observe a significant mismatch between the resistance of the
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polymer solution against probe diffusion and rotation. The great facilitation of the latter is certainty ascribed to the depletion layer effect, while quantitative evaluation of this phenomenon in nonentangled systems seems an interesting perspective for further investigation.
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Acknowledgments K.S. acknowledges financial support of the Ministry of Science of Poland within the Diamond Grant program DI2011 015341. R.H. and A.W. thank the National Science Center for funding the project from the funds granted on the basis of the decision number 2011/02/A/ ST3/00143 (Maestro grant). T.K. thanks the Ministry of Science of Poland for support within the Iuventus Plus program IP2012 015372. S.J. thanks the Ministry of Science of Poland for support within the Iuventus Plus program IP2012 015172. Appendix A. Supplementary data Supplementary data related to this article can be found at http:// dx.doi.org/10.1016/j.polymer.2014.07.029. References [1] Broseta D, Leibler L, Lapp A, Strazielle C. EPL 1986;2:733e7. [2] Heo Y, Larson RG. J Rheol 2005;49:1117e28. [3] Holyst R, Bielejewska A, Szymanski J, Wilk A, Patkowski A, Gapinski J, et al. Phys Chem Chem Phys 2009;11:9025e32. [4] Kalwarczyk T, Ziebacz N, Bielejewska A, Zaboklicka E, Koynov K, Szymanski J, et al. Nano Lett 2011;11:2157e63. [5] de Gennes P-G. Scaling concepts in polymer physics. Cornell University Press; 1979. [6] Ying Q, Chu B. Macromolecules 1987;20:362e6. [7] Pan C, Maurer W, Liu Z, Lodge TP, Stepanek P, Vonmeerwall ED, et al. Macromolecules 1995;28:1643e53. [8] Rauch J, Kohler W. J Chem Phys 2003;119:11977e88. [9] Chapman CD, Lee K, Henze D, Smith DE, Robertson-Anderson RM. Macromolecules 2014;47:1181e6. [10] Teraoka I. Polymer solutions: an introduction to physical properties. John Wiley & Sons, Inc; 2002. [11] Flory P. Principles of polymer chemistry. Cornell University Press; 1971. [12] Zimm BH. J Chem Chys 1956;24:269e78. [13] Wu C. J Polym Sci Pol Phys 1994;32:1503e9. [14] Cheng RS. Macromol Symp 1997;124:27e34. [15] Dondos A, Tsitsilianis C, Staikos G. Polymer 1989;30:1690e4. [16] Dondos A, Tsitsilianis C. Polym Int 1992;28:151e6. [17] Cooper EC, Johnson P, Donald AM. Polymer 1991;32:2815e22. [18] Doi M. Introduction to polymer physics. Oxford University Press; 1997. [19] Graessley WW. Polymer 1980;21:258e62. [20] Cotton JP, Nierlich M, Boue F, Daoud M, Farnoux B, Jannink G, et al. J Chem Phys 1976;65:1101e8. [21] Berry GC, Nakayasu H, Fox TG. J Polym Sci Pol Phys 1979;17:1825e44. [22] Daoud M, Cotton JP, Farnoux B, Jannink G, Sarma G, Benoit H, et al. Macromolecules 1975;8:804e18. [23] Wool RP. Macromolecules 1993;26:1564e9. [24] Zdyrko B, Varshney SK, Luzinov I. Langmuir 2004;20:6727e35.
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Scaling of activation energy for macroscopic flow in poly(ethylene glycol) solutions: Entangled e Non-entangled crossover ski a, Tomasz Kalwarczyk a, Agnieszka Wisniewska a, Krzysztof Sozan lik a, Christoph Pieper b, Stefan A. Wieczorek a, Sławomir Jakieła a, Karolina Ke˛ dra-Kro b €rg Enderlein , Robert Hołyst a, * Jo a b
Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland €ttingen, Friedrich-Hund-Platz 1, D-37077 Go €ttingen, Germany III. Institute of Physics, Georg August University Go
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 June 2014 Received in revised form 16 July 2014 Accepted 18 July 2014 Available online 25 July 2014
We postulate an empirical scaling equation, which accurately describes flow of polymer solutions, complimenting the paradigm of length-scale-dependent viscosity. We investigated poly(ethylene glycol) aqueous solutions and observed an exponential dependence of viscosity on the hydrodynamic radius of a single coil Rh divided by the correlation length x. Properties of the system changed abruptly with the onset of chain entanglement at concentration corresponding to x ¼ Rh. We propose a single equation valid for all the investigated systems, analyze the physical meaning of parameters appearing therein and discuss the impact of chain entanglement. Viscous flow is treated as an activated process, following the Eyring rate theory. We show that the difference of activation energy for flow between pure solvent and polymer solution, DEa, is a function of concentration, whose derivative has a discontinuity at the crossover concentration. For dilute PEG solutions DEa takes values of up to several kJ/mol and is proportional to the intrinsic viscosity. We successfully apply the scaling approach to the diffusive motion of a protein (aldolase) in solutions of 25 kg/mol PEO (concentrations of 2e20%), investigated by fluorescence correlation spectroscopy (FCS). A significant difference in the influence of crowding on translational and rotational motion of the protein is revealed. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Viscosity scaling Polymer solutions PEG
1. Introduction Transport properties of complex liquids are often determined not only by the chemical composition and molecular characteristics of their constituents, but also by the characteristic length scales of the system [1,2]. In polymer solutions, the key ones are the hydrodynamic and gyration radii of single coils (Rh and Rg, respectively) and the correlation length x. These parameters are the basis for estimation of the effective viscosity experienced by diffusing probes, which may be lower than the bulk viscosity by orders of magnitude [3,4]. The intrinsic microstructure of the solution depends heavily on the molecular weight and concentration of macromolecules: increasing congestion of coils leads to qualitative changes in the structure of the liquid, shifting the balance of predominant interactions and severely affecting the dynamics [5e9]. In this paper, we demonstrate a scaling approach allowing for
* Corresponding author. Tel.: þ48 22 343 3123. E-mail address: [email protected] (R. Hołyst). http://dx.doi.org/10.1016/j.polymer.2014.07.029 0032-3861/© 2014 Elsevier Ltd. All rights reserved.
unified description of viscosity of polymer solutions at different concentration regimes, fully consistent with the paradigm of length-scale-dependent viscosity. We show that the intrinsic viscosity is a function of activation energy for viscous flow. We provide an explicit definition of this activation energy and quantitatively describe its changes upon the onset of entanglement of polymer chains, which marks the crossover between different concentration ranges. Polymer handbooks define three main concentration regimes: dilute, semi-dilute, and concentrated [5,10]. In the dilute regime polymer coils are separated and behave similarly to hard spheres with respect to each other [11], while the dynamics of the system are well approximated by the Zimm model and predominated by the hydrodynamic interactions [2,12]. Various subdivisions of the dilute regime have been suggested, based on changes in the interactions between coils [13e16]. Although the implications of the existence of the extremely dilute regime are potentially important, this concentration range is in fact approaching the infinite dilution limit and is not accessible in most experimental setups nor practically exploited.
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In the semi-dilute regime polymer coils start to overlap, gradually loosing their individuality to constitute a uniform mesh e a toy-model depicting the shifts occurring in the solution is presented in Fig. 1a. With further increase of the volume fraction of macromolecules, the solution approaches the polymer melt conditions, where chains are assumed to be ideal. This situation corresponds to the concentrated regime. The critical concentration separating dilute and semi-dilute regimes, c*, is usually defined as the concentration at which the whole volume is occupied by polymer coils, so that they touch, but not interpenetrate each other significantly [5,10,17,18]. Therefore, it can be simply defined as
Mw
c ¼ . ; 4 3pR3g NA
The dynamic, internal structure of complex liquids is of key importance for description of probe diffusion therein. Numerous experimental observations of unexpectedly high mobility of small probes in such systems [3,27e31] suggested a breakdown of the Stokes-Sutherland-Einstein relation (D ¼ kBT/6phrp, where: D e diffusion coefficient, h e viscosity, rp e probe radius) for rp < Rg [31,32]. In our recent works on semi-dilute systems [3,4], we demonstrated that the application of de Gennes' scaling framework [5] to the description of size-dependent viscosity [33] allows to resolve this issue. In this approach, the notion of effective radius Reff is introduced, where 2 2 R2 eff ¼ Rh þ rp :
(1)
where Mw is the polymer molecular weight, Rg is the polymer gyration radius and NA is the Avogadro number. We shall use this definition further on, although it must be noted that it does not constitute a sharp boundary [6,19,20]. Several definitions of c* other than Equation (1) have also been proposed [10,19,21] that give slightly different values. Moreover, even if we assume the polymer coils at c* to be impenetrable spheres of radius Rg, including different theoretically possible sphere packing arrangements (Fig. 1) could shift the predicted overlap concentration value even by a factor of 2 from the value given by Equation (1). Another uncertainty to this definition comes from Refs. Rg, which usually is determined for very dilute solutions and may shift upon concentration changes [10,13,22]. The theoretical and experimental difficulties concerning the intermediate concentration range may conceal important changes occurring in the system. Graessley et al.19 proposed to divide the semi-dilute region into distinct subregimes: entangled and non-entangled [2]. It has been shown that the onset of entanglement between chains crucially influences the properties of the system e e.g. its rheology and elasticity [23], grafting efficiency [24], or fiber formation [25,26]. We shall use this notion further on to explain the changes in transport properties of polymer solutions.
(2)
Viscosity experienced by the probe is then given by
" a # R h ¼ h0 exp b eff ; x
(3)
where: h0 e solvent viscosity, a and b e parameters of the order of unity. x is the correlation length of local monomer concentration fluctuations, interpreted also as the mesh or blob size in entangled systems [5,10,18]. It is estimated as
x ¼ Rg
c b c
;
(4)
where b is a scaling exponent given by Ref. b ¼ n(1 3n)1 [5,34,35]. The n parameter relates the coil size to the Mw, accounting for repulsive excluded volume interactions within the mean-field approach. According to Flory [11], n should be equal to 3/5 for a three-dimensional polymer coil in a good solvent, which gives b ¼ 3/4. Equation (3) is in line with the well established paradigm of stretch exponential dependence of effective solution viscosity on the polymer concentration [36e39]. It has been proven functional for different complex liquids, including systems other than flexible polymer solutions, e.g. suspensions of rigid micelles [4] or bacterial cytoplasm [40]. Recently [41], it was merged with Eyring's rate theory [42,43] to describe mass transport in complex systems as an activated process [44e50]. It was found that parameter b from Equation (3) is temperature-dependent, and the viscosity scaling can be written in terms of the Arrhenius equation as [41]
DEa ; h ¼ h0 exp RT
(5)
where
a R DEa ¼ g eff ; x
Fig. 1. a) Schematic changes of the internal structure of a polymer solution with increasing concentration. c*, overlap concentration, is usually treated as the threshold separating dilute and semi-dilute regimes. b) Different possible packing models of polymer chains treated as impenetrable hard spheres at c*. Real polymer coils are not that well defined in terms of shape, and the onset of overlap as well as entanglement is gradual. Therefore, strict determination of critical crossover concentration is somewhat dubious.
(6)
which is the excess activation energy for viscous flow over the one observed for pure solvent. g is a system-dependent parameter expressed in terms of energy and equal 4.0 ± 0.4 kJ/mol for PEG (PEO) aqueous solutions [41]. It should be noted that in the large probe limit (rp [ Rh) the effective radius reduces to the polymer hydrodynamic radius and therefore both Equations (3) and (5) apply also to the macroscopic viscosity and viscous flow of the polymer solution. However, most of the aforementioned scaling considerations refer to the case of heavily entangled systems. In this paper we attempt to extend this reasoning to dilute systems, investigating the experimentally observed changes of scaling parameters. We provide a simple, qualitative physical explanation of the observed changes and crossover between concentration regimes, along with
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a semi-empirical quantitative formulation based on extensive rheological measurements and literature data. We prove that intrinsic viscosity is directly proportional to DEa and can be used to estimate the latter. Finally, we apply the obtained formulas to translational and rotational diffusion of probes in dilute polymer solutions, demonstrating the universality of the viscosity scaling approach and its relevance to both probe mobility as well as bulk flow in complex systems. 2. Materials and methods 2.1. Polymer systems The presented research utilized a common model polymer system e aqueous solutions of poly(ethylene glycol) e PEG and poly(ethylene oxide) e PEO.1 Due to very good solubility of the polymers in water, we were able to perform measurements in a broad range of molecular masses (1 e 8000 kg/mol), concentrations (0.1 e 50% w/w) and temperatures (278 e 323 K), while the good solvent conditions were satisfied in all cases. As the investigated polymer system is well-described, we could estimate hydrodynamic and gyration radii of the coils (denoted as Rh and Rg, respectively) form accurate, empirical formulas available in the literature: [51,52] 0:583 Rg ¼ 0:0215Mw
0:571 Rh ¼ 0:0145Mw :
(7)
PEG (PEO) aqueous solutions are broadly used in polymer research as convenient model systems, while due to their availability and biocompatibility they are of much interest for the industry, e.g. in cosmetics or pharmaceutics [53,54]. We used both molecular weight standards of low polydispersity, obtained from Polymer Standard Service GmbH, Mainz, Germany (molecular weights: 1, 3, 6, 12, 18, 500, and 1000 kg/mol e see Supporting Information for exact Mw and PDI values), as well as more polydisperse polymers obtained from Fluka and SigmaeAldrich (molecular weights: 6, 8, 12, 20, 600, 2000, and 8000 kg/ mol). All solutions were prepared in deionized water and stirred for at least 24 h to ensure uniformity of the samples. The overall range of viscosities covered in the experiments was ca. 1 e 1000 mPa s. 2.2. Rheometry and fluorescence correlation spectroscopy Viscosity measurements at 298 K for high-PDI samples were performed using an Anton Paar rotational viscometer with a coneplate geometry. For low-PDI samples, Anton Paar AMVn rolling ball/ falling ball microviscometer was used to decrease the total solution volume necessary for each measurement. Densities of solutions, required in the latter method, were obtained using an Anton Paar DSA-48 vibrating tube densimeter. The temperature dependence of viscosity was investigated with a Malvern Kinexus rotational rheometer in the range of 278 e 323 K. Temperature was controlled within ±0.1 K. In case of dilute polymer solutions, the measured viscosities were close to the solvent viscosity. To provide additional, even more accurate data in this region, we performed experiments based on estimating the flow rate in a capillary. Custom-assembled equipment, controlled by an application written in LabVIEW, consisted of a microfluidic pump, a loosely coiled stainless steel capillary (2 m long, inner radius of 2 mm) and a receiver beaker
1 PEG and PEO are commonly used, historically justified names for poly(oxyethylene), the only difference being the Mw: customarily, PEG is used for polymers of Mw < 20 kg/mol, while PEO e for longer chains.
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placed on an electronic balance. The whole setup was enclosed in a thick-walled box made of expanded polystyrene, wherein the temperature was controlled within ±0.1 K. The viscosity values were calculated from the flow rates using the Poiseuille equation. It should be noted that a variety of methods was used for measurements of macroscopic viscosity in this study and all the results were fully consistent. We applied fluorescence correlation spectroscopy (FCS) to validate the applicability of obtained scaling to molecular mobility. We measured coefficients of translational and rotational diffusion (denoted as Dt and Drot, respectively) of fluorescently labeled aldolase in dilute and semi-dilute polymer solutions. Highprecision FCS measurements of lateral diffusion where conducted with dual-focus FCS; see Refs. [55,56] for the technical details. Rotational diffusion measurements where also performed with fluorescence correlation spectroscopy as described in Ref. [57]. Aldolase was labeled with bis-functional Cy5 dye which binds at both ends to lysin residues on the protein surface and thus ensures co-rotation of the dye with the protein. The results could be easily compared with the macroviscosity data, since in the paradigm of size-dependent viscosity the inverse proportionality of D and h is conserved across all length scales and therefore [4]
D h ¼ 0: D0 h
(8)
This relation holds for both translation and rotations of the protein, on condition that Dt is only referred to D0 measured for translation and Drot is referred to D0 measured for rotation (since coefficients of translational and rotational diffusion in pure solvent are obviously different). This also means that effective viscosity experienced by the protein may be different for translational and rotational motion. 3. Results and discussion 3.1. Measurements at 298 K First, we performed extensive viscometric experiments at a constant temperature of 298 K for samples falling in both dilute and semi-dilute concentration regimes. According to the de Gennes' general scaling idea, we plotted all the results against c/c* (Fig. 2a), where both c and c* were put in terms of mass of polymer per volume unit of solvent. As could be expected, a clear dependence was observed. We applied an extension of the viscosity scaling paradigm (Equation (3)), originally developed for semi-dilute systems [3,4]. The result is shown in Fig. 2. For solutions of relatively high concentration the relationship is fully satisfied. However, for low-viscosity samples a significant, systematic deviation is observed. Although monotonicity and general trend is conserved, which suggests validity of the basic c/c* scaling, some rapid change in the particular form of the equation seems obvious. Interestingly, the crossover does not occur at c ¼ c* (as defined in Equation (1)), but at higher concentration, namely when (Rh/x) ¼ 1. If we put Rh/ Rg ¼ 0.6, which is roughly accurate across a broad Mw spectrum, and a, b parameters established for semi-dilute PEG solutions, we obtain a shift of the observed crossover from the c ¼ c* value by a factor of 2. Parameter b ¼ 3/4 is based on Flory's calculation [11] of the exponent n and is essential for estimation of the correlation length x. Although the blob size in the case of isolated coils is equal to Rg, it should not be still identified with the correlation length, as it is in the entangled regime. x does not describe the range of correlation of fluctuations of local monomer density any more: with decreasing concentration, polymer coils becomes hard-sphere-like, while in
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description of a is also valid for other polymer systems e for example, thus calculated value of a for solution of polystyrene in acetophenone (theta solvent) [62] reproduces the results of viscosity data fitting within a 7% error [4]. In the non-entangled regime, polymerepolymer interactions become less pronounced and the mesh structure recedes. The polymer solution approaches then a model of freely diffusing separate hard spheres. Since in this approximation the chains do not interpenetrate each other and are simply treated as spheres of radius of Rh, the swelling-related factor 1 RhR1 g can be dropped in this situation, leaving a ¼ b . For dilute polymer solutions in good solvent this produces a ¼ 4/3, which is in good agreement with the fitted value of 1.29. Although such interpretation of the scaling parameters would require a thorough validation in other polymer systems to be considered as exact and universal, it may for now serve as a useful approximation. In Fig. 3 we present data from rheological measurements at 298 K plotted according to the two-regime scaling. For fitting of the parameters we used only the results obtained for molecular mass standards, while data for polymers of broad Mw distribution were overlayed upon the fitted curve. b was treated as a free fitting parameter, common for both regimes. A value of 1.615 was obtained. b is related to the activation energy for the viscous flow and is discussed in detail in the next section of this manuscript. 3.2. Activation energy for viscous flow
Fig. 2. Results of viscosity measurements for aqueous polymer solutions: a) Relative viscosity plotted against concentration reduced by overlap concentration; b) Same data, plotted according to Equation (3), with one set of fitted parameters across the whole concentration range (a ¼ 0.75, b ¼ 1.75). All data concern macroscopic viscosity (limit of infinitely large probe), therefore Rh is used instead of Reff (according to Equation (2)). Inset shows the non-entangled regime in detail, where a significant, systematic offset appears. Apparent low quality of the fit suggests that a more sophisticated approach is needed. Legend applies to both panels; asterisk denotes polymers of broad Mw distribution.
the limit of infinite dilution they can be treated as material points. Therefore, a definition of correlation length useful for dilute solutions should account for the mean distance between centers of mass of neighboring coils. Such approach can be generalized by looking at x as the mean distance from a given monomer, at which a monomer belonging to a different coil can be found, which is in line with the definition postulated by Rubinstein [58] and retains validity across all concentration ranges, reproducing the blob size in semi-dilute solutions. Parameter a is related to the internal structure of the complex liquid, while its value is characteristic for a given polymer/solvent system [4]. Therefore, it is natural to expect a change of a upon crossover from non-entangled to entangled solutions [5,19,23,59,60]. The systematic deviation from Equation (3) observed for dilute solutions (Fig. 2b) confirms that it is actually the case. We analyzed the data, taking into account the effects of the qualitative change of the structure of the complex liquid occurring at x ¼ Rh. From fitting, we obtained values of a equal 0.78 for entangled PEG systems (which is congruent with our previous results) [41] and 1.29 for dilute solutions. The latter value reproduces the result obtained for solutions of hard spheres [61], which represent a good model for dilute polymer solutions. On the basis of fitting of data presented here and previously [4], 1 we propose a possible interpretation of a ¼ RhR1 for entangled g b solutions, which matches the obtained value within a 3% error. Such
Following the recently proposed [41] application of Eyring's rate theory [42,43] to size-dependent transport properties of complex liquids, we performed viscosity measurements in a broad temperature range (278 e 323 K). The objective of these experiments was to extend the applicability of the proposed scaling of activation energy for viscous flow to dilute polymer solutions and include it in the interpretation of intrinsic viscosity of such systems. We found that parameter b appearing in Equation (3) is temperaturedependent, which was in line with the previously described results [41]. We therefore implemented Equation (5) to the description of viscosity of dilute polymer solutions. In the discussed limiting case of macroscopic viscous flow (which corresponds to an infinitely large probe), effective radius Reff is reduced to the polymer hydrodynamic radius Rh. By analogy to Equation (6), the excess of activation energy for flow of solution over the activation energy for flow of pure solvent can therefore be described as
Fig. 3. Results of viscosity measurements at 298 K plotted according to Equation (3), including the shift of parameters at the non-entangled e entangled crossover (marked with a vertical dashed line). Asterisk denotes polymers of broad Mw distribution.
A. Wisniewska et al. / Polymer 55 (2014) 4651e4657
DEa ¼ g
Rh x
a :
(9)
Following the above definition, we analyzed both new and literature [63,64] data on the temperature dependence of viscosity of PEG (PEO) solutions. The results are presented in Fig. 4. Parameter g is kept constant and equal to the value obtained previously [41] for aqueous PEG (PEO) solutions, i.e. 4.0 kJ/mol g is an empirical parameter stemming from polymerepolymer and polymer-solvent interactions. It is therefore expected to change from system to system, depending on the chemical composition. The values of a and b parameters applied here are same as those established in experiments at 298 K. All the numeric parameters are compiled in Table 1. DEa directly reflects the contribution of the polymer to the overall resistance of solution against flow. For a given polymersolvent system, activation energy for viscous flow is a continuous
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Table 1 Compilation of parameters appearing in the scaling equations. b is the de Gennes' exponent used for estimation of correlation length, calculated as n(1 3n)1 [5,34,35].a concerns the structure of the liquid and can be approximated as b1 in 1 non-entangled solutions and RhR1 in the entangled ones. g, expressed in energy g b units, is related to the polymerepolymer and polymer-solvent interactions.
Entangled Non-entangled
a
b
g [kJ/mol]
0.78 1.29
0.75 0.75
4.0 4.0
function of concentration e an exemplary curve for PEG (18 kg/ mol)-water solutions is plotted in Fig. 5. First derivative of DEa with respect to c presents a discontinuity at the point of crossover between non-entangled and entangled regimes, when Rh ¼ x and the a parameter changes. If we apply the interpretation of a ¼ b1 suggested for nonentangled solutions, we may write Equation (9) as
a g Rh c : h ¼ h0 exp RT Rg c
(10)
If we now approximate the above formula by the first term of the expansion of the exponential function (which is valid at low concentration), we obtain
g Rh a c : h ¼ h0 1 þ RT Rg c
(11)
This form is equivalent to the basic equation for viscosity of polymer solutions, h ¼ h0(1 þ c[h]), with intrinsic viscosity [h] given as
½h ¼
g Rh a 1 : RT Rg c
(12)
It is generally accepted [66e68] that, according to the MarkeHouwink equation, intrinsic viscosity is inversely proportional to the overlap concentration: [h] ¼ a(c*)1 a is often assumed to be 1 [10], although other values have also been suggested a [68,69]. From Equation (12), we get a ¼ g(RT)1(RhR1 g ) , which for dilute aqueous PEG solutions at 298 K gives a value of a ¼ 0.85. Equation (12) allows for easy estimation of the intrinsic viscosity of dilute polymer solutions. An even more accurate formula could be obtained if we added the next term of the exponential expansion to Equation (11), i.e. [h]2c2/2. We would then obtain an analog of the frequently used form of the Huggins equation [70]. However, the quality of currently available data is not sufficient to validate such minor corrections and we shall confine ourselves to the more coarse estimation of [h] given by Equation (12). Following this simplification, a direct relationship between DEa and [h] can be postulated for non-entangled polymer solutions:
DEa ¼ ½hcRT:
(13)
3.3. Applicability to probe diffusion
Fig. 4. Viscosity scaling including the temperature dependence. Panel b) shows the data for the non-entangled regime in more detail. The notion of activation energy for viscous flow is introduced through Equation (6). Single asterisk denotes polymers of broad Mw distribution. Data marked in a legend with y, z, and x are adapted from Refs. [63e65], respectively.
In our earlier work we have stated that the length-scaledependent viscosity approach can be used for uniform description of both probe diffusion and macroscopic viscous flow of complex liquids [3,4,41]. To test the validity of this statement in reference to the hereby postulated formulas for non-entangled solutions we performed fluorescence correlation spectroscopy measurements. We used a labeled protein, aldolase (rp ¼ 4.5 nm), as a probe freely diffusing in solutions of 25 kg/mol PEO. We covered both the non-entangled and entangled concentration regimes. We measured translational and rotational diffusion coefficients,
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phase boundary, originating from an entropic background (conformational freedom of a chain near an interface is lower than in bulk solution. This effect diminishes its entropy and makes such location of a chain unfavorable). The DL typically spans up to a few nm away from the interface and may appear e.g. at the surface of diffusing microspheres or nanoparticles [71]. It has been shown that the DL may have a significant influence on both rotational and translational diffusion in crowded environment [72,73]. However, probe rotation is usually facilitated by the DL much more than translation [73]. Such theoretical predictions are strongly supported by the experimental results presented hereby. 4. Conclusions
Fig. 5. Excess of activation energy for flow of polymer solution, DEa (which is the difference between Ea for solution and for pure solvent) plotted against polymer concentration. Curve drawn for aqueous solution of PEG, Mw ¼ 18 kg/mol, at 298 K. The plot is continuous across the whole concentration range; an inflection is observed at the entangledenon-entangled crossover.
translating them into effective viscosity using Equation (8). Effective radius was estimated according to Equation (2). In case of probe diffusion, parameter g may differ from the one observed for macroscopic flow (roughly reproducing it in the large probe limit) [41]. It is justified by the influence of probe-polymer interactions on the average probe velocity. On the basis of simple fitting of viscosity data, g for aldolase diffusion in PEG is estimated to be 3.7 kJ/mol. All other parameters are kept the same as for macroscopic flow. The results are plotted in Fig. 6. Solid line refers to the scaling function given by Equation (9), where effective radius Reff is used instead of Rh [4]. Good agreement between Equation (9) and the data for translational diffusion can be observed. On the other hand, rotational viscosity experienced by the probes is significantly lower than the translational across the whole concentration range. Such observation is an experimental confirmation that the depletion layer (DL) effect occurs not only in entangled, but also in highly diluted polymer solutions. The DL is a region of decreased polymer concentration in the vicinity of a
Fig. 6. Effective translational and rotational viscosities experienced by a labeled protein, aldolase (rp ¼ 4.5 nm), diffusing in 25 kg/mol PEO aqueous solutions. Polymer concentration ranged from 2 to 20% w/w. Values for translation match the hereby presented scaling (solid line) well. Rotational diffusion is significantly facilitated due to spontaneous formation of a depletion layer around an object immersed in a polymer solution.
We have demonstrated that viscosity scaling for PEG (PEO) aqueous solutions, previously reported for entangled systems, can be extended to the dilute regime. The clearly observed crossover between the regimes occurs at (Rh/x)a ¼ 1, i.e. at polymer concentration about twice higher that c*. This finding may be interpreted as indication of an intermediate concentration regime between dilute and semi-dilute: semi-dilute non-entangled. However, no change of scaling parameters is observed at the c* threshold, which would be the dilute e semi-dilute non-entangled border. We may therefore interpret the onset of entanglement as the determining factor inducing changes in system properties instead of coil overlap. This is in line with two earlier observations: that the onset of chain entanglement happens at concentration significantly higher than c* [19,60,74], and that properties of polymer systems heavily depend on the presence of chain entanglements [9,24,75]. What influences the rheology of complex systems is not just the volume fraction occupied by macromolecules, but also the topology of the dynamic supramolecular structures they form. It should also be noted here that there is no general agreement on the exact definition of c* and the observed entanglement-related crossover may in fact be close to the dilute e semi-dilute shift, if not even equivalent. In Equation (9) we explicitly formulate an applicable description of viscosity of polymer solutions across a broad concentration range. The proposed approach is based on notions commonly used for characterization of polymer systems: hydrodynamic and gyration radii, correlation length and the Flory exponent. We offer an interpretation of the parameters appearing in the equations and analyze their shifts at the crossover between different concentration regimes. Expansion of the exponential formula (Equation (11)) allows to estimate the intrinsic viscosity of polymer solutions in the dilute regime (Equation (12)), providing a reference to Einstein's simplistic approach to hard sphere solutions [76]. The notion of activation energy for viscous flow can be successfully introduced into the scaling formula, allowing to consider viscous flow in terms of an activated process (Equation (9)). This may open new ways to explore the interactions in polymer solutions and understand the principals of transport properties of such systems. Moreover, also intrinsic viscosity of dilute solutions can be easily described in terms of the activation energy (Equation (13)). The presented conclusions are based on precise viscosity measurements performed using various methods for a model good solvent system: PEG and PEO aqueous solutions. We have already shown previously that scaling equations of this kind can be easily adapted to different complex liquids (including even biological systems) [4,40]. Validity of the hereby proposed physical description is supported by literature data, which fall on the curves obtained from our experimental results. Furthermore, we prove the applicability of the scaling approach to description of probe diffusion. It is another important step towards a uniform and universal description of transport of mass in complex systems. We also observe a significant mismatch between the resistance of the
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polymer solution against probe diffusion and rotation. The great facilitation of the latter is certainty ascribed to the depletion layer effect, while quantitative evaluation of this phenomenon in nonentangled systems seems an interesting perspective for further investigation.
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