Can percolation theory explain the gelation behavior of diblock copolymer worms?

Physical gelation by block copolymer worms can be explained in terms of multiple inter-worm contacts using percolation theory, suggesting that worm entanglements are irrelevant in this context.


Introduction
It is well known that certain surfactants (or binary mixtures thereof) can form highly anisotropic worms in aqueous solution. [1][2][3][4] These systems have potential applications as thickeners, 5,6 in drag reduction, 7 and for enhanced oil recovery. 4,8 The 'living' nature of these self-healing systems has been demonstrated and sophisticated techniques such as contrast variation neutron scattering have been utilized to characterize their structure. 9,10 Surfactant worms typically exhibit mean contour lengths of the order of 1-10 mm. The concept of worm entanglements as a physical mechanism for gelation has been suggested on the basis of a combination of rheological and theoretical studies. [10][11][12] Diblock copolymer worm gels have been recognized for almost two decades. 13 Over the last ve years or so, the development of polymerization-induced self-assembly (PISA) has enabled the rational, reproducible synthesis of a wide range of diblock copolymer worm gels directly in water, polar solvents (e.g. ethanol) or non-polar solvents (e.g. nalkanes). [14][15][16][17][18][19][20][21][22] In particular, diblock copolymer worms prepared via dispersion polymerization oen exhibit thermoresponsive gelation, undergoing a reversible worm-tosphere morphological transition either on heating in ethanol or n-alkanes 18,23,24 or on cooling in aqueous solution. 20,25 In each case, this morphological transition appears to be the result of surface plasticization of the core-forming block, which leads to a subtle change in the packing parameter for the diblock copolymer chains. 26,27 Typically, the mean worm width is well-dened, is of the order of a few tens of nm and is dictated by the mean degree of polymerization (DP) of the core-forming block. In contrast, the mean worm length is rather ill-dened and is typically of the order of hundreds of nm. Compared to the dimensions reported for surfactant worms, diblock copolymer worms appear to be too short to account for the observed formation of free-standing gels via a worm entanglement mechanism. Percolation theory has been used for many years to account for the substantial differences in conductivity thresholds observed for many types of conductive particles dispersed in electrically insulating matrices. [28][29][30][31][32][33][34][35] Typically, spheres exhibit a percolation threshold volume fraction of around 0. 16,29,30 whereas highly anisotropic rods (e.g. polyaniline needles or carbon nanotubes) form fully-connected conductive networks at signicantly lower volume fractions, sometimes below 0.01. [31][32][33][34][35] Recently, percolation theory has been extended to include polydisperse rods exhibiting a wide range of rod lengths, 36,37 which is oen the case encountered experimentally. More specically, for cylindrical rods with a high aspect ratio (i.e. length/width ratio), Chatterjee 36 has used mean eld theory to show that the critical volume fraction for the percolation threshold, f c , can be estimated using eqn (1): where L w is the weight-average rod length, R is the numberaverage rod cross-sectional radius and s R is the standard deviation of the rod cross-sectional radius. As noted by Chatterjee, for populations of rods with narrow width polydispersities, relatively high aspect ratios, and uncorrelated variations in the widths and lengths, the percolation threshold is governed by the ratio of the number-average radius to the weight-average (rod) length. As noted above, the average worm cross-sectional radius R is well-dened, so s R tends to zero. Hence eqn (1) can be simplied to give: [N.B. It can be shown that the approximation made when deriving eqn (2) leads to a small systematic underestimation of f c (see ESI †)]. Otten and co-workers drew similar conclusions to that of Chatterjee using a somewhat different mathematical approach. 37 We postulated that the percolation threshold required for the formation of an extended 3D network of inter-connected electrically conductive rods randomly dispersed in an insulating matrix to produce macroscopic electrical conductivity 38,39 should be equivalent to that required for formation of a macroscopic physical gel by a colloidal dispersion of rods. Herein, we evaluate to what extent eqn (2) provides a useful description of the gelation behavior observed for two examples of diblock copolymer worms. 36,37 For this approach to be valid, gelation should occur as a result of multiple interworm contacts (see Scheme 1), which would provide an alternative gelation mechanism to the inter-worm entanglements model previously (and correctly) invoked for surfactant worms. The two diblock copolymer systems studied herein were chosen because they represent relatively long, highly exible worms 25 and relatively short, stiff worms, respectively. 40 Thus they represent two limiting copolymer morphologies for which contrasting experimental data might be anticipated.

Results and discussion
Initially, we sought literature data for colloidal dispersions of rigid rods to support our hypothesis. Recently, Nordenström et al. reported an interesting study of the aqueous gelation behavior of various cellulosic nanorods of varying dimensions and surface charge. 41 More specically, a series of six cellulose nanorods were prepared with varying length/diameter ratios (or aspect ratios) and their (de)gelation behavior was characterized using dynamic light scattering (DLS). The critical volume fraction for gelation was shown to be inversely proportional to the aspect ratio. Furthermore, it was postulated that gelation was simply a result of multiple contacts with neighboring nanorods, which arrests their translational diffusion in solution. However, no specic link was made to the recent mathematical advances Scheme 1 Chemical structures for (a) poly(glycerol monomethacrylate)-block-poly(2-hydroxypropyl methacrylate) (PGMA-PHPMA) diblock copolymers prepared by RAFT aqueous dispersion polymerization and (b) poly(methacrylic acid)-block-poly(styrene-alt-N-phenylmaleimide) copolymers prepared by RAFT dispersion polymerization in a 50/50% w/w ethanol/1,4-dioxane mixture. (c) Schematic cartoon illustrating formation of a continuous 3D network of worms above the critical gelation concentration (CGC) owing to multiple inter-worm contacts. In contrast, these inter-worm contacts are broken on dilution below the CGC, resulting in a free-flowing dispersion rather than a gel. developed to describe the percolation behavior of polydisperse rods. 36,37 Of the six types of cellulose nanorods reported by Nordenström et al., 41 the most relevant to the present study of neutral worms is that with the lowest surface charge, which had a mean length of 520 nm (determined by DLS) and a mean radius of 3.35 nm (measured by AFM studies). Using eqn (2), we calculate the theoretical percolation volume fraction, f c , for such cellulosic nanorods to be 0.0064, which is in reasonably good agreement with the experimental f c of 0.0073 reported by Nordenström and co-workers. 41 Thus our hypothesis of physical equivalence between the respective critical percolation thresholds required for solid-state electrical conductivity and physical gelation appears to have some merit.
The poly(glycerol monomethacrylate) 56 430 ] worm gels evaluated in this study were prepared using PISA as described by Blanazs et al. 22 and Yang and co-workers 40 respectively (see Scheme 1 for the relevant chemical structures). More specically, the highly exible PGMA 56 -PHPMA 155 worms were synthesized via reversible addition-fragmentation chain transfer (RAFT) aqueous dispersion polymerization of 2-hydroxypropyl methacrylate (HPMA), and are clearly highly anisotropic as judged by transmission electron microscopy (TEM, see Fig. 1a). In contrast, the relatively short, stiff PMAA 81 -P(St-alt-NMI) 430 worms were prepared by RAFT dispersion alternating copolymerization of styrene with N-phenylmaleimide using a 1 : 1 ethanol/1,4-dioxane mixture. These latter worms are much less anisotropic (see Fig. 1b). In both cases, the diblock copolymer chains possess relatively narrow molecular weight distributions as determined by gel permeation chromatography (GPC) and comparison to their respective macro-CTAs indicates high blocking efficiencies (see Fig. S1 †). The mean aspect ratio (i.e. length/width ratio) for each type of worm can be determined using small-angle X-ray scattering (SAXS), as described below. 42 TEM analysis conrms that the worm cross-sectional radius is well-dened in both cases. More specically, the mean core radius, r c , for PGMA 56 -PHPMA 155 and PMAA 81 -P(St-alt-NMI) 430 is estimated to be 11.1 AE 1.3 and 19.2 AE 2.1 nm, respectively.
In contrast, the worm contour length, L w , is clearly rather ill-dened. This is because such worms are formed via stochastic 1D fusion of multiple spheres during PISA. 44 In principle, SAXS is a powerful technique for characterizing block copolymer nano-objects, not least because X-ray scattering is averaged over many millions of particles and hence much more statistically robust than TEM studies. 45 Accordingly, SAXS patterns were recorded for the two worm dispersions at 1.0% w/w copolymer concentration, see Fig. 2. Both SAXS patterns exhibit a gradient of approximately À1 at low q, which is indicative of highly anisotropic rods (or worms). Fitting such patterns using an established worm model 46 provides detailed and robust structural information, including the weight-average L w and crosssectional worm radius R. For example, SAXS indicates an L w of approximately 1100 nm for PGMA 56 -PHPMA 155 worms, with a corresponding core radius, r c , of 8.5 AE 0.9 nm (see Fig. 2a). However, the highly hydrated stabilizer chains also contribute to the overall effective worm dimensions. Given the mean DP of the PGMA 56 chains, the thickness of this additional stabilizer layer is estimated to be 3.6 nm by SAXS analysis. Thus the  effective worm cross-sectional radius, R, for these 'hairy' PGMA 56 -PHPMA 155 worms is calculated to be 12.1 AE 0.9 nm (see ESI † for calculation details). In contrast, SAXS analysis of the PMAA 81 -P(St-alt-NMI) 430 diblock copolymer worms suggests an L w of approximately 296 nm (see Fig. 2b). These latter worms have an r c value of 20.0 AE 2.7 nm and a stabilizer thickness of 6.6 nm, giving an overall R value of 26.6 AE 3.0 nm. Hence the mean aspect ratios (or L w /R values) for the PGMA 56 -PHPMA 155 and PMAA 81 -P(St-alt-NMI) 430 worms are 89 and 11, respectively. These strikingly different aspect ratios are useful in the context of the present study because they enable a more rigorous test of the percolation theory recently developed for polydisperse rods. 36,37 Thus, according to eqn (2), the critical percolation volume fraction, f c , required to form a 3D gel network comprising PGMA 56 -PHPMA 155 worms is expected to be signicantly lower than that required for gelation when using the PMAA 81 -P(St-alt-NMI) 430 worms.
It is well-known that semi-concentrated dispersions of such PGMA-PHPMA worm gels exhibit thermoresponsive behavior, with degelation occurring on cooling below the critical gelation temperature (CGT) as a result of a worm-to-sphere transition. 20,25 When applying percolation theory to such thermosensitive systems, it is important to determine the characteristic temperature that corresponds to long, linear worms (as opposed to branched worms or worm clusters). This is readily achieved using shear-induced polarized light imaging (SIPLI), as recently reported by Mykhaylyk and co-workers. 43 Briey, an aqueous worm dispersion is subjected to applied shear using an optorheometer, which enables simultaneous interrogation of the sample using polarized light. The appearance of a distinctive Maltese cross motif indicates shear-induced alignment of the highly anisotropic worms. If such experiments are performed as a function of temperature, the temperature at which the brightest Maltese cross is observed corresponds to the formation of the most linear (i.e. longest) worms. Such measurements are shown in the inset of Fig. 2a and S2 † and indicate an optimum temperature of 17 C, which is very close to that at which the SAXS studies were performed (18 C). It is noteworthy that the PMAA 81 -P(St-alt-NMI) 430 diblock copolymer worms do not exhibit such thermoresponsive behavior, so the temperature at which SAXS analysis is conducted is not particularly important in this case.
Utilizing the structural information provided by SAXS in combination with eqn (2), the theoretical critical volume fraction (f c ) required for the percolation threshold (and hence macroscopic gelation) is predicted to be 0.011 AE 0.001 and 0.090 AE 0.009 for the PGMA 56 -PHPMA 155 and PMAA 81 -P(St-alt-NMI) 430 worms, respectively. This approximate eight-fold difference simply reects the substantial difference in aspect ratio for these two types of worms.
Experimental f c values can be estimated from tube inversion tests, which were performed at ambient temperature (17-18 C) for varying copolymer volume fractions (see obtained from oscillatory rheology, see Fig. 3. In this case, degelation is indicated by the point of intersection of the storage modulus (G 0 ) and the loss modulus (G 00 ) curves, and these latter experiments are considered more reliable.
In polymer physics, the worm-like chain model is used to describe the behavior of semi-exible polymers. 47 For such worm-like chains, the Kuhn length is equal to twice the persistence length, where the latter parameter quanties the chain stiffness. In principle, the behavior of the long, exible diblock copolymer worms described in our study is analogous to that of an individual polymer chain. 48 The Kuhn lengths derived from SAXS studies of both types of worms are included in Table 1. The Kuhn length for the short, stiff worms is simply equal to the weight-average worm contour length (L w ). In contrast, the Kuhn length for the long, exible worms is much lower than L w , which implies signicant exibility.
Clearly, there is some discrepancy between the theoretical and experimental f c values summarized in Table 1. However, percolation theory is derived assuming rigid rods, whereas the PGMA 56 -PHPMA 155 worms clearly exhibit signicant exibility (see TEM images in Fig. 1). This necessarily reduces the effective weight-average worm length L w , which in turn leads to a higher f c value. Given this important caveat, the fair agreement observed between the experimental and theoretical f c values supports our hypothesis that such worm gels form a 3D network simply via multiple contacts between neighbouring worms. In contrast, the PMAA 81 -P(St-alt-NMI) 430 worms are much stiffer (the glass transition temperature for the core-forming P(St-alt-NMI) 430 block is around 208 C. 40 Thus, better agreement between experimental and theoretical f c values is expected, and indeed observed.
From Table 1, the theoretical f c for stiff worms is approximately eight times greater than that for exible worms. In contrast, the corresponding experimental f c ratio is approximately four. This suggests that eqn (2) can be rewritten as: where the proportionality constant k varies by at least a factor of two depending on the degree of worm exibility. It is perhaps also worth emphasizing here that, given the worm dimensions indicated by SAXS studies, the 'worm entanglements' mechanism invoked to account for the gelation of surfactant worms does not appear to account satisfactorily for the physical gelation observed for these much less anisotropic diblock copolymer worms.

Conclusions
In summary, recent advances in percolation theory for polydisperse rods provide an improved understanding of the gelation behavior exhibited by diblock copolymer worms. Combined with experimental data, this suggests that a 3D gel network forms primarily via multiple contacts between neighbouring worms, rather than as a result of worm entanglements.

Conflicts of interest
There are no conicts to declare.  (2). b Determined by oscillatory rheology.