Effect of homopolymer solubilization on triblock gel structure and mechanical response



The linear modulus, swelling behavior, and high strain response of a set of well-characterized model triblock gels were investigated to understand the effect of homopolymer solubilized within the micelle core on gel structure and mechanical properties. Structural parameters were obtained from small-angle X-ray scattering (SAXS) as well as from self-consistent field theory (SCFT) calculations. Experimental results are compared with Neo-Hookean and exponentially strain hardening models for gel behavior and rigid filler effects are discussed. The main conclusion is that the addition of homopolymer to the micelle core increases the chain stretching in both the core and coronal blocks. The total extension of a chain for a given external load is fixed by its length; however, the initial prestretch imparted to the chain due to micellization changes with the size of the micelle core and can greatly reduce the amount of extension observed for a given external force. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 1395–1408, 2010


The ability for amphiphilic molecules such as block copolymers to act as surfactants has long been recognized.1 Materials, such as small molecules, homopolymers, or nanoscale particles,2, 3 can be selectively solubilized into block copolymer domains and there is interest in using block copolymer micelles shuttles for the delivery of therapeutic agents such as drugs or genes4 or nanoparticles for medical imaging.5 Although diblock micelles are discrete objects, ABA triblock copolymer micelles can either be discrete or form a network depending on whether the A or B blocks form the micelle core. In the former case, the midblocks can form linking bridges between micelle cores giving rise to elastic behavior. Elastic gels with material solubilized into the micelle cores could be useful as biodegradable scaffolds with controlled release during degradation.

In a first approximation, coronal chains in block copolymer micelles can be treated as polymer brushes grafted to a curved surface. Micelle structure can be roughly quantified by comparing the thickness of the corona, H, to the radius of the core, r. For “hairy” micelles, H/r is large while the aggregation number is small. For “crew-cut” micelles, H/r is small and the aggregation number is large. In the “hairy” case, coronal chains have density profiles similar to that of star-polymers while coronal chains in the “crew cut” case behave more similarly to densely grafted brushes. The perturbation of coronal conformation due to packing constraints set by the core size has implications for the intermicellar potential and is manifested in the type of long range order adopted in solutions of diblock micelles.6, 7

The work described in this article is aimed at understanding and quantifying how changes in core size affect the number and conformation of bridging midblocks in triblock gels as well as gel properties such as modulus, swelling, and strain hardening. Additionally, this article will present predictions for the behavior of ideal polymer gels based on Neo-Hookean and exponentially strain hardening models and compare them with the behavior of the triblock gel system studied here. Independent control of the micelle core size and the number of elastically active chains in triblock gels can be achieved either by synthesizing a series of triblocks with fixed midblock length and variable endblock lengths or by adding a species that will preferentially swell the micelle core. We choose the latter method both because it does not require multiple syntheses and because solubilization of material into micelle cores has implications for nanocomposites,2, 3 copolymer/homopolymer blends,8–11 and drug delivery applications.12

This article is laid out as follows. It begins with a discussion of the relevant literature on solubilization in micellar copolymer systems followed by an introduction to the acrylic triblock alcogels studied here as model triblock gels. Next, predictions from Neo-Hookean and exponentially strain hardening strain energy density functions for gel modulus, equilibrium swelling, and uniaxial compression behavior are presented. Experimental results measuring the effect of homopolymer solubilization on the linear modulus, swelling behavior, and gel structure for a range of triblocks and gel concentrations are presented. To better understand in detail the effect of homopolymer addition, we then focus on a single triblock and gel concentration, obtaining structural parameters from self-consistent field theory (SCFT) calculations and characterizing the high strain mechanical behavior. We close with a discussion of how ideas about chain stretching can be used to rationalize the changes in gel behavior with homopolymer addition as well as considering the implications of the finite size of the crosslinking aggregates in micellar gels.


The ability of block copolymer micelles to solubilize homopolymers in their cores has been studied experimentally and theoretically. Whitmore and Smith developed a theory to describe a system in which AB diblock micelles solubilize B homopolymer in their cores and are dispersed in a matrix of A homopolymer.11 They found good agreement between experiments and the theoretical prediction that if the B homopolymer is shorter than the B block of the copolymer, the solubility limit for the B homopolymer occurs when its volume fraction is approximately equal to the volume fraction of B from the block copolymer in the system. Experimental13, 14 and theoretical15 studies have also explored the solubilization of homopolymer into the diblock micelle cores in selective solvents and find that the homopolymer solubility limit scales linearly with the concentration of copolymer. For a fixed copolymer length and endblock fraction, the total amount of homopolymer able to be solubilized is roughly constant as the homopolymer length increases until a critical length is reached and solubility drops steeply. This is understood because solubilization of large homopolymer chains causes strong perturbation of the micelle dimensions.15 SCFT calculations by Leermakers et al. of AB diblocks in B-selective solvent showed that when the micelle core was swollen by the addition of A homopolymer, the segment density profile for the B blocks was perturbed relative to an unswollen micelle.16

A few studies have investigated the effect of adding homopolymers to thermoplastic elastomer gels.8, 10, 17, 18 Quintana et al. studied the effect of adding polystyrene homopolymer to poly[styrene–(ethylene/butylene)–styrene] (SEBS) gels10; however, these experiments were complicated by the high polydispersity of the homopolymers. Jackson et al. looked at the addition of poly(2,6-dimethylphenylene oxide) (PPO), a styrene compatible homopolymer, to SEBS gels.8 The PPO chains were shorter than the endblocks and selectively stained TEM images indicated complete solubilization of the PPO into the micelle cores, despite a high polydispersity of the PPO chains. For PPO/SEBS gels, the micelle core size increased linearly with PPO content. The addition of 1.5 wt % PPO to a 10 wt % triblock gel increased the shear modulus by ∼23%; however, no further increase in modulus was observed when the PPO content was increased to 3 wt %. The strain at failure decreased with PPO addition while the failure stress was unaffected. Walker et al. studied SEBS gels with added syndiotactic homopolymer PS (shPS) that was ∼10 times longer than the endblocks.17 They compared the increase in modulus with increasing shPS to that achieved with PPO as well as to nanofillers such as fumed silica and organoclays. For a given wt % of filler, shPS lead to the greatest increase in modulus. For example, 1.5 wt % of shPS and PPO increases the gel modulus by factors of ∼3 and ∼1.2, respectively. Unlike the relatively short PPO chains, the long shPS chains were not solubilized into the micelle cores and instead formed nanofilaments. The greater reinforcement ability of the long unsolublized shPS over short solubilized PPO is attributed to absorption of micelles to the shPS nanofilaments. Flanigan et al. investigated effect of adding short isotactic poly(methyl methacrylate) (PMMA) homopolymer to alcogels formed from triblocks with PMMA endblocks and poly(tert-butyl acrylate) (PtBA) midblock.18 The homopolymer greatly improved the thermostability of the gels due to stereocomplexation between the isotactic homopolymer and the syndiotactic endblocks; however, isotactic homopolymer also increases the time necessary for the gel structure to equilibrate. They also noted that homopolymer addition increased the average distance between micelle cores [as evidenced by shifts of small-angle X-ray scattering (SAXS) interference peaks to lower q], but the increase in micelle separation was not quantified.

These previous studies have focused on changes in the gel's linear response and have not explored the effects of homopolymer on gel deformation behavior at high strains, where strain hardening would be expected to occur; therefore, it is not known what effect, if any, homopolymer addition has on the high strain response. While the average micelle core size measured from TEM images was shown to increase with increasing homopolymer content,8 a quantitative understanding of how homopolymer solubilization affects the average distance between micelles does not exist. It is also not known how the number of bridging midblocks is affected by homopolymer addition.

Model System: Acrylic Triblock Copolymer Gels

To better understand the effect of homopolymer solubilization on triblock gel structure and properties, acrylic triblock copolymer gels were selected as a model system. To avoid complications from homopolymer chain length, polydispersity, and homopolymer/endblock interactions and to facilitate SCFT calculations, micelle cores were swollen by a low polydispersity homopolymer much shorter than, but chemically identical to, the endblocks.

Acrylic triblock copolymers with PMMA endblocks and poly(butyl acrylate) (PBA) midblocks form thermoreversible gels in alcohols19–21 due to the strong temperature dependence of χAS, the interaction parameter between the endblocks and the solvent. At high temperature the triblock is molecularly dissolved. Between the critical micelle temperature (CMT) and the glass transition temperature (Tg) of the endblock aggregates, the gel is viscoelastic with a relaxation time defined by rate of endblock exchange that varies rapidly with temperature.21 Eventually, the still slightly swollen endblocks reach their Tg and then are effectively locked in place resulting in an elastic solid.19 The very rapid, thermally reversible liquid–solid transition makes PMMA–PBA–PMMA triblocks excellent model systems for studying the link between gel structure and mechanical properties. Homopolymer PMMA can be easily incorporated into the gel by mixing above the CMT and then cooling below Tg. For PMMA–PBA–PMMA triblocks in alcohols, the CMT is generally less than 100 °C and the endblock Tg is typically above room temperature, although both transition temperatures have block length and concentration dependence.

At intermediate temperature, the exchange of endblocks is quite rapid21, 22 and endblock aggregates can achieve their equilibrium aggregation number; however, as temperature decreases, the endblock exchange rate also decreases and aggregates can be kinetically hindered from reaching equilibrium at low temperatures.22, 23 Because the gels are prepared by cooling from an equilibrium solution, the size and composition of the aggregates can vary in response to triblock and homopolymer concentration. At room temperature, the observed gel structure reflects the equilibrium structure at the temperature where the endblock exchange rate is no longer shorter than experimental time scales and the aggregates are kinetically hindered from reaching equilibrium. The implications of kinetic trapping on selecting appropriate parameters for SCFT calculations are discussed later.

A schematic of the gel structure at low temperature with and without added PMMA homopolymer is shown in Figure 1. It should be noted that the aggregation numbers for these materials are quite high21 (∼100) and that only a few midblocks are depicted for clarity. The network structure can be described by the average distance between micelle cores, d, and the relative fraction of midblocks that form bridges instead of loops.

Figure 1.

Schematic of gel structure at low temperature with homopolymer solubilized in the micelle cores. It should be noted that the aggregation number of the gels is much higher than is depicted and that in this study the endblocks and homopolymer are chemically identical. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]


Behavior of Ideal Polymer Gels

The behavior of neat elastomers is a useful starting point for a discussion of polymer gels since they can be thought of as elastomers swollen with solvent. For a Neo-Hookean material, the strain energy density, U, is given by24

equation image(1)

where λx, λy, and λz are the principal extension ratios (for incompressible materials λmath image λmath image λmath image = 1), J1 is I1 − 3 where I1 is the first strain invariant, and E is Young's modulus corresponding to the undeformed state, for example, where J1 = 0. For uniaxial compression, λx = λz = λmath image where λy is the compression ratio and the nominal stress, σmath image, (force divided by undeformed cross-sectional area) is equal to equation image. Thus, using eq 1 σmath image for Neo-Hookean materials can be written as

equation image(2)

The Young's modulus for an unentangled, unsolvated (dry) polymer network, denoted by subscript d, can be calculated using simple rubber elasticity theory

equation image(3)

where vd is the number density of chains, Rd is the average distance between crosslinks, and Rmath image is the mean-squared end-to-end distance for equivalent Gaussian chains between crosslink points.25

By assuming that a polymer gel can be thought of as an elastomer that is affinely swollen with solvent, an equivalent expression for a gel can be written by replacing Ed, vd, and Rd with values in the swollen state (denoted by the subscript s). Both vs and Rs can be related to the values in the dry state through the gel's polymer concentration, ϕp, using vs = vdϕp and Rmath image = Rmath imageϕmath image. The modulus of a polymer gel can be written as

equation image(4)

As vd = ρ/MNav and only those chains that are elastically active contribute to the modulus, eq 4 can be rewritten as

equation image(5)

where Nav is Avogadro's number, f is the fraction of elastically active molecules, M is the polymer molecular weight, and ρ is the polymer density.

The total osmotic pressure for a polymer gel, Π, can be expressed as Π = Πel + Πs where Πel is the elastic contribution that resists swelling and Πs is the contribution from polymer/solvent contacts that drives swelling. Πel is defined by

equation image(6)

where V is the swollen volume and V0 is the initial gel volume. For isotropic swelling,

equation image(7)


equation image(8)

Combining eqs 1, 6, and 8 yields

equation image(9)

Thus, for an elastomer swollen to equilibrium

equation image(10)

where Esw is the modulus of the gel in its equilibrium swollen state, Eref is the gel modulus in the reference state and λswell is the linear extension ratio required to go from the reference state to the equilibrium swollen state. As λswell is related to the change in polymer concentration, we can write

equation image(11)

where ϕmath image and ϕmath image are the polymer volume fractions in the initial reference state and equilibrium swollen state, respectively.

An estimate of Πs for a polymer gel can be obtained by assuming that the osmotic pressure of the gel is equivalent to that of a solution of high molecular weight polymer at the same polymer volume fraction as the gel. In the semidilute case, scaling theory26 gives

equation image(12)

where ζ is the screening length and a0 is a characteristic monomer size.

For a gel swollen to equilibrium in pure solvent Π = 0. Thus Πel = −Πs, which for a Neo-Hookean materials yields

equation image(13)

Solving for the equilibrium polymer volume fraction yields

equation image(14)

This coupling of the network elasticity to the osmotic pressure means that network chains stretched due to the formation of the gel structure will have a reduced ability to expand to accommodate solvent, limiting the gel's ability to swell. In extreme cases where network chains are highly extended during network formation, syneresis is observed and the gel expels solvent to gain entropy by allowing less-extended chain conformations.

Although eq 1 is mathematically simple and can be used to described elastomers and gels at low strains, these materials often exhibit significantly more strain hardening than is predicted by the Neo-Hookean model.27–29 This behavior can be described by modifying the Neo-Hookean model with a term that accounts for finite chain extensibility. The response of acrylic triblock alcogels is well described by a strain energy density function28

equation image(15)

which is equivalent to the model developed by Fung30 that is often used to describe deformation of blood vessels.31 The only free fitting parameter, J*, accounts for the finite extensibility of the chains in a way that is similar to a model proposed by Gent.32 Comparison of this strain energy density function with the strain energy density for a Neo-Hookean material ( equation image) reveals that they are equivalent at low strain but the exponential term leads to strain hardening as J1 approaches J*.

Using the expression for the strain energy density given in eq 15, the nominal stress during uniaxial compression can be written as

equation image(16)

where λy is the compression ratio. This is equivalent to the Neo-Hookean response given in eq 2 multiplied by the term equation image which accounts for strain hardening.

Πel for strain hardening materials described by eq 15 can be calculated in an analogous manner to the Neo-Hookean case and yields

equation image(17)

Similarly to the uniaxial compression response, Πel for the exponentially strain hardening case is equivalent to the Neo-Hookean solution multiplied by the term equation image, which becomes significant as J1 approaches J*. Therefore, the significance of the strain hardening term for a given system will depend both on the extent of swelling, λ, as well as J*.

It is also true for a strain-hardening gel that Πel + Πs = 0 after swelling to equilibrium in pure solvent and ϕmath image can be calculated in an analogous manner to the Neo-Hookean case. Using eqs 8, 12, and 17 yields,

equation image(18)

As equation image, it is not possible to easily arrive at an algebraic expression for ϕmath image as it was in the Neo-Hookean case. However, it is clear that as J1 approaches J* the predictions of both models for ϕmath image will diverge with the Neo-Hookean model predicting more swelling (smaller ϕmath image) than the exponentially strain hardening model. Finally, we note here that the two models presented above assumed ideal polymer networks, for example, that the crosslink points are infinitely small.

Micellar Triblock Copolymer Gels

Micellar triblock gels differ from idealized gels with point like crosslinks in a few important ways. Depending on the relative block fractions and polymer lengths, the volume of the micelle cores, and thus of the crosslinks, can be significant. However, as long as the endblock aggregates remain discrete objects, it is expected that the material response of the gel will be dominated by the deformation of the bridging midblocks. Therefore, rubber elasticity has been used to rationalize the modulus of micellar gels. Micellar triblock gels are not formed via swelling of a dry network and thus, unlike swollen elastomers, Rd is not an experimentally relevant variable. The average distance between micelle centers (d), equivalent to the average distance between crosslinking points, is easily measured from scattering and has been shown to vary as a function of gel concentration.21 Therefore, eq 4 can be recast as

equation image(19)

and has been used to understand the modulus of micellar triblock gels with unentangled midblocks.33 In the case of triblock gels, f is the fraction of molecules which form bridges rather than loops. It has been shown that the combined variable equation image and the micelle aggregation number increase with increasing gel concentration for acrylic triblock alcogels.21

In the previous discussion of ideal polymer gels, it was assumed that all the polymer in the gel was osmotically active. For micellar triblock gels which have glassy endblock aggregates, the amount of solvent in the aggregates is insensitive to total gel concentration, which suggest that the endblock should be considered as osmotically inactive when considering gel swelling.23 If we assume as a first approximation that only the midblock is osmotically active, we can rewrite eq 12 for a micellar gel as

equation image(20)

where α is the volume fraction of midblock in the triblock.

Rigid Crosslink Volume Effects

It is important to also consider implications of the nonpoint-like crosslinks in these systems. Several approaches that have been advanced for accounting for the contribution of the rigid aggregates in phase separated ABA triblock melts.34, 35 The most common treats the aggregates as rigid fillers that reinforce the elastomeric matrix in a way analogous to classic filled rubbers. It is well known that filling a viscous liquid or an elastomer with rigid particles causes, respectively, the viscosity or the modulus to increase with increasing filler volume fraction.36–40 At relatively low volume fraction of filler, E can be modeled using the Guth–Gold relationship

equation image(21)

where E0 is the modulus of the unfilled elastomer and ϕfiller is the volume fraction of rigid filler. The major drawback to this approach is that eq 21 is derived using the assumption that continuum mechanics holds (e.g., that the size scale of the filler is larger than molecular dimensions); as noted by Bard and Chung, this is not the case for phase separated triblock melts.35

Application of this model to triblock melts is complicated by the fact that the low strain modulus in these systems is generally dominated by midblock entanglements.34, 35 However, for the triblock gels studied in this article, the midblocks are unentangled21 removing this difficulty. It should also be noted that while the ratio of gel modulus at a given filler content to the unfilled modulus is relatively straight forward for an elastomer to which rigid particles are added; for the triblock systems, a completely unfilled system is one that contains no aggregated endblocks and, therefore, has no crosslinks and no modulus. Thus, all the gels investigated in this study are “filled” to various extents. The volume fraction occupied by the glassy, yet still solvated, PMMA aggregates in triblock gels can be calculated using

equation image(22)

where ϕPMMA and ϕS are the volume fraction PMMA in the gel and the volume fraction of solvent in the core of the micelle, respectively. The last term accounts for the fact that the PMMA will be slightly solvated even at low temperatures and eq 26 can be used to calculate ϕS (at 25 °C, ϕS = 0.24). Lastly, as eq 21 is written, the matrix modulus, E0, is treated as independent of filler content. However, if the topology of the elastomer network is affected by amount of rigid phase, E0 should not be treated as a constant and should be replaced with eq 4. For these reasons, we feel that a treatment based on eq 19 provides a more appropriate description of the gel mechanical response than the simple rigid filler prediction based on eqs 21 and 22.



Three different symmetric triblock copolymers with PMMA endblocks and poly(n-butyl acrylate) (PnBA) midblock were obtained from Kuraray Co. (Japan) and were used as received. The naming convention uses A and B to refer to the PMMA and PnBA blocks, respectively, with subscript numbers referring to each block's molecular weight in kg/mol. The triblocks all had polydispersity indexes of ∼1.16. Triblock details are listed in Table 1. Homopolymer PMMA of 9 kg/mol and a polydispersity index of 1.09 (P1938-MMA) was obtained from Polymer Source and was used as received. It will be denoted as A9.

Table 1. Description of Polymers Studied
inline image

Samples were prepared by dissolving the copolymer in 2-ethylhexanol (2EH), a selective solvent for PnBA, in sealed vials above ∼90 °C and then cooling to room temperature. Gels with added homopolymer were prepared by adding A9 to an already formed neat gel so that a fixed weight ratio of 1:4, 1:2, 1:1, or 2:1 homopolymer to endblock was achieved. Samples were then sealed, reheated to ∼90 °C to obtain a clear solution, and cooled. They remained clear, indicating no macrophase separation of the homopolymer occurred. All the added homopolymer is assumed to be solubilized in the micelle core because 2EH is a poor solvent for PMMA at room temperature. Gels with homopolymer additions will be referred to in the text by the vol % of the base gel to which homopolymer was added and the ratio of homopolymer:endblock in the sample.

Indentation Experiments

Indentation measurements were performed on a homemade apparatus consisting of a cylindrical flat punch indenter of radius 0.39 mm, connected to a load cell attached to an inchworm motor. Displacements were measured via a fiber optic displacement sensor. Details of the setup can be found in Lin et al.41 Samples were loaded at a fixed velocity of 10 μm/s until a predetermined maximum load was reached and the indenter was immediately retracted at the same rate. The Young's modulus, E, was calculated from the linear portion of the loading curves using:42

equation image(23)

where σavg is the average stress under the indenter, P is the measured load, a is the radius of the indenter, δ is the measured displacement, h is the thickness of the sample (1.6 mm or 3.2 mm for these experiments), and fc is a function that accounts for the geometric confinement of the sample. It is given by43

equation image(24)

Swelling Experiments

Swelling experiments were conducted by casting gels in washers between glass slides. Gels were cooled, removed from the washers, weighed, and placed in baths of pure solvent. Once the gels reached equilibrium, they were removed from the solvent bath, excess solvent was removed with filter paper and then they were weighed. The equilibrium swelling ratio was calculated by dividing the swollen weight, W(t), by the initial weight, W(0). It was assumed that all change in mass was related to movement of solvent, which is consistent with the endblock aggregates being in a glassy state at room temperature. To convert from weight to volume fractions densities of 1.19, 0.83, and 1.04 g/cm3 were used for PMMA, PnBA, and 2EH, respectively.

Small-Angle X-Ray Scattering

SAXS experiments were performed at the DND-CAT beamline 5ID-D at Argonne National Lab's Advanced Photon Source. Gel samples were made by casting a heated solution in a washer (1.6 mm thick, 7.6 mm diameter) between two Kapton layers and then cooling to room temperature prior to testing. The scattering vector (q = 4π sin (θ/λ)) range covered 0.1–1.0 nm−1 using a beam energy of 17 keV. Isotropic patterns were collected on a 2D detector and then integrated over all azimuths to generate one dimensional data.

The low q (q < 0.5 nm−1) region of the patterns were fit using the Kinning–Thomas disordered hard-sphere model that incorporates the Percus–Yevick correlation function.44–47 The details of the procedure can be found in Seitz et al.21 and Bras.22 The average distance between micelle cores, d, is calculated with d = (Vm)1/3, where Vm is the volume per micelle calculated from the fit parameters.

Compression Experiments

Compression experiments were conducted on gel cylinders 3.5 mm thick and 8 mm in diameter using a Sintech 20G universal testing machine with a 220 N Sensotec load cell. Gels were placed between glass plates coated with a thin layer of silicon oil to reduce friction and adhesion between the gel and the glass. Initially a gel was compressed at 0.5 mm/min to a maximum load of 0.2 N with immediate retraction to 0.02 N to ensure complete contact with gel at the start of a test. Samples were then compressed at a fixed crosshead speed of 25 μm/s to a maximum load of 50 N. Load and displacement data were recorded. The extension ratio, λy, is defined as h/h0 where h is the deformed gel thickness and h0 is its undeformed thickness. The nominal compressive stress is defined as the load divided by the undeformed area of the sample. Assuming the gels are incompressible, the biaxial extension ratio is given by λbiax = λmath image.

SCFT Calculations

SCFT calculations were performed using the methodology described by Bras23 and readers are directed to this reference for a full explanation of the procedure used. Triblock, homopolymer, and solvent degrees of polymerization, N, were chosen to give the correct molar volumes of the polymers when normalized by the molar volume of 2EH (156.9 g/cm3). Table 2 summarizes the values of N of the various components in the system and their interaction parameters, χ. Because the relaxation times, and thus the time scale for micelle equilibration, increases rapidly with decreasing temperature,21 the value of χAS was selected to correspond to a temperature where the time scale of equilibration becomes longer than experimental time scales. Gels in 2EH and butanol have very similar behavior and thus a published48 temperature dependence of the interaction parameter between PMMA and butanol was used to estimate the temperature corresponding to the value of χAS. For PMMA in butanol, χAS = 0.9, corresponds to a temperature of 48 °C. At this temperature, the gels have relaxation times on the order of a few minutes.21 For lower values of χAS, the simulations correspond to equilibrium structures that the experimental system is kinetically hindered from reaching.23

Table 2. SCFT Parameters
inline image


Effect of Homopolymer Solubilization on Gel Structure and Properties

Figure 2 plots E/vskBT, as a function of the total amount of PMMA in the gel (endblock plus homopolymer). Values of E/vskBT < 3 indicate a low bridging fraction since equation image is always greater than 1. An ideal network in which all midblocks are in a bridging conformation and have Gaussian configuration, corresponds to E/vskBT = 3 (shown as the dashed line). From Figure 2 and eq 19, it is clear that as homopolymer is added to a gel the combined variable equation image increases.

Figure 2.

Reduced modulus, E/vskBT, as a function of the total PMMA concentration in the gel (endblock plus homopolymer). The dashed line represents E/vskBT = 3, which corresponds to an ideal network where equation image. Circle, square, and triangle symbols correspond to samples with nominally 7, 14, and 28.3 vol % triblock, respectively. Closed symbols correspond to neat gels whereas open symbols represent gels with added A9 homopolymer. Blue, purple, and green data points correspond, respectively, to A25B116A25, A23B31A23, and A9B53A9 gels. Sample series with the same nominal triblock vol % but varying homopolymer amount are connected with a line to guide the eye.

For acrylic gels with glassy endblock aggregates, it is assumed that the endblocks are osmotically inactive at room temperature. Thus, the volume fraction of midblock in the gel is the concentration driving swelling.22 Figure 3 plots (a) the equilibrium swelling ratio and (b) average distance between micelles, d, as measured from SAXS against the volume fraction of midblock in the gel. (A more detailed discussion of the SAXS data will be presented below.) The swelling ratio decreases dramatically with increased homopolymer addition even though the total amount of midblock in the gel, and thus the osmotic driving force for swelling, is nearly constant. Therefore, the elastic force resisting swelling must be becoming more significant, which is consistent with the increase in modulus with homopolymer addition. Increased modulus and decreased ability to swell suggest that the chains are being forced to adopt a more extended conformation as homopolymer is added to the gel. The increase in average distance between micelles with increasing homopolymer content is shown in Figure 3(b). However, it should be noted that the relative contributions from chain stretching and bridging fraction are unable to be decoupled from either modulus or swelling measurements.

Figure 3.

Swelling ratio and average distance between micelles (from SAXS) versus initial volume fraction of PnBA for A25B116A25 (circles) and A23B31A23 (triangles) gels. Closed and open symbols correspond to neat gels and gels with added A9 homopolymer, respectively. Lines are guides to the eye only.

Fixed Gel Concentration

To more fully understand the effect of homopolymer solubilization on gel structure and properties, we now focus on a single triblock and concentration. The details of this series of samples are given in Table 3.

Table 3. Details of Nominally 14 vol % A25B116A25 Gels
inline image

Although the changes in the linear modulus with homopolymer addition, shown in Figure 2, are striking, equally interesting are the effects on the nonlinear response. Figure 4 shows nominal compressive stress versus biaxial extension ratio for nominally 14 vol % A25B116A25 gels measured in uniaxial compression. All samples show strong strain hardening as λbiax increases. Shown in dashed lines in Figure 4 are the predictions for Neo-Hookean materials with E equal to that of the samples. It is clear that none of the gels are well described by a Neo-Hookean model. Additionally, as homopolymer content increases the deviation from Neo-Hookean occurs at smaller values of λbiax. The solid lines are fits to eq 16 using the values of J* and E listed in Table 3. Even with a single free fitting parameter, eq 16 does a much better job capturing the gel response than the Neo-Hookean model.

Figure 4.

Nominal compressive stress versus biaxial extension ratio for nominally 14 vol % A25B116A25 gels with increasing homopolymer content. Solid lines correspond to fits to eq 16 whereas dashed lines are equivalent to Neo-Hookean models. Parameters for the model are listed in Table 3.

SAXS patterns for nominally 14 vol % A25B116A25 gels are shown in Figure 5. The lack of increased low q scattering indicates the homopolymer is solubilized in the micelle cores and is consistent with the lack of turbidity in the samples (the upturn seen at the lowest q is beamstop scatter). In these system, scattering arises from electron density differences between the highly solvated corona and the weakly solvated PMMA cores. The main peak (q = 0.15 nm−1) arises from interference between micelles and reflects the average distance between scattering centers while the undulations at higher q arise from the form factor of the micelle core. The lack of higher order reflections indicates a random distribution of micelles rather than crystalline arrays. As homopolymer is added the structure factor peak shifts toward lower q indicating increased distance between micelles. Following the methodology in Seitz et al.21 we modeled the scattering using a Kinning–Thomas model45, 47 for the random distribution of hard spheres in a liquid. By fitting the low q data, values for the hard sphere radius (Rhs) and volume fraction of spheres (η) are determined. The average distance between scattering centers (e.g., the average distance between micelles, d) can be calculated using21

equation image(25)

and is plotted in Figure 7(b).

Figure 5.

Normalized intensity versus scattering vector for SAXS patterns of nominally 14 vol % A25B116A25 gels with increasing homopolymer content. Curves have been off-set vertically for clarity.

Although SAXS provides information about the average distance between micelles, it cannot provide details about the distribution of homopolymer in the micelle core. SCFT techniques allow determination of detailed volume fraction profiles for block copolymer melts and solutions.49 SCFT calculations for this series of gels were performed to further elucidate the effect of homopolymer on micelle structure and help understand the implications for gel response. All the SCFT results discussed are for systems with χAS = 0.9 which corresponds to an experimental temperature of ∼48 °C.23 Below this temperature, gel relaxation times become much larger than the experimental time scales and the micelles will not be able to equilibrate. For temperatures below ∼50 °C, the gel storage modulus and X-ray scattering intensity plateau indicating that further structural evolution does not occur as the gel cools.21 Selecting a lower or higher value for χAS will affect the exact values for micelle size, solvation, and bridging fraction; however, the trends for increasing homopolymer content are unchanged.

Figure 6 plots SCFT volume fraction profiles for the nominally 14 vol % A25B116A25 gels as a function of the micelle radius, R, normalized by triblock chain's unperturbed dimension, R0. Spherically symmetric boundary conditions were used; thus, the bridging chains cross the reflective boundary linking micelle cores. Several features are worth noting. The homopolymer segregates to the micelle core and the total size of the micelle increases with homopolymer addition. If the total volume fraction of PMMA (homopolymer plus endblock fraction) is plotted, it becomes clear that homopolymer addition is equivalent to an increase in the size of the micelle core. The amount of solvent in the micelle core is insensitive to homopolymer addition. In fact the amount of solvent in the micelle core, ϕs, at a given temperature is nearly identical to the solvent fraction in a high molecular weight polymer that is equilibrated with pure solvent. As the solvent chemical potential, μS, is equal to zero when in equilibrium with pure solvent and the core contains only solvent and PMMA (e.g., ϕS + ϕPMMA = 1), the solvent concentration in the core23 can be calculated using the Flory–Huggins expression

Figure 6.

Radial volume fraction profiles from SCFT calculations with χAS = 0.9 for nominally 14 vol % A25B116A25 gels with increasing homopolymer content. Each is labeled with the ratio of homopolymer:endblock [g:g] along the right side.

equation image(26)

The value of χAS for PMMA and 2EH can be estimated using the known temperature dependence for PMMA in butanol.48 At χAS = 0.9, the solvent fraction in the core is ∼0.4 while at room temperature it is ∼0.24.

The average distance between micelles can be calculated from the SCFT micelle size in an analogous manner as the SAXS data using

equation image(27)

where VM and Req are, respectively, the micelle volume and radius at equilibrium. However, in the SCFT calculations all distances are defined in terms of the dimensionless parameter R/R0 and to convert to true distances the value R0 must be defined. For homopolymer melts Rmath image is unambiguously defined as equation image where a and N are the statistical segment length and the degree of polymerization, respectively.26 In a semidilute solution with polymer concentration, ϕp, chain dimension Rmath image is given by26

equation image(28)

Our system has the additional complication that ϕp is not uniform throughout the system due to differing solvation between the endblocks and the midblock. Thus, we need to use a value corresponding to some average polymer concentration. Because d is experimentally determined from SAXS, we use Rmath image as a fitting parameter to adjust dSCFT to agree with dSAXS [these data are plotted in Fig. 7(b)]. This yields Rmath image equal to 29.6 nm. Using eq 28 with a = 0.7 nm and N = 1404, this would correspond to a solution with ϕp ≈ 0.38, which is reasonable since it falls between the solvation of the coronal and core chains.

Figure 7.

Bridging fraction (a), average domain spacing (b), modulus (c), strain hardening parameter (d), and equilibrium swelling ratio (e) as a function of homopolymer addition for nominally 14 vol % A25B116A25 gels.

Although dielectric spectroscopy has been used to experimentally measure bridging fraction in triblock melts and gels, it is extremely challenging and is limited to chains with a component of their dipole moment aligned along the backbone.50, 51 One of the greatest strengths of SCFT calculations for these triblock gels is their ability to calculate the bridging fraction. The bridging fraction as a function of homopolymer addition is shown in Figure 7(a). The slight decrease in bridging molecules as the average distance between micelles increases is expected because entropic considerations make looping rather than bridging more favorable as the length the bridge must span increases. However, for this system, the change in bridging fraction over the range of homopolymer concentrations studied is relatively small (0.57–0.49). As expected for a solvated system, these values are lower than what is predicted for triblock melts.52

Figure 7 summarizes the effects of homopolymer addition on gel structure and properties for nominally 14 vol % A25B116A25 gels. The top two plots show how f and d, which reflect the topology of the gel network, change as the micelle cores are swollen with homopolymer. Swelling the core forces the micelle centers further apart but only weakly affects the fraction of bridging chains. The changes in E mirror increases in d suggesting that changes in chain conformation dominate the increase in modulus with homopolymer addition. In contrast to the micelle spacing and linear modulus which increase with homopolymer addition, J*, which describes the extension prior to strain hardening, decreases. Likewise the equilibrium swelling ratio also decreases. Ideas about how to understand these observations in terms of chain conformation are presented in the discussion section.

Modulus Models

With the experimental results presented above, we are now in a position to determine whether the changes in gel modulus with homopolymer addition can be understood through changes in network topology described by rubber elasticity (eq 19) or whether the reinforcing effect of the glassy aggregates must also be accounted for. Typically for filled elastomers, the elastic modulus of the matrix that contains the rigid particles is taken to be invariant with filler concentration. We treat E0 as a constant with values selected such that the Guth–Gold model (eq 21) matches the experimentally determined modulus for the sample with the lowest ϕfiller. This yields a value of 7.1 kPa for E0. The comparison between the model prediction and the gel modulus measured from the indentation experiments are shown in Figure 8, which plots Young's modulus, E, as a function of the volume fraction of glassy, slightly solvated PMMA domains (ϕfiller). The right hand axis is E/E0 and represents the increase in modulus due to the rigid filler effect. While the Guth–Gold expression seems to do a good job describing the experimental data (there is some deviation at the highest PMMA content sample), it is important to remember that this model is based on the assumption of continuum mechanics being valid. It should also be noted that for triblock gels some rigid aggregates are necessary to act as crosslinks and a value of ϕfiller = 0 represents a hypothetical rather than a realizable, case. Therefore, while the Guth–Gold approach is not inconsistent with the experimental data, the agreement is more coincidental than definitive.

Figure 8.

Young's modulus, E, as a function of volume fraction of rigid filler, ϕfiller, for nominally 14 vol % A25B116A25 gels. The right hand axis is E/E0 and represents the increase in modulus due to the rigid filler effect. Data from indentation measurements are shown as open squares. Filled diamonds are modulus values calculated from eq 19 with R0 = 29.7 nm, f from SCFT, and d from SAXS. Solid line is Guth–Gold model (eq 21 with E0 = 7.1 kPa). Rigid filler content, ϕfiller, equal to volume fraction associated with solvated PMMA domains (eq 22).

We are also in a position to calculate the modulus of the triblock gels using rubber elasticity (eq 19) because the combination of SCFT calculation and SAXS experiments has allowed us to measure f and d for each of these samples and vS is known from the amount of triblock in the sample. The only variable left to determine the appropriate value of is R0 and this can be done by comparing the predictions of SCFT and SAXS measurements (see above discussion of R0). Also included as filled symbols in Figure 8 are the results of eq 19 using the SCFT and SAXS data. These values are in excellent agreement with the modulus values obtained from indentation experiments for all samples measured. Indeed, when changes in f and d are accounted for in this manner, it is unnecessary to apply any rigid filler correction. In reality, the increase in modulus is likely due to a combination of changes in network topology as well as rigid filler effects. However, these results indicate that it is important to remember that for micellar networks the matrix properties are not invariant with rigid aggregate content and that applying a Guth–Gold-type expression likely overestimates the rigid filler contribution to the modulus.

Swelling Models

From the experiments, we are also in a position to compare the behavior of the triblock gels with the equilibrium swelling predictions of the Neo-Hookean and exponentially strain hardening models. Taking the as-prepared state as our reference state, the equilibrium swelling ratio allows us to calculate λswell, indentation experiments determine Eref, and uniaxial compression experiments determine J*. As discussed above, we assume that only PnBA midblocks are osmotically active at room temperature (e.g., the amount of solvation in the glassy PMMA domains is invariant during swelling experiments). A value of 1.0 nm for a0 was used to calculate the models as it yielded the best agreement to the experimental data for the neat gel and is consistent with osmotic pressure data of Noda et al. for semidilute polymer solutions.53 Plotted in Figure 9 are the extension ratio due to swelling, λswell, experimentally determined from swelling experiments as well as predictions from the Neo-Hookean (eq 13) and strain hardening (eq 18). Although the strain hardening model predicts less swelling than the Neo-Hookean model, given the inherent error the in swelling measurements, it is difficult to differentiate with confidence between the models.

Figure 9.

Extension ratio due to swelling, λswell, as a function of homopolymer addition for nominally 14 vol % A25B116A25 gels. Open diamonds are data from swelling experiments while filled circles and triangles are the swelling predictions of the Neo-Hookean (eq 13) and exponentially strain hardening models (eq 18), respectively, with a0 = 1.0 nm.


As discussed earlier, simple rubber elasticity is able to account for the changes in gel modulus as a function of homopolymer if the bridging fraction of chains and the average distance between micelles are considered. Therefore, this section explores how ideas about chain stretching can be useful in rationalizing the changes in modulus, swelling, and strain hardening as a function of homopolymer addition for these triblock gels.

The total extensibility of a chain is set by its fully extended length divided by its unperturbed end-to-end distance, R0; however, the ability of a chain in a gel network to extend in response to an external force will depend on how its conformation was affected by network formation. Incorporation in a micelle causes a chain to be stretched relative to R0 and the magnitude of the stretching depends on the core size and coronal chain length. Thus, the formation of a micellar network imparts a prestretch to the chains. The ability of a triblock gel to deform in response to an external force (be it an external load, P, as in compression or an osmotic pressure, Πs, as in swelling) is limited by the magnitude of the prestretch imparted during micelle formation.

A gel's behavior can be understood, in the first approximation, by the behavior of a single chain between two nodes (in triblock gels the nodes correspond to the micelle centers) and it is helpful to define several uniaxial extension ratios which are summarized in Table 4. The prestretch imparted during micellizaiton can be described by λmicelle which compares the average distance between micelle cores, d, to the unperturbed chain length, R0, and is plotted in Figure 10(d). We use d measured by SAXS and R0 = 29.7 nm, the value that generated the best agreement between the SCFT and SAXS for all samples. Obviously this is a simplification because R0 in solution scales with the polymer concentration; however, the gel concentration varies little across these samples so the error associated with this assumption is small. Clearly as more homopolymer is solubilized in the micelle core, the amount of prestretch imparted to the chains increases. Equation 19 can be rewritten as E = 3vskBTfλmath image making explicit the link between stretching due to micellization and the linear modulus.

Figure 10.

Extension ratios, λmath image, λ*, λswell, and λmicelle, as a function of homopolymer addition for nominally 14 vol % A25B116A25 gels. Schematic insets illustrate the relevant lengths for each extension ratio.

Table 4. Extension Ratio Definitions
inline image

To define extension ratios for swelling and strain hardening, we first must define the relevant length scales. Assuming that a gel swells isotropically, the average distance between cores at equilibrium, dswell, is given by equation image, where equation image is the volumetric swelling ratio. Thus, λswell = dswell/d. As the volume fraction of midblock is roughly constant with increased homopolymer, the osmotic pressure, Πs, driving swelling is nearly constant for all samples in the series. The decrease in λswell shown in Figure 10(c) is due to the differing amounts of prestretch of the chains.

The onset of strain hardening occurs when J1 approaches J*, which defines an equivalent uniaxial extension ratio, λ*, given by J* = λ* + (2/λ*) − 3. Prior to external loading, the chain dimension is given by the average distance between micelles, d, while at the onset of strain hardening it is given by an extended length, d*, such that d* = dλ*. As λswell and λ* depend on the average distance between micelles in the as-prepared gel, anything that serves to increase the stretch of chains due to network formation will cause a decrease in swelling and an earlier onset of strain hardening.

Because the total triblock chain length is unaffected by homopolymer solubilization, the total amount of chain extension should be constant for all these samples. Full chain extension will not occur because the glassy PMMA micelles yield at stresses lower than the stresses required to fully extend the chain.28 However, we can define a normalized extension ratio, λmath image, which compares the amount of chain extension at the onset of strain hardening, d*, to the unperturbed dimension, R0. As shown in Figure 10(a), within experimental error λmath image is invariant with homopolymer addition as is expected from the constant triblock length. The variation with amount of homopolymer solubilization for all the extension ratios are summarized in Figure 10 along with schematics of the relevant length scales for each ratio. The on-set of strain hardening occurs at a value that can be quantitatively predicted by accounting for the prestretch induced during micellization.


We have investigated a set of well-characterized model triblock gels experimentally and with SCFT calculations to understand the effect of homopolymer solubilized within the micelle core on gel structure and mechanical properties. Over the range of concentrations studied, homopolymer PMMA shorter than the endblocks was successfully solubilized into the micelle cores. The main conclusion is that the addition of homopolymer to the micelle core increases the chain stretching in both the core and coronal blocks. The total extension of a chain for a given external load is fixed by its length; however, the initial prestretch imparted to the chain due to micellization changes with the size of the micelle core and can greatly reduce the amount of extension observed for a given external force. This idea helps to rationalize the observations made in this study, enumerated below.

  • 1Increasing homopolymer content increases the average distance between micelles and the linear modulus while decreasing the equilibrium swelling in pure solvent and the extension at which strain hardening becomes significant.
  • 2SCFT calculations allow quantitative determination of the bridging fraction of chains and show that it weakly decreases with homopolymer content. SAXS experiments were used to quantitatively measure how the distance between micelle centers varied as a function of homopolymer addition. There was good agreement between the average distance between micelles predicted from SCFT and that measured from SAXS experiments.
  • 3Rubbery elasticity is sufficient to explain the increase in modulus with homopolymer in this system addition if changes in bridging fraction and the average distance between crosslinks are taken into consideration.
  • 4While the Neo-Hookean model is sufficient to describe gel behavior at low strain (such as during swelling), it is insufficient to capture the strain hardening which occurs at larger deformations due to finite extensibility of the chains. A simple exponential strain hardening model which accounts for the finite extensibility of the chains captures the gel behavior extremely well.


We are grateful to Kuraray Co. of Japan for providing the triblocks used in these studies and to Steven Weigand for his help in collecting the SAXS data. This material is based upon the work supported under a National Science Foundation Graduate Research Fellowship and by the Northwestern University Materials Research Center, through the NSF MRSEC program DMR-0520513. Portions of this work were performed at the DuPont-Northwestern-Dow Collaborative Access Team (DND-CAT) Synchrotron Research Center located at Sector 5 of the Advanced Photon Source. DND-CAT is supported by the E.I. DuPont de Nemours and Co., The Dow Chemical Company, the US National Science Foundation through Grant DMR-9304725 and the State of Illinois through the Department of Commerce and the Board of Higher Education (Grant IBHE HECA NWU 96). Use of the Advanced Photon Source was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract W-31-10.