Defects, Solvent Quality, and Photonic Response in Lamellar Block

Jan 22, 2014 - ... Brown School of Engineering, Rice University, Houston, Texas 77251, United States ... Atsushi Noro , Yusuke Tomita , Yushu Matsushi...
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Defects, Solvent Quality, and Photonic Response in Lamellar Block Copolymer Gels Yin Fan,*,† Joseph J. Walish,‡ Shengchang Tang,† Bradley D. Olsen,† and Edwin L. Thomas*,§ †

Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States § Department of Materials Science and Nano Engineering, Department of Chemical and Biomolecular Engineering, Brown School of Engineering, Rice University, Houston, Texas 77251, United States ‡

S Supporting Information *

ABSTRACT: Stimuli-responsive photonic gels are made from the lamellar block copolymer (BCP) poly(styrene-b-2-vinylpyridine) (PS−P2VP), where the photonic responses are triggered by swelling/deswelling of the P2VP block with a selective solvent. When compared to isotropic swelling in chemically cross-linked homopolymer gels, the P2VP block in the lamellar BCP shows significantly lower degrees of swelling in alcohol−water cosolvents. The glassy PS layers completely constrain the lateral expansion of the P2VP gel layers and the dislocation defect network that develops during BCP self-assembly provides a counter force to vertical swelling. A model based on Flory−Huggins mixing and dislocation network strain energy is proposed to capture the swelling behavior of the BCP and is then used to estimate the dislocation network density in the lamellar BCP. This work establishes the quantitative relationship between the reflective color of the photonic gel, the effective χ parameter of the swellable block and the solvent, and the defect density of the BCP film and demonstrates the potential utility of these photonic materials as a quick means to measure solvent quality or defect density.

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regime.13,14 The color of BCP photonic crystals can be tuned by choice of the molecular weight or by blending low molecular weight homopolymers.15 Broadband dynamic tunability of the reflective color can be achieved in BCP photonic gels by swelling one block with a selective solvent2 or both blocks with a neutral solvent.8 The rapid solvent transport and corresponding large volume changes in the gel layers of the lamellar BCP gels contribute to the sensitive and reversible photonic responses on the subsecond time scale to stimuli such as pH, salt concentration, temperature, and mechanical strain.2,4−7 Swelling of the gel layers within a lamellar BCP is different from swelling of cross-linked homopolymer gels. Swelling in cross-linked gels is isotropic, driven by the free energy of mixing and hindered by the entropy decrease due to subchain network stretching as described by the Flory−Rehner theory.16 In lamellar BCPs, the chains are tethered at a fixed junction density to the interfaces and since one block is glassy, the selective swelling is only along the layer normal. BCPs with perfect lamellar morphology are expected to swell infinitely in a good solvent due to the entropy-driven unbinding of bilayers into planar micelles. Indeed, annealed BCP films of a low molecular weight (Mn 7.8/10.0 kg mol−1 for PS/P2VP) dissolved in methanol, a selective solvent for P2VP, demonstrating layer unbinding. Dissolution or unbinding in selective solvents was not

INTRODUCTION Sensors, smart coatings, and tunable camouflage employ materials that are capable of changing color in response to stimuli.1 Photonic gels made by block copolymer self-assembly display responsive reflective color over a broad wavelength range to a large variety of stimuli.2−7 The photonic responses result from the reflection of light waves by the periodic lamellar structure of the BCP gels. Such structural color can be varied by swelling/deswelling of one or both blocks in the BCP.2,8 Polymer solution thermodynamics triggers swelling and is key to understanding the photonic responses to solvent quality. The responsive lamellar design of the BCP photonic gels is inspired by the camouflage of cephalopods that are capable of changing skin color within milliseconds.9,10 The parallel layers in lamellar BCPs resemble the parallel protein platelets comprising the iridophores in the cephalopod skin. Such multilayers composed of materials with different layer refractive indices and thicknesses are a 1D photonic crystal or Bragg stack. Reflection from a Bragg stack depends on the thicknesses and refractive indices of both types of layers, the wavelength, polarization and direction of the incident light and can be tuned via these parameters. BCP microphase separation11,12 provides a facile method to fabricate such photonic crystals. BCPs of roughly symmetric block composition self-assemble into a lamellar morphology. With the proper choice of block chemistry and molecular weights, long-range ordered lamellar BCPs on a flat substrate reflect light and display colors across the visible spectral © 2014 American Chemical Society

Received: November 5, 2013 Revised: January 9, 2014 Published: January 22, 2014 1130

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temperature. The as-annealed BCP films were transparent with no reflection or scattering of visible light. The film swelled into a photonic gel when the cuvette was filled with the pure alcohols or the alcohol− water cosolvents. UV−vis transmission spectra were collected on a Varian Cary 6000i UV−vis−NIR spectrophotometer using the 1 cm path-length quartz cuvettes. A clean quartz cuvette was used for the baseline. The transparent dry films quickly showed color upon the addition of the cosolvent and the reflection peak positions stayed constant after several seconds. An equilibration time of 3 min was allowed at each sampling point to ensure steady-state swelling. The reflectance was calculated from the transmittance by R = 1 − T, assuming zero absorption or diffuse scattering. All spectral data presented in this work for a given compositional series were collected on the same film. The film was blown dry with nitrogen before swelling again. The error bars were estimated by repeatedly swelling the same sample five times. Because of the limit of the wavelength range of the UV−vis spectrophotometer and the absorption by the BCP, spectra below 300 nm could not be obtained. P2VP Gels. 2VP was distilled under vacuum before use. The 2 % mol DVB (purified by passing through a neutral alumina column) was added to the 2VP as a cross-linker, the mixture was dissolved at a concentration of 40% in 1:1 ethanol−water. 225 μL AMPS aqueous solution (10% by weight), and 10 μL TEMED was added per 30 mL reaction mixture to initiate the polymerization at room temperature. The reaction mixture was poured into a mold with two glass slides separated with 5 mm rubber spacers. A clear gel formed overnight. After 24 h, the gel was taken out from the mold and soaked in pure ethanol. Solvent was changed every 12 h for 3 days to extract unreacted monomers and low molecular weight oligomers, after which the gel has reached swelling equilibrium in ethanol. A total of five pieces of gel were made in separate molds from the same batch of reaction mixture. Uniaxial compression experiments were performed on a Zwick/Roell Z2.5/TS1S materials testing machine using TestX-pert V10.1 master software (Ulm, Germany) with a 20 N load cell. All tests were performed at room temperature. A cylindrical gel (25 mm diameter and approximately 8 mm thickness) was cut out by a custom-made cutter. A preload of 0.01 N was applied to the specimen and compression was performed at a nominal strain rate of 10% sec−1 to 10% nominal strain. The compression modulus was calculated from the linear region between 5% and 10% nominal strain. The shear modulus was taken as 1/ 3 of the compression modulus. Five samples per mold were cut and tested to determine the cross-link density of the gel in each mold. Gel pieces (approximately 2 g each) were then dried in a hood and then in a vacuum oven, both at room temperature. The dried gel pieces were then soaked in various alcohol−water mixtures for 3 days. The mass swelling ratios for all the various alcohol−water cosolvents were taken as the ratio of the swollen gel weight and the dry P2VP network weight. The swelling ratios error bars were estimated from five replicates, one from each mold.

observed with a much higher molecular weight BCP (Mn 102/97 kg mol−1 for PS/P2VP) used for the photonic gels in this paper. The difference likely results from the unbinding kinetics and especially the degree of perfection in the lamellar order. Thin films of a lamellar BCP tend to organize with the layers parallel to the substrate. Both edge and screw types of dislocations are frequently observed in such films.15,17 For example, cross-sectional transmission electron microscopy (TEM) (see Supporting Information, Figure S1 and S2) and 3D stimulated emission depletion (STED) microscopy18 show both types of line defects in lamellar PS−P2VP samples. Dislocations in 1D periodic layered media can only have their Burgers vector along the layer normal with magnitude equal to the period. For the in-plane ordered, periodic BCP bilayer film, the Burgers vector in the dry film is equal to the lamellar period. The edge dislocations have their line vectors in the plane of the film, whereas the screw dislocations have their line vectors along the normal to the film direction. Screw dislocations pin together multiple layers along the core region oriented normal to the layers whereas edge dislocations only pin two adjacent layers along a horizontally oriented core region. Dislocation lines in layered media must form continuous loops, end at external surfaces and/or form networks as first suggested by Bouligand.19 We find the swelling/deswelling behavior reproducible over multiple cycles, which suggests that the network of glassy PS layers does not undergo fracture or large plastic deformation during swelling which would irreversibly alter the BCP structure. Thus, in the PS−P2VP photonic gels, the glassy PS block layer network serves as a retarding force during selective swelling, while the interconnected P2VP regions serve as an important rapid solvent transport pathway across the PS layers.17 Thus, both P2VP solution thermodynamics and PS layer network deformation need be considered in the BCP swelling model. In this paper, we report the photonic responses of PS−P2VP/ alcohol−water gels to the cosolvent composition and develop a quantitative model for selective swelling of lamellar BCPs. We propose a quantitative relationship between the reflective color of the BCP gel, the effective Flory−Huggins χ parameter between the gel block and the solvent, and the dislocation line defect network of the BCP film. The results also demonstrate the potential utility of the BCP gels as sensing materials for the χ parameter or the defect density.

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EXPERIMENTAL METHODS

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Materials. A PS−P2VP diblock copolymer with a number-average molecular weight of 102 kg mol−1 for the PS block, 97 kg mol−1 for the P2VP block and polydispersity of 1.12 was purchased from Polymer Source Inc. Propylene glycol monomethyl ether acetate (PGMEA, Alfa Aesar) and chloroform (Mallinckrodt) were used for the copolymer film preparation. Methanol (Mallinckrodt), absolute ethanol (Koptec), 1propanol (Aldrich), and Millipore Milli-Q water were used to make the alcohol−water solutions. 2-vinylpyridine (2VP) monomer and divinylbenzene (DVB) cross-linker were purchased from Aldrich. Ammonium per-sulfate (AMPS, Mallinckrodt), and N,N,N′,N′tetramethylethylenediamine (TEMED, IBI Scientific) were used for P2VP gel synthesis. All chemicals were used as received unless otherwise noted. The alcohol−water cosolvents were mixed by volume. The cosolvent marked with ϕalcohol in composition is a mixture of alcohol and water with the original volumes by the ratio of ϕalcohol/(1 − ϕalcohol). PS−P2VP Photonic Gels. The BCP films were cast inside a 1 cm quartz spectrometer cuvette. A 15 μL PS−P2VP solution (5% weight fraction in PGMEA) was spread to cover one wall of a cuvette and the film was allowed to slowly dry in air. The film was then annealed overnight in saturated chloroform vapor and allowed to dry at room

RESULTS AND DISCUSSION Photonic Responses of PS−P2VP. PS−P2VP is a wellstudied BCP material with a large χ between the two blocks.20,21 PS and P2VP have similar glass transition temperatures (∼100 °C) and refractive indices (∼1.6), while their different chemical properties provide the opportunity for selective swelling. The cosolvents employed are binary mixtures of methanol, ethanol or 1-propanol with water and are strongly selective for P2VP at room temperature, leaving PS in a glassy state. The cosolvent quality for P2VP can be changed at a fixed temperature by choice of the alcohol and by adjusting the volume fraction of water in the cosolvent because of the very different solubilities of P2VP in water and in alcohol (see Table 1). As expected, the reflectivity peaks of the PS−P2VP/alcohol− water photonic gels depend on cosolvent composition. Figure 1 shows the reflection spectra of the PS−P2VP gels in methanol− water and the wavelengths of the reflectivity peaks in the three alcohol−water cosolvents. Pure alcohols swell P2VP, and the 1131

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Table 1. Solubility Parameters at 25 °C and Refractive Indices at a Wavelength of 589 nm at 20 °C22−28 PS P2VP water methanol ethanol 1-propanol

δ (MPa1/2)

n

17.5 23.3 47.9 29.7 26.0 24.3

1.60 1.60 1.33 1.33 1.36 1.38

this range. Interestingly, the overall peak shift behaviors with cosolvent composition are in all cases nonmonotonic with a broad maximum in the wavelength of the peak reflectivity at ϕalcohol ∼ 0.9. Modeling the reflection spectra using the transfer matrix method (TMM)29,30 reveals that the peak shifts are mainly due to changes in P2VP block swelling ratios. The refractive index of each alcohol−water cosolvent is nearly constant at all compositions because of the close refractive indices values of water and the alcohols (Table 1). The P2VP layers cannot expand laterally so the swelling ratios are equal to the thickness ratios of swollen P2VP to dry P2VP layers. The details of the TMM modeling can be found in the Supporting Information. TMM modeling suggests a roughly linear relationship between the reflectivity peak wavelengths and the P2VP swelling ratios αP2VP as a comprehensive result of changes in both thickness and effective refractive index of the swollen P2VP layers. The modeling captures the P2VP swelling ratio trends with a maximum between 0.8 and 0.9 ϕalcohol for all three alcohol− water cosolvents, consistent with the reflectivity peak shifts in Figure 1. Alcohol−water mixtures show nonlinear cosolvency for several polymers,31−34 i.e., the solubility of the polymers in the cosolvent shows a minimum or maximum instead of following a monotonic relation between the solubilities in the two pure solvents. Although not reported in the prior literature, a similar maximum cosolvency effect of the alcohol−water mixtures to P2VP is likely the cause for the nonmonotonic peak shift in the photonic gels. Homopolymer P2VP Gels. Chemically cross-linked P2VP were synthesized by copolymerizing 2VP and DVB (2 mol %). The cross-link densities of the P2VP gels were measured to be ∼27 kg mol−1 by uniaxial compression tests (see Supporting Information). Neglecting the influence of cross-linkers as a different type of monomer, the effective Flory−Huggins parameters of P2VP in alcohol−water cosolvents χeff can be calculated from Flory−Rehner theory in eq 1: χeff ρ Vs ⎛ 1 ⎛ 1⎞ 1 1 ⎞ ⎟=0 ln⎜1 − ⎟ + + 2 + P2VP ⎜ 1/3 − ⎝ α⎠ α 2α ⎠ Mc ⎝ α α (1)

α is the volume swelling ratio of the gel and ρP2VP is the density of dry P2VP, 1.15 g cm−3.35 Vs is the molar volume of the cosolvent and can be calculated from the molecular weight and density of the cosolvent using handbook data (see Supporting Information). Mc is the molecular weight between cross-links measured by compression tests of surface lubricated gel discs. As shown in Figure 3, the swelling ratios of the P2VP homopolymer gel in alcohol−water cosolvents vary with cosolvent composition in a similar way with those of the PS− P2VP BCPs in Figure 2 arising from the nonmonotonic cosolvency of alcohol−water mixtures for P2VP. Figure 3 demonstrates a clear similarity of all three alcohol−water cosolvents to swell the P2VP homopolymer network. The χeff values indicate a slight decrease in cosolvent quality for P2VP in the order of 1-propanol, ethanol, and methanol. This observation is in accord to the relative solubility parameters with the 1propanol−water cosolvents having the closest solubility parameter values to P2VP and therefore the lowest χeff. We also confirmed the aforementioned nonmonotonic solubility trend of the P2VP blocks in the BCP/alcohol−water cosolvent systems with the swelling ratios of cross-linked P2VP homopolymer gels in Figure 3.

Figure 1. (a) Reflection spectra of a PS−P2VP photonic gel in methanol−water cosolvents of varying methanol volume fraction ϕmethanol; (b) reflectivity peak wavelengths of PS−P2VP photonic gels in three different alcohol−water cosolvents. As ϕalcohol decreases from pure alcohol, the reflectivity peak initially red-shifts and then blue-shifts toward lower ϕalcohol with a broad maximum at ϕalcohol ∼ 0.9 in all three cosolvents.

photonic gels reflect visible color due to the increased P2VP layer thickness and the increased refractive index mismatch between the PS and the swollen P2VP layers. As a nonsolvent, addition of water is expected to shrink the P2VP gel layers and to blue-shift the reflectivity of the photonic gel. The reflection spectra show increasing blue-shifts with increasing water concentration for ϕalcohol < 0.9 and the visual color of the photonic gels changed from green to blue to clear. The sensitive peak shifts between ϕalcohol 0.4 and 0.7 (highlighted in Figure 1b) suggest that the photonic gels may be used as cosolvent composition indicators in 1132

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measured from the cross-linked P2VP gels can be used in the model. We also assume that PS is not plasticized by the cosolvents so that it is valid to use the modulus of amorphous glassy PS at room temperature in the model. The free energy of mixing between P2VP and the cosolvent is the driving force for selective swelling. We neglect the tethered brush configuration of the P2VP block and assume that the mixing is homogeneous within the P2VP gel layers. The free energy of mixing per n BCP molecules (and thus n P2VP molecules) is thus assumed to be given by the Flory−Huggins mean-field theory:36 ⎡ χn ⎤ ⎛ 1⎞ 1 ΔFmix = kT ⎢ns ln⎜1 − ⎟ + n ln + s ⎥ ⎝ ⎣ α⎠ α α ⎦

Figure 2. TMM-calculated P2VP block swelling ratios αP2VP in PS− P2VP/alcohol−water photonic gels of varying alcohol volume fraction ϕalcohol.

(2)

ns is the number of cosolvent molecules swelling the n P2VP chains and can be obtained by ns = n(α − 1) MP2VPρs/ρP2VPMs, with MP2VP and Ms the molecular weights of the P2VP block and the solvent, respectively. Since MP2VP is much larger than Ms, therefore n is much smaller than ns and the second term in eq 2 can be neglected. We assume the contribution of the defect networks to the free energy of BCP swelling is equal to the difference of the strain energy of dislocation networks before and after swelling. The strain energy of the dislocation networks depends on both the core energy and the surrounding strain field energy.

BCP Swelling Model. The favorable contributions to layer swelling include the free energy of mixing between P2VP and the cosolvent, the increase in system entropy by layer unbinding and layer shape fluctuations. The opposing force is the enthalpic strain energy due to the glassy PS layer network deformation driven by the P2VP swelling. We use the balance of the free energies of mixing and the change in the dislocation line energies (elastic strain fields) to determine the swelling equilibrium. First, we assume that the effective interaction parameters χeff between P2VP and the cosolvent are the same in cross-linked homopolymer gels and in BCP gels, so that the χeff values

Fdefects = (Ecore , s + Efield , s)Ls + (Ecore , e + Efield , e)Le

(3)

Figure 3. Swelling ratios of cross-linked P2VP homopolymer gels in (a) methanol, (c) ethanol, and (e) 1-propanol−water cosolvents and the effective Flory−Huggins interaction parameters χeff between P2VP and (b) methanol, (d) ethanol, and (f) 1-propanol−water cosolvents. 1133

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Here Ei are the energies per unit dislocation line length, and Li are the dislocation line lengths. The subscripts “core” and “field” stand for the core energy and the strain field energy, and “s” and “e” for the screw or edge dislocation types, respectively. The dislocation lines tend to form networks as depicted in Figure 4.

and P2VP have equal moduli and Poisson’s ratios, so we use the shear modulus G0 and Poisson’s ratio of amorphous PS for the BCP film, Gdry = 1 GPa and νdry = 1/3.39 In the swollen state, the modulus and the Poisson’s ratio can be calculated by treating the lamellae as a composite of glassy PS and gel P2VP layers. The modulus and the Poisson’s ratio of a lamellar composite is the average of the in-plane and normal components, which can be calculated by the parallel or in-series models:40 Gswollen = =

2 1 Gin − plane + Gnormal 3 3 −1 φ ⎞ 2 1⎛ φ (G1φ1 + G2φ2) + ⎜ 1 + 2 ⎟ 3 3 ⎝ G1 G2 ⎠

=2

νswollen =

=

Swelling of the layers in the dislocation core region depends on the type of dislocation. For the left- and right-handed helicoidal screw dislocations, the core region connects alternating layers of PS to PS and P2VP to P2VP.18,37 Swelling expands the P2VP layers but is retarded by the helix-like springs formed by the connected glassy PS layers. Edge dislocations are comprised of either PS or P2VP terminating layers. A P2VP terminating layer results in joining of the two adjacent PS layers, locally inhibiting the terminating layer from swelling. A PS terminating layer does not inhibit swelling as the adjacent P2VP layers are free to expand. Assuming that there are equal numbers of terminating PS and P2VP layers, half of the edge dislocations are ineffective at retarding swelling. Overall, the defects provide variations in the degree of P2VP layer swelling, consistent with the observed broad reflectivity peak profiles. The complicated swelling behaviors of the dislocation cores also make it difficult to estimate the Ecore terms in eq 3. The screw dislocation cores are partially swollen due to the helical connectivity of PS and P2VP layers; the cores of the P2VP terminating edge dislocations are swollen, while the PS terminating ones are not swollen. In order to have a tractable analysis, we neglect the difference in the total defect core energies between the swollen and dry states. Therefore, the total strain energy of dislocations in eq 3 can be simplified as: 1 Fdefects = Efield , sLs + Efield , eLe (4) 2 The strain field energy per unit length of dislocation line is given by38 Gbs 2 R ln , 4π R0

Efield , e =

Gbe 2 R ln 4π (1 − ν) R0

(6)

2 1 νin − plane + νnormal 3 3

2 1 = (ν1φ1 + ν2φ2) + 3 3

Figure 4. Schematics of a defect region in a lamellar PS−P2VP film with dislocation networks before and after selective swelling. Adapted from ref 19.

Efield , s =

2 G0 (α + 1)

φ1 ν G1 1 φ1 G1

+ +

φ2 ν G2 2 φ2 G2

9α + 7 18(α + 1)

(7)

φ1 and φ2 are the volume fractions of the glassy PS or gel P2VP layers in the lamellae. The shear modulus of P2VP gels was measured as ∼6 kPa in the compression tests of the cross-linked P2VP gels. The shear modulus of the homopolymer P2VP gels is a few kPa. As GPS, dry ≫ GP2VP, gel, eqs 6 and 7 can be simplified. The Possion’s ratio of the gel layers νP2VP, gel is approximately 1/ 2. To estimate the dislocation line lengths in the BCP film, we assume that the line vectors of edge dislocations lie along the x or y axis and the screw dislocations along z. We also assume that the average distances between dislocations are the same for both the screw and edge dislocations, Rs = Re = R. For simplicity, the dislocation lines are assumed to decorate a cubic lattice. The average lengths of dislocations lines are taken to be R, the same as the average distance between dislocations. The dislocation line lengths per unit volume are 1/R2 for screw and 2/R2 for edge dislocations respectively and the total dislocation line lengths in the film of n BCP molecules in the dry state are Ls =

2nMBCP nMBCP V = , Le = 2 2 R NAρBCP R NAρBCP R2

(8)

MBCP and ρBCP are the molecular weight and density of the BCP, and NA is Avgadro’s number. Then the dislocation strain energy at the dry state can be calculated and is given by: dry Fdefects =

5nMBCPG0d0 2 8πNAρBCP R2

ln

R d0

(9)

The Burgers vectors for both screw and edge dislocations are equal to the lamellar periodicity, which is d0 in the dry state and increases to (1 + α)d0/2 when the P2VP layers swell. The lengths of screw dislocation lines also increase (1+α)/2 times during swelling while the lengths of the edge dislocations stay constant. The dislocation strain energy in the swollen state is:

(5)

Here b is the Burgers vector, R the average distance between dislocations, and R0 the size of the dislocation cores. G and ν are the shear modulus and the Poisson’s ratio. In the dry state, PS 1134

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Figure 5. Fitting of swelling ratio αP2VP versus cosolvent composition ϕalcohol for the swelling model. (a−c) P2VP swelling ratios in PS−P2VP gels in cosolvents of water and (a) methanol, (b) ethanol, and (c) 1-propanol. Red circles are estimated by the model using the average dislocation-dislocation distance fitting parameter R ∼ 3 μm and black squares are experimental data measured from the photonic responses. (d) Fitting parameter R independently calculated for the three alcohol−water cosolvents. The error bars show the overlapping 95% confidence intervals. swollen = Fdefects

⎞ nMBCPG0d0 2 (1 + α)2 ⎛ 1 9 R ⎜ ⎟ ln + 12 ⎝ 4 9α + 11 ⎠ d0 πNAρBCP R2

measured by homopolymer gel swelling and Flory−Rehner theory; R is an unknown constant because the same BCP sample was used in measurements of photonic repsonses. R can be fitted using the α and χ data for all three cosolvents and was done separately due to the difference in the molecular volume Vs (see Figure 5). The calculated swelling ratios using the fitted parameter and the model equation (red) agree with the experimental values (black). The estimation of R ∼ 3 μm seems reasonable with respect to a sampling of cross-sectional TEM images of the as-annealed lamellar PS−P2VP samples. The overlapping 95% confidence intervals suggest that the model is effective in capturing the role of the dislocation networks in the BCP sample. An alternative is the Alexander brush model which considers the balance between the entropy loss of the P2VP brushes by the displacement of the swollen blobs and the free energy of mixing calculated by the excluded volume effect.41 This brush model predicts equilibrium swelling ratios that are lower than the experimental data at all alcohol concentrations.

(10)

Therefore, the strain energy difference between the dry and the swollen states is ΔFdefects =

nMBCPG0d0 2 ⎡ (1 + α)2 ⎛ 1 9 5 ⎞⎤ R ⎜ + − ⎟⎥ ln 2 ⎢ 9α + 11 8 ⎠⎦ d 0 πNAρBCP R ⎣ 12 ⎝ 4 (11)

The free energy of swelling for n PS−P2VP molecules is: ΔF = ΔFmix + ΔFdefects = nkT +

⎡ ⎛ χ⎤ 1⎞ (α − 1)⎢ln⎜1 − ⎟ + ⎥ ⎣ ⎝ α⎠ α⎦ MsρP 2VP MP 2VPρs

⎞ nMBCPG0d0 2 ⎡ (1 + α)2 ⎛ 1 9 5⎤ R ⎜ ⎟ − + ⎥ ln 2 ⎢ 9α + 11 ⎠ 8 ⎦ d0 πNAρBCP R ⎣ 12 ⎝ 4

■

(12)

At swelling equilibrium, the total free energy of the BCP gel should be minimized. The condition of ∂ΔF/∂α = 0 gives an equation for α and χ with the average distance between dislocations R as a fitting variable:

A model describing the selective swelling of a lamellar BCP was constructed based on Flory−Huggins mean-field mixing and the line defect strain energy of the interconnected dislocation network comprised of glassy PS layers. The model quantitatively relates the photonic gel’s reflectance and the P2VP swelling ratio to the effective Flory−Huggins χ parameter between P2VP and the cosolvent and the average defect density. The results indicate that the defects in BCP self-assembly impact selective swelling and suggest that the lamellar BCP photonic gels have a potential application as sensing materials for χ or defect density.

χ⎤ 12πRT ⎡ ⎜⎛ 1⎞ 1 + 2⎥ ⎢⎣ln⎝1 − ⎟⎠ + α α G0Vs α ⎦ +

(1 + α)(α 2 + 4.44α + 4.38) ln(R /d0) =0 (α + 1.22)2 (R /d0)2

CONCLUSION

(13)

In eq 13, α is the swelling ratio of P2VP layers and is calculated by TMM from the reflection spectra of the photonic gels; χ is 1135

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S Supporting Information *

Cross-sectional TEM images of lamellar PS−P2VP films, calculation and table of the refractive indices and molecular volumes of alcohol−water cosolvents, TMM variables, and molecular weight between cross-links of homopolymer P2VP gels by uniaxial compression tests. This material is available free of charge via the Internet at http://pubs.acs.org. Corresponding Authors

*E-mail: [email protected] (Y.F.). *E-mail: [email protected] (E.L.T.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank the National Science Foundation (DMR 0804449), the Air Force Office for Scientific Research (AOARD FA238611-1-4095), and the U.S. Army Natick Soldier Research, Development, and Engineering Center for funding support of this research. The authors also thank the MIT Institute for Soldier Nanotechnologies (ISN) for use of the UV−vis spectrophotometer, the ellipsometer, and the TEM, and Dr. Simona Socrate for use of the Zwick mechanical tester.

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dx.doi.org/10.1021/ma402287x | Macromolecules 2014, 47, 1130−1136