Transition between Phantom and Affine Network Model Observed in

Jan 14, 2013 - Faculty of Advanced Life Science, Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan. ABSTRACT: The elastic ...
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Transition between Phantom and Affine Network Model Observed in Polymer Gels with Controlled Network Structure Yuki Akagi, Jian Ping Gong, Ung-il Chung, and Takamasa Sakai* †

Department of Bioengineering, Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan ‡ Faculty of Advanced Life Science, Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan ABSTRACT: The elastic moduli of elastomeric materials are predicted by the affine or phantom or junction affine network models. Although these models are often used, we do not know the requirement conditions for each model or even the validity of each model. The validation of these models is difficult because of the network heterogeneity. In this study, we tried to evaluate these models using Tetra-PEG gel, which has extremely homogeneous network structure. We performed the stretching and tearing tests, and for the first time, observed the transition between the phantom and affine network models around the overlapping concentration of prepolymers.

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INTRODUCTION The elastic modulus is one of the simplest physical properties of polymer gels and elastomers. The elastic modulus is correlated with the concentration of elastically effective chains by three popular models; the affine,1 phantom,2 and junction affine3,4 network models. The difference between these models is the fluctuation of cross-links. The affine model assumes that cross-links are firmly connected to the macroscopic body and network strands deform in the same manner with the macroscopic deformation. On the other hand, the phantom network model assumes that cross-links fluctuate, and the deformation of the network strand is suppressed by the fluctuation. The junction affine model is the intermediate of these models. For a perfect tetra-functional polymer network, the elastic modulus predicted by the affine network model is double of that predicted by the phantom network model. Despite this discrepancy, both models are often used in order to evaluate the polymer networks without mentioning the validity. At this point, we do not know the requirement conditions for each model or even the validity of each model.5 The validation of these models is hampered by the network heterogeneity, which is intrinsically introduced to the network during the cross-linking process, and complicates the network structure.6 Because the effect of heterogeneity is not predicted qualitatively by any measurement or feed condition, the examination of these models is extremely difficult. Recently, we have succeeded in fabricating a near-ideal polymer network called Tetra-PEG gel, which is formed by A-B type cross-end coupling of two tetra-arm poly(ethylene glycol) (TetraPEG) units that have mutually reactive amine (TetraPEG-NH2) and activated ester (TetraPEG-OSu) terminal groups, respectively.7 Our previous small angle neutron © 2013 American Chemical Society

scattering measurements (SANS) suggest that Tetra-PEG gel has practically no spatial inhomogeneities in the range below 200 nm.8,9 Furthermore, a series of Tetra-PEG gels with different concentrations and molecular weights of network strands obeyed the phantom network model.10 However, our previous investigation had two problems. One was the limited experimental range. We only investigated below and around the overlapping concentration. Flory predicted that the phantom network model is applicable for dilute state, because the fluctuation of a cross-link enlarges in that situation.3 Thus, it is expected that the model shifts from the phantom network model to affine network model with increasing concentration. The other is the restricted experimental method. In order to evaluate these models, we should confirm the absence of trapped entanglements, which is one of the heterogeneities and acts as pseudocross-links increasing the elastic modulus.11 If there are trapped entanglements, we cannot distinguish whether the deviation from the phantom network model prediction is originated from trapped entanglements or from the change in models. Although we did check the power law of elastic modulus and neo-Hookean behavior of stress− elongation curve, these results were not the direct evidence. Thus, it is important to directly investigate the concentration of elastically effective chains by another measurement. Therefore, in this study, we extend the experimental condition to above the overlapping concentration and add tearing measurement. The fracture energy obtained by tearing Received: November 2, 2012 Revised: December 4, 2012 Published: January 14, 2013 1035

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velocity of 40 and 500 mm/min, while the other arm was maintained stationary. The tearing force F was recorded.

measurement is another method to obtain the concentration of elastically effective chains.12 Because the fracture occurs in ultimate elongation, the fracture energy is hardly affected by the fluctuation of cross-links. We fabricated the Tetra-PEG gels from different molecular weights of prepolymers (Mn = 5, 10, 20, 40 kg/mol). Then, we measured the reaction efficiency, elastic modulus, and fracture energy. Finally, we discuss the models predicting the elastic modulus and the fracture energy.

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RESULTS AND DISCUSSION Characteristics of TetraPEG-NH2 and TetraPEG-OSu. TetraPEG−OH was synthesized by anionic polymerization. The Mn’s were successfully set to approximately 5K, 10K, 20K and 40K, and the polydispersities were narrow (Table 1). The

EXPERIMENTAL PROCEDURE

Table 1. Molecular Weight, Polydispersity, End-Group Functionality (X0), and Overlapping Polymer Volume Fraction (ϕ*) of 5K, 10K, 20K, and 40K Tetra-PEGs

Synthesis of Prepolymers. Tetraamine-terminated PEG (TetraPEG-NH2) and tetra-OSu-terminated PEG (TetraPEG-OSu) were prepared from tetrahydroxyl-terminated PEG (TetraPEG−OH) having equal arm lengths. Here, -OSu stands for N-hydroxysuccinimide. The detailed preparation methods were reported previously.7 The molecular weights of TetraPEG-NH2 and TetraPEG-OSu were matched to each other (Mn = 5, 10, 20, 40 kg/mol). Characterization of Tetra-PEG Modules. The 1H NMR spectra were obtained on a JEOL JNM-AL (300 MHz) or JEOL Alpha series (500 MHz) spectrometer with tetramethylsilane (TMS) as the internal standard and CDCl3 as the solvent. The molecular weight and polydispersity were determined using a gel permeation chromatography system (TOSOH HLC-8220) equipped with two TSK gel columns (G4000HHR and G3000HHR). The columns were eluted with DMF containing lithium chloride (10 mM) with a flow rate of 0.8 mL/min at 40 °C. The molecular weights were calibrated with poly(ethylene glycol) standards (Polymer Laboratories, Ltd., Church Stretton, UK). The relative viscosity of the solutions was measured with a rheometer (MCR501; Anton Paar, Graz, Austria), using the cone−plate geometry at a constant shear rate of 100 s−1 at 25 °C. Fabrication of Tetra-PEG gels. Constant amounts of TetraPEGNH2 and TetraPEG-OSu (“ϕ”: 0.034−0.12) were dissolved in phosphate buffer (pH7.4) and phosphate-citric acid buffer (pH5.8), respectively. In order to control the reaction rate, the ionic strengths of the buffers were chosen. In the case of Tetra-PEG gel (10K and 40K), 25 mM of buffer solution was used for lower polymer volume fraction (“ϕ”: 0.034−0.066) and 50 mM was used for higher polymer volume fraction (“ϕ”: 0.081−0.096). In the case of Tetra-PEG gel (5K and 20K), 50 mM of buffer solution was used for lower polymer volume fraction (“ϕ”: 0.034−0.066) and 100 mM was used for higher polymer volume fraction (“ϕ”: 0.081−0.096). Two solutions were mixed, and the resulting solution was poured into the mold. At least 12 h was allowed for the completion of the reaction before the following experiment was performed. Infrared (IR) Measurement. The gels were prepared as rectangular films (height: 20 mm, width: 20 mm, thickness: 10 mm). Prepared gel samples were soaked in H2O for 2 days at room temperature and then dried. The dried samples were cut into thick films (thickness: 40 μm) using a Microtome (SM2000R, Leica). First, these samples were soaked in D2O, then soaked in a mixture solvent of D2O and PEG (Mw = 0.40 kg/mol) with volume ratio of 1:1. IR spectra of these samples were obtained at 25 °C using a JASCO FTIR-6300 equipped with a deuterated triglycine sulfate (DTGS) detector, in which 128 scans were coadded at a resolution of 4 cm−1 for samples. More than 2 samples were tested for each network concentration. Stretching Measurement. The stretching measurement was carried out on dumbbell shape standardized as JIS K 6261−7 sizes (2 mm thick) using a mechanical testing apparatus (Autograph AG-X plus; SHIMADZU, Kyoto, Japan) at a constant velocity of 60 mm/ min. The gel samples were used in an as-prepared state. More than 10 samples were tested for each network concentration, and the observed moduli were arithmetically averaged. Tearing Test. The tearing tests for the Tetra-PEG gels was carried out using a commercial test machine (Tensilon RTC-1150A, Orientec Co.). The gels were cut using a gel cutting machine (Dumbbell Co., Ltd.) into the shape specified by JIS K 6252 as 1/2 sizes (50 mm ×7.5 mm ×1 mm, with an initial notch of 20 mm). The two arms of the test sample were clamped and one arm was pulled upward at a constant

5K TetraPEG-NH2 5K TetraPEG-OSu 10K TetraPEG-NH2 10K TetraPEG-OSu 20K TetraPEG-NH2 20K TetraPEG-OSu 40K TetraPEG-NH2 40K TetraPEG-OSu

Mn [g/mol]

PD [−]

X0 [−]

ϕ* [−]

× × × × × × × ×

1.04 1.03 1.05 1.01 1.04 1.02 1.07 1.10

0.96 0.98 0.94 0.96 0.96 ≈1.0 0.92 0.94

0.097

5.33 5.34 9.62 1.11 2.04 1.94 4.51 4.16

103 103 103 104 104 104 104 103

0.062 0.035 0.013

end groups were then changed to NH2 or OSu. The end-group functionalities (X0) were higher than 0.9. We measured the viscosity (η) of 5K, 10K, 20K, and 40K TetraPEG-NH2 at several polymer concentrations (c). The viscosities of the TetraPEG-OSu were similar to that of the corresponding TetraPEG-NH2 (data not shown). We calculated the specific viscosity (ηsp) as η − η0 ηsp = η0 (1) where η0 is the viscosity of the solvent. Figure 1 shows the reduced viscosity (ηred = ηsp/c) as a function of c. By

Figure 1. Reduced viscosity as a function of ϕ0 for TetraPEG-NH2 (5K, rhombus; 10K, circle; 20K, square; 40K, triangle). Dotted lines show the results of the line fitting.

extrapolating c to zero, the intrinsic viscosity ([η]) was obtained from the intercept. lim

c→0

ηsp c

= lim ηred = [η] c→0

(2)

Then, using the Flory−Fox equation, the gyration radius (Rg) was obtained as13 1036

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[η] = Φ

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elasticity theory,16 we estimated the elastic modulus (G) from the initial slope of the stress (σ)−elongation (λ) curve. Here, we assumed incompressibility of the Tetra-PEG gels, which has been confirmed by our previous study.17 The elastic moduli were then predicted under the affine network model (Gaf) and the phantom network model (Gph) as

(6R g 2)3/2 Mn

(3)

where Φ is the constant (2.7 × 1023 mol−1). Although this equation was originally proposed for linear chains, this equation can be utilized to star-shaped polymers.14 We obtained the overlapping polymer volume fraction (ϕ*) by assuming that the intramolecular polymer volume fraction is the same as the macroscopic polymer volume fraction (Table 1 and Figure 2). The ϕ* obeyed the scaling prediction, i.e., ϕ* ∼ Mnβ, β = −0.80 = −4/5.15

Gaf = νkBT (affine network model)

(4)

Gph = (ν − μ)kBT (phantom network model)

(5)

where kB is the Boltzman’s constant and T is the absolute temperature. The number densities of elastically effective chains (ν) and active cross-links (μ) for tetra-functional network are predicted according to the tree-like approximation. Although Tetra-PEG gel is AB-type coupling of tetra-arm polymers, we can reduce it to AA-type coupling of tetra-arm polymer when we consider the stoichiometric condition.18 As for AA-type coupling of tetra-arm polymer, the probability that an arm does not lead to an infinite network (P∞) is correlated with p as19 P∞ = p ·P∞3 + (1 − p)

(6)

Using P∞, ν and μ are predicted as follows:

Figure 2. ϕ* as a function of Mn of TetraPEG-NH2. The dotted line is the fitting curve showing the relationship, ϕ* ∼ Mn−0.8.

Reaction efficiency. The Tetra-PEG gels were formed by mixing two equimolar prepolymer solutions with varying initial polymer volume fraction (ϕ0) and Mn. The Tetra-PEG gels were formed from prepolymers with Mn = 5K, 10K, 20K, and 40K are named 5K, 10K, 20K, and 40K Tetra-PEG gel, respectively. The reaction efficiency (p) was estimated by FTIR measurement for gels swollen in the mixture solution of D2O and PEG (Mw = 0.4 kg/mol). The detailed calculation method of p was shown in our previous paper.10 The variation of p against ϕ0 is shown in Figure 3. The value of p was almost

⎧ 3 ⎛4⎞ ⎫ 4 ⎛4⎞ ν = Φ⎨ ·⎜ ⎟(1 − P∞)3 P∞ + ⎜ ⎟(1 − P∞)4 ⎬ 2 ⎝4⎠ ⎩ 2 ⎝3⎠ ⎭

(7)

⎧ 1 3 ⎛4⎞ ⎫ 1 4 ⎛4⎞ ⎜ ⎟(1 − P∞)4 ⎬ ⎜ ⎟(1 − P∞)3 P∞ + μ = Φ⎨ 2 2 ⎝4⎠ ⎩ 3 2 ⎝3⎠ ⎭

(8)

where Φ is the number density of the Tetra-PEG prepolymers. The variation of G, Gaf, and Gph against ϕ0 is shown in Figure 4. Gaf and Gph increased linearly with increasing ϕ0, reflecting the constant p. As for the 10K and 20K Tetra-PEG gels, Gph and G corresponded well with each other in a wider range than the other gels, suggesting that their elasticities are roughly predicted by the phantom network model. In the higher concentration region, however, G was slightly larger than Gph. As for the 5K Tetra-PEG gel, the downward deviation of G from Gph was increasingly pronounced with decreasing ϕ0, suggesting that elastically ineffective loops were formed in the low ϕ0 range.20 However, it seems that G approaches Gph asymptotically in the high ϕ0 region. The 40K Tetra-PEG gel shows distinct behavior; G was above Gph and close to Gaf. Taken together, the phantom-like behavior was observed in the low Mn and ϕ0 condition, on the other hand, the affine-like behavior was observed in the high Mn and ϕ0 condition. In order to discuss the whole tendency, we plotted G/Gaf against ϕ0/ϕ* (Figure 5). In this figure, phantom and affine network model predictions are the flat lines showing G/Gaf = 0.5 (dashed line) and 1 (dotted line), respectively. Surprisingly, all of the data fall onto a single curve. In the range from ϕ to 3.0ϕ*, the elastic moduli are well predicted by the phantom network model. The downward deviation below ϕ* is due to the formation of elastically ineffective loops. In this region, G/ Gaf scales with ϕ0/ϕ* as G/Gaf ∼ (ϕ0/ϕ*)2.0. In the range above 3.0ϕ*, G/Gaf increased with increasing ϕ0 and approached to 1.0. In this region, G/Gaf scales with ϕ0/ϕ* as G/Gaf ∼ (ϕ0/ϕ*)0.5. This power is smaller than that predicted for the effect of trapped entanglements.21,22 These data strongly suggest that trapped entanglements are introduced to the network or that the model shifts to the affine network model or both. Only from these results, we cannot distinguish whether the deviation from the phantom network model prediction is

Figure 3. The reaction efficiency (p) as a function of initial polymer volume fraction in the Tetra-PEG gels (5K, rhombus; 10K, circle; 20K, square; 40K, triangle).

constant against ϕ0. p was in the range 0.82−0.95 in 5K, 10K, and 20K Tetra-PEG gels, and was between 0.7 and 0.8 in 40K Tetra-PEG gel. The decrease in p may be caused by the decrease of the reaction rate with increase in Mn and will be discussed in our forthcoming paper. Elastic modulus. We performed a stretching test for the 5K, 10K, 20K, and 40K Tetra-PEG gels. According to the linear 1037

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Figure 4. Value of G (squares) estimated from the stretching measurement, Gaf (circles) and Gph (triangles), estimated from the reaction efficiency (p) as a function of ϕ0 in the (a) 5K, (b) 10K, (c) 20K, and (d) 40K Tetra-PEG gels. The dashed line is the guide showing the relationship, G ∼ ϕ0.

tearing measurements at a constant velocity of 40 and 500 mm/ min, and confirmed T was constant in this velocity regime. Thus, we successfully estimated T0 from these measurements. In the Lake-Thomas model,12 T0 is estimated as the energy needed to break the chemical bonds on the fracture surface and is represented as

T0 =

ν∼

originated from trapped entanglements or from the change in models. Thus, we tried to investigate the effect of trapped entanglements by the tearing measurement. Tearing Test. We performed the tearing measurement for the 5K, 10K, 20K and 40K Tetra-PEG gels, in order to investigate the fracture energy (T). The trouser-shaped gel sample is used for the tearing measurements. The two arms of the sample were clamped and one arm is pulled upward at a constant velocity, while the other arm was maintained stationary. T is represented as23 2F w

(10)

where L is the displacement length, N is the degree of polymerization of network strand, and U is the energy required to rupture a monomer unit. Assuming there is no trapped entanglements, and the elastically effective chains are formed only by chemical cross-links, ν and L are represented as follows:

Figure 5. The value of G/Gaf as a function of ϕ0/ϕ* for the TetraPEG gels (5K, rhombus; 10K, circle; 20K, square; 40K, triangle). The dashed and dotted lines are the guides showing G/Gaf = 0.5 and 1.0, respectively.

T=

1 LνNU 2

ϕ0 N

L ∼ aN1/2

(11) (12)

where a is the monomer length. Thus, T0 scales with the ϕ0 and N as T0 ∼ ϕ0N1/2

(13)

The values of T0 for the 5K, 10K, 20K, and 40K Tetra-PEG gels are shown against “ϕ” and N in Figure 6, parts a and b, respectively. Here, we used the calculated values of N for perfectly reacted polymer networks. T0 increased with increasing “ϕ” and N, showing the relationship, T0 ∼ “ϕ”1.0 and T0 ∼ N1/2. Then, we replotted the all data of T0 against ϕN1/2 in Figure 7. As expected, all the data fell onto a master curve showing the relationship, T0 ∼ “ϕ”N1/2. These results corresponded well with the Lake−Thomas model prediction, suggesting the validity of the assumption that there are no trapped entanglements regardless of ϕ0 and N.

(9)

where F is the tearing force and w is the thickness of the specimen. It is widely known that T depends on the tearing speed; T is constant at below a specific tearing speed and called the intrinsic fracture energy (T0).24,25 We performed the 1038

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that the elastically effective chains are dominantly formed by chemical cross-links. (iii) The model predicting the elastic modulus shifts from the phantom network model to the affine network model at around the overlapping polymer concentration. This transition is expected to occur also in other conventional polymer gels, in which the transition will be observed around the three times overlapping concentration of virtual polymer having the same degree of polymerization with the network strand.

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AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been financially supported by the Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST) from the Japan Society for the Promotion of Science (JSPS) and Grant-in-Aids for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology (No. 23700555 to T.S. and No. 24240069 to U.C.).

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REFERENCES

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Figure 6. (a) Fracture energy as a function of ϕ0 for the Tetra-PEG gels (5K, rhombus; 10K, circle; 20K, square; 40K, triangle). (b) Fracture energy as a function of N for the Tetra-PEG gel (ϕ0 = 0.034, rhombus; ϕ0 = 0.050, circle; ϕ0 = 0.066, square; ϕ0 = 0.081, triangle; ϕ0 = 0.096, right triangle).

Figure 7. Fracture energy as a function of ϕ0N1/2 in the Tetra-PEG gels (5K, rhombus; 10K, circle; 20K, square; 40K, triangle). The dashed line is the master curve.

The validation of this assumption allows us to discuss the model of elastic modulus. We had two hypotheses explaining the deviation of elastic modulus from the phantom network model prediction; (i) the existence of trapped entanglements and (ii) the change in models. Now, we can eliminate the possibility of hypothesis (i), confirming the validity of hypothesis (ii). Thus, the model predicting the elastic modulus shifted from phantom to affine network model at around 3ϕ*. As far as we know, this transition is observed for the first time.

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CONCLUSION The major findings of this paper are as follows: (i) The reaction conversion of Tetra-PEG gel was constant against ϕ0 but depends on N. (ii) The fracture energy of Tetra-PEG gel was explained by the Lake-Thomas model under the assumption 1039

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(24) deGennes, P. G. Langmuir 1996, 12 (19), 4497−4500. (25) Tanaka, Y.; Kuwabara, R.; Na, Y. H.; Kurokawa, T.; Gong, J. P.; Osada, Y. J. Phys. Chem. B 2005, 109 (23), 11559−11562.

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