Supersoft and Hyperelastic Polymer Networks with Brushlike Strands

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

Supersoft and Hyperelastic Polymer Networks with Brushlike Strands Heyi Liang,† Sergei S. Sheiko,‡ and Andrey V. Dobrynin*,† †

Department of Polymer Science, University of Akron, Akron, Ohio 44325, United States Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599-3290, United States

‡

S Supporting Information *

ABSTRACT: Using a combination of the scaling analysis and molecular dynamics simulations, we study relationship between mechanical properties of networks of graft polymers and their molecular architecture. The elastic response of such networks can be described by replacing the brushlike strands with wormlike strands characterized by the effective Kuhn length which is controlled by the degree of polymerization of the side chains nsc and their grafting density 1/ng. In the framework of this approach we have established relationships between the network structural shear modulus G, strands extension ratio β, and architectural triplet [nsc, ng, nx], where nx is the degree of polymerization of the backbone strand between cross-links. Analysis of the simulation data shows that G could increase with β (G ∝ β), which reflects the “golden rule” of elastomers: softer materials are more deformable. However, networks of graft polymers can also break this rule and demonstrate an increase of the modulus G with decreasing extension ratio β such as G ∝ β−2. This can be achieved by changing the grafting density of the side chains 1/ng and keeping nx and nsc constant. This peculiar mechanical response of graft polymer networks is in agreement with experimental studies of poly(dimethylsiloxane) graft polymer elastomers.

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INTRODUCTION Attachment of side chains to a linear backbone produces new classes of polymers (Figure 1) with distinct physical properties.1−12 Depending on the grafting density and side chain length, we distinguish comb and bottlebrush macromolecules.13 Increasing the grafting density 1/ng and degree of polymerization of the side chains nsc suppresses the entanglement threshold12,14−19 and thus empowers the creation of supersoft elastomers with modulus as low as 100 Pa.20−30 Further, synthetic control over three architectural parameters [nsc, ng, nx] (see Figure 1) allows for decoupling the conventional correlation G ∝ λmax−2 between shear modulus G and maximum elongation at break λmax established for linear chain networks. This ability in independent control over stiffness and strain stiffening of graft polymer networks makes it possible to mimic the entire deformation behavior of biological tissues and synthetic gels in a single-component network just through variation in molecular architecture of the network strands.30 Using the scaling approach and computer simulations, we present detailed analysis of how molecular architecture of network strands influences mechanical response of graft polymer networks. Our study is based on the previously developed diagram of states of un-cross-linked graft polymers in a melt state, which determines different conformational regimes in terms of the fraction of the backbone monomers.12,13,31 φ = ng /(ng + nsc)

volume interactions between densely grafted side chains expel monomers of neighboring macromolecules from the pervaded volume of a given graft polymer. In a bottlebrush regime, in order to maintain a constant monomer density in a melt, bottlebrush backbones or side chains have to stretch. Bottlebrushes with extended backbones belong to the stretched backbone (SBB) regime, whereas macromolecules with stretched side chains belong to the stretched side chain (SSC) regime. Bottlebrushes with fully stretched side chains define the rodlike side chain (RSC) regime. In the different conformational regimes, the effective Kuhn length, bK, can be expressed in terms of the degree of polymerization of the side chains nsc and number of bonds between grafting point of the side chains, ng, and system molecular parameters as summarized in Table 1.13 The rest of paper is organized as follows: (i) using expressions for the effective Kuhn length of graft polymers obtained for different conformation regimes (Table 1), we present a model of graft polymer networks; the model predictions are compared with results of (ii) molecular dynamics simulations and (iii) uniaxial tensile tests of poly(dimethylsiloxane) (PDMS) graft polymer networks with the accurately controlled architectural triplet [nsc, ng, nx].

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RESULTS AND DISCUSSION Networks of Combs and Bottlebrushes. Consider networks of combs and bottlebrushes made by cross-linking their backbones in a melt state (Figure 1). Graft polymers

(1)

The diagram of states shown in Figure 2 identifies (i) the comb regime, where both side chains and backbones of graft polymers interpenetrate and remain in their unperturbed ideal chain conformations, and (ii) the bottlebrush regime, where excluded © XXXX American Chemical Society

Received: December 1, 2017

A

DOI: 10.1021/acs.macromol.7b02555 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

Figure 1. (a) A graft polymer chain with degree of polymerization of backbone nbb, number of bonds between grafted side chains ng, and degree of polymerization of the side chains nsc. (b) A graft polymer network with degree of polymerization of the backbone between cross-links nx. Cross-link bonds are shown in red.

σ (λ ) =

−2 ⎞ ⎛ ⎛ β(λ 2 + 2λ−1) ⎞ ⎟ G 2 (λ − λ−1)⎜⎜1 + 2⎜1 − ⎟ ⎟ 3 3 ⎝ ⎠ ⎠ ⎝

(2)

where G is structural shear modulus and β is strand extension ratio. While G determines material stiffness, the β parameter controls the elongation at break and onset of strain stiffening due to finite chain extensibility. These parameters are determined by concentration and molecular architecture of the network strands. The strand extension ratio is defined as β = ⟨R in 2⟩/R max 2

(3)

⟨Rin2⟩

where is the mean-square end-to-end distance of the undeformed network strands in as-prepared networks and Rmax = nxl is the contour length of a fully extended strand. By considering graft polymers as wormlike chains with the effective Kuhn length bK, ⟨Rin2⟩ can be written as35 ⎛ ⎛ 2R ⎞⎞⎞ bK ⎛ ⎜⎜1 − exp⎜ − max ⎟⎟⎟⎟⎟ ⟨R in 2⟩ = bK R max ⎜⎜1 − 2R max ⎝ b K ⎠⎠⎠ ⎝ ⎝

Thus, the strand extension ratio β is a function of the number of Kuhn segments per strand α−1 ≡ Rmax/bK

Figure 2. Diagram of states of graft polymers in a melt with bond length l, Kuhn length of the backbone and side chains b, and monomer excluded volume v. SBB − stretched backbone regime, SSC − stretched side chain regime, and RSC − rodlike side chain regime. Logarithmic scales. Adapted with permission from ref 13.

⎛ ⎛ 2 ⎞⎞⎞ α⎛ β = α⎜1 − ⎜1 − exp⎜ − ⎟⎟⎟ ⎝ α ⎠⎠⎠ 2⎝ ⎝

regime

bottlebrush SBB SSC RSC

Kuhn length, bK

regime boundaries vφ−1 ≤ (bl)3/2nsc1/2, for nscl/b > 1 vφ−1 ≤ l3nsc2, for nscl/b < 1

b b

(bl)3/2nsc1/2 ≤ vφ−1 ≤ bl2nsc bl2nsc ≤ vφ−1 ≤ l3nsc2 l3nsc2 ≤ vφ−1

vl−3/2b−1/2φ−1nsc−1/2 (v/φl)1/2 lnsc

(5)

For flexible strands, bK ≪ Rmax, the extension ratio is equal to the inverse number of Kuhn segments per strand β ≈ α. This case is relevant to conventional linear chain networks. However, brushlike networks can be made of stiffer strands (bK ∼ Rmax), which results in β < α. The structural shear modulus of the network, G, is proportional to the number density of the stress-supporting strands ρs

Table 1. Effective Kuhn Length of Graft Polymers in Different Regimes and Regime Boundariesa comb

(4)

G = C1kBTρs

l − bond length, b − Kuhn length of the linear polymer strand, and v − monomer excluded volume.

a

⟨R in 2⟩ bK R max

(6)

where C1 is a numerical constant that depends on the functionality of the cross-links and network topology, kB is the Boltzmann constant, and T is the absolute temperature. In order to express ρs in terms of the architectural parameters of graft polymers, we have to eliminate contribution of the dangling ends which do not support network stress upon deformation and effectively reduce density of stress supporting

between cross-links can be modeled as linear chains with effective Kuhn length, bK.13,31 For a broad range of synthetic and biological networks undergoing uniaxial deformation at constant volume, it has been validated that the true stress increases with elongation ratio λ as32−34 B


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substituting the corresponding expressions for bK (Table 1) into the definition of parameter α ≡ bK/Rmax. The expressions for structural shear modulus and other parameters of networks of combs and bottlebrushes are summarized in Table 2.

strands. As shown in Figure 1, cross-linking graft polymers with degree of polymerization of the backbone nbb produce two chain ends of varying length. This gives the following expression for the density of the stress-supporting strands ⎛1 2 ⎞ ρs = ρφ⎜ − ⎟ nbb ⎠ ⎝ nx

Table 2. Structural Shear Modulus, G, and Number of Effective Kuhn Segments per Network Strands, α−1, in Different Conformation Regimes

(7)

where ρ is the monomer number densities in a melt. For densely cross-linked networks, nx ≪ nbb, eq 7 reduces to a conventional expression for ρs ≅ ρφ/nx. Substituting eqs 3 and 7 into eq 6, we obtain G = C1ρkBTβα −1φ(n x −1 − 2nbb−1)

(8)

Equation 8 is a general expression for the structural modulus of graft polymer networks. Its explicit form as a function of the architectural parameters of the graft polymers can be obtained by using corresponding expressions for β and α in different regimes of diagram of states (see Figure 2). Before proceeding further, it is important to point out the difference between the structural shear modulus G and the apparent shear modulus at small deformations σ (λ ) G G0 ≡ lim 2 = (1 + 2(1 − β)−2 ) −1 λ→ 1 λ − λ 3

(9)

β = C2n x −1

(10b)

G = C1ρkBTβα−1φ(nx−1 − 2nbb−1)

Cαnx−1φ−1nsc−1/2 Cαnx−1φ−1/2 Cαnx−1nsc

ρkBT nx

λmax =

(12)

R max 2 1/2

⟨R in ⟩

= β −1/2 =

ln x bln x

≈ n x1/2

(13)

From eqs 12 and 13, we conclude that for linear chain networks G ∝ nx −1 ∝ β ∝ λmax −2

(14)

This equation represents a “golden rule” of the materials sciencesofter materials (networks) are more deformable. For networks of combs and bottlebrushes this golden rule can be broken through independent variation of the architectural parameters [nsc, ng, nx] as follows from eqs 8−11. As such, one can design networks with more complex relationships between modulus and elongation at break (or extension ratio) depending on the molecular architecture of the graft polymers.30 In the next section, we show how this model of the nonlinear network deformation can be applied to networks made by cross-linking graft polymer strands. Comparison with Simulations and Experiments. Predictions of the scaling model of comb and bottlebrush networks are verified in coarse-grained molecular dynamics simulations.37 In our simulations macromolecule backbones and side chains are modeled as bead−spring chains38 composed of beads with diameter σ interacting through truncated shifted Lennard-Jones (LJ) potential. Connectivity of the monomers into graft polymers and cross-linking bonds are represented by the combination of the FENE and truncated shifted LJ potentials. We performed simulations of macromolecules with spring constants of the FENE potential being equal to 30kBT/ σ2 and 500kBT/σ2. The set of structural parameters for studied systems is summarized in Table 3. In the case of ng = 0.5, two side chains are grafted to each backbone monomer. For all

where C2 is a numerical constant. Equations 10a and 10b suggest that G and β can be controlled independently, e.g., by maintaining constant nx and varying φ. Bottlebrushes. Stretched Backbone (SBB) Regime. At high grafting density, steric repulsion between side chains results in backbone extension. This results in larger effective Kuhn segments (Table 1) and alters expression for the structural modulus G given by eq 8. The strand extension ratio β is given by eq 5 with parameter α = Cαn x −1φ−1nsc−1/2

C2nx−1 (≅β)

Here we neglect the dangling end contribution, which is appropriate for densely cross-linked networks with nx ≪ nbb (see eq 10a). The strand extension ratio β is related to the elongation at break λmax of a network strand defined as the ratio of the maximum strand end-to-end distance, Rmax, to the initial size of the strand, ⟨Rin2⟩1/2. For flexible network strands, we obtain the following scaling relation between λmax and nx:

which is commonly used to characterize stiffness of linear chain networks. Both moduli (G and G0) have the same value only in the case of coiled networks strands characterized by β ≪ 1 (β ≈ α). For networks with pre-extended strands (e.g., gels and bottlebrushes), the structural and apparent moduli are different as follows from eq 9. Combs. The loose grafting of side chains in the comb regime does not perturb the ideal polymer chain conformation of both the side chains and backbone and leads to bK ≈ b. The presence of the side chains merely results in dilution of the stresssupporting backbones, which in turn lowers G. For networks with flexible strands (β ≪ 1, β ≅ α), we obtain the following expressions for G and β (eqs 3 and 8): (10a)

G = C1ρkBTφ(nx−1 − 2nbb−1)

To conclude this section, we compare mechanical properties of the linear chain networks with that of comb and bottlebrush networks. For linear chain networks with the degree of polymerization between cross-links nx below the entanglement threshold, the network structural shear modulus is35,36 G≈

G = C1ρkBTφ(n x −1 − 2nbb−1)

α

G

regime comb bottlebrush SBB SSC RSC

(11)

where Cα is a numerical constant. Similar to comb systems, G and β can be controlled independently by coordinated variation of nx and φ. Further increase of grafting density causes extension of the side chains and results in two additional regimes (subregimes of the bottlebrush regime) with distinct mechanical behaviors. In the stretched side chain (SSC) and rodlike side chain (RSC) regimes (Figure 2), the expression for G remains the same as in eq 8. However, a new equation for β (eq 5) is obtained by C


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Macromolecules Table 3. Summary of Studied Systems

studied systems, the monomer density is set to ρσ3 = 0.8. The simulation details are described in the Supporting Information. All simulations are performed using the LAMMPS39 simulation package. In our simulations, the graft polymers are cross-linked through the end monomers of the side chains, which corresponds to the experimentally studied systems.30 Note that such cross-linking scheme results in a hybrid network composed of brushlike and linear network strands, which requires explicit consideration of the elastic response of both strands populations. This can be done using the phantom network model as discussed in detail in the Supporting Information. However, in a wide range of the strands’ architectures the elastic response of graft polymer networks is dominated by the deformation of the graft polymer strands. Therefore, such networks can be approximated by graft polymer networks cross-linked through their backbones as shown below. Note that this approximation of the elastic properties of networks of graft polymers is also successfully used in analysis of the experimental data in refs 24, 30, and 34. The mechanical properties of networks of graft polymers are obtained by analyzing stress as a function of the elongation ratio λ of networks undergoing a uniaxial elongation along xaxis at a constant volume. The tensile stress σxx is calculated from the pressure tensor Pii as follows:40−42

σxx =

3 1 Pxx − 2 2

∑ Pii i

(15)

The typical stress−deformation curves of graft polymer networks are shown in Figures 3a−c (the remaining set of the stress−deformation curves can be found in the Supporting Information). In each figure, only one part of the architectural triplet [nsc, ng, nx] is varied. In Figures 3a and 3b, the decrease of the side chain length and strand length between cross-links both lead to the higher shear modulus and lower extensibility. Note that to make connection with experimental studies of the graft polymer networks, we used experimentally established correlation between strands extension ratio β and λmax ≈ β−1/2.24,30 In Figure 3c, however, both shear modulus and extensibility increase with the decrease of grafting density. This demonstrates that elastomers made by cross-linking graft polymers could break the locked relationship between shear modulus and elongation at break given by eq 14. It is important to point out that the trends in network deformations observed in Figures 3a−c are in a good qualitative agreement with experimental results for uniaxial deformation of the PDMS networks made of comb and bottlebrush strands.30 For each set of networks, the reduced shear modulus αG/β linearly increases with the increase of reduced density of strand φ/nx (Figures 3d−f). In Figures 3d and 3f, the intercept of linear fit is positive D


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Figure 3. Dependence of the tensile stress σxx on the deformation ratio λ for networks of graft polymers (a−c). (a) Illustration of the increase in the network structural modulus G and decrease in network extensibility with increasing DP of side chains nsc for networks with ng = 8, nx ≈ 16. (b) Decrease in network structural modulus G and increase in network extensibility with increasing DP of backbone between cross-links nx for networks with ng = 2 and nsc = 8. (c) Nonlinear deformation of the networks with nsc = 8, nx ≈ 16, and different values of ng. The value of the bond spring constant is equal to kspring = 30kBT/σ2 in (a) and (b) and kspring = 500kBT/σ2 in (c). The dashed lines are the best fit to eq 2 with structural modulus G and extension ratio β as fitting parameters. Correlations between network mechanical properties and strand architecture illustrated by linear relationship between reduced shear modulus and reduced density of stress supporting strands (d) for networks shown in (a), (e) for networks shown in (b), and (f) for networks shown in (c). Symbol notations are summarized in Table 3. Dashed lines in (d−f) illustrate linear correlation between αG/β and φ/nx.

because the network structure is not altered by the change of side chain length or grafting density. In Figure 3e, the reduced shear modulus vanishes at finite reduced density of strand. In this case, the length of strand increases due to the decrease in the number of cross-links per chain ncr. The network structure breaks down when the number of cross-links per graft polymer strand ncr,0 is equal to 2. This corresponds to the intersection on reduced density of strand axis (φ/nx)0 = φ/(nbb/(ncr,0 + 1)) = 0.0047. To highlight universality of the relationship between chemical structure of networks and their mechanical properties, in Figure 4 we show dependence of the reduced shear modulus αG as a function of parameter βφ/nx for all studied network systems listed in Table 3. The universality in the network mechanical properties is not only seen in correlations with structural network parameters but also in an excellent fit of the simulation data to the universal function describing stress in deformed networks undergoing

uniaxial elongation (see eq 2). To highlight this fact, we introduce the deformation dependent shear modulus:32 G(I1) ≡

σxx(λ) λ 2 − λ −1

=

−2 ⎛ βI1(λ) ⎞ ⎞ G ⎛⎜ ⎟ 1 2 1 + − ⎜ ⎟ ⎝ 3 ⎜⎝ 3 ⎠ ⎟⎠

(16)

where I1(λ) is the first invariant for uniaxial network deformation I1(λ) = λ2 + 2/λ. Figure 5 combines data for the reduced deformation-dependent network shear modulus G(I1)/ G as a function of βI1/3. The dashed lines in both figures are given by the equation32 −2 ⎛ G(I1) βI1(λ) ⎞ ⎞ 1 ⎛⎜ = ⎜1 + 2⎜1 − ⎟ ⎟ ⎝ G 3⎝ 3 ⎠ ⎟⎠

(17)

The good collapse of both simulation and experimental data provides further confirmation that graft polymer networks could be described as networks of linear chains with the E


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supporting strands, which results in a decrease of the structural shear modulus of networks by a factor φ (see Table 2). This dilution effect works well for networks with comblike strands of low grafting density. In this so-called comb regime (Figure 2), the network shear modulus is φ times smaller than corresponding shear modulus of the networks of linear chains of the same degree of polymerization between cross-links. However, in the case of bottlebrush strands, there is an additional contribution to G due to stiffening and extensibility of the network strands. Interplay between the dilution and stiffening effects allows breaking down the “golden rule” G ∝ λmax−2 ≈ β of linear chain networks (Figure 6) and design Figure 4. Universal relationship between mechanical properties of networks of graft polymers (structural modulus G and strand extension ratio β) and structural parameters of the network strands, described by nx and fraction of the backbone monomers φ, illustrated by linear scaling between αG and βφ/nx. Symbol notations are summarized in Table 3. Logarithmic scales.

effective Kuhn length in which the concentration of the stress supporting strands should be corrected by the fraction of the backbone monomers φ. Note that the deviation of the simulation data from the universal plot is observed in the range of the network stresses where bonds of graft polymer backbones and cross-linking bonds begin to deform.

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CONCLUSIONS Using a combination of the analytical calculations and computer simulations, we show that networks of graft polymers can be described as networks of linear chains made of macromolecules with effective Kuhn length. The latter is controlled by the degree of polymerization of the backbone between side chains ng and number of monomers in the side chains nsc (see Table 1). This representation of the network strands follows from the analysis of the comb and bottlebrush macromolecules in a melt.13 Partitioning of monomers between side chains and backbones causes effective dilution of the stress

Figure 6. Breaking the “golden rule” of the materials design, G ∝ λmax−2. The values of the λmax for this plot are calculated as λmax = β−1/2. The “golden rule” is shown as the dashed line with a slope −2 in logarithmic scales. Data for linear chain networks are shown by brown symbols: nx ≈ 4 (rhombs), nx ≈ 5 (triangles), nx ≈ 6 (inverted triangles), and nx ≈ 9 (squares). Filled green triangles show correlations between G and λmax for data sets from Figure 3a for which DP of side chains nsc = 2, 4, 8, 16, and 32, ng = 8, and nx ≈ 16. Filled blue symbols represent correlations for data shown in Figure 3b and correspond to graft polymer networks with DP of backbone between cross-links nx = 8, 11, 16, 20, and 34, ng = 2, and nsc = 8. Open triangles of different colors show correlations for data in Figure 3c.

Figure 5. Comparison between experimental and simulation data for nonlinear network deformations. (a) Simulation data for the reduced deformation dependent network shear modulus G(I1)/G as a function of parameter βI1/3 for networks of graft polymers with nx ≈ 16. Symbol notations for simulation data are summarized in Table 3. (b) Experimental data for the reduced deformation dependent network shear modulus (G(I1)/G vs βI1/3) of the bottlebrush networks with ng = 1, side chain length nsc = 14, and different number of monomers between cross-links nx = 50 (purple squares), 67 (blue triangles), 100 (light blue inverted triangles), 200 (orange diamonds), and 400 (red hexagons) (data are adapted from ref 30). The dashed lines are given by eq 17. F


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materials with simultaneously increasing stiffness and extensibility. Indeed, one can observe is a new peculiar scaling relation, G ∝ λmax4 ≈ β−2, demonstrating an increase of the structural modulus with elongation at break λmax ≈ β−1/2 (open symbols). This scaling reflects a crossover from the bottlebrush regime to the comb regime through variation of the grafting density ng−1 of the side chains keeping the degree of polymerization between cross-links nx almost constant. Thus, by changing the molecular architecture of the network strands, it is possible to make more rigid networks that at the same time are hyperelastic. This peculiar behavior of networks of graft polymers observed in our simulations is in agreement with recently published experimental data on PDMS networks.30

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.7b02555. Simulation details, model of hybrid networks (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (A.V.D.). ORCID

Heyi Liang: 0000-0002-8308-3547 Sergei S. Sheiko: 0000-0003-3672-1611 Andrey V. Dobrynin: 0000-0002-6484-7409 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are grateful to the National Science Foundation for the financial support under Grants DMR-1407645, DMR1624569, and DMR-1436201.

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REFERENCES

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Supersoft and Hyperelastic Polymer Networks with Brushlike Strands

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