Dynamics of Bottlebrush Networks - Macromolecules (ACS Publications)

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Dynamics of Bottlebrush Networks Zhen Cao,† William F. M. Daniel,‡ Mohammad Vatankhah-Varnosfaderani,‡ 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 at Chapel Hill, Chapel Hill, North Carolina 27599-3220, United States

‡

S Supporting Information *

ABSTRACT: The deformation dynamics of bottlebrush networks in a melt state is studied using a combination of theoretical, computational, and experimental techniques. Three main molecular relaxation processes are identified in these systems: (i) relaxation of the side chains, (ii) relaxation of the bottlebrush backbones on length scales shorter than the bottlebrush Kuhn length (bK), and (iii) relaxation of the bottlebrush network strands between cross-links. The relaxation of side chains having a degree of polymerization (DP), nsc, dominates the network dynamics on the time scales τ0 < t ≤ τsc, where τ0 and τsc ≈ τ0(nsc + 1)2 are the characteristic relaxation times of monomeric units and side chains, respectively. In this time interval, the shear modulus at small deformations decays with −1/2 time as GBB . On time scales t > τsc, bottlebrush elastomers behave as networks of filaments with a shear modulus GBB 0 (t) ∼ t 0 (t) −1/4 −1/2 ∼ (nsc + 1) t . Finally, the response of the bottlebrush networks becomes time independent at times scales longer than the Rouse time of the bottlebrush network strands, τBB ≈ τ0N2(nsc + 1)3/2, where N is DP of the bottlebrush backbone between cross-links. In this time interval, the network shear modulus depends on the network molecular parameters as GBB 0 (t) ∼ (nsc + 1)−1N−1. Analysis of the simulation data shows that the stress evolution in the bottlebrush networks during constant strain-rate deformation can be described by a universal function. The developed scaling model is consistent with the dynamic response of a series of poly(dimethylsiloxane) bottlebrush networks (nsc = 14 and N = 50, 70, 100, 200) measured experimentally.

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INTRODUCTION Bottlebrush macromolecules1−12 can be viewed as filaments where polymer side chains play a dual role of diluent and rigidity mediator of the bottlebrush backbone.1,10,13−20 Their unique combination of large diameter and high flexibility results in disentanglement of bottlebrush melts manifested in a corresponding decrease of the entanglement plateau modulus.1,11,21−24 This intrinsic disentanglement allows for the synthesis of the supersoft and superelastic solvent-free elastomers with shear modulus down to ∼100 Pa and tensile strains at break up to 800% in a solvent free state.1 Along with the degree of polymerization (DP) of the backbone between cross-links (N), mechanical properties of disentangled bottlebrush networks are determined by two independently controlled parameters: DP of side chains (nsc) and DP of the spacer between the neighboring side chains (ng). The ratio of ng/nsc controls the molar fraction of the backbone in the network (dilution of the backbone) and the rigidity (Kuhn length) of bottlebrush filaments with diameter Dsc. For bottlebrush networks with ng = 1, when the strands between cross-links are longer than bottlebrush Kuhn length bK ≈ Dsc ∼ nsc + 1 ,25 the bottlebrush network can be viewed as networks of flexible filaments.26−28 In this so-called backbone dilution regime, the equilibrium shear modulus of a bottlebrush network at small deformations decreases with increasing the side chain and backbone DPs of a network strand −1 25 as GBB 0 ∼ [N(nsc + 1)] . This scaling behavior continues until © XXXX American Chemical Society

the contour length of a bottlebrush strand becomes smaller than its Kuhn length bK. For systems with shorter strands, the network’s elasticity is controlled by the bending modes of the bottlebrush macromolecules on the length scales smaller than bK. The properties of such networks are similar to those of semiflexible filaments29−32 and demonstrate a weaker dependence of the shear modulus on the side chain DP as GBB 0 ∼ nsc−0.5.25 Therefore, by changing nsc (at the same N), we can access different network deformation regimes in chemically identical polymeric networks. Although the description of equilibrium properties of bottlebrush networks is relatively straightforward, the behavior at short and intermediate length and time scales remains unclear. In this regard, it is important to realize that bottlebrush macromolecules represent a hierarchical multiscale system capable of demonstrating both molecular and particulate (filament-like) features (Figure 1). On length scales of a single side chain, the system can be viewed as a melt of linear chains,33 while on the longer length scales, bottlebrushes behave as thick filaments. However, unlike the biological filaments,29 bottlebrush macromolecules in a melt are significantly more flexible with the effective Kuhn length being proportional to their diameter bK ≈ Dsc. While our understanding of this dual nature Received: June 24, 2016 Revised: September 23, 2016

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DOI: 10.1021/acs.macromol.6b01358 Macromolecules XXXX, XXX, XXX−XXX

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a melt state. We consider a monodisperse polymeric networks with DP between cross-links, N, shorter than the corresponding DP of entanglement strand,33,40 Ne (N < Ne). In the framework of the scaling model of polymeric networks, on the time scales t smaller than the strands’ Rouse time, τR ≈ τ0N2 (τ0 is characteristic monomer relaxation time), the dynamics of the network strands is considered to be identical to those in a melt of polymer chains with the same DP.33,40 At such time scales, the chain’s dynamics is not influenced by cross-links and network relaxation occurs through relaxation of short sections of the network strands with the number of monomers, g(t) < N, whose Rouse time, τ0g(t)2, is on the order of time t (see Figure 2a). The number density of such strands in a melt with monomer number density ρ is equal to ρ/g(t) ≈ ρ(τ0/t)1/2. Each strand contributes on the order of thermal energy kBT (kB is the Boltzmann constant and T is the absolute temperature) to the network modulus. Therefore, modulus at small deformations decays with time as

Figure 1. Simulation snapshots of hierarchical structure of bottlebrush macromolecules illustrating different length scales: bottlebrush network (a), single bottlebrush macromolecule with bottlebrush backbone shown in green and side chains colored in blue (b), and bottlebrush chain segment (c).

of bottlebrushes is beginning to emerge,1,11,21,22,34 its role in network dynamics and viscoelastic properties, which is vital for many applications including implants, membranes, and actuators, is still unclear. In this paper, we use a combination of the scaling analysis, computer simulations, and experimental studies to elucidate factors controlling the dynamic response of bottlebrush networks on different time and length scales. Particularly, we are interested in the crossover between the chain-like and filament-like regimes. We identify two Rouse-like relaxation regimes33,35 of bottlebrush network dynamics. In the first regime, at time scales shorter than the Rouse time of the side chains, the relaxation of the bottlebrush networks is similar to the melt of side chains. At longer times confined between the Rouse times of the side chains and the bottlebrush strands between cross-links, the relaxation behavior of the bottlebrush network displays a second Rouse-like regime similar to relaxation of filaments with thicknesses determined by the bottlebrush diameter and with a transition zone in between. Finally, at time intervals longer than the Rouse time of bottlebrush strands between cross-links the network displays a purely elastic response. The rest of the paper is organized as follows: we first present a scaling analysis of the bottlebrush network dynamics, the scaling model predictions are compared with results of the molecular dynamics simulations of uniaxial deformation of bottlebrush networks at different deformation rates, and finally we present results of the experimental studies of the poly(dimethylsiloxane) (PDMS) bottlebrush networks to confirm the network relaxation picture observed in molecular dynamics simulations and predicted by the scaling approach.

⎛ τ ⎞1/2 G0L(t ) ≈ ρkBT ⎜ 0 ⎟ , ⎝t ⎠

for τ0 < t ≤ τR

(1)

However, at time scales longer than the Rouse time of polymeric strands, t > τR ≈ τ0N2, the network response is pure elastic with a constant shear modulus26 G0L(t ) ≈ ρkBT /N ,

for τR < t

(2)

In Figure 3, we summarized different relaxation regimes of networks of linear chains in accordance with eqs 1 and 2. Bottlebrush Networks. As mentioned in the Introduction, molecular bottlebrushes can be viewed as filaments with a brush interior structure (see Figures 2b,c).25 Therefore, assuming sufficient separation of the time scales and sufficiently large molecular dimensions, we consider three main relaxation processes in bottlebrush networks: (i) relaxation of the side chains (see Figure 2b), (ii) relaxation of the bottlebrush backbone on length scales shorter than the bottlebrush Kuhn length, bK, and (iii) relaxation of an entire bottlebrush filament between cross-links (see Figure 2c). Below we develop a scaling model incorporating these relaxation processes to describe dynamics of bottlebrush networks on different time and length scales. We will illustrate this approach on bottlebrush networks with Nb ≫ bK, where b is the bond length assumed to be the same for backbone and side chains. The side chains undergo the fastest relaxation. Here we consider a case when each bottlebrush backbone monomer has one side chain attached to it (ng = 1). Assuming that the side

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RESULTS AND DISCUSSION Scaling Analysis of Network’s Dynamics. Linear Chain Networks. We begin our discussion with a brief overview of the Rouse dynamics of linear chain networks33,36−39 cross-linked in

Figure 2. Schematic representation of different Rouse modes of network strand relaxation: (a) segments of linear chain made of blobs with number of monomers g(t), (b) side chain of a bottlebrush macromolecule considered as being made of blobs with gsc(t) monomers, and (c) blob structure of the bottlebrush backbone with each blob having gBB(t) backbone monomers. Cross-links are shown by green squares. B


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ρkBT ⎛ τ0 ⎞1/2 K ⎜ ⎟ ≈ , ξ 2b gBB(t )2 (nsc + 1)1/4 ⎝ t ⎠

kBT

for τsc < t ≤ τK

(5)

where we take into account the space-filling condition ρb ≈ 1. 5/4 Note that at t ≈ τsc the modulus GBB 0 (τsc) ≈ ρkBT/(nsc + 1) given by eq 5 is lower than the modulus GBB (τ ) ≈ ρk T/(n 0 sc B sc + 1) obtained from relaxation of the side chains in eq 3, which results in the two parallel lines in Figure 3 different by a factor of (nsc + 1)−1/4. This disparity indicates that the simple scaling consideration of the two dominant relaxation processes neglects coupling between relaxation modes of the backbone and side chains. This coupling should dominate bottlebrush relaxation at time scales t ≈ τsc. However, for the currently available bottlebrush systems nsc ∼ 10, the ratio between moduli given by eqs 3 and 5 is a weak function of the side chain degree of polymerization, (nsc + 1)1/4 ∼ 1, of the order of unity. Note that polydispersity of the side chains may impose additional difficulty in resolving crossover between the side-chain and bottlebrush Rouse-like relaxation regimes. At time scales τK < t, we can consider bottlebrushes as linear chains made of Kuhn monomers with effective monomer relaxation time, τK. By applying the Rouse model to a melt of chains of Kuhn monomers eq 1, we obtain 3

Figure 3. Time dependence of shear modulus at small deformations G0(t) of the networks of linear chains (dashed line) and networks of cross-linked bottlebrush macromolecules (solid line). The networks have monomer number density ρ, degrees of polymerization of the side chains nsc, and backbones between cross-links N, respectively. For networks of linear chains, nsc = 0. Logarithmic scales.

chains relax independently and are unaware of their connectivity to the backbone, we can write the following expression for the Rouse-like relaxation of network modulus, which is identical to the relaxation of linear chain melts (eq 1) ⎛ τ ⎞1/2 G0BB(t ) ≈ ρkBT ⎜ 0 ⎟ , ⎝t ⎠

for τ0 < t ≤ τsc

ρkBT ⎛ τ0 ⎞1/2 ⎛ τ ⎞1/2 ⎜ ⎟ G0BB(t ) ≈ ρK kBT ⎜ K ⎟ ≈ , ⎝ t ⎠ (nsc + 1)1/4 ⎝ t ⎠ for τK < t ≤ τBB

(3)

where ρK ≈ ρ/(nsc + 1)3/2 is the number density of the effective Kuhn monomers each containing (Dsc/b)3 ≈ (nsc + 1)3/2 chemical monomeric units. Correspondingly, the characteristic Rouse relaxation time of the bottlebrush network strand can be written as

where τsc ≈ τ0(nsc + 1)2 is the Rouse time of the side chain, and in the expression (nsc + 1), factor 1 accounts for the backbone monomer to which the side chain is attached. Note that at time scales t ≈ τsc the network modulus decreases by a factor of (nsc + 1)−1 (see Figure 3). At longer time scales, τsc ≤ t ≤ τK, where τK is the relaxation time of the bottlebrush Kuhn length, the bottlebrush can be modeled as a semiflexible polymer29,30 having effective bending D b constant K ≈ bsc ≈ bK ≈ nsc + 1 ,25 where bK ≈ b nsc + 1 is bottlebrush Kuhn length. On this time scale, relaxation of the shear modulus is dominated by transversal bending of bottlebrush sections containing gBB(t) ≈ (Kt/(nsc + 1)τ0)1/4 backbone monomers. This expression for gBB(t) follows from the corresponding expression for the mode relaxation time for the semiflexible polymers29,30 in which one has to substitute a monomeric friction coefficient by the friction coefficient of the whole side chain attached to a backbone monomer. This effectively increases friction coefficient nsc + 1 times. At t ≈ τK, we observe the longest bending mode of a bottlebrush filament involving gBB(τK ) ≈ nsc + 1 backbone monomers. By equating the two last equations at t ≈ τK, we obtain the relaxation time of this longest bending mode as τK ≈ τ0(nsc + 1)5/2

(6)

⎛ Nb ⎞2 τBB ≈ τK ⎜ ⎟ ≈ τ0(nsc + 1)3/2 N 2 ⎝ bK ⎠

(7)

It is interesting to point out that the expression for the modulus eq 6 in this regime is identical to eq 5. Finally, for τBB < t the response of the bottlebrush networks is time independent with modulus G0BB(t ) ≈ ρkBT /N (nsc + 1),

for τBB < t

(8)

Note that eqs 3−8 describing dynamics of the bottlebrush networks are reduced to eqs 1 and 2 describing the timedependent modulus of linear chain networks when nsc = 0. It is also instructive to comment on how these relations will transform with varying degree of polymerization N between cross-links. The crossover to a regime where bottlebrush networks behave as networks of semiflexible chains takes place at bN ≈ bK ≈ b (nsc + 1) . In this regime, the Rouse time of the bottlebrush network strand between cross-links is equal to τBB ≈ τ0N4(nsc + 1)1/2 and network modulus saturates at G0BB ≈ ρkBT /N 2 (nsc + 1) reproducing results obtained in ref 25. In the next sections we will compare scaling model predictions with results of computer simulations and experiments. Simulations of Dynamics of Bottlebrush Networks. To verify the scaling predictions for the existence of different network relaxation regimes, we have performed molecular

(4)

Using the expression for the modulus of the semiflexible networks with distance between filaments ξ being on the order of bottlebrush diameter, ξ ≈ Dsc ≈ b nsc + 1 , we can write the following expression for the time-dependent modulus of the bottlebrush networks as C


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Figure 4. Computer simulations of the strain rate dependence of stress−strain curves for bottlebrush networks with degrees of polymerizations of the side chains: nsc = 0 (linear chains) (a), nsc = 2 (b), nsc = 5 (c), and nsc = 20 (d) for different strain rates 10−3 τLJ−1 (filled black squares), 2.5 × 10−4 τLJ−1 (dot center red circles), 10−4 τLJ−1 (filled red circles), 2.5 × 10−5 τLJ−1 (dot center blue pentagons), 10−5 τLJ−1 (filled blue pentagons), 2.5 × 10−6 τLJ−1 (dot center purple rhombs), 10−6 τLJ−1 (filled purple rhombs), and 10−7 τLJ−1 (filled wine triangles). The FENE potential spring constant is equal to 30 kBT/σ2, and the temperature is 1.0 in reduced LJ units.

dynamics simulations41,42 of bottlebrush networks undergoing uniaxial deformation43,44 at a constant strain rate, ε̇. Bottlebrush macromolecules are modeled as bead−spring chains composed of beads with diameter σ interacting through truncated shifted Lennard-Jones (LJ) potential.45 The connectivity of monomers into bottlebrush macromolecules is maintained by the combination of the FENE and truncated shifted LJ potentials. In our simulations, each monomer belonging to the bottlebrush backbone has a side chain attached to it. The number of monomers belonging to the side chains, nsc, is varied between 0 and 20. The bottlebrush macromolecules are randomly crosslinked by the same bonds as the ones connecting monomers into bottlebrush macromolecules (see ref 25 and Supporting Information for simulation details). The relaxed bottlebrush networks with monomer number density 0.8 σ−3 are uniaxially deformed at a constant volume and a strain rate ε̇ varying between 10−7 and 10−3 τLJ−1, where τLJ is the standard LennardJones time. In the simulated bottlebrush networks, the numberaverage degree of polymerization between cross-links ⟨N⟩ is varied between 17.2 and 19.7 for a FENE potential spring constant equal to 30 kBT/σ2, and 15.7 to 16.7 for a FENE potential spring constant equal to 500 kBT/σ2 (see Supporting Information for details). Figure 4 summarizes our results for true stress dependence on the strain at different strain rates varying between 10−7 and 10−3 τLJ−1. According to the Boltzmann superposition principle,46,47 stress evolution in polymeric networks undergoing small uniaxial elongation at a constant strain rate ε̇ is σxx(t ) =

∫0

t

E0(t − t ′) dε(t ′) = ε ̇

∫0

Using the stress relaxation shear modulus for linear chains given by eqs 1 and 2, integration of eq 9 results in σxx(t ) ≈ 2E0Lετ̇ R 1/2t 1/2 ,

(10)

where = ≅ 3ρkBT/N is the Young’s modulus of network of linear chains at small deformations. For networks undergoing uniaxial deformation at a constant strain rate, we can substitute t = ε/ε̇ into eq 10 to express σxx(t) as a function of strain ε EL0

3GL0

̇ ε1/2 , σxx(ε) ≈ 2E0LτR 1/2ε1/2

for t < τR

(11)

In the opposite limit, t > τR σxx(ε) ≈ 2E0Lετ̇ R + E0Lε(̇ t − τR ) ≈ E0Lε ,

for t ≥ τR (12)

Equation 11 suggests nonlinear behavior of the network shear modulus at high strain rates, which is verified by stress− strain curves of both linear chain (see Figure 4a) and bottlebrush networks (see Figure 4b−d). (The log−log version of this plot is shown in the Supporting Information to highlight curves overlap.) This indicates that such deformation rates probe chain dynamics at the time scales shorter than the Rouse time. Overlap of the stress−strain curves obtained at slow strain rates suggests that at such deformations the system response is both purely elastic and rate (time) independent in accordance with eq 12. Note that the stress−strain behaviors of the linear chain and bottlebrush systems are qualitatively similar, with the only difference being the crossover values of the strain rates and values of the equilibrium (rate independent) Young’s modulus. This similarity is consistent with Figure 3, which displays the relaxation moduli of linear and bottlebrush networks as shifted parallel lines both having a slope of −0.5. Since time dependent stress σxx(t) is an integral characteristic of the modulus (see eq 9), this shift is further smoothed by the integration.

t

E0(Δt ) dΔt

for t < τR

(9)

where E0(t) is a time-dependent Young’s modulus measured at small deformations (ε → 0). Note that E0(t) = 3G0(t) for incompressible networks with Poisson ratio equal to 0.5.47

D


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Figure 5. Time dependence of the normalized true stress of bottlebrush networks with degree of polymerization of the side chains nsc = 0 (linear chains) (a), nsc = 2 (b), nsc = 5 (c), and nsc = 20 (d) for different strain rates 10−3 τLJ−1 (filled black squares), 10−4 τLJ−1 (filled red circles), 10−5 τLJ−1 (filled blue pentagons), 10−6 τLJ−1 (filled purple rhombs), and 10−7 τLJ−1 (filled wine triangles). The FENE potential spring constant is equal to 500 kBT/σ2, and the temperature is 3.0 in reduced LJ units. Solid line corresponds to the best fit to eq 14 with τ̃R of the network as a fitting parameter.

⎧ 2ε1/2 ̇ τ01/2ε1/2 , for τ0 < t ≤ τsc ⎪ ⎪ ⎪ ̇ τ01/2 1/2 2ε1/2 σxx(ε) ≈ Em⎨ 2ετ̇ 01/2τsc1/2 + (ε − (ετ̇ sc)1/2 ), (nsc + 1)1/4 ⎪ ⎪ ⎪ ε(N (n + 1))−1 , for τ < t ⎩ sc BB

(13)

time scales τsc < t if we replace τ̃R with τ̃BB and substitute the degree of polymerization N between cross-links of the bottlebrush backbone by its average value ⟨N⟩ in the corresponding expressions for the Young’s modulus and backbone Rouse time as EBB 0 ≈ Em/(⟨N⟩(nsc + 1)) and τ̃BB ≈ τ0(nsc + 1)3/2⟨N⟩2, respectively. Below we will apply eq 14 to analyze data shown in Figure 4 for linear chains and those for bottlebrush networks. In Figure 5, we replot the stress−strain data from Figure 4 using universal coordinates in accordance with eq 14. For this plot, we have used data sets with 0.05 < ε < 0.8 and excluded data with ε < 0.05 that show large stress fluctuations. The solid lines in Figures 5 correspond to the best fit of the small deformation data with eq 14, using the Rouse time as a fitting parameter and the modulus obtained from equilibrium network deformation simulations.25 There is a very good agreement between simulation data and eq 14. For our longest side chains nsc = 20, the simulation data first follow the universal line then begin to deviate due to nonlinear deformation of the bottlebrush strands between cross-links. In these networks, the Kuhn length of the bottlebrush strands is on the order of the distance between cross-links. Therefore, small deformations are sufficient to reach a nonlinear network deformation regime, where the network deformation is controlled by bottlebrush bending modes.25 To further illustrate the robustness of the fit,

where we introduce Em = 3ρkBT. Note that for monodisperse bottlebrush networks with Nb ≫ bK the value of the Young’s modulus at small deformations is EBB 0 ≈ Em/(N(nsc + 1)). However, real networks are polydisperse which results in a distribution of the Rouse times for individual network strands between cross-links such that crossover from the melt-like relaxation of strands to network relaxation regime depends on distribution of the chain’s DP, N, value. In the case of the exponential distribution of the chain degree of polymerization, the normalized probability to find a strand with a degree of polymerization N can be written as w(N) = ⟨N⟩−1e−N/⟨N⟩, where ⟨N⟩ is the number-average degree of polymerization between cross-links. This allows for analytical calculation of the corresponding averaged time-dependent modulus with distribution function w(N) which results in the following expression (see Supporting Information for derivation details) σxx(t ) ≈ E0Lετ̇ R̃ (t /τR̃ + 2(1 − exp( − t /τR̃ ))

for τsc < t ≤ τBB

(14)

Thus, the stress evolution in a polydisperse network of linear chains undergoing uniaxial constant rate deformations depends on the Rouse relaxation time of an average strand between cross-links with relaxation time τ̃R ≈ τ0⟨N⟩2 and the Young’s modulus at small deformations EL0 ≈ Em/⟨N⟩. Note that eq 14 can describe stress evolution in the bottlebrush networks on the E


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filaments. Deviation from the linear scaling is an indication of crossover into the regime when bottlebrush strands between cross-links behave as semiflexible filaments, i.e., ⟨N ⟩ < (nsc + 1) . For such networks, the longest relaxation time scales with the system parameters as τ̃BB ≈ τ0⟨N⟩4(nsc + 1)1/2. This behavior is in agreement with our simulations of the static properties of the bottlebrush network, which demonstrate two different dependencies of the network shear modulus.25 Experimental Studies of Dynamics of Bottlebrush Networks. The trends seen in Figures 3 and 6 are corroborated through a series of mechanical tests of covalently cross-linked poly(dimethylsiloxane) (PDMS) bottlebrush elastomers (see the Supporting Information for details on synthesis, characterization and mechanical measurements of bottlebrush networks). The first test was a series of uniaxial extensions at different strain rates of a PDMS bottlebrush elastomer (nsc = 14, ng = 1, N = 200, Poisson’s ratio ≈0.5), where nsc is chemical DP of the side chain, ng is number of chemical backbone monomers per side chain, and N is chemical DP of the backbone between cross-links as determined through stoichiometry.1 Figure 8 displays a normalized stress σxx/ε̇

we use the values of the Rouse times obtained from fitting simulation data with eq 14 to collapse all our simulation curves in linear network deformation regime with 0.05 < ε < 0.3 into one universal plot as shown in Figure 6.

Figure 6. A master curve for bottlebrush networks undergoing uniaxial deformation at constant strain rates ε̇. E0 is the Young’s modulus at small deformations, and τ̃R is the Rouse time. Simulations of the bottlebrush networks with different degrees of polymerizations of the side chains are shown by changing symbol colors: nsc = 0 (black), nsc = 2 (red), nsc = 5 (blue), nsc = 10 (purple), and nsc = 20 (wine); simulations with different FENE potential spring constants are shown by using different symbol shapes: 30 kBT/σ2 (squares) and 500 kBT/σ2 (circles); simulations at temperature 1.0 (filled symbols) and 3.0 (open symbols) in reduced LJ units. Dashed line is given by the equation y = x + 2(1 − exp(−√x)).

The corresponding values of the Rouse time obtained from the fit of the deformation of different bottlebrush networks are shown in Figure 7 as a function of the parameter τ0(nsc + 1)3/2⟨N⟩2. (The value τ0 for this plot was estimated from monomer mean-square displacement.) The linear scaling of the Rouse time with this parameter confirms the scaling relation given by eq 7 for the Rouse model of bottlebrush strand. Such bottlebrush networks can be considered as networks of flexible

Figure 8. Experimentally measured time dependence of the normalized tensile stress for a PDMS bottlebrush network (N = 200, nsc = 14, ng = 1) subjected to uniaxial extension at constant strain rates: 0.0026 s−1 (black squares), 0.0127 s−1 (red circles), 0.0250 s−1 (blue triangles), 0.1010 s−1 (pink inverted triangles), 0.2392 s−1 (yellow rhombs), 0.2907 s−1 (wine left triangles), 0.6773 s−1 (cyan right triangles), 1.1069 s−1 (violet hexagons), 0.0127 s−1 (orange stars), 0.0127 s−1 (navy pentagons). Solid line corresponds to the best fit to eq 14 with EBB 0 = 6.61 kPa and τ̃BB = 0.046 s. Inset: schematic representation of bottlebrush network illustrating primary structural parameter (N, nsc, and ng) and cross-linking mechanism (black dots represent cross-linking separate backbones through mutual incorporation of short difunctional linear cross-linker).

versus ε/ε̇ master curve, which was constructed by combining the multiple sets of the stress/strain curves recorded at different strain rates. On time scales shorter than the Rouse time of the network, τ̃BB (achieved at high strain rates), the normalized stress displays a characteristic slope of 0.5 with respect to time in accordance with eq 14. At lower rates, the sample is deformed on time scale longer than τ̃BB and the time dependence of normalized stress increases to a power of 1. Both characteristic slopes are consistent with the computer simulation results shown in Figures 5 and 6. By fitting the experimental data with eq 14, we obtained a relaxation time of τ̃BB = 0.046 s and a bottlebrush network Young’s modulus of EBB 0 = 6.6 ± 0.1 kPa. The Rouse time extracted from Figure 8 is conceptually the same as the Rouse time given by eq 7; however, eq 14 accounts for the inherent polydispersity of real networks using number-average values of N.

Figure 7. Dependence of the Rouse time of the bottlebrush network strands, τ̃R, obtained from fitting simulation data for time-dependent normalized true stress (see Figure 5) as a function of the estimated Rouse relaxation time of the bottlebrush network τ0⟨N⟩2(nsc + 1)3/2 (eq 7) for systems with different FENE potential spring constants ksp and temperatures (reduced LJ units): 30 kBT/σ2 and 1.0 (filled red squares); 30 kBT/σ2 and 3.0 (open red squares); 500 kBT/σ2 and 1.0 (filled blue circles); 500 kBT/σ2, and 3.0 (open blue circles). F


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Zone B (linear chains) contains the last two sections of dynamic behavior and shows internal relaxation of highly correlated side chains connected through the backbone followed by a Rouse-like relaxation of linear chains’ segments (marked by the orange line in regime IV in Figure 9), followed by the glass transition zone universal to neat polymer materials (regime V). A shoulder-like feature can be seen in regime IV where the frequency dependence of modulus becomes weaker than the Rouse one, G0′ ∼ ω0.5, and the storage modulus dominates over the loss modulus (Supporting Information Figure S3). Because the side chain molar mass (1000 g/mol) is about 10 times below the entanglement molar mass, this feature cannot be attributed to the entanglement of side chains. Similar behavior has been observed in previous studies of bottlebrush polymers11 and was ascribed to orientation dynamics of bottlebrush molecules. Note that this behavior could also be explained by coupling in relaxation dynamics of neighboring side chains through bottlebrush backbone. A closer look at regime III (transition between filament-like and linear chain behavior) displays a visible increase in the slope of G0′ versus ω with respect to the Rouse-like behavior (slope 0.5) of both bottlebrush filaments (regime II) and linear side chains (regime IV). This behavior is consistent with the scaling model predictions shown in Figure 3; namely, a side chain Rouse relaxation is followed by a drop in modulus and then Rouse relaxation of filamentous bottlebrush segments. However, due to the weak dependence of the drop in modulus (nsc + 1)−1/4 (see Figure 3) along with the relatively small DP (nsc = 14) and the inherent dispersity (Đ = 1.01) of the side chains in the currently available bottlebrush elastomers, we are not able to resolve a clean transition between the two Rouse regimes. It would be desirable to synthesize a new set of bottlebrushes with significantly longer side chains (nsc in the hundreds), but such materials would also require a simultaneous and nontrivial increase in backbone length, which represents a substantial synthetic challenge for future studies. However, unlike the ambiguous behavior in regime III, the rubbery plateaus in regime I demonstrate a clear effect of the cross-link density on the plateau modulus. The values of the shear modulus GBB 0 for the bottlebrush networks obtained from the plateau shear moduli are consistent with those obtained BB from tensile tests EBB 0 = 3G0 (see Table 1). In our scaling analysis of the plateau modulus of the bottlebrush networks, we consider two pure asymptotic regimes for shear modulus dependence on the bottlebrush molecular parameters. However, our bottlebrush systems are in a crossover; therefore, to take this fact into account, we have to use a general expression for the shear modulus at small deformations (ε → 0)25,31

In order to confirm the power law relaxations and transition regimes predicted in Figure 3, Figure 9 displays master curves

Figure 9. Master curves of the storage modulus of PDMS bottlebrush networks (nsc = 14, ng = 1) with four different degrees of polymerizations between cross-links N = 200 (green), 100 (red), 70 (blue), and 50 (black). The curves display five relaxation regimes: I, the elastic network plateau; II, the Rouse-like relaxation of bottlebrush segments and transition to elastic plateau. The Rouse-like zones are marked by the series of light blue lines with slopes of 0.5: III, the transition between filament-like (regime I and II) and linear chain (regimes IV and V) behaviors; IV, Rouse-like regime of linear side chains represented by the orange line (the indistinct plateau is a result of side chain relaxations being coupled through the polymer backbone); and V, the glass transition regime. Inset: dependence of the structural shear modulus GBB of the bottlebrush networks on the DP between cross-links N.

of shear storage modulus, G0′ , versus the radial frequency, ω, which were measured for a series of covalently cross-linked PDMS bottlebrush elastomers with nsc = 14 and N values of 200, 100, 70, and 50 estimated from stoichiometry (see Supporting Information for details). The modulus relaxation spectrum can be broken down into two distinct zones. Zone A (bottlebrush filaments) corresponds to the relaxation of filament-like strands and consists of two regimes (regime I and II). Regime I represents the time independent rubbery plateau due to chemical cross-links resulting in permanent stress storage. Regime II corresponds to the Rouse-like relaxation of the filament-like bottlebrush network strands (marked by light blue lines in Figure 9). Because of the relatively small length of the strands, this regime is truncated and only allows clear observation of the characteristic Rouse slope of 0.5 for the network with the longest strands (N = 200). The bottlebrush polymers then enter a crossover regime (regime III), where the bottlebrush relaxation mechanisms transition from filament-like to linear chain Rouse relaxations.

Table 1. Structural Parameters and Mechanical Properties of PDMS Bottlebrush Networks Na

nscb

c GBB 0 [kPa]

200 100 70 50

14 14 14 14

2.3 7.3 15.5 20.1

d EBB 0 [kPa]

9.9 24.3 45.0 66.0

± ± ± ±

βf

GBB [kPa]e

0.3 0.3 0.6 0.9

2.8 6.2 10 13.5

± ± ± ±

0.1 0.2 0.4 0.5

0.11 0.17 0.23 0.28

± ± ± ±

τ̃BB [s]g 0.01 0.01 0.01 0.02

4.6 5.2 1.6 8.1

× × × ×

10−2 10−3 10−3 10−4

a

Targeted DP of the backbone between cross-links calculated from the mole fraction of the difunctional cross-linker (see Supporting Information). Degree of polymerization of the side chains. cShear modulus of the bottlebrush samples at small deformations as obtained from the master curves. d Young’s modulus obtained from uniaxial extension stress−strain curves at small strain (ε → 0). eGBB structural shear modulus determined by fitting the tensile stress−strain curves. fβ extension ratio of bottlebrush backbone in bottlebrush elastomers determined by fitting the tensile stress−strain curves. gRouse times of the bottlebrush networks. b

G


Article

Macromolecules G0BB = GBB(1 + 2(1 − β)−2 )/3

order of t ≈ τ0N4(nsc + 1)1/2 and network plateau modulus −1/2 −2 GBB N . 0 (t) ∼ (nsc + 1) These different modes of the bottlebrush network relaxation within a hierarchy of the time scales could have vital implications for the design of supersoft and superelastic materials for applications such as adhesive surfaces, implants, and actuators.

(15)

where GBB is the structural modulus of the network which is proportional to concentration of the stress supporting strands, and β = ⟨Rin2⟩/Rmax2 is the extension ratio of the network strands prior to deformation, ⟨Rin2⟩, and the square of the endto-end distance of the fully extended bottlebrush backbone strand, Rmax = Nb. For networks made of flexible strands for which ⟨Rin2⟩ ≈ bKbN and bN ≫ bK the values of the parameter β = bK/bN ≪ 1 the structural modulus GBB and shear modulus G0BB at small deformations are the same, and both are determined by the concentration of the stress supporting strands (see for discussion in refs 25 and 31). However, for the finite values of the parameter β the shear modulus GBB 0 at small deformations is a function of the strand extension ratio.25,31 This additional contribution should be eliminated (see eq 15) to obtain system structural modulus GBB. This requires knowledge of the value of the parameter β. These values for studied networks were obtained from fitting equilibrium stress/ strain curves (see Supporting Information Figure S3). The inset in Figure 9 shows the structural network modulus GBB as a function of N. The data points for this plot were obtained from the corresponding values of the network shear modulus GBB 0 at small deformations using eq 15. The observed GBB ∼ N−1 trend is consistent with the scaling prediction for structural modulus GBB ∼ (N(1 + nsc))−1.25 The estimation of the Rouse time from the G0′ (ω) curves is ambiguous since we do not observe a pure filament relaxation regime (filament Rouse regime) with a characteristic scaling dependence G′0(ω) ∼ ω0.5. Therefore, to estimate the Rouse time of the bottlebrush networks, we use a point where the ratio of the storage modulus G0′ (ωBB) to its plateau value is the same as for sample with N = 200, for which we have independently obtained the Rouse time and is approximately equal to 1.45. This gives the estimated Rouse times τ̃BB = 2π/ ωBB to be 5.2 × 10−3, 1.6 × 10−3, and 8.1 × 10−4 s for bottlebrush networks with N = 100, 70, and 50, respectively.

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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.6b01358. Simulation details, dynamics of polydisperse networks, synthesis, characterization, and mechanical properties of PDMS bottlebrush networks (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail [email protected] (A.V.D.). *E-mail [email protected] (S.S.S.). 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 the Grants DMR-1409710, DMR1407645, DMR-1624569, and DMR-1436201.

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REFERENCES

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CONCLUSIONS We have demonstrated the hierarchical nature of the bottlebrush network relaxation. This multiscale relaxation process is confirmed by using a combination of the scaling analysis as well as computational and experimental studies. In particular, our analysis shows that at time scales shorter than the Rouse time of the side chains, τ0 < t < τ0(nsc + 1)2, the relaxation of the bottlebrush network is that of a melt of linear −1/2 . chains with the time-dependent shear modulus GBB 0 (t) ∼ t 2 At the time scale t ≈ τ0(nsc + 1) , the relaxation process of bottlebrush backbone and side chains are coupled, resulting in a crossover to filament-like behavior. This filament-like relaxation process continues up to time scales t ≈ τ0N2(nsc + 1)3/2. In this interval the time-dependent bottlebrush shear modulus has a weak dependence on the side chain degree of polymerization, −1/4 −1/2 t . Finally, at the time scales longer GBB 0 (t) ∼ (nsc + 1) than the Rouse time of the bottlebrush strand, τ0N2(nsc + 1)3/2, the response of the bottlebrush networks is purely elastic with −1 −1 for networks network shear modulus GBB 0 (t) ∼ (nsc + 1) N made of bottlebrush strands with bN ≫ bK ≈ b (nsc + 1) . When the opposite inequality holds, bN ≤ bK ≈ b (nsc + 1) , the bottlebrush strands between cross-links behave as semiflexible filaments with longest relaxation time being on the H


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Dynamics of Bottlebrush Networks - Macromolecules (ACS Publications)

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