Engineering the Mechanical Behavior of Polymer Networks with

Apr 3, 2018 - angles in space, and (2) the internal angle allows relative rotations between the two arms. As the chain forms, the monomers deform and ...
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Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

Engineering the Mechanical Behavior of Polymer Networks with Flexible Self-Assembled V‑Shaped Monomers Noy Cohen,*,†,‡ Omar A. Saleh,† and Robert M. McMeeking†,‡,§ †

Department of Materials and ‡Department of Mechanical Engineering, University of California, Santa Barbara, Santa Barbara, California 93106, United States § School of Engineering, University of Aberdeen, King’s College, Aberdeen AB24 3UE, Scotland S Supporting Information *

ABSTRACT: Recent works demonstrated that the structure of selfassembled monomers can be engineered. Subsequently, performance in polymer-network assemblages of such monomers can be enhanced through microstructural design. This work presents a multiscale analysis of polymers comprising flexible V-shaped monomers. We show that two mechanisms enable the deformation of a chain: conformational entropy and elastic energy stemming from the monomers deformation. The mechanical behavior of a polymer network is obtained through an integration from the chain to the network level. We demonstrate that the shear modulus, the extensibility limit of the network, and the force−elongation response can be tuned by manipulation of the V-shaped monomer structure. As an example, the range of properties that can be achieved in an elastomer comprising 4-arm nanostar monomers subjected to uniaxial extension is examined. We find that appropriate manipulation of the monomer structure can lead to anywhere between an ∼100% decrease to an ∼40% increase in the shear modulus and the extensibility limit.

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INTRODUCTION Tailoring the properties of polymers by microscopic design allows the enhancement of their functionality in modern applications. For example, the coupled response of electroactive polymers can be significantly improved by modifying the molecular structure of the monomers.1 Programming DNAbased hydrogels to swell in response to specific biomolecules (such as proteins and glucose) is another example, producing materials that are applied in smart drug delivery systems.2−4 Several works also proposed to tailor the microstructure of DNA-based hydrogels such that they swell upon exposure to specific external oligonucleotide sequences. It has been shown that such hydrogels can serve as logic gates.4−8 Microscopic design is enabled thanks to modern synthesis techniques.9 For example, molecular scale self-assembly has been exploited to synthesize polymers with DNA monomers, resulting in a wide range of responses.10 It was also demonstrated that the design of the geometry of DNA origami nanostructures allows the tuning of their mechanical properties.11,12 Recent works also showed that the behavior of selfassembling, synthetic DNA nanostar (NS) hydrogels can be engineered by manipulation the stiffness of the NS.13,14 The stretching response of semiflexible polymers has been analyzed in previous works.15−20 These analyses assumed chains comprising rigid monomers and employed a combination of the freely jointed chain (FJC) model, the freely rotating chain (FRC) model, and the wormlike chain (WLC) model to determine the chain response. Other works considered the extension of monomers in a polymer chain in response to sufficiently large forces and extended the FJC model.21,22 In this © XXXX American Chemical Society

work we explore the range of behaviors that can be engineered into a polymer network with chains comprising flexible Vshaped monomers. The flexibility of the monomers provides an additional degree of freedom to the chain which can be exploited to control the mechanical properties of the network. Specifically, manipulation of the monomer structure allows control of the stiffness of the chain and, in turn, the shear modulus and the extensibility limit of the network. It is highlighted that as opposed to the WLC model, the stiffness of a chain according to the proposed model stems from the stiffness of the individual V-shaped monomers. With the aim of guiding the design, we conduct a rigorous multiscale analysis that exploits the hierarchical structure of the polymer network. We show that the properties and the macroscopic behavior are determined by two parameters: the angle between the two arms of the V-shaped monomers and the stiffness of the monomer. We demonstrate that an appropriate monomer design can lead to a significant increase in the shear modulus and the extensibility limit. Consequently, the force−elongation response of the polymer network can be controlled. Additionally, the proposed framework is robust since it can be applied to various length scales. We show that the force−elongation behavior of the network can be controlled by tuning the structural properties of the V-shaped monomers. Specifically, the force required to attain moderate deformations can be reduced by increasing the compliance of the monomers. Received: January 11, 2018 Revised: April 3, 2018

A

DOI: 10.1021/acs.macromol.8b00065 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules The work starts by analyzing the behavior of a flexible Vshaped monomer. As a practical example, we consider a chain comprising 4-arm NS monomers, as proposed in ref 14 where two arms in each NS are hybridized. A Boltzmann-type probability density function (PDF) describing the deformed state of the NS monomer is determined by maximizing the entropy of a chain that is subjected to appropriate constraints or equivalently minimizing the Helmholtz free energy. A summation over the monomers in a chain leads to the force− elongation relation of the chain. We show that as a result of the monomer flexibility, two factors contribute to the extension of the chain: the conformational entropy and the deformation of the monomers. The numerical microsphere method is employed to determine the response of the network.23,24 The analysis reveals that the response of the chain and the network is determined by a characteristic ratio that averages the structural properties of the monomers. This ratio is of paramount importance to the design since it dictates the shear modulus and the extensibility limit of the network. To illustrate the merit of the proposed framework, we investigate the range of behaviors exhibited by a polymer network with chains comprising 4-arm NS monomers subjected to uniaxial extension. Interestingly, we find that manipulation of the structure of the junction connecting the 4 arms can result in an ∼15% increase in the stiffness and the lock-up stretch. In a general polymer with V-shaped monomers, we show that programming the stiffness and the angle between the two arms of the V-shaped monomers can lead to anywhere between an ∼100% decrease to an ∼40% increase in the shear modulus and the extensibility limit. We emphasize that the proposed framework is not limited to a specific monomer structure and can be easily extended to any monomer design by appropriate energetic modifications.

Figure 1. Schematic of a polymer chain comprising 4-arm NS monomers.

corresponds to the most probable distribution of monomer orientations, maximizing entropy.25,15 As opposed to standard polymers in which the monomers are considered rigid, the compliance of the V-shaped monomers provides an additional degree of freedom. Specifically, an external force and thermal fluctuations may lead to a change in the angle α and, consequently, in the monomer length. In the following, we make two conjectures. First, the arms can rotate independently such that the internal angle determines the monomer length. Second, the interactions between the monomers in the chain are negligible. Specifically, the orientational angles of the Vshaped monomer structure are determined by thermal fluctuations and maximization of entropy (or equivalently, minimization of the Helmholtz free energy). Consequently, the elastic energy of the monomer is

Kk bT (α − α0)2 (1) 2 where kb is the Boltzmann constant, T is the absolute temperature, and K is a springlike stiffness describing the energetic cost of deforming the V-shaped structure. We underscore that the behavior of monomers with a different structure can be captured by an appropriate modification to the internal energy (eq 1). Before proceeding, we point out that the configuration of the monomer is determined by either the internal angle or the effective monomer length. Practically, the degree of freedom is chosen and programmed into the molecular structure during the fabrication process. Recent works showed that this degree of freedom can be controlled. For example, the synthesis of scaffolded DNA origami allows to fabricate specific molecular structures.9 It is also possible to manipulate a DNA structure with salt-induced flexibility by changing the solution conditions.26 These two techniques allow to fabricate monomers with a specific structure distribution. Following Bai et al.,26 this work considers a polymer comprising monomers in which the internal angle is the independent parameter that governs the macroscopic response. The internal angle is uniformly distributed in the range 0 ≤ α ≤ π/2. For convenience, we present the main results for the case in which the effective monomer length b is controlled in Supporting Information section 4. It is emphasized that there are no qualitative differences in the predictions between the two cases. The main quantitative differences are seen in monomers with a low stiffness and stem from the statistical weights corresponding to the different distributions. In an equilibrium state, the probability density function (PDF) of the NS monomer orientations is given by (see Supporting Information section 1) UV =

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MODEL OF THE CHAIN Consider an aggregate of n randomly oriented flexible V-shaped monomers comprising two rigid arms of length l/2 with an angle 2α0 between them. The ends of two arms attach to form a chain. The gap between consecutive arms, consisting of unbound nucleobases, is large enough to provide the monomers three degrees of freedom to rotate: (1) the Vshaped monomer structure can rotate about two orientational angles in space, and (2) the internal angle allows relative rotations between the two arms. As the chain forms, the monomers deform and rotate about the three degrees of freedom to ensure thermal equilibrium and maximization of entropy or equivalently minimization of the Helmholtz free energy. Consequently, the current angle between the arms is 2α, where 0 ≤ α ≤ π/2. The length and the direction of the line segment connecting the monomer to its neighbors are denoted b = l sin α and b̂, respectively, where b̂ is a unit vector. The vector connecting the two ends of the chain is thus r ⃗ = ∑ni=1b(i)b(̂ i). As an illustration, Figure 1 depicts a chain comprising 4-arm NSs, where only two of the arms are hybridized. Such monomer structures were manufactured and characterized in a recent work.14 Next, we fix one end of the chain and exert a force f ⃗ of magnitude f on the other end. Two mechanisms define the deformation of the chain: (1) the conformational entropy of the chain and (2) the deformation of the monomer lengths resulting from rotation of the internal angle. From an entropic viewpoint, it is assumed that the configuration of the chain B

DOI: 10.1021/acs.macromol.8b00065 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules p (θ , α ) =

K

(

exp χ sin α cos θ − 2 (α − α0)2

) (2)

Z

where 0 ≤ θ ≤ π is the polar angle relative to the extension direction and Z=

⎛

⎞

∫ exp⎝χ sin α cos θ − K2 (α − α0)2 ⎠ dΓ dα ⎜

⎟

(3)

denotes the partition function. Here, χ = f l/(kbT), dΓ = sin θ dθ dϕ is the solid angle, and 0 ≤ ϕ < 2π is the azimuthal orientation angle of the monomer direction relative to f.⃗ Note that with the above definitions the PDF is normalized such that ∫ p dΓ dα = 1. Accordingly, the force−extension relation can be obtained via r = nl

∫ sin α cos θ p dΓ dα

(4)

Figure 2. PDF of the NS monomers aligned along the end-to-end vector for various ratios r/(nl). The continuous, the dashed, and the dotted curves correspond to the compliant, the intermediate, and the stiff monomers.

Note that in the highly stretched chain limit α → 90°, and the well-known Langevin behavior of chains comprising rigid monomers is obtained. Before proceeding, we point out that several works demonstrated that the proposed chain structure can be realized and optimized. Zhou et al. demonstrated that the internal angle and the stiffness can be controlled in a simple DNA V-shaped structure.12 An assembly of 4-arm NS monomers can also form a polymer chain by hybridizing only two of the 4 arms. Such a structure was characterized in a recent work, where it was shown that the initial angle α0 and the stiffness K of the junction connecting the four arms can be engineered during the fabrication process.14 Generally, the stiffness of the monomers can range from rigid (K → ∞) to fully flexible (K → 0). This stiffness, for example, can be controlled by tuning the electrostatic repulsion between arms through solution salt concentration.26

prefer larger NS angles, and the number of monomers aligned to the end-to-end direction increases, as seen by the PDF values. These effects are explained as follows: due to the compliance of the NS, the extension of the chain causes the NSs to open up, thus increasing the average monomer length along θ = 0, and forces the monomers to rotate toward the r ⃗ direction. The stiff monomers aligned to the end-to-end vector are most likely to have an initial angle α0 in an unstretched chain. Roughly, the internal forces maintaining the NS structure are stronger than the thermal forces. As the chain stretches, the energy required to deform the NSs is significantly higher than the entropic cost of reorientation, and accordingly we find that the profile of the distribution of the NS angles does not vary. Near the fully extended state of the chain, most of the monomers point along the θ = 0 direction, and further extension is only possible by deforming the NSs. This effect is depicted by the narrowing of the bell-shaped distribution, describing a decrease in the variance. Specifically, the influence of the thermal fluctuations diminishes, and it is mostly the mechanical forces that determine the monomer lengths. As expected, the distribution of the intermediate NS monomers falls between the stiff and the compliant behaviors. Here, the internal forces in the NS are comparable to the thermal forces. Figure 3 plots the PDF of monomers that are perpendicular (θ = 90°) and antialigned (θ = 180°) with respect to the endto-end vector for various stretches. The distribution of the perpendicular NS monomers remains unaffected by the stretch. This is expected since they are also perpendicular to the external force. However, it is important to note that the values of the PDF decrease significantly as we extend the chain, which indicates that the population of these monomers decreases. Next, we examine the distribution of the antialigned monomers. Compliant antialigned NSs tend to close up, as evident by the PDFs at the bottom of Figure 3. This occurs due to the low energetic cost of deforming these monomers in a chain that is subjected to moderate extensions. On the other hand, the moderate deformations of a chain with stiff NS monomers is entropy-driven. This can be seen by the

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PROBABILITY DENSITY FUNCTION OF THE KUHN LENGTH To better understand the microscopic behavior of the V-shaped monomers, we study the distribution of monomers with various K values in a polymer chain. To this end, we consider the specific 4-arm NS monomer design that has been recently realized.14 Consider a chain comprising 4-arm NSs, where two of the four arms are hybridized. Upon construction, the positions of the four arms can be visualized as four lines pointing from the center of a tetrahedron to its four vertices, and accordingly the initial angle is α0 = 0.5 arccos(−1/3) ≈ 54.7°. We examine the response of polymer chains comprising three types of NS monomers: stiff (K = 20), intermediate (K = 5), and compliant (K = 0.1). Figure 2 plots the PDF p (θ = 0,α) (given by eq 2) of the NS angles in monomers that are aligned along the end-to-end vector r ⃗ for several chain extensions. Note that the PDF in eq 2 depends on the quantity χ, which is proportional to the force. The relation between the for χ and r/ (nl), or equivalently between the force and the elongation, is given by eq 4. We note that the compliant monomers fluctuate and explore a wide variety of monomer lengths. In the undeformed chain (r/(nl) = 0) the thermal forces are significantly stronger than the internal forces maintaining the NS structure, leading to an equally probable distribution of the NS angles. As the chain stretches, two effects are observed: the compliant monomers C

DOI: 10.1021/acs.macromol.8b00065 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

Figure 4. Characteristic ratio c0 as a function of α0 for various monomer stiffness values.

Figure 3. PDF of NS monomers that are perpendicular and antialigned to the end-to-end vector for various ratios r/(nl). The continuous, the dashed, and the dotted curves correspond to the compliant, the intermediate, and the stiff monomers.

seen that c0 monotonically increases with the angle. Practically, this observation signifies that the stiffness of a chain can be increased by decreasing α0 while keeping K constant. In the following, we investigate the behavior of the function c0 at high and at low K values. The initial response of chains with high K values is mostly entropic. In other words, the V-shaped monomers tend to reorient rather than deform. The decrease in the stiffness of the chain with an increase in the angle stems from the elongation of the monomers. To see this, recall that the extension of the chain is proportional to the length of the monomers that reorient. Additionally, note that in the limit K → ∞ the monomers are rigid and c0 → sin(α0) to recover the Langevin behavior, as expected. The extension of chains comprising compliant V-shaped monomers, i.e., small K, is enabled due to two mechanisms: the elastic deformations and the reorientation of the monomers. Despite significant thermal effects, the V-structures exhibit a slight tendency toward their engineered structure. Specifically, V-structures with large α0 tend to be slightly open on average. As a consequence, the extension of such chains can be attained with relatively weak forces. In chains comprising V-shaped monomers with small α0 the additional cost of deforming the monomers leads to the stiffening of the chain. Interestingly, in the limit K → 0, c0 → sin(π/4) and the stiffness of the chain is 3/c02 = 6. This result stems from the total collapse of the Vstructure in response to thermal fluctuations, which leads to a uniform distribution of V-angles. Figure 4 also reveals that the characteristic ratio is independent of the monomer stiffness in monomers with an angle α0 = 45°. In this case, the sum of the two energy contributions associated with the extension of the chain are equal in magnitude for any K value. The large force/elongation response of the chain can also be controlled by tuning the properties of the monomer structure. To illustrate the effects of the monomer stiffness, we once again consider a chain comprising 4-arm NSs. Figure 5 plots the normalized force χ as a function of the relative extension r/(nl) for several values of K. This relation is given in eq 4. The dotted and the dashed curves correspond to the FJC model, i.e.

distribution profile of these NS monomers, which remains unchanged. It is emphasized that the population of the monomers along the θ = 90° and the θ = 180° decreases while the number of monomers along θ = 0° increases. Specifically, at zero extension, the probability p(θ) is equally likely for all θ. At an extension r/(nl) = 0.6, p(θ = 0) increases about 20-fold, while p(θ = 90°) and p(θ = 180°) decreased by 50% and 90%, respectively.

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DESIGN OF A POLYMER CHAIN WITH V-SHAPED MONOMERS In this section we explore the range of responses which can be programmed into a polymer chain. As a reference, we note that in the limit of a chain with rigid monomers, i.e., K → ∞, eq 4 can be analytically evaluated. Such chains display the classical behavior of a FJC, i.e., r /(nl) = sin α0 3(χ sin , α0), where 3(x) = coth(x) − 1/x is the Langevin function. The sin α0 factor accounts for the initial length of the monomer. The design of the initial angle and the stiffness of the V-shaped structure allows the tuning the behavior of the chain. First, we examine the range of initial chain stiffness values that can be engineered. A first-order Taylor series of eq 4 about χ = 0 reveals that χ = 3 r/(c02nl). We define the characteristic ratio c0 =

∫ sin2(α)2p(χ = 0) dα

(5)

where p(χ = 0) is the PDF (eq 2) in the absence of an external force. The characteristic ratio is bounded by 0 ≤ c0 ≤ 1 and accounts for the variations in the length of the monomers stemming from thermal fluctuations. We emphasize that there is a subtle difference between Flory’s characteristic ratio and c0. The characteristic ratio can be easily computed for two limiting cases. In the limit K → ∞ the rigid monomers are not affected by the thermal sources and therefore c0 → sin α0. In the limit K → 0 the V-shaped structure collapses and c0 → 1/2. The quantity 3/c02 is henceforth referred to as the chain stiffness. Figure 4 plots c0 as a function of the V-angle for several monomer stiffness values. The dotted and the dashed curves provide bounds for the characteristic ratio values, and it can be

χ=

⎛r 1 1 ⎞ 3−1⎜ ⎟ sin(α0) ⎝ nl sin(α0) ⎠

(6)

and D

DOI: 10.1021/acs.macromol.8b00065 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

Figure 6. Average NS angle ⟨α⟩ versus r/(nl) for chains comprising NS monomers with K = 0.1 (star symbols), K = 5 (diamond symbols), K = 20 (circle symbols), and K = 100 (triangle symbols). The dashed curve corresponds to ⟨α⟩ = α0.

Figure 5. χ versus r/(nl) for chains comprising NS monomers with K = 0.1 (star symbols), K = 20 (circle symbols), K = 50 (square symbols), and K = 100 (triangle symbols). The dotted and the dashed curves correspond to eqs 6 and 7, respectively.

⎛r⎞ χ = 3−1⎜ ⎟ ⎝ nl ⎠

■

(7)

POLYMER NETWORK The behavior of a polymer network is directly governed by the response of the individual chains. In the following, we study the range of macroscopic responses that can be engineered into the polymer. To this end, consider an incompressible polymer network with N0 chains per unit volume, where each chain comprises n = 100 4-arm NS monomers. Initially, the chains are randomly oriented and uniformly distributed. We prescribe a global coordinate system in which the ith referential end-to-end vectors is

respectively, and serve as limiting cases for a chain subjected to large forces. Two regimes are observed: Deformations smaller than r/(nl) ≈ 0.6 lead to the behavior discussed above. The response of the chains under larger deformations differs and is the focus of the following discussion. The deformation of chains with stiff NSs is inhibited, and therefore the elongation is initially enabled due to the rotation of the monomers. Subjected to sufficiently large forces, the entropic cost of further rotating the monomers is comparable to the elastic energy required to stretch the NS. As a consequence, large forces are required to attain deformations near the fully stretched state, as illustrated by the curves with the square and triangular symbols, corresponding to K = 50 and K = 100, respectively. Chains comprising compliant NS monomers deform and rotate simultaneously. Accordingly, the forces required to attain large deformations in compliant chains are weaker than those in stiff NS chains. This effect can be appreciated from the curve with the star symbols, corresponding to K = 0.1. To illustrate the interplay between the elastic and the entropic energies, Figure 6 depicts the average NS angle ⟨α⟩ = ∫ α p dΓ dα as a function of r/(nl) for several NS stiffness values. For comparison, the dashed curve corresponds to the engineered NS angle. In the limit of compliant NS monomers, i.e. K → 0, we find that at r/(nl) = 0 the angle ⟨α⟩ → 45°. As previously discussed, this stems from the thermal forces that eliminate the initially engineered preferred angle. The two mechanisms that govern the extension of compliant chains are again observed. Specifically, ⟨α⟩ experiences an increase in compliant monomers throughout the extension of the chain. This pertains to an increase in the elastic and the entropic energies of the chain. On the other hand, Figure 6 shows that the initial angle ⟨α⟩ → α0 as the NS stiffness increases. Furthermore, as K increases, smaller variations are observed in ⟨α⟩ for a longer range of stretches. This indicates that the reorientation of the NS monomers gives rise to the deformation of the chain. Only under a sufficiently large force do the NSs begin to deform.

r0⃗(i) = l n c0r0̂

(8)

where r̂0 = {cos(Ψ), sin(Ψ) cos(Φ), sin(Ψ) sin(Φ)}, and 0 ≤ Ψ ≤ π and 0 ≤ Φ ≤ 2π are the polar and the azimuthal angles, respectively. Since the chains are randomly oriented and uniformly distributed in the reference configuration, it can be shown that ⟨r/⃗ nl⟩ = 0. As customary in polymer physics, it is assumed that the magnitude of r0⃗ equals the root mean square of the end-to-end distance.15,27 The characteristic ratio c0 accounts for the elastic energy that is stored in the chain upon formation. It is highlighted that the classical result r0 = l sin(α0) n is only obtained in the limit K → ∞ and provides an upper bound. The polymer is subjected to a second-order deformation gradient tensor F. Because of the incompressibility of the network det F = 1. It is assumed that the chains undergo affine deformations,15,27 and therefore the deformed end-to-end vector of the ith chain is r ⃗ = l n c0t,⃗ where t ⃗ = Fr̂0. Accordingly, the stress that develops in the network is (see Supporting Information section 2)

σ ̅ = μ∞ n

χ t ⃗t ⃗ λC

N

(9)

where ⟨·⟩N denotes the average of a chain-related quantity · over the polymer network, λC = t ⃗· t ⃗ is the deformation of a chain, and μ∞ = N0kbT. Equation 9 can be approximated in the two limiting cases to yield the initial shear modulus μ = μ∞/c0 and the stretch at which the network reaches its extensibility limit E

DOI: 10.1021/acs.macromol.8b00065 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules λlu =

ln n = l n c0 c0

bounded by 1/sin(α0) ≤ μ/μ∞, λlu/ n ≤ 2 . However, the range of polymer properties can be significantly broadened through modification of the NS angle.

(10)

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The quantity λlu is henceforth referred to as the lock-up stretch. These limits highlight the importance of the characteristic ratio. Specifically, the shear modulus and the lock-up stretch increase as c0 decreases. We recall Figure 4, which demonstrates that in order to reduce the characteristic ratio we need to decrease the NS angle, vary the NS stiffness, or do both. It is emphasized that the increase in the lock-up stretch stems from the shorter initial average end-to-end distance while the increase in the shear modulus is attributed to the variations in the elastic and the entropic energy contributions in the network. To illustrate the predictions of the proposed model, consider a polymer film comprising 4-arm NS monomers subjected to the deformation gradient 1 (I − xx̂ )̂ F = λxx̂ ̂ + (11) λ

CONCLUSIONS This work presents a rigorous multiscale analysis of polymers with flexible V-shaped monomers. The monomers possess two structural degrees of freedom which can be controlled in the fabrication process: the stiffness and the V angle. The analysis reveals that properties of the chain and the polymer network depend on a characteristic ratio which accounts for the Vshaped monomer structure. We show that manipulation of the monomer structure enables the design and the optimization of the response of the chain and, consequently, the polymer network for a specific functionality or application. These findings pave the path to the enhancement of polymer-based devices. From a practical viewpoint, we highlight that this framework is motivated by recent works which demonstrated that it is possible to engineer the structure of the monomers using advanced fabrication techniques.9,11,12,14 We also emphasize that while this work focused on a specific monomer structure, the proposed framework can be extended to account for any polymer comprising elastic monomers by appropriate modifications to the elastic energy.

corresponding to a uniaxial extension, where λ is the ratio between the current and the initial length of the film along the x̂ direction. To account for the incompressibility, a work-less pressure-like term is added to the stress and enforces boundary conditions. Accordingly, the stress along the x̂ direction is σx = σx̂̅ ·x̂ − σŷ̅ ·ŷ (see Supporting Information section 3). To integrate from the chain to the network level and compute the stress σ̅, we employ the microsphere technique with 42 representative directions, as explained in Supporting Information section 3.23,24 Figure 7 plots σx/μ∞ versus λ for various values of K. The dotted and the dashed curves serve as a comparison and

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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.8b00065. Section 1: derivation of the probability density function in eq 2; section 2: derivation of the stress in eq 9; section 3: details of the numerical microsphere technique used to integrate from the chain to the macroscopic level; section 4: the response of a polymer with length-controlled monomers (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (N.C.). ORCID

Noy Cohen: 0000-0003-2224-640X Notes

The authors declare no competing financial interest.

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Figure 7. Stress σx/μ∞ as a function of the stretch λ for chains comprising NS monomers with K = 0.1 (star symbols), K = 5 (diamond symbols), K = 20 (circle symbols), and K = 100 (triangle symbols). The dotted and the dashed curves correspond to Langevin behavior (K → ∞) with α0 = 90° and α0 = 45°. The inset shows a plot of 1/c0 vs K.

ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under Award DESC0014427. O.A.S. thanks the Alexander von Humboldt Foundation for support.

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correspond to the Langevin behavior (K → ∞) with α0 = 90° and α0 = 45°, respectively. We find that various stress−strain curves can be obtained by designing the microstructure. The inset of Figure 7 depicts the range of shear moduli and lock-up stretches that are available for the engineered NS angle α0 ≈ 54.7°. Since α0 > 45°, increasing the stiffness of the NS monomers leads to a decrease in the shear modulus and the lock-up stretch. It is important to note that for the chosen NS angle the available shear moduli and lock-up stretches are

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DOI: 10.1021/acs.macromol.8b00065 Macromolecules XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.macromol.8b00065 Macromolecules XXXX, XXX, XXX−XXX