Computationally Driven Design of Soft Materials with Tissue-like

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Chapter 3

Computationally Driven Design of Soft Materials with Tissue-like Mechanical Properties Heyi Liang,1 Mohammad Vatankhah-Varnosfaderani,2 Sergei S. Sheiko,2 and Andrey V. Dobrynin*,1 1Department

of Polymer Science, University of Akron, Akron, Ohio 44325, United States 2Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 27599-3220, United States *E-mail: [email protected]. E-mail: [email protected].

Mimicking the mechanical properties of soft materials and biological tissues is crucial for novel materials development for medical implants, tissue engineering, soft robotics, and wearable electronics. Unfortunately, the required combination of softness, strength, and toughness is difficult to replicate in synthetic materials. Modern design strategies are predominantly Edisonian in nature and are based on exploratory mixing of assorted polymers, variation in cross-linking schemes, and solvents. However, it was recently demonstrated that it is possible to encode mechanical properties of soft tissues in solvent free synthetic elastomers by varying architecture of the network strands. This approach is based on the theoretical and computational studies of correlations between mechanical properties and architecture of networks with brush-like strands. Different types of graft polymers such as combs and bottlebrushes were modeled as ideal chains or filamets with effective Kuhn length. This representation of graft polymers allows for a precise mapping of network’s mechanical properties in both linear and nonlinear deformation regimes into molecular architecture of the network strands.

© 2018 American Chemical Society

This approach to materials design was tested by reproducing mechanical properties of assorted biological gels and tissues using poly(dimethylsiloxane) (PDMS) and poly(n-butyl acrylate) (PBA) graft polymer elastomers. This technique lays the foundation for a computationally driven materials design that will be capable of encoding mechanical properties of soft materials in solvent free elastomers.

Introduction Materials (synthetic and biological) (1–7) demonstrate an astounding variety of the mechanical properties which is reflected in different shapes of their stress-deformation curves (see Figure 1a). Polymer gels like jellyfish are soft and highly deformable while bone is rigid and brittle. Biological tissues (8) like heart and skin demonstrate unique strain adaptability manifested in a sharp increase in their instantaneous modulus, dσ/dλ, with deformation. However, despite this variety of mechanical responses, there is a general trend correlating the value of the Young’s modulus at small deformations E0 with an elongation-at-break λmax or strain-at-break εmax = λmax − 1 for materials undergoing uniaxial deformation (see Figure 1b) (1). This relationship is known as a “Golden rule” of the materials science and identifies a general trend that more rigid materials are less deformable. For synthetic elastomers or rubber both the modulus and elongation-at-break are uniquely determined by the degree of polymerization between cross-links nx. This results in a universal relationship, , confining a large fraction of polymeric networks into a single trend line with a lower bound imposed by the entanglements of the network strands, ne (9–11). For solvent free elastomers an entanglement modulus is on the order of Eent ~ 1 MPa and elongation-at-break λmax ~ 5 if no specific routes in network preparations are undertaken (12–16). In the case of brittle materials the universal follows from the fact that for a material to break it scaling relation requires to store a particular amount of the energy density, . However, there is a large class of biological and soft composite materials that occupy space below the “Golden rule” trend – referred to as the “Biological triangle” in Figure 1b. This poses a challenge of how to create synthetic replicas that will be capable of reproducing mechanical properties of materials from the “Biological triangle”. The conventional approach is to use multicomponent systems by utilizing Edisonian approach thorough exploratory mixing of assorted polymers, cross-linking schemes, and solvents. However, this approach is imperfect in property control (8, 17–23). For example, solvent can leak under applied stress or evaporate when the environmental conditions are changed. Furthermore materials properties are matched only in the linear deformation regime thus restricting synthetic materials design to the range of small deformations. 34

Figure 1. Diversity in materials mechanical properties is illustrated by uniaxial tensile stress-strain curves. b) “Golden rule” of materials science establishes an inverse relationship between elongation-at-break λmax and modulus E0. To display materials with different λmax in a single plot, E0/ρ is shown as a function of

, where ρ is the mass density. Data replotted from ref (1).

A new approach to the design of a wide class of polymeric materials capable of mimicking mechanical properties of tissues is based on variation of architecture of the network strands such as graft polymers (see Figure 2a) (1). This allows replication of materials’ (tissue or composite) mechanical properties in solvent free elastomers. For example, changes in the degree of polymerization (DP) of the side chains, nsc, and their grafting density, 1/ng, provides flexibility in control over the concentration of the stress supporting backbones (dilution of the backbone monomers). In addition to the dilution effect, by varying 1/ng and nsc one can change the effective Kuhn length of the graft polymers, making a network strand more rigid or flexible (1, 9, 24). The following strategy for replication of the mechanical properties of a soft material or tissue using graft polymer elastomers was outlined (1) (see Figure 2b): (i) fitting of the entire stress-strain curve to the macroscopic network model expressed in terms of the network modulus and elongation-at-break; (ii) using established correlations between macroscopic network parameters and chemical structure of graft polymers defined by architectural triplet [nsc, ng, nx] to determine feed ratios for network synthesis; (iii) synthesis of the polymer network with a given chemical structure of the network strands; and (iv) measurement of the mechanical properties of the replica and verification of the replication procedure. In the rest of the chapter we overview steps of the materials design and replication process. To put this approach on solid footing we begin our discussion with solution of the forward problem and establish relationships between mechanical properties of the networks of graft polymers and their chemical structure.

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Figure 2. a) Schematic representation of the network made of graft polymer strands. b) Flow chart for replicating mechanical properties of a selected tissue in graft polymer elastomers.

A Model of Polymeric Networks and Gels A starting point of the replication procedure is fitting the materials stress-strain curve in the linear and nonlinear deformation regimes to a network deformation model (see Figure 2b). In our approach we use a network model which bridges a network deformation with the deformation of the individual network strands (25, 26). In particular, this model takes advantage of universality in individual strands’ deformation by representing them as worm-like chains with the effective Kuhn length bK (27). In the framework of this model the true stress generated in a network undergoing uniaxial elongation with elongation ratio λ at a constant volume is equal to

where G is the structural shear modulus, β is the strand extension ratio, and the first strain invariant for network undergoing a uniaxial deformation is I1(λ) − (λ2 + 2/λ). These two parameters G and β uniquely determine a network’s stress-deformation curve. In particular, the structural modulus G controls material stiffness and β parameter is responsible for the onset of strain-stiffening due to finite extensibility of the network strands and is related to the elongation-at-break. The strand extension ratio is defined as a ratio of the mean square end-to-end distance of the undeformed network strands in as-prepared networks, square of the contour length of a fully extended strand, length l, as follows

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, and

, with a bond

For network strands described by worm-like chains with the effective Kuhn length bK,

is written as (11)

It is convenient to express the strand extension ratio β in terms of the number of the Kuhn segments per strand α−1 ≡ Rmax/bK

Two limiting cases are clearly identified. For flexible strands with bK >Rmax, β ~1. This is usually the case for biological networks and gels (28, 29). Structural shear modulus of the network, G, is proportional to the number density of the stress-supporting strands ρs

where C1 is a numerical constant that depends on the functionality of the crosslinks and network topology, kB is the Boltzmann constantan and T is the absolute temperature. It is important to point out the difference between the structural modulus G and the shear modulus at small deformations

used to characterize stiffness of linear chain networks (11). It follows from eq 6 that these two moduli G and G0 are only equal in the case of flexible networks strands for which β 1 (30). It is important to point out that the value of the crowding parameter Φ > 1 corresponds to a hypothetical system, where bottlebrush macromolecules maintain ideal conformations of side chains and backbones even at infinitely (unreasonably) large grafting density. In real systems, however, in the range of system parameters with Φ > 1, the backbone and side chains will stretch to maintain the melt density (ρ ≈ ν−1). The explicit expression for location of the crossover between combs and bottlebrushes given by Φ ≈ 1 is obtained by solving eqs 8 and 9 for a composition parameter

Figure 6. 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 – rod-like side chain regime. Logarithmic scales. Adapted with permission from ref (30). Copyright 2017 American Chemical Society.

This parameter describes partitioning of monomers between a side chain and backbone spacer between two neighboring side chains and characterizes “dilution” of the backbone. 41

Figure 6 shows diagram of states of graft polymers as a function of their chemical structure and composition (30). There are two main regimes: (i) Comb regime, where both side chains and backbones of graft polymers interpenetrate and remain ideal and (ii) Bottlebrush regime, where excluded volume interactions between densely grafted side chains expel monomers of the neighboring macromolecules from the pervaded volume of a given graft polymer. In a Bottlebrush regime, bottlebrush backbones or side chains have to stretch to maintain a constant monomer density in a melt. Bottlebrushes with extended backbones are in the Stretched Backbone (SBB) regime and those with stretched side chains belong to the Stretched Side Chain (SSC) regime. Bottlebrushes with fully stretched side chains define the Rod-like Side Chain (RSC) regime. Table 1 presents the effective Kuhn length, bK, in terms of the degree of polymerization of the side chains nsc and the number of bonds between grafting points of the side chains, ng, and system molecular parameters (30, 32).

Table 1. Effective Kuhn Length of Graft Polymers in Different Regimes and Regime Boundaries (30)

Scaling relations for dependence of the effective Kuhn length on the macromolecular architecture summarized in Table 1 have been verified in the coarse-grained molecular dynamics simulations (33, 34) of the graft polymers in a melt (30). Figure 7 combines simulation data for the reduced effective Kuhn length bK/b as a function of the crowding parameter, Φ. In a comb regime the effective Kuhn length saturates at b. With increasing value of the crowding parameter, Φ, the interactions between side chains result in stiffening of macromolecules. This leads to increase of the effective Kuhn length bK. In the range of the values of the crowding parameter Φ>1, simulation data confirm a scaling dependence for the effective Kuhn length of the bottlebrushes, bK ≈ bΦ. 42

Figure 7. Dependence of the normalized Kuhn length, bK/b, of the graft polymers (see eq 8) for graft polymers with degree of on the crowding parameter polymerization of the side chains nsc varying between 2 and 32 and number of bonds on the backbone between side chains ng in the interval 0.5 and 32 (ng=0.5 corresponds to two side chains attached to each backbone monomer). See ref 30 for symbol notations. Thin solid lines show scaling predictions in comb and bottlebrush regimes. Adapted with permission from ref (30). Copyright 2017 American Chemical Society.

Model of the Graft Polymer Networks The next step is to use results for the dependence of the effective Kuhn length bK on the graft polymer architecture to develop a model of the graft polymer networks (31, 32, 35). Consider graft polymer networks made by cross-linking precursor macromolecules with the backbone degree of polymerization nbb in a melt with the monomer number density ρ. This cross-linking procedure produces two dangling ends of varying length per chain reducing the number of stress supporting strands (32)

Substituting eq 11 into eq 5, one obtains the following general expression for the structural shear modulus of graft polymer networks

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The explicit form of eq 12 in terms of the architectural parameters of the graft polymers is derived by using corresponding expressions for the Kuhn length bK in different regimes of diagram of states shown in Figure 6 and listed in the Table 1. The results of such derivation are summarized in Table 2 (32).

Table 2. Structural Shear Modulus, G, and Inverse Number of Effective Kuhn Segments Per Network Strands, α, in Different Conformation Regimes (32)

The scaling expression for structural modulus and number of effective Kuhn segments per network strand have been tested in the coarse-grained molecular dynamics simulations of the graft polymer networks. In these simulations, the graft polymers have been cross-linked through the end monomers of the side chains, which corresponds to the experimentally studied systems (1, 9). This cross-linking scheme results in a hybrid network composed of brush-like and linear network strands and requires explicit consideration of the elastic response of both strands’ populations. 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. Thus, such networks can be approximated by graft polymer networks cross-linked through their backbones (see for detailed discussion ref (32)). The stress-deformation curves of graft polymer networks of different chemical sructures are shown in Figures 8a-c. As evident from Figures 8a and 8b, the decrease in the side chain length or strand length between cross-links both leads to the higher shear modulus and lower strand extensibility. However, we can achieve both shear modulus and strand extensibility increase with the decrease of grafting density as shown in Figure 8c. The linear relationships (see Figures 8d-f) between network parameters α, β, and G describing its mechanical properties and structural parameters [nsc, ng, nx] of the network strands allows for a simple calibration procedure for creation of libraries of the graft polymers for replication of soft materials and biological tissues using soft solvent free networks with predetermined stress-deformation curves. 44

Figure 8. 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 crosslinks 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 dashed lines are the best fit to eq 1 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). Dashed lines in figures (d-f) illustrate linear correlation between αG/β and φ/nx. Adapted with permission from ref (32). Copyright 2018 American Chemical Society. 45

Figure 9. Breaking the “Golden rule” of the materials design, . The values of the λmax for this plot are calculated as λmax = β−1/2. The “Golden Rule” is shown as the dash 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 blue symbols show correlations between G and λmax for data sets from Figure 8 a for which DP of side chains nsc = 2, 4, 8, 16, 32 and ng = 8, nx ≈ 16. Filled green triangles represent correlations for data shown in Figure 8 b and corresponds to graft polymer networks with DP of backbone between crosslinks nx= 8, 11, 16, 20, 34, ng = 2 and nsc = 8. Open triangles of different colors show correlations for data in Figure 8 c. Adapted with permission from ref (32). Copyright 2018 American Chemical Society. (see color insert)

Figure 9 shows correlations between structural modulus and elongation-atbreak defined in terms of the strand extension ratio β as λmax ≈ β−1/2 (32). For networks made of comb-like strands, their structural modulus is φ times smaller than the corresponding modulus of the networks of linear chains with the same degree of polymerization between cross-links. In the case of bottlebrush strands, there is an additional contribution to the structural modulus G due to stiffening and finite extensibility of the network strands. Interplay between the dilution and stiffening effects results in simultaneous increase of stiffness and extensibility. , demonstrating This is manifested in a new scaling relation, an increase of the structural modulus with elongation-at-break (open symbols). This scaling corresponds to a crossover from the Bottlebrush regime to the Comb of the side chains keeping regime through variation of the grafting density the degree of polymerization between crosslinks nx almost constant. Thus the 46

graft polymer networks can break the “Golden rule” and occupy the “Biological triangle” (see Figure 1b). This feature is crucial for soft materials and tissue replication. In the next section we discuss implementation of the procedure for replication of the mechanical properties of soft and tissue-like materials in solvent-free graft polymer elastomers.

Experimental Implementation of the Replication Procedure The outlined approach for replication of tissue properties was successfully tested by synthesizing poly(dimethylsiloxane) (PDMS) and poly(n-butyl acrylate) (PBA) graft polymer elastomers with accurately controlled sets of [nsc, ng, nx] (1). The mechanical properties of such networks were obtained in the broad range of strains to verify correlations between structural triplet [nsc, ng, nx] and resultant network mechanical properties. Analysis of stress-strain curves of the network samples has confirmed that graft polymer networks violate the “Golden rule” populating the “Biological triangle” (see Figure 10a). The synthesized libraries of PDMS and PBA graft polymer elastomers prime the way for replication of tissues in this class of polymeric materials through calibration of relationships between network parameters α, β, and G (or E) describing its mechanical properties and structural parameters [nsc, ng, nx] (see for detail ref (1)). These correlations are used in the synthesis step of the protocol outlined in Figure 2b to create mechanical replicas of representative tissues.

Figure 10. a)Combined (λmax) plot for comb and bottlebrush elastomers. Eent entanglement modulus. b) Stress-strain data (□) for alginate gel, jellyfish tissue, and poly(acrylamide-co-urethane) gel are shown together with fitting analysis of the data by eq. 1 (dashed lines), and curves for PDMS bottlebrush and comb mimics synthesized via the fitting analysis with the corresponding architectural triplets [nsc, ng, nx] as indicated (solid lines). Adapted with permission from ref (1). Copyright 2017 Nature. Figure 10b displays successful replication of the stress-strain curves of alginate gel, jellyfish tissue, and composite poly(acrylamide-co-urethane) gel, with [nsc, ng, nx] combinations of [14, 1, 67], [28, 2, 100], and [14, 4, 1200] . 47

Conclusions and Outlook We overview a universal materials design platform that accurately encodes targeted physical properties in network architecture. This approach eliminates the current disconnect between material formulation and function which is compounded by imprecise control of network architecture and empirical descriptions of hierarchic structure-property relations. It employs the closed-loop integration of chemistry, physics, and biology to decode the structure-property relations of targeted networks and then program them in precise molecular architectures. Since the resultant materials are solvent-free, they will neither freeze in the Arctic nor dry in the Sahara. No liquid components will exist to be squeezed out in subsurface environments or to evaporate. However, despite of all these advantages the library of the graft polymers limits the types of the biological tissues for replication. It is currently impossible to replicate materials with extension ratio β > 0.3 (1). The tissues with such values of the parameter β are heart tissues, arteries, and skin. The required combination of the initial materials’ softness and strong strain hardening can be achieved in thermoplastic elastomers composed of liner-bottlebrush-linear ABA triblock copolymers (1). This additional complexity in the chemical structure of the network building blocks can provide additional degree of freedom for tissue mimicking. Furthermore, self-assembly of the linear blocks creates moldable physical networks that could be reprocessed on demand. We hope future studies will explore the full potential of this class of copolymers for tissue replication.

Acknowledgments The authors are grateful to the National Science Foundation for the financial support under the Grants DMR-1407645, DMR-1624569 and DMR-1436201.

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