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Microscopic origins of the mechanical response of nanostructured elastomeric materials Parker, Amanda J. 2017

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Microscopic origins of the mechanicalresponse of nanostructured elastomericmaterialsbyAmanda J. ParkerB.Sc. Hons., Victoria University of Wellington, 2008M.Sc., Victoria University of Wellington, 2011A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December, 2017c© Amanda J. Parker, 2017iiAbstractWe use a molecular dynamics (MD) framework to study the mechanical propertiesof triblock copolymer materials which form thermoplastic elastomers (TPEs). Thesematerials form physical, rather than chemical, cross-links as a result of their phase-separated nano-structure. It is difficult, or impossible, to measure the details ofnetwork chains and monomers experimentally. However, it is these microscopicfeatures that give rise to the material’s elastomeric properties. We use a coarse-grained bead-spring model which retains the vital details of the chain network andthe nano-structured regions while removing unnecessary atomistic detail.We first present a simulation strategy for the equilibration of nano-structuredcopolymer melt morphologies. MD simulations with a soft pair potential that allowsfor chain crossing result in efficient modelling of phase segregation. We successfullyreintroduce excluded volume pair interactions with only a short re-equilibration ofthe local structure, allowing configurations generated (with this method) to be usedfor studies of structural and mechanical properties.We then study the plastic deformation of triblock TPEs, probing the microscopicmechanisms operative during deformation and how they connect to the macroscopicstress response. We compare two deformation modes, uniaxial stress and strain,which emulate experimental tests and conditions around material failure. We findthat triblocks’ stress response exhibits a significant increase in strain hardeningcompared to homopolymeric chains. We analyse several microscopic properties,including: the chain deformation, monomer displacement, deformation and divisionof glassy domains, and void formation.We introduce an entropic network model for the stress response utilising mi-croscopic information about chain configurations and their topological constrains.The model assumes additive contributions from chain stretch and the stretch be-tween chain entanglement points and results in quantitative prediction of the stressresponse. Only one parameter fit is required to describe both triblock and ho-mopolymers systems. We compare our model to recent entropic models developedfor vulcanised rubbers and probe its limitations and more general applicability. Ex-tensions to more complicated architectures are possible (e.g. stars).iiiLay SummaryWe study materials that combine properties of both rubbers and plastics. Thesematerials are made of polymers, extended chain-like molecules. The key to howthey perform under material tests depends on how these chains are folded and howthey interact with each other. We use a computational model to study polymerson this small scale while the material is being stretched or deformed. We canmake measurements and collect statistics that are difficult or impossible to obtainexperimentally. Using the results of this computational modelling we develop atheory that connects how the chains behave to how the material behaves.ivPrefaceThe material in this thesis is the original work of the author, Amanda J. Parker.All work was supervised by Jo¨rg Rottler including guidance in analyses and projectdirection. Several portions of this thesis have been published as original manuscriptsin peer-reviewed journals with Amanda J. Parker as the primary author: section2.3 and Chapter 3 are published in [69], a version of Chapter 4 in [81] and the up tosection 5.6 of Chapter 5 in [99]. All other chapters are original work, first publishedin this document. Manuscripts were edited by J. Rottler.We obtained an implementation of the Z1-code from Martin Kro¨ger [66]. Allmolecular dynamics simulations are performed in LAMMPS [35] or HOOMD [36].vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Block copolymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Amorphous states . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Thermoplastic elastomers (TPEs) . . . . . . . . . . . . . . . 31.2 Describing polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Isolated chains . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Interacting chains . . . . . . . . . . . . . . . . . . . . . . . . 91.2.3 Entropic network models . . . . . . . . . . . . . . . . . . . . 121.2.4 Elastomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.5 Block copolymer elastomers . . . . . . . . . . . . . . . . . . . 161.3 Computational modelling . . . . . . . . . . . . . . . . . . . . . . . . 161.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Modelling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1 Molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.1 Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.1.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 212.2 Microscopic polymer model . . . . . . . . . . . . . . . . . . . . . . . 232.2.1 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.2 Equilibrated configurations . . . . . . . . . . . . . . . . . . . 242.3 Soft Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.4 Equilibrating configurations . . . . . . . . . . . . . . . . . . . . . . . 272.4.1 Random-walk configurations . . . . . . . . . . . . . . . . . . 27vi2.4.2 Brute force equilibration of microscopic model . . . . . . . . 282.4.3 Soft model equilibration method . . . . . . . . . . . . . . . . 292.4.4 Setting temperature and pressure . . . . . . . . . . . . . . . . 292.5 Mechanical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.6 Primative path analysis (PPA) and entanglements . . . . . . . . . . 302.6.1 Z1-method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 A Soft Potential Method For Equilibrating Block Copolymers . 333.1 Methods and models . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.1.1 Soft model implementation . . . . . . . . . . . . . . . . . . . 353.2 Comparison of short chain conformations . . . . . . . . . . . . . . . 393.2.1 Phase separation of 5-90-5 triblocks . . . . . . . . . . . . . . 393.2.2 Equilibration of N=100 chain conformations . . . . . . . . . . 413.2.3 Dynamics of short chains . . . . . . . . . . . . . . . . . . . . 423.3 Longer chains - equilibration with soft model only . . . . . . . . . . 423.3.1 Phase-separation of longer chains . . . . . . . . . . . . . . . . 453.3.2 Chain conformations of longer chains . . . . . . . . . . . . . . 453.3.3 Dynamics of longer chains . . . . . . . . . . . . . . . . . . . . 453.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 Molecular Mechanisms of Plastic Deformation in TPEs . . . . . . 504.1 Modelling methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.1.1 Glassy clusters . . . . . . . . . . . . . . . . . . . . . . . . . . 524.1.2 Applied deformations . . . . . . . . . . . . . . . . . . . . . . 534.2 Macroscopic stress response . . . . . . . . . . . . . . . . . . . . . . . 534.3 Microscopic response . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.3.1 Chain end-to-end distances . . . . . . . . . . . . . . . . . . . 564.3.2 Monomer dynamics . . . . . . . . . . . . . . . . . . . . . . . 604.3.3 Clusters and bridging chains . . . . . . . . . . . . . . . . . . 624.3.4 Void formation . . . . . . . . . . . . . . . . . . . . . . . . . . 684.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 A Molecular Simulation Based Network Model . . . . . . . . . . . 715.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.2 Molecular dynamics implementation . . . . . . . . . . . . . . . . . . 725.2.1 Microscopic model potential parameters . . . . . . . . . . . . 735.2.2 Defomation simulation details . . . . . . . . . . . . . . . . . . 745.3 Relating chain stretch to stress response . . . . . . . . . . . . . . . . 745.3.1 Effective stretch . . . . . . . . . . . . . . . . . . . . . . . . . 745.4 Entanglement Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.5 Entropic elasticity model development . . . . . . . . . . . . . . . . . 775.6 Application of the entropic network model . . . . . . . . . . . . . . . 815.7 Comparison to other network models . . . . . . . . . . . . . . . . . . 81vii5.7.1 Linking stretch length scales . . . . . . . . . . . . . . . . . . 835.7.2 Stretch components . . . . . . . . . . . . . . . . . . . . . . . 856 Limitations and Extensions . . . . . . . . . . . . . . . . . . . . . . . . 916.1 Entropic model at larger strain . . . . . . . . . . . . . . . . . . . . . 916.2 Does the network model generalise? . . . . . . . . . . . . . . . . . . 936.3 Star Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.3.1 Star morphology results . . . . . . . . . . . . . . . . . . . . . 966.3.2 Star deformation results . . . . . . . . . . . . . . . . . . . . . 1007 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103A Deformation Snapshots . . . . . . . . . . . . . . . . . . . . . . . . . . 113B Supplementary Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118C Freely-jointed Chain Bond Vector Components Calculation . . . 121D Faster Rate Deformations . . . . . . . . . . . . . . . . . . . . . . . . . 123viiiList of Tables3.1 Summary of potential parameters used for soft model equilibration(κ, χ), microscopic model deformation (εAA, εBB, εAB), chain lengths(N), and total simulated monomers (N ×M) in later chapters. . . . 47A.1 Pure stress simulation snapshots where minority monomers are or-ange, and majority monomers are purple. Shorter chains under purestress deformation as described in Chapter 4 . . . . . . . . . . . . . 114A.2 Pure strain simulation snapshots N = 100 where minority monomersare orange, and majority monomers are purple. As described in Chap-ter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115A.3 Pure strain simulation snapshots N = 300 where minority monomersare orange, and majority monomers are purple. As described in Chap-ter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116A.4 Pure strain simulation snapshots N = 500 where minority monomersare orange, and majority monomers are purple. As described in Chap-ter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117D.1 Summary of deformation parameters, microscopic model (εaa, εbb,εab), strain rate and chain lengths (N). . . . . . . . . . . . . . . . . . 123ixList of Figures1.1 (a) A TPE material (b) [1] Transmission electron microscopy imageof SBS (stryrene-butadiene-styrene) copolymers. Image covers a di-ameter of 0.5 µm, styrene domains are 10-24 nm in diameter. (c)Polymer chain structure of TPE (d) Chemical structure of polystyrene 11.2 (a) Diblock and (b) ABA linear symmetric triblock with 50% A-typecontent. A lamellar structure is formed and (c) diblock chains cannotform bridges (d) triblock chains forming bridges . . . . . . . . . . . 21.3 (a) Chemical crosslinks in a traditional rubber (b) physical crosslinksin a thermoplastic elastomer. . . . . . . . . . . . . . . . . . . . . . . 31.4 Diagram of polymer length-scales (a) bond-length b (b) end-to-enddistance Ree (c) MSID(N-2), a demonstration that calculating themean squared internal distance MSID(n) requires averages for all por-tions of the chain with length n = |i− j| . . . . . . . . . . . . . . . 61.5 Force extension curves in term of end-to-end distance (Ree) scaled bymaximum chain extension (Rmax). Red: eq. (1.13) infinite maximumextension. Blue eq. (1.16) finite maximum extension . . . . . . . . . 101.6 (a) Entanglement points in a system of polymers (marked with squares)(b) Tube model: tube (grey) around polymer (red line) with primitivepath (grey dashed line) . . . . . . . . . . . . . . . . . . . . . . . . . 111.7 Monomer mean-square displacement for tube model dynamics regimesinteracting polymer chains based on Doi and Edwards [13] figure 6.10 121.8 Relaxation modulus G(t) after step strain with the plateau modu-lus and Rouse and reptation time-scales marked. Based on Doi andEdwards figure 7.3 [13] . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 a) Periodic boundary conditions: central simulation cell stored andminimum image convention applies b) Unwrapped boundary condi-tions c) Wrapped boundary conditions. . . . . . . . . . . . . . . . . 222.2 Potentials in microscopic model. (a) Blue: Lennard-Jones pair po-tential truncated and shifted. Red: FENE bond potential. Green:Sum of potentials - applies to bonded monomers. (b) Blue: Lennard-Jones pair potential truncated and shifted. Purple: Lennard-Jonespair potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24x2.3 a) Representation of LJ excluded volume beads on two chains, chainscannot cross. (b) Representation of soft model beads on two chains,chains are able to pass through each other. (c) Soft (red) and Lennard-Jones (blue) pair potentials. (d) Radial distribution functions for soft(red) and microscopic (blues) models. . . . . . . . . . . . . . . . . . 262.4 500 5-90-5 triblock copolymers. Blue A-type monomers, Red B-typemonomers shown for coordinates x + y < L/2. (a) Initial randomwalk chain arrangement (b) Final equilibrated chain arrangement . 282.5 Determining the glass transition temperature . . . . . . . . . . . . . 292.6 a) Configuration of two entangled chains b) the chain is considered asan infinitely thin line and the chain ends are fixed c) the length of thepath is monotonically decreased and the number of kinks decreasesd) converged final state [65] figure 1. . . . . . . . . . . . . . . . . . . 312.7 Moves used in Z1-method multiple disconnect path minimisation. a)Nk → Nk − 1 b) Nk constant c) Nk → Nk + 1. Based on [66] figure 1. 323.1 Triblock copolymer morphologies obtained from 100 bead-spring poly-mers in a 5-90-5 configuration with Lennard-Jones interactions (toprow) and the corresponding soft potential (bottom row) for two dif-ferent segregation strengths (left column: ǫAB = 0.8ǫ/χ0 = 1, rightcolumn: ǫAB = 0.5ǫ/χ0 = 2, see text for details). Minority phasebeads are shown in blue. . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 (a) Structure factors and (b) radial distribution functions for A-beadsfrom simulation of 500 chains with N=100. Solid symbols show resultsfrom the soft polymer simulations after equilibration for 105τ withsoft potential model, open symbols correspond to the LJ polymerafter 106τ . The segregation strengths for the parameter pairs areχ = 1.9(ǫAB = 0.8ǫ), χ = 2.0(χ0 = 1), χ = 7.3(ǫAB = 0.5ǫ), χ =7.0(χ0 = 2) , χ = 8.6(ǫAB = 0.4ǫ), χ = 9.1(χ0 = 2.5). In panel (b),dashed lines show additionally the soft polymer before reinsertion ofthe excluded volume (LJ) interactions. . . . . . . . . . . . . . . . . . 383.3 Equilibration of mean squared internal distances for chains with N=100beads. (a) LJ polymer with ǫAB = 0.5ǫ, (b) Soft polymer with χ0 = 2where the additional black curve is after 200τ LJ reintroduction (c)MSID of final configurations of homopolymers. The insets give therelative deviation of the final MSID for the soft and LJ polymers. . . 403.4 Monomer mean square displacement for chains of length N=100 withseveral phase separation strengths. . . . . . . . . . . . . . . . . . . . 413.5 Morphology of 800 triblock copolymer chains in a 25-450-25 configu-ration after equilibration with soft pair potential for 105τ and χ0 = 2.Minority monomers are shown in blue. . . . . . . . . . . . . . . . . . 43xi3.6 Structure factor between A-beads (a) after equilibration for 105τ withsoft potential for varying chain lengths N=100,200,300,500, (b) forchains of length N=500 during equilibration with soft potential and(c) for chains N=200 during equilibration with soft potential. . . . . 443.7 Mean square internal distances during equilibration of long chainsN=500 with (a) LJ polymer (100 chains) and (b) soft polymers (800chains) and (c) soft homopolymers (800 chains). . . . . . . . . . . . . 483.8 Mean square displacement during equilibration. The upper threecurves correspond to the soft potential equilibration of chain lengthsN=100 and N=500 with χ0 = 2. For comparison the behavior ofN=500 homopolymers is also shown. The lower three curves show theequivalent curves for equilibration of the LJ polymers with ǫAB = 0.5ǫfor the triblocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.1 Snapshots of morphologies before deformation in a box of size L =80σ. Top: 1,600 triblock copolymer chains in a 5-90-5 configuration.Bottom: 960 triblock copolymer chains in 25-450-25 configuration.Minority monomers are in blue and majority monomers in red. . . . 544.2 True stress vs. engineering strain for (a) pure uniaxial strain and (b)pure uniaxial stress deformation. Solid lines: triblocks. Dashed lines:homopolymers. Chains lengths N are given by the colour key. . . . 554.3 Snapshots of the 25− 450− 25 triblock system during pure uniaxialstrain deformation at strains 1, 3, 5, 7. Minority monomers are in blueand majority monomers in red. Here the average density decreasesas ρ(ǫ) = ρinitial/(ǫ+ 1). . . . . . . . . . . . . . . . . . . . . . . . . . 574.4 Snapshots of the 25− 450− 25 triblock system during pure uniaxialstress deformation at strains 1, 3, 5, 7. Minority monomers are in blueand majority monomers in red. Minority monomers are not displayedin the upper portion. . . . . . . . . . . . . . . . . . . . . . . . . . . 584.5 Non-affine strain of the end-to-end distance during (a) pure uniaxialstrain deformation (b) pure uniaxial stress deformation. Solid lines:triblocks. Dashed lines: homopolymers. Chains lengths N are givenby the legend. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.6 Non-affine monomer displacement magnitudes calculated between snap-shots separated by strains of 2.5%. Top (a)(b): Filled symbols - ho-mopolymers, open symbols - triblocks. Bottom (c)(d): Crosses: Atype beads in triblock, Filled triangles: B type beads in triblocks.Left (a)(c): Pure uniaxial strain. Right (b)(d): Pure uniaxial stress. 61xii4.7 Asphericity of minority monomer domains (a,b), number of minoritymonomer domains (c,d) and proportion bridging chains (e,f) duringpure uniaxial strain deformation (left - a,c,e) and pure uniaxial stressdeformation (right - b,d,e). Green: 25 − 450 − 25 triblocks, red:15 − 270 − 15 triblocks, blue: 5 − 90 − 5 triblocks. All data reflectaveraging over three simulation boxes containing 480,000 monomers. 634.8 Percentage of chains which switch from (a,b) bridges to loops and(c,d) loops to bridges in a strain interval of 0.5 in pure uniaxial straindeformation (a,c) and pure uniaxial stress deformation (b,d). Greensquares: N = 500 triblocks, red circles: N = 300 triblocks, bluetriangles: N = 100 triblocks. All data reflect averaging over threesimulation boxes containing 480,000 monomers. . . . . . . . . . . . 654.9 Mean-squared end-to-end distance scaled by chain lengthN for chainsthat were (a,b) bridges at t = 0 and (c,d) loops at the beginningof the deformation for pure strain deformation (a,c) and pure stressdeformation (b,d). Green: N = 500 triblocks, red: N = 300 triblocks,blue: N = 100 triblocks. All data reflect averaging over a simulationbox containing 480,000 monomers. . . . . . . . . . . . . . . . . . . . 664.10 Monomer density relative to average density (black,red,yellow scale)and minority monomer density relative to average density (white toblue scale) in a plane of width ∼ 4σ located at x = 0. Five differentvalues of strain = 2, 3.5, 5, 6.5, 8 are shown during pure uniaxialstress deformation of a 25− 450− 25 triblock system. Voids (black)begin to appear for strains ≥ 6.5 and reach a volume fraction of ∼ 1%for ǫ = 8. Red dashed lines highlight one minority monomer domainbreaking apart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.11 Minority monomer density relative to average density (white to bluescale) and monomer density relative to average density (black,red,yellowscale) during the early stages of pure strain deformation of the 25−450 − 25 triblock (a) and (b) and in the N = 500 homopolymer (c).Densities are shown in a plane of width ∼ 4σ in the x = 0 plane atstrains = 0, 0.05, 0.1, 0.15, 0.2. . . . . . . . . . . . . . . . . . . . . 705.1 (a) Stress response in terms of the entropic elasticity factor g(λ) =λ2−1/λ. (b) Effective stretch geff = λ2cz−λ2cx in terms of componentsof the chain end-to-end stretch vs. g(λ), black line g(λ) = geff . (c)Stress vs. effective stretch. Green: triblocks, red: homopolymers,blue: cut chains for N = 300 (), N = 500 (◦), and N = 800 (△). . 755.2 Entanglement length (from Z1 analysis) vs effective stretch. Green:triblocks, red: homopolymers, blue: cut chains for N = 300 (),N = 500 (◦), and N = 800 (△). . . . . . . . . . . . . . . . . . . . . . 77xiii5.3 Ratios of the chain end-to-end distance to the maximum possibleend-to-end separations for (a) the chains segments between entangle-ments and (b) the midblock segments in the triblock chains. Green:triblocks, red: homopolymers, blue: cut chains for N = 300 (),N = 500 (◦), and N = 800 (△). . . . . . . . . . . . . . . . . . . . . . 795.4 MD simulation stress vs. stress calculated from models for triblocksand homopolymers/cut chains Green: triblocks, red: homopolymers,blue: cut chains for N = 300 (), N = 500 (◦), and N = 800 (△). . 805.5 Comparison of model (points) to simulation data (lines) in terms ofg(λ) = λ2 − 1/λ. Green: triblocks, blue: homopolymers, red: cutchains for N = 300 (), N = 500 (◦), and N = 800 (△). . . . . . . . 825.6 Non-affine strain model. Green: triblocks, red: homopolymers forN = 300 (), N = 500 (◦), and N = 800 (△). Black line: mastercurve for our model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.7 Comparison of chain stretch λc to entanglement stretch λk. Green:triblocks, red: homopolymers for N = 300 (), N = 500 (◦), andN = 800 (△). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.8 Entropic elasticity factors for chain (g(λc)) and entanglement stretches(g(λc)). Green: triblocks, red: homopolymers for N = 300 (),N = 500 (◦), and N = 800 (△). . . . . . . . . . . . . . . . . . . . . . 845.9 Two extremes of the deformation of entanglements under pure uni-axial strain deformation in two dimensions: affine and like a freely-jointed chain. The relative change to the components of Rk is showin red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.10 Stretch components compared to expected volume conserving stretchλ2x = 1/λz. Top: Triblocks. Bottom: Homopolymers. Closed sym-bols: chain end-to-end stretch. Open symbols: entanglement stretch.N = 300 (Red), N = 500 (Green), and N = 800 (Orange) . . . . . . 875.11 Chain and entanglement entropic elasticity factors, comparison tobounding predictions. Open symbols: simulation results. Black line:affine prediction. Closed symbols: freely-jointed chain (FJC) predic-tion. N = 300 (Red), N = 500 (Green), and N = 800 (Orange) . . . 896.1 Agreement between model and simulated stresses. Open symbols:Entanglement vectors recalculated at each point. Closed symbol: ini-tial entanglement vectors used to calculate stretch. N = 300 (red ),N = 500 (green ◦), and N = 800 (orange △). The vertical lines markthe change in slope for each curve with the corresponding colour,where the simulated results start to deviate from the model. . . . . . 92xiv6.2 Volume conserving uniaxial strain deformation, initial equilibratedconfigurations as described in Chapter 5 strained at a rate 10−5 (a)Asphericity of glassy clusters (b) Normalised number of glassy clusters(c) stress response. Red(◦) N = 300, green() N = 500 orange (△)N = 800. Coloured vertical lines indicate where the model becomesa poorer fit for the corresponding chain length. See slope change in 6.1. 946.3 Agreement of simulated stress and model as applied to pure uni-axial strain deformations presented in chapter 4 N = 100 (blue×)N=300(red), N = 500 (green ◦), and. Inset: homopolymers,main plot triblocks. Coloured vertical lines correspond to onset ofmore pronounced strain hardening as observed in 4 for the corre-sponding coloured chain lengths . . . . . . . . . . . . . . . . . . . . 956.4 (a) 3 arm star polymer. Arms are different colours and the glassyend regions are highlighted in a lighter shade. Each arm is N=300monomers connected to one central monomer. The outer 30 beadsare glassy. (b) Phase separated morphology with minority monomers:blue, and majority monomers: red . . . . . . . . . . . . . . . . . . . 976.5 Static structure factor for star polymers with varying numbers ofarms; green: linear triblock (2 arms), red: 3 arms, blue: 4 arms,orange: 5 arms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.6 Probability that a glassy cluster containing one arm from a star willcontain a total of x arms from that star. Star polymers with varyingnumbers of arms; green ×: linear triblock (2 arms) , red : 3 arms,blue ◦: 4 arms, orange △: 5 arms. Black lines are reproduced fromfig. 4 in ref. [106],  3 arms, △ 5 arms. . . . . . . . . . . . . . . . . 996.7 Probability that a star’s arms will be in x different glassy clusters.Star polymers with varying numbers of arms; green ×: linear triblock(2 arms) , red : 3 arms, blue ◦: 4 arms, orange △: 5 arms. Blacklines are reproduced from fig. 4 in ref. [106],  3 arms, △ 5 arms. . 996.8 Volume conserving uniaxial strain deformation for star polymers withvarying numbers of arms; green: linear triblock (2 arms) , red: 3 arms,blue: 4 arms, orange: 5 arms. Each arm is 300 monomers the outer30 are hard, the inner 270 are soft. . . . . . . . . . . . . . . . . . . 100B.1 Determining the statistical independence of Rc, Rk. Green: triblocks,red: homopolymers, blue: cut chains for N = 300 (), N = 500 (◦),and N = 800 (△). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118B.2 Test of the 8 chain model assertion relating stretches to their compo-nents. (a) Entanglement stretch (b) Chain stretch . Green: triblocks,red: homopolymers, blue: cut chains for N = 300 (), N = 500 (◦),and N = 800 (△). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119xvB.3 Comparison Cartesian components of (a) Entanglement stretch (b)Chain stretch . Green: triblocks, red: homopolymers for N = 300(), N = 500 (◦), and N = 800 (△). . . . . . . . . . . . . . . . . . . 120D.1 Volume conserving uniaxial strain deformation, identical equilibratedconfigurations to Chapter 5 but strained more quickly at a rate 10−4(a) Asphericity of glassy clusters (b) Normalised number of glassyclusters (c) stress response. Red(◦) N = 300, green() N = 500orange (◦) N = 800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124D.2 Volume conserving uniaxial strain deformation, identical equilibratedconfigurations to Chapter 5 but strained more quickly at a rate 10−4.Ge = 0.018 Red N = 300, green N = 500, orange N = 800. Dashedlines: triblocks, solid lines: homopolymers . . . . . . . . . . . . . . . 1251Chapter 1IntroductionThis thesis focuses on the multi-scale problem of modelling block copolymers. Weendeavour to contribute to the understanding a class of interesting, economicallyrelevant materials: thermoplastic elastomers (TPE). In doing so we advance compu-tational modelling approaches and theoretical descriptions of these materials: whichoffer possibilities of further generalisation and application. Figure 1.1 shows an ex-ample of the scales of detail within a TPE material; firstly there is a photo of TPEmedical tubing followed by a transmission electron microscopy (TEM) image of thematerial. In this image there is an array of white spots (diameter 10-24nm) on agrey background, showing the periodic nanostructure of the TPE material. On asmaller scale again we can see that circles are in fact made up of the ends of polymerchains and finally that those chains are made up of repeated chemical units.(a) (b) (c) (d)Figure 1.1: (a) A TPE material (b) [1] Transmission electron microscopy image ofSBS (stryrene-butadiene-styrene) copolymers. Image covers a diameterof 0.5 µm, styrene domains are 10-24 nm in diameter. (c) Polymer chainstructure of TPE (d) Chemical structure of polystyrene1.1 Block copolymersBlock copolymers form tunable nanostructured materials. This feature is exploitedcommercially to produce a wide range of materials for specific applications. Poly-2(a) (b)(c) (d)Figure 1.2: (a) Diblock and (b) ABA linear symmetric triblock with 50% A-typecontent. A lamellar structure is formed and (c) diblock chains cannotform bridges (d) triblock chains forming bridgesmers are extended chain molecules made up of repeat units called monomers. If allthe monomers in a chain have the same chemical structure it is defined as a ho-mopolymer, but of course far more complicated combinations of differing monomersare possible. Block copolymers are made up of connected blocks. The simplestincarnation is a linear diblock-copolymer with the two block-types noted as A andB. These materials are particularly interesting because different blocks can phase-separate to form a nanostructured material.The shape of the nano-structured regions is dictated by: the strength of thephase segregation (χ), the number of monomers in each chain (N) and the propor-tion of the chain that is made up of the differing blocks. The morphologies for alinear symmetric ABA triblock copolymer vary from spherical clusters centred on abody centered cubic (bcc) lattice, cylindrical columns on a square lattice, gyroidal,and lamellar phases [2, 3]. Diblock copolymers have a very similar phase diagramto symmetric ABA triblocks. However, the triblocks have additional complexity intheir chain topology since single chains can form a bridge across two phase-separatedregions. Figure 1.2 shows representation of a lamellar structure for these two poly-mers, highlighting this difference in topology.1.1.1 Amorphous statesThe polymer systems we focus on are amorphous, ie. there is no crystalline order.Above the glass transition temperature Tg, polymer molecules form a viscoelastic.3(a) (b)Figure 1.3: (a) Chemical crosslinks in a traditional rubber (b) physical crosslinksin a thermoplastic elastomer.The polymers can still flow past each other, but are constrained in their motion byinteractions with other chains. Below Tg the viscosity increases by orders of magni-tude, below which long range motion ceases. However, the amorphous arrangementof monomers remains - forming a glass, a hard and sometimes brittle material [4].Notably, this glass transition is fully reversible once Tg is again exceeded. Whilecrystalline polymer phases with significant chain alignment can form at lower tem-peratures or in response to deformations, we will be focussing on amorphous systems.Some common polymers are in the glass state at room temperature (e.g. polystyrene),others are soft (e.g. polybutadiene, polyisoprene). If the constitutive parts of a blockcopolymer have different Tg, one region can be hard while the other remains soft.Therefore, not only are block copolymers nanostructured, but these regions can havevastly different material properties. The combination creates unique materials withproperties which are tunable depending on the desired application.1.1.2 Thermoplastic elastomers (TPEs)Traditional rubbers are homopolymers which have undergone vulcanisation - a slow,irreversible curing process. They are known as thermosets. Rubbers were used asmaterials long before their molecular structure started to be understood. Polyiso-prene forms naturally in rubber trees. In terms of macroscopic properties a rubber;stretches rapidly and considerably under tension, exhibits high tensile strength andstiffness when stretched, and retracts rapidly upon the release of stress, regainingits original shape.On a microscopic scale, thermosets are made up of long polymers (high molecularweight) in an amorphous state, above Tg. The vulcanisation process forms chemicalcrosslinks between polymers which restrict their overall mobility (see figure 1.3a).4ABA triblock copolymer TPEsAs mentioned above, triblock copolymers are particularly interesting since an in-dividual chain can bridge from a hard region through a soft one and end up in ahard region (and vice versa). In the regime where a low percentage of the chains areA-type ends, spherical phase-separated regions are formed (see figure 1.3b). Whenthere are soft spheres in a hard background, the material can have enhanced ductilityand toughness [5]. For example, when the majority phase is made of polystyrene thematerial is a clear flexible thermoplastic [6]. The reverse scenario (hard spheres in asoft background) encompasses a large class of materials that have been commerciallyavailable for more than 50 years, thermoplastic elastomers [6, 7]. The hard regions(typically polystyrene) act as physical cross-links between chains. The soft regionsare most commonly polybutadiene (SBS) or polyisoprene (SIS) and more recentlypolyisobutylene (SIBS). They are utilised in applications as diverse as vibration ab-sorption, running shoes, yoga mats, and medical applications where flexibility andstrength are desirable (eg. tubing, valves) [8].Compared to typical thermoset rubbers, TPEs have notable advantages. Ratherthan undergoing a chemical curing process, thermoplastic elastomers obtain similarelastic properties by the physical, reversible glass transition. They can be rapidlyreformed (due to the dynamic nature of the glass transition). They are thereforerecyclable (unlike vulcanised rubbers) and can be processed using typical plastictechniques such as injection moulding and extrusion [5, 6].The mechanical response of TPEs can be tested easily by deforming the mate-rial, and the phase-separated regions can be resolved using transmission electronmicroscopy (TEM) or through small/wide angle Xray scattering (SAX/WAX). Ona microscopic level measurements become more limited. Computational modellingallows the study of microscopic details and statistics that are difficult or impossibleto obtain experimentally, bridging between theory and experiments.Other block-copolymer TPEsThere are several other polymer systems which demonstrate bridging between phase-separated regions and can therefore form TPEs. Multiblock copolymers with anABABA... structure are commercially available as thermoplastic elastomers [6].Typically one block forms crystalline regions while the other remains amorphous.ABC polymers can form more complex phase-separated structures including coreshell systems which also allow bridging given all three phases are inclined to separate[9]. Non-linear or star polymers are being looked to for the creation of strongerelastomers with a higher styrene content [10–12].Differing chain-morphologies and block combinations offer a huge array of possi-bilities to continue to develop tunable materials. However, there can be a significantcost to attempting to create new materials in a lab - with no certainty of their prop-erties. Here too we see a place for computational modelling, contributing to the5guiding of material development.1.2 Describing polymersThe theory of polymer melts is well developed and covered in several texts [13–16].This section expands upon details of polymer physics and elastomeric materials toprovide a clear understanding of the choice of simulation and theoretical methodsused to understand TPEs. We will start with the simpler case of describing a singlechain, covering: the relevant terminology, chain statistics, monomer dynamics andthe force resulting from stretching. We then cover the same topics for the morecomplex case of interacting chains and the resulting material properties, includingdescriptions of entropic network models that could be applied to TPEs.1.2.1 Isolated chainsIdeal freely-jointed chainWe start by considering how to describe a single isolated polymer chain. In a simpledescription of a polymer there are two key length-scales to consider: the separationb of monomers along the backbone of the chain, and the direct distance Ree betweenthe chain’s first and last monomers, also called the end-to-end distance. Both arevisualised in figure 1.4. A freely-jointed chain has b =const. fixed and the chain canbe considered as a random walk. For a chain with N monomers these two lengthsare on average related byR2ee = (N − 1)b2 (1.1)It should be noted that this relationship assumes an isotropic environment (alldirections are equivalent), so it ceases to hold if the chain is stretched or otherwisedeformed.We can also consider the distribution of bonds angles between each set of 2 bondsalong the chain. If there are some constraints or angle dependent interactions suchthat monomers cannot completely backfold (achieve a bond angle of 0), the aboveequation becomes modified to an effective bond-length in terms of the characteristicratio c∞,c∞ =R2ee(N − 1)b2 , (1.2)where c∞ = 1 by definition for a freely jointed chain. In the limiting case of arod (where all bond angles are 180) Ree = (N − 1)b so the maximum value thecharacteristic ratio can take is the number of bonds cinfty = (N − 1).This latter description is more appropriate for real polymer chains because theexcluded-volume of monomers restricts backfolding, therefore c∞ > 1. However,Kuhn and Gru¨n [17] showed that an ideal freely-jointed chain can approximate areal chain with a rescaled bond length. A number of bonds are combined in a Kuhn6b(a)MSID(1) = b2(b)MSID(N) = R2eeRee(c)MSID(n) =〈R(|i− j|)2〉Figure 1.4: Diagram of polymer length-scales (a) bond-length b (b) end-to-end dis-tance Ree (c) MSID(N-2), a demonstration that calculating the meansquared internal distance MSID(n) requires averages for all portions ofthe chain with length n = |i− j|segment with the end-to-end distance of each segment defining a new bond-length,the Kuhn-length. The chain is then well described as a random walk with chainstatistics governed by equation (1.1).The full internal conformation of the chain can be characterized by the mean-squared internal distance of monomersMSID(n) =〈R(|i− j|)2〉 (1.3)where R(|i − j|) is the distance between two monomers with indices i < j. Theaverage is taken over all chain segments of size n = |i − j| over an ensemble ofchains. According to eq. (1.2), MSID(n)/nb2 → c∞ for n → N . Figure 1.4c)demonstrates this averaging.Dynamics - The Rouse ModelA common simple model for the dynamics of non-interacting polymers is the Rousemodel [18]. A polymer is considered to be made up of N monomer beads linearlyconnected by N − 1 simple harmonic bonds. Choosing the spring constant k suchthatk =3kBTb2(1.4)7gives a Gaussian distribution of bond-lengths consistent with an entropic restoringforce. In this way the harmonic springs are connected to the restoring force of astretched fixed bond length freely-jointed chain. The Rouse model accounts for howthis connectivity restricts the movement of monomers.The monomers undergo Brownian motion. There is a friction force contribution,with friction coefficient ζ constant for all monomers, and random forces ~fn whichsatisfy the equipartition theorem with zero mean variance that simulates coupling toa heat bath (as we will see in section 2.1.1). The monomer motion is over-damped.Therefore, the spring forces on each bead balance the friction and stochastic forces,giving the following set of coupled differential equations for the position ~Rn of thenth monomerζd~Rndt= −3kBTb2(2~Rn − ~Rn−1 − ~Rn+1) + ~fn. (1.5)and for the first and last monomers,ζd~R1dt= −3kBTb2(~R1 − ~R2) + ~f1 ζ d~RNdt= −3kBTb2(~RN − ~RN−1) + ~fN . (1.6)Taking a Fourier transform gives uncoupled equations for the amplitudes ~Xp forp = 0 . . . N −1 normal modes. The monomer position vectors can then be expressedas a linear combination of these modes~RN = ~X0 +N∑p=1~Xp cos(pπN(N +12)), (1.7)where ~X0 is the centre of mass.The uncoupled equations for the normal modes are solved in terms of the stochas-tic forces and then averaged to give time correlation functions for the Fourier am-plitudes〈 ~Xp(t) ~Xp(t)〉 = kBTkpexp(−t/τp), (1.8)where p > 0, and kp is the wave-number of that Fourier mode. The longest relaxationtime-scale τ1 relates to how long the chain takes to diffuse a distance equal to Ree(called the Rouse time τr):τr = τ1 =ζN2b23π2kBT(1.9)and scales with N2. This time-scale marks the cross-over from Rouse to diffusivedynamics. The monomer mean-square displacement (MSD) during the so-called8Rouse regime scales as t1/2〈(~Rn(t)− ~Rn(0))2〉=(4kBTb23πζ) 12 √t, t≪ τr(6kBTNζ)t, t≫ τr(1.10)Beyond τ1 the mean-square displacement is dominated by the center of mass diffu-sion.Mechanical ResponseGiven that we are interested in elastomers, we want to consider the effects of stretch-ing out the polymer chains. Elastomers and polymers in general are in a very differ-ent regime than the majority of materials. While metals might typically be deformedby a few percent before failure, rubbers can be stretched up to 1000% [19, 20]. Tak-ing an ideal chain and stretching its ends leads to a restoring force. The chain actsas an entropic spring; the probability distribution P of configurations is a functionof Ree and therefore entropy S is as well,P (Ree) ∝ exp(− 3R2ee2Nb2)(1.11)S(Ree) = kBln(P ) + const (1.12)From the configurational free energy we find the forcef(Ree) =dFdRee=3kBTNb2Ree (1.13)As the chain is stretched, it is forced into fewer configurations, resulting in arestoring force. When a strain is applied to a polymeric material this entropic effectcauses strain-hardening. This entropic force is intuitive if one considers the extremecase of a freely-jointed chain with a fixed bond length stretched to its limit, there isonly one way for the bonds to be arranged. As the ends are brought closer togetherthere are more possible paths for the polymer to follow between them.Finite extensibilityChains with simple harmonic bonds can be stretched even when they are limited toone configuration. However, an ideal chain with a fixed b has a maximum extensionRmax = (N − 1)b, and so the force further diverges as this limiting chain-length9is approached. Considering the end-to-end vector to be in the z-direction, then inpolar coordinates we can project the lengthRee =N−1∑i=1b cos(θi) (1.14)where θi is the angle from z-axis of the ith bond. This allows the partition functionto be formulated in such a way that it becomes the product of identical integrals.The free energy can then be calculated, and differentiated to give〈Ree〉 = b(N − 1)[coth(fbkBT)− 1fb/kBT](1.15)or in terms of the forcef(Ree) =kBTbL−1(ReeRmax)(1.16)where L−1 is the inverse of the Langevin functionL(x) = coth(x)− 1x(1.17)Figure 1.5 demonstrates that the difference in these two force extension curvesbecomes significant at Ree/Rmax > Interacting chainsAs soon as we consider an ensemble of interacting chains, topology becomes a con-sideration, and descriptions of the system quickly become more complex. Figure1.6a shows a cartoon of a system of chains, highlighting the points where they cross,called entanglement points. Chain-uncrossability means that the presence of otherchains restricts the movement of every chain.The tube modelThe Edward’s tube model is a mean field picture that describes the dynamics ofpolymers with such constraints [21, 22]. A tube is caused by the other polymerssurrounding a chain which restricts its motion. Figure 1.6(b) shows a single high-lighted chain and tube within a background of other chains. The axis of the tube,called the primitive path [13], is the shortest path between the ends of a polymerchain that has the same topology. The entanglement length Ne is defined as thenumber of monomers between entanglement constraints. If all primitive paths areminimised in length, while preserving the topology, the entanglements appear askinks. Chains that are short relative to the entanglement length are consideredweakly or not entangled and adhere to Rouse model dynamics.10 0 2 4 6 8 10 0  0.2  0.4  0.6  0.8  1f [b/KbT]Re / RmaxFigure 1.5: Force extension curves in term of end-to-end distance (Ree) scaled bymaximum chain extension (Rmax). Red: eq. (1.13) infinite maximumextension. Blue eq. (1.16) finite maximum extensionDynamicsThere are several regimes describing the dynamics of this model. On short time-scales polymer segments (Ne monomers) move freely within the tube displayingRouse dynamics. When they move on a length-scale where they are impacted bythe presence of the polymer network, they are then constrained to move along thetube. Only the ends explore new space, moving the tube along - this motion is calledreptation [23, 24]. On this time-scale, the mean-square displacement of monomers〈(r(t)− r(0))2〉 ∝ t1/4. Reptation follows from a combination of the Rouse con-straints and the tube constraints (chain friction scales linearly with chain length).The following regime is governed by only the tube constraints, with the MSD t1/2.Finally there is a cross-over to diffusive dynamics with a disentanglement time whichscales as N3. See figure 1.7 for the MSD covering these regimes. The entanglementlength Ne characterises the crossover from the Rouse to reptation regime and canbe related to both the tube diameter and cross-over time.Mechanical responseTaking this knowledge of entanglements, we can also qualitatively understand thedifferent regimes of response for polymers under deformation: viscous flow, rubber-like elasticity, viscoelasticity, and glassy behaviour. Above Tg the material willexhibit viscous flow, chains can completely slip past each other. For longer chains,entanglements inhibit this slippage, so the material does not flow. The entangle-11(a)(b)Figure 1.6: (a) Entanglement points in a system of polymers (marked with squares)(b) Tube model: tube (grey) around polymer (red line) with primitivepath (grey dashed line)12Figure 1.7: Monomer mean-square displacement for tube model dynamics regimesinteracting polymer chains based on Doi and Edwards [13] figure 6.10ments form a transient network of restraints similar to the cross-links in a rubber.The response depends on the rearrangement of chain segments between entangle-ments, resulting in rubber-like elasticity. Closer to Tg, the response remains similarbut time-dependent and is therefore considered viscoelastic. Below Tg the motionof chain segments becomes very limited. The material is a glass, first displaying alinear response to deformation followed by yielding, softening and hardening.The response to step stressing can be related to the bond vectors in the meltwhich are described by the tube model. The plateau modulus G(0)N = σplateau/γcan be measured in the melt experimentally from the stress at the plateau observedafter initial yielding σplateau scaled by the rate of shear γ, see figure 1.8. From thetube modelG(0)N =4ρkT5Ne(1.18)where ρ is the monomer density. This theory, relating entanglements to an experi-mentally measurable quantity, is one of the great achievements of polymer physics.1.2.3 Entropic network modelsThe tube model describes chain entanglement effects. Elastomeric materials havelong polymers that experience these constraints from other chains but also havecross-links restraining their motion. Entropic network models consider the effect ofsuch constraints to calculate a stress response to deformation. The simpler modelsonly take into account either entanglements or cross-links.13Figure 1.8: Relaxation modulus G(t) after step strain with the plateau modulus andRouse and reptation time-scales marked. Based on Doi and Edwardsfigure 7.3 [13]Affine network modelThe affine network model is the simplest approach for an unentangled polymer withno finite length constraints and often a starting point for more complex models. Itis set up like a Kuhn model with Ne defining the bond length of a freely-jointedchain. The free energy and therefore the force-extension expression for a freely-jointed chain (eq. 1.16) can then be expressed in terms of chain density (ν = ρ/Ne).Elastic stress is then defined in terms of the cross-correlation of end-to-end vectors:σelαβ = ν3kBTb2Ne〈RαRβ〉 (1.19)where α,β are Cartesian directions and the angle brackets average over all chains.This model takes a deformation in terms of a global stretch(λx, λy, λz) =(LxLx0,LyLy0,LzLz0)(1.20)and assumes Ree deforms affinely by the same stretches, essentially assuming aperfect coupling of chain ends to the elastic background. If the starting chain con-figuration obeys random walk statistics the initial cross-correlation〈Rα0Rβ0〉 = 13b2Nδαβ (1.21)14and after stretching〈RαRβ〉 = 13b2Nλ2αδαβ (1.22)The full stress terms on the diagonal of the stress tensor also have a hydrostaticpressure contributionσαβ = σelαβ − Pδαβ (1.23)This is typically removed by subtracting two diagonal stress components. The affinenetwork model stress is thenσzz − σxx = νkBT (λ2z − λ2x) (1.24)A common test of elastic properties is volume conserving (λxλyλz = 1) uniaxialstrain, λz = λ, λx = λy =1√λfor whichσzz − σxx = νkBT (λ2 − 1λ) (1.25)The pre-factor defines the elastic modulus G = νkBT and the rest of the expressionis known as the entropic elasticity factor g(λ) = (λ2 − 1λ).Phantom-Network modelThe phantom-network model [25] allows fluctuations of the chain end-points. Thismodel is closer to a representation of a real cross linked system. Chain endpointsare treated as if attached to each other at cross links and the elastic backgroundwith additional chains. This modifies the stress from the affine network model bya factor (1− 2/φ) where φ is the network functionality, i.e. the number of chainsthat meet at each cross link8-chain modelThe Arruda-Boyce 8-chain model [26] again assumes an affine deformation of chainends, but also takes into account the finite extensibility of chain.Rather than considering all possible chain configurations this model assumesthat the chain stretch can be calculated by the stretching of a representative system.They take 8 chains, each running from the centre of a cube to one of its vertices andthereby derive a relationship linking the end-to-end chain stretch to the componentstretches,λchain =ReeRee0=√13(λ2x + λ2y + λ2z) (1.26)This allows the derivation of a stress formulation that includes Langevin hardeningeffects.σzz − σxx = GL−1(h)3h(λ2z − λ2x). (1.27)15here the h is the ratio of the end-to-end distance and its maximum value h =Ree/Rmax and the L−1 is the same inverse Langevin function in the force extensi-bility eq. (1.16).1.2.4 ElastomersElastomers are subject to both entanglement and cross link constraints, along withfinite length constraints at large deformations. The stress response of an elastomeris a usually assumed to be a linear combination of the contributions from entangle-ments (σe) and cross links (σc).σ = σc + σe (1.28)Ideally, we want to relate the microscopic behaviour of chains to the experimentallymeasurable stress response [27]. There has been significant work in this area. Herewe summarise some simple entropic models for stress response to an applied strain.The slip-tube modelIn the tube model, entanglement constraints are allowed to fluctuate, but remainat fixed points along the chain. The slip-tube model [28] relaxes this constraint,allowing ‘slip-links’ that can slide along chains but not pass through each other.The constitutive law derived from these assumptions is well approximated on therange 0.1 < λ < 10 byσzz =(Gc +Ge0.74λ+ 0.61λ−1/2− 0.35)(λ2 − 1λ)(1.29)whereGc =ρkBTNe(1− 2φ)(1.30)as in the phantom-network model andGe =47ρkBTNe(1.31)The slip-tube model agrees well with experimental data for vulcanized rubbers whenGc and Ge are fit to uniaxial stress-strain curves [20], as well as uniaxial strainsimulations for copolymer TPEs (to λ = 2) [29]. There is no finite chain lengthconsideration, so this model is inapplicable in the regime Ree → Rmax.The non-affine strain modelA recent model developed for cross-linked rubbers from Davidson et al. [30] com-bines the 8-chain model and the non-affine tube model [28, 31] so that the latter is16applicable over larger strains.σi = Gcλ2i13(∑j λj + 9λ2max∑j λj + 3λ2max)+Ge(λi − 1λi)+ p (1.32)The cross-link term is equivalent to the 8-chain model, taking non-affine behaviourinto account. Where the maximum stretchλmax =1g√N1− 2/φ,and the cross-link modulusGc =kbTρNeg2(1− 2/φ),depend on the unspecified function g which interpolates between affine network(g =√1− 2/φ)and phantom network behaviour (g = 1).The model successfully fits experimental data for natural and silicone rubbersunder uniaxial and biaxial strain. However, this requires fitting three parametersGc, Ge and λmax with a Monte-Carlo algorithm across both deformation modes.1.2.5 Block copolymer elastomersThe above elasticity models for elastomers are all developed for chemically cross-linked materials. However, triblock copolymer elastomers differ from typical ther-mosets in some key ways. The cross-links are not merely points but glassy inclusionsof a finite size. Therefore, the deformation of the glassy regions should also be con-sidered to fully understand the material properties. In addition, if there is anyirreversible deformation of the glassy regions then the recoverability of the elasticmaterial will be compromised. It has been suggested that normal entanglementsin the soft component act as effective cross-links because the ends are securely an-chored [7]. Chain pull-out (from the styrenic spherical ends) can occur at largestrains, which essentially makes the physical cross-links caused by the hard spher-ical domains breakable [32]. The network functionality φ can also be significantlyhigher than typical rubbers (φ = 4 usually assumed), which makes the PhantomNetwork model correction negligible.1.3 Computational modellingModelling physical systems is a delicate balancing act. The key is deciding whichdetails to include in the model and what to remove; while still obtaining results thatprovide physical meaning and insight. Coarse graining - the removal of degrees of17freedom - makes a model less precise, but also more generally applicable. In addition- less detail garners a lower computational resource cost.Multiscale modelling as a field focusses on developing material models with the‘right’ level of detail and connects models with different time and length scales. Weperform simulations using a coarse grained bead-spring model within a moleculardynamics framework, the details of which are given in Chapter 2. Simulating blockcopolymers is a multiscale problem, presenting a set of unique challenges. Chainscannot cross through each other due to hard excluded volume interactions so thetime-scale of the system is determined by reptation dynamics with a disentanglementtime that scales cubically with chain length [13]. Additionally, if we start with arandom distribution of chains, formation of phase separated regions requires matterto move on the length scale of these domains.1.4 OverviewIn this thesis we study the microscopic origins of the material properties of triblockcopolymer TPEs. To do so we perform molecular dynamics (MD) simulations ofdeformations, measuring the macroscopic stress response whilst also tracking andinterpreting microscopic features. Ultimately we develop a theoretical model linkingthe microscopic chain and entanglement stretches to the stress response, in excellentagreement with our simulation results.We choose MD implemented with a coarse grained bead-spring model as themethodology we use to study TPEs. This allows us keep the details of chains andmonomers without the cost of fully atomistic simulations. However, we still cannotreach macroscopic time and length scales in our modelling. Therefore, within theseconstraints, we do not try to quantitatively reproduce macroscopic stress magni-tudes. Instead we take steps to confirm that our results qualitatively match exper-imental results. These steps include;• simulating multiple chain lengths to confirm trends as chain length increases• deforming with multiple strain rates and using the slowest where possible• using the dynamics of monomers to confirm that chains are long enough to beentangled• using large enough simulation cells that periodic boundary conditions do notimpact results• ensuring well equilibrated chain conformations and phase-separated morpholo-gies• comparing and relating the triblock (macroscopic and microscopic) responsesto homopolymeric or other appropriate reference systems.18• comparing stress response qualitatively to experimental resultsMany of the simulation method challenges we face and successes we achieve can becouched in terms of the above list. These steps give us robust simulation resultswhich build upon and surpass the details of previous simulations of triblocks TPEs.The specific simulation methods used are introduced in Chapter 2 and further elab-orated upon in the relevant chapters that follow.In chapter 3 we implement a soft potential equilibration methodology whichsuccessfully equilibrates entangled polymer chains with the desired phase-separatedstructure within the MD framework. The methodology is motivated by the differencein scaling for constraint relaxation in the Rouse and reptation regimes of motion, N2vs N3. No pre-knowledge of the structure must be included in the model, the phasesself-segregate. We compare brute-force and soft model equilibration directly andexcellent consistency in chain conformations and phase-separated region structureis achieved. The computational savings are of at least a factor of 10 for unentangledchains. Using this method we can equilibrate entangled triblock chains allowing usto investigate the material properties of an entangled block-copolymer system.We simulate and compare two deformation modes in Chapter 4. One approx-imates an experimental uniaxial strain test and the other probes material failurethrough cavitation. We study the stress response for both triblock and homopoly-mer chains of varying lengths. Microscopic measures tracked during deformationinclude: how the glassy regions deform and separate, the response of bridging andlooping chains between glassy regions, void formation, and monomer dynamics bytype. We gain significant insight in to deformation details that cannot be studiedexperimentally, and confirm agreement with experiments for those that can.In Chapter 5 we introduce a model of entropic network elasticity that utilisessimulation insight into microscopic stretch. We study both the stretch of chain endsand the stretch between entanglement points (identified using the Z1-method, seebelow for details) in relation to the global stretch. We confirm the vital importanceof the triblocks’ network connectivity structure to the stress response using a ‘cutchain’ systems. Our model provides an excellent quantitative agreement betweenpredicted and simulation stretch. Only one parameter is fit, compared to 2-4 inother models, to successfully fit triblock, homopolymer and cut-chain stress results.We also compare this model success to other recent entropic models and establishwhich assumptions in these models hold or fail for our simulation data.Finally Chapter 6 probes the limitations of the applicability for our entropicmodel by comparing the simulation results from Chapters 4 and 5. These limitsin themselves provide insight into the link between microscopic quantities and thestress response. We also consider promising future directions for this work, includingpreliminary results for the equilibration and mechanical response of nanostructuredstar polymers.19Chapter 2Modelling Methods2.1 Molecular dynamicsMolecular dynamics (MD) essentially allows us to conduct experiments within acomputer, with a level of detail dictated by our choice of model and computationalresources. MD iterates the classical equations of motion with forces acting on eachparticle that are calculated from potentials acting between particles in the system[33]. Time is discretised into steps ∆t and velocity and position are updated itera-tively every time-step. This integration time-step is the limiting time-scale for MD.It must be small enough to capture all smallest relevant vibrations, but as large aspossible to improve efficiency [34].We use two molecular dynamics packages; LAMMPS [35] and HOOMD [36].Both can be parallelised across central processing units (CPUs) and graphical pro-cessing units (GPUs) though HOOMD is optimised for the latter. Both MD packagesuse the velocity-Verlet algorithm below to perform the time integration [37].Velocity-Verlet algorithmDenoting the velocity ~v, acceleration ~a, force f , potential U and time t the algorithmis applied simultaneously to all particles in the simulation as follows:1. Calculate the velocities at an intermediate (half) time-step:~v (t+ t/2,∆t) = ~v(t) + ~a(t)∆t/22. Update the positions using these velocities~r(t+∆t) = ~r(t) + ~v (t+∆t/2)∆t3. Calculate the new forces and update accelerations by Newton’s 2nd lawf(t+∆t) =dUd~r∣∣∣∣~r(t+∆t)4. Complete the velocity iteration~v(t+∆t) = ~v (t+∆t/2) + ~a(t+∆t)∆t/2202.1.1 EnsemblesMD applied with the velocity-Verlet algorithm is a powerful methodology. Vastarrays of physical situations and molecular models can be defined by choosing thepotentials from which the accelerations are calculated. Integrating Newton’s classi-cal equations of motion as above evolves the microcanonical ensemble (NVE) witha deterministic trajectory. Different thermodynamic ensembles can be simulatedwithin an MD framework by defining additional forces or altering the equationsof motion that are integrated. This allows interactions with the surrounding en-vironment that are more realistic than a perfectly energy conserving ensemble. Athermostat couples with velocity, while a barostat couples with the simulation boxdimensions. In either case this coupling is local. For example, to fix a temperatureheat is controlled on a particle-by-particle basis, rather than allowing heat diffusionfrom boundaries (as would be the case for an experimental set up). However, ifthe timescale of for heat diffusion is considered much shorter than the simulatedtime scales both a local and a boundary coupling to a heat bath become equivalent.Hence the more computationally efficient local coupling is typically used.Langevin dynamicsApplying a Langevin thermostat to the NVE ensemble adds two forces on a particleby particle basis;f =dUdr− mτdv + fr (2.1)The term proportional to velocity simulates friction or viscous drag with τd adamping timescale. fr is a random force with a zero mean that simulates couplingto a localised heat bath. fr satisfies the fluctuation dissipation theorem such thatthe variance of fr ∝√kBTm∆tτdconnects the force to a particular temperature T . Thissamples from the canonical ensemble with Langevin (i.e. stochastic) dynamics.Nose´-Hoover ThermostatThe canonical (NVT) ensemble can be implemented using a Nose´-Hoover thermostat[38, 39]. The equations of motion are modified with an additional term ξ thatsimulates a heat bath with associated mass and friction. The new variable ξ entersinto the equation of motion as the strength of a ‘frictional force’ on the particle sothat the acceleration ism~a = −∂U∂~r−mξ~v (2.2)The heat flow then has its own equation of motion [40]dξdt=1Q(T − 3NkBT)(2.3)21where T is total kinetic energy N is the total number of atoms and Q can beinterpreted as a ‘heat bath mass’ and is of the form Q = 6NkBTτ2T . τT is the ‘tem-perature relaxation time’. Now, the energy of the heat bath and the system togetheris conserved, rather than the latter in isolation. This thermostat is deterministic.Nose´-Hoover BarostatThe Nose´ Hoover barostat [41] is analogous to the Nose´ Hoover thermostat forthe isothermal-isobaric ensemble (NPT). Now two additional degrees of freedomare added to the equations of motion, one for temperature and one for pressure.LAMMPS uses a modified version to account for the box changing in shape [42].The particle positions are rescaled to set the pressure and this rescaling can becoupled to particular spatial dimensions (i = x, y, z).dLidt= νiαLi (2.4)where νi is an ‘effective strain rate’. The equation of motion for position is modifiedto incorporate νdridt=vi + νi(ri −RCM,i) (2.5)mai = −∂U∂ri−mνivi (2.6)where RCM,i is the i’th Cartesian component of the center of mass of the system.The variable ν has the following dynamicsdνidt=V (t)τ2BNkBT(Pii − P ) . (2.7)Here, Pij are the elements of the total pressure tensor, P is the fixed externalpressure, and τB is the pressure relaxation time.2.1.2 Boundary conditionsWe use a cuboid simulation cell - its initial dimensions are cubic but can change viathe application of a barostat or applied deformation. There are no additional inter-actions at the simulation cell boundaries. Rather, periodic boundary conditions areemployed and the polymer chains can freely cross them, modelling a bulk material.A minimum image convention ensures monomers interact with the closest image ofthe remaining system, see figure 2.1(a).The exception is calculations regarding full polymer chains, for example Ree. Wethen unwrap the boundary conditions so that full chains are considered at once sothat these lengths are physically meaningful. Figures 2.1(b,c) demonstrate wrappedand unwrapped boundary conditions for chains in two dimensions.22(a)(b) (c)Ree6= ReeFigure 2.1: a) Periodic boundary conditions: central simulation cell stored andminimum image convention applies b) Unwrapped boundary conditionsc) Wrapped boundary conditions.232.2 Microscopic polymer modelWe use a well established bead-spring model described by Kremer and Grest [43].This model has been used extensively, and has been found to give material responsesin good agreement with experimental results, in polymer melts [44, 45] and glassyphases [46–48] as well as chemically cross-linked elastomers [49, 50] and triblocksystems [51–53]. In these studies each monomer bead in the simulations representssome molecular weight of a real polymer. A single monomer in this coarse grainedmodel represents more than one repeat unit of styrene for example.All beads interact via a 6-12 Lennard-Jones potential,Vij(r) = 4ǫij[(σr)12−(σr)6](2.8)that is truncated at rc = 1.5σ and shifted for continuity, see figure 2.2(b). Theindices i, j run over the two types of monomer A and B, which is how we set controlphase separation and the material properties of the block copolymer components.We could instead vary the length scale σ to drive phase separation [57]. How-ever, given that rescaling the relevent ǫii directly rescales Tg for each polymer blockspecies, we found varying ǫ to be more intuitive.Neighbouring beads along the chains interact additionally with a non-linearspring potentialVFENE(r) = −12KR20 ln[1−(rR0)2](2.9)with R0 = 1.5σ and K = 30ǫ/σ2. These choices ensure that the balance betweenthe excluded volume of the pair potential and the strength of the spring potentialis such that chains do not pass through each other - an accurate representationof physical polymer chains. The spring potential is independent of monomer type.Figure 2.2(a) shows the combination of FENE and LJ potentials experienced bymonomers connected along the backbone of chains. Note that part of the attractivetail of the LJ potential is included so that we can study glassy phases.2.2.1 UnitsOur simulations are not conducted in regular physical units. Instead the energyand length units are defined by the Lennard-Jones potential values ǫ and σ. TheBoltzman constant (kB) and the mass of each monomer are set to one. This fixesthe dimensions for other units and are referred to as Lennard-Jones reduced units.It is possible to map LJ units to a molecular weight of polystyrene [13]. But,with this research we are trying to understand block copolymers in a more gener-alised way: focusing on the physics of polymer chains rather than the chemistry ofparticular monomers. All we need to know about the chemistry is encompassed inthe relative values of Tg or elastic moduli and the length of the polymer chains in24(a) (b)Figure 2.2: Potentials in microscopic model. (a) Blue: Lennard-Jones pair potentialtruncated and shifted. Red: FENE bond potential. Green: Sum ofpotentials - applies to bonded monomers. (b) Blue: Lennard-Jones pairpotential truncated and shifted. Purple: Lennard-Jones pair potential.relation to the entanglement length. The relative Tg are set by the like monomer in-teractions ǫAA, ǫBB. The segregation strength is controlled by the energy parameterfor the interaction between unlike monomers, ǫAB. To model experimental rubberswe need entangled chains, so we need simulate chains with lengths numerous timeslonger that the entanglement length Ne.2.2.2 Equilibrated configurationsWhile the above microscopic model gives us an effective means to study the me-chanical response of TPEs, simulating block copolymers remains a multiscale prob-lem. Obtaining phase-separated morphologies with equilibrated chain conformationspresents a set of unique challenges. Chains cannot cross through each other due tohard excluded volume interactions so the timescale of the system is determined byreptation dynamics with a disentanglement time that scales cubically with chainlength [13]. Additionally, if we start with a random distribution of chains, forma-tion of phase separated regions requires matter to move on the length scale of thesedomains.There are successful methods which deal with these issues separately. In ho-mopolymers, by combining MD with chain rebridging Monte Carlo, long chain sys-tems can be equilibrated [54, 55]. This method samples network chain configurationsmore efficiently by cutting and redefining chains. However, this technique cannotbe extended to copolymers as it does not help with the mass diffusion required forphase separation. Similarly, phase diagrams of copolymers can be efficiently stud-25ied with field theoretic methods that use free energy functionals wherein individualchains are replaced with continuous density fields [3]. There are also approacheswhich build the systems with the phase separated regions [56, 57] which requiresknowledge of a predetermined arrangement of the phase-separated regions.2.3 Soft ModelIn chapter 3 we develop and establish the efficacy of using a softer pair potentialin the initial equilibration stage of our simulations. Our approach is similar tothat currently pursued by several research groups [58–60], which retain the notionof discrete chains but also allows for thermodynamically driven phase segregation.The equilibrium morphology is achieved by considering soft interactions that allowchains to cross through each other, while hard sphere monomer interactions arerestored for subsequent deformation studies.Figure 2.3(a) shows a representation of the microscopic model with hard monomerbeads that disallow chains passing through each other and 2.3(b) a softer modelwhich allows chain crossing. This soft pair potential would give unphysical resultsif used to study material properties. However, compared to a brute force approachwith the microscopic model, it allows for a significantly faster equilibration of thechain conformations and triblock phase morphologies, a requirement to performmeaningful mechanical tests.The interaction energy of the purely repulsive soft potential is proportional tothe overlapping volume from spheres centred on each particle [61]Vij(r) =3kBT(κ0 ± χ02)8πr3sρ0(2 +r2rs)(1− r2rs)2(2.10)and truncated at rc = 2rs. The degree of phase separation is set by χ0 where + (−)applies to unlike (like) particles. The parameter κ0 can be related to the inverseisothermal compressibility κT [62], and determines the softness of the beads. Thesoft model results in a vastly different local packing structure to the bead-springmodel. We can study the the local monomer density by calculating the radialdistribution function (rdf)g(~r) =1ρ〈∑α 6=0δ (~r − ~rα + ~r0)〉. (2.11)Figure 2.3c) demonstrates the difference in local packing and at small r for eachmodel in relation to the pair potentials in figure 2.3(d). While the LJ pair potentialdiverges, creating an excluded volume region, the soft potential allows monomers toeven sit directly on top of each other.The microscopic model LJ pair potential defines beads with a significant ex-cluded volume; the potential becomes divergently repulsive below r = 216 . The26 0  0.5  1  1.5  2  2.5V(r)r [σ]0-ε 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0  1  2  3  4  5  6g(r)r(a) (b)(c) (d)Figure 2.3: a) Representation of LJ excluded volume beads on two chains, chainscannot cross. (b) Representation of soft model beads on two chains,chains are able to pass through each other. (c) Soft (red) and Lennard-Jones (blue) pair potentials. (d) Radial distribution functions for soft(red) and microscopic (blues) models.27soft pair-potential remains finite even at r = 0, see figure 2.3(c). Combined withthe choice of spring potential, this means soft-model chains can pass through eachother and sample configuration phase space more like unentangled chains. Monomermovement is therefore governed by Rouse rather than reptation dynamics.The same spring potential is used along the back-bone of the chains as in the mi-croscopic model. However, since the microscopic model chain cannot fully backfolddue to excluded volume constraints [55], we impose an additional angle potentialwhich penalizes the chain folding back on itself,V (θ) = Kθ [1 + cos(θ)] (2.12)where θ is the angle between adjacent bonds.The choice of the soft model parameters (χ0,κ0,rs,Kθ) are dictated by matchingthe structural properties of each model which will be elaborated on in the followingchapter.2.4 Equilibrating configurationsBefore testing the macroscopic material properties of TPEs, we need an equilibratedsystem. The definition of equilibration here comes back to the key details that re-main in our model: we want to ensure our simulations are representative of a realsystem while retaining the level of detail we are interested in. Therefore, we needto ensure that the microscopic chain conformations and phase-separated regions arewell equilibrated. We consider our systems equilibrated when there is no change inthese two properties on a logarithmic time-scale, see section 3.1 for details. Thisis not true equilibrium but the same can be said of experimental systems. Glassypolymer systems are in an amorphous metastable state [4]. The following sectionssummarise the steps used to obtain equilibrated configurations appropriate for me-chanical tests.2.4.1 Random-walk configurationsThe simulation box volume is defined with a fixed monomer density ρ = 0.85,the length of chains N and the number of chains in the simulation. Our startingconfigurations are generated using a random walk algorithm. The step-length of therandom walk is set as the average bond length b expected for the microscopic modelb = 0.97. We also can indirectly set the expected end-to-end distance by setting aminimum distance criteria dmin for the distance between each monomer pair n and(n + 2). This is necessary because the full model has excluded volume constraintsand is therefore best generated by a non-ideal random walk.The initial random walk chains have the correct average bond length and end-to-end distance by construction. However, the intermediate length-scales are notequilibrated, since the chains have no interactions when they are generated they can28(a) (b)Figure 2.4: 500 5-90-5 triblock copolymers. Blue A-type monomers, Red B-typemonomers shown for coordinates x+ y < L/2. (a) Initial random walkchain arrangement (b) Final equilibrated chain arrangementbe in unphysical arrangements initially (e.g. directly on top of each other) and thereis no phase-separated organisation of A-type regions, see figure 2.4. Therefore, anequilibration in the high temperature liquid state, where chains have high mobility,is required.2.4.2 Brute force equilibration of microscopic modelWe equilibrate the brute force simulations following the method in reference [55]:1. Setup random walk chains at a monomer density ρ = 0.85, b = 0.97, anddmin = 1.022. Perform a slow pushoff (that deals with particles sitting very close to eachother) with only the bond potential (eq. 2.9) and a soft pair potential (differentto soft model 2.10),Uij(r) = A[1 + cos(πrrc)]with cutoff rc = 21/6 ramping A from 1−100 over 50τ , NVT, T = 1.0, τd = 0.53. Replace the soft pair potential with the LJ pair potentials (eq. 2.8), resetvelocities to 0 every 50 simulation timesteps, NVT, T = 1.0, and dampingtime-scale τd = 0.25. These steps ensure that monomers’ local packing can berearranged within what will become the excluded volume region without thedivergent Lennard-Jones pair potential driving monomers rapidly apart.4. Equilibrate with Langevin dynamics (section 2.1.1), T = 1.0, τd = 1.0.29 40000 45000 50000 55000 60000 65000 0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1VolumeTemperature  ε = 1.0Tg = 0.389 Figure 2.5: Determining the glass transition temperature2.4.3 Soft model equilibration methodThe soft potential equilibration is implemented with the following method:1. Setup random walk chains at a monomer density ρ = 0.85, b = 0.97, anddmin = 1.022. Equilibrate soft model (section 2.3) with Langevin dynamics (section 2.1.1) atT = 1.0.3. Reverse coarse grain by changing to the microscopic model pair potentials.For the first 10τ , explicitly rescale the velocities of all beads at every step andlimit the step size to 0.03σ (see Ch.3).2.4.4 Setting temperature and pressureEquilibrations are performed at high temperature (T=1.0) where the system is amelt and can more readily sample phase-space. For the material to act as a TPEwe need to quench to a temperature in between Tg of the different homopolymericblocks, so that the spheres become hard while the background remains soft.To determine Tg, we perform a constant volume (or constant pressure) temper-ature ramp on a homopolymeric system. On a plot of the conjugate variable vs.temperature, a change in slope gives an approximation of Tg, see figure 2.5. Wemake a neccesary assumption that the Tg of the constituent blocks of a triblock is30the same as Tg in an equivalent homopolymer systems. For the range of N we studyTg is insensitive to chain length. We hold the LJ energy scale ǫAA = 1.0 constantand set ǫBB by scaling ǫAA by the desired difference in Tg. After equilibration ofa triblock the temperature is quenched (at the same rate) to a chosen temperatureTgB < T < TgA.Some care must be taken with pressure and densities. Differing pressures affectthe mechanical properties of polymer glasses [63]. Also, if the pressure becomesnegative at any point, cavitation can occur. If a pressure rescaling is required adensity ramp is performed before the temperature quench. Density appears in theentropic model formulas so must either be kept constant for comparability or anyvariation kept in mind.2.5 Mechanical testsWe test the mechanical properties of our nanostructured triblock copolymers byperforming simulated deformations while tracking relevant properties. We applyuniaxial strain with a constant engineering strain rate in the z directionǫzz =L− L0L0(2.13)where L is the box length in the deformation direction. The strain is related to thestretch used in the network models by λz = ǫzz +1. We deform up to a large strain(ǫzz=8) to test elastic properties. Different deformations are applied as appropriate:• Pure stress: σzz is the only non-zero component of the stress tensor. Abarostat is applied to the x and y directions. This is the closest simulation ofan experimental test of uniaxial strain.• Volume conserving: The diagonal components of the strain tensor are suchthe the box volume V = L30 is conserved, (ǫxx,ǫyy,ǫzz)=(1√λ+1, 1√λ+1,λ+1).The off-diagonal strain tensor components are zero. This is equivalent to apure stress deformation if the material remains incompressible. It is the mostcommon approach for assessing entropic models.• Pure strain: The cross-sectional area perpendicular to the deformation di-rection is conserved. Only the z-component of the strain tensor is non-zero(ǫxx,ǫyy,ǫzz)=(0,0,λ+1). This is an extreme deformation, with a large increasein volume, designed to test the material close to failure and study cavitation.2.6 Primative path analysis (PPA) and entanglementsWe use several metrics to track the glassy regions, polymer chains and monomersduring applied deformations. The majority are fairly intuitive and based on sets31(a) (b)(c) (d)Figure 2.6: a) Configuration of two entangled chains b) the chain is considered asan infinitely thin line and the chain ends are fixed c) the length of thepath is monotonically decreased and the number of kinks decreases d)converged final state [65] figure 1.pre-defined in the geometric input, e.g. calculated from all A-type monomers orthe end-to-end distance of each chain. Identifying entanglement points and theentanglement length Ne is more subtle because the entanglements are not predefinedfeatures of the chain.From the tube model as described in section 1.2.2, each chain can be assigneda topologically equivalent primitive path (pp). This path represents a random walkwith the Kuhn-length approximating the walk’s step-length. If the primitive pathsfor all chains are found, the entanglement points appear as kinks within otherwisestraight paths (with segment lengths Rk), separated on average by Ne monomers.The location of these kinks does not need to be explicitly pin-pointed to find Ne.Given the statistics of the ideal chain (eq. 1.1) the entanglement length can beextracted from the end-to-end distance (Ree) and contour-length of the primitivepath NeR2k = NR2ee. This common approach can be implemented within an MDframework [45, 64].However, since the Gaussian statistics break down if the chains are deformed, weneed to explicitly find each entanglement point to track Ne during a deformation.The Kro¨ger Z1-method achieves this by using a geometric minimisation to identifythe primitive paths. We utilised an implementation of the Z1-method received fromKro¨ger.32(a) (b) (c)Figure 2.7: Moves used in Z1-method multiple disconnect path minimisation. a)Nk → Nk − 1 b) Nk constant c) Nk → Nk + 1. Based on [66] figure Z1-methodThe Z1-method [65–68] performs a contour reduction topology geometric minimi-sation to find primitive paths. Topologically, there are no closed loops with linearpolymer chains; if the end-points of two chains are fixed, the chains can alwaysbe untangled if stretched far enough. The Z1-method avoids this problem by min-imising all chain paths simultaneously, see figure 2.6. Firstly the end points of allchains are fixed and each chain is considered infinitely thin. Each chain is made upof straight line segments and bending points called kinks. Initially there is a largenumber of kinks per chain Nk. The polymer chains paths are collectively referredto as the multiple disconnected path. The length of the multiple disconnected pathis minimised using steps that can add or remove kinks, see figure 2.7. Ultimatelywhen the minimisation is complete there are far fewer kinks than in the initial set-up. The monomer indices of these kinks, which define each chain’s primitive pathare recorded and the entanglement length is calculated from the average numberof kinks per chain: Ne = N/Nk. As with the tube model, self entanglements areexcluded.33Chapter 3A Soft Potential Method ForEquilibrating Block CopolymersIn this chapter we develop a simulation strategy for the creation of equilibratednanostructured copolymer melt morphologies 1. Molecular dynamics simulations ofbead-spring chains with a soft pair potential are used for efficient modelling of phaseseparation, while preserving Gaussian chain statistics and chain conformations of anunderlying microscopic model. In a second step, hard excluded volume interactionsare reintroduced that match the copolymer segregation strength but only requirereequilibration of local packing structure. We show that substantial computationalgains can be achieved for equilibrating moderately entangled bead-spring polymers.The resultant configurations can be used for further studies of structural and me-chanical properties in melts or glasses. This is the method we use for generatingconfigurations in later chapters.Computer simulations of copolymeric systems present a set of unique challenges.In both atomistic and coarse-grained force fields, chain uncrossability enforced byhard excluded volume interactions requires the chains to undergo reptation dynamicswith a disentanglement time that grows cubically with chain length [13]. Addition-ally, phase separation takes places on essentially diffusive time scales, as matterhas to be transported on the length scales of the nm-sized domains. The combi-nation of these two effects creates a truly multiscale problem in space and time.Successful strategies exist to tackle each issue individually. By combining moleculardynamics (MD) with chain rebridging Monte Carlo simulations, long chain systems(many entanglement lengths) can be routinely equilibrated [54, 55]. Equilibriumphase diagrams of copolymers can be efficiently studied with field theoretic meth-ods that tend to invoke phenomenological free energy functionals, but individualmonomers are replaced with continuous density fields [3]. Only few methods ex-ist that retain the notion of discrete chains but also treat the thermodynamics ofsegregation. One promising avenue, currently pursued by several research groups,is the use of soft interactions in coarse-grained models. In these approaches, chainconnectivity is preserved, but monomer interactions are replaced with weakly re-pulsive potentials that allow chain crossing [58, 59, 70]. A key point is that thepolymer chain is now subject to Rouse dynamics with a characteristic relaxationtime that only grows quadratically with chain length. Several alternatives exists for1This Chapter has been published as ref [69]34mediating the now weak excluded volume interactions. In an elegant formulationthat interpolates monomer densities on a grid and uses Monte Carlo simulationswith ”field-theoretically inspired” Hamiltonians [51, 62], efficient phase separationof symmetric diblock and triblock [71] copolymers was achieved. The same physicscan also be described via pair potentials that capture the overlap of soft spheres[61]. While computing pairwise interactions can be less efficient than a grid basedmethod for dense systems, it offers the advantage of easy and seamless implementa-tion in a molecular dynamics setup. In this formulation the approach is technicallyvery close to the well-known dissipative particle dynamics (DPD) method [72] thatwas already used 15 years ago in studies of block copolymers [73, 74].We have adopted the latter approach in the context of triblock copolymers with asymmetric ABA structure that form spherical nanodomains. We present a method-ology to equilibrate such copolymer chains from an initially mixed random stateinto a desired phase separated morphology with correct chain statistics and con-formations. In a first step, we implement molecular dynamics simulations with arepulsive soft pair potential to allow for shorter time scales for rapid equilibrationof the nanostructured morphology. We then reverse coarse grain, introducing aLennard-Jones pair potential for monomer interactions in order to create configu-rations that can be used as a starting point for further investigations of mechanicalproperties [57, 75].We present a comparison of the properties of equilibrated melts for short unen-tangled chains between both models. In particular, we show that our soft model/reversecoarse-graining equilibration strategy yields melts with static structure factors, ra-dial distribution functions and mean squared internal distances (chain conforma-tions) identical to those from direct simulation of the microscopic model. We thenapply our strategy to longer (weakly entangled) chains and explore the equilibrationof the phase separated structure as well as the dynamics of the system. Our ap-proach achieves equilibrated morphologies on typical MD time scales where directsimulations are computationally infeasible.3.1 Methods and modelsWe consider symmetric ABA triblock copolymers, where the A-type constitutes 10%of a linear chain, and model those using standard coarse-grained bead-spring chainsas described in section 2.2. All beads interact via a 6-12 Lennard Jones potential (eq.2.8). The segregation strength can be controlled by varying the energy parameterǫAB for interaction between unlike beads relative to the interaction ǫAA = ǫBB = ǫbetween like beads. Neighbouring beads along the chain interact additionally withthe nonlinear spring potential (eq. 2.9) with R0 = 1.5σ and K = 30ǫ/σ2. Thenumber of beads along the chains is varied from N = 100 (unentangled chains)to N = 500 (weakly entangled chains) [45]. Initial configurations are preparedusing standard methods as described in section 2.4.2. All simulations are conducted35at segment density ρ0 = 0.85σ−3 and temperature kBT = 1ǫ. As usual the LJparameters determine the fundamental unit of time τ =√mσ2/ǫ.3.1.1 Soft model implementationFor chain lengths N > 100, brute force equilibration with molecular dynamics be-comes prohibitively slow due to the ensuing reptation dynamics. For homopolymers,a standard approach is to couple MD with double-bridging Monte Carlo moves toimprove phase space sampling. Here we follow a different strategy and replace thehard excluded volume interactions of the LJ potential with softer interactions, seesection 2.3. Since these soft chains now can pass through each other, entanglementsare no longer present and the dynamics becomes Rouse-like.The equations of motion for both polymer models are integrated with a velocityVerlet algorithm with time step 0.01τ and thermostatted with Langevin dynamics,which adds a velocity dependent damping term with damping rate 1τ−1 and randomforces satisfying the fluctuation-dissipation theorem to the deterministic forces.SoftnessWe implement the purely repulsive soft pair potential given by eq. 2.10. As sug-gested by Mu¨ller [61], we set the size of the soft beads to rs = (3π/4)13 b = 1.25σ,where b = 0.97σ is the equilibrium bond length. κ0, which determines the softnessof monomer beads, can be related to the isothermal compressibility κT , which wecan calculate from our microscopic model simulations, giving κ0 ≈ 30. However,here we use it as a free parameter that we adjust in order to balance between twocompeting goals: softer interactions imply faster dynamics, but make subsequentreverse coarse graining more difficult due to large differences in local packing. Fig-ure 2.3(d) in the previous chapter highlights this variation. With a LJ potential,the radial distribution indicates an excluded volume region, there are essentially nomonomers within r = 1σ of each other. Whereas for the soft potential there is anon-zero density even at r = 0. This difference in local packing requires carefulhandling to allow switching between the soft and microscopic model. It is not pos-sible to simply switch potentials from soft to LJ pair potential without inducinghuge particle velocities. In a less extreme case, the local rearrangement resultingfrom switching pair potentials can result in a change to the intermediate lengthchain topology. Given that the reason we implement the soft model is to equilibratethese intermediate length scales such a rearrangement negates the benefit of the softmodel equilibration. We have determined a value of κ0 = 10 as an optimal com-promise (see also below). Experimentally, local packing can be studied via neutronscattering.36Consistent chain conformationsAn important requirement for consistency is that the Gaussian chain statistics ofthe two polymer models are the same. By equation 1.2 the end-to-end distance ofhomopolymeric chains is proportional to c∞ asymptotically at large length scales.For the fully flexible LJ bead-spring model, we find c∞ = 1.84, since the chaincannot fully backfold due to excluded volume constraints [55].In order to compensate for the lack of excluded volume constraints in the softpolymer, we impose the additional angle potential eq. 2.12. For a given compress-ibility κ0 = 10, we find that Kθ = 0.4ǫ matches the c∞ above for homopolymericchains.Reverse coarse grainingOnce we obtain an equilibrated structure with the soft pair potential, we reversecoarse grain and switch back to the LJ model with hard excluded volume interaction.For the first 10τ after this change in potential we explicitly rescale the velocities ofall beads at every step and limit the step size to 0.03σ. We show below that thisprotocol allows neighbouring particles to move away from each other with minimalstretching of the covalent bonds or distortion of the chain configurations. After thisstep, we are able to continue the simulation with normal integration of the equationsof motion.Phase separationFor triblocks, in addition to chain conformations, our equilibration procedure mustpreserve the thermodynamics of the block copolymer system. The strength of phasesegregation drives the nanostructured morphology. We first find for an interac-tion parameter ǫAB the corresponding segregation strength χ0 in the soft potentialmodel by matching the long wavelength behavior of the structure factor (see below).The driving force for phase separation is commonly quantified with the segregation(Flory-Huggins) parameter χ. We then estimate the segregation strength via theexpression [76]χ = ρ0∫d3r[gAB(r)VAB(r)− (gAA(r)VAA(r) + gBB(r)VBB(r))/2], (3.1)where gij(r) are the radial distribution functions (see eq. 2.11) of the differentmonomer species, at the end of the equilibration procedure (i.e. after reverse coarsegraining). This value can then be compared to results from direct equilibration ofthe LJ polymer model for short chains to confirm thermodynamic consistency.37Figure 3.1: Triblock copolymer morphologies obtained from 100 bead-spring poly-mers in a 5-90-5 configuration with Lennard-Jones interactions (top row)and the corresponding soft potential (bottom row) for two different seg-regation strengths (left column: ǫAB = 0.8ǫ/χ0 = 1, right column:ǫAB = 0.5ǫ/χ0 = 2, see text for details). Minority phase beads areshown in blue.3810-310-210-1100 0.1  1  10S AA(q)q [σ-1] (a) χ0 =εAB  = 0 1 2 3 4 5 6 7 8 0  1  2  3  4  5  6g AA(r)r [σ](b) χ0 =εAB  = 3.2: (a) Structure factors and (b) radial distribution functions for A-beadsfrom simulation of 500 chains with N=100. Solid symbols show resultsfrom the soft polymer simulations after equilibration for 105τ with softpotential model, open symbols correspond to the LJ polymer after 106τ .The segregation strengths for the parameter pairs are χ = 1.9(ǫAB =0.8ǫ), χ = 2.0(χ0 = 1), χ = 7.3(ǫAB = 0.5ǫ), χ = 7.0(χ0 = 2) ,χ = 8.6(ǫAB = 0.4ǫ), χ = 9.1(χ0 = 2.5). In panel (b), dashed linesshow additionally the soft polymer before reinsertion of the excludedvolume (LJ) interactions.393.2 Comparison of short chain conformationsWe begin by studying short chain systems of N = 100 beads that can still beequilibrated by direct MD simulation of the LJ bead spring model. In this waywe can verify that, although the interactions and dynamics differ, equivalent finalmorphologies are obtained. Images in the top row of Fig. 3.1 show chain configura-tions after direct simulation of the LJ model for 106τ and two different segregationstrengths. With ǫAB = 0.8ǫ the A-type ends are just beginning to phase separate,but for ǫAB = 0.5ǫ we start to see ordering and clustering. The larger clustershave a roughly spherical shape and there is a significant number of smaller clustersdistributed in between. Equilibration with the soft potential model for 105τ withχ0 = 1 and χ0 = 2 followed by reverse coarse graining produces visually indistin-guishable configurations.3.2.1 Phase separation of 5-90-5 triblocksIn order to characterize the phase separation quantitatively, we calculate the staticstructure factor for A-beads,SAA(q) =1NA〈∣∣∣∣∣NA∑i=1ei~q·~ri∣∣∣∣∣2〉, (3.2)where the sum runs over only the A-type beads. Figure 3.2(a) compares structurefactors for the soft polymer and direct equilibration of the LJ bead-spring polymerfor three different segregation strengths. Both models were simulated until we seeno change to SAA(q) on logarithmic timescales. We therefore consider these config-urations equilibrated, although the true equilibrium configuration is a bcc lattice ofspherical micelles. Even in experiments, these are rarely achieved. In all cases, thereis a small peak at q = 8σ−1 which corresponds to the typical interbead distance thatis essentially set by the LJ particle diameter. A second, more interesting peak ap-pears at a lower wavevector q = 0.5σ−1 and signals the formation of nanodomainsseparated on much longer length scales. The position of this peak in SAA(q) aswell as their amplitudes are in excellent agreement for the three sets of parametersin both models. Values for the segregation parameter χ computed from eq. (3.1)(see caption) agree better than 6% in all cases. The peak amplitude increases withincreasing phase separation and is an order of magnitude larger for the system withvisible phase separation in the right panels of Fig. 3.1 compared to the left panels.While Fig. 3.2(a) confirms that the morphologies on large scales agree, it is alsoimportant to verify that the same local packing structure is achieved. Figure 3.2(b)shows the radial distribution function (rdf) of the A-type beads, which highlightsthe difference in the structures at short length scales. Once the LJ pair potentialis reinserted into the simulation and equilibrated for a short time, we find rdfs thatare indistinguishable from direct simulation of the LJ model. We also plot in Figure40Figure 3.3: Equilibration of mean squared internal distances for chains with N=100beads. (a) LJ polymer with ǫAB = 0.5ǫ, (b) Soft polymer with χ0 = 2where the additional black curve is after 200τ LJ reintroduction (c)MSID of final configurations of homopolymers. The insets give the rel-ative deviation of the final MSID for the soft and LJ polymers.41100101102103104105101 102 103 104 105 106〈(r - r0)2〉  [σ2 ]t [τ]χ0 = 1.0χ0 = 2.0χ0 = 2.5εAB = 0.8εAB = 0.5εAB = 0.4slope 0.5slope 1.0Figure 3.4: Monomer mean square displacement for chains of length N=100 withseveral phase separation strengths.3.2(b) the rdfs obtained at the end of equilibration with the soft potential, butbefore reverse coarse graining. As expected, the soft model features a very differentstructure as beads can overlap with little energy penalty and the oscillatory structureof gAA(r) is absent. The key point of the strategy is, however, that this discrepancyis rectified after the short LJ reintroduction step without altering the longer lengthscale structure of the phase separated regions.3.2.2 Equilibration of N=100 chain conformationsA meaningful equilibration strategy for polymeric systems must also guarantee cor-rect chain conformations. An established benchmark that characterizes the internalstructure of the chains is the mean-squared internal distance of monomers, see equa-tion 1.3. According to eq. (1.2), MSID(n)/nb2 → c∞ for n → N . The evolutionof the MSID curves during equilibration is shown for both the soft and LJ pairpotentials in Fig. 3.3. We can see that the initial curves start from the bond lengthsquared, but instead of asymptotically increasing towards c∞ there is a bump aroundn = 20 that has its origin in the initial chain preparation procedure. This bumpdecreases in magnitude and moves to higher n until the curves stop changing and42we consider the internal chain conformations equilibrated. While for the presentshort chains N = 100 both methods perhaps require a similar equilibration time(104τ − 105τ), the soft potential method takes dramatically less time to smooththe bump at intermediate length scales, 103τ compared to 105τ . Above n = 90 anupturn in the MSID curve develops, which is not seen in homopolymeric systems(Fig. 3.3(c)). For the soft potential equilibration this upturn takes more time toequilibrate than the bump at intermediate length scale (104τ) while for the LJ po-tential the time scales are the same. Panel (b) of Fig. 3.3 shows additionally theMSID after the final reinsertion of the LJ potential. We find only minimal devia-tions at intermediate length scales as the beads adapt locally to the excluded volumeconstraints. The bead softness κ0 has been chosen precisely to minimise this effectwithout too much slowing down of the equilibration dynamics.3.2.3 Dynamics of short chainsFurther insight into the dynamics of the two polymer models can be obtained fromthe monomer mean-squared displacement,g1(t) =〈∑i|~ri(t)− ~ri(0)|2〉, (3.3)where the sum is over Cartesian coordinates i. Figure 3.4 shows g1(t) for varyingdegrees of phase separation. For both models we see Rouse dynamics at shorttimescales (g1(t) ∼ t0.5) that eventually switches to diffusive dynamics (g1(t) ∼ t).Reptation dynamics is absent as the chains are too short. The dynamics of the softpotential equilibration are faster in two respects. Firstly, on all timescales the meansquared displacement during the soft equilibration is an order of magnitude largerthan for the LJ equilibration as the chains are more compressible and experience lessfriction. Secondly, the crossover from Rouse to diffusive dynamics occurs an orderof magnitude sooner (104τ compared to 105τ) for the soft potential equilibration. Inboth models the crossover to diffusive dynamics is slowed by the phase separation.The slope of the mean squared displacement curves changes an order of magnitudelater for the systems with the strongest phase separation.3.3 Longer chains - equilibration with soft model onlyWe now consider longer chains N = 200 − 500 that we equilibrate using the softpair potential method. Figure 3.5 shows an equilibrated structure for 800 chains oflength N = 500. We see that the minority A-type particles have formed sphericalclusters. They are of similar size and well separated with no smaller clusters foundbetween the large ones. Compared to the right panel of Fig. 3.1, which containsN = 100 chains with the same potential parameters, the clusters appear larger andbetter defined.43Figure 3.5: Morphology of 800 triblock copolymer chains in a 25-450-25 configura-tion after equilibration with soft pair potential for 105τ and χ0 = 2.Minority monomers are shown in blue.4410-210-1100101102 (a) N=10020030050010-210-1100101102S AA(q)(b) 102τ103τ104τ105τ106τ10-210-1100101102 0.1  1q [σ-1](c)Figure 3.6: Structure factor between A-beads (a) after equilibration for 105τ withsoft potential for varying chain lengths N=100,200,300,500, (b) forchains of length N=500 during equilibration with soft potential and (c)for chains N=200 during equilibration with soft potential.453.3.1 Phase-separation of longer chainsFigure 3.6(a) shows the structure factor between A-beads in the final configurationsof systems with varying chain lengths. The peak in SAA(q) occurs at smaller q-valuesas the chain length increases, consistent with the increase in size and spacing of thespherical micelles observed in Figure 3.5. Since the central block of B-type particlesis becoming longer, it becomes much easier for the chains to bridge between A-typespheres. The peak in SAA also increases in height as the chains become longer,indicating that the phase separated regions become sharper.The lower panels of Fig. 3.6 show the evolution of SAA(q) during the equilibra-tion. We see that at short wavevectors the structure factor is initially flat. A peakthen develops at q = 0.5σ−1 and moves to lower q in tandem with the increasingamplitude of SAA(q). We consider the phase separated structure to be equilibratedwhen the height of this peak ceases to change. Note that this equilibration time in-creases by only a factor of 10 from 104τ to 105τ when comparing the lower (N = 200)and middle (N = 500) panels.3.3.2 Chain conformations of longer chainsFigure 3.7 compares the equilibration of the internal chain structure for longer chains(N=500) between the soft and LJ pair potentials. In contrast to Fig. 3.3, the MSIDcurves now show a significant difference in the evolution of the chain conformationsthat depends on the potential used. There is clearly no equilibration in the LJpolymer case (panel (a)), since after 106τ the MSID is still changing significantly.Although the overshoot at intermediate length scales shifts to the right, there is nodecrease in its magnitude. There is only a small upturn in the graph at the longestinternal lengths, indicating that little phase separation has occurred. On the otherhand, in the soft potential case (panel (b)) the bump is smoothed completely after104τ and there is an upturn at the end of the curve developing from 103τ onward.For comparision we also show in panel (c) the same calculation for homopolymermelts. Here equilibration is already reach are 103τ and the MSID curves reach aplateau as phase separation is absent.3.3.3 Dynamics of longer chainsThe polymer dynamics during equilibration of these longer chains are shown inFig. 3.8 in form of g1(t). It is instructive to first inspect the N=500 homopoly-mer, which reveals a distinct difference between the two polymer models. In thesoft potential case we see Rouse dynamics at short time scales, switching to diffu-sive dynamics about one order of magnitude later in time than the shorter chains(N=100). With the LJ potential, rather than an upturn in the mean square displace-ment, there is a decrease in slope around 104τ consistent with reptation dynamics,g1(t) ∼ t1/4. For the triblocks, we find a gradual decrease from the Rouse slope of460.5 starting at 104τ with both pair potentials. Since this decrease is absent for thehomopolymer with soft potential we can take it to be an effect of phase separationlimiting the movement of particles. Still, diffusion in the soft model is about oneorder of magnitude faster than in the LJ polymer model.3.4 DiscussionWe have presented a simulation strategy to achieve equilibrated copolymer mor-phologies for short to moderately entangled linear polymers. The approach uses softpair potentials for the equilibration step and can therefore be implemented with min-imal effort in commonly used molecular dynamics packages such as LAMMPS andHOOMD. While here our underlying target model was a coarse-grained bead-springmodel, there are no fundamental obstacles to generalize the method to united atomor all-atom polymer force fields. We placed special emphasis on achieving correctc∞-ratios and mean-squared internal distances, as these macromolecular propertiesare of crucial importance for any subsequent investigation of mechanical properties.Entanglement effects are critical in the competition between chain pullout and chainscission during deformation. Considerations of numerical efficiency were less impor-tant in the design of our approach, as the future fracture simulations will requiremore computational resources than the equilibration itself. If a rapid overview of theequilibrium phase behavior is to be obtained, the grid-based Monte Carlo methodsmentioned in the Introduction are likely a more efficient choice.Variations of the present approach to use configurations produced by soft-coarsegrained interactions as a starting point for polymers with hard excluded volumeshave already been considered in other contexts. For homopolymers, Vettorel et al.[77] and Zhang et al. [78] developed a scheme in which each coarse-grained softsphere corresponds to a segment of an underlying fine-grained bead spring model,and interactions are computed either on- or off-lattice with Monte Carlo. A verypromising extension consists in formulating a hierarchy of coarse-grained modelsthat reinsert short-scale details only gradually over multiple steps. Such a strategyseems necessary for the equilibration of ultralong (N ≥ 103) chains, where evenRouse-dynamics is too slow due to N2-scaling of the relaxation times. Along similarlines, Carbone et al. [79] suggested to begin with conformations from continuousrandom walk conformations with appropriate Kuhn lengths, and then to fine-grainby reinserting atomistic segmental interactions. For unentangled random diblockcopolymers, Steinmu¨ller et al. first used single-chain in mean-field Monte Carlosimulations for the phase separation with varying segregation strengths and thencompared to results from direct MD simulations of a bead-spring model with LJinteractions [52, 80]. They also performed a reverse coarse-graining step and showedthat the local packing structure (rdfs) agree after short reequilibration, but mean-squared internal distances or entangled chains were not investigated. We hope thatour results here expand upon and further demonstrate the utility and promise of47this modelling philosophy.The key outcome here is that we have a method which can satisfactorily equili-brate moderately entangled chains into the desired spherical phase separated mor-phology, this work has been published [69]. We could consider extending this methodto equilibrate ultralong chains (N > 1000). There are methods which combine MDand Monte-Carlo to this end in homopolymeric systems [77, 78] and recently anextension to a hierarchy of coarse-grained models that reinsert shorter scale detailsonly over a number of steps [60]. The latter method in particular could be adaptedusing these soft potentials. However, we now choose to apply this soft model methodto obtain equilibrated, entangled TPE systems upon which to perform mechanicaltests. Table 3.1 gives a summary of the potential parameters used in our studies.εAA εBB εAB κ χ N N ×MChapter 3 1.0 1.0 varied 10 varied 100, 200, 300, 500 50,000 , 400,000Chapter 4 1.0 0.7 0.3 10 2 100, 300, 500 480,000Chapter 5 1.0 0.5 0.2 10 3 300, 500, 800 480,000Table 3.1: Summary of potential parameters used for soft model equilibration (κ,χ), microscopic model deformation (εAA, εBB, εAB), chain lengths (N),and total simulated monomers (N ×M) in later chapters.48Figure 3.7: Mean square internal distances during equilibration of long chainsN=500 with (a) LJ polymer (100 chains) and (b) soft polymers (800chains) and (c) soft homopolymers (800 chains).49100101102103104105101 102 103 104 105 106〈(r - r0)2〉  [σ2 ]t [τ]Triblock 100Homopolymer 500Triblock 500Triblock 100Homopolymer 500Triblock 500slope 0.5Figure 3.8: Mean square displacement during equilibration. The upper three curvescorrespond to the soft potential equilibration of chain lengths N=100and N=500 with χ0 = 2. For comparison the behavior of N=500 ho-mopolymers is also shown. The lower three curves show the equivalentcurves for equilibration of the LJ polymers with ǫAB = 0.5ǫ for thetriblocks.50Chapter 4Molecular Mechanisms ofPlastic Deformation in TPEsIn this chapter we study plastic deformation of sphere-forming triblock TPEs withMD simulations in order to elucidate microscopic mechanisms operative in nanos-tructured macromolecular materials. 1 We compare two deformation modes of bothuniaxial stress and strain, respectively, and vary the polymer chain length from un-entangled to moderately entangled chains. Our simulations show that triblocks ex-hibit significant increase of strain hardening compared to homopolymer elastomers.We analyze several properties such as global chain deformation and local monomermotion, number and shape of spherical domains and the evolution of the fractionof chains bridging between domains. Results confirm the notion of improved me-chanical properties through effectively crosslinking chain ends. Void nucleation atdifferent stages of deformation is observed to occur either at the interface betweenglassy and rubbery phases or immediately following the breakup of glassy domainsand is therefore intimately related to the elastic heterogeneity of the material.We continue to focus on linear symmetric ABA triblock copolymers, where thehard component is polystyrene while the soft component is most commonly polybu-tadiene or polyisoprene [6]. As described in section 1.1.2 when in the sphere-formingregime, chain ends are confined to hard glassy minority regions, allowing a rubberyphase to be effectively crosslinked. This forms a TPE combining desirable materialproperties of rubbers and plastics.Experimental results show highly non-linear stress-strain response in styrenicthermoplastic elastomers. Early studies by Holden et al. fit experimental data forSIS and SBS triblocks with a constitutive equation that accounts for the effectsof rigid fillers and entanglements [7]. This treats the glassy spherical inclusions ashard discrete particles and describes the data up to an extension ratio of 290%,while material failure occurs at much greater elongation of 1000%. At these largerextensions the tensile strength of the triblock copolymers is much greater than vul-canised rubbers. In keeping with this assumption, Prasman and Thomas find thatin an SIS/mineral oil blend deformation is affine up to strains of 300%, and PSdomains keep their shape under extension [82]. However, other studies find that themicroscopic structure can be altered during deformation by chain pullout at temper-atures below the glass transition temperature of polystyrene [32], while hysteresis1A version of this work has been published in ref. [81]51[83] and cavitation [84, 85] were also observed. The proportion of chains bridgingbetween glassy regions controls the structure of the network. This proportion canbe varied by introducing diblocks [20, 86, 87] or cyclic copolymers [88, 89] to thetriblock system and has been shown to affect the macroscopic stress-strain response.On the whole, it is difficult to get a comprehensive picture of how the microstruc-ture of these TPEs determines the macroscopic stress response experimentally. Forinstance, it is important to understand changes to the morphology during extensionto large strains and their dependence on molecular and deformation parameters. Weapproach this question using molecular dynamics simulations.A few molecular dynamics simulations have studied deformation of phase-separatedtriblock copolymers so far, but these have relied on building copolymer systems withan imposed morphology. This requires knowledge of a predetermined arrangementof the phase-separated regions. For instance, in the works of Leonforte [90] andMakke et al. [75, 91] samples were constructed by growing individual chains in afixed lamellar geometry using biased random walk techniques (radical like poly-merization) [92]. These authors varied the fraction of triblocks bridging betweendifferent glassy lamellae and explored their role for yield and strain hardening inuniaxial tensile deformation. While decreasing the amount of bridging moleculesleads to a rapid reduction of strain hardening, replacing them with loop moleculesthat return to the same lamella was found to hardly alter the mechanical response.These simulations also show that the lamellar phase exhibits buckling upon yieldingand develops a chevron like structure [75] due the spatially varying elastic propertiesof the layered material. Deformation of sphere-forming short triblock copolymerswas also studied by Aoyagi et al. [57], however these authors generated a spheri-cal morphology first using self-consistent field (SCF) simulations before performingmolecular simulations with bead-spring chains. Uniaxial elongation of short tri-blocks shows failure at 350% strain where minority phase domains break up [57].In an approach similar to ours, Chantawansri et al. [29] simulated a dilute gel ofphysically cross-linked triblock copolymers under uniaxial tension by coupling poly-mer models with and without chain crossing [93], and determined the chain lengthdependence of the modulus in accordance with predictions from rubber elasticitymodels. Other aspects of the plastic deformation of filled elastomers, such as thereinforcement with increasing filler fraction, have also been reproduced with sim-pler models that consider hard particles modelling the spherical inclusions in a fillermatrix modelled implicitly by harmonic springs [94].We consider both unentangled and moderately entangled chains under uniax-ial deformation up to engineering strains of 800%. We compare their mechanicalresponse to homopolymers of the same length. First we give the details of the imple-mentations of the microscopic bead-spring model and the soft model equilibrationfollowed by analysis of the response of the chains, monomer beads and phase sepa-rated regions to deformation. We also investigate density fluctuations that lead tocavitation in each deformation mode and correlate the location to the surrounding52microstructure.4.1 Modelling methodsWe model generic linear symmetric ABA triblock copolymers, where the A-typeminority monomers make up 10% of the chain, with the coarse-grained bead-springmodel as described in section 2.2. We set the pair potential (eq. 2.8) parametersǫAA = ǫ and ǫBB = 0.7ǫ so that the minority A-particles form hard spheres in asoft matrix. ǫAB then controls the segregation strength and is fixed at ǫAB = 0.3ǫ.All chains in a simulation have the same number of beads N , which is varied fromshort unentangled chains N = 100 to longer weakly entangled chains N = 300and N = 500 [45]. The entanglement length of the homopolymer melt has beendetermined via Primitive Path Analysis (PPA) [45] as Ne ≈ 55. We compare thetriblocks to homopolymers of the same chain length N and the same interactionstrength as the majority B-type monomers of the triblock, set at 0.7ǫ. All valuesare reported in reduced simulation units, where length is measured in units of σ,stresses in units of ǫ/σ3 and temperatures in units of ǫ/kB. Similarly, all times aremeasured in units of τ =√mσ2/ǫ, where m is the mass of a bead.The internal chain conformations and nano-structured morphology are first equi-librated using the soft model coarse-graining strategy, following the method in sec-tion 2.4.3. We use the same parameters κ0 = 10 for the pair potential softness (eq.2.10) and the matching Kθ = 0.4 for the angle potential (eq. 2.12). However, the ǫijparameters now differ to those used in chapter 3. So, we need to recalculate χ0 bycomparing brute force and soft model equilibrations of short chains and matchingχ (3.1). We find χ0 = 2 to be the appropriate choice.After equilibration and reverse coarse-graining we quench the temperature from1.0 to 0.3 with quench rate 10−4 at fixed positive pressure. This temperature lies be-tween the glass transition temperature Tg determined for homopolymers consistingof A (Tg = 0.39) and B (Tg = 0.27) type monomers. Values for Tg were estimatedfrom the intersection of linear fits to the temperature dependence of the specific vol-ume during cooling as shown in figure 2.5. This results in glassy A-type inclusions,while the B-monomers are still in the rubbery phase.4.1.1 Glassy clustersFigure 4.1 shows equilibrated morphologies before deformation for the N=100 andN=500 triblock systems. Hard glassy clusters are identified with a cluster algorithmthat investigates the local neighbourhood of each particle and links all particles thatare separated by less than a cut-off of 1.2. The cut-off is chosen to be larger than thenearest neighbour shell and less than the shortest separated clusters. The shorter thechains the greater the number Nc of these phase-separated regions, Nc = 318, 76, 31for N = 100, 300, 500 respectively. For all chain lengths the simulation box contains53480,000 monomers so that there are at least 30 phase separated minority monomerregions to ensure good configurational averaging. Results are averaged over threedifferent replicas for each chain length, and the number of glassy spheres varies byless than 5%. The static structure factors (eq. 3.2) reveal a characteristic spacingof 27 (N=500) but give no indication of long range order.4.1.2 Applied deformationsWe perform two uniaxial deformations, pure strain and pure stress, both with aconstant engineering strain rate ˙εzz = 10−4 in the z-direction. Experimentally, auniaxial strain test might take on the order of seconds. The deformation timeswe simulate are orders of magnitude smaller, They are limited by computationalconstraints. The simulations presented here take 3 days to calculate on 32 CPUcores or 1 day on 8 CPU cores and 1 GPU when implemented in LAMMPS. Slowerdeformations, ˙εzz = 10−5, are presented in later chapters. Observing trends withdecreasing deformation rate is key to relating these simulations to those performedexperimentally.In the pure uniaxial strain deformation only the εzz component of the straintensor is non-zero and the box dimensions in the two perpendicular directions areunchanged. As a result, ǫxx = ǫyy = 0 and the cross-sectional area perpendicularto the deformation is conserved. By contrast, in the pure stress deformation theonly non-zero stress component is σzz while σxx and σyy are held at zero usinga Nose´ Hoover barostat with barostat time constant 7.5τ At the same time, thetemperature is controlled via a Langevin thermostat with time constant 1τ . In eachcase the equations of motion are integrated with a velocity Verlet formulation with atime step of 0.005τ . The box resize introduces a streaming velocity to every particle,which is removed from the velocity considered for thermostatting. We simulate largestrains up to εzz = 8 to investigate polymeric effects in the strain hardening regime.4.2 Macroscopic stress responseWe first consider the macroscopic response during deformation before relating backto the evolution of the microstructure. Figure 4.2 shows the stress-strain curvesfor each deformation mode. In the pure stress case (panel 4.2a) at small strainwe observe initially a cavitation peak followed by softening and drawing for bothhomopolymers and triblocks. The peak stress at cavitation is highest for the ho-mopolymers, where it is also independent of chain length. For both the unentangledN = 100 triblock and homopolymers, the stress tends to zero at large strains indi-cating material failure through chain pullout. As the chain length and hence numberof entanglements increase, we observe an increasing amount of strain hardening asan upturn in the stress above εzz = 4.5 (εzz = 6) for the N = 500 (N = 300) chains.This trend is enhanced once chain ends are physically cross-linked by the glassy54Figure 4.1: Snapshots of morphologies before deformation in a box of size L = 80σ.Top: 1,600 triblock copolymer chains in a 5-90-5 configuration. Bot-tom: 960 triblock copolymer chains in 25-450-25 configuration. Minoritymonomers are in blue and majority monomers in red.55 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35σzz(a) 100300500 0 0.5 1 1.5 2 2.5 3 0  1  2  3  4  5  6  7  8σzzStrain(b) 0 0.1 0.2 0.3 0  0.25  0.5 0 0.1 0.2 0  1  2  3Figure 4.2: True stress vs. engineering strain for (a) pure uniaxial strain and (b)pure uniaxial stress deformation. Solid lines: triblocks. Dashed lines:homopolymers. Chains lengths N are given by the colour key.56spheres. As a result, the stress-strain curves of the triblock systems lie above thosefor the corresponding homopolymers. These trends are in qualitative agreementwith experiments [83].In the case of pure stress deformation (panel 4.2b) the cavitation peak is absent asthe simulation box can adjust laterally to the applied strain, and both the entangledhomopolymers and triblocks display strain hardening. However, this hardening ismore dramatic in the triblock system. Firstly, the stress response is steeper atintermediate strains. Secondly, there is again a further increase of the stress atlarge strains. This hardening is related to the deformation and break-up of hardminority phase-separated regions (see below).Typically we only compare the triblock stress response to majority B-type ho-mopolymer stress responses. However, we can also simulate deformation of ho-mopolymers made up of the hard A-type beads to obtain an estimate for the elasticcontrast present in regions of the triblock system. The small strain (≤ 1%) elas-tic response in the pure stress protocol permits the direct measurement of Young’smodulus Y and the Poisson ratio ν for the different polymers. For the homopoly-mers we find a ratio YA/YB ≈ 100, while νA = 0.41 and νB = 0.48 independent ofchain length.4.3 Microscopic responseFigures 4.3 and 4.4 show several snapshots of the N = 500 triblock system duringeach deformation mode (snapshots for all chain lengths and pure strain homopoly-mers are shown in Appendix A) In figure 4.3 we observe cavitation and drawingin the pure strain deformation. During the formation of the fibril network, thereis significant breakup of the glassy inclusions. In figure 4.4 by contrast, we see theminority clusters primarily stretch and deform during pure stress deformation. Wenow quantify what is happening to the microstructure during these deformations.4.3.1 Chain end-to-end distancesWe can consider how the chains are deforming globally by studying the end-to-end distance Ree = r(N) − r(1), where r(1) and r(N) are the positions of thefirst and last monomer on a given chain. In order to see whether the chains arebeing deformed more or less than the simulation box as a whole, we subtract theaffine deformation ε¯Ree0, where Ree0 is the end-to-end vector in the undeformedconfiguration and ε¯ is the global strain tensor. Figure 4.5 shows the non-affine changeof Ree relative to Ree0 in the undeformed material for different deformations. Firstwe observe that in all cases, the deformation is more affine in the triblock than inthe corresponding homopolymer case. For each chain length there are similar trendsfor the corresponding triblock and homopolymer. In the pure strain deformations(figure 4.5a) none of the systems deform affinely, which is consistent with the large57Figure 4.3: Snapshots of the 25 − 450 − 25 triblock system during pure uniaxialstrain deformation at strains 1, 3, 5, 7. Minority monomers are in blueand majority monomers in red. Here the average density decreases asρ(ǫ) = ρinitial/(ǫ+ 1).58Figure 4.4: Snapshots of the 25 − 450 − 25 triblock system during pure uniaxialstress deformation at strains 1, 3, 5, 7. Minority monomers are in blueand majority monomers in red. Minority monomers are not displayedin the upper portion.59-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5(Re- εRe0 -Re0)/R e0(a) 100300500-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0  1  2  3  4  5  6  7  8(Re- εRe0 -Re0)/R e0Strain(b)Figure 4.5: Non-affine strain of the end-to-end distance during (a) pure uniaxialstrain deformation (b) pure uniaxial stress deformation. Solid lines:triblocks. Dashed lines: homopolymers. Chains lengths N are given bythe legend.60increase in box volume and consequent void formation. For the entangled chainsthe non-affine strain of |Ree| is positive while for the N=100 homopolymer it isnegative for strains larger than two. At these strains, the untangled elastomer isfailing through chain pullout as the stress drops monotonically, see Fig. 4.2(a).In the pure stress deformations (figure 4.5b) the entangled triblock systems de-form affinely as well as the N=500 homopolymer. In general the deformation be-comes more affine with increasing chain length, which is consistent with the notionof a deforming entanglement network. The presence of glassy inclusions promotesthis trend further as chain ends are spatially immobilized.4.3.2 Monomer dynamicsTo understand the dynamics of the soft and hard components on a more locallevel, we consider the mean non-affine displacement (NAD) Dna of the monomersthemselves. The Cartesian components Dnam are given byDnam = rm − r0m − ǫmnr0n, (4.1)where r0 denotes the monomer position at an earlier time and m,n denote Carte-sian components. Here we focus on the NAD between small strain intervals of 2.5%.Panels (a) and (b) of figure 4.6 show the magnitude of the NAD averaged over allmonomers as function of strain. This quantity serves as an indicator for plastic ac-tivity. For pure strain deformation (fig. 4.6(a)) the NAD rapidly increases from zeroto a maximum value over a strain of 10%, and even faster for pure stress deformation(fig 4.6(b)). Note than in amorphous materials, non-affine displacements already oc-cur in the elastic (reversible) loading regime due to local elastic heterogeneities. Inboth cases, Dna decreases as the deformation progresses and the decrease is morepronounced for longer chain lengths. For the unentangled homopolymer, Dna re-mains close to constant while for the longer N = 500 chains Dna decreases by 20%for the homopolymers and 40% for the triblocks at εzz = 8. The magnitude ofDna is always less for the triblock monomers than for monomers of the equivalentchain length homopolymers. This difference becomes more pronounced during thedeformation.Panels (c) and (d) of figure 4.6 averageDna over majority and minority monomersin the triblock systems. For the pure strain deformation (panel (c)) we see thatinitially the soft majority beads (triangles) have a higherDna ≈ 1.3 which is the samefor all chain lengths. However, the monomers in the hard beads (crosses) show alower Dna the longer the chains become, but this value increases during deformation.For the N = 500 (N = 300) triblocks, there is a crossover at εzz = 7.5 (εzz = 7)where the minority beads first have a larger Dna than the majority beads. For theshorter N = 100 chains no crossover is reached. This crossover point indicates thatthe glassy regions are now deforming more the soft background. A similar patternis seen in the pure stress deformation (panel (d)). Here the crossover happens latersince Dna for the minority monomers increases more slowly.61 0.2 0.4 0.6 0.8 1 1.2 1.4Dna(a)500300100 0.2 0.4 0.6 0.8 1 1.2 1.4 0  1  2  3  4  5  6  7DnaStrain(c)(b) 0  1  2  3  4  5  6  7  8Strain(d)Figure 4.6: Non-affine monomer displacement magnitudes calculated between snap-shots separated by strains of 2.5%. Top (a)(b): Filled symbols - ho-mopolymers, open symbols - triblocks. Bottom (c)(d): Crosses: A typebeads in triblock, Filled triangles: B type beads in triblocks. Left (a)(c):Pure uniaxial strain. Right (b)(d): Pure uniaxial stress.624.3.3 Clusters and bridging chainsWe now analyse changes in shape, number and connectivity of the minority monomerphase-separated regions during deformation.Shape of minority clustersThe average asphericity of minority monomer clustersA = λ1 − 12(λ2 + λ3) (4.2)is calculated from an average of the ordered eigenvalues λi of the mean radius ofgyration tensor for each cluster j,Smn =1NjNj∑i=1(r(i)m − r(j)CMm)(r(i)n − r(j)CMn). (4.3)The sums run over the number of beads in each cluster Nj , and coordinates r(j)CMare with respect to the centre of mass for each cluster. Figure 4.7(a) shows that theasphericity increases during pure strain deformation, however a threshold is reachedwhere this value plateaus or decreases. This indicates that the clusters can onlydeform to a threshold point before they break up into more spherical clusters. In thepure stress deformation shown in Figure 4.7(b), A increases much more dramaticallythan the pure strain case, i.e. the spheres are being deformed much more stronglyas they are staying intact longer. This effect can also be seen qualitatively in thesnapshots Figures 4.3 and 4.4.Number of minority clustersThe number of clusters Nc shown in panels (c) and (d) exhibits opposite trends.Nc rises much faster to up to 3.5 times their initial number Nc0 during pure straindeformation, while Nc rises to only about 1.5 ×Nc0 during pure stress deformation.Interestingly, in the latter case Nc first decreases before rising at larger strainsas clusters that are close in the xy−plane first merge and then break up later.Clusters therefore stay more spherical but break up more easily during pure straindeformation, while they tend to stay intact but deform more strongly in the purestress deformation.Proportion of bridging chainsWe also consider the proportion of bridging chains pb in panels 4.7(e) and (f). Chainends terminating in the same cluster are said to loop, while chain ends ending intwo different clusters are said to bridge. The bridging fraction ranges between 80-90%, which is very close to values reported in a previous simulation study [57]. In63 0 20 40 60 80 100 120 140 160A(a) 0.5 1 1.5 2 2.5 3 3.5Nc/Nc0(c) 0.78 0.82 0.86 0.9 0  1  2  3  4  5  6  7p bStrain(e)(b)(d) 0  1  2  3  4  5  6  7  8Strain(f)Figure 4.7: Asphericity of minority monomer domains (a,b), number of minoritymonomer domains (c,d) and proportion bridging chains (e,f) during pureuniaxial strain deformation (left - a,c,e) and pure uniaxial stress defor-mation (right - b,d,e). Green: 25−450−25 triblocks, red: 15−270−15triblocks, blue: 5−90−5 triblocks. All data reflect averaging over threesimulation boxes containing 480,000 monomers.64experiments values ranging between 40% [95] and 90% [88] are reported and maybe specific to material and preparation protocol. Interestingly, the bridging fractionchanges little throughout the deformations. Despite the changes in the number ofclusters they remain connected by bridging chains. In general there is an increasingtrend of pb when the number of clusters is increasing and vice-versa. This meansthat loops are turning into bridges if clusters divide rather than isolated chainspulling out of clusters.Behaviour of bridging vs. looping chainsTo further understand the variation in A,Nc, and pb, we consider the rate at whichbridging chains are changing to looping chains and vice-versa. Figure 4.8 shows thepercentage of chains that switch in strain intervals of 0.5. In panels 4.8(a) and (b)we see that the rate of bridges switching to loops is fairly constant but non-zero forboth pure strain (panel 4.8(a)) and pure stress (panel 4.8(b)) deformations. The rateof chains changing from loops to bridges (panels 4.8(c,d)) shows more pronouncedtrends. For both deformation modes there is initially a low rate of transition fromloops to bridges which eventually increases, but the increase happens at larger strainsfor longer chains. For the pure stress deformations (panel 4.8(d)), the increase intransition from loops to bridges happens at strain of approximately 2,4.5, and 6 forN = 100, 300, 500, respectively. These strains correspond well to the upturn in thenumber of A-type clusters seen in panel 4.7(d) as well as the decrease of asphericityin panel 4.7(b). The behaviour of the transition rates also explain the observednon-monotonic trends of the bridging fraction pb. Since the transition rate for loopsto bridges is always smaller than the transition rate for bridges to loops up to athreshold strain, pb initially decreases. However, at large strains loops transforminto bridges faster than vice-versa, and pb rises again.We also consider how the bridging and looping chains are deforming separately.Figure 4.9 shows the mean-squared end-to-end distance scaled by chain length,R2ee/N , for chains that were classified as bridges (panel 4.9(a,b)) or loops (panel4.9(c,d)) in the undeformed systems. Initially, we find R2ee/N ≈ 1.8− 2.0 for bridg-ing chains, while for looping chains R2ee/N ≈ 0.12. These differences reflect thefact that the loop chain ends by definition are located in the same cluster. Duringdeformation, the end-to-end distance of the bridges increases monotonically for allchain lengths in both pure strain (4.9(a)) and pure stress deformations (4.9(b)). Forthe loop chains by contrast, R2ee/N initially remains constant and increases onlylater once a threshold strain is reached. By comparison with Fig. 4.8(c) and (d) andFig. 4.7(c) and (d), respectively, we see that these strains again closely correspondto those where the rate of loops switching to bridges and the number of clusters Ncincrease.65 0 0.5 1 1.5 2 2.5% chains(a) bridges → loops  0 0.5 1 1.5 2 2.5 0  1  2  3  4  5  6  7% chainsStrain(c) loops → bridges (b) bridges → loops  0  1  2  3  4  5  6  7  8Strain(d) loops → bridges Figure 4.8: Percentage of chains which switch from (a,b) bridges to loops and (c,d)loops to bridges in a strain interval of 0.5 in pure uniaxial strain defor-mation (a,c) and pure uniaxial stress deformation (b,d). Green squares:N = 500 triblocks, red circles: N = 300 triblocks, blue triangles:N = 100 triblocks. All data reflect averaging over three simulationboxes containing 480,000 monomers.66 0 10 20 30 40 50 60 70Ree2 /N(a) 500300100 0 1 2 3 4 5 6 7 8 9 0  1  2  3  4  5  6  7Ree2 /NStrain(c)(b) 0  1  2  3  4  5  6  7  8Strain(d)Figure 4.9: Mean-squared end-to-end distance scaled by chain length N for chainsthat were (a,b) bridges at t = 0 and (c,d) loops at the beginning of thedeformation for pure strain deformation (a,c) and pure stress deforma-tion (b,d). Green: N = 500 triblocks, red: N = 300 triblocks, blue:N = 100 triblocks. All data reflect averaging over a simulation boxcontaining 480,000 monomers.67ρ/ρ¯ρA/ρ¯Figure 4.10: Monomer density relative to average density (black,red,yellow scale)and minority monomer density relative to average density (white toblue scale) in a plane of width ∼ 4σ located at x = 0. Five differentvalues of strain = 2, 3.5, 5, 6.5, 8 are shown during pure uniaxialstress deformation of a 25 − 450 − 25 triblock system. Voids (black)begin to appear for strains ≥ 6.5 and reach a volume fraction of ∼ 1%for ǫ = 8. Red dashed lines highlight one minority monomer domainbreaking apart.684.3.4 Void formationAt large strains, where the break up of hard minority monomer regions has occurredwe observe some void formation in the long chain triblock systems undergoing purestress deformation. This is not seen in the rubbery homopolymer systems. Tovisualise this process we divide the simulation box volume into ∼8000 cubic bins(∼ 4 monomer diameters in length) and consider the local density of monomers ρrelative to the average density ρ¯ in each. Figure 4.10 shows snapshots of the densitybinned in the x = 0 plane. We see the development of voids at εzz = 6.5. Alsoshown is the density of the minority monomers ρA relative to the total density.In these snapshots we can see that hard minority clusters start to deform aboveεzz = 3.5 and break up at εzz = 6.5. The voids appear to form primarily whereminority clusters are being pulled apart, while little correlation was found with thelocal pressure distribution. This is consistent with earlier simulation studies oncavitation in homopolymers, which identified instead low local elastic moduli as thebest predictor for the formation of cavities [96].Figure 4.11 shows the same visualisation of the density during the early stagesof pure strain deformation where cavitation occurs. Figure 4.11(c) shows that in theN=500 homopolymer there is no variation in density at εzz = 0.05 but at greaterstrains localised round voids form. Figure 4.11(b) shows the same snapshot for theN=500 triblock. Here variations in density already become visible at εzz = 0.05.Voids develop first at the surfaces of the minority regions (shown in figure 4.11a).The minority regions remain undeformed and the voids develop in a non-symmetricalway around them. Indeed the stress-strain curves of figure 4.2(a) indicate that cavi-tation occurs at smaller strains in the 25-450-25 triblock than in the homopolymers.Irrespective of chain length all homopolymers display the same behaviour at smallstrain, while for the triblocks the stress response is chain length dependent.4.4 DiscussionWe have performed molecular simulations of the structure and response of modelABA triblock copolymers in a spherical phase-separated morphology under two uni-axial deformation modes. Model parameters were chosen so that the A(B)-phaseswere below (above) the glass transition temperatures, therefore mimicking the sit-uation in a thermoplastic elastomer. Under pure stress uniaxial deformation, bothunentangled homopolymer and triblock chains show yielding and chain pullout atlarge strains, which is consistent with previous simulation results for short chains[57]. For entangled polymers, we find stress-strain curves that are qualitatively simi-lar to styrenic block copolymer experiments [7, 85, 97]: the triblocks exhibit a muchstronger hardening response with stresses that exceed those in the correspondinghomopolymer elastomer by up to six times. The global chain deformation becomesaffine as the chain length increases. An analysis of the NAD as a measure of the69degree of local plastic activity reveals a gradual transition of plastic activity fromthe rubbery regions to the glassy regions at very large strains. This transition co-incides with the strain at which the asphericity of the spheres is maximal. Onlyupon further straining do spheres begin to break up and their number increases,while the bridging fraction remains constant during the entire deformation. Theseobservations are consistent with the interpretation that the trapping of chain endsin the hard glassy regions is key to forming a cross linked rubbery phase in whichentanglements in the soft background are preserved.Under pure strain deformation by contrast, the elastomer first fails through cav-itation followed by a fibril drawing process. Systems composed of short chains failthrough chain pullout following cavitation, in close similarity to simulations of craz-ing in glassy polymers [98]. Longer triblock chains exhibit a stress plateau followedby strain hardening. In contrast to the pure stress deformation, glassy spheres dobreak up much earlier in the deformation process, and therefore remain more spher-ical rather than extremely extended along the deformation axis. The slight increaseof the bridging fraction with sphere breakup indicates that chain pullout is not dom-inant in this deformation mode either. Plastic activity is initially concentrated inthe rubbery phase, but increases in the glassy phase as strain hardening progresses.Localised density variations during deformation of triblock elastomers have beenobserved in experiments. Inoue et al. have found using small angle x-ray scatteringof SIS polymers that at higher elongations regions of low density form in the iso-prene region, stating that microvoids could lead to material failure [85]. Our studyshows that void formation both in the early stages of pure strain deformation andin the late stages of pure stress deformation is intimately related to the presence ofglassy inclusions. Cavities are observed preferentially either at the interface betweenglassy and rubbery regions, or between a glassy region that fragmented just beforecavitation. Our simulations are consistent with experimental trends and provide in-sight into microscopic deformation mechanisms of nano-structured macromolecularmaterials. The simulation methodology is flexible and can be extended beyond lin-ear polymers to more complex topologies such as star or miktoarm polymers. Thepotential of these polymers to further improve mechanical properties is currentlybeing explored in experiments [10, 97].70ρ/ρ¯ρA/ρ¯(a) (b) (c)Figure 4.11: Minority monomer density relative to average density (whiteto blue scale) and monomer density relative to average density(black,red,yellow scale) during the early stages of pure strain defor-mation of the 25 − 450 − 25 triblock (a) and (b) and in the N = 500homopolymer (c). Densities are shown in a plane of width ∼ 4σ in thex = 0 plane at strains = 0, 0.05, 0.1, 0.15, 0.2.71Chapter 5A Molecular Simulation BasedNetwork ModelIn this chapter, we introduce an entropic network model for copolymer elastomersbased on the evolution of microscopic chain conformations during deformation. Weshow that the stress results from additive contributions due to chain stretch at theglobal as well as entanglement level. When these parameters are computed withmolecular simulations, the theory quantitatively predicts the macroscopic stress re-sponse. The model requires only one elastic modulus to describe both physicallycrosslinked triblock networks and uncrosslinked homopolymers and fits the simula-tion data excellently. 15.1 MotivationAs introduced in section 1.2.3, entropic network models of elastomers attempt, withvarying degrees of complexity, to connect mechanical response to the behaviourof polymer chains during deformation. However, the microscopic chain configura-tions cannot easily be measured experimentally. Therefore, theoretical models aretypically fit to at least the two elastic moduli Ge and Gc relating to the stress con-tributions of entanglements and crosslinks. The recent non-affine strain model [30]fits to experimental data for vulcanized rubber well into the large strain regime.It combines the non-affine tube [28] and the Arruda-Boyce 8-chain [26] models sothat the former becomes applicable for larger strains. The 8-chain model takes intoaccount the effect of finite chain length, but only considers crosslinks, while the non-affine tube model accounts for the free energy cost of an entangled chain confinedto a deforming tube.Previous MD simulations of coarse-grained polymers reproduce the experimentalstress response of elastomers qualitatively [81, 100], while also providing insight intothe deformation at chain level. However, even here, fitting to a model is required toseparate the stress contributions from crosslinks and entanglements between chains[29]. In addition, while crosslinks are easy to identify and track during a moleculardynamics simulation, entanglements are not. Methods based on primitive pathanalysis (PPA) [45] that rely on Gaussian chain statistics no longer apply duringsignificant chain deformation. Progress has been made recently utilising Kro¨ger’s1Parts of this Chapter have been published in reference [99]72Z1-method (see section 2.6) to identify changes to the entanglement length and tubediameter during deformation [29, 100, 101].We introduce here an entropic network description of the stress response oftriblock copolymers to volume-conserving uniaxial strain in terms of the change inseparation of chain ends and entanglement points. We use MD simulations to trackboth of these parameters throughout the deformation, and use the microscopic chainlevel deformation as input into the macroscopic constitutive law. Our descriptionrequires only one elastic modulus to describe the contribution of both entanglementsand crosslinks to the stress.As in previous chapters we focus on the common ABA triblock copolymer elas-tomers, which when made with a rubbery midblock and ∼10-20% styrenic end-blocks forms a spherical morphology, see Figure 4.1. The glassy styrenic regionsact as physical crosslinks. Early experimental stress-strain curves were fit to em-pirical models [7, 83]. More recently, the slip-tube model developed for vulcanizedrubbers has been applied to describe both experimental and simulation results fortriblock copolymer elastomers. It was found to be a good description for uniaxialdeformation of SIS triblock polymers for intermediate stretch ratios (λ = 2.25− 4)[20]. Chantawansri et al. [29, 102] applied the same model to dissipative parti-cle dynamics (DPD) simulations both with and without crosslinks up to a stretchλ < 1, fitting Gc and Ge that differ by up to an order of magnitude. However, thetriblock polymer network differs in important ways from typical rubbers: all chainsare end-linked by the glassy regions so there are no dangling ends, the full chainlength remains a relevant length scale, and the network functionality is higher (forour systems ∼ 50 rather than 4).5.2 Molecular dynamics implementationAs is Chapter 4, we apply a deformation test of mechanical properties to ABAtriblock copolymer systems. We use the same equilibration method and microscopicbead-spring model (described in Chapter 2) but with implementation parametersthat are consistent with the goals of this Chapter - to study the connection betweenmicroscopic properties and entropic model predictions of stress response.To summarise the implementation details that remain consistent: we use thestandard Kremer-Grest bead-spring model with a Lennard-Jones (LJ) pair potentialacting between beads, and a FENE-spring potential acting along the backbone of thechains [55]. The LJ potential is truncated at r0=1.5 to include an attractive regime,and all results are quoted in reduced LJ units. To obtain initial configurations, thechain conformations and phase-separated regions are equilibrated with HOOMD[36, 103] using the soft potential following the method described in section 2.4.3before the above model with hard excluded volume interactions is introduced.There are 480,000 monomers in each simulation box and each chain is 10% A-type and we compare the triblocks to homopolymers N = 300, 500, 800, made up of73the majority monomers. We also have a reference ‘cut’ system which has the samemorphology as the triblocks, but chains are cut into separate A,B,A parts after theequilibration.We require a positive pressure through-out the deformations. However, applyingan NVT volume conserving strain results in a pressure decrease. At a densityρ = 0.85 (obtained from the equilibration steps) the pressure drops below zero sothe number density after equilibration is ramped ρ = 1.0 using a box resize equalin all dimensions. Following this, the temperature is quenched from T = 1.0 to 0.29at a rate of 10− Microscopic model potential parametersIn all chapters the minority pair potential parameter is kept consistent ǫAA = 1.0,this sets the scale of temperature. However, here we alter the majority pair potentialparameter to ǫBB = 0.5 compared to ǫBB = 1.0 in Chapter 3 and ǫBB = 0.7 inChapter 4. This variation in ǫBB changes the elastic contrast between the A and Bcomponents of the triblock. For ǫBB = 0.5 the A and B glass transition temperaturesdiffer by a factor of two, approximating the ratio of glass transition temperatures inpolystyrene and polybutadiene. Similarly, room temperature is approximately halfway between the Tg’s for polystyrene and polybutadiene. Therefore, for mechanicaltests we set the temperature to T = 0.29, the midpoint of Tg = 0.19 and Tg = 0.38(determined as described in section 2.4.4).In this chapter we have chosen our simulation parameters with close reference toa particular physical system. The parameter choices in previous chapters were morerestricted by model development. Firstly, in Chapter 3 we established the equilibra-tion protocol with ǫBB = ǫAA (no-elastic contrast between regions) to isolate howǫAB drives the phase segregation. Secondly, in Chapter 4 short chains (N = 100)were included in our study (connecting our simulations to previous triblock TPEstudies with weakly entangled chains [57]). We found that the phase segregation ofN = 100 chains cannot be discerned by eye when ǫBB = 0.5, only by calculatings(q) or χ. This remains true for vanishingly small values of ǫAB (smaller valuesdrive more pronounced phase separation); there is no visible formation of sphericalregions. Hence, we used potential parameter values that could successfully formspherical regions in short chains (ǫBB = 0.7, ǫAB = 0.2, figure 4.1 has snapshots ofphase segregation).However, the deformation results of Chapter 4 indicate that N = 100 chainsare a somewhat pathologic case. They are only weakly entangled (N ≈ 2Ne) andthe stress response does not display strain hardening such as in the N = 300, 500cases, but rather indicates material failure. So in this chapter, while we still useN = 100 chains simulations for matching the soft and microscopic model parameters,we exclude them from the mechanical tests. We instead take N = 300, 500, 800which gives N = 7− 18Ne. Upon visual and calculated confirmation of satisfactoryspherical phase-separated morphologies forming at these chain lengths, we can use74the lower value of ǫBB without concern.5.2.2 Defomation simulation detailsWe then use LAMMPS [35, 104] to apply a volume conserving uniaxial strain at anengineering strain rate of 10−5.This strain rate is an order of magnitude slower than chapter 4. Given thatwe want to study entropic network models, which give no consideration to viscouseffects, our deformations should be as slow as computationally feasible. Configura-tional entropy is linked to the plateau stress (section 1.2.2). A typical way to findthe plateau stress is to perform a step strain and then allow the system time to relax(see figure 1.8). Our simulations cannot respond to a step strain to a large stretch(bonds are deformed too far for the FENE potential). However, we can relax theconfigurations at particular stretches.5.3 Relating chain stretch to stress responseUpon deformation the global stretch varies as λ = λz = 1/√λx = 1/√λy. Figure5.1(a) gives the stress response of the material to this deformation. In terms ofthe entropic elasticity factor g(λ) = λ2 − λ−1 (section 1.2.3) the stress is linearfor the longest homopolymers when g(λ) > 3.5. This indicates that the simpleaffine-network model (see section 1.2.3 ) is a good descriptor in this strain regime.However, the slope of this curve decreases with decreasing chain length, which wouldindicate a changing value of the elastic modulus G which we expect to be a fixedmaterial property independent of chain length. The cut chains show a very similarresponse to the homopolymers, markedly different from that of the triblocks, wherethe hardening is more pronounced. Again the response is mostly linear for thelongest chains, and the hardening varies with chain length. Here the shortest chains(N = 300) exhibit the strongest hardening response. We find that at a strain rate10−4 results are qualitatively identical and are presented in appendix D.5.3.1 Effective stretchThe entropic stress factor we discuss is specific to a volume conserving uniaxialstrain, more generally it give the relationship between cartesian components of theglobal stretch tensor g(λ) = λ2−λ−1 = λ2z−λ2x. Given the chain length dependenceof the stress, we consider the stretch of chain end-to-end vectors Rc rather than theglobal stretch. We compute an effective stretchgeff = λ2cz − λ2cx, (5.1)where λcz and λcx are the rms component-wise stretches of the chain [47]. This iscalculated for the relevant soft part of the network: the full chain for homopolymers,7500.10.20.3σz-σx(a)1020304050 10 20 30 40 50λcz2-λcx2λ2-1/λ(b) 10 20 30 40 50σz-σxλcz2-λcx2(c)Figure 5.1: (a) Stress response in terms of the entropic elasticity factor g(λ) =λ2 − 1/λ. (b) Effective stretch geff = λ2cz − λ2cx in terms of componentsof the chain end-to-end stretch vs. g(λ), black line g(λ) = geff . (c)Stress vs. effective stretch. Green: triblocks, red: homopolymers, blue:cut chains for N = 300 (), N = 500 (◦), and N = 800 (△).76the mid-blocks of the triblocks and the B-type chains in the cut systems. Figure5.1(b) shows how the effective chain stretch varies in relation to the global stretch.All homopolymers and cut chains display sub-affine stretch throughout the defor-mation but the longest chains approach the affine limit. The triblock chains, bycontrast, display super-affine behavior for the shorter chains, which is due to thestrong anchoring of the midblock between the almost rigid glassy spheres.The relationship between the deformation of glassy regions and the degree ofaffinity of chain stretch has been explored in detail in Chapter . 4. Here we focuson the regime without significant deformation of glassy regions - so that entropicnetwork models can be applied. We do take into account the trends with increasingchain length, towards affine in all cases. Further comparisons of the difference inglassy phase responses at larger strains will be made in Chapter 6.Figure 5.1(c) shows the stress response in terms of the effective stretch geff ratherthan the global stretch. The data collapses onto two curves: one for triblocks andone for both the cut and homopolymer chains. The presence of the glassy spheresis therefore not the dominant cause of the difference in the stress response of thetriblocks compared to the homopolymers. Some models include a correction for thestyrenic end-blocks acting as a inert filler [7, 20]. They invoke the Guth-Smallwoodequation, which increases the elastic modulus by a factor (1+2.5φ+14.1φ2), whereφ is the volume fraction of end-blocks. Since we find no difference between cutand homopolymer chains, this correction is unnecessary. We now focus on thepolymer network deformation to explain why the triblock and homopolymer/cut-chain responses differ.Accounting for any non-affinity in the deformation of chain ends significantlydecreases the chain length dependence of the triblock stress response and removes itfor the homopolymers. The homopolymer stress-strain relationship remains linearfor all data collected and all chain lengths, but the triblocks still exhibit a strongernonlinear hardening. Though all chains are now described as having a linear stressresponse at least over some range of stretch, two different elastic moduli would berequired to describe the deformation for triblocks and homopolymers.5.4 Entanglement LossWe next consider if entanglement loss could describe the differing stress responses oftriblocks and homopolymers. We use an implementation of the Z1-method providedby Kro¨ger to monitor the entanglement length Ne during deformation (section 2.6).This algorithm geometrically minimizes a shortest minimally connected path (SP)for each chain simultaneously. Kinks are entanglement points where chains interact,and their number decreases during the minimization process. The final number ofkinks per chainNk is related to the entanglement length byNk = N/Ne. Throughoutthe deformations we find in Figure 5.2 that the entanglement length increases forall chains by less than 10%. There is no difference in entanglement loss when77 40 42 44 46 48 50 0  10  20  30  40  50 Ne λzx2-λcx2Figure 5.2: Entanglement length (from Z1 analysis) vs effective stretch. Green:triblocks, red: homopolymers, blue: cut chains for N = 300 (), N =500 (◦), and N = 800 (△).comparing the homopolymers, triblocks and cut chains other than small variationsin initial values Ne ≈ 42.Given this minimal entanglement loss, we now investigate how the polymer net-work responds to deformation on the length scale of the entanglements. We takethe initial set of monomers defined by the kinks of the SP from the Z1 analysis.The end points of the chains are included as kinks only for triblocks. The triblockmid-blocks are tethered to the glassy region while the homopolymer end-points areuntethered. We track these monomers during deformation and define entanglementvectors Rk between those along each chain.5.5 Entropic elasticity model developmentTypically network models focus on deriving the relationship between entanglementsand crosslinks with the applied deformation. However, we can determine this di-rectly through simulation. This allows us to formulate a model for the stress interms of the entanglement and crosslink stretches. Here, the entanglement lengthNe (and therefore number of kinks Nk) is approximated as constant at the initialvalue.The distances Rk and Rc are statistically independent variables. We check thisby confirming 〈Rk〉〈Rc〉 = 〈RkRc〉 throughout the deformations (see SI Figure 1).78The entropies related to the number of possible configurations at a given Rc andRk are therefore additive, and we can consider the triblock network to be the su-perposition of two networks on different length scales. The first, consisting of thechain segments between entanglements, is also applicable to homopolymers. Eachsegment between two entanglement points has Ne beads, with end-to-end distanceRk that is initially R2k(0) = c∞Neb2, where b is the FENE bond length. The longercrosslink network enforced by the glassy regions on the triblock chains, has initialend-to-end separation R2c(0) = c∞Nb2. We do not differentiate between chainsthat bridge between two glassy regions and those that loop back. While the initialend-to-end distances are dramatically different for looping and bridging chains, wefind that both their stretches are equal as well the number of kinks found by theZ1-analysis.For a network of chains of length N crosslinked at their ends, the (purely en-tropic) free energy density can be written as [26]W = GN(hL−1(h) + ln L−1(h)sinh(L−1(h))), (5.2)where G = νkBT with the chain density ν, h is the ratio of the chain end-to-enddistance to the maximum possible end-to-end separation, and L−1 is the inverseLangevin function. Given that Rc and Rk are statistically independent, the to-tal energy density can be written as the sum of contributions from entanglementsand crosslinks. We use the 8-chain model to link the end-to-end chain stretchλchain = h√N to the component stretches, λchain =√13(λ2x + λ2y + λ2z). We confirmthis relationship holds for both chain stretch λc = Rc/Rc(0) and the entanglementstretch λk = Rk/Rk(0) (figure B.2).The stress contributions can then be found by differentiating the energy densitywith respect to the deformation of the respective chains,σi = λidWdλi+ c = GL−1(h)3hλ2i + c. (5.3)Entropic network models typically have this form, with contributions due toentanglements and crosslinks, and fit the two elastic moduli Ge and Gc to experi-mental or simulation data. However, the moduli are related and we can avoid fittingboth. In terms of monomer number density ρ, Ge = ρkBT/Ne and Gc = ρkBT/N =Ge/Nk. In the triblock system, both entanglements and crosslinked ends contributeadditively, henceσz − σx = Ge[L−1(hk)3hk(λ2kz − λ2kx) +1NkL−1(hc)3hc(λ2cz − λ2cx)], (5.4)where hc = Rc/Nb and hk = Rk/Neb. Since the homopolymers and cut chainsonly display Gaussian hardening in Fig. 5.1 (i.e. the stress vs elasticity factor curve79 0 0.2 0.4 0.6 0.8 1 0  10  20  30  40  50 h e λzx2-λcx2(a) 0 0.2 0.4 0.6 0.8 1 0  10  20  30  40  50 h c λzx2-λcx2(b)Figure 5.3: Ratios of the chain end-to-end distance to the maximum possible end-to-end separations for (a) the chains segments between entanglementsand (b) the midblock segments in the triblock chains. Green: triblocks,red: homopolymers, blue: cut chains for N = 300 (), N = 500 (◦),and N = 800 (△).is linear), we propose to take the simpler affine network model in terms of the80entanglement stretch for these systems,σz − σx = Ge(λ2kz − λ2kx). (5.5)Figure 5.3 shows the dependence on chain stretch of both the entanglement andchain proportions of maximum extension, he and hc. Homopolymers only haveentanglements not cross links so we only consider he. For all the homopolymershe ≤ 0.4 such that the Inverse Langevin function is in the small stretch limit L1(x) =3/he and the 8-chain model is equivalent to the affine network model (see figure 1.5).In the case of chain ends, figure 5.3(b), hc becomes larger than this small stretchlimit. hc has a chain length dependence with the shortest chain approaching 60%of the maximum extension while for the longest chain hc plateaus closer to 30%.The global affine stretch applied is longer in units of the shortest chain maximumextension (300b) compared to the longest chain maximum extension (800b). 0 0.05 0.1 0.15 0.2 0.25 0  5  10  15  20  25  30σz - σxModel/GeFigure 5.4: MD simulation stress vs. stress calculated from models for triblocks andhomopolymers/cut chains Green: triblocks, red: homopolymers, blue:cut chains for N = 300 (), N = 500 (◦), and N = 800 (△).815.6 Application of the entropic network modelIn order to test these models, we plot the MD measured stresses against the valuespredicted by the model in Figure 5.4. Here, the values for the stretch components,hk, and hc are input into eqs. (5.6) and (5.5) directly from the simulations. For allsystems, the data collapses onto a single curve that indicates a nearly perfectly linearrelationship. This excellent agreement is strong evidence that we have correctlyidentified all relevant microscopic chain deformations. The slope of this mastercurve furthermore has a value of Ge = 0.009, which agrees well with the expectedvalue of Ge = ρkBT/Ne = 0.007.As a further illustration of the success of our description, we compare in Figure5.5 model and simulated stresses in terms of the global stretch λ. Again, we findexcellent full quantitative agreement between model prediction and stress-straincurves for all systems and chain lengths studied. Our theory uses only microscopicdeformation variables, and establishes an almost fit-free description of the non-linearmechanical response of nanostructured polymeric elastomers.5.7 Comparison to other network modelsThe non-affine strain model (eq.(1.32)) represents the most recent entropic model tosuccessfully fit the experiment stress response of vulcanised rubbers with the leastfit parameters (3 rather than 4 [30]). We expect similar mechanisms to be releventin triblock TPEs, so we compare our simulation results and entropic model to thisstandard. They are very similar in spirit, each considers the stress contributionsfrom the global deformation of crosslinks and from entanglements which add anadditional length scale to the problem.The non-affine strain model still has 3 parameters, which are fit to experimentalresponse curves across two deformation modes. The topology of triblock TPEssimplifies this model. A large number of chain ends meet at each glassy crosslink(> 20 compared to 4 in vulcanised rubbers). Therefore, with regard to the parameterg, which interpolates between phantom network and affine network behaviour ofcrosslinks, we can take the latter limit. This simplifies the first term of eq. (1.32)to the 8-chain model. Our cross link contribution term is identical to this (keepingin mind that we apply it to chain stretch rather than global stretch). Also givenN/Ne = Nk the elastic moduli are related by Gc = Ge/2Nk. For our simulationseq.(1.32) simplifies toσz − σx = Ge[λcz − λcx + 1λcz− 1λcx+12NkL−1(hc)3hc(λ2cz − λ2cx)]. (5.6)However, these simplifications also mean there is no longer a free parameter to fitthe stress variation for different triblock chain lengths. We are able to calculatethe maximum stretch directly as Nb. We could fit the remaining parameter Ge,82 0 0.05 0.1 0.15 0.2 0.25 0.3 0  5  10  15  20  25  30  35  40                        σz-σx g(λ)Points - modelLines  - simulation stressN=800 0 0.05 0.1 0.15 0.2 0.25 0.3 N=30010 20 30N=5000 10 20 30Figure 5.5: Comparison of model (points) to simulation data (lines) in terms ofg(λ) = λ2−1/λ. Green: triblocks, blue: homopolymers, red: cut chainsfor N = 300 (), N = 500 (◦), and N = 800 (△).however the Nk prefactor is insufficient to account for the variation in triblockstress responses. Figure 5.6 reproduces figure 5.4 for the non-affine model. Wedo not see the same data collapse to a master curve as we did with our model.The curves are not linear, indicating increasingly poor agreement with the model,which overestimates strain hardening. There is a strong chain length dependencein the triblock curves. The shortest triblock chains show the best agreement withthe model. The simulation and model stresses are linearly related up to an applied83 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0  2  4  6  8  10  12  14  16  18  20σz - σxModel/GeFigure 5.6: Non-affine strain model. Green: triblocks, red: homopolymers for N =300 (), N = 500 (◦), and N = 800 (△). Black line: master curve forour modelstretch ∼ 4.5, and the slope is close to that of our master curve, indicating thatin this case both models provide the same Ge. In the following section, we explorethe relationship between entanglement and chain stretch to understand why thenon-affine strain model does not agree well with our simulations.5.7.1 Linking stretch length scalesTypically entropic network models focus on relating experimentally measurablequantities to a model and backing out the microscopic parameters through fitting.However, we have the details of microscopic scale stretches that can test not onlymodel fits, but the assumptions that they are built on. We find that chain stretchis not an ideal predictor of entanglement stretch. This results in other entropicnetwork models giving a poorer fit to the stress data than taking the entangle-ment stretch as an input directly. The network models we have discussed assumesome relationship between macroscopic deformation and entanglement deformationto ultimately calculate the resulting stress. We equate our chain stretches withexperimental stresses and investigate the relationship in our deformations betweenentanglement and chain stretches.We first find that the magnitude of the chain stretches are linearly related to84 1 1.5 2 2.5 3 3.5 4 1  1.5  2  2.5  3 λ c λkFigure 5.7: Comparison of chain stretch λc to entanglement stretch λk. Green:triblocks, red: homopolymers for N = 300 (), N = 500 (◦), andN = 800 (△). 0 5 10 15 20 25 0  5  10  15  20  25  30  35  40  45g(λk)g(λc)Figure 5.8: Entropic elasticity factors for chain (g(λc)) and entanglement stretches(g(λc)). Green: triblocks, red: homopolymers forN = 300 (), N = 500(◦), and N = 800 (△).85the magnitudes of the entanglement stretches, see figure 5.7. Both triblocks andhomopolymers have the same slope λc = 1.5λk. If deformation of entanglementswas purely affine (and without fluctuations), we would expect the slope to be one.Therefore, the deformation of the entanglements is not affine, but the linear depen-dence is consistent with both the assumptions of the non-affine strain model andthe 8-chain model.However, this linear relationship between stretch magnitudes does not translateto a linear relationship with a consistent slope between the entropic stress factorsg(λc) and g(λk), see figure 5.8. So the common assumption of a direct linear rela-tionship of chain and entanglement stretches fails (see figure B.3 for the individualstretch components). The assumed entanglement component stretches do not havea constant slope, which is why the non-affine strain model over-estimates strainhardening.For the triblock chains there is also a chain length dependence in the relationshipbetween g(λk) and g(λc). Though it is a small variation, this microscopic level detailentirely accounts for the fact that our model predicts the same Ge for all chainlengths. None of the other input parameters in our model eq. (5.6) display a chainlength dependence. The N = 300 chains have the closest to a linear relationshipbetween chain and entanglement elasticity factors, and indeed the same systemshows the best agreement between the non-affine model and ours. Ultimately thedifference in g(λk) between triblocks and homopolymers determines the variation instress response curves. This is why, although the cross-link terms are identical, weobtain a better fit to our data with our model that uses microscopic input ratherthan the non-affine strain model.5.7.2 Stretch componentsWe now probe the relationship between entanglement and chain stretch in our simu-lations further to understand why their components are not linearly related. We con-sider two limits of how the entanglement points could be deforming: either affinelywith the chain stretch or as a freely-jointed chain with rigid bonds (FJC). Figure5.9 clarifies that the complete chain is considered in terms of how the entanglementpoints deform. A simple stretch is applied in one dimension (λz, λx) = (λ, 0). Wesee that in the affine case the bonds Rk stretch and the Cartesian components ofRk are by definition linearly related by the stretch. In the freely-jointed chain caseRkx decreases and Rkz increases as the fixed-length bonds rotate to align with thedeformation axis. Of course our deformations have a more complex volume con-serving stretch applied, but the key observations here hold. In one limit chain andentanglement vector components deform proportionately, while in the other theyrotate instead.86Figure 5.9: Two extremes of the deformation of entanglements under pure uniaxialstrain deformation in two dimensions: affine and like a freely-jointedchain. The relative change to the components of Rk is show in red.87 0.1 0.3 0.5 0.7 0.9 0.1  0.3  0.5  0.7  0.91/λ zλx2Triblocks 0.1 0.3 0.5 0.7 0.9 0.1  0.3  0.5  0.7  0.91/λ zλx2HomopolymersFigure 5.10: Stretch components compared to expected volume conserving stretchλ2x = 1/λz. Top: Triblocks. Bottom: Homopolymers. Closed symbols:chain end-to-end stretch. Open symbols: entanglement stretch. N =300 (Red), N = 500 (Green), and N = 800 (Orange)88Affine stretch componentsAffine deformation of entanglement points is a common assumption. The physicalinterpretation is that ‘slack‘ between entanglements is taken up before the entan-glements points start to deform as monomers of FJC. Recalling figure B.2 we haveshown our average chain and entanglement stretches both follow the 8-chain modelassumption that λ2 = λ2z + 2λ2x. However, this doesn’t mean that both stretchlength-scales deform in the same way as the global volume conserving stretch whichimplies λ2x = 1/λ2z. Figure 5.10 shows that the chain stretches, in both homopoly-mers and triblocks, deform completely affinely (λ2x = 1/λ2z) while the entanglementstretches do not. This non-affinity of the entanglement stretch explains why therelationship between chain and entanglement stretch magnitudes is not one to one,(figure 5.7). However, there is no chain length dependence in the non-affinity ofentanglement stretch components indicating that we still have not accounted forthe chain length dependence of the rubber elasticity factor for the triblock g(λk).Freely-jointed chain stretch componentsWe can consider the extreme opposite to affine deformation, the FJC, to explainthe chain length dependence in our entanglement stretch components. By the Kuhnhypothesis, the chain defined by the entanglement points obeys the same statisticsas a FJC (see figure 5.9 for an illustration). We introduced the force-extensionrelation for a single FJC in section 1.2.1. In terms of our simulation results, we canalready say that the FJC model does not strictly apply because the entanglementstretch increases by a factor ≈ 2. However, it can provide a bound for the chainstretch behaviour.While an affine deformation preserves the ratio of chain and entanglement stretchcomponents, the deformation of a FJC is somewhat more complex. A simple exam-ple is a two-dimensional uniaxial deformation. In the affine case the bonds becomelonger but the magnitude of the component perpendicular to the deformation di-rection remains the same, while in the FJC the monomers move closer to the chainaxis (see figure 5.9).In terms of the Kuhn chain (setting the bond length b = Rk), we show inappendix C that the entropic elasticity factor readsg(λkx, λkz) =3β2cos 2θ(3 + β2 − 3β cothβ), (5.7)where θ = tan−1Rx/Rz and β = fb/kT . See Appendix C for calculation details.Figure 5.11 give a comparison of our simulated stretches to the affine and FJCdeformation limits. We find that our results are bounded by these two limits andthat the chain length dependence of the FJC stretch response is echoed by oursimulation data.89 0 5 10 15 20 25 0  5  10  15  20  25  30  35  40  45g(λk)g(λc)Affine limitFJC limitFigure 5.11: Chain and entanglement entropic elasticity factors, comparison tobounding predictions. Open symbols: simulation results. Black line:affine prediction. Closed symbols: freely-jointed chain (FJC) predic-tion. N = 300 (Red), N = 500 (Green), and N = 800 (Orange)90These results suggest that the cumulative effect of the entanglement tube con-finement and anchored end-points is perhaps best approximated as a spring withequilibrium length given by the initial entanglement separation Rk(0). With theforce f = −K(Rk − Rk(0)) we have the correct limiting cases: as the spring con-stant K → ∞ then Rk = const. corresponding to the FJC limit, and as K → 0 wereach the affine limit. The chain length dependence of the stretch components intriblocks is not predicted or accounted for by current network models. Our simu-lation guided network model is successful at taking all relevant microscopic effectsinto account.91Chapter 6Limitations and ExtensionsWe have presented methods to equilibrate, study and interpret the material prop-erties of linear triblock copolymer materials, along with an analysis of those results.In particular, we have demonstrated that we can account for all relevant microscopiccontributions to the stress over the range of strains to which our model was applied.Now we can consider the most impactful way to apply or generalise these methodsand results.We first present results of the model from Chapter 5 for larger applied strains.We observe the eventual breakdown of our model, and relate that to the glassyregion and polymer network responses. We apply our entropic model to the sim-ulation results of the pure stress deformations in Chapter 4. We thereby establishthat this network model generalises to a system with a different ratio of glassy tobackground softness than the simulations with which it was developed. Finally, weconsider future areas that our models and methods could be applied to. We presentpreliminary results for the equilibration of star polymers with a varying number ofarms, a potential avenue for studying bridging chain effects.6.1 Entropic model at larger strainIn the previous chapter, we specifically chose a strain regime where the deformationof glassy regions is small, the range over which we expect our network model tohold. We have not included in our model a stress contribution from the deformationof the glassy regions themselves. We do, however, account for how Rc changes asa result of triblock mid-block chains being ‘attached’ to the glassy regions. In thissection, we re-examine the cluster deformation and stress data from Chapter 5 atlarger strains. We discuss the limitations of our model and examine the microscopiceffects which lead to its eventual failure. In itself, this failure provides insight intothe link between the glassy regions and deformation and the polymer network.Figure 6.1 plots the predicted model stress against the simulated stress. A linearresult indicates good agreement with the model. This agreement holds until thepenultimate stretch value for both N = 500 and N = 800 triblock chains (λ = 7.5,marked with a vertical line). The stress from the N = 300 chains deviates, showingstronger strain hardening at a shorter global stretch (λ = 7). So, our model hasgood agreement with this simulation data over a large range of stretches.Additionally, the dashed lines in figure 6.1 show the model fit with the entangle-92 0 0.2 0.4 0.6 0.8 1 1.2 0  10  20  30  40  50  60σz - σxModel/GeFigure 6.1: Agreement between model and simulated stresses. Open symbols: En-tanglement vectors recalculated at each point. Closed symbol: initial en-tanglement vectors used to calculate stretch. N = 300 (red ), N = 500(green ◦), and N = 800 (orange △). The vertical lines mark the changein slope for each curve with the corresponding colour, where the simu-lated results start to deviate from the model.93ment vectors recalculated every time-step. We determined, based on figure 5.2, thatthere is little change to the entanglement length up to a strain of 5. In this rangethere is also no difference in entanglement length between triblock and homopoly-mer chains. So, we used the set of entanglement vectors determined by applyingthe Z1-method to the undeformed system to calculate entanglement stretches λk forthe full deformation. Now that we are applying larger strains, we return to thatassumption and test it. We recalculate the entanglement vectors for every snapshot(using the Z1-method). Since these vectors are now constantly redefined we mustcalculate the stretch from averaged lengths rather than averaging stretches. In figure6.1 we observe little difference in the model agreement between the updating andinitial entanglement vector curves. This is a further indication that entanglementloss is not a driver of the stress response, even at these large strains.We now consider a microscopic justification for the breakdown of our model.Figure 6.2 shows the change in asphericity and number of clusters as compared tothe stress response. Up to a strain of 5, the data to which we fit out model, thereis little change in either of the cluster metrics. There is some change to asphericitycorresponding directly with the number of clusters decreasing. So, clusters whichjoin together are less spherically symmetric. Note however that the maximum as-phericity reached here is < 25. At the marked strains, where deviation from ourmodel occurs, we see a small upturn in the asphericity and little change to thenumber of clusters. Note that this is a dramatically different response of the glassyregions than seen for the deformations at faster strain and smaller elastic contrastin figure 4.1, with minimal change to the glassy regions by comparison.6.2 Does the network model generalise?To facilitate a more generalised discussion of the entropic model, we also fit it to thedeformation data from Chapter 4. We compare the pure stress deformations fromChapter 4 to the constant volume deformations in the Chapter 5. We restrict theformer to data where the volume remains constant (to a strain of ǫ ≈ 7), so thatdensity variation isn’t a concern in the model. The simulations differ in strain rate,phase segregation strength, and also the ratio of glass transition temperatures. Asummary of these parameters is given in table D.1. The chain flexibility and generalspherical morphology remains the same.We can also compare to slower deformations (strain rate 10−4) of the same systemas in Chapter 5 (shown in Appendix D). A more marked increase in asphericity isseen for all chains from ǫ ≈ 5 (D.1) which corresponds to strain at which our modelbreaks down (D.2). The fitted Ge is also larger (Ge=0.018), which is expected asthere is more impact from viscous effects at a faster strain rate. A commonalitybetween all three simulation sets, is that they have the same initial bridging ratioof approximately 85%, which changes little during deformation.In chapter 5 we noted that there is a marked up-turn in the stress response as94 0 5 10 15 20 25 30 35 40A(a) 0.6 0.8 1 1.2 1.4 1.6 1.8Nc/Nc0(b) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0  1  2  3  4  5  6  7σz - σxStrain(c)Figure 6.2: Volume conserving uniaxial strain deformation, initial equilibrated con-figurations as described in Chapter 5 strained at a rate 10−5 (a) As-phericity of glassy clusters (b) Normalised number of glassy clusters(c) stress response. Red(◦) N = 300, green() N = 500 orange (△)N = 800. Coloured vertical lines indicate where the model becomes apoorer fit for the corresponding chain length. See slope change in 6.1.95 0 0.2 0.4 0.6 0.8 1 0  10  20  30  40  50σz - σxModel/Ge 0 0.1 0.2 0.3 0.4 0.5 0  10  20  30Figure 6.3: Agreement of simulated stress and model as applied to pure uni-axial strain deformations presented in chapter 4 N = 100 (blue×)N=300(red), N = 500 (green ◦), and. Inset: homopolymers, mainplot triblocks. Coloured vertical lines correspond to onset of more pro-nounced strain hardening as observed in 4 for the corresponding colouredchain lengthsǫ =(6, 3.5) for the N=(300, 500). There is also a downturn from the longer chainbehaviour for the poorly entangled N=100 chain stress at ǫ =3. We identify a similarupturn in the figure 6.2 stress at ǫ =(4.5, 6, 6.5) for the N=(300, 500, 800) chains.Figure 6.3 reproduces figure 5.4 for these simulation results, testing the successof our entropic model, eq (5.6). Notably, we find that again, both the triblocksand homopolymers collapse to the same master curve. The inset shows that the ho-mopolymers agree with this master curve over the full deformation. Fitting the slopegives Ge = 0.013. Of our earlier fit results Ge = 0.007 for ε˙zz = 10−5, Ge = 0.018for ε˙zz = 10−4 (see Appendix D) this Ge is closer to the latter. Therefore, for thefaster strain rates the entanglement modulus is higher and for the slower strain ratesimulations it is lower. This trend is consistent with a material with a viscoelasticresponse. The triblock results deviate from the model fit at smaller strains in thismodel. However, recalling figure 4.1 the onset of glassy sphere deformation and96division occurs at much smaller strains. The coloured lines in figure 6.3 mark thestrains discussed in chapter 4 at which there is an upturn in the strain hardeningand which correspond to deformation and division of the glassy regions.This confirms that our entropic network model successfully applies to two tri-block TPE simulations with differing elastic contrasts and two strain rates. Withoutany refitting it applies across varied chain lengths and to homopolymer and triblocks.Deviation from this model corresponds to significant deformation and break downof the spherical glassy regions.6.3 Star PolymersThere has been some recent interest in block copolymer materials with a branchedstructure [12]. Shi et al. [10] find that asymmetrical three-arm block copolymerscan form TPEs with a much higher styrene content (up to 82%) with a ‘bricksand mortar’ structure. They posit that this could lead to stronger, but still elas-tomeric materials. There are experimental works which find that symmetric stars arestronger than linear triblock elastomers [97, 105]. Spencer and Matsen [106] studythe variation of bridging ratios in star copolymers that form spherical glassy regionsusing self-consistant field theory (SCFT). (AB)M star polymers have M arms ra-diating from a central point and our soft model approach offers a very promisingmethod to study nanostructured star polymer materials. The time-scale for theirequilibration is even more challenging than for reptating linear chains, the armsessentially need to retract to the centre point of the star to explore a new path, andtherefore center of mass diffusion is very slow [107].Soft model equlibration of starsIt is not feasible to brute force equilibrate star polymers even at short chain lengthsfor setting parameters of the soft potential. We therefore use the same microscopicand soft model parameters as in Chapter 5. We implement (AB)M stars, with Marms each of 300 monomers in length, and the outer 10% of each arm is glassy.We equilibrate stars with 3-5 arms and a linear triblock (M = 2) for comparison.We apply the soft model equilibration method in section 2.3 to obtain equilibratedconfigurations.6.3.1 Star morphology resultsFigure 6.4(a) demonstrates a single three arm star polymer in our simulations while6.4(b) shows the corresponding spherical phase-separated structure. To quantita-tively analyse any variation in phase separated morphology we calculate the staticstructure factor, eq. (3.2). Figure 6.5 demonstrates that there is no variation in thestructure factor for stars with 2-5 arms. This result indicates an excellent oppor-tunity to use star polymers to study diverse network topologies with a consistent97(a) (b)Figure 6.4: (a) 3 arm star polymer. Arms are different colours and the glassyend regions are highlighted in a lighter shade. Each arm is N=300monomers connected to one central monomer. The outer 30 beads areglassy. (b) Phase separated morphology with minority monomers: blue,and majority monomers: red98 0.1 1 10 100 0.1  1S(q)qFigure 6.5: Static structure factor for star polymers with varying numbers of arms;green: linear triblock (2 arms), red: 3 arms, blue: 4 arms, orange: 5arms.phase-separated morphology. We can thereby isolate the effects of changing bridgingratios from changing morphology.We analyse the more complex bridging statistics of glassy polymers by two met-rics. Firstly, we compute the probability p(x) that a glassy cluster containing onearm from a star will contain a total of x arms from that star, figure 6.6. We find thatthe probability of there being a single arm in each cluster drops from 88% for thelinear triblock (the percentage of bridging chains) to 60% for M = 5 stars. We candirectly compare this measure to the self-consistent field theory (SCFT) results ofSpencer and Matsen [106]. We find that forM = 3 the probabilities for (1,2,3) armsin each cluster are (75%,21%,3%) compared to their (64%, 30%, 5%). This is verygood agreement considering the two completely different equilibration methods. Wefind a greater p(1) (75% compared to 64%) and a steeper drop off in probability.M = 5 stars also show good agreement to these SCFT results, though again p(1) ishigher in our case (60% compared to 48%).We also consider the probability p(y) that a star’s arms will be in y differentglassy clusters, figure 6.7. This indicates that for a small number of arms M = 2, 3the most likely configurations are with all arms in separate clusters. However, starswith a larger number of arms M = 4, 5 reach a peak in probability at y = 3, 4clusters. The values for p(y)=1 give the probability of all chains in a star looping99 0 0.2 0.4 0.6 0.8 1 1  2  3  4  5ProbabilityNumber of armsFigure 6.6: Probability that a glassy cluster containing one arm from a star willcontain a total of x arms from that star. Star polymers with varyingnumbers of arms; green ×: linear triblock (2 arms) , red : 3 arms, blue◦: 4 arms, orange △: 5 arms. Black lines are reproduced from fig. 4 inref. [106],  3 arms, △ 5 arms. 0 0.2 0.4 0.6 0.8 1 1  2  3  4  5ProbabilityNumber of clustersFigure 6.7: Probability that a star’s arms will be in x different glassy clusters. Starpolymers with varying numbers of arms; green ×: linear triblock (2arms) , red : 3 arms, blue ◦: 4 arms, orange △: 5 arms. Black linesare reproduced from fig. 4 in ref. [106],  3 arms, △ 5 arms.100 0 0.4 0.8 1.2 1.6 2 1  2  3  4  5  6  7σzz-σxxλFigure 6.8: Volume conserving uniaxial strain deformation for star polymers withvarying numbers of arms; green: linear triblock (2 arms) , red: 3 arms,blue: 4 arms, orange: 5 arms. Each arm is 300 monomers the outer 30are hard, the inner 270 are soft.back to the same cluster. This drops dramatically from 12% looping chains in thelinear triblock to 3% for 3 arm stars. So, even a star with 4 arms changes thetopology of our network so that there are no longer any completely looping chains.6.3.2 Star deformation resultsGiven this change in the bridging statistics as a result of increasing the numberof star arms, we compare the corresponding stress responses for these stars. Weperform a volume conserving uniaxial strain at ε˙zz = 10−4, see figure 6.8. Despitesome small variation in the stress response we see no dependence on the number ofstar arms. This result is counter to the expectations that larger bridging ratios willresult in stronger materials. However, in this range of strain there is little change tothe glassy regions - an exploration of larger strain or different strain regimes couldyield an M dependence. Recent experimental results [105] corroborate our findingthat there is no difference in the phase separated structure due to an increasing M(they compare a linear triblock to a 6 arm star). They also find that the stressresponses are consistent for small strains, but that the star displays stronger strainhardening.101Chapter 7ConclusionThis work marks a significant contribution to both computational modelling meth-ods and to understanding the underlying mechanisms of mechanical response intriblock TPEs. In both these areas we have established results that could be ap-plied more broadly than the scope of this thesis. Our focus has been on linking themechanical response of nanostructured materials to microscopic scale mechanisms.This includes statistics regarding chains, monomers, and glassy regions which arenot readily measurable experimentally.Our computational modelling approach was determined by this motivation, inparticular we wanted to retain the chains’ topological network structure. We imple-mented MD with a coarse-grained bead spring model that is effective at modellingglassy and soft polymer phases. The initial equilibration requires chain-ends to un-dergo significant diffusive motion to form segregated glassy regions. We equilibratedentangled chains by applying a soft pair potential model during the equilibrationphase. With this method we have successfully equilibrated TPE like morphologies oftriblock copolymers and simple symmetric stars within an MD framework withoutrequiring prior knowledge of the phase separated structure.Other studies of block copolymers have used SCFT methods to obtain phase-separated structures, which is particularly effective for calculating phase diagrams[2]. Guided growth of polymer chains has allowed for several studies of the defor-mation of lamellar phases [56]. However, the particular system of spherical regionswith interconnecting chains, is difficult to obtain using these methods. If a meanfield method is applied, chains must be reintroduced. Previous studies into triblockTPE deformations have been limited to short chains [57] or smaller deformations[29]. We confirmed that the presence of entanglements is vital to the elastomericmaterial properties, requiring long chains to be simulated. We also observed thatlonger chains result in larger glassy spheres but cause little change to ratio of bridg-ing and looping chains. The equilibration of triblock chains up to 17Ne in length is akey achievement that enables the deformation studies that follow. In Chapter 6 wealso equilibrate stars polymers with a varying number of arms, demonstrating thebroader applicability of this equilibration strategy. We were able to reintroduce themicroscopic model without undoing either of our equilibration measures, allowingus to use these structures in studies of material properties.In Chapter 4 we studied two deformation modes, pure stress and pure strain.The first approximated an experimental uniaxial strain and the second large dilata-tional deformations. We found that in both cases voids formed around glassy regions102splitting apart, strain hardening was stronger in triblocks compared to homopoly-mers, and that there is a chain length dependence of the stress response. Studiesof the glassy regions indicated that their deformation and break-up is connectedto an upturn in the strain hardening stress response. This regime is also markedby a sharp decrease in the displacement of soft monomers, while the glassy regionmonomers remained mobile. In Chapter 6 we confirmed these connections betweenthe strain hardening and the response of the glassy regions were also displayed in aTPE with a different relative softnesses of the glassy and soft regions.In Chapter 5 we found that the chain length dependence of the stress response inhomopolymers is removed by taking into account the chain end-to-end stretch ratherthan the global deformation. Using an implementation of the Kro¨ger-Z1 method wewere also able to track entanglement points during deformation. Typically entan-glement information is backed out from network models or from Gaussian statistics(which do not hold at large strains). We identified that entanglement loss was notsignificant in either homopolymers or triblocks and confirmed in Chapter 6 that ourmodel fit is not impacted by using the initial or updating entanglement vectors.In keeping with recent entropic network models we considered the network struc-ture of triblocks on two length-scales; one which relates to entanglements and oneto cross-links. We confirmed the typical assumption that the evolution of theselength-scales is statistically independent, implying stress contributions are additive.We then developed an entropic network model by applying the Arruda-Boyce 8-chain model to each of these length-scales. The key point to the success of thismodel is that we use both the chain stretch and entanglement stretch determinedfrom simulation rather than the global stretch. We found excellent agreement ofour stress results to our model, demonstrating the successful identification of allrelevant microscopic contributions to the stress. Our model is almost fit free. Itrequires only a determination of Ge, for which our fit was in good agreement withthe calculated value. It describes both homopolymer and triblocks and cut chains ofvarying lengths without refitting this parameter. Typical entropic network modelsdeveloped for vulcanised rubbers [28] and applied to triblocks [29] require fitting 3-4parameters. We confirmed that this model fits excellently to two systems with differ-ent Ge. We also found that the model deviates from simulated stresses when glassyregions start to deform significantly, as is expected from our model construction.In terms of future work, our network model could be generalised. There areopportunities for future application to different deformation modes, varying chainstiffness and to chains with more complex structures than a linear backbone, such asstars. Our preliminary deformation results for stars polymers indicate one promisingavenue for further study. 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As described in Chapter 4116e=N=300 Triblock0137e=N=300 Homopolymer0137Table A.3: Pure strain simulation snapshots N = 300 where minority monomers areorange, and majority monomers are purple. As described in Chapter 4117e=N=500 Triblock0137e=N=500 Homopolymer0137Table A.4: Pure strain simulation snapshots N = 500 where minority monomers areorange, and majority monomers are purple. As described in Chapter 4118Appendix BSupplementary Plots 0 500 1000 1500 2000 2500 0  500  1000 1500 2000 2500<Rc><Rk><RcRk>Figure B.1: Determining the statistical independence of Rc, Rk. Green: triblocks,red: homopolymers, blue: cut chains for N = 300 (), N = 500 (◦),and N = 800 (△).119 1 2 3 4 5 6 7 8 1  2  3  4  5  6  7  8λ k2(λ2kx+λ2ky+λ2kz)/3(a) 1 3 5 7 9 11 131 3 5 7 9 11 13λ c2(λ2cx+λ2cy+λ2cz)/3(b)Figure B.2: Test of the 8 chain model assertion relating stretches to their compo-nents. (a) Entanglement stretch (b) Chain stretch . Green: triblocks,red: homopolymers, blue: cut chains for N = 300 (), N = 500 (◦),and N = 800 (△).120 1 1.5 2 2.5 3 3.5 4 4.5 5 1  2  3  4  5  6  7λ kzλcz 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0.4  0.5  0.6  0.7  0.8  0.9  1λ kxλcxFigure B.3: Comparison Cartesian components of (a) Entanglement stretch (b)Chain stretch . Green: triblocks, red: homopolymers for N = 300 (),N = 500 (◦), and N = 800 (△).121Appendix CFreely-jointed Chain BondVector Components CalculationIn terms of a fixed force f the partition function for an extended chain isZ =[4π sinhββ]N= [Z]N (C.1)where β = fb/kBT , and Z is the partition function for a single bond. Given that wehave assumed a constant force we can take the partition function for a single bondZ in terms of that force.Z =[4π sinhββ](C.2)We now find the component of each bond perpendicular to the chain. Given thatbond length is fixed we use spherical coordinates for the integration. The z-directionis defined as the chain axis - the direction of the end-to-end vector ~Rk.bx = b cosφ sin θ (C.3)In terms of the probability distributionp(θ, φ) =e−U/kBTZ〈b2x〉 =∫ πθ=0∫ 2πφ=0p(θ, φ)b2x sin θ dθdφ (C.4)=βb24π sinhβ∫ πθ=0∫ 2πφ=0exp (β cos θ) cos2 φ sin3 θ dθdφ (C.5)=βb24 sinhβ∫ πθ=0exp (β cos θ) sin3 θ dθ (C.6)=βb24 sinhβ[4β coshβ − 4 sinhββ3](C.7)=b2β2(β cothβ − 1) (C.8)122Similarly in the z-directionb2z = b2 cos2 θ (C.9)〈b2z〉 =∫ πθ=0∫ 2πφ=0p(θ, φ)b2z sin θ dθdφ (C.10)=b2β2(2 + β2 − 2β cothβ) (C.11)Global cartesian co-ordinates:Rather than in the coordinate system of their particular chain (x, y, z) we wantto know the components of the bond vectors in the global Cartesian co-ordinates(x′, y′, z′), averaged over all chains. Each chain rotates to align more with the z′deformation axis with the applied volume conserving strain. The angle θ betweenthe z and z′ axes for each chain is given bytan θ = Rcx′/Rcz′ = λcx′/λcz′We can then write the bond vector components in the global co-ordinate frame as〈b2z′〉 = 〈b2x〉 sin2 θ + 〈b2z〉 cos2 θ (C.12)〈b2x′〉 = 〈b2x〉 cos2 θ + 〈b2z〉 sin2 θ (C.13)assuming that all chains deform on average like a chain with initial chain vectorRc(0) = (|Rc|/√3, |Rc|/√3, |Rc|/√3) (the same assumtion made by the 8-chainmodel). Applying this reasoning to bonds b2x′(0) = b2/3 we then have the bondcomponent stretches in the global deformation frame,λ2x′ = 3〈b2x′〉/b2. (C.14)and calculate the elasticity factorg(λx′ , λz′) =3b2[〈b2z′〉 − 〈b2x′〉] (C.15)=3b2(cos2 θ − sin2 θ) [〈b2z〉 − 〈b2x〉] (C.16)=3β2cos 2θ(2 + β2 − 2β cothβ − β cothβ + 1) (C.17)=3β2cos 2θ(3 + β2 − 3β cothβ) (C.18)123Appendix DFaster Rate Deformationsεaa εbb εab ˙εzz N TPure stress Chapter 4 1.0 0.7 0.3 10−4 100,300,500 0.3Volume conserving Chapter 5 1.0 0.5 0.2 10−5 300,500,800 0.3Volume conserving Chapter D 1.0 0.5 0.2 10−4 300,500,800 0.3Table D.1: Summary of deformation parameters, microscopic model (εaa, εbb, εab),strain rate and chain lengths (N).124 0 20 40 60 80 100 120 140A(a) 0.5 1 1.5 2 2.5Nc/Nc0(b) 0 0.5 1 1.5 2 2.5 3 3.5 4 0  1  2  3  4  5  6  7σz - σxStrain(c)Figure D.1: Volume conserving uniaxial strain deformation, identical equilibratedconfigurations to Chapter 5 but strained more quickly at a rate 10−4 (a)Asphericity of glassy clusters (b) Normalised number of glassy clusters(c) stress response. Red(◦) N = 300, green() N = 500 orange (◦)N = 800125 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0  10  20  30  40  50  60σz - σxModel/GeFigure D.2: Volume conserving uniaxial strain deformation, identical equilibratedconfigurations to Chapter 5 but strained more quickly at a rate 10−4.Ge = 0.018 Red N = 300, green N = 500, orange N = 800. Dashedlines: triblocks, solid lines: homopolymers


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