Strain localization and yielding dynamics in disordered collagen networks
Swarnadeep Bakshi, Vaisakh VM, Ritwick Sarkar, Sayantan Majumdar
SStrain localization and yielding dynamics in disordered collagen networks
Swarnadeep Bakshi a , Vaisakh VM a,b , Ritwick Sarkar a , and Sayantan Majumdar a ∗ a Soft Condensed Matter Group, Raman Research Institute, Bengaluru 560080, India b Department of Physics, HKUST, Clear Water Bay, Hong Kong (Dated: February 19, 2021)Collagen is the most abundant extracellular-matrix protein in mammals and the main structuraland load-bearing element of connective tissues. Collagen networks show remarkable strain-stiffeningproperties which tune the mechanical functions of tissues and regulate cell behaviours. Linearand non-linear mechanical properties of in-vitro disordered collagen networks have been widelystudied using rheology for a range of self-assembly conditions in recent years. However, a oneto one correlation between the onset of macroscopic network failure and local deformations insidethe sample is yet to be established in these systems. Here, using shear rheology and in-situ high-resolution boundary imaging, we study the yielding dynamics of in-vitro reconstituted networks ofuncrosslinked type-I collagen. We find that in the non-linear regime, the differential shear modulus( K ) of the network initially increases with applied strain and then begins to drop as the networkstarts to yield beyond a critical strain (yield strain). Measurement of the local velocity profile usingcolloidal tracer particles reveals that beyond the peak of K , strong strain-localization and slippagebetween the network and the rheometer plate sets in that eventually leads to a detachment. Wegeneralize this observation for a range of collagen concentrations, applied strain ramp rates, as well asdifferent network architectures obtained by varying the polymerization temperature. Furthermore,by fitting the stress vs strain data with a continuum affine network model, we map out a statediagram showing the dependence of yield-stain and -stress on the reduced persistence length andmesh size of the network. Our findings can have broad implications in tissue engineering, particularly,in designing highly resilient biological scaffolds. I. INTRODUCTION
Biopolymer networks, the major structural component of intra- and extracellular environment in animal body,show strikingly different mechanical properties compared to synthetic polymer gels [1–8]. Properties like non-linearstrain stiffening, negative normal stress in biopolymers are related to the ‘semi-flexibility’ of the filaments as indicatedby their relatively high bending rigidity ( κ ). In polymeric system, the filament rigidity is generally expressed in termsof thermal persistence length ( l p = κk B T ) that indicates how tangent-tangent angular correlation decays along thefilament due to thermal fluctuations. For biopolymers, l p is significantly larger than that for the synthetic polymers,yet, much smaller compared to a rigid rod. This indicates that despite of their rigidity, biopolymers show significantthermal bending fluctuations [8–10]. ∗ [email protected] a r X i v : . [ c ond - m a t . s o f t ] F e b Type-I collagen is the most abundant protein in the extracellular matrix (ECM) of mammalian cells. Besidesproviding a scaffold for connective tissues, the mechanical and structural properties of ECM governs crucial cellularfunctions like cell proliferation, adhesion, migration, wound-healing, signaling etc [11–14]. Abnormalities in ECMstiffness gives rise to various pathological conditions [15, 16].To better understand the effect of linear and non-linear mechanics on various cellular and tissue functionali-ties, in-vitro reconstituted assemblies of disordered isotropic collagen networks have become very popular in recentyears, particularly, in the context of bio-physics, mechano-biology and tissue engineering [17–19]. Such studiesconsider both simple shear [20, 21] and extensional deformations [22]. Non-linear mechanics of collagen is complex andhighly architecture dependent [23]. Collagen networks with similar linear moduli can have widely different non-linearstrain stiffening response [24]. The complexity mainly arises from the presence of hierarchical length scales in thesystem. Entropic elasticity of individual filaments, non-affine deformations, network heterogeneity, stress inducedchanges in network architecture contribute to the non-linear mechanics in these systems. Interestingly, collagennetworks can show stability and finite elasticity for the average local network connectivity (cid:104) z (cid:105) varying approximatelybetween 3 and 4 which is well below the isostaticity ( (cid:104) z (cid:105) = 6 in three dimension) as predicted by Maxwell criterion[25]. Theoretical models demonstrate that bending rigidity of filaments stabilizes such sub-isostatic networks andstrain stiffening originates from a bending to a stretching dominated response of the individual filaments/bundles[4, 26, 27]. Continuum unit-cell models (e.g. 3-Chain and 8-Chain Models) of semi-flexible filaments, provide compactanalytical expressions effectively describing the non-linear strain stiffening in different biopolymer networks [28, 29].Rheology and in-situ microscopy techniques like, confocal fluorescence (CFM), confocal reflectance (CRM)and boundary stress microscopy (BSM) provide important insight into the correlation of visco-elasticity, networkdeformation, failure and stress heterogeneity with the local network structure in these systems [30–32]. Stressrelaxation in these systems strongly influenced by the magnitude of applied strain [20]. Similar to other biopolymers,non-linear mechanics of collagen networks also demonstrates striking hysteric effects [33–35].Although, non-linear strain stiffening in disordered isotropic collagen networks has been extensively studied,much less attention has been paid to the dynamics of yielding and network failure. It is found that networkarchitecture affects the magnitude of yield strain in collagen. Networks polymerized at higher temperature havingfiner fibrils with smaller pore sizes show a much larger yield strain compared to that corresponding to more bundlednetworks obtained at lower polymerization temperatures [23, 24]. Non-linear mechanics and yielding in these systemsalso show interesting system-size dependence over the length scales much larger than the network mesh size [36]. Avery recent study attempts to correlate the fracture strain with the local connectivity and plasticity of the collagennetwork using rheology and in-situ CFM. They directly probe the network failure over the length scales of singlefilaments to a few mesh size [31].Despite of these studies, a direct correlation between the onset of bulk network failure and microscopic defor-mations still needs to be established. Moreover, in all the studies probing change in network structure using CFMand CRM in conjugation with rheological measurements, the imaging is done in the flow-vorticity plane compatiblewith standard microscopy set-up. Particularly, high resolution imaging in the flow-gradient plane is difficult due tothe presence of air-sample interface in a plate-plate or cone-plate geometries used for the rheology measurements.Interestingly, the possibility of strain localization in the flow-gradient plane has been speculated in the context ofsystem-size dependent stiffening and yielding in collagen networks [36]. However, to our knowledge, in-situ straindistribution over mesoscopic to macroscopic length scales in flow-gradient plane has not been explored in these systems.Here, we study non-linear strain stiffening and yielding behaviour in uncrosslinked type-I collagen networksusing rheology and in-situ boundary imaging over a range of concentrations and polymerization temperatures. Wefind that the network softening or yielding starts well below the breaking/rupture strain. We also observe that highstrain accumulates in the sample near the shearing boundaries when the applied strain crosses the yield strain. Suchlocalized strain initiates a slippage between the network and the rheometer plate eventually leading to a detachment.Furthermore, fitting the stress-strain curves with a unit cell based network model for affine deformations, we map outa state diagram connecting the yielding behavior with the reduced persistence length and mesh-size of the network. II. RESULTS AND DISCUSSIONS
We reconstitute networks of Type-I collagen starting from collagen monomers. Collagen monomers are formed bytriple-helical polypeptides of length ≈
300 nm and width ≈ o C, the network is very heterogeneous with thick parallel bundles, whereas, more homogeneous network withfiner fibrils are observed at 25 o C. The distributions of fibril/bundle diameter for different temperatures are obtainedfrom the freeze fracture SEM data (sample size: N ∼ o C - 35 o C, both the mean and standard deviation filament diameter decreases with increasing temperature,as also reported in earlier studies [23, 39, 46].Rheology measurements are carried out on a MCR-702 stress-controlled rheometer (Anton Paar, Graz, Austria)using cone-plate geometry having rough sand-blasted surfaces (Materials and Methods). Such rough surfaces minimizewall-slippage of the sample under shear. The polymerization process of collagen is monitored by applying a smalloscillatory shear strain (amplitude γ = 2%, frequency f = 0.5 Hz) and measuring the linear visco-elastic moduli G (cid:48) and G (cid:48)(cid:48) as a function of time. After an initial increase, G (cid:48) and G (cid:48)(cid:48) reaches a plateau value (Fig. S1), indicating thatthe network is polymerized. We carry out the rheology measurements over a concentration ( φ ) range of 1 mg/ml - 3mg/ml. In all cases, we find that over a wide range of frequency G (cid:48) ( f ) is much larger than G (cid:48)(cid:48) ( f ) (Fig. S2), indicatingthat the networks behave like a visco-elastic solid. To probe the response of the network as a function of strain, weapply a constant strain ramp rate ˙ γ = 1%/s on the network and measure the stress response. We show the variationof shear stress ( σ ) and differential shear modulus K = dσdγ as a function of γ (Fig. 1(d)) for φ = 2 mg/ml and T =35 o C. We find that after a mild strain-weakening regime, K increases rapidly beyond an onset strain γ o indicating thenon-linear strain stiffening of the network, when the shear modulus of the network increases with increasing strain.At larger strain values, K reaches a maximum before starting to drop beyond γ y , the yield-strain of the network, o C 25 o C 17 o C 4 o C (c1) (c2)
20 25 30 3510 (c)(d) (e) K [ P a ] σ [ P a ] N ( d ) d [nm] γ c [ % ] 𝜎 c [ P a ] γ[%] T [°C] (a) (b) Figure 1: Freeze-fracture SEM image of collagen polymerized at (a) 4 o C (Scale = 1 µ m and (b) 25 o C (Scale = 2 µ m). (c)Bundle fibril diameter distribution. For Fig. 1(a),(b) and (c), the collagen concentration φ = 1 mg/ml. (d) Variation ofdifferential shear modulus (top panel) and stress (bottom panel) with applied strain for polymerization temperature T = 35 o C.Critical stress and strain points corresponding to the onset, yield and breaking are marked with the dashed lines. (e) Variationof critical strain and stress as a function of polymerization temperature (squares: onset stress/strain, circles: yield stress/strainand stars: breaking stress/strain). Error bars represent standard deviations for two independent measurements. For Fig. 1(d)and (e) the collagen concentration φ = 2 mg/ml. indicating a network weakening under large strain. Interestingly, the peak stress is reached at a higher strain valuecompared to γ y . We define the strain value at which the stress reaches the peak as the breaking/rupture strain γ b forthe network. Beyond γ b the stress weakening starts. Mathematically, d σdγ | γ y = 0 and dσdγ | γ b = 0 defines the yieldingand breaking points of the network, respectively. From Fig. 1(d) and (e) we note that γ y ≈
50% where as, γ b ≈ γ o , γ y and γ b and the corresponding stress values ( σ o , σ y , σ b ) as a function of polymerizationtemperature in Fig. 1(e) (top and bottom panels). We find that both the critical strain and stress values increase withincreasing temperature. Similar trends are also observed for φ = 1 mg/ml and 3 mg/ml (Fig. S3). For temperaturesbelow 20 o C, we observe condensation of small water droplets on the rheometer plates. This can change the momentof inertia of the shearing plates and introduce measurement artifacts. Thus, for rheology measurements we do notprobe any temperature below 20 o C.To get a deeper insight into the yielding behaviour of collagen networks, we map out the spatio-temporal evolutionof the velocity field in the sample using particle imaging velocimetry (PIV) technique. We illuminate the sample usinga LED light source (Dolan-Jenner Industries) and image the diffused scattering from the sample boundaries in theflow-gradient plane using a digital camera (Lumenera) fitted with a 5X long working distance objective (Mitutoyo).We do not get any appreciable scattering from the pure collagen and the sample appears almost transparent. We put (b) (c) 𝑣 𝐱 [ μ m / s ] 𝑧 [μm] (a) γ[%] 𝑣 𝐱 [ μ m / s ] 𝑧 [μm] +𝑣 𝐩 −𝑣 𝐩 𝑑 𝑑 ’ 𝑥𝑧 Figure 2: (a) Typical boundary image with superposed velocity profile for the collagen sample seeded with 2.8 µ m polystyreneparticles (1% v/v). d and d (cid:48) respectively represent the gap between the plates and the sample width used for PIV analysis. ± v p denote the plate velocities. (b) and (c) show the velocity profiles obtained from the PIV analysis for T = 30 o C and 25 o C,respectively. Color gradient represent increasing applied strain ( γ ) with a ramp rate of 1%/s. The red dots indicates the platevelocities. The dashed lines represent the velocity profile predicted assuming affine deformations. We see strong non-affinedeformation for larger strain values close to yielding. Here, φ = 2 mg/ml.
1% (v/v) polystyrene tracer particles ( d = 2.8 µ m) [47] in the collagen samples (Fig. 2(a)) to enhance the scatteredintensity required for the PIV measurement. We find that ∼
1% (v/v) is the minimum amount of tracer particlesrequired to get a fairly uniform distribution of speckle pattern. Freeze fracture SEM images (Materials and Methods)show that the particles have some affinity to stick to the networks (Fig. S4), however, FTIR spectra (Materialsand Methods) points out that no chemical bonds between the particles and collagen fibers are formed (Fig. S5).We confirm that introducing such low concentration of tracer particles does not modify the rheological behaviour ofcollagen networks (Fig. S6). A typical boundary image displaying the speckle pattern with velocity vectors obtainedfrom PIV analysis is shown in Fig. 2(a). The plates are moving with velocities + v p and − v p in a counter-rotationconfiguration giving an applied shear rate ˙ γ = v p d , where d is the gap between the plates. Evolution of the spatially-averaged velocity profiles across the gap [ v ( z ) vs z ] with increasing strain values are shown in Fig. 2(b) and 2(c) forpolymerization temperatures of 30 o C and 25 o C, respectively. We see that for all strain values γ < γ y , the profilesremain almost linear, indicating an affine deformation. The average shear rate inside the sample below γ < γ y is givenby v d (cid:48) , where ± v indicate the velocities inside the PIV window at z = 0 and z = d (cid:48) (Fig. 2(a)). However, for γ ≥ γ y the velocity profiles evolve rapidly, in particular, near one of the plate boundaries (Fig. 2(b) and 2(c)) and finallya detachment happens at γ = γ b (also see Movie 1). Such evolution of velocity profile indicates strain localizationand slippage near the boundary. Beyond the detachment, the shear strain in the network becomes negligible andentire network moves with a constant velocity with the plate remains attached to. Sometimes, the network can alsodetach from both the plates (Movie 2). Interestingly, for few rare occasions we observe that some part of the networkcan reattach with the moving plates after the initial detachment giving rise to more complex fracture patterns (datanot shown). The boundary near which velocity profiles rapidly evolve and finally sample detachment takes place,randomly varies for different experimental runs and most likely depends on the attachment of the network with therheometer plates. Such trend is observed for all sample concentrations, temperatures and applied shear rates weconsider in the present study. Our observations indicate a close connection between the yielding onset and boundary T = 30 o C 2 mg/ml, 1%/s 2 mg/ml, 5%/s 2 mg/ml, 10%/s 2 mg/ml, 20%/s 3 mg/ml, 1%/s 3 mg/ml, 5%/s 3 mg/ml, 20%/sT = 25 o C 2 mg/ml, 1%/s 2 mg/ml, 10%/s 3 mg/ml, 10%/s K [ P a ] γ [%] γ 𝑦 [%] 𝑣 S [ μ m / s ] (a) (b) γ 𝑠 [ % ] γ 𝑠 Figure 3: (a) Variation of shear modulus K (top panel) and slip velocity v s obtained from PIV analysis (bottom panel) for φ = 2 mg/ml and T = 30 o C as a function of applied strain. Slippage between the sample boundary and the rheometr plateshow a significant increase beyond the yield strain ( γ y ), as shown by the data in the shaded region. Otherwise, such slippage isnegligible. We mark the onset strain beyond which the slippage increases ( γ s ) by an arrow in the bottom panel. (b) Variationof γ s as a function of γ y for a range of sample conditions and strain ramp rates, as marked in the legend. The dashed red-linecorresponds to γ s = γ y . dynamics of the collagen network.To further explore the correlation between the yielding and failure dynamics of the network, we define the slip-velocity v s = | v p − v | and plot it as a function of applied strain ( γ ) in Fig. 3(a) for T = 30 o C. We find from Fig.3(a) that v s remains small as a function of γ till γ = γ y , for both the top and bottom plates. This indicates that,over a range of strain values, both in the linear and non-linear regime, the slippage between the sample and the platesis not significant. As γ crosses γ y (indicated by the peak of K ), v s starts to increase rapidly, particularly, for oneof the plates and reaches a maximum near γ = γ b . The maximum value of v s ∼ v p , however, in some cases thevalue of slip-velocity can be even larger due to strong elastic retraction of the network after the rupture. The appliedstrain value beyond which v s increases rapidly defines the slip-strain ( γ s ), as shown in Fig. 3(a) (bottom panel). Toestablish the relation of boundary slippage with the yielding behaviour, we plot γ s as a function of γ y in Fig. 3(b) forvarying sample conditions as well as, strain ramp rates. Remarkably, we observe a strong correlation between γ s and γ y , indicating that boundary slippage plays an important role in yielding of collagen networks.Our observation points out the generality of boundary failure dynamics in governing the yielding behaviour of thenetworks over a wide range of parameters. Moreover, the appearance of significant boundary slippage only deep insidethe strain-stiffening region indicates that such detachment is triggered by internal stresses generated in the systemdue to strong non-linear deformation of the network. We indeed observe a significant negative normal stress in thesystem before yielding (Fig. S7). Currently, we do not fully understand why the network failure/rupture always takesplace close to the sample boundaries. There can be a possible connection with network rarefaction under non-lineardeformation as observed for highly cross-linked actin networks [34]. Such rarefaction results in lower average number ofcontacts between the filaments close to the boundaries giving rise to a local weakening of the sample. For some cases,we observe a drop in the scattered intensity from the sample near the shearing plates close to yielding (Fig. S8). Suchintensity drop near the plates increases further beyond the breaking strain. This observation coupled with the fact
20 25 30 350.60.7 (a) (b)(c)
T [°C]
T = 20°C γ [%]
T = 28°CT = 35°C L p / L c ξ / L c σ / G Figure 4: (a) Symbols indicate the variation of normalized stress as a function of applied strain for different polymerizationtemperatures as shown. Error bars represent the standard deviations obtained from two independent experiments under thesame sample condition. Solid lines represent the fits to the 8-chain model as described in the main text. We get very goodagreement in all cases upto intermediate strain values, however, significant deviation from the fitted model is observed dueto non-affine deformations close to yielding. Variation of reduced persistence length (b) and mesh size (c) as a function oftemperature obtained from the fitting. Here, φ = 2 mg/ml. that tracer particles have an affinity to stick to the collagen fibers (Fig. S4), also supports the local rarefaction picturementioned earlier. However, more studies are required to confirm such deformation induced network heterogeneity incollagen systems.Next, we turn to the temperature dependent microscopic network architecture that controls the yielding behaviourin these systems. Although, the mechanical properties of a single biopolymer filament is well described by Worm LikeChain model (WLC) [48], a network formed by many such filaments have random structures that make the derivationof analytical expression for free energy and stress-strain relation extremely complex [28]. To simplify this problem,several lattice-based models have been proposed. These models essentially use the concept of a ‘mesh cell’ or, anelemental volume, such that, the entire sample can be constructed just by 3-D translation of this elemental volume.The main assumption of this model is affine or uniform deformation in the system spanned by the deformation of the‘mesh cell’. As mentioned earlier, we observe affine deformations over a wide range of strain values below yielding.Thus, such lattice based models should capture the mechanics for collagen networks for γ < γ y . One such models forthe ‘mesh cell’ is , where the cell is body-centered cubic. Eight chains connect the eight corners of thecube to the center. The mesh size ( ξ ) of the network is given by the edge-length of the cubic cell. The stress ( σ ) vsstrain ( γ ) relationship for this network model under simple shear is given by [28], σ = 23 nk B T γa (cid:20) cπ [3 − (3 + γ ) a ] − cπ (cid:21) (1)where, the parameters a = ξL c and c = l p L c are the reduced mesh-size and persistence length, respectively. Here, n is the number of entanglement points per unit volume. A dimensionless form of the above equation can be obtained . . . . . . . . . . . . . . . . (a) (b) Figure 5: Variation of (a) yield strain ( γ y ) and (b) yield stress ( σ y ) with reduced persistence length and mesh size. Tetrahedron,cube and sphere shaped symbols correspond to 1 mg/ml, 2 mg/ml and 3 mg/ml collagen concentrations, respectively. Dark tolight color represents the decreasing magnitude of γ y or σ y . Error bars represent the standard deviations of two independentmeasurements. Here, φ = 2 mg/ml. if we divide both sides by the average plateau modulus ( G = √ G (cid:48) + G (cid:48)(cid:48) ) obtained from the frequency sweep data(Fig. S2). We use the above equation to fit σ/G vs γ data obtained experimentally for different temperatures andcollagen concentrations. For clarity, we show the fitted data only for φ = 2 mg/ml concentration at three differenttemperatures in Fig. 4(a). We see very good agreements for strain values γ < γ y in all cases. For the fittinginitialization, we use n values similar to those obtained in the recent simulations for collagen networks [49] (also seeTable of Parameters in S.I.). The parameters l p L c and ξL c obtained from the fitting are plotted in Fig. 4(b) and (c)as a function of temperature. We find that both of these parameters decrease with increasing temperature. Thisresult also agrees with the variation of filament diameter obtained from the SEM images, where we observe that fibrilthickness decreases with increasing temperature (Fig. 1(c)). We should also note that for 8 chain model, it is assumedthat each chain is a single collagen filament, however, bundle formation is always present in biopolymers [50]. Also,8-chain model can not capture the strain softening behaviour at smaller strain values coming predominantly fromthe filament buckling [9].We show the dependence of yield strain/stress ( γ y or σ y ) on microscopic network parameters (reduced mesh-sizeand persistence length) by generalized three-dimensional state diagrams in Fig. 5(a) and (b). The variation ofmicroscopic network architecture is obtained by varying the polymerization temperature (20 C - 35 C) as well ascollagen concentration (1 mg/ml - 3 mg/ml). We see from Fig. 5(a) that with increasing value of reduced mesh sizeand persistence length γ y monotonically drops. This is remarkable, since, it indicates that despite of the complexnetwork architecture the yield strain is essentially governed by two microscopic network parameters related to themesh size and persistence length. Networks with finer fibrils are more resilient. Similar trend is also observed for σ y (Fig. 5(b)). We note that, although, γ y shows similar dependence on the reduced mesh size and persistence lengthfor all concentrations, σ y values for 1 mg/ml collagen networks are much lower. This observation is in line withthe results mentioned in [31], where networks with lower connectivity show higher plasticity and larger fracture strain. III. CONCLUSION
In conclusion, we study the yielding and in-situ strain localization under steady shear in disordered networks of type-I collagen over a range of network architectures obtained by varying the polymerization temperatures and collagenconcentrations. Using an unit cell based affine network model we attempt to generalize the yielding behaviour underdifferent sample conditions in terms of microscopic network parameters. For rheology experiments, we use a cone-plategeometry that ensures homogeneous shear-stress field in the sample. This indicates that the observed strain localizationdoes not originate due to any imposed stress heterogeneity in the system. We find that network weakening or yieldingstarts well below the fracture strain, signifying that yielding is a gradual process in disordered collagen networks.Remarkably, we find a strong correlation between the onset of network slippage and the yield strain for differentimposed strain ramp rates, polymerization temperatures and collagen concentrations. Such observation points outthe generality of strain localization and boundary failure dynamics in controlling yielding in these systems. Similarto earlier studies [23, 31], we also observe that the networks with higher onset strain values for stiffening (observedfor higher polymerization temperatures) also show higher values of the yield and breaking strain as indicated in Fig.1(e). However, in contrary to the observation of network fracture at random locations [31], we find that boundaryslippage leads to detachment/rupture. Since, we image the entire gap between the shearing plates in-situ, our spatialresolution is only ∼ µ m which is much larger than the network mesh size [36]. This indicates that, althoughboundary failure predominantly controls the yielding and fracture of the bulk network, local plasticity and fractureover few mesh-sizes can not be ruled out. Furthermore, due to low spatial resolution, we can not confirm whetherthere is a slippage or, formation of a narrow band of high shear rate close the boundary. To resolve such issues, directimaging of the network deformation in the flow-gradient plane using fluorescence microscopy or very fast z-scanning(along the gradient direction) of the network using a high speed laser scanning confocal microscope tracking thein-situ deformations, will be an interesting future direction to explore. Nevertheless, due to the roughness of thesand blasted plates, the velocity field very close to the boundary can be quite complex. Our observation of localizednetwork rarefaction correlates well with the strain localization and slippage originating at the sample boundaries.However, more experimental as well as theoretical insights are needed to confirm such mechanism. Our study canprovide strategies for delaying network failure and thus help in designing more resilient collagen based scaffolds forvarious engineering and biomedical applications. We hope that our work will motivate further studies on microscopicmechanism of failure and strain localization in collagen and other biopolymer networks. IV. AUTHORS CONTRIBUTIONS
S.B., V.V.M. and S.M. designed the research, S.B. and V.V.M. performed the experiment, S.B., R.S. and S.M.performed the data analysis. All the authors contributed in writing the manuscript.
V. ACKNOWLEDGEMENT
S.M. thanks SERB (under DST, Govt. of India) for a Ramanujan Fellowship. We thank Ivo Peters for developingthe Matlab codes used for PIV analysis. We also thank Gautam Soni, Pramod Pullarkat, Reji Philip and Ranjini0Bandyopadhyay for allowing us to use their lab/common instrument facilities. We thank K M Yatheendran for helpwith the SEM imaging and Sachidananda Barik for synthesising the polystyrene particles. We thank Madhu Babufor the help with the FTIR measurements and RRI workshop facility for machining the humidity chamber for thesample.
VI. MATERIAL AND METHODSA. Chemicals Used:
The collagen hydrogels are synthesised from acid-soluble rat tail collagen Type-I [Conc. 8.70 mg/ml in 0.02 NAcetic acid], (Corning, Bedford, MA). The collagen monomers are polymerized using the phosphate buffered saline(PBS) solution (1X, pH 7.34) prepared by uniform mixing of 2 g of NaCl, 50 mg of KCl, 0.36 g of Na HPO and60 mg of KH PO in 250 mL of deionized water, with pH adjusted to 7.4 by adding 0.01 M HCl. The Polystyrenemicrobeads (PS) are used as the tracer particles are synthesized in the laboratory by a procedure described previously[47]. B. Characterization of Collagen hydrogels:
1. Rheology:
For rheology experiments we use a 2018 made MCR-702 Twin-Drive stress-controlled rheometer (Anton Paar, Graz,Austria). A 25 mm diameter top cone geometry (cone angle: 2 ◦ ) with a 25 mm diameter parallel bottom plate (bothmade of stainless steel and both are sandblasted) are used. We mix the collagen monomers of a desired concentrationwith 1X PBS buffer and transfer the sample on the cold Peltier controlled rheometer bottom plate, immediately. Thenthe Peltier temperature is increased to the desired value. We polymerize collagen networks between the rheometerplates maintained at a fixed temperature. We also use a humidity chamber to prevent solvent evaporation during therheology experiments. For in-situ imaging studies, we use a thin layer of 5 cSt silicone oil (Merck) around the sampleto prevent solvent evaporation.
2. Freeze-fracture Electron Microscopy:
To probe the collagen network architecture, we carry out SEM imaging using an Ultra Plus Cryo-SEM (Zeiss,Germany) set-up. The collagen hydrogels of different concentrations (1 to 3 mg/ml) are prepared by adding the stockcollagen to PBS solution in 1.5 ml centrifuge tubes, followed by a thorough mixing. Different samples of hydrogels weremade by varying the polymerization temperature over the range 4 ◦ - 35 ◦ C. Polymerized samples appear translucentunder the ambient light. After polymerization, the samples were transferred to the sample holder for freeze-fractureSEM using a micro-pipette, as soon as possible. The sample holder is dipped in liquid nitrogen to freeze the samplesinstantly. Next, the frozen samples are mounted on the SEM sample stage using carbon tape and are sputter coatedwith platinum to a thickness of ∼
3. Fourier Transform Infrared Spectroscopy (FTIR):
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Movie DescriptionsMovie1:
In this movie, we show the in-situ deformation of the sample surface in flow-gradient plane correlatingwith stress vs strain response of collagen network (2 mg/ml) seeded with 1% (v/v) PS. The applied strain ramprate is 1%/s. The images are captured using a digital camera (Lumenera) fitted with a 5X long working distanceobjective (Mitutoyo) at a frame rate of 1 Hz with a resolution of 1200 X 1800 (pixel) . The images are then croppedand compressed such that the spatial resolution is ∼ µ m/pixel. The movie is sped up 14 times as compared to thereal time. As mentioned in the main text, we see that network slippage with one of the shearing plates (here, topplate) appears well before the peak stress/breaking stress is reached. The onset of such slippage correlates well withthe yield strain of the network (as indicated in the movie). Beyond the peak stress, the detachment (characterized byan elastic retraction of the whole network) takes place from the top plate. The drop in scattered intensity near thetop plate close to the yield strain indicates network rarefaction (also see Fig. S8 and the main text) before detachment. Movie2:
In this movie, we show the in-situ deformation of collagen network (2 mg/ml) seeded with 1%(v/v) PS for an applied strain ramp rate of 10%/s. The movie is sped up 1.4 times as compared to the real time.Here also, we find similar correlation of the boundary dynamics with the yield and breaking strains of the network(data not shown). Interestingly, as opposed to Movie1, the network detachment happens from both the plates in thiscase. Also, we see a clear signature of network rarefaction (drop in scattered intensity) near the shearing boundariesclose to yielding (also see Fig. S8). Beyond the network breakage/rupture from both the plates, a clear signature ofnetwork contraction away from the plates is seen.6
Table of parameters
Parameters obtained from the 8-chain model fitting for three different collagen concentrations. φ =1mg/mlTemperature ( ◦ C) n k B T(J m − ) l p L c ξL c
20 80 0.421 0.76025 180 0.400 0.73830 200 0.235 0.48835 270 0.234 0.47937 380 0.229 0.470 φ =2mg/mlTemperature ( ◦ C) n k B T(J m − ) l p L c ξL c
20 100 0.390 0.75025 335 0.366 0.71728 350 0.306 0.64630 370 0.284 0.60635 470 0.252 0.554 φ =3mg/mlTemperature ( ◦ C) n k B T(J m − ) l p L c ξL c
20 300 0.463 0.79025 350 0.308 0.65828 380 0.273 0.60730 420 0.255 0.57935 500 0.253 0.5747
Supplementary figures -1 -1 (a) (b) G ′ [ P a ] t [s] G ′′ [ P a ] t [s] Figure S1: Variation of storage ( G (cid:48) , panel (a)) and loss ( G (cid:48)(cid:48) , panel (b)) moduli as a function of time during the polymerizationof collagen networks. The applied oscillatory strain amplitude is 2% and frequency is 0.5 Hz. As indicated, different symbolsrepresent different concentrations of collagen. The plateau reached after the jump in G (cid:48) or G (cid:48)(cid:48) values represents the polymerizedstate of the network. The error bars are the standard deviations of two independent measurements under the same condition.Here, polymerization temperature (T) is 25 ◦ C in all cases. -2 -1 -2 -1 (a) (b) G ′ [ P a ] f [Hz] G ′′ [ P a ] f [Hz] Figure S2: Frequency dependent storage ( G (cid:48) , panel (a)) and loss ( G (cid:48)(cid:48) , panel (b)) moduli for polymerized collagen networks.The applied oscillatory strain amplitude is 2%. Different symbols represent different concentrations of collagen. T = 25 ◦ C. Theerror bars are the standard deviations of two independent measurements under the same condition. We see that, G (cid:48) >> G (cid:48)(cid:48) over the entire range of frequencies probed.
20 25 30 35 4010 γ c [ % ] 𝜎 c [ P a ] T [°C]
20 25 30 35 4010 -1 γ c [ % ] 𝜎 c [ P a ] T [°C] (a) (b)
Figure S3: Variation of critical stress and strain values with polymerization temperature for collagen networks. Collagenconcentration, φ = 1 mg/ml (panel (a)) and φ = 3 mg/ml (panel (b)). Here, squares represent onset strain/stress, circlesrepresent yield strain/stress and stars represent breaking strain/stress. The error bars are the standard deviations of twoindependent measurements under the same condition.
200 nm (a)
400 nm (b)
Figure S4: Freeze fracture SEM image of collagen network ( φ = 2 mg/ml) seeded with 1% PS of mean diameter of 2.8 µ m.The collagen fibrils seem to have some affinity to stick to the PS surface. PS+Collagen
Collagen T r a n s m i tt a n c e [ % ] Wavenumber [𝑐𝑚 −1 ] PS T r a n s m i tt a n c e [ % ] Wavenumber [𝑐𝑚 −1 ] (a) (b) Figure S5: FTIR spectra of pure collagen (shown by the red line in (a)) and collagen mixed with 1% PS (shown by the blackline in (a)). FTIR spectrum for only PS particles are shown in (b). As indicated in (a), troughs shown in both the spectra(pure collagen and collagen + PS) are very similar. This indicates that there are no chemical bonding interactions betweencollagen and PS. Without Ps With PS γ [%] K [ P a ] -1 Without PS With PS γ [%] σ [ P a ] (a) (b) Figure S6: Comparison of network response with and without added PS (1%). We see that stress ( σ , panel (a)) and differentialshear modulus ( K , panel (b)) show very similar behaviour with applied strain ( γ ) for both the cases. The error bars are thestandard deviations of two independent measurements under the same condition. Here, φ = 2 mg/ml and T = 30 ◦ C. -2 -1 -2 -1 σ [ P a ] − σ N [ P a ] γ[%] Figure S7: Typical variation of shear stress ( σ , shown by the black squares) and negative normal stress ( σ N , shown by the redcircles) as a function of applied strain γ for collagen network with φ = 2 mg/ml and T= 25 ◦ C. We see that significant negativenormal stress (comparable to the shear stress) develops in the network in the strain stiffening regime and reaches a maximumnear the yielding. γ[%] Τ 𝐼𝐼 𝑖 𝑛 (a) (b) (c) (d) (e) (f) γ[%] Τ 𝐼𝐼 𝑖 𝑛 Top plateBottom plateTop plate
Bottom plate γ b γ y γ b γ y Figure S8: Panels (a), (b), (d) and (e) show typical boundary images of the sample with dashed horizontal lines parallel tothe plates indicating the positions where the average intensity (normalized by the initial intensity) as a function of appliedstrain ( γ ) are calculated as shown in (c) and (f). Positions near the top and bottom plates indicated by blue and black lines,respectively. Red line indicates a position near the midway between the plates. (a), (b), (c) correspond to a strain ramp rate of10%/s and (d), (e), (f) correspond to 1%/s. For both ramp rates, the intensity near both the plates (c) or, one of the plates (f)drops significantly beyond the yield ( γ y ) and the breaking ( γ b ) strains as indicated. Panels (a) and (d) correspond to the initialunstrained state ( γ = 0) of the sample, whereas, (b) and (e) represent that after the network rupture. The drop in scatteredintensity near yielding indicates network rarefaction leading to detachment. Here, φ = 2 mg/ml and T = 30 ◦◦