Where $f_D$ is the dilution factor, and $C_{hp}$ is the hydroxyproline concentration.

### 2.2.1.6 Collagen solutions purity

The purity of the solution is determined by electrophoresis under denaturing conditions on polyacrylamide gels. This allows to identify the proteins present as a function of their molecular weight.

#### Preparation of polyacrylamide gels

There are two types of gels: a concentration gel and a migration gel. The amount of polyacrylamide will define the size of the pores in the gel. The concentration gel at 4% of polyacrylamide and the migration gel at 6% allow to separate the different polypeptide chains from the collagen solution. The gels contain ammonium persulfate and TEMED: the first induces the polymerization and the second one accelerates this process. The two gels are cast one after the other between two glass plates held vertically, the migration gel being cast at the bottom.

<figure><img src="image_43.png" /><figcaption>Figure 2.26: Electrophoresis gel in denaturing conditions. On the left, the rainbow molecular weight marker and the respective weights in kDa. On the right, the most marked bands are characteristic of the collagen. The two top bands correspond to the dimers $\beta_{11}$ and $\beta_{12}$ and the two most important bands are the $\alpha_1$ and $\alpha_1$ strands. Concentrations decreasing from left to the right.</figcaption></figure>

#### Preparation of samples

The samples have a volume of 20 µL and correspond to a mixture of 10 µL of a collagen solution more or less diluted with 10 µL of sample buffer. Three different dilutions are used: 2.5, 5 and 10. The sample buffer contains SDS which will denature the collagen molecule when heated for 5 minutes at 100 °C in boiling water. The whole is then cooled for 5 minutes on ice and the samples are placed in the wells of the concentration gel. The first well is reserved for the molecular weight marker (SDS-6H, SIGMA). It enables to relate the migration distance of the molecules present in the collagen solution to a molecular weight.
**Migration**

The whole system (gels + samples) is immersed in a migration buffer and a potential difference is applied between the base of the migration gel and the top of the concentration gel. As long as the samples are in the concentration gel, the applied voltage is 80 V. When the migration front reaches the second gel, the voltage is fixed at 120 V.

**Fixing and staining**

To fix the proteins and to color them, the gel is placed for 1 hour in a solution of Coomassie blue. Then, the colored gel is placed in various bleaching baths containing water, ethanol and acetic acid in order to remove the Coomassie Blue excess. A gel obtained for one of the purifications is presented in Figure 2.26. We observe the most marked bands, which are characteristic of the collagen: the two top bands corresponding to the dimers β₁₁ and β₁₂ and the two most important bands being the α₁ and α₁ strands. The presence of contaminants are revealed by marked bands at other weights, which is not the case. The collagen batch is thus assumed to be pure.

**2.2.1.7 Collagen solutions concentration**

After purification, the stock collagen concentration c is determined by hydroxyproline dosage as previously described. To concentrate this stock solution, centrifugal concentrators (Vivaspin) are used. They are composed of two parts: an upper structure covered with a filter membrane with a 300 kD cut-off (Fig 2.27 dashed lines) and a lower part which is a traditional tube. First, the upper structure is filled with the stock collagen solution (Fig.2.27, left 1) and the introduced volume mass is recorded. Then, the two structures are combined and spinned at 4400g. By centrifuging, a solvent flux occurs (Fig.2.27 left, black arrows) through the membrane that will retain the molecules larger than the cut-off.

<figure><img src="image_44.png" /><figcaption>Figure 2.27: Collagen concentration scheme. The stock collagen solution (1) is put in the centrifugal concentrator tube. After centrifugation at 4400g during several hours, the solvent (2) passes through the filter membrane and the concentrated collagen solution (3) can be collected.</figcaption></figure>

The pass-through solvent (Fig.2.27, right 2) is regularly removed from the lower part. Moreover, the collagen solution is frequently homogeneized to avoid high concentration gradient close to the filter membrane. This whole process takes several hours or days according to the concentration c' to achieve. This concentration is reached when the desired
mass of collagen is weighed.

## 2.2.2 Model system: sodium alginate

### 2.2.2.1 Structure

Alginate is a biopolymer, more precisely a polysaccharide, extracted from cell walls of brown seaweeds [16]. It is a block copolymer composed of mannuronic acid (M) and guluronic acid (G) (Fig. 2.28, A). These two acids have the same molecular formula, but their spatial conformations are different. Indeed, the carboxyl group is either in the equatorial position (M) or in the axial position (G). An alginate chain can be composed of homo-blocks (MM and GG) (Fig. 2.28, B) and block heteropolymer (GM or MG) (Fig. 2.28, C).

<figure><img src="image_45.png" /><figcaption>Figure 2.28: Structures of: A) the mannuronic (M) and guluronic (G) acids, B) the MM and GG homo-blocks and C) hetero-blocks. Adapted from [17]</figcaption></figure>

The molar mass of the alginates depends on several parameters regarding the origin, the age and the part (top, bottom or center) of the algae from which it is extracted. The molar mass is very important as it will affect the rheological behaviors and the physico-chemical properties of the polymer. The chemical structure of the alginate chain is also modified by the origin of the algae: the blocks M and G is arranged in an order and proportion specific to each alga.

### 2.2.2.2 Chemical properties

#### Solubility, stability and charge

- The solubility of alginates depends on several parameters as the pH, ionic strength of the solvent or the "hardness" of the water (= calcium ions content). The pKa values of mannuronic acid and guluronic acid are respectively 3.38 and 3.65 [18]. For
$pH < pKa$, alginate is converted to alginic acid, which is insoluble in water. If the pH is decreased abruptly, it causes the alginic acid molecules precipitation and if it is done slowly, a gel is formed due to the controlled release of protons. The accurate pH value that causes precipitation of the alginates is dependent on the composition and the structure of the polymer [19]. In our study conditions, sodium alginate is soluble in deionized water and for the preparation of high alginate solution concentrations (> 1% (w|w)), the salts dissolution may take a few days.

- The stability of alginates depends on temperature, pH and the presence of contaminants. The glycosidic bonds may rupture when the alginate is used in acidic or basic medium. For instance, at pH below 5, the acidic hydrolysis of those bonds induces a decrease in the molar mass by cleavage of the chains. Alginate is considered stable in the pH range 5-10. The polymer degradation is thus minimal around neutral pH and increases by deviating from this pH stability area.

- Alginate is negatively charged. At low pH (around 2), it remains uncharged due to the complete protonation. While increasing the pH, the charge becomes negative and at pH 6.5, alginate is completely deprotonated. [20].

<figure><img src="image_46.png" /><figcaption>Figure 2.29: A: Total persistence lengths calculated from the Benoit-Doty equation as a function of the ionic strength $I$ for different alginate concentration: Black square: 0.05 g/L, White square: 0.1 g/L and Triangle: 0.1 g/L with $MgCl_2$. B: Influence of pH on the calculated persistence lengths for alginic acid 0.1 g/L at an ionic strength of 50 mM. Three different regimes describing the behavior of the AA molecules are thought to occur: $pH < 4$: aggregation of highly protonated alginic acid molecules; $4 < pH < 8$: increase in molecular rigidity due to an increased intramolecular electrostatic repulsion; $pH > 8$: depolymerisation. [21]</figcaption></figure>

## Flexibility and persistence length

In the literature, the persistence length ($P$) of alginate has been determined by several techniques: rheology, electron microscopy, atomic force microscopy etc. Values of $P$ range between 5 to 17 nm ([22] [23]). Avaltroni et al. investigated the influence of both ionic strength ($I$) and pH on $P$ [21]. Their experimental results are shown on the figure 2.29.

At low ionic strength ($I = 1mM$), $P$ is about $5.5nm$ and decreases until $4nm$ for highest $I$ values (Fig. 2.29, A). Such a length corresponds to 9-10 monomers [21].

Regarding the pH influence (Fig. 2.29, B), a slight decrease of $P$ is observed from $pH 8$ to 4 while approaching the alginate $pKa$ (around 3.2 – 3.65). As previously described, alginate is thus protonated which decreases electrostatic repulsion [21]. This increases the
molecular flexibility which is consistent with the decrease of P. From this work, we can also quantify the alginate flexibility through the parameter α which is equal to 8 for an estimated contour length of 50 nm. The alginate is assumed to be also semi-flexible even rigid for lowest P values. Moreover, M monomers chains are more flexible than G ones. As a consequence, the persistence length depends on the ratio M/G of the alginate solution.

## Ionic binding of alginates/Gelation

One of the most interesting features of alginates is their ability to selectively form ionic bonds to divalent cations such as $Ca^{2+}$, $Sr^{2+}$ and $Ba^{2+}$ in order to form hydrogels. They bind more or less easily to those cations depending on their chemical nature. For example, the affinity increases following the order: $Mg^{2+} < Mn^{2+} << Ca^{2+} < Sr^{2+} < Ba^{2+} < Cu^{2+} < Pb^{2+}$ [18]. This affinity depends not only on electrostatic forces, but also on steric hindrance.

While sodium alginate solution is put in contact with divalent ions such as $Ca^{2+}$, an expanded network is formed. The structure conformation changes: alginate chains and ions form an arrangement called "egg-box" structure (Fig. 2.33), in which the ions representing the "eggs" are entrapped between two chains:

<figure><img src="image_47.png" /><figcaption>Figure 2.30: Schematic representation of the "egg-box" structure. Top: egg-box dimer, Bottom: coordination of $Ca^{2+}$ in a cavity created by a pair of guluronate sequences along alginate chains. Adapted from [24]</figcaption></figure>

For steric hindrance, calcium ions can only bind to carboxylic functions in axial positions which is the case for G-block. This mechanism is improved if the G-block exceeds 20 residues. In the case of M-blocks, their conformation assimilated to a flat ribbon, does not allow complexation with calcium ions. Thus, alginate gel formation depends on the ratio of M and G-blocks present in the polymer. From Fang et al. studies [24], this mechanism also depends on the alginate chains length. In the case of short and rigid chains, dimers
aggregate laterally through an inter-chain association, whereas in the case of longer and more flexible chains, the association occurs through intra-chain association (single chain level). Moreover, in both cases, the binding of calcium ions to alginate can be divided into three main sequential steps upon increasing calcium ions content (where R is the feeding ratio of Ca²⁺ to G residue) as shown in Figure 2.31 below:

<figure><img src="image_48.png" /><figcaption>Figure 2.31: Schematic illustration of the multiple-step binding of Ca²⁺ to alginate, in the case of short (Top) and long-chain alginate (Bottom). The zigzag lines, smooth lines, and black dots stand for G blocks, M blocks, and calcium ions, respectively. Adapted from [24]</figcaption></figure>

- **Step 1 (R < 0.25):** monocomplexation = calcium ions interaction with a single G-unit forming monocomplexes

- **Step 2 (0.25 < R < 0.55):** dimerization = propagation and formation of the egg-box dimers via pairing of the monocomplexes.

- **Step 3 (R > 0.55):** multimers formation = lateral association of the egg-box dimers.

### 2.2.2.3 Solution preparation

Sodium alginate was purchased from Kalys Gastronomie (average molecular mass $M_w$= 246 kDa). Salts are dissolved in ultrapure Millipore water in order to obtain solutions at the desired concentration in % ($w|w$). A small amount of sodium azide solution at 3% is added to avoid bacterial growth. The water is added progressively in order to avoid the formation of clots. We have made use of solutions with alginate concentrations ranging from c = 4, 6 to 10 wt.% (equivalent of 40, 60 and 100 mg/mL), referred to in the following sections as ALG4, ALG6 or ALG10. At the lowest concentration < 4%, the solution was magnetically stirred overnight. For higher concentrations > 4%, alginate powder was left to swell and dissolved for at least 2 days under periodic gentle manual stirring. Homogeneity of the solutions is assessed from its optical aspect. For rheological measurements, a solution previously prepared at 4% is diluted to reach concentrations between 0.025 and 3%.

To induce gelation, alginate solutions were extruded in a cross-linking bath containing divalent ions (Ca²⁺). Calcium chloride dihydrate (M=147.02 g/mol) was purchased from Prolabo. It is also dissolved in ultrapure Millipore water to obtain solution at 0.1, 0.5 or 1M and stirred until the complete dissolution of salts.
### 2.2.3 Collagen-alginate mixtures

To improve threads properties at the structural and mechanical levels compared to pure collagen based ones, we decided to mix collagen and alginate. The idea is to produce threads from the collagen-alginate mixture to benefit from key strengths of each species.

#### 2.2.3.1 Chemical properties

While mixing proteins and polysaccharides together, different kinds of interactions may occur: steric exclusion, hydrogen bonding, hydrophobic or electrostatic interactions. In our case, the collagen and the alginate have an electrical charge under the experimental chosen conditions (pH, ionic strength...). The main interaction is thus electrostatic and may be either attractive or repulsive depending on the charge of the species involved [25] (Fig. 2.32).

<figure><img src="image_49.png" /><figcaption>Figure 2.32: Schematic representation of the types of structural arrangements of the molecules involved that can occur when proteins and polysaccharides are mixed together. Adapted from [25], [26]</figcaption></figure>

If they are oppositely charged, attractive interaction is predominant and can lead to soluble (1-phase) and/or insoluble (2-phase) complexes. This 2-phase system is composed of a phase rich in both biopolymers and a second one depleted in both species. Depending on the attraction strength and the chemical properties of the polymers, the rich phase can be a coacervate or a precipitate.

If the proteins and polysaccharides have similar electric charges or if one or both are uncharged, repulsive interaction occurs. At low biopolymer concentrations, the two species
are cosoluble and form a 1-phase solution. While increasing the concentration, segrega-
tive phase separation occurs and a 2-phase solution is then formed in which each phase
is rich in one biopolymer.

To control this interaction and move from attractive to repulsive ones, it is necessary to
adjust the pH, ionic strength, biopolymer ratio and other parameters [26].

**2.2.3.2 Solution preparation**

<figure><img src="image_50.png" /><figcaption>Figure 2.33: 1:Collagen 60 mg/mL, 2:Collagen 30 mg/mL, 3: Collagen/alginate (5:1) mixture; 4: Alginate 4%.</figcaption></figure>

Fresh and sterile sodium alginate salt (as previously described) is dissolved in 500 mM acetic acid to obtain solution at 4 % (w/w) (Fig. 2.33, 4). A mass m1 of collagen stock solution at 3mg/mL (500mM AA) and a mass m2 of sodium alginate solution at 4 % (w/w) are added to a concentrator centrifugal tube. The mixture is then centrifuged at 4400 g until reaching the desired ratio Coll:Alg. Three different ratios Collagen:Alginate were tested: 1:1, 2:1, and 5:1 (Fig. 2.33, 3). The collagen-alginate mixture solution before extrusion is quite opaque that may already testify the interaction between the two species.

To induce gelation of alginate inside the mixture, solutions were extruded in a cross-
linking bath containing divalent ions ($Ca^{2+}$) at 1M as previously detailed.
## 2.3 References

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[20] Severian Dumitriu. *Polysaccharides: structural diversity and functional versatility*. CRC press, 2004. 80

[21] Fabrice Avaltroni, Marianne Seijo, Serge Ulrich, Serge Stoll, and Kevin J Wilkinson. Conformational changes and aggregation of alginic acid as determined by fluorescence correlation spectroscopy. *Biomacromolecules*, 8(1):106–112, 2007. 80

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# Chapter 3

# Rheological properties of collagen/alginate solutions

"Sweet scents of honey float through the air."

Winnie the Pooh

## Abstract

In this chapter, oscillatory and steady state shear measurements were performed on collagen at 30 mg/mL and on three alginates solutions. First, for collagen solution at 30 mg/mL, our results are presented and compared with the work of Gobeaux et al. (2010)[1]. From this study, we extracted the order of magnitude of viscosities of dense collagen solutions involved in our work. Then, selected alginate concentrations comparable with collagen ones and exhibiting quite similar behaviors were characterized. These two complex fluid systems show strong visco-elastic and shear-thinning properties.

## Contents

<table><tr><td><strong>3.1</strong></td><td><strong>Collagen solution rheological characterization</strong></td><td><strong>88</strong></td></tr><tr><td>3.1.1</td><td>Oscillatory shear flow</td><td>88</td></tr><tr><td>3.1.1.1</td><td>Complex modulus measurement</td><td>88</td></tr><tr><td>3.1.1.2</td><td>Comparison with Gobeaux et al. work [1]</td><td>89</td></tr><tr><td>3.1.2</td><td>Steady shear flow</td><td>90</td></tr><tr><td>3.1.2.1</td><td>Viscosity</td><td>90</td></tr><tr><td>3.1.2.2</td><td>First normal stress difference</td><td>92</td></tr><tr><td>3.1.2.3</td><td>Typical physical quantities for other concentrations</td><td>93</td></tr><tr><td><strong>3.2</strong></td><td><strong>Alginate solutions rheological characterization</strong></td><td><strong>94</strong></td></tr><tr><td>3.2.1</td><td>Viscosity analysis</td><td>94</td></tr><tr><td>3.2.1.1</td><td>Alginate concentration choice</td><td>94</td></tr><tr><td>3.2.1.2</td><td>Shear-thinning nature</td><td>95</td></tr><tr><td>3.2.2</td><td>Complex modulus analysis</td><td>96</td></tr><tr><td>3.2.3</td><td>First normal stress difference and Weissenberg number</td><td>97</td></tr><tr><td><strong>3.3</strong></td><td><strong>Conclusion</strong></td><td><strong>99</strong></td></tr><tr><td><strong>3.4</strong></td><td><strong>References</strong></td><td><strong>100</strong></td></tr></table>
During the extrusion, solutions will be forced to pass through a die of a few hundred microns diameter and be subjected to high shear stresses. It is thus necessary to fully characterize the rheological properties of our complex fluids systems before extruding them in order to understand the different flow phenomena.

A complete study of collagen rheological properties at concentrations up to 112 mg/mL was already performed by Gobeaux et al. (2010) [2]. We compare our results based on the data analysis approach used in this study and Gobeaux PhD thesis [3].

Sodium alginate is our model system and will be also characterized to choose solution concentrations exhibiting similar properties as collagen.

## 3.1 Collagen solution rheological characterization

Rheological characterization was performed on collagen solutions at 30 mg/mL. Our results were both compared and supplemented with those of Gobeaux et al. [1] in which a study on collagen solutions at different concentrations was performed. As reported, studying complex fluids demands much attention as their rheological properties depend on their history. The solutions structure can be disturbed by strong stresses and it takes a very long time for them to recover their "rest" structure. This may explain why few investigations of collagen rheological properties at such high concentrations have been performed.

### 3.1.1 Oscillatory shear flow

#### 3.1.1.1 Complex modulus measurement

Shear rheological measurements were carried out using a rheometer imposing strain. Cone-plate geometry was used in dynamic mode and allowed to probe angular frequencies ranging from 0.01 to 100 rad/s. The viscous and elastic components of the complex modulus were then measured. Results are presented in Figure 3.1.

To have systems comparable with the previous study [1], we brought the collagen solutions to the same initial state before measurements: first, the sample (30 mg/mL) was sheared during 300s at a constant shear rate of 8 s⁻¹ and then left at rest for one hour. The collagen solution exhibited a viscoelastic behavior depending on the stress frequency (Fig. 3.1). Shear storage G' and loss G'' moduli increased with frequency. According to Gobeaux analysis [3], we also modeled the two curves by power law written as:

$$G'(\omega) = G'_{0} \left(\frac{\omega}{\omega_{0}}\right)^{\alpha}$$

$$G''(\omega) = G''_{0} \left(\frac{\omega}{\omega_{0}}\right)^{\beta}$$

where $\omega_0 = 1$ rad/s, $G'_0 = G'(\omega_0)$ and $G''_0 = G''(\omega_0)$. The respective exponents and prefactors are: $\alpha = 0.092 \pm 0.06$ and $\beta = 0.20 \pm 0.12$; $G'_0 \approx 486 \pm 10$ Pa and $G''_0 \approx 91 \pm 20$ Pa.
<figure><img src="image_51.png" /><figcaption>Figure 3.1: Shear storage G' (black ◦) and loss G" (red ▽) moduli against angular frequencies for collagen solution at 30 mg/mL</figcaption></figure>

### 3.1.1.2 Comparison with Gobeaux et al. work [1]

According to Gobeaux et al. results (Fig. 3.2) exponents are equal to 0.3 and prefactors to $G'_0 \approx 460$ Pa and $G''_0 \approx 230$ Pa. In our case, those exponents are lower and closer to those found for very high concentrations (300 mg/mL) (Fig 3.2, red results). The previous power law fit might not be suitable for our solution. However, the shear and loss moduli are almost in the same order of magnitude.

<figure><img src="image_52.png" /><figcaption>Figure 3.2: Prefactors $G'_0 (○)$, $G''_0 (Δ)$ and power-law exponent $β (◦)$ as a function of the collagen concentration. Our experimental results are added in the respective orange filled markers. The study concentration region of interest the closest to our concentration is shaded. Adapted from [1]</figcaption></figure>
3.1.2 Steady shear flow

The material was sheared between the cone and the plate in which shear rate is con-
stant within the whole sample. We measured the physical quantities of interest: viscosity,
normal forces as a function of the imposed shear rate.

**3.1.2.1 Viscosity**

The collagen solution viscosity was measured for constant shear rates ranging from 0.1 to 100 s⁻¹. Upon increasing the shear rate, the viscosity decreases (Fig 3.3), attesting to the shear-thinning behavior of the collagen at 30 mg/mL. It could be interpreted as the entangled molecules network rupture or simply as the molecules alignment due to the imposed shear.

<figure><img src="image_53.png" /><figcaption>Figure 3.3: Comparison of the complex (or dynamic) viscosity η* measured in oscillatory regime and the viscosity η measured in steady state for collagen solution at 30mg/mL. The shear-thinning behavior of collagen in dense solution is clearly observed. Moreover, Cox-Merz rule is verified.</figcaption></figure>

Cox-Merz rule application

The Cox-Merz rule is an empirical correlation between dynamic and steady flow vis-
cosity found by their respective authors [4] and was verified for almost all polymer melts,
concentrated and semi-dilute polymer solutions. It is assumed that the complex viscosity
η* measured in oscillating dynamic regime and the viscosity η measured in steady state
are equal while assimilating the angular frequency ω to the shear rate γ̇:

$$
|\eta^*(\omega)| = \eta(\dot{\gamma})|_{\omega=\dot{\gamma}}
$$

$$
\text{with: } |\eta^*| = \sqrt{\left(\frac{G'}{\omega}\right)^2 + \left(\frac{G''}{\omega}\right)^2}
$$
In our case, the complex viscosity η* was slightly overestimated compared to the viscosity η, but the trends were very similar. Thus, it is assumed that Cox-Merz rule is verified for our collagen solution at 30 mg/mL

Viscosity curves fitting

According to the further Gobeaux analysis, we modeled in the same way the viscosity linear domain plotted in figure 3.3 by a power law as proposed by Ostwald [5].

$$ \eta \propto K \dot{\gamma}^{-n} $$

where K is a constant and n is an exponent to be determined. In the case of Newtonian fluids, n = 0. The exponent $n_1$ was measured in the steady state ($\eta = f(\dot{\gamma})$) and $n_2$ in the dynamic regime ($\eta^* = f(\omega)$). The results are gathered in Table 3.1 and compared with the values obtained previously at 31 mg/mL [3].

<table><caption>Table 3.1: Power law fit exponent values $n_1$ for η and $n_2$ for η* at collagen concentration of 30 mg/mL and comparison with Gobeaux previous results [3].</caption><thead><tr><th>Study</th><th>C (mg/mL)</th><th>n<sub>1</sub> (η)</th><th>n<sub>2</sub> (η*)</th></tr></thead><tbody><tr><td>Picaut</td><td>30</td><td>0.83</td><td>0.91</td></tr><tr><td>Gobeaux</td><td>31</td><td>0.79</td><td>0.75</td></tr></tbody></table>

Our experimental results are in good agreement with Gobeaux PhD thesis results [3]. In the range of shear rates tested, the Newtonian plateau is not reached (Fig. 3.3) and we only observe the shear-thinning part.

In fact, when the collagen solution is submitted to lower shear rates, it is possible to see the transition between the "plateau" and the linear domain and so to deduce the critical crossover shear rate $\dot{\gamma}_0$ from which the shear-thinning behavior starts [3]. From this, a characteristic time $\tau_c$ is defined as: $\tau_c = 1/\dot{\gamma}_0$, below which the collagen chains do not have time to come back to their initial entangled configuration. Gobeaux characteristic disentanglement time for high collagen concentrations are given in the figure 3.4.

<figure><img src="image_54.png" /><figcaption>Figure 3.4: Characteristic disentanglement time $\tau_c$ as a function of the collagen concentration. The concentration corresponding to our experimental condition is pointed out by the red arrow. Results from [3]</figcaption></figure>
For collagen solution at 31 mg/mL, close to our experimental condition, this time is about 200s (respectively $\dot{\gamma}_0=5 \times 10^{-3} s^{-1}$) as shown by the red arrow in Figure 3.4. This time can reach a few hours for the highest concentrations.

### 3.1.2.2 First normal stress difference

Under steady shear flow, we can also measure the normal forces $F_N$ for shear rates in the range of 0.1 to $100 s^{-1}$ and calculate the associated first normal stress difference $N_1$ defined as: $N_1 = F_N/S$ where S is the rheometer geometry surface. It is then possible to estimate the Weissenberg number $Wi$ which is the ratio between the first normal stress difference and the shear stress:

$$Wi = \frac{N_1}{\sigma}$$

$$\text{where } \sigma = \eta(\dot{\gamma}) \dot{\gamma}$$

$Wi$ is a dimensionless number which compares the relaxation time of the fluid to a specific process time. We calculate this number for collagen solution at 30 mg/mL from measured $F_N$ values (Fig. 3.5, purple).

<figure><img src="image_55.png" /><figcaption>Figure 3.5: Shear rate dependent Weissenberg numbers for collagen solution at 30 mg/mL. Purple: $Wi$ calculated from normal forces experimental measurements in steady shear flow. Blue: $Wi$ calculated from empirical Laun relationship in oscillatory shear. The dashed line indicates the value of $\dot{\gamma}_0$ and the full line is $(\dot{\gamma}/\dot{\gamma}_0)^m$ with $m \approx 0.3$.</figcaption></figure>

We also attempted to extract $Wi$ number by using the empirical relationship proposed by Laun [6] as it is classically performed for polymer melts. It connects the first normal stress $N_1(\dot{\gamma})$ measured using steady shear flow to the storage modulus $G'(\omega)$ and loss modulus $G''(\omega)$ measured in oscillatory shear for $\dot{\gamma} = \omega$ by:
$$N_{1,Laun}|_{\omega=\dot{\gamma}} = 2G'(\omega) \left(1 + \left(\frac{(G'(\omega))}{G''(\omega)}\right)^2\right)^{0.7}$$

We thus have:

$$Wi_{Laun} = \frac{N_{1,Laun}}{\sigma}$$

$$\text{where } \sigma = \eta^*(\omega) \omega$$

Figure 3.5 shows the shear rate dependence of the Weissenberg numbers: 1) $Wi$ calculated from the normal forces measurements and 2) $Wi_{Laun}$ from the oscillatory shear with $G'(\omega)$ and $G''(\omega)$ (Fig. 3.5, blue).

The two Weissenberg numbers calculated from the steady and oscillatory shear experiments are not in good agreement. In fact, collagen solutions display strong anisotropic behaviors and this kind of empirical law is not suitable to model such specific fluids.

However, we attempt to describe the Weissenberg number $Wi(\dot{\gamma})$ from our experimental data by a power law:

$$Wi = \left( \frac{\dot{\gamma}}{\dot{\gamma}_0} \right)^m$$

We extract from this curve fitting the physical quantities: a) the exponent: $m \approx 0.3$, b) the crossover shear rate: $\dot{\gamma}_0 \approx 1.2s^{-1}$ and $Wi(\dot{\gamma}_0) \approx 0.097$. The characteristic disentanglement time $\tau_c$ is equal to 0.83 s, which is about 200 times smaller than the one found in Gobeaux et al. study (Fig. 3.4).

### 3.1.2.3 Typical physical quantities for other concentrations

In chapter 5, we describe the production of collagen threads at different concentrations in order to find the best condition (swelling, collagen fibrils alignment, mechanical properties etc) for the further studies. Three concentrations are tested: 15, 30 and 60 mg/mL. Based on the Gobeaux et al. work [3], we can already have an idea of the physical quantities for those conditions.

In this concentration range, different regimes may be observed. Indeed, $c^*$ which is the overlap concentration separating the dilute and the semi-dilute regime was estimated from the specific viscosity. $c^*$ is defined as the concentration at which polymer molecules start to contact with one another, which induces changes of the solution viscoelastic properties. For collagen solution, it is assumed to be around 3-5 mg/mL [2]. In our case, concentrations are higher than 5 mg/mL which means that even at 15 mg/mL, solutions are in the semi-dilute regime. At 30-60 mg/mL, collagen solutions are in entangled state. It would be also interesting to have an order of magnitude of the "plateau" viscosity, the crossover shear rate and the exponents of the shear-thinning linear region $n_1$. To do so, we extracted those quantities from the following figure and the results presented in [3]:
<figure><img src="image_56.png" /><figcaption>Figure 3.6: Viscosity of collagen at different concentrations as a function of the shear rate. The concentrations of interest are highlighted by colored arrows: 16.4 (orange), 31 (red) and 64 (purple) mg/mL. Eye guidelines in dashed lines are given to estimate η₀. From [3]</figcaption></figure>

The concentrations studied close to ours are highlighted by colored arrows. From this figure 3.6, we could obtain the "plateau" viscosity η₀. We deduced the crossover shear rate from Figure 3.4 given the characteristic disentanglement time τ_c. The table 3.2 shows those quantities as a function of concentration.

<table><caption>Table 3.2: Characteristic physical quantities of collagen solutions of interest, from [3].</caption><thead><tr><th>C (mg/mL)</th><th>η₀ (Pa.s)</th><th>γ̇₀ (/s)</th><th>n₁</th></tr></thead><tbody><tr><td>16.4</td><td>50</td><td>1.5 · 10⁻²</td><td>0.65-0.72</td></tr><tr><td>31</td><td>5000</td><td>5 · 10⁻³</td><td>0.79</td></tr><tr><td>64</td><td>50 000</td><td>4 · 10⁻⁴</td><td>0.81</td></tr></tbody></table>

## 3.2 Alginate solutions rheological characterization

In this section, the same rheological characterizations as collagen solutions is performed on alginate ones. The choice of the concentrations of alginate solutions was substantially motivated by the possibility to transpose the results obtained on alginate extrusion to collagen.

### 3.2.1 Viscosity analysis

The viscosity of sodium alginate solutions in pure water was measured at temperature T = 20 °C for shear rates ranging from 0.01 to 200 s⁻¹.

#### 3.2.1.1 Alginate concentration choice

For the lowest concentrations (< 1 wt %), we used the double gap Couette cell geometry and above, the cone-plate geometry. Measurements at c = 1 wt % were done with the two
geometries to ensure the results continuity. Figure 3.7 displays the specific viscosity η<sub>sp</sub> defined as η<sub>sp</sub> = (η<sub>0</sub> − η<sub>s</sub>)/η<sub>s</sub> as a function of the concentration c, where η<sub>0</sub> is the viscosity of the solution in the vanishing shear rate Newtonian regime and η<sub>s</sub> is the solvent (water) viscosity. While increasing the alginate concentration, the specific viscosity increases gradually with three characteristic slopes corresponding to the usual regimes observed for polymer solutions [7]:

- c < c*: dilute regime

- c < c* < c<sub>E</sub> : semi-dilute unentangled regime

- c > c<sub>E</sub> : entangled regime

where c*, as previously mentioned, is the overlap concentration (c*=0.35%) and cE is the entanglement concentration (cE=1%). The overlap concentration for alginate was quite similar to that of collagen: c*<sub>alg</sub> ≈ c*<sub>coll</sub> ≈ 3 − 5mg/mL. Collagen solutions of 15 to 60 mg/mL exhibit zero-shear viscosity ranging between 50 and 50 000 Pa.s. To study alginate solutions with the same order of viscosity, we choose to use concentrations above 4 wt % (4, 6 and 10 wt %) which also correspond to the entangled state (Fig. 3.7 grey region).

<figure><img src="image_57.png" /><figcaption>Figure 3.7: Specific viscosity of alginate solutions as a function of concentration (wt %), measured at T = 20 °C. The boundaries between the three scaling regimes define the overlap c* = 0,35 wt % and entanglement c<sub>E</sub> = 1 wt % concentrations. The grey zone indicates the concentration domain relevant to our work, corresponding to strongly entangled solutions.</figcaption></figure>

3.2.1.2 Shear-thinning nature

Figure 3.8 shows the viscosity evolution as a function of the shear rate for the chosen solutions ALG4, ALG6 and ALG10. Each solution exhibit a marked shear-thinning behavior which can be characterized by the Cross law [8], [9]:

$$
\eta(\dot{\gamma}) = \frac{\eta_0}{1 + (\dot{\gamma}/\dot{\gamma}_0)^{1-m}}
$$

We found a single exponent m equal to 0.29 ± 0.02 for the three concentrations. This exponent value is compatible with that usually reported (m ≈ 0.24) for entangled and
random coil polysaccharide solutions [10]. Moreover, in the case of collagen solutions, we showed that: $\eta \propto \dot{\gamma}^{-n}$ with $0.65 < n < 0.81$. From the Cross law fit, it is assumed $\eta \propto \dot{\gamma}^{m-1}$ with $m=0.29$ which corresponds to an exponent $n = 1 - m = 0.71$. Thus, collagen and alginate solutions used during our study had almost the same shear-thinning behaviors.

<figure><img src="image_58.png" /><figcaption>Figure 3.8: Shear-thinning behavior of the alginate solutions at 4, 6 and 10%. The solid lines are a fit by a Cross law with a common shear-thinning exponent $m = 0.29$. The dashed lines mark the crossover shear rates $\dot{\gamma}_0$.</figcaption></figure>

The characteristic physical quantities ($\eta_0$, $\dot{\gamma}_0$ and the exponent $n$) found from the Cross law fits for each concentration are gathered in the following table 3.3:

<table><caption>Table 3.3: Characteristic physical quantities of chosen alginate solutions: ALG4, ALG6 and ALG10.</caption><thead><tr><td>C<sup>0</sup> (wt%)</td><td>η<sub>0</sub> (Pa.s)</td><td>γ̇<sub>0</sub> (/s)</td><td>τ<sub>c</sub> (s)</td><td>n</td></tr></thead><tbody><tr><td>4</td><td>30</td><td>8</td><td>0.13</td><td>0.71</td></tr><tr><td>6</td><td>110</td><td>3</td><td>0.33</td><td>0.71</td></tr><tr><td>10</td><td>1900</td><td>0.3</td><td>3.3</td><td>0.71</td></tr></tbody></table>

**NB:** The characteristic times $\tau_c$ for alginate ranging from 130 ms to 3.3 s was very low compared to collagen ones ($80s < \tau_c < 30$ min for $16.4$ mg/mL < $c < 60$ mg/mL).

### 3.2.2 Complex modulus analysis

Storage (G') and loss (G") moduli were measured with cone-plate geometry under oscillatory shear for angular frequencies between 0.1 and 300 rad/s. Figure 3.9 shows the results for the three alginate concentrations. Here again, it was clear that the three alginate solutions displayed viscoelastic behaviors.

Same analyses as collagen moduli were performed to assess the exponents of those two quantities in the linear region. Moreover, the crossover angular frequency allowed us to estimate the relaxation time $\lambda$ of each solution. All those computed values are presented in Table 3.4.

The complex viscosity $\eta^*$ was also calculated from G' and G" measurements with the Cox-Merz rule. It was verified for those three concentrations.
<figure><img src="image_59.png" /><figcaption>Figure 3.9: Shear storage G' (empty circle) and loss G" (filled circle) moduli against angular frequencies for: ALG4 (black), ALG6 (red) and ALG10 (blue). The dashed lines show the crossover angular frequency for each condition.</figcaption></figure>

<table><caption>Table 3.4: Fit exponents α and β for the respective storage and loss moduli; the crossover angular frequency ωc and the relaxation time λ estimated from ωc values for the three alginate concentrations: ALG4, ALG6 and ALG10.</caption><thead><tr><td>Solution</td><td>α (G')</td><td>β (G")</td><td>ω<sub>c</sub> (rad/s)</td><td>λ (ms)</td></tr></thead><tbody><tr><td>ALG4</td><td>1.16</td><td>0.72</td><td>30</td><td>33</td></tr><tr><td>ALG6</td><td>1.05</td><td>0.72</td><td>20</td><td>50</td></tr><tr><td>ALG10</td><td>0.65</td><td>0.53</td><td>2.5</td><td>400</td></tr></tbody></table>

### 3.2.3 First normal stress difference and Weissenberg number

The viscoelastic properties were also characterized by the measurement of the first normal stress difference N₁ as a function of the shear rate. The experimental results are shown in Figure 3.10 (empty circles). As previous analysis performed for collagen, we used Laun empirical relationship to also compute first normal stress difference (N₁,Laun) from the shear and loss moduli measurements (Fig 3.10 filled circles). The results from the two ways of measuring N₁ were in good agreement for ALG10 which was the most concentrated solution. For ALG4 and ALG6, the N₁,Laun was overestimated compared to the direct measurements of N₁. Here again, it is probably due to the N₁ measurement difficulty as the normal forces may be too low for this concentration range. However, the N₁,Laun allowed us to estimate the Weissenberg number Wi for alginate (Fig. 3.11)
<figure><img src="image_60.png" /><figcaption>Figure 3.10: First normal stress difference measured in steady state shear (empty circles) and computed from Laun empirical relationship through G' and G" measurements (filled circles). The dashed lines show the crossover shear rate for each condition.</figcaption></figure>

For $\dot{\gamma} > \dot{\gamma}_0$, Wi number can be described by a simple power law as previously done for col-
lagen. A common exponent equal to 0.3 was found for the three alginate concentrations.
The Weissenberg number estimated for collagen at 30mg/mL had the same exponent but
was lower even compared to ALG4.

<figure><img src="image_61.png" /><figcaption>Figure 3.11: Sublinear, shear rate dependent Weissenberg number for alginate solutions. The dashed lines indicate the values of $\dot{\gamma}_0$ and the full lines are the power fit with a common exponent equal to 0.3.</figcaption></figure>
3.3 Conclusion

Collagen and alginate solutions exhibit strong viscoelastic and shear-thinning behav-
iors which are quite similar in the concentration region of interest. Moreover, those com-
plex fluids will be studied in the same "entanglement" state. The empirical rules and
relationships are verified and in good agreement with the literature for alginate system.
In the case of collagen, Cox-Merz rule can be applied but further analysis are needed to
better model such specific fluid.

This rheological study makes us confident to have chosen comparable concentration con-
dition with two different systems. It will probably help us to transpose extrusion results
from one system to the other.
**3.4 References**

[1] Frédéric Gobeaux, Emmanuel Belamie, Gervaise Mosser, Patrick Davidson, and So- phie Asnacios. Power law rheology and strain-induced yielding in acidic solutions of type i-collagen. *Soft Matter*, 6(16):3769–3777, 2010. [iv, 87, 88, 89]

[2] Frédéric Gobeaux, Emmanuel Belamie, Gervaise Mosser, Patrick Davidson, and So- phie Asnacios. Power law rheology and strain-induced yielding in acidic solutions of type I-collagen. *Soft Matter*, 6(16):3769, 2010. [88, 93]

[3] Frédéric Gobeaux. *Phases denses de collagène de type I: transition isotrope/cholestérique, fibrillogénèse et minéralisation*. PhD thesis, Université Pierre et Marie Curie-Paris VI, 2007. [88, 91, 93, 94]

[4] W Cox and E Merz. Rheology of polymer melts—a correlation of dynamic and steady flow measurements. In *International Symposium on Plastics Testing and Standardization*. ASTM International, 1959. [90]

[5] Wolfgang Ostwald. Ueber die geschwindigkeitsfunktion der viskosität disperser systeme. i. *Colloid & Polymer Science*, 36(2):99–117, 1925. [91]

[6] HM Laun. Prediction of elastic strains of polymer melts in shear and elongation. *Journal of Rheology*, 30(3):459–501, 1986. [92]

[7] Liang Guo, Ralph H Colby, Charles P Lusignan, and Andrew M Howe. Physical gelation of gelatin studied with rheo-optics. *Macromolecules*, 36(26):10009–10020, 2003. [95]

[8] Malcolm M Cross. Rheology of non-newtonian fluids: a new flow equation for pseudoplastic systems. *Journal of colloid science*, 20(5):417–437, 1965. [95]

[9] R Byron Bird, Robert C Armstrong, and Ole Hassager. Dynamics of polymeric liquids. vol. 1: Fluid mechanics. 1987. [95]

[10] ER Morris, AN Cutler, SB Ross-Murphy, DA Rees, and J Price. Concentration and shear rate dependence of viscosity in random coil polysaccharide solutions. *Carbohydrate polymers*, 1(1):5–21, 1981. [96]
# Chapter 4

## Extrusion instabilities of polymer solutions

> "It's up to you how far you'll go. If you don't try, you'll never know."

Merlin in *The Sword in the Stone*

### Abstract

The purpose of this chapter is to study instabilities affecting the extrusion of a viscoelastic polymer solution. Two different configurations were studied depending on the presence or not of gelling ions in the exit solution.

We have first performed an extensive study using sodium alginate solutions. The non-gelling configuration is akin to the industrial one, hence widely studied polymer melt extrusion process. Here, however, we have made use of capillaries ("dies") with unusually large length to radius ratios. We have been able to prove that in that case, the helical extrusion instability observed at high enough Weissenberg number is the mere consequence of a bulk pipe flow instability, predicted theoretically for such highly shear-thinning fluids but never observed experimentally.

In the presence of gelling $Ca^{2+}$ ions, more complex phenomena are involved and lead to unexpected and spectacular instabilities that we will describe.

Finally, we have made use of our results on alginate to narrow the domain of control parameter in our quest of an analogous helical instability. We show that such an instability actually occurs and we give hints of a "crimp like" helical ordering of the collagen fibrils.

### Contents

<table><tr><td><strong>4.1 Quick literature overview: polymer melt extrusion instabilities</strong></td><td><strong>103</strong></td></tr><tr><td>4.1.1 Surface instability: the sharkskin</td><td>104</td></tr><tr><td>4.1.2 Spurt-flow defect</td><td>104</td></tr><tr><td>4.1.3 Bulk distorsion</td><td>104</td></tr><tr><td>4.1.3.1 Helical</td><td>104</td></tr><tr><td>4.1.3.2 Gross melt fracture</td><td>105</td></tr><tr><td>4.1.4 Conclusion</td><td>105</td></tr><tr><td><strong>4.2 Extrusion in a bath of non-binding ions</strong></td><td><strong>105</strong></td></tr><tr><td>4.2.1 Context and motivations</td><td>105</td></tr><tr><td>4.2.1.1 Conclusion</td><td>108</td></tr><tr><td>4.2.2 Results: Experimental evidence of helical, supercritical instability in pipe flow of shear thinning fluids</td><td>108</td></tr><tr><td>4.2.3 Implementation on other systems: In the "crimp" quest</td><td>122</td></tr></table>
<table><tr><td>4.2.3.1 Quick literature overview</td><td>122</td></tr><tr><td>4.2.3.2 Helical instability: from alginate to collagen</td><td>123</td></tr><tr><td><b>4.3 Extrusion in a bath of cross-linking ions</b></td><td><b>125</b></td></tr><tr><td>4.3.1 Overall observations</td><td>125</td></tr><tr><td>4.3.2 Smooth region</td><td>126</td></tr><tr><td>4.3.3 Die swelling</td><td>128</td></tr><tr><td>4.3.4 Wavy, striated patterns and smooth alternation</td><td>130</td></tr><tr><td>4.3.4.1 Volumetric waves and surface striations</td><td>130</td></tr><tr><td>4.3.4.2 Oscillating flow</td><td>134</td></tr><tr><td>4.3.5 Gel clogging</td><td>135</td></tr><tr><td><b>4.4 Conclusion</b></td><td><b>137</b></td></tr><tr><td><b>4.5 References</b></td><td><b>138</b></td></tr></table>
**4.1 Quick literature overview: polymer melt extrusion instabilities**

It has taken a long time of sustained experimental efforts to classify the different instabilities occuring, depending on the nature (linear, branched...) of the polymers, on the geometry (die length, radius) and the throughput. Several expressions have been coined to describe these instabilities which are now recognized as misleading. For instance, melt fracture expression was used to gather and describe all polymer melt instabilities as it was first thought that they came all from local ruptures of the material. However, there is no evidence that those phenomena are all related to the rupture of polymer melt and the term "melt fracture" is not suitable but still used [1].

<figure><img src="image_62.png" /><figcaption>Figure 4.1: Examples of polymer melt extrudates exhibiting: 1) Sharkskin, 2) Spurt-flow, 3) Helical, and 4) Chaotic defects. Adapted from [2]</figcaption></figure>

It is thus meaningful to distinguish between surface and bulk instabilities, depending on the extension of the profile perturbation with respect to the average extrudate radius: 1) the sharkskin, 2) the "spurt-flow", 3) helical and 4) chaotic defects [2]. As shown in Figure 4.1, these instabilities spoil in a more or less pronounced way the extrudate: either affecting the surface, or the volume. For each characteristic defect, we review the main observations and possible mechanisms which may initiate those instabilities.
4.1.1 Surface instability: the sharkskin

Sharkskin affects the surface of the extrudate and takes its name from its typical pattern
that looks like the skin topological structure of sharks. It generally occurs at low shear
rates and it is not impacted by the length of the die or the angle of the inlet. Moreover, the
velocity field at distances larger than the capillary radius is not disturbed while noticing
sharskin instability. All those observations point to the idea that this defect is produced at
the die exit. Moreover, surface treatment promoting wall slip in the exit vicinity has been
shown to suppress sharkskin [3]. In fact, the die exit displays a singular point which is the
site of high stresses and elongation.

A possible mechanism was proposed and thought to be responsible for the sharkskin pattern [2]. In fact, due to the abrupt transition from no slip to no stress boundary conditions at the capillary exit, the flow is singular at the rim, i.e. the stress field inside the pipe and the shear rate outside should diverge. This unphysical unbounded behavior must lead to localized but important changes, e.g. wall slip (e.g. via chain disentanglement) or fracture of the viscoelastic extrudate.

4.1.2 Spurt-flow defect

Several names are used to designate this instability: cycling melt fracture, stick-slip, spurt or plug flow etc. It appears as a regular succession of different morphologies areas, generally a smooth one alternating with sharkskin and this phenomenon may be very periodic [4], [5]. While imposing the flow rate, this instability is also revealed through pressure jumps [6]. It is very sensitive to the geometry of the flow and so the ratio of the pipe length L over its diameter D. If this ratio is close to 0, oscillations vanish.

From those observations, it is assumed that this defect occurs in an unstable region where
it can go from a smooth regime to a surface instability one. Here again, two main expla-
nations are given [7]:

- At first it was thought that this instability was only due to the viscoelastic nature of the material.

- A second mechanism is the succession of sliding and adhesion of the polymer chains to the wall. In the sliding phase, polymer chains keep their entangled configuration and the extrudate is smooth. In the adhesive phase, the chains are first elongated and then recover their initial entangled configuration at the outlet which leads to the "skin" fracture as previously explained.

The first hypothesis was discarded and the second one is assumed to represent the real mechanisms. Thus, the best way to kill this oscillating defect is to induce a permanent sliding on the wall by changing the roughness of the die.

4.1.3 Bulk distorsion

The more spectacular instabilities are the bulk distorsion which impacts the whole vol-
ume of the extrudate.

4.1.3.1 Helical

This instability affects the entire volume of the extrudate and exhibits helical shape which
can be very periodic. The amplitude of the defect increases with the flow rate. It is damped
while increasing the capillary length [7] and also while decreasing the angle of the die en-
try [8]. By observing the velocity and the stress fields in the reservoir, many authors at-
tributed the phenomena to the contraction at the die entry where destabilization of the
elongational flow occurs [9], [10]. However, it should also be considered that this instabil-
ity may develop itself along the pipe [11].

4.1.3.2 Gross melt fracture

The gross melt fracture or chaotic defect gives to the extrudate an extremely disturbed ap-
pearance [1]. For all materials, it is observed at very high flow rates. This instability is quite
difficult to quantify but is always ascribable to too severe flow and die entry conditions.

**4.1.4 Conclusion**

The instabilities in the case of polymer melt extrusion can spoil either the surface or the
volume of the extrudate. In some cases, both types can even be superimposed. Most of
the mechanisms involved are understood but some of them are still contested and need
to be further investigated. Regarding industrial concerns, it is necessary to fully improve
this knowledge to be able to kill their defects and produce smooth extrudate in a wide
range of velocity.

In the following sections, we study the extrusion of alginate solutions (ALG4, ALG6 and ALG10) in the presence or not of binding ions. We report for both conditions the instabilities obtained. Based on the polymer melt extrusion knowledge, we strive to give possible mechanisms involved in our case.

**4.2 Extrusion in a bath of non-binding ions**

**4.2.1 Context and motivations**

Alginate extrusion in a bath without binding ions is the simplest way to study flow
stability. In fact, it avoids all parasitic phenomena which occur with divalent ions as gelation. Moreover, the presence of the solution allows to get rid of the surface tension and
especially the gravity, which adds stretching due to the weight of the extrudate.
We will present here the preliminary results that initiate the thorough and quantitative
investigation of such extrusion presented in 4.2.2.

At the beginning, we naively wanted to know if the instabilities observed for polymer
melt were also visible with polymeric solutions. We decided to play with the different
affordable parameters, one at a time: alginate concentration, velocity, die material or di-
mensions, and observe the flow at the die exit.

First, we decided to use the lowest concentration and extruded alginate solutions at
4% (ALG4) with glass capillary. At low throughput, the extrudate was smooth (Fig. 4.2,
a). While increasing the velocity, a very fine pattern appeared (Fig. 4.2, b) black ar-
row) compared to the smooth one. This reminded us of the sharkskin instability but the
amplitude was very low. We thus tested higher alginate concentration (ALG10) with the
same material (glass). The sharkskin clearly appeared with a higher contrast which sug-
gested a stronger amplitude. It seemed to disappear while moving away from the die exit.
<figure><img src="image_63.png" /><figcaption>Figure 4.2: Morphologies observed during the extrusion of: ALG4 in water with a glass capillary at low a) and increased velocity; c) ALG 10 in water with a glass capillary and d) ALG 10 in $MgCl_2$ 0.5M with metallic needle. b), c) and d) extrudates exhibit sharkskin instability. White scale bar = 200 µm</figcaption></figure>

NB: The extrusion bath was filled with deionized water or $MgCl_2$ 0.5M and thus no thread is formed. $Mg^{2+}$ ions were assumed to induce very weak binding with alginate chains and we verified that results in water or $MgCl_2$ were the same. Thus, the extrudate defects could only be observed in real time as it dissolved in such bath.

However, it was only due the alginate dissolution in water. In the rest of the study, we thus extruded alginate in a solution of $MgCl_2$ as $Mg^{2+}$ are non binding ions and alginate (strongly charged) is poorly soluble in such strongly saline solutions (I ≈ 0.5M). We verified that the same results were obtained for extrusion in NaCl solution of same ionic strength (I).

After changing the concentration, the question of the influence of the die material arose. We then extruded then ALG10 with stainless steel needle (Fig. 4.2, c)). Here again, the sharkskin defect was clearly visible and became quite chaotic for higher throughputs (Fig. 4.2, d)).

With those qualitative observations, we already had an idea of the impact of different parameters on the **sharkskin instability**: the amplitude seemed to be higher for ALG10 compared to ALG4 and the same observation was made for glass capillary compared to the metallic needle. The metallic needle inner surface may be rougher than the glass capillary one which can explain the higher skin facture in the first case. Indeed, as previously explained, in this case, the polymer chains adhesion is stronger and leads to a higher elongational stress at the die exit.
As this first instability was obtained, we decided to further explore the **velocity parameter**. We used exactly the same conditions as presented in Figure 4.2 but we increased the velocity. For all conditions (ALG4, ALg10) and even while changing the die material, the extrudate exhibited a wavy shape (Fig. 4.3). It looked quite similar to the **helical pattern** observed for polymer melts. However, here, simple observations were not sufficient to extract a trend. In fact, it was quite difficult to observe the "birth" of such instability. When it was clearly seen, the instability was already well established and the amplitude evolution was not fully assessed even after increasing the camera acquisition frame rate.

<figure><img src="image_64.png" /><figcaption>Figure 4.3: Morphologies observed during the extrusion of: a) ALG4 in water with a glass capillary; b) ALG 10 in water with a glass capillary and c) ALG 10 in $MgCl_2$ 0.5M with metallic needle, for higher velocities compared to the first set of observations 4.2. a), b) and c) extrudates have a wavy shape. White scale bar = 200 µm</figcaption></figure>

We therefore faced several questions: How can we track the beginning of this instability? Has this wavy instability an helical geometry? And finally, does this instability come from the die entry or develop itself along the pipe?

Fortunately, some key "particles", which were not at first expected to be so useful, helped us to answer several questions. Those "particles" were the air bubbles entrapped in our solutions. Some may consider them as parasitic particles, but in our case, they were essential tracers of this wavy defect. As shown in the figure 4.4, some **train of bubbles** could appear during the extrusion. While the wavy instability was observed, the train path was sinuous and periodic. More interesting, we observed that a smooth extrudate contained first a straight and then a sinuous path of air bubbles while increasing the velocity. This meant that we could track the birth of this instability by looking at the form of the tracers paths.

The profile of the extrudate could then be systematically followed and analyzed from the instability "birth" in order to extract its undulation amplitude as a function of the velocity. Glass capillaries allowed us to observe those tracer inside the pipe. Thus, we could observe how this instability evolved all along the pipe.

The question of the die entry effect still remained and contrary to previous polymer melt studies, where authors directly saw this effect by using a transparent slit die, it was not possible to observe this region in our case due to the set-up configuration. However,
<figure><img src="image_65.png" /><figcaption>Figure 4.4: Path of train of bubbles during the extrusion of ALG10 in MgCl₂ 0.5M while wavy shape is observed. White scale bar = 200 µm</figcaption></figure>

the different capillaries could insert into each other leading to a contraction or an expansion configuration. In other words, we produced a **second "die entry"** which is transparent and allows us to simulate the event taking place at the die entry. We can either put a stable flow or an unstable flow in the first capillary which stood as the die entry and observe what it became in the second one (Fig. 4.5)

<figure><img src="image_66.png" /><figcaption>Figure 4.5: Interlocking of two capillaries of radii 250 and 150 µm which act at a die entry configuration where contraction of the flow occurs. Train of bubbles allow to see the die entry effect. Wavy to wavy state is presented. Flow direction is given by the grey arrows.</figcaption></figure>

### 4.2.1.1 Conclusion

Without binding ions, two main instabilities observed for polymer melts are found here: the sharkskin and the helical defects. For the first, we clearly noticed the die material impact which leads to severe skin fractures in the case of metallic needles. Helical instability is more difficult to track, however, several solutions came to us and allowed us to quantify precisely its shape, amplitude and to understand the mechanisms behind this instability. The results of this thorough analysis are presented in the following article.

## 4.2.2 Results: Experimental evidence of helical, supercritical instability in pipe flow of shear thinning fluids

In this first study, we report the extrusion of strongly shear thinning alginate solutions through long glass capillaries (aspect ratios L=R up to 200), in a bath of non-binding ions solution. The extrudate shape and the tracer paths (e.g. Fig. 4.4) are observed together in order to analyze the flow stability.

From those experiments, we provide proofs of: (i) the existence of an instability associated with the helical distorsion of the extrudate, (ii) the supercritical nature of the bifurcation and (iii) concomitant emergence of undamped velocity oscillations extending along the whole length of the capillary. [12]
Experimental evidence of a helical, supercritical instability in pipe flow of shear thinning fluids.

L. Picaut, O. Ronsin, C. Caroli and T. Baumberger

Institut des NanoSciences de Paris, CNRS, Sorbonne Université - Pierre et Marie Curie,
UMR 7588, 4 place Jussieu, 75005 Paris, France

We study experimentally the flow stability of entangled polymer solutions extruded through glass capillaries. We show that the pipe flow becomes linearly unstable beyond a critical value ($W_{i,c} \approx 5$) of the Weissenberg number, via a supercritical bifurcation which results in a helical distorsion of the extrudate. We find that the amplitude of the undulation vanishes as the aspect ratio $L/R$ of the capillary tends to zero, and saturates for large $L/R$, indicating that the instability affects the whole pipe flow, rather than the contraction or exit regions. These results, when compared to previous theoretical and experimental works, lead us to argue that the nature of the instability is controlled by the level of shear-thinning of the fluids. In addition, we provide strong hints that the non-linear development of the instability is mitigated, in our system, by the gradual emergence of gross wall slip.

INTRODUCTION

Extrusion, which consists in forcing viscoelastic polymer melts or solutions out of a reservoir through an axisymmetric pipe, or “die”, is an important shaping process, unfortunately prone to a variety of instabilities which spoil the surface or the bulk of the extrudate [1–3]. Among these, an ubiquitous one, usually referred to as “melt fracture”, results in a distortion of the extrudate into a more or less pronounced and regular helical shape. Visualization of the velocity and stress fields in the reservoir [3–6] has lent strong support to the idea that melt fracture was ascribable to the well documented destabilization of the elongational flow upstream of the sudden contraction at the entry of the die [4, 7]. Due to viscous damping, such velocity oscillations are expected to decay over a finite length as the fluid is advected through the pipe. Hence, it is not surprising that helical instabilities were reported by many authors as being more severe when using short dies, rather than pipes with larger length/radius ($L/R$) aspect ratios [3]. However, as pointed out by Larson [1], one cannot rule out that the flow in a long pipe may itself develop a bulk helical instability, in which case the amplitude of the extrudate distorsion should not decrease upon increasing the die aspect ratio.

The stability of the flow in an infinitely long pipe, in the limit of vanishing Reynolds numbers, has been the subject of extensive theoretical studies, in the case of viscoelastic fluids obeying either the Oldroyd-B [8, 9] or the White-Metzner [10, 11] constitutive equations. In the first case, dealing with the peculiar class of non shear thinning fluids, it has been shown that the Poiseuille flow, although *linearly stable*, exhibits a *weakly non-linear* subcritical instability: beyond a critical value of the Weissenberg number $W_i$, the flow is unstable against a finite but small noise amplitude, therefore resulting in a hysteretic behavior [8, 9]. The existence of such an instabil-

ity has later been confirmed by the experimental studies of Bertola et al. [12] and Pan et al. [13] on axisymmetric pipe and 2D channel flows.

A wider class of great practical importance deals with polymeric fluids which exhibit both a large first normal stress difference and strong shear thinning in the same range of shear rates. It has been studied theoretically by Wilson and coll. [10, 11, 14] within the framework of the White-Metzner equation, under the assumption of a power law dependence of the shear-rate dependent viscosity $\eta(\dot{\gamma})$. They have shown that the base flow is *linearly unstable* provided that the shear thinning exponent $n = 1 - |d \log \eta/d \log \dot{\gamma}|$ is smaller than 0.3 [10, 11]. The non-linear behavior has not been investigated, however, so that the super- or subcritical nature of the instability remains unpredicted. Strong hints that a bulk flow instability actually occurs beyond a finite $W_i$ value in strongly shear-thinning fluids ($n \approx 0.2$) have been provided by Bodiguel et al. [15] who observed the emergence of sinuous tracer paths in microfluidic 2D channels. This emergence is associated with that of strong fluctuations of the velocity field, as confirmed by Poole [16] using larger scale pipe flows with $Re \approx 100$. However, due to the difficulty of quantifying the spatio-temporal behavior of the velocity field inside the die, these studies did not permit a full characterization of the nature of the bifurcation, in terms of amplitude evolution and possible presence of hysteresis.

Here, we report on a study of the extrusion through long glass capillaries (aspect ratios $L/R$ up to 200) of a strongly shear thinning ($n \approx 0.3$) polymer solution exhibiting a rheology well described by a White-Metzner equation. A detailed quantitative analysis of the extrudate shape, together with the observation of tracer paths in the glass capillaries, provide experimental proof of: (i) the existence of an instability associated with the helical distorsion of the extrudate, (ii) the supercritical nature of the bifurcation, which occurs beyond a critical Weissenberg $W_{i,c} \approx 4–6$, with a wavelength $\lambda_c \approx 3R$, (iii) the
concomitant emergence of undamped velocity oscillations extending along the whole length of the capillary,

On this basis, we provide what we think is the first evidence and characterization of a helical linear instability of the axisymmetric base flow in a long pipe, in the vanishing Reynolds number limit. We are therefore able to conclude that, at least for shear thinning viscoelastic fluids, upstream destabilization of the contraction flow is not the sole route to the helical extrudate instability which, though an ubiquitous phenomenon, has attracted much less attention in the theoretical litterature than the “sharkskin” or “spurt flow” ones [1, 2, 17].

## MATERIALS AND METHODS

### Polymer solutions: preparation and rheological characterization

Polymer solutions were prepared by dissolving sodium alginate (average molecular mass $M̄ = 246$ kDa, Kalys S.A., France), a biopolymer extracted from seaweed, in deionized water. 500 ppm sodium azide were added to prevent bacterial development. We have made use of solutions with alginate concentration $c = 4, 6$ or $10$ wt.%, referred to in the following as ALG4, ALG6 or ALG10. At the lowest concentration (ALG4), overnight magnetic stirring was sufficient to obtain a homogeneous solution. For higher concentrations, alginate powder was left to swell and dissolve for several days under periodic gentle manual stirring. Homogeneity of the solution was assessed from its optical aspect.

Rheological measurements were performed using a stress-controlled rheometer (MCR 501, Anton Paar) equipped with a 50 mm diameter, 2° cone-plate, sand-blasted cell geometry.

The concentration dependence of the zero-shear specific viscosity (see data in supplemental material) yields a concentration at the onset of the entanglement regime $c_E = 1$ wt.%, so that the three systems tested in this work are well entangled polymer solutions. Their viscoelastic properties were characterized by the measurement of both the steady flow viscosity $\eta(\dot{\gamma})$ and the first normal stress-difference $N_1(\dot{\gamma})$ for shear rates $\dot{\gamma}$ in the range $10^{-1}-10^2$ s$^{-1}$.

As seen on Fig.1, the three solutions exhibit a marked shear thinning behavior that we have characterized using a Cross law:

$$ \eta = \frac{\eta_0}{1 + (\dot{\gamma}/\dot{\gamma}_0)^{1-n}} \quad (1) $$

Fitting values of the plateau viscosity $\eta_0$ and crossover shear rate $\dot{\gamma}_0$ are given in Table I. We found that a single shear-thinning exponent $n$ was able to account for the behavior of the three solutions, namely $n = 0.29 \pm 0.02$, a value compatible with the “universal” one ($n \sim$

0.24) reported for a broad class of entangled, random coil polysaccharide solutions [18] to which alginate belongs.

<figure><img src="image_67.png" /><figcaption>FIG. 1: Shear-thinning behavior of the alginate solutions. The solid lines are a fit by a Cross law with a common shear-thinning exponent $n = 0.29$. The dashed lines mark the crossover shear rates $\dot{\gamma}_0$. Parameters of the fit are listed in Table I.</figcaption></figure>

<table><caption>TABLE I: Plateau viscosity $\eta_0$ and crossover shear rates $\dot{\gamma}_0$ values for the three alginate concentrations $c$ used in this study. The last two columns give the critical values of the average shear rate and the corresponding Weissenberg number estimated at the capillary wall at the onset of the helical instabiliy.</caption><thead><tr><th></th><th>c [wt.%)</th><th>η₀ [Pa.s]</th><th>γ₀ [s<sup>-1</sup>]</th><th>V/R|c[s<sup>-1</sup>]</th><th>Wᵢₐ</th></tr></thead><tbody><tr><td>ALG4</td><td>4</td><td>30</td><td>8</td><td>340-500</td><td>5-5.6</td></tr><tr><td>ALG6</td><td>6</td><td>110</td><td>3</td><td>60</td><td>4</td></tr><tr><td>ALG10</td><td>10</td><td>1900</td><td>0.3</td><td>8-14</td><td>4.3-5.1</td></tr></tbody></table>

Fig. 2 shows the shear rate dependence of the Weissenberg number $W_i$ defined as:

$$ W_i = \frac{N_1(\dot{\gamma})}{\eta(\dot{\gamma})\dot{\gamma}} \quad (2) $$

As commonly observed with shear-thinning viscoelastic fluids [15] $W_i(\dot{\gamma})$ exhibits a markedly sublinear growth indicative of an extended spectrum of viscoelastic relaxation times. This is reasonably well described for $\dot{\gamma} > \dot{\gamma}_0$ by a simple power law:

$$ W_i \approx (\dot{\gamma}/\dot{\gamma}_0)^m \quad (3) $$

with $m \sim 0.3$. From $m \sim n$, we conclude that entangled alginate solutions are well described by a White-Metzner model, yielding $\eta = G\tau$ and $N_1 = 2G(\dot{\gamma}\tau)^2$, with a constant elastic modulus $G$ and a shear-rate dependent relaxation time $\tau \sim \dot{\gamma}^n$ for $\dot{\gamma} > \dot{\gamma}_0$.
<figure><img src="image_68.png" /><figcaption>FIG. 2: Sublinear, shear rate dependent Weissenberg number for alginate solutions. The dashed lines indicate the values of $\dot{\gamma}_0$ and the full lines are $(\dot{\gamma}/\dot{\gamma}_0)^m$ with $m \simeq 0.3$.</figcaption></figure>

Extrusion set-up

<figure><img src="image_69.png" /><figcaption>FIG. 3: Schematic representation of the extrusion setup.</figcaption></figure>

Polymer solutions were extruded through cylindrical capillaries of inner radius $R$ and length $L$ using commercial 1 mL syringes. The body of the syringe was filled with the viscous polymer solution using a spatula. The volumetric throughput $Q$ was prescribed by driving

the piston velocity through a home-made syringe pump operated by a stepping motor. As usual when working with microfluidic circuits, their finite compressibility associated with their large hydraulic impedance results in long transients. These were monitored by measuring the force on the piston with an on-line transducer. Fortunately, thanks to shear thinning, the impedance decreases markedly as the throughput increases so that for all $Q$ values used in this study, the volume of fluid extruded during a transient remained negligibly small. In the following, we will characterize the steady state flow by the average velocity in the capillary $V = Q/(\pi R^2)$.

Except when specified, we have made use of cylindrical borosilicate glass capillaries with radii 100, 150 and 250 µm $\pm$10% (VitroCom, USA). Each capillary, used as purchased, was first diamond-cut to the prescribed length, then inserted into a shortened commercial steel needle, with an inner radius chosen to tightly fit the outer radius of the capillary. The assembly was finally epoxy-glued together, with the capillary end just emerging out of the steel needle, into the tappered plastic hub (see Fig. 3).

In order to minimize surface tension and gravity-induced drawing effects on the extrudate, extrusion was carried out into a bath of aqueous liquid. We have made use of a 0.5 M $MgCl_2$ saline solution into which alginate is poorly soluble so that, although $Mg^{2+}$ ions do not crosslink alginate molecules, the extrudate preserves its shape and sharp interface long enough for its distortions to be quantified.

The whole set-up (syringe, capillary and bath) was thermostated at $T = 20 \pm 0.5$°C.

The extrusion bath being contained in a glass cell, significant lengths (typically $\simeq 20R$) of the capillary and the extrudate on either side of the exit plane can be monitored under direct, quasi-parallel illumination, using a video camera equipped with a macro objective and operating at speeds up to 300 frame/s. Sequences where acquired for each value of the control parameter $V$ and stored for analysis.

Profile analysis

The undulation amplitude $\Delta R$ of the extrudate (see left panel Fig.4) is measured from the reconstruction of its profile, using the section visible at each time $t$ within the field of view. First of all, we select one edge and determine its position $y(t)$ at a distance $x = D$ from the exit plane of the capillary (see Fig.4 for the definition of $x$ and $y$). A value $D = 4R$ is chosen so as to be out of the zone of viscoelastic recovery responsible for the “die swelling” phenomenon. The edge position in the fixed framework linked to the capillary evolves with time due to the advection of the distorted extrudate along the capillary axis but also perpendicular to it due e.g. to the flagellar motion associated with the sinuous distor-
<figure><img src="image_70.png" /></figure>

FIG. 4: Left panel: Snapshots of the extrudate exiting from a glass capillary with internal radius $R = 250~\mu\text{m}$ (see dashed white lines) for increasing average velocities $V$. At a critical value $V_c$ sinusoidally, indicated by white arrows on (c) and (d), see text. Right panel: Characteristic paths of microbubbles tracked in the end portion of the capillary extending $\simeq 20R$ upstream of the exit plane. The velocity scale is common to both panels. We have expanded the vertical scale in order to magnify the bubble displacement perpendicular to the axis.

<figure><img src="image_71.png" /></figure>

FIG. 5: Power spectrum of a sinusoidal extrudate profile $R(x)$ for a velocity $V \simeq 1.2 \times V_c$ with $V_c$ the velocity at the onset of oscillations (see text).

<figure><img src="image_72.png" /></figure>

FIG. 6: Azimuthal isotropy of the rms amplitude of the extrudate undulation ($V = 2 \times V_c$) as measured by a fixed camera but for different angular positions $\theta$ of the whole extrusion setup ($R = 100~\mu\text{m}$) around its symmetry axis.

Evidence of a bulk flow instability inside the capillary

Oscillations of the extrudate necessarily result from an oscillating velocity field inside the die, at least in the close vicinity ($\simeq R$) of the exit plane. In order to check this,

we have tracked microbubbles advected along the glass capillary in the end region extending $\simeq 20R$ upstream of the exit plane. The right panel of Fig.4 shows representative paths selected at different average velocities below and above the threshold $V_c$. As expected, whereas for
<figure><img src="image_73.png" /><figcaption>FIG. 7: Evolution of the oscillation amplitude and spatial frequency upon first increasing, then decreasing the control parameter $V$ in the vicinity of the threshold $V_c$, for a capillary with $R = 150~\mu\text{m}$, showing no measurable hysteresis.</figcaption></figure>

<figure><img src="image_74.png" /><figcaption>FIG. 8: Evolution of the oscillation amplitudes and spatial frequencies $R = 100~\mu\text{m}$ (red full circles), $150~\mu\text{m}$ (black squares) and $250~\mu\text{m}$ (blue open circles). The dotted lines are guides for the eyes.</figcaption></figure>

<figure><img src="image_75.png" /><figcaption>FIG. 9: Amplitude $\Delta R_{max}$ at apparent saturation, wavelength $\lambda = V_c/f_0$ of the oscillations at the onset of the oscillations, and threshold velocity $V_c$ with respect to the capillary radius $R$.</figcaption></figure>

<figure><img src="image_76.png" /><figcaption>FIG. 10: Characteristics of the instability in the vicinity of its threshold, plotted in reduced form. Same symbols as on Fig. 8.</figcaption></figure>

$V < V_c$ the paths are straight, for $V > V_c$ their projection onto the observation plane are quasi-sinusoidal.

At this stage, we must assess whether the flow instability responsible for the extrudate one is an entry, exit or bulk one. For this purpose, we have monitored for $V > V_c$ bubble paths at different locations along the capillary, up to a distance $\approx 80R$ upstream of the exit. The observation of quasi-sinusoidal paths with comparable amplitudes and periods, whatever their location along $x$, illustrated on Fig. 11, indicates the presence of undamped oscillations of the velocity field, i.e. an instability which, in contradistinction to the sharkskin one [1], is *not* localized in the vicinity ($\approx R$) of the exit plane.
Moreover, note that the wavelengths of the bubble paths are ≈ 8R, i.e. significantly larger than the period (≈ 3R) of the extrudate distorsion. This was already pointed out by Wilson and Loridan [11] in their analysis of Bodiguel et al. data [15]. Such an apparent wavelength dilatation is akin to a Doppler effect. The analysis performed in supplemental material enables us to estimate that the magnitude of the phase velocity $v_φ$ of the oscillating perturbation is comparable with V.

<figure><img src="image_77.png" /><figcaption>FIG. 11: Paths of lagrangian tracers recorded at different locations along a $R = 250~\mu\text{m}$ capillary. The portions of the capillary corresponding to each path are shown on the upper drawing.</figcaption></figure>

Finally, the fact that the extrudate instability is observed for capillaries with aspects ratios $L/R$ as large as 200 strongly suggests that it cannot originate from a destabilization of the flow upstream of the abrupt contraction. As pointed out by Larson [1], some unstable pipe modes must be at work in order to pump energy from the base flow and compensate for the viscous losses associated with the velocity oscillations.

In order to make a more precise statement, we have constructed capillaries with a 5:3 contraction by glueing a $R_− = 150~\mu\text{m}$ capillary inside a $R_+ = 250~\mu\text{m}$ one and cutting the narrow portion at different lengths $L_{150}$ (see inset of Fig.12). The throughput range was chosen so that the flow in the wider capillary remained stable while enabling to cross the instability threshold — as revealed by the undulations of the extrudate — in the narrower one. Fig.12 shows the maximum amplitude $\Delta R_{max}$ of the undulations as a function of $L_{150}/R_−$. At large enough aspect ratios, i.e. $L_{150}/R_− \gtrsim 40$, $\Delta R_{max}$ becomes insensitive to the length. This definitely establishes that the helical deformation of the extrudate is the outer manifestation of an instability which affects the whole flow inside the capillary. The results presented so far were obtained for $L/R \simeq 200$, hence correspond to the instability of

an “infinite” pipe. For $L/R < 40$, the amplitude decreases markedly and, when extrapolated to the “orifice die” limit ($L/R \to 0$) the amplitude vanishes (Fig.12). As will be discussed in the next section, this is in strong contradistinction with the usually reported phenomenology for polymer melt extrusion where shorter dies, mostly used in practice, are also the most prone to the so-called “melt fracture” helical instability.

<figure><img src="image_78.png" /><figcaption>FIG. 12: Maximal amplitude of undulation of a extrudate exiting a $R_− = 150~\mu\text{m}$ capillary at a distance $L_{150}$ downstream a 5:3 contraction with $R_+ = 250~\mu\text{m}$ (see sketch on inset). The broken lines are guides for the eyes.</figcaption></figure>

# DISCUSSION

## Comparison with previous studies

*Influence of shear-thinning on the nature of the instability* — The results presented so far are in clear contradiction with those of Bertola et al. [12] and Pan et al. [13] who evidenced a pipe instability controlled by the Weissenberg number but exhibiting a large hysteresis — an obvious *subcritical* feature, while the instability is, in our case, *supercritical*. Since the two cases differ essentially by the steady flow characteristics of the fluids under study, namely their being shear-thinning or not, we are led to conclude that it is this rheological feature which primarily controls the nature of the pipe instability. This is fully supported by the theoretical prediction that planar channel flows of “weakly” shear-thinning White-Metzner fluids ($n > 0.3$) are linearly stable [10] while “strongly” shear-thinning ones ($n \le 0.3$) are linearly unstable. The theoretical limiting value of $n$ for axisymmetric pipe flows is unknown but our results indicate that, in this respect, entangled alginate solutions with $n \simeq 0.3$ belong to the “strongly shear thinning” class of fluids.

Beyond proving the existence of a *linear* pipe flow instability, our results reveal that: (i) it corresponds to a
direct Hopf bifurcation — a feature which would require
a *non-linear* analysis of the flow dynamics to be theoreti-
cally assessed — and (ii) the first unstable mode is *helical*
— hence a relevant theoretical study of the flow stability
should be *fully 3D* [17].

As far as symmetry breaking is concerned, our pipe
flow helical instability is strongly reminiscent of the *sin-
uous* one reported by Bodiguel et al. [15] for 2D micro-
channel flows. Although the high molecular weight, par-
tially hydrolyzed polyacrylamide (PAa) solutions used in
their study exhibited an exponent *n* ≃ 0.2, i.e. were
highly shear thinning, the results could not be directly
compared to the theoretical predictions of [10]. Indeed,
on the one hand, for technical reasons [14], the theoretical
flow stability was restricted to “varicose” axisymmmmet-
ric modes. On the other hand, PAa does not obey the
White-Metzner constitutive equation with a single expo-
nent ruling power laws (1) and (3). In [15], *n* = 0.21 and
*m* = 0.43 indicates that the elastic modulus itself is rate
dependent. The consequence of this extra complication
has been studied by Wilson & Loridan [11] using an ex-
tended constitutive law. Interestingly, albeit they have
restrained their analysis to varicose modes, they have
shown that the flow stability, i.e. the critical value of the
Weissenberg number, is noticeably increased by the rate
dependence of the modulus. This is an important exam-
ple showing that rheological details may have quantita-
tive consequences on a flow instability while preserving
its qualitative characteristics. This would be worth be-
ing confirmed experimentally by swapping the two fluid
systems — ALG and PAa — in each of the two flow
configurations of [15] (2D channel flow) and the present
study (axisymmetric pipe flow).

*Competition between pipe and contraction flow insta-
bilities* — In the field of extrusion instabilities, a touch-
stone is the broad phenomenology pertaining to the so-
called “melt fracture” of polymer melts [3, 5, 20]. Here we
focus on the extensive study by Combeaud et al. [6, 20]
of the helical extrusion instability of linear polystyrene
(PS). Their polymer melts and our entangled polymer so-
lutions exhibit a common rheological characteristic since
the flow viscosity of PS is strongly shear thinning [20]
with a power law exponent *n* ≃ 0.2, a plateau viscosity
*η*₀ ≃ 5 × 10⁴ Pa·s and a cross-over shear rate *γ*₀ ≃ 1 s⁻¹.
It appears therefore meaningful to compare the results
obtained on PS and ALG, taking advantage that in both
studies, the extrudate together with the upstream (pipe
or contraction) flow have been characterized.

At first sight, the two systems behave similarly since
in both cases the extrudate exhibits at large enough
throughput a transition from a cylindrical to a fairly
regular helicoidal shape. The authors report that for
PS this instability occurs at shear rates far beyond the
range accessible in a cone-plate rheometer. Therefore,
the critical value $W_i^c$ of the Weissenberg could only be
roughly estimated by extrapolation of their rheological

data to range between 2 and 10. Since our own estimate
is $W_i^c = 5 \pm 1$, the comparison is not conclusive, even
though both ranges are compatible.

Much more specific is the behavior of the pitch $\lambda$ of
the distorted extrudate upon increasing the shear rate.
In this respect, both systems show antagonistic trends:
while for ALG it is the *spatial* frequency $\lambda^{-1}$ which re-
mains constant (Fig.8), for PS it is the *temporal* one $V/\lambda$
which exhibits but small variations. This strongly sug-
gests that both systems develop instabilities of different
origins. This is confirmed in [6] by the direct visualization
of the velocity field in the reservoir, upstream of the inlet,
which clearly indicates that emergence of the extrudate
instability coincides with the destabilization of the con-
traction flow, a phenomenon which has been extensively
studied by McKinley et al. for non shear-thinning Boger
fluids [7]. Correlatively, as the die aspect ratio is reduced,
the extrudate undulation becomes more pronounced for
PS, whereas it vanishes for ALG (Fig.12).

Although for practical reasons extrusion of polymer
melts is usually performed through short dies, Combeaud
et al have extended their study to aspect ratios up to
$L/R = 64$. They show that the helicoidal undulations
persist up to this aspect ratio value, although strongly
reduced (see Fig. 3.7 of [20]). The gradual attenuation
of the helix amplitude however strongly slow down for
$L/R \gtrsim 32$ (see Fig. 3.8.b. of [20]). Interestingly, this
value, which could be considered a priori quite large for
an “infinite die” regime to be reached, is compatible with
our own finding that the “infinite pipe” limit is reached
for $L/R \simeq 40$ (12). From this comparison we conclude
that the main body of experiments on PS (resp. ALG)
are concerned with “short” (resp. “long”) dies in which
reservoir (resp. pipe) flow instabilities are favored.

In light of the present results, it should be extremely
useful to focus experimentally on a contraction flow in-
stability of ALG and a possible pipe flow instability of
PS, although the high viscosity of the melts probably pre-
cludes using the long pipes required for this instability to
fully develop.

Non-linear instability development vs. wall slip

The standard extrusion phenomenology of polymer
melts involves the gradual transition from a quasi-
periodic helical disturbance to a more or less chaotic one
as the throughput is increased beyond a critical thresh-
old. With this in mind we have systematically explored
the evolution of the extrudate undulations beyond $W_i^c$.
No widening of the oscillation spectrum (see Fig.5) was
observed up to the experimental limit of $V \simeq 10 \times V_c$
(however only a two-fold increase in the slowly varying
Wi). Surprisingly, although one could expect a slow-
ing down of the amplitude growth due to the build-up of
stabilizing non-linear terms in the flow dynamics, we sys-
<figure><img src="image_79.png" /><figcaption>FIG. 13: Evolution with respect to V of (a) the swelling ratio Q, (b) the sharkskin contrast index (see text) and (c) the scaled oscillation amplitude. Different radii are represented by the same symbols as in Fig.8.</figcaption></figure>

tematically observe a marked decrease of the amplitude, as shown on Fig.13.c. This means that the plateauing trend observed close above $V_c$ (Fig.8) signals, in fact, the existence of a *maximum* of the amplitude. Such a behavior is, to our knowledge, quite unusual in non-linear dynamics.

Since the extrudate undulations involve some motion of the surrounding bath fluid, we have checked whether viscous damping could be involved. We have changed the saline solution (viscosity $\eta_s \simeq 10^{-3}$ Pa.s) for a 1 wt.% solution of sodium alginate (viscosity $\eta_{1\%} \simeq 10^{-1}$ Pa.s). No significant difference in the $\Delta R(V)$ curve was observed,

neither close to onset nor well beyond threshold, suggesting that the viscosity of the bath is not relevant to the undulation development.

Some further insight into the underlying physical mechanisms responsible for this behavior can be gained from the simultaneous measurement of two characteristics of the extrudate, namely the (die-)swelling ratio $Q$ and the sharkskin amplitude. The swelling ratio in the presence of oscillations is determined by measuring the average axial velocity of the extrudate $V_{out}$ well downstream the capillary exit. It is defined [26] as $Q = (V/V_{out})^{1/2}$. The sharkskin amplitude is estimated from the contrast of the resulting pattern, determined under constant illumination and video acquisition parameters.

Fig.13.a reveals an unusual behavior of the swelling ratio which is expected to grow as some power law of the shear rate with a small exponent [21]. Here, $Q$ first increases with the throughput but hardly builds up to values larger that 1.3 and then decreases to a plateau value $\lesssim 1.2$ (close to the value 1.1 for a purely newtonian fluid). The same trend is exhibited by the sharkskin contrast (see Fig.13.b and Fig.4): although always superimposed to the volume oscillations, the sharkskin is less and less severe as the throughput is increased beyond threshold.

Both die swell and sharkskin are qualitative indicators of the validity of the no-slip condition at the capillary wall. Die swell is a manifestation of the sudden relaxation of the first normal stress difference which itself requires viscous shear to build up. It is therefore mitigated by wall slip. Sharkskin results from the stress singularity which develops at the exit rim where boundary conditions abruptly switch from no-slip to no-stress [22]. It is attributed to the accommodation of the singularity via tearing of the viscoelastic material [23]. Wall slip clearly smoothes out the singularity hence limits the development of sharkskin. This suggests that the strong attenuation of $\Delta R$ is due to the concomitant build up of wall slip.

Furthermore, the data of Fig.13.a show that, quite unexpectedly, the relevant variable when the capillary radius is changed is not the shear rate but rather the average velocity $V$. This can be understood as a hint for the occurrence of gross slip with a velocity $V_s \sim V$. The no slip condition is then replaced by a tribological relationship $\sigma_w = f(V_s)$ which depends on “microscopic” characteristics of the fluid-wall interface, and not on the “macroscopic” radius. So, the residual shear rate at the wall $\dot{\gamma}_w(V_s) \sim \sigma_w^{1/n}$, must be determined by $V_s$, hence by $V$ rather than $V/R$. Accordingly, we have plotted the relative oscillation amplitudes for different radii as a function of $V$ (Fig.13.c). It is clear that, although in the growing part of the amplitude development this variable is not the relevant one, in the decreasing part it leads to a reasonable collapse of the data.
These results, though preliminary, indicate that the non-linear development of the instability is probably mitigated by the gradual build-up of wall slip. We cannot conclude at this point whether the fact that both phenomena occur at neighbouring values of the wall shear rate is a mere coincidence or signals some correlation. Interestingly, non-trivial coupling between flow instability and wall slip can be found in the previously discussed study of Bodiguel et al. [15] (their Fig.2) although the authors have rather focussed on the drag reduction associated with the occurrence of elastic turbulence in their microfluidic channel flow.

## CONCLUSION

We have studied in detail the helical instability which occurs when extruding an entangled polymer solution through glass capillaries with large aspect ratios. Coupling the quantitative analysis of the extrudate undulations with the visualization of tracer paths inside the capillaries we were able to demonstrate that this extrusion instability is the downstream manifestation of the destabilization of the base flow occurring everywhere inside the pipe. The instability exhibits the characteristics of a supercritical Hopf bifurcation occurring above a critical Weissenberg number $Wi_c = 5 \pm 1$.

As suggested by the seminal theoretical work of Wilson & Rallison [10], a key rheological characteristic of our fluid system, as regards its flow stability, is certainly its rather strong shear-thinning behavior. Recent studies have also brought out the essential role played by shear thinning in the stability of viscoelastic flows at zero [24] or moderate [25] Reynolds numbers. In serpentine micro-channels, shear thinning is found to stabilize the flow against the onset of elastic turbulence [24]. In large pipes, it yields a new type of symmetry-breaking supercritical transition [25]. In our case, as well as probably in microchannel planar flow [15], “strong” shear-thinning is responsible for the linear instability of the base flow, whereas Boger fluids designed to exhibit negligible shear-thinning are only non-linearly unstable [12, 13].

The present work therefore suggests to make use of fluid systems with increasing shear-thinning exponents $n$ in order to study experimentally the evolution of the pipe flow instability, from weakly non-linear subcritical at small $n$ to linear supercritical at larger ones.

An intriguing issue that the present work leaves open is the origin of the large aspect ratio ($L/R \simeq 40$) required for an “infinite” pipe regime to be reached. To our knowledge, it is the first time that this phenomenon is put forward and we hope that it will stimulate further theoretical and numerical studies.

Finally, preliminary results show that, at least for glass capillaries, the non-linear development of the helical undulations is mitigated by the gradual build-up of wall slip.

In the case studied here, the helical instability and the onset of gross slip occur in a narrow bracket of control parameter, leaving open the possibility of some causal relationship. Our set-up makes it possible to change the die material from glass to metal or even PTFE, and will therefore be suitable for future investigation of the role of wall slip in pipe flow instabilities.

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[26] The amplitude of the extrudate undulation remains smaller than 0.1 so that the volumetric flow of the extrudate can be approximated to $V \times (\pi R_{out}^2)$ with $R_{out}$ the radius of the thread, to within a few percent.
Supplemental material

CONCENTRATION DOMAINS OF ALGINATE
SOLUTIONS

The viscosity of sodium alginate solutions in pure wa-
ter was measured, at temperature T = 20°C, using a
double gap Couette cell geometry . Figure 1 shows the
specific viscosity η<sub>sp</sub>(c) = (η(c) − η<sub>s</sub>)/η<sub>s</sub> as a function of
the concentration c, where η is the viscosity of the solu-
tion in the vanishing shear rate Newtonian regime and
η<sub>s</sub> is the solvent (water) viscosity.

The three usual regimes of polymer solutions are ob-
served [1]:

* dilute regime where η<sub>sp</sub> ~ c at low concentrations
c < c*

* semi-dilute unentangled regime at intermediate concentrations c* < c < c<sub>E</sub>

* entangled regime where η<sub>sp</sub> ~ c<sup>3.5</sup> at high concentrations c > c<sub>E</sub>.

From the above results, we conclude that the c = 4–10
wt.% solutions used in the present work are in a rather
strongly entangled state.

<figure><img src="image_80.png" /><figcaption>FIG. 1: Specific viscosity of alginate solutions vs concentra-
tion, measured at T = 20°C. The boundaries between the
three scaling regimes define the overlap c* = 0.35 wt.% and
entanglement cE = 1 wt.% concentrations. The grey zone in-
dicates the concentration domain relevant to this study, cor-
responding to strongly entangled solutions.</figcaption></figure>

CALCULATED BASE FLOW VELOCITY
PROFILE AND WALL SHEAR RATE

As a consequence of the strong shear-thinning rheol-
ogy, well described by a Cross law:

$$
\eta = \frac{\eta_0}{1 + (\dot{\gamma}/\dot{\gamma}_0)^{1-n}} \quad (1)
$$

the calculated velocity profile in an infinite capillary with
no-slip condition at the wall differs markedly from the
Poiseuille solution for a newtonian fluid.

Whereas for V/R ≪ ˙γ₀ the profile is indistinguishable
from the Poiseuille one, when V/R ≫ ˙γ₀, it becomes that
of a power law fluid (for which σ = η₀ ˙γ₀(˙γ/ ˙γ₀)ⁿ). This
profile [2] is very flat over a large part of the capillary,
therefore exhibiting a central quasi-plug flow with shear
localized close to the wall (Fig.2).

<figure><img src="image_81.png" /><figcaption>FIG. 2: Velocity profiles for the flow of a fluid obeying a Cross law with n = 0.3. Lower curve: the Poiseuille solution for a newtonian fluid corresponding to the limit V/(Rγ₀) ≪ 1. Upper curve: solution for a power-law fluid, corresponding to V/(Rγ₀) ≫ 1. Intermediate curves (bottom to top) are for: V/(Rγ₀) = 2, 17, 160, the latter being almost indistinguishable from the power law one. Note the plug-like nature of the power-law profile in a wide core zone (0 ≤ r/R ≲ 0.5).</figcaption></figure>

<figure><img src="image_82.png" /><figcaption>FIG. 3: Evolution of the parameter α = ˙γ<sub>w</sub>/˙γ<sub>0</sub> with the re-
duced shear rate. The dashed line is the pure power law limit
α = 6.3. The grey area corresponds to the experimental range
of shear rates.</figcaption></figure>
It is customary to estimate the Weissenberg number $Wi(\dot{\gamma})$ at the wall, where the shear rate $\dot{\gamma} = \dot{\gamma}_w$ is maximum. We express it as:

$$ \dot{\gamma}_w = \alpha V/R $$

with $\alpha$ a number depending on the velocity profile. In the low and high shear rate limits:

$$ \alpha = \begin{cases} 4 & \text{Poiseuille } (n = 1), \\ 3 + \frac{1}{n} & \text{Power law} \end{cases} $$

We have computed the dependence of $\alpha$ on $V/R$ by straightforward numerical integration of the flow equation for the Cross law (1) under a no-slip condition at the wall. This was used to estimate the wall shear rate $\dot{\gamma}_w$ and the corresponding $Wi$ for any value of the throughput. As seen on Fig.3, in the range $10 < V/(R\dot{\gamma}_0) < 10^3$ s$^{-1}$ relevant to this study, $\alpha \simeq 6$. We have used this value to compute the expression of the Weissenberg number for our experiments:

$$ Wi \approx 1.7 \left( \frac{V}{R\dot{\gamma}_0} \right)^{0.3} $$

## ANALYSIS OF LAGRANGIAN TRACER PATHS

When a passive tracer moves along with an axisymmetric base flow $v_x(r)\vec{e}_x$ perturbed by a propagating harmonic mode of pulsation *omega* and wavelength $\lambda_v$, it experiences an oscillating velocity field which is itself moving with respect to the capillary frame at the phase velocity $v_\varphi$ of the mode. Accordingly, the wavelength $\Lambda$ of the tracer's path is different from $\lambda_v$. The effect is analogous to the classical Doppler effect and, accordingly, the following relationship holds [3] between the period $T$ of the tracer oscillations and that, $2\pi/\omega$, of the velocity field:

$$ T(r) = \frac{2\pi/\omega}{|1 - v_x(r)/v_\varphi|} $$

Here, it is assumed that the radial velocity perturbation is small with respect to $v_x$, so that the tracers remain at

a quasi-constant distance $r$ from the capillary axis (as is actually the case for the traces of Figs. 4 and 11 of the main article).

Since $\Lambda = v_x T$ and $2\pi/\omega = \lambda_v/v_\varphi$, one gets straightforwardly:

$$ \Lambda = \frac{\lambda_v}{|1 - v_\varphi/v_x(r)|} $$

We want to compare $\Lambda$ with the measured wavelength $\lambda$ of the extrudate undulations. Extrudate incompressibility yields: $\lambda R_{out}^2 = (2\pi/\omega)VR^2$ where we have written that the pulsation $\omega$ of the velocity perturbation at the capillary exit is also that of the undulation. In terms of the die-swelling ratio $Q = R_{out}/R$, one gets:

$$ \frac{\lambda}{\lambda_v} = \frac{V}{v_\varphi} \frac{1}{Q^2} $$

And finally:

$$ \frac{\Lambda}{\lambda} = \frac{v_x}{V} \frac{Q^2}{|v_x(r)/v_\varphi - 1|} $$

Knowing the imposed average flow velocity $V$ and measuring the average axial velocity along the path $v_x$, we have estimated the radius $r$ of the path using the calculated velocity profile of Fig.2. All the traces shown on Figs. 4 and 11 of the main article correspond to $r/R \simeq 0.8-0.9$, meaning that the corresponding tracers travelled in the near-wall zone where they could experience significant shear rate.

With typical values, $Q \simeq 1.3$, $v_x/V \simeq 0.7$, $\Lambda/\lambda \simeq 2.7$, one gets $|v_x/v_\varphi - 1| \simeq 0.5$, indicating that the phase velocity of the mode probed by the tracers is on the order of their average velocity, i.e. of $V$ itself.

[1] L. Guo, R. H. Colby, C. P. Lusignan, and A. M. Howe, *Macromol. 36*, 10009 (2003).

[2] R. Byron Bird, R.C. Armstrong and O. Hassager, *Dynamics of Polymeric Liquids*, Vol.1 *Fluid Mechanics*, 2nd ed. John Wiley & Sons, New-York 1987.

[3] H.J. Wilson & V. Loridan, *J. Non-Newton. Fluid Mech.* **223**, 200 (2015).
### 4.2.3 Implementation on other systems: In the "crimp" quest

#### 4.2.3.1 Quick literature overview

Collagen fibrils within the tendon fascicles sheaths exhibit alternating oblique orientations referring to as "crimp", as presented in Chapter 1 (1.1.1.2.1 p 9). This tendon wavy shape was suggested to act as spring or a shock-absorber, which may improve tendon mechanical properties [13]. Thus, several emerging strategies were developed in order to confer wavy shape to scaffolds and in particular to threads. We briefly report three main techniques used for this purpose.

### Microtransfer molding

<figure><img src="image_83.png" /></figure>

### Sacrificial process

<figure><img src="image_84.png" /></figure>

### Electrohydrodynamic jet printing

<figure><img src="image_85.png" /><figcaption>Figure 4.6: Example of protocols used to produce wavy scaffolds. Top: Microcrimping method and scanning electron microscopy of crimped collagen microfibers [14]. Scale bar = 200 µm, Middle: Sacrificial process and a representative SEM image of fabricated scaffolds containing two fiber populations. [15] Scale bar = 1 µm, Bottom: Electrohydrodynamic jet printing set-up and example of moving path and resulting fiber. [16] Scale bar = 200 µm</figcaption></figure>

A first method to produce wavy microstructures is the microtransfer molding, used by Caves et al. [14]. The system is composed of a lead-screw assembly, a base membrane on which synthetic collagen fibers are aligned parallel and a microridged membrane. After several steps of extension, compression, freezing, drying, chemical treatment, they obtained wavy sheet of collagen fibers with a wavelength of about 100 µm.

A second method consists in producing a construct made of two polymer thread (elec-
trospinning) populations from which one will be removed or "sacrificed". Szczesny et al. used this method with PLLA and PEO polymers. As shown in the figure (4.6), PEO population threads were removed by wash step [16]. The "crimped" shape is given to the remaining population by heating them.

Recently, Wu et al. used an electrohydrodynamic jet printing device to print wavy polymer (PCL) fibers (Fig. 4.6) [15]. The resulting fibers have a diameter of about 100µm, a pattern wavelength of 360µm and crimp angle about 45°.

In both studies, authors showed that the mechanical properties of the scaffolds were improved by using wavy fibers compared to their respective controls. However, those techniques used synthetic polymer solutions and may be not suitable for collagen ones. Indeed, electrospinning could denature the collagen and the post-processing such as chemical cross-linking may impact the biocompatibility of the constructs. Moreover, these methods are quite sophisticated and the whole procedure may take time. Finally, the resulting scaffolds exhibit more a "2D" wavy pattern than a spring-like shape as for native tendon crimp.

**Our strategy** consists in transposing the helical instability observed during the extrusion of alginate solutions (without binding ions) to the collagen one and thus produced "crimped" threads.

### 4.2.3.2 Helical instability: from alginate to collagen

<figure><img src="image_86.png" /><figcaption>Figure 4.7: Collagen at 30 mg/mL extrusion in A) PBS 1X and B) PBS 5X observed in real time and post extrusion by polarized light microscopy with (top corner) and without quarter wave-plate (bottom corner). Eye guidelines in blue dashed lines helps the helical pattern visualization. Scale bars = 200 µm</figcaption></figure>

The alginate solution at 10 wt % exhibited rheological properties close to the collagen solution at 30 mg/mL (Chapter 3, 3.1.2.3 p 93). From the study of alginate extrusion with
glass capillaries in the case of non-binding ions, we have found a critical velocity above
which the helical instability starts appearing.

Thus, we decided to test the extrusion in almost the same conditions with collagen solu-
tion at 30 mg/mL. The idea was to produce collagen threads with inner structure present-
ing an helical geometry. The final goal would be to mimic the "crimp" and so to improve
collagen threads properties (Chapter1, 1.1.1.2.1 p 9).

First, to induce slow fibrillogenesis of collagen molecules, phosphate buffer saline PBS (1X) was put in the extrusion bath. The helical pattern revealed quite rapidly over a large range of velocity. At higher ionic strength, with PBS 5X, this instability was also observed. Even if the helix was less pronounced than the one observed with alginate solutions, the whole inner structure followed the helix geometry as shown by polarized light microscopy.

**Conclusion:**

By a simple extrusion process, we induce the helical alignment of collagen inside the threads. It should be noted here that this analysis of instabilities took place in parallel to the biological tests and its application to collagen came very recently. Therefore those helical threads were not checked for their mechanical properties neither for the cell culture at the time of the writing up of my thesis. Nevertheless, those results are very promising in the "crimp" quest and may improve the quality of artificial tendon-like collagen threads.
4.3 Extrusion in a bath of cross-linking ions

Without cross-linking ions, sharkskin and helical instabilities were observed down-
stream. While we have shown that the latter affects the whole flow along the capillary,
the former was known to be attributable to the stress and strain rate singularity generated
in the exit zone due to the no-slip to no-stress boundary condition abrupt transition [12].
In the presence of binding ions in the exit zone, it is legitimate to think that sharkskin
could be modified by the ions while the helical instability would remain unaffected. In
addition new instabilities could appear as well. In this section, we will focus on the in-
stabilities "zoology" obtained for alginate solution ALG4 extrusion with glass capillary in
a cross-linking ions (calcium) bath while varying the velocity. The different effects due to
the calcium ions or alginate concentrations and needles materials will be discussed for
each instability.

**4.3.1 Overall observations**

The results obtained for ALG4 extruded with glass capillary in chloride calcium 0.5M
bath are gathered in Figure 4.8. During the extrusion of such solutions, several instabili-
ties are observed:

<figure><img src="image_87.png" /><figcaption>Figure 4.8: Morphologies observed during the extrusion of ALG4 in CaCl₂0.5M with a glass capil-
lary (R=150µm) for exit velocities ranging from 0.2 to more than 132 mm/s : A)Smooth, B) Wavy,
C) Alternating, D) Striated, E) Smooth, F) Narrowed. White scale bar = 200 µm</figcaption></figure>

Upon increasing the velocity from rest up to 0.2 mm/s, the alginate extrudate is smooth
(Fig.4.8 A) and then different patterns appear one after the other (Fig.4.8 B, C, D). First,
the alginate flow has a wavy shape (B), then, tightened striations (D) appear and those
two may alternate between smooth and wavy/striated pattern (C). For higher velocities
(around ten times more), the alginate flow seems to be smooth again (Fig.4.8 E) but with
an external diameter higher than in the first smooth region. Finally, for the highest attain-
able velocities, we observe a flow narrowing at the die exit and an opaque region (Fig.4.8
F)red arrow) inside the capillary is growing. Those phenomena are observed for the dif-
ferent capillary radii. It is quite surprising that the sharkskin defect does not appear as
previously observed in deionized water. This means that the presence of binding ions
avoids the formation of sharkskin.

In the following, each instability is described in more details and discussed.

NB: For the highest velocities, camera acquisition frame rate did not allow the observation of the flow surface in real-time. In such cases, the alginate extrudate was quickly removed from the bath and observed by optical microscopy "post-extrusion".

**4.3.2 Smooth region**

As previously mentioned, at very low velocities, the extrudate surface was smooth (Fig. 4.8 A). Moreover, the flow diameter (D') seemed to be smaller than the die exit. To quantify this observation, D' was measured from the real-time motion pictures for different velocities. The swelling ratio D'/D was plotted as a function of the shear rate γ̇ for the available stainless steel needles of inner diameters D (Fig. 4.9):

<figure><img src="image_88.png" /><figcaption>Figure 4.9: Ratio diameters D'/D plotted as a function of the shear rate for the extrusion of ALG4 in CaCl₂0.5M with stainless steel needles. D' is the measured extrudate diameter and D the inner needle diameter. In the region of interest (shaded area), the ratio is below 1 which shows that the extrudate has a smaller diameter than the needle one.</figcaption></figure>

For shear rates ranging between 0.1 to 10 s⁻¹, we had: D'/D < 1 (the region of interest is shaded). This meant that the extrudate diameter was significantly smaller than the imposed inner diameter. This shrinkage could even reach 40% of the initial diameter for the lowest shear rates.

NB: The same measurements were performed with glass capillaries with the same order of magnitude.

How can we explain thus this remarkable shrinkage? As the shrinkage was not observed
in the previous study while there was no cross-linking ions, this phenomenon can only be
due to the presence of calcium ions.
Indeed, when the alginate extrudate came in contact with the calcium ions, a gel was then formed through the "egg box" mechanism previously described (2.2.2.2 p 81). The gel was formed from the outside and the gelation continued progressively to reach the extrudate core. The gelation front was visible: fine area where the contrast changes (Fig. 4.10 left). Further the die exit, the extruded thread was thus completely gelled.

However, this gelation is more complex. In fact, several mechanisms occur simultaneously: ion exchanges ($Na^{2+} \leftrightarrow Ca^{2+}$) which allow alginate/calcium binding, calcium ions diffusion through the new formed network and sodium alginate solution emergence from the die exit.

A first approach to estimate the dominant phenomena consists in analyzing dimensionless number. The two forms of species transport in a flow, that is convective and diffusive, are generally compared by the Peclet number $Pe$, defined as:

$$Pe = \frac{t_{diff}}{t_{conv}} = \frac{\bar{v} R}{D}$$

It relates to the fact that the typical convection time over a distance $R$ is defined by $t_{conv} = R/\bar{v}$ whereas the diffusion time over this same distance is expressed as $t_{diff} = R^2/D$, where $D$ is the diffusion coefficient of the species considered in the respective solvent. Alginate is a macromolecule and has a lower diffusion coefficient than calcium ions, implying that alginate should not diffuse significantly before reacting with the ions. The considered diffusive species are thus the calcium ions.

In the direction of the flow at a velocity $\bar{v} = 0.2$ mm/s, with $D \sim 10^{-9} m^2/s$, and a capillary radius $L$ of 150 $\mu m$, we obtain $Pe_{||} \sim 30$. Since $30 \gg 1$, it means that the diffusion is negligible and convection is dominant. However, in the radial direction, since the flow is longitudinal, the Peclet number $Pe_{\perp}$ is equal to zero as there is "no velocity". In such case, the diffusion of calcium ions is the dominant phenomenon.

Little reaction takes place in the gel, since the concentration of free alginate is very low. As the boundary between sodium alginate (liquid) and calcium alginate (gel) is continuously moving, the gel layer is growing and the liquid core decreases. We can estimate the diffusion length $L_{diff}$ over which the gelation front is converging (Fig. 4.10). This length is related to the Peclet number by:

$$L_{diff} = \bar{v} \times t_{diff} = Pe \times R$$

As $Pe \approx 30$, the diffusion length from this estimation is equal to: $L_{diff} \approx 30R$. While measuring this length on the real time extrusion figure 4.10 (left), we find: $L_{diff} \approx 8R$. This means that the effective diffusion of calcium ions is faster than the predicted one.

At the die exit (Fig. 4.10, insert), if we suppose that the diffusion and convection phenomena are comparable ($Pe \sim 1$), the physical quantity $\delta$ called the "skin depth" which corresponds to the gelled thickness at a distance $h$ from the die exit can be estimated as:

$$Pe = 1 = \frac{\bar{v} \delta}{D} \\ \delta = \frac{D}{\bar{v}} \approx 20 \mu m$$
<figure><img src="image_89.png" /><figcaption>Figure 4.10: Steady and homogeneous alginate flow (left) and the scheme representation of the syneresis process occuring (right): calcium ions diffusion and water expulsion. The insert focuses on the nearly die exit zone. Black scale bar = 200 µm</figcaption></figure>

While measuring this length on Figure 4.10 (left), we obtain: $\delta \approx 25 \pm 5 \mu m$ which is in good agreement with this estimate.

In parallel of calcium ions diffusion, a last phenomenon called syneresis occurs [17]. This is characterized by the contraction of alginate chains network while local divalent ions concentration is increasing. During the syneresis, the water of the gel is exuded, which has the effect of decreasing the dimensions of the extrudate and increasing locally the polymer concentration. This reorganization of the network increases the stability of the hydrogel and reduces its porosity which may impact the calcium ions diffusion speed. Moreover, the composition of alginate plays here an important role in this mechanism as alginate solutions with long G-type sequences contract less than the alginates which possess shorter ones [17].

### 4.3.3 Die swelling

From the first observations, at higher velocities, the extrudate seems to be smooth (Fig.4.11, right) with a larger diameter than the die and compared to the smooth region while syneresis occurs. If we go back to the diameter ratio D'/D plot, we observe that this ratio is greater than 1 for shear rates ranging from almost 10 to 1000 $s^{-1}$ (Fig. 4.11, shaded area). It means that the extrudate diameter is higher than the inner needle diameter which is thus called the die swelling. The swelling increases with the velocity and exceeds 1.5 times the internal diameter of the needle.

This phenomenon is due to the strong viscoelastic properties of the alginate solutions.
<figure><img src="image_90.png" /></figure>

Figure 4.11: Left: Die swelling image of ALG4 extrusion in CaCl₂0.5M with metallic needle D=780µm (Scale bar = 200µm. Right: Ratio diameters D'/D plot as a function of the shear rate. In the region of interest (shaded area), the ratio is larger than 1 which shows that the extrudate has a higher diameter than the needle one.

Such fluids display also other spectacular demonstrations as the Weissenberg effect [18]:

<figure><img src="image_91.png" /></figure>

Figure 4.12: The Weissenberg or rod-climbing effect imaging [18]. The fluid is high molecular weight PIB in low molecular weight polyisobutylene. It can be compared to newtonian fluids (ex: water) behavior, where the swirling effect causes a depression vortex rather than a climbing effect as observed here.

NB: In the case of newtonian fluids, the extrudate has the same diameter as the die and the rotation of the rod creates a vortex and not a fluid climbing.

Those two effects are directly related to the normal stresses developed when such systems are submitted to a shear flow [19]. In fact, during extrusion process, the solution composed of long, entangled and coiled polymer chains, is forced to pass through a narrow die channel which elongates those chains in the flow direction and compresses them in the normal direction (Fig. 4.13). At the die exit, the release of the compressive forces and the "memory effect" of such viscoelastic fluid make the polymer chains go back to
<figure><img src="image_92.png" /></figure>

Figure 4.13: Scheme of die swelling mechanism: in the narrow channel, the polymer chains are elongated in the flow direction and compressed in the normal direction. At the die exit, the polymer chains return to the "coiled" configuration which leads to the extrudate swell. Adapted from [2].

their original configuration. Thus, there is a contraction of the chains in the flow direction and an expansion in the normal direction leading to the extrudate swell.

Moreover, from the rheological properties presented in Chapter 3, it has been shown that the alginate solutions are shear-thinning where the viscosity is constant (newtonian Plateau) for $\dot{\gamma} < \dot{\gamma}_0$ and decreases for $\dot{\gamma} > \dot{\gamma}_0$. In our study case, $\dot{\gamma} > \dot{\gamma}_0$ meaning that the viscosity function depends on the shear rate and is described by a power law. When increasing the shear rates, the viscosity decreases as the polymer chains tend to align along the shear. Thus, the swelling phenomenon is not due to the cross-linking ions but to the strong shear-thinning of alginate solutions.

### 4.3.4 Wavy, striated patterns and smooth alternation

#### 4.3.4.1 Volumetric waves and surface striations

##### 4.3.4.1.1 General observations

The wavy and striated patterns were not observed without cross-linking ions whatever extrusion speed tested. Therefore, a first assumption was that those instabilities were due to calcium ions. An easy way to verify this hypothesis was to change the calcium ions concentration. The same extrusion of ALG4 was thus performed in a calcium ions bath at 0.5M and 0.1M at the same velocity ($\bar{v} = 0.1 \text{ mm/s}$).

At $[Ca^{2+}] = 0.5\text{M}$, the extrudate showed high periodic rough striations whereas at $[Ca^{2+}] =$
<figure><img src="image_93.png" /><figcaption>Figure 4.14: Extrusion of ALG4 in CaCl₂0.5M (left) and CaCl₂0.1M (right) with glass capillary R=250µm at v̄ = 0.1mm/s. Striations are periodic at 0.5M and sporadic at 0.1M.</figcaption></figure>

0.1M those patterns were more sporadic and more difficult to observe as the roughness
amplitude of those striations may be reduced. Therefore, striations are promoted by the
gelling salt and so, the calcium ions are responsible for those instabilities.

4.3.4.1.2 Striations and waves analysis

It was very surprising to see how periodical the striated patterns were. In order to track
in real-time their presence and assess their spatial wavelengths, we decided to analyze
the striation using a laser beam. Unfortunately, this procedure was not suited for the real-
time tracking of this instability as the laser beam illumination would suffer from numer-
ous reflexions due to the glass tank and the solution in the bath. We therefore analyzed
"post extruded" alginate threads outside of the set-up when striations were observed. The
diffraction pattern is shown in Figure 4.15.

The diffraction pattern is composed of several fringes spaced at a length L. Those fringes
are perpendicularly oriented to the striations found on the alginate thread surface. The
striated patterns act as a diffraction grating. It is thus possible to estimate the distance
between two striations:

*Numerical application:* From the diffraction relationship between the different physical quantities, we have:

$$
a = \frac{2 \lambda D}{L}
$$
<figure><img src="image_94.png" /><figcaption>Figure 4.15: Laser diffraction experiment on an alginate thread which exhibits periodic striations and the respective diffraction pattern observed directly on a screen.</figcaption></figure>

where *a* is the striations wavelength on alginate thread, $\lambda$ is the laser beam wavelength $\lambda = 660$ nm, D the length between the thread and the screen; and L the length measured on the screen between two fringe patterns ($L = 4$ cm). From the relationship, we obtain: $a \approx 12\mu$m which is consistent with the microscopical observations.

<figure><img src="image_95.png" /><figcaption>Figure 4.16: Eye guidelines of the possible mechanism which leads to wavy (left) and striated pattern (right). The gelation may occur at the die exit and propagate along the die circumference. At low throughput (left), the gelation front sticks and debonds while the solution still flows (protrusion). At higher throughput, this mechanism occurs more rapidly: the protrusions do not form and the gelation travels the entire circumference quasi instantaneously (right). Scale bar = 200µm.</figcaption></figure>

**Possible mechanism:** It is assumed that wavy and striated patterns both come from the presence of calcium ions and may be due to the same instability but at different time scales. Calcium ions might diffuse and penetrate at the die exit vicinity over a small length, but long enough to create a gelled area which sticks and moves radially away from the die
surface. In the "wavy" case, the throughput is slow enough to give to the annular gel front the time to propagate over ONE die circumference while solution continues flowing (protusion). As a consequence, the pattern is helical (Fig 4.16 left). When the throughput increases, this phenomenon occurs more rapidly and the gel "annulus" debonds from the die over the all circumference at almost the same time (Fig 4.16 right), leading to quasi entire rings which appear very periodically.

The time period T between two striated events were measured at different throughputs. The striations velocity $v_s$ at which the striations progress, was estimated as follows:

$$v_s = \frac{2 \pi R}{T}$$

where the numerator represents the die inner periphery with R its radius and T the time period. Figure 4.17 displays the striations velocity evolution as a function of the shear rate $\dot{\gamma}$ for three different capillary radii.

<figure><img src="image_96.png" /><figcaption>Figure 4.17: Striations velocity evolution as a function of the shear rate for three capillaries of radii: Red: 100 µm, Green: 150 µm and Purple: 200 µm</figcaption></figure>

The striations velocity tend to values of about: $v_s = 0.1-1m/s$ for high shear rates. As a possible line of reflection, we can calculate the sound velocity of a plane transverse shear wave in a material. This velocity is related to the shear modulus G and the mass density ρ by:

$$v_{sound} = \sqrt{\frac{G}{\rho}}$$

Here, for the high shear rates, the shear modulus is about 200 Pa.s. The mass density of sodium alginate is 1600 kg/m³. The resulting shear wave velocity is estimated: $v_{sound} \approx 0.35 m/s$. The striations velocity and the shear wave velocity are of the same order of magnitude. This might suggest that the striation is an adhesive fracture process which, as such, has a limit velocity of the order of the transverse sound velocity. (Rayleigh wave speed for mode I brittle fracture, [20])
#### 4.3.4.2 Oscillating flow

Between the wavy and striated regimes, an unstable region is observed. While the velocity is close to the lower or upper ranges, the extrudate surface alternates with smooth and respectively wavy (low) or striated (high) patterns. An example of alginate extrudate with oscillating surface is shown in figure 4.18:

<figure><img src="image_97.png" /><figcaption>Figure 4.18: Alginate extrudate from ALG4 extrusion in CaCl₂ 0.5M, with a glass capillary R=150 µm at $\bar{v}$= 1mm/s. This extrudate displays two morphologies: right: smooth region and left: striated pattern. Red dashed lines are eyes guidelines to see the die swelling superimposed with striations defect. Scale bar = 200 µm</figcaption></figure>

**NB:** Those instabilities were observed for glass, metallic and PTFE capillary. It was more surprising in the last case as Teflon is hydrophobic and does not promote adhesion.

We can also notice that the striations appear superimposed with a diameter increase whereas the smooth part has a diameter comparable with the inner capillary diameter (Fig. 4.18,4.19). Based on polymer melt interpretation for the stick-slip instability, it is assumed that as the swelling and striations appear together, polymer chains stick to the wall surface leading to their disentanglement and so to the die swelling at the exit where chains entangle. On the contrary, when chains are sliding on the wall, there is no die swelling and the extrudate is smooth [1].

<figure><img src="image_98.png" /><figcaption>Figure 4.19: Spatio-temporal image of ALG4 extrusion with stainless steel needle of radii 760µm in the oscillating regime. Stick is associated to die swelling and Slip to the smooth region. Red arrow shows the flow direction. Scale bar = 200 µm.</figcaption></figure>
Finally, regarding the time period of each phenomenon, we observe that, generally, the stick phase has shorter durations than the slip phase (Fig. 4.19). Moreover, both appear quite periodically. Further investigations must be performed to better quantify this oscillating regime.

### 4.3.5 Gel clogging

The last instability is the most surprising. While increasing extrusion velocity at its maximum, gel clogging near the die exit is observed. We first noticed this incredible instability with a metallic needle. With this material, the gel clogging can reach 50% of the die diameter! To better understand the gel clogging formation, we used glass capillaries and repeated the experiment. It was quite difficult to find it but it seems that after several experiments with the same capillary, it favored the instability. Moreover, the instability must appear on one capillary "side" in the field of view to be seen.

<figure><img src="image_99.png" /><figcaption>Figure 4.20: Gel clogging steps: A) The gel nucleation site appears and B) grows until C) beginning to go out and D) being expulsed. The flow diameter decreases progressively. Extrusion of ALG4 in $CaCl_2$ 0.5M with glass capillary 500µm. Needle diameter=390µm.</figcaption></figure>

Figure 4.20 displays the different steps of the gel clogging formation. Thanks to the glass transparent capillary, we can follow the development of the gel clogging. First, a nucleation point is observed and grows inside the capillary up to some critical size. Then, it begins to go out of the capillary and it is finally expulsed. Moreover, one gel clogging event is often followed by series of other events which appear quite periodically (Figure 4.21, A) from the same "nucleation" site. In order to further investigate this gel clogging, we have used needles with different angles.

**Angle effect:** for an angle of 30° at the die exit (Figure 4.21, B), in the case of stainless steel needle, the gel clogging is also observed. The instability is always located on the top angle side (side where the solution is "first" in contact with the bath)

**Die material effect:** as previously mentioned, the gel clogging was larger for stainless steel needles than for glass capillaries. It was not yet observed in the case of PTFE tubing.

**Calcium ions effect:** we performed the same extrusion experiment at the highest velocity in chloride calcium solution at 0.1M (5 times lower). No gel clogging was noticed.

All those qualitative observations suggest that calcium ions are essential to induce quick gelation at the die exit and that rough material improves this instability. We can hypothesize that the roughness acts as an ion entrance favorable parameter which makes easier ion diffusion at the die exit vicinity.

**Short comment:** As this bulk defect spoils the extrudate and may clog half of the die (Fig. 4.22 B)), we would like to know how this large barrier impacts the force signal and then
<figure><img src="image_100.png" /><figcaption>Figure 4.21: Spatio-temporal images of ALG4 extrusion with stainless steel needle of radii 460µm of A) blunt exit and B) exit angle 30° of several gel clogging events.</figcaption></figure>

if it is possible to predict the gel clogging. The force signal was monitored during the extrusion of alginate solutions at exit velocity of 40 mm/s (Fig. 4.22, A)). However, even if gel cloggings were amazing and blocked almost the middle of the capillary exit, the force signal is not impacted enough to be detected. Indeed, the two events showed by the purple arrows are the gel clogging expulsions and both forces signals do not stand out from the signal noise.

<figure><img src="image_101.png" /><figcaption>Figure 4.22: A) Force and velocity signal of the extrusion of ALG4 in CaCl₂0.5M with a glass capillary of 500µm. Purple arrows point out gel clogging expulsions. B) A corresponding example of gel clogging defect on an extrudate is observed post-extrusion. Scale bar: 200µm.</figcaption></figure>

The gel clogging defect is quite inconsistent with the velocity range at which it appears. Indeed, it is assumed that, at high velocity, calcium ions may not diffuse inside the cap-
illary. However, they reach the internal die wall probably through singular die exit defect area which constitutes an ion entrance point and thus a gel nucleation point.

## 4.4 Conclusion

Sodium alginate solutions were chosen to be our model system to study the extrusion instability and harness our experimental set-up before using it with collagen solutions. First, we studied the extrusion of alginate in the absence of binding ions. Two characteristic instabilities already reported for polymer melt extrusions were observed and quantified: the sharkskin perturbing the extrudate surface and the helical instability which disturbed the extrudate volume. We proved that the die characteristic dimensions had an influence on the second one which was assumed to affect the whole pipe flow. Those observations had never been reported before in the case of polymer solutions.

In the second study, we extruded alginate solutions in a bath containing calcium ions. Quite surprisingly, instabilities were observed such as the highly periodic striations on threads surface or the gel clogging which appeared at high velocities. Indeed, in the presence of divalent ions, the gelation of alginate occured and several phenomena could appear at the same time: calcium ions diffusion, alginate network formation, syneresis etc, making the results interpretation more complicated. Several potential mechanisms were anyway described but further experiments were needed to better understand those unique instabilities.

Finally, according to our initial idea, we attempted to transpose results obtained for alginate solutions to collagen ones. We have succeeded in reproducing helical distorsion on collagen threads. This last result offer interesting prospects for providing a tendon-like structure to our collagen threads, akin the "crimp".
**4.5 References**

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[3] JM Piau, N El Kissi, and B Tremblay. Influence of upstream instabilities and wall slip on melt fracture and sharkskin phenomena during silicones extrusion through orifice dies. *Journal of non-newtonian fluid mechanics*, 34(2):145–180, 1990. 104

[4] Laurent Robert. *Instabilité oscillante de polyéthylènes linéaires: observations et interprétations*. PhD thesis, Nice, 2001. 104

[5] H Münstedt, M Schmidt, and E Wassner. Stick and slip phenomena during extrusion of polyethylene melts as investigated by laser-doppler velocimetry. *Journal of Rheology*, 44(2):413–427, 2000. 104

[6] Christelle Combeaud. *Etude des instabilités volumiques en extrusion de polystyrène et polypropylène*. PhD thesis, École Nationale Supérieure des Mines de Paris, 2004. 104

[7] J-F Agassant, Dawn R Arda, Christelle Combeaud, Armin Merten, Helmut Muenstedt, Malcolm R Mackley, Laurent Robert, and Bruno Vergnes. Polymer processing extrusion instabilities and methods for their elimination or minimisation. *International Polymer Processing*, 21(3):239–255, 2006. 104

[8] Christelle Combeaud, Yves Demay, and Bruno Vergnes. Experimental study of the volume defects in polystyrene extrusion. *Journal of non-newtonian fluid mechanics*, 121(2):175–185, 2004. 104

[9] H Nguyen and DV Boger. The kinematics and stability of die entry flows. *Journal of non-Newtonian fluid mechanics*, 5:353–368, 1979. 105

[10] Gareth H McKinley, William P Raiford, Robert A Brown, and Robert C Armstrong. Nonlinear dynamics of viscoelastic flow in axisymmetric abrupt contractions. *Journal of fluid mechanics*, 223:411–456, 1991. 105

[11] Ronald G Larson. Instabilities in viscoelastic flows. *Rheologica Acta*, 31(3):213–263, 1992. 105

[12] Lise Picaut, Olivier Ronsin, Christiane Caroli, and Tristan Baumberger. Experimental evidence of a helical, supercritical instability in pipe flow of shear thinning fluids. *Physical Review Fluids*, 2017. 108, 125

[13] James H.-C. Wang. Mechanobiology of tendon. *Journal of Biomechanics*, 39(9):1563–1582, January 2006. 122

[14] Jeffrey M Caves, Vivek A Kumar, Wenjun Xu, Nisarga Naik, Mark G Allen, and Eliot L Chaikof. Microcrimped collagen fiber-elastin composites. *Advanced Materials*, 22(18):2041–2044, 2010. 122
[15] Yang Wu, Jerry YH Fuh, and Yoke San Wong. Crimped fiber printing via e-jetting for tissue engineering. In *ASME 2017 12th International Manufacturing Science and Engineering Conference collocated with the JSME/ASME 2017 6th International Conference on Materials and Processing*, pages V004T05A014–V004T05A014. American Society of Mechanical Engineers, 2017. 122, 123

[16] Spencer E Szczesny, Tristan P Driscoll, Hsiao-Yun Tseng, Pang-Ching Liu, Su-Jin Heo, Robert L Mauck, and Pen-Hsiu G Chao. Crimped nanofibrous biomaterials mimic microstructure and mechanics of native tissue and alter strain transfer to cells. *ACS Biomaterials Science & Engineering*, 2016. 122, 123

[17] Kurt Ingar Draget, Olav Gåserød, Ingrid Aune, Peder O Andersen, Bente Storbakken, Bjørn Torger Stokke, and Olav Smidsrød. Effects of molecular weight and elastic segment flexibility on syneresis in ca-alginate gels. *Food Hydrocolloids*, 15(4):485–490, 2001. 128

[18] David V Boger and Kenneth Walters. *Rheological phenomena in focus*, volume 4. Elsevier, 2012. 129

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[20] Jay Fineberg and M Marder. Instability in dynamic fracture. *Physics Reports*, 313(1):1–108, 1999. 133
# Chapter 5

# Preparation and characterization of collagen scaffolds

"In every job that must be done, there is an element of fun."

Mary Poppins in *Mary Poppins*

## Abstract

In this chapter, we first tested the reproducibility of collagen type I extrusion while changing the stock solution batch. We then assessed the ionic strength influence of the extrusion bath on the collagen thread stability and structure ageing. Based on several criteria, this study allowed us to select a single condition that was used to analyze the effect of collagen solution concentration. As a complementary experiment and to study the benefits of their added properties, collagen-alginate mixtures were extruded and the resulting threads were compared with the pure collagen ones. Finally, in view of cell culture experiments, the stability of the collagen based threads in culture medium was characterized

## Contents

<table><tr><td><b>5.1</b></td><td><b>Threads structure characterization techniques</b></td><td><b>143</b></td></tr><tr><td>5.1.1</td><td>Photonic microscopies</td><td>143</td></tr><tr><td>5.1.1.1</td><td>Optical (bright field) and polarized microscopies</td><td>143</td></tr><tr><td>5.1.1.2</td><td>Second Harmonic Generation (SHG) Microscopy</td><td>146</td></tr><tr><td>5.1.2</td><td>Electron microscopies</td><td>152</td></tr><tr><td>5.1.2.1</td><td>Scanning Electron microscopy (SEM)</td><td>153</td></tr><tr><td>5.1.2.2</td><td>Transmission Electron microscopy (TEM)</td><td>154</td></tr><tr><td>5.1.3</td><td>Differential Scanning Calorimetry (DSC)</td><td>155</td></tr><tr><td><b>5.2</b></td><td><b>Collagen batch quality criterion</b></td><td><b>156</b></td></tr><tr><td><b>5.3</b></td><td><b>Fibrillogenesis buffer (Ionic strength) effect</b></td><td><b>157</b></td></tr><tr><td><b>5.4</b></td><td><b>Pure collagen vs collagen/alginate mixtures threads study</b></td><td><b>180</b></td></tr><tr><td>5.4.1</td><td>Pure collagen threads of different concentrations</td><td>180</td></tr><tr><td>5.4.1.1</td><td>Extrusion and maturation</td><td>180</td></tr><tr><td>5.4.1.2</td><td>Structural characterization</td><td>183</td></tr><tr><td>5.4.2</td><td>Collagen/alginate threads</td><td>188</td></tr><tr><td>5.4.2.1</td><td>Ratio choice</td><td>188</td></tr><tr><td>5.4.2.2</td><td>Structural characterization</td><td>189</td></tr></table>
<table><tr><td>5.4.3</td><td>Mechanical properties comparison</td><td>193</td></tr><tr><td><b>5.5</b></td><td><b>Preliminary study to cell culture</b></td><td><b>194</b></td></tr><tr><td>5.5.1</td><td>General morphology</td><td>194</td></tr><tr><td>5.5.1.1</td><td>Optical observations</td><td>194</td></tr><tr><td>5.5.1.2</td><td>Topography analysis</td><td>194</td></tr><tr><td>5.5.2</td><td>Diameter stability</td><td>195</td></tr><tr><td>5.5.3</td><td>Structural characterization</td><td>196</td></tr><tr><td>5.5.3.1</td><td>Pure collagen threads</td><td>196</td></tr><tr><td>5.5.3.2</td><td>Collagen:alginate threads</td><td>197</td></tr><tr><td>5.5.3.3</td><td>Collagen stability: thermal analysis</td><td>199</td></tr><tr><td>5.5.4</td><td>Mechanical properties</td><td>200</td></tr><tr><td>5.5.5</td><td>Discussion</td><td>200</td></tr><tr><td><b>5.6</b></td><td><b>Conclusion</b></td><td><b>201</b></td></tr><tr><td><b>5.7</b></td><td><b>References</b></td><td><b>202</b></td></tr></table>
The collagen-based threads have been studied as a function of several parameters as the time, fibrillogenesis buffers, material nature and concentration and extrusion velocity. We have characterized each thread from the nanometric to the sample scale by using different techniques. The mechanical properties of the threads that were assessed by tensile testing methods and have already been described in Chapter 2 (2.1.3 p 69). In this chapter we first present the other techniques techniques of characterization that were used, their underlying principles, as well as our main results.

# 5.1 Threads structure characterization techniques

Two main categories of structure characterization techniques have been used: those involving photon radiation and those involving electron radiation. The first category allowed us to characterize the collagen 3D organization and distribution inside the threads. Electron microscopy techniques have revealed the topography of the threads, the inner structure at the nanometric scale and have allowed us to measure the diameter of the fibrils and to verify their integrity. Combining those two types of techniques, we determined the most advantageous threads for the project according to the different physicochemical conditions.

## 5.1.1 Photonic microscopies

### 5.1.1.1 Optical (bright field) and polarized microscopies

Optical bright field microscopy is used to observe the threads evolution and stability as a function of several parameters (maturation time, concentration, buffer etc). It allows us to gather informations about their structural shape at the thread scale. Polarized light microscopy is performed to assess qualitatively the main orientation of collagen inside the threads.

#### Bright field

A light wave is an electromagnetic wave, which is composed of an electric field $\vec{E}$ and a magnetic field $\vec{B}$ (Fig. 5.1). These fields are perpendicular to each other and in a plane perpendicular to the direction of propagation: the wave plane. The polarization of a wave is characterized by the direction of $\vec{E}$. For natural lighting, the vector $\vec{E}$ varies randomly over time, and so, this light is not polarized.

#### Linear polarization

When the direction of $\vec{E}$ remains constant in the wave plane, the polarization is linear. In a (Oxy) coordinates system (Fig. 5.1), the vector $\vec{E}$ is written as: $\vec{E} = E_x \vec{e}_x + E_y \vec{e}_y$ If $\alpha$ is the angle between the axis (Ox) and the vector $\vec{E}$, it is then expressed as:

$$ \begin{cases} E_x = E_0 \cos(\alpha) \cos(\omega t) \\ E_y = E_0 \sin(\alpha) \cos(\omega t) \end{cases} $$

#### Elliptical polarization
<figure><img src="image_102.png" /><figcaption>Figure 5.1: Scheme of a linearly polarized sinusoidal electromagnetic wave travelling in the positive x-direction. $\vec{E}$ and $\vec{B}$ are the electric and the magnetic fields respectively of the light wave, which propagates at the celerity $c$. (From: https://www.miniphysics.com/, "Electromagnetic Spectrum and Sinusoidal EM Plane Waves")</figcaption></figure>

In the case of elliptical polarization, the tip of the vector $\vec{E}$ describes an ellipse in the wave plane. If $a$ and $b$ are the two half-axes of the ellipse, the ellipticity is then defined by: $\tan(\psi) = b/a$.

The ellipse is said to be "right" if the tip of $\vec{E}$ turns clockwise and "left" if it rotates in the trigonometric direction. The components of $\vec{E}$ are given by:

$$ \begin{cases} E_x = E_0 a \cos(\omega t) \\ E_y = E_0 \varepsilon b \sin(\omega t) \end{cases} $$

with $\varepsilon = \pm 1$.

If the major axis of the ellipse has an angle $\alpha$ with the axis (Ox) we have:

$$ \begin{cases} E_x = E_0(a\cos(\alpha)\cos(\omega t) - b\sin(\alpha)\sin(\omega t)) \\ E_y = E_0(a\sin(\alpha)\cos(\omega t) + b\cos(\alpha)\sin(\omega t)) \end{cases} $$

## Circular polarization

Circular polarization is a special case of elliptic polarization. It is obtained when $a = b$, we thus have:

$$ \begin{cases} E_x = E_0 a \cos(\omega t) \\ E_y = E_0 \varepsilon a \cos(\omega t) \end{cases} $$

*Microscope parameters:* All threads observations were performed on a Nikon Eclipse E600L microscope equipped with a digital camera DXM 1200 CCD. The samples were placed horizontally between two crossed polarizers. Between them, a collagen thread with an isotropic structure will appear black whereas if it has an anisotropic organized structure, it will be revealed colored and textured. This is due to the **birefringence** of the collagen.

## Birefringence

In an isotropic material, the light propagates at the same velocity in all directions of space. The medium can therefore be characterized by a unique refractive index $n$, defined by the
relation $n = c/v$, where c is the light celerity in the vacuum and $v$ the light propagation velocity in the medium.

The birefringence of a medium results from its anisotropy. Here, we focus only on an uniaxial anisotropy. Such a medium can not be characterized only by a single index of refraction. The ellipsoid of the indices is defined by its surface composed of three axes: $n_o$, the ordinary index (along (Ox) and (Oy) axes) and the extraordinary index $n_e$ (Fig. 5.2). In a positive uniaxial medium, we have: $n_e > n_o$.

<figure><img src="image_103.png" /><figcaption>Figure 5.2: Index ellipsoid for a positive uniaxial medium: $n_o$ and $n_e$ are the respective ordinary and extraordinary index where $n_e > n_o$ leading to the ellipsoid.</figcaption></figure>

The birefringence is determined by placing this medium between two crossed polarizers at 45°. The first polarizer selects linearly the incident light in a given direction. In the absence of a sample (or in the presence of an isotropic sample), the analyzer (second polarizer) oriented at 90° avoids the transmission of this light, causing an extinction. In the case of birefringent material, it is possible to decompose the propagating light into two components along the axes of the index ellipsoid. The transmitted light (at the analyzer output) is the combination of those two components and has an intensity $I'$ given by:

$$I' = Isin^2(2\alpha)sin^2(\frac{\pi\delta}{\lambda})$$

with: I the incident light intensity (at the polarizer input); $\alpha$ the angle formed between the polarizer and $n_e$; $\delta$ the delay between the two parallel vibration components of the polarizer and the analyzer; and $\lambda$ the light wavelength.

The path delay $\delta$ defined as $\delta = \Delta n \times e$ (with a the medium thickness) is a function of the anisotropic medium traversed. The variations of the intensity at a sample point depend on the single parameter $\alpha$. The transmitted intensity is maximum for values of $\alpha= \pm \pi/4$ and is minimal or equal to zero for $\alpha= \pm \pi/2$. We can quantify the birefringence through the subtraction of the two index: $\Delta n = n_e - n_o$. The birefringence is therefore positive when $n_e > n_o$, which means that the light propagation velocity ($v = c/n$) is greater along the ordinary axis of the ellipsoid.
In practice, when an anisotropic material is observed under a polarized light microscope, the rotation of the sample plate will vary the angle $\alpha$. In the case of type I collagen, its extraordinary index occurs in the axial direction of both fibrils and the molecule itself.

## Sample preparation and observations parameters

All samples were put in optical observation chambers in which several drops of extrusion buffer were added to keep the samples in wet conditions at room temperature. The threads diameters were measured over two weeks for stability analysis. For polarized light microscopy observations, a wave plate ($\lambda$) was sometimes added between the crossed polarizers in order to change the linearly polarized light into circularly polarized light. This operation made easier the observations of collagen molecules orientation despite the thickness of the threads.

### 5.1.1.2 Second Harmonic Generation (SHG) Microscopy

With the two first optical techniques, we can easily quantify the threads stability over time and have an idea about the collagen orientation inside the thread. However, it is not precise enough to observe the possible structural changes and organization at the collagen molecule scale. To do so, more advanced and sophisticated techniques must be used such as the Second Harmonic Generation (SHG) microscopy [1]. This technique presents a much better axial resolution than the polarized light microscopy and allows to image the texture of the collagen-based threads in 3D. Two contrast modes are used on the same focal volume: SHG and 2PEF (two photon excited fluorescence).

#### 5.1.1.2.1 Principle

Multiphoton microscopy uses non-linear interactions between light and matter as a source of contrast. SHG is a coherent process which consists in the conversion of an intense pulse laser source of frequency $\omega$ into its harmonic at $2\omega$, by the generation of 1 single photon at $2\omega$ from 2 photons at $\omega$ [2]. This process is not very effective and requires focusing the energy by using femtosecond pulses. This increases the signal intensity while maintaining an average power compatible with biological samples. The signal intensity is then proportional to the square of the density of the molecules generating the harmonic response.

SHG process is due to the oscillation of polarizable electrons at the molecular scale and to the electric field of induced dipoles that will interfere at the macromolecular scale. The three-dimensional organization of the molecules in the focal volume is thus important for obtaining an SHG signal.

The interaction of an electromagnetic wave with a molecular bond or a single molecule induces the oscillation of an electron of this molecule. The potential of the excited electron evolves in the same way as the one of a harmonic oscillator according to the Lorentz model. Thus the frequency at which this electron oscillates is equal to that of the excitation wave (Rayleigh scattering, Fig. 5.3, a). In reality, the potential is not exactly parabolic, which leads to the generation of harmonics (Fig. 5.3, b). In the case of symmetric potential, only the uneven multiple harmonics of the exciter wavelength are generated. This is why, a centro-symmetric molecule can not create a SHG signal.
<figure><img src="image_104.png" /><figcaption>Figure 5.3: Excitation of molecules with a radiation of frequency ω. a) While exciting a symmetrical molecule, the frequency of the excitation radiations are equal to that of the scattered radiation: this is the Rayleigh scattering. b) If the molecule is non-centrosymmetric, we have both Rayleigh scattering and harmonic scattering (double frequency radiation: 2ω) at high excitation intensity. b1. If two non-centrosymmetric molecules are parallel, we have constructive interferences of the radiations at the double frequency 2ω. b2. On the contrary, for two non-centrosymmetric antiparallel molecules, the radiation is in phase opposition and there is then destructive interference with zero scattering intensity. The scattered intensity thus varies quadratically with the number of parallel non-centrosymmetric molecules. (Adapted from [3])</figcaption></figure>

The polarization $\vec{p}$ is then given by the relation:

$$ \vec{p} = \alpha \vec{E} + \beta \vec{E} \cdot \vec{E} + \gamma \vec{E} \cdot \vec{E} \cdot \vec{E} + \dots $$

where $\alpha$ is the linear polarizability, $\beta$ and $\gamma$ the first and second order hyperpolarizabilities and $\vec{E}$, the electric field is written as:

$$ \vec{E} = \vec{E}_0 e^{-i\omega t} + cc $$

with $\omega$ the wave pulsation and $cc$ the complex conjugate.

If two non-centrosymmetric molecules, separated by a small distance (< $\lambda$ : incident wavelength), are aligned parallel in the same direction, the generated harmonic radiations are the same for the two molecules. The polarization of order 2 which is identical for the two molecules is given by:

$$ \vec{p}^{(2)}(2\omega) = \beta \vec{E}(\omega) \cdot \vec{E}(\omega) $$

Thus, these harmonic radiations interfere constructively and the total field is twice that of a single dipole (Fig. 5.3, b1). Since the scattered intensity is the square of the total
field, the intensity is 4 times greater than that obtained in the case of a single molecule. However, if the polarities of the molecules are opposite, destructive interference occurs resulting in zero signal (Fig. 5.3, b2).

### 5.1.1.2.2 Origin of the collagen SHG signal

The collagen SHG signal comes from the peptide bonds at the molecular level. As previously described, SHG is sensitive to non-centrosymmetric molecules assemblies. The collagen molecule, due to the alignment of its numerous peptide bonds (Fig. 5.4 a), b) top), has a non-negligible first hyperpolarizability (Fig. 5.4 b) bottom).

<figure><img src="image_105.png" /><figcaption>Figure 5.4: Schemes of: a) a peptide bond which is responsible for the SHG signal of the collagen molecule [4]. In fact, this bond acts as an electric dipole (arrow). b) Top: The origin of the SHG signal in collagen is related to the hyperpolarizability $\beta$ of the peptide bonds present in each collagen helix. Bottom: Thus, the SHG signal will depend on the organization of these bonds in the excitation volume, therefore, of the 3D organization of the triple helices. The triple helix has $C_{\infty}$ symmetry. Adapted from [5]</figcaption></figure>

Because of its three $\alpha$-helices, collagen has $C_3$ symmetry, but to simplify, it is considered as a $C_{\infty}$ symmetry. The compactness and stiffness of the triple helix ensure that a large number of harmonophores (peptide bonds) are well aligned along the molecular axis, which allows the coherent built-up of SHG signal. Moreover, the coherent addition of the second harmonic signal of each triple helix within a fibril leads to an important SHG signal of the fibrillar collagen.
5.1.1.2.3 Polarization-resolved Second Harmonic Microscopy

In order to assess the three-dimensional mean orientation of collagen in the threads, we
used polarization-resolved SHG. This further technique allows determining the orienta-
tion of the collagen helices in the excitation volume (Fig. 5.5 a), b)).

<figure><img src="image_106.png" /></figure>

Figure 5.5: The polarization-resolved SHG consists in exciting the sample with a linearly polarized beam which angle θ is changed. a. Schematic representation of the experimental setup. The polarization angle θ of the excitation is controlled by two achromatic wave plates. The excitation field E^ω is focused through an objective. The second harmonic radiation is detected forward with a condenser (Numerical Aperture NA ≈ 1) and the 2PEF signal backwards through the focusing lens. b. Area focus of the excitation volume. The black arrow gives the mean orientation of the triple helices present in the excitation volume. (x, y, z) is the corresponding reference with the x-axis in the mean direction of the triple helices. c. Top: (XYZ) is the laboratory coordinates reference frame. The incident polarization E^ω is represented by the red arrow and is assumed to be located in the plane of the image (XY). c. Bottom: Example of SHG intensity polarimetric diagram related to the orientation of collagen molecules within the focal volume. Adapted from [5]

It consists in exciting the sample with a linearly polarized field but whose orientation θ in
the focal plane is changing. It is then possible to plot the SHG intensity as a function of θ
(Fig. 5.5 c). Several structure characteristics can be deduced from this function:

- The triple helices average orientation in the image plane, which is represented by the black arrow (Fig. 5.5 b)

- The mean SHG intensity which depends on the triple helices density and organiza-
  tion.

- The triple helix orientation disorder which is given by a parameter ρ which corre-
sponds to the square root of the ratio of the detected intensities for, respectively,
an incident polarization parallel to the triple helices direction and a perpendicular
one.

**Theory**

The SHG response of a medium is characterized by its second order nonlinear suscepti-
bility tensor $\chi^{(2)}$, related to the the polarizability $P^{2\omega}$ to the incident electric field $E^{\omega}$ by:

$$P_i^{(2\omega)} = \chi_{ijk}^{(2)} E_j^{\omega} E_k^{\omega}$$

where i, j and k correspond to the directions x, y and z. The triple helices of collagen are assumed to be aligned along x in the excitation volume with a minimal disorder. Assuming that the triple helices have cylindrical symmetry and by applying Kleinman symmetry [6], only seven tensor components are nonzero and take only two different values:

$$\chi_{xxx} \text{ and } \chi_{zzz} = \chi_{zxz} = \chi_{yxy} = \chi_{zzx} = \chi_{yyx}$$

The susceptibility tensor $\chi$ corresponds to the averaging of the hyperpolarizability tensor $\beta$ in the focal volume. This implies that the polarizability at the harmonic frequency $2\omega$ is then written:

$$\left\{ \begin{aligned} P_x^{(2\omega)} &= \chi_{xxx}^{(2)} E_x^2 + \chi_{xyy}^{(2)} E_y^2 + \chi_{xyy}^{(2)} E_z^2 \\ P_y^{(2\omega)} &= 2\chi_{xxy}^{(2)} E_x E_y \\ P_z^{(2\omega)} &= 2\chi_{xxy}^{(2)} E_x E_z \end{aligned} \right.$$

If (X, Y, Z) is the laboratory reference frame, the mean orientation of the triple helices can be fully described with the polar $\varphi$ and azimuth $\psi$ angles (Fig. 5.6). In the following figure 5.6, the two angles are represented as follows:

<figure><img src="image_107.png" /><figcaption>Figure 5.6: The two angles $\varphi$ and $\psi$ describe the orientation of the collagen molecules in the excitation volume: the orientation in the plane of the image $\varphi$; and the out-of-plane orientation given by $\psi$, which can be measured from two parameters: the mean SHG intensity $I_{mean}$ and the anisotropy parameter $\rho$.</figcaption></figure>

where $\varphi$ is the triple helix orientation in the image plane (XY) and is directly measured by polarization-resolved measurements; and where $\psi$ is the off-plane angle.

For polarization-resolved SHG (P-SHG) measurements, the incident electric field is perpendicular to the propagation direction Z of the light in the laboratory reference frame (XYZ), with an angle $\theta$ with respect to the X axis. Thus its components are: $E_X^{\omega} = E_0 \cos(\theta)$; $E_Y^{\omega} = E_0 \sin(\theta)$; $E_Z^{\omega} = 0$

The total SHG intensity $I^{2\omega}$ is then calculated by transposing the susceptibility tensor of the triple helix (xyz) coordinates system to the laboratory reference frame (XYZ). Indeed,
the symmetries are legitimate in the reference of the triple helix but it is necessary to express the incident electric field in the laboratory reference frame. The triple helices are not necessarily in the focal plane and the total SHG intensity is written as:

$$I^{2\omega}(\theta) = \cos^2(\psi)A(\psi) \cdot \cos[4(\theta - \varphi)] + B(\psi) \cdot \cos[2(\theta - \varphi)] + C(\psi)$$

where A(ψ), B(ψ) and C(ψ) are directly related to the components of the susceptibility tensor in the frame (XYZ) and can be expressed as a function of the susceptibility components of the triple helices in the (xyz); and ψ the off-plane angle of their mean orientation.

Finally, we can extract three quantitative parameters from this total SHG intensity (measured for different excitation angles), which will allow us to obtain a variety of informations on the 3D orientation of the triple helices.

- the orientation φ of the triple helix projections in the image plane (XY)

- the mean SHG intensity $I_{mean}$, which is directly connected to the angle ψ in the first order by: $I_{mean} \propto \cos^2(\psi)$. If the triple helices are in the plane of the image, the signal is more intense than if they are perpendicular to this plane.

- and, the anisotropy parameter ρ, which quantifies the "disorder" within the focal volume. It is related to the SHG intensity by: $\rho = \sqrt{I(\theta)/I(\theta + \pi/2)}$.

### 5.1.1.2.4 Experimental system

The second harmonic imaging was carried out at the Ecole Polytechnique in the Laboratory of Optics and Biosciences by Guillaume Ducourthial (Post Doctoral researcher under the supervision of Marie-Claire Schanne-Klein, CNRS) with a laser scanning microscope as already presented in the figure 5.5. The excitation is provided by a femto-second titanium-sapphire laser (Mai-Tai, Spectra-Physics) set at 860nm (2.60W). A circular polarization was used to image all the structures independently from their orientation in the focal plane or a linear polarization was used for P-SHG. The collagen threads imaging was performed using a 20x 0.95 NA objective with lateral resolution of 0.6 µm and an axial resolution of 3.5 µm. For P-SHG experiments, SHG and two-photon excited fluorescence (2PEF) are detected in three channels equipped with photomultiplier Tubes (P25PC, Electron tubes) and spectral filters:

- Forward SHG: transmission (1 Semrock 680SP + 1 Semrock 720SP + 1 Semrock 427/10)

- Backward: 2PEF (1 GG5 + 2 Semrock 680SP)

- Backward SHG: epidetection (1 Semrock 680SP + 1 Semrock 720SP + 1 Semrock 427/10)

The polarization was changed from 0° to 360° with a 10° step.

For the imaging of collagen threads with cells, second harmonic generation was coupled with fluorescence in order to detect both SHG signal of collagen and fluorescence of cells.

To do so, the different signals are detected in three channels (CH) equipped with photomultiplier tubes (P25PC, Electron tubes) and spectral filters:

- Forward: SHG (1 Semrock 680SP + 1 Semrock 720SP + 1 Interferential Filter Semrock FF01-427/10)

- Backward: Alexa (2 Semrock 680SP + Alexa 488 Filter + dichroic Filter),

- Backward: DAPI (1 Semrock 680SP + 1 Semrock 720SP + DAPI filter + dichroic Filter)

Fine plane by plane images sampling was performed every microns over the first 50µm from the thread surface and then, every 10µm until reaching the thread center. At least
two areas were observed in order to check the sample structure homogeneity. The 3D image reconstructions were done with *Imaris* software.

### 5.1.1.2.5 Sample preparation

Samples without cells were kept in their fibrillogenesis buffer. When cultured with cells, the threads were first fixed in paraformaldehyde 4% (PBS 1X). Then, for both conditions, they were put in an observation chamber composed of round cover glass sides separated with two Teflon torus shaped spacers (thickness of 225 µm). Few drops of PBS 1X were added in the chamber to keep the sample in wet conditions and avoid its drying.

## 5.1.2 Electron microscopies

Electron microscopy is based on the same principle as optical microscopy, except that the radiation used comes from electrons and not from photons (Fig. 5.7). The low wavelength of this radiation, which is of the order of the picometer ($10^{-12}$ meter), allows us to reach an atomic-level resolution. In the following section, two types of electron microscopies used in our work are described: scanning (SEM) and transmission (TEM).

<figure><img src="image_108.png" /></figure>

Figure 5.7: Schemes of optical and electron microscopes: The sample is illuminated by a source (photons or electrons). Several lenses (optical or magnetic) allow to focus the beam. For optical and transmission electron microscopy (TEM), the beam passes through the sample, which is observed by transmission. For scanning electron microscopy, the electrons of the focused beam interact with the sample atoms leading to the emission of secondary electrons very close to the specimen surface. Those electrons are then detected. An example of photos obtained on our samples is given for each technique. (From https://microbiologyinfo.com, "Differences between Light Microscope and Electron Microscope")
**5.1.2.1 Scanning Electron microscopy (SEM)**

Scanning Electron Microscopy is based on electron-matter interactions. It allows high-resolution images of the sample surface.

**5.1.2.1.1 Principle**

The main components of a Scanning Electron Microscope are: an electron source (gun) and a high voltage device, a column under a secondary vacuum, a set of electronic lenses to obtain a fine beam, a turntable for the installation of the samples, an electron detector coupled to a signal amplifier and finally an image display system (Fig. 5.7 right).

The interactions between the sample and the incident beam electrons are of different natures and induce several emissions, summarized in Figure 5.8. In this work, only the secondary electronic emissions will be of interest.

<figure><img src="image_109.png" /><figcaption>Figure 5.8: Electron-matter interaction volume scheme. (From: http://gems-inclusions.com "Scanning electron microscopy")</figcaption></figure>

**Secondary electron** emissions are generally caused by inelastic scattering between primary electrons of the beam and those of the sample. These electrons have low energy (5 to 50 eV max) and the associated volume of interaction is about few *nm*³. When an incident electron interacts with the electrons of an atom of the sample, it emerges with a loss of energy. A secondary electron is then emitted and the atom is ionized by the ejection of the latter. The secondary electrons are weakly bound to the conduction band. An incident electron can cause the ejection of several secondary electrons, but their low energy causes only those near the surface (less than 10 nm) to emerge from the sample. They give a characteristic image of the surface since a topographic change will modify the number of secondary electrons emitted, useful for the visualization of the topography.

The **backscattered electrons** correspond to the primary electrons re-emitted after having undergone elastic shocks with the nuclei of the atoms as well as inelastic shocks with the orbital electrons. They have an energy close to the primary electrons (up to 50eV) and an associated interaction volume of 0.5 to 1 µm³. This volume is closely related to the operating conditions. The backscattered electronic emission is sensitive to the topography
of the sample but also to its chemical composition (elements with high atomic numbers appear more clear). However, the contrast of the topography remains predominant and the differences in gray levels due to the chemical composition will only be perceived for perfectly flat samples. Thus, they provide information on the chemical composition of the sample.

**X-ray photon** is emitted when a primary electron, colliding with an atom, ejects an electron from an inner layer. The latter is replaced by an electron of a higher layer and thus a photon whose energy is equal to the difference of the levels of electronic energy is emitted. The associated volume of interaction is of the order of $\mu m^3$ Quantitative chemical analysis of the sample is possible by using this type of interaction.

### 5.1.2.1.2 Sample preparation and observation system

All threads were fixed using 3.63 % glutaraldehyde in a cacodylate/saccharose buffer (0.05 M/0.3 M, pH 7.4) for 1 hour at 4°C. Following fixation, samples were washed three times in a cacodylate/saccharose buffer (0.05 M/0.3 M, pH 7.4) and dehydrated through successive ethanol baths with increasing concentrations from 70% to 100% alcohol. Thereafter, samples were dried at the carbon dioxide critical point and gold sputtered (20 nm) for analysis. Samples were observed with Hitachi S-3400N SEM operating at 10 kV.

#### 5.1.2.2 Transmission Electron microscopy (TEM)

Transmission Electron Microscopy allows to image ultrafine cross-sections of the samples. With this technique, we obtain very local information about the collagen organization inside threads at the nanometric scale.

##### 5.1.2.2.1 Principle

A transmission electron microscope is composed of several parts: the source from which the electrons are emitted, condenser lens which focus the source beam, an objective stage where the image of the sample is formed and finally a projection area where the image of the sample is enlarged. The entire column is under a high vacuum (Fig. 5.7 middle). The different parts are detailed:

The electrons are emitted from the **canon** by heating a filament of metal which corresponds to the cathode. The extraction can be carried out by heating the metal to a temperature close to its melting temperature (2400 °C. for tungsten): this is a thermoionic emission. The electrons emitted by the gun are used by the **lenses** to create the electronic probe. The axis of the lenses is parallel to the path of the electrons and allows to focus them. The focal length of the lenses is related to the ampere-turns numbers of the coil. Thus, by varying the current which passes through the coil, the focal distance of the lens can be modified and the magnetic field is produced. The coils are often made of soft iron elements in order to increase the magnetic field. The electromagnetic lenses are directly used to focus the beam on the sample after passing the electrons through a field diaphragm.

The electron beam then passes through the **sample** where various electron-matter inter-
actions occur as previously described. There are the elastic (energy-free) and the inelastic interactions. In this case, when the electrons interact with the sample atoms, they lose energy. They will be more or less absorbed and diffracted. The resolution is limited by the optical aberrations which come essentially from the objective lens. The three main types of aberrations encountered are the following: defocusing, astigmatism and spherical aberrations. The final image is formed on a screen associated with a CCD camera which allows to image the sample section.

### 5.1.2.2.2 Sample preparation and observation system

Collagen threads with or without cells were fixed as described above. After fixation, samples were postfixed with 2% osmium tetroxide in cacodylate/saccharose buffer solution. They were washed in cacodylate/saccharose solution, dehydrated in successive ethanol baths (from 50° to 100°), and finally embedded in araldite. Embedded samples were sectioned on an Ultracut Reichert Jung. Thin sections (0.5 µm) were stained with toluidine blue and observed on a Nikon Eclipse E600Pol microscope equipped with a 40x (0.65 NA) objective and a DXM 1200 CCD camera. As biological samples are essentially formed by light atom, they little interact with electrons. Thus, in order to observe biological samples such as collagen, it may be useful to contrast them with heavy metals such as uranyl acetate. For this purpose, the ultrathin sections (70 nm) were contrasted with uranyl acetate and observed on a transmission electron microscope FEI Tecnai spirit G2 operating at 120 kV. Images were recorded on a CCD Camera (Orius Gatan 832 digital).

## 5.1.3 Differential Scanning Calorimetry (DSC)

The differential scanning calorimetry (DSC) technique allows characterizing the stability of the material by measuring the heat exchanges between the sample to be studied and a reference (here: air). Several physical quantities can then be extracted from those exchanges analysis: the glass transition ($T_g$), melting ($T_m$) and crystallization ($T_c$) temperatures; or the reaction enthalpies to assess the degree of crosslinking of certain polymers.

In our case, while heating the sample at a constant speed, conformation changes and thermal denaturation of the collagen molecules may occur. DSC measures the enthalpy ($\Delta H$) resulting from the conformational changes induced by the temperature rise. Collagen can indeed be more or less organized and structured in the threads and thus, the more stable the molecules, the higher the melting temperature.

### Experimental set-up and parameters

The thermograms of the collagen-based threads were recorded using a DSC Q20 (ThermoAnalytic) device. Since the denaturation of collagen is irreversible, the threads are subjected to only one cycle of temperature rise and decrease. For this, samples of 2cm length were cut before measurement and placed in hermetic aluminum capsules. They were then transferred to the DSC device and stabilized for 5 minutes at 20°C. Then, the temperature was increased up to 90°C at a speed of 5°C/min. For each condition, at least 3 samples were analyzed. Each sample was previously weighed to determine the mass rate of heat flow during denaturation processes.
## 5.2 Collagen batch quality criterion

One collagen purification of almost 30-40 rat tail tendons leads to a collagen volume of about 2 liters at 2-4 mg/mL. As we extruded dense collagen solutions of concentration range between 15 and 60 mg/mL, we "consumed" large quantities of the dilute stock solution to be concentrated. As a consequence, several collagen batches were needed to cover the totality of the different studies. It was thus necessary to verify the reproducibility of the extrusion for different batches. Figure 5.9 shows the results obtained for extrusion of 3 different collagen batches. First, batches A and B at 60 mg/mL were extruded with exactly the same parameters and buffer conditions. The two threads (Fig. 5.9 A1, B1) look very similar: they are straight and their surface and volume are homogenous. Then, the same experiment was performed between batches A and C (60mg/mL) (Fig. 5.9 A2,C2). In this case, the morphology difference is obvious: on the right, "thread C" displays non-homogeneous areas, which are more or less transparent and seem to reveal severe extrusion disturbance. On the contrary, "thread A" is still straight and homogeneous.

<figure><img src="image_110.png" /><figcaption>Figure 5.9: Collagen threads observed post-extrusion. Different batches (A,B,C) are tested and compared: Condition 1: PBS 10x, D=650µm; Condition 2: PBS 5x, D=390µm. A and B collagen threads show similar morphologies. A and C collagen threads are different: C displays perturbations at the surface and inside. Symbols refer to the corresponding rheological measurements. Black scale bars are 200µm</figcaption></figure>

**Remark:** one collagen batch serves for several users. For example, batch A was used for cornea or wound project. While using batch C, some users observed also different behaviors. In their case, they have to form collagen disks or sheets. However, with batch C, the collagen spreading was not a homogeneous manner and it was not possible to produce regular matrices. This kind of batch was considered as "bad" batch and the batch A as a "good" one.

As extrusion technique is very sensitive to this difference, we can rapidly conclude if a batch is good or not. However, we wanted to go further and find a physical criterion to quantify the batch quality after its purification. As the spreading of collagen was different, and the extrudate disturbed, the idea was that the rheological behaviors varied from one batch to another.

Since large quantities of collagen were needed during rheological measurements. In order to use small volumes, we performed the tests with collagen solutions at 5mg/mL rather than at high concentration with the smallest geometry available. Seven batches were analyzed and their respective concentrations were verified by hydroxyproline titration so as
to have the same for all. The results are presented in figure 5.10:

<figure><img src="image_111.png" /><figcaption>Figure 5.10: Viscosity as a function of the shear rate for: left: seven collagen batches at 5mg/mL. Two groups (1, 2) of batches can be distinguished corresponding to the "good" and "bad" respectively; right: Gobeaux PhD thesis results [7] for collagen at 4.9 mg/mL.</figcaption></figure>

From Gobeaux PhD thesis rheological analysis [7], "plateau" viscosity η₀ of about 0.18 Pa.s is found for collagen solution at 4.9 mg/mL. Based on this result, we can set a viscosity threshold and distinguish two groups of batches: 1) η₀ around 0.2 Pa.s and 2) η₀ > 0.8 Pa.s. The previous batches A and B tested by extrusion are in the region 1 whereas C is in the region 2. Therefore, C has a higher viscosity than the batches A and B. For the same velocity (thus the same shear rate), rheological behaviors are different. For example, at γ̇ around 2-3 s⁻¹, batches of A and B are in the newtonian region whereas batch C exhibits shear-thinning behaviors.

By measuring viscosity at low concentration, we can already assess the quality of the collagen batch. It is assumed that several parameters as the rat age, race, or the purification process may impact the quality of a batch. But for now, no clear origin of those difference arises from this rheological characterization.

## 5.3 Fibrillogenesis buffer (Ionic strength) effect

The aim of this first study is to determine the best conditions to induce a stable sol-gel transition while preserving the alignment of the molecules and obtaining the best mechanical properties. For this purpose, 8 extrusion buffers of different ionic strength from 150 mM to 1.65 M are tested. They are used for the extrusion and during the ageing of the collagen threads. Therefore, all along this study, we narrow down the conditions to determine those giving threads of optimized characteristics.

The results of the present study are presented in the following article. Briefly, collagen solutions at 60 mg/mL were extruded at a constant exit speed of 3.4 cm/s in order to form threads of homogeneous diameter. The extrusion solution was varied to assess the effect of the ionic strength. It was the only parameter to be changed. One of the eight buffers led to collagen threads that fulfilled the majority of the requirements: extrusion characteristics, fibrillogenesis, storage stability and mechanical properties.
# ARTICLE

## Fibrillogenesis buffer influence on pure dense collagen threads ageing

Lise Picaut<fnref n="1" />,<fnref n="2" />, Léa Trichet<fnref n="2" />, Olivier Ronsin<fnref n="1" />, Bernard Haye<fnref n="2" />, Isabelle Genois<fnref n="2" />, Tristan Baumberger<fnref n="1" /> and Gervaise Mosser<fnref n="2" />.

## Abstract

A home-made set up built to extrude dense and highly viscous solutions was used to extrude 60 mg/mL acidic collagen solutions into meters of regular threads. Out of an analysis of eight different extruding and fibrillogenesis buffers, we have pinned down a condition (PBS 5X) in which the diameter of the extruded thread was stable throughout extrusion, fibrillogenesis and storage. In this condition, the threads observed between cross-polarizers show good anisotropy suggesting that the majority of collagen fibrils are aligned parallel to the thread axis. Transmission and scanning electron microscopy reveal the 67 nm "D" period of the fibrils, whose diameters are around 20 and 80 nm. The threads present a Young modulus of 1.66 ± 0.45 MPa and an ultimate tensile strength of 0.33 ± 0.08 MPa without the use of any cross-linkers or dehydration process, thus under conditions mimicking physiological ones.

**Key words:** Extrusion, Dense Collagen, Ionic Strength, Maturation.

<footnote n="1">¹ Institut des Nanosciences de Paris, Sorbonne Universités, UPMC Univ Paris 6 and CNRS-UMR 7588, F-75005 Paris, France</footnote>

<footnote n="2">² Laboratoire de Chimie de la Matière Condensée de Paris, Sorbonne Universités, UPMC Univ Paris 6 and CNRS-UMR 7574, F-75005 Paris, France</footnote>
# 1 Introduction

Tendons can present chronic or acute injuries due to intrinsic or external factors which can also be combined [25]. The etiology of tendinopathies is indeed often recognized as being multifactorial [36],[45],[28] with causes including work and sports activities, drugs [21], age and biomechanical imbalance[25]. Spontaneous regeneration is limited due to a poor vascularization and a low number of cells. Treatments can be conservative, surgical, or both, with or without rehabilitation programs[36],[22],[34],[1]. Surgical treatments go from sutures to insertion of metallic implants or grafts, which can be auto, xeno or allografts. Synthetic grafts made of polyester (Dacron®), polytetrafluoroethylene (PTFE or Teflon®)[33], silicone rubber[42], polypropylene (Marlex® mesh)[6], polyethylene teraphtalate (PET, devices Orthotape® and Orthocoupler®)[26],[3] are also used. Depending on the system, more or less promising physiological results were obtained with, in some cases, no re-rupture occurrence observed at the level of the graft consolidation. Alongside with those synthetic grafts, grafts made out of biopolymers are also being developed with in mind the goal to mimic as well as possible the biochemistry and three-dimensional organization of tendon in order to optimize the biocompatibility (i.e. absence of immune rejection or toxicity[23], proper host-response[2], promotion of cell adherence, proliferation and differentiation). Tendons and ligaments belong to connective tissues that are characterized by the extra-cellular matrices (ECM) in which cells are embedded. The ECM is composed of many biochemical components such as collagens, non-collagen proteins, and proteoglycans[46],[32].

In tendon, collagens reach a concentration of 60-85% of the dry weight of the tissue[32], collagen I being the major structuring bio-component. Tendons and ligaments are organized in a hierarchical manner from collagen I molecules to fascicles [4]. Molecules, fibrils and fascicles are aligned in a preferred orientation in relation to the function of those tissues. For instance, excised rat-tail tendons present an overall alignment of the bundles along the tendon axis. This is clearly seen by cross-polarized optical microscopy. They also reveal units, ranging from 10 to 100 µm, of alternating oblique orientations to the tendon axis. The repetition of those units, referred to as "crimp" [27],[10], is a highly compliant element and was suggested to be a natural shock-absorber[40]. Elaborations of tendon-like scaffolds are inspired by this hierarchical construct and this anisotropy [29],[41]. Another important characteristic of tendons is their mechanical properties defined by the Young moduli, the ultimate tensile strength (UTS) and the strain at break as assessed along their longitudinal axis. Those vary from 0.5 to 1.9 GPa for the Young moduli and from 50 to 120 MPa for the UTS [19],[16]. Such values have not been yet achieved with wet simple extruded collagen-based thread scaffolds, with highest values obtained around 46 MPa for Young Modulus and 766 MPa for UTS with dehydration and stretching steps ([8],[18], [31], [11]). Higher values reaching 766 MPa and 110 MPa for Young modulus and UTS respectively are only achieved when chemical cross-linkers are being applied ([8],[18], [31], [11]). Thus future developments of scaffolds for tendon augmentation, and even more so for tendon replacements, will need to achieve higher mechanical prop-
erties. Indeed, the better the tendon struc-
tural characteristics and mechanical proper-
ties will be approached, the better the scaf-
fold will resist solicitations and promote cell
differentiation [38]. In order to address this
subject, we made use of the lyotropic prop-
erties of acidic-collagen I solutions (i.e: col-
lagen molecules self-organize into liquid crys-
tal phases as soon as a critical concentration
in collagen is reached) [12],[13]. These liquid
crystals can be stabilized thanks to a pH in-
crease to create organized fibrillar scaffolds.
The pH increase triggers the fibrils forma-
tion while preserving the long-range liquid-
crystalline organization [14]. We combined
this approach with an extrusion process to
generate high-density, anisotropic, collagen
threads. Using a homemade extruding ap-
paratus, we focused on determining the best
conditions to induce a stable sol-gel transi-
tion while preserving the alignment of the
molecules and achieving optimal mechanical
properties.

2 Material and Methods

Collagen type I preparation

Type I collagen was purified from young Wis-
tar or Sprague-Dawley rat tail tendons as
previously described[37]. Extracted tendons
were washed several times with phosphate-
buffered saline (PBS) to remove blood and
fat traces. They were subsequently soaked
and washed in 4 M NaCl solution to remove
cells. Finally, tendons were extensively rinsed
again with PBS and solubilized in 0.5 M acetic
acid. After successive and selective precipita-
tions, respectively at 0.3 M and 0.6 M NaCl,
the final collagen I pellets were solubilized in
500 mM acetic acid and thoroughly dialyzed
against the same solvent to completely remove

salt content. The solutions were kept at 4°C
and centrifuged at 41,000xg (Beckman Coul-
ter J26-XP) for 4 h before use. Sample purity
was assessed by SDS-PAGE electrophoresis.
Collagen concentration was determined from
the acidic solution by titrating the amount
of hydroxyproline. Collagen solutions were
concentrated using centrifugal filtration units
(Vivaspin®, Sartorius, with a 100kD cutoff)
spun at 3000xg, 10°C, until reaching the de-
sired final concentrations (60 mg/mL). Solu-
tions were then placed in 1 mL syringes and
degassed by centrifugation at 3000xg for 30
min at 10°C.

Extrusion of dense collagen solu-
tions

The home-made extrusion set-up, schema-
tized in Figure 1, was built to allow in-sight
visualization, (ii) application of great forces
compatible with highly viscous solutions, (iii)
precise monitoring of the extrusion speed and
buffer exchange during extrusion. Briefly, a 1
mL syringe, filled with a 60 mg/mL collagen
solution in 500 mM acetic acid, was mounted
with a 23-gauge (inner diameter of 390 µm)
blunt stainless-steel needle, and loaded verti-
cally. A motorized stage was used to bring
down a rod in contact with the syringe pis-
ton and to impose its speed. The extruded
fluid velocity was about 50 mm/s. The col-
lecting cuvette was filled with the fibrillogen-
esis buffer (pH 7.4). At the blunt needle exit,
extruded threads, plunging into the fibriloge-
nesis buffer, were observed and followed by an
optical through-hole mounting. Prior to their
use, the threads were kept and left for mat-
uration in fibrillogenesis buffer at room tem-
perature for 2 weeks under gentle agitation to
prevent their sticking together.
<figure><img src="image_112.png" /><figcaption>Figure 1: Extruding homemade experimental set up. A syringe (Terumo®) of 1 mL contains the collagen solution to be extruded. It has an inner diameter Dp equal to the piston diameter of 4.45 mm. Stainless steel blunt needles (Terumo®) of 390 or 650 µm inner diameters D, respectively 23G and 20G, are set at the syringe end. A steel rod is clamped to a stepping motorized stage, which imposes a velocity vp ranging from 0 to 2000 µm/s. The average output velocity V is calculated as follows: $V = vp \times (\frac{Dp}{D})^2$. In order to apply the same shear condition for different inner diameters, the output velocity must be the same. The collagen solutions are extruded at $V=50 \pm 2$ mm/s. A steel structure holds the syringe with its needle as well as an optical glass cell (Hellma®) containing the extrusion bath. Holes are drilled to create optical windows. The whole set-up is put inside a temperature-controlled enclosure. The extrusion bath can be removed at any time during the process to collect the extruded thread, while keeping the syringe in place. A quasi-parallel beam is created with a light source (S) limited by a diaphragm placed at the focal length of a convergent lens. The light beam illuminates the glass cell and the extrusion flow is followed thanks to a video camera equipped with a macro-objective and placed on the other side. Acquisition speeds of 600 frames/s can be reached. This optical system allows us to obtain extrusion images of a high contrast level as shown in Figure 2AB.</figcaption></figure>

Several fibrillogenesis buffers were tested with ionic strengths ranging from 0.15 M to 1.65 M (Table 1). For each buffer, the pH was adjusted prior to use to pH 7.4 with HCl or

NaOH solutions depending on the initial pH measured. The same buffer was used for the extrusion and the maturation. Ionic strength was calculated by taking into account all the
ions present in the solution according to the equation

$$ \sum_{i=0}^{n} c_{i} z_{i}^{2} $$

, where $c_i$ is the molar concentration of each ion and $z_i$ the charge of that ion, assuming ideal solution.

<table><caption>TABLE 1</caption><thead><tr><th>No</th><th> buffers</th><th>Concentrations</th><th>Ionic strength</th><th>D/D<sub>0</sub></th></tr></thead><tbody><tr><td>1</td><td>Na<sub>2</sub>HPO<sub>4</sub></td><td>50 mM</td><td>150 mM</td><td>1,4 +/- 0,1</td></tr><tr><td rowspan="4">2</td><td>PBS 1X (home made)</td><td></td><td></td><td></td></tr><tr><td>NaCl</td><td>137 mM</td><td></td><td></td></tr><tr><td>KCl</td><td>2.7 mM</td><td>165 mM</td><td>1,70 +/- 0,08</td></tr><tr><td>Na<sub>2</sub>HPO<sub>4</sub></td><td>8 mM</td><td></td><td></td></tr><tr><td></td><td>NaH<sub>2</sub>PO<sub>4</sub></td><td>1.5 mM</td><td></td><td></td></tr><tr><td rowspan="4">3</td><td>PBS 1X (commercial)</td><td></td><td></td><td></td></tr><tr><td>NaCl</td><td>137 mM</td><td></td><td></td></tr><tr><td>KCl</td><td>2.7 mM</td><td>166 mM</td><td>1,53 +/- 0,09</td></tr><tr><td>Na<sub>2</sub>HPO<sub>4</sub></td><td>8 mM</td><td></td><td></td></tr><tr><td></td><td>KH<sub>2</sub>PO<sub>4</sub></td><td>2 mM</td><td></td><td></td></tr><tr><td>4</td><td>Na<sub>2</sub>HPO<sub>4</sub></td><td>100 mM</td><td>300 mM</td><td>1,7 +/- 0,1</td></tr><tr><td rowspan="2">5</td><td>Na<sub>2</sub>HPO<sub>4</sub></td><td>50 mM</td><td></td><td></td></tr><tr><td>NaCl</td><td>150 mM</td><td>300 mM</td><td>1,43 +/- 0,06</td></tr><tr><td rowspan="2">6</td><td>Na<sub>2</sub>HPO<sub>4</sub></td><td>50 mM</td><td></td><td></td></tr><tr><td>KCl</td><td>150 mM</td><td>300 mM</td><td>1,32 +/- 0,05</td></tr><tr><td>7</td><td>PBS 5X (home made)</td><td></td><td>825 mM</td><td>1,00 +/- 0,05</td></tr><tr><td>8</td><td>PBS 10X (home made)</td><td></td><td>1650 mM</td><td>0,75 +/- 0,04</td></tr></tbody></table>

<caption>Table 1: *Fibrillogenesis buffers and collagen threads diameters evolution upon ageing*</caption>

## Maturation

Collagen threads (60 mg/mL in collagen), were kept over two weeks at room temperature in the different buffers listed in Table 1. Their diameters D were measured along the maturation process. We define the swelling ratio as $Q = D/D_0$ where D0 is the inner diameter of the needle. D is measured and

plotted over time. Standard Deviation of this ratio was calculated by:

$$ \Delta(\frac{D}{D_0}) = \sqrt{(\frac{\Delta D}{D_0})^2 + (\frac{D\Delta D_0}{(D_0)^2})^2} $$

where $\Delta D$ and $\Delta D_0$ are the respective standard deviation of the thread diameter and the needle inner diameter measurements.