Alginate, a natural polyelectrolyte
Alginate is a natural polysaccharide extracted from brown seaweed and its basic structure consists of an alternation of D-Mannuronic acid (M-block) monomers and L-Guluronic acid (G-block) monomers (see Figure 2.1). With carboxylate functions -COO, alginate is a polyelectrolyte. Its molecular weight can range from 50 g/mol to 106 g/mol. The alternation and length of G-blocks and M-blocks depend on the geographical origin of alginates and can be determined by 1H-NMR spectroscopy (see Section 2.2.4.a)).
Penman and Sanderson  reported that brittle gels are obtained from alginates with low M/G ratio while elastic gels are formed from alginates with high M/G ratio. However, it must be noted that the gel properties are also related to the presence of homopolymeric block structures (FGG or FMM) and alternating blocks (FMG).
Rheology of polyelectrolytes
Definition and conformational properties Polyelectrolytes are polymers with ionizable groups. In solution they are dissociated into polyvalent macro-ions (ions on the polymeric chain) and counterions of opposite charge. The high charge of the polyion induces a strong electric field which attracts these counterions, resulting in their partial condensation onto the polymer backbone. This phenomenon is referred to as the Manning-Oosawa counterion condensation [2, 3] and is illustrated in Figure 2.2. This interaction between the polyvalent macro-ion and counterions is the origin of characteristic properties of polyelectrolytes remarkably dierent from solutions of uncharged polymers. Most of the polyelectrolyte chains are long and flexible but their conformation depends on their charge and interaction with counterions . In dilute salt-free solution, the counterions are homogeneously distributed in the solution volume and so the charges on the polymer chains interact via the unscreened Coulom potential. De Gennes et al.  implemented the concept of « blobs » to polyelectrolytes: the polyion can be separated into parts called ’blobs’ of characteristic size D. For length scales smaller than D, the polymer conformation is similar to the one of a neutral
polymer in solution. Therefore, within a blob, the statistics of the chain is determined by the thermodynamic interaction between the neutral polymer and the solvent. This means that the total energy of the electrostatic interaction between charges inside the blob is of the order of the thermal energy kT [4–6]. On larger length scales (l > D), the blobs are subjected to electrostatic repulsion so they tend to align thus forming a fully extended chain of electrostatic blobs of length L as illustrated in Figure 2.3 (a).
In semi-dilute solution we can define a correlation length . For length scales smaller than , the dilute solution scaling applies. On larger length scales (l > ), the chain can be described by a random walk of correlation blobs as shown in Figure 2.3 (b).
With increasing charge or concentration, the flexible polymer chain changes its shape from a contracted random coil to a fully extended conformation. Therefore, when studying the rheology of polyelectrolyte solutions, we need to distinguish between the case of a solution without salt and the case where the polyelectrolyte is in the presence of salt.
Osmotic pressure in polyelectrolytes
Another important property of polyelectrolytes which is remarkably dierent from neutral polymers is the osmotic pressure. In fact, the osmotic pressure of a semi-dilute solution of neutral polymers is essentially the thermal energy kT per correlation volume (3): p kT 1 3 c > c.
Polyelectrolyte solutions have an additional contribution to their osmotic pressure because of ions i, such that the total osmotic pressure is p + i. Donnan equilibrium  requires charge neutrality on both sides of the membrane separating the polyelectrolyte solution from pure solvent. To calculate the ion contribution, we must take into account the counterions (of concentration c in number density) and the salt ions (of concentration cs in number density) . In the case of polyelectrolyte solutions with many more counterions than salt ions (i.e. c >> 2Acs, A being the number of monomers between uncondensed charges), the ion contribution can be written as: i kT c A c >> 2Acs (2.1) In the other limit, where salt concentration is high, the counterions are almost uniformly distributed on both sides of the membrane and the salt redistributes to maintain charge neutrality. This redistribution gives a contribution to osmotic pressure and can be expressed as follows: i kT c2 4 A2 cs c << 2Acs.
All sodium alginates used during this study are purchased from AGI company. We mainly focus on one batch in the entire manuscript which we characterize below.
M/G ratio determination 1H-NMR spectroscopy is performed on 1 g/L sodium alginate solution in D2O. In order to increase the signal-to-noise ratio, the analyses are achieved at a temperature of 80C with a Bruker Avance III 600 MHz. The NMR experiments are recorded with a spectral width of 6009 Hz, a relaxation delay of 20 s and a number of scans of 64.
For sake of consistency, our NRM spectrum is compared with other data collected by Fertah et al.  and Fenoradosoa et al. . We obtain FG = 0.65 and aM/G ratio equal to 0.55. This ratio is quite low, which is characteristic of strong and brittle gels according to .
Molecular weight determination Size exclusion chromatography (SEC) is performed on alginate samples to determine their average molecular weight. Solutions of sodium alginate are prepared at 2 mg/mL in NaNO3 0.2 M. The measurements give an average molecular weight of Mw = 250 kg/mol and Mn = 125 kg/mol which results in a polydispersity index of 1.7 0.3.
Experimental protocol to reduce alginate Mw
We have decided to reduce the chain molecular weight (Mw) in order to be able to add more alginate in the suspension without increasing considerably its viscosity while ensuring a larger bead stiness after gelation, to prevent its deformation at the end of the drip-casting process.
Holme et al.  studied the thermal depolymerization of alginates in the solid state. Following this idea, we manage to have a robust and reproducible potocol to cut the alginate chains. We use a glass container (20 mL) in which we put sodium alginate powder (5 g), close it and put it in a drying oven at 120C for 24 hours. After this heating step, the container is transferred in an iced bath for several minutes and then stored in the fridge.
We analyze the resulting powder by size exclusion chromatography and 1H-NMR, giving a molecular weight of Mw = 60 kg/mol and a polydispersity index of 1.7 0.3. Whenever this protocol is used to reduce the sodium alginate chain size, a SEC analysis is performed to be sure we obtain the same molecular weight value of 60 kg/mol.
Table of contents :
Résumé en Français
1.1 Industrial context
1.1.1 Ceramic beads applications
1.1.2 Ceramic beads manufacturing
1.2 Scientific questions and challenges to overcome
I Alginate solutions and suspensions: rheology and gelation
Introduction of Part I
2 Suspension preparation & rheological properties
2.1 Polyelectrolytes solutions & particle suspensions: state of the art
2.1.1 Alginate, a natural polyelectrolyte
2.1.2 Rheology of polyelectrolytes
2.1.3 Osmotic pressure in polyelectrolytes
2.1.4 Interaction with particles
2.2.1 Calcium chloride
2.2.3 Dispersing agent (PAA)
a) Chemical characterization
b) Experimental protocol to reduce alginate Mw
2.3 Suspension preparation
2.3.1 Zirconia milling
2.3.2 Addition of alginate and suspensions studied
2.4 Zirconia – PAA (dispersant) interaction
2.4.1 Total Organic Carbon (TOC) principle
2.5.1 Apparatus and protocol
2.5.2 Alginate solutions
a) Long alginate chains (Mw = 250 kg/mol)
b) Shorter alginate chains (Mw = 60 kg/mol)
2.5.3 Zirconia – alginate suspensions
a) Influence of alginate concentration on flow behavior
b) Understanding the yield-stress behavior
c) Controling the yield-stress behavior with alginate Mw
3 Gelation mechanism & bead characterization
3.1 Polyelectrolyte gels: state of the art
3.1.1 Alginate beads
3.1.2 Alginate gelation properties
3.2 Experimental investigation of the gelation process
3.2.1 Preliminary observations of syneresis
3.2.2 Quantification of syneresis
a) Eect of alginate concentration
b) Eect of zirconia addition
c) Eect of calcium concentration
d) Eect of alginate molecular weight
e) Evolution of gel front with time
3.3 Bead mechanical characterization
3.3.2 Experimental protocol
3.3.3 Experimental results
a) Bead behavior upon loading and unloading
b) Influence of gelling time
c) Influence of calcium concentration
d) Influence of zirconia content
e) Influence of alginate concentration
f) Elastic modulus estimation
II Physics of drip casting process
Introduction of Part II
4.1 On the spreading of liquid drops: state of the art
4.1.1 Newtonian drops impacting a solid surface
4.1.2 Yield-stress fluids impacting a solid surface
4.1.3 Droplets impacting a liquid bath
4.1.4 Impacts of solidifying droplets
4.2 Experimental results: impacts of alginate-zirconia suspensions
4.2.1 Experimental observations
4.2.2 Influence of impact speed
4.2.3 Influence of calcium concentration
4.2.4 Influence of rheology
4.3 Simulation & comparison with experiments
4.3.2 Numerical results
4.3.3 Comparison with experimental data
5.1 Droplet relaxation: state of the art
5.2 Driving force of the relaxation
5.2.1 Experimental observations
5.2.2 Hypothesis for the driving force
5.3 Obstacles to relaxation
5.3.1 Suspension viscosity
5.3.2 Gel mechanical properties
5.4 Stress estimation
5.5 Simulation & comparison with experiments
5.5.2 Model results and comparison with data
a) Influence of calcium concentration
b) Influence of suspension viscosity
c) Influence of gel modulus
5.5.3 Limitations of the model
6 Conclusion and perspectives
A Relaxation of alginate droplets without zirconia
A.1 Experimental observations
A.2 Estimating forces
B Relaxation of suspension droplets with low Mw alginate
B.1 Experimental observations