# Structure–Elastic Properties Relationships in Gelling Carrageenans

## Abstract

**:**

## 1. Introduction

**G**-units), and 4-linked α-D-galactopyranose (

**D**-units) or 4-linked 3,6-anhydrogalactose (

**A**-units). Figure 1 presents the chemical structures of the gelling carrageenans that are most utilized commercially. Iota-carrageenan (Iota) is nearly a homopolymer of

**G4S-DA2S**disaccharide units, containing a very low amount (of the order of 5 mol%) of

**G4S-DA**disaccharide units [6]. Kappa-carrageenan (Kappa) is a little more heterogeneous, with up to 10 mol% of

**G4S-DA2S**disaccharide units discontinuing blocks of

**G4S-DA**disaccharide units [6]. A third gelling carrageenan, commercially known as kappa-2 or weak kappa (Kappa2), has progressively gained industrial impetus. Kappa2 replaces mixtures of Kappa and Iota in niche applications where intermediate gelling properties between the hard and brittle gels formed by Kappa, and the softer but deformable gels formed by Iota, are needed [7]. Kappa2 is a random block copolymer made of sequences of

**G4S-DA**(making up 45–80 mol% of the hybrid chain [7]) and

**G4S-DA2S**[8,9]. Depending on the seaweed and the extraction route used to isolate the polysaccharide,

**G4S-DA**and

**G4S-DA2S**blocks can be separated by more sulfated disaccharide units, such as

**G4S-D6S**(mu-carrageenan) and

**G4S-D2S,6S**(nu-carrageenan) [6,10].

## 2. Structure and Elastic Properties of Carrageenan Gels

**G4S-DA**units remains present as impurities in Iota [28]. Thus, Table 1 and Table 2 report the details of the carrageenan samples used when available, namely, the type of cations used to gel the carrageenan and the type of carrageenan.

**Table 1.**Structural and elastic properties of Kappa gels prepared with salt conditions and carrageenan samples specified in the first column “System”. Mesh size ε, length L and diameter d of strands were obtained from electronic microscopy (EM), fluorescence recovery after photobleaching (FRAP), particles tracking (PT), atomic force microscopy (AFM), nuclear magnetic resonance (NMR), small angle X-ray scattering (SAXS) or wide angle X-ray scattering (WAXD). Column G

_{0}reports the range of gel linear shear storage modulus G

_{0}measured at temperatures between 10 and 25 °C. The last column lists the referenced literature for the corresponding row.

Sample | Structural Features | Rheology | Ref. | |||
---|---|---|---|---|---|---|

Carrageean Form ^{1} | Salt | ε (nm) | L (nm) | d (nm) | G_{0} (kPa) | |

Com. | No salt | 500–10^{4} (EM ^{3}) | 5 × 10^{4} (EM ^{3}) | 100–500 (EM ^{3}) | 0.02–20 | [29] |

Com. | No salt | 2 × 10^{3}–10^{4} (EM ^{3}) | 3 × 10^{3}–10^{4} (EM ^{3}) | 2.34 (SAXS) | 2.6 | [30] |

Na | CaCl_{2} | Up to 5 × 10^{4} (EM ^{3}) | Up to 5 × 10^{4} (EM ^{3}) | 10^{2}–10^{3} (EM ^{3}) | 0.5–3.5 | [31] |

Na | NaCl | 6 (WAXD) | >100 (AFM ^{3}) | 1.1 (AFM) | 0.1 | [32] |

K | KCl | 4.3 and 6 (WAXD) | >100 (AFM ^{3}) | 1.5 (AFM) | 1.4 | [32] |

Na | KCl | <10 (NMR ^{4}) | n.a. ^{2} | n.a. ^{2} | 40 | [33] |

Na | KCL | <80 (FRAP) | n.a. ^{2} | n.a. ^{2} | 1 | [34] |

Na | CaCl_{2} | <80 (FRAP) | n.a. ^{2} | n.a. ^{2} | 0.01 | [34] |

Na | KCl | <100 (PT) | n.a. ^{2} | n.a. ^{2} | 2 | [35] |

Com. | KCl | 9.5 (SAXS) | n.a. ^{2} | 25.1 (SAXS) | 11 | [36] |

Com. | CaCl_{2} | 17.7 (SAXS) | n.a. ^{2} | 41 (SAXS) | 130 | [36] |

Na | KCl | n.a. ^{2} | <200 (AFM ^{3}) | 1.5–2 (AFM) | 2 | [16] |

Na | CaCl_{2} | n.a. ^{2} | <500 (AFM ^{3}) | 1–1.5 (AFM) | 0.2 | [16] |

^{1}Com.: commercial carrageenan used as received; Na: sodium form; K: potassium form; Ca: calcium form.

^{2}n.a.: non available.

^{3}Estimates from pictures reported in the corresponding reference.

^{4}Estimate from the hydrodynamic radius of the polymer probe used in NMR.

**Table 2.**Elastic and structural properties of Iota gels. Same column labelling and notes as in Table 1.

Sample | Structural Features | Rheology | Ref. | |||
---|---|---|---|---|---|---|

Carrageenan Form ^{1} | Salt | ε (nm) | L (nm) | d (nm) | G_{0} (Pa) | |

Na | NaCl | 6 (WAXD) | <100 (AFM ^{3}) | 2 (AFM) | 15 | [32] |

K | KCl | 6 (WAXD) | <100 (AFM ^{3}) | 2 (AFM) | 100 | [32] |

Na | KCl | >>10 (NMR) ^{4} | n.a. ^{2} | n.a. ^{2} | 400 | [33] |

Na | KCl | <90 (FRAP) | n.a. ^{2} | n.a. ^{2} | 20 | [34] |

Na | CaCl_{2} | <90 (FRAP) | n.a. ^{2} | n.a. ^{2} | 80 | [34] |

Na | KCl | >>100 (PT) | n.a. ^{2} | n.a. ^{2} | 100 | [35] |

Com. | KCl | n.a. ^{2} | n.a. ^{2} | 14–62 (SAXS) | 500 | [36] |

Com. | CaCl_{2} | n.a. ^{2} | n.a. ^{2} | 7–48 (SAXS) | 310 | [36] |

Na | NaCl | n.a. ^{2} | <200 (AFM ^{3}) | 1 (AFM) | 10 | [16] |

Na | KCl | n.a. ^{2} | <200 (AFM ^{3}) | 0.8 (AFM) | 18 | [16] |

Na | CaCl_{2} | n.a. ^{2} | >10^{3} (AFM ^{3}) | 1 (AFM) | 18 | [16] |

Com. | NaCl | 1.2–1.5 nm (SAXS) 2 × 10 ^{4}–3 × 10^{4} (EM) | 27.5 (SAXS) | 7.5 (SAXS) | 130–160 | [37] |

Com. | No salt | <2 × 10^{3} (EM ^{3}) | <2 × 10^{3} (EM ^{3}) | 10^{2}–10^{3} (EM ^{3}) | 270 | [38] |

^{1}Com.: commercial carrageenan used as received; Na: sodium form; K: potassium form; Ca: calcium form.

^{2}n.a.: non available.

^{3}Estimates from pictures reported in the corresponding reference.

^{4}Estimate from the hydrodynamic radius of the polymer probe used in NMR.

_{0}of gels measured with small amplitude oscillatory shear. Data compiled in Table 1 and Table 2 confirm that overall Kappa forms gels which are one order of magnitude more elastic than Iota gels. Only a few works listed in Table 1 and Table 2 presented the large strain behavior of carrageenan gels. The strain for onset of nonlinear behavior γ

_{NL}was reported to depend on the gelling conditions (both salt and carrageenan concentrations), varying between 10% [30] and 1% [32] for Kappa, and between 15% [32] and 100% [38] for Iota. This again confirms the well documented brittleness of Kappa gels when compared with more strain-resistant Iota gels. However, none of these studies commented on the qualitative hardening or softening of gels under large strains. Similarly, the concentration dependence of G

_{0}is virtually ignored in the set of references compiled in Table 1 and Table 2. A single work reported on the power law scaling G

_{0}~c

^{n}where c is the carrageenan concentration in the gels. An exponent n = 2.7 was found for Kappa gels [29]. Such scaling being of theoretical relevance, it has, however, been studied in dedicated rheological research (see Section 3, below). Gels are usually formed in the shearing geometry of the apparatus, and the gelling conditions were different from those used to gel the samples for structural characterization. This brings additional complexity for the identification of structure–elasticity relationships, since both properties are known to depend on the gelling route (see e.g., [10,26]). Structure–elasticity relationships in Kappa2 gels have not received much attention in the open literature [10], in contrast to the study of relationships in gelled mixtures of Kappa and Iota [20,33,43]. Section 4, below, will start to fill this gap of knowledge.

_{0}values. Aggregates of filaments and further clustering of aggregates were also reported recently for Kappa [42]. Similar clustering was also reported for Iota gels thereby explaining their weaker elasticity when compared to Kappa gels [35]. Thus, cluster elasticity, as well as elasticity associated with cluster–cluster interactions, could also play a role in gels rheology, as it is in colloidal systems [44,45]. Returning to the macromolecular scale, dynamic heterogeneity has recently been highlighted, as free carrageenan chains with larger mobility in the solvent [33] can be released from the Iota or Kappa gel matrix [34]. However, the impact of such free chains on the gels’ elasticity remains unclear. Differences in the persistence length of Kappa and Iota filaments has also been inferred from AFM imaging [12,32]. The effect of cations on the persistence length of Kappa filaments was earlier suggested by EM pictures which related the presence of more flexible filaments with the weaker elasticity of the gels [26]. As explained below, in the theoretical section, the persistence length affects the rigidity of the corresponding filament and thus has a direct impact on the network’s elasticity.

## 3. Theoretical Analysis of the Elasticity of a Network of Semi-flexible Filaments

_{f}of strands between two crosslinks [47], whereas the starting point for the strain expansion of the shear modulus is the phenomenological Blatz–Sharda–Tschoegl equation [48]. Equation (2a) was used with some success to reproduce the strain hardening of gelatin gels. More important, it incorporates the concentration dependence or the elastic shear modulus at small strain, G

_{0}(see Equations (2c) and (2d)). The latter are scaling models originally proposed by Jones and Marques [49] to describe networks of rod-like polymers with constant crosslink functionality (only strand length is allowed to vary). These scaling laws were claimed to give a good account for carrageenan gels if one considers pure rod strands with d

_{f}= 1 [49]. In addition to Equation (2a), Groot et al. [47] also tested another molecular model where the strain hardening stems from the specific geometrical aspects of the flexible chains connected to rod-like strands making up the network (see Equation (3a)). This phenomenological model thus describes a certain degree of structural heterogeneity in the gel.

**Table 3.**Theoretical predictions for the shear elastic modulus G and its nonlinear dependence G(γ) with shear strain γ. G

_{0}is the (linear) elastic shear modulus at small strain. References to the original papers where the G(γ) expressions have been first introduced are gathered in the last column labelled “Ref.”.

Equations | G(γ) | Structural Parameters | Refs. |
---|---|---|---|

(1a) (1b) | $G\left(\gamma \right)={G}_{0}\mathrm{exp}\left({\left(\gamma /{\gamma}^{\ast}\right)}^{2}\right)$ ${\gamma}^{\ast}={\lambda}_{max}-{\left({\lambda}_{max}\right)}^{-1},\text{}{\lambda}_{max}=\frac{{l}_{max}}{{l}_{0}}$ | ${\gamma}^{\ast}$: Critical value of shear strain at which stiffening is dominant; l _{max}: Length of a fully extended strand;l _{0}: Length of a strand at rest. | [46] |

(2a) (2b) | $G\left(\gamma \right)=\frac{2{G}_{0}}{{n}_{BST}}\frac{{\lambda}^{{n}_{BST}}-{\lambda}^{-{n}_{BST}}}{{\lambda}^{2}-{\lambda}^{-2}}$ $\lambda =\frac{1}{2}\gamma +\sqrt{1+\frac{1}{4}{\gamma}^{2}},\text{}{n}_{BST}=\frac{{d}_{f}}{{d}_{f}-1}$ | d_{f}: Fractal dimension of strands making up the network;c: Polymer volume fraction; Enthalpic elasticity: strands are connected by rigid (frozen) crosslinks [50]; Entropic elasticity: strands are connected by mobile crosslinks [50]. | [47,49] |

(2c) | ${G}_{0}~{c}^{\frac{3+{d}_{f}}{3-{d}_{f}}}$ enthalpic | ||

(2d) | ${G}_{0}~{c}^{\frac{3}{3-{d}_{f}}}$ entropic | ||

(3a) (3b) (3c) | $G\left(\gamma \right)={G}_{0}A+{G}_{0}B\text{}\frac{{\gamma}^{2}}{1+0.125{\gamma}^{2}}$ $A=1.12+0.24\frac{l}{{R}_{0}}+0.068{\left(\frac{l}{{R}_{0}}\right)}^{2.5}$ $B=0.095+0.15\frac{l}{{R}_{0}}+0.114{\left(\frac{l}{{R}_{0}}\right)}^{2.5}$ | l: Length of a rod-like strand connected to swollen chains in good solvent with radius of gyration R_{0}. | [47] |

(4a) | $G(\gamma )=\frac{{G}_{0}}{3}(1+2{(1-\beta \frac{{\gamma}^{2}+3}{3})}^{-2})$ | β: Chain elongation ratio given by $\beta =\frac{\langle {R}_{in}{}^{2}\rangle}{{R}_{max}{}^{2}}$, where $\langle {R}_{in}{}^{2}\rangle $ is the mean-square average end-to-end distance of a strand in the unstrained network and ${R}_{max}{}^{2}$ is the square of the end-to-end distance of the fully extended strand; E _{bend}: Bending rigidity of a strand;c: Polymer volume fraction; | [50] |

(4b) | ${G}_{0}~{E}_{bend}{c}^{2}$ | ||

(5) | $G\left(\gamma \right)=\frac{2nkT}{3}{x}^{2}(\frac{1-{x}^{4}}{e\pi {\left[1-\left(2+{\gamma}^{2}\right){x}^{2}+{x}^{4}\right]}^{2}}-e{\pi}^{2})$ | n: Crosslink density in the network; k: Boltzmann constant; T: Temperature; e: Dimensionless strand stiffness parameter comparing bending and thermal energy; $x=\frac{\epsilon}{{L}_{c}}$, where ε is the distance between crosslinks, and L _{c} is the contour length of a strand joining the crosslinks. | [51] |

(6a) | $G\left(\gamma \right)=\frac{{G}_{0}}{2}{\left[1-{\left(1+\frac{1}{2}\gamma \left(\gamma -\sqrt{{\gamma}^{2}+4}\right)\right)}^{\frac{1}{2}}\right]}^{5}\times \frac{\left({\gamma}^{2}+2\right)\sqrt{{\gamma}^{2}+4}-\gamma}{{\left[\gamma +\frac{\left(\gamma -\sqrt{4+{\gamma}^{2}}\right){\gamma}^{2}}{2}\right]}^{1/2}}$ | E_{bend}: bending rigidity of a strand with diameter d;c: polymer volume fraction. | [17] |

(6b) | ${G}_{0}~\frac{{E}_{bend}}{{d}^{4}}{c}^{5}$ |

_{max}in Table 3). Note that for stiff chains (rod-like strands), this theory indicates that the network modulus, G

_{0}, depends on the bending elasticity of the chains and shows a quadratic scaling with the polymer concentration (Equation (4b)), which is a special case of an enthalpic network with d

_{f}= 1 (see Equation (2c)). Equation (5) builds on two network characteristics to describe the strain hardening [51]. One relates to the topology of the network. This is described by the ratio x of the distance between two crosslinks (i.e., the mesh size of the network whose topology is modeled by a cubic structure where each edge crosslinks three wormlike chains) over the contour length of the wormlike chain joining the two crosslinks. The other network characteristic relates to a stiffness parameter e which balances the contribution of chain bending and thermal elasticity. Equations (4a) and (5) successfully described the strain hardening of collagen, actin or fibrin networks [50,51]. However, the rationalization of the network’s elasticity by two parameters, as in Equation (5), offers the possibility of predicting the first normal stress difference N

_{1}of sheared networks.

_{1}is found to be negative. This is in contrast with a finer network of more flexible strands which exhibits a positive N

_{1}under shear [51].

_{0}, which also presents a strong dependence with the polymer concentration (see Equation (6b)).

_{0}. Table 4 gives a partial account of such studies performed with rotational rheometry, where power law relationships were documented or identified after re-plotting and fitting the data to a power law equation.

_{f}= 1) and the elasticity of the network as purely entropic (Equation (2d)). On the other limit, taking d

_{f}= 2.5 for the fractal dimension of chains in an incipient gel [58] and an enthalpic network (Equation (2c)), one reaches n = 11. Nonetheless, the majority of the rheological studies compiled in Table 4 suggests that $2\le n\le 3$, which is in harmony with an enthalpic network of rod-like filaments (Equations (2c) and (4b)) with bending rigidity ruling the network’s elasticity. This was expected from the L and d values listed in Table 1 and Table 2. Clearly, testing of the nonlinear elastic properties is needed to extract more structural information than simply d

_{f}and also validate the consistency of a theoretical treatment of a rheological data set within a carrageenan concentration range. Such an exercise is presented in the following sections.

## 4. Linear–Nonlinear Elastic Properties of COMMERCIAL KAPPA and Iota and of a Selected Kappa2

**G4S-DA**was detected in the commercial Iota. The carrageenan datasheets indicate that K

^{+}cations are predominantly present in these commercial samples. However, gels formed by Kappa in KCl salts are prone to significant water syneresis [59], inherently leading to rheometrical issues. Kappa and Iota gels were thus prepared in 0.1 M NaCl by mixing the corresponding amount of carrageenan with 0.1 M NaCl at 80 °C for 30 min. Note that under such salt conditions nearly similar intrinsic viscosities were measured for both Kappa and Iota [60]. Kappa2 was selected from a series of hybrid carrageenans extracted from Mastocarpus stellatus seaweeeds [61]. This sample was isolated in the Na

^{+}form, and the copolymer chain is made of 51.2 mol% of

**G4S-DA**, 31.7 mol% of

**G4S-DA2S**and 17.1 mol% of non-gelling carrageenan disaccharide (mu- and nu-carrageenans). Though this extract showed the best gelling properties [61], it requires more than 3 wt.% Kappa2 to give gels with sufficient elasticity in 0.1 NaCl to allow rheological testing. Thus, with a view to testing a wider range of Kappa2 concentrations, Kappa2 gels were formed in 0.1 M KCl.

_{0}= 24 kPa in Figure 3a) to the denser network with smaller pores in the corresponding EM picture. Thus, network connectivity seems to be related to the larger elasticity of Kappa gels, whereas differences in the elasticities of Kappa2 and Iota gels can only be rationalized so far by differences in the E

_{bend}of filaments (see Equations (4b) and (6b) in Table 3) or by the enthalpic or entropic natures of the corresponding networks (see Equations (2c) and (2d) in Table 3).

_{NL}is shifted to smaller strains as the concentration in Kappa is increased, spanning a range between 10% and 1% for the range of concentrations tested. This is in harmony with the values for γ

_{NL}reported in a few studies [30,32,63], which also document the abrupt decay seen in Figure 4 for both G′ and G″ beyond γ

_{NL}. A strain softening, characterized by a smoother drop in G′ coinciding with a local maximum in G″, is also found at the smallest and largest concentrations tested before a more acute drop in the shear moduli at greater strains. The inset to Figure 4a reports the concentration dependence of G

_{0}for all gels tested and extracted from the plateau in G′(γ) at small strains. The fit of the G

_{0}data to a power law returns an exponent of n = 3.5 ± 0.4. The latter is in fair agreement with those figures listed in Table 4.

_{0}data is indicative of the difficulty of achieving reproducible experiments, which calls for additional nonlinear testing with different protocols and shearing geometries.

_{max}, where a maximum in the loss modulus G″ is located. A qualitatively similar strain hardening has recently been reported for an Iota gel (2 wt.% with no salt) [38], whereas strain softening has been documented when Iota is gelled in the presence of salts [32,63]. Indeed, the strain hardening is less evident for the gels formed at higher carrageenan concentrations for both Iota and Kappa2.

_{0}. The scaled curves highlight the power law dependence G

_{0}= Ac

^{n}which is displayed in the insets to Figure 7, with n = 2.01 ± 0.08 for Iota and n = 3.1 ± 0.1 for Kappa2. This concentration scaling is in harmony with the theoretical treatments reviewed in Table 3: the structural and elastic features of worm-like filaments remain the same for all concentrations (see Equations (2c), (2d) and (4b)).

## 5. Rationalizing Structural and Elastic Results by Mechanical Models

_{e}is needed to delay the strain hardening towards larger strain amplitudes [64]. Equation (6a) gives a too steep hardening which cannot accommodate the nonlinear elastic properties of the carrageenan gels tested here. The fact that this theory cannot describe the networks built by Iota and Kappa2 was expected since the power law exponents n computed from the concentration dependence of G

_{0}largely differ from 5 and also because structural analysis has long established the existence of crosslinks in Iota gels and pre-gels [19]. Similarly, Figure 8a indicates that Equation (3a) cannot reproduce the experiments with Iota gels. In contrast to this, Figure 8b suggests that the Kappa2 gel is made of rod domains which are 100 times larger than the coils connecting them (see parameter L/R

_{0}in Table 5).

_{max}/l

_{o}, β or x in Table 5), and this inherently explains why nonlinearity is reached at smaller strains for Kappa2 when compared to Iota. Based on the established differences in the persistence lengths of Kappa and Iota [12], one can conjecture that the copolymeric nature of Kappa2 entails a persistence length between those of Kappa and Iota, giving rise to self-assembly into straighter filaments than in Iota. The structural information conveyed by the pictures displayed in Figure 3 is that both Kappa2 and Iota gels exhibit similar network density, i.e., they show similar ε values. Taking on board the theoretical meaning of parameter x in Equation (5), this suggests that L

_{c}is larger for Iota, as expected from the above arguments about persistence length. Interestingly, parameter e in Equation (5) is found to be 0.55 for both types of carrageenans, thus suggesting that self-assembly occurs with filaments of similar elasticity (see Table 3), resulting from a balance between their bending rigidity and their conformation [51].

_{f}computed from the nonlinear elastic behavior of Kappa2 gels (Equation (2a)) does not comply with the power law exponent n = 3.1 ± 0.1 inferred from their linear elastic behavior. n values computed from d

_{f}using either the enthalpic or entropic hypothesis are significantly smaller (2.3 and 1.65, respectively; see Table 5). The same issue arises for Iota gels but is less dramatic since the entropic n estimated from d

_{f}is 2.4, which is closer to the value (n = 2.01 ± 0.08) estimated from the concentration scaling of G

_{0}. Note here that the more entropic nature of the Iota gels complies with the looser nature of the network inferred from the data analysis with Equations (1a), (4a) and (5), and the fact that a Gaussian elasticity G

_{e}needs to be added to Equation (4a) to reproduce its elastic behavior [50]. However, Kappa2 gels require larger contributions from Gaussian elasticity to reproduce the data with Equation (4a). This is at odds with the strain hardening attributed to the presence of more stretched filaments between crosslinks. In addition, the scaling of G

_{0}with the concentration in Kappa2 is significantly larger than the quadratic prediction of Equation (4b). Thus, Gaussian chains are needed to describe the larger linear elasticity of Kappa2 gels, whereas straighter filaments (see parameter β) are required to impart strain hardening. This is, indeed, the second issue of the data analysis presented here. Additional structural parameters in Equation (4a) are thus needed to better describe the elastic behavior of networks whose elastic nature is between entropic and enthalpic [24,50]. Note here that Equation (5) partially helps in settling this issue to some extent. It incorporates the filament flexibility (parameter e) to give a more consistent description of the nonlinear elasticity of both Kappa2 and Iota, which only differ by their structural parameter x.

## 6. Conclusions and Perspectives

_{0}on the carrageenan concentration and the strain hardening behavior of Iota gels. Although not sufficiently documented in the literature, the strain hardening is established in the present study and rationalized by theoretical models. The latter explain the quadratic concentration scaling of the strain hardening which stems from the rod-like shape of the filaments (with fractal dimensions of the order of 1.7) imparting more enthalpic elasticity than entropic elasticity to the network.

_{0}. Due to its larger persistence length, Kappa self-assemble in straighter and more connected strands when compared with Iota. However, a review of the literature indicates that both systems show nearly identical strand lengths (of the order of 100 nm) and radii (of the order of 1 nm). Mesh sizes in the network are also reported to be nearly identical. Therefore, the superior elasticity of Kappa gels has been assigned to the greater stiffness of Kappa strands (or filaments), as was suggested by authors who established that Kappa forms straighter strands [26,32]. Several recent theoretical treatments of filamentous networks’ elasticities, briefly reviewed here, suggest that in this case Kappa gels should be strain hardening. However, strain thinning followed by an abrupt gel yielding is consistently reported in the limited sample of the literature reviewed here. The present study confirms this high strain behavior.

- -
- Effects of stress build-up in the network and the systematic study of normal stresses under large strains. A volume change often accompanies the sol–gel transition during cooling. The latter triggers the build-up of stresses within the gel (pre-stress), which can be monitored or even removed by controlling normal stresses [65]. In addition, such pre-stress influences the phase diagram of filamentous networks [51] and thus their nonlinear elastic response, which can show negative or positive Poynting effects. Thus, measuring normal stresses generated by gel-setting and large strains is important to help distinguish between entropic or enthalpic elasticity in filamentous networks [51] and to clarify the absence of strain hardening in Kappa gels.
- -
- In rheometer structural characterization during carrageenan gel build-up, e.g., with in situ birefringence measurements [66], should be re-visited to avoid differences in the thermal history of gels prepared for structural characterization and rheology. Further, such rheo-optical measurements will help to assign structural changes to large strain behavior.
- -
- Nonlinear elasticity should be studied with additional methods which incorporate the time of deformation [67], not restricting studies to the dynamic oscillatory testing reviewed here. In particular, such time needs to be taken into account during the thixotropic study of carrageenan gels, a topic of industrial interest not touched on here.
- -
- Most importantly of all, and as pointed out by Picullel [19], efforts should be made for studying model carrageenans, i.e., with established disaccharide composition and a single type of counter-ion. Thus, the tailored extraction of carrageenans from selected seaweeds seems preferable to converting commercial samples into a single cation form, since such processes are known to degrade the polysaccharide [68].

## Funding

## Conflicts of Interest

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**Figure 2.**The most popular models of the self-assembling of carrageenan single helices (

**A**,

**B**) or double helices (

**C**,

**D**) giving rise to a three dimensional network showing elasticity: (

**A**,

**C**) represent self-assembling at the helical (or secondary structure) level, whereas (

**B**,

**D**) represent self-assembling at the super-helical level (tertiary or quaternary structures).

**Figure 3.**CryoSEM pictures and mechanical spectra (insets) of gels formed with 2 wt.% carrageenan at 20 °C for: (

**a**) Kappa in 0.1 NaCl; (

**b**) Iota in 0.1 NaCl; (

**c**) Kappa2 in 0.1 KCl. White bars in CryoSEM pictures indicate 30 microns.

**Figure 4.**Response of Kappa gels (squares: 0.55 wt.%; circles: 0.75 wt.%; triangles 1.1 wt.%; diamonds: 1.5 wt.%; stars: 2 wt.%) to a sweep in the dynamic shear strain γ at 1 Hz and 20 °C: (

**a**) storage modulus G′; (

**b**) loss modulus G″. Inset: concentration dependence of the storage modulus G

_{0}measured in the linear regime; the line is a power law fit to the data, returning an exponent of n = 3.5 ± 0.4.

**Figure 5.**Response of Iota gels (squares: 0.3 wt.%; circles: 0.7 wt.%; up triangles 1.2 wt.%; down triangles: 1.7 wt.%) to a sweep in the dynamic shear strain γ at 1 Hz and 20 °C: (

**a**) storage modulus G′; (

**b**) loss modulus G″. γ

_{max}signals the strain where G″ passes through a local maximum for the gel with 0.3 wt.% Iota.

**Figure 6.**Response of Kappa2 gels (squares: 0.5 wt.%; circles: 0.75 wt.%; up triangles 1.1 wt.%; stars: 1.5 wt.%) to a sweep in the dynamic shear strain γ at 1 Hz and 20 °C: (

**a**) storage modulus G′; (

**b**) loss modulus G″. γ

_{max}signals the strain where G″ passes through a local maximum for the gel with 0.5 wt.% Kappa2.

**Figure 7.**Strain dependence of the shear storage modulus G′ scaled by the linear shear modulus G

_{0}of the corresponding gels: (

**a**) Iota gels with 0.3 wt.% (squares), 0.5 wt.% (circles), 0.7 wt.% (up triangles), 0.8 wt.% (down triangles), 1.1 wt.% (diamonds), 1.25 wt.% (stars) and 2 wt.% (line); (

**b**) Kappa2 gels with 0.5 wt.% (squares), 0.75 wt.% (circles) and 1.1 wt.% (triangles). Insets present the concentration dependence of G

_{0}and lines are power law fits to the data.

**Figure 8.**Strain dependence of the shear storage modulus G′ scaled by the linear shear modulus G

_{0}, plotted as lines and measured with: (

**a**) Iota gel with 0.3 wt.%; (

**b**) Kappa2 gel with 0.5 wt.%. The symbols represent the following equations computed with the parameters listed in Table 5: Equation (1a), squares; Equation (2a), circles; Equation (3a), triangles; Equation (4a), diamonds; Equation (5), stars; Equation (6a), crosses.

**Table 4.**Exponents n of the power law equation G

_{0}= Ac

^{n}describing the dependence of the linear shear elastic modulus G

_{0}on the carrageenan concentration c in gels formed under the conditions detailed in the column “Sample”. The values of constant A used in the fitting of the power equation to the data are not listed.

Sample | n | Ref. | |||
---|---|---|---|---|---|

Carrageenan Form ^{1} | Salt | Temperature (°C) | Concentration Range (wt.%) | ||

Kappa | |||||

Com. | No salt | 10 | 0.8–3.5 | 2.7 | [29] |

Com. | No salt | 40 | 0.8–2.5 | 2.7 | [29] |

Na | KCl + NaCl | 15 | 0.03–0.2 | 2.1 | [52] |

K | No salt | 20 | 0.7–1.4 | 7.7 ± 0.2 ^{2} | [53] |

Na | KCl | 5 | 0.25–3 | 3.2 ± 0.3 ^{2} | [54] |

Com. | KCl | 25 | 1.1–2 | 2.4 ± 0.1 ^{2} | [55] |

Com. | NaCl | 20 | 0.5–2 | 3.5 ± 0.4 | t.w. ^{4} |

Iota | |||||

Na | KCl | 5 | 0.5–3 | 3.4 ± 0.3 ^{2} | [54] |

Com. | No salt | 25 | 7–7 | 3.2 | [56] |

Com. | KCl | 25 | 1–2.5 | 2.1 | [56] |

Na | NaCl | n.a. ^{3} | 2–3 | 10 ± 1 ^{2} | [57] |

K | KCl | n.a. ^{3} | 1–2 | 3.7 ± 0.4 ^{2} | [57] |

Ca | CaCl_{2} | n.a. ^{3} | 0.25–1.5 | 1.6 ± 0.3 ^{2} | [57] |

Com. | NaCl | 20 | 0.3–2 | 2.01 ± 0.08 | t.w. ^{4} |

^{1}Com.: commercial carrageenan used as received; Na: sodium form; K: potassium form; Ca: calcium form.

^{2}Computed after re-plotting original data and fitting to a power law.

^{3}n.a.: not available.

^{4}t.w.: this work (see Section 4).

Iota | Kappa2 | Structural Parameters | Equations |
---|---|---|---|

4.7 | 0.8 | l_{max}/l_{0} | (1a) |

1.75 | 1.18 | d_{f} | (2a) |

3.8 | 2.3 | $n=\frac{3+{d}_{f}}{3-{d}_{f}}$ | (2c) |

2.4 | 1.65 | $n=\frac{3}{3-{d}_{f}}$ | (2d) |

0.01 | 100 | L/R_{0} | (3) |

0.07 | 0.5 | β | (4a) |

0.1 | 2 | G_{e}/G_{0} | (4a) ^{1} |

0.15 | 0.15 | e | (5) |

0.09 | 0.55 | $x$ | (5) |

2.5 | 0.01 | G_{e}/G_{0} | (6a) ^{1} |

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Hilliou, L.
Structure–Elastic Properties Relationships in Gelling Carrageenans. *Polymers* **2021**, *13*, 4120.
https://doi.org/10.3390/polym13234120

**AMA Style**

Hilliou L.
Structure–Elastic Properties Relationships in Gelling Carrageenans. *Polymers*. 2021; 13(23):4120.
https://doi.org/10.3390/polym13234120

**Chicago/Turabian Style**

Hilliou, Loïc.
2021. "Structure–Elastic Properties Relationships in Gelling Carrageenans" *Polymers* 13, no. 23: 4120.
https://doi.org/10.3390/polym13234120