mechanical properties of alginate gels: empirical characterisation

Mechanical properties of alginate gels: empirical characterisation

Marco Mancini, Mauro Moresi *, Roberto Rancini

Istituto di Tecnologie Agroalimentari, Universit�a della Tuscia, Via S. C. de Lellis, I-01100 Viterbo, Italy

Abstract

The compressive engineering stress (rE)±strain (eE) relationship up to rupture of several mannuronic and guluronic alginate gels

was studied by varying the alginate e�ective concentration (ceff ) from 0.8 to 1.7% w/w and was correlated by using the power model

�rE � kenE� with coe�cients of determination (r2) greater than 0.95. Whereas the degree of concavity (n) was about constant

(2.17�0.07) for all the gels examined, the rigidity constant (k) was found to increase with alginate intrinsic viscosity [g], guluronic

residues fraction (xG), and ceff , this constant being generally greater for the G-rich alginate gels at ceff�const. The rigidity constant,

as well the rupture stress (rER) and deformation work (LER), allowed the new indicators [k/(ceff )1:4, rER/(ceff )

1:4, and LER/(ceff )1:4] to be

calculated and used to discriminate e�ciently the gelling ability of M-rich alginates from that of G-rich ones. Stress relaxation

testing yielded a constant relaxation time spectrum with dimensionless viscoelastic coe�cients practically independent of ceff , the

latter showing clearly that high-M alginate gels were more elastic than the high-G ones Ó 1999 Elsevier Science Ltd. All rights

reserved.

Keywords: Alginates; Concentration; Gels; Gel strength; Guluronic fraction; Intrinsic viscosity; Mechanical properties; Modelling; Rupture strength;

Speci®c deformation work; Stress relaxation; Uniaxial compression

1. Introduction

Formation of alginate gels is quite a complex opera-tion. It depends on the type of alginate used (i.e. high-

Journal of Food Engineering 39 (1999) 369±378

Notation

a integer index as de®ned by Eq. (17)

Ai generic dimensionless viscoelastic coe�cient of a

generalised Maxwell body

ai generic empirical coe�cient of Eq. (7)

c alginate concentration in the jellied mass (% w/w)

dp degree of polymerisation of any sodium alginate

(�Mn/198)

Ei modulus of elasticity of the ith spring (Pa)

F(t) compression charge applied (N)

G* dimensionless relaxation stress [�F(t)/F(0)]

G(0) initial relaxation stress [�rE(0)/eE] (Pa)

H instantaneous height of the specimen under

compression (m)

k rigidity constant (Pa)

LE deformation work per unit gel volume (J/m3)

Mn number-average molecular mass (Dalton)

n degree of concavity (dimensionless)

nca2� g-equivalents of calcium ions to saturate

theoretically all the carboxylic groups present in

any alginate molecule (� 1/2 nNaAlg dp)

ni overall number of experimental data referred to the

ith alginate class

nNaAlg molar mass of sodium alginate (mol)

q empirical exponent (dimensionless)

r2 coe�cient of determination

S0 initial cross-sectional area of any specimen (m2)

t time (s)

* Corresponding author. Tel.: +39-761-357494; fax: +39-761-357498;

e-mail: [email protected]

t0:05;df single tail t-Student value at 95% con®dence level

and df degrees of freedom

Vt cross-head speed (m/s)

xi generic block fraction in the alginate chain

(dimensionless)

y generic dependent variable

Greek symbols

e generic strain (dimensionless)

eE engineering strain as de®ned by Eq. (1)

(dimensionless)

eT Hencky strain as de®ned by Eq. (2) (dimensionless)

é initial diameter of any cylindrical specimen (m)

[g] intrinsic viscosity at 0.1 M ionic strength (dL/g)

li viscosity of the ¯uid ®lling the ith dashpot (Pa s)

r generic stress (Pa)

rE engineering stress as de®ned by Eq. (1) (Pa)

rT true stress as de®ned by Eq. (2) (Pa)

si relaxation time of the ith Maxwell element (s)

Subscripts

e� e�ective

G guluronic

M mannuronic

nom nominal

0 initial

R referred to the rupture point of the compressive r±ecurve

0260-8774/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved.

PII: S 0 2 6 0 - 8 7 7 4 ( 9 9 ) 0 0 0 2 2 - 9

guluronic ``G'' or high-mannuronic ``M'' alginates),degree of conversion into calcium alginate (this beingtheoretically equal to 100% when the molar ratio be-tween Na� and Ca�� ions is 0.5), source of calcium ions(viz. calcium chloride, phosphate, lactate or acetate) andmethods of preparation (Moe, Draget, Skj�ak-Braek &Smidsrùd, 1995; Sime, 1990).

The gelling capability of jellifying agents is usuallymeasured by means of an empirical, yet not well de®ned,parameter known as gel strength. It is determined byseveral empirical test methods involving the submissionof specimens with di�erent shape and dimensions touniaxial compression testing (Mitchell, 1980; Moe et al.,1995). Therefore, the value of such a parameter relies onthe deformation applied, cross-head speed used and ifthe test is prolonged up to specimen rupture (Mitchell,1980). Moreover, the rate at which the specimens aredeformed during the measurement may a�ect the con-cavity of the resulting compressive stress±strain rela-tionship as a consequence of stress decay due toviscoelastic relaxation of cross-linked alginate chains,development of internal hydrostatic pressures that causewater exudation through the network, as well fracture ofthe air-containing cells (Peleg, 1997), or to junctionzones breakdown (Pines & Prins, 1973).

A great number of works have dealt with the assess-ment of the rheological behaviour of alginate gels byusing static (Amici, Mancini & Moresi, 1996; Mitchell &Blanshard, 1976; Nussinovitch, Peleg & Normand, 1989)or dynamic (Doublier, Launay & Cuvelier, 1992) tests.With speci®c reference to static testing, the results ofstress-relaxation allowed the alginate gels to be describedas viscoelastic solids by means of a mechanical modelconsisting of two Maxwell elements in parallel with onespring (Amici et al., 1996; Nussinovitch et al., 1989). Onthe contrary, the results of creep experiments evidenced aliquid-like viscoelastic behaviour as described by anothermechanical model consisting of one Maxwell element inseries with two Kelvin±Voigt elements (Mitchell &Blanshard, 1976). These results clearly show that all or

the majority of the cross-links in alginate gels are notpermanent, but move or break when the gels are sheared.

By testing gels at di�erent alginate concentrations (c),their normalised relaxation curves were found to exhibitpractically the same relaxation-time spectrum, theirinitial relaxation stress G(0) increased with c, but nostrain limits were given to assure the linear viscoelas-ticity of the gel samples (Amici et al., 1996; Nussinovitchet al., 1989).

Moreover, quite limited information is available toestimate the jellifying strength of the commercial high-mannuronic (M) or high-guluronic (G) alginates, as welltheir rheological response to di�erent stress±strain con-ditions as a function of the concentration, chemicalcomposition and number-average molecular mass of thealginate used.

The main aim of this work was to determine anddescribe univocally the mechanical properties of a seriesof high-M or high-G alginate gels by means of empiricalmodels based on uniaxial compression and stress re-laxation tests.

2. Materials and methods

Sodium alginate [C6H7O6Na]n from several algalsources (Table 1) was used for gel preparations ac-cording to the internal setting method (Onsùyen, 1992),by varying the alginate nominal concentration (cnom) inthe range 0.75±1.5% w/w. Any alginate powder waspreviously characterised (Clementi, Crudele, Parente,Mancini & Moresi, 1999), by determining the percentageof mannuronic (M) and guluronic (G) blocks by NMRspectroscopy, and intrinsic viscosity ([g]) at 0.1 M ionicstrength. By using a Mark±Houwink regression{[g]� 1.228´10ÿ4´Mn

0:963}(Clementi, Mancini & Mor-esi, 1998), it was possible to predict the number-averagemolecular mass (Mn) of any sodium alginate, and con-sequently, its degree of polymerisation (dp�Mn/198), asshown in Table 1.

Table 1

Main chemical (mannuronic- and guluronic-block fractions, xM and xG) and physical properties {intrinsic viscosity [g] at 0.1 M ionic strength;

number-average molecular mass, Mn; degree of polymerisation, dp} of the sodium alginate powders used in this work, as extracted from Clementi et

al. (1999)

Alginate Type No. Supplier xG (%) XM (%) [l] (dL/g) Mn (kDa) dp

Macrocystis

pyrifera

High-M 1 LV s Sigma, St. Louis, USA 0.38 0.62 5.9 72.7 367

2 MV D Sigma, St. Louis, USA 0.35 0.65 9.6 119.9 606

3 HV h Sigma, St. Louis, USA 0.37 0.63 10.7 134.5 679

4 CE e C. Erba, Milan, Italy 0.35 0.65 9.3 116.2 587

Laminaria

hyperborea

High-G 5 BDH r BDH, Poole, UK 0.63 0.37 5.9 73.1 369

6 LF10/60 m Savini, Milan, Italy 0.57 0.43 6.8 84.6 427

7 SF60 d Savini, Milan, Italy 0.63 0.37 17.0 217.7 1099

370 M. Mancini et al. / Journal of Food Engineering 39 (1999) 369±378

Once assigned the alginate concentration (cnom), itwas possible to estimate: (i) the molar mass of sodiumalginate (nNaAlg) to be dissolved in demineralised water;(ii) the g-equivalents (nca�� 1/2 nNaAlg dp) of calciumions to be added (to saturate theoretically all the car-boxylic groups present in any alginate molecule, thisinvolving a Na�:Ca2� ratio of 0.5); (iii) the mass ofcalcium acid phosphate dihydrate (CaHPO:

42H2O) asnca2� times the molecular mass of the calcium source (i.e.172.09 Da); (iv) the mass of glucono-d-lactone as equalto that of the calcium source used. As an example, toprepare 350 g of a gel consisting of sodium alginate no. 1(Table 1) at cnom� 1% w/w, 3.5 g of the above sodiumalginate and 1.52 g of the calcium salt were charged intoa mixer and thoroughly dispersed in 343.46 g of de-mineralised water at ambient temperature under vacu-um to minimise entrapment of air bubbles. By adding1.52 g of glucono-d-lactone under vigorous mixing, pHreduced, thus liberating the Ca2� ions. The resultingdispersion was quickly poured into a 1-dm3 beaker,which was weighed and stored at room temperature forabout 24 h. Then, the gel mass was dripped, gentlywiped with ®lter paper and weighed to estimate the ef-fective alginate concentration (ceff ).

Prior to compression, the jellied mass was segmentedinto cylindrical specimens using a stainless steel corkborer (inside diameter� 25 mm and thickness� 0.5mm), that were reduced to an height of ca. 25 mm with avery sharp cutter.

All mechanical tests were performed using an Instron4301 Universal Testing Machine (Instron International,UK) equipped with a 100 N load cell and connected to aPC Olivetti mod. PCS33 via an analogic/digital IEEEPCB EXA 504-148 converter (Instron International,UK). A specially designed programme (Series, XII V2,2004 Cyclic Test, Instron International, UK) enabledthe dynamometer to be operated from the computer,and its continuous voltage and external displacementreadings vs. time output to be acquired.

Subsequently, the instantaneous values of specimenheight H(t) and compression charge applied F(t) wereconverted into engineering (rE) or true (rT) stress vsengineering (eE) or Hencky (eT) strain according to thefollowing de®nitions:

rE � F �t�S0

; eE � H0 ÿ H�t�H0

; �1�

rT � rE�1ÿ eE�; eT � ÿ ln�1ÿ eE�; �2�where S0 and H0 are the initial cross-sectional area andheight of each specimen, while its instantaneous cross-sectional area was estimated on the hypothesis of con-stant volume.

Other specimens were submitted to stress relaxationtesting for as long as 30 min by applying a deformationof 25% of their initial height using a constant cross-head

speed (Vt) of 2 mm/s, to minimise the loading periodwith respect to the smallest relaxation time. During alltests both extremities of any specimen were covered with30 mm ®lter paper disks to avoid slippage between thesample surfaces and dynamometer plates.

All the tests were replicated ®ve times, thus showingtheir mean values together with their correspondingstandard deviations.

3. Results and discussion

3.1. Compression testing

After 24 h gelation at room temperature, all gelsunderwent syneresis, thus yielding di�erent amounts offree water, that allowed the e�ective alginate concen-tration (ceff ) to be estimated (Table 2). The ratio betweenthe e�ective and nominal concentrations was found tobe practically constant and equal to 1.050�0.006 re-gardless of the number-average molecular mass and G-block fraction of the alginates used.

As an example, Fig. 1 shows the compressive stress(r)-deformation (e) relationships for high-M and high-Galginate gels at 1% w/w nominal concentration.

In all tests, the general shape of such curves wascharacterised by an upward concavity. This compressivebehaviour (strain hardening) is generally attributed tostructure densi®cation (which is typical of compressiblecellular materials, such as bread and sponges, Peleg,1997) or to highly crosslinked polymer systems. Rubberelasticity theory predicts strain hardening as chainsjoining adjacent crosslinks become stretched, thus in-volving reorientation of chain segments longer than thedistance between crosslinks and probably the movementof whole molecules relative to one another (Mitchell &Blanshard, 1976).

Under small strains, their structure is practically un-altered and returns to their original shape once thecharge applied is removed. At larger strains, crosslinks(Mitchell & Blanshard, 1976) or the air-containing cells(viz. those formed during the gelation process, Mancini,Moresi & Rancini, 1998) may collapse or fracture, thuslimiting the mechanical resistance of the structure.When much of the crosslinks or cells are collapsed, thecompacted structure o�ers a progressively increasingresistance, that resembles the deformation behaviour ofan incompressible solid.

In all tests, the upward concavity was maintainedindependently of the engineering (Fig. 1(a)±(c)) or true(Fig. 1(b)±(d)) r±e values used, this being an indicationof true compressibility (Peleg, 1997). On the contrary,when incompressible materials are tested, the upwardconcavity of r±e curves generally disappears providedthat the cross-sectional area expansion and non-lin-earity of strain are taken into account by using

M. Mancini et al. / Journal of Food Engineering 39 (1999) 369±378 371

Eq. (2). However, this was not the case for the gelsunder study.

By referring to the engineering stress and strain dataonly, the yield stress (rER) at failure increased with thee�ective alginate content, being generally greater for thehigh-G alginates at constant ceff and Mn values (Fig. 2).The corresponding rupture strain appeared to be inde-pendent of alginate concentration and practically con-stant

eER � 0:536� 0:041 �nM

� 16� for high-M alginates �xG < 0:5�;eER � 0:503� 0:042 �nG

� 12� for high-G alginates �xG > 0:5�;

�3�

where nM and nG are the overall number of experimentaldata for the two classes of alginates used. However,the di�erence between the above average eER valuesresulted to be statistically signi®cant at the 5% level(t0:95;26� 2.06).

To further discriminate the outcomes of compressiontesting, the deformation work per unit gel volume (LE)was estimated as

LE �Z

rE deE; �4�

and plotted against eE. Fig. 3 shows such curves for upto ®ve cylindrical specimens obtained from M- and G-rich alginate gels at 1% w/w nominal concentration.

Table 2 lists the experimental mean values, togetherwith their corresponding standard deviations, of theengineering stress (rER), strain (eER) and speci®c defor-mation work (LER) at failure for ®ve cylindrical speci-mens with initial diameter (é) and height (H0), asderived from M- and G-rich alginate gels at four dif-ferent nominal concentrations (cnom).

3.2. Modelling of compressive stress±strain curves

Among the numerous empirical models proposed inthe literature (Masi, Sepe & Cavella, 1997; Peleg, 1997),the power model (Peleg & Campanella, 1989) was cho-sen to describe the experimental non-linear stress±strainrelationships shown in Fig. 1.

rE � kenE; �5�

Table 2

Main results of the uniaxial compression testing on ®ve cylindrical specimens with initial dimensions (é and H0) of di�erent calcium alginate gels at

di�erent nominal (cnom) and e�ective (ceff ) alginate concentrations: e�ect of the alginate type (see Table 1) on the average values (together with their

corresponding standard deviations) of the engineering stress (rER), strain (eER) and speci®c deformation work (LER) at failure, and empirical pa-

rameters (k and n) of the power model (5) together with their corresponding coe�cients of determination (r2)

Alginate Cnom

(% w/w)

Ceff

(% w/w)

é

(mm)

H0

(mm)

rER

(kPa)

eER LER

(kJ/m3)

log (k#)

(# kPa)

n r2

No. Type

1 LV 0.75 0.82 22.9 24.7�0.8 42.3�1.3 0.46�0.01 5�0 2.24�0.03 2.18�0.09 0.98

1.00 1.05 23.3 25.0�0.5 64.8�3.3 0.49�0.01 8�1 2.28�0.06 2.13�0.10 0.96

1.25 1.33 23.2 25.3�0.3 76.2�1.6 0.46�0.01 9�0 2.48�0.02 2.12�0.08 0.97

1.50 1.54 23.0 25.5�1.1 118.8�4.0 0.48�0.02 14�1 2.63�0.02 2.18�0.08 0.97

2 MV 0.75 0.79 23.4 24.7�0,4 83.7�5.2 0.54�0.01 11�1 2.29�0.03 2.08�0.11 0.96

1.00 1.08 23.7 25.3�0,1 136.5�4.4 0.54�0.00 18�1 2.51�0.05 2.10�0.06 0.95

1.25 1.31 23.9 25.1�0,6 175.7�3.3 0.56�0.00 24�1 2.62�0.01 2.22�0.03 0.96

1.50 1.54 23.4 24.7�0,2 201.3�0.2 0.55�0.01 28�2 2.68�0.02 2.08�0.04 0.97

3 HV 0.75 0.80 22.8 25.4�0.7 101.7�4.8 0.57�0.01 14�1 2.37�0.01 2.16�0.07 0.96

1.00 1.08 23.4 25.3�0.6 146.5�4.2 0.55�0.01 20�1 2.60�0.05 2.18�0.09 0.98

1.25 1.40 23.1 24.6�0.0 229.9�6.6 0.59�0.01 32�1 2.72�0.03 2.29�0.04 0.97

1.50 1.65 25.7 24.3�0.4 260.0�10.1 0.54�0.01 37�2 2.84�0.02 2.10�0.02 0.97

4 CE 0.75 0.76 23.5 24.0�0.5 69.2�1.6 0.55�0.01 10�1 2.27�0.02 2.22�0.06 0.97

1.00 1.01 23.8 24.3�0.6 110.4�9.1 0.56�0.02 17�2 2.46�0.03 2.15�0.06 0.97

1.25 1.27 23.8 25.0�0.5 130.6�6.3 0.57�0.01 18�1 2.52�0.03 2.28�0.07 0.97

1.50 1.51 22.9 24.5�0.3 165.9�1.1 0.56�0.00 24�0 2.65�0.02 2.24�0.07 0.98

5 BDH 0.75 0.80 23.1 24.7�0.5 100.5�5.1 0.46�0.01 13�1 2.63�0.02 2.13�0.05 0.98

1.00 1.09 23.9 25.1�0.4 164.5�4.0 0.49�0.00 21�1 2.76�0.04 2.15�0.05 0.97

1.25 1.33 24.0 25.1�0.5 192.8�5.8 0.48�0.00 25�1 2.89�0.02 2.22�0.08 0.98

1.50 1.57 24.1 25.4�0.6 236.6�15.4 0.48�0.01 32�2 2.97�0.01 2.17�0.03 0.99

6 LF 0.75 0.79 22.6 24.7�0.2 109.0�2.7 0.46�0.00 14�0 2.63�0.02 2.05�0.04 0.98

1.00 1.04 23.3 25.1�0.5 134.8�3.9 0.47�0.01 18�1 2.71�0.01 2.05�0.02 0.98

1.25 1.31 25.3 25.1�0.7 221.3�3.3 0.49�0.01 29�1 2.92�0.03 2.18�0.04 0.98

1.50 1.55 23.3 24.4�0.7 291.2�10.6 0.48�0.01 39�2 3.03�0.05 2.04�0.12 0.98

7 SF 0.75 0.81 21.5 25.2�0,5 187.8�5.19 0.55�0.00 26�0 2.71�0.02 2.22�0.04 0.98

1.00 1.04 22.5 24.6�0.4 268.5�13.5 0.55�0.01 38�2 2.87�0.02 2.21�0.10 0.98

1.25 1.32 23.4 24.4�0.5 343.1�7.8 0.54�0.00 46�1 2.96�0.01 2.18�0.05 0.98

1.50 1.55 23.4 24.7�0.6 506.4�10.2 0.58�0.01 72�2 3.14�0.03 2.34�0.14 0.99


where k is the rigidity constant that represents a measureof sti�ness, while n is the degree of concavity, that ac-counts for the deviation from linearity. For n� 1, Eq.(5) reduces to HookeÕs law and k coincides with the

modulus of elasticity; while for n smaller or greater thanone, a downward or upward concavity was accountedfor.

Such an empirical model allowed the experimentalrE±eE curves to be reconstructed with coe�cients ofdetermination (r2) greater than 0.95 (Table 2).

Fig. 2. Mean engineering compressive stress at failure (rER) vs. algi-

nate e�ective concentration (ceff ) for di�erent high-M (open symbols)

and high-G (closed symbols) alginate gels. Same symbols as in Table 1.

The continuous lines were calculated using Eq. (8) and the average k

and n values listed in Table 2.

Fig. 3. Engineering deformation work (LE) per unit volume of up to 5

cylindrical specimens of a guluronic (BDH type: a) and mannuronic

(MV type: b) alginate gels at 1% w/w nominal concentration, submitted

to uniaxial compression (Vt � 120 mm/min) as a function of the cor-

responding engineering deformation (eE). The continuous lines were

calculated using Eq. (9) and the average k and n values listed in Table 2.

Fig. 1. Compressive stress (r)±strain (e) relationships (expressed as

engineering, a and c, and true, b and d, stress±strain values) of cylin-

drical specimens (diameter and height equal to ca. 25 mm) of high-G

(a±b: closed symbols) and high-M (c±d: open symbols) alginate gels at

a nominal concentration (cnom) of 1.0% w/w. Same symbols as in Table

1.


In particular, the degree of concavity (n) was practi-cally constant and equal to

n � 2:17� 0:07; �6�whatever the alginate type and concentration (Fig. 4(a)).On the contrary, not only did the rigidity constant (k)increase with ceff , but also depended on the alginate type(Fig. 4(b)).

Generally speaking, the rigidity constant might beexpressed as a power function of the alginate concen-tration (ceff ), G-block fraction (xG) and intrinsic vis-cosity [g]

k � a0 xa1G �g�a2 ca3

eff ; �7�where ai are empirical coe�cients to be estimated byusing the least-squares method upon bilogarithmictransformation of the dependent (k) and independent(xG, [g], ceff ) variables.

Knowledge of such a relationship would allow thestress (rER) and deformation work (LER) at failure (thatis the strength and energy required to break a gel withunitary surface and volume, respectively) to be esti-mated as follows:

rER � k�eER�n; �8�

LER �Z rER

0

rE deE � ken�1

ER

n� 1: �9�

The continuous lines in Figs. 2 and 3 allow a directcomparison between the experimental rER and LER val-ues and those calculated via Eqs. (8) and (9), respectively,using the average values of k and n reported in Table 2.

Despite Eq. (7) led to a mean percentage error be-tween the experimental and calculated k values of13.7%, the reconstruction of rER and LER via Eqs. (8)and (9) was quite poor, their associated mean percentageerrors being about 30%.

A better data ®tting was obtained by partitioning thek, rER and LER values into two classes according as thealginates were mannuronic or guluronic. More speci®-cally, any of the dependent variables (i.e. the rigidityconstant, k; stress, rER, and volumetric deformationwork, LER, at failure) was divided by the alginate ef-fective concentration (ceff ) raised to an empirical expo-nent q. Then, the bilogarithmic plots (Fig. 5) of thesenew dependent variables [y/(ceff )

q] vs. the alginate in-trinsic viscosity [g] exhibited linear trends with slopesde®nitively di�erent depending on the predominance ofguluronic or mannuronic residues in the alginate chain.In this way, by minimising the least squares among ex-perimental and calculated values of all the three newdependent variables with respect to q, it was possible toidentify three indicators [k/(ceff )

1:4, rER/(ceff )1:4 and LER/

(ceff )1:4], representing, respectively, the speci®c sti�ness,

failure stress and breakage energy, the empirical regres-sions of which are listed in Table 3.

These indicators appeared to be capable of discrimi-nating more e�ciently the gelling ability of M-rich al-ginates from that of G-rich ones than the test methodsgenerally used by alginate suppliers and users. Thesemethods simply measure the force required to rupture agel of given alginate content and are sometimes unableto rank appropriately the gels tested. In fact, as shown inFig. 5(b), the speci®c failure stress of M-rich alginategels appeared to be quite similar to that of G-rich algi-nate gels for log10[g]�1.2, that is when their number-average molecular mass was about 200 k Dalton. ForMn values of this order of magnitude, even the speci®cbreakage energy for both types of the alginates studiedcoincided (Fig. 5(c)).

When M- or G-rich alginates with intrinsic viscositiesof about 101:2 dL/g are examined, their selection has tobe based upon other characteristics, such as their dif-ferent stress relaxation abilities.

3.3. Stress relaxation test

To assess the e�ect of stress relaxation on the com-pressive stress±strain relationship, square cylindrical

Fig. 4. Parameters n (a) and k (b) of the power model (5) of di�erent

high-M (open symbols) and high-G (closed symbols) alginate gels

against alginate e�ective concentration (ceff ). Same symbols as in Table

1. The continuous line in (a) refers to the average n value, while the

broken lines indicate its standard deviation range. The continuous and

broken lines in (b), respectively, refer to G- and M-rich alginate gels

and were calculated using the speci®c regressions shown in Table 3.


specimens of alginate gels at 1.0 and 1.5% w/w nominalconcentrations were initially loaded at a constant de-formation rate of 2 mm/s. Once a strain of 25% ± about50% of the average strain at failure for the gels tested (3)± was generated, the cross-head was stopped and thestress relaxation was monitored for as long as 30 min,

this being much longer than the time required (0.05 min)to deform the specimens.

During all tests the stress exhibited an asymptoticallydecaying trend with a residual value di�erent from zero,thus pointing out a solid-like viscoelastic behaviour forthe di�erent alginate gels studied. Fig. 6 shows the timecourse of the dimensionless relaxation stress, G*(t), de-®ned as the ratio between the actual F(t) and initial F(0)reactions, for any gels at 1.0% w/w nominal concentra-tion. Such an evolution was then associated with theresponse of a generalised Maxwell body (a spring inparallel to n Maxwell elements) (Rao, 1992):

G��t� � A0 �Xn

1

Ai exp�ÿt=si�; �10�

with

A0 � 1ÿXn

1

Ai �11�

to ful®l the initial condition: G��0� � 1;

Ei � AiG�0� for i � 0; 1; . . . ; n; �12�

li � Eisi for i � 1; 2; . . . ; n; �13�where Ai represents the generic dimensionless viscoelas-tic coe�cient; G(0) the initial relaxation stress and E0 themodulus of elasticity of the spring; while Ei, li and si

refer to the ith Maxwell element and are the modulus ofelasticity of its spring, the viscosity of the ¯uid ®lling itsdashpot and its relaxation time, respectively.

By determining the values of the unknown visco-elastic coe�cients (Ai, si) by a non-linear regressionmethod, it is di�cult to ®t all the experimental condi-tions using the same relaxation time spectrum, this beingalso reported for instance by Bertola, Bevilacqua andZaritzky (1991) and Masi (1989).

By applying an operating procedure practically basedon the method of Nussinovitch et al. (1989), the timecourse of the relaxation stress experimentally observedwas reconstructed by assigning initially the followingarbitrary spectrum of relaxation times:

si � a� 103ÿi min for i � 0; 1; . . . ; 5; �14�where a is an integer index ranging from 1 to 9.

In this way, it was possible to discard all the Maxwellelements the contribution of which to the reconstruction

Fig. 5. Speci®c sti�ness [k/(ceff )1:4: a], failure stress [rER/(ceff )

1:4: b], and

breakage energy [LER/(ceff )1:4: c] for di�erent high-M (h,- - -) and high-

G (n ± ) alginate gels as a function of alginate intrinsic viscosity [g].

The continuous and broken lines were calculated using the regressions

shown in Table 3.

Table 3

Empirical regressions of the speci®c sti�ness [k/(ceff )1:4], failure stress [rER/(ceff )

1:4], and breakage energy [LER/(ceff )1:4] as a function of the mannuronic

or guluronic nature of the alginate used, together with their corresponding coe�cients of determination (r2) and mean percentage errors

Parameter Mannuronic alginates Guluronic alginates Mean percentage error

y/(ceff )q (%)

k/(ceff )1:4 0.017´[g]0:70 (r2� 0.75) 338.84´[g]0:25 (r2� 0.74) 8.5

rER/(ceff )1:4 0.209´[g]1:40 (r2� 0.94) 45.71´[g]0:60 (r2� 0.89) 7.6

LER/(ceff )1:4 0.363´[g]1:67 (r2� 0.97) 6.17´[g]0:59 (r2� 0.89) 8.5


of a G*±t relationship was statistically insigni®cant atthe con®dence level of 95%. Whatever the relaxation testperformed, the minimum number of statistically signi-®cant Maxwell elements was equal to 2, their corre-sponding relaxation times being:

s1 � 5� 10ÿ2 min � 3 s; s2 � 5 min � 300 s: �15�It was therefore possible to describe the viscoelastic

response of all the gels considered by using a 5-elementmodel, its dimensionless coe�cients Ai being estimatedvia the least-squares method (Table 4).

Not only was the spectrum of relaxation times inde-pendent of alginate concentration, but also irrespectiveof the mannuronic or guluronic alginate type. The di-mensionless viscoelastic coe�cient Ai, though una�ectedby cnom, resulted to be conditioned by the predominanceof mannuronic or guluronic residues in the alginatechain (Fig. 7). In particular, the A0 and A1 values at

cnom� 0.5 and 2% w/w were extracted from a previouswork (Amici et al., 1996).

In analogy to previous observations on other gelsmade of agar, alginate and j-carrageen (Nussinovitchet al., 1989), the general shape of the normalised relax-ation curves remained practically unchanged, while thestrength of such gels increased with the gum concen-tration.

By averaging the Ai values in listed Table 4, it waspossible to derive the following:

A0 � 0:34� 0:05; A1 � 0:31� 0:05; A2 � 0:35� 0:05

for high-M alginates

A0 � 0:09� 0:04; A1 � 0:53� 0:06; A2 � 0:38� 0:03

for high-G alginates

Therefore, M-rich alginates formed gels with agreater residual stress (A0), practically constant andequal to about 34% of the initial relaxation stress G(0),while G-rich ones yielded gels with a smaller residualstress of the order of ca. 9% of G(0). Moreover, the ®rstanelastic component (A1) involved a rapid stress decayof about 53 or 31% of G(0) in the G- or M-rich alginategels, respectively.

As compared to the high-M alginate gels, high-Galginate gels were stronger and brittle owing to thehigher G(0) and lower A0 values. This was also apparentfrom creep data obtained by Mitchell and Blanshard(1976).

It is however worthy noting that the smallest relax-ation time (s1) for all the gels tested was of the sameorder of magnitude of the loading time and this con-trasts with the general rule that the relaxation curves are

Table 4

Main results of the stress relaxation testing (eE� 0.25 and Vt � 120 mm/min) on ®ve cylindrical specimens with initial dimensions (é and H0) of

di�erent calcium alginate gels at two nominal (cnom) and e�ective (ceff ) alginate concentrations: e�ect of the alginate type (see Table 1) on the initial

relaxation stress G(0), and dimensionless viscoelastic constants (Ai) of Eq. (10) together with their corresponding coe�cients of determination (r2)

Alginate cnom

(% w/w)

ceff

(% w/w)

é

(mm)

H0

(mm)

G(0)

(kPa)

A0 A1 A2 r2

No. Type

1 LV 1.0 1.05 22.4 24.8�0.4 39.7�4.4 0.35�0.01 0.28�0.01 0.37�0.00 0.975

1.5 1.54 23.7 24.9�0.2 70.0�5.4 0.31�0.00 0.25�0.00 0.44�0.00 0.947

2 MV 1.0 1.08 23.3 25.1�0.5 49.0�1.4 0.33�0.00 0.24�0.01 0.43�0.01 0.975

1.5 1.54 23.0 25.1�0.4 61.4�0.9 0.49�0.00 0.21�0.00 0.30�0.01 0.976

3 HV 1.0 1.07 23.2 25.3�0.5 56.2�4.9 0.26�0.01 0.31�0.02 0.43�0.01 0.975

1.5 1.61 23.3 25.5�0.6 100.1�9.3 0.27�0.02 0.32�0.01 0.41�0.01 0.975

4 CE 1.0 1.01 22.5 24.1�0.3 42.1�1.6 0.30�0.01 0.36�0.01 0.34�0.00 0.968

1.5 1.51 23.4 24.2�0.2 66.4�0.6 0.32�0.00 0.35�0.02 0.33�0.02 0.960

5 BDH 1.0 1.09 23.6 24.4�0.2 83.5�3.1 0.06�0.01 0.53�0.00 0.41�0.01 0.922

1.5 1.57 24.1 25.0�0.4 136.8�2.7 0.09�0.01 0.52�0.01 0.38�0.00 0.941

6 LF 1.0 1.04 23.8 24.3�1.5 95.0�4.9 0.11�0.00 0.49�0.00 0.39�0.01 0.957

1.5 1.55 22.9 25.1�0.6 189.9�11.7 0.12�0.01 0.48�0.01 0.40�0.02 0.956

7 SF 1.0 1.04 23.7 25.1�0.2 110.2�1.3 0.12�0.00 0.50�0.00 0.38�0.00 0.961

1.5 1.55 23.4 25.4�0.4 164.9�8.8 0.16�0.01 0.46�0.02 0.38�0.02 0.962

Fig. 6. Dimensionless relaxation stress G*(t) vs. time (t) of di�erent

cylindrical specimens of 1% w/w alginate gels submitted to an initial

25% deformation (eE). Same symbols as in Table 1.


valid only for t greater than 10 times the loading period(Rao, 1992).

Therefore, by analysing the stress-relaxation datacollected at times greater than 30 s only, the aboverheological model reduced to a 3-element model, char-acterised by a unique relaxation time of 300 s and by anelastic component (AI

0) practically constant as previouslyrevealed.

AI0 � 0:51� 0:07 for high-M alginates;

AI0 � 0:21� 0:11 for high-G alginates;

the greater dispersion observed for the G-rich alginatesbeing due to the alginate no. 5 (BDH type), its AI

0 valuesranging from 0.03 to 0.13. Even in this simpli®ed case,the residual stress of high-M alginate gels was generallygreater than that of high-G ones, thus con®rming theaforementioned de®nitive di�erence between the twoalginate types examined.

4. Conclusions

Despite the determination of the mechanical proper-ties of alginate gels using uniaxial compression andstress relaxation testing is a�ected by uncertainty anddiscrepancy because of unavoidable stress decay due to

viscoelastic relaxation, water exudation, fracture ofcrosslinks or of the air-containing cells, such simple testscan help at characterising the di�erent gelling ability ofcommercial G- or M-rich alginates.

More speci®cally, the response to stress relaxation ofalginate gels yielded a constant spectrum of relaxationtimes, independent of alginate concentration in therange 0.5±2% w/w and irrespective of the mannuronic orguluronic alginate type. The associated dimensionlessviscoelastic coe�cients Ai, though una�ected by cnom,resulted to be conditioned by the predominance ofmannuronic or guluronic residues in the alginate chain.Finally, whatever the e�ective gum concentration, high-M alginate gels appeared to be softer and more elasticthan high-G ones, in consequence of the smaller G(0)and greater A0 values in agreement with previous ®nd-ings by Mitchell and Blanshard (1976).

Uniaxial compression testing up to rupture of thesegels allowed the rigidity constant (k), and stress (rER)and deformation work (LER) at failure to be easily es-timated, thus resulting in a straightforward discrimina-tion of the gelling ability of M-rich alginates from thatof G-rich ones via the following new indicators: speci®csti�ness [k/(ceff )

1:4], speci®c failure stress [rER/(ceff )1:4 ],

and breakage energy [LER/(ceff )1:4].

As an example, provided that di�usion-set gels are tobe produced by co-extruding a fruit pur�ee mix with analginate solution and dropping the resulting beads intoa calcium setting-bath and that the jellied beads are toexhibit a certain rupture strength, elastic behaviour,and low-brittle tendency, the above results indicates®rstly that high-M alginates are to be chosen. In fact,their gels are generally more elastic than the high-Gones owing to their greater asymptotic stress relaxationconstant A0. Among the several mannuronic alginatescommercially available, the combined evaluation of thestress and deformation work at failure, or their esti-mation via the empirical regressions listed in Table 3,would allow the appropriate intrinsic viscosity (that isnumber-average molecular mass) to be selected so as toguarantee the gel strength required with the minimumalginate concentration, and consequently, minimumformulation costs.

Acknowledgements

This research was supported by a grant from theItalian Ministry of Research and University: specialgrant COFIN98.

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