Modeling capillary formation in calcium and copper alginate gels

Materials Science and Engineering C 58 (2016) 442–449

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Modeling capillary formation in calcium and copper alginate gels Diego Caccavo a,⁎, Anna Ström b,c, Anette Larsson b,c, Gaetano Lamberti a a b c

Department of Industrial Engineering, University of Salerno, 84084 Fisciano, SA, Italy SuMo BIOMATERIALS, VINN Excellence Center, Chalmers University of Technology, Göteborg, Sweden Pharmaceutical Technology, Chemistry and Chemical Engineering, Chalmers University of Technology, Göteborg, Sweden

a r t i c l e

i n f o

Article history: Received 15 May 2015 Received in revised form 5 August 2015 Accepted 22 August 2015 Available online 1 September 2015 Keywords: Alginate Ionotropic gelation Gel capillaries Modeling

a b s t r a c t Alginate solutions in the presence of bivalent ions can form ionic cross-linked gels. In particular gelation conditions the gel structure can be characterized by great anisotropy with the presence of straight capillaries along a preferential direction. These materials can find applications mainly in high-tech sectors, like tissue engineering, where the gel characteristics play a crucial role. Despite the need of mastering the capillary formation and properties, the process remains a poorly known problem, and its development is left to trial and error procedures. In this work a quantitative approach to the description of the capillary formation process has been developed. The theory proposed by Treml et al. (2003) has been implemented and extended to an alginate different from the one used in that study and two different ions (calcium and copper). Some of the model parameters have been derived through simple measurements; others have been scaled using proper scaling equations. Experiments have been performed in different gelation conditions, varying alginate and ionic solution concentrations, to highlight the effects of these parameters on the anisotropic structure and to validate the model. In all the analyses done, the model has performed nicely showing a good reliability in the prediction of gel characteristics like capillary formation, capillary length and process time. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Alginate is a renewable polysaccharide commonly extracted from algal plants (brown algae or Phaeophyceae). It is a linear polymer consisting of (1–4)-linked β-D-mannuronic acid (M) and α-L-guluronic acid (G) [1]. The polymer has been extensively used as thickener and gelling agent in the food and pharmaceutical industries. It is increasingly studied for use in biomedical applications partly related to its ability to form gels under mild conditions e.g. encapsulation of cells or microorganisms. Alginate form gels in the presence of di- and tri-valent ions, at room temperature and over a wide range of pH. The ionically crosslinked alginate gel is formed by bi and trivalent ions and G-unit complexes in a structure described by the “egg-box” model [2]. It is predominantly G units that are involved in the junction zones, the gel properties are thus dependent on the G/M ratio of the polymer [1,3,4]. An alginate high in G units give a stronger and more brittle gel compared to alginates rich in M units [1,3,4]. The ratio and arrangement of the M and G units depend on the botanical source and the state of maturation of the plant [1]. As the gelation kinetics is fast, slow introduction of di- and tri-valent cations, e.g. calcium ions is needed to create an isotropic gel. This can be achieved by dispersing an insoluble calcium source in the alginate solution, followed by the addition of a slowly hydrolyzed acid. Such calcium ⁎ Corresponding author. E-mail address: [email protected] (D. Caccavo).

http://dx.doi.org/10.1016/j.msec.2015.08.040 0928-4931/© 2015 Elsevier B.V. All rights reserved.

alginate gels are of isotropic character and the gelation methodology is commonly referred to as the internal method [1,5]. Another gelation route involves the anchoring of alginate to the container surface and the creation of a top gelled alginate layer, over which an ion solution composed of ions enabling crosslinks is added. The ions diffuse in this way across the gelled membrane into the alginate solution, creating a highly anisotropic structure where macroscopic channels (Fig. 1) run from the top to the bottom of the gel [6]. Various routes aiming at preparing hydrogels with macroscopic pore structures are explored for the use in biomedicine and tissue engineering where successful growth of cells require gels containing pores with 100 μm in diameter or larger [7,8]. The anisotropic structure of ordered capillaries of defined diameters obtained using external gelation of alginate has therefore received an interest as potential scaffold material [9–11] but also as use as membranes [12] or to simply study the structure for better understanding of biological processes [6,13]. Studies have been devoted to the control of capillary diameter, height and number as well as mechanical and microstructural properties of the gel [14]. The mechanism behind the formation of ordered and repeated capillaries within the alginate gel is however still not fully understood. The phenomena of obtaining ordered capillaries within alginate gels was first reported by Thiele who proposed that the capillaries originate from a phase separation mechanism of droplet segregation [6]: Sol + Electrolyte(A) ↔ Gel + Electrolyte(B) + Water.

D. Caccavo et al. / Materials Science and Engineering C 58 (2016) 442–449

Nomenclature C0A ~g C A C0ion D DA Deff A Deff ion erf kB kr ~r k kdiff r l lm Mw NA Nag NAV t T vc vcrit c vr ~r v z0 ~z0 zr α γ ε η⁎ θ κ

initial alginate sol concentration [mol/m3] alginate concentration of the gel [mol/m3] initial ionic sol concentration [mol/m3] general definition of diffusivity [m2/s] alginate diffusivity close to the gelation front [m2/s] effective alginate diffusivity [m2/s] effective ion diffusivity [m2/s] error function [−] Boltzmann constant [J/K] reaction rate constant [m3/(mol s)] dimensionless reaction rate constant [−] reaction rate constant in the case of diffusion limitation [m3/(mol s)] length of a Kuhn segment of alginate [m] diameter of an alginate monomer [m] molar mass [g/mol] alginate degree of polymerization [−] numbers of monomers forming a Kuhn segment [−] Avogadro constant [1/mol] process time [s] temperature [K] contraction velocity [m/s] critical contraction velocity [m/s] gel formation front velocity [m/s] dimensionless gel formation front velocity [−] time dependent thickness of the contraction layer [m] dimensionless thickness of the contraction layer [−] reaction front [m] friction constant between contracting chains and fluid [kg/(mol s)] displacement parameter [m2/s] degree of contraction [−] viscosity of the fluid surrounding the alginate chains [Pa s] structure formation coefficient [−] coefficient of convective transport [mol/m4]

It was stated that the gelation process is accompanied by dehydration so that the finely distributed drops of water are trapped within the zone of sol–gel-transition. Further delivered water molecules will accumulate and be pushed by the gelation front towards the sol creating alginate free pore channels [11]. The research group of Kohler proposed later the theory of chemically fixed dissipative structure formation [15]. This idea was based on the observation [16] of convective movement, similar to Benard cells, close to the already gelled alginate. The phenomenon was explained considering the contraction of the alginate chains during the complexation by the bivalent cations. This induces a movement of solvent, close to the gelation front, that gives rise to periodic variations of the alginate gel density in the gelation plane (horizontal plane). This theory was implemented in a mathematical model and semi-qualitatively validated to a system of Cu-induced alginate gel [17]. The molecular dynamics measured using magnetic resonance (MR) upon the diffusion of Cu ions into an alginate solution suggest however that capillary formation is related to spinodal decomposition mechanism rather than the hydrodynamic convection roll model described by Kohler and co-workers [13]. This is concluded due to the lack of coherent transverse convective velocities within the Cu-alginate sample. Transverse convection would be necessary for the presence of “roll-like” convective movements. Capillary type structures have been predicted in two dimension model based on total free energy containing an elastic stress contribution and a dynamic coupling between this

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stress and diffusion [18]. The molecular dynamics of the calcium containing alginate gel was less conclusive than the copper containing gel potentially related to the lack of consistent capillary formation in the case of calcium as diffusive ions. Nevertheless, Maneval and coworkers propose that a model based on spinodal decomposition approach may be more fruitful owing to the measured anisotropic dynamics observed in their study. Independent of the exact mechanism (hydrodynamic convection or spinodal decomposition) being responsible for the capillary formation in ion induced alginate gel, we show in this work that, through the generalization of the model suggested by Treml and co-workers (chemically fixed dissipative structure) it is possible to predict the minimum ionic strength required of the counterions for the formation of capillaries in alginate solution of different concentrations. The predictions give further the time required to obtain specific capillary lengths. The counterions chosen in this work are Ca2+ and Cu2+, both extensively used in biomedicine and tissue engineering, however the present approach could be easily extended to other (multivalent) ions. We present a quantitative analysis of prediction and experimental data obtained for the Cu-alginate system (previously only a semiqualitative analysis was done) and the Ca-alginate system (which as far as we are aware has not been previously done). Such analysis will aid and fine tune the experimental preparation of capillary alginate gels and the work presented here highlights, by combined experimental and modeling approach, the conditions for the capillary formation process and their effect on the resulting structures. 2. Materials and methods 2.1. Materials Alginate Protanal RF6650 was kindly provided by FMC Biopolymers (UK). It contains 70% guluronate according to the supplier. Calcium chloride dihydrate was purchased from Sigma–Aldrich (Germany) and copper chloride dihydrate was purchased from Apotek Produktion & Laboratorier (Sweden). 2.2. Methods Alginate solutions were prepared by careful addition of alginate powder to deionized water at room temperature under vigorous stirring. The dispersion was thereafter heated to 353 K in a water bath and kept at this temperature for 30 min or until dissolution was obtained. The pH of the polymeric solution was adjusted from pH 7.3 to pH 7 using 0.1 M HCl. The solution was thereafter deaerated, to remove the trapped air bubbles, by leaving the solution in an open flask for 5–10 min inside a vacuum chamber. 2.2.1. Gel preparation Externally set alginate gels were prepared by coating a glass beaker (V = 50 ml and d = 40 mm) brushing the internal wall with alginate solution, which then was allowed to dry in an oven set at 383 K for 30 min. The proceeding was repeated three times and the beaker was preheated in the oven at 393 K prior the first application. A known amount of alginate solution at 293 K was poured into the glass beaker. The surface of the polymeric solution was sprayed with 2 ml of bivalent ionic solution until a gel membrane was formed on top of the alginate solution. The gel membrane was left to set for 30 min after which other ionic solution at 293 K was carefully poured on top of the membrane. The ionic solution was left to diffuse through the membrane, and the beaker was covered with parafilm and allowed to rest for the desired gelation time. The coating procedure is necessary to assure capillary formation. Since the amount and concentration of both alginate and ionic solutions are experiment dependent, their values will be specified later on in the text.

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Fig. 1. On the top left, a scheme of the anisotropic gel with the cross sections highlighted. On the bottom left part, the perpendicular section to the ions diffusion (Ruthenium Red was used to stain the alginate gel and enhance the contrast), that correspond to the red section in the scheme. On the right part, the parallel section to the ions diffusion (blue section in the scheme). The alginate gel with capillaries was obtained with 1.5% w/w of alginate solution and 0.5 M of calcium chloride. The ionic solution was placed at z = 0 and let to diffuse through the alginate solution (“z” direction). The capillaries appear close to the upper surface (z = 0) and are recognizable until z* (dashed line). Below z*, despite the gel formation, capillaries are not present. (For interpretation of the references to color in this figure legend, the reader is referred to the online version of this chapter.)

2.2.2. Gels and capillary length measurement The length of the gel and capillaries were determined using a calibrated Digital Pro-Max caliber (0–150 mm) from Fowler Tools and Instruments. The gel was removed from the container and carefully separated from the remaining alginate solution. The measures have been carefully done paying attention to not deform the gel in any phase of its handling (readers interested in the mechanical properties of the capillary alginate gels are referred to reference [14]). To measure the capillary length the gels have been dissected with a cutter along their height as shown in Fig. 1. 2.2.3. Determination of intrinsic viscosity An automated Ubbelohde viscometer (Schott-Geräte, Germany) with a capillary of 531 0a was used to determine the intrinsic viscosity of the alginate used in the present study. The capillary was immersed into a water bath set at 293 K. The average of the flow-through time of the solvent (0.05 M Na2SO4) and dilute samples of alginate was determined for the calculation of relative and specific viscosity, ηrel and ηspec, respectively. The flow-through time of each sample was repeated five times. The Hagenbach corrections were applied on the running times before calculating the relative viscosity (ηrel).

ηrel ¼

η τ ¼ η0 τ0

ð1Þ

where τ equals corrected flow-through time and τ0 is the corrected flow-through time of the solvent. The specific viscosity is given by

ηspec ¼

η−η0 ¼ ηrel −1: η0

ð2Þ

The intrinsic viscosity, [η] in [dl g−1], was determined by plotting ηspec/c and ln(ηrel)/c against the concentration, (c in [g dl− 1]), and extrapolating to zero concentration [19]. The molar mass of the alginate

was calculated using the Mark–Houwink–Sakurada equation ½η ¼ K Mw a

ð3Þ

where K = 0.011 [ml mol g−1] and a = 0.93 [20]. 3. Modeling The theory proposed by Treml et al. [17] is briefly introduced and the methods developed to derive generalized parameters are shown. 3.1. The model The model proposed by Treml and co-workers [17] is built from the momentum and mass balance and called “hydrodynamic model”. It consists of Navier–Stokes equations for the hydro-dynamical part [16], coupled mass transport and instantaneous/irreversible reaction for the process [21] and results from random walk simulations of a phantom chain [22]. Solving the Navier–Stokes equations accounting for friction forces, coupled with a mass balance on the alginate chains in the contraction layer [16,17], it was possible to derive an equation for the critical that is the minimum contraction velocity to contraction velocity vcrit c yield a stationary Benard-like pattern (see Introduction section), i.e. the minimum vc that allows to obtain capillaries. C ~ vr ð44Þη vr
þ 7:9  10−3 A 3 z0 κ αz0 κ g

vcrit c ≈

ð4Þ

Where η⁎ is the water viscosity (1× 10−3[Pa s]), vr is the gel formation front velocity, α is the friction constant between contracting alginate chains and fluid, z0 is the thickness of the contraction layer, κ is the coefficient of convective transport of the alginate chains from ~ g is the alginate concentrathe alginate sol to the contraction layer and C A

tion in the gel.

D. Caccavo et al. / Materials Science and Engineering C 58 (2016) 442–449

~ g were obtained from a macroscopic diffuThe parameters vr and C A sion reaction model. The simplifying hypothesis was a very fast reaction (instantaneous) between ions and alginate, limited by diffusion of both the reactants from their bulk solutions (similar to Barba et al. [23] where a first order reaction was considered). It was considered that the diffusion is a “long-ranged” process (the concentration profile extends over distances much larger than z0) in comparison with the convective motion of the solution in the immediate neighborhood of the reaction front. This means that convective motion can be neglected in the length eff scale of the diffusive transport. Effective diffusivities, Deff A and Dion, were utilized for the alginate molecules in the alginate sol and for the ion in the gel/ionic solution respectively (with “effective” are indicated the values of the diffusivity resulting from the transformation of the Nernst–Plank equation into purely concentration gradient driven flux [21]).

rffiffiffiffi γ vr ¼ t γ¼

ð5Þ

z2r 4t

ð6Þ

sffiffiffiffiffiffiffiffiffi ! g 1 1 Deff γ 0 A ~ ¼ pffiffiffi 0 1 C C exp − A A sffiffiffiffiffiffiffiffiffi γ π Deff A γ A 1−erf @ eff DA

ð7Þ

Table 1 Model parameters from the original model [17].

The coefficient of convective transport, κ, was obtained coupling the macroscopic diffusion reaction model to the alginate mass balance close to the gel formation front where, under the influence of the hydrodynamic movement, a convective transport is expected [21]. C0 vr κ ¼ 0:3 A DA

ð9Þ

where DA is the alginate diffusivity close to the gelation front. The expression for the contraction layer thickness was derived considering that in the microscopic scale the reaction does not occur in the precise plane individuated by zr but in a small volume, with height z0 in the neighborhood of zr. The mass balances on the ion and alginate molecules, considering a second order reaction between alginate (CA) and ion (Cion), with the introduction of two new variables: the kinetic constant kr and the degree of polymerization NA, allowed to obtain the z0 expression. z0 ¼ ð1:31Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi eff NA 3 Dion DA 2 ln 4:23 kr C0A vr

ð10Þ

The friction coefficient α was found from results of random walk simulations of a phantom chain.

α ¼ NAv

  38 πη lm   15 ln Nag

ð11Þ

Value

[m3/(mol s)] kdiff r l [m] lm [m] Nag [−]

1 × 107 1.52 × 10−8 4.6 × 10−10 68

eff parameters of the type of alginate and ion (DA ; Deff A ; Dion ; kr ; NA ) are known. A scheme of the calculation procedure is reported in Fig. 2. The degree of contraction ε goes to relate the gel formation front velocity to the contraction velocity:

vc ¼ εvr :

ð12Þ

Its expression, derived from results of random walk simulations of a phantom chain, is the following: "

qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffi ! ! 2 C0ion Deff C0A Deff γ γ A ion 0sffiffiffiffiffiffiffiffiffi1 exp − 0sffiffiffiffiffiffiffiffiffiffi1 exp − ¼ : ð8Þ Deff Deff A ion γ A γ A @ @ 1 þ erf 1−erf Deff Deff A ion

Parameter

where lm is the diameter of an alginate monomer (Table 1), NAv is the Avogadro constant, Nag is the number of monomer forming a Kuhn segment (Table 1). At this point, using all the previous equations, it is possible to calculate the critical contraction velocity once that the initial alginate and ionic solution concentration (C0A , C0ion) are chosen and the representative

ε¼ 1þ where zr is the position of the reaction front, t is the process time and C0A is the initial alginate sol concentration. The displacement parameter, γ, is a constant that can be derived from the following equation:

445

~r 3:17 v ~r ~z2 =3 k

!#−1 ð13Þ

0

~r, ~z are dimensionless quantities introduced ~ r, k where the variablesv 0 to describe ε: 2

19πη l   15kB Tln Nag ~ r ¼ kr k diff kr z0 z~0 ¼ l

~r ¼ vr v

ð14Þ

where l is the length of a Kuhn segment (Table 1), kB is the is the reaction Boltzmann constant, T the system temperature and kdiff r rate constant in the case of diffusion limitation (Table 1). The model was completed introducing the so called “structure formation coefficient” θ that is the simple ratio between vc and vcrit c : θ¼

vc vcrit c

ð15Þ

when θ is greater than 1, the capillary formation occurs since the contraction velocity vc is greater than the minimum contraction velocity and therefore it assured the presence of convective pattern. On the contrary, when θ is lower than one, being vc smaller than vcrit c there will not be the convective pattern and hence no capillary formation. 3.2. Model parameter estimation New model parameters have been derived to describe the gelation process with the material used in this work. The intrinsic viscosity of the alginate used was determined to be 950 ml g−1, resulting in an average molar mass of 203 × 103 [g mol−1]. The degree of polymerization (NA) has been derived from the average molecular mass combined with the molecular mass of a monomeric unit, giving a degree of polymerization of about 103 (monomers/molecule) (Table 2).

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Fig. 2. Scheme of the calculation procedure of the critical contraction velocity.

The alginate diffusivities (DA , Deff A ) have been scaled with a power law relation valid for a random chain [24]: 1 ≈ M0:57 w D

ð16Þ

The effective diffusion coefficients for the ions (Ca2+, Cu2 +) have been determined experimentally using Eqs. (6) and (8). The alginate concentration was fixed to 1.5% w/w (C0A) and the ionic solutions concentration to 0.5 M (C0ion). The gelation was stopped at 3 h (t = tproc) and the thickness of the gel disk obtained has been measured with the caliber (zr(tproc)). In this way the displacement parameter γ for the specific case was determined from Eq. (6) and the Deff ion calculated from Eq. (8). It has to be noted that the displacement parameter depends on the initial concentrations (C0A and C0ion), instead the diffusivities are independent of the initial concentrations and the value of Deff ion could be thus extended to different gelation conditions. The kinetic constant kr used as fitting parameter by Treml et al. [17] has been obtained with the same approach in this work. The gelation

Table 2 Model parameters derived in this work. Parameter

Alginate (Protanal RF6650)

NA [−]

1000 1.018 × 10−10

2 Deff A ½m =s DA ½m2 =s

8.250 × 10−12 2+

zr |3h [mm] γ [m2/s] 2 Deff ion ½m =s kr [m3/(mol s )]

Ca

Cu2+

7.796 1.407 × 10−9 1.165 × 10−9 7000

7.206 1.202 × 10−9 9.983 × 10−10 350

has been done with 40 ml of alginate 1.5% w/w (C0A) using a calcium solution concentration of 0.5 M (CCa2+0) for the calcium alginate gel, and with a copper solution concentration of 0.05 M (CCu2+0) for the copper gels. The capillary formation, as expected, stopped at a certain depth while the gelation continued till the bottom (Fig. 1). Knowing the depth that the capillaries reached, it has been possible to tune the model, acting on kr to assure a θ value greater than one in the conditions experimentally observed. 4. Results and discussion The tunable process parameters that would lead to the formation of the anisotropic structure are mainly, using this approach, the concentrations of alginate and ionic solutions. Their effects are treated separately in the next paragraphs. 4.1. Effect of ionic solution concentration In Fig. 3 the effect of the calcium and copper solution concentration on the capillary formation and length, with an alginate solution concentration fixed to 1.5% w/w, is shown. These graphs report the structure formation coefficient, θ, versus the decimal logarithm of the gel/capillary formation front, zr. The capillary formation starts at θ N 1 and lasts until it remains greater than the unity. From the second intersection of θ with the unity straight line (the intersection at larger zr) it is possible to derive the final capillary characteristics in terms of depth and time of formation (through the Eq. (6)). From these graphs it is possible to see that the minimum ionic solution concentration to obtain straight pores is about 0.012 M for calcium ions and 0.013 M for copper ions. However, it has to be considered that even using a concentration of 0.025 M the capillaries would appear in a very thin layer, smaller than 1 mm for both the ions. This probably would not permit even their recognition since in this first zone the membrane takes over. The model prediction has been

D. Caccavo et al. / Materials Science and Engineering C 58 (2016) 442–449

447

Fig. 3. Modeling prediction of the effect of different ionic solution concentrations on the capillary formation, with an alginate solution concentration at 1.5% w/w (the logarithm argument has been normalized for 1 m). The calcium solution concentration effect on the left and copper solution concentration effect on the right.

confirmed experimentally by the preparation of gels with calcium solution at 0.025 M that, below the membrane, despite the gel formation, have not shown capillaries. From these graphs it is even possible to observe that with the same ionic solution concentration, different ions bring to different results. Indeed the capillary length in the Cu-gel will be greater than the capillaries in the Ca-gel. An analogous reasoning could be done on the time of the capillary formation where, for the copper, the process should last longer with respect to the calcium, generating at the end longer capillaries. Therefore increasing the ionic strength of the crosslinking ions that diffuse into the alginate solution, longer capillaries should be generated. This is perfectly reasonable and in agreement with a concentration gradient driven process. 4.2. Effect of alginate solution concentration To show the negative effect of high alginate sol concentration on the capillary formation and growth, an experiment has been set up in which the calcium solution concentration has been kept constant and the alginate solution concentration has been varied. In particular the ionic concentration has been chosen on a threshold value: 0.025 M. It was shown that this concentration, with an alginate solution of 1.5% w/w,

does not lead to anisotropic structure. Nevertheless, this same ionic concentration sol in contact with a less concentrated alginate sol, considering the modeling results in Fig. 4, should be able to generate macroscopic capillaries. On the left of Fig. 4 is reported the effect of the alginate solution concentration on the capillary formation in terms of gelation time, whereas on the right in terms of position of the reaction front. Decreasing the alginate sol concentration the formation of capillaries should take place, lasting for a longer time (Fig. 4 left) and generating longer capillaries (Fig. 4 right). Two concentrations have been tested: 1% w/w and 0.5% w/w, where the predictions of the final capillary lengths were 1.5 mm (in 1.75 h) and 8 mm (in 15 h), respectively. Experimentally it has been found that no capillaries were present below the membrane in the first case whereas in the second case (alginate solution 0.5% w/w) a capillary length of 7 mm has been recorded (8 mm predicted) (Fig. 5). Therefore, the alginate concentration plays an important role in the capillary formation. According to the model prediction, increasing the alginate concentration leads to a reduced velocity of the gel formation front. This is physically explainable considering that more G-units, in the same volume, will be available for the complexation phenomenon. In this manner even the contraction velocity of the alginate chains/complexes will be lowered, bringing a reduction of the convective pattern phenomenon and thus reduced driving force for capillary formation.

Fig. 4. Modeling prediction of the alginate solution concentration effect on capillary formation with a fixed calcium solution concentration at 0.025 M. On the left the structure formation coefficient versus the gelation time (the logarithm argument has been normalized for 1 s), on the right the structure formation coefficient versus the position of the reaction front (the logarithm argument has been normalized for 1 m).

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reach depth of centimeters (10− 2 m) in the gel. Since the gelation time and reaction front (or gelation front) are related through Eq. (6), it is possible to quantify the effective time needed to reach the fully developed anisotropic structure that for the Ca2+ gel has been predicted of about 9 h whereas for the Cu2+ has been predicted of about 124 h. In terms of capillary length, for the calcium gels the model has predicted 0.7 cm and experimentally the length was 1 cm; the prediction for the copper gels was 2.5 cm and it has been found experimentally 2.7 cm. The comparison, despite considering the number of simplifications and approximations, shows a good agreement between the actual capillary length obtained and that predicted for both Ca- and Cu-alginate systems.

4.4. Validating predictions of capillary and gel front position during formation

Fig. 5. Picture of capillary length in a gel made with alginate solution at 0.5% w/w and CaCl2 solution at 0.025 M. The capillaries appear close to the upper surface (z = 0) and are recognizable until z* (dashed line). Below z*, despite the gel formation, capillaries are not present.

4.3. Validating capillary length predictions at equilibrium conditions To test the model predictions at the equilibrium with different gelation conditions (from those one used for the calibration), the ionic solution of both ions (Ca2+, Cu2+) has been fixed to 0.1 M, the alginate concentration has been kept at 1.5% w/w and the gelation process has been allowed to take place for long time (1 week). In this way it has been possible, like in Fig. 1, to measure the depth reached by the capillaries. The comparison between model and experiments (done in triplicate) are shown in Fig. 6. From the modeling point of view, it has to be considered that only when θ N 1 there will be capillary formation. In both cases the capillary creation should start close to 1 μm below the gel surface and should

Fig. 6. Capillary length at infinite gelation time with alginate 1.5% w/w and ionic solution at 0.1 M (the logarithm argument has been normalized for 1 m). Comparison between modeling predictions (lines) and experimental results (marks). The solid line and the circular mark describe the calcium alginate gel, the dashed line and the square mark describe the copper gel.

The model so formulated has the potential to predict the capillary length not only at the equilibrium conditions, as shown in the previous paragraph, but even during the gelation process. Indeed once the Eq. (8) is solved for the displacement parameter, the velocity of the gelation front (Eq. (5)) and the capillary length (Eq. (6) backward) are readily calculated. As it can be seen the gelation front grows, in agreement pffiffiffiffiffiffiffiffi with [25–27], with the time to the power of 0.5: zr ¼ 4γt. It has to be once more stressed that the gelation front is equal to the capillary formation front as long as θ N 1, as soon as θ become smaller than 1 the capillary formation front stops while the gel front continues. Experiments, performed in triplicate, have been set up to monitor the movement of the capillary and gel fronts and compared to modeling results. For this, alginate solution at 1.5% w/w and ionic solution at 0.5 M were used. Practically the gelation was stopped after a certain time and the gel thickness, as well as the capillary length, was determined. The modeling and experimental results are shown in Fig. 7. As it can be seen from both experiments and modeling predictions, the calcium gels show a faster gelation velocity that let them form longer capillaries at a given time interval compared to the copper gels. Despite the agreement between predictions and experiments is not perfect, slightly overestimating with the model the gelation fronts of both ions, the description can be considered satisfactorily with these degrees of approximation. It has to be noticed that in the first 24 h for both types of gels, the gelation front coincides with the capillary formation front, meaning that the anisotropic structure is being formed along with the

Fig. 7. Gelation and capillary fronts at different process times, compared with the modeling predictions. Alginate gel at 1.5% w/w and ionic solutions at 0.5 M.

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gel. As the process time passes the gelation front moves forward deeper parts where the driving force for capillary formation, the ion concentration, starts to significantly decrease. Despite the gel formation is still possible, the capillary formation is not, therefore, at a certain depth, there will be the separation of these two fronts. As represented in Fig. 7, the modeling results show that the capillary formation for the calcium gels should stop after 35 h, forming capillaries with a maximum length of 2.7 cm. After that point the gelation will continue but without the formation of capillaries. Experimentally it has been found that at process time of 48 h this difference is net, about 3 mm, going to confirm the modeling results in terms of maximum capillary length as well as capillary formation time. For the copper instead the model has predicted the possibility to reach capillary length of about 10 cm in a time of 20 days. 5. Conclusions In this work a quantitative approach to the description of straight capillary formation process in alginate gels has been developed. A theory from literature [17] has been generalized via simple steps based on scaling equations and length measurements, thus allowing for a straightforward model extension to different kind of alginates and crosslinking ions. The modeling predictions have been tested experimentally using calcium and copper as crosslinking ions, showing a good reliability and thus validating the modeling approach. In particular, it has been possible to quantitatively predict the ion and alginate concentration effect on the capillary formation. The calculations, validated against experimental results, showed that the higher the ionic concentration, the longer the capillaries, and underlined the existence of a minimum ionic solution concentration able to generate capillaries (at a given alginate concentration). It has been confirmed that a higher initial alginate sol concentration reduces (and abolishes) the tendency of capillary formation. Even in this case it has been possible to estimate the maximum alginate sol concentration that would lead to an anisotropic structure (at a certain ionic solution concentration). Furthermore the process kinetics has been described by calculation and validated with experiments, giving information on the time required to obtain a certain capillary length. The time point at which the capillary formation ends (while the gel front continues) has been derived by coupling the process kinetics with equilibrium information. In conclusion, the main variables that influence the capillary formation have been highlighted showing their effect on the gel structure in terms of capillary formation and length. With this modeling approach the use of trial and error procedures in the setting up this peculiar process could be reduced. Conflicts of interest The authors confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. Acknowledgements The authors are grateful for the financial support from VINNOVA to the VinnExcellence center SuMo Biomaterials and Vinnmer grant to A.S. Part of this work was supported by the “Ministero dell’ Istruzione dell'Università e della Ricerca” (contract grant PRIN 2010/2011 20109PLMH2). References [1] K. Draget, Alginates, in: G.O. Phillips, P.A. Williams (Eds.), Handbook of Hydrocolloids, 2nd ed.Elsevier Science 2009, pp. 807–828.

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