Yousef Murtaja

Lubomir Lapčík

Barbora Lapčíková

Shweta Gautam

Martin Vašina

Lubomir Spanhel

Jakub Vlček

Barbora Cz

Lapčíková

Intelligent high-tech coating of natural biopolymer layers

Keywords: biopolymers, crosslinking, diffusion coefficient, hydrogel swelling, thermodynamics

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The technology of intelligent materials paves way for the development of sustainable products via their acute flexibility. Their ability to adapt to environment changes in response to a stimulus, enhances their multi-functionality in daily applications as well as in all the engineering fields, medical, defence, and automobile industries. Some of the examples of intelligent materials include memory alloys, hydrogels, thermochromic pigments, photochromic pigments, biopolymer coatings, shape memory polymer etc. [START_REF] Mishra | Smart Materials For Clean And Sustainable Technology For Smart Cities[END_REF]. The structureproperty relationships permit the significant variation of important physicochemical and biological properties, which broaden their application prospects in material science, engineering, and industrial sectors. The expeditious growth in the science and engineering sectors demands the development of novel and sustainable materials, thereby placing new requirements on material engineers to prepare highly sophisticated mono and composite systems [START_REF] Lv | Bio-based episulfide composed of cardanol/cardol for anti-corrosion coating applications[END_REF].

In the field of materials, the cost and energy consumption for the preparation and processing of heavyweight building materials (mainly transport systems) is very high, thereby limiting their scope for future applications. To avoid this, a constant search has been focused on development of lightweight polymer composites endowing excellent mechanical properties by using low-cost and straightforward synthesis processes. It is also worth mentioning the increasing demand to reduce the environmental wastage and carbon footprint, particularly in the materials industries. Therefore, it is required by researchers to focus on enhancing the properties and functions of the biopolymers. In the view of the same, significant research has been done to produce biopolymers and bio-composites with comparable properties and functionalities to those of traditional bio-based packaging [START_REF] Hu | Biopolymer-based lightweight materials for packaging applications, Lightweight materials from biopolymers and biofibers[END_REF]. Some of the biopolymers that have been studied extensively are: Polylactic acid (PLA), polyhydroxyalkanoates (PHA) [START_REF] Bourbonnais | Application of polyhydroxyalkanoate granules for sizing of paper[END_REF], starch, cellulose, pectin, chitosan among others. It is expected that in the next decade, the weight of cars will reduce to half the original weight owing to the use of lightweight biopolymer composites; consequently, reducing the fuel consumption by 50%. The advantages of lightweight polymer composites and polymer-based ceramics will be applied further to design and manufacture advanced lightweight engines. In this context, natural biopolymers have been drawing more attention because of the following reasons: (i) construction of living organisms´ bodies can be done strategically; (ii) different properties in solid, semi-solid, and solution phases; (iii) can be applied directly without requirement of further processing; (iv) permitting the simple design of thin coatings on the surface of solids by the existing methods, which are conventional in printing arts, textile, and packaging material industries [START_REF] Lapcik | Paper-based composite planar material[END_REF][START_REF] Lapcik | Method of increasing polypropylene biocompatibility by formation of a structured surface layer based on native or modified collagen[END_REF] and thus improving hydrophobicity and hydrophilicity. Cross-linked hydrogel coatings for surface enhancement of contact lenses have also been reported [START_REF] Nicolson | Extended wear ophthalmic lens, Official Gazette of the United States Patent and Trademark Office Patents[END_REF][START_REF] Chabrecek | Surface modification of extended wear contact lenses by plasma-induced polymerization of vinyl monomers, Fundamental and Applied Aspects of Chemically Modified Surfaces[END_REF].

The literature analysis indicates a dominant position of carbon based polymers in the form of nanoscale fibres and dispersions, cyclic fullerenes, alongside silicon and boron polymers. Of the aforementioned biopolymers, the review highlights cellulose and its derivatives viz, oxycellulose, carboxymethyl cellulose, nitrocellulose and hydroxyethyl cellulose.

Cellulose is the most plentiful, renewable, and natural polymer on the planet, with yearly output ranging from 75 to 100 billion tonnes for various industrial and biological uses [START_REF] Lapcik | Study of Penetration Kinetics of Sodium Hydroxide Aqueous Solution into Wood Samples[END_REF]. It is a glucose-based linear polysaccharide that gives mechanical support to plant cells, bacteria, algae, and fungi (Fig. 1(a)) [START_REF] Marchessault | All things cellulose: a personal account of some historic events[END_REF]. Most commercial cellulose fibres are obtained as pulp (natural fibres) or derivative from the natural cellulose of wood, barks, cotton, leaves, and jute. Carboxymethyl cellulose (CMC) was the first cellulose (Fig. 1 CMC is used in the paper, textile, pharmaceutical, cosmetic, and food industries, just like cellulose and other cellulose derivatives. Nitrocellulose (NC) is also one of the most commercially relevant and widely utilized natural cellulose derivatives [START_REF] Shao | Structure and properties of cellulose nitrate[END_REF]. Its nitrogen content determines the properties and applications of NC. NC with nitrogen content below 12.0 wt% can be used for coatings, printing inks, varnishes, filtration membranes, microfluidic technology, immunoassays, and biochemical analyses [START_REF] Zhang | Direct laser initiation and improved thermal stability of nitrocellulose/graphene oxide nanocomposites[END_REF][START_REF] Li | Use of nitrocellulose membranes as a scaffold in cell culture[END_REF][START_REF] Gao | Dual-scaled porous nitrocellulose membranes with underwater superoleophobicity for highly efficient oil/water separation[END_REF]. The formation of NC aerogel significantly increases NC's surface area and broadens its applications in separations, purification science, and biomedicine [START_REF] Zhang | Alcogel and aerogel of nitrocellulose formed in nitrocellulose/acetone/ethanol ternary system[END_REF]. Oxycellulose (OC) is a semi-synthetic derivative of cellulose that undergoes chemical depolymerization and enzymatic hydrolysis under in vivo conditions [START_REF] Dimitrijevich | Biodegradation of oxidized regenerated cellulose[END_REF][START_REF] Dimitrijevich | In vivo degradation of oxidized, regenerated cellulose[END_REF]. It is a non-toxic, biodegradable, biocompatible, and bioresorbable polymer [START_REF] Stilwell | Oxidized cellulose: chemistry, processing and medical applications, Drug Target[END_REF][START_REF] Kumar | HNO3/H3PO4-NANO2 mediated oxidation of cellulosepreparation and characterization of bioabsorbable oxidized celluloses in high yields and with different levels of oxidation[END_REF]. It is a well-established surgical material commercially available as a knitted fabric or staple fibres used to stop bleeding and prevent undesirable tissue adhesion during post operative healing [START_REF] Bajerová | Oxycellulose beads with drug exhibiting pH-dependent solubility[END_REF]. Hydroxyethyl cellulose (HEC) is non-ionic, hydrophilic, tasteless, and non-toxic cellulose derivative used extensively in pharmaceuticals, cosmetics, and food products (Fig. 1(c)) [START_REF] Kanmani | Properties and characterization of bionanocomposite films prepared with various biopolymers and ZnO nanoparticles[END_REF][START_REF] Noreen | Hydroxyethylcellulose-g-poly (lactic acid) blended polyurethanes: Preparation, characterization and biological studies[END_REF]. For instance, HEC crosslinked with citric acid is used as probiotic entrapment material [START_REF] Singh | Cellulose-based edible films for probiotic entrapment[END_REF]. HEC coatings in combination with other polysaccharides such as chitosan are used and seldom investigated [START_REF] Franca | Chitosan/hydroxyethyl cellulose inserts for sustained-release of dorzolamide for glaucoma treatment: In vitro and in vivo evaluation[END_REF]. It has excellent film-forming ability, biocompatibility, and biodegradability, making it suitable for biomedical applications [START_REF] Alekseeva | Effect of the bentonite filler on structure and properties of composites based on hydroxyethyl cellulose[END_REF]. Composites of natural biopolymers with aluminosilicates are used as packaging materials in the food industry [START_REF] Tunç | Preparation and characterization of biodegradable methyl cellulose/montmorillonite nanocomposite films[END_REF].

The primary aim of this review is to characterize the aforementioned biopolymers (particularly CMC and HEC) followed by an extensive evaluation of their diffusion and diffuse process of swelling and dissolution of polymers. The review briefly covers the diffusion and coefficient or cross-linking of biopolymers, followed by a meticulous focus on mechanical properties, surface tension for spreading, diffusion coefficient, kinetics, and thermodynamics of diffusion, swelling, and dissolution [START_REF] Mikula | Kinetic dissolution measurement of polymers by solution viscosity recording[END_REF]. The review also evaluates the preparation of varieties of industrial auxiliaries for printing arts and textile industries, such as offset print humidifying solutions, offset printing plate's protective solutions, and means for improving colouring of textiles [START_REF] Kalisek | Evaluation of the fixation of polyester based textile materials. III. Influence of the fixation conditions on swelling time and degree of fixation[END_REF][START_REF] Pelikan | Applications of numerical methods in molecular spectroscopy[END_REF][START_REF]Fotochemické procesy[END_REF].

Characterisation of biopolymers

The characterisation evaluation of biopolymers (CMC and HEC) was conducted on the basis of crystal/solid-state and liquid/cross-linked state. The dynamics of the macromolecular chain movement was studied by temperature-dependent dielectric spectra in the frequency range of 20 Hz to 1 MHz [START_REF] Liedermann | Study of the Molecular Mobility of Polysaccharide Solid Thin Layers by Dielectric Relaxation Spectroscopy[END_REF] (Table 1) [START_REF] Liedermann | Dielectric relaxation spectroscopy of some polysaccharides[END_REF]. The characterisation of CMC with corn starch on the basis of crystallinity, mean length, percent purity and mean width was also reviewed (Table 2) [START_REF] Yu | Characterization and Solubility Effects of the Distribution of Carboxymethyl Substituents Along the Carboxymethyl Cellulose Molecular Chain[END_REF]. The data indicates the use of CMC biopolymer as a thickener, emulsion binder, and stabilizer [START_REF] Liedermann | Dielectric relaxation spectroscopy of some polysaccharides[END_REF][START_REF] Mali | Citric acid crosslinked carboxymethyl cellulose-based composite hydrogel films for drug delivery[END_REF] in the detergent, cosmetic [START_REF] Tirpude | Rabeprazole sodium delayed-release multiparticulates: Effect of enteric coating layers on product performance[END_REF], food, pharmaceutical, drilling, and textile industries [START_REF] Kausar | Polymer coating technology for high performance applications: Fundamentals and advances[END_REF]. It can also act as a binding granule to form drug tablets in the pharmaceutical industries. The degree of substitution does not affect the percentage of purity.

The percentage of crystallinity varies directly with the mean width [START_REF] Choi | Biomimetic flow control based on morphological features of living creatures[END_REF]. HEC is a non-ionic cellulose ether whose water solution properties are very useful in the pharmaceutical, alimentary, and industrial fields [START_REF] Marcelo | Determination of molecular parameters of hydroxyethyl and hydroxypropyl celluloses by chromatography with dual detection[END_REF]. HEC can be used as thickening agent, protective colloid, binder, stabiliser, and suspending agent in a variety of industrial applications including pharmaceuticals, textiles, paper, adhesives, coatings and emulsion polymerisation [START_REF] Guo | Pharmaceutical applications of naturally occurring water-soluble polymers[END_REF].

Sodium salt of carboxymethyl cellulose (NaCMC) is the most extensively used CMC polymer. The rheological analysis for NaCMC indicated that the degree of substitution (DS) affects the flow properties of NaCMC samples. At higher concentration, heterogeneously substituted samples have the ability to form gels [START_REF] Lopez | Electrostatic and hydrophobic interactions in NaCMC aqueous solutions: Effect of degree of substitution[END_REF] and may present thixotropic or shear aging [START_REF] Lopez | Electrostatic and hydrophobic interactions in NaCMC aqueous solutions: Effect of degree of substitution[END_REF][START_REF] Barba | Synthesis and characterization of carboxymethylcelluloses from non-wood pulps II. Rheological behavior of CMC in aqueous solution[END_REF]. As the DS decreases below a certain threshold (around 0.9), the interchain aggregates and formation of fringed micelles with crystalline core takes place [START_REF] Lopez | Electrostatic and hydrophobic interactions in NaCMC aqueous solutions: Effect of degree of substitution[END_REF][START_REF] Lopez | Oscillatory rheology of carboxymethyl cellulose gels: Influence of concentration and pH[END_REF][START_REF] Liebert | Exploitation of reactivity and selectivity in cellulose functionalization using unconventional media for the design of products showing new superstructures[END_REF][START_REF] Xiquan | Kinetics of the carboxymethylation of cellulose in the isopropyl alcohol system[END_REF]. These micelles are responsible for the thixotropic and gelling properties of the weakly substituted NaCMC. The rheological properties of the physical gels of CMC without the addition of cross-linking agents depends on hydrophobic interactions between individual macromolecular chains [START_REF] Dürig | Sur la structure des solutions aqueuses de carboxymethylcellulose[END_REF][START_REF] Debutts | Rheology of sodium carboxymethylcellulose solutions[END_REF][START_REF] Hermans | Investigation of the elastic properties of the particle network in gelled solutions of hydrocolloids. I. Carboxymethyl cellulose[END_REF]. This study revealed that CMC samples with DS ≈ 0.7-0.8 can be separated into two fractions by centrifugation. Fraction A displayed pseudo-plastic, non-thixotropic and non-gelling behaviour in salt free aqueous solution whereas fraction B displayed some solubility in water and good solubility in concentrated NaOH. The substituent distribution of fraction B was heterogeneous which led to the formation of thixotropic gel at higher concentration by hydrophobic inter-chain associations.

The rheological properties of aqueous solution of HEC can be controlled by modifying the polymer backbone with side chain achievements such as ionic or hydrophobic groups [START_REF] Patruyo | Shear and extensional rheology of solutions of modified hydroxyethyl celluloses and sodium dodecyl sulfate[END_REF]. In hydrophobically modified HEC [START_REF] Sau | Synthesis and solution properties of hydrophobically modified (hydroxyethyl) cellulose[END_REF][START_REF] Tanaka | Interaction of hydrophobically modified hydroxyethyl cellulose with various added surfactants[END_REF][START_REF] Kästner | Hydrophobically and cationically modified hydroxyethyl cellulose and their interactions with surfactants[END_REF], the thickening action is caused by the association of the hydrophobic side chains into an intermolecular transient network. On the other hand, cationically modified HEC tend to behave like typical poly-electrolytes [START_REF] Kästner | Interactions between modified hydroxyethyl cellulose (HEC) and surfactants, Colloids Surf[END_REF]. The viscosity enhancement is due to repulsion of the charged groups along the chain which expands the macromolecule. As the surfactant concentration increases, the number of side chains per micelle decreases to the point that intermolecular associations of polymer molecules are inhibited due to the electrostatic repulsion between attached micellar aggregates. As a consequence, the shear viscosity of the solution reaches a maximum and then decreases as surfactant concentration is further increased [53,[START_REF] Thuresson | Influence of cosolutes on phase behavior and viscosity of a nonionic cellulose ether. The effect of hydrophobic modification[END_REF]. The inhibition of interchain interactions through hydrophobic side chains eventually might lead to a viscosity that is even lower than that of the pure polymer solution.

Tensile strength (TS) is the film's retainable maximum tensile force. It explains the film's maximum force occurring during measurement. Various ratios of CMC produce different tensile strength results. In general, the tensile strength values reduce with the addition of a substrate. In a study conducted on CMC produced by carboxymethylation using sodium monochloroacetate and varying concentrations of sodium hydroxide, it was found that the NaOH concentration affected the TS but not the elongation at break (EB) of the CMC films [START_REF] Rachtanapun | Suwaporn Luangkamin, Krittika Tanprasert and Rungsiri Suriyatem. Carboxymethyl cellulose film from durian rind[END_REF]. TS increased with NaOH concentrations of 20 and 30 g/100 mL (Table 3) [START_REF] Rachtanapun | Suwaporn Luangkamin, Krittika Tanprasert and Rungsiri Suriyatem. Carboxymethyl cellulose film from durian rind[END_REF].

Beyond the 30 g/100 mL concentration, the TS of the films decreased. The more carboxymethyl groups there are, which is directly related to the DS, the more the intermolecular forces between the polymer chains increased, resulting in a higher TS. Somewhat similar results for TS and EB were recorded in another studies [START_REF] Rachatanapun | Synthesis and characterization of carboxymethyl cellulose from Mimosa Pigra peel[END_REF][START_REF] Wahyuningtyas | Combination of carboxymethyl cellulose (CMC)-Corn starch edible film and glycerol plasticizer as a delivery system of diclofenac sodium[END_REF]. The studies involved the formation of films with CMC and corn starch. It was found that combining four grams of CMC in the absence of corn starch yielded the highest tensile strength whereas, combination of corn starch with CMC in the ratio of 3:1 yielded the lowest tensile strength (Fig. 2). The percent of elongation increased with a lower concentration of CMC-corn starch (Fig. 3). In general, it was found that the tensile strength reduces with increase in the percentage elongation [START_REF] Wahyuningtyas | Combination of carboxymethyl cellulose (CMC)-Corn starch edible film and glycerol plasticizer as a delivery system of diclofenac sodium[END_REF]. Similar results for tensile strength and elongation at break analysis of HEC were studied [START_REF] Lu | All cellulose composites prepared by hydroxyethyl cellulose and cellulose nanocrystals through the crosslink of polyisocyanate[END_REF][START_REF] Khan | Mechanical and barrier properties of nanocrystalline cellulose reinforced chitosan based nanocomposite films[END_REF][START_REF] Huq | Nanocrystalline cellulose (NCC) reinforced alginate based biodegradable nanocomposite film[END_REF]. The study evaluated the tensile strength and elongation at break for pure HEC films vs the films prepared with the addition of cellulose nanocrystals (CNC). The addition of 4% of CNC increased the tensile strength by 120.2% but beyond 8%, the addition of CNC did not enhance the tensile strength. The elongation at break reduced by 95.6% in comparison to pure HEC film. The results indicated that the addition of CNC can improve the tensile strength but beyond a threshold value, the elongation at break decreases [START_REF] Lu | All cellulose composites prepared by hydroxyethyl cellulose and cellulose nanocrystals through the crosslink of polyisocyanate[END_REF].

Diffusion of polymers and surface tension

The term 'diffuse actions' is understood as whole processes that include the transport of substances because of the gradients in chemical potential. Temperature, pressure, electric potential, and magnetic field have significant contributions to chemical potential gradients.

Many authors have comprehensively studied this topic [START_REF] Mali | Citric acid crosslinked carboxymethyl cellulose-based composite hydrogel films for drug delivery[END_REF][START_REF] Kassem | Phosphoric acid-mediated green preparation of regenerated cellulose spheres and their use for all-cellulose cross-linked superabsorbent hydrogels[END_REF][START_REF] Kirsch | Gelatin-methacryloyl (GelMA) formulated with human platelet lysate supports mesenchymal stem cell proliferation and differentiation and enhances the hydrogel's mechanical properties[END_REF].

Two direct relations for explaining the diffusion phenomena are known as Fick's laws. For binary system applying the one-dimensional diffusion Fick's laws: i. Fick´s law: 𝐽 𝑖 = -𝐷 𝑖 𝛿𝑐 𝑖 𝛿𝑥 [START_REF] Mishra | Smart Materials For Clean And Sustainable Technology For Smart Cities[END_REF] ii. Fick´s law:

𝛿𝑐 𝑖 𝛿𝑡 = 𝛿 𝛿𝑥 (𝐷 𝑖 𝛿𝑐 𝑖 𝛿𝑥 ) (2)
where t, x, J i , D i , and c i are time, coordinates, diffusion flow, diffusion coefficient, and concentration of component i, respectively. For known initial and peripheral conditions by solving the given differential equations, the value of diffusion coefficient (D i ) can be determined. For unit concentration gradient, diffusion coefficient determines the flow of a substance per unit area perpendicular to the direction of flow. The values of diffusion coefficient for low molecular weight polymers are within 10 5 to 10 14 cm 2 s 1 . R. M. Barrer et al. [START_REF] Barrer | Some properties of diffusion coefficients in polymers[END_REF] divided the systems of polymer-penetrant into four different categories:

a) ∂c ∂t = D 𝜕 2 c 𝜕x 2 D = const. c = k.c ' (3) b
)

𝜕c 𝜕t = 𝜕 𝜕x [D(c) 𝜕c 𝜕x ] D = f (c) c = k.c ' (4) c
)

𝜕c 𝜕t = 𝜕 𝜕x [D(c) 𝜕c 𝜕x ] D = f (c) c = f '(c ' ) (5) d
)

𝜕C 𝜕t = 𝜕 𝜕x [D(c) 𝜕C 𝜕x ] D = f (c, t) c = f '(c ' ) (6)
'Category a' is typical for the diffusion of rare and most di-atomic gases in polymers at the viscoelastic state. 'Category b' describes the diffusion in systems of rubber and paraffinic hydrocarbons. 'Category c' is applicable for the diffusion of heavier hydrocarbons in rubber layers. 'Category d' is very common, and the diffusion coefficient depends on concentration and time in this category. Notably, this case involves the strong mutual interactions between chain and penetrant molecules. The mutual interactions usually lead to slow relaxations in the crystalline polymer.

Many technological fields depend on the exact knowledge of the effect of solid/liquid interface interactions on final spreading of liquid drops. The experiment observed the surfactant solutions behaviour at critical micellar concentration (CM conc × 10) (Fig. 4) [START_REF] Mourougou-Candoni | Influence of dynamic surface tension on the spreading of surfactant solution droplets impacting onto a low-surface-energy solid substrate[END_REF] and water drops from the impact moment to one second after the impact [START_REF] Lapčík | Gelová forma hmoty jako základ materiálově-inženýrských prvků : teze přednášky k profesorskému jmenovacímu řízení v oboru Materiálové vědy a inženýrství[END_REF]. All the solutions have the same process of spreading (Table 4) [START_REF] Mourougou-Candoni | Influence of dynamic surface tension on the spreading of surfactant solution droplets impacting onto a low-surface-energy solid substrate[END_REF]. The surface tension is the difference between the foaming medium's surface tension, the defoamer's surface tension, and both substances interfacial tension for the spreading diffusion coefficient (S); thus, mathematically, we can write : 𝑆 = 𝜎 𝑓 -𝜎 𝑑 -𝜎 𝑑𝑓 [START_REF] Nicolson | Extended wear ophthalmic lens, Official Gazette of the United States Patent and Trademark Office Patents[END_REF] where σ f is the surface tension of the foaming medium, σ d is the surface tension of the defoamer and σ df is the interfacial tension of both substances.

The defoamer's surface tension reduction causes the spreading coefficient to become increasingly positive. The process shows the defoaming's thermodynamic tendency.

Equation 3 is applicable for insoluble liquid defoamers in bulk. However, deforming effectiveness can be increased by certain dispersed hydrophobic solids. The process of spreading depends mainly on surfactants.

Diffuse process of swelling and dissolution of polymers

Diffuse process of swelling

The conversion of solid polymer in solution at temperatures below the second-order turning point (T g ) usually comprises swelling and dissolution processes. In the swelling process, the solvents come to introduce at the solid polymeric phase and expand the solid surface in the liquid phase. This effect is known as Kirkendahl´s effect [START_REF] Thompson | Solvent-free polymer lithography via the Kirkendall effect, Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms[END_REF]. It is the impact that leads to loosening of the polymer structure thereby enhancing the movement of macromolecular chain segments, and enhancing the activity of free volume in the system alongside the replacement polymer-polymer contacts by the polymer-solvent contacts. It results in the creation of swollen surface layer with defined thickness and structure [START_REF] Abou-Yousef | High efficiency antimicrobial cellulose-based nanocomposite hydrogels[END_REF]. The movement of phase interface, i.e., the interface between swollen layer and the liquid solvent is necessary to understand as its own swelling. The diffusion process of polymer dissolution has been studied in detail by Lapčik and Valko [START_REF] Lapčík | Kinetics of swollen surface layer formation in the diffusion process of polymer dissolution[END_REF]. According to them, the diffusion of liquid solvents into polymer foils can be evaluated on the basis of parameters describing the formation of a surface diffusion layer. Furthermore, the polymer matrix swells due to the solvent effect resulting in the developed surface diffusion layer being donated as a surface swollen layer (SSL) [START_REF] Ueberreiter | [END_REF]. The swelling process is unsteady in nature resulting formation of the swollen surface layer.

Swollen surface layer

SSL of polymer represents interphase resistance of a given value influencing the diffusion in both directions from the interface. Ueberreiter K. and Asmussen F. [START_REF] Ueberreiter | [END_REF][START_REF] Ueberreiter | Die auflösungsgeschwindigkeit von polymeren. 1. Formulierung des vorganges und seine temperaturabhängigkeit[END_REF][START_REF] Asmussen | Die auflösungsgeschwindigkeit von polymeren, 2. Mitteilung. Untersuchung der stabilität der quellschicht durch konzentrationsund viskositätsmessungen[END_REF][START_REF] Ueberreiter | Velocity of dissolution of polymers. Part I[END_REF][START_REF] Asmussen | Velocity of dissolution of polymers. Part II[END_REF][START_REF] Asmussen | Die Auflösungsgeschwindigkeit von Polymeren[END_REF][START_REF] Ueberreiter | Die auflösungsgeschwindigkeit von polymeren. 4. Mitt. Der einfluß der strömung des lösungsmittels[END_REF] reported that SSL comprises complex structures, mostly in temperatures below T g . Its thickness  depends primarily on molecular weight, temperature, the kinetic-thermodynamic activity of solvent and minutely on the thermal history of polymer samples. The other four layers within SSL were reported by the authors mentioned (Fig. 5) [START_REF] Ueberreiter | [END_REF][START_REF] Ueberreiter | Velocity of dissolution of polymers. Part I[END_REF][START_REF] Asmussen | Die Auflösungsgeschwindigkeit von Polymeren[END_REF][START_REF] Pisarcik | The influence of hydroxyethylcellulose on the diffusion of acetylsalicylic-acid trihydromagnesium salt in aqueous-solution[END_REF]. If the melting temperature drops below T gel , i.e., T L < T <T gel , the rubbery layer   disappears and then only layers      , and   are retained. When melting temperature decreases below T L , i.e., T < T L , layers   and   disappear, so the SSL is formed only by layer   . The latter two cases can be well clarified by the following reasons: dissolution in the range of temperatures (T gel ) and (T L ), i.e., T L < T < T gel leads to weakening of the rubbery layer   to a value that is not visible; it indicates that it cannot affect the swelling process. The transport of solvated polymers into the coexisting liquid phase of the solvent is the rate-determining step at higher dissolution temperatures. The liquid layer consists of solvent molecules accumulated by adsorption on the surface of polymeric substances. Even if the dissolution in flowing solvent is the rate of flowing fluid in this layer virtually zero, the concentration of polymer in this layer grows from almost zero value, which is on the interface with flowing solvent at a value   that is on the interface with the rubbery layer. The thickness of this layer depends decisively on the rate of the solvent flow. With increasing rate, the thickness decreases to a certain limit value and does not change anymore. The rubbery layer has the character of a liquid phase with a fixed structure. The content of solvent molecules is large, and the thickness of the layer is comparable with the thickness of the liquid layer. The concentration gradient from one interface to another is almost the same as the liquid layer. A sudden increase in the concentration gradient was observed at the rubbery and tough swollen layer interface. Polymer chains in this layer do not move in the same fashion as in the liquid phase and the thickness of the layer is significantly smaller than in the previous two regions. The system of this layer is composed of the continuous liquid phase, which is continuously permeated by a dispersed polymer (with a gel-like character). The concentration of the solvent is markedly smaller compared to the rubbery phase. The infiltration layer is characteristically different from the previous. The movement of solvent molecules in the direction of concentration gradient takes place in this layer, probably in the exact mechanism as the gas transport through polymeric membrane, i.e., jumping from one vacancy to another without any significant movement of polymer segments. The number of polymer-polymer cross -links is comparable with the number in the original sample. As mentioned before, it is clear that the diffusion of solvent molecules controls the kinetics of polymers into a solid polymer sample.

Diffusion and diffusion coefficient of cross-linking

Diffusion involves the movement of atoms, molecules, and ions from a higher concentration region to a lower one. The concentration gradient is the major driving force of diffusion.

Various fields, such as chemistry, widely use the diffusion concept to characterise swelling kinetics [START_REF] Peterkova | Determination of the diffusion coefficient of water into atelocollagen type I thin films by attenuated total reflection Fourier transform infrared spectroscopy[END_REF] and measurements of diffusion of macromolecules [START_REF] Xiong | Rectangle FRAP for Measuring Diffusion with a Laser Scanning Microscope[END_REF]. A mean square displacement's long-time slope gives the diffusion coefficient (D) of cross-linking:

𝐷 = 1 6 Iim t→∞ 𝑑 𝑑𝑡 ⟨|𝑟(𝑡) -𝑟(0)| 2 ⟩ (8)
where r(0) and r(t) are the centres of the mass's vectors at time = 0 and t, respectively, in the biopolymer CMC, an increase in the polymer concentration first reduces the diffusion coefficient. However, a larger polymer concentration than a critical value increases the diffusion coefficient [START_REF] Wu | Effect of cross-linking on the diffusion of water, ions, and small molecules in hydrogels[END_REF]. The diffusion coefficient in a CMC relates to the excess mixing volume and biopolymer fraction in the solution. Within the concerned region, the concentration of the solution density is given by the formula:

𝑝 = 𝑝 𝐵 0 (1+𝑎𝑊𝑝) (9)
where 𝑝 𝐵 0 and a𝑊𝑝 are the pure solvent density (0.998 g/cm 3 ) and the polymer concentration (𝑤𝑡/𝑤𝑡), respectively. For CMC, the value of "a" is 0.0047.

𝐷𝑔 𝐷0 = exp [-π ( ̇𝑟𝑠+𝑟𝑓 𝑘𝑠𝑎𝜑 -0.75𝐶∞-0.25 (1-2) -0.253 +2𝑟𝑓 ) 2 ] ( 10
)
where a is the equivalent bond length of the monomer, φ is the polymer volume fraction, r s is the radius of solute, r f is the radius of polymer chain, C  is the characteristic ratio of polymer,  is the Flory-Huggins polymer/solvent interaction parameter, D g is the diffusion coefficient of solute in gel, and D 0 is the diffusion coefficient of solute in water calculated from MD simulation and verified by literature [START_REF] Wu | Effect of cross-linking on the diffusion of water, ions, and small molecules in hydrogels[END_REF]. Since the fitting parameter for the different polymers and solutes considered, Amsden proposed this model as a "universal" model for solute diffusion in hydrogels.

An increase in the cross-linking density causes the diffusion of sodium ions to reduce at 0.5 M. The dynamic and theoretical predictions helped obtain the particles' normalized diffusion [START_REF] Wu | Effect of cross-linking on the diffusion of water, ions, and small molecules in hydrogels[END_REF]. The water content variation also changes the rate of diffusion [START_REF] Wahyuningtyas | Combination of carboxymethyl cellulose (CMC)-Corn starch edible film and glycerol plasticizer as a delivery system of diclofenac sodium[END_REF]. There is a need to estimate the Rhodamine's rotational relaxation time to calculate the translational diffusion coefficient [START_REF] Wu | Effect of cross-linking on the diffusion of water, ions, and small molecules in hydrogels[END_REF]. The increase in the cross-linking density causes the diffusion coefficient of small. The rise in the relaxation times of the hydrogen bonding correlates with the reduction in water diffusion.

The kinetics of swelling and dissolution of CMC and HEC

The swelling and dissolution mechanism for CMC and HEC are not homogeneous because of its multi-scale and complex structures. The ballooning phenomena are the most spectacular effects of the heterogeneous swelling and dissolution. Some selected zones along the fibre provide places for swelling to take place. It is assumed that the structural origins of the ballooning effect relate to the morphological variations among the walls. Each of the balloons presents a swollen state of the biopolymers. The quality of the solvent strongly influences the swelling and dissolution mechanisms (Fig. 6) [START_REF] Lapčík | Kinetics of swollen surface layer formation in the diffusion process of polymer dissolution[END_REF]. The swelling and dissolution characteristics can be revealed at the different cellulose structures' lengths from the macromolecule to the fibre's walls.

Kinetics of swelling

In practical applications, a higher swelling rate and a higher swelling capacity are required. It is well known that the swelling kinetics for the absorbents is significantly influenced by various factors such as swelling capacity, the size distribution of powder particles, specific size area, and composition of the polymer (Fig. 7, 8) (Table 5) [START_REF] Jamlang | Developing a physically cross-linked hydroxyethyl cellulose hydrogel for wound dressing applications[END_REF][START_REF] Ghanbarzadeh | Physical properties of edible modified starch/carboxymethyl cellulose films, Innovative food science & emerging technologies[END_REF][START_REF] Seki | Carboxymethylcellulose (CMC)-hydroxyethylcellulose (HEC) based hydrogels: synthesis and characterization[END_REF]. Various authors [START_REF] Joshi | Graft copolymerization of 2hydroxyethylmethacrylate onto carboxymethyl chitosan using CAN as an initiator[END_REF][START_REF] Pandey | Graft copolymerization of acrylic acid onto guar gum initiated by vanadium (V)-mercaptosuccinic acid redox pair[END_REF] have investigated the influence of these parameters on the swelling capacity.

For example, Metz et al. [START_REF] Metz | Transport of water vapor and inert gas mixtures through highly selective and highly permeable polymer membranes[END_REF] analysed the dependency of water absorbency of superabsorbent polymers on particle size. Results indicated that the absorption rate increased as the particle size became smaller. This may be attributed to an increase in surface area with decreasing particle size of samples [START_REF] Soleimani | Studies of swelling kinetics of carboxymethyl cellulose-g-PMAAm-co-PNIPAm superabsorbent hydrogels[END_REF].

In a study conducted on the formation and subsequent analysis of swelling kinetics of a CMC-N-isopropylacrylamide based hydrogel, the effect of particle size on the swelling kinetics was confirmed. The rate of water absorbance increased sharply and then began to level-off (Fig. 9) [START_REF] Soleimani | Studies of swelling kinetics of carboxymethyl cellulose-g-PMAAm-co-PNIPAm superabsorbent hydrogels[END_REF]. Considering the second -order kinetics, the swelling rate at any time is expressed as:

𝑑𝑊 𝑑𝑡 = 𝐾(𝑊 ∞ -𝑊) 2 ( 11
)
where W is the water content of the superabsorbent at time t, W ∞ is the water content before the equilibrium has been reached and K is the constant. Substituting equation 11 with limits t = 0 and W = 0 to W, and rearrangement, the equation can be re-written as:

𝑡 𝑊 = 1 𝐾𝑊 ∞ 2 + 1 𝑊 ∞ 𝑡 (12)
The swelling of the super-absorbent composites obeys second order kinetics (Fig. 10) [START_REF] Seki | Carboxymethylcellulose (CMC)-hydroxyethylcellulose (HEC) based hydrogels: synthesis and characterization[END_REF][START_REF] Soleimani | Studies of swelling kinetics of carboxymethyl cellulose-g-PMAAm-co-PNIPAm superabsorbent hydrogels[END_REF].

Similar results were found for the hydrogels developed from HEC [START_REF] Wang | Synthesis, swelling and responsive properties of a new composite hydrogel based on hydroxyethyl cellulose and medicinal stone[END_REF]. Composite hydrogels with co-polymerization with HEC, sodium acrylate and medicinal stone (MS) were created.

The analysis of kinetic swelling (Fig. 11) [START_REF] Wang | Synthesis, swelling and responsive properties of a new composite hydrogel based on hydroxyethyl cellulose and medicinal stone[END_REF], depicted the effects of MS content, particle size and saline solution on the swelling kinetics. It was found that the swelling rate was faster at initial 600 seconds. With prolonging swelling time, the swelling rate decreased until a plateau was reached. Schott's second-order swelling kinetics model [START_REF] Schott | Swelling kinetics of polymers[END_REF] (equation 13) was introduced to evaluate the hydrogels' kinetic swelling behaviours.

𝑡 𝑄 𝑡 = 1 𝐾 𝑖𝑠 + 1 𝑄 ∞ 𝑡 (13)
were, Q t (g/g) is the swelling capacity of the hydrogel at time t (s), Q  (g/g) is the theoretical equilibrium swelling capacity, and K is (g/g s) is the initial swelling rate constant.

Kinetics of dissolution

The dissolution of polymeric materials has some typical features. The rate of dissolution is controlled by several factors, the determining one is the diffusion rate into the polymer matrix. The diffusion rate depends on the chemical structure of the polymer, solvent and functional groups present on the polymer chain (dissolving parameter density of cohesive energy), solvent molecular volume, dissolution temperature, molecular weight of the polymer, structural ordering of the polymer etc. By getting the polymer into contact with the solvent, the diffusion of solvent molecules into the polymer matrix proceeds, creating thus the SSL (δ) [START_REF] Kalisek | Evaluation of the fixation of polyester based textile materials. III. Influence of the fixation conditions on swelling time and degree of fixation[END_REF][START_REF] Mracek | The diffusion process of sodium hyaluronate (Na-HA) and Na-HA-n-alkyl derivatives films swelling[END_REF]. The thickness of this layer depends on the temperature of dissolution, mixing rate, molecular weight of the polymer, kinetic and thermodynamic activity of the solvent, and thermal history of polymers [START_REF] Ueberreiter | Die auflösungsgeschwindigkeit von polymeren. 1. Formulierung des vorganges und seine temperaturabhängigkeit[END_REF][START_REF] Asmussen | Die auflösungsgeschwindigkeit von polymeren, 2. Mitteilung. Untersuchung der stabilität der quellschicht durch konzentrationsund viskositätsmessungen[END_REF][START_REF] Ueberreiter | Diffusion in polymers, Diffusion in Polymers[END_REF][START_REF] Lapcik | The Internal Pressure in Thin Polymer Layers in Contact with Liquids, 41.Mezinárodní konference o nátěrových hmotách[END_REF].

Surface swollen layer of a polymer represents interfacial resistance of definite matter, which influences diffusion of molecules in one or another direction from the interface. Ueberrreiter and Asmussen have found that it has a complicated structure with respect to the dissolving temperature [START_REF] Kalisek | Evaluation of the fixation of polyester based textile materials. III. Influence of the fixation conditions on swelling time and degree of fixation[END_REF].

If temperature of dissolution (T) ranges between gel point temperature (T gel ) and glass transition temperature (T g ) (T gel < T < T g ), then the structure of the surface layer consists of liquid (δ 1 ), rubbery (δ 2 ), solid swollen (δ 3 ), and infiltration layer (δ 4 ). Thickness and distribution of particular layers in swollen polymer layer are represented in Fig. 12 [89].

Cross-linkers for creating gel-polymer

The gel in general is consisted of the cross -linked networks of hydrophilic water-soluble polymers. Gel polymers can swell after absorbing water. It is possible to prepare natural polymer-based hydrogels by mechanical and physical cross-linking [START_REF] Drnovska | Hyaluronate derivatives and their applications[END_REF]. In the case of polymeric networks, their structure is fully dependent on covalent bonds or physical inter-molecular interactions between individual macromolecules. However, each connecting point of the network are mutually separated by a linear secondary chain. In the case when these secondary chains are sufficiently long, and their chemical structure allows adoption of various conformations, these create soft and flexible materials. Contrary to this, rigid secondary chains cause the creation of hard and glassy materials. The characteristic property of the stable elastomeric network is its swelling ability when in contact with proper solvent systems. Therefore, because the polymeric network is, in fact, structuralised solid matter that cannot be dissolved with an exemption of the cases when there is observed loss of their coherent macroscopic structure [START_REF] Lapčík | Gelová forma hmoty jako základ materiálově-inženýrských prvků : teze přednášky k profesorskému jmenovacímu řízení v oboru Materiálové vědy a inženýrství[END_REF]. Thus, when such cross-linked structure is immersed into the solvent, which is miscible with the polymer chains of the network, the solvent is absorbed, and the network is swollen. The increase of the volume can be up to one thousand percent [START_REF] Lapcik | Hyaluronan: Preparation, structure, properties, and applications[END_REF]. Chemical reactions initiated by pressure, heat, irradiation, and change in the pH level can facilitate cross-links formation [START_REF] Uyanga | Citric acid crosslinked natural bi-polymer-based composite hydrogels: Effect of polymer ratio and beta-cyclodextrin on hydrogel microstructure[END_REF]. The latter cross-linking can be achieved by chemical reactions such as polycondensation, polyaddition or by physical entanglements of macromolecular chains. Physical entanglements and disentanglements of macromolecular chains were reported [START_REF] Kupska | The viscometric behaviour of sodium hyaluronate in aqueous and KCl solutions, Colloids and surfaces: Aphysicochemical and engineering aspects[END_REF][START_REF] Lapcikova | Rheological Properties of Food Hydrocolloids based on Polysaccharides[END_REF] as induced by temperature and pH of the dissolved polymer system known as helix-coil transition. For example, mixing certain chemical agents, such as cross-linking reagents with partially polymerized or unpolymerized resin creates cross-links [START_REF] Toledo | Carboxymethyl cellulose/poly (acrylic acid) interpenetrating polymer network hydrogels as multifunctional adsorbents[END_REF]. Cross-linking is commonly used to improve starch performance for a variety of applications [START_REF] Raab | Materiály a člověk:(netradiční úvod do současné materiálové vědy[END_REF]. Cross-linking agents, such as sodium tri metaphosphate, phosphorus oxychloride, epichlorohydrin, citric acid, diphenyl sulphone, anthracene endo-peroxide [100, 101, 102], sodium tripolyphosphate, and 1,2,3,4-diepoxybutane have been used for the improvement of the materials´ mechanical properties and the starch products´ water stability.

Divinyl sulfone (DVS) is a non-zero length cross-linker that has been widely used to chemically modify polymers to improve the mechanical properties and enhance the performance according to the suitability (Fig. 13) [START_REF] Maleki | Effect of shear on intramolecular and intermolecular association during cross-linking of hydroxyethylcellulose in dilute aqueous solutions[END_REF]. It has been extensively used in combination with cellulose and its derivatives, especially in the pharmaceutical industries.

The concentration of DVS added to the biopolymer directly influences its rheological properties. It has been found that higher concentrations of DVS increases the viscosity thereby indicating a higher cross-linking in the biopolymer [START_REF] Maleki | Effect of shear on intramolecular and intermolecular association during cross-linking of hydroxyethylcellulose in dilute aqueous solutions[END_REF].

Polymers may act as a basic building component of the larger systems namely of block co-polymers and micelles. The latter micelles can be organised in the presence of the surface-active agents to liquid crystalline systems [105].

Conclusions

Viewing the state of polymeric materials, the reviewed biopolymers possess enhanced structural diversity, biodegradability, thin coat-forming ability, and diverse flexibility, allowing them to be used in varied application prospects in technologically influential intelligent applications. The lightweight and biocompatibility of cellulose derivatives have been essential in biomedical and non-biomedical fields. The reviewed biopolymers have a high potential in the preparation of intelligent high-tech materials and in unlocking new possibilities for natural biopolymer layers to be better suited to the environment and the needs of end-users. This review provides an attractive route for the scientific community to research the technologically important and intelligent coatings for varied purposes. ref. [START_REF] Lapčík | Kinetics of swollen surface layer formation in the diffusion process of polymer dissolution[END_REF] with the permission of Springer Nature licence no. 5274210286515 Fig. 7 Water uptake of HEC hydrogel samples with respect to time (Reproduced from ref.

[80] with the permission of Trans Tech Publications Ltd. Request ID 600074454)

Graphical abstract

Graphics Abstract

  (b)) to attain commercial and industrial relevance in the 1920s, and it will continue to have a dominant position in the global market. CMC can be easily made via simple processes and low-cost ingredients. The vast market potential of CMC is the main advantage compared to other cellulose derivatives.

Figure chapter Fig. 1 (Fig. 2 Fig. 3 Fig. 4 Fig. 5 Fig. 6

 chapter123456 Figure chapter

Fig. 8 Fig. 9 Fig. 10 Fig. 11 (Fig. 12 Fig. 13

 8910111213 Fig. 8 Moisture absorption of starch films as a function of CMC content (Reproduced from

  permittivity of the sample, ԑ s static permittivity of the sample, (ԑ s -ԑ ∞ ) relaxation strength, τ 0 position of the relaxation on the same axis, α slope of the lowfrequency side of the relaxation curve, product α × β describes the slope of the highfrequency side of the relaxation curve, A' is sample surface area, and n is purely D.C. conductivity mechanism.

Figure 4

 4 Figure 4

Figure 5

 5 Figure 5

Figure 6

 6 Figure 6

Figure 7

 7 Figure 7

Figure 8

 8 Figure 8

Figure 9

 9 Figure 9

Figure 10

 10 Figure 10

Figure 11

 11 Figure 11

Figure 12

 12 Figure 12

Figure 13

 13 Figure 13

Table chapter Table 1

 chapter1 Calculated parameters of measured dielectric spectra of studied polysaccharides (Courtesy of[START_REF] Liedermann | Dielectric relaxation spectroscopy of some polysaccharides[END_REF])

Table 2

 2 Different characteristics of CMC sample (Reproduced from ref.[START_REF] Yu | Characterization and Solubility Effects of the Distribution of Carboxymethyl Substituents Along the Carboxymethyl Cellulose Molecular Chain[END_REF] with the courtesy of BioResources, North Carolina State University)

Table 3

 3 Mechanical properties for CMC films synthesized with various NaOH concentrations at 25 ºC, 75% RH (Courtesy of [55])

Table 4

 4 Characteristics of surfactant solutions. Critical micellar concentration (CM conc) and equilibrium surface tension at CM conc × 10. (Reproduced from ref.[START_REF] Mourougou-Candoni | Influence of dynamic surface tension on the spreading of surfactant solution droplets impacting onto a low-surface-energy solid substrate[END_REF] with the permission of Academic Press, Elsevier Science licence no. 5270931090431)

Table 5

 5 Maximum water uptake of hydrogels (Reproduced from ref.[START_REF] Jamlang | Developing a physically cross-linked hydroxyethyl cellulose hydrogel for wound dressing applications[END_REF] with the permission of Trans Tech Publications Ltd. Request ID 600074454)

Table 1 .

 1

	Parameter	Compound
		CMC	HEC
	10 3 (ԑ s -ԑ ∞ )	37.60	2.28
		2.18
	α	0.0598	0.676
		0.970
	β	1.000	1.000
		1.000
	10 -3 τ 0 /s	68.100	5.120
		3.810
	A'	2.48 × 10 -19 1.08 × 10 -18

Table 2 .

 2

	Degrees of substitution	Mean length	Mean width	Percentage crystallinity	Purity
	(DS)	(mm)	(μm)	(%)	(%)
	0.75	2.28	48.1	18.03	95
	0.76	2.13	55.7	19.60	95
	0.80	1.94	52.8	23.51	95
	0.73	1.43	51.7	25.85	95

Table 3 .

 3

	Types of film	Tensile strength	Elongation at break
		(MPa)	(%)
	20g/100mL NaOH-CMC	141.16 ± 8.32	2.32 ± 0.26
	30g/100mL NaOH-CMC	255.54 ± 4.13	2.20 ± 0.81
	40g/100mL NaOH-CMC 140.77 ± 14.43	2.07 ± 0.50
	50g/100mL NaOH-CMC	53.42 ± 8.21	2.95 ± 1.13
	60g/100mL NaOH-CMC	79.62 ± 9.76	2.28 ± 0.98

Table 4 .

 4

	Surfactant CMC	Equilibrium surface tension
		(g/litre)	at CM conc × 10
			(mN/m)
	NPOEOP	0.06	36.9
	C13OE18	0.08	27.7
	Silvett L77	0.10	20.4
	C10OE6	0.80	26.1
	TTAB	1.18	37.2
	CTAC	0.42	34
	DOS	0.92	27
	SDS	2.38	36.5
	SDG	0.60	25
	NPOEOP: nonylphenyl polyoxypropylene; C13OE18: isodecyl polyoxyethylene;
	Silvett L77: trisilioxane oxypropylene polyoxyethylene; C10OE6: isodecyl polyoxyethylene;

TTAB: tetradecyl trimethylammonium bromide; CTAC: cetyltrimethylammonium chloride; DOS: sodium dioctyl sulfosuccinate; SDS: sodium dodecyl sulfate; SDG: sodium decylgalacturonate

Table 5 . 12.5% HEC 13.75% HEC 15.0 % HEC

 5

	% Maximum Water Uptake 160.5	173.8	156.9

Acknowledgement

The authors would like to express their gratitude for financing this research by the internal grant of Palacky University in Olomouc (project no. IGA_PrF_2022_020) and to the internal grant of Tomas Bata University in Zlin (project no. IGA/FT/2022/005). Financial support to the author YM by Fischer scholarship of the Faculty of Science, Palacky University in Olomouc in 2021. The authors would also like to express their gratitude to Mgr. Martin Pykala, PhD. (Catrin, Palacky University Olomouc) for preparation of graphical abstract.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have influenced the work reported in this paper.


Audrey Douinot

Hélène Roux

Pierre-André Garambois

Kévin Larnier

Denis Dartus

Which hydrological activity in weathered bedrock during flash floods?

- = ∂ ∂ + ∂ ∂ ( )       - + = ∂ ∂ I S 1 K t I f 0 s θ θ ( ) ( ) ( )       - - = = m exp T T dx h grad T Q s 0 out θ θ

Conclusion & Perspectives

Subsurface f. WR f.

Flows through deep horizons were estimated to represent up to 29 % and 46% of the total hydrograph according to MARINE hydrological modelling and hydrograph decomposition respectively. These results are consistent with previous studies (Vannier et al, 2013) that underline hydrological activity into weathered bedrock layers with significant contribution to flash flood processing.

Similar results are provided by both independent hydrological modeling and hydrograph decomposition methods, that give them consistency.

For more accurate results the hydrological model should detail subsurface flows and flows into deeper horizons with appropriate physical modeling. Hydrograph decomposition could help to find different flow contributions. The method has to be improved, by calibrating BFI parameter and computing the recession constant by focusing on flash flood events.

A physical process oriented model with 5 parameters to calibrate ( , soil depth: C Z )

Agly: sedimentary catchment Beaume:

granitic catchment

  Open Archive TOULOUSE Archive Ouverte (OATAO)OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 13870 To link to this article : URL : http://www.hymex.org/?page=public/workshops/8/programme To cite this version : Douinot, Audrey and Roux, Hélène and Garambois, Pierre-André and Larnier, Kevin and Dartus, Denis Which hydrological activity in weathered bedrock during flash floods? The case of Mediterranean catchments. (2014) In: 8th HyMeX Workshop -Hydrological cycle in Mediterranean Experiment -HyMeX 2014, 15 September 2014 -18 September 2014 (Valetta, Malta). (Unpublished) Any correspondance concerning this service should be sent to the repository administrator: staff-oatao@listes-diff.inp-toulouse.fr Which hydrological activity in weathered bedrock during flash floods? The case of Mediterranean catchments Audrey Douinot, Hélène Roux, Pierre-André Garambois, Kévin Larnier, Denis Dartus Université de Toulouse ; INPT, UPS ; IMFT (Institut de Mécanique des Fluides de Toulouse) ; F-31400 Toulouse, France CNRS ; IMFT ; F-31400 Toulouse, France Objectives Assessing hydrological dynamic into weathered rock layer with: hydrological modeling: MARINE (Roux et al., 2011) baseflow separation (Eckhardt, 2005) Case study: 8 catchments of French mediterranean area Estimation of flow and storage into weathered rock layers through MARINE

  processing : MARINE forcing with soil depth from BD-sol: subsurface flow estimations MARINE forcing with likehood calibration: flow into weathered rock layers (WR F.) computed making recession analysis.The BFI parameter is fixed to 0.8 as(Eckhardt, 2005) suggests.

  to apply the decomposition method Roux, H., Labat, D., Garambois, P. A., Maubourguet, M.-M., Chorda, J. & Dartus, D. 2011. A physically-based parsimonious hydrological model for flash floods in Mediterranean catchment Nat. Hazards Earth Syst. Sci. J1 -NHESS, 161, 2567-2582. Eckhardt, K. 2005. How to construct recursive digital filters for baseflow separation. Hydrological Processes, 19, 507-515.Vannier, O., Braud, I., Anquetin, S. 2013. Regional estimation of catchment-scale soil properties by means of streamflow recession analysis for use in distributed hydrological models. Hydrological Processes.


Rafael Accácio Nogueira
email: rafael-accacio.nogueira@centralesupelec.fr

Romain Bourdais
email: romain.bourdais@centralesupelec.fr

Hervé Guéguen
email: herve.gueguen@centralesupelec.fr

Detection and Mitigation of Corrupted Information in Distributed Model Predictive Control Based on Resource Allocation

In distributed predictive control structures, communication among agents is required to achieve a consensus and approach an optimal global behavior. Such negotiation mechanisms are sensitive to attacks on these exchanges. This paper proposes a monitoring scheme that detects and mitigates these attacks' effects in a resource allocation framework. The performance of the proposed method is illustrated through simulations of the temperature control of multiple rooms under power scarcity.

I. INTRODUCTION

 to integrate optimality and constraints.

 techniques are a promising way to handle the optimization problem's complexity. In these structures, there is no longer a single controller for all systems. Instead, we use a set of local communicating controllers. These strategies thus reduce the computing burden while increasing confidentiality.

Many works use distributed optimization techniques, such as Lagrangian relaxation [3], Alternating Direction Method of Multipliers (ADMM) [4], primal decomposition [5], dual decomposition [6], [7], [8], and others [9], [10]. In these methods, local agents interact with a coordinator who uses an iterative process to ensure convergence towards the solution of the initial problem.

Usually, it is assumed that all agents work in perfect cooperation. However, when it is not the case, these uncooperative behavior have pernicious effects on the overall system, and their impact can be studied. The cause of this disruptive behavior can be either involuntary due to hacking or malfunctioning, or voluntary, by developing selfish behavior. Recent work has begun to explore this issue. In the article [6], the authors are interested in the vulnerabilities induced when distributed predictive control is built on dual decomposition. They analyze the impact of the deception depending on where it occurs: either in the followed references

or directly in the local cost functions or coupling constraints. The same authors propose defense strategies against these attacks, either by using secure scenarios based on reliable historical data [START_REF]Scenario-based defense mechanism for distributed model predictive control[END_REF] or by ignoring extreme values of control signals [START_REF] Velarde | Vulnerabilities in lagrange-based distributed model predictive control[END_REF]. Then [START_REF] Chanfreut | Vulnerabilities in distributed model predictive control based on Jacobi-Gauss decomposition[END_REF] extends the initial work to analyze the vulnerabilities of the Jacobi-Gauss decomposition method.

Another way of dealing with these changes in behavior can be using robust distributed control principles, coupled with hierarchical identification of the attack [START_REF] Braun | Hierarchical attack identification for distributed robust nonlinear control[END_REF], or the introduction of probabilistic models that implement a resilient strategy if the information exchanged is outside the confidence interval [START_REF] Ananduta | Resilient distributed model predictive control for energy management of interconnected microgrids[END_REF].

In this work, we analyze the exchange among agents controlled by DMPC using primal decomposition, which is perfectly adapted for agents that share resources. More specifically, we are interested when malicious agent steers these exchanges. By exploiting the nominal structure that characterizes the communication between the agents and the coordinator, we propose a monitoring scheme that detects an attack, and if necessary, corrects it.

The remainder of this paper is organized as follows. First, in Section II, the primal decomposition-based DMPC is introduced. In Section III, we discuss a model of the agents' selfish behavior that exploits the vulnerabilities of this DMPC structure. Then, in Section IV, we discuss the structure of the DMPC and how we can exploit it to construct a defense scheme to counteract the selfish agent. At last, we present a particular mechanism to detect the agents' selfish behavior and mitigate its effects. Moreover, in Section V, an application is given to illustrate and evaluate the algorithm's performance. Finally, in Section VI, we conclude, and we give an outlook of future works.

II. PRELIMINARIES AND PROBLEM STATEMENT

Notation: In this paper, ∥ • ∥ and ∥ • ∥ F represent the ℓ 2 and Frobenius norms. ∥v∥ Y is the weighted norm, ∥Y 1 2 v∥. P T (•) is the Euclidean projection onto set T. ⊗ represents the Kronecker product. 1 m,n is a m × n matrix filled with 1. I c is a c × c identity matrix. π v denotes the number of elements in v. A vector v i , correspond to the i-th agent, and these vectors can be stacked in a vector v.

A. Model Predictive Control

Our primary purpose is to control a system composed of M subsystems using MPC. The dynamics of the state x i (k) of i-th agent w.r.t input u i (k) are described by the following linear discrete-time systems:

x i (k + 1) = A i x i (k) + B i u i (k) (1)
The M subsystems are coupled under linear input constraints. We assume as an interesting case when these constraints prevent the subsystems from meeting the systems' needs. Consequently, the constraints will always be active, yielding the same results from equality constraints [START_REF] Boyd | Convex Optimization[END_REF]:

M i=1 Γ i u i (k) = u max (2)
where Γ i : R π u i (k) ×π u i (k) and u max : R π u i (k) ×1 . A known formulation of the MPC structure [START_REF] Camacho | Model Predictive Controllers[END_REF], [START_REF] Velarde | Vulnerabilities in lagrange-based distributed model predictive control[END_REF], [START_REF]Scenario-based defense mechanism for distributed model predictive control[END_REF], [START_REF] Simon | Reference tracking mpc using terminal set scaling[END_REF], [START_REF] Chanfreut | Vulnerabilities in distributed model predictive control based on Jacobi-Gauss decomposition[END_REF] with finite prediction horizon N p is the following:

Problem 1: Global MPC Problem. minimize ui(k:k+Np-1|k) J G (k) M i=1 J i (k) Np j=1 ∥v i (k + j|k)∥ 2 Qi + ∥u i (k + j -1|k)∥
Q i ≥ 0, R i > 0. v i (k) represents a control objective. It can either be v i (k) = w i (k) -x i (k) for reference tracking, where w i (k) is a state reference, or v i (k) = x i (k) for disturbance rejection.
The optimal value of the problem 1 is denoted by J ⋆ , and the optimal control sequences are represented by

u ⋆ i (k : k + N p -1|k).
At each time k, the problem is solved, and the u ⋆ i (k|k) are applied in each respective i subsystem, following a receding horizon strategy.

One can see that if the subsystems were not coupled by (2), the overall system could be decomposed into M parts, solvable in parallel. Multiple decomposition methods solve this problem [START_REF] Boyd | Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers[END_REF], [START_REF] Paulen | Primal and dual decomposition for distributed MPC -theory, implementation, and comparison in a SoS simulation framework[END_REF], [START_REF] Pflaum | Comparison of a primal and a dual decomposition for distributed MPC in smart districts[END_REF], [START_REF] Chanfreut | Vulnerabilities in distributed model predictive control based on Jacobi-Gauss decomposition[END_REF]. Still, since we are interested in resource constraints and the dual decomposition does not enforce local feasibility [START_REF] Boyd | Notes on decomposition methods[END_REF], the primal decomposition is chosen.

B. Distributed Model Predictive Control

The technique consists of decomposing the coupling constraints (or complicating constraints [START_REF] Boyd | Notes on decomposition methods[END_REF]) of the original optimization problem into local versions with additional variables that are shared among them, negotiating the value of these variables until a consensus is reached.

Problem 1 is decomposed into multiple subproblems (3a), solvable in parallel, and a master problem (3b), which is equivalent to the original problem and uses information of the subproblems [START_REF] Boyd | Notes on decomposition methods[END_REF]:

J ⋆ i (θi(k)) = minimize u i (k:k+Np-1|k) Ji(k) s.t. (1) Γiui(k) = θi(k) : λi(k)        ∀i ∈ {1, . . . , M } ∀j ∈ {1, . . . , Np} (3a)
J ⋆ = minimize θ(k:k+Np-1|k) M i=1 J ⋆ i (θi(k)) s.t. M i=1 θi(k) = umax (3b)
The subproblems (3a) are formed by the local objectives J i (k) and a set of local constraints, with a sequence of allocations θ i (k : k + N p -1|k) and associated sequence of dual variables (Lagrange multipliers) λ i (k : k + N p -1|k). For brevity's sake, we drop the (k : k + N p -1|k) sequence notation, using only where pertinent.

The variables θ i represent the resource or the "quantity" allocated for each subproblem; thus, the names "quantity decomposition" and "resource allocation" are also given for this decomposition [START_REF] Cohen | Optimization by decomposition and coordination: A unified approach[END_REF].

The master problem shown in (3b) can be solved using an iterative method that updates the allocation sequence θ i .

Due to the form of the constraints, we use the projected sub-gradient method whose recurrence equation is:

θ (p+1) = P H (θ (p) -ρ (p) g (p) ) (4)
where

H = {θ | M i=1 θ i = u max }, g (p)
is a sub-gradient of J ⋆ (θ (p) ) at the instant p and ρ (p) is an iteration step, wellchosen, so the method converges.

The sum M i=1 θ i can also be represented by the matrix multiplication I M c θ, where

I M c = 1 M,1 ⊗ I c . Where c = π ui(k:k+Np-1|k) = N p π ui(k) .
Assuming strong duality holds, we can use the sensitivity analysis of the problem [13, § 5.6.2], and we can conclude that the opposite of the sequences of optimal dual variables, -λ ⋆ i , which are θ

(p) i dependent, is a sub-gradient of J ⋆ i (θ (p)
i ), which can be used in (4) to solve the problem (3b). Applying the Euclidean projection onto H [16] and using -λ ⋆ i (θ 4) results in the complete expression for the allocation's update [10, §VI-C]:

(p) i ) in (
θ (p+1) i = θ (p) i + ρ λ ⋆ i (θ (p) i ) -I M c (I M c T I M c ) -1 I M c T λ ⋆ (θ (p) )
(5) In each step (p), the subproblems receive a sequence of allocation of the total resources. Then they return their corresponding sequence of dual variables so the master problem can be solved by updating the allocations, recommencing the negotiation. Once a consensus is reached, the negotiation is finished, each subsystem takes the last sequence of inputs u ⋆ i (k : k + N p -1|k) calculated and applies the first element u ⋆ i (k|k), following a receding horizon strategy. Delegating the iterative process of allocation update to an agent with the coordinator's role, we have the scheme in Fig. 1 that illustrates the negotiation. Observe that each block negot solves (5) for a respective agent i. This way, the only interaction that the coordinator has with the subsystems is via the variables λ

(p) i and θ (p+1) i
, increasing the privacy of the subsystems.

Algorithm 1 resumes the distributed control problem solved to calculate the optimal input sequence at each time k using quantity decomposition.

III. ATTACK IN DMPC SCHEME

As expected [START_REF] Boyd | Notes on decomposition methods[END_REF], [START_REF] Cohen | Optimization by decomposition and coordination: A unified approach[END_REF], this decomposition method works well when each agent cooperatively calculates its λ i correctly. Here we study the effects when an ill-intentioned agent exploits the scheme for its interest. Coordinator initializes θ (0) p := 0 repeat Subsystems solve (3a), and send λ ⋆ i (θ (p) ) Coordinator updates allocations ( 5) [START_REF]Scenario-based defense mechanism for distributed model predictive control[END_REF], [START_REF] Chanfreut | Vulnerabilities in distributed model predictive control based on Jacobi-Gauss decomposition[END_REF], [START_REF] Ananduta | Resilient distributed model predictive control for energy management of interconnected microgrids[END_REF] present 4 types of attacks, which can be divided into 2 principal groups: changes in the optimization parameters (selfish attack -multiply the objective function by a scalar α, fake reference, and fake constraints) and nonagreed control (liar agent). In the decomposition scheme used in this work, the coordinator allocates the resources. So we can discard the last kind of attack.

λ i → λ negot negot negot λ (p) M λ (p) 2 λ (p) 1 θ (p+1) 2 λ (p) θ (p+1) 1 θ (p+1) M
p := p + 1 until ∥θ (p) -θ (p-1) ∥ ≤ ϵ [6],
Although we could make the same analysis from the mentioned works, we are interested in the coordinator's point of view, so any of these attacks will reflect as a change on the λ i received. Therefore, we propose that any selfish agent sends a corrupted λi = γ i (λ i )

to the coordinator instead of sending the agreed λ i . We give a unidimensional example where γ i (λ i ) = τ i λ i to illustrate such an attack. Here, 4 agents negotiate with the coordinator, and agent 1 attacks the system (τ 1 ̸ = 1).

In Fig. 2, we see that when

τ i > 1, agents 1's local cost J ⋆
1 decreases while all other costs, including the overall J ⋆ , increase. This attack is comparable to the selfish attack portrayed in [START_REF] Velarde | Vulnerabilities in lagrange-based distributed model predictive control[END_REF]. This decrease in the cost justifies the attack since the attacking agent has more comfort than all others. On the other hand, when τ 1 tends to 0, J ⋆ 1 increases and all others J i decreases, while still degrading the overall cost J ⋆ . Such an agent could be considered as a benevolent agent or an agent attacked by a malevolent one.

From this variation in the values of λ 1 caused by τ 1 , we can interpret its role in the negotiation: the values of λ i Costs J and J i for different values of τ 1

J J 1 J 2 J 3 J 4
Fig. 2. Change of J ⋆ with respect to the non-cooperative coefficient τ 1 .

represent the dissatisfaction with the given allocation θ i . Since the negotiation (5) finds its stability when

λ (p) -I M c (I M c T I M c ) -1 I M c
T λ (p) = 0, that means when all λ i are equal to the mean of the λ i . We can interpret that the coordinator's role is to minimize the overall dissatisfaction. This way, the selfish agent can lie about its dissatisfaction (increasing λ i by using an adequate γ i (•)), driving the negotiation to a value of θ i that "satisfies" it more (lower optimal value J ⋆ i ). Another effect we can expect from the observation of ( 5) is that the negotiation may not converge for some values of λ i . We can find those values by the analysis of the eigenvalues of the iterative process. This effect is illustrated in the hatched area in Fig. 2.

IV. SECURE DMPC BASED ON RESOURCE ALLOCATION

As seen, a malicious agent can deviate the allocations for its benefit, driving the negotiation or even destabilizing it. Hence, it is needed to find a way to lessen the effects caused by this agent. To fill this gap, we propose a detection and mitigation mechanism to reduce the effects of any agent malfeasance in the negotiation. However, before presenting the mechanism, we need to analyze the problem structure to sustain the proposition.

A. Quadratic Case -Formal Analysis

Another known form to represent the problems (3a) is using matrix representation [START_REF] Chanfreut | Vulnerabilities in distributed model predictive control based on Jacobi-Gauss decomposition[END_REF]:

minimize Ui(k) J i (θ i ) 1 2 U i (k) T H i U i (k) + f i (k) T U i (k) s.t. Θ i U i (k) = θ i : λ i (7)
If we take reference tracking, for instance, we have:

Hi = D T i QiDi + Ri fi(k) = D T i Qi(Mixi(k) -Wi(k)) (8)
The input and setpoint predictions for times k to k + N p calculated in time k are adequately stacked in vectors U i (k) and W i (k). M i and D i are the prediction matrices of the MPC. Qi , Ri , and Θ i are block diagonal matrices built repeating N p times Q i , R i , and Γ i respectively. Notice that the matrices H i are not only symmetric positive definite, but they are also time-invariant, unlike the f i (k), which depend on x i (k) and W i (k).

Observe that since J i (θ i ) is quadratic, we can get an explicit solution for its dual variables λ i , which are affine with respect to θ i :

λ i = -P i θ i -s i (k) (9)
where

P i = (Θ i H -1 i Θ T i ) -1 and s i (k) = P i Θ i H -1 i f i (k)
. We can observe that P i are symmetric and depend only on Θ i and H i , which are time-invariant.

B. Detection and mitigation

In this secure scheme, the exchange between coordinator and agents is divided into two parts: first, to detect any misbehavior, and second, the negotiation itself, which limits the effects of eventual attacks.

Assumption 1: γ i (•) is the same during the negotiation phase for a given time k (it does not depend on p).

Assumption 2: We suppose the agent chooses a linear function such as [START_REF] Cohen | Optimization by decomposition and coordination: A unified approach[END_REF] and we define Pi (k) = T i (k)P i and si (k) = T i (k)s i (k).

λi = γ i (λ i ) = T i (k)λ i = -T i (k)P i θ i -T i (k)s i (k),
Given that P i does not change from time to time, we can use the relation between θ i and λ i , shown in (9), to find estimates P i (k) such as:

λi = γ i (λ i (θ i )) = -Pi (k)θ i -si (k) (11)
Remark 1: If the estimation does not converge, necessarily there has been a change in P i since the relation between λ i and θ i has ceased to be affine.

If we estimate P i (k) for two different times k and they differ, then there has been a change in behavior in agent i.

Assumption 3: We have access to the nominal value of P i , denoted Pi , from reliable attack-free historical data.

Using this strategy, we can detect a deviation from nominal behavior using

E i (k) = ∥ P i (k) -Pi ∥ F , where ∥ • ∥ F is the Frobenius norm. Let d i ∈ {0, 1} be an indicator that detects the attack in agent i. If the disturbance E i (k) respects an arbitrary bound E i (k) ≤ ϵ P , (12)
then d i = 0, and no attack is detected. Otherwise, d i = 1, and a change in behavior of agent i is detected. If the attack is detected and we want to counteract the change in λ i , one strategy would be to recover λ i from an inverse of γ i (•).

Assumption 4: We suppose λi = 0 only if λ i = 0, which implies T i (k) invertible.

Using these assumptions, we can try to estimate the inverse of T i (k) as in

T i (k) -1 = Pi Pi (k) -1 , (13)
and from ( 9), we can derive a method to reconstruct λ i :

λ irec = T i (k) -1 λi = -Pi θ i -T i (k) -1 si (k). (14)
Notice that we also need si (k) to use this reconstruction. This reconstructed λ irec can be used in [START_REF] Paulen | Primal and dual decomposition for distributed MPC -theory, implementation, and comparison in a SoS simulation framework[END_REF]. Observe that, as [START_REF] Simon | Reference tracking mpc using terminal set scaling[END_REF] does not depend on λi , the rest of the negotiation process takes place without taking the attacking agent's responses into account.

In case no attack is detected, the coordinator can use the λi during the negotiation phase. This mechanic of detecting and choosing which version of λ to use during the negotiation, corresponds to the inclusion of a supervisor for each agent (Fig. 3), Observe in Fig. 3 that the coordinator sends θi to the agents. These values may be the ones from the negotiation or other. The reason to send different values is discussed in the following subsection.

C. Considerations about parameter estimation

As seen, we need to estimate Pi (k) and si (k). This estimation is achieved by the relation between θ i and λ i shown in [START_REF] Chanfreut | Vulnerabilities in distributed model predictive control based on Jacobi-Gauss decomposition[END_REF]. As we suppose there is no noise in the communication, we propose to use Recursive Least Squares (RLS) with a forgetting coefficient ϕ to find simultaneously unbiased estimates of Pi (k) and si (k).

If we try to estimate during the negotiation, the estimation will fail since consecutive values of λ p i and θ p i are necessarily linearly dependent [START_REF] Paulen | Primal and dual decomposition for distributed MPC -theory, implementation, and comparison in a SoS simulation framework[END_REF], and estimators become badly scaled. This fact is known and is described as low input excitation [17, §5]. As a counter-measure, to enrich the input excitation, the coordinator sends a sequence of random values of θ i until the estimation converges. It then resumes the typical negotiation, eventually using the mitigation mechanism if an attack is detected. 3. Scheme for Secure DMPC Assumption 5: Since P i is expected to be symmetric [START_REF] Chanfreut | Vulnerabilities in distributed model predictive control based on Jacobi-Gauss decomposition[END_REF], we suppose that the attacker chooses a T i (k) that does not change the structure of the resulting matrix, so it can not be discovered. In this case, we assume Pi (k) symmetric and invertible. As Pi (k) is symmetric, we estimate only the upper triangle, reducing the number of estimated parameters from π Pi(k) to

λ(p+1) 2 θ2 (p) λ2 (p+1) Agent 1 Supervisor negot negot λ(p+1) 1 θ1 (p) λ1 (p+1) θ (p) 2 λ(p) θ (p) 1 Agent M Supervisor negot M . . . λM (p+1) θ (p) M θM (p) λ(p+1) M Fig.
π Pi (k) + √ π Pi (k)

2

, and consequently the length of the estimation sequence [START_REF] Åström | Adaptive Control, ser. Addison-Wesley series in electrical and computer engineering: Control engineering[END_REF].

We stack the elements of P i (k) and si (k) estimated in a step h in vectors η h i . The estimation converges when ∥η h iη h-1 i ∥ ≤ ϵ, with ϵ arbitrarily small.

D. Secure DMPC

After all the reflections about parameter estimation and the detection and mitigation mechanism, we can finally propose a secure DMPC based on the reconstruction of λ i .

Algorithm 2 summarizes the process used to find the optimal inputs u ⋆ i (k|k) to be applied at each time k. We can see the two phases: the detection phase, where the coordinator detects if the system is attacked and by which agent. And the second phase, where the usual negotiation in algorithm 1 takes place, using different values of λ i depending on if the respective agent is an attacker.

In the next section, we present an example to illustrate the performance of the mechanism. Subsystems solve (3a), and send λ ⋆ i (θ (h) ) Coordinator estimates P i (k) (h) and si (k) (h) h := h + 1 until ∥η h iη h-1 ∥ ≤ ϵ Coordinator computes d i using [START_REF] Ananduta | Resilient distributed model predictive control for energy management of interconnected microgrids[END_REF] Negotiation Phase:

Coordinator initializes θ (0) p := 0 repeat Subsystems solve (3a), and send λ ⋆ i (θ (p) ) Coordinator updates allocation (5) using adequate versions of λ i for each agent:

λ ⋆ i (θ (p) ), if d i = 0 and λ irec , if d i = 1 p := p + 1 until ∥θ (p) -θ (p-1) ∥ ≤ ϵ V. EXAMPLE: TEMPERATURE CONTROL
In this example, we want to control the temperature of 4 distinct rooms (called I, II, III, and IV) under power scarcity using quantity decomposition. The systems are modeled as continuous-time linear time-invariant systems using the 3R-2C model [START_REF] Gouda | Building thermal model reduction using nonlinear constrained optimization[END_REF].

The state-space model of each subsystem is given by:

˙ x Ai x W i = ẋi = A ci x i + B ci u i y i = C ci x i (15)
where

A ci = -1 Cres i Rfi - 1 Cres i Rii 1 Cres i Rii 1 CsiRii -1 CsiRoi -1 CsiRii B ci = 10 Cres i 0 T C ci = 1 0 (16)
We can see the meaning and the values of its parameters in tables I and II.

The states x Ai and x W i represent the mean temperatures of the air and walls inside room i. The input u i is the heating power for the corresponding room. The global coupling constraint is

4 i=1 u i (k) = 4kW.
The subsystems are discretized using the zero-order hold discretization method with sampling time T s = 0.25h and the quantity decomposition-based DMPC is implemented using prediction horizon N p = 4.

Three scenarios are simulated for a period of 5 hours: 1) Nominal behavior.

2) Agent I presents constant non-cooperative behavior T I (k) = 4 I √ π P I for k ≥ 6, without correction. 3) Agent I presents constant non-cooperative behavior T I (k) = 4 I √ π P I for k ≥ 6, with correction, ϵ P = 10 -4 .

In Fig. 4, first, we compare the output of the agent I (air temperature in the room) with its reference (20 • C), and then the decision variable E I (k) with the threshold ϵ P . All the 3 scenarios above are represented with indices N (for nominal), S (for selfish), and C (for corrected).

Observe that in the nominal behavior, the reference w I is not reached due to power scarcity since we deliberately set a total power not sufficient to satisfy the needs of each agent. As expected, the decision variable lies under the threshold ϵ P = 10 -4 with values of order E N I (k) ≈ 10 -10 . When the agent presents a selfish behavior, the tracking error w Iy I is reduced but insufficient to attain the reference. In this case, the detection variable surpasses ϵ P , E S I = 9.762, indicating the change of behavior of agent I.

When the correction is activated in the system, we see that the corrected y C I approaches the nominal value y N I , illustrating the good performances of our proposition.

We can also evaluate the performance of the proposed mechanism by comparing the local and global costs calculated using the initial cost function presented in (7) using N as the total period of simulation, N = 20. The same 3 scenarios are compared in table III.

As in Section III, when agent I is selfish, we see the decline of its cost at the expense of increasing all other costs. This increase in cost degrades the global objective. When the correction mechanism is activated, the differences between costs are minimal, and the global cost stays close to the nominal value, highlighting the mechanism's performance.

VI. CONCLUSION AND FUTURE WORKS

In this paper, an algorithm for monitoring and correcting exchanges between agents in a resource-sharing system has been proposed. The algorithm exploits the particular structure of exchanges, part of which must be constant over time. The first phase consists of identifying this constant part and checking if an attacker has modified it. From this identification, it is possible to reconstruct the original mechanism and find the centralized optimality. This principle should be generalized to other types of decomposition structures, and this is what we plan to do in the near future.

Fig. 1 .

 1 Fig. 1. Scheme of DMPC using a coordinator and M agents.

Algorithm 1 :

 1 Quantity decomposition based DMPC.

Algorithm 2 :

 2 Secure DMPC.

© 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See https://www.ieee.org/publications/rights/index.html for more information.

VII. ACKNOWLEDGMENTS

The authors would like to acknowledge C. R. Sorgho for her preliminary results.