Modeling oil paint network formation and the characterization of the molecular topology

Bachelor Thesis Chemistry

Modeling oil paint network formation

for characterization of the molecular

topology

Jorien Duivenvoorden

Supervisors: dr. I. Kryven

, prof. dr. P.D. Iedema

and dr. C. Fonseca Guerra

1

_{Paint Alterations in Time,}

Van ’t Hoff Institute for Molecular Sciences, University of Amsterdam

and The Rijksmuseum, Amsterdam

2

_{Computational Biochemistry and Molecular Recognition}

Division of Theoretical Chemistry

Free University Amsterdam

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Bachelor Thesis Scheikunde

Modeling oil paint network formation and the

characterization of the molecular topology

door

Jorien Duivenvoorden

2 juli 2015

Studentnummer

10431810

Onderzoeksinstituut

HIMS

Onderzoeksgroep

PAinT

Verantwoordelijk docent

Prof. dr. P.D. Iedema

Abstract

Oil paintings and their degradation is a popular topic of research. However, cured, or polymerized, oil paint is a strongly cross-linked polymer and analysis of the structure of the network is practically impossible because of this. Insight in the structure is necessary to explain several degradation processes, such as the migration of metal ions, that are not fully understood thus far. In this study the only option to obtain structural information about the molecular topology of an oil paint network after polymerization is employed; an advanced model is developed that simulates the formation of an oil paint network. Because this challenge has not been addressed until now, we looked for the methodology to other fields of research. We employed a kind of Gillespie Monte Carlo method in combination with graph theory, a mathematical approach to describe the connectivity of a network. The monomers of an oil paint polymer are triacylglycerides (TAG-units), which contain double bonds that are responsible for the formation of the cross-links. The starting point of the model is the assumption that the reactivity of these monomers only depends on their number of double bonds, or their functionality. This is the main assumption from polymer reaction engineering and the application to oil paint is already a novelty. Oil paint monomers can have a functionality up to 9, which is considerably more than most industrial monomers. Because of this and when no other effects that influence the reactivity of the monomers are taken into account, the network becomes too dense. Therefore, we added three novel advanced routines to approach a realistic chemical system more accurately. Preferential coupling, the first effect, ensures that monomers that are close together will connect with a high probability. Secondly, intramolecular bonds within a TAG-unit are allowed. The third effect is steric hindrance and implies that shielded monomers have a low probability of connecting. The final part of this research is about the validation of the model. An experiment is designed within the research group and we wrote an algorithm to simulate this experiment. The goal of the experiment is to break the network into pieces by hydrolyzing the TAG-units. The original cross-links are retained in these fragments that can be analyzed. The experimental results can be coupled to the model to find the original network and all its structural information.

Dutch summary

Er wordt tegenwoordig veel onderzoek gedaan naar olieverfschilderi-jen, omdat men ze wil bewaren voor volgende generaties. Het probleem is dat olieverf geen statisch systeem is, maar in grote mate verandert in de tijd. Deze veranderingen in olieverf zijn verantwoordelijk voor de degra-datie ervan. Om de degradegra-datie te kunnen voorkomen, is het belangrijk om de chemie van olieverf beter te begrijpen. Olieverf is een polymeer-systeem, dat wil zeggen dat het is opgebouwd uit bouwstenen, die we monomeren noemen. Deze monomeren binden met elkaar en vormen een ingewikkeld netwerk tijdens het drogen van de verf. Een olieverfschilderij is dus eigenlijk niets meer dan een ingewikkeld polymeernetwerk. Dit is goed te zien in de figuur. Als je inzoomt op een schilderij, zie je eerst de verschillende verflagen en de pigmenten zitten. Zoom je vervolgens nog verder in, tot op een moleculaire schaal, dan wordt dit polymeer netwerk zichtbaar.

De tuin van Daubigny van Van Gogh op microscopische en moleculaire schaal. Meer inzicht in de topologie, of de moleculaire architectuur, van dit netwerk is belangrijk om degradatieprocessen te kunnen begrijpen. Ex-perimenteel is het namelijk vrijwel onmogelijk om de structuur van een gepolymerizeerd netwerk te kunnen bepalen. Daarom wordt in dit onder-zoek het olieverfnetwerk gesimuleerd. We hebben een geavanceerd model ontwikkeld dat de formatie van een olieverfnetwerk simuleert om zo aller-lei eigenschappen te kunnen ophelderen. We zijn begonnen met een al-gemene aanname: de reactiviteit van de monomeren wordt bepaald door de hoeveelheid bindingen die ze kunnen maken. Sommige monomeren in olieverf kunnen wel negen bindingen maken, wat in vergelijking met de meeste industrile monomeren erg veel is. Daarom krijgt het netwerk een erg hoge dichtheid. Zo hoog zelfs, dat dit fysisch niet mogelijk bli-jkt te zijn. Daarom zijn drie effecten toegevoegd aan het model, die ervoor zorgen dat het model steeds meer een echt chemisch polymeer-netwerk benaderd. Ten slotte is er in dit onderzoek ook rekening mee gehouden dat een link tussen het model en de realiteit noodzakelijk is om de resultaten te kunnen gebruiken. Daarom is er een experiment ontworpen ter validatie en een algoritme geschreven dat dit experiment simuleert. De experimentele resultaten kunnen teruggekoppeld worden naar het model om zo informatie over de structuur van het netwerk te krijgen. Deze informatie is cruciaal om olieverf beter te kunnen begrijpen en de huidige conserveringsmethodes te verbeteren. Naast olieverf, kan ons model ook benut worden om andere polymeersystemen te simuleren; een onverwachte bijkomstigheid die erg interessant kan blijken te zijn

Contents

1 Introduction 5

1.1 Oil paint network and its characterization . . . 5

1.2 Research context . . . 6

1.3 Introduction to graph theory . . . 7

2 Standard model 10 2.1 Simulation details . . . 10

2.2 Gillespie Monte Carlo . . . 11

2.3 Post-processing . . . 13

2.3.1 Component size distribution . . . 13

2.3.2 Gel fraction . . . 13

2.3.3 Gel point . . . 13

2.3.4 Degree distribution . . . 14

2.3.5 Cyclomatic number . . . 14

2.3.6 Average shortest path . . . 14

2.3.7 Clustering coefficient . . . 14

2.3.8 Number of triangles . . . 15

2.3.9 Linear size distribution . . . 15

2.3.10 Community distribution . . . 16

2.3.11 Dispersity . . . 16

2.4 Discussion of the results . . . 17

3 Advanced network formation 22 3.1 Preferential coupling . . . 22

3.1.1 Discussion of the results . . . 23

3.2 Intramolecular bonds . . . 30

3.2.1 Discussion of the results . . . 30

3.3 Shielding . . . 32

4 Validation of the model 33

5 Conclusion 37

6 Prospects 39

7 Acknowledgments 39

Appendices 40

1

Introduction

1.1

Oil paint network and its characterization

Since oil paint has first been discovered by Van Eyck in the early fif-teenth century, it is has been a widely used paint medium for many years to come. Whether Van Eyck really was the first to use oil paint is dis-putable, but art historians do agree on the boost that the use of oil paint experienced from this moment onwards.1 Historically, oil paintings are painted with drying oils such as linseed, walnut and poppy seed oils.2In the drying process of these oils no solvent evaporates, but the chemical structure of the oil changes.3_{Would Van Eyck have known that oil paint} is actually a dense, entangled network made from cross-linked building blocks that we call monomers? Probably not.

Figure 1: A cross-section of Daubigny’s Garden by Van Gogh4 and a schematic representation of the oil paint network on a molecular level.

Nowadays, it is clear that oil paint forms a cross-linked polymer net-work during the drying process. The oil monomers consist of a glycerol molecule linked to three fatty acids through ester bonds. They are tri-acylglycerides, or TAG-units, and are shown in figure 2. Fatty acids can contain double bonds in their hydrocarbon chains that we call un-saturations. These double bonds are responsible for the drying process, because they react with oxygen to form cross-links. These double bonds are called functionalities and they correspond to the maximum number of cross-links one TAG-unit can make. Slightly different, but not less important, is the degree of a TAG-unit. This corresponds to the current number of cross-links one TAG-unit has made.

Figure 2: A TAG-unit is the abbreviation of triacylglyceride, which is built up from a glycerol molecule and three fatty acids.

Unfortunately, the reactivity of oil paint does not disappear once it has dried. Several side-reactions can still occur which lead to the degra-dation of the painting, such as hydrolysis of the ester bonds, formation of new oxygen containing functional groups and oxidative cleavage of the fatty acid hydrocarbon chains.5 6 The details of most of these pro-cesses are not known; an unfortunate fact, because knowledge about these degradation processes could help to improve existing conservation meth-ods. Not only specific details of reactions within oil paint are interesting, but also the physical properties of the oil paint network itself. More insight into the structure of this interconnected network could explain migration of mobile paint components, visible but yet not completely understood processes.

The problem with dried oil paint, however, is that it is a dense, in-terconnected network and therefore the experimental characterization of a polymer network is rather limited. It is possible to measure the mass fraction of the insoluble fraction, the swelling caused by a specific solvent, and the weight distribution of molecules in the soluble fractions.7 8_Also, the fatty acid composition of the gel can be analyzed with GC-MS by hydrolyzing the TAG-units.2 9 The molecular topology, or structure, can not be experimentally determined. Therefore, the only option to obtain this structural information which is so highly valuable for conservation scientists, is to simulate the oil paint network.

1.2

Research context

Simulations in art research are not common, but since oil paint is actually a polymer, other fields of research can provide us with more knowledge about simulating polymers. Because modeling has been an important part of polymer reaction engineering for 60 years now, it can provide suitable methodology to simulate polymers. Considerable contribution to this field has been done by Duˇsek who used population balance equations to describe network formation, already in the sixties and seventies.10 By considering all possible structures of a polymer network, he could derive mean properties even without the computational power that is available nowadays. Like the name indicates, mean field models do not obtain

spe-cific topologies, but only mean properties, because no distinction is made between different monomers.11 12 13 Importantly, Duˇsek did adjust the probability of connecting monomers in such a way that not all monomers have the same chance of connecting.14 _{A mean field model is one way} to describe a network, but there are several alternatives. They can be subdivided in in three groups: mean field models, percolation models and kinetic Monte Carlo models. Compared to mean field, percolation mod-els make more use of topologies, although constrained on a grid. This does mean that percolation models have three dimensions and therefore provide spatial information.

The third group is the kinetic Monte Carlo method, which is a prob-abilistic method that allows the following of the growth of individual chains as a function of time and/or conversion by jumping from one state to another according to a certain probability. A drawback of this method is its high computational requirements. The definition of such a state can be a property, such as a size distribution of polymer units.15 Recently, Hamzehlou et al. have made a contribution to this field by taking a topology as the state on which the simulation is based. Using their model the conversion evolution of the entire molecular weight distri-bution, the cross-linking density distribution and available double bond density distribution before and after gel point can be determined in con-trast to earlier models that could only give detailed information before the gel point. Surprisingly, the only structural information this model offers is the chain length between two branching points, the distribution of chains with a given chain length and cross-linking density.7 Particu-larly, this structural information that can be extracted from topologies is highly interesting for conservation scientists, since it cannot be extracted from actual oil paint and, more importantly, it explains several processes in the oil paint that lead to degradation such as the formation of metal soaps.

The hiatus in the field of theoretical descriptions of polymer networks lies in the combination of the work of Duˇsek and Hamzehlou et al. and this is exactly where this study is focused on. The topologies obtained by a kinetic Monte Carlo model employed by Hamzehlou et al. provide structural information that can be used to adjust the probability of cer-tain monomers to connect, just as Duˇsek did. Only now, the monomers can be treated separately, because all the information needed for this is provided by the topologies. A start has been made by Meimaroglou and Kiparissides who take steric hindrance into account with their combined stochastic kinetic/topology algorithm.16 _{This study will advance further} into the field of kinetic Monte Carlo topology models with the use of graph theory, which proves to be very suitable for this.17

1.3

Introduction to graph theory

In this study the oil paint network is described using graph theory. Graph theory employs mathematical structures to describe connections in a net-work. These networks can range from your friends on Facebook, to the World Wide Web and to neurological networks in your brain.18 _{In the}

field of sociology, graph theory is a widely applied method to study net-works and to obtain structural information about netnet-works. A network is made up of points that are called nodes. The connection between the nodes is called an edge. All the information about the connectivity of the network is stored in an adjacency matrix. This matrix is an N by N matrix and N represents the number of nodes. A one in the matrix indicates a connection or edge between the two nodes. Figure 3 shows a network that is described by the adjacency matrix on the right.

In this study graph theory will be used to research the molecular topology of an oil paint network and therefore the chemical concepts have to be translated into graph theory concepts. Table 1 shows the most important concepts. A more elaborate list of terms is shown in Appendix 1.

Chemical concept

Graph theory concept

TAG-unit

Node

Cross-link

Edge

Functionality

Maximum number of neighbours

Degree

Current number of neighbours

Table 1: Most important concepts in this study

The goal of this study is to write an advanced model in MatLab R _that

describes an oil paint polymer network accurately and that will make the characterization of the molecular topology of this network possible. The model is based on a random graph, described for the first time by Erd˝os and R´enyi in 1951. In a random graph the edges are placed at random between a fixed number of nodes n and these edges are present with an equal probability p. This simple model, also called the Erd˝os-R´enyi model, or G(n, p), can be solved analytically.19 The model described in this report is also a random graph, but the probability p depends on several factors and the most important one is the fatty acid composition of linseed oil, shown in Appendix 2. The fatty acid composition deter-mines the functionality of the TAG-units, which means that a TAG-unit can have a functionality up to 9. This is considerably more than most industrial polymers.11

In this study the general assumption from polymer reaction engineer-ing has been applied to oil paint; the reactivity of the monomers depends only on their functionality.11 _{However, not only the maximum degree of} the monomers is extended in this study, but also novel advanced routines are added to the standard model in order to provide detailed structural information about a dried oil paint network. This report will describe the mathematics of the model and it will discuss the first results obtained by the model.

2

Standard model

2.1

Simulation details

We developed a model, in which we applied the main assumption from polymer reaction engineering to oil paint and we call it the standard model. The information about the simulated network is stored in adja-cency matrix A. Matrix A describes the connectivity of the network and it is important to realize that matrix A and the topology of the network contain exactly the same information.

A = ai,j= (

1 if nodes i and j are connected

0 if nodes i and j are not connected (1) Every node has an index i = 1, . . . , N assigned to it. The fatty acid composition of linseed oil determines the maximum number of connec-tions node i can form, which is the functionality, or maximum degree, dmax

i = 0, . . . , 9. The maximum degrees dmaxi are sampled form a prede-fined distribution, Wl= X i+j+k=l i,j,k>0 EiEjEk, l = 0, . . . , 9, 0 < Wl< 1 (2)

where Ei = 0, . . . , 3 denotes concentrations of fatty acid with i unsatu-rations. The nodes with maximum degree 0 are called singles, because they will never be able to connect to another node. To save calculation time, the singles are omitted in the network formation process until the end, where the number of singles will be added to the total number of nodes again. dmax

i is a vector that assigns every node their functionality. An important constraint in the process of network formation is that this maximum degree can never be exceeded.

Next, a distribution ri,j is created that describes the possibilities of making connections between every couple of nodes (i, j). These probabil-ities are only determined by the functionality of the nodes. Every time a node i makes a connection to node j, the functionality of these two nodes decreases with one. Therefore, the functionality has to be recalculated every time by subtracting the number of bonds a node has made (dcurr

i ) from their original functionality (dmax

i ). This can be found in matrix A that holds all the information about the network. By summating values in all the columns and adding the values in the diagonal, the current degree (dcurr i ) can be found. dcurr_j = 1 2 _XN i=1 ai,j+ aj,j  , j = 1, ..., N (3)

df ree_i = dmaxi − dcurri (4) The chance of node i connecting to node j is calculated by multiplying the functionality of the nodes. All chances are stored in ri,j.

ri,j = (

df ree_j · df ree_i , i 6= j

An important assumption that is made in the standard model is called the well-stirred assumption. It means that it is assumed that TAG-units are diffusing through the whole system so that every monomer is in principle able to form a connection to any other monomer. In Appendix 3 an flowchart of the algorithm is shown. Figure 4 shows a visualisation of a network obtained with the standard model at different stages of formation. However, for a better description of an actual oil paint network, more factors have to be taken into account. Unfortunately, this quickly becomes complicated. In section 3 novel strategies for the formation of an advanced network and their implementations into the model are discussed.

2.2

Gillespie Monte Carlo

The method of the aforementioned generation of a network is based on the Gillespie Monte Carlo algorithm.20 _{The Monte Carlo methods are} a broad class of computational algorithms that are based on repeated random sampling. Below is a summary of the steps to run the Gillespie’s algorithm:

Initialization: Initialize the number of molecules in the system, reac-tion constants, and random number generators. In this case the num-ber of nodes (N ) and the functionality or maximum degree distribution (dmax_{) are initialized.}

Monte Carlo step: Generate random numbers to determine the next reaction to occur as well as the time interval. The probability of a given reaction to be chosen is proportional to the number of substrate molecules. The before mentioned matrix ri,j describes all the possibil-ities of making connections between every couple of nodes. To get a probability distribution, ri,j has to be normalized:

Pi,j= ri,j N P i,j=1 ri,j (6)

From this probability distribution couples can be randomly sampled to connect.

Update: Increase the time step by the randomly generated time in Step 2. Update the molecule count based on the reaction that occurred. This model does not contain time, but only degree of polymerization or conversion. Even though time plays no role, the formation of the connections still happen in the right order.

Iterate: Go back to Step 2 unless the number of reactants is zero or the simulation time has been exceeded. In this model the conversion indicates the progress of the network formation. The conversion is indicated with the symbol χ and corresponds to the degree of polymerization. Thus, when the conversion is one, the maximum number of bonds has formed. Subsequently, the number of possible reactants is zero and the network formation has terminated.

(a) χ = 0.16 (b) χ = 0.18

(c) χ = 0.23 (d) χ = 0.36

Figure 4: Visualisation of networks at different values of conversions (χ). The colors indicate individual molecules, also called components, but have no further meaning.

2.3

Post-processing

The next step in this research is to analyze the generated network. There are several methods to post-process the data and they result in different interesting properties. In order to have as little fluctuations as possible, we generate a large number of networks with the same settings and av-erage these networks. This number varies from around 10 to a couple thousands. It depends on the property, because some are more sensitive to fluctuations than others.

2.3.1

Component size distribution

The network consists of independent groups of connected nodes, called components (c). From a chemical point of view, these components are molecules. Single nodes are components with size 1. The largest compo-nent is also called the giant compocompo-nent (cmax_{). To find the components} of a network, a breadth-first search is executed.21_{This is a standard} al-gorithm that goes over all undiscovered nodes and finds their neighbours. With this information, the distribution of the nodes over the components can be found.

2.3.2

Gel fraction

Before the gel fraction can be discussed, first two very important concepts (sol and gel) have to be explained. During the drying process of oil paint, or the network formation, monomers start to connect and they form small molecules. While this happens, the system is still in the liquid phase, or the so-called sol regime. At some point, the giant molecule reaches a critical size and it becomes likely that it will consume all the other molecules until all monomers are connected to each other. This point is called the gel point and the system after this point is in the solid phase, or gel regime. The gel fraction is the fraction of nodes involved in the gel.

2.3.3

Gel point

The gel point marks the transition from sol to gel. Experimentally, the gel point is visible as a jump in viscosity.22 _{To find the gel point in} this model, two networks are generated for two values of N and the size of the giant component and of the second largest component are calculated. When N is increased before gel point, the ratio between the giant component and the second largest component stays the same or goes down. This is because both these components are not part of the gel and grow independently from each other with a increasing number of nodes. However, once the giant component starts to grow faster than the second largest component when N is increased, a gel is formed and the gel point is reached. Therefore, by looking at the ratio between the size of the giant component and the second largest component, the gel point can be identified.

2.3.4

Degree distribution

The degree distribution shows the degree of the nodes, which corresponds to their number of neighbours. This value can be calculated by summat-ing the ones in each column of matrix A as is shown in equation 3.

2.3.5

Cyclomatic number

The cyclomatic number G corresponds to the number of independent loops in the network. It is calculated by looking at the number of edges (m), number of nodes (N ), and the number of disconnected components (molecules) M.

G = m − N + M (7)

Another way to interpret this number is as the maximum number of edges that can be removed before the network does not contain any cycles, or in other words, before it becomes a tree network. If one edge in a tree network is removed, the network falls apart and is not connected anymore. Therefore, the cyclomatic number can be used as a measure for the resilience or strength of the network.

2.3.6

Average shortest path

The minimum number of edges between all the combinations of nodes is called the average shortest path and it is calculated using the Dijkstra algorithm,23 _{which finds the shortest path between an initial node to a} certain node Y . The average value of the shortest paths between all the nodes is a measure of the connectivity of the network. If the network is heavily connected, by all sorts of cycles, the average shortest path increases, since it is not possible anymore to get from one side of the network to the other in a few steps. The average shortest path can also be seen as a measure for the radius of a network.

2.3.7

Clustering coefficient

The clustering coefficient is the number of triangles connected to one node divided by the maximum number of possible triangles in the network. An example is shown in figure 5.

Figure 5: An example of a node (node 1) with a clustering coefficient of 1/6.

2.3.8

Number of triangles

A property of an adjacency matrix is that the matrix to the power of p returns all nodes that are reachable in p steps. This is used to describe the triangles in the network (Ti), shown here, and also later on, in section 3.1.

N X

i=1

Ti= A3/6 (8)

2.3.9

Linear size distribution

This distribution shows the sizes of the linear parts of the components in the network. A linear strand is defined as a group of nodes that have a degree of 2. In other words, it counts the edges between two branching points. To find these linear edges, first all the nodes with degree other than 2 are removed and a new subgraph is created with all the nodes involved in a linear piece. Subsequently, the number of nodes is counted and every time 1 is added to this number. This is because the removal of nodes involves breaking of edges. When in a linear strand of three nodes (see figure 6) the nodes with degree other than 2 are removed, only one node remains. When 1 is added to this number, the original number of edges of the linear strand is regained. The linear size distribution is related to the elasticity of the network. A high number of linear strands means that the network is elastic and vice versa.24 25In addition to the linear size distribution, the number of triangles and the clustering coefficient are also measures for the elasticity.

Figure 6: Linear strand with the numbers corresponding to the degree.

2.3.10

Community distribution

Once the network starts forming a gel, it does not necessarily have to happen in a homogeneous way. It is quite likely that the formation of components starts at different points in the network, like nuclei, and ex-pands from there on. Eventually, these small clusters, or communities, will end up connecting to each other and the gel is formed. These com-munities are still visible within the gel, until the number of edges becomes too high and the network becomes too dense. A network can be cut up into pieces to find these communities, but it is important to know which way to cut is the best way. Therefore, in 2004 Newman introduced a quality function Q, or modularity, that can be assigned to these different ways of cutting.26_{The maximum Q indicates the optimal way to cut the} network into a certain number communities, which could have been the original communities in the sol.

2.3.11

Dispersity

In polymer science the dispersity is the ratio of the mass-average degree of polymerization to the number-average degree of polymerization.27 In this study dispersity can be viewed as a measure of dispersion or spread of sizes of the components. For a uniform polymer, the dispersity is 1, which means that all the chains or components have the same size.27_The value goes up for more disperse distributions of sizes.

2.4

Discussion of the results

In this section results of the standard model are discussed. Figure 7 shows the gel fraction as a function of the conversion. At the gel point, that lies at χ = 0.18, the gel fraction starts to increase up until the moment when all the nodes are involved in the gel. The line in figure 7 increases with a large slope, because at the gel point the formed clusters merge with the giant component. Before the gel point, nodes are already involved in a giant component, but this is not called gel yet and this is why the value of gel fraction is 0. The gel point is found with the method described in section 2.3.3, which means that the gel point is dependent on the chosen measuring points. Conversion, χ 0 0.2 0.4 0.6 0.8 1 G el fr ac ti on 0 0.2 0.4 0.6 0.8 1

Figure 7: The fraction of nodes involved in the gel is shown as a function of the conversion. The gel point is visible at χ = 0.18.

The gel point is also visible in figure 8 that shows the fraction of edges that needs to be broken to obtain a tree topology as a function of the conversion. The value is zero until the gel point and increases after it. The maximum value is 0.68, which means that at full conversion a little more than two thirds of the bonds have to be broken before the network falls apart.

Conversion, χ 0 0.2 0.4 0.6 0.8 1 C y cl om at ic fr ac ti on 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Figure 8: The cyclomatic fraction or the fraction of edges that needs to be broken to obtain a tree topology is shown as a function of the conversion.

In addition, the gel point marks a transition in figure 9 in which the size distribution is shown with a logarithmic scale on both axes. It is important to notice that the sizes are shown of the components in the sol, which means that the giant component is omitted. The maximum shifts towards larger values until the gel point (the black line). This is because the sizes of the components increase, until they start to be consumed by the giant component. When this happens, the maximum decreases again. In the end, all the nodes will be involved in the giant component. Component size 100 101 102 103 104 F re q u en cy 10-4 10-2 100 0 0.2 0.4 0.6 0.8 1 Conversion, χ

Figure 9: The size distribution is shown of the components in the sol with a logarithmic scale on both axes. The colors correspond to the conversion and the black line is at gel point (χ = 0.18).

In figure 10 the influence of the gel point is also visible. The average shortest path length is shown as a function of the conversion. The cut

off is at χ = 0.18. At this point the average shortest path increases very fast, because the components connect to form the gel. The line is cut off, however, because there are too little measuring points. The line indicates that the shortest paths increase until cross-links start to appear within the giant component and as a result the average shortest path decreases again. Conversion, χ 0 0.2 0.4 0.6 0.8 1 Av er age sh or te st p at h le n gt h 0 5 10 15 20 25

Figure 10: The average shortest path length is shown as a function of the conversion. The cut off is at χ = 0.18.

Next, in figure 11 the dispersity, or the dispersion of sizes, is showed as a function of conversion. The dispersity goes up once components start to form and has a maximum around the gel point. After gel point, the dispersity goes down dramatically and the value approaches 1. This is because in gel there is only one component, so there is no dispersion of sizes.

Conversion, χ 0 0.2 0.4 0.6 0.8 D is p er si ty 0 5 10 15 20 25 30

Figure 11: The dispersity as a function of the conversion is shown. The peak lies at χ = 0.18. Finally, figure 12 shows the degree distribution of the standard model

for an increasing conversion. At low conversion most nodes only connect to a few other nodes, so the maximum lies at low degrees. It shifts to higher degrees with an increase in conversion until it starts to overlap with the maximum degree distribution (the black line) that corresponds to the functionality of the nodes. The maximum of this black line lies at 7. This is an important value, because it tells us that most nodes at full conversion have 7 neighbours. This indicated that are network is very dense, which is also visible in figure 13.

Degree 0 1 2 3 4 5 6 7 8 9 F re q u en cy 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.2 0.4 0.6 0.8 1 Conversion, χ

Figure 12: The degree ( N P i=1

dcurr

i ) distribution of the standard model is shown. The colors corre-spond to the conversion. The black line represents the maximum degree distribution (

N P i=1 dmax i ), or functionality.

Figure 13: A network at full conversion that consists of 1000 nodes obtained from the standard model.

3

Advanced network formation

The standard model is based on a simple principle and it results in a very dense network. Whether this network is too dense and therefore unphysi-cal, will be discussed later on in this section. First, the implementation of three factors, intramolecular bonds, preferential coupling and shielding, are discussed. We modified the standard model and included these three factor so that will make the model approach a realistic chemical system more accurately and we call this modified model the advanced model.

3.1

Preferential coupling

An important factor in the formation of the network is called preferential coupling. In a real chemical network nodes are not randomly connected, but certain couples are preferred. Preferential coupling indicates the influence of the topology of the network. The definition is simple: when two nodes are close to each other, they are more likely to connect. The implementation, however, is not as simple. This is because position is not specified in this model. Therefore, we have to look at the connections that the nodes have already formed. The probability of two nodes that are linked through a common neighbour is increased and this effect decreases when the path between the two nodes becomes longer. To find the nodes that are connected by a path of length p, it is enough to consider the pth _{power of the adjacency matrix. Indeed, an i, j-element of matrix A}p equals to 0 unless there exists a link between nodes i, j of length p. In this operation, however, the edges can be counted more than one time and this is not desired. Here we will define a matrix Ap. An i, j-element of this matrix is equal to 0 if the nodes are connected by a link of length p but are not connected with a shorter link:

Ap= (Ap> 0) · (1 − Ap−1) (9) Here (Ap_{> 0)}

i,jis 1 if i, j-element of Apis greater than 0, and operation ”·” should be understood as the element-wise multiplication.

The distribution ri,j, obtained from the standard model, is modified by multiplying it with a certain factor. In this factor, matrix Ap is mul-tiplied with path length p with a certain exponent. This exponent p−3/2 indicates that probability for connecting two nodes decays algebraically as topological distance p increases.14 _{The parameter α determines the} strength of the influence of the preferential coupling. Equation 11 shows the normalization step.

r_i,j0 = ri,j· (1 + α pmax X p=1 App−3/2) (10) Pi,j= r0_i,j N P i,j=1 r0 i,j (11)

Also, more than one bond between the nodes is allowed. The function-ality of the nodes is updated every time a connection is formed. When

a couple is sampled that already made a connection, the 1 in matrix A stays unchanged, but the functionality of these nodes decreases. There-fore, matrix A is still only filled with zeroes and ones.

3.1.1

Discussion of the results

By changing α the influence of the preferential coupling on some of the properties of the network can be measured. It is important to mention that the value of α has to be multiplied with the number of nodes N . Therefore α = α · N . A very important result of the preferential coupling is the change in density of the network. As mentioned before, a network obtained from the standard model has a very high density. As is shown in figure 14, a network with preferential coupling looks more sparse. This is because the network is built up from clusters, caused by the preferential coupling, visible in figure 15.

Figure 14: A network at full conversion for α = 5. The colors correspond to the communities. The network originally consisted of 1000 nodes, but here only the giant component is shown.

In figure 15 for small α the modularity goes up with conversion, but decreases at some point. This is also illustrated by the network in the black circles. It appears that the preferential coupling has a effect on the modularity. In the standard model communities are formed around the gel point, but the nodes connect after the gel point in such a disorderly way that in the end the communities are not recognizable anymore. The networks with preferential coupling do not form communities as quickly, but in the networks at full conversion the communities are clearly visible. The reason for this change is the tendency of nodes to connect with their close neighbours. In short, all networks with different values of α experience some degree of clustering, but for low values of α this is lost

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Figure 15: The modularity, or Q-value, is shown as a function of conversion. The colors represent different values of α ranging from 0.1 (blue) to 10 (pink). All the values of α have to be multiplied with N . In the circles the giant components corresponding to the lines are shown.

Another indication of the change in density because of preferential coupling is shown in figure 16. It shows the degree distribution of a net-work with α = 5. The maximum at full conversion lies at 3, which means that most nodes in a fully polymerized network have only three neigh-bours. In a network obtained from the standard model this maximum lies at 7, thus resulting in a very dense network.

Degree 0 1 2 3 4 5 6 7 8 9 F re q u en cy 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Conversion, χ

Figure 16: The average degree ( N P i=1

dcurr

i ) distribution of a network with α = 5 is shown. The colors correspond to the conversion.

The definite proof that a network obtained from the standard model is too dense, and therefore unphysical, is shown in figure 17. The optimal shape to pack nodes in is a sphere, which means that the maximum

density is reached. When you double the volume of a sphere, the radius does not double, but scales with a factor of √3

2: (2V ) = 4 3πr 3 ₍₁₂₎ r = 3 s 3 4π(2V ) (13)

The number of nodes for networks with different values of α, or the volume, is doubled. The radius cannot be calculated, because the model does not have any spatial dimensions. However, we can look at the average shortest path. This is a measure of the size of the network. When the average shortest path of the smaller network (a1) is divided by the average shortest path of the bigger network (a2) we obtain a scaling factor that we can compare to the one of a sphere. For low values of α the scaling factor is below the lower bound of √3

2, which means that these networks are too dense. Also, for α = 0 (the standard model) the network is too dense and this confirms to our hypothesis. From figure 17 we can extract a range of values of α that corresponds to realistic networks. This range is from α = 5 to α = 8.

Figure 17: The scaling factors as a function of α are shown. One of the main properties, the gel point, is affected by a change in α. This is shown in figure 18. Two effects are visible. The first and the most important is the postponement of the gel point by preferential coupling. The more α increases, the more the gel point is shifted towards higher conversion. Furthermore, fluctuations are visible in the data that are caused by a higher value of α. This is problematic, because these fluctu-ations make the gel point unclear. The overall trend of a postponed gel point, however, is visible and this explains several other results obtained with preferential coupling.

Conversion, χ 0 0.2 0.4 0.6 0.8 1 G el fr ac ti on 0 0.2 0.4 0.6 0.8 1

Figure 18: The gel fraction is shown as a function of conversion. The color represent different values of α ranging from 0.1 (blue) to 10 (pink). All the values of α have to be multiplied with N .

Figure 19 shows the cyclomatic fraction for different conversions as a function of α. For high conversion, the cyclomatic fraction goes down with an increasing α. This decrease is again because of the shift in gel point caused by the preferential coupling. As can be seen in figure 8, the cyclomatic fraction only starts to increase after the gel point, so if the gel point is postponed because of an increasing α, the cyclomatic fraction decreases. However, at low conversion the shape of the curve changes and the cyclomatic fraction goes up with an increasing α. This is because at higher alphas loops are more preferred and the gel point does not play a role yet, because the network is in sol regime. In short, two phenom-ena are visible: with preferential coupling loops are preferred and in sol regime this causes an increase in cyclomatic fraction with increasing α and preferential coupling postpones the gel point which causes a decrease in the cyclomatic fraction with increasing α. These two phenomena both happen at the same time and this is why the lines gradually change shape.

α 0 2 4 6 8 10 C y cl om at ic fr ac ti on 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.2 0.4 0.6 0.8 1 Conversion, χ

Figure 19: The cyclomatic fraction is shown as a function of α. All the values of α have to be multiplied with N . The different colors represent different conversions that range from just before χ = 0.18 to full conversion.

Next, in 20 that shows the clustering coefficient as a function of α all the lines increase, since triangles are preferred more when α is increasing. In the standard model, when α is zero, some triangles are formed after gel point, although by coincidence, and this is why the clustering coefficients starts at a low value.

α 0 2 4 6 8 10 C lu st er in g co effi ci en t 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Conversion, χ

Figure 20: The clustering coefficient is shown as a function of α. All the values of α have to be multiplied with N . The different colors represent different conversions that range from just before χ = 0.18 to full conversion.

Figure 21 shows the average path length in the network as a function of conversion and for different values of α. For low α it is very similar to figure 10, although the cut-off is not visible here, because the data is

not modified manually. However, when α increases the maximum shifts towards higher conversion. This is because the nodes first connect to their neighbours with preferential coupling and start to make long range connections later. The peak of the line has a correlation to the gel point, although it can not be said that is corresponds completely. But the same trend is visible in this figure as in figure 18.

Conversion, χ 0 0.2 0.4 0.6 0.8 1 Av er age p at h le n gt h 0 5 10 15 20 25 30

Figure 21: The average path length is shown as a function of conversion. The colors represent different values of α ranging from 0.1 (blue) to 10 (pink). All the values of α have to be multiplied with N .

The average linear sizes in figure 22 increase with α. This is slightly counter intuitive, since preferential coupling prefers loops and not linear sizes. However, preferential coupling does postpone the gel point. This means that the network in the end will become less connected than with-out preferential coupling and therefore will have larger linear pieces. It is also important to notice that the difference on the y-axis between the starting and end point is only 0.6.

α 0 2 4 6 8 10 Av er age li n ear si ze s 2 2.1 2.2 2.3 2.4 2.5 2.6

3.2

Intramolecular bonds

Another factor that influences the distribution ri,jis the formation of in-tramolecular bonds. Inin-tramolecular bonds are cross-links formed between fatty acids within the same TAG-unit. In matrix A this is represented by a one on the diagonal. The chance of a node forming a connection with itself is a certain constant β times a binomial coefficient that indicates all the possible intramolecular bonds. The binomial coefficient n_k
tells you how many groups of size k can be formed from n things. To make sure that the intramolecular bonds are only made between the different fatty acids in a TAG-unit, and not within the same fatty acid, the binomial coefficient is not based on the functionality of the node. The trick is to look at the individual fatty acids in the TAG-units (df a_i ) and take for the n of the binomial coefficient the functionality of each fatty acid in one TAG-unit and for k 2. This is added on the diagonal of matrix ri,j. To get probability distribution Pi,j, ri,j has be normalized.

ri,j=      (df ree_j ) · (df ree_i ), i 6= j βd f a i 2  , i = j (14)

An important step in the intramolecular bonds is the update step. Once an intramolecular bonds has formed, the functionality of the fatty acids within the TAG-unit has to be updated. This is executed as follows. An array is made that contains the functionality of the three fatty acids per node. If a one is placed in the matrix at the diagonal at node i, the functionality of the fatty acids of this node will be updated. Two of the fatty acids from node i are sampled and the functionality of these fatty acids will be lowered with 1. Because normal connections also change the functionality of the fatty acids, this updating procedure is also applied on normal bond formation between two nodes. In that case one of the fatty acids of node i is sampled and one of the fatty acids of node j and 1 is subtracted from their functionality.

3.2.1

Discussion of the results

The influence of β on the gel point, shown in figure 23 is clear: it shifts the gel point towards higher conversion as expected. Thus, the formation of intramolecular bonds postpones the formation of a gel.

Conversion, χ 0 0.2 0.4 0.6 0.8 1 G el fr ac ti on 0 0.2 0.4 0.6 0.8 1

Figure 23: The gelfraction as a function of conversion is shown. The colors represent different values of β ranging from 0.001 (red) to 32 (yellow). All the values of β have to be multiplied with N .

This postponed gel point is also visible in figure 24 that shows the cyclomatic fraction of a network for different values of β. For low conver-sion, the cyclomatic fraction is zero, because the network is in sol regime and β does not change this. For higher conversion, the cyclomatic fraction goes up, because more bonds are formed. Just as expected, the cyclo-matic fraction decreases for higher values of β, because less connections with other nodes can be formed, since more connections are consumed within the TAG-units themselves.

β 0 5 10 15 20 25 30 C y cl om at ic fr ac ti on 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.2 0.4 0.6 0.8 1 Conversion, χ

Figure 24: The cyclomatic fraction is shown as a function of β. The different colors indicate the conversion.

3.3

Shielding

Shielding or steric hindrance implies that if the surroundings of a node are too crowded, the probability of making a connection to another node is very low. At first, we developed a simple method to describe the shield-ing. Nodes that are involved in one or more triangles are defined as being shielded and experience a negative effect on the probability of making a new connection. To calculate this effect, first the local triangles (Ti), or the triangles each node is involved in, are calculated by multiplying the clustering coefficient by the number of possible triangles around every node, which can be calculated by a binomial coefficient. In this case, n represents the number of edges connected to this node, or dcurr, and k is 2. The local clustering coefficient (Ci) is the number of triangles around a certain node divided by the possible triangles around this node (see section 2.3.7). Now, the effect (S) can be calculated and it should be negative. This is why Ti is in the denominator multiplied by a cer-tain constant δ that determines the strength of the shielding effect. The shielding effect Si is multiplied with ri,j.

Ti= Ci· dcurr i 2  (15) Si= 1 (Ti+ 1)δ (16) ri,j= Si· ri,j, i 6= j (17) However, this description is too simple, because only triangles are taken into account. A more elegant way to describe the shielding effect is with the use of an eigenvector. This method is based on the page ranking method that Google R _uses.28 _{The idea is that the shielding of} node i depends on the number of neighbours j and on their shielding. But their shielding again depends on the number of neighbours and their shielding, which includes the shielding of node i. This is comparable to the eigenvalue problem of the ranking of page i, which is based on the ranking of all the pages that link to page i. Also, this effect is tuned by parameter δ. In this report, no results of the influence of δ will be discussed, because of time reasons.

4

Validation of the model

In order to use the results obtained from the model, some kind of vali-dation is necessary. Within the PAinT group an experiment is designed that will form the link between the model and reality. The experiment is based on the hydrolysis of the TAG-units. As mentioned before, the problem with obtaining structural information about the molecular topol-ogy of oil paint, is the high connectivity and complexity of the network. Only before the gel point, the structure of the loose components can be analyzed. Therefore, the solution would be to do an experiment that postpones the gel point. This is exactly what happens when the network is hydrolyzed. The ester bonds between the glycerol molecule and the fatty acids in the TAG-units are broken. However, the cross-links are retained and therefore, we obtain fragments that still have the original cross-links. The fragments can be analyzed with GC-MS and a size dis-tribution can be measured. When this is compared to the simulated size distribution, the original network can be found through the model. In this section the algorithm that we developed to simulate this experiment is discussed.

1. Creating distribution array: for nodes i to N three degrees are sampled from the fatty acid distribution and these values are stored in an array, while it is made sure that the three values, a, b, and c, together do not exceed the maximum degree, or functionality, of every node. This is done because the nodes will be split up into three pieces, a, b and c.

2. Sampling: for every node i its neighbours are found. For neighbour j piece a, b or c is sampled. If a is sampled, it means that neighbour j is connected to piece a of node i and nothing will change in matrix A. This is because after hydrolysis piece a will be seen as a new node.

3. Edge removal and creation of new node: When piece b or c is sampled, it means that neighbour j is connected to piece b or c. Now this edge between node i and j is removed and a new node is created. In other words, node i is divided into three parts and the degrees of these three parts together will be equal to the degree of the original node. It is important to take into account that matrix A has to stay symmetrical. Therefore, the actions described in equation 18 have to be executed as well for A(i, j) as for laA(j, i), but when i = j only one edge has to be removed.

if piece b is sampled A(i, j) = A(i, j) − 1, A(j, (n + 1)) = 1 if piece c is sampled A(i, j) = A(i, j) − 1, A(j, (n + 2)) = 1 (18) 4. Iterate: this procedure is executed for nodes i, ..., N .

In case of intramolecular bonds, the hydrolysis algorithm has to be modified, because the TAG-unit will not be split up completely, when two fatty acids are connected via an intramolecular bond. This problem is solved by making a distribution of the probabilities of the possible

intramolecular bonds between the three fatty acids: p1= dcurrf a1 · d curr f a2 p2= dcurrf a2 · d curr f a3 p3= dcurrf a3 · dcurrf a1 (19)

In step 2 not only a value from the first distribution array is sampled, but also a value from the second distribution array (equation 19). Now for every neighbour j of every node i it is known to which part of node i it is connected and it is known which two fatty acids are connected inside node i. If neighbour j connects to piece a or if it connects to piece b and this piece is connected via intramolecular bonding to piece a or if it connects to piece c and this piece is connected to piece a nothing will happen. For the other possibilities, a new node is created. This modified algorithm will only hold as long as only one intramolecular bond is formed, which is fixed in the network formation algorithm. To improve the model, more than one intramolecular bond has to be allowed. However, the model described in this report does not allow more than one intramolecular bond. if      a is sampled b is sampled, b connects to a c is sampled, c connects to a continue

if else A(i, j) = A(i, j) − 1, A(j, (n + 1)) = 1 (20) A schematic representation of the algorithm is shown in figure 25. Also, in appendix 4 a flow chart of the algorithm is shown.

The influence of β on the response on hydrolysis is researched. As is visible in figure 26, the sizes of the components in a non-hydrolized network (on the left) with β = 0.001 increase until the gel point (indicated by the black line in the conversion bar) and after that, the components are consumed by the giant component and the sizes decrease again. This is similar to the size distribution of a network obtained from the standard model. When the network is hydrolyzed, the sizes still experience an increase and a decrease after the gel point, but the gel point is shifted towards higher conversion.

Figure 26: Size distribution of a non-hydrolyzed network (left) with β = 0.001 and a hydrolyzed network (right) with β = 0.001.

For a network with a higher value of β (β = 4) something different happens. On the left a non-hydrolized network is shown and just like before, the sizes of the components increase until the gel point and de-crease after that. The gel point is again shifted, although this time the cause is the higher value of β that also results in a postponed gel point. Now, when the network is hydrolyzed, the sizes do not decrease anymore. This means that no gel is formed. Apparently, the intramolecular bonds modify the network in such a way that it becomes less connected, and therefore easier to break up by hydrolyzing.

Figure 27: Size distribution of a non-hydrolyzed network (left) with β = 4 and a hydrolyzed network (right) with β = 4.

5

Conclusion

In this report several examples are shown of how the parameters α and β influence the properties of the network. These properties are straight-forward, such as the density and gel point, and also more complex, such as the elasticity, strength and modularity, or inhomogeneity. For the val-idation of the model, the response to hydrolysis is the most important property that is influenced by the parameters. The results of the ex-periment, which is currently in progress, have to be compared to results from the model. By changing α, β and δ the response to hydrolysis can be tuned in such a way that the simulated results approach the exper-imental results closely. In the ideal case, the optimal parameter set is found, but probably it will be more likely that an optimal range is found. This is because the parameters individually have an effect on the size distribution, as is shown for β in section 3.3, but combined this effect changes. Once the optimal parameter set or range has been found, the original network can be found through the model. Now, all the desired structural information about the molecular topology of the network can be extracted from the model. A schematic representation of this concept is shown in figure 28.

The model that we developed and that is discussed in this report is a new model that incorporates novel advanced routines. In the advanced model, the system is completely represented by the topology and it is also the topology that defines the reactivity of the monomers, in stead of only the functionality. Because the system is represented by the topology, there is no space involved. But it is like a trade-off; because there are no spatial dimensions, it makes the model time and space scalable. Space scalable means that it is applicable to other systems. The size of the monomers of other systems is not of importance, because they can still be represented as nodes. Time scalable means that paint alterations can be studies with our model either in microseconds or in centuries.

Our model provides an important step towards understanding the physical properties of oil paint and these insights will help to explain the interaction of pigments with the network and the degradation processes in oil paintings; highly valuable knowledge for conservations scientists. Furthermore, not only oil paint, but also industrial polymer networks can be studied using our model.

6

Prospects

In this report the development of a novel model and its first results are discussed. Also, a start is made for the validation of the model. In the PAinT group, progress is being made in the design of the hydrolysis experiment. It turns out that the full hydrolysis of the network is quite difficult and also the solving of this hydrolyzed network. There are exper-iments running with loose fatty acids that are cured and analyzed. These hydrolysis experiments are promising and important for the validation of the model. It will be important to design the optimal experiment to find the optimal range of parameters.

Also, more routines can be added to the advanced model, to simulate a real chemical system even more accurately. The problem is, however, that more unknown parameters need to be added. Therefore, it is important consider the necessity of these routines.

Finally, it will be interesting to investigate certain processes on a smaller scale with for example DFT calculations. Density Functional Theory (DFT) is a very accurate method to simulate chemical reactions on an atomic scale.29 _{To see whether for example pigments affect the} polymerization of the oil paint, DFT can be employed. The majority of pigments are metal salts and their reactivity is an intensely studied subject within art research and the PAinT group.30 31 _{Results of these} calculations could predict the value of parameters that can be added to our model.

7

Acknowledgments

I would like to thank my daily supervisor dr. Ivan Kryven (HIMS, UvA) for his help and support. I would also like to thank Prof. dr. Piet Iedema (HIMS, UvA) and dr. C´elia Fonseca Guerra (Division of Theo-retical Chemistry, VU) for their supervision. Furthermore, I would like to thank the Computational Chemistry Group at the UvA and the Paint Alterations in Time Group, a collaboration between the HIMS Institute of the UvA and the Rijksmuseum. Lastly, I would like to acknowledge the MIT Strategic Engineering Research Group for their on line database of Matlab R _{tools for network analysis.}

Appendix 1: Terminology

Community

Group of nodes that is strongly connected to each

other and weakly connected to the rest of the gel

Component

Independent group of connected nodes

Connectivity

Whether and how nodes are connected to one another

in the network

Conversion

Degree of polymerization

Edge

Connection between two nodes

Degree

Current number of neighbours of a node

Functionality

Maximum number of neighbours of a node, also called

maximum degree

Gel

Highly interconnected polymer network that is in solid

phase

Gel point

Transition point from sol to gel, also called percolation

threshold

Giant component

The largest component in the network

Graph / Network

Collection of nodes that is connected by edges

Graph theory

The use of mathematical structures to describe

con-nections in a network

Hydrolysis

Chemical reaction that splits the TAG-units into three

pieces.

Node / Vertex

Point in a network

Path

Number of edges that connect two nodes

Resilience

Ability to withstand removal of edges

Sol

A loosely connected polymer system that is in liquid

phase

Appendix 2: Fatty acid composition

Fatty acid

Chemical composition

% in linseed oil

Palmitic acid

C16:0

6,58%

Stearic acid

C18:0

4,43%

Oleic acid

C18:1

18,51%

Lineoleic acid

C18:2

17,25%

Linolenic aicd

C18:3

53,21%

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