From the ateliers of great masters to the labs of the Rijksmuseum

Imagine that you are a young artist living in the 19th century. You study a lot and buy cheap paints and canvases from the local art supply shop. You rent a room in the attic and you like to experiment with paints. Suddenly an important collector buys your painting and you become famous. Your work is exhibited in the best salons of the city and people are even willing to buy your old paintings and sketches.

All these young artists created our cultural heritage, thus we want to preserve their work for the future generations. A lot of scientists nowadays, including myself, develop methods that can be used in the restoration and preservation of paintings. The artist from the paragraph above is not an easy case for the restorers and conservators. Bad quality paints and poor storage conditions make the degradation of paint unpredictable and the restoration becomes more complex.

When we want to start with the restoration of a painting the material we are going to use for the restoration will react with the paint in the painting. Before starting we would like to know what the impact will be of some treatment with solvents on the condition of the painting. Environmental conditions like humidity and temperature also contribute to paint degradation. Understanding these chemical processes helps understand how the materials in oil paint react with each other and how fast degradation occurs.

An example where knowledge of such chemical processes is crucial is described in an article of Joen Hermans for The Analytic Scientist:

“The formation of metal soaps in oil paint is a good example. If you mix a lead- or zinc-containing pigment with oil and let it age for a few months under humid conditions, metal soaps (complexes of metal ions and saturated fatty acids) will spontaneously form.”

And from his personal blog we read:

“The presence of metal soap phases has been linked to many types of paint deterioration, such as brittleness, delamination and transparency of paint layers, as well as the formation of disfiguring protrusions on paint surfaces.”

Figure 1: Johannes Vermeer, Gezicht op Delft, 1660-1661, oil on canvas, Mauritshuis, The Hague. Metal soaps are visible on the right image. Credits: Mauritshuis/A. van Loon.

My goal in my research is to understand these chemical processes. This helps design restoration methods that have a relatively low risk of damaging the painting in the future.

From paintings to networks

At this point you may ask the question: “Wait, what is this article doing on the Network Pages? Does it have to do anything with networks?

The answer is: oil paint is a network! Oil paint mainly consists of pigment and binding medium. Pigment particles are responsible for the color of a paint. Oil is a binder that keeps pigment particles together. In liquid paint oil molecules are disconnected. When paint is exposed to the air, oil molecules start reacting with oxygen and form radicals: highly reactive atoms with unpaired electrons. Molecules with radicals rapidly form connections between each other. As time passes, all the oil molecules become connected and this state of material corresponds to dry paint. Such a transition in the state of the material, from liquid oil to dry paint is called a phase transition.

To speak the language of chemistry, this process can also be viewed as a formation of a polymer network: oil molecules (monomers) make connections with each other (cross-links) and form a giant polymer (see Figure 2).

Figure 2: Oil molecules (monomers) make connections with each other (cross-links) and form a giant polymer.

 

In what follows I want to explain, without many technical details, how some mathematical analysis can help understand the chemical processes responsible for the drying of oil paint. In the last part of the article we explain in more depth how these methods work for the readers who want to have a more detailed look into this stream of research.

When trying to understand the chemical process behind oil paint drying there are two major challenges, complexity and size. In the case of linseed oil, monomers (oil molecules) undergo a lot of chemical transformations before they reach their final stable state. This shows that the chemical process is rather complex, to tackle this issue we use an algorithm called Automated Reaction Mechanism (ARM) in order to “program” chemistry. When we want to run this algorithm on the computer we are confronted with the second challenge, namely the size of the chemical compounds. Chemical compounds like polymers are in general huge. In order to deal with this complication we developed a methodology called the Monomer Approach. The automated reaction mechanism combined with the monomer approach allows us to keep track of the various chemical reactions taking place until the linseed oil monomers reach their final stable state, a state corresponding to dry paint.

The algorithm that deals with complexity – Automated reaction mechanism

In the field of polymer reaction engineering scientists deal with complexity by keeping track of the concentrations of various configurations of monomers. However, in classical polymers, monomers are identical units that form cross-links with each other. Monomers only differ by the number of adjacent cross-links which is not the case with oils. In the case of linseed oil, monomers undergo a lot of chemical transformations before the whole compound reaches its final stable state. Moreover, one monomer contains up to three different functional groups (reactive sites) and it can form up to three cross-links of three different types (see also this article for a more detailed description). This yields a large amount of possible configurations of monomers which makes it practically infeasible to reconstruct the exact reactions that took place manually. For this reason we developed an algorithm, the automated reaction mechanism, and, as I like to call it, we “programed” chemistry. Using the ARM we are able to keep track of how monomers transform during the various reaction steps until the final product is obtained.

At this point the second big challenge emerges, polymers are composed of a huge amount of monomers, which is always an issue when you want to simulate a system. Simply because these simulations are run on a computer. Such simulations become very slow when the systems we are simulating get very large.

The method that deals with the size – Monomer approach

The monomer approach combines the automated reaction mechanism with some ideas from polymer engineering. In the monomer approach we are interested only in information concerning the monomer’s adjacent cross-links (connections) and functional groups (reactivate sites). In network theory this representation of the network is called the ‘annealed network’ (Newman, see Figure 3 below). Such an approach is often used in epidemiology to capture the structure of a social network to model epidemic outbreaks. Using the monomer approach we essentially look simultaneously at all possible polymer networks that can be formed by the monomers. This type of network is similar to the configuration model, a random graph model which has received a lot of attention during the last thirty years.

Figure 3: An example of the annealed network representation.

Joining Forces

The automated reaction mechanism combined with the monomer approach yields the reaction network corresponding to the drying process of the linseed oil. The monomers represented as molecular graphs and the possible reactions are encoded as transformations on the patterns (reactive sites) of these graphs. When all possible molecular graphs of the monomers are uncovered, the reaction network is built and further transformed into the kinetic model describing the time evolution of the concentrations of all monomers. The solution of the kinetic model is the main result of our methodology. From the solution of the kinetic model we derive two types of results: results coming solely from the annealed representation and global properties of the polymer network obtained by applying tools from random graph theory

From the annealed representation one can extract the concentrations of monomers having a specific functional group to get a better understanding of the chemical reaction. As for the global properties of the polymer network, the solution also contains information about the number and type of the adjacent cross-links per monomer in the annealed representation of the network. These concentrations can further be used as an input for the configuration model to predict global properties of the resulting polymer network (Kryven 2016), such as gel point (the moment corresponding to the phase transition of the material), component size distribution (molecular weight distribution for polymers) and the average size of connected components that are not part of the giant component.

From networks back to paintings

This comprehensive modeling methodology is a bottom-up approach that allows us to understand the dynamics of chemistry in the binding medium of oil paint. Having detailed information about the chemical composition of oil paint at different points of drying time may help to interpret experimental data as well as answer questions as:

– What is the age of the painting?

– What is the composition of the mixture of oils used by one or another artist?

– How various pigments influence the degradation process?

All in all, paintings are not stable objects. While you are reading this article, the degradation reactions are happening in the museums all around the world. And if we don’t want art to vanish completely, we should take care of it and make sure that we understand the way it’s ageing.

Yuliia Orlova is a PhD student at the University Of Amsterdam In the Computational Chemistry Group. Her research focuses on combining knowledge from the fields of mathematics and chemistry to predict drying and degradation of oil paint. Yuliia is highly interested in science behind the preservation of cultural heritage, often visits museums and does a bit of painting in her spare time.

The featured image is by Ankhesenamun 96 on Unsplash.

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