David Simoncini

Sébastien Verel

Philippe Collard

Manuel Clergue

Centric selection: a way to tune the exploration/exploitation trade-off

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Introduction

The exploration/exploitation trade-off is an important issue in evolutionary computation. By tuning the selective pressure on the population, one can find an optimal (or near-optimal) tradeoff between exploitation and exploration. In cellular Evolutionary Algorithms (cEAs), the population is embedded on a bidimensional toroidal grid and each solution interacts with its neighbors thanks to a certain neighborhood. The convergence rate of the algorithm is then dependent of the shape and size of the grid and of the neighborhood. The smallest symetric neighborhood that can be defined is the well-known Von Neumann neighborhood of radius 1. It guarantees a slow isotropic diffusion of genetic information through the grid. But when solving complex multimodal problems, it is necessary to slow down even more the propagation speed of the best solution because the algorithm still often converges over a local optimum.

Our goal in this paper is to establish a relation between the selective pressure on the population and the effects of recombination and mutation operators, in order to find an optimal exploration/exploitation trade-off. To do so, we propose a new selection scheme able to control the selective pressure and a theoretical model which takes into account the effects of stochastic variations on an optimization problem. In section 2 we define a selection scheme able to tune the selective pressure and present the algorithm used in the experiments. In section 3, we analyze the selective pressure with respect to the selection method and present a new model which takes into account the stochastic variations. In section 4 we present performances on Quadratic assignment problem instances and on NK-Landscapes and we explain the results with the model proposed.

Cellular Evolutionary Algorithms

A cellular Evolutionary Algorithm (cEA) [START_REF] Whitley | Cellular genetic algorithms[END_REF] is an EA in which the population is embedded on a bidimensionnal toroidal grid (see figure 1). Each cell of the grid contains a solution. Embedding the solutions on a grid allows defining a neighborhood between the cells. The most commonly used one in cEAs is the Von Neumann neighborhood (shown on figure 1). At each generation, every cell on the grid is updated by selecting parents in its neighborhood and applying stochastic operators such as crossover and mutation. Several strategies exist, synchronous and asynchronous, to update the cells. The small overlapped neighborhoods guarantee the diffusion of solutions through the grid [START_REF] Spiessens | A massively parallel genetic algorithm: Implementation and first analysis[END_REF]. Such algorithms are especially well suited for complex problems [START_REF] Jong | On decentralizing selection algorithms[END_REF] and are of advantage when dealing with dynamic problems [START_REF] Tomassini | Spatially Structured Evolutionary Algorithms: Artificial Evolution in Space and Time (Natural Computing Series)[END_REF].

Selective pressure

One of the main properties that differs between EAs and cEAs is the rate of convergence (propagation speed of the best solution) : It is exponential for EAs and quadratic for cEAs. Therefore, the selective pressure on the population is weaker for a cEA than for an EA. Controlling the selective pressure is critical since it can avoid premature convergence of the algorithm when solving complex multimodal problems. Several parameters related to the selective pressure can prevent the algorithm from getting stuck in a local optimum. The topology of the grid, the local neighborhood, the properties of the selection operator are such parameters. By correctly tuning these parameters for a given problem, one can find a good exploration/exploitation tradeoff and minimize the risks of premature convergence. Sarma et al. established a link between the radius of the neighborhood and the radius of the grid : changing this ratio directly affects the selective pressure on the population [START_REF] Sarma | An analysis of the effects of neighborhood size and shape on local selection algorithms[END_REF]. Alba et al. analyzed performances of cEAs with a fixed size neighborhood and different grid shapes. They arrived to the conclusion that thin grids are well-suited for complex multi modal problems and large grids are well-suited for simple problems. The main explanation is that thinner grids give lower selective pressure [START_REF] Alba | Cellular evolutionary algorithms: Evaluating the influence of ratio[END_REF]. Takeover times and growth curves analysis are useful to measure the selective pressure on a population, but it is not sufficient to decide of a trade-off between exploration and exploitation: it is necessary to include effects of the stochastic variations due to the operators in the analysis. Janson et al. proposed a hierarchical cEA which allows achieving different levels of exploration /exploitation tradeoff in distinct zones of the population simultaneously [START_REF] Janson | Hierarchical cellular genetic algorithm[END_REF].

A standard technique to study the induced selective pressure without introducing the perturbing effect of variation operators is to let selection be the only active operator, and then monitor the number of best solution copies in the population [START_REF] Goldberg | A comparative analysis of selection schemes used in genetic algorithms[END_REF]. The takeover time is the time needed for one single best solution to colonize the population with selection as the only active operator. Let λ be the size of the population, t the number of generations and N (t) the number of best solution copies at generation t. The population is initialized with one solution of good fitness and λ -1 solutions of null fitness. Since no other evolution mechanism but selection takes place, the good fitness solution spreads over the grid. The takeover time is then the smaller time t such that N (t) = λ. Analysing the growth of N (t) as a function of t also gives an indication on the selective pressure. It shows the convergence rate of the algorithm when selection is the only active operator. When the slope of the growth curve of N (t) is low, the convergence rate is low and the takeover time is high. On the other hand, a high slope of the growth curve leads to a short takeover time. In the first case, the selective pressure on the population is weaker than in the second case.

Growth curves and takeover time models

Characterizing the growth curves and the takeover time is an important issue in the study of the selective pressure [START_REF] Goldberg | A comparative analysis of selection schemes used in genetic algorithms[END_REF]. Many models have been proposed to define the behaviour of structured population evolutionary algorithms. Sarma and De Jong proposed a logistic model in which the coefficient of the growth curve of the best solution is shown to be an inverse exponential of the ratio between radii of the neighborhood and the underlying grid [START_REF] Sarma | An analysis of the effects of neighborhood size and shape on local selection algorithms[END_REF]. This conclusion was guided by an empirical analysis of the effects on the convergence rate and the takeover time of several neighborhood sizes and shapes. Sprave proposed a hypergraph based model of population structures and a method for the estimation of growth curves and takeover times based on the probabilistic diameter of the population [START_REF] Sprave | A unified model of non-panmictic population structures in evolutionary algorithms[END_REF]. Gorges-Schleuter proposed a study about takeover time and growth curves for cellular evolution strategies. She obtained a linear model for ring populations and a quadratic model for a torus population structure [START_REF] Gorges-Schleuter | An analysis of local selection in evolution strategies[END_REF]. Several authors wrote about theoretical or empirical models of growth curves and takeover time. Giacobini et. al. proposed a model for cellular evolutionary algorithms with asynchronous update policies [START_REF] Giacobini | Selection intensity in asynchronous cellular evolutionary algorithms[END_REF]. He summarized his results and proposed models for synchronous updates [START_REF] Giacobini | Selection intensity in cellular evolutionary algorithms for regular lattices[END_REF] that will be evoked later in this paper. Alba proposed a model for distributed evolutionary algorithms consisting in the sum of logistic definition of the component takeover regimes [START_REF] Alba | Growth Curves and Takeover Time in Distributed Evolutionary Algorithms[END_REF]. In his paper, he made an interesting review of existing models and compared two of them (the logistic model and the hypergraph model) with his newly proposed one. For a detailed state of the art of cEAs, see [START_REF] Alba | Cellular Genetic Algorithms[END_REF].

Centric selection

In this section we present a new selection scheme for cEAs that allows tuning accurately the selective pressure.

Algorithm 1 Centric Selection algorithm

CentricSelectionindex: int, β:

double neighbors ←-GetNeighborhood(index) candidate 1 ←-Select(neighbors, β) candidate 2 ←-Select(neighbors, β) Best(candidate 1 , candidate 2 )
The centric selection (CS) idea is to change the probability of selecting the center cell of the neighborhood. This scheme allows slowing down the convergence speed while keeping an isotropic diffusion of good solutions through the grid. The CS is a determinist tournament selection. But unlike the standard deterministic tournament, cells in the neighborhood may have different probabilities of being selected for the competition. The anisotropic selection [START_REF] Simoncini | Anisotropic selection in cellular genetic algorithms[END_REF] is another selection scheme which modifies the probability of selection a cell for a deterministic tournament. With the anisotropic selection, the diffusion of solutions is not isotropic, so we propose the CS which is easier to study. We have p c = β the probability of selecting the center cell and p n = p s = p e = p w = 1 4 (1 -β) the probability of selecting either north, south, east or west cell. When β = 1 5 , all cells have the same probability of being selected for the competition: this particular case of CS is the standard binary tournament selection. When β = 1, only the center cell can be selected for the tournament: in this particular case where the same solution is selected two times, the crossover operator is not applied in the cEA. Only mutations are applied to the solution, and with an elitist replacement strategy, the algorithm behaves as the parallelisation of as many hill climbers as there are solutions in the population. The CS is described in algorithm 1. The candidates compete in a deterministic tournament returning the best one. For each cell on the grid, two parents are selected per generation, as we can see in the algorithm 2. Stochastic variations operators are applied to the parents, generating two children. The replacement strategy is elitist: the best child replaces the current solution on the grid if it has a better fitness. The use of a temporary grid is necessary for a synchronous update of the cells.

Algorithm 2 Description of our cEA cEApopulation: vector, β: double tempGrid: vector while continue()

do for i = 1 to GridSize do parent 1 ←-CentricSelection(i, β) parent 2 ←-CentricSelection(i, β) (child 1 , child 2 ) ←-Crossover(parent 1 ,parent 2 ); Mutate(child 1 ) Mutate(child 2 ) tempGrid[i] ←-Best(population[i], child 1 , child 2 ) end for
Replace(population, tempGrid) end while

Modeling cEAs

In this section, we present two models of the search dynamic in cEAs. In the first one, the Equilibrium Model (EM) we consider that the optimal solution has been found and observe how it colonizes the grid. This model is classical in the studies on the selective pressure and one the exploration/exploitation tradeoff. The informations given by this model are takeover times and best solution growth curves. As the stochastic variations operators are not taken into account, the same dynamic occurs in experimental runs when the recombination and mutation operators are ineffective: when the system has reached an equilibrium.

In the second one, we consider that a better solution can be found with a certain probability and observe the frequency of apparition of this new solution with respect to our algorithm's parameters. It is a model of the transition between two periods of fitness stability. We call this new model the Punctuated Equilibria Model (PEM).

Equilibrium model

In order to measure the selective pressure induced by the CS, we observe what happens when no more solution improvement is possible. In this case, crossover and mutation are no longer useful and the evolution process has reached an equilibrium. Hence, we observe the time needed for a single best solution to conquer the whole grid, and look at the growth curve obtained and the takeover time.

We measure the effects of CS on selective pressure by observing these growth curves and takeover times on a square grid of side 64. Figure 2 shows the takeover time as a function of β. The takeover time is not defined for β = 1. The selective pressure drops when the value of β increases. We can see on figure 3 the growth of the number of copies of the best solution in the population (top) and its growth rate (bottom). There are two stages in the shape of the curve. The growth rate is linear in the first part and quadratic in the second part. When using this selection scheme, the diffusion of the best solution is still isotropic. So the best solution roughly propagates describing an obtuse square as long as no side of the grid is reached. This corresponds to the first part of the growth rate curve. Once the sides are reached by some copies of the best solution, the dynamic changes as we can observe on the second part of the growth rate curves.

Punctuated equilibria model

In this section, we propose a new model which will help in the understanding of the search dynamics of an Evolutionary Algorithm. This model was first designed for a cellular EA but can be easily extended to any kind of evolutionary algorithm. We consider a cEA initialized with random solutions. We make sure that the best solution in the population is unique. Our goal is to simulate an evolutionary run: We simulate recombination and mutation operators with probabilities that the mating is efficient or not (i.e. produces a new best solution). We consider three different types of matings: between two copies of the best solution (mating 11), between one copy of the best solution and one suboptimal solution (mating 01) and between two sub-optimal solutions (mating 00). We introduce probabilities P 11 , P 01 and P 00 that matings of type 11, 01 and 00 produce a new best solution, fitter than the previous best one. Figure 4 is an example of evolutionary run on some optimization problem (minimisation task). We can see that there are some stagnation periods where the best solution don't improve. Then, an amelioration occurs and the population enters another stability period. An evolutionary run is a sum of stagnation periods and punctual improvements. Our punctuated equilibria model computes the probability of improving the best solution in the population according to the variables described above. With this model, the probability of finding a new best solution at a given generation t is :

p(t) = 1 -(1 -P 00 ) n00(t) (1 -P 01 ) n01(t) (1 -P 11 ) n11(t)
where n 00 (t), n 01 (t) and n 11 (t) are the number of matings of each type for the generation t. The average time to find a new best solution is given by :

E = t≥1 tp(t)
The performance of an algorithm can be measured by the time E needed to find a new best solution but also by the probability P of improvement in a preset time T . We have the probability of improving the best solution in T generations :

P = 1 - T t=1 (1 -p(t)) P = 1 -(1 -P 00 ) Σ00(T ) (1 -P 01 ) Σ01(T ) (1 -P 11 ) Σ11(T )
with Σ ij (T ) = T t=1 n ij (t) the sum over T of mating of each type. The parameters P ij are problem dependent and the values of Σ ij are given by the selection scheme used. The selection process is usually controlled by a parameter such as the tournament size or in the case of the CS: β. This parameter should be used to maximize the probability1 P . Intuitively, the ideal selection process maximizes the Σ ij which have the higher P ij . More precisely, assuming that the control parameter of the selection process is β, the parameter β * which maximizes the probability P (T ) verifies:

dP dβ (β * ) = 0 which gives: log(1 -P 00 ) ∂Σ00 ∂β (β * ) + log(1 -P 01 ) ∂Σ01 ∂β (β * ) + log(1 -P 11 ) ∂Σ11 ∂β (β * ) = 0 (1)
If it is possible to have a model of Σ ij (β), it would be possible to calculate the optimal β as a function of P ij .

In this model, the exploration/exploitation tradeoff is given by the number of each possible matting (00, 01 and 11). The model could be used to explain the probability and the time to find a new best solution according to the selective pressure, and also to tune the value of parameters which have an impact on the selective pressure, such as β, to have the highest probability to evolve toward a new best solution. Equation 1gives precisely the best exploration and exploitation tradeoff and allows computing the optimal value of β (in our case) for this trade-off. In the following, we will show the validity of the PEM on some optimization problems.

QAP and NK landscapes

In this section, we study the effect of selective pressure on performances through experiments of a cEA with CS on two well-known classes of problems. The optimal exploration / exploitation tradeoff found will be explained thanks to the PEM presented in the previous section.

Problems presentation

The problems proposed, Quadratic Assignment Problem and NK landscapes, are known to be difficult to optimize. The important number of instances of the Quadratic Assignment Problem and the tunable parameters of the NK landscapes allow managing the difficulty of the problems.

Quadratic Assignment Problem

This section presents the Quadratic Assignment Problem (QAP) which is known to be difficult to optimize. The QAP is an important problem in theory and practice as well. It was introduced by Koopmans and Beckmann in 1957 and is a model for many practical problems [START_REF] Koopmans | Assignment problems and the location of economic activities[END_REF]. The QAP can be described as the problem of assigning a set of facilities to a set of locations with given distances between the locations and given flows between the facilities. The goal is to place the facilities on locations in such a way that the sum of the products between flows and distances is minimal. Given n facilities and n locations, two n × n matrices D = [d kl ] and F = [f ij ] where d kl is the distance between locations k and l and f ij the flow between facilities i and j, the objective function is :

Φ = i j d p(i)p(j) f ij
where p(i) gives the location of facility i in the current permutation p.

Nugent, Vollman and Ruml proposed a set of problem instances of different sizes noted for their difficulty [START_REF] Nugent | An experimental comparison of techniques for the assignment of techniques to locations[END_REF]. The instances they proposed are known to have multiple local optima, so they are difficult for an evolutionary algorithm.

The best algorithm known is the fast hybrid evolutionary algorithm [START_REF] Misevicius | A fast hybrid genetic algorithm for the quadratic assignment problem[END_REF] which combines an evolutionary algorithm with an improvement of the fast tabu search of Taillard.

Set up

We use a population of 400 solutions placed on a square grid (20 × 20). Each solution is reprensented by a permutation of N where N is the size of a solution.

The algorithm uses a crossover that preserves the permutations:

• Select two solutions p 1 and p 2 as genitors.

• Choose a random position i.

• Find j and k so that p 1 (i) = p 2 (j) and p 2 (i) = p 1 (k).

• exchange positions i and j from p 1 and positions i and k from p 2 .

• repeat N/3 times this procedure where N is the size of an solution.

This crossover is an extended version of the UPMX crossover proposed in [START_REF] Migkikh | Combined genetic and local search algorithm for the quadratic assignment[END_REF]. The mutation operator consists in randomly selecting two positions from the solution and exchanging those positions. The crossover rate is 1 and we do a mutation per solution. We perform 200 runs for each tuning of the two selection operators. An elitism replacement procedure guarantees the solutions stay on the grid if they are fitter than their offspring.

NK landscapes

The NK landscapes were proposed by Kaufmann to model the boolean network and used in optimisation in order to explore how epistasis is linked to the ruggedness of search spaces [START_REF] Kauffman | The Origins of Order[END_REF]. Epistasis corresponds to the degree of interactions between the "loci" of a solution and ruggedness is the number of local optima of the search space. The main characteristic of NK Landscapes is that they allow tuning the epistasis level with a single parameter K. The parameter N determines the length of the solutions.

The fitness of solutions for a NK landscape is given by the function

f : {0, 1} N → [0, 1]
defined on binary strings of length N . Each binary string is a solution with N locations. An atom with fixed epistasis level is represented by a fitness component

f i : {0, 1} K+1 → [0, 1]
associated to each bit i. It depends on the value of the bit i and on the value of K other bits of the string (K must fall between 0 and N -1). The fitness f (x) of x ∈ {0, 1} N is the average of the values of the N fitness components f i :

f (x) = 1 N N i=1 f i (x i , x i1 , ..., x ik )
where {i 1 , ..., i k } ⊂ {1, ..., i -1, i + 1, ..., N }. Many ways have been proposed to choose the K other locations from the N of the solutions. The mainly used ones are adjacent and random neighborhoods. With the first one, the K nearest locations of the location i are chosen (the solution is taken to have periodic boundaries). With the random neighborhood, K locations are randomly selected from the solution. Each fitness component f i is specified by extension, ie a random number y i,(xi,xi1,...,x ik ) from [0, 1] is associated with each element (x i , x i1 , ..., x ik ) from {0, 1} K+1 . Those numbers are uniformly distributed in the interval [0, 1].

Set up

The size of the population is 400 solutions. The crossover used is a one point crossover, applied with a probability of 1. The mutation is a bit flip applied with a probability of 1 n where n is the size of a solution. We perform 200 runs for every parameter set, and each run stops after 1500 generations. Runs are performed on instances of sizes N = 32, with K ∈ 2..12.

Performances

Figure 5 and table 1 show performances of a cEA using CS on some QAP instances of various sizes. The instance in figure 5 is a well-known instance of size 30. The first fact that we notice when looking at these results is that there is an optimal setting, different from the extreme values 0 and 1 for the parameter β. This indicates that for a certain setting of the parameters, and thus for a certain selective pressure, the search dynamic leads to optimal results. Curves representing the instances summarized in table 1 have the same shape as figure 5. On each instance, the optimal value of β is around 0.85. The performances increase up to these values and then decrease. Performances of CS are significantly better than the one obtained with a cEA using standard binary tournament selection. The standard cEA is observable on the curve at the points β = 0.2 and is reported in the table 1. We can also notice in table 1 that the standard deviation is lower for the optimal value of β than with a cEA with binary tournament selection. 2 present performances of a cEA with CS on some instances of NK landscapes. Parameters of the landscapes are N = 32 and K = 10 for the figure 6 and are summarized in table 2 for the other instances.

We can see that the shape of the performances' curve is different from the QAP curve. The performance increases until β reaches its maximum value. The same results are obtained for all the instances in table 2. The parameter K tunes the difficulty of the instance. We can see that for K = 2, there is no optimal value for β. The reason is that the optimum is always found. For K = 4, the standard cEA sometimes get stuck in a local optimum, and with β = 1 our algorithm always find the optimum. On every instance, except K = 2, the optimal value for β is 1.

However, this value β = 1 is a particular one, since it breaks all communications on the grid. As long as the value of the parameter increases, the chances of selecting two different solutions for recombination decrease. For β = 1, the algorithm is the parallelisation of as much hill climbers as there are cells on the grid : It constantly selects the center cells of the neighborhoods, so there is no crossover and any amelioration is due to a bit flip.

So the best setting for CS can be compared to the parallelisation of as many hill climbers as there are cells on the grid. The parallelisation of hill climbing seems to be a good algorithm for solving NK landscapes problems, which could be explained by the size of basins of attraction [START_REF] Verel | The connectivity of nk landscapes' basins: a network analysis[END_REF] and [START_REF] Ochoa | A study of nk landscapes' basins and local optima networks[END_REF].

Probabilities of discovering better solutions

In order to explain the optimal values of β for QAP and NK-Landscape and to validate the PEM, we compute P , the probability of discovering a new best solution in the population taken from the PEM, with real data. We calculated it for one instance of QAP and one instance of NK-Landscapes. With this calculation we want to find the value of β that maximizes the probability of discovering a new solution. This probability depends on the value of Σ ij , and thus on time: if at a generation t no new solution is discovered, the actual best solution spreads in the population according to the selective pressure. If during an interval of time corresponding to the takeover time no new solution is discovered then the population converges. We can compute the ideal β value for a given number of generations T because Σ ij (T ) relies on β and on time: after T generations Σ ij (T ) is different according to β, and for the optimal value of β Σ ij (T ) leads to the best probability P . We estimated the Σ ij with the same experiments done to compute growth curves and takeover time. We averaged the number of matings of each type at each generation over 10 3 runs. Then, we needed to know the probabilities P 00 , P 01 and P 11 . We estimated these probabilities using a Bayesian process during the runs. We averaged the values obtained by generations over 500 runs. Figure 7 shows the result of the estimation of probabilities on the QAP instance Nug30. The ordonate scale is logarithmic because of the variations of probabilities. The curves representing the P ij intersect, so the value of β which maximizes P may change during a run. We computed P with estimated values of P ij taken by steps of 50 generations. The values of Σ ij are also generation dependent. For each value of β, we took the Σ ij value after 100 generations: that is Σ ij (100). During a run, it would correspond to allowing a stagnation period of 100 generations before stopping the run, which is reasonnable. The figure 8 shows the optimal values of β as a function of generations for the QAP instance Nug30. During the first 700 generations, the optimal value is β = 0.2. Then, there is a transition of approximatively 150 generations. During this phase, the ideal value of β grows until it reaches 1. In our experiments, we observe optimal values of β between 0.8 and 0.9 according to the QAP instance. Values of β are constant during the runs. But the PEM shows that the selective pressure should be strong at the beginning (low values of β) and then weak (high values of β). If β is constant, intermediate values in the range [0.8, 0.9] give the best average selective pressure for QAP instances.

The figure 10 shows the optimal values of β as a function of generations for a NK-Landscape with N = 32 and K = 10. We can see that the optimal β value increases fastly and reaches 1 in the early generations. The ideal selective pressure is weak, and it is not surprising that the best performances are obtained when β = 1 in our experiments. Figure 9 shows the estimation of P ij as a function of generations. We can see that the curve representing P 11 drops down very fast. With a negligible probability of improving the current best solution with mutations, there is no sense in spreading this solution. With β = 1, the current best solution in the population cannot spread.

The PEM has been used in order to explain the exploration / exploitation trade-off on two different classes of problems. Coupled with the centric selection, it showed the ideal selective pressure along the search process. This model can be used to tune any parameter which has some influence on the number of matings of each type defined in the previous section. The computation cost is low, since the estimation of probabilities by a Bayesian process is precise: we averaged the estimation on 500 runs but the standard deviation was low (≈ 10 -6 ).The Σ ij are only computed once since they are independent from the optimization problem tackled.

Conclusion

The exploration/exploitation trade-off is an important issue in evolutionary algorithms. In this paper, we propose a model that takes into account stochastic variations and improvement of the quality of the solutions, the punctuated equilibria model. In order to study the exploration / exploitation trade-off we propose a tunable selection operator: the centric selection. By monitoring the probability of selecting the center cell of neighborhoods for a tournament selection, this selection operator allows tuning accurately and continuously the selective pressure with one single parameter (β). The performance results on QAP instances and NK-Landscapes showed different optimal settings of the centric selection, and thus different ideal selective pressures. Using the punctuated equilibria model, we put in evidence the optimal values of the centric selection's control parameter observed on QAP instances and NK-Landscapes. The punctuated equilibria model also put in evidence that the ideal selective pressure is not constant during the search process in the case of QAP instances.

In this paper, we used the PEM in order to explain experimental results. In future works, we will use it in order to predict optimal exploration / exploitation trade-offs and to adapt the selective pressure during the runs. To do so, we will both estimate the P ij and tune the selection operator online during the search process. The centric selection will be used in auto-adaptative algorithms with the advantage of modifying the exploration / exploitation ratio with a single parameter. It will also be applied on real problems and compared to other optimization methods.

Figure 1 :

 1 Figure 1: Representation of a cEA and Von Neumann neighborhood in dashed line.

Figure 2 :

 2 Figure 2: Average takeover time as a function of β.

9 Figure 3 :

 93 Figure 3: Growth curves of N (t) (top) and its growth rate (bottom) for different values of β.

Figure 4 :

 4 Figure 4: Example of evolutionary run

Figure 5 :

 5 Figure 5: Performances on nug30 for a cEA using CS.

Figure 6 :

 6 Figure 6: Performances on NK with N = 32 and K = 10 for a cEA using CS.

Figure 7 :

 7 Figure 7: Estimated P ij on QAP instance nug30

Figure 8 :Figure 9 :

 89 Figure 8: Optimal values of β on QAP instance nug30

Figure 10 :

 10 Figure 10: Optimal values of β on NK-Landscape N = 32, K = 10

Table 1 :

 1 Avg. results and std.dev.on QAP instances

	Instance	Std cGA	Best avg. results Opt. β
	Nug30 Tai40a	6178 [28] 3.23 × 10 6 [14343]	6144 [14] 3.21 × 10 6 [12000]	0.88 0.84
	Sko42 Tai50a	15969 [75] 5.092 × 10 6 [20721]	15909 [34] 5.080 × 10 6 [13372]	0.82 0.82
	Tai60a	7429118 [27760]	7385390 [19391]	0.86
	Figure 6 and table

Table 2 :

 2 Avg. performances and std.dev.on NK instances with N = 32

	K	Std cGA	Best avg. results Best β
	2	0.734329 [0]	0.734329 [0]	[0, 1]
	4 0.79597 [0.003]	0.798197 [0]	1
	6 0.782934 [0.01]	0.799124 [0.003]	1
	8 0.771277 [0.01]	0.789103 [0.004]	1
	10 0.763510 [0.01]	0.785115 [0.003]	1
	12 0.750043 [0.01]	0.774479 [0.009]

In the following equations, we only denote the dependance on β for P and Σ ij for readability.


domain: chim sde

text:
Julien Morin

Gregory Brochard

Virgine Bergé

Aurélie Rosset

Sébastien Artous

Simon Clavaguera

Rafal S Strekowski

Henri Wortham
email: henri.wortham@univ-amu.fr

Uptake of m -xylene and VOC emissions by mineral photocatalytic paints of indoor air building interest

Keywords: Indoor air, Uptake coefficient, VOC emissions, Photocatalytic paints, PTR-MS

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INTRODUCTION

On average, humans spend over 90% of their lives indoors [1][2][START_REF] Robinson | USEPA Res[END_REF] . Recent studies [START_REF] Geiss | [END_REF]5 have shown that the breathed indoor air has indoor-to-outdoor environment VOC concentration ratios up to 14.3. This is worrisome since some of the VOCs commonly found within indoor environments are known to cause adverse human health effects. For example, some of the pollutants commonly found within indoor environments include known carcinogens such as benzene, dichloromethane, and formaldehyde 6 . Indoor VOC emission sources include household activities such as cooking and smoking, carpentry, household furnishing, and cleaning products, among others. Given the multitude of indoor emission sources of VOCs, the current focus of building authorities and construction industries has been to develop new technologies that include adsorption, biofiltration, condensation and/or photocatalytic oxidation (PCO) to improve indoor air quality and provide a saner indoor environment for its occupants [7][8][9] . As a result, a new generation of "cleaning" materials has been in development based on the PCO technique. The PCO technique takes advantage of the paints' photocatalytic properties to oxidize and/or reduce air pollutants that adsorb on the semiconductor surfaces. This technique has been hailed as a promising new zeroemission technology that improves indoor air quality 10 by destroying all the VOCs.

To better understand the essence of indoor air pollution, a series of model compounds known as BTEX, namely, benzene, toluene, ethylbenzene and o-m-p xylenes, have been traditionally used to study indoor air quality [11][12][13][START_REF]INRS, Xylènes[END_REF] . For example, Morin et al. (2020) has recently reported on the photocatalytic degradation of gas-phase m-xylene on organic paints as a function of different experimental conditions that included nano-TiO2 content and nature, irradiance, relative humidity (RH), surface temperature, pigment volume concentration (PVC) and aging in the lab [START_REF] Morin | [END_REF] . In their work, Wortham and coworkers [START_REF] Morin | [END_REF] observed that the m-xylene degradation on organic photocatalytic paints was strongly dependent on environmental parameters that included solar irradiance, %RH, and surface temperature in addition to the paint composition, namely, the nano-TiO2 content and nature, PVC and aging. M-xylene has often been used to study photochemical degradation processes within indoor settings because (1) it is observed in indoor air in trace amounts 16 , (2) it has well documented negative human health effects [START_REF]INRS, Xylènes[END_REF] , (3) it is a relatively inert compound due to its aromatic ring structure and (4) it is not used as a binder nor a solvent in paint preparation, therefore, it is not emitted by studied paints 17 , minimizing any unwanted experimental artifacts.

Even though the engineering motivation behind the development of photocatalytic paints has been to solve the problem of indoor air pollution, many studies have reported contradictory or nuanced results at best. That is, it is now well documented that photocatalytic paints do emit VOCs themselves in the presence of UV radiation [START_REF] Geiss | [END_REF][17][18][19][20] . It has been proposed that under UV irradiation the active radical species that include OH and O radicals generated on nano-TiO2 surfaces react with the binder molecules present within the paint matrix to produce and emit VOCs. The VOC emission by photocatalytic paints under given environmental conditions has become a major 'green chemistry' problem because all paint materials do contain organic solvents or organic binders to a certain degree. However, the presence of organic binders in paints is structurally important because it is the binder material itself that keeps the pigments in place after the paint dries. Unfortunately, an important increase in emissions of different VOCs such as formaldehyde, acetaldehyde, and acrolein under UV irradiation conditions has been observed for selected organic photocatalytic paints 17,19,20 . Further, the photocatalytic reactions between the radical species and binder molecules are important because they affect paint durability, have a negative impact on indoor air quality, and limit the paint's photocatalytic properties 21 . The photo-degradation of the binder may reduce the oxidative advantages of the PCO method as a 'green-chemistry' tool to better indoor air.

One of the current engineering solutions to overcome this problem is to reduce the organic binder content to 5% and to replace the remaining organic binder with a mineral one. Such mineral paints have been studied under laboratory conditions by Morin et al. (2019). These investigators observed a 43% decrease in total VOC emissions 17 when mineral paints were employed under given experimental conditions, a very promising result. Considering these promising results, a new generation of mineral photocatalytic paints have been developed to better indoor air quality.

In this laboratory work, the uptake coefficients of m-xylene and VOC surface emission fluxes for mineral paints containing three different photocatalysts, namely, nano-TiO2, PEG 3350 and CNC, within the binder matrix were determined in the presence of UV radiation. The influence of aging and the type of photocatalyst was evaluated using the comparative approach for uptake and emissions. The obtained uptake and emission results were compared with the reference paints uptake and emission results that did not contain any nano-TiO2 photocatalyst material to determine the impact of nano-TiO2 on each photocatalytic paint. The obtained results will help to improve future binder formulation materials to increase the paints' photocatalytic activities while minimizing any corresponding VOC emissions from the photocatalytic paints. The obtained results will help the building industry community to foster the development of paints and coatings that will contribute to a better indoor quality for the occupants.

MATERIALS AND METHODS

Experimental study

The experimental approach used to study the uptake coefficient (γ) of m-xylene on photocatalytic and reference mineral paints is similar to the one used in a previous study on m-xylene photocatalytic degradation on selected organic paints [START_REF] Morin | [END_REF] . The schematic representation of the experimental set-up and the experimental procedure have been published elsewhere [START_REF] Morin | [END_REF]17,19 . Only the experimental details that are particularly relevant to this work are given below.

Briefly, a borosilicate glass double-wall horizontal-flow-tube-reactor with an internal volume of about 131 cm 3 was employed to study the uptake of m-xylene using the High-Sensitivity-Proton-Transfer-Reaction Mass Spectrometer (HS-PTR-MS) (IONICON Analytik, Austria) detection (see supporting information, Fig 1). The temperature within the reactor was maintained at a constant temperature, T(K) = ±1, by circulating deionized water through the outer jacket using a temperature-controlled circulating bath (RC6 LAUDA, Germany). The experiments were carried out at surface temperature values ranging from T(K) = 292 to 298. All experiments were carried out at p = 1 atm total pressure of air (SF2, Atlas Copco, Sweden) that was scrubbed using an activated carbon cartridge. The geometry of the flow-tube reactor was such that it allowed for the scrubbed air carrier gas flow to enter at one end and the HS-PTR-MS detection system to be located downstream at the opposite end.

The VOC impurity content of the scrubbed air carrier gas used in the m-xylene uptake experiments was compared with the VOC content of a certified synthetic air cylinder (Linde gas, France, >99.999% stated purity). The comparison of the VOC impurity content measured in the scrubbed air and in the Linde synthetic air cylinder is shown in Figure 1. The only exception was the measured mixing ratio value of methanol. Here, the maximum methanol mixing ratio measured in the scrubbed air flow was 5.2 ppb, a value that was about twice the methanol mixing ratio value measured in the Linde gas cylinder. All the other measured mixing ratio values were observed to be < 1 ppb for the selected 14 VOCs, including m-xylene (0.04 ppb).

Further, additional emission experiments using the scrubbed air gas flow source and the Linde air cylinder carrier gas flow source were carried out in the presence of photolyzing radiation. No additional emission sources have been observed in the presence of the photolyzing light.

A typical experiment involved the m-xylene reactant that was allowed to enter the horizontal reactor through a movable injector that permitted to vary the contact time between the m-xylene and mineral paint surface to determine the uptake kinetics of the given heterogeneous reactions.

The selected paint was applied only on one side of the Leneta (black plastic-vinyl chloride/acetate copolymer, Leneta company, U.S.A.) plate that was 26 cm long and 2 cm wide based on a standard operating procedure developed by the manufacturer ALLIOS 17 . The plate covered by the selected mineral paint was placed within the horizontal reactor with the painted side oriented up.

The reactor was placed within a stainless-steel box equipped with four UV fluorescent lamps (Philips TL-D 18W Actinic BL, 340-400 nm, λmax = 368 nm, length 60 cm) (see supporting information, Fig 1). The four fluorescent lamps were placed side-by-side on the upper side of the box and could be operated individually. The spectral irradiance of the lamps has been determined previously by Wortham and coworkers (2014) 22 . The integrated UV spectral irradiance was estimated to be 8.8 W m -2 and 18.6 W m -2 for 2 and 4 lamps, respectively, between l ~ 340 nm to 400 nm. The experimental spectral irradiance was 8.8 W m -2 because this value represents an averaged integrated irradiance from l ~ 340 nm to 400 nm of solar light that is known to enter an indoor environment (Bartolomei et al, 2014). The m-xylene uptake coefficient experiments carried out at 18.6 W m -2 were performed to increase the MS detection sensitivity of the m-xylene signal and to better determine the impact of the nano-TiO2 photocatalyst.

Gas-phase m-xylene was generated using the gas-phase generator (PUL 200, Saint Chamas, France) which consisted of a sealed Teflon permeation tube (Perfluoroalkoxy, 4.0 x 6.4 mm, 8 cm long) filled with pure m-xylene (Sigma-Aldrich, ≥ 99.5%) [START_REF] Morin | [END_REF] . The flow out of the generator was 10 sccm (Brooks SLA Series mass flow controller, in the range 0 to 10 sccm, ± 1% stated accuracy) and was allowed to enter the movable injector of the horizontal reactor. A sheath air flow of 190 sccm (Brooks SLA Series mass flow controller, in the range 0 to 500 sccm, ± 1% stated accuracy) was allowed to enter the reactor to dilute the m-xylene flow. The resulting experimental m-xylene mixing ratio was ~40 ppb.

Relative humidity was adjusted by allowing a portion of the sheath airflow to enter a bubbler containing deionized water (resistivity > 18 MΩ cm). The %RH was measured continuously using the "Hygrolog NT2" (Rotronic) hygrometer with "HygroClip SC04" probe located at the exit of the reactor. The experiments were carried out at relative humidity values that ranged from 1% to 42% and the experimental uncertainty in the relative humidity value measurements was ± 1.5%RH.

The m-xylene mixing ratio was measured online using the HS-PTR-MS by following its mother 1 ion pic m/z 107. Similar to our previous work on VOC emissions on selected organic photocatalytic 2 paints, 16 other signature VOC model compounds were measured continuously and online using 3 the HS-PTR-MS analyzer 17 . The list of all the VOCs identified using the HS-PTR-MS system 4 including the corresponding mass-to-charge (m/z) ratios of the corresponding mother ions are 5 given in Table 1 The compounds listed in Table 1 have been chosen as model compounds by the scientific Similar to our previous work on organic paints, the HS-PTR-MS analyzer drift pressure and drifttube temperature were fixed at p(mbar) = (2.23±0.01), T(K) = (333±1), respectively 17 . The drift tube voltage was 500 V. This value corresponded to the electric field (E, V cm -1 ) to ambient air number density (N, cm -3 ) ratio (E/N) value of 124 Townsend (1 Townsend = 10 -17 V cm 2 ).

Preparation of paints

Mineral paints were formulated and produced by a local paint manufacturer ALLIOS (Marseille, France). Three types of nanoparticles used to formulate the selected paints included (1) nano-TiO2 PC500 in anatase form, ( These newly synthesized TiO2 NPs were physically incorporated into the mineral matrix-based paint by the manufacturer ALLIOS (France) using a standard operating procedure used in paint industry. First, a mixture of ground additives (slurry) was prepared with each of the active nanosized TiO2 form listed above. Then, the paint of mineral matrix-based was mixed with the slurry to reach the desired quantity of nano-TiO2 within the paint matrix. The nano-TiO2 percent weight obtained in the slurry, paint, and finally in the dry paint are presented in Table 2. The obtained slurry nano-TiO2 percent composition, slurry in paint and the paint nano-TiO2 percent compositions are presented in Table 2. Mineral binders contained 5% of the organic binder to help the drying process of the paint on the Lenata surface so that only non-volatile content of paint remained after the coating solidified. We underline that no nano-TiO2 particle content was present in the reference paint. The paints were applied only on one side of Leneta supports according to a standard operating procedure developed by the manufacturer ALLIOS 28 . This procedure allowed for production of homogeneous, uniform and reproducible wet films that were 100 µm thick. Then, the paints were stored for 21 days at T(K) = 296 while a humidified (55%RH) synthetic air was allowed to flow over the plates. The resulting paint thickness was 40 µm. To assure proper fit into the reactor, the Leneta support was fixed on a glass plate with wire to both ends prior to experiments.

Reference and the three photocatalytic paints have been aged in an accelerated aging chamber (model QUV accelerated weathering, Q-Lab, U.S.A.) for 500 hours and 1000 hours according to the ISO norm 16474-3 29,30 . Aging time of 500 hours corresponded to a well-aged paint and the aging time of 1000 hours corresponded to a paint at the end of its lifetime.

Before and after the process of artificial paint weathering, the paints' surface morphology was investigated using the scanning electron microscopy (SEM-FEG, LEO 1530, Germany). In addition, the surrounding environment of elements present at the near paint surfaces (first 10 µm)

were investigated using the X-ray photoelectron spectroscopy (XPS, Versaprobe II ULVAC-PHI spectrometer, Japan). The experimental protocol and the characterization results have been published in detail by Boutry et al., (2017) and Rosset et al., (2021).

Kinetic measurements

One on the aims of this study was to evaluate the uptake coefficient (γ) values of m-xylene on photocatalytic and reference mineral paints. The uptake coefficients were calculated using Equation 1 [START_REF] Finlayson-Pitts | Chemistry of the upper and lower atmosphere : theory, experiments, and applications[END_REF] :

𝛾 = !" !"# # $ % Eq. 1
where ω is the mean molecular velocity (cm s -1 ), V/S is the ratio of the volume of the reactor to surface area of the Lenata plate (cm) and k1st the pseudo-first order rate constant of m-xylene loss (s -1 ).

The rate constant (k1st) was calculated using the linear regression of the slope of ln(C0/Cx) plotted as a function of the residence time (t) [START_REF] Morin | [END_REF]22,[START_REF] Bedjanian | [END_REF] . The C0 was the m-xylene initial concentration at the position P0, that is, when the injector was all the way in the horizontal flow reactor and the mxylene was not allowed to encounter the mineral paint surface. The Cx was the m-xylene concentration at different positions of the injector that ranged from P1 to P4 (Table 3). More precisely, the P1 to P4 injector positions refer to different distances of the injector as it was moved back within the horizontal flow-tube reactor. The difference between C0 and Cx corresponds to the amount of m-xylene consumed by different physical and reactive processes that occur on the mineral paint surface that include loss by reaction, dry deposition, adsorption, and reactive loss by reaction with impurities, albeit, considered to be negligible. In a typical experiment, each injector position was held for at least 30 minutes to stabilize the signal and to reduce the impact of signal fluctuation and variability and to minimize the effects of noise factors. After each Px; the movable injector was pushed in to its original P0 position to obtain, again, the C0 value.

Blank experiments were carried out to ensure that no other reactive or physical wall loss processes occurred during the experiments. Further, the Cooney-Kay-Davis method was used to correct the obtained uptake coefficient values from diffusion limitation in the flow tube 33,34 .

VOCs mixing ratio and estimation of surface emission fluxes

Based on the original work of Lindinger and coworkers 35,36 and our previous work on photocatalytic paint emissions [START_REF] Morin | [END_REF]17 , the mixing ratio (ppb) was calculated using the Equation 2.

𝐶 & = 1.65 × 10 '(( × * $%&'# × , $%&'# ( "×-$%&'# ( × .[0] .[2 ) 3 * ]405×.[2 ) 3 * (2 ( 3)]
Eq. 2

where Udrift is the drift tube voltage (V), Tdrift is the drift tube temperature (K), k is the protonproton-transfer reaction constant (molecule cm 3 s -1 ), Pdrift is the pressure within the drift-tube (mbar), i[X], i[H3O + ] and [H3O + (H2O)] are the ion signal counts per second of the target compound (X), hydronium ion (H3O + ) and hydronium ion cluster (H3O + (H2O)), respectively, normalized by the corresponding transmission efficiency.

The m/z specific relative transmission efficiency was experimentally determined over the mass range of 21-181 with a calibration gas standard, consisting of a mixture of 14 aromatic organic compounds (TO-14A Aromatic Mix, Restek Corporation, Bellefonte, USA, 100±10 ppb in Nitrogen).

The factor Xr is compound specific and has been described in detail by de Gouw et al. 25 and reflects both the difference in the rate coefficient for the proton-transfer-reactions H3O + + R and H3O(H2O) + + R and the difference in transmission efficiencies for the two reagent ions within the quadrupole mass filter region of the mass spectrometer. Different values for k and Xr used in this work have been taken from our previous work on VOC emissions from indoor photocatalytic paints 17 .

The surface emission fluxes (molecule cm -2 s -1 ) were determined using the following Equation 3. Eq. 3

Surface Emission Fluxes 𝑋 =

where the internal reactor volume, the residence time within the reactor and the exposed paint surface area are expressed in units of cm 3 , s and cm 2 , respectively, The Cx is the mixing ratio (ppb) measured at the exit of the reactor as a function of time for the target compounds. The 2.46×10 10 value is the ppb to molecule cm -3 s -1 conversion factor calculated using the ideal gas law under normal pressure and standard temperature conditions.

RESULTS AND DISCUSSION

In a typical investigation of the uptake coefficient of m-xylene on a given photocatalytic mineral paint, the Leneta support with a painted surface up was first placed within the horizontal reactor and the injector was pushed through completely (position P0) so that no contact was allowed between the m-xylene flow and the painted surface. The studied paint was then allowed to rest within the horizontal flow reactor under "dark" conditions, that is, with the UV lamps off, for 2 hours under given experimental conditions of temperature and flow dynamics. At this time, since the photocatalytic properties of the paint were not activated in the absence of UV irradiation, the "dark" VOC emissions were determined. Furthermore, during this time, since m-xylene was allowed to flow into the reactor without any contact with the painted surface, the m-xylene initial mixing ratio value, C0, was measured. After a given time, the UV lights were turned on. Here, in the presence of UV irradiation, the photocatalytic properties of the paint were activated and the photocatalytically formed reactive species, namely hydroxyl radicals (˙OH), superoxide radicals (˙O2-) and hydrogen peroxide radicals (˙OOH) 37 , reacted with m-xylene and binder organic constituents present on the surface of the TiO2 catalyst 21,30 .

As shown in Figure 2, upon exposure of the painted surface to UV irradiation, a formation of formaldehyde was observed. More precisely, an intense response to the gas-phase formaldehyde ion by the mass analyzer was observed as soon as the UV lamps were turned on. The formation of gas-phase formaldehyde was observed independent of the injector position with respect to the exposed (or not) painted surface within the horizontal flow-tube reactor. For example, the initial injector position, P0, with the UV lamps on allowed to check if there was any photo-degradation of m-xylene within the movable injector itself. After, the m-xylene mixing ratio, Cx, was measured on-line at a given injector position, Px, over the painted surface within the horizontal flow-tube.

Finally, when the injector was all the way "out" and the entire painted surface within the reactor was exposed to the m-xylene flow (position P4), the UV lights were turned off and the selected VOC emissions were monitored continuously for 2 hours. Here, the comparison of the m-xylene mixing ratio in position P4 while the UV lamps were on and P4 while the UV lamps were off was used to verify if the m-xylene degradation in position P4 was due to photocatalytic destruction or other chemical or physical phenomena. As may be seen in Figure 2, the typical m-xylene temporal profile obtained for mineral photocatalytic paint PEG 3350 was contrary to the expected results where the gas-phase m-xylene was expected to degrade once in contact with the painted surface. In fact, no degradation of mxylene was observed as the injector position was varied, thus, allowing the m-xylene to come in contact with the painted surface. Such an expected decrease in the m-xylene mixing ratio was observed in our previous study on photocatalytic destruction of selected VOCs on organic photocatalytic paints 15 . To our great surprise and contrary to our previous work on reactivity of organic photocatalytic paints, the photocatalytic activity of mineral photocatalytic paints seemed to be non-existent or negligible at best.

As shown in Figure 2, contrary to the m-xylene signal profile, the measured formaldehyde temporal profile was observed to differ. In fact, when the UV lights were turned on, the formaldehyde mixing ratio signal was observed to increase significantly. On the other hand, once the UV lamps were switched off, the formaldehyde mixing ratio was observed to decrease immediately to reach its initial value within few minutes. This important impact of irradiance showed the presence of photocatalytic activity in the binder. Further, the observed emissions of low molecular carbonyl compounds such as formaldehyde is consistent with other studies on UV irradiation of photocatalytic paints 38,39 .

Based on the m-xylene and formaldehyde temporal profile results described above, it may be concluded that the PEG 3350 mineral paint was photocatalytically active. However, the m-xylene was unable to reach the radical species on the photocatalytic paint surface to react. We propose several reasons for the observed phenomenon. One of the reasons is that the water vapor competed with the gas-phase m-xylene for adsorption on the active site of the photocatalyst [40][41][42] . Another possibility is that a thin layer of water formed on the painted surface, thus, hindering the interaction between the gas-phase m-xylene and the radical species 43 . The formation of this thin layer of water on nano-TiO2 is facilitated because mineral binder is more porous and more permeable to water vapor 44 than the organic binder. Additional experiments were performed to better understand the observed lack of photocatalytic destruction of m-xylene on selected mineral paints under the experimental conditions employed.

RH impact

Investigations into the relationship between the photocatalytic activity of TiO2 organic binder paints and relative humidity has already been investigated by Martinez et al. and Wortham and coworkers [START_REF] Morin | [END_REF]45 . These investigators have reported that the photocatalytic activity of selected TiO2 organic binder paints decreased by a factor of two when relative humidity was increased. Similar to the investigations into the relationship between the photocatalytic activity of TiO2 organic binder paints and relative humidity cited above, m-xylene uptake rate experiments were performed on TiO2 mineral binder paints used in this work as a function of relative humidity. The effect of the RH on measured uptake coefficient values of m-xylene on selected TiO2 mineral binder paint is shown in Figure 3. As shown in Figure 3, an increase in relative humidity was observed to lead to a decrease in the measured uptake coefficient values of m-xylene on selected TiO2 mineral paints. For example, the uptake coefficient of m-xylene was calculated to be 𝛾 = (8.8 ± 1.5) × 10 '; at 1% RH and 𝛾 = (4.6 ± 2.0) × 10 'Q at 40% RH. The observed almost two orders of magnitude decrease in the measured uptake coefficient value with increasing relative humidity may have been a result of physical competition for access to the radical species present on the paint's nano-TiO2 surface between the water molecules and m-xylene. More precisely, at low humidity, m-xylene may not have had to compete for access to the active surface sites of the photocatalytic paint, resulting in a more effective degradation of m-xylene; an argument that is supported by a higher uptake coefficient value. On the other hand, as the experimental RH was allowed to increase, the competition between the water molecules and the gas-phase m-xylene to access the active sites of the photocatalytic paint surface likely resulted in a decrease in the overall photocatalytic heterogeneous reactivity of the paint 8 ; a result that is supported by lower m-xylene uptake coefficient values in the presence of water vapor. In view of the obtained relative humidity effect results and to better compare the m-xylene uptake on selected paints, experiments were carried using low relative humidity values within the horizontal flow-tube reactor.

Uptake coefficients

To better characterize the photocatalytic properties of paints, m-xylene uptake coefficients were measured on weathered paint supports. Here, the selected paints were exposed to fluorescent UV lamp radiation, heat and water under controlled laboratory conditions to reproduce the aging effects that occur under real-environment conditions 29,30 . The effects of paint aging on the measured m-xylene uptake coefficients are shown in Figure 4. The aged paints included a reference paint and three photocatalytic paints. The reference paint did not contain any nano-TiO2 content and only micro-TiO2 pigments used in paint formulations. On the other hand, in addition to the micro-TiO2 pigment content present in all paints, the nano-TiO2 and PEG 3350 photocatalytic paints contained 3.5% of the nano-TiO2 content and the CNC photocatalytic paint contained 0.5% of the nano-TiO2 content. [START_REF] Morin | [END_REF]21,30 . These investigators reported that with aging, as paints were exposed to aging conditions that degraded paint' binder material, nano-TiO2 particles became physically more accessible to react with gasphase m-xylene, hence, increasing the paints' photocatalytic activities. That is, as the paint binder content was observed to degrade with "age" under taxing environment conditions, the uptake coefficient values were observed to increase. For example, while the nano-TiO2 paints were manufactured using nano-TiO2, PEG 3350 paints were fabricated using the nano-TiO2 coated with bio-inspired ligand polyethylene glycol. The presence of the bio-inspired ligand polyethylene glycol coating likely modified the physical and chemical characteristics of the nano-TiO2 compared to the nano-TiO2 paint formulation that lacked such a coating. This conjecture is supported by a recent work carried out by Laisney et al. (2021) on the spatial repartition of nanoparticles within the binder matrix, particle size and the nano-TiO2 aggregation-toagglomeration ratio 46 . On the other hand, the nano-TiO2 present in the PEG 3350 paints were more dispersed within the binder, the particle size was bigger, and the aggregation-toagglomeration ratio was lower compared to the nano-TiO2 paints. This modification allowed to increase the quantity of active sites present on the paint surface resulting in an increased photocatalytic activity.

As may be seen in Figure 4, the measured uptake coefficient values were observed to be independent of the aging process and no photocatalytic activity has been observed for reference and CNC paints. As shown in Figure 4, the presence of micro-sized TiO2 particles within the reference paint's formulation did not have any observable effect on its photocatalytic properties.

Further, the absence of photocatalytic activity in the CNC paints may be explained (1) by the hybrid CNC-nano-TiO2 protective properties of the binder staving off the photocatalytic degradation of m-xylene and (2) the low quantity of nano-TiO2 particles present within the CNC paint matrix to allow any observable m-xylene degradation. To better examine these two hypotheses, an increase of the nano-TiO2 content of the CNC paint binder may be imagined.

Unfortunately, at this point such higher nano-TiO2 formulation of CNC paints is not industrially possible.

VOC emissions

VOC emissions from reference and three photocatalytic paints have been studied as a function of weathering laboratory conditions that mirror a real environment. As may be seen in the supporting information Table S1-S4, no emissions due to a photocatalytic activity have been observed for methanol, formic acid, acrolein, acetic acid, MVK, propionic acid, benzene, butanoic acid, TMB and benzoic acid. VOC-emission results obtained for other compounds for the selected paints that have not been weathered are in Figure 5, at t = 500 hours and t = 1000 hours of aging time are shown in Figure 6 and Figure 7, respectively. As may be seen in Figure 5, the VOC surface emission flux values for reference and photocatalytic paints were observed to be equivalent for all listed compounds. As a result, it was assumed that any reactive interactions between the organic compounds present within the binder and the nano-TiO2 particles did not take place that would prevent or limit VOC emissions.

For reference paints, VOC emissions were observed to be independent of the aging process for most of the listed compounds. However, as may be seen in Table S1, an increase of 75% in acetaldehyde emission has been observed between 0 and 1000 hours of aging. The observed increase in acetaldehyde emission may be due to a low quantity of the photocatalyst present within the paint's binder 21 . Overall, VOC photocatalytic emissions values were observed to be very low.

This observation demonstrates that micro-TiO2 pigments have a negligible effect on the paint's photocatalytic properties.

As shown in Table S4, an increase in VOC emissions has been observed at t = 1000 hours of aging time for CNC paints. An increase of 34% in formaldehyde emissions, 99% in acetaldehyde emissions, 204% in pentanal emissions and 253% in octanal emissions have been observed at 1000 hours of aging time compared to paints that have not been weathered. The observed increases in VOC emissions are likely the result of a presence of 0.5% of nano-TiO2 within the paint's binder.

At t = 1000 hours of aging time, the binder is likely degraded to result in an enhanced interaction between the paint's organic fraction present within the binder and the nano-TiO2 particles. As shown in Figure 6 and Figure 7, The CNC paint VOC emissions are observed to be negligible compared to an increase in VOC emissions observed for nano-TiO2 and PEG 3350 paints.

Shown in the supporting information, nano-TiO2 (Table S2) and PEG 3350 paint (Table S3) VOC emissions are observed to increase significantly between t = 0 hour and t = 500 hours of aging time and become constant between t=500 hours and t=1000 hours of aging time apart from acetaldehyde and octanal emissions which are observed to increase. The observed difference in VOC emissions between the photocatalytic paints and the reference paint demonstrates that the paint VOC emissions are due to a photocatalytic effect resulting from a presence of nano-TiO2 content. For example, an observed increase in acetaldehyde emissions of a factor 5.4 and 4.3 has been observed at t = 1000 hours of aging time for paints photocatalytic paints nano-TiO2 and PEG 3350, respectively, compared to the reference paint (Figure 7).

The VOC emissions are observed to be lower for PEG 3350 paints compared to nano-TiO2 after t = 500 hours and 1000 hours of aging time. A decrease of 21%, 34% and 13% of VOC emissions has been observed for formaldehyde, acetaldehyde, and acetone, respectively, for PEG 3350 paints compared to nano-TiO2 paints at t=500 hours of aging time (Figure 6). A decrease of 11%, 7% and 31% for formaldehyde, acetaldehyde, and acetone emissions, respectively, has been observed

for PEG 3350 paints compared to nano-TiO2 paints at t=1000 hours of aging time (Figure 7). That is, the PEG 3350 paints are observed to emit less VOCs than nano-TiO2 paints. All surface emission fluxes are listed in Table S1-S4 in Supporting Information.

CONCLUSION

Uptake coefficients and VOC emissions have been measured for one reference mineral paint containing three different kinds of TiO2 nanoparticles, namely, nano-TiO2, PEG 3350 and CNC.

Their performances were compared to the same industrial paint containing no nanoTiO2 whatever its form. The obtained laboratory results contribute to improve the understanding and knowledge on the impact of nano-TiO2 photocatalysts present within paints on VOC loss and emissions within indoor environments. The obtained results are important to better indoor air quality and/or limit indoor VOC sources that may induce adverse effects on human health or well-being.

The laboratory results obtained in this work showed that the reference paint's micro-TiO2 pigment content did not result in m-xylene degradation and VOC emissions were measured to be negligible, a result that is in agreement with our previous work on micro-TiO2 pigment organic paints [START_REF] Morin | [END_REF]30 .

The absence of photocatalytic activity of the new CNC photocatalytic mineral paints may be explained by different binder effects such as (1) the hybrid CNC-nano-TiO2 protective role of the binder staving off the photocatalytic degradation of m-xylene and (2) the 0.5% nano-TiO2 content may be too low to allow any m-xylene uptake or loss.

While the m-xylene uptake coefficient experiments carried out using nano-TiO2 and PEG 3350 paints resulted in an interesting photocatalytic activity, these paints were shown to emit VOCs.

Globally, the photocatalytic activity was observed to be more important for the newer PEG 3350 photocatalytic paint than the older nano-TiO2 phototcatalyst. Further, the newer PEG 3350 photocatalytic paint was observed to result in less VOC emissions compared to the older nano-TiO2 photocatalytic paint. Clearly, more studies and engineering work is warranted to improve future paint formulations using the PEG 3350 photocatalyst content to better indoor air environment.

The mineral photocatalytic paints are a promising, albeit challenging, innovation for the building industry involved in paint manufacturing processes to produce "cleaner" paint formulations that limit indoor air pollution and better indoor air environment. A photocatalytic activity of mineral paints and the VOC emissions of mineral photocatalytic paints are observed to be lower than that of photocatalytic organic paints. However, to better quantify the photocatalytic effects in addition to environmental benefits and disadvantages, of mineral and organic paints, experiments need to be carried out under real-environment conditions.

the French National Research Agency (ANR) through the PIA (Programme Investissement d'Avenir).

ASSOCIATED CONTENT

Supporting Information

Reference paint surface emission fluxes (Table S1), nano-TiO2 paint surface emission fluxes (Table S2), PEG 3350 paint surface emission fluxes (Table S3) and CNC paint surface emission fluxes (Table S4), schematic presentation of the horizontal flow-tube reactor (Fig 1).

Notes

The authors declare no competing financial interest.

Figure 1 :

 1 Figure 1: VOC mixing ratios (ppb) obtained from air compressor (pink) and synthetic air bottle (green). Mixing ratios of formaldehyde to acetic acid are on the left axis and acrolein to octanal are on the right axis. Errors bars are ±1σ precision based on 70 experimental point values As shown in Figure 1, most of the obtained mixing ratio values of the selected VOCs measured in

9

  community to represent major chemical families typically observed within indoor air settings, 10 namely, family of acids, carbonyl compounds, aromatics, and alcohols.

-

  TiO2 coated by bio-inspired ligand polyethylene glycol (PEG 3350) and (3) nano-TiO2 grafted onto cellulose nanocrystals (CNC). The synthesis of the new types of photocatalytic TiO2 NPs as well as their characterization has been published in detail by Rosset and coworkers (2021) (Rosset et al., 2021).

8 +

 8 ×9.!;×(< !, ×Internal reactor volume (Residence time ×Exposed paint surface)

Figure 2 :

 2 Figure 2: Typical temporal profile of m-xylene (red line) and formaldehyde (blue line) mixing ratio at 298 K, RH = 40%, irradiance 8.8 W m -2 , aging: 500 hours and nano-TiO2 PEG 3350. Px corresponds to the different position of the movable injector during the experiment.

Figure 3 :

 3 Figure 3: The observed effect of relative humidity on the measured uptake coefficient of m-xylene on selected TiO2 mineral binder paints. Experimental conditions: Active ingredient = nano-TiO2 PEG 3350, [m-xylene] = 40 ppb, T = 292 K, light intensity = 18.6 W m -2 , aging = 500 hours. Error bars are derived from uncertainties associated to uptake coefficient.

Figure 4 :

 4 Figure 4: The effect of aging and photocatalyst on uptake coefficient of m-xylene. m-xylene mixing ratio: 40 ppb, temperature: 292 K, light intensity: 18.6 W m -2 , RH: 1%, reference paint (red square), paint nano-TiO2 (black circle), paint PEG 3350 (green lozenge), paint CNC (blue lozenge). Error bars are derived from uncertainties associates to uptake coefficient. As shown on the Figure 4, prior to aging, reference and photocatalytic paints m-xylene uptake

Figure 5 :

 5 Figure 5: VOCs surface emission fluxes (cm -2 s -1 ) obtained for selected compounds under UV irradiation. Surface emission fluxes for formaldehyde, acetaldehyde, acetone and acrylic acid on the left axis and pentanal and octanal on the right axis. Experimental conditions: T = 298 K, 40% RH, solar irradiance = 8.8 W m -2 , aging = 0 hours. Error bars are 1σ precision based on twenty experimental data point values.

Table 1 .

 1 Compounds detected and quantified using the HS-PTR-MS technique.

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	Compounds	m/z	References
	Formaldehyde	31	23 ; 24
	Methanol	33	25 ; 24 ; 26
	Acetaldehyde	45	23 ; 25 ; 24 ; 26
	Formic Acid	47	24
	Acrolein	57	24
	Acetone + Propanal	59	23 ; 25 ; . 24 ; 26
	Acetic Acid	61	25 ; 24 ; 27 ; 26
	Methyl Vinyl Ketone	71	25 ; 24
	Acrylic Acid	73	19
	Propionic Acid	75	27
	Benzene	79	23 ; 25 ; 24 ; 26
	Pentanal + Vinyl Acetate	87	23 ; 19
	Butanoic Acid	89	27
	Ethylbenzene + Xylene	107	23 ; 25 ; 24
	Trimethylbenzene + Ethyltoluene	121	23 ; 25 ; 24
	Benzoic Acid	123	26 ; 19
	Octanal	111 & 129

17

. 6 19 8

Table 2 : Percent compositions of nanoparticles in slurry (%), paint (%) and dry paint (%) used in this work.

 2

	Paint	Nano-TiO2 in slurry (%)	Slurry in paint (%)	Nano-TiO2 final dry
		(w/w)	(w/w)	content (%) (w/w)
	Reference	0	0	0
	nano-TiO2	35	10	3.5
	PEG 3350	35	10	3.5
	CNC	13.7	3.65	0.5

Table 3 .

 3 Distance to the end of reactor (cm), residence time (s) and exposed surface (cm 2 ) between gaseous m-xylene and mineral paint as a function of movable injector position.

	Injector	Distance (cm)	Exposed surface (cm 2 )	Residence time (s)
	position
	P0	0	0	0
	P1	4	8	5.4
	P2	8	16	10.9
	P3	15	30	20.4
	P4	22	44	29.9

ACKNOWLEDGMENTS

The authors thank and acknowledge Isabelle Michaud-Soret (CEA, Grenoble) for the synthesis of PEG 3350 nanoparticles, Isabelle Capron (INRA, Nantes) for the synthesis of CNC nanoparticles.

The authors acknowledge gratefully LABEX SERENADE (no.ANR-11-LABX-0064) funded by