Multiple Subject Learning for Inter-Subject Prediction Sylvain Takerkart * † and Liva Ralaivola † * Institut de Neurosciences de la Timone UMR 7289, Aix Marseille Université, CNRS, Marseille, France † Laboratoire d'Informatique Fondamentale UMR 7279, Aix Marseille Université, CNRS, Marseille, France Abstract-Multi-voxel pattern analysis has become an important tool for neuroimaging data analysis by allowing to predict a behavioral variable from the imaging patterns. However, standard models do not take into account the differences that can exist between subjects, so that they perform poorly in the intersubject prediction task. We here introduce a model called Multiple Subject Learning (MSL) that is designed to effectively combine the information provided by fMRI data from several subjects; in a first stage, a weighting of single-subject kernels is learnt using multiple kernel learning to produce a classifier; then, a data shuffling procedure allows to build ensembles of such classifiers, which are then combined by a majority vote. We show that MSL outperforms other models in the inter-subject prediction task and we discuss the empirical behavior of this new model.

I. INTRODUCTION

Background. In recent years, the use of machine learning approaches in neuroimaging has gained in popularity. The most prominent application of machine learning is the socalled multi-voxel pattern analysis (MVPA), that consists in predicting a behavioral variable from functional MRI data. The appeal of these multivariate methods relies on their increased sensitivity compared to standard univariate models. However, their generalization power on data recorded in new subjects suffers from the large variability that exists within a population.

Contribution. We specifically focus on the so-called intersubject prediction problem and we propose a method that is precisely aimed at giving reliable predictions for data of any subject for which no data was accessed to during the learning process. Our method builds upon a hierarchical probabilistic setting, and it makes use of two well-known tools from the machine learning artillery to deal with heterogeneous and variance-inducing data : multiple kernel learning (MKL [START_REF] Kloft | Lp-norm multiple kernel learning[END_REF], [START_REF] Lanckriet | Learning the kernel matrix with semidefinite programming[END_REF], [START_REF] Rakotomamonjy | Simple mkl[END_REF]), and data sampling/shuffling [START_REF] Breiman | Bagging predictors[END_REF], [START_REF] Wolpert | Stacked generalization[END_REF].

Related work.

The standard MVPA paradigm to address the inter-subject prediction task consists in pooling together all samples from the subjects available at training time. A single classifier is estimated in a supervised manner on this dataset to be later tested on data from new subjects. This paradigm, which is used by default in the literature, largely ignores the various sources of inter-subject variability, namely the differences in anatomy and functional organization across subjects, and the fact that all samples are not drawn from the same probability distribution. Therefore there is a crucial need for more elaborate models specifically tuned for the intersubject prediction task, like those recently proposed in [START_REF] Lorbert | Kernel hyperalignment[END_REF], [START_REF] Marquand | Bayesian multi-task learning for decoding multi-subject neuroimaging data[END_REF], [START_REF] Takerkart | Graphbased inter-subject classification of local fmri patterns[END_REF] and the new model we introduce in this paper.

II. METHODS

A. Setting and Addressed Problems

Training dataset. We consider the following setting

D .
= ∪ S s=1 D s . We additionally assume that each subject has participated in exactly the same fMRI protocol, so that ∀n ∈ {1 . . . N }, y1 n = y 2 n = . . . = y S n and, in what follows, y n will be used to denote the category associated with the n-th sample. We will use y as a compact notation for the vector y

. = [y 1 • • • y N ] ∈ C N .
Probabilistic setting. A hierarchical -hierarchical, because distributions on distributions are considered-probabilistic setting that may be associated with the generation of D is as follows. There is an unkown and fixed distribution L s defined on the space of distributions on X × C. A realization

L o |s : X × C → R
D s ∼ N ⊗ n=1 L o |s;yn . (1)
The probabilistic setting we have just described allows us to say that training sample D (given y) is so that:

D ∼ S ⊗ s=1 L s N ⊗ n=1 L o |s;yn ,
and the core random pair (s, (X s , Y s )) that defines our learning problem is therefore distributed according to 1

L . = L s ⊗ L o |s . (2)
Addressed problems. The main problem that we address is then to learn from D a predictor f :

X → C with risk R(f ) . = P (s,(X s ,Y s ))∼L (f (X s ) = Y s )
as small as possible. It is essential to understand that the inter-subject nature of the problem stems from s being an independent random variable (of law L s ): being able to make reliable predictions for subjects not seen during training is ultimately the goal conveyed by the minimization of R(f ). This general problem may be tackled by considering two subproblems: i) that of identifying, among the training subjects, those that provide the most representative behaviors so as to more heavily rely on them in the learning of f and, ii) that of reducing the variance of the learned predictor that may be due to the scarceness of data and the natural inter-subject variability of the data [START_REF] Kuncheva | Random subspace ensembles for fMRI classification[END_REF]. In II-C, we propose a learning model to address both subproblems (and thus, the general problem) which echoes the hierarchical decomposition of L as in (2).

B. Simple Strategies for Inter-Subject Prediction with SVC

We here briefly recall naive strategies to deal with learning from data coming from various subjects. In order to lighten the exposition, we now consider that C is reduced to the set C = {-1, +1} of two labels. A real-valued function f : X → R is readily associated with the thresholded decision function x → sign(f (x)), which predicts the label of x according to the sign, -1 or +1, of f (x); with a slight abuse, any function f will also denote its associated thresholded predictor. The specific class F ⊆ R X of functions that we will consider throughout, is that of kernel-based functions: we consider that we have at hand some positive definite kernel (see [START_REF] Scholkopf | Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond[END_REF]) k : X × X → R, and that F is defined as

F . = f : f (•) = m i=1 α i k(x i , •), (x 1 , . . . , x m ) ∈ X m .
The core predictor that we will use are Support Vector Classifiers (SVC). If trained on D s , an SVC f s writes as:

f s (x) = N n=1 α s n k(x s n , x), (3)
where the kernel used are centered on the training data x s n . However, this is intuitive that if a classifier is aimed at the inter-subject prediction task it should take advantage of all available data of D. This can be achieved by combining single subject classifiers (f s ) s=S s=1 through a vote, to give classifier

f vote (x) = S s=1 β s f s (x) = S s=1 β s N n=1 α s n k(x s n , x), (4)
where the (β s ) s=S s=1 is a sequence of (nonnegative) voting weights. Among the various ways to learn those weights, we will simply use the most trivial with:

β 1 = . . . = β S = 1/S.
Another strategy that uses all the data from D consists in pooling all the training samples together, i.e assuming that they are independent realizations of the same fixed random variable. This way, an SVC is learned on the whole training set D, with no regards to the hierarchical probabilistic decomposition discussed previously, and in particular, no distinction between data from different subjects. The SVC classifier f pool obtained with such an approach takes the form

f pool (x) = N n=1 S s=1 α s n k(x s n , x). (5)
Note that if the form of f pool encompasses that of f vote (see ( 4)), which is recovered for α s n = β n α s , the loss of the hierarchical structure suggested by the probabilistic model is here lost, as both indices s and n play the same role.

C. Multiple Subject Learning

We now describe the approaches that we promote: Multiple Subject Learning (MSL) which, as indicated by its name, is built upon the idea of Multiple Kernel Learning [START_REF] Lanckriet | Learning the kernel matrix with semidefinite programming[END_REF].

MSL. When using data from all available subjects to train a classifier, it is implicitely hoped that the subjects are a representative sample of the distribution L s of subjects. However, it seems intuitive that some subjects in D might contribute in a stronger manner to a good generalization, i.e. a small risk. Weighting the contribution of each subject in the learning process is therefore a natural way to account for this belief, as already mentioned previously, when the voting scheme was discussed (see f vote ). Here, we take advantage of the fact that the prediction functions under consideration are kernel-based classifiers to have the weighting intervene at the kernel level, and we propose to look for a classifier f msl of the form

f msl (x) = N n=1 α n S s=1 β s k(x s n , x), (6)
where the sequence (β s ) 1≤s≤S of nonnegative coefficients, which weighs the contributions of data from the different subjects, has to be learned from the training examples D together with the sequence (α n ) 1≤n≤N . Introducing the following notation will help us make a clearer connection with MKL: i)

x n . = [x 1 n • • • x S n
] ∈ X S denotes the vector made of the concatenation of the x s n 's, ii) likewise, for x ∈ X , x ∈ X S is the vector x . = [x • • • x] of S concatenations of x, iii) Π s : X S → X is the orthogonal projector such that Π s x extracts the s-th block (of size F -recall that X = R F ) of coordinates from x, so that Π s x n = x s n and Π s x = x, and iv) k s : X S × X S → R is the positive definite kernel such that k s (x, x ′ ) . = k(Π s x, Π s x ′ ). Classifier f msl of ( 6) can now be rewritten as:

f msl (x) = N n=1 α n S s=1 β s k s (x n , x) . = f msl (x) (7)
(where f β ∈ R X S ), or,

f msl (x) = N n=1 α n K β (x n , x), if K β : X S × X S → R is the positive kernel defined by K β . = S s=1 β s k s (K β is positive definite because it is a nonnegative combination of positive kernels).
Given model [START_REF] Marquand | Bayesian multi-task learning for decoding multi-subject neuroimaging data[END_REF], the problem we face is therefore to both learn the coefficients (α n ) 1≤n≤N of a kernel classifier and the weights (β s ) 1≤s≤S of a relevant combination of kernels: this is exactly the problem of Multiple Kernel Learning (see [START_REF] Lanckriet | Learning the kernel matrix with semidefinite programming[END_REF]). Here, each kernel k s to be combined is specifically associated to subject s, hence the name of Multiple Subject Learning.

Ensembles of MSL. We observe from [START_REF] Marquand | Bayesian multi-task learning for decoding multi-subject neuroimaging data[END_REF] and, in fact, from the definition of x n , that we have grouped together the data x 1 n , . . . , x S n . However, standard fMRI protocols are designed to measure repetitions of responses that are stationary in time, such that, from a probabilistic point of view x s n and x s n ′ are identically distributed provided that y n = y n ′ (see how D s is distributed according to (1)). The association of x 1 n , . . . , x S n into x n we assumed so far is therefore arbitrary and several other matchings are just as valid as long as they group together vectors x 1 n1 , . . . , x S n N such that y n1 = . . . = y n N . We may take advantage of that remark to build ensembles of MSL classifiers that get rid of this arbitrary grouping and make it possible to lower the variance of the predictor learned. The strategy to do so is based on the use of permutations, which allow various groupings of the data that, in turn, give various MSL predictors. Given such ensemble of M MSL classifiers (f msl m ) m=M m=1 , we may compute their combined prediction by

f msl * (x) = M m=1 f msl m (x), (8)
Here, the learning of f msl m is the result of i) a random and uniform draw of S permutations 2 σ m 1 , . . . σ m S over {1, . . . , N } such that for any n, y σ m 1 (n) = . . . = y σ m S (n) = y n and ii) the solving of MSL problem [START_REF] Marquand | Bayesian multi-task learning for decoding multi-subject neuroimaging data[END_REF] with, for n = 1, . . . , N, the vector x n ∈ X S defined as

x n = [x 1 σ m 1 (n) • • • x S σ m S (n) ],
which is where some shuffling of the training data occurs. As announced earlier, the strategy to learn f msl * better echoes the probabilistic decomposition discussed above.

III. EXPERIMENTS

A. Dataset

To evaluate our MSL framework, we analyzed data from an fMRI experiment during which nine subjects listened to auditory stimuli centered around five frequencies while performing tapping with a finger of the left hand, matched to the audio frequency, thus defining five classes of trials. The acquisitions comprised a high resolution T1, as well as EPIs (TR = 2.4s, voxel size = 2x2x3mm) recorded during five functional sessions; each session included six trials per condition presented in a pseudo-randomized order.

The functional data was analyzed in SPM8, with motion and slice timing corrections, followed by a GLM comprising one regressor per trial. The corresponding beta maps served as estimates of the response size for each trial. Freesurfer was then used to perform cortical reconstruction and registration to a spherical atlas. Each beta map was projected onto this atlas, thus providing a vertex to vertex mapping across subjects. Three cortical regions of interest were defined on the atlas to respectively cover the right and left auditory cortices and the right somato-sensory cortex.

The goal of the inter-subject prediction task was to guess the class of the stimulus from the response pattern (chance level = 0.2). We examined the different strategies that were defined in Section II i) the SVC classifiers learned on singlesubject data (defined in eq. ( 3) and hereafter called s-SVC); ii) the classifier obtained by a majority vote on the set of s-SVCs (eq. ( 4), hereafter vote-SVC); iii) the SVC learned by pooling data from all subjects (eq. ( 5), hereafter pool-SVC); iv) the multiple subject learning classifiers (eq. ( 6) and 2 A permutation over {1, . . . , N } is a bijection from {1, . . . , N } to itself. eq. ( 7), hereafter MSL and MSL*). The algorithms were run with the linear kernel and evaluated with a leave-one-subjectout cross-validation. For the first three strategies to perform at their best, a univariate feature selection was performed before learning the classifier; the percentage of selected features and the regularization constant C were chosen in a nested crossvalidation scheme. MSL and MSL* were estimated with C = 1. on the full feature set, using the l 2 norm implementation of MKL available in the Shogun toolbox.

B. Results

For each region, 500 MSL classifiers were learnt, each from a random set of within-subject shufflings. To generate an MSL* classifier of size M , we randomly drew M MSL classifiers from these 500. For each value of M , we generated 100 MSL* classifiers and averaged the results. Comparing learning strategies: Fig. 1 shows the accuracy rate of each classifier type, averaged on all folds of the cross-validation as well as on each single fold.

First, we qualitatively analyse these results. We observe that i) all classifiers perform very differently across folds of the cross-validation; ii) within each fold (i.e each column in Fig. 1), a large variation of the performances of the s-SVCs is also present; and iii) in most cases, at least one of the s-SVCs outperformed pool-SVC and vote-SVC. These qualitative observations respectively confirm our initial intuitions that i) the inter-subject variability weights heavily on the results; ii) some subjects will be more informative than others to generalize over the population; and iii) the pooling strategy vastly used in the litterature is clearly sub-optimal.

Secondly, we quantitatively assess the performances of the different classifiers by performing t-tests with paired samples corresponding to the performances of classifiers in each fold of the cross-validation. There was no significant differences between vote-SVC and pool-SVC classifiers (p > 0.2 for all three regions). Our MSL* classifiers (with M = 50) outperformed the other strategies in all cases (p < 0.05 for all three regions), which supports the effectiveness of this model.

2) Influence of the size of MSL ensembles:

We further examined the influence of the size of the MSL* ensembles. The results are summarized on Fig. 2, in which each color corresponds to the analysis conducted in one of the three ROIs, and the x axis of all subplots corresponds to varying size M of the MSL* ensembles. Fig. 2.A presents the MSL* accuracy rates. One can clearly see the accuracy gain offered by the construction of such ensembles and that their performances reaches an asymptotic maximum when M is of the order of a few dozens. The curves on Fig. 2.B represent the number of MSL* classifiers (amongst the 100 generated for each M ) for which the performance was significantly better (paired t-test, p < 0.05) than vote-SVC (dashed lines) and pool-SVC (solid lines). Those curves quickly reach 100% (except for the solid blue line), which means that the gain of accuracy offered by MSL* is robust to the choice of the ensemble itself.

IV. DISCUSSION AND CONCLUSION

We have presented several models that combine the information from multiple subjects to learn a classifier aimed at performing predictions on data from subjects not available at training time. We have demonstrated that our multiple subject learning framework makes it possible to i) combine information from different subjects in a multiple kernel learning fashion, and ii) build ensembles of classifiers by shuffling the ordering of examples in each subjects. The resulting MSL* classifiers vastly outperform other strategies in the inter-subject prediction task in an fMRI experiment designed to study the organization of the auditory and somato-sensory cortices. Future work will focus on the interpretation of the results provided by MSL. Beyond the use of the outcome of the predictions, our MSL framework provides weights for each subject. We plan on studying their consistency, both across folds of the cross-validation and across the different MSL classifiers used in a MSL* ensemble. Furthermore, similarly to the use of MKL weights, the MSL weights could further be used to detect which subjects contribute the most strongly in such classification task, or even to perform subject selection/rejection: on can imagine setting up a recursive subject elimination by removing the subject with the smallest weight at each iteration to maximize the generalization performance.

We also plan on attempting to improve the construction of the MSL* ensembles to further increase the performances of our framework. Indeed, using a purely random selection of classifiers is known to result in sub-optimal ensembles, in the sense that one can build an equally efficient ensemble of smaller size by appropriately choosing the classifiers. This could be done by maximising the information diversity that is provided by each classifiers of the ensemble [START_REF] Ruta | Classifier selection for majority voting[END_REF].

Fig. 1 .

 1 Fig. 1. Comparing classifiers. Accuracy rates on test subject s1, ..., s9, or averaged by cross validation (right column). Chance level is 0.2.

Fig. 2 .

 2 Fig. 2. Influence of the ensemble size M (x axis) on A: the accuracy of MSL*; and B: the proportion of MSL* classifiers that outperform pool-SVC (solid lines) and vote-SVC (dashed lines). Each color refers to one ROI.

Note that the work in[START_REF] Agarwal | Generalization Bounds for the Area under the ROC Curve[END_REF], theoretically shows how learning from data sampled according to a distribution conditioned on y provides results regarding learning from the corresponding unconditional distribution.

ACKNOWLEDGMENT

Thanks to the CNRS Neuro-IC program for funding, and to the Centre IRMf de Marseille and its staff for data acquisition.


Christophe Drouet

Pierre Alphonse

José Luis

García Fierro

Abel Rousset

Equilibrium and Kinetics of NO and CO Chemisorptions on Nonstoichiometric Nickel-Copper Manganites

Keywords: NO, CO, chemisorption, nickel-copper manganites

The equilibrium and kinetics of adsorption of NO and CO on nonstoichiometric nickel-copper manganites have been investigated through volumetric measurements. The adsorption isotherms were satisfactorily fitted to the Freundlich equation. The equilibrium coverages at 298 K were found to depend closely on the chemical composition of the oxide; thus, a decrease in the coverage beyond a maximum copper extent was observed. The adsorption isotherms of NO at various temperatures in the range from 298 to 473 K showed that the equilibrium coverage decreases with increasing temperature. This behavior enabled us to follow the logarithmic decrease of the heat of adsorption of NO on such surfaces. The adsorptions of NO and CO on surfaces preadsorbed with CO and NO, respectively, were also studied. These experiments showed the ability of NO to displace CO preadsorbed molecules whereas the contrary did not hold, suggesting the existence of common adsorption sites as well as some specific CO adsorption sites. Finally, some kinetic data are reported showing that the experimental adsorption results fit the Elovich equation (with t 0 ≈ 0), although two distinct rate processes could be identified.

INTRODUCTION

Environmental concerns over air pollution have led to a particular interest in CO oxidation catalysts. An example is the car exhaust engine which produces CO emissions contributing directly to air pollution and smog. The catalytic converters employed today contain a catalytic material which oxidizes CO to CO 2 and also reduces the amount of NO X and hydrocarbons. The catalytic components included in the cartridges of these converters are platinum, rhodium, and palladium, expensive metals which are deactivated by thermal reactions, cycling operations, and poisons. Thus, the development of new materials with high performance for the above-mentioned reactions are required.

Nonstoichiometric nickel and nickel-copper manganites have already been investigated in previous works (1)(2)(3). It was reported (3) that the substitution of part of the manganese cations by copper reinforces the thermal stability of such metastable phases. Recent catalytic experiments have shown that such manganites exhibit high activity toward the CO/O 2 and CO/NO reactions, which is closely related to the nickel and copper contents of the oxides (results to be published).

The aim of the present paper is to investigate the equilibrium and kinetics of the adsorption of NO and CO molecules on nickel-copper manganites. For such a purpose, the adsorption isotherms of NO and CO have been recorded for various chemical compositions to determine the effect of nickel and copper contents on the adsorption properties. The study of NO adsorption at various temperatures was also examined to establish the dependence of the heat of adsorption on surface coverage.

MATERIALS AND METHODS

Preparation of Samples

The oxide systems studied in this work are nonstoichiometric nickel-copper manganites corresponding to the general formula Ni x Cu y Mn 3-x-y O 4+δ . These are cation-deficient oxides (δ ranging from 0.2 to 0.7) and crystallize in the cubic spinel structure (3). These compounds were synthesized in two steps: (i) coprecipitation of a mixed Ni-Cu-Mn oxalate precursor of desired composition and (ii) thermal decomposition of this precursor at 623 K in air for 6 h (the detailed synthesis is described elsewhere (3)). Such a preparation route permits one to prepare oxides with high specific area (S w > 100 m 2 /g) as shown in Table 1. Manganese oxide was prepared according to the same procedure as above from thermal decomposition of manganese oxalate (α allotropic form), and the phase obtained was Mn 5 O 8+δ (3,4), which does not exhibit the spinel structure (5).

Chemisorption Measurements

A conventional volumetric Pyrex-glass high-vacuum system with greaseless stopcocks capable of maintaining a dynamic vacuum of 10 -6 Torr (1 Torr = 133.3 Pa) was used. Temperatures were measured with a chromel-alumel thermocouple. Pressure measurements were made by means of an MKS capacitance transducer comprised of a sensor head 310 BHS1000, an electronic unit, and a meter unit. The apparatus had a measuring range of 10 -3 to 10 3 Torr and an accuracy of 10 -2 Torr. The total volume of the measuring cell and reactor was about 10 cm 3 .

Prior to each adsorption experiment, the sample (50 mg) was outgassed under high vacuum for 1 h at 573 K, and the temperature was then set to the desired value. Such a pretreatment was compulsory to clean the surface; however, it could have led to a slight reduction of the surface. Each adsorption experiment was carried out, after pretreatment, introducing in several successive steps a given pressure of adsorbate in the calibrated volume containing the sample and noting the equilibrium pressure reached after a stabilization time of 3 h. The application of Boyle's law then led easily to the determination of the quantity of gas adsorbed from the value of the equilibrium pressure and that of the calibrated volume.

The adsorption experiments of NO and CO on surfaces preadsorbed with CO and NO respectively were carried out from clean surfaces as follows: (i) exposure of the sample to the first adsorbate at the desired pressure (calculated from adsorption isotherms to correspond to surface saturation), (ii) evacuation of the remaining gas phase under a primary vacuum (ca. 10 -2 Torr) for 15 min, and (iii) introduction of the second adsorbate at successively increasing pressures.

The kinetics of NO and CO adsorption on the surface of the oxides was investigated following the variation of the pressure measured by the MKS pressure transducer as a function of time after introduction of a given pressure of adsorbate on a clean surface.

RESULTS AND DISCUSSION

Equilibrium of Adsorption

The adsorption isotherms of NO and CO on the nonstoichiometric nickel-copper manganite Ni 0.25 Cu 0.80 Mn 1.95 O 4+δ at 298 and 373 K are reported in Fig. 1 as N a = f (P eq ), where N a is the number of molecules adsorbed per gram of oxide and P eq is the equilibrium pressure of the adsorbate reached. At first glance, the shape of the three isotherms, (a), (b), and (c) corresponding respectively to the adsorptions of NO and CO at 298 and of NO at 373 K, are of type I in the BDDT classification, reaching an equilibrium asymptote from low pressures (P ≈ 50 Torr). In contrast, the adsorption isotherm of CO at 373 K (curve (d)) deviates clearly from type I. Such a behavior can be explained easily by recalling the exceptional reactivity of these oxides toward CO oxidation, even at room temperature. Therefore, to discriminate between the pure chemisorption and the oxidation process, CO-adsorption experiments on these oxides must be recorded at temperatures not higher than room temperature. However, the isotherm corresponding to CO adsorption at 298 K (curve (b)) still belongs to type I of BDDT classification, despite the activity of the oxide for CO oxidation. It is presumed that the vacuum pretreatment might have reduced the oxide surface to a sufficient extent as to stop its activity toward CO oxidation.

If the data of isotherm (a) in Fig. 1 are replotted in terms of ln(N a ) = f (ln(P eq )), a correlation coefficient of 0.9988 is obtained, showing the linearity of this plot which is indicative of a Freundlich-type isotherm:

N a = k • P 1/n eq , [1]
where n and k are temperature-dependent constants, the values of which can be directly deduced from the slope and intercept of the ln(N a ) vs ln(P eq ) plot.

If the same mathematical treatment is repeated for isotherms (b) and (c), a similar linearity between ln(N a ) and ln(P eq ) is obtained, showing again that the Freundlich equation precisely describes the experimental results. Such a correspondence with the Freundlich equation points out the heterogeneity of the surface, and various other studies on chemisorptions of NO and CO on transition metal oxides (6)(7)(8) have also shown the agreement with this isotherm. As expected, isotherm (d) does not fit the Freundlich equation since the catalytic oxidation of CO is overimposed to the adsorption process.

As the catalytic properties of the nickel-copper manganites depend closely on the chemical composition of the oxide, the study of the effect of the nickel and copper contents on the quantities of NO and CO chemisorbed appear to be of great relevance. The adsorption isotherms of both adsorbates at 298 K on manganese oxide Mn emphasized here that the vacuum pretreatment of Mn 5 O 8+δ led to a slight change in color of the manganese oxide from black to dark brown, certainly indicative of the reduction, at least partial, of Mn 5 O 8+δ to Mn 3 O 4 . However, the results corresponding to the isotherm obtained for manganese oxide, whichever it was, could bring interesting data to compare to nickel and nickel-copper-containing oxides. Moreover, Mn 3 O 4 possesses the spinel structure which is a common point with the other oxides and can thus be taken as good reference oxide.

In the case of NO adsorption, the equilibrium coverage reached on manganese oxide is much smaller than that on nickelcontaining manganites, which underlines the influence of nickel on NO chemisorption (Fig. 2a). Furthermore, the comparison of the three nickel-containing oxides with different copper contents shows a slight decrease of the quantity of NO adsorbed with increasing copper content. The little difference among the specific areas of the three oxides (see Table 1) cannot explain, at least completely, this behavior. Figure 2b shows the same conclusion as above, concerning the important influence of nickel, for CO chemisorption. In contrast, the effect of copper on the quantity of CO chemisorbed is different from the case of NO chemisorption. Indeed, the incorporation of copper into the nickel mangan- As stated above, the high reactivity toward CO oxidation of the oxides studied in this work impeded us from monitoring the evolution of the equilibrium coverage of CO at temperatures above room temperature, and the equipment used in this study did not allow us to work at lower temperatures. In contrast, NO chemisorption could be investigated at temperatures higher than room temperature, and the results reported hereafter concern experiments conducted in the temperature range from 298 to 473 K.

Figure 3a reports the adsorption isotherms of NO on Ni 0.25 Cu 0.80 Mn 1.95 O 4+δ at 298, 333, 373, and 473 K. As can be seen, the NO equilibrium coverage decreases progressively when the temperature increases. The shape analysis of these isotherms shows that Eq. [1] is still valid, and the corresponding parameters of Freundlich's isotherm (n and k) are given in Table 3. As a general trend, both n and k decrease with increasing temperature; in particular, n tends to unity, approaching Henry's law. The data of Fig. 3a were replotted in Fig. 3b in terms of ln(N a ) vs ln(P eq ), and the intersection point of all the isotherms, deduced by extrapolation, gives the value of the monolayer coverage N m ∼ = 1.05 × 10 21 molecules of NO/g. It can be noted that this value leads to a surface area covered by NO of ≈130 m 2 /g (taking a value of 0.125 nm 2 for the cross section of the NO molecule ( 9)) which is close to the specific surface area (150 m 2 /g) derived from the nitrogen adsorption isotherms at 77 K.

The variation of the adsorption heat of NO with coverage can be directly determined from Freundlich's theory by the relation

H = -H m • ln θ, [ 2
]
where θ = N a /N m and H m can be calculated from n with the relation

H m = n RT/(1 -r T ), [ 3
]
where r is a correcting parameter introduced by Halsey (10), taking into account the small variation of H m with T . The values of H m and r are also reported in Table 3.

On the other hand, the heat of adsorption, at constant coverage θ , can be derived from the Clausius-Clapeyron equation

d ln P d 1/T θ = - H R . [4]
Figure 4 shows the variation of H with the NO coverage obtained by both methods (using Eqs. [2] and [4]), taking a mean value of H m = 20 kJ/mol. As expected in the case of Freundlich-type isotherms, the experimental heat of adsorption decreases logarithmically with the coverage and the two curves are in quite good agreement within the experimental errors.

From the adsorption isotherms, a set of isobars was determined and some of them are plotted in Fig. 5, showing that the experimental data in the temperature range 298-473 K correspond to the descending branch of the isobars, i.e., the chemisorptive part. These descending branches begin near room temperature, which recalls the results obtained by Peña et al. (11) for the chemisorption of NO on the manganese-containing perovskite LaMnO 3 .

To investigate the eventual competitive behavior of NO and CO molecules for the surface adsorption sites, the adsorption of each gas was studied in the same conditions as before, but on surfaces saturated with the other adsorbate. The results presented below are relative to competitive adsorption at 298 K on the mixed oxide Ni 0.25 Cu 0.80 Mn 1.95 O 4+δ . Figures 6a and6b give the evolution of the equilibrium pressure P eq of NO and CO respectively as a function of the initial pressure P ini (before admission of the gas into the sample chamber) in the cases of a clean and a CO (NO respectively) preadsorbed surface. The theoretical variation of pressure, i.e., without adsorption (only taking into account the gas expansion), is also reported in each figure with a dashed line for comparison.

As can be seen, both cases are distinct and will be dealt with separately.

In the case of a CO-preadsorbed surface (Fig. 6a), the experimental curve corresponding to NO adsorption may be divided into two successive steps: a first one (up to P ini ≈ 60 Torr) for which the experimental points fit the theoretical curve, meaning that the total number of gaseous molecules remains as if no adsorption occurred, and a second step where the curve slowly reaches the level obtained without CO preadsorption (which means that the NO uptake is no longer affected by the previous CO preadsorption). Moreover, in contrast to the adsorption on a clean surface where the pressure constantly decreases with time after admission of the adsorbate into the sample chamber, the pressure follows here a U-type variation with time as shown in Fig. 7. This behavior was observed for each experimental point of the first step of the P eq = f (P ini ) curve of Fig. 6a. The variation of the pressure before reaching equilibrium shows that there is a first rapid uptake of NO (descending branch in Fig. 7) followed by a slower release of gaseous molecules up to equilibrium (ascending branch), corresponding to a return to the initial pressure. This implies that the number of gaseous molecules at equilibrium is the same as the initial one, and that the experimental points of the first step of Fig. 6a follow the theoretical curve as if there had not been any adsorption. Such a behavior could be explained by the rapid formation of a surface complex between CO preadsorbed molecules, as carbonyl or carbonate species (which are visible by IR) and NO adsorbate molecules. Such a complex has already been envisaged from IR studies by Panayotov et al. (12) in the case of NO and CO coadsorption on the spinel CuCo 2 O 4 . This complex, relatively unstable, could then decompose, leading to final NO adsorption and release of a carbon-containing molecule in the gas phase (CO or CO 2 ). However, step 2 in Fig. 6a shows that the equilibrium pressure finally reaches that obtained without CO preadsorption, showing apparently that the overall amount of the CO preadsorbed molecules has been replaced by NO. These results are also in accordance with the data of Table 2, which show a higher equilibrium coverage for NO than for CO. Thus, it can be inferred that (i) at least part of the NO and CO adsorption sites are common and (ii) NO is able to displace CO-preadsorbed molecules, which is in agreement with several other studies on NO/CO competitive adsorption (6,13,14).

CO adsorption on a NO-preadsorbed surface is completely different from the previous case. Indeed, Fig. 6b shows that the isotherm exhibits the classic shape of the adsorption on a clean surface, corresponding however to a CO uptake clearly inferior to that obtained without NO preadsorption. Moreover, in the present case, the adsorption kinetic curve P = f (t) decreases, in a monotonous way, up to equilibrium like for adsorptions on clean surfaces, and does not display any U-type behavior, which allows us to exclude the formation of a complex surface intermediate between NO and CO molecules. These findings emphasize the difficulty, or even inability for CO to remove NO-preadsorbed molecules, as already observed by Yao and Shelef (6) on cobalt-containing spinels. However, as shown in Table 2, at the same temperature (298 K) the NO equilibrium coverage is higher than that of CO. The inability for CO to displace NO-preadsorbed molecules (up to saturation) should thus make it impossible for CO to adsorb, which is nevertheless the case. This apparent contradiction could be explained by the presence of some specific adsorption sites for CO. An in situ FT-IR study is presently in progress to determine the surface species formed upon adsorption of NO and/or CO on such surfaces and to better evaluate the interaction between the surface sites and these adsorbate molecules.

Kinetic Results

To determine the rate of the adsorption process of NO or CO molecules on the surface of such spinel oxides, the pressure drop taking place upon exposure of the samples to the adsorbate molecules was monitored as a function of time. The evolution of the number of NO or CO molecules adsorbed per gram of oxide (N a ) with time was then directly determined from the calibration of the volumes of the adsorption cell. The adsorption rates observed for NO and CO on a given oxide at constant initial pressure are of the same order of magnitude and give the same shape of kinetic curves. Figure 8a The linearity of the plot ln(d N a /dt) vs N a (Fig. 8b) shows that the adsorption rate process follows the Elovich equation

d N a dt = a • exp(-b • N a (t)), [ 5]
where a and b are two constants. The integration of this equation gives a useful equation to analyze kinetic results:

N a (t) = 1 b ln(ab) + 1 b ln(t + t 0 ). [6]
This integrated form introduces the time t 0 related to the pre-Elovichian period that often precedes the Elovichian rate process. Aharoni and Ungarish (15) have proposed a method for the determination of t 0 based on the extrapolation of t when Z tends to zero (where Z stands for the reciprocal of the adsorption rate). The application of this method to our kinetic data gave t 0 ≈ 0, showing that this time can be neglected in a first approximation.

Figure 9 gives the plot of N a vs ln(t + t 0 ) corresponding to the same example as above and taking t 0 = 0. Equation [6] shows that, for a rate process that follows an Elovich equation, this plot must be linear. However, two distinct linear branches are observed, revealing the existence of two distinct rate processes. If the same mathematical approach is run for CO adsorption, similar conclusions can be drawn, in particular, concerning the two Elovich linear branches for the plot of N a vs ln(t + t 0 ). The occurrence of two adsorption rate processes has already been observed on similar adsorbate-adsorbent systems (16). In the present case, this could be interpreted assuming the existence of two kinds of adsorption sites with different surface accessibility or reactivity. Such hypotheses will have to be confirmed by additional experimental results.

CONCLUSION

The equilibrium adsorption of NO and CO molecules on nonstoichiometric nickel-copper spinel manganites was studied, showing that these adsorptions follow the Freundlich model. The chemical composition of the oxides was found to have a great influence on the equilibrium coverage reached (N a ), and the logarithmic decrease of the adsorption heat of NO with N a was determined from isotherms obtained at different temperatures. The adsorption of NO and CO on surfaces preadsorbed with CO and NO, respectively, were also investigated and the results showed the ability of NO to remove preadsorbed CO molecules whereas the contrary did not hold. Although an important part of the adsorption sites were found to be common for CO and NO, CO appeared to possess some specific sites. The transitory formation of a surface complex between CO-preadsorbed molecules and NO adsorbate molecules was assumed from kinetic results when the order of adsorptions was CO preadsorption-NO adsorption. Finally, the kinetics of adsorption of NO or CO molecules could precisely be described by the Elovich equation, with t 0 ≈ 0, and two distinct rate processes were identified.

FIG. 1 .

 1 FIG. 1. Adsorption isotherms on Ni 0.25 Cu 0.80 Mn 1.95 O 4+δ of NO at 298 K (a) and 373 K (c) and CO at 298 K (b) and 373 K (d).

FIG. 2 .

 2 FIG. 2. Adsorption isotherms at 298 K of NO (a) and CO (b) on Mn 5 O 8+δ , Ni 0.25 Mn 2.75 O 4+δ , Ni 0.25 Cu 0.38 Mn 2.37 O 4+δ , and Ni 0.25 Cu 0.80 Mn 1.95 O 4+δ .

FIG. 3 .

 3 FIG. 3. Adsorption isotherms of NO on Ni 0.25 Cu 0.80 Mn 1.95 O 4+δ at 298, 333, 373, and 473 K as (a) N a = f (P eq ) and (b) ln(N a ) = f (ln(P eq )).

FIG. 4 .

 4 FIG. 4.NO heat of adsorption on Ni 0.25 Cu 0.80 Mn 1.95 O 4+δ calculated from Eqs.[2] and[4] (see text).

FIG. 5 .FIG. 6 .

 56 FIG. 5. NO adsorption isobars on Ni 0.25 Cu 0.80 Mn 1.95 O 4+δ .

FIG. 7 .

 7 FIG. 7. Evolution of the pressure with time in the case of NO adsorption on a CO-preadsorbed surface of Ni 0.25 Cu 0.80 Mn 1.95 O 4+δ at 298 K.

FIG. 8 .

 8 FIG. 8. Kinetic data for NO adsorption at 298 K on Ni 0.22 Cu 0.57 Mn 2.21 O 4+δ as (a) N a = f (t) and (b) ln(d N a /dt) = f (N a ).

FIG. 9 .

 9 FIG. 9. Elovich plot N a = f (ln(t + t 0 )) for NO adsorption on Ni 0.22 Cu 0.57 Mn 2.21 O 4+δ at 298 K, taking t 0 = 0 (see text).

TABLE 1 Specific Surface Area, S w , of the Samples

 1

	Oxide composition	S w (m 2 /g)
	Mn 5 O 8+δ	125
	Ni 0.25 Mn 2.75 O 4+δ	165
	Ni 0.22 Cu 0.57 Mn 2.21 O 4+δ	150
	Ni 0.25 Cu 0.38 Mn 2.37 O 4+δ	155
	Ni 0.25 Cu 0.80 Mn 1.95 O 4+δ	150

TABLE 2 Equilibrium Coverages for NO and CO Adsorption at 298 K

 2 structure leads to an equilibrium coverage of CO more than 3 times larger than that in the absence of Cu. However, a further increase of copper content leads to a decrease of the quantity of CO adsorbed, as can be seen for the Ni 0.25 Cu 0.80 Mn 1.95 O 4+δ sample. A comparison of the results of Figs.2a and 2breveals that, for the same oxide, the NO equilibrium coverage is always higher than that obtained for CO (for the same temperature, here, 298 K). The corresponding values of the equilibrium coverages (given by the value of the intercept of the asymptote reached) are reported in Table2. These results are similar to those of Yao and Shelef (6) obtained for cobalt-containing spinel systems.

		Equilibrium coverage (molecules/g)
	Oxide	NO	CO
	Mn 5 O 8+δ	0.6 × 10 20	0.2 × 10 20
	Ni 0.25 Mn 2.75 O 4+δ	2.1 × 10 20	0.5 × 10 20
	Ni 0.25 Cu 0.38 Mn 2.37 O 4+δ	2.2 × 10 20	1.8 × 10 20
	Ni 0.25 Cu 0.80 Mn 1.95 O 4+δ	1.9 × 10 20	1.2 × 10 20

ite

TABLE 3 Freundlich Parameters (n, k) and Heat of Adsorption for Monolayer Coverage H m in the Case of NO Adsorptions on Ni 0.25 Cu 0.80 Mn 1.95 O 4+δ at 298, 333, 373, and 473 K

 3

	T (K)	298	333	373	473
	n	5.4	3.2	2.5	1.9
	k (molecules/g)	10.1 × 10

19

4.70 × 10 19 2.81 × 10 19 0.85 × 10 19 H m = n RT/(1r T ) 23.0 16.6 16.2 22.1 (kj/mol) r r ∼ 1 . 4 × 10 -3