J Mabon

M Ortner

J Zerubia

CNN-BASED ENERGY LEARNING FOR MPP OBJECT DETECTION IN SATELLITE IMAGES

Keywords: small object detection, remote sensing, marked point process, convolutional neural network

This article presents a method combining marked point processes and convolutional neural networks in order to detect small objects in optical satellite images. In this setting, objects are scattered densely: the energy based formulation of a point process allows us to factor in priors to account for object interactions. Classical marked point process approaches use contrast measures to account for object location and shape, which prove limited in complex scenes. Instead, we use convolutional neural networks to learn energy terms that are more resilient to object and context visual diversity. Finally we present a procedure to learn the relative weights of prior and likelihood terms. We test our approach on remote sensing images and compare it to contrast based approaches. Code is available at https://github.com/Ayana-Inria/MPP CNN RS object detection.

INTRODUCTION

Small object detection in optical satellite images is often a challenging task; limited spatial resolution makes the objects of interest only a few pixels large, leaving less visual information to use. Moreover objects such as vehicles are often scattered densely, introducing interactions between neighboring objects, and increasing the difficulty in separating instances.

Small object size in images with resolutions around 0.5m/px, makes raster to raster segmentation impractical. This issue furthers the need for vectorisation of the extracted information.

Numerous methods based on Convolutional Neural Networks (CNN) such as Faster R-CNN [START_REF] Shaoqing Ren | Faster R-CNN: Towards real-time object detection with region proposal networks[END_REF], YOLO [START_REF] Redmon | You Only Look Once: Unified, Real-Time Object Detection[END_REF] or Reti-naNet [START_REF] Lin | Focal loss for dense object detection[END_REF] propose to detect objects in "natural" images, where objects are large and interactions between instances are scarce. Also, the size of detection boxes and their lim-Thanks to BPI France (LiChiE contract) for funding, and to the OPAL infrastructure from Université Côte d'Azur for providing computational resources and support. ited parametrisation, make these approaches less reliable in remote sensing applications.

On the other hand, Marked Point Process (MPP) based approaches rely on a stochastic geometry model. They jointly solve the detection and selection of objects while vectorizing the extracted information. These models also account for interaction models through priors. Such approaches classically use likelihood terms built upon contrast measures [START_REF] Descombes | Multiple objects detection in biological images using a marked point process framework[END_REF][START_REF] Craciun | Joint detection and tracking of moving objects using spatio-temporal marked point processes[END_REF][START_REF] Lacoste | Point processes for unsupervised line network extraction in remote sensing[END_REF] which perform great in images where the objects of interest are clearly contrasted with respect to the background.

However, complex image backgrounds, varying illumination conditions as well as object diversity make the contrast measures less effective. While it is possible to design more elaborate measures, this proves tedious and increases computational cost.

In this article we solve this issue by combining CNN based energy terms with a MPP that models priors and object interactions. Our main contributions are as follows:

• we formulate the detection task as an energy minimisation problem, while introducing priors.

• we build likelihood measures on simple CNN models to replace contrast measures.

• we propose a method to learn the weights for the combination of energy terms on the data.

In section 2 we present the marked point process framework and the energy model that drives it in sections 3 and 4. We then detail in section 5 the training and inference procedures. Finally, in section 6 we evaluate this approach on satellite imagery and compare results with contrast based approaches, along with other methods such as [START_REF] Yi | Oriented object detection in aerial images with box boundary-aware vectors[END_REF].

POINT PROCESS FRAMEWORK

We consider the image space as S ⊂ R 2 . A configuration of points Y is a finite non-ordered set of elements of S×M , with M the mark space. A mark can be any random variable from the radius of a circle to a discrete categorization of the object. In our case, an object y ∈ Y is comprised of coordinates i, j

j i α b a s = a + b 2 r = a b
Fig. 1. Rectangle parametrization in S, and three marks that describe a rectangle : size s, ratio r and angle α (see Fig. 1).

A configuration of points can be modeled as the realization of a non-uniform Marked Point Process (MPP); the density of which is defined by h relative to the uniform point process [START_REF] Van Lieshout | Markov Point Processes and Their Applications[END_REF]. The model of selection and interaction of points derives from an energy U , through a non-normalized Gibbs density :

h(Y ) ∝ exp(-U (Y )) (1)
To infer the best fitting configuration of objects Y from an image X, it needs an energy U tot (X, Y ), the minimum of which is reached for the optimal configuration Y = Y .

ENERGY MODEL

Given an image X and a configuration of points Y , the total energy is defined, as the sum for each point y ∈ Y , of the combination of energy terms listed in Table 1.

Energy term Notation

Eq. # position likelihood U p (X, y)

2 mark likelihood U m (X, y) 3 size prior U s (y) 4 overlap prior U o (y, N y ) 5 alignment prior U a (y, N y ) 6
Table 1.

The likelihood energy terms U p and U m measure the conformity of the point y regarding the image X. The prior terms U s , U o , U a measure the coherence of the point configuration itself considering the known properties of the studied objects. The latter depends on the point y ∈ Y and its neighborhood in Y , N y = {ỹ ∈ Y |ỹ = y, y -ỹ < d max }.

In this section we detail each energy term. The combination of these energy terms into a total energy U tot (X, Y ) is discussed in section 4.

CNN based likelihood measurement

To circumvent the limitations of contrast measures as likelihood energy terms, we devise U p and U m sampled from energy maps inferred with CNN models.

Position likelihood term

A first Unet [START_REF] Ronneberger | U-Net: Convolutional Networks for Biomedical Image Segmentation[END_REF] 1 infers a probability map of object centers in order to extract U p (X, y), the energy relative to the position of the object.

The localisation of object centers is similar to a keypoint detection task [START_REF] Cao | Realtime multi-person 2D pose estimation using part affinity fields[END_REF]. However, these keypoints are often so close that a heatmap inference approach -ie. learning a center probability map dilated by a Gaussian filter (see Fig. 2b)creates connectivity between objects that makes them harder to separate. Moreover, reducing the variance on the Gaussian around centers, reinforces the unbalance in labels, and contradicts the imprecision of annotations (ground truth object locations are often noisy).

Thus, we learn this center map through a proxy task of learning a vector field. The model is trained to infer for an image X of size (h, w), a vector field V of size (h, w, 2), which unitary vectors point towards the closest object center (Fig. 2c). By computing the divergence of the vector field, object centers become negative divergence locations, and separations between instances are mapped to positive values (Fig. 2d and2f).

The position energy is then :

U p (X, y) = -σ b + a.div( V ) yi,yj (2)
where σ is the sigmoid function and div( V ) yi,yj is the value of the divergence of V inferred from the image X, at the location of y. Parameters a and b are estimated while training the neural network.

Mark likelihood term

The energy U m (X, y) associated to the marks of a point y (s, r, α, defining the shape of a rectangle) is computed as follows: for each mark m, we discretize its value range into N m intervals. From the Unet, for a given image X of size (h, w), we extract a tensor M m of size (h, w, N m ).

We use Softmax( M m yi,yj ) as discrete approximation of the likelihood of the values of mark m at the position of y. Thus for each y and mark m ∈ {s, r, α}, given a value y m , and index(y m ) the index of the interval containing y m : Softmax( M m yi,yj ) index(ym) is trained to estimate the probability P (y m |X, y i , y j ).

While a regression model would output a single mark value per pixel, our classification-like approach allows to assign a probability to all possible values of a mark at a given pixel, allowing to inform on ambiguous situations with multimodal estimated likelihood densities.

U m (X, y) = m∈{s,r,α} -Softmax( M m yi,yj ) index(ym) (3)

Priors on configurations

We use various prior terms to regularize point configurations, given the expected properties of the studied objects:

Size prior

Enforces size limits on rectangles. This helps avoiding zero area objects. With a 0 and a 1 the minimum and maximum area hyperparameters:

U s (y) = max{a 0 -area(y), area(y) -a 1 , 0} (4)

Overlap prior

Penalizes the overlapping of objects [START_REF] Descombes | Multiple objects detection in biological images using a marked point process framework[END_REF] :

U o (y, N y ) = max ỹ∈Ny area(ỹ ∩ y) min{area(ỹ), area(y)} (5)

Alignment prior

Rewards configurations where objects are aligned (eg. cars often drive and park aligned), with y α the angle of y relative to the image horizontal axis (see Fig. 1):

U s (y, N y ) = min ỹ∈Ny {-| cos(|y α -ỹα |)|} (6)

COMBINING ENERGY TERMS

In order to compute the total energy U tot (X, Y ) we combine for each point all energy terms defined previously (Ta-ble 1). We formalize this energy combination as a function f parametrised by θ, with U X,y the vector of energies defined in Table 1:

U tot (X, Y, θ) = y∈Y f θ (U X,y ) (7)

Weighted-sum combination

In classical point process applications, such as [START_REF] Descombes | Multiple objects detection in biological images using a marked point process framework[END_REF][START_REF] Craciun | Joint detection and tracking of moving objects using spatio-temporal marked point processes[END_REF][START_REF] Lacoste | Point processes for unsupervised line network extraction in remote sensing[END_REF], the energies are combined as a weighted sum. This weighted sum can be written as follows 2 :

f θ (U X,y ) = θ p U p + 1 Up<0 .(θ m U m + θ s U s + θ o U o + θ a U a ) (8)
The indicator term 1 Up<0 forces to omit any prior or mark contribution when the point is not well positioned. Weights θ p , . . . , θ a are usually set manually. This approach also requires finding a threshold for every energy term so that valid points have a negative energy contribution, and non-valid ones have positive energy contributions: ie. one would have to set a threshold t p for U p (X, y) = σ(b + a.div( V ) yi,yj ) -t p such as valid points y have U p (X, y) < 0 (see [START_REF] Lacoste | Point processes for unsupervised line network extraction in remote sensing[END_REF]).

Logistic combination

We introduce a formulation inspired by logistic models, denoting θ w the weights vector and θ b a scalar:

f θ (U X,y ) = 2σ (θ w • U X,y + θ b ) -1 (9)
This formulation allows for the following :

• the sigmoid σ maps valid points to an energy close to -1 and non-valid one close to 1, thus keeping energy per point bounded.

• removing the indicator term from equation 8 makes f θ continuous and allows to learn θ more consistently (see subsection 5.2).

• introducing a bias term θ b , allows to learn all subenergies threshold terms at once, subtracting the need to fit thresholds on individual energies.

TRAINING AND INFERENCE

Our model runs as follows:

1. learning underlying CNN models for likelihood terms U p and U m on the training set.

2. learning θ for the energy combination model f θ , given the sub-energies and the training set (see Section 5.2).

3. at inference, for an image X, computing vector field V and map M . Since U p and U m only depend on y, V and M , we write U tot as a function of V , M , Y and θ. This way we don't have to recompute V and M for every Y we want to compute U tot for.

4. then inference of Y through the simulation of the point process derived from U tot (see subsection 5.3).

Learning likelihood terms

Position

The position energy model is trained to minimize the following cost function:

L pos (X, Y ) = MSE( V , V ) + BCE(M, σ(a.div( V ) + b)) (10
) where V is the vector field built from the ground truth configuration Y , M a heatmap of object centers (binary center map derived from Y , passed through a Gaussian filter with variance 0.6). MSE and BCE are respectively the Mean Square Error and the Binary Cross Entropy.

Marks

The energy model on marks is trained to minimize the following cost function for every mark m (s, r and α):

L m (X, Y ) = 1 |P | p∈P CE(Softmax( M m p ), Softmax(M m p )) (11
) where P is the set of pixels in X, and CE the Cross Entropy between the estimated distribution Softmax( M m p ) and the ground truth Softmax(M m p ) for every pixel. Both models are trained on 2/3 of the dataset, from which we sample smaller patches (2/3 around objects of interest, 1/3 uniformally).

Learning θ, the energy combination model

To learn the function f θ that combines the sub-energies, we proceed to enforce an ordering of configuration energies: given a ground truth configuration Y for an image X, we perturbate Y to get Y with a perturbation kernel Q (by removing, adding, translating, rotating or scaling points randomly). Thus the point configuration Y is less fitting to X than Y ; this means U tot (X, Y, θ) < U tot (X, Y , θ).

The works in [START_REF] Craciun | Joint detection and tracking of moving objects using spatio-temporal marked point processes[END_REF][START_REF] Yu | Multiple-target tracking by spatiotemporal Monte Carlo Markov chain data association[END_REF] set a number of these constraints, then solve for the parameters with linear programming. This can lead to over-constrained problems because of imperfect ground truth or setting too many constraints.

We instead formulate these constraints as a loss to minimize over all examples, allowing to consider more samples without any over-constraint issue:

L θ (X, Y ) = U tot (X, Y, θ) -U tot (X, Y , θ) (12)
Algorithm 1 shows the learning procedure for the energy model from equation 9; gradient descent is performed on parameters θ w and θ b , with Q a perturbation kernel introduced in section 5.3.

Algorithm 1 θ learning procedure

θ w ← [1, . . . , 1] T , θ b ← 0 while not converged do Y ∼ Q(Y → •) ∆θ ← ∆ θ (U tot (X, Y, θ) -U tot (X, Y , θ))
Update θ based on ∆θ using Adam optimizer end while

Inferring configurations: point process simulation

Once the total energy U tot is fully defined by θ, V and M , we look for the configuration that minimizes this energy:

Y = argmin Y ∈S×M U tot (X, Y ) (13)
We simulate the point process through a Reversible Jump Monte Carlo Markov Chain (RJMCMC) [START_REF] Peter | Reversible jump Markov chain Monte Carlo computation and Bayesian model determination[END_REF]. This method extends the Metropolis Hastings algorithm allowing to explore a state space of varying dimension. The convergence is ensured as long as a uniform birth-death kernel in S × M is implemented. To speed up convergence [START_REF] Peter | Reversible jump Markov chain Monte Carlo computation and Bayesian model determination[END_REF] uses translation/rotation/scaling kernels. Moreover we add birth-death and translation/rotation/scaling kernels that propose points based on a non-uniform density [START_REF] Lacoste | Point processes for unsupervised line network extraction in remote sensing[END_REF], derived from the potential U p (X, •) + U m (X, •). These densities are easy to sample from as they derive from pre-computed maps V and T . The modified configuration Y is obtained by simulation of the MPP of density proportional to exp(-U tot (X, •)/T ). We use simulated annealing for which the temperature T decreases geometrically (see Algorithm 2). Here Q is the perturbation kernel produced by the random choice of one of the above mentioned kernels.

EXPERIMENTAL RESULTS

Data: DOTA gsd50

Our goal is the detection of small objects in images from satellites such as Pléiades3 or CO3D 4 , which have a typical spatial resolution (or ground sampling distance) of 50cm/px. We use the DOTA [START_REF] Xia | DOTA: A large-scale dataset for object detection in aerial images[END_REF] dataset, containing images and ground truth from various sources and resolutions. We set the spatial Algorithm 2 MPP inference procedure

V ← Position-CNN(X) M ← Marks-CNN(X) Y ← {}, T ← T 0 while not converged do Y ∼ Q(Y → •) r ← min 1, Q(Y →Y ) Q(Y →Y ) e U tot ( V , M ,Y, θ)-U tot ( V , M ,Y , θ) T
With probability r: Y ← Y T ← αT end while resolution to 50cm/px by subsampling the images with higher resolutions. We only keep the small-vehicle class as we are only interested in ground vehicles detection for this application.

MPP based on contrast mesures

We also build MPP models using some contrast measures found in literature [START_REF] Lacoste | Point processes for unsupervised line network extraction in remote sensing[END_REF][START_REF] Maria S Kulikova | Extraction of arbitrarily-shaped objects using stochastic multiple birth-and-death dynamics and active contours[END_REF]. The framework remains similar but we replace U p and U m with a single contrast energy U c . In Table 2 we denote µ y σ 2 y the empirical mean and variance of pixels inside shape y, µ s σ 2 s mean and variance on shape contour s, n y normals of contour s, and ∇X the image gradient.

First, to assert the viability of such measures for these data, we compute U c for ground truth rectangles and for randomly scattered rectangles. Suited measures should allow to classify easily between ground truth points and random points. We plot precision-recall curves for several of those contrast measures (see Table 2) on one sample image of DOTA and in synthetic high-contrast data (see Fig 3). We compare with the proposed detection energy given by U p + U m and show that it outperforms contrast based measures. This is to be expected as those measures were designed for usecases where objects of interest are well contrasted with respect to their background. 2)

Measure

measures [START_REF] Lacoste | Point processes for unsupervised line network extraction in remote sensing[END_REF][START_REF] Maria S Kulikova | Extraction of arbitrarily-shaped objects using stochastic multiple birth-and-death dynamics and active contours[END_REF]. We also compare to a full deep learning method BBA-Vectors [START_REF] Yi | Oriented object detection in aerial images with box boundary-aware vectors[END_REF]. Detection metrics are shown in table 3. We perform detection using the weighted-sum f θ with manual weights (see 4.1) and denote it MPP+CNN . The model using the logistic f θ and learned weights (see 4.2) is denoted MPP+CNN † . For BBA-Vectors [START_REF] Yi | Oriented object detection in aerial images with box boundary-aware vectors[END_REF] detection threshold is set at argmax s∈[0,1] F1(s) 5 .

Our method performs better than contrast based MPP that miss closely packed cars or detect high contrast elements as positives; the over-detection of irrelevant objects leads to low precision and F1 scores. Those methods would require further adaptations to function properly on the current dataset.

The proposed method matches state of the art fully deep learning approach [START_REF] Yi | Oriented object detection in aerial images with box boundary-aware vectors[END_REF] in terms of metrics, while having more visually coherent inferred configurations: the priors included in the model regularize inferred configurations as shown in Fig. 4.

Method

F1

Precision Recall MPP+T-test [START_REF] Lacoste | Point processes for unsupervised line network extraction in remote sensing[END_REF] 0.03 0.01 0.59 MPP+Gradient [START_REF] Maria S Kulikova | Extraction of arbitrarily-shaped objects using stochastic multiple birth-and-death dynamics and active contours[END_REF] 3. Detection metrics for several models (IOU threshold: 0.25).

CONCLUSION AND PERSPECTIVES

In this paper we solve the shortcomings of contrast mesures in visually complex images by replacing those with CNN learned measures, while maintaining the MPP formulation thus allowing regularization over the output using priors. Additionally we show how energy weights can be learned over the training data in a more robust manner than [START_REF] Craciun | Joint detection and tracking of moving objects using spatio-temporal marked point processes[END_REF]. Finally, as the use of priors in our model incrementally improves detection results compared to state of the art, we aim at applying our method to image sequences where object dynamics require stronger priors.

Fig. 2 .

 2 Fig. 2. Closeup image with centers overlay (a), centers heatmap (b), vector field (c) and divergence (d) (red> 0, blue< 0)

Fig. 4 Fig. 3 .

 43 Fig.4shows results on image samples for our method and various others: we compare the proposed CNN and MPP combination with models based on contrast or image gradient

Fig. 4 .

 4 Fig. 4. Inference results for various models. For example 1: false positives, 2-3: misaligned objects

Table 2 .

 2 Various contrast measures from literature and Average Precision (AP) computed on the DOTA image (AP D ) and the synthetic image (AP s ) from Fig.3

		U c equation	AP D AP s
	Ours	U p (X, y) + U m (X, y)	0.97	0.96
	T-test[6]	|µy-µs| σ 2 y ns ny + s σ 2			0.40	0.99
	Gradient[14]	n y (t) •	√	∇X(s(t)) |∇X(s(t))| 2 +	dt 0.48	0.99

6.3. MPP with CNN based energies

as described in[START_REF] Ronneberger | U-Net: Convolutional Networks for Biomedical Image Segmentation[END_REF], but with a limited depth (3 pooling operations instead of 4) and 32 channels in the first convolutional layer.

for better readability we shorten Up(X, y) as Up, as for Um, . . . , Ua

Constellation Pléiades [pleiades.cnes.fr]

Constellation Optique 3D [intelligence-airbusds.com]

F1(s): F-score for a given detection threshold s


Nicolas Jouault

Perrine Vallat

Florent Dalmas

Sylvère Said

Jacques Jestin
email: jacques.jestin@cea.fr

François Boué

Well dispersed fractal aggregates as filler in polymersilica nanocomposites: long range effects in rheology

We are presenting a new method of processing polystyrene-silica nanocomposites, which results in a very well-defined dispersion of small primary aggregates (assembly of 15 nanoparticles of 10 nm diameter) in the matrix. The process is based on the use of a high boiling point solvent, in which the nanoparticles are well dispersed, and a controlled evaporation procedure. The filler's fine network structure is determined over a wide range of sizes, using a combination of Small Angle Neutron Scattering (SANS) and Transmission Electronic Microscopy (TEM) experiments. The mechanical response of the nanocomposite material has been investigated both for small (ARES oscillatory shear and Dynamical Mechanical Analysis) and large deformations (uniaxial traction), as a function of the concentration of the particles in the matrix. Our findings show that with a simple tuning parameter, the silica filler volume fraction, we can investigate in the same way the structure-property correlations related to the two main reinforcement effects: the filler network contribution, and a filler-polymer matrix effect. Above a silica volume fraction threshold, we were able to highlight a divergence of the reinforcement factor, which is clearly correlated to the formation of a connected network built up from the finite-size primary aggregates, and is thus a direct illustration of the filler network contribution. For a silica volume fraction lower than this percolation threshold, we obtain a new additional elastic contribution of the material, of longer terminal time than the matrix. This cannot be attributed to the filler network effect, as the filler is well dispersed, each element separated from the next by a mean distance of 60 nm. This new result, which implies the filler-matrix contribution of the reinforcement, must include interfacial contributions. Nevertheless, it cannot be described solely with the concept of glassy layer, i.e. only as a dynamic effect, because its typical length scale extension should be much shorter, of the order of 2nm. This implies a need to reconsider the polymer-filler interaction potential, and to take into account a possible additional polymer conformational contribution due to the existence of indirect long range bridging of the filler by the chains.

I Introduction

challenge, both from the industrial and fundamental points of view. According to the characteristics of the filler (nature, shape, matrix structure, concentration) and of the elastomer matrix (interaction potential with the filler), many systems, with many associated mechanical properties can be found 1 .

More recently, a specific interest has developed in the size effect, due to a better availability of much smaller particles. The first, simple idea was that the enhancement of the composite properties could be induced by increasing the specific contact area between the filler and the matrix; i.e., by reducing the size of the particle filler. This reduction could be made by downscaling to the polymer network mesh size, allowing further interaction 2,3 . The second idea was that using nanosized particles would produce a convenient extent of clustering. Controlled clustering could make it possible to control the hierarchical organization of the matrix's filler, resulting in specific filler-matrix interactions opening the way to new mechanical properties. These ideas, together with advances in characterization methods, lead to the creation of elaborate advanced composites with nanofillers, from classical carbon black 4 , to silica 5,6,7,8,9 and up to carbon nanotubes 10 . Most studies used rheological approaches of various mechanical solicitations, and showed various kinds of rheological transition related to reinforcement.

In this context, two main contributions to reinforcement stand out, borrowing both from earlier research and from more recent theory. The first contribution is linked to the filler network structure: beyond the classical hydrodynamic description of Einstein 11 , Guth and Smallwood 12 , Batchelor 13 , the idea is that in a further range of particle concentration, the particles can form a connected network (as in the percolation model), whatever their shape: simple spheres, more anisotropic rods, platelets 9,10 or complex shapes like fractals 14 . The second contribution is attributed to the interface between the matrix polymer chains and the particle filler. Beyond earlier ideas of adsorption-desorption and occluded rubber confinement, the idea that the chain dynamics could be modified in the area close to the filler object was reformulated in terms of a kind of "glassy layer", through earlier 15,16,17,18 and more recent 19 works, including NMR evidence on mobility and signature on the temperature dependence of mechanical response.

The experimental distinction between these two classes of contributions to the reinforcement mechanisms, filler network and interface, is difficult because they are often correlated:

-when starting from already complex fillers (such as carbon black or fumed silica) the structure of the filler affects the build-up, the geometry, the connectivity of the filler network, and the related interfacial interaction. The interaction potential between the chain and the filler is also associated to the filler's form factor. As discussed by Vilgis 20 , the problem is multi-length and multi-time scale so that many characteristics lengths must be identified, from the nanometer range to the macroscopic one.

-when starting from individual nanoparticles, once again, an (often spontaneous) build-up of clusters brings the problem back to the former situation. However, particularly in industry, dispersion control implies a use of surfactants, grafted short linear molecules, coupling agents, which greatly and simultaneously modify -and thus correlate -the filler structure and the local interface. Trying to reduce this correlation, this same idea of controlling the filler's local dispersion, in particular the nanofillers, has been used in academic research, through more or less sophisticated chemistries like in-situ polymerization after dispersion of nanoparticles 21 , or grafting polymerization on individual nanoparticles 22,23 . Other methods implied simple control of electrostatics repulsion 24 , or of magnetic attraction 25 , by mixing the particles with polymer beads (latex) -instead of polymer chains like in the great majority of tire processing (at least up to very recently).

In the present paper, we are presenting another strategy, which turns out to be very simple, based on a new method of nanocomposite processing, using a polystyrene matrix filled with silica nanoparticles. First, we will present the method, based on the use of a particular solvent, with a high boiling point, above the glass transition temperature T g of the polymer, and a controlled-evaporation procedure which appears to limit the attractive Van der Waals interactions effect, and thus makes it possible to obtain a particularly well defined dispersion of the silica particles in the matrix. Secondly, we will describe the fine characterization of the filler dispersion in the matrix on a very wide length range, through a combination of Small-Angle Neutron Scattering (SANS) and Transmission Electronic Microscopy (TEM). Thirdly, we will report the mechanical response of the material at low (ARES oscillatory shear plate-plate rheometer, and Dynamical Mechanical Analysis, DMA) and large deformation (uniaxial stretching). Finally, we will discuss the link between the filler structure inside the matrix at the local scale, with the mechanical behavior of the nanocomposites at the macroscopic scale.

II Material and methods

Sample preparation

The colloidal solution of silica nanoparticles is a gift from NISSAN (DMAC-ST). In this solution, the native particles, of an average radius of 5.2 nm, are electrostatically dispersed in the dimethylacetamide (DMAc), a polar solvent which is also a good solvent for the polymer used as a matrix, polystyrene (PS, Aldrich, M w =280 000 g/mol, I p =2, used as received). The glass transition T g of pure PS is around 100°C. A concentrated solution of PS in DMAc (10 % v/v) is mixed with a solution of silica in DMAc at various fractions, ranging from 0 to 30 % v/v. The mixtures are stirred (using a magnetic rod) for two hours. They are then poured into Teflon moulds (5 cm  5 cm 2.5 cm), and let cast in an oven at constant temperature T cast = 130 °C. This yields dry films of dimension of 5 cm  5 cm  0.1 cm (i.e. a volume of 2.5 cm 3 ). The specific conditions of film formulations, solvent titration, etc… are discussed in the Results section of the article. We cut some disks out of the films (diameter 1cm, thickness 1mm) for the plate-plate oscillatory shear cell, and 2 cm  0.5 cm rectangular 1mm thick films, for DMA as well as for uniaxial stretching.

SANS experiments

Measurements were performed at the Laboratoire Léon Brillouin (LLB) on the SANS spectrometer called PACE. Two configurations were used: one with wavelength 17 Å, sample-to-detector distance of 4.70 m, and a collimation distance of 5.00 m, and one with wavelength 6 Å, sample-to-detector distance of 3m, and a collimation distance of 2.50 m corresponding to a total q range of 2.10 -3 Å -1 to 0.1 Å -1 .

Data processing was performed with a homemade program following standard procedures 26 with H 2 O as calibration standard. Small deviations, found in the spectra at the overlap of two configurations, are due to different resolution conditions and (slight) remaining contributions of inelastic, incoherent, and multiple scattering. To get the cross-section per volume in absolute units (cm -1 ), the incoherent scattering cross section of H 2 O was used as a calibration. It was estimated from a measurement of the attenuator strength, and of the direct beam with the same attenuator. The incoherent scattering background, due to protons of the matrix chains, was subtracted using a blank sample with zero silica fractions.

Transmission Electronic Microscopy

In order to complete on a larger scale the SANS analysis of the nanocomposite structure, conventional TEM observations were also performed on composite materials. The samples were cut at room temperature by ultramicrotomy using a Leica MZ6 Ultracup UCT microtome with a diamond knife. The cutting speed was set to 0.2mm.s -1 . The thin sections of about 40 nm thickness were floated on deionized water and collected on a 400-mesh copper grid. Transmission electron microscopy was performed on a Philips Tecnai F20 ST microscope (field-emission gun operated at 3.8 kV extraction voltage) operating at 200 kV. Precise scans of various regions of the sample were systematically done, starting at a small magnification which was then gradually increased. The slabs observed were stable under the electron beam. The sample aspect was the same in every area of every piece. Apart from a few cutting scratches, moderate buckling, very rare bubbles, and impurities, the pictures given below are completely representative of the single aspect of the sample, which thus appears homogeneous.

Oscillatory Shear Small Deformation Tests

Shear tests, corresponding to low deformation levels (0.5%), were carried out in the dynamic mode in strain-controlled conditions with a plate-plate cell of an ARES spectrometer (Rheometrics-TA) equipped with an air-pulsed oven. This thermal environment ensures a temperature control within 0.1 °C. The samples were placed between the two plate (diameter 1mm) fixtures at high temperature (160 °C), far above the glass transition, put under slight normal stress (around 0.5 N), and temperature was decreased progressively, while gently reducing the gap to maintain a constant low normal stress under thermal retraction. The zero gap is set by contact, the error on sample thicknesses is thus minimal; estimated at +0.010mm with respect to the indicated value. Artifacts slipping are noticeably reduced by this procedure, as checked by its reproducibility, and also by a sweeping in amplitude, at constant pulsation, which also made it possible to determine the limit of the range of linear deformation. To stay below this limit, the shear amplitude was fixed to 0.5%. Samples are stabilized at the temperature for 30 minutes before starting measurements. The reproducibility was first tested on pure PS samples with an average of 5 repetition measurements, permitting an estimatation of variations of the order of 10%. In dynamic mode, the frequency range was from 0.5 to 100 rad/s for different temperatures (from 160 to 120 °C), and time-temperature superposition was applied. The obtained multiplicative factor can be adjusted to WLF law 27 as follows

ref 2 ref 1 T T T C ) T T .( C ) a log(     (1)
where a T is the multiplicative factor, T ref is the reference temperature of the master curve (in our case 143 °C), T the temperature of the measurement, C 1 and C 2 the WLF parameters. We found C 1 =6.72 and C 2 =98.03 °C, which is commonly obtained for PS samples [START_REF] Ferry | Viscoelastic properties of polymers[END_REF] . For this plate and plate technique, we only focused on the samples with low silica concentration (from 0 to 5% v/v), because for higher silica fractions (> 5% v/v) the measurement reproducibility was not insured (risk of slippage).

Dynamic Mechanical Analysis

Rectangular pieces of film 2cm long  0.5 cm wide were sanded down to a constant thickness of 0.8 mm. Dynamic mechanical analysis (DMA) measurements were performed on a TA DMA Q800 device in oscillatory tension mode, at fixed deformation rate (0.1%) and fixed frequency (5 Hz), at temperatures ranging from 40 °C to 300 °C with a heating rate of 10 K/min. Analysis of the oscillatory stress response is done by the software provided by TA; a preloading of 0.04 N is applied to avoid buckling. Note that 5 Hz corresponds to a short timespan, 0.2 sec. However, a measurement at 200 °C, applying a time-temperature superposition factor of 296, corresponds when using Eq. 1. to 60 sec at the reference temperature of 143 °C (equivalent to 2  / 60 = 0.1 rad/s). Two identical experiments (for pure PS samples) were performed to check the reproducibility of the measurements, and correct obtained values. The storage (E') and the loss (E'') moduli of the complex Young modulus (E*), and the loss factor tan δ = (E''/E') at 5 Hz, are extracted from the measurements. The temperature at which tan δ shows a maximum is noted T α .

6 Stress-strain Isotherms Samples for stress-strain isotherms cut from films are carefully sanded down to a constant thickness ~ 1mm within a few microns. A grid of lines is drawn on the sample with a felt pen. Samples are stretched up to a pre-defined deformation value in a controlled constant-rate deformation (d/dt = 0.005 sec -1 , λ being the elongation ratio L(t)/L(t=0)), at temperature T = 120 °C i.e. ~ 20 °C above the glass transition temperature of the polymer matrix (T g = 98 °C for pure PS). This corresponds to a typical time (d/dt) -1 = 200 sec, which after applying the time-temperature superposition factor (0.0087 from 120 °C to 143 °C), corresponds at 143 °C to 1.8 sec. The tensile force F(), where = L/L 0 defines the elongation ratio with respect to the initial length L 0 , measured with a HBM Q11 force transducer, and converted to (real) stress inside the material  by dividing the force by the assumed cross-section during elongation. Indeed the deformation of the film is assumed to be homogeneous, and at constant volume; thus the crosssection decreases as (1/√.) 2 . Apart from stress-strain curves, we analyze our data in terms of the nanocomposite reinforcement factor R () = ()/ matrix (), where  matrix () is the stress-strain curve for the polymer with a zero silica fraction.

7 Thermal Characterizations Differential Scanning Calorimetry (DSC) measurements were performed on TA DSC Q100 under helium flow to characterize the T g of the nanocomposites. 5-10 mg of samples were put into a hermetic aluminium pan. An empty cell was used as reference. Samples were heated from 25 °C to 140 °C at a heating rate of 10 °C/min and kept at this temperature for 15 min to erase the thermal history of the materials. Then they were cooled to 25 °C at the same speed. This cycle was repeated once; reported T g was from the second heating, determined as the mid point of the heat flow step. Thermal Gravimetry Analysis (TGA) measurements were performed on TA TGA Q50 under nitrogen flow to evaluate the quantity of residual solvent and check silica volume fraction. A mass of 15 mg of the sample was heated from 25 °C to 600 °C at a heating rate of 10 °C/min and weight loss was measured. The weight fraction of residual solvent was found to be between 100 °C and 200 °C.

III Results

Composite films formulation and processing

As we said in our Introduction, we feel it is very important in nanocomposite science to obtain welldefined model systems, in which one knows how the filler is dispersed inside the matrix, to discriminate the polymer chain interfacial effects (including the dynamic ones) from effects of the network filler structure in the composites' mechanical properties. Nanocomposites of PS filled with silica nanoparticles have been extensively studied in the past decade [START_REF] Bansal | [END_REF]30 . The results show the difficulty of obtaining a good dispersion of the silica particles at the nanometer scale, which formed in most cases large compact aggregates (of the order of a hundred nanometers or larger) in the PS matrix. One cause of this dispersion problem may be a difficulty in controlling, during the filmification process, the evaporation speed of the solvent used (toluene, THF, methylethylketone (MEK) [START_REF] Bansal | [END_REF] …), and the temperature with respect to boiling point and T g . Indeed various dispersions can be obtained with the same solvent [START_REF] Bansal | [END_REF]30 . Anyway, in our case, Dimethylacetamide (DMAc), the solvent in which silica particles are electrostatically stabilized (and which is also a good solvent for the PS), has a high boiling point, 167 °C. The casting temperature could be fixed at 130 °C, which corresponds to a relatively controlled evaporation regime, while remaining above the T g of the bulk polymer. The drying time has been determined, using the dry film reference (without silica particles), as the one for which DSC measurements showed a glass transition temperature stabilizing at T g = 98 °C , i.e. very close to the one of native polymer powder (T g = 100 °C). The residual quantity of solvent inside the composite film is found to plateau around 0.4 %wt, as determined by thermogravimetry.

Local structure by SANS

The structure of the silica-PS composite films has been explored by SANS. The evolution of the SANS signal as a function of the silica volume fraction  SiO2 in the films, respectively 6.6, 10.5, 15.7, 19.8 and 29.4% v/v, is presented Fig. 1. After the standard corrections and normalizations, the intensities (1/V. dΣ/dΩ, in cm -1 ) are normalized by the volume fraction and corrected from the incoherent scattering of the non-filled PS matrix as follows:

  matrix I SiO film I Q I     ) 1 ( ) ( 2  (2)
Let us now discuss the aspect of the different spectra, which evidence different Q ranges:

-at high Q, all curves superimpose perfectly after dividing by the silica fraction, indicating a good normalization of the concentration and the thickness of the samples. The scattering intensity decreases like Q -4 , which is characteristic of the scattering of a sharp well-defined interface between the native silica particles (R 0 = 5 nm) and the polymer matrix. At the intermediate Q range, we note an oscillation, highlighted in a Q 3 .I versus Q representation (see insert in Fig. 1) where it gives a maximum. Since the position of this maximum does not vary with the concentration, we can assume that it corresponds to a privileged distance inside every object, more precisely between initial spheres in contact inside the objects (in other words an internal structure factor).

[Figure 1]

-at low Q, the remarkable point of the whole set of data in Fig. 1 is the absence of any upturn in intensity, for Q tending to 0. This indicates that if larger agglomerates existed, it would require lower Qs to detect them; in practice, TEM at low magnification (see below) shows, in the direct space, a complete absence of such agglomerates. For small concentrations (6.6 and 10.5% v/v), at low Q, the curves exhibit a plateau which is the signature of finite-size objects in the probed length scale. These objects, which we will now call "primary aggregates", are the results of the aggregation of a finite number of the native silica beads. The form factor of the primary aggregate can be approached from the most diluted scattering curve (6.6 %v/v, neglecting inter-aggregate correlations). We propose to model it by the form factor of fractal objects with the following equation 31 :

      0 ) , ( 3 0 3 ) , ( ) , ( . ) ( dR R L R dR R R L R Q beads native P D Q agg N Q agg P f   (3)
where agg N is the number of native beads inside the primary aggregate, D f is the fractal dimension of the primary aggregates,

) R , Q ( P beads native
is the form factor of a sphere, and L(R, ) is the log-normal distribution of the radius, with a variance ). As we are dealing with the most diluted film, neglecting inter-aggregates correlations as explained above, we assume the structure factor between aggregates er agg S int to be equal to 1. The total scattering intensity can thus be expressed as:

) Q ( S ) Q ( P ) Q ( I ra int agg agg 2      (4) where 2 
 is the contrast difference between the silica and the PS matrix. The intra-aggregate structure factor, due to the repulsive interaction between initial silica beads in contact, can be expressed on the basis of the Percus -Yevick relation 32 , which depends only on the radius and the concentration of the particles. We then use this structure factor ra int agg S to divide the measured intensity of the film containing 6.6 % v/v of silica particles; Fig. 2 shows the result and its fit to Eq.3 for P agg (Q). In the insert, one can see the superposition of the intensity and the calculated structure factor before division by S aggintra .

[Figure 2]

In the calculation of Eq.3, we use the radius and the log-normal polydispersity distribution () of the native particles, determined from a SANS measurement of a diluted solution of particles (0.1% v/v) and fixed here to R 0 =5nm, =0.36. The concentration is also fixed at the nominal value, so only the aggregation number and the fractal dimension are fitted. The result of the analysis, N agg =15, D f =2.5 (corresponding to objects of a mean radius around 30 nm), agrees well with the experimental curve. The difference between the calculation and the experimental data around 4 10 -2 Å -1 comes mainly from the fact that we have not taken into account the polydispersity in the expression of the PY structure factor.

The fractal dimension of the primary aggregates (2.5) extracted from the analysis should be also a little bit over estimated by the presence, but not clearly visible and accessible in the experimental Q range, of the inter-aggregates structure factor.

For larger silica fractions in the film, we see at low Q a second remarkable feature: a maximum appears. At first sight, the presence of this "peak" confirms the good dispersion of the particles in the polymer matrix: there is a privileged distance in the system. Let us now discuss its origin, using the evolution of the peak position as a function of the concentration (Fig. 3).

Two situations can be considered. The first situation is described by the well-known hard sphere model: the primary aggregates are distributed at random, on the condition that they cannot overlap; when increasing their concentration, they come closer to each other without being connected. This would lead to a liquid-like order showing repulsive interaction between the mass centers of the primary aggregates, and the variation of the peak position as the function of the concentration should be 33 Q* (1/3) . The second situation is that the fractal primary aggregates connect with each other, which can be analogous to a percolation transition. In this case, Q* should correspond to the characteristic size of the network mesh, and display a Q* 0.88 scaling 34 . Plotting Q* versus  (Fig. 3), we obtain a slope of 0.87 +/-0.05, suggesting that we are in this second situation for Φ SiO2 > 10% (below this value no maximum is seen in our Q range). Thus SANS measurements give us a global but relevant view of both the form factor of the primary aggregates, and the organization of the filler network in the polymer matrix.

However, to correlate the rheological macroscopic properties with the local filler structure, we need to make a final check, and verify that there is no additional organization of the objects at a larger scale than the one probed with SANS. At the same time, it will also be interesting to confirm the local structure deduced from scattering, i.e. from correlations in reciprocal space, with a picture in real space.

This is what we will show in the next section, using Electronic Transmission Microscopy.

[Figure 3] 3 Structures on a Larger Scale by TEM Two concentrations of silica particles, 6.6 and 15.7 % v/v, have been investigated with TEM. Fig. 4 presents the results obtained at two magnifications (see bars for 200 nm and 1 μm in the top and bottom

Figures respectively).

The most important result is that the homogeneity of the two samples is extremely good from the 50 nm scale to the macroscopic scale.

At Φ SiO2 = 6.6%, the size distribution of the objects appears centered on a mean value which agrees well with the mean radius of primary aggregates deduced from the SANS analysis, i.e. 30 nm. The width of the distribution is enough to let objects of 200-300 nm at most appear. The origin of these objects is briefly discussed below. However, a further analysis of TEM would be out of this paper's scope, and we will keep using SANS results as a safe ensemble average of correlations. At Φ SiO2 = 15%, the real space picture confirms the model deduced from SANS. The structure keeps an "open" shape, id est does not show larger compact lumps: it shows a connected structure. We observe a characteristic net mesh size of the same order as D = 2 Π/ Q* = 2Π/ 5.3 10 -3 Ǻ -1 ~ 120 nm, corresponding to the abscissa Q* of the repulsion peak in the SANS spectrum for 15%. Thus the important result here is that on a larger scale, up to macroscopic scale, the silica spatial distribution is homogeneous, with no connectivity signature at 5% and with connectivity at 15%. Such changes from one picture to the next, at larger silica fractions, raise the question of the evolution of the structure with silica volume fractions. Is it related to an aggregation process specific to each concentration? Or to the progressive aggregation or percolation, according to the concentration, of initial building bricks which would be the primary aggregates, until one reaches a connected network?

We will not attempt to answer this question here.

For Φ SiO2 = 6.6%, to evaluate the mean rim-to-rim distance between two primary clusters, we have to return to SANS data. Due to the absence of a maximum in the available range 2 to 3. 10 -3 Ǻ -1 in the SANS curve for the diluted regime, this cannot be calculated it directly. We can make two different estimates. On the one hand, in the hypothesis of a progressive connection of primary aggregates, D is larger than the minimal distance D min , at a larger concentration, extracted when a maximum is detectable: in figure 2, the peak appears for Φ SiO2 = 15%, and gives D min ~ 120 nm, as shown above. On the other hand, ignoring this hypothesis, we can, on a general basis, use the absence of a visible maximum at Q > 3. 10 -3 Ǻ -1 . This suggests, in a less accurate way, that this distance is larger than D = 2/ 3. 10 -3 Ǻ -1 ~ 200 nm. Knowing the mean radius of the primary aggregates, 30nm, we can deduce that the minimal distance between the rims of two primary aggregates, for the diluted regime (Φ SiO2 = 6.6%), can be estimated around 60 nm in the first case, and 100 nm in the second case. This order of magnitude is also confirmed by a careful examination of a large number of real space picture (an example being figure 4(a)). Since the thickness of the slab is of the order of 50 nm here, a value of 60 nm can safely be drawn for the 3 d space.

[Figure 4] 4 Low deformation mechanical measurements Low deformation measurements are interesting because the microstructure of the films shows much less alteration The first set was obtained using the shear plate-plate rheometer ARES (amplitude 0.5%).

In Fig. 5(a) the variation of the elastic modulus G' is presented for a fraction of silica increasing from 0 to 5 % v/v, as a function of the product of the pulsation  (rad/s), by the factor a T of time-temperature superposition to a reference temperature 143 °C. For further discussion, we note that the lowest value of .a T , 10 -2 rad/s, corresponds to a maximum accessible time 2  / 10 -2 rad/s = 600 sec at 143 °C. For pure PS, the curve has the usual shape in a log-log plot. At a high frequency, it shows a slope of -1/2 associated to the Rouse modes of the chains. At an intermediate pulsation 0.1 rad/s, a drop is observed which is characteristic of a terminal time ("creep zone", around 2 / 60 sec, at T = 143 °C ). At a lower pulsation, a decrease with G'() ~  2 can be, as usual, associated to a liquid state of the polymer. The terminal time measured is in agreement with the molecular weight used here for the matrix. The height of the plateau is of order of 1.10 5 Pa, i.e. close to the value G 0 N, well known for the entangled rubbery plateau of PS (2.10 5 Pa). The behavior of G'' () is given in Fig. 5 (b), and is also characteristic of entangled polystyrene of this molecular weight.

[Figure 5]

When silica is introduced, both G' () and G''() show clearly that, at high pulsation .a T >100 rad/s, whatever the silica volume fraction, the behavior of the nanocomposite is very close to the one of the PS matrix: the Rouse modes of the chains are still observed. At intermediate and low frequencies, differences appear. At .a T >10 rad/s, this is limited to a progressive increase of G' with silica fractions, analogous to elastic reinforcement as expected. But in the lowest pulsation regime, .a T < 1 rad/s a much more differentiated behavior is observed: adding silica greatly increases the terminal times. For the lowest silica volume fraction, it is only slightly increased compared to the one for pure PS; the creep zone stays in the same pulsation range, but a more accurate value can be extracted from the  abscissa of the crossing of G' and G''. As soon as the silica fraction reaches 2% v/v, the terminal relaxation time become so long that the creep zone is no longer visible in the experimental window. The existence of a long elasticity time is accompanied, predictably, by a low dissipation, so that the G'' values become inferior to the G' values in the low frequency range. In such a low  range, the slopes n, of the G' and G'' curves in log-log plot become parallel -in a rather narrow window in practice -and the ratio G'/G'', written as tan  () is equal to tan (n/2), as observed in a gelling process 8 . For higher silica concentrations (from 3 to 5%), this additional (with respect to pure matrix) elastic behavior is even more marked in the same low  range. It seems at >3% that the curve decay for  tending to 0 is slower. The trend of a plateau at lower  is more and more apparent at  = 4% and 5%. Comparison of Fig. 5 (a) and Fig. 5 (b) enable us to see the parallel evolution of G' and G''. We can note, on the curve corresponding to 4%, the same type of shoulder at the same abscissa as for pure PS, suggesting that we see a contribution from chain reptation before (i.e. at higher  than) the final slow decay. The height at low of what we could call the "secondary plateau" is close to 1.10 5 Pa. Values at intermediate pulsations have also increased, becoming of the order of 5.10 5 up to 10 6 Pa. This makes measurements at higher  SiO2 more delicate, due in particular to risks of slippage.

For high concentrations, we turned to DMA experiments performed at fixed frequency (5 Hz), with an amplitude deformation ratio of 0.1%. As detailed in Section II, using time-temperature superposition, measurements between 143 °C and 200°C correspond to a time range at 143°C between 0.2 sec and 60 sec. In Fig. 6, we show the evolution of the storage modulus E' and loss modulus E" for different volume fraction of silica (from 0% v/v to 20% v/v), as a function of temperature. For pure PS (without silica), at low temperature, i.e. for T < T α , the curve is typical of amorphous polymer with a high elastic modulus plateau of about 2.5 GPa. It corresponds to the glassy regime. The value obtained for T α is the usual one retained for the glass temperature of PS.

[Figure 6]

When T increases above T α , a steep decrease of E' is observed, followed by a shoulder corresponding to the entanglement rubbery plateau, very narrow before to be cut at slightly larger temperature by a strong decrease corresponding to the terminal relaxation. Measurements were stopped above the temperature at which the modulus becomes too weak, because the samples were flowing over during the measurement time.

In the presence of silica, different behaviors appear successively as the volume fraction increases.

First, the glassy plateau keeps the same height over the whole range of Φ SiO2 . Second, at high temperatures (T > 130 °C), three successive types of behavior can be distinguished:

-for the lowest value of Φ SiO2 (1% v/v), the behavior remains close to that of pure PS: the creep zone shoulder at higher temperatures keeps the same shape; it shifts slightly when passing from pure PS to 1% silica-by less than 10 °C-, after which the sample flows. Note the height of the inflexion point is of the same magnitude as the modulus obtained on the oscillatory shear plate-plate ARES rheometer (i.e. of the order of 0.1 MPa; we recall that E = 3 G in linear deformation range).

-for higher  SiO2 range, 3% v/v and 5% v/v, the most important fact is that at high T (T ~ 150 °C), the characteristic creep zone of pure PS disappears completely: the curvature is reversed, so that E' decreases slowly and seems to tend toward a plateau (~ 1MPa) at high temperatures (which however it does not reach).

-for volume fractions of 10% v/v, this plateau becomes clearly visible, and much higher (10 7 Pa).

-for 10 and 15 % v/v, definitely, high elastic plateaus are present, at the 5Hz frequency (10 MPa and 100 MPa) -for 20% v/v, the plateau height has jumped to more than 10 8 Pa, only one decade below the glassy plateau height. The slight decrease visible at the highest temperature may be an artifact. We did not measure at larger temperatures than 300 °C because this would pass the limit of matrix's thermal stability. These results will be examined further in the Discussion section.

5 High deformations: Stress-Strain Isotherms [Figure 7] Fig. 7(a) shows the evolution of real stress σ as a function of the elongation ratio λ for  SiO2 = 0, 1, 2, 3, 4, 5, 7.5, 10, 12.5, 15% v/v. For higher concentrations (> 15% v/v), samples did not stretch homogenously at high elongation ratios: the stress was extreme, and the samples escaped it by forming local shear bands in different zones along the sample, leading to a somehow disorganized deformation.

We do not present those curves and only focus on homogenous and well-stretched samples. In these elongation measurements, all shorter processes will not produce an elastic reversible contribution, though they will contribute to the effective viscosity, but this contribution is not dominant. The set of stretching curves can be summarized by the progressive increase with  SiO2 of an initial stress jump at the beginning of the elongation, while the shape at large  is similar, except for the shift due to the initial jump. This initial stress jump can be expressed in terms of an initial slope of stress vs. deformation  = -1 in a low elongation ratio (λ<1.1) regime, giving an effective Young modulus E nanocomposite in this range of deformation. We have to be aware of the limitations of this analysis: the deformation  and the initial velocity gradient d/dt at  close to 1 are not accurately known, so that the sample may initially be subjected to a stronger and faster deformation than the nominal values. This may, in particular, alter the filler network. However, we will see in the Discussion below that these values compare well with the DMA values at equivalent instrumental times.

For very high silica concentration ( SiO2 > 10% v/v), a maximum is observed in the stress-strain curves, more and more pronounced when  SiO2 increases. Most of the time, this peak appears simultaneously with local necking, which is not discussed here. Actually, to compare with the pure matrix, a simplistic way to represent the reinforcement effect in nanocomposites is to plot the ratio of nanocomposite stress over the pure matrix stress versus . In Fig. 7(b) this reinforcement factor R () increases notably with  SiO2 for small  (from 1 for 1% v/v to 100 for 15% v/v). At large , all curves show a rapid decrease toward what seems to be a constant value. This final "value" is a more slowly increasing function of  SiO2 ; no strong divergence with  SiO2 is seen here. Note however that the reinforcement factor remains 4 for 15%, at an elongation ratio of 2.

6 Differential Scanning Calorimetry Fig. 8(a) shows the Differential Scanning Calorimetry thermograms for the pure polymer and the different nanocomposites (from 1% to 20% v/v). All curves exhibit similar behavior: a slow decrease of heat flow followed by a steeper descent, provoking a step and corresponding to the glass transition temperature (T g ). We did not observe change of step height, i.e. of specific heat capacity ΔC p (normalized by weight fraction of polymer) at T g which is constant as a function of silica content. The width ΔT is also constant with the increase of filler concentration. So magnitude and width of the step associated with the glass transition remain constant with the increase of silica fractions. In every case, we observe a single transition, in agreement with the spatial homogeneity of the sample.

[Figure 8] Fig. 8(b) shows the change in such measurement of T g compared to that of pure PS for several nanocomposites (1, 3, 5, 10, 15, 20% v/v).The T g determined by DSC increases with the addition of silica. The positive shift is 6 °C for Φ SiO2 = 20%, i.e. of same order than for T α (maximum of tan δ(ω) )

and for the maximum of E". An opposite shift (negative) has been found for other PS-Silica composites prepared in different conditions [START_REF] Bansal | [END_REF] , but in further investigations of the same group, leading to better dispersion, this negative shift vanished 30 .

IV Discussion

The precise description of the key parameter which governs the reinforcement mechanisms of nanocomposites depends on a thorough knowledge of the filler dispersion on the local scale. This is not always accessible, due to difficulties of sample processing (reproducibility, complexity) and suitability of the samples to characterization methods. Here, a reliable and reproducible processing technique allowed us to obtain well-defined nanocomposites, on which we were able to apply a combination of SANS and electronic microscopy to picture the nano-morphology of the silica particle inside the PS matrix. The dispersion can be summarized as follows: the system presents a silica volume fraction threshold (located around 7% v/v) between two main domains of the silica organization in the matrix.

For silica volume fractions lower than this critical value, the silica particles gather in finite-sized fractal clusters called "primary aggregates", of a small mean radius  30 nm, which are homogenously dispersed on a larger scale inside the matrix; they have no direct connectivity since they are separated from each other by a mean typical rim-to-rim distance of 60 nm, which represents the minimal estimated value. For silica volume fractions larger than the critical silica volume fraction, similar primary aggregates still exist and, due to the increase of their number per volume unit, they percolate into a directly connected and continuous network. The next step is now possible: to identify the mechanical responses of the composite with the two different structural organizations, for various characteristic time ranges (at 143°C) and rates of deformation depending on the technique: shorter times (0.5-60 s) for DMA (low deformation) and (2 s) for stretching (large deformation), and longer times (200 s) for small deformation for ARES.

For the shorter times, the results are summarized in Fig. 9. The moduli E nanocomposite extracted from the initial slopes of the stretching curves (see Section II) have been represented in the form of the reinforcement ratio E nanocomposite / E matrix , as a function of filler concentration. We see in this plot a first linear part, then a fast divergence around 7%.

[Figure 9]

This behavior is confirmed on the same Fig. 9 with moduli extracted from DMA measurements at T = 1.3 T  , which corresponds to instrumental times at 143 °C equivalent to stretching times. The two curves show parallel behavior. Assuming a fully elastic contribution, the crossover of the reinforcement factor, which corresponds to the critical volume fraction of the connected filler network's formation, can be interpreted in terms of connectivity, more precisely of percolation between the primary aggregates. Several classical equations 11,12,13 have attempted to describe such effects in the literature, but are known to fail to reproduce the divergence. This behavior was formerly observed for the rubbery modulus obtained from DMA of polymers reinforced by fillers of higher aspect ratio (fibers, platelets 9,10 ), and also formulated by Heinrich et al 14 for fractal aggregates. We have attempted to fit data to such a formula 14 , but the fit is very sensitive to the value of the different fractal dimensions, and easily gives unphysical values. If we restrict ourselves to the crossing between the low  SiO2 and high  SiO2 asymptotes of the curve in Figure 8, we obtain an indicative value of * SiO2 ~ 7 to 10%. There is thus a clear direct correlation here between such mechanical behavior in nanocomposites and the local structure of the filler in the polymer matrix: the divergence of the reinforcement factor appears at the critical silica volume fraction at which the isolated primary aggregates are closer to each other and form a connected network.

For the longer times, a second and more surprising mechanical response of the composite appears in the silica volume fraction domain, in which the filler elements are not directly connected to each other, i.e. below the silica volume fraction threshold, for  SiO2 < * SiO2 . Although the rims of primary aggregates are separated by a mean minimal distance of 60 nm, the oscillatory shear experiments (figure 5) and the DMA experiment at high temperature (figure 6) show an additional process corresponding to a longer terminal time, greater than the terminal time of the matrix. This means that after a decay of most of the elasticity at the matrix terminal time, an elastic fraction stays. At  SiO2 ≥ 4%, the complete relaxation of this elastic fraction is beyond our accessible time range. The occurrence of such elasticity between aggregates not directly connected, but separated by very long paths through a polymer matrix of shorter terminal time, which should shortcut any additional elasticity, is, to our knowledge at least, an unexpected and original result.

The origin of this additional elastic contribution can first be discussed along the idea of the existence of a "glassy zone" around the particle filler. This has been proposed from earlier studies 7,15,16,17,18 , up to more recent research 35 from 1 H NMR and rheological measurements. It is suggested that this zone increases the effective volume of the filler around each particle 35 , and also that it shifts the threshold for percolation between the hard regions. The EHM model [START_REF] Eisenberg | Introduction to Ionomers[END_REF] developed for ionomers assumes the existence around the hard zones of intermediate mobility regions of a given thickness, which merge into low mobility regions when they percolate. The EHM model has been extended to nanofillers in polymer 7 . It has also been suggested that the hard coronas act on the steric repulsion ("hard sphere model") between the filler particles, hence on the spatial correlations as observed using SANS characterization 23 . In these systems, the filler is mostly spherical particles, and volume fractions are often higher than 10% 7,35,[START_REF] Eisenberg | Introduction to Ionomers[END_REF] . A comparison with these studies addresses the problem of the value of the thickness ζ of the glassy region around the filler. It should be equal to 30 nm in our case, considering that the lowest distance between the closest rims of two primary aggregates is 60 nm. Several of the earlier studies on 1 H NMR proposed an estimate for ζ of a few nm (for temperatures usually above T g by several tens of °C). Similar values were proposed from the volume fraction dependence of DMA measurements in ionomers [START_REF] Eisenberg | Introduction to Ionomers[END_REF] . More recently, Berriot et al 35 Moreover, we face another discrepancy with the measurements of Berriot et al 19,35 . In their case, the dependence of ζ over T-T g results in the failure of the usual William-Landel-Ferry-Vogel-Fulcher timetemperature superposition. As a consequence, the reinforcement factor R (T,ω) does not obey the same time temperature superposition as the matrix. In our case, conversely, the time-temperature superposition applies correctly for oscillatory shear measurements, with the same coefficients as for the pure PS matrix. If a value of 25 nm is required for ζ to explain our Φ SiO2 = 6.6 % data, this value must be kept at 120 °C for stretching, and above for shear. This is also imposed by the DMA measurements at 5 Hz which display high reinforcements for Φ SiO2 = 6.6 %, up to T -Tg = 70 °C. So far from T g , ζ NMR has in most studies been estimated to a few nm. Keeping much larger values would mean a particularly strong interaction between the filler and the polystyrene. Though it is in principle possible, since the filler and the polymer are different from the ones in ref. 35 and other NMR investigations, the existence of such an interaction is in contradiction with the DSC measurements: even at = 20%, DSC gives a shift of the average T g of 6 °C only. In summary, attributing through-polymer connectivity between our aggregates to glassy zones around the nanofillers implies thicknesses larger by more than a factor 10 to commonly accepted values, and without visible temperature dependence.

Along the same lines, the origin of the additional elastic contribution could be the result of the existence of continuous glassy paths between aggregates. This has been proposed as a theory to explain shifts in T g and other dynamics in ultra-thin films 38 for which some shift in T g appears, for typical film thickness of 50 nm 39 . This analogy with thin film has also been proposed by Kumar et al. [START_REF] Bansal | [END_REF]30 to explain the shift in T g in some nanocomposites (which is abated in [START_REF] Bansal | [END_REF] but not in 30 , as it can depend on the matrix polymer interaction and the processing). Note that if the orders of magnitude of the characteristic distance are the same in our system and in thin films, a marked difference appears in the sign of the shifts in T g which are negative in thin film for cases corresponding to no or weak polymer-surface interaction, while it is slightly positive in our case (figure 8). This observation implies the need to reconsider the polymer-filler interaction in our case which appears to be possibly stronger than expected. Apart from specific surface chemistry considerations, which we are not in a position to study in our system, the origin of a stronger polymer-filler interaction could be attributed to the shape of our primary aggregates; at scale between the size of the individual particles (5 nm) and the size of the aggregate (30nm) the surface is contorted. This can be associated with earlier proposals that the interaction potential between polymer and a fractal filler surface is much enhanced by its roughness 14,20 .

This assumption of stronger polymer-filler interaction opens the way to an additional conformational contribution, via the bridging of the primary aggregates by some matrix chains. Saverstani et al. 40 have proposed a model and some simulations based on the formation of a mixed polymer-filler network by adsorption of the chain on the filler objects. They have showed that an additional elastic contribution appears in the modulus when the mixed network is formed for a typical rim-to-rim distance equal to twice the gyration radius of the polymer chains. For our PS chains (Mw = 280 000 g/mol, I p =2,) the average R g should be ~15nm; thus a distance of 30 nm is expected, still lower than our minimal rim-torim distance of 60 nm, but molecular weight polydispersity could play a role in this phenomenon.

To end this Discussion, we would like to recall that secondary relaxation times were formerly observed in filled systems of well-dispersed particles 23 ; in this case, a model assuming no specific lower-mobility regions, but only spatial correlations at large distances was used 41 . However, the plateau moduli obtained in this case are much lower than in the present study.

V Summary and Conclusion

In summary, we have been able to synthesize, over a wide range of volume fraction, polymer systems filled with very small particles (5 nm in size), aggregated at nanometer size only, and well dispersed at all larger sizes. We were then able to characterize the structure precisely over the full relevant size range, from nanometer to sample size, owing to a combination of Small Angle Neutron Scattering and Transmission Electronic Microscopy. The fillers form small primary aggregates of tens of particles, of elongated, slightly ramified shape, with a radius around 30 nm, distributed homogenously in the matrix and separated by a mean rim-to-rim distance of 60 nm. When the silica volume fraction is increased, a threshold * SiO2 appears, at which the primary aggregates percolate in a directly connected filler network. Two very different nanocomposite structures can thus be tuned with the simple silica volume fraction parameter. The resulting mechanical response of the material was then analyzed through different mechanical tests. Initially, with short time-spans, both stretching and DMA experiment were a simple illustration of the influence of the filler network connectivity on reinforcement properties. A direct correlation between the divergence of the reinforcement factor and the formation of the primary aggregates' connected filler network was seen. Secondly, with longer time spans (than the matrix terminal time) and small deformation (1%), a more surprising mechanical signature appeared below the silica volume fraction threshold for connectivity: the material exhibited an additional elastic contribution, with very long terminal times (not accessible). Comparing this effect with the dynamic effect attributed to a glassy layer model around the filler, the discrepancy is that the typical extension layer was around 2 nm, much lower than our rim-to-rim distance of ~ 60 nm. Similar values were however seen in T g shifts of thin polymer films (the typical thickness threshold being around 50 nm).

We must also reconsider the polymer-filler interaction, which can be stronger due to the higher adsorption capacity of fractal filler compared to single spheres. In this case, the mechanical behavior could be the result of an additional conformational effect through the formation, for a typical distance of 2.R g  30 nm, of a mixed polymer-filler network by a bridging effect on the part of the polymer chains.

  obtained by NMR an increasing thickness with decreasing T a bit closer to T g : ζ NMR increasing from 2 nm at T-T g = 100 °C to 3 nm for T-T g ~ 50 °C. These values of ζ NMR agreed with the values of effective volume with respect to nominal volume fraction extracted from mechanical reinforcement. Using a formula extracted from the thickness dependence of T g in thin films 37 , the same authors could extrapolate this dependence down to lower T, in the range T-T g < 50 °C, and this dependence agreed with the values of ζ mech obtained from mechanical reinforcement. The latter passes 25 nm only for T very close to T g only (ζ mech ~ 30 nm at T-T g ~3 °C35 ). This is much closer to T g than the temperature of our different mechanical measurements, i.e. between T-T g = 20 °C and 70 °C. Since ζ results.

Figure 1 : 1 )Figure 2 :Figure 3 :Figure 4 :

 11234 Figure 1: Scattering of the PS nanocomposite as a function of the filler concentration. The curves are normalized by the volume fraction of silica. In insert, the I.Q 3 versus Q representation for the same scattering curves. The oscillation, corresponding to initial spheres in contact, is highlighted around 4.10 - 2 Å -1 .

Figure 5 :

 5 Figure 5: Elastic shear modulus G' (a) and G'' (b) as a function of pulsation ω, using time -temperature superposition (T 0 =143°C) coefficient a T defined in the text, for different volume fractions of silica in the composite (0, 1, 2, 3, 4, and 5 % v/v).

Figure 6 :

 6 Figure 6: DMA measurements of elastic modulus E' as a function of temperature for different volume fraction of silica (from 0% to 20% v/v) (note for comparison with Fig. 5 that E'=3.G).

Figure 7 :

 7 Figure 7: (a) Real stress as a function of elongation ratio for different silica volume fractions, (b) reinforcement factor as function of the elongation ratio. The experiments have been performed at T g +20°C

Figure 8 :Figure 9 :

 89 Figure 8: (a) Heat flow vs temperature for different silica content v/v %: 0, 1, 3, 5, 10, 15, 20 (To clarify, all data have been shifted along the heat flow axis). (b) T g composite -T g bulk PS as a function of silica content. Vertical errors bars are standard deviations from measurements (+/-1°C).

  Table Of Contents Use Only"

Acknowledgments

We would like to thank the Région Bretagne and the CEA for N. Jouault's PhD Grant. The Laboratoire Léon Brillouin is a common facility of the CEA (IRAMIS) and CNRS.


Jérémie Gummel

Fabrice Cousin

François Boué

come

Counterions release from electrostatic complexes of polyelectrolytes and proteins of opposite charge : a direct measurement

Counterions release from electrostatic complexes of polyelectrolytes and proteins of opposite charge : a direct measurement

Jérémie Gummel, Fabrice Cousin* and François Boué Laboratoire Léon Brillouin, CEA Saclay 91191 Gif-sur-Yvette Cedex France RECEIVED DATE (automatically inserted by publisher); e-mail: fabrice.cousin@cea.fr

There has been recently a large interest [1] for the understanding of the mechanisms governing the complexation of polyelectrolytes and proteins of opposite charges due to the growing potential of applications (proteins fractionation [START_REF] Izumrudov | [END_REF], controlled drug release [3], biosensors [4]). This is also linked to the emergence of new polyelectrolyte assemblies such as spherical brushes or multilayers. While an obvious and confirmed driving force of complexation is electrostatic attraction, the role of the counterions (e.g. the Sodium cations of a polyanion chain, noted c.i. below) is a key point. Being many and of small mass, free c.i. have a high translational entropy: e.g., it increases strikingly the osmotic pressure of polyions solutions. If they are dispersed into the solvent after complexation, the entropy gain will be important, and could balance the loss of conformational entropy of the polymer. To estimate this entropy, one has to account for the "condensation" phenomenon: if the distance a between two charges is shorter than the Bjerrum length above which thermal agitation is higher than electrostatic attraction, l B , a large fraction (1-a / l B ) of c.i. is trapped close to the polyion [5] ; condensation is relevant also for spherical objects [6] like globular proteins. During complexation, the charge neutralization of opposite charged species can lead to a release of the condensed c.i. The released c.i regain as much translational entropy as the free ones [7,8]. Such effects could differentiate strongly charged synthetic polyelectrolyte (a/l B ~2.4Å /7.2 Å ~ 1/3) where condensation is important, from less charged chains (f=1) like polysaccharides. In numerical simulations [9], condensed and free c.i. release is observed. But experiments gave until now only indirect indications: calorimetric measurements during complexation (proteins/polycation [10] or polyanion/polycation [11]) evidence an endothermic entropically driven contribution. Protein penetration within a polyelectrolyte brush [12] is also indirectly attributed to c.i. release. The present paper reports the first direct structural observation, at our best knowledge. We present here a 'yes -no' experiment, unambiguously linked to the presence or not of the chain c.i. in proteins-polyelectrolyte complexes. The protein is lysozyme, positively charged at low pH, the polyion is polystyrene sulfonate (PSS), with one negative charge per unit. This experiment is based on SANS measurements where we take advantage of the fact that hydrogenated PSS chains and lysozyme have exactly the same neutron density length. Thus the scattering from both species can be switched off simultaneously using a solvent, here a 57%H 2 O/43%D 2 O mixture, which has the same neutron density length: this is called "matching". The remaining signal is the scattering of the counterions only for such "CI" labeling. We chose 4 samples who should, or not, have only released counterions, hence who should scatter, or not.

Our choice of samples is based on already known structures, observed by SANS [13] in complexes of lysozyme and short Sodium PolystyreneSulfonate (PSSNa) which are in dilute regime.

When the ratio of negative to positive effective charges brought by components, [-]/[+] intro , is close to 1, the system is made of dense globular primary complexes (radius ~ 10 nm) organized at higher scale in a fractal way. Using this time deuterated PSS chains, we could switch off either the protein scattering (PE labeling) or the d-PSS scattering (Prot labeling). For such globular structure, comparison of data for these two contrasts gives accurately the inner composition of the globules. Typical spectra will be given below in the case of TMA c.i., which gives very similar results. Let us first summarize three points: (i) whatever [-]/[+] intro , globules have a neutral dense core , i.e. [-]/[+] inner ~ 1 (note that the negative charge concentration [-] intro comprises all units of the polyion, so core neutrality suggests the release of condensed c.i.). (ii) The core is in addition surrounded by a polymer corona in excess of PSS (hairy core). (iii) This gives four typical structures (Figure 1 Such possibility to tune from "naked" to "hairy" cores permits to compare the scattering where all c.i. should be released -"naked" cores, with the one where some c.i. are still present -"hairy" cores where c.i. are condensed on the dangling polyions of the shell. For a strong conclusion the second should be detectable, while the first should be much lower: using the "CI" labelling conditions, the c.i. scattering would be the one of dispersed very small species, very low [START_REF]The low q scattering of counterions condensed on free proteins and free PSS chains is also negligible (low q scattering of free proteins and PSS is negligible whatever the labeling[END_REF].

Experimental success relies on the efficiency of the "CI labelling", i.e. on good contrast of counterions: they have to be visible when lysozyme and hydrogenated PSS are masked by the 57%H 2 O/43%D 2 O solvent. The c.i. scattering length density must be very different from the solvent one (in other words have a high contrast). For this we use deuterated tetramethylammonium c.i., (CD 3 ) 4 N + (noted d-TMA), which contains 12 non labile deuterium atoms (conversely Na + has -accounting for solvatation, a very poor contrast in this solvent [START_REF] Dubois | [END_REF]).

We firstly checked that replacing Na by TMA does not change the structure of complexes: we performed SANS measurements (PAXY spectrometer, LLB, Saclay, France), with same concentrations as in [13], on samples with d-PSS-TMA. The latter is obtained by Makowski sulfonation [16] of deuterated d-PS chains, like for PSSNa. In the last step, polystyrene sulfonic acid (d-PSSH) is neutralized by tetramethylammoniumhydroxyde (h-TMA(OH)) instead of sodium hydroxide NaOH. The chains weight average polymerization degree N is 40. The complexes are made at pH 4.7 (the protein has a net charge +11) in acetic acid/ acetate buffer of ionic strength 5.10 -2 M. Lysozyme is at 40 g/L for all 4 samples, and PSS at 0.02M, 0.03M, 0.05M and 0.1M, corresponding to [-]/[+] intro : 0.65, 1, 1.66 and 3.33 respectively. Solvent is either a 57%H 2 O/43%D 2 O mixture that "matches" protein, yielding d-PSS signal, or a 100% D 2 O mixture that "matches" d-PSS and yields the protein signal. Results for [-]/[+] intro = 1.66 and 3.33 are shown on figure 2. For a given [-]/[+] intro , PSS and lysozyme signals present the same features, indicating that the two species are spatially organized in the same way in the system [13]. We observe: a correlation peak at 0.2 Å -1 corresponding to the contact distance between two proteins, a q -4 decay at intermediate q due to surface scattering of the globules (which look dense at the corresponding larger scale), and an upturn towards a q -2.1 decay at low q due to the fractal organization of the globules at even larger scale. For [-]/[+] intro > 1 , the upturn onset is at lower q for the PSS signal because the globule looks larger when the visible species is PSS and a shell is present. Values are lower for d-PSS signal than lysozyme signal because the volume fraction inside the globules when [-]/[+] inner = 1 is lower for PSS than for lysozyme. Thus features are all the same for h-TMA c.i. than with Na + [13]. The only small difference is the primary complexes size which is slightly larger with TMA counterions.

Then, the specific signal of TMA c.i. was measured using four samples in the 57%H 2 O/43%D 2 O solvent at same concentrations, but replacing d-PSS h-TMA by h-PSS d-TMA. The latter is obtained by neutralization of h-PSSH by d-TMA(OH) (Eurisotop France). On Figure 1, at first sight, samples containing 'naked' globules ([-]/[+] intro = 1 and 0.65) do not scatter at low q, while samples with 'hairy' globules ([-]/[+] intro = 1.66 and [-]/[+] intro = 3.33) do scatter. A control sample with Na + c.i. has been measured for [-]/[+] intro = 1.66. It is compared on figure 1 with the d-TMA and does not show any scattering at low q; this confirms that the low q scattering from hairy globules comes from the d-TMA c.i..

In order to check whether this scattering comes from the decoration of the shell by the c.i., we compare it to the PSS chains scattering on figure 2 Applying a renormalization factor F of the order of 10 (11 for [-]/[+] intro = 1.66 and 12 for [-]/[+] intro = 3.33), the c.i. signal fits quite well the PSS signal at low q. One should expect the shape of the curves to be different since corona (for counterions) and sphere (for PSS) have different form factor; however the differences should lie at larger q-range (q > 0.02 Å -1 ), where the c.i. signal is too noisy to be significant. A quick calculation gives an order of the magnitude of the normalization factor F. It should be proportional to the squared ratio of the d-TMA and d-PSS contrasts (~ 1, the neutron density length of (CD 3 ) 4 N + and d-PSS are close) times the square ratio of their volume fraction ((1-f). V c.i. /V PSS ) 2 . Using 1 -f = 1 -(a / l B ) ~ 2/3), and our estimate of V c.i. /V PSS , the calculated F is not very far from the experimental value of about 10. This is a good support of the decoration of the PSS shell by counterions. In summary, we have shown by a specific labelling experiment that in a system where proteins and polyelectrolyte chains strongly interact to form dense globules, the inner charge stoichiometry of the globules is accompanied by a complete release of all counterions (condensed as well as not condensed) from the core of the globules (same concentration everywhere in the sample). The only significant scattering signal from c.i. comes from the ones trapped in the shell surrounding the complexes when they exist.

Abstract

Though often considered as one of the main driving process of the complexation of species of opposite charges, the release of counterions has never been experimentally directly measured. We present here the first structural determination of such a release by Small Angle Neutron Scattering in complexes made of lysozyme, a positively charged protein and of PSS, a negatively charged polyelectrolyte. Both components have the same neutron density length, so their scattering can be switched off simultaneously in an appropriate "matching" solvent; this enables determination of the spatial distribution of the single counterions within the complexes. The counterions (including the one subjected to Manning condensation) are expelled from the cores where the species are at electrostatic stoichiometry.

  .a): for [-]/[+] intro < 1, "naked" cores plus free protein, for [-]/[+] intro = 1, naked cores, for [-]/[+] intro > 1, hairy cores, for [-]/[+] intro > 2 hairy cores plus free polyions.

FIGURE 1 :

 1 FIGURE 1: Structures of globules made of lysosyme (green), h-PSS (green), and (CD 3 ) 4 N + counterions (red) in 100%H 2 O solvent (left ) or in 57%H 2 O/43%D 2 O solvent (right) and corresponding scattering of (CD 3 ) 4 N + in the 57%H 2 O/43%D 2 O solvent. (a) and (b) : 'naked cores' (([-]/[+] intro =

FIGURE 2 :

 2 FIGURE 2: Comparison of lysozyme scattering (in 100%D 2 O solvent), PSS scattering (d-PSS in 57%H 2 O/43%D 2 O solvent) and c.i. (CD 3 ) 4 N + in 57%H 2 O/43%D 2 O solvent with h-PSS chains) in hairy globules. The c.i. scattering is also compared with PSS one after normalization. (a) [-]/[+] intro = 1.66, (b) [-]/[+] intro = 3.33

  H.S. Makowski, R.D. Lundberg, G.S. Singhal US Patent 3 870 841, 1975, to EXXON Research and Engineering Co.