Umar Farooq
email: umar.farooq@univ-lyon2.fr

Tej Prasad Dhamala
email: tejpdhamala@live.com

Antoine Nongaillard
email: antoine.nongaillard@univ-lille1.fr

Yacine Ouzrout
email: yacine.ouzrout@univ-lyon2.fr

Muhammad Abdul Qadir
email: aqadir@jinnah.edu.pk

A Word Sense Disambiguation Method for Feature Level Sentiment Analysis

Keywords: Sentiment Analysis, Contextual Polarity, Word Sense Disambiguation

Sentiment analysis is an automatic method used to determine that the opinion of a person about a subject is positive or negative. One of the most important tasks in sentiment analysis is to disambiguate the sense of words according to context. Most errors in sentiment analysis are because of improper sense disambiguation. Few methods for this purpose have been proposed in literature. However, they are not able to properly determine the context of word in a sentence. In addition, the lexicon dictionaries used by these methods lack word senses and also do not provide a context matching technique. These issues need to be addressed in order to improve the performance of sentiment analysis so that it can be used by customers and manufacturers for decision making. In this paper, we propose a feature level sentiment analysis system, which produces a summary of opinions about product features. A word sense disambiguation method is introduced which accurately determines the sense of a word within a context while determining the polarity. In addition, a heuristic based method is proposed in order to determine the text where opinion about a product feature is expressed. The results show that the proposed methods achieve better accuracy than existing methods.

I. INTRODUCTION

Sentiment analysis also known as opinion mining refers to the process of determining opinions or emotions expressed in text about a subject. Sentiment analysis is a wide research area at the intersection of different domains: natural language processing, computational linguistics and text mining. Although sentiment analysis is a recent research area, which has been introduced in 2001 [START_REF] Das | Yahoo! for amazon: Sentiment extraction from small talk on the web[END_REF], [START_REF] Tong | An operational system for detecting and tracking opinions in on-line discussion[END_REF]. It arouses a lot of interest and numerous standalone applications are developed to identify sentiment and opinions, e.g., in product reviews, news, twitters and blogs [START_REF] Dave | Mining the peanut gallery: Opinion extraction and semantic classification of product reviews[END_REF]- [START_REF] Shi | An attribute-based sentiment analysis system[END_REF]. Several sentiment analysis systems have been proposed in literature in order to classify the polarity in a document, sentence, phrase or about a product feature. In this paper we focus on feature level sentiment analysis where we determine that the opinion about a product feature is either positive or negative. The results of feature level sentiment analysis are very useful for both customers and manufacturers in order to make decisions. The customers can used the results in order to compare products based on different features to make purchase decision. Similarly, manufactures can use the results of feature level sentiment analysis to know the strengths and weaknesses of the product in order to improve it and to launch different marketing strategies.

One of the issues in sentiment analysis is word sense disambiguation. Word sense disambiguation is an automatic way to determine sense of the word within a context. Generally, word sense disambiguation is used to identify the intended meaning of a word in a sentence. However, in sentiment analysis it refers to the process of determining the sentiment orientation sense (i.e. positive, negative and neutral sense) of a word in a sentence within a context. Therefore, in sentiment analysis, words have three different senses, i.e. positive, negative or neutral. Similarly, the context of a word refers to other words around on which the sense of this word depends. Indeed, a word polarity may vary according to its context. As illustrated in Example 1, the word 'small' appears in both positive and negative senses depending on context. In hotel domain 'small' expresses negative opinion, however the same word expresses positive opinion in mobile and laptop domains. Countless descriptive words face similar situations like large, short, lightweight, long, removed, cheap etc. Most errors in sentiment analysis are due to the word appears in sentence in one sense but the polarity obtains from lexicon is of another [START_REF] Akkaya | Subjectivity word sense disambiguation[END_REF]. In short, word sense disambiguation is very important to accurately determine a sentence polarity.

Example 1: This mobile is small enough that it comes easily in my jeans packet. The hotel is very small. I have very small laptop which I can carry easily.

Most of the sentiment analysis systems determine the polarity without any sense disambiguation. Few authors proposed sense disambiguation methods [START_REF] Wilson | Recognizing contextual polarity in phrase-level sentiment analysis[END_REF], [START_REF] Khan | Sentiment classification using sentence-level lexical based semantic orientation of online reviews[END_REF], however determining the context in a sentence and identifying it exact match in lexicon do not provide good results. Most of the sentiment analysis systems are based on pre determined lexicon dictionary of opinionated words. However, most of these lexicon dictionaries contain only the words with their associated polarities, rather than the words polarities with different senses [START_REF] Akkaya | Subjectivity word sense disambiguation[END_REF]. Some lexicon building methods are proposed which obtained senses of words from WordNet. However, these lexicons do not support word sense disambiguation and also lack word senses. In addition, there are several words such as intensifiers and reducers which increases or decreases the polarity of other words. Such words include very, little, enough, deeply, entirely etc. These words can only be handled if the relationship between intensifier/reducer and modified word are identified accurately and if the polarity of modified word are correctly disambiguated. However, most of the existing methods are not able to handle properly such kind of words.

On the other hand, an issue concerned with feature level sentiment analysis is to determine the exact boundaries of the text where opinion about a product feature is expressed. In literature, three different methods, i.e. clause based [START_REF] Popescu | Extracting product features and opinions from reviews[END_REF], sentence based [START_REF] Poria | Sentic demo: A hybrid concept-level aspect-based sentiment analysis toolkit[END_REF], [START_REF] Hu | Mining and summarizing customer reviews[END_REF] and adjective based [START_REF] Hu | Mining opinion features in customer reviews[END_REF], [START_REF] Zhang | Feature-level sentiment analysis for chinese product reviews[END_REF] methods are used for this purpose. The clause based method only considers the clause in which the product feature appears even if the next clause is also about the same feature. The sentence based method considers the whole sentence where a feature appears. However, when two or more than two features appear, this method considers the complete sentence for both/multiple features, although the opinion about one feature can be opposite to another. The adjective based method considers only the adjective used with product feature to determine the polarity while ignoring other opinion bearing parts of speech such as verb, adverb and noun. All these methods either ignore some portion of the text about a feature or consider irrelevant text for a feature while determining the polarity. If a sentiment analysis method is not able to accurately locate the exact text about a feature then how we can trust on the classification results.

Considering these issues, in this paper we propose a feature level sentiment analysis method, which extracts product reviews from several web sources and produces a summary of opinions about different features of the product. A heuristic based feature scope identification method is proposed in order to determine the exact text where opinion about a feature is expressed. In addition, a word sense disambiguation method is proposed which first builds a lexicon dictionary of words senses for a particular domain and then locates the context of a word in a sentence and obtains polarity from the lexicon accordingly.

This paper is organized as follows. In Section II, related works are presented and analyzed. Section III explains the proposed word sense disambiguation and feature scope identification methods. Finally, Section IV evaluates the performance of the proposed methods.

II. RELATED WORK

The majority of sentiment analysis systems determine the polarity without any sense disambiguation. Although word sense disambiguation is one of the most important tasks in sentiment analysis, still few methods are proposed in literature for this purpose. Plenty of errors in sentiment analysis are because of improper sense disambiguation [START_REF] Wilson | Recognizing contextual polarity in phrase-level sentiment analysis[END_REF] , i.e. a word appears in one sense however the polarity obtained from the lexicon is of another sense. Some researchers such as Kim and Hovy [START_REF] Kim | Determining the sentiment of opinions[END_REF], Yu and Hatzivassiloglou [START_REF] Yu | Towards answering opinion questions: Separating facts from opinions and identifying the polarity of opinion sentences[END_REF], Hu and Liu [START_REF] Hu | Mining and summarizing customer reviews[END_REF] and Grefenstette et al. [START_REF] Grefenstette | Coupling niche browsers and affect analysis for an opinion mining application[END_REF] begin by first creating lexicon dictionaries where words are associated with the prior polarity out of context. However. the contextual polarity of a word appeared in a phrase may be different than the word's prior polarity because a word may appears in different senses [START_REF] Wilson | Recognizing contextual polarity in phrase-level sentiment analysis[END_REF]. In addition, there are several words which prior polarity is difficult to define because they do not carry specific polarity by itself, such words include long, short, small, feel, think, deeply, entirely, practically etc.

In [START_REF] Wilson | Recognizing contextual polarity in phrase-level sentiment analysis[END_REF] a method to determine the contextual polarity of phrase is proposed. This approach is based on the fact that neutral clauses are frequent, therefore polar clauses are identified first. The context of word (which is a bag of three words: the previous word, the word itself, and the next word) is used with linguistic features such as modification features, sentence feature and structure features to determine the polarity of polar clauses. However, in this method the word context is used to determine the effect of enhancers, modifiers and negation instead of disambiguating the word sense. The MPQA (Muti perspective Question Answering Opinion Corpus) is used which list down only the words and phrases with their associated polarities rather than words with different senses.

A word sense disambiguation method at sentence level is proposed in [START_REF] Khan | Sentiment classification using sentence-level lexical based semantic orientation of online reviews[END_REF] which is based on parts of speech pattern matching. While determining the polarity, parts of speech pattern of the sentence is extracted and compared with the WordNet glossaries in order to identify the appropriate sense in SentiWordNet. However, the parts of speech pattern matching do not achieve satisfying results because a word used in the same parts of speech pattern may not have the same sense. For example, in sentences: "the mobile is small" and "the hotel is small" the word "small" appeared in the same parts of speech pattern. However, in first sentence "small" is used in positive sense while in second sentence the same word is used in negative sense. This method also depends on SentiWordNet lexicon dictionary which have several issues as discussed later in this Section.

Most of the sentiment analysis systems are based on prebuilt lexicon dictionaries of opinionated words. Several dictionaries such as SentiSense [START_REF] De Albornoz | Sentisense: An easily scalable concept-based affective lexicon for sentiment analysis[END_REF], SentiWordNet [START_REF] Baccianella | Sentiwordnet 3.0: An enhanced lexical resource for sentiment analysis and opinion mining[END_REF], Micro-WNOp [START_REF] Cerini | Micro-wnop: A gold standard for the evaluation of automatically compiled lexical resources for opinion mining[END_REF] and WordNet-Affect [START_REF] Strapparava | Wordnet affect: an affective extension of wordnet[END_REF] are built which used WordNet information in order to automatically annotate the synsets with polarity scores. The issue in these lexicon dictionaries is that they list down only the common senses of a words which is based on WordNet. Indeed, the senses in WordNet are according to intended meaning and it is difficult to transformed them into sentiment orientation sense. In addition, these lexicon dictionaries do not support any sense matching mechanism in order to disambiguate the word sense. Some authors used the WordNet glossary for this purpose [START_REF] Khan | Sentiment classification using sentence-level lexical based semantic orientation of online reviews[END_REF], however it does not achieve accurate results.

A word sense disambiguation method will only be effective if the following conditions are satisfied.

• An appropriate method is required to determine the context of word in a sentence.

• The context of a word not only depends on the words around but also on the topic discussed which may not be explicit. Therefore, it is also necessary to identify the topic discussed and its boundaries within the text.

• The lexicon dictionary must have all senses of words according to contexts with their associated polarities instead of only listing words with polarities.

• The senses must be according to sentiment orientations context instead of based on intended meaning (as extracted from WordNet in most existing lexicon dictionaries).

• An appropriate context matching method method is required, which the lexicon dictionary must support in order to determine the appropriate context in the lexicon so that the polarity (sense) of a context can be obtained.

Indeed, it is very difficult and time consuming to obtain all senses of words manually for a domain which will be repeated again for another domain [START_REF] Ng | Getting serious about word sense disambiguation[END_REF]. Similarly, it is also challengeable task to automatically extract all words used in different contexts in distinct domains. However, it can be possible to extract words with different contexts automatically only for a particular domain. Therefore, in this research work we propose a mechanism which obtains all the senses of words in a particular domain and also satisfies the above conditions.

III. WORD SENSE DISAMBIGUATION

In sentiment analysis it is very important to disambiguate the context of the word in order to improve the accuracy. A word may appear in positive, negative or neutral senses if used in different application domains. Even in the same domain a word may appear in different senses if used with different product features. As illustrated in Example 2, the word 'long' used in mobile domain, expresses different opinions (i.e. positive and negative) when used with different features (i.e. battery and application). A sentiment analysis method which is not capable to disambiguate the sense will classify these sentences wrongly.

Example 2:

The battery works for a long time. The application takes long time to load.

The existing word sense disambiguation methods have two main problems. 1) These methods are not able to properly determine the context of word in a sentence. 2) In addition, the lexicon dictionaries (on which the methods depend) do not support the sense disambiguation because of two reasons. First, these lexicon dictionaries either lack word senses or only contain words with their associated polarities rather than the words polarities with different senses [START_REF] Akkaya | Subjectivity word sense disambiguation[END_REF]. Secondly, they do not provide a context matching method in order to obtain the polarity of the accurate sense from the lexicon. Based on these issues, we propose a method which determines the context of word in a sentence and builds a lexicon dictionary which contains most of words senses in a particular domain. In addition, this method also provides a context matching mechanism in order to obtain the polarity of the corresponding context from the lexicon.

The context of word in a sentence depends on four things, i.e. global topic, local topic, global context and local context. We need to disambiguate both global and local context. The global context is the context of a word in a specific domain of application, for example, electronics, sport goods and automobiles etc. Similarly, the Local context is the context of a word within a sentence which depends on the words around. Therefore, the proposed method for word sense disambiguation is divided into two steps (i.e. Building Lexicon Dictionary and Local Context Disambiguation). The global context is disambiguated by developing a lexicon dictionary only for a specific application domain. In addition, the local context is disambiguated by considering local topic discussed, linguistic characteristics, and the semantic relationships between different parts of speech within the sentence.

A. Building Lexicon Dictionary

Word sense disambiguation relies on lexicon dictionary of words senses. If a lexicon dictionary either lacks word senses or only contains a list of words instead of senses, then word sense disambiguation method will not be effective. We propose a method to identify the senses of words in a specific domain to build a lexicon dictionary. The following steps are followed to build sense-tagged lexicon. Reviews about a specific product are extracted from different review sites. Product features are extracted from the reviews using existing feature extraction method [START_REF] Hu | Mining opinion features in customer reviews[END_REF]. The text where opinions are expressed about different features are located in reviews. All opinionated words used to express opinions about features are extracted with the contexts in order to obtain seed set of word senses. The polarities of the seed of word senses are annotated and expanded by obtaining synonyms and antonyms from WordNet. All these steps are explained in the remaining part of this section.

1) Product Reviews Extraction:

In order to build a lexicon dictionary for specific domain, the reviews about a product are extracted from Web. For this purpose, a wrapper is developed which extracts reviews automatically from e-commerce and review sites such as Ebay, Cnet and Amazon. The wrapper is fully automatic, the user simply need to select a product and the wrapper automatically locates the web pages (containing reviews about that product) and extract reviews from all those web pages.

2) Product Feature Extraction: In this step the proposed method in [START_REF] Hu | Mining opinion features in customer reviews[END_REF] is implemented in order to determine the product features. The users are also allowed to modify the results in order to improve the accuracy. The users can select only the most frequent features, map features identified by the system as separate features which are actually single feature, however different names are used (e.g. battery life and battery, application, program and app etc.) and even can define their own features. However, for building lexicon dictionary we considered all the features about which the customers expressed opinions.

3) Feature Scope Identification: The chunk of text where opinion about a feature is expressed is called the feature scope. The feature scope can be a single clause, multiple clauses or even the whole sentence. Feature scope identification is very important because sense disambiguation not only depends on the topic discussed (i.e. product feature) but also on the boundaries of text where this topic is discussed. In fact, more than one features can be discussed in a single sentence. Similarly, a feature can also be discussed in multiple clauses. It is very important not only to separate opinions about one product feature from another but also to identify all those consecutive clauses where opinion about a single feature is expressed. In literature, three different methods, i.e. clause based [START_REF] Popescu | Extracting product features and opinions from reviews[END_REF], sentence based [START_REF] Poria | Sentic demo: A hybrid concept-level aspect-based sentiment analysis toolkit[END_REF], [START_REF] Hu | Mining and summarizing customer reviews[END_REF] and adjective based [START_REF] Hu | Mining opinion features in customer reviews[END_REF], [START_REF] Zhang | Feature-level sentiment analysis for chinese product reviews[END_REF] are used to identify the scope of feature. However, all these methods are not able to identify the feature scope correctly. We propose a heuristics based feature scope identification method which gives more accurate results. This method uses the following heuristics rules which are based on: the clause in which the feature appears, parts of speech pattern, noun clause, conjunction used, and the reference words used at the beginning of succeeding clause.

• The scope of a product feature starts from the clause where a feature appears and it may or may not be extended to the succeeding clauses.

• If the succeeding clause does not contain a noun (i.e. not a noun clause) then the scope of the feature is extended to the next clause because no other topic or feature is discussed.

• However, if the succeeding clause contains another product feature then it means that the scope of feature discussed in preceding clause is limited only to that clause.

• On the other hand, if the succeeding clause is a noun clause but the noun is not a product feature, then we have three choices. i) If the conjunction used is either 'AND' or 'OR' and the same parts of speech pattern up to three words is followed on either sides of conjunction then it is most likely that the same feature is also discussed in the next clause. ii) If there is a reference word at the beginning of the succeeding clause then there is highly likelihood that the scope is extended to the next clause even if it is a noun clause.

Reference words (e.g. it, this, that, these, those, which, such etc.) are used to refer back to the object or idea that have been mentioned earlier. iii) If both cases i) and ii) do not exist, then it is highly likelihood that the scope of the feature is only limited to the clause in which it appeared.

The algorithm 1 is proposed to identify the feature scope which is based on the heuristics rules. The inputs to the algorithm are the Product features set P F , the reviews set R, the reference words set RW and conjunctions set C. The output of the algorithm is the feature scope F S. First, the clause F CL in which the feature appeared is identified which is assign to the candidate feature scope CF S (Line 1-2). The remaining steps identify that whether the feature scope F S only contains the CF S or extended to the succeeding clause SCL . If the SCL is not a noun clause N CL then both CF S and SCL are included into the feature scope (L 3-5). If the succeeding clause is N CL and the noun N N which appeared in this clause belongs to the feature set P F , then only CF S becomes the F C (L 6-8). If the SCL is N CL , the conjunction C used is neither 'AND' nor 'OR' and SCL does not have reference word at the beginning, then F S is limited only to CF S (L 9-11). Similarly, if the SCL is N CL and either the conjunction used is OR/AND or SCL contains RW at the beginning then both CF S and SCL are included into the F S (L 12-14). If there is another SCL then all steps from 3-14 are repeated in order to check that weather this succeeding clause is also included in the F S or not. In this way, the algorithm identifies the individual clause, multiple clauses or even the whole sentence where opinion about a product feature is expressed. CF S = CF and Repeat Step 3 to 14 17: End 4) Building Seed Set of Word Senses: In order to build seed of word senses, feature scopes identified for all product features in the previous step are considered to automatically extract words in different contexts. Only the opinionated parts of speech which carry opinions (i.e. adjectives, verbs, adverbs and nouns) are extracted with their contexts. As the context of a word depends on other words around with which the word has some semantic relationship. Therefore, for each opinionated part of speech we proposed a semantic relationship in order to identify the context. The following four semantic relationships i.e. for adjective JJ, verb VB, adverb RB and nouns NN are used to identify the context and to extract these words with their contexts automatically.

1) The most important part of speech is adjective which describes a noun. This noun is usually the product feature or some other noun which further describe the product feature PF. This means that the polarity of an adjective depends on the product feature which it describe (i.e. JJ, NN/PF).

2) The verb describes some action related to product feature and hence the polarity of verb can be determine while considering the product feature in mind (i.e. VB, NN/PF). 3) Similarly, the adverb describes or modifies either an adjective or a verb which further describes a noun or product feature. The adverb do not have a direct relationship with product feature and hence the effect or polarity of adverb can be determine by considering the whole sequence of relationship (i.e. RB, JJ/VB, NN/PF). 4) Sometime the noun (other than the product feature) which carries opinion also describes some aspect of the product feature (i.e. NN, PF).

The polarities of words in different contexts in seed set are annotated by a human annotator. For this purpose a user interface is built where each word is assign either positive, negative or neutral polarity according to the context, within the rang of 0 to 1, based on the strength of positive or negative opinion expressed. Manual annotation is used because the automatic annotation methods are not able to accurately annotate the words within a context. In addition, one single incorrect annotation in the seed set can affect large quantities of synonyms and antonyms in the target lexicon in seed expansion.

5) Seed Expansion:

In order to build lexicon dictionary, the seed set is further expanded automatically by obtaining the synonyms and antonyms from WordNet. The synonyms set obtained for a word used in a specific context is given the same polarity as that word has. On the other hand, the antonyms are assign the opposite polarity of the word for which they are obtained. For example, if a word appeared in single context and five synonyms and antonyms are obtained, then all synonyms in the same context are assigned the same polarity and antonyms within in the same context are assign with opposite polarity. On the other hand, if there is a word in three different contexts and five synonyms and antonyms are obtained, then by using this method thirty other words with these three senses are obtained (i.e. 5*3=15 by synonyms set and 5*3=15 for antonyms set). In this way, the seed set of word senses is expanded in order to build the lexicon dictionary. The lexicon dictionary contains the opinionated words with different contexts (i.e. semantic relationships) as well as the senses (i.e. polarities).

B. Local Context Disambiguation

This section explains that how the local context of a word within a sentence or feature scope F S is disambiguated while determining the polarity. Before disambiguating the local context first the product features and feature scopes are identified using the corresponding methods. While determining the polarity of a F S or sentence, first the context of every opinionated word are determine using the four semantic relationships proposed in section III-A4. Second, the word with their context is searched in the lexicon and the polarity of the match context is obtained from the lexicon dictionary. If exact match is not found in the lexicon then the polarity of the partial match context is obtained. Lets we want to find the polarity of sentences in Example 3. In first sentence, only one opinionated part of speech that is bright/jj is used to express opinion. In second sentence, three opinionated parts of speech, i.e. works/VB, long/JJ and time/NN are used to express opinion about battery P F . First, the contexts of these words are determined using the following three semantic relationships.

1) The JJ 'bright' describes an aspect of feature (i.e.

screen/PF bright/jj+Screen/PF).

2) The VB 'work' describes an action related to feature battery/PF (i.e. work/VB+battery/PF).

3) The JJ 'long' describes the NN 'time' which further describes an aspect of battery PF (i.e. long/JJ+time/NN+battery/PF).

In second step, these words are searched according to these contexts (i.e. bright/jj+Screen/PF, work/VB+battery/PF and long/JJ+time/NN+battery/PF) in the lexicon. The polarities of the match contexts are obtain from the lexicon in order to determine the polarity of the whole sentence.

Example 3

The screen is bright.

The battery works for a long time.

It is easy to identify the semantic relationships between parts of speech when each part of speech appears only once in a sentence. Conversely, it is challengeable task to identify the relationships when each part of speech appears multiple times in a single sentence or even in a single clause. On the other hand, it becomes more difficult if multi topics discussed in a single sentence. Therefore, in order to make our task easier we already make sure that the feature scope is only about one topic (i.e. about a product feature) using feature scope identification method. We used the following heuristic rules to automatically determine the semantic relationships between opinionated parts of speech.

• Usually, parts of speech have relationships within a clause. Only in the following cases the relationship is extended to the succeeding clause. a) When there is no noun in succeeding clause then the opinionated parts of speech have relationships with feature discussed in preceding clause. b) Similarly, when there is a noun but the noun is not product feature the opinionated words in succeeding clause have relationships with the words in preceding clause.

• Adverb is usually used before an adjective or a verb, which it modifies.

• The adjective and verb have relationship with nearby noun (which may be product feature) within a clause. If there is no noun in the clause then the adjective and verb will have relationship with the noun discussed in preceding clause.

IV. RESULTS

We performed three sets of experiments in order to evaluate the performance of our proposed methods. The first set of experiments is performed to measure the performance of feature scope identification method. The second set of experiments is performed to evaluate local context disambiguation method. The last set of experiments is performed to evaluate the performance of overall feature level sentiment analysis method. As, the word sense disambiguation method depends on both feature scope identification and local context disambiguation, thus both first and second set of experiments collectively evaluates the performance of word sense disambiguation. The third set of experiments shows how much the word sense disambiguation contributes to the overall performance of feature level sentiment analysis.

A. Evaluation of Feature Scope Identification

In order to measure the efficiency of the feature scope identification method, reviews about mobiles are extracted using the reviews extraction method from e-commerce and review sites. The existing feature extraction method [START_REF] Hu | Mining opinion features in customer reviews[END_REF] is used in order to extract mobile features about which opinions are expressed. Seven hundred reviews which had good English are selected. The proposed heuristic based method is used to identify the feature scopes from these reviews. There were 1505 sentences in these reviews where 2721 opinions about different mobile features are expressed.

The results of the proposed method are compared with clause based [START_REF] Popescu | Extracting product features and opinions from reviews[END_REF], sentence based [START_REF] Poria | Sentic demo: A hybrid concept-level aspect-based sentiment analysis toolkit[END_REF], [START_REF] Hu | Mining and summarizing customer reviews[END_REF] and adjective based [START_REF] Hu | Mining opinion features in customer reviews[END_REF], [START_REF] Zhang | Feature-level sentiment analysis for chinese product reviews[END_REF] methods in Fig. 1. The results show that the accuracy of the proposed method is 88 %, which is better than existing methods. Conversely, the accuracy of the sentence based method is the worst of all these methods, i.e. the accuracy is only 32 %. The accuracy is worst because in most cases it was observed that opinions about multi features are expressed in a single sentence. However, the sentence based method is not able to distinguish opinion of one feature from another. On the other hand, the accuracy of the clause and adjective based methods are close to each other, i.e. 47.7 and 47.2 % respectively. The accuracy of clause based method is low because it does not consider the succeeding clauses where opinion about the same feature can be expressed. On other hand, the accuracy of adjective based method is not good because it only considers adjective but other parts of speech (i.e. verb, adverb and noun) also carry opinion. An important aspect is that the feature scopes wrongly identified by the proposed and clause based methods are not completely false because at least one clause about a feature is identified instead of all the clauses.

B. Evaluation of Local Context Disambiguation

Another set of experiments is performed to evaluate the performance of local context disambiguation method. As the local context depends on the four semantic relations therefore this experiment is performed in order to evaluate that how accurately our system identifies these semantic relationships. For this purpose 2381 feature scopes are extracted from 257 reviews using the proposed feature scope identification method. There were 2128 adjective and feature/noun relationships (JJ and PF/NN), 841 verb and feature/noun relationships (VB and PF/NN), 541 adverb and verb relationships (RB and VB), 351 adverb and adjective relationships (RB and JJ) and 134 noun and feature relationships (NN and PF). The proposed heuristics based method is used to identify these semantic relationships. Fig. 2 shows the accuracy of the proposed method for different semantic relationships. The results shows that the overall accuracy of semantic relationships is 87.3 %. The accuracy of the JJ and PF/NN semantic relationship is the best, i.e. 97.7 %. Similarly, the accuracy of RB and VB, VB and PF/NN, JJ and PF/NN and NN and PF are 92.3 %, 88.2 % , 83.7 % and 90.2 % respectively. It was observed that adverb usually appears before the part of speech with which it has semantic relationship, within the same clause. Therefore, the accuracy of semantic relationships where an adverb was involved is better than the other semantic relationships. In order to evaluate the performance of overall feature level sentiment analysis the mobile application domain is considered. In first step, a lexicon dictionary for mobile domain is built by using the proposed method. For this purpose 2500 reviews about mobile which have good English are extracted from e-commerce sites using the review extraction method. Product features are identified and the proposed feature scope identification method is used, which identify 3146 feature scopes in these reviews. The proposed method of building seed of word senses is used which determined 3517 seed words. All these 3517 words which include 1492 adjectives (JJ, PF), 1092 verbs (VB, PF), 460 nouns (NN, PF) and 473 adverbs ( i.e. 307 (RB, JJ, PF) and 166 (RB, VB, PF)) are extracted with the corresponding contexts. The polarities are assign to the seed set by an annotator. The seed is expanded by obtaining 51848 synonyms and antonyms from WordNet, i.e. 37914 synonyms and 10417 antonyms are obtained. In this way, a lexicon dictionary which contains 55365 opinionated words with contexts and senses are built.

In the second step, the lexicon built is used in order to determine the feature level sentiment analysis. For this purpose 3000 reviews are extracted. However, these reviews are different from the reviews extracted for building lexicon dictionary. Table I shows the performance of the proposed

A(C) = tp + tn tp + tn + f p + f n × 100 (1)
P (C) = tp tp + f p (2)
R(C) = tp tp + f n (3)
F = 2 × P.R P + R (4)
Table II compares the results of the proposed method with existing methods. The comparison is made with feature level sentiment analysis [START_REF] Hu | Mining and summarizing customer reviews[END_REF], [START_REF] Yang | Context-aware learning for sentence-level sentiment analysis with posterior regularization[END_REF], sentence level sentiment analysis [START_REF] Khan | Sentiment classification using sentence-level lexical based semantic orientation of online reviews[END_REF] and phrase level sentiment analysis [START_REF] Wilson | Recognizing contextual polarity in phrase-level sentiment analysis[END_REF] methods. All these existing methods either used a word sense disambiguation method or incorporated some features to determine contextual polarity. The results show that the proposed method improved the performance of feature level sentiment analysis considerably, hence outperform the existing methods based on all evaluation measures. The accuracy of the proposed method is very good, i.e. 93.2 %. In addition, it also offered the best recall and precision. The significant improvement in performance is because of three main reasons. First, a lexicon dictionary is used which contains most of the words senses for a particular domain. Second, an appropriate word sense disambiguation method is adopted. Finally, all the opinionated parts of speech (i.e. adjective, verb, adverb and noun) are considered while determining the polarity.

V. CONCLUSION

Feature level sentiment analysis is very important for customers and manufacturers to make decisions. The two issues and causes behind errors in feature level sentiment analysis are: improper sense disambiguation and inability to locate the text where opinion about a feature is expressed. In order to address these issues, in this paper we proposed a feature level sentiment analysis method which determines the opinions about different product features. A word sense disambiguation method is proposed which performed the following sub tasks in order to accurately disambiguate the sense of word while determining the polarity. a) Determines the exact boundaries of the text where opinion about a feature is expressed. b) Uses an appropriate method to identify the context of word in a sentence. c) Provides a context matching mechanism in order to obtain the polarity of the corresponding context from the lexicon. d) Builds a lexicon dictionary which not only contains the senses of words in a particular domain but also support a context matching mechanism. The results shows that the proposed methods considerably improved the overall performance of feature level sentiment analysis.

As future works, an argument mining method will be proposed to further analyzed the results of feature level sentiment analysis in order to determine the reasons behind customer preferences, likes and dislikes. This will allows the manufactures to make decisions more effectively. Furthermore, feature level sentiment analysis system will be integrated with PLM (Product Life Cycle Management System) in order to allow the organizations to make decisions throughout product life cycle.

Algorithm 1 2 :

 12 Algorithm for Feature Scope Identification Inputs:P F = {f 1 , f 2 , f 3 . . . . . . f n } , R = {r 1 , r 2 ,r 3 . . . . . . r n }, RW , C Outputs: F S 1: Search the clause in which feature appeared. CF S = F CL 3: IF SCL ⊂ N CL Then 4: F S = CF S + SCL 5: End IF 6: IF (SCL ⊂ N CL) && (N N ⊂ P F ) Then 7: F S = CF S 8: End IF 9: IF (SCL ⊂ N CL) && (C = 'AND' 'OR') && (RW / ∈ SCL) Then 10: F S = CF S 11: End IF 12: IF (SCL ⊂ N CL) && ((C = 'AND' 'OR') (RW ∈ SCL)) Then 13: F S = CF S + SCL 14: End IF 15: IF There is another SCL Then 16:

Fig. 1 :Fig. 2 :

 12 Fig. 1: Accuracy of Feature Scope Identification

TABLE I :

 I Performance of Feature Level Sentiment Analysis On the other hand, the precession P for a class C is defined as the total number of true positive tp (i.e. the instances correctly classified as belonging to class C) divided by total number of instances classified as belonging to class C (i.e. true positive and false positive f p, which are instances incorrectly classified as belonging to class C ). Similarly, the recall for a class C is the number of true positive divided by the number of instances that actually class C in truth. Whereas, F-measure is the harmonic mean of precession and recall. In other words, accuracy, precession, recall and F-measure are computed by Equations 1, 2, 3 and 4 respectively.

		Positive	Negative	Neutral	Overall
	Instances	2378	1250	186	3558
	Accuracy	94.1	94.2	98.0	93.2
	Precession	0.95	0.90	0.82	0.98
	Recall	0.94	0.92	0.77	0.94
	F-measure	0.95	0.91	0.80	0.96
	feature level sentiment analysis, which is measured in term
	of accuracy, precession, recall and F-measure. Accuracy is
	actually the percentage of total number of instances accurately
	classified.

TABLE II :

 II Comparison of Feature Level Sentiment Analysis with Existing Methods

	Methods	Precision	Evaluation Measures Recall F-measure	Accuracy
	Supervised Method [22]	-	-	-	82.7
	Semi-Supervised [22]	-	-	-	64.4
	Hu and Liu SL [12]	0.69	0.64	0.66	84.2
	Phrase Level [9]	0.63 0.72	0.67 0.82	0.65 0.77	65.7
	WSD POS Pattern Matching [10]	-	-	-	86.6
	Proposed Method	0.98	0.94	0.96	93.2

ACKNOWLEDGMENT

This project has been funded with support from the European Commission (cLink Project: 372242-1-2012-1-UK-ERA MUNDUS-EMA21). This publication reflects the views only of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein.


C Stolz
email: stolz@lms.polytechnique.fr

Thermodynamical description of running discontinuities

Keywords: Hamiltonian, shock-waves, jump, discontinuity, energy-release rate

come

SOME GENERAL FEATURES

In order to explain and to predict the motion and the equilibrium of bodies or structures subjected to various physical interactions, a kinematical description of the motion is first performed. Usually one looks for the motion of a material point M from a reference configuration by describing its displacement u(M, t).

After the kinematical description of the body, one has to deal with the mechanical interactions. Many statements permit the description of these interactions ; we can use for example the virtual-power statement. This describes the mechanical interaction between each material point of the body with respect to a given loading distribution. For sake of simplicity and conciseness of this presentation, a thermodynamical description of interaction is adopted.

First, the local state is defined by a set of state variables such as the strain a set of internal parameters and the absolute temperature The local interaction is defined by a thermodynamical potential or the free energy per unit of mass. The thermodynamical forces are defined by the state equations :

is the reversible stress, A is the thermodynamical forces associated to the internal vari-ables, and is the entropy. In the case of reversibility, the knowledge of the free energy is sufficient to determine the local state of equilibrium defined by the stress In nonlinear mechanics, the internal state is generally associated with irreversibility. Then the fundamental inequality of thermodynamics implies that the internal production of entropy must be non negative. The equations of state do not provide the full constitutive equations; some complementary laws are necessary to describe the irreversibility. In the total dissipation, we distinguish the part due to the conduction and the part due to internal forces. The two parts are assumed to be separately non negative. The mechanical part has the form:

The state of stress during the evolution has been decomposed as Let us assume that the behavior belongs to the class of the so-called generalized standard materials. This ensures the existence of a potential of dissipation

The potential d is a convex function of the variables, with a minimum value at the origin. The evolution of the internal state is given by the normality rule : this means that the subdifferential of d is the set of state such that:

for all admissible fields

The existence of such a potential for the dissipation ensures the positivity of the entropy production:

EQUILIBRIUM AND QUASISTATIC EVOLUTION

As time evolves the displacement field and the internal state must satisfy the following :

• the state equations :

• the conservation of the momentum:

• the evolution equations of the state variables. • all the boundary conditions. For the overall system the rule of the free energy is replaced by the global free energy:

In a global description the equations of state possess the same form as in the local one, but the state of the system is defined by fields of state variables. The equations of state are relationships between fields of state variables:

for which the evolution is given in terms of fields of state variables as :

where the set is defined obviously by : then we have for regular functions :

Let us consider that the external loading derives from a potential given in terms of traction applied on the external surface of the body. Then, the global free energy can be replaced by the potential energy of the system : By combining all the equations in terms of fields of state variables, we can state the quasistatic evolution in a global manner by the variational system :

These equations are defined on a set of admissible fields. The displacement is subject to boundary conditions over These equations are general; they contain the essential structure of a problem of quasistatic evolution. The first equation of this system explains the conservation of the momentum taking into account the constitutive law, the second explains the thermodynamic forces associated with the internal parameters.

THE DYNAMICAL CASE

The Hamiltonian is the total energy of the system [Stolz, 1988]:

These relations are obtained by the following definition:

By using the properties of the characterization of the evolution of the internal state and integration over the body, we can define the dissipative function:

The first term is the kinetic energy is the momentum, the second is the internal energy with the density:

and the last is the potential energy due to prescribed loading. The equation of motion are given by Taking account of the decomposition of the stress of the conservation of the momentum inside the volume and of the boundary conditions we modify the expressions : then with the relations : we obtain the conservation of momentum in the Hamiltonian form:

Finally, the Hamiltonian formulation of the evolution of the system is obtained:

A conduction law must be given and the positivity of the entropy production must be verified to determine the evolution of the system : For the real motion, the value of the Hamiltonian is the sum of the kinetic energy, of the internal energy and of the potential energy of the external (given) load. The conservation of the energy of the system can be easily rewritten as where q is the heat flux, and we consider that no external volume heat source are prescribed.

The definition of the Hamiltonian can be extended to generalized media. The proposed description can be performed in all the cases in which we can define the behavior by two potentials: a global free energy and a dissipative function. If some particular internal constraints exist, this description must be revised.

MECHANICAL TRANSFORMATIONS ALONG MOVING SURFACES

We consider a moving interface along which mechanical transformation occurs [START_REF] Pradeilles-Duval | Mechanical transformations and discontinuities along a moving surface[END_REF]. Two materials coexist at any tune in the structure and the body is heterogeneous. The evolution of the surface along which the transformation takes place is characterized in the energy analysis. Some connections can be made with the notion of configurational forces, [Gurtin, 1995, Maugin, 1995,Truskinovski, 1987,Grinfeld, 1980, 1991].

Let denote the domain, composed of two distinct volumes and which are occupied by two materials with different mechanical properties. The perfect interface between them is assumed to be regular and is denoted by Material 1 changes to materials 2 along by an irreversible process. Hence moves with a normal velocity positive along

The state of the system is characterized by the displacement field u, from which a strain field is derived The other parameters are the temperature the internal parameters and the position of the boundary When the surface propagates, with velocity (N normal outward to mechanical quantities can have a jump, and all volume integrals have rates defined by :

The mass conservation is defined by the continuity of the mass flux :

Then the conservation of energy and the entropy production are rewritten as By the momentum conservation we have This is true for all volume then we deduce the local equations of conservation :

The entropy production takes the form:

The interface is perfect, then the displacement is continuous along the interface and the temperature is assumed to be continuous Some dissipation can occur along the moving surface. The mechanical discontinuities must satisfy some constraints, they must be kinematically admissible and ensure the positivity of the entropy production. The jumps of the mechanical quantities must verify the Hadamard equations: and the momentum equation:

Combining the local equations of conservation, we rewrite the production of entropy in terms of volume and surface contribution:

where the quantity Along the surface, the production of entropy is defined in a similar form as in a quasistatic thermomechanical coupled evolution by replacing the tension along the surface by the mean tension, and as in an isothermal analysis the results of Abeyaratne and Knowles [Abeyaratne et al.,1990].

The behavior of each phase is defined by the tree energy as a function of strain temperature and a set of internal variables the evolution of which is governed by a pseudo potential of dissipation a convex function of As before, the state equations are where A, the thermodynamical forces associated with the internal parameter satisfies :

The dissipation due to conduction is given by a conduction law. As before, the variation of the Hamiltonian H determines the power heat supply. We decompose the volume into the two volumes the normal outward to the position of the surface is an internal variable and the Hamiltonian takes the form: where The equations of motion are then Each volume has a contribution to the global Hamiltonian:

And for the overall system we obtain the equation:

Making the distinction between volume and surface terms, we find

The gradient of the Hamiltonian with respect to the position of the moving surface determines the intensity of the heat source due to the propagation of mechanical discontinuities. This gradient takes the form of a release rate of internal energy :

The production of entropy has a surface term which is explained as In the same spirit, the intensity of the source of entropy production is a release rate of free energy divided by the temperature. Without more hypotheses this term is not directly connected with the variation of the Hamiltonian.

In the isothermal case, the dissipation and the heat sources are related by the uniform temperature, the heat flux is zero and

The dissipation is a jump of entropy along the moving surface given by defining the isothermal Hamiltonian by the dissipation is expressed by the variation of the isothermal Hamiltonian:

By considering the surface and volume terms, we see that the energy release rate is the ther-modynamical force associated with the velocity of and is the derivative of the Hamiltonian with respect to the position of the surface. The expression is recovered.

QUASISTATIC EVOLUTION

In isothermal evolution, complementary relations must be considered to describe irreversibility. An energy criterion is chosen as a generalized form of the well known Griffith theory. Then, we assume otherwise. This is a local energy criterion. At each equilibrium state, the interface can be decomposed into two subsets where the propagation is either possible or not Let denote by the subset of where the critical value is reached. The evolution of the interface is governed by the consistency condition, during the evolution of if, at the geometrical point the criterion is reached then the derivative of G following the moving surface vanishes This leads to the consistency condition written for all point inside otherwise With Hadamard relations this derivative takes on the final form where In that case, the evolution is determined by the functional :

Then the evolution is given by for all v * kinematically admissible field, and

The discussion of the stability and bifurcation along an evolution process can be now investigated as proposed by Pradeilles and [START_REF] Pradeilles-Duval | Mechanical transformations and discontinuities along a moving surface[END_REF]. Consider the rate of displacement v solution of the boundary value problem for any given velocity of propagation, v satisfies: discontinuity and the jump of internal parameters. One must determine the loading path or the history of all the quantities inside the shock; this describes the internal structure of the shock.

To solve this problem we can consider the discontinuity surface as a layer normal to the direction of the propagation of the shock ; one has to study the inner expansion of all the quantities in a continuous process in the frame translating with the shock surface, assuming that all quantities depend only on the local normal coordinate For the inner expansion, X varies from to We are interested in a one dimensional motion.

A constitutive law and a pseudo-potential of dissipation being given to describe the internal behavior, the dissipation is known inside the shock and therefore the jump of entropy is given as Using the other jump conditions expressed in terms of the given constant of the shock, we can rewrite the jump of entropy as This defines the shock generating function P: in the steady state analysis, and P has the following form:

The jump of P is the total dissipation:

The function P is supposed to be a continuous function of X, and the value of P on a section dX has the following form:

Then P is related to the Lagrangian defined on the section dX :

Defining the dissipative function D as previously :

we adopt a Fourier law for the thermal conduction

The shock structure is determined by the canonical equations :

and we obtain the local property If the temperature is continuous i.e. the jump of P is exactly related to the definition of G s . If a continuous process governs the internal structure of the discontinuity, the quantity G s is a global characteristic of the moving discontinuity. When local behavior is given in term of w and D, the behavior of all the jumps are given. This is a way to build global behavior for shock waves or moving discontinuities inside a continuum. It is the analogue of homogenization for dynamical behavior.

A typical example has been given in plasticity. In this case the determination of compatible states (+) and (-) is not easily studied from a general viewpoint. For propagation of longitudinal waves, if we suppose that the loading process is monotonic, we obtain a curve which gives the relation between the jump of the quantities, as in the adiabatic Hugoniot curve in gas dynamics [Mandel, 1978]. This result is obtained by the assumption of a radial loading path during the shock. This is the structure of the shock. But in a general case, the internal structure for shock wave in plasticity must be studied as proposed by Germain and Lee [Germain et al., 1973].

CONCLUSIONS

We have studied the thermodynamics of running discontinuities from a global point of view and related all the quantities to a Hamiltonian principle generalized to nonlinear behavior. In quasistatic evolution we have given a boundary value problem for the evolution of a moving surface, the motion being governed by an energy criterion. Some connections with fracture mechanics can be also given in the same formalism [Stolz et al, 1996;Bui et al, 1987]. Finally the internal structure of discontinuities has been investigated to provide a global behavior of the moving surface in agreement with the positivity of the production of entropy.

and non classical boundary conditions on Consider the value W of F for this solution and the stability of the actual state is determined by the condition of the existence of a solution and the uniqueness and non bifurcation is characterized by 6. THE CANONICAL EQUATIONS OF SHOCK WAVES.

The local equations of shock waves will be derived from a thermodynamical potential and a dissipation pseudopotential. The canonical equations rule the jump relations and the constitutive behavior in a section of the shock [START_REF] Nguyen | [END_REF]. Let us consider a shock wave, we propose to study the evolution of the shock. Locally the surface of discontinuity is replaced by its tangent plane, and one considers a frame moving with a velocity which is the normal speed of the shock. Along the line of discontinuities, some relationships between the jumps of quantities must be verified:

• for the momentum :

• for the energy :

• for the entropy production :

The quantity m denotes the mass flux and accounts for the mass conservation. The shock is governed by the constants which are related respectively to the flux of momentum and to the flux of energy. The jump conditions are rewritten as

The main problem is to determine the state if the state and the constant are given while respecting the positivity of the entropy production in an anelastic material. The jump conditions give us only the jump of entropy, but no direct relation between this


E Chabert
email: chabert@lms.polytechnique.fr

M Bornert

E Bourgeat-Lami

J.-Y Cavaillé

R Dendievel

C Gauthier

J L Putaux

A Zaoui

Filler-filler interactions and viscoelastic behavior of polymer nanocomposites

Keywords: Nanocomposites, Mechanical properties, Percolation, Discrete modeling approach, Self-consistent modeling, Cryo TEM

This work presents the main results obtained within a project on mechanical properties of polymer based nanocomposites. The specific point was how to analyze and model the filler-filler interactions in the description of the viscoelastic behavior of these materials. This paper aims at presenting the general strategy used by the different partners to address this question, together with original experimental results and micro-mechanical modeling. Different nanocomposite materials were fabricated using the latex route, leading to random dispersions of rigid submicronic particles (PS = polystyrene, silica) in a flexible polybutylacrylate matrix at various volume fractions. In addition, encapsulated silica particles in a styrene-acrylate copolymer were produced, leading, after film formation, to a limited number of contacts between silica fillers. The processing route of these encapsulated particles was optimized and the resulting morphology was analyzed by TEM experiments. In the case of random mixtures, a strong effect of reinforcement appears in the rubbery field of the soft phase when the filler content is above a critical fraction (percolation threshold). The reinforcement in the rubbery plateau can be still exacerbated in the case of the PS particles if the material undergoes a heat treatment above the main relaxation of the PS phase. These experimental results illustrate the difference between geometrical percolation (when particles are just in contact) and mechanical percolation (with strong interactions between the fillers). The comparison of the results for PS and silica fillers shows once more that the strength of the interactions plays an important role. To account for the whole set of experimental data, two ways of modeling were explored: (i) homogenization methods based on generalized self-consistent schemes and (ii) a discrete model of spheres assembly which explicitly describes the ability of the contacts to transmit efforts.

Introduction

Among the several reasons to incorporate fillers into polymers (cost reduction, improvement of some physical properties such as flame retardancy or barrier properties) mechanical reinforcement is expected [START_REF] Rothon | Particulate-filled polymer composites[END_REF]. Traditional fillers display average characteristic sizes in the range of several microns. However, due to the development of nanosized fillers, the specific influence of the nanometric size in the reinforcement mechanisms has to be addressed. Composite materials based on nano-sized fillers, the so-called nanocomposites, are presently studied especially because they may have unusual combinations of properties [START_REF] Sumita | [END_REF][3][4][5][START_REF] Chazeau | Relationships between microstructural aspects and mechanical properties of polymer-based nanocomposites[END_REF]. These unusual properties may be a consequence of the extremely large specific interfacial area (hundreds of m 2 /g). They may be also related to the very short distances between the reinforcing fillers (about 10 -8 m), that may become close to the characteristic size of the macromolecular coils. In addition, some years ago, we showed that drastic reinforcing effects may be observed at very low volume fractions for fillers with very large aspect ratio, when the percolation of the fillers occurs [START_REF] Favier | [END_REF].

However, the processing of composite materials that only differ by the size of the fillers (keeping constant the dispersion state and surface properties) is not a simple task. In or-der to by-pass this difficulty, experimental data may be compared to theoretical predictions that are developed to account for the mechanical behavior of heterogeneous materials. Parameters which play a role for the mechanical properties of a filled polymer (elastic modulus for instance) are the following: (i) the (visco)elastic properties of its constitutive phases, (ii) the volume fraction of filler, (iii) the morphology (i.e., shape, aspect ratio, and distribution of the filler within the polymeric matrix) and (iv) the interactions between fillers and between filler and matrix. Various models have been proposed in the literature to understand the complex interplay between these parameters and to display a prediction of the elastic moduli of polymer composites. Moreover, it is generally accepted that these elastic calculations can be extended to the description of the viscoelasticity of filled polymers through the correspondence principle of Hashin [8]. In fact, the classical models, at least in their former developments, ignore direct interactions between fillers: fillers are considered as a homogeneous phase which interacts with the matrix or, at most, with the effective medium, and thus these models may be unefficient when the interactions between the fillers themselves rule the mechanical response of the composite. Moreover, they cannot simply account for the percolation of rigid fillers within a soft matrix.

The work presented in the following has been performed in the framework of a collaboration between several laboratories. The general strategy was first to elaborate model nanocomposites with controlled morphologies, to characterize their mechanical behavior in the linear domain and to develop mechanical models adapted to describe their behaviors. The first part of the paper presents the fabrication route to process polymers filled with spherical particles with diameters in the nanometer range. In order to understand reinforcement mechanisms depending on connectivity or on the aggregation state, we have focused on spherical fillers to avoid orientation effects. Materials were produced with different dispersion states: nanocomposites obtained from a mixture of hard/soft latex particles and nanocomposites made of encapsulated hard particles surrounded by a soft polymer shell. In the first part (Section 2), we provide details on sample preparation and characterization of the obtained materials. In a second part (Section 3), the viscoelastic behavior of the nanocomposites, characterized by dynamic mechanical measurements, is described. These experimental data illustrate the influence of filler-filler interactions on the linear mechanical behavior of these materials. At last (Section 4), different routes to model the (visco)elastic responses are discussed.

Processing of nanocomposite systems

A convenient way to process nanocomposite materials is based on the mixture of various aqueous suspensions (colloids) [9][10][11][12]. Emulsion polymerization is well known to provide in a simple way polymer colloidal suspensions with typical particle size in the range of ten to a few hundred nanometers. When both colloids are film forming, a co-continuous material can be expected, provided that the fraction of each component is large enough. Blending of hard and soft particles would lead, after film formation, to a random distribution of hard particles in the continuous soft matrix, with a certain probability to form aggregates depending on the volume fraction of filler. On the contrary, if the particles are structured, with a stiff core and a soft shell, then the material should consist, after shell coalescence, of a soft matrix with regularly dispersed stiff spherical domains, without contact between each other.

Nanocomposites obtained from PS-PBA and silica-PBA latex blends

Different composites, based on the mixture of a film forming latex (matrix) with aqueous suspension of fillers in required proportions, were prepared. Choice was made of a poly(butyl acrylate) matrix (PBA, glass transition temperature T g around -47 • C) filled with spherical particles of either polystyrene (PS, T g around 97 • C) or silica. Homopolymer latexes of PS or PBA were obtained through batch emulsion polymerization process. A typical recipe was as follows: deionized water = 900 g, monomer = 90 g, NaHCO 3 = 0.75 g, initiator = 0.75 g (K 2 S 2 O 8 for PS and (NH 4 ) 2 S 2 O 8 for PBA), emulsifier = 2.91 phm of NC12(3-(dimethyl dodecylammonium) propane-1-sulfonate) for PS, and 0.33 phm of sodium dodecyl sulfate for PBA. The temperature of polymerization = 70 • C. Latexes were cleaned using ion exchange resins removing most of the surfactant, in order to avoid a possible effect on film forming process, film microstructure or film properties. They were cleaned in a non-diluted state (around 10 wt.% solid content) until conductivity remains constant. Particle sizes were around 110 nm for PS and 135 nm for the PBA latex (as measured by dynamic light scattering, Malvern Autosizer Lo-C instrument). Silica nanoparticles have been synthesized according to the Stöber process [13]. In a typical procedure, absolute ethanol (1555 g, Acros Organics), de-ionized water (24.9 g) and ammonia (12.75 M, 96.6 g, Laurylab) were filled into a 5 L polypropylene flask. The mixture was stirred at 160 rpm to be homogenized and tetraethoxysilane (91.65 g, Fluka) was introduced at once. Reaction occurred at room temperature under continuous stirring for 2 weeks. The alcoholic silica suspension was dialyzed against water before use. Particle size was determined to be 125 nm by dynamic light scattering (DLS). The solid content of the resulting suspension was determined gravimetrically.

After blending different amounts of filler (PS or silica) (from 15 to 45 vol.%) and PBA latexes, films were made by evaporation under controlled atmosphere during a delay long enough to achieve a complete maturation, i.e., 2 weeks at 35 • C and 90% relative humidity. Although PBA matrix is transparent after maturation, highly filled composite films were rather opaque, suggesting aggregation of the hard phase [14], as it has been previously observed on similar PS/PBA systems by small angle neutron scattering [15]. Films were tested after the film formation process (and called "as-dried films") or after a thermal treatment at 140 • C for 4 h.

Nanocomposites obtained from encapsulated silica in P(S-BA) copolymer

The second nanocomposite system was based on encapsulated silica in a P(S-BA) copolymer. The choice of this copolymer (with glass transition close to 0 • C was motivated by the possibility to test the mechanical behavior. As a matter of fact, we tried first with films issued from latex of encapsulated silica in PBA, but these films were very difficult to handle. One reason could be the formation of a high degree of crosslinking in these systems due to both chain transfer to polymer and bimolecular termination by recombination [16]. Such a mechanism is expected to significantly affect film formation and mechanical properties.

Since silica is initially hydrophilic, its surface needs to be modified to make possible anchoring and polymerization of the hydrophobic butyl acrylate and styrene (co)monomers through an emulsion polymerization process. This was achieved by the chemical grafting of an alkoxysilane bearing a polymerizable methacryloyl end group (see Fig. 1). The grafting was performed as described previously by direct addition of 3-trimethoxysilyl propyl methacrylate (MPS, Acros Organics) to an aqueous silica suspension [START_REF] Bourgeat-Lami | Organic/Inorganic Hybrid Materials[END_REF]. The silica sol (30N50), with a mean hydrodynamic diameter of 68 nm as determined by DLS and 32 wt.% solid content, was kindly supplied by Clariant S.A. (France) and diluted in de-ionized water before use. A fixed amount of MPS (corresponding to 16 mol/m 2 silica surface) was introduced in the diluted silica sol (10 g L -1 ) containing 0.25 g L -1 sodium docedyl sulfate surfactant (SDS, Acros Organics). The reaction was conducted at room temperature for 1 week. The suspension was concentrated using an evaporating rotator before use.

The grafted silica particles were further engaged in a free radical polymerization process to make the polymer grow from their surface. The emulsion polymerization reaction was performed in batch at 70 • C in a 250 mL double wall glass reactor fitted with a condenser. The reactor was charged with 100 g of the aqueous suspension containing the grafted silica beads (37 g L -1 ) and the surfactant (a mixture of SDS : 1.25 g L -1 and poly(oxyethylene) isooctyl cyclohexyl ether : TX-405, 0.75 g L -1 , Aldrich). After degassing, the monomers from Aldrich (styrene : 36 g L -1 and butyl acrylate : 63 g L -1 ) and the initiator (potassium persulfate : KPS, 0.5 g L -1 , Acros Organics) were successively introduced at 70 • C under stirring to start polymerization. The monomer to polymer conversion was of 86% as determined gravimetrically.

The morphology of the composite particles was characterized using cryo-transmission electron microscopy (cryo-TEM). Using the method described elsewhere [18,19], thin liquid films of the particle suspension were formed on carbon membranes and rapidly frozen into liquid ethane. The particles were then observed at low temperature embedded in a preserving film of vitreous ice, using a Philips CM200 'Cryo' microscope operated at an accelerating voltage of 80 kV. The TEM image in Fig. 2 clearly attests for a successful encapsulation of the nanometric silica particles by the P(S-BA) copolymer. Although the silica beads appear to be uncentered in the polymer shell, they are expected to be homogeneously distributed in the coalesced latex film. However, a limited number of silica/silica contacts may be present.

For comparison purpose, blends of silica and P(S-BA) copolymer latexes were prepared by mixing together the silica sol and a latex of P(S-BA). This copolymer latex was obtained from the previous latex of encapsulated silica in P(S-BA), after having eliminated the grafted silica by centrifugation. Silica/P(S-BA) nanocomposites were obtained after maturation of latex blends or encapsulated latex during 2 weeks at 35 • C and 90% relative humidity.

Viscoelastic behavior

The dynamic shear moduli (G , G ) of various nanocomposites were measured as a function of temperature with a homemade inverted pendulum working in helium atmosphere [20]. Experiments were performed in the temperature 0,00001 0,0001 0,001 0,01 0,1 range [-170 to 150 • C] with a heating rate of 1 • C/min and at a fixed frequency of 0.1 or 1 Hz.

PS-PBA nanocomposites

The dynamic mechanical responses of PBA and PS/PBA "as-dried systems" (i.e., just after PBA maturation) are shown in Fig. 3. The properties of a film of pure PS (obtained by freeze-drying/hot pressing) are also reported. A significant reinforcement of the shear modulus is observed in between PS and PBA main (or α) relaxations (-32 • C ≤ T ≤ 118 • C). One can notice that this reinforcement is not linear: the faster increase is observed around 20 vol.%, i.e., near the geometric percolation threshold. In the same temperature range, the height and the width of the tan δ peak (associated with the PBA glass transition) decrease, with a slight shift of the relaxation to lower temperature. Nevertheless, differential scanning calorimetry measure- ments performed on these systems showed that the PBA glass transition temperature remains constant and equal to -47 • C (for a heating rate of 10 • C/min) whatever the filler content. Hence, the shift of T α to lower temperature as the filler content increases is probably due to a mechanical coupling effect. In addition to a higher level of the relaxed modulus, the temperature dependence of composite moduli differs from that observed for the pure matrix and for the filler phase. For the composite materials, the relaxed modulus first decreases until 70 • C and then increases before falling again above the PS main relaxation. These two counteracting phenomena lead to the presence of a bump in the the tan δ curves, located between the peaks associated with the PBA and PS main relaxations. As discussed in [21], these two phenomena are a consequence of the evolution of filler-filler interactions with temperature: in the low temperature range (T ≤ 70 • C), the decrease of relaxed composite moduli can be attributed to a decrease of filler-filler interactions via an immobilized matrix, whereas above 70 • C the increase of composite moduli should come from the very beginning of the coalescence between PS particles leading to an enhancement of filler-filler interactions.

The formation of a rigid network starts to occur but, as temperature approaches the PS glass transition, PS aggregates soften until complete transformation into rubbery behavior.

Annealing effects

Similar tests were performed on PS filled composites heated at 140 • C (well above the PS glass transition) during 4 h, to achieve the complete coalescence of neighboring PS particles. Fig. 4 shows G and tan δ versus temperature for sample filled with 35% PS before and after the thermal treatment. The annealing treatment leads to a higher relaxed modulus and a lower value of tan δ between the two relax-Fig. 6. Real shear modulus and loss factor vs. temperature (1 Hz) for PBA samples filled with PS (᭺) compared to silica filled nanocomposites ( ).

ations. Moreover, the decrease of G with temperature follows an evolution similar to that measured for pure PS. This indicates that the temperature dependence of the modulus is now governed by the stiff connected phase (PS). The very low value of the loss factor can be attributed to the presence of a rigid network preventing particle friction. Similar experiments performed with other filler concentrations are reported in Fig. 5. After thermal treatment, the increase of the rubbery plateau is observed even for the lowest filler volume fraction (15%). That indicates that particle aggregation may lead to an increase of the shape factor, and consequently has an effect on reinforcement even below the percolation threshold. These results obtained for "as-dried" and annealed samples underline the differences in mechanical responses between a system in which the particles are just in contact and a system in which the particles are strongly bonded. T/ Ta matrix T/ Ta matrix 25% encapsulated silica 25% blended silica matrix of encapsulated silica matrix of blended silica tan d Fig. 7. Real shear modulus and loss factor vs. normalized temperature for P(S-BA) samples filled with 25% silica (0.1 Hz) : (᭺) film obtained from blend of silica and matrix suspensions, ( ) film obtained from core(silica)/shell(matrix) latex.

Comparison with silica-PBA nanocomposites

The dynamic mechanical spectrometric responses of PBA-silica filled composites are compared with those of PBA-PS composites in Fig. 6. Two interesting points can be stressed: (1) In the glassy state, a slight reinforcement is observed for silica fillers (whose shear modulus is about thirty times higher than that of the glassy matrix), as expected from a simple mixture effect. On the contrary, no reinforcement is observed in the glassy state in case of PS-PBA systems since both phases have similar moduli.

(2) In the rubbery plateau, composites filled with silica particles exhibit a higher modulus and a weaker temperature dependence than those filled with PS particles. These results suggest once more that the strength of filler-filler interactions (Hydrogen bonds between silica particles versus weaker Van der Waals bonds between PS particles) has an effect on the level of the relaxed modulus and the temperature dependence.

Silica-P(S-BA) nanocomposites

The mechanical study was extended to P(S-BA) films filled with 25% silica. In Fig. 7 the viscoelastic behavior of nanocomposites made from a mixture of silica and P(S-BA) suspensions is compared with that of a nanocomposite fabricated from a core (silica)/shell P(S-BA) latex. In this figure, temperatures are normalized in order to take into account the shift of temperature of main relaxation observed between the two P(S-BA) matrices (8 • C), despite precautions that have been taken to have the same matrix in both films. These experiments show clearly that the reinforcement in the rubbery state decreases when the short range interactions between fillers are decreased. All these results, which show very different responses of well controlled materi-als and morphologies, are a challenge for micro-mechanical modeling.

Modeling approaches

Such approaches aim at modeling the mechanical interactions between the constituents of the considered composite materials so as to reproduce the main experimental observations. As a first step, this can be done in a simple way by using phenomenological models, such as the serial-parallel model of Takayanagi [22], which does not take explicitly into account any morphological characteristics but gives a direct uniaxial representation of the presumed mechanical coupling through adequate serial or parallel connections and some adjustable parameters. The serial-parallel model has been extended by taking into account the percolation concept [23,24]. In that case, the fraction of filler acting in parallel, corresponding to the percolated network of filler, is derived from percolation theory. It depends on two parameters, namely the percolation threshold (f i c ) and the critical exponent (t). Fig. 8 a shows the evolution of the shear modulus predicted by this model, calculated with parameters given in [24], i.e. a percolation threshold f i c = 0.154 for spheres, and a critical exponent t = 1.8. Such a model accounts well for the reinforcement in case of strong filler interactions (annealed PS-PBA systems) whereas it is too stiff for weaker filler interactions (as-dried PS-PBA systems) (see Fig. 8a). In this figure and as for the other modelings realized in this section, the reinforcement obtained at 60 • C has been considered, supposing that filler-filler interactions via immobilized matrix were lower at this temperature [21](case of as-dried PS/PBA nanocomposites).

More predictive treatments can be developed in the framework of homogenization theory by proposing an explicit connection between given morphological informations on the microstructure of a "representative volume element" (RVE) of the considered material and the overall mechanical response of the associated "equivalent homogeneous medium" (EHM). The classical way to describe the morphology is to use point correlation functions (of increasing order) of the local moduli. Bounds for the overall quantities can then be derived from variational approaches, when available: this is possible for elasticity or, for specific loading paths such as those used above (DMA), and also for linear viscoelasticity. However, in the case of polymer based composites, there is a huge difference between filler and polymer moduli (e.g., shear modulus of silica ≈ 30 GPa, glassy polymer ≈ 1 GPa, rubber ≈ 1 MPa), so that the obtained bounds (of first, second and third order using, respectively, first, second and third-order correlation functions) are gen- erally too far apart to be of practical use for comparison with experimental data (as illustrated for Hashin-Shtrikman bounds in Fig. 8b). In addition, point correlation functions are not a very convenient tool to express, as wished here, the connectivity of a given phase, which would need correlation functions of very high order, out of reasonable reach from experiment. An alternative method is related to the so-called "morphologically representative pattern-based approach" (MRP) [25], which describes the microstructure with use of finite composite patterns instead of points; the material content of such patterns, which are mapping the whole RVE, is supposed to be known as well as some statistical characteristics of their spatial distribution. Several typical morphologies can be modeled in this way, such as the matrix/inclusion situations with isolated particles in a continuous matrix, and vari-ational methods, similar to those related to classical "point" approaches, can be developed. Since the corresponding morphological information is richer, the resulting bounds are sharper. For instance, the real shear moduli of silica/P(S-BA) nanocomposites are compared in Fig. 9a to the upper bound for particulate composites described by the Hashin composite sphere assemblage (the so-called HSZ bound [26]). Since experimental data lie above the bound, even those for the nanocomposite issued from encapsulated silica, a third phase of additional chemical agent used for the process (non grafted MPS) or of immobilized matrix might play a role.

Estimates can also be derived from specific adequate assumptions, both for elastic and viscoelastic moduli. When applied to the Hashin composite sphere (or cylinder) assemblage, former models can be recovered (e.g., Kerner's model [27], the 3-phase model-Christensen and Lo's [28]-or the n-phase model-Hervé and Zaoui [29]). The latter generalized self-consistent estimate is obtained by an iteration technique, embedding the different phases or more specific patterns in the EHM itself [30]. Three-phase predictions are plotted in Fig. 8b and compared to Hashin-Shtrikman bounds [31] and the experimental data. This treatment is satisfactory when particles are well dispersed in the matrix and far enough apart from each other. However, as soon as particle aggregation occurs and especially when the rubbery plateau modulus is concerned, the 3-phase model yields far too soft predictions. Improvements of this treatment can be developed in order to better take into account aggregation effects by combining, as authorized by the MRP theory, different elementary patterns, as in [32]. For instance, Fig. 9 shows estimates calculated using two patterns, namely a stiff core in a thin soft shell and a soft shell, that makes possible to account for locally higher concentration of the filler phase. Another possibility is to embed the particles in an additional shell of stiffer matrix which could represent that part of the polymer matrix with reduced molecular mobility which is blocked within particles [21].

To go further, two different treatments have been investigated: both of them focus on the mechanical effects of contact between neighboring particles. The first one (Section 4.1) still refers to the MRP approach whereas the second one (Section 4.2) makes recourse to a discrete simulation.

MRP approach

The patterns were defined in order to enhance the continuity of the filler phase while the matrix still remains connected. Two internal pattern contents were tested: inside a spherical shell of matrix phase, the filler phase occupies either a cylinder or a disc touching the pattern border. The cylinder radius and the disc thickness depend on the filler concentration (Fig. 10). In the assemblage, the orientation θ of the disc or cylinder is assumed equiprobable and the spatial distribution of the centers of the patterns is isotropic. The self-consistent estimate of the effective elastic properties (C eff ) of such materials is obtained from the FEM computation of the mechanical response of a MRP embedded in an infinite medium with moduli (C eff ) and subjected to homogeneous strain conditions at infinity. This localization problem has been solved for all filler inclusions oriented with an angle θ = 0. Due to the MRP axisymmetric geometry, the average localization tensors A r θ=0 of each phase r (matrix or inclusion in the pattern with the orientation θ = 0) exhibit transversely isotropic symmetry. They are characterized by six components that are obtained by the computation of the average stress and strain tensors in the composite inclusion for four applied strain conditions (longitudinal and transverse shear, plane compression and uniaxial load). These calculations have been conducted in 2D FEM models using Fourier and axisymmetric modes. For these computations, the radius of the spherical "infinite" medium was four times larger than the radius of the composite inclusion (as shown by other calculations, a ratio between the two radii larger than 3 is sufficient [30]), the number of eight-noded elements was between 700 and 900, the degrees of freedom around 5000 [START_REF] Aussant | Internal Report LMS[END_REF].

In principle, one should solve the localization problems for all orientations θ and compute the average tensors A r θ θ ; the self-consistent estimate C eff is then given by:

C eff = C m + f i (C i -C m ) : A i θ θ (1)
where the superscripts i and m refer to the inclusion and matrix, respectively and f i is the filler concentration. However, the recourse to isotropy invariants allows one to compute A i θ θ from A i θ=0 only, which makes the computation easier. The implicit self-consistent equation given above (A i θ=0 depends on C eff ) is solved iteratively by means of a simple fixed point algorithm. Detailed equations are given in [START_REF] Aussant | Internal Report LMS[END_REF]. With a convergence criterion of the order of 10 -5 , depending on the morphology of the pattern, 5 to 60 iterations were necessary to obtain the self-consistent estimate.

Fig. 11 compares the self-consistent estimates for the shear modulus of the composite for filler inclusions being either cylinders, discs, spheres (this third case corresponds to the classical "3-phase" model where only the matrix is continuous). One can observe that the use of cylinders and discs leads to the appearance of a quasi-mechanical percolation located around 30% and 50%, respectively. These percolation thresholds are governed by the surface fraction of emerging filler within the MRP, which depends itself on the filler inclusion shape. For the 3-phase model, no percolation of the filler phase is observed. Thus, the use of adequate MRP seems to be a promising approach for modeling mechanical percolation through the self-consistent type homog- enization technique. Further improvements of this technique could be obtained by using various MRPs, with more realistic filler geometries and accounting for the actual statistics of contacts between fillers. As a whole, this method offers the advantage of dealing with different possible morphologies of particles and of predicting some moderate mechanical percolation effects for adequate pattern contents. Nevertheless, it suffers from the fact that only averaged pair interactions between patterns far away from each other can be taken into account through their interactions with the effective medium, while pair interaction between first neighbor patterns are smoothed out in a way that might not be an accurate description of their actual contribution to the effective behavior. Any improvement would depend on a more detailed description of the pattern contents, involving several fillers in close interaction in various configurations, and this could make the numerical computations more difficult. That is the reason why a different approach has been developed too, in order to enhance the influence of short range interactions between fillers through a discrete treatment.

Discrete simulation

Another attempt for a better understanding of elastic properties of nanocomposites is to focus on the short range interactions between filler particles and between matrix and filler. In this scope, a discrete numerical model taking explicitly into account the microstructure and the nature of contacts between phases has been developed [21,[START_REF] Chabert | Nanomaterials for Structural Applications[END_REF]. Calculations are made on a close-packing of hard and soft monodisperse spheres (e.g. Jagota [35]), which represent either the filler particles or the matrix particles (≈ 1000 spheres with a packing density ≈ 0.57). At each contact between neighbouring spheres, three forces and three moments are transmitted. These efforts are expressed as a linear function of the displacements and the rotations of the connected spheres, exhibiting different "contact stiffnesses" (normal, tangential, flexural and torsional) which depend both on the modulus of each connected sphere and on the contact area, characterized by the parameter α = (R c /R) 2 , where R c and R are the contact radius and the sphere radius, respectively. The larger the contact area (i.e. α), the higher the transmission of moments through the contact. Three kinds of contacts were considered: soft/soft, hard/hard and soft/hard contacts. As the matrix is continuous and totally wets the filler particles, the contact area radius of soft/soft and hard/soft contacts is approximated by the radius of the spheres themselves. For hard/hard contacts, we used a variable contact area (i.e. α hh ) in order to quantify the effect of the strength of filler-filler interactions on the effective modulus which is numerically deduced by applying adequate boundary conditions. Fig. 12 shows the effect of the strength of hard/hard contact (i. e. α hh ) on the evolution of the packing modulus with the filler ratio. When filler-filler interactions are stronger (α hh = 1), the effective modulus increases suddenly at the geometrical percolation threshold φ c = 18.5 vol.%. When filler-filler interactions are weaker (α hh ≤ 10 -2 ), i.e., when hard particles are allowed to roll on each other, there is no discontinuity at the percolation threshold. The weaker the strength of filler-filler interactions, the lower is the packing modulus. Experimental values obtained for PS-PBA nanocomposites at 60 • C in "as-dried" and "annealed" states are also reported in Fig. 12. After the annealing treatment, PS particles are strongly bonded. As a consequence, calculations made with α hh = 1 (forces and moments are transmitted as in bulk PS) are in very good agreement with experimental data. Before the annealing treatment, the strength of filler-filler interaction is weaker and the value α hh = 1.35 × 10 -2 , determined by inverse method, allows one to account very well for all PS volume fractions. The same approach could be used for modeling the whole temperature evolution of "as-dried" composite moduli. However, adjusting a new value of the contact area at each temperature is not quite satisfactory. In the detailed paper [21] a physical law describing the temperature evolution of the contact area is proposed.

Conclusion

As a conclusion, the study of various model systems, which makes possible to vary the contrast between filler and matrix, the filler ratio and the short range interactions between filler particles, has shown that the reinforcement increases when the filler ratio increases and when the strength of filler interactions increases. In case of strong short range filler interactions (for instance chemical bonds between PS particles after the annealing treatment), the relaxed modulus increases significantly above the geometric percolation threshold. This can be attributed to the formation of a quasi-rigid network. In case of weak filler interactions (for instance PS-PBA systems before the annealing treatment), the relaxed modulus decreases faster with temperature when PS concentration increases, suggesting a temperature dependence of filler interactions. Such a temperature dependence may not be invoked for silica-PBA systems and PS-PBA annealed systems. At last, it has been shown that the decrease of filler interactions thanks to an encapsulation process leads to a lower reinforcement. The challenge of modeling these various responses by micromechanical approaches has only been partly answered at the moment. The most promising treatments refer to the introduction of refined statistical information on the contacts between fillers in the MRP approach on the one hand and to the development of a physically sound modeling of the rate and temperature dependence of the contact interactions in the discrete simulation on the other hand.

(Fig. 1 .

 1 Fig. 1. Chemical formula of the 3-trimethoxysilyl propyl methacrylate silane coupling agent used in the silica encapsulation reaction.

Fig. 2 .

 2 Fig. 2. Cryo-TEM image of grafted silica particles (dark) encapsulated with P(S-BA) through emulsion polymerization.

Fig. 4 .

 4 Fig.4. Real shear modulus and loss factor for PBA samples filled with 35% PS, before and after thermal treatment (1 Hz).

Fig. 5 .

 5 Fig. 5. Real shear modulus and loss factor vs. temperature for "annealed" PS-PBA blends with different filler amounts (1 Hz).

Fig. 8 .

 8 Fig. 8. Reinforcement at 60 • C vs. volume fraction of filler for PS-PBA nanocomposites : (ᮀ) as-dried, () after thermal treatment (4 h at 140 • C). (a) Percolation approach with percolation exponent = 1.8 and geometrical percolation threshold = 0.154, (b) Generalized (3 phase) self-consistent estimation, classical HS bounds and experimental data.

Fig. 9 .

 9 Fig. 9. (a) P(S-BA) samples filled with 25% silica (0.1 Hz), comparison between experimental data and the upper HSZ bound. (b) Self-consistent estimates for the real shear moduli of PS/PBA nanocomposites using two MRPs.

Fig. 10 .

 10 Fig. 10. (a) Arrangement of spherical "morphologically representative patterns". (b) Internal structure of patterns: the filler phase occupies either a cylinder or a disc. Contacts between fillers are possible in the arrangement.

Fig. 11 .

 11 Fig. 11. Shear modulus of composites with inclusions being either cylinders, discs or spheres. For the computation, the matrix and filler phases have the properties of PBA (K = 1 GPa ; G = 2 MPa) and PS (K = 2.4 GPa ; G = 669 MPa) at 60 • C.

Fig. 12 .

 12 Fig. 12. Real shear modulus of PS-PBA nanocomposites at 60 • C before (ᮀ) and after () the annealing treatment. (-×-) Modelling with the discrete numerical simulation varying the strength of filler-filler interactions (α hh ).

Acknowledgements

This work was performed under the auspices of "Programme Matériaux" from CNRS under contract no. 113. One of the authors (E. Chabert) was supported through the Brite-Euram project BE 97-4448 administrated by the European Community. C. Graillat is also gratefully acknowledged for providing technical help.