Célia Da Costa Pereira

Didier Dubois
email: dubois@irit.fr

Henri Prade
email: prade@irit.fr

Andrea G B Tettamanzi
email: andrea.tettamanzi@univ-cotedazur.fr

Parsimonious Representation of Knowledge Uncertainty Using Metadata About Validity and Completeness

Keywords:

We investigate how metadata about the uncertainty of knowledge contained in a knowledge base can be expressed parsimoniously and used for reasoning. We propose an approach based on possibility theory, whereby a classical knowledge base plus metadata about the degree of validity and completeness of some of its portions are used to represent a possibilistic belief base. We show how reasoning on such belief base can be done using a classical reasoner.

INTRODUCTION AND RELATED WORK

In general, the process of getting a piece of information from a Knowledge Base (KB) is driven by practical purposes-such a piece of information will be used to justify certain decisions, for example. Its quality has therefore an important role to play in the success of the decisions made. The quality of a piece of information can be measured by considering different dimensions. Most contributions in the literature concentrate exclusively on the amount of true (known) facts in a KB for assessing its quality. [START_REF] Wick | Assessing confidence of knowledge base content with an experimental study in entity resolution[END_REF], for example, propose several algorithms for estimating a value of confidence based on the probability of a fact in a KB being true. In [START_REF] Dong | From data fusion to knowledge fusion[END_REF], the authors studied the applicability of data fusion techniques to solve the problem of knowledge base feeding. The criterion they used to construct quality knowledge bases was to identify the true values of data items among multiple observed values provided from different (and maybe unknown) sources with different reliabilities. However, as it has been pointed out for example by [START_REF] Razniewski | But what do we actually know?[END_REF],

In other words, a knowledge base is in general incomplete. This obviously has an impact on the overall quality of a KB-the more it is incomplete the lesser is its quality and the more the pieces of information extracted from it have to be considered (used) with caution.

The problem of representing both validity and completeness has begun to be dealt with for information stored in databases many years ago before being addressed for knowledge bases (KBs). For example, we can consider the model of database integrity proposed by [START_REF] Motro | Integrity = validity + completeness[END_REF] and the work by [START_REF] Demolombe | Answering queries about validity and completeness of data: From modal logic to relational algebra[END_REF], who used modal logic for reasoning about validity and completeness of information stored in relational databases as precursors of some ideas that have later been adopted for KBs. However, the representation of the incompleteness in information stored in the databases has been inspired by earlier works on the representation of incompleteness in knowledge bases as the ones proposed by [START_REF] Levesque | Incompleteness in knowledge bases[END_REF][START_REF] Levesque | The logic of incomplete knowledge bases[END_REF] and the one proposed by [START_REF] Collins | Reasoning from incomplete knowledge[END_REF] for reasoning with this kind of knowledge bases.

Recent work on annotating KBs with metadata about their completeness has been done, in the context of the semantic Web, by [START_REF] Darari | Completeness statements about RDF data sources and their use for query answering[END_REF][START_REF] Razniewski | But what do we actually know?[END_REF], who studied the way in which statements about completeness can be used when answering queries. According to their approach, it is then possible, given a statement about a topic, to specify if information about it in the base is complete or not. However, the gradual view of completeness in data sources has not been considered.

Solutions to construct a possibilistic belief base from a crisp KB using topical validity and completeness metadata, like [START_REF] Da Costa Pereira | Handling topical metadata regarding the validity and completeness of multiple-source information: A possibilistic approach[END_REF], suffer from some limitations, mainly due to the fact that, in order to guarantee consistency, they have to sacrifice much of the expressive power of the knowledge representation language. In particular, the "facts" that can be recorded in the knowledge base are restricted only to ground formulas without negation and disjunction. Indeed, negative information (i.e., facts that do not hold) is critical for the correctness of queries involving negation [START_REF] Razniewski | But what do we actually know?[END_REF]. Completeness and negative information are closely related: if we know that a portion of a KB is complete, it is as if we knew an infinity of negated facts (all those relevant to that portion that are not in the KB).

Motivated by the above considerations, we want to answer the following research question: is it possible that a classical knowledge base plus metadata information on the (gradual) validity and completeness with respect to a few configurations (groups, portions, subjects, topics of statements it contains), enables one to represent a possibilistic belief base and perform possibilistic inferences by using a classical reasoner? This research question leads us to the following subquestion: which should be a suitable definition for such a configuration which in turn will allow us to define appropriate validity and completeness functions?

We propose a framework based on possibility theory to represent and reason about gradual notions of validity and completeness in KBs. Since it would be impractical to associate values of possibility and necessity to each single assertion in a knowledge base (KB), we show that, thanks to validity and completeness metadata, it is possible to express degrees of possibility/necessity for formulas entailed by the KB in a parsimonious way (i.e., without having to associate a weight to each single formula) and to perform possibilistic inferences on top of a classical KB, considering, in addition to possibilistic uncertainty, also negative information.

In particular, we present a way to represent validity and completeness information (with respect to particular slices of information) in a knowledge base in a way that permits said validity and completeness information to simulate the knowledge base as a possibilistic knowledge base, where the possibilistic information is derived exclusively from the validity and completeness. The advantage of this approach is that the validity and completeness is only assessed at the slice level, where in a possibilistic knowledge base, the possibility distribution needs to be defined for all facts. Therefore, when the number of slices is much smaller than the number of facts, the proposed representation is much more parsimonious.

The paper is organized as follows: Section 2 gives then some background about the formal tools we use. We present our proposal in Section 3, which explains how (gradual) validity and completeness are related to the beliefs of an agent. Finally, Section 4 discusses some possible applications of our proposal.

BACKGROUND

We first provide a brief refresher on possibility theory, before recalling the basics of possibilistic logic, a logic where classical formulas are weighted in terms of certainty.

Possibility Theory

Fuzzy sets [START_REF] Zadeh | Fuzzy sets[END_REF] are sets whose elements have degrees of membership in [0, 1]. Possibility theory [START_REF] Dubois | Possibility Theory-An Approach to Computerized Processing of Uncertainty[END_REF]) is a mathematical theory of uncertainty that relies upon fuzzy set theory, in that the (fuzzy) set of possible values for a variable of interest is used to describe the uncertainty as to its precise value. At the semantic level, the membership function of such set, π, is called a possibility distribution and its range is [0, 1]. A possibility distribution can represent the available knowledge of an agent. π(I ) represents the degree of compatibility of the interpretation I with the available knowledge about the real world if we are representing uncertain pieces of knowledge. By convention, π(I ) = 1 means that it is totally possible for I to be the real world, 1 > π(I ) > 0 means that I is only somehow possible, while π(I ) = 0 means that I is certainly not the real world.

A possibility distribution π is said to be normalized if there exists at least one interpretation I 0 s.t.

π(I 0 ) = 1, i.e., there exists at least one possible situation which is consistent with the available knowledge.

Definition 1. (Possibility and Necessity Measures) A possibility distribution π induces a possibility measure and its dual necessity measure, denoted by Π and N respectively. Both measures apply to a classical set S ⊆ Ω and are defined as follows:

Π(S) = max I ∈S π(I ); (1)
N(S) = 1 -Π( S) = min I ∈ S {1 -π(I )}. (2)
In words, Π(S) expresses to what extent S is consistent with the available knowledge. Conversely, N(S) expresses to what extent S is entailed by the available knowledge. It is equivalent to the impossibility of its complement S-the more S is impossible, the more S is certain. A few properties of Π and N induced by a normalized possibility distribution on a finite universe of discourse Ω are the following. For all subsets A, B ⊆ Ω:

1. Π(A ∪ B) = max{Π(A), Π(B)}; 2. Π(A ∩ B) ≤ min{Π(A), Π(B)}; 3. Π( / 0) = N( / 0) = 0; Π(Ω) = N(Ω) = 1; 4. N(A ∩ B) = min{N(A), N(B)}; 5. N(A ∪ B) ≥ max{N(A), N(B)}; 6. Π(A) = 1 -N( Ā) (duality); 7. N(A) > 0 ⇒ Π(A) = 1; Π(A) < 1 ⇒ N(A) = 0; A consequence of these properties is that max{Π(A), Π( Ā)} = 1. In case of complete ig- norance on A, Π(A) = Π( Ā) = 1.

Possibilistic Logic

Before going into details about possibilistic logic, we would like to put forward our motivation for such a logic for handling uncertainty in this work. Information is often pervaded with uncertainty, and it may be convenient to associate pieces of information with certainty levels. These certainty levels can often be qualitatively assessed only using a finite completely ordered scale ranging from "fully certain" to "not certain at all", with intermediary levels such as "almost certain", or "somewhat certain". Possibility theory offers such a qualitative setting, when a finite subset of [0, 1] including 0 and 1 is used and then only the ordering of the degrees in [0, 1] is meaningful, in agreement with the use of max and min operators. Moreover, the inverse mapping 1-(•) exchanges the necessity scale with a possibility scale, such as "fully possible", "quite possible", "somehow possible", "not possible at all (= impossible)". In the following, the pieces of information are associated with certainty levels which are viewed as lower bounds of necessity measures. Then, the min-decomposability of necessity measures with respect to conjunction acknowledges the fact that to be certain at least at some level α that a conjunction of facts is true, we should be certain at least at level α that the truth of each fact is certain at least at level α.

Possibilistic logic [START_REF] Dubois | Possibilistic logic[END_REF] has been originally motivated by the need to manipulate syntactic expressions of the form (φ, α) where φ is a classical logic formula, and α is a certainty level, with the intended semantics that N(φ) ≥ α, where N is a necessity measure. It is then possible to consider that all the propositions of the considered language can be totally ordered on a given scale. In our case, propositions are formulas. Besides, in possibilistic logic, a level of inconsistency can be associated with a knowledge base as recalled now.

A possibilistic knowledge base B is a set of possibilistic logic formulas

{(φ i , α i ) | i = 1, . . . , m}. Clearly, B can be layered into a set of nested classical bases B α = {φ i | (φ i , α i ) ∈ B and α i ≥ α} such that B α ⊆ B β if α ≥ β. Proving syntacti- cally B ⊢ (φ, α)
amounts to proceeding by refutation and proving B ∪ {(¬φ, 1)} ⊢ (⊥, α) by repeated application of the resolution rule

(¬φ ∨ ψ, α), (φ ∨ ν, β) ⊢ (ψ ∨ ν, min(α, β)). Moreover, B ⊢ (φ, α) if and only if B α ⊢ φ and α > inc(B), where inc(B) is inconsistency level of B defined as inc(B) = max{α | B ⊢ (⊥, α)}. It can be shown that inc(B) = 0 iff B * is consistent, with B * = {φ i | (φ i , α i ) ∈ B}.
Thus reasoning from a possibilistic base just amounts to reasoning classically with subparts of the base whose formulas are strictly above the certainty level.

A possibilistic knowledge base

B = {(φ i , α i ) | i = 1, . . . , m} encodes the constraints N(φ i ) ≥ α i .
B is thus semantically associated with a possibility distribution [START_REF] Dubois | Possibilistic logic[END_REF]

π B (I ) = min i=1,...,m max(φ I i , 1 -α i ),
where φ I i = 1 if I is a model of φ i , and φ I i = 0 otherwise. As it can be seen, π B (I ) is all the larger as the interpretation I makes false only formulas with low certainty levels. π B is the largest possibility distribution, i.e., the least committed distribution assigning the largest possibility levels in agreement with the constraints N(φ i ) ≥ α i for i = 1, . . . , m. The distribution π B rank-orders the interpretations I of the language induced by the φ i 's according to their plausibility on the basis of the strength of the pieces of information in B. If the set of formulas B * is consistent then the distribution π B is normalized (i.e., ∃I , π B (I ) = 1). The semantic entailment is defined by

B |= (φ, α) iff ∀I , π B (I ) ≤ π {(φ,α)} (I ).
Reasoning by refutation in propositional possibilistic logic is sound and complete, applying the syntactic resolution rule. Namely, it can be shown that

B |= (φ, α) iff B ⊢ (φ, α) and inc(B) = 1 -max I π B (I ).
Algorithms for reasoning in possibilistic logic and an analysis of their complexity, which is similar to the one of classical logic, multiplied by the logarithm of the number of levels used in the necessity scale, can be found in [START_REF] Lang | Possibilistic logic: complexity and algorithms[END_REF].

As it was hypothesized by [START_REF] Motro | Integrity = validity + completeness[END_REF] for the case of relational databases, here, to formalize the concepts of validity and completeness in the the case of knowledge bases, we shall assume the existence of a hypothetical knowledge base that captures a designated environment of the real world perfectly. The knowledge base K mentioned in the paper is then an approximation of such hypothetical knowledge base.

When dealing with relational databases, only the statements explicitly present in the database are considered as true (valid). The others are considered as false (closed world assumption). When dealing with sets of formulas, the true statements are those explicitly represented in the dataset, plus those which can be inferred thanks to a reasoner. However, due to the open world assumption, we cannot suppose that the other statements are false-the truth status of some statements may be unknown in case of incomplete knowledge.

In this section, we recall the notions of validity and completeness, first introduced in [START_REF] Demolombe | Answering queries about validity and completeness of data: From modal logic to relational algebra[END_REF], and made gradual in the setting of possibilistic logic [START_REF] Dubois | Valid or complete information in databases -A possibility theory-based analysis[END_REF] for dealing with relational databases and adapt them to the more general setting of knowledge bases, where (i) the open world assumption holds, (ii) implicit knowledge can be inferred by logical deduction, and (iii) negative information is also taken into account unlike what was proposed in [START_REF] Da Costa Pereira | Handling topical metadata regarding the validity and completeness of multiple-source information: A possibilistic approach[END_REF].

It is often the case that the knowledge contained in a knowledge base is not all certain to the same degree. There will be statements whose truth is absolutely certain. This might be the case of ontological axioms or integrity constraints. Other groups of statements, obtained for example from the same source or covering the same subject, might have the same degree of certainty, but statements from different portions of the knowledge base might be believed with greater or lower certainty.

Our working hypothesis is that, as suggested by [START_REF] Da Costa Pereira | Handling topical metadata regarding the validity and completeness of multiple-source information: A possibilistic approach[END_REF], the degree of certainty of every piece of information depends on the degree to which the knowledge base is valid and complete with respect to all groups, portions, subjects, topics (or whatever else we wish to call them) of statements it contains. We think that an intuitive name for this notion of a semantically determined homogeneous portion of a knowledge base may be a slice and we will stick to this term from now on.

While it is true that the term slice might lead to confusion with the same term as used in the hypercube data model [START_REF] Gray | Data cube: A relational aggregation operator generalizing group-by, cross-tab, and sub totals[END_REF], as a matter of fact, the suggestion that a slice may be a subset defined by fixing one or more dimensions constitutes a good and useful intuition. Indeed, if we interpret slices as knowledge "topics" or "domains", then this is exactly what slices are, with the specificity that here every "dimension" can be viewed as a binary truth assignment to a formula, e.g., in a hypotetical knowledge base about travel,"x is a flight and x departs from London", thus giving the slice of the knowledge base that provides information about flights departing from London.

Postulates

To be able to talk about the validity and completeness of information stored in a knowledge base with respect to a particular slice, we need a formal way of defining the latter. We begin with the most general and neutral definition, whereby a slice T is just a set of formulas. A more precise definition is deferred to when we will have discussed the properties that a slice must satisfy.

A few basic postulates for slices, based on common sense arguments, are the following.

P1 Slices are non-empty. We assume slices are defined by the designer or a user of a knowledge base in order to state metadata about the validity and completeness of portions of knowledge in the base; defining an empty slice would defeat its purpose.

P2 Slices are all distinct. Defining two equivalent slices would be redundant and of no practical use; therefore, we can safely bar this possibility.

P3 For every slice not contained in another slice (we may call it a "top-level" slice), there exists a formula entailed by the knowledge base that only belongs in that slice and in no other slice.

What justifies stating Postulate P3 is that for every portion of a knowledge base a knowledge engineer might want to define in order to state metadata on it, one would expect that either that portion is a proper subset of another portion (i.e., a sort of sub-topic or sub-domain), or, if it is not, then its very definition is motivated by the existence of some facts that are not covered by other slices. For instance, in a knowledge base about travel, I might want to define a slice about "airports" because there are assertions involving airports, like "London Heathrow (LHR) has four operational terminals", that do not deal with any other possible slices, like "flights", "airlines", "aircraft", and so on. Or I might choose to define "aviation", which includes all of them, including the assertion about LHR, which is not covered by any other existing slice. Let K be a set of formulas in a decidable logical language L, for which there exists a reasoner capable of performing inferences and deduce other formulas which are not explicitly contained in K.

Under the closed-world hypothesis typical of databases, which is the setting in which [START_REF] Dubois | Valid or complete information in databases -A possibility theory-based analysis[END_REF] was stated, it would be reasonable to admit that what cannot be deduced from an agent's knowledge base corresponds to what the agent believes to be false.

However, in the case of a knowledge base, the open-world assumption holds and the agent is capable of performing logical inferences (e.g., thanks to a reasoner). Therefore, we must think in terms of logical entailment of formulas.

Without loss of generality, we will assume a Herbrand semantics for L. Definition 2. The Herbrand base of L is the set H L of all ground atoms in L. An interpretation (or model) is a function I : H L → {0, 1}, which can also be viewed as a subset of the Herbrand base, I ⊆ H L (the set of all atoms φ such that φ I = 1). We denote Ω 2 H L the set of all interpretations.

We write K |= φ to denote the fact that formula φ is a logical consequence of all the formulas in K. Using a sound and complete reasoner, if K |= φ, φ can also be deduced from K by the agent (which we write K ⊢ φ), whereas if K ̸ ⊢ φ (φ cannot be deduced from K), this means that K ̸ |= φ (φ is not a logical consequence of K). Finally, given a set S of formulas, K |= S if and only if ∀φ ∈ S, K |= φ.

Graded Validity and Completeness

The purpose of a knowledge base is to store axioms and assertions that summarize an agent's knowledge about the world or, at least, a limited portion of the world, which is relevant to the problem the agent is intended to deal with. We will take an objectivist stance by assuming that there exists, among all the possible interpretations on language L, one that reflects the actual state of affairs. Let us denote such interpretation by I * . Then we may say that the objective truth of any formula φ ∈ L is given by φ I * . To be absolutely clear about that, we are assuming that I * is real and an objective truth exists for every formula, independently of a knowledge base K and of the agent using it. As a matter of fact, the knowledge represented in (a slice of) K might (and will, in general) not reflect reality perfectly or accurately.

Given this premise, the notions of validity and completeness of a knowledge base K with respect to a slice may be defined as follows:

• K is valid with respect to a slice iff, for every formula φ in that slice, K |= φ implies that I * |= φ, i.e., φ is objectively true; • K is complete with respect to a slice iff, for every formula φ in that slice, K ̸ |= φ implies that I * ̸ |= φ, i.e., φ is objectively false. As pointed out in Section 1, we will use the term beliefs to refer to (possibly partial, incomplete, or invalid) information held by an agent. An agent may then believe something to different degrees. We suppose that these degrees depend on both the degree of completeness of the sets of statements and on the reliability or trustworthiness of the information source. For example [START_REF] Da Costa Pereira | Handling topical metadata regarding the validity and completeness of multiple-source information: A possibilistic approach[END_REF], information related to an Air France flight should be complete if the source is the Air France carrier itself. However, the completeness could be lower if the source is a private travel agency with a partial coverage about the current flights from the different companies including those of Air France. Similarly, the degree of trust to be associated with information fed by a clerk should be lower than the one to be associated with information fed by a supervisor. Still, we would like to stress that the way in which such degrees are obtained is out of the scope of this paper.

As pointed out in Section 2, possibility theory is well suited to model degrees of certainty or, dually, degrees of possibility. Besides, possibility theory, unlike other theories of uncertainty like probability theory, is well -suited to model total ignorance which is necessary to represent situations in which we have, for example, both K ̸ |= φ and K ̸ |= ¬φ. This is the reason why, here, we adopt this theory to represent the gradual property of both the reliability of an information source as well as the completeness of information regarding a particular slice.

We assume that K is a consistent, classical (as opposed to possibilistic) knowledge base, i.e., K contains statements (such as axioms and assertions) expressed in one of the decidable logical languages usually employed to represent knowledge in practical applications (examples might be Datalog, description logics, RDF + RDFS, or one of the profiles of OWL).

We assume that, in addition to K, metadata about validity and completeness of information stored in K are given in the form of two functions, Val and Comp, which associate a degree of validity and completeness, respectively, to a number of slices defined on K. Let S K ⊂ 2 L the set of such slices. In practice, these two functions might be implemented by a lookup table, listing their values for each defined slice. Definition 3. Let Val : S K → [0, 1] be such that, for each slice T ∈ S K , Val(T ) is the degree to which K contains valid information about slice T , which means, for all formulas φ such that K |= φ and φ ∈ T , N(φ) ≥ Val(T ).

Intuitively, if we can deduce φ from the knowledge base K, and φ belongs in slice T , and we know that the source who fed K is (somehow) reliable for domain of T then the agent should believe φ at least as much as the degre to which the source is reliable. If the source is fully reliable then φ will be certain for the agent.

Definition 4. Let Comp : S K → [0, 1] be such that, for each slice T ∈ S K , Comp(T ) is the degree to which K contains complete information about slice T , which means, for all formulas φ such that K ̸ |= φ and φ ∈ T ,

Π(φ) ≤ 1 -Comp(T ).
Intuitively, if we cannot deduce φ from the knowledge base K, and φ belongs in slice T , and we know that information in K about slice T is (somehow) complete, then φ should be certainly false.

It is reasonable to assume that, given two slices T and T ′ ,

T ⊆ T ′ ⇒ Val(T ) ≥ Val(T ′ ), (3)
T ⊆ T ′ ⇒ Comp(T ) ≥ Comp(T ′ ). (4)
Indeed, if we are told that K contains reliable (resp. complete) information about a broader domain T ′ to a given degree α, then K cannot be less reliable (complete) about a narrower (i.e., more specific) domain T ; if anything, it might be more reliable (complete) about T if a more reliable (complete) source is available just for T . The extent to which the agent believes φ depends on (i) what is supposed to be known about φ -can we deduce φ from K? -, and (ii) on the validity and completeness of K with respect to the slices that contain φ. That being the case, K, together with the metadata provided by Val and Comp, should allow us to compute the degree of possibility and necessity for any arbitrary formula φ, as follows:

Π(φ) = 1, if K |= φ, min T :φ∈T {1 -Comp(T )}, otherwise;
(5)

N(φ) = max T :φ∈T Val(T ), if K |= φ, 0, otherwise. (6)
Let us call π the hypothetical possibility distribution that induces the possibility and necessity measures of Equations 5 and 6 and let B a hypothetical possibilistic belief base corresponding to it. Furthermore, among all possibility distributions compatible with Π and N, we will select the one that makes the least commitment, i.e., the maximal (most general) one.

Existence Conditions

We now derive a necessary condition for the existence of such a possibility distribution π.

Let φ be such that K |= φ; if K is consistent, K ̸ |= ¬φ. By the duality property of the possibility and necessity measures, it must be Π(φ) = 1-N(¬φ); this is satisfied by Equations 5 and 6, since Π(φ) = 1 and N(¬φ) = 0. Proposition 1. Let us assume that K is consistent and that there exists a possibility distribution π that induces the possibility and necessity measures of Equations 5 and 6. Then, for all φ such that K |= φ, max

T :φ∈T Val(T ) = max T ′ :¬φ∈T ′ Comp(T ′ ). (7)
Proof. If measures N and Π are induced by the same possibility distribution π, it must be N(φ) = 1 -Π(¬φ); therefore, by Equations 5 and 6, we can write max

T :φ∈T Val(T ) = N(φ) = 1 -Π(¬φ) = 1 -min T ′ :¬φ∈T ′ {1 -Comp(T ′ )} = max T ′ :¬φ∈T ′ Comp(T ′ ),
which proves the thesis.

A formula φ such that K ̸ |= φ and K ̸ |= ¬φ poses no problem, because Comp and Val do not interact:

Π(φ) = min T :φ∈T {1 -Comp(T )}, Π(¬φ) = min T ′ :¬φ∈T ′ {1 -Comp(T ′ )}, N(φ) = N(¬φ) = 0.
We now prove that a formula and its negation cannot belong in the same slice, unless the two functions Val and Comp are identical.

Proposition 2. Either Val(T ) = Comp(T ) for all slice T , or, for all formula φ and for all slice T , φ ∈ T ⇒ ¬φ ̸ ∈ T .

Proof. By contradiction: we show that if φ ∈ T and ¬φ ∈ T , a contradiction can be derived. Let φ be a formula belonging in just one slice T . If the slices for which Val or Comp are defined as all distinct, there will always be at least one such formula. If not, the equivalent slices can be merged together so that all slices are distinct. By the assumption, ¬φ ∈ T too. Now, we apply Equations 5 and 6 to compute the possibility and necessity of both φ and ¬φ: without loss of generality, let us assume K |= φ, and therefore, K ̸ |= ¬φ; then

Π(φ) = 1, Π(¬φ) = 1 -Comp(T ), N(φ) = Val(T ), N(¬φ) = 0.
By the duality property,

Val(T ) = N(φ) = 1 -Π(¬φ) = Comp(T ).
We can observe that the case in which Val = Comp defeats the very purpose of having the two complementary notions of validity and completeness; if that were the case, it would suffice to call the function into which those two notions would confound themselves "trust", because it would reflect a general notion of reliability of information about a given slice. Therefore, since we are interested in investigating the use of validity and completeness as two distinct notions, in what follows, we will make the assumption that, in general, Val ̸ = Comp. As a consequence, we now know that any acceptable definition of what a slice is will have to satisfy the postulate that a formula and its negation cannot belong in the same slice: formally, for every slice T and formula φ, φ ∈ T ⇒ ¬φ ̸ ∈ T . For instance, a slice might be defined as the set of (ground) formulas that are satisfied by a formula with free variables (i.e., a query).

With this notion of slice (i.e., such that if a formula belongs to a slice, then the negation of that formula does not belong in the slice), we are able to prove possibilistic generalizations of results obtained by [START_REF] Demolombe | Database validity and completeness: Another approach and its formalisation in modal logic[END_REF] in a KD doxastic logic.

First of all, the fact that a formula and its negation cannot belong to the same slice motivates the definition of the dual of a slice. Definition 5. Let T be a slice. The dual of T , denoted ¬T , is the slice such that φ ∈ T iff ¬φ ∈ ¬T .

The dual of a slice is thus a sort of complement, but not in the set-theoretic sense, because there may exist a formula ψ such that ψ ̸ ∈ T and ψ ̸ ∈ ¬T ; therefore, in general, ¬T ̸ = T . A straightforward consequence of Definition 5 is that ¬(¬T ) = T .

The intuition behind the next proposition is that if the knowledge base is consistent and if information is complete concerning both T and ¬T , then, for each formula φ in T or in ¬T , either the agent believes φ or it believes ¬φ.

N(φ) = 1 -Π(¬φ) ≥ Comp(¬T ); N(¬φ) = 1 -Π(φ) ≥ Comp(T ); hence, max{N(φ), N(¬φ)} ≥ ≥ max{Comp(¬T ), Comp(T )} ≥ min{Comp(T ), Comp(¬T )}.
T ′ :φ∈T ′ Val(T ′ ) ≤ Val(T ),
which, together with Equation 9, yields the thesis.

No other case being possible, this concludes the proof.

It has been proven [START_REF] Demolombe | Database validity and completeness: Another approach and its formalisation in modal logic[END_REF] that, for a consistent base K, if K is complete about ¬T (the complement of slice T ), then K is valid about slice T ; the following an extension of that result to the gradual case. Proposition 5. If K is consistent, for all slice T , Comp(¬T ) ≤ Val(T ).

Complexity of Reasoning

The results that have been derived above show that one can "simulate", as it were, a possibilistic belief base by means of a crisp base K together with metadata about the validity and completeness of K with respect to a number of "slices" (i.e., sets of formulas). It is not important to know a least-commitment possibility distribution that induces the possibility and necessity measures of Equations 5 and 6 or to represent one of its corresponding possibilistic bases B explicitly, since K, together with its associated metadata Val and Comp, is sufficient to compute any possibilistic inference using any available classical reasoner, as demonstrated by the algorithm shown in Figure 1, adapted from (da Costa Pereira et al., 2017).

Require: K ⊂ L: a consistent KB; φ ∈ L: a formula; Ensure: N(φ).

1: α ← 0 2: if K |= φ then end for 14: end if 15: return α. Property 2. The cost of Algorithm 1, is at most ∥S K ∥ + 2 classical inferences, where ∥S K ∥ is the number of slices defined for K.

Proof. Algorithm 1 needs first of all to execute at most two classical inferences: the one in Line 2 and, in case K ̸ |= φ, the one in Line 8. Then, checking whether a formula belongs in a topic costs at most one classical inference and it has to be done for all the slices defined for K.

Notice that, according to this result, while the cost of a possibilistic inference is higher then the cost of a classical inference, it is so only by a factor which depends on the number of slices defined on the KB. It is to be expected that this number will, in general, be much smaller than the number of facts contained in the KB. In other words, the overall complexity of possibilistic inference will be in the same class as classical inference.

DISCUSSION AND CONCLUSION

We have shown that a classical knowledge base plus metadata information on the (gradual) validity and completeness of its "slices" enables one to represent a possibilistic belief base and perform possibilistic inferences by using a classical reasoner at a cost which, albeit larger than the classical counterpart by a multiplicative factor proportional to the number of slices, lies in the same complexity class. All of our results are valid for the general case of a decidable fragment of first-order logic and thus they can be readily transferred to state-of-the-art and popular knowledge representation languages, like Datalog and RDF + OWL and their reasoners. This also means that our suggestion to use gradual metadata about validity and completeness may be applied to representing and reasoning with possibilistic uncertainty on top of the standard infrastructure of the semantic Web, without requiring any ad hoc extension and at a reasonable cost. In that setting, one way of implementing the notion of a slice might be through RDF named graphs.

Future work includes demonstrating how our proposal can be deployed on the semantic Web infrastructure to represent and reason about uncertain knowledge with a proof-of-concept implementation.

  Assuming the usual definition of satisfaction (given an interpretation I ∈ Ω and a formula φ ∈ L, I |= φ if and only if φ evaluates to true in I ), we define the notion of entailment as follows: K |= φ if and only if, for every interpretation I ∈ Ω, I |= K implies I |= φ.

Proposition 3 .

 3 If K is consistent, for all slice T , min{Comp(T ), Comp(¬T )} ≤ min φ∈T max{N(φ), N(¬φ)}. Proof. We prove this proposition by showing that, for all formula φ ∈ T , min{Comp(T ), Comp(¬T )} ≤ max{N(φ), N(¬φ)}, from which the thesis follows. Given a formula φ ∈ T , there are just three mutually exclusive cases: Case I. K |= φ and, therefore, by the consistency of K, K ̸ |= ¬φ; we thus have, by Def. 4, N(φ) = 1 -Π(¬φ) ≥ Comp(¬T ); hence, max{N(φ), N(¬φ)} ≥ ≥ N(φ) ≥ Comp(¬T ) ≥ min{Comp(T ), Comp(¬T )}. Case II. K ̸ |= φ and K ̸ |= ¬φ; we thus have, by Def. 4,

  Case III. K |= ¬φ and, therefore, by the consistency of K, K ̸ |= φ; we thus have, by Def. 4,N(¬φ) = 1 -Π(φ) ≥ Comp(T ); hence, max{N(φ), N(¬φ)} ≥ ≥ N(¬φ) ≥ Comp(T ) ≥ min{Comp(T ), Comp(¬T )}.Since the three above cases are exhaustive for every formula φ ∈ T , this concludes the proof. Proposition 4. If π is the least commitment possibility distribution such that Equations 5 and 6 hold, for all slice T , Val(T ) = min φ∈T :K|=φ N(φ). (8) Proof. By Def. 3, Val(T ) ≤ N(φ) for all φ ∈ T such that K |= φ; therefore, we can write Val(T ) ≤ min observe that the set of slices {T ′ : φ ∈ T ′ } always includes T as one of its elements, because φ ∈ T ; therefore, max T ′ :φ∈T ′ Val(T ′ ) ≥ Val(T ); furthermore, • either T is a top-level slice and, by Postulate P3, there exists a formula φ * ∈ T that does not belong to any other slice, in which case max T ′ :φ * ∈T ′ Val(T ′ ) = max{Val(T )} = Val(T ), whence we can conclude that min φ∈T :K|=φ max T ′ :φ∈T ′ Val(T ′ ) = Val(T ), which is the thesis; • or there exists another slice T such that T ⊂ T , for which, by Equation 3, Val( T ) ≤ Val(T ), which leads us to conclude that min φ∈T :K|=φ max

  Proof. By Def. 4, Comp(¬T ) ≤ 1 -Π(¬ψ) = N(ψ), for all ¬ψ ∈ ¬T (and, therefore, ψ ∈ T ) such that K ̸ |= ¬ψ; let β = min ψ∈T :K̸ |=¬ψ N(ψ); then we can write Comp(¬T ) ≤ β. Clearly, β ≤ min φ∈T :K|=φ N(φ), because {φ ∈ T : K |= φ} ⊆ {ψ ∈ T : K ̸ |= ¬ψ}, since K |= φ ⇒ K ̸ |= ¬φ. Therefore, by Proposition 4, we have Comp(¬T ) ≤ β ≤ min φ∈T :K|=φ N(φ) = Val(T ), which proves the thesis.

Figure 1 :

 1 Figure 1: An algorithm that "simulates" a possibilistic inference from B using K, Val, and Comp.Property 1. Algorithm 1 is correct (i.e., it computes N(φ)).Proof. If K |= φ, Equation 6 is applied; otherwise, Equation 5 together with duality: N(φ) = 1 -Π(¬φ).

ACKNOWLEDGEMENTS

This work has been partially supported by the French government, through the 3IA Côte d'Azur "Investments in the Future" project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002.


Clément Freymond

Alain Guinault

Matthieu Gervais

Mirosla Pluta

Tomasz Makowski

Ewa Piorkowska

Cyrille Sollogoub
email: cyrille.sollogoub@lecnam.net

Crystallization behavior and morphological features of ethylene-vinyl alcohol 44 copolymer

Keywords: thermal properties, crystallization, EVOH, ethylene-vinyl alcohol copolymer, crystalline morphology Research

This work is a first attempt to study the crystallization behavior of ethylene-vinyl alcohol copolymer with 44 mol% of ethylene units (EVOH44) and to observe the supermolecular structures developed during its crystallization. Thermal analysis has evidenced a very fast crystallization and the formation of different crystal populations during isothermal crystallization. In-situ wide-angle X-ray scattering experiments using a synchrotron radiation source have shown a unique orthorhombic morphology, independently of the crystallization conditions. Small spherulites (with an average radius of about 1 µm) were observed using polarized optical microscopy and confirmed by typical four-leaf patterns obtained by smallangle light scattering. In-situ atomic force microscopy has revealed stacked lamellae growing from common centers to form small spherulitic entities. Finally, studies of isothermal crystallization have evidenced the existence of partial crystallization (especially at high crystallization temperatures) and the need for higher undercooling to complete crystallization.

Introduction

Ethylene-vinyl alcohol copolymers (EVOH) are semi-crystalline polymers widely used in food packaging industry due to their outstanding barrier properties to gases and hydrocarbons [START_REF] Okaya | Ethylene vinyl alcohol copolymers[END_REF]. Those properties have widely been assigned on the one hand to the inherent high degree of crystallinity of those copolymers, and on the other hand, to the numerous hydrogen bonds coming from the presence of hydroxyl groups, that leads to a reduction of free volume hindering the permeant diffusion [START_REF] Ito | Crystallinity dependence of freevolume size and its distribution of ethylene-vinyl alcohol copolymer evaluated by positronium lifetime measurement[END_REF][START_REF] Ito | Free-volume study of ethylene -vinyl alcohol copolymer evaluated through positronium lifetime measurement[END_REF][START_REF] Ito | Correlation study between oxygen permeability and free volume of ethylene-vinyl alcohol copolymer through positronium lifetime measurement[END_REF]. Produced via a saponification reaction of a parent ethylene-co-vinyl acetate copolymer (EVAc), ethylene-vinyl alcohol copolymer displays an irregular succession of monomer units [START_REF] Moritani | Carbon-13 nuclear magnetic resonance study on sequence distribution and anomalous linkage in ethylene-vinyl alcohol copolymers[END_REF]. Despite this random structure, EVOH crystallizes irrespective of the copolymer composition [START_REF] Takahashi | Crystal structure of ethylene-vinyl alcohol copolymers[END_REF][START_REF] Matsumoto | The crystallinity of ethylene-vinyl alcohol copolymers[END_REF]. Bunn and coworkers [START_REF] Bunn | Mixed crystal formation in high polymers[END_REF][START_REF] Bunn | Molecular structure and the crystallinity of long-chain polymers[END_REF] suggested that, since hydroxyl groups and hydrogen atoms are not too different in size and shape, crystallization may occur whatever their position along the chains, inducing the formation of a 'mixed crystal', as opposed to a 'pure crystal' where all crystal meshes have the exact same composition. Further investigations using thermal analysis, IR and NMR spectroscopies and X-ray diffraction confirm the fact that both ethylene and vinyl alcohol residues are incorporated into the crystalline regions of the copolymer [START_REF] Takahashi | Crystal structure of ethylene-vinyl alcohol copolymers[END_REF][START_REF] Matsumoto | The micro-structure of ethylene-vinyl alcohol copolymer[END_REF][START_REF] Nakamae | Elastic moduli of the crystalline regions in the direction perpendicular to the chain axis of ethylene-vinyl alcohol copolymers[END_REF][START_REF] Su | Spectroscopic investigation of the polymorphism and side group location of ethylene copolymers[END_REF][START_REF] Su | Polymorphism and side group location of ethylene copolymers characterized by FTIR and NMR spectroscopy[END_REF]. Thus, the EVOH crystalline structure was found to adopt the poly(vinyl alcohol) (PVOH) lattice (i.e., monoclinic) for low ethylene content, the polyethylene (PE) lattice (i.e., orthorhombic) for high ethylene content and an intermediate lattice (i.e., pseudo-hexagonal) for intermediate compositions. This latter can be seen as a slightly distorted orthorhombic lattice, in which the hydroxyl groups form 'point defect' [START_REF] Bunn | Mixed crystal formation in high polymers[END_REF][START_REF] Wunderlich | Thermodynamics of crystalline linear high polymers. I. Comparison of the melting transitions of solution and melt-crystallized polyethylene[END_REF]. Cerrada et al. [START_REF] Cerrada | Wide-angle X-ray diffraction study of the phase behavior of vinyl alcohol-ethylene copolymers[END_REF] pointed out the effect of both copolymer composition and thermal history on crystal structure. Besides, using NMR spectroscopy, VanderHart et al. [START_REF] Vanderhart | Solidstate 13 C nuclear magnetic resonance spectroscopy of ethylene/vinyl alcohol copolymers: Morphological partitioning of hydroxyls[END_REF] studied samples in the ethylene-rich composition ranges and concluded that there was no significant partitioning in EVOH. Nevertheless, they observed rejection of some defects (ethyl and butyl branches from polymerization side reactions or acetate branches from the incomplete hydrolysis of the EVAc parent copolymer) from the crystalline zones. Franco-Urquiza and coworkers [START_REF] Franco-Urquiza | Influence of processing on the ethylene-vinyl alcohol (EVOH) properties: Application of the successive self-nucleation and annealing (SSA) technique[END_REF][START_REF] Franco-Urquiza | Influence of the melt extrusion process on the mechanical behavior and the thermal properties of ethylene vinyl alcohol copolymer by applying the successive self-nucleation and annealing thermal fractionation[END_REF], using the successive self-nucleation and annealing (SSA) technique developed by Müller et al. [START_REF] Müller | Successive self-nucleation/annealing (SSA): A novel technique to study molecular segregation during crystallization[END_REF], showed the existence of a strong segregation during EVOH crystallization possibly related to the structural modifications induced by repetitive melt extrusions. In a quite comprehensive study [START_REF] Su | Two-dimensional correlation infrared spectroscopy reveals the detailed molecular movements during the crystallization of poly(ethylene-co-vinyl alcohol)[END_REF], the leading role of hydrogen bonds in the crystallization process of EVOH32 with 32 mol% of ethylene units was evidenced via 2D correlation infrared spectroscopy. In particular, two different crystallization processes were identified, both dominated by the generation of hydrogen bonds: a primary crystallization (around 160 °C) leading to the separate crystallization of vinyl alcohol units and ethylene units; a secondary perfection process (around 100 °C) characterized by a local rearrangement and the formation of imperfect mixed crystals. Different strategies have been proposed to modify and tune the crystallization process of EVOH by mixing it with another polymer or additive that may interact with the numerous -OH groups: polymer blending [START_REF] Kalfoglou | Compatibilization of poly(ethylene-co-vinyl alcohol) (EVOH) and EVOH-HDPE blends: Structure and properties[END_REF][START_REF] Petris | Study of blends of nylon 6 with EVOH and carboxyl-modified EVOH and a preliminary approach to films for packaging applications[END_REF], additive addition [START_REF] Vannini | Strategy to modify the crystallization behavior of EVOH32 through interactions with low-molecular-weight molecules[END_REF][START_REF] Zs | Decreased oxygen permeability of EVOH through molecular interactions[END_REF][START_REF] Luzi | Antioxidant packaging films based on ethylene vinyl alcohol copolymer (EVOH) and caffeic acid[END_REF], chemical modification [START_REF] Sánchez-Chaves | Novel glycopolymers containing aminosaccharide pendant groups by chemical modification of ethylene-vinyl alcohol copolymers[END_REF], nanocomposite preparation [START_REF] Aktzi | EVOH/clay nanocomposites produced by melt processing[END_REF][START_REF] Orr | Effect of processing method on cellulose nanocrystal/polyethyl eneco-vinyl alcohol composites[END_REF][START_REF] Wang | Montmorillonite@chitosan-poly(ethylene oxide) nanofibrous membrane enhancing poly(vinyl alcoholco-ethylene) composite film[END_REF]. Despite a rather extensive characterization of the crystalline structure of EVOH copolymers, there is very little understanding of the crystallization mechanism and behavior of these copolymers. In particular, no data are available in the literature concerning the supermolecular structures developed during EVOH crystallization, still an inherent feature of the morphology of crystalline polymers. In the aim of gaining more understanding on the crystallization behavior of EVOH, an EVOH44 which does not exhibit polymorphic behavior. was chosen for this study. Thermal analysis in conjunction with real-time wideangle X-ray scattering (WAXS) at a synchrotron radiation source, were performed. Basic morphologies of bulk melt-crystallized EVOH44 were also observed using different characterization techniques: polarized optical microscopy (POM), small-angle light scattering (SALS), atomic force microscopy (AFM), and small angle X-ray scattering (SAXS).

Materials and methods

Materials

Ethylene vinyl-alcohol (EVOH) E171B with 44 mol% of ethylene unit (EVOH44) was supplied by Kuraray. Low density polyethylene (LDPE) 2426F was supplied by PTT Global Chemical.

Preparation of EVOH44 film

An EVOH44 film was prepared by extrusion from pellets that have been dried for 24 h at 80 °C in a Somos dryer prior to processing. Since EVOH is sensitive to thermal degradation and difficult to extrude, it was coextruded with LDPE with no tie layer for easy delamination of the different layers. PE/EVOH/PE (35/30/35 wt%) films were prepared by coextrusion using two single-screw extruders of 20 mm diameter and length to diameter ratio of 20 (L/D: 20:1) from Scamex (France), combined in a classical A/B/A feedblock. The temperature profile was set at 220/220/220/220/220 °C and 220 °C for the barrel and the die, respectively, with a screw speed of 30 rpm. The chill-roll temperature was adjusted to 40 °C, and the final thickness of the EVOH44 film was around 300 µm.

Characterization methods 2.3.1. Thermogravimetric analysis (TGA)

TGA was carried out on a Q500 apparatus from TA Instruments (USA) under an oxygen or nitrogen flow of 60 ml/min. A first heating ramp from room temperature to 200 °C at 10 °C/min followed by a subsequent isothermal step of 10 min have been performed. The samples were then cooled to 155 °C and held at that temperature for 240 min. Finally, the sample was cooled down to 50 °C and heated up to 200 °C at 10 °C/min.

Differential scanning calorimetry (DSC)

DSC was performed using a Q1000 apparatus from TA Instruments (USA), calibrated with an indium standard scanned at 10 °C/min. Samples were cut from the extruded EVOH44 films, and their weights ranged typically from 2 to 8 mg. For non-isothermal and isothermal crystallization, all the samples were first heated to 200 °C and held at this temperature for 10 min to erase their thermal history. For non-isothermal crystallization, samples were then cooled to 20 °C at different cooling rates (2, 5, 10 and 50 °C/min) and heated again to 200 °C at 10 °C/min. For isothermal crystallization, they were cooled at 50°C/min to 138, 142, 147, 150, and 155°C, then maintained at these temperatures for various times of 15, 15, 30, 60, and 240 min respectively. For isothermal crystallization at 142 °C, additional times were investigated [START_REF] Ito | Crystallinity dependence of freevolume size and its distribution of ethylene-vinyl alcohol copolymer evaluated by positronium lifetime measurement[END_REF][START_REF] Moritani | Carbon-13 nuclear magnetic resonance study on sequence distribution and anomalous linkage in ethylene-vinyl alcohol copolymers[END_REF][START_REF] Matsumoto | The micro-structure of ethylene-vinyl alcohol copolymer[END_REF][START_REF] Su | Two-dimensional correlation infrared spectroscopy reveals the detailed molecular movements during the crystallization of poly(ethylene-co-vinyl alcohol)[END_REF][START_REF] Goderis | Primary and secondary crystallization in a homogeneous ethylene-1-octene copolymer: Crystallinity heterogeneity studied by SAXS[END_REF], and 240 min). After completion of the crystallization process, the samples were heated at 10 °C/min to 200 °C, both directly after crystallization and after an intermediate cooling step at 5 °C/min to 20 °C.

Polarized optical microscopy (POM)

15 µm thin slices of EVOH44 were cut from the cross-section of the film using an LKB microtome equipped with a glass knife. The samples were put between two glass substrates, set in a Linkam THMS600 hot stage (Waterfield, UK) connected to a liquid nitrogen cooling pump. The heating stage was put under an Axio imager 2 microscope from Zeiss (Germany), and images in transmission mode were recorded with a camera (AxioCam CCD-2Mc) each 1 second. Isothermal crystallization was performed by heating sample to 200 °C at high speed (150 °C/min), holding the temperature at this value for 10 min to erase the thermal history, then cooling at 150 °C/min to 155 or 145 °C, and held at these temperatures for 60 and 30 min, respectively.

Atomic force microscopy (AFM)

Multimode atomic force microscopy (AFM) equipped with Nanoscope V controller from Veeco (USA) was used in the tapping mode. The tips with a spring force of 40 N/m and an oscillation frequency of 300 kHz, were obtained from BudgetSensors. The EVOH44 film was placed on a mica substrate, introduced in a heating stage, heated to 200 °C for 10 min, and then cooled to 150 or 145 °C with no precise control of the rate of the cooling ramp.

Small angle light scattering (SALS)

100 µm thick extruded films were sandwiched between microscope glasses and crystallized isothermally in Linkam THMS600 hot stage (Waterfield, UK) by heating samples at 150 °C/min to 200 °C, holding at this temperature for 10 min, then cooling at 150 °C/min to 138, 145, 147 and 150 °C, and crystallizing at these temperatures for 15, 15, 30 and 60 min, respectively. The scattering patterns of the films were recorded under Hv polarization conditions (crossed polarizers) using a He-Ne laser with a wavelength (λ) of 632.8 nm.

Wide angle X-ray scattering (WAXS)

WAXS measurements were performed on the SWING beam line of the Soleil Synchrotron facility (St. Aubin, France), with a monochromator set at 17 keV [START_REF] David | Combined sampler robot and highperformance liquid chromatography: A fully automated system for biological small-angle X-ray scattering experiments at the Synchrotron SOLEIL SWING beamline[END_REF]. Using a CCD detector at 0.553 m from the sample, diffraction patterns were recorded for reciprocal spacing q = 4πsin θ/λ (where 2θ is the scattering angle and λ = 0.077 nm is the X-ray wavelength) varying between 0.0019-0.61 nm -1 with an exposure time of 5 s. Foxtrot software was used to obtain 1D WAXS curves by circular averaging of the full 2D images (0-360°). Isothermal crystallization was carried out in Linkam (UK) THMS600 hot stage according to the same protocol as used for the films prepared for SALS analysis. Diffractograms were treated with Fityk software for deconvolution of the crystalline and amorphous phases and the different crystalline peaks.

Small angle X-ray scattering (SAXS)

SAXS-2D was used for the determination of long periods of lamellar structure and lamellae thickness of EVOH44 samples. 200 µm thick extruded films were sandwiched between Kapton foils and mica plates and crystallized isothermally in Linkam THMS600 hot stage, according to the same protocol as used for the films prepared for SALS and WAXS analysis. A 1.2 m long Kiessig-type SAXS camera was coupled to GeniX Cu-LD Xenocs (France) Xray system (Cu Kα, wavelength 0.154 nm, operating at 50 kV and 1 mA). The patterns were recorded with a Pilatus 100K solid-state detector from (Dectris, Switzerland).

Results and discussion

Thermal analysis

TGA was first conducted to verify the thermal stability of the EVOH44 during the different thermal treatments carried out with the different techniques. TGA curves (not shown) show no significant modification of the evolution of the mass during the analyses under nitrogen or oxygen, which seems to indicate a stability of the copolymer under those conditions. Figure 1a displays the DSC scans of non-isothermal crystallization for different cooling rates (2, 5, 10, and 50 °C/min). Classically, the crystallization peak shifts towards the lower temperatures (from 146 to 135 °C) as the cooling rates increases from 2 to 50 °C/min. A second very small exothermic peak around 90 °C is also present, similarly to what was observed by Su et al. [START_REF] Su | Two-dimensional correlation infrared spectroscopy reveals the detailed molecular movements during the crystallization of poly(ethylene-co-vinyl alcohol)[END_REF]. Preliminary experiments of POM observation of nucleation and growth of the crystals have evidenced a very fast crystallization. This rapid crystallization may be due to preexisting nuclei, and several conditions were tested for the annealing pretreatment to erase them (up to 250 °C during 10 min) with no substantial observed differences. Since the crystallization of EVOH44 is very fast, the highest cooling rate (5 °C/min) was chosen to reach the selected temperature of isothermal crystallization in order to avoid the beginning of the crystallization during the cooling. According to Figure 1a, when cooling the sample at 50 °C/min, the lowest isothermal crystallization temperature is 138 °C. Figure 1b shows the variation of enthalpy as a function of time during isothermal crystallization at different temperatures (138, 142, 145, 147, 150, and 155 °C). It appears that for the lowest temperatures (138 and 142 °C), i.e., for the highest undercooling ΔT, the exotherm peak is truncated, suggesting that crystallization begins during cooling before reaching the target temperature. When the temperature increases, crystallization slows down significantly, leading to a very broad exothermic crystallization peak hardly distinguishable from the baseline. This indicates that the temperature range for isothermal crystallization is very narrow.

The melting endotherms of the samples after isothermal crystallization at different temperatures scanned at 10 °C/min, are shown in Figure 2a (both with and without subsequent cooling after isothermal crystallization). The melting curves reveal the presence of several peaks (two or three), designated by T m1 , T m2 and T m3 , respectively, from the lowest to the highest temperatures. Such multiple melting peaks are not observed after continuous cooling from the melt, i.e., for non-isothermal crystallization (dynamic crystallization), whatever the cooling rate. Figure 2b presents the evolution of the multiple peaks as a function of the isothermal crystallization temperature, and values of the melting enthalpy associated with the peaks are given in Table 1. Similar multiple melting peaks have been widely observed for both homopolymers (PEEK, PET, PLA, iPP, PBS, iPS …) [START_REF] Blundell | On the interpretation of multiple melting peaks in poly(ether ether ketone)[END_REF][START_REF] Marand | Influence of structural and topological constraints on the crystallization and melting behavior of polymers. 2. Poly(arylene ether ether ketone)[END_REF][START_REF] Alamo | Structural and kinetic factors governing the formation of the γ polymorph of isotactic polypropylene[END_REF][START_REF] Alamo | A morphological study of a highly structurally regular isotactic poly(propylene) fraction[END_REF][START_REF] Pan | Polymorphous crystallization and multiple melting behavior of poly(L-lactide): Molecular weight dependence[END_REF][START_REF] Wang | Multiple melting behavior of poly(butylene succinate)[END_REF][START_REF] Schick | The origin of annealing peaks in semicrystalline polymers: Enthalpy recovery or melting?[END_REF][START_REF] Jonas | Synchrotron X-ray scattering studies of crystallization of poly(ether-etherketone) from the glass and structural changes during subsequent heating-cooling processes[END_REF] and copolymers [START_REF] Pan | Polymorphous crystallization and multiple melting behavior of poly(L-lactide): Molecular weight dependence[END_REF][START_REF] Okui | Crystallization of ethylene/vinylacetate random copolymers[END_REF][START_REF] Crist | Crystallization and melting of model ethylene-butene copolymers[END_REF][START_REF] Pérez | Crystallization behaviour of fractions of a copolymer of propene and 1-hexane[END_REF][START_REF] Lee | Secondary crystallization behavior of poly(ethylene isophthalate-co-terephthalate): Time-resolved smallangle X-ray scattering and calorimetry studies[END_REF]. It generally appears difficult to ascertain the cause of this phenomenon that largely depends on the polymer and crystallization conditions. It can be attributed to several different factors: melting, recrystallization, and remelting during heating, molar mass segregation during crystallization, formation of several crystalline populations with different morphologies (such as lamellar thickness, perfection, or stability) or crystal structures (polymorphism). In our case, the lowest melting temperature peak (T m1 ) always appears slightly above the crystallization temperature. When the crystallization temperature is low, i.e., below 145 °C, the T m1 peak does not exist or is barely discernable. As the crystallization temperature increases, this peak becomes more pronounced. In our case, the peak can be assigned to the EVOH44 fraction that did not crystallize isothermally but during subsequent cooling. This is confirmed by the fact that when the samples were heated directly after isothermal crystallization, the first melting peak disappeared (dotted curves). It points out the fact that when crystallization temperature is higher than 145 °C, a complete crystallization is not achieved and requires a higher supercooling, which will be discussed later. As shown in Figure 2a, a small shoulder (T m2 ) is observed in the upper melting peak temperature (T m3 ) for the isothermal crystallized samples. Such a shoulder has already been observed by some authors for annealed EVOH [START_REF] Fonseca | Microhardness and thermal study of the annealing effects in vinyl alcohol-ethylene copolymers[END_REF][START_REF] Ramakrishnan | Well-defined ethylene-vinyl alcohol copolymers via hydroboration: Control of composition and distribution of the hydroxyl groups on the polymer backbone[END_REF] and has been attributed to the formation of non-perfect lower melting crystallites or associated with the melting of crystallites with different lamella thicknesses. This small shoulder appears systematically around 10 °C above the crystallization temperature and could correspond to the socalled annealing peak, whose origin is widely discussed in the literature [START_REF] Schick | The origin of annealing peaks in semicrystalline polymers: Enthalpy recovery or melting?[END_REF]. Additional isothermal crystallization performed at 142 °C for various times (from 1 to 240 min) show that this peak does not appear from the beginning of the crystallization but after a longer crystallization time of 5 min (Figure 3). Another noteworthy point is the slight shift to higher temperatures of the upper-temperature peak (T m3 ) that could be the signature of a classical lamella thickening during the isothermal crystallization.

WAXS results

Wide angle X-ray scattering (WAXS) experiments were performed to gain information about the structure of EVOH44 and its evolution during isothermal crystallization. The diffractogram of an extruded film (see Figure 4) exhibits orthorhombic lattice typically observed for this copolymer composition by previous authors [START_REF] Matsumoto | The crystallinity of ethylene-vinyl alcohol copolymers[END_REF][START_REF] Cerrada | Wide-angle X-ray diffraction study of the phase behavior of vinyl alcohol-ethylene copolymers[END_REF][START_REF] Pérez | Rapid determination of comonomer content, crystallinity, and long spacing by multiple-pulse proton NMR in ethylene-vinyl alcohol copolymers[END_REF], with mainly three characteristic peaks corresponding to the following crystallographic planes: (110) and ( 200) from the orthorhombic PE lattice and (100) from the PVOH monoclinic lattice.

Isothermal experiments between 138 to 150 °C were followed by WAXS using a synchrotron radiation source. The short exposure time (5 s) and the quick rate of the heating stage (150 °C/min) allowed following the isothermal crystallization from the beginning. Figure 5a shows the typical WAXS patterns as a function of time during isothermal crystallization of EVOH44 at 142 °C. While the beginning of the annealing highlights no diffraction peak, indicating that the sample was amorphous, diffraction peaks appear after several tens of seconds and start to grow simultaneously with time, suggesting a mixed crystalline form. Those peaks correspond exactly to the ones obtained after non-isothermal crystallization.

Still, a slight shift towards higher angles was observable for all three peaks (Figures 5b and5c) during crystallization, though more pronounced for the (200) peak. This shift is the signature of a slight contraction of the lattice, already observed previously by some authors when cooling the samples at slower rates [START_REF] Cerrada | Wide-angle X-ray diffraction study of the phase behavior of vinyl alcohol-ethylene copolymers[END_REF] or thermally annealing the samples at higher temperatures [START_REF] López-Rubio | Morphological alterations induced by temperature and humidity in ethylene-vinyl alcohol copolymers[END_REF]. It was interpreted as an improvement of crystalline morphology. In the case of EVOH32, a clear transition was observed from a defective orthorhombic morphology to a more thermodynamically stable monoclinic unit cell, with a slow cooling rate or throughout annealing at high temperature. This transition was reflected first, by a broadening of the (110) peak and a shift to higher angles of the (200) peak, and then by a splitting of the (110) peak into the two monoclinic ( 101) and (10 ̅ 1) peaks [START_REF] Cerrada | Wide-angle X-ray diffraction study of the phase behavior of vinyl alcohol-ethylene copolymers[END_REF]. Still, in those previous studies, no such clear transition was observed for EVOH44, for which the orthorhombic unit cell was maintained regardless of the cooling rate and the crystallization temperatures. Moreover, by measuring lattice parameters 'a' and 'b' as a function of time during isotherm, no significant variation has been evidenced (see Figure 6). Since EVOH44 is located, with a composition of 56 mol% of vinyl alcohol, very close to the orthorhombicmonoclinic transition [START_REF] Matsumoto | The crystallinity of ethylene-vinyl alcohol copolymers[END_REF], it is possible that we observe, during this isothermal crystallization, the beginning of the transition towards the more stable monoclinic lattice. By comparing the diffraction patterns at the end of isothermal experiment for the different crystallization temperatures, as shown in Figure 7a, it appears clearly that the crystallization is not completely finished at the highest temperatures. This would suggest a significant slowing-down of the crystallization or rather a partial crystallization at high temperatures, which will be addressed later. Moreover, during the cooling step or the isothermal crystallization, the evolution of the (100) peak of monoclinic lattice (Figure 7b) follows the same shift as the (110) and ( 200) peaks (Figure 7c). This observation suggests that VOH units and ethylene units are involved in the same crystalline structure regardless of the thermal treatment.

Morphological analysis

To provide support for the previous conclusions, several complementary morphological analyses have been conducted using different techniques.

Polarized optical microscopy (POM)

Polarized optical microscopy (POM) was first used to follow the isothermal crystallization at 155 and 145 °C. In both cases, the nucleation process and the growth of the supermolecular structure could not be observed. Figure 8 displays the images obtained using POM after isothermal crystallization at both temperatures and after cooling to 20 °C. In both cases, the formed crystalline structures are very small (one can distinguish those with sizes from around 1 to a few micrometers), revealing an approximately spherical symmetry and exhibiting a rather uniform Maltese Cross, suggesting a crystallization with the formation of spherulitic and lamellar morphologies. It is to notice that no substantial differences are observed between the structures obtained at different crystallization temperatures, neither in the spherulite type nor in their dimension.

Small-angle light scattering (SALS)

The observations performed with an optical microscope were complemented by SALS (Figure 9). The four-leaf patterns obtained for all the films are typical of a spherulitic structure. The intensity profiles at μ = 45° azimuthal direction are shown in Figure 10. For The values of r av are collected in Table 2, revealing spherulites with about 1 µm radii and evidencing no significant influence of the crystallization temperature on the dimensions of the spherulites, which is globally consistent with previous observations using POM. It is worth noticing that r av calculated according to Equation ( 1) is, in fact, the fifth-order average for 3D-spherulites [START_REF] Bartczak | Homogeneous nucleation in polypropylene and its blends by small-angle light scattering[END_REF]. Moreover, the scattering intensity at μ = 0 ° (azimuthal direction) was strong, especially at 147 and 150 °C, which indicates strong isotropic scattering and evidences the presence of a less ordered crystalline phase.

Atomic force microscopy (AFM)

Because of the very small dimensions of the spherulites, techniques with higher resolution (like SEM or AFM) could be more relevant to identify the crystalline morphology and lamellar structure of EVOH44. At first, SEM was tried using a permanganate etching. Despite different etching times and etchant concentrations, the etched surface was full of holes and no crystalline morphology was observed. Observations using atomic force microscopy (AFM) offered better results to follow EVOH44 crystallization. First, we probed at room temperature the surface of samples previously crystallized at different temperatures for 2 h and then slowly cooled to room temperature. The images of Figure 11 display spherulite-like structures composed of stacked radial lamellae. Although it seems that the lamellae grew from nucleation centers, the contours and limits of the spherulites do not appear clearly, and some randomly arranged lamellae are also visible. Secondly, we observed in-situ the crystallization and the formation of spherulites during the isothermal crystallization. We show in Figure 12 several series of AFM images taken at 145 and 150 °C at various times. At 145 °C, the early stages of crystallization were captured with the melt in the upper part and the beginning of the crystallization in the bottom part of Figure 12a, and after 5 minutes, all the volume was filled with spherulites. The crystallization behavior appeared, however, different at 150 °C: in some zones (see zone B, Figure 12e), the crystallization was not complete even after 2 h, leaving some areas without any crystals, as if the crystallization was greatly slowed down or even stopped. A closer look at spherulites created during isothermal crystallization (see the inset image in Figure 12e) reveals that the lamellae are grown during the isothermal annealing from a relatively coarse and open structure. The unfilled volume was completely filled after cooling to 20 °C subsequent to isothermal crystallization (Figure 12f). Unfortunately, the contours of the spherulites are not well defined, and it is impossible to determine whether the lamellae previously formed during isothermal treatment continue to grow and/or to thicken during subsequent cooling, or some others appear in the unfilled space. On this last image, it was impossible to distinguish the lamellae formed during the isothermal crystallization from those formed during the subsequent cooling.

Small-angle X-ray scattering (SAXS)

SAXS was used to characterize the lamellar structure and investigate the impact of the crystallization temperature on the lamellae thickness. Figure 13a shows the SAXS-2D patterns obtained for EVOH44 samples crystallized isothermally between Kapton foils at different temperatures. Figure 13b shows the Kratky plots, Iq 2 vs. q, where I is the intensity and q is the scattering vector. The mean long period (LP) values, deduced from the position of the peaks of the Kratky plots, according to the Bragg law, are given in Table 3. In addition, the mean values of lamellae thickness and LP determined based on the correlation function [START_REF] Goderis | Use of SAXS and linear correlation functions for the determination of the crystallinity and morphology of semi-crystalline polymers. application to linear polyethylene[END_REF] are collected in Table 3. The crystallinity level of isothermally crystallized EVOH44 specimens, 30-35% as shown further, supports the assignment of L c to lamellae thickness rather than to amorphous layer thickness. It must also be mentioned that, according to Figure 7, orthorhombic lattice and monoclinic lattice were formed in EVOH44. However, a very small content of the latter was determined by comparing peak areas (around 2%), and no evidence of the second long period was found on the Kratky plots. LP and L c values shown in Table 3 for films crystallized at different temperatures are similar. They slightly increase with increasing crystallization temperature, indicating, quite classically, slightly thicker lamellae at a higher temperature. However, smaller values are then obtained for the specimen crystallized at 150 °C. These slightly smaller values could result from incomplete crystallization at this crystallization temperature and crystals formed during subsequent cooling. It is worth mentioning that very similar results were obtained for films crystallized isothermally between mica plates (not shown). However, it must be emphasized that the differences in LP and L c values are very small, corresponding to a few bonds and indicating the predominant influence of chemical structure rather than crystallization temperature.

Crystallization after isothermal step

The previous observations (DSC and WAXS results, AFM images) have pointed out the fact that EVOH44 cannot always fully crystallize in isothermal conditions, particularly at high temperatures. It was observed as well that further crystallization upon subsequent cooling completed the partial isothermal crystallization. We have tried here to tackle this question, adopting a more quantitative approach by using in-situ WAXS and DSC analysis.

In particular, in-situ WAXS allowed us to follow the crystallization from the beginning and to estimate the degree of crystallinity by separating the amorphous and crystalline contributions (see Figure 4). Plotting the evolution of the degree of crystallinity as a function of time, as shown in Figure 14, is an interesting means to track the progress of crystallization. Obviously, the isothermal crystallization of EVOH44 was characterized by a sharp increase of the degree of crystallinity, followed by stabilization on a plateau value, on which crystallization actually seemed to continue but extremely slowly. The lower the crystallization temperature, i.e., the higher the undercooling, the faster the isothermal crystallization and the higher the degree of crystallinity at which the stabilization occurs. After the isothermal crystallization and stabilization of the crystallization process, only the cooling to ambient temperature allows the continuation of the crystallization. Finally, the degree of crystallinity reached after an isothermal crystallization at 150°C and subsequent cooling is slightly lower (around 30%) than those of samples crystallized at lower temperatures (around 35%). Similar results and tendencies were obtained with DSC measurements which allowed determining the extent of crystallization in isothermal conditions, depending on time and temperature, and the complementary crystallization upon subsequent cooling (see Figure 15). Moreover, it is worth noticing that the crystallization enthalpy measured for the subsequent crystallization after the isothermal step (i.e., T c1 ) is in good agreement with the melting enthalpy determined for T m1 (see Table 1) corresponding to the crystal melting formed during this subsequent cooling. This behavior is unexpected and deviates from the crystallization process of a regular homopolymer, for which the crystallization should be complete or nearly complete in isothermal conditions below the melting temperature [START_REF] Hoffman | The rate of crystallization of linear polymers with chain folding[END_REF]. It is to notice that the upper crystallization temperature (150 °C) is rather high for EVOH44. This temperature is located in the melting endotherm peak observed when heating the polymer after quenching. We expect that at such a temperature, not the entire polymer is capable of crystallizing, and especially the small crystals, e.g., the thinnest lamellae are unstable and are not formed. Still, theoretically, only the equilibrium melting temperature should be of concern in governing the crystallization kinetics. The equilibrium melting temperature of EVOH44 was estimated to be around 175 °C, by Alvarez et al. [START_REF] Alvarez | Isothermal crystallization of poly(vinyl alcohol-co-ethylene)[END_REF]. Though, the graph of Figure 2b points out the difficulty to determine the equilibrium melting temperature as far as multiple melting peaks appear, as raised by several authors [START_REF] Okui | Crystallization of ethylene/vinylacetate random copolymers[END_REF][START_REF] Cheng | Glass transition and melting behavior of poly(ethylene 2,6-naphthalenedicarboxylate)[END_REF][START_REF] Liu | Multiple melting behavior in isothermally cold-crystallized isotactic polystyrene[END_REF], which is often the case for copolymers. Such multiple-step crystallizations have already been observed previously for different polymer systems: branched polyethylene [START_REF] Glotin | Crystalline morphology of isothermally crystallized branched polyethylene[END_REF][START_REF] Mandelkern | Supermolecular structure and thermodynamic properties of linear and branched polyethylenes under rapid crystallization conditions[END_REF], blends of unbranched and short-chain branched polyethylene [START_REF] Stein | Some scattering studies of polymer microstructures, surfaces, and interfaces[END_REF], copolymers [START_REF] Crist | Crystallization and melting of model ethylene-butene random copolymers: Thermal studies[END_REF][START_REF] Alizadeh | Influence of structural and topological constraints on the crystallization and melting behavior of polymers. 1. Ethylene/1-octene copolymers[END_REF], and even for some homopolymers [START_REF] Marand | Influence of structural and topological constraints on the crystallization and melting behavior of polymers. 2. Poly(arylene ether ether ketone)[END_REF][START_REF] Cheng | Glass transition and melting behavior of poly(oxy-1,4-phenyleneoxy-1,4-phenylenecarbonyl-1,4-phenylene) (PEEK)[END_REF]. For example, Glotin and Mandelkern [START_REF] Glotin | Crystalline morphology of isothermally crystallized branched polyethylene[END_REF] have noticed that the crystallization of branched polyethylene is very sensitive to temperature. Besides, many studies on the crystallization of random copolymers (generally with one non-crystallizable comonomer) have shown that the crystallizable sequence distribution plays an important role in the crystallization process. In this case, the lamellar thickness is not controlled only by the undercooling but also by the sequence of crystallizable units [START_REF] Alizadeh | Influence of structural and topological constraints on the crystallization and melting behavior of polymers. 1. Ethylene/1-octene copolymers[END_REF]. The sequences that are not long enough to crystallize at a given temperature may crystallize at lower temperatures. A phenomenon of segregation occurs to gather the sequences of the same lengths allowing them to reach the critical dimensions. However, it was observed that usually for those random copolymers, this segregation is hindered by different restrictions (nucleation, overcrowding, pinning) inducing reduced mobility and limited transport of chain segments and leading to incomplete and lower crystallinity [START_REF] Crist | Crystallization and melting of model ethylene-butene copolymers[END_REF][START_REF] Goderis | Primary and secondary crystallization in a homogeneous ethylene-1-octene copolymer: Crystallinity heterogeneity studied by SAXS[END_REF]. Further crystallization requires cooling to lower temperatures and leads generally to less organized structures that have fringed micelle character.

In the case of EVOH44, at medium vinyl-alcohol contents, a strong rejection of short-chain branches has been observed by VanderHart et al. [START_REF] Vanderhart | Solidstate 13 C nuclear magnetic resonance spectroscopy of ethylene/vinyl alcohol copolymers: Morphological partitioning of hydroxyls[END_REF] leading to a significant partitioning. The SSA fractionation technique proposed by Franco-Urquiza et al. [START_REF] Franco-Urquiza | Influence of processing on the ethylene-vinyl alcohol (EVOH) properties: Application of the successive self-nucleation and annealing (SSA) technique[END_REF] confirmed the existence in EVOH32 of a strong segregation linked with the presence of the short chain branching that may generate a crystalline distribution during crystallization. The role of the numerous hydrogen bonding and their impact on this segregation are yet to be understood. They could possibly act as an additional physical hindrance to crystallization, as it has been evidenced in PVOH crystallization [START_REF] Peppas | Crystallization kinetics of poly(vinyl alcohol)[END_REF]. This behavior echoes the results of Su et al. [START_REF] Su | Two-dimensional correlation infrared spectroscopy reveals the detailed molecular movements during the crystallization of poly(ethylene-co-vinyl alcohol)[END_REF], who have pointed out the existence of two regions (temperature range) in the crystallization process of EVOH32 associated to different molecular movements dominated by hydrogen bonds.

Conclusions

The crystallization habit of EVOH44 has been for the first time investigated combining DSC, X-ray scattering, and other techniques (POM, SALS, AFM). POM observations revealed the presence of very small spherulites, with a mean radius around 1 µm, but it was impossible to observe the nucleation and the spherulite growth due to a very fast crystallization. However, in-situ AFM observations enabled tracking the early crystallization stage and observing the spherulitic morphology, confirmed by SALS analysis, presenting many stacked lamellae of various thicknesses. Isothermal crystallization has evidenced the existence of a two-step crystallization, resembling the process of crystallization in other random copolymers. A first rapid primary step (primary crystallization) corresponds to the spherulite growth that may be inhibited especially at high crystallization temperatures leading to a partial crystallization. Higher undercooling is needed to complete crystallization with a secondary crystallization, generating thinner lamellar structures. Further investigations are needed to explain this behavior and in particular to investigate the role of short chain-ends and hydrogen bonds in this complex crystallization behavior.

Figure 1 .

 1 Figure 1. DSC thermograms of EVOH44 a) non-isothermal crystallization at different cooling rates (2, 5, 10 and 50 °C/min), b) isothermal crystallization at different temperatures (138, 142, 147, 150, 155 °C) after cooling from 200 °C at 50 °C/min.

Figure 2 .

 2 Figure 2. DSC thermograms showing the melting peaks of isothermally crystallized samples (heating rate of 10 °C/min, endo -down), both directly after crystallization (dashed line) and after an intermediate cooling step at 5 °C/min to 20 °C (full line) a); evolution of the melting peak temperatures as a function of crystallization temperature b).

Figure 3 .

 3 Figure 3. Heating scan (at 10 °C/min) applied to EVOH44 after crystallization at 142 °C for different times and subsequently cooled to 20 °C at 50 °C/min.

Figure 4 .

 4 Figure 4. EVOH44 WAXS diffractogram, including the deconvolution of the crystalline and the amorphous part.

Figure 5 .

 5 Figure 5. Evolution of EVOH44 WAXS patterns as a function of time during annealing at 142 °C for 15 min a), focus on the characteristic peak corresponding to the (100) plane of PVOH b), focus on the characteristic peaks corresponding to the (110) and (200) planes of PE c).

Figure 6 .Figure 7 .

 67 Figure 6. Evolution of the lattice parameters (a) and the spacing lattice (b) as a function of time during isothermal crystallization of EVOH44 at 142 °C.

Figure 8 .

 8 Figure 8. Optical microscopy images of EVOH44 superstructure at room temperature, a) after crystallization for 30 min at 145 °C and b) for 60 min at 155 °C.

Figure 9 .

 9 Figure 9. SALS patterns of EVOH44 films crystallized isothermally at various temperatures.

Figure 10 .

 10 Figure 10. Intensity profiles vs. scattering angle θ determined from SALS patterns at the azimuthal direction of 45° for isothermally crystallized EVOH44 films at different temperatures.

Figure 11 .

 11 Figure 11. AFM phase images of the free surface of EVOH44 samples cooled to room temperature after crystallization for 2 hours at a) 145 °C, b) 147 °C and c) 150 °C. The scale bar corresponds to 500 nm.

Figure 12 .

 12 Figure 12. AFM phase image of EVOH44 during annealing at 145 °C a) and b) and cooling to room temperature c), at 150 °C, two different zones d) and e) and after cooling f). The scale bar corresponds to 1 µm.

Figure 13 .

 13 Figure 13. 2D-SAXS patterns a) and Kratky plot (Iq 2 vs. q) b) recorded for isothermally crystallized EVOH44 films at different temperatures.

Figure 14 .

 14 Figure 14. Evolution of EVOH44 degree of crystallinity as a function of time during isotherm (filled symbol) and during subsequent cooling (open symbol) measured by WAXS.

Figure 15 .

 15 Figure 15. DSC scans of EVOH44 during the subsequent cooling of several isothermal crystallization at a) 138 °C, b) 145 °C, c) 147 °C and d) 150 °C.

Table 1 .

 1 Values of melting peak temperatures of the different peaks and values of the associated total melting enthalpy, as well as T m1 and T m3 enthalpy.

	T c [°C]	T m1 [°C]	T m2 [°C]	T m3 [°C]	Total ΔH m [J/g]	ΔH (T m1 ) [J/g]	ΔH (T m3 ) [J/g]
	138	-	145.38	161.7	65.4±1.0	2.9	68.8
	142	141.5	149.3	162.3	71.2±2.3	6.0	65.3
	147	149.5	154.9	162.9	65.3	17.4	53.2
	150	152.2	157.9	164.8	62.9±4.4	27.5	45.7
	155	159.5	-	167.4	62.4	59.4	7.4

Table 2 .

 2 Average radii of spherulites in EVOH44 films crystallized isothermally at various temperatures.

	Crystallization temperature [°C]	r av [µm]
	138	1.05
	142	1.08
	145	1.13
	147	1.14
	150	1.18

Table 3 .

 3 Mean long period values (LP 1 ) calculated using the Bragg equation from the position of the peak maximum of the Kratky plot, mean long period values (LP 2 ) and mean lamella thickness (L c ) calculated based on the correlation function for isothermally crystallized EVOH44 films at different temperatures between Kapton foils.

	Crystallization temperature [°C]	LP 1 [nm]	LP 2 [nm]	L c [nm]
	138	16.3	15.4	4.05
	142	16.8	15.5	4.13
	145	16.9	15.5	4.20
	147	17.1	15.7	4.27
	150	15.6	15.7	4.19

Acknowledgements

This study was achieved within the context of a French FUI project (NOXY II). The authors acknowledge the synchrotron facility SOLEIL -beamline SWING (project n°2 0171204) and especially J. Perez (beamline manager) and T. Bizien (beamline scientist) for their precious help and advice. The authors would like to thank Dr. S. Tencé-Girault and Dr. S. Roland for fruitfull discussion.