domain: spi phys math info

text:
Stéphanie Roussel
email: stephanie.roussel@onera.fr

Xavier Pucel
email: xavier.pucel@onera.fr

Valentin Bouziat
email: valentin.bouziat@onera.fr

Louise Travé-Massuyès
email: louise@laas.fr

Model-Based Synthesis of Incremental and Correct Estimators for Discrete Event Systems

State tracking, i.e. estimating the state over time, is always an important problem in autonomous dynamic systems. Run-time requirements advocate for incremental estimation and memory limitations lead us to consider an estimation strategy that retains only one state out of the set of candidate estimates at each time step. This avoids the ambiguity of a high number of candidate estimates and allows the decision system to be fed with a clear input. However, this strategy may lead to dead-ends in the continuation of the execution. In this paper, we show that single-state trackability can be expressed in terms of the simulation relation between automata. This allows us to provide a complexity bound and a way to build estimators endowed with this property and, moreover, customizable along some correctness criteria. Our implementation relies on the Sat Modulo Theory solver MONOSAT and experiments show that our encoding scales up and applies to real world scenarios.

Introduction

For autonomous systems, state estimation and tracking is a critical task because it strongly influences the decisions that are made and that can be essential to the life of the system. It provides the means to diagnose the faults and to react to the various hazards that can affect the system.

In this paper, we focus on discrete event systems [START_REF] Zaytoon | Overview of fault diagnosis methods for discrete event systems[END_REF]]. When the system state is partially observable, the number of candidate estimates can quickly become too large to be usable. In embedded or distributed systems, memory and communication limitations can also become a problem, even with symbolic representations techniques such as in [START_REF] Torta | An on-line approach to the computation and presentation of preferred diagnoses for dynamic systems[END_REF]. These limitations lead us to propose an estimation strategy that retains only one state out of the set of candidate estimates at each time step, as in [START_REF] Bouziat | [END_REF]. This can be seen as an extreme strategy but nevertheless efficient to feed the decision system with a clear input and consistent with several works that select a limited number of best candidates according to some preference criterion, for example probabilities like in [Williams and Nayak, 1996;Kurien and Nayak, 2000]. However, the more we limit the number of estimates, the more we may be confronted with the problem of dead-ends related to the fact that next observations may prove that previous estimates were wrong and there is no continuation in the estimated trajectory. Because backtracking [Kurien and Nayak, 2000] is not a viable solution with real time constraints, we propose to build single-state estimators that are guaranteed to avoid dead-ends. The ability of a system to support the construction of a single-state dead-end free estimator has been defined as single-state trackability in [START_REF] Bouziat | [END_REF].

Several properties have been investigated for discrete event systems. A system is (strongly) detectable if one can determine its state along some (all) trajectories of the system [Shu et al., 2007]. Diagnosability [START_REF] Lafortune | [END_REF] and manifestability [START_REF] Dague | [END_REF] amount to strong detectability and detectability for faulty trajectories. Singlestate trackability is somewhat orthogonal to these properties.

In this paper, we recall the necessary and sufficient condition for single-state trackability as it is derived in [START_REF] Bouziat | [END_REF] based on equality of regular languages. A first contribution is to show that this condition can be expressed in terms of the simulation relation between automata [Holík et al., 2018]. This allows us to provide a complexity bound for the single-state trackability problem and a way to build estimators endowed with this property. A second contribution of the paper is to propose two customizable correctness criteria that lead the estimator to constrain the acceptable estimator states towards the real states of the system. A third contribution is an implementation that relies on the Sat Modulo Theory (SMT) solver MONOSAT for synthesising estimators. Intensive experiments show that our implementation is able to synthesize dead-end free estimators on real world scenarios.

The paper is organised as follows. Related work is discussed in Section 2. Section 3 introduces incremental singlestate estimation and how it is performed in our framework. The property of single-state trackability is defined and characterized in Section 4 and the associated decision problem is proved to be in NP. Customisable correctness of single-state estimators is formalized in Section 5. Section 6 presents the SMT encoding for synthesizing single-state estimators given a system specification. Experimental results are presented in Section 7, and future work is discussed in Section 8.

A related research domain is the well-known controller synthesis problem, as described in [START_REF] Ramadge | [END_REF]. Given a specification of the system, controller synthesis consists in finding a decision procedure that produces a command to apply to the system, given observations as input. Under partial observability, a trend of work proposes to synthesize finite-state controllers, which use a size-limited memory to record information related to the past. These were first introduced for POMDP [Meuleau et al., 1999] and shown to be useful for non-deterministic planning [START_REF] Bonet | [END_REF][START_REF] Pralet | [END_REF]. Restricting the information recorded from the past is also one important feature of our approach. However, there are several differences compared to our incremental estimation approach. The above works design controllers guaranteed to satisfy some properties related to planning, in particular the evolution of the controlled system must terminate in a goal state. In our estimation framework, we are rather interested in correctness properties, i.e. guaranteeing that the estimated state is "as close as possible" to the real state of the system (see Section 5).

Controller synthesis has also been approached within the game theory framework. In particular, [Doyen and Raskin, 2011] consider observation-based strategies for two-player turn-based games. Similar to our approach, observationbased strategies rely on imperfect information on the past observation sequence, hence defining partially observable games. Such games occur in the synthesis of a controller that does not see the private state of the plant. The main difference with our approach is that game theory considers games as complete graphs. Such a hypothesis means that strategies cannot encounter dead-ends in the execution, i.e. there is always a feasible action from any state of the game.

Incremental Single-State Estimation

In this section, we present the process of state estimation for partially observable systems modelled by non-deterministic Moore machines with an empty input alphabet, and whose output alphabet represents the system observations. Definition 1 (System). A system M is defined by a 5-tuple (S, s 0 , O, ∆, obs) consisting of the following:

• a finite set of states S;

• an initial state s 0 ∈ S;

• a finite set of observations O (output alphabet in the Moore machine);

• a transition relation ∆ ⊆ S 2 ;

• an observation function obs : S → O mapping each state to its output.

Without loss of generality, we consider that all the states of the system are reachable from the initial state s 0 .

We define cands as the function from S × O to 2 S such that for all state s in S, for all o in O, cands(s, o) represents the set of successors of s that have observation o. Formally, ∀s ∈ S, ∀o ∈ O, cands(s, o) = {t|(s, t) ∈ ∆ and obs(t) = o}.

We also define nextObs the function S → 2 O such that nextObs(s) is the set of observations that can be observed just after the system is in state s. Formally, ∀s ∈ S, nextObs(s) = {o|cands(s, o) = ∅}.

Notation A state sequence seq is a list (s 0 , s 1 , . . . , s n-1 ) where each s i is a state in S; |seq| = n is the length of the sequence and seq[i] = s i is the ith state in the sequence; last(seq) designates the last state of seq; if s is a state, seq • s is the sequence of length |seq| + 1 that begins with seq and ends with s. Similarly, we define an observation sequence. We also extend the function obs to a state sequence: if seq is a state sequence, obs(seq) is the observation sequence seq obs such that |seq obs | = |seq| and seq obs

[i] = obs(seq[i]) for all i ∈ [0, n -1].
Definition 2 (Language, observation language). The language associated with a system M = (S, s 0 , O, ∆, obs) is the set of state sequences accepted by the system and starting with s 0 . Formally

L(M ) = {seq ∈ S + |seq[0] = s 0 and ∀i ∈ [1, |seq| -1], (seq[i -1], seq[i]) ∈ ∆}.
The observation language is the language accepted by the system projected on the observations. Formally, L obs (M ) = {obs(seq)|seq ∈ L(M )}.

We now focus on the estimation part for systems defined above. Since we consider non-deterministic, partially observable systems, there may be several state sequences that explain a given observation sequence. We adopt an incremental approach to select a unique explanation, called an estimation strategy. Then, we show that for a given system, any estimation strategy can be represented by a finite state machine called an estimator.

Definition 3 (Estimation strategy). An incremental singlestate estimation strategy for a system M = (S, s 0 , O, ∆, obs) is a function estim : S × O → S such that for all s in S, for all o in nextObs(s), estim(s, o) represents the estimated state of the system at time step n if it was estimated in state s at time step n -1 and if o is observed at time step n. We impose the estimation strategy to be consistent both across time (i.e. estim is a function) and with the system behaviour (i.e. estim(s, o) belongs to cands(s, o)).

For conciseness purposes, we use "estimation strategy" to refer to "incremental single-state estimation strategy". Given a system and an estimation strategy, the estimation process takes place as follows.

Definition 4 (Estimated sequence). Let M be a system, seq obs be an observation sequence in L obs (M ) and estim an estimation strategy for M . The estimated sequence for seq obs is the state sequence seq ∈ L(M ) such that seq

[0] = s 0 and for all i in [1, | seq| -1], seq[i] = estim( seq[i -1], seq obs [i]).
Example 1. Let M = (S, s 0 , O, ∆, obs) be the system represented in Fig. 1. We have S = {s 0 , . . . , s 5 }, O = {a, b}, ∆ is represented by the arrows in the figure, obs(s 0 ) = obs(s 3 ) = obs(s 5 ) = a and obs(s 1 ) = obs(s 2 ) = obs(s 4 ) = b.

In this system only two pairs in S × O have more than one estimation candidate: (s 0 , b) and (s 1 , a). Let estim 1 be the estimation strategy such that estim 1 (s 0 , b) = s 1 and estim 1 (s 1 , a) = s 3 , and the unique candidate is selected estimated state time step system state observation otherwise (for example, cands(s 3 , a) = {s 3 } so necessarily estim 1 (s 3 , a) = s 3 ).

0 s 0 a 1 s 2 b 2 s 5 a 3 s 5 a s 0 s 1 s 3 s 3 estim1(s0, b) estim1(s1, a) estim1(s3, a)
When the system produces the observation sequence (a, b, a, a), it can be explained by three state sequences in the system: (s 0 , s 1 , s 3 , s 3 ), (s 0 , s 1 , s 5 , s 5 ) and (s 0 , s 2 , s 5 , s 5 ). With the strategy estim 1 , we select the first one. The estimation process is illustrated in Figure 2 using this strategy.

In Example 1, one can observe that the state s 2 cannot be part of any estimated sequence, as the estimation strategy chooses a branch of the system states that does not contain s 2 . The set of states that the estimation strategy actually uses is denoted S and is formally defined by the set of states in S that are part of an estimation sequence associated with a sequence of observation in L obs .

An estimation strategy estim for a system M = (S, s 0 , O, ∆, obs) can also be represented by an estimator, i.e. a 5-tuple M = ( S, s 0 , O, ∆, obs) composed of the reachable states and transitions in the system with the strategy. Definition 5 (Estimator). Let M = (S, s 0 , O, ∆, obs) be a system and estim an estimation strategy. The estimator induced by estim is a deterministic Moore machine M = ( S, s 0 , O, ∆, obs) such that:

• S ⊆ S is the set of states that belong to some estimated sequence for the given estimation strategy;

• ∆ is the smallest subset of ∆ such that ∀ s ∈ S, ∀o ∈ nextObs( s), ( s, estim( s, o)) belongs to ∆.

In Figure 1, M 1 is the estimator associated with the estimation strategy estim 1 introduced in Example 1.

Definition 5 indicates how for a given system, every estimation strategy can be represented by an estimator. Note that the reverse also holds: for a given system M = (S, s 0 , O, ∆, obs), any deterministic Moore machine M = ( S, s 0 , O, ∆, obs) such that S ⊆ S and ∆ ⊆ ∆, is in fact an estimator that implements an estimation strategy estim such that estim(ŝ, o) is the unique t such that (ŝ, t) ∈ ∆ and obs( t) = o, for ŝ ∈ S and o ∈ O. In the rest of the paper, we use estimators to represent estimation strategies.

Single-State Trackability

Dead-ends

In this section, we first recall the notion of dead-end as defined in [START_REF] Bouziat | [END_REF]. During estimation, the estimator may choose a state sequence different from the one taken by the system. This can be particularly problematic when the following conditions happen: a) the system is in state s, the estimator estimates that it is in state s with s = s; b) the system moves in state t and produces observation o; c) there exists no candidate t in S such that obs( t) = o and ( s, t) belongs to ∆.

Definition 6 (Dead-end). A dead-end for an estimator M of a system M is an observation sequence that belongs to L obs (M ) but does not belong to L obs ( M ). An estimator M is dead-end free if there is no dead-end for it, i.e. L obs (M ) = L obs ( M ). Example 2. We consider the example illustrated in Figure 1. Let M 2 be the estimator induced by the estimation strategy estim 2 defined by estim 2 (s 0 , b) = s 2 (no other choices necessary). The sequence of observations (a, b, b, b) belongs to L obs (M ) as it is produced by the state sequence (s 0 , s 1 , s 4 , s 4 ) but it does not belong to L obs ( M 2 ). This observation sequence is therefore a dead-end for M 2 . Note that M 1 is dead-end free.

In [Kurien and Nayak, 2000], dead-ends are handled by backtracking in time in order to modify previous estimation choices. The main drawback of this technique is the lack of control over both the computational time and memory. Moreover, some actions may have been taken based on the previous estimated states that cannot be undone. Backtracking in time and revising estimations may turn past actions sub-optimal or even inconsistent.

Another way to tackle dead-ends is to keep a fixed number k of past observations and reason on this window [START_REF] Su | Window-based diagnostic algorithms for discrete event systems: What information to remember[END_REF]. However this approach is also subject to dead-ends as the estimator and system paths may diverge earlier beyond the memorized window.

Our goal is to design estimators able to follow the execution of a system while keeping just a unique state in memory and never encountering dead-ends. Such deterministic incremental estimators may be feasible depending on the system specifications. We express that possibility by defining singlestate trackability for discrete event systems.

Trackability Problem

In [START_REF] Bouziat | [END_REF], a system is considered as singlestate trackable if there exists a dead-end free single-state incremental estimator for this system. They show that this is the case if and only if there exists an estimator with an observation language equal to that of the system. In this paper, we take the language equivalence as a definition of Single-State Trackability. Definition 7 (Single-State Trackability). A system M is single-state trackable if there exists an estimator M for M such that L obs (M ) = L obs ( M ).

Determining whether a system is single-state trackable or not is called the TRACKABILITY problem.

The problem of deciding if two languages are equivalent (also called trace-equivalence) is, in the general case, a PSPACE-complete problem [START_REF] Hüttel | Hans Hüttel and Sandeep Shukla. On the Complexity of Deciding Behavioural Equivalences and Preorders[END_REF]. This leads to the definition of a rather complex algorithm for the TRACKABILITY problem in [START_REF] Bouziat | [END_REF]. This approach can be improved because the system and the estimator, for which language equivalence is considered, are not independent from each other. In fact, the estimator is a submachine of the system's machine. A contribution of this paper is to prove that in this case, the language equivalence problem is reduced to the simulation preorder problem, which is decidable in polynomial time [Shukla et al., 1996].

We first adapt the definition of simulation to the case of a system and an estimator. Definition 8 (Simulation for estimators). Let M = (S, s 0 , O, ∆, obs) be a system and M = ( S, s 0 , O, ∆, obs) be an estimator for M . M simulates M if there exists a relation R ⊆ S × S such that:

(s 0 , s 0 ) ∈ R (1) ∀(s, s) ∈ R, obs(s) = obs( s) (2) ∀(s, s) ∈ R, ∀t ∈ S s.t. (s, t) ∈ ∆, ∃ t ∈ S s.t. ( s, t) ∈ ∆ and (t, t) ∈ R (3)
Informally, if s is a state in S and s a state in S, then s simulates s (or (s, s) is in R) represents the fact that M can estimate that the system is in state s while it really is in state s without encountering a dead-end later in the execution.

Proposition 1. Let M be a system and M be an estimator for M . M is dead-end free (or L obs (M ) = L obs ( M )) if and only if M simulates M .

Proof. (⇐) L obs ( M ) ⊆ L obs (M ) as the states and transitions of M are subsets of those of the system. Moreover, let seq obs ∈ L obs (M ) be produced by seq ∈ L(M ). We prove that for all k < |seq|, there exists a sequence seq ∈ L( M ) of length k such that obs(seq) = obs( seq) and

(seq[k], seq[k]) ∈ R, which entails L obs ( M ) ⊇ L obs (M ).
(⇒) We consider the following relation R: ∀s ∈ S, s ∈ S, (s, s) ∈ R iff ∃seq obs ∈ L obs (M ), seq ∈ L(M ), seq ∈ L( M ) s.t. obs(seq) = obs( seq) = seq obs , s = last(seq) and s = last( seq). We show that R is a simulation relation. (1,2) By construction of R, we have (s 0 , s 0 ) ∈ R, and for all (s, s) ∈ R, obs(s) = obs( s). (3) Let (s, s) ∈ R and t ∈ S such that (s, t) ∈ ∆. By construction of R, ∃seq obs ∈ L obs (M ), seq ∈ L(M ), seq ∈ L( M ) s.t. obs(seq) = obs( seq), s = last(seq) and s = last( seq). (s, t) ∈ ∆ so seq.t ∈ L(M ) and produces the sequence of observation seq obs .obs(t) ∈ L obs (M ). By the equality of languages and definition of L obs ( M ), there exists seq ∈ L( M ) s.t. obs( seq ) = seq obs .obs(t). As M is deterministic, seq obs is generated by the same state sequence seq. So, seq is equal to seq. t and ( s, t) ∈ ∆. By construction of R, (t, t) ∈ R.

The following corollary presents a necessary condition for estimators. It indicates that any observation sequence that can follow a state s can also follow the state s by which s is estimated. This corollary is used in Section 6 to increase the efficiency of estimator synthesis.

Corollary 1. Let M be a system, M a dead-end free estimator for M , s a state in S, s a state in S, and R a simulation relation such that (s, s) ∈ R. For any observation sequence seq obs , if seq obs can follow s then seq obs can follow s. Corollary 2. The TRACKABILITY problem is in NP.

Proof. Checking if M is a dead-end free estimator for M is polynomial since it comes to checking whether M simulates M , which can be encoded into HORN-SAT clauses as shown in [Shukla et al., 1996].

Estimator Correctness

Correctness is the property of an estimator that estimates the actual system state. So far we provided a way to generate dead-end free estimators, which can be seen as a very weak form of correctness as a dead-end free estimator cannot be proven wrong by the system's observations. However, depending on the operational context, total correctness may be required or an incorrect pessimistic or optimistic estimation may be satisfying [START_REF] Bouziat | [END_REF][START_REF] Couto | [END_REF].

In this section, we model correctness as constraints and optimization criteria that allow some end-user to generate interesting estimators. Correctness constraints allow one to restrict the possible estimator states associated with some given system state. Correctness criteria let one associate a cost with each possible estimator, with the lowest cost being associated with the "most correct" estimators. Definition 9 ((φ 1 , φ 2 )-correctness). Let M be a system, φ 1 ⊆ S and φ 2 ⊆ S be two sets of states of M , M a dead-end free estimator for M , and R the associated simulation relation.

M is (φ 1 , φ 2 )-correct if whenever the system is in a state of φ 1 , then the estimator is in a state of φ 2 , i.e. ∀(s, s) ∈ R, if s ∈ φ 1 then s ∈ φ 2 .
In a symbolic model where states are represented by Boolean variables, correctness constraints can be represented by propositional formulae. They can also be deduced from a proof of reachability of an undesirable pair (s, ŝ) as produced in [START_REF] Couto | [END_REF]. With such constraints, one can require that whenever some critical fault is present in the system state, it is also present in the estimated state. This ensures the estimator is correct with respect to this critical fault.

Example 3. Let us consider the system M presented in Example 1 and illustrated on Figure 1. Let f 1 be a fault only present in states φ f1 = {s 1 , s 3 , s 4 } and f 2 a fault only present in φ f2 = {s 5 }. The estimator M 1 is (φ f1 , φ f1 )correct as all the states in φ f1 are estimated by states in φ f1 . However M 1 is not (φ f2 , φ f2 )-correct as s 5 is estimated by s 3 which does not belong to φ f2 .

While it is very flexible, in some systems, (φ 1 , φ 2 )correctness might be either too strong or too tedious to specify. For example, in Example 3, there exists no estimator that satisfies both (φ f1 , φ f1 ) and (φ f2 , φ f2 )-correctness. In order to relax this requirement, a first laborious approach is to enumerate the possible estimated states for each system state. Another approach consists in expressing a weaker form of correctness, for instance that (φ 1 , φ 2 )-correctness should be satisfied "as much as possible". In this regard we introduce a cost function that lets one model this kind of requirement.

Definition 10 (Correctness cost). Let M be a system, M an estimator for M , and R their simulation relation. Let cost : S × S → N be a cost function such that cost(s, s) represents how much we want to avoid estimating that the system is in state s when its real state is s. The correctness cost for M is cost( M ) = (s, s)∈R cost(s, s) Definition 11 (Correctness order). Let M be a system, M 1 and M 2 be two dead-end free estimators for this system, and cost a cost function. M 1 is strictly more correct than M 2 with respect to cost if cost( M 1 ) < cost( M 2 ).

There are several ways to define a cost for pairs of states. A straightforward one is to check if the real state of the system is correctly estimated. Such a cost c = is formally defined by ∀(s, s) ∈ S 2 , c = (s, s) = 0 if s = s, 1 otherwise. One could also consider a cost based on (φ 1 , φ 2 )-correctness: ∀(s, s) ∈ S 2 , c φ1,φ2 (s, s) = 0 if s ∈ φ 1 and s ∈ φ 2 , 1 otherwise. Symbolic models make it easier to specify cost functions. The Hamming distance ( [Hamming, 1950]) is an example of a cost function for Boolean variables. Weighted sums, or lexicographical aggregation of costs can also be used to emphasize some critical faults for example. Example 4. Let us consider the system M presented in Example 1 and illustrated on Figure 1. We define the dead-end free estimator M 3 similar to M 1 except that estim 3 (s 1 , a) = s 5 . We consider a correctness cost c such that ∀i, c(s i , s

i ) = 0, c(s 1 , s 2 ) = c(s 2 , s 1 ) = 1, c(s 5 , s 3 ) = 1 and c(s 3 , s 5 ) = 2.
This cost represents that it is preferable to estimate that the system is in state s 3 when it really is in state s 5 than the opposite. We have c( M 1 ) = 2 and c( M 3 ) = 3. This means that M 1 is more correct than M 3 with respect to cost c.

Estimator Synthesis

In this section, we describe a procedure to solve the TRACK-ABILITY problem. To decide if a system M is single-state trackable, we try to synthesize a dead-end free estimator for it. More precisely, we consider that a system M can be seen as a graph and that a dead-end free estimator M is a sub-graph that satisfies several constraints, that can be encoded into SMT provided the underlying theory provides graph predicates. We specifically target the solver MONOSAT ( [Bayless et al., 2015]), that allows to handle classical SAT constraints but also graph constraints. Dead-end free estimator synthesis Let M = (S, s 0 , O, ∆, obs) be a system and M = ( S, s 0 , O, ∆, obs) the dead-end free estimator we try to synthesize for M . If M exists (i.e. M is single-state trackable), then states and transitions of M are subsets of those of M and following Prop. 1, there exists a simulation relation R between the states of M and M . We associate the system M with a directed graph G whose vertices are the states S and whose edges are the transitions ∆. Similarly, G is the graph associated with an estimator M .

First, we instantiate the following Boolean decision variables:

• for each s ∈ S, v s is true iff s belongs to S;

• for each (s, t) ∈ ∆, e s,t is true iff (s, t) belongs to ∆;

• for each (s, s) in S 2 , r s, s is true iff (s, s) belongs to R.

We next instantiate the following constraints.

∀(s, t) ∈ ∆, e s,t → v s ∧ v t (4) ∀ s ∈ S, v s ↔ reachable( G, s 0 , s) (5)
∀s ∈ S, ∀o ∈ nextObs(s),

t∈S|obs(t)=o e s,t ≤ 1 (6) ∀s ∈ S, ∀o ∈ nextObs(s), v s → t∈S|obs(t)=o e s,t (7)
r s0,s0 (8)
∀s ∈ S,

We first encode that the estimator is a well-built automaton. Constraint (4) states that if an edge is selected in G, its vertices are selected as well. Constraint (5) enforces the states of the estimator to be exactly the ones reachable from s 0 in the graph G. We use here the MONOSAT predicate reachable.

Constraints ( 6) and ( 7) force M to be deterministic: for every state s and every successor observation o, exactly one outgoing transition leads to a state with observation o. We use two constraints, as in SAT/SMT the "at most one" cardinality constraint (6) cannot easily be combined with the implication of Constraint (7).

We next consider constraints related to the simulation relation. Constraint (8) states that the initial state simulates itself (see Definition 8). Constraint (9) encodes that every state of the system is simulated by at least one state of the estimator. Constraint (10) encodes the simulation relation from Definition 8 and also states that the edge associated with the simulation relation must belong to M1 .

Constraint (11) makes the simulation relation minimal as the estimator states must all simulate at least one system state.

Using a SMT solver let us use the predicate reachable in constraint (5). This constraint can be expressed in SAT clauses using for instance the Floyd-Warshall algorithm [START_REF] Cormen | [END_REF] with a complexity of O(|V | 3 ). MONOSAT achieves a complexity of O(|V | • |E|) [Bayless et al., 2015].

From Definition 8 and Corollary 1, the number of variables r s, s can be reduced by creating variables only for pairs (s, s) for which s accepts all the observation sequences of length 2 (or more) that s accepts.

Correct estimator synthesis The encoding presented above allows to synthesize dead-end free estimators. It is possible to enrich it to synthesize correct estimators, as defined in Section 5. (φ 1 , φ 2 )-correctness can be encoded as a hard constraint the following way: ∀s ∈ φ 1 , ∀ s ∈ S \ φ 2 , ¬r s, s . The second correctness requires a cost function c as input, and is encoded as the following pseudo Boolean criterion: minimize (s, s)∈R r s, s • c(s, s).

Experiments

Our implementation uses the SCALA programming language, and each experiment was performed on a computer with a Intel R Xeon R CPU E5-2699 v3 @ 2.30GHz processor and a limit of 16GB of memory.

Examples from the literature We have encoded benchmarks from three system models from the literature:

• valve controller [Sampath et al., 1995] that models a valve controller for heating, ventilation and air conditioning units with a valve that can be stuck closed or stuck open, a pump that can fail in modes on and off and a controller; each component has 4 possible states; this system is not trackable.

• valve driver [Williams and Nayak, 1996] that models a valve driver that has 4 modes, 6 command inputs and 3 command outputs; this system is trackable.

• baggage transfer [START_REF] Pencolé | [END_REF] that models a baggage transfer system composed of 2 conveyors, 1 pis-

Random examples

In order to test our prototype intensively, we randomly generated a set of systems as follows.

First, we generated a set of small systems of 5 states, 3 observations, and 0.5 transition probability for any two states. Second, we aggregated them by Cartesian product (of states and observations), with a 0.1 probability that any two states are incompatible (in this case their pair is removed from the aggregated states). The benchmark contains systems obtained by aggregation of 4 and 5 small systems. Finally, we selected only "interesting" systems, such that: 1. the system is trackable; 2. the estimator has less than half the number of states of the system (so as to reject trivially trackable systems); 3. there exists a state s such that the system has no ({s}, S\ {s})-correct estimator. This method generated systems from 10 to 1461 states, with more small systems than large ones.

In all our experiments, most of the computation time is used for generating SMT clauses whereas the time required by MONOSAT to solve the SMT instance is negligible. This explains why the computation time is nearly the same for the trackable and untrackable versions of random examples. We have yet to properly assess the impact of a cost function on the solving time. Figure 3 shows that on random benchmarks, the solving time grows regularly in the number of states. The valve driver and valve controller systems require similar computation time as random systems. For the baggage transfer system, the computation time is higher, because of the large number of symmetries in the system.

Conclusion

This paper describes a new approach for checking single-state trackability as defined in [START_REF] Bouziat | [END_REF]. As a system is simulated by its estimator, we prove that the TRACKABIL-ITY problem is in NP. We also define two customizable correctness types for modelling relevant estimators. We solve the TRACKABILITY problem by reducing it to a MONOSAT SMT query, along with correctness requirements. The approach is validated on a set of benchmarks both from the literature and from randomly generated problems.

Studying NP-completeness with and without correctness constraints is a natural next step in our work. Allowing for delayed estimation, or using a bounded number of memorized states at each time step could help apply this approach to a larger range of systems, including real-world systems, and will be considered in future work.

Symbolic representation of estimators with preferences as in [START_REF] Bouziat | [END_REF] is also a possible extension.

Figure 1 :

 1 Figure 1: A simple system M , with one possible estimator M1 depicted in solid red, and another M2 in dashed blue.

Figure 2 :

 2 Figure 2: Estimation process in Example 1.

Figure 3 :

 3 Figure 3: Computation time w.r.t. number of system states. Both scales are logarithmic.

Note that Constraint (9) is redundant with Constraints (8) and (10), but improves the solver's performance.


Julen Vadillo

Izaskun Larraza

Aitor Arbelaiz

Maria Angeles Corcuera

Maud Save

Christophe Derail
email: christophe.derail@univ-pau.fr

Arantxa Eceiza
email: arantxa.eceiza@ehu.es

Influence of the addition of PEG into PCL-based waterborne polyurethane-urea dispersions and films properties

Keywords: emulsion polymerization, morphology, polyurethane, synthesis and processing techniques

In this work, segmented waterborne polyurethane-urea (WBPUU) dispersions containing hydrophobic polycaprolactone (PCL) and hydrophilic poly(ethylene glycol) (PEG) in different molar ratios are synthesized and used for the subsequent preparation of films by casting. The molar ratio of PEG is gradually increased up to 0.20 in order to analyze the effect of low hydrophilic PEG content (from 1.7 to 7.1 wt %) on the properties of resulting dispersions and films. Regarding the dispersions, the addition of PEG results in an increase of the particle size, from 86 AE 1 in the case of pure PCL-based system to 112 AE 15 for systems containing 7.1 wt % of PEG, as well as in the formation of a core-shell structure in the particles. Films show different behaviors depending on their PEG content. WBPUUs containing just PCL or very low content of PEG in the soft segment present higher stiffness. However, the addition of PEG in a content of 3.4 wt % or higher hinder the ability of the short-range ordering of the hard segment, increasing the elongation at break from 842 AE 102 MPa for PCL-based WBPUU to 1312 AE 84 MPa for the system with the highest PEG content. Systems with higher PEG content form nanoparticles with more segregated core-shell structures inducing to the film a higher hydrophilicity. Hence, the addition of PEG to a PCL-based WBPUU allows to tune the properties of the resulting film increasing the range of application of these materials.

INTRODUCTION

Thermoplastic-segmented polyurethanes and polyurethane-ureas (PU and PUUs) are part of a wide family of polymeric materials. Their thermal, thermomechanical, and specific mechanical properties as well as their biocompatibility 1,2 make them remarkably attractive materials for their use in different application fields such as automotive industry, biomaterials, and coatings. 3 These materials are formed by a soft segment (SS) and a hard one (HS) constituted by a long chain diol, and by both an isocyanate and a chain extender, respectively. In the case of PU, the chain extender is a low=molecular-weight diol, whereas in the case of PUU is a low-molecular-weight diamine. The variation of the nature of the reagents as well as the ratio between HS and SS, which usually are thermodynamically immiscible resulting in a microphase-separated morphology, leads to materials with tunable final properties. 4,5 One of the major drawbacks of PU and PUU is that due to their hydrophobic nature, their manipulation in liquid phase implies the use of organic solvents. The recent awareness on the environmental issues as well as the increasingly stricter environmental legislation, which limited atmospheric emissions of volatile organic compounds (VOCs), have drawn the interest of many researchers in finding more environmental friendly alternatives for the manipulation of polymeric systems. [START_REF]Adoption of the Paris Agreement[END_REF] One of these alternatives comes from aqueous polymeric systems, like PU and PUU dispersions in water (WBPU and WBPUU), which have experienced increasing attention in the last years since their manipulation does not involve VOC emissions. Additionally, water is a cheap, nontoxic, environmentally benign solvent and it is relatively easy to manipulate, so WBPU and WBPUU have become excellent candidates to substitute classic PU and PUU, respectively, in many applications. [START_REF] Zhang | [END_REF]8 Inherently, PU and PUU are no miscible in water, but in case of WBPU and WBPUU, ionic groups and hydrophilic nonionic chains contained in the polymer backbone could act as internal emulsifiers enabling the dispersion in water. 9 Ionic groups, completely or partially neutralized by the presence of counterions, can be cationic-like quaternary ammonium groups and anionic-like carboxylated or sulfonated groups, whereas nonionic can be hydrophilic polymer chains as, for example, poly(ethylene glycol) (PEG), behaving usually as part of the SS. 10,11 Hence, by using these ionic groups or nonionic highly hydrophilic polyols as emulsifiers, stable waterborne dispersions constituted by PU or PUU nanoparticles can be obtained. When the dispersion is formed, the ionic groups of the particles, surrounded by the counterions forming the electrical double layer, as well as the nonionic hydrophilic domains are arranged in the surface forming the shell of the particle. 12,13 Conversely, hydrophobic domains will be situated in the center of particles forming the core-shell of the particles. 14 Attending to the HS, apart from the aforementioned emulsifier, it is composed by a diisocyanate and a short-chained diol or diamine. In the case of diisocyanates, aliphatic isophorone diisocyanate (IPDI) is the most commonly used in the synthesis of waterborne dispersions due to its lower reactivity with water compared to aromatic ones, which allows a better control of the synthesis procedure. 15 Additionally, its asymmetric structure leads to less ordered structures and thus, eases the diffusion of water during the dispersion step. 16 Chain extender wise, the most used ones are 1,4 butanediol (1,4-BD) in the case of WBPU and ethylene diamine (EDA) in WBPUU. 17 Regarding the SS, in the literature, different types of polyols have been reported in the WBPU and WBPUU synthesis; concretely, most of the authors reported systems containing polyester type polycaprolactone (PCL), [18][19][20][21] or polyether type polytetramethylene ether glycol 8,22 ; but there are also many works involving other types of polyols as, for example, castor oil, 23,24 polyisobutyl lactate diol, 25 polycarbonate of 1,6-hexanediol, 26 or even polyols synthesized from CO 2 . 27 Aside from these polyols, which are hydrophobic, other authors reported the use of hydrophilic polyols in WBPUU synthesis, mainly, in order to obtain hydrogels 28 or as explained before to use them as a nonionic emulsifier. It has been observed that dispersions containing PEG, also known as poly(ethylene oxide) (PEO), with high percentage of solid content (above 30 wt %) tends to form physical gels. Yoo et al. synthesized hydrogels based on a 40 wt % solid content WBPUU dispersion with an SS composed by PEG for wound healing dressings, 29 whereas Young-Kim et al. prepared a coating coming from a PEG-based WBPUU dispersion with low solid content for waterproof breathable fabrics. 30 Block copolymers containing PEG hydrophilic block as SS have been also investigated. Yen et al. synthesized a series of WBPUU using poly(ε-caprolactone-b-ethylene glycol-b-ε-caprolactone) as polyol. The molecular weight of PEG in the SS and SS content were varied. 31 In a recent work developed in our group poly(ethylene glycol-b-propylene glycol-b-ethylene glycol) and poly(propylene glycol-b-ethylene glycol-b-propylene glycol) triblock copolymers with PEG blocks with different molecular weights and different PEG contents were also investigated as SS of WBPUU and compared with only PEG-based WBPUU (Díez-García et al.). The presence of triblock copolymers as SS resulted into a decrease of Young modulus and elongation at break compared with PEG-based WBPUU. Additionally, it was observed that the use of hydrophobic-hydrophilic triblock copolymer promotes the microphase separation. 32 In the last years, there has been an increasing interest in using more than one polyol, in order to modify the properties of the final material as well as to incorporate new functionalities. Fuensanta et al. combined two hydrophobic polyols and studied the properties of films prepared from WBPUU synthesized varying the ratio of polycarbonate of 1,6-hexanediol/polyadipate of BD in the SS. 33 However, the combination that probably has drawn more attention is the one formed by hydrophilic and hydrophobic polyols. The possibility of producing hydrogels containing a hydrophobic polyol such as PCL has been an interesting field of investigation, concretely for biomedical applications. Tsai et al. prepared hydrogel from a WBPUU, and studied different ratios of PCL with two different copolymers: poly(L-lactide-co-PEO) (PLLA-PEO) and poly(Dlactide-co-PEO) (PDLA-PEO) for its use as a new cell printing material. 34 Hsu et al. used a combination of PCL and PLLA-PEG copolymer to prepare a WBPUU with thermoresponsive gelation. 35 The aforementioned literature has proved the versatility of synthesizing WBPUUs by using combinations of polyols for many applications; but in case of hydrophilic/hydrophobic diol combinations, most of the works are focused on the properties of gels instead of films. Regarding the latter, in a previous work developed in our research group, different ratios of a hydrophobic biobased polyol poly(trimethylene ether glycol) (P3MG) and PEG (both M w = 1000 g mol -1 ) were used in order to prepare solvent and catalyst free WBPUUs; varying the PEG molar ratio from 0.1 to 0.4 (Díez-García et al.). It was seen that the decrease of the P3MG content result into an increase of both Young modulus and elongation at break, as well as a higher phase separation in the nanoscale. 13 However, based on our knowledge, there is only a few research works where mixtures of PCL and hydrophilic diols were been used to synthesize WBPU, and there is not research works on using PCL and PEG as SS in the synthesis of WBPUU. Cacik et al. analyzed the influence of the molecular weight of PEG over thermal properties and crystallinity of PCL/PEG-based WBPU. 36 Hao et al. in contrast studied the behavior in a WBPU varying the PEG content from 0.17 to 0.45 of molar ratio. 37 Therefore, this work focuses on studying the influence of the variation in the ratio of a hydrophobic diol-like PCL and a hydrophilic diol-like PEG into the properties of dispersions of WBPUU and films prepared from the synthesized dispersions by doing a comprehensive analysis of the influence of low PEG content (from 0.05 to 0.20 M ratio). It is expected that the addition of PEG will increase the hydrophilicity of the film, 30 as well as modify its morphology and thus its mechanical and thermal properties due to the different natures of PEG compared to the more hydrophobic PCL. However, it is important to determine at which PEG content the effect into the properties of the systems are modified, or in other words, what is the minimal content of PEG to achieve this modification of the properties. In this way, the aim of this work was to determine the minimum PEG content in the synthesized WBPUU dispersion at which the properties of the final material change significantly, in order to demonstrate that the incorporation of different contents of PEG could be a suitable strategy to modulate the final properties of the dispersions and films. The synthesized dispersions could be employed in VOCs free coatings or adhesives, as well as base for the preparation of environmentally friendly functional coatings by means of the incorporation of water dispersible nonentities and additives. [38][39][40] As explained before, the solid content of the dispersion is a key parameter in order to obtain either dispersions or gels, hence in this work the solid content was maintained low enough (25 wt %) to avoid gelification so as to study both dispersions and films. The particle size, distribution, and stability of the colloidal dispersions as well as the physicochemical, thermal, mechanical, and thermomechanical properties, morphology, and hydrophilicity of the films were evaluated and compared to the literature.

EXPERIMENTAL

Materials

Waterborne PUU dispersions were synthesized using a difunctional PCL diol and a difunctional PEG (molecular weight from the supplier M n = 2000 and 1000 g mol -1 , respectively) as SS, both provided by Sigma-Aldrich. Hydroxyl index, determined by titration according to ASTM D4274-05 was used to determine the molecular weight of the polyols, considering that they are difunctional. As a result, molecular weights of 1957 and 992 g mol -1 were obtained for PCL and PEG, respectively. 2,2-Bis(hydroxymethyl)propionic acid (DMPA, 98% purity) and EDA (99% purity), used as internal emulsifier and chain extender, respectively, provided also by Sigma-Aldrich, and IPDI (99.5% purity), supplied from Covestro, were used as HS. Dibutyl tin dilaurate, provided by Sigma-Aldrich, was used as catalyst. The polyols and the DMPA were dried under vacuum at 60 C for 4 h prior to their use. Triethylamine (TEA, 99.5% purity) was used as neutralizer of the carboxylic groups of DMPA (provided by Fluka) whereas butanone (supplied by Panreac) was used to transfer the neutralized DMPA into the reaction medium and also adjust viscosity.

Synthesis of Waterborne PUUs

The synthesis of the WBPUU has been carried out using a 250 mL four-necked flask equipped with a mechanical stirrer, thermometer, condenser, and nitrogen inlet within a thermostatized bath. The synthesis consists on a two-step process where in the former the formation of the prepolymer is carried out, whereas the latter consist on the phase inversion and the chain extension. In the first step of the reaction, PCL, PEG, and IPDI were reacted in bulk at 80 C for 3 h using a 0.1 wt % of DBTL, and then, the reaction is cooled until 50 C where DMPA previously neutralized with TEA diluted in 5 mL of butanone was added. After 1 h, the system is cooled until room temperature, where the phase inversion step was carried out by adding deionized (DI) water drop by drop, under vigorous stirring; the decrease of temperature before the phase inversion leads to avoid the reaction of the isocyanate with water. Finally, the chain extender (EDA) dissolved previously in DI was added drop by drop at 35 C at which the chain extension reaction between the isocyanate groups of the nanoparticles and the diamine-type chain extender is promoted, and left to react for 2 h to form the final WBPUU. Each step reaction progress was evaluated by dibutylamine back titration method according to ASTM D 2572-97. Once the reaction was completed, the remaining solvent was removed by vacuum distillation at 60 C. The pH of the dispersion was measured after the preparation, obtaining values in the range of 7-8 for all samples. Additionally, ionic strength has been calculated for all systems, obtaining values in the range of 0.075-0.078 mol L -1 .

Systems with different SS compositions were prepared, varying the molar ratio of PEG from 0.05 to 0.2. The theoretical solid content was fixed at 25 wt % in order to guarantee stable dispersions. All systems were prepared with the molar ratio of polyol/ diisocyanate/emulsifier/chain extender of 1/3.5/1/1.5. HS content, which is calculated as the percentage of the diisocyanate, emulsifier and chain extender, and the emulsifier percentage remain almost the same for all the synthesized WBPUU. The scheme of the reaction is displayed in Figure 1 whereas the composition of the synthesized WBPUUs is displayed in Table I. The dispersions were storage under room temperature and no presence nor of aggregates nor of deposited material have been observed during time.

Films Preparation

The dispersions were poured in Teflon molds and dried in a climatic chamber at 25 C with a 50% of relative humidity for 1 week, followed by 3 days in a vacuum oven at 25 C (400-420 mbar). After this time, PU films with a thickness of about 0.5 m were obtained from WBPUU dispersions. The solid content of the prepared dispersion has been taken into account in order to ensure the desirable thickness of the film. The films have been storage at room temperature in a desiccator in order to keep them away from humidity.

Characterization Dynamic Light Scattering. The particle size of the dispersions was measured by dynamic light scattering (DLS) using a BI-200SM goniometer, from Brookhaven. The intensity of dispersed light was measured using a luminous source of He-Ne laser (Mini L-30, wavelength = 637 nm, 400 mW) and a detector (BI-APD) placed on a rotary arm which allows measuring the intensity at 90 . Samples were prepared mixing a small amount of the synthesized aqueous dispersion with ultrapure water. Six measurements were performed per system. Zeta Potential. The stability of the dispersions was analyzed by zeta potential using a ZetaSizer Nano Series ZEN3600, from Malvern Instruments. Samples were prepared diluting the dispersions with ultrapure water. Solid Content. Solid content was calculated by drying 2 mL of each dispersion in an oven at 100 C for 8 h. The solid content, which was calculated by duplicate, was taken as the ratio between the weight of the dispersion and the weight of the dried sample following eq. ( 1):

%Solid content = Dried sample weight Weight of dispersion × 100 ð1Þ
Transmission Electron Microscopy. Transmission electron microscopy (TEM) measurements were performed on a TECNAI G2 20 TWIN operated at 80 kV and equipped with LaB6 filament. Samples used for TEM analysis were diluted into water. A drop of suspension was spread onto a TEM cooper grid (300 Mesh) covered by a pure carbon film and dried at room temperature. The grid was glow discharged before put the drop of suspension.

Gel Permeation Chromatography. Weight-average molecular weight (M w ) and dispersity (Ð) of the prepared films were determined by gel permeation chromatography (GPC) using a Thermo Scientific chromatograph, equipped with an isocratic Dionex UltiMate 3000 pump and a RefractoMax 521 refractive index detector. The separation was carried out at 30 C within four Phenogel GPC columns from Phenomenex with 5 μm particle size and 105, 103, 100, and 50 Å porosities, respectively, located in an UltiMate 3000 thermostated column compartment. Tetrahydrofuran (THF) was used as mobile phase at a flow rate of 1 mL min -1 . Samples were prepared dissolving the obtained PUs films by casting in THF at 1 wt % and filtering using nylon filters with 2 μm pore size. M w and Ð were reported as weight-average polystyrene standards.

Water Contact Angle Measurement. Static water contact angle (WCA) of the surface of the film was measured using a Phoenix-300 goniometer of SEO. For the measurement, drops of DI water of 0.1 μL have been used. Ten measurements were performed per sample.

Fourier Transform Infrared Spectroscopy. The characteristic functional groups of WBPUUs were analyzed by Fourier transform infrared (FTIR) using Nicolet Nexus spectrometer provided with a MKII Golden Gate accessory (Specac) with a diamond crystal at a nominal incidence angle of 45 and ZnSe lens. Spectra were recorder in attenuated total reflection (ATR) mode between 4000 and 650 cm -1 averaging 64 scans with a resolution of 8 cm -1 .

Proton Nuclear Magnetic Resonance. The chemical structure of WBPUUs was analyzed by proton nuclear magnetic resonance ( 1 H NMR). The measurement was carried out in a Bruker Avance 500 spectrometer, equipped with a BBO probe with gradient in z axis and using a resonant frequency of 125.77 MHz. Acquisition times were established at 3 s averaging 64 K scans, using an interpulse delay of 2 s. A time domain of 64 K was used in a spectral width of 10,000 Hz. Thermogravimetric Analysis. The thermal stability of the films was determined by thermogravimetric analysis (TGA) using a Mettler Toledo equipment. Between 5 and 10 mg of samples were introduced in ceramic pans. The samples were heated from 25 to 800 C in nitrogen atmosphere at a scanning rate of 10 C min -1 .

Thermomechanical Analysis. The thermomechanical behavior of the films was determined by using an Eplexor 100 N analyzer Gabo equipment. The measurements were carried out in tensile mode from -100 to 150 C at a constant scanning rate of 2 C min -1 and a fixed frequency equal to 1 Hz. The constant strain was established as 0.05% to keep the linear behavior.

Mechanical Testing. Tensile analysis was performed on an Instron 5967 testing machine provided with a 500 N load cell and pneumatic grips to hold the samples. Samples were cut in strips of 2.8 mm in width and 0.5 μm in thickness and tested at a constant rate of 20 mm min -1 at room temperature with a distance between clamps of 10 mm. Young modulus (E), stress at break (σ b ), and strain at break (ε b ) were averaged from stressstrain of eight specimens of each series.

Atomic Force Microscopy. The morphology of the different WBPUU systems was analyzed by atomic force microscopy (AFM) in tapping mode, using a Nanoscope IIIa scanning probe microscope (MultimodeTM Digital Instruments) with an integrated force generated by cantilever/silicon probes, applying a resonance frequency of about 180 kHz. The cantilevers were 125 μm long and had a tip radius of 5-10 nm. Samples were prepared by drying a drop of PUU dispersion on glass supports via spin coating (Spincoater P6700) at 200 rpm for 130 s. Height and phase images were taken simultaneously.

RESULTS AND DISCUSSION

Dispersions Characterization

The particle size of the synthesized WBPUU dispersions and their average diameter measured by DLS, the zeta-potential measurements as well as the real solid content are summarized in Table II.

DLS reveals a particle size of 86 AE 0.1 nm for the PCL100 system, with a polydispersity of 0.05 AE 0.02; these results are similar to those reported by other authors, which reported values in the range between 70 and 100 nm. 33 Comparing this obtained particle size with the dispersions containing PEG, an increase of both particle size and polydispersity can be observed. The hydrophilic nature of PEG allows the swelling of the particles by interactions with water; thus, the particle present not only higher particle size, but also higher polydispersity with all systems presenting monomodal distributions with the absence of aggregates. This increase of the particle size leads to the reduction of the specific surface area, [41][42][43] which can affect to the stability of the dispersion; however, in this case, no modification of the stability was observed as the particle size increased.

As far as the stability of the dispersions is concerned, all the systems present zeta potentials below -30 mV, so hence, the stability of the prepared dispersions is proved since stable dispersions have a zeta potential above +30 mV and below -30 mV. 44 The solid content was also determined by the aforementioned method. The synthesized WBPUUs showed a solid content close to the theoretical, suggesting that all the reagents were adequately integrated in the stable nanoparticles. The WBPUU dispersions were stored under room temperature and no presence nor of aggregates nor of sediment material has been observed during time.

The dispersions were also analyzed by TEM. Images of the WBPUUs with different PEG contents are shown in Figure 2. The obtained images confirm the higher particle size of the WBPUUs containing PCL/PEG in their SS compared with the WBPUU based on PCL. Additionally, the hydrophilic PEG in the SS tends to arrange at the nanoparticle shell. As can be observed in the inset of each image, at the intermediate PEG content the core and the shell start begin to differentiate and at the highest PEG content, a more defined core-shell structure seems to be observed in PCL80PEG20 system, suggesting that the addition of PEG promotes structural changes into the nanoparticle.

Films Characterization

The molecular weight of the films was determined by gel permeation chromatography. The obtained results are displayed in Table III. Results showed that PCL100 is the WBPUU with the higher molecular weight. The addition of PEG results into lower molecular weight, probably owing to the lower molecular weight of PEG polyol (992 g mol -1 ) compared with PCL polyol (1957 g mol -1 ). This decrease of molecular weight is noticeable and greater in systems with higher amounts of PEG. Similar molecular weight values are reported in the literature for waterborne PUs containing PEG in their SS, obtaining values between 25,000 and 15,000 g mol -1 with different contents of PEG. 32 Despite this decrease, the aforementioned measured pH range of 7-8, confirmed that the reaction has been performed successfully.

Regarding dispersity, all system present similar values with the addition of PEG but higher compared with systems only containing PCL. Additionally, the hydrophilic-hydrophobic nature of the surface of the prepared films has been determined by WCA measurement (Table III). Analyzing the obtained values, the addition of hydrophilic PEG to the WBPUUs resulted in a decrease of the contact angle as expected from the hydrophilic PEG shell observed for the particles. In films with higher amount of PEG it tends to arrange in the shell of the particle, showing more affinity with water compared with systems based only on PCL or with small amounts of PEG.

The characteristics functional groups of PUs were studied by FTIR. Figure 3(a) shows the FTIR spectra of the WBPUU synthesized with only PCL in the SS, PCL100, and the WBPUU synthesized with the highest content of PEG, PCL80PEG20. Both spectra show a peak in the 3350-3320 cm -1 interval, where hydrogen-bonded and nonhydrogen-bonded N─H of urethane and urea groups appear. 4 In this case, the observed signal, situated at 3330 cm -1 suggests that most of the N─H groups are involved in hydrogen bonds. The peak around 1735 cm -1 corresponds to the C O vibration of PCL and urethane group, 45 whereas a signal situated at 1640 cm -1 is attributed to hydrogen bonded C O of urea groups. 46 The peak situated at 1550 cm -1 in Amide II region is assigned to the C─N stretching vibration and N─H bending of urethane groups. The band located around 1460 cm -1 corresponds to ─CH 2 ─ stretching band and the signal situated at 1024 cm -1 is assigned to C─O stretching vibration. Additionally, the absence of bands in the 2260-2280 cm -1 interval (─NCO group) confirmed that the diisocyanate had completely reacted during the synthesis, and thus, means that the polymerization reaction has happened successfully. 47 Comparing both spectra, no differences were observed, neither shifting of existing bands nor apparitions of new signals have been observed despite of their different SS compositions.

The chemical structure of the prepared WBPUUs was also analyzed by 1 H NMR. Spectra displayed in Figure 3(b) showed differences between PCL100 and PCL80PEG20 WBPUUs. Spectrum of PCL100 showed typical chemical shifts related to the different protons of PCL polyol at 4.06 (Ha), 2.4 (Hb), 1.67 (Hc), and 1.36 (Hd) ppm. The spectrum of PCL80PEG20, in addition to peaks of PCL, showed a peak at 3.6 ppm, which is assigned to the methylene protons of the backbone of PEG (He). The presence of this new signal confirms the incorporation of the PEG. A peak related with methylene protons of DMPA at 3.9 ppm (Hf) is also observed in both spectra. Taking into account the molecular weight of both polyols and the areas of Ha peak of PCL and He peak of PEG, the PEG content in PCL80PEG20 WBPUU is determined, obtaining a value of 0.22 which is similar to the theoretical.

Thermal behavior of the films has been studied by DSC. The thermograms of synthesized WBPUUs and neat polyols for comparison purposes are displayed in Figure 4 and the values of the calculated transitions are displayed in Table IV. A transition was observed in all samples around -55 C, which is related with the transition of the SS (T g SS). Comparing to the glass-transition temperature of pure polyols, an increase was observed respect to PCL T g at -66.7 C (no T g was observed for PEG in this interval), due to mobility restrictions when they were integrated in the PU structure. In addition, no melting peaks of SS were observed, which indicates that this integration prevents PCL or PEG rich domains crystallization. Neat PEG and PCL showed a melting peak at 42.7 and 46.7 C, respectively. X-ray diffraction analysis has been performed in order to confirm the lack of SS crystallization. The obtained results that are displayed on the inset of Figure 4 does not show any signs of crystallization in all studied WBPUUs agreeing with DSC results. Regarding the HS, an endothermic transition can be observed in all samples in the 70-80 C interval attributed to the short-range ordering HS domains. 48 This endothermic transition temperature measured for PCL100 is in the range of the values reported previously for other WBPUUs containing only PCL in the SS. 18 The temperature of this transition seems to decrease with the addition of PEG. Additionally, the enthalpy assigned to this phenomenon also seems to be lower with the addition of PEG. The tendency of PEG to arrange in the shell of nanoparticles could hinder the ability of interactions between urethane-urea groups in the HS, thus resulting in a decrease of both melting temperature and enthalpy. This decrease is more appreciable starting from a 3.4 wt % of PEG.

The thermal stability of WBPUU has been studied by TGA. The thermal degradation of PUUs is believed to be a depolycondensation process which starts above 200 C with the degradation of HSs, and finished with the degradation of the soft ones. 49 In general, it consists on a first two degradations corresponding to urea and urethane linkages around 250 and 300 C, respectively, and a third one around 350-400 C assigned to the SS. 50 However, the temperature of the different degradation steps can be modified since the thermal stability of the PUUs is highly depending on the composition of the hard and SS, the type of diisocyanate, the ionic group content as well as the microphase separation and crystallinity of the phases. 51 The TGA and DGT curves of the WBPUUs and polyols are displayed in Regarding the thermomechanical properties, the evolution of storage modulus (E 0 ), loss modulus (E 00 ) and tan δ with temperature of the different WBPUUs have been determined by dynamic mechanical analysis (DMA), and the obtained results of the different systems are presented in Figure 6. As can be seen, the PCL100 WBPUU showed the higher storage modulus at glassy state. The decrease in E 0 and the maximum of tan δ and E 00 observed at low temperatures are related to the glass-transition temperature of the SS. For all the synthesized WBPUUs, the addition of PEG does not significantly change the T g SS as also observed by DSC. Notwithstanding, a flattening and broadening of the tan δ peak is observed at the highest PEG content, owing to the restrictions in the molecular mobility from PEG chains assembled in the nanoparticles shell. Above T g SS all systems showed a decreasing of the elasticity part E 0 , presenting PCL100 and PCL95PEG5 higher modulus at room temperature. Similar behaviors were observed in the literature with maximum tan δ values, which is related with the T g SS, in the same interval. 13,18,36 As far as the thermomechanical stability is concerned, the addition of PEG seems to decrease the thermomechanical stability compared to the PCL-based polymer (PCL100). Both PCL100 and PCL95PEG5 showed higher stabilities whereas the other systems containing more amount of PEG experienced a significant decrease, showing a progressive drop of their E 0 and E 00 at lower temperatures. The assembly of the PEG into the shell of the particles hampers the formation of short-range interactions between urethane-urea groups. Mechanical properties of the WBPUU were evaluated by tensile testing. Figure 7 shows the stress-strain curves of the different systems, whereas mechanical parameters such as Young modulus, tensile stress, and elongation at break are summarized in Table V. The behavior of the different systems agrees with previous DSC and thermomechanical analysis results. A decrease of the urethane and urea short-range ordering in the HS with the addition of PEG results in lower tensile modulus and stress at break, but higher strain at break. Similar thermal and mechanical behaviors were found for PCL100 and PCL95PEG5 WBPUUs. Comparing these values with those reported in previous works, one can notice higher values of Young modulus (23 AE 2 MPa against 6.4 AE 0.4 MPa), as well as similar strain at breaks (842 AE 102% vs. 891 AE 51%). 38 Other systems presenting higher amount of PEG exhibited similar Young modulus values as well as similar strain at break suggesting that the addition of only 3.4 wt % is enough to modify the behavior of the WBPUU. Moreover, it is important to indicate that no remarkable differences were observed at higher PEG content.

AFM analysis was carried out in order to analyze the morphology of films formed from the dispersions. Figure 8 shows the phase and three-dimensional representation of height images of different PUU systems. High dissipating soft domains give dark contrast while stiffer phases appear brighter due to the different energy dissipation by the cantilever, enabling the possibility of differentiate the different domains. 52 Analyzing the obtained images, separated microphase segregation can be observed in PCL100 system. These phases, which are formed by the coalescence of the nanoparticles, thus presenting a spherical microstructure, are formed by dark domains corresponding to the core PCL and lighter zones, which are composed by the HS and represent the shell of the particles. The formation of these microdomains leads to better mechanical properties as well as higher thermomechanical stability as was obtained in mechanical testing and DMA. The addition of PEG to the WBPUU, however, seems to result into a modification of the morphology which can be observed with the addition of very small quantities of PEG (3.4 wt %); above this quantity, no remarkable differences are observed in the morphology. Due to its hydrophilic nature, the PEG tends to arrange at the shell of the particle, becoming slightly thicker, as was also observed by TEM. As can be observed for the WBPUU with the highest amount of PEG, the thick of the particles shell increases whereas the dark domains corresponding to the PCL decrease their size, leading to the formation of a slightly different microstructure to that observed in PCL100. Hence, PCL90PEG10 and PCL80PEG20 present similar morphology as was also observed by TEM, both due to the arrangement of PEG in the shell of the particles. The analyzed microstructures corroborate that hydrophilic PEG act as nonionic emulsifier facilitating the formation of the nanoparticles.

It can conclude that the morphology changes observed by AFM as well as the core-shell morphology highlighted by TEM both confirm the assembly of PEG in the shell of the particle. This result into a hamper of the formation of short-range ordering between urea and urethanes, which leads to systems with different mechanical, thermal, and thermomechanical properties. Above this modification of the properties, the inclusion of the PEG to the backbone eases the transformation of the dispersion into a physical gel, which opens the possibility of using these materials in other fields such as the biomedical one widening the application field of this type of materials.

CONCLUSIONS

In this work, stable waterborne PUU (WBPUU) dispersions varying PCL/PEG molar ratio in the SS were successfully synthesized. The increase of the PEG content in the WBPUU resulted into an increase of the particle size and polydispersity, obtaining values of 112 AE 15 for PCL80PEG20 compared with the 86 AE 1 of PLC100. Moreover, zeta-potential measurements confirmed the stability of all dispersions. The particle morphology analyzed by TEM revealed that the addition of PEG led to a more segregated core-shell structure which is not observed in PCL100 particles.

Regarding film properties, PCL100 system showed higher Young modulus, thermomechanical stability, and higher enthalpy assigned to the short-range ordering of hard domains due to the microstructure observed in that system. Systems containing PEG in their SS, however, showed higher elongation at break owing to the observed microstructure where PEG hampers the formation of short-range ordered domains between urea and urethanes. Concretely, the elongation at break increased 500% comparing the PCL100 system with the one containing the higher amount of PEG. Additionally, films containing hydrophilic PEG present also higher affinity with water obtaining decreasing WCAs when PEG is added, from 90 AE 3 of PCL100 to72 AE 4 of PCL80PEG20.

Finally, a low amount of 3.4 wt % of PEG is enough to modify the properties and higher PEG content does not result in a noticeable variation of the properties of films.

Figure 1 .

 1 Figure 1. Scheme of the WBPUU synthesis. Synthesis conditions and chemical structure of the reagents. [Color figure can be viewed at wileyonlinelibrary.com]

Differential

  Scanning Calorimetry. The thermal properties of the films were determined by differential scanning calorimetry (DSC) using Mettler Toledo 822 equipment provided with a robotic arm and an electric intracooler as refrigerator unit. Between 5 and 10 mg of samples were encapsulated in aluminum pans and heated from -50 to 200 C at a scanning rate of 10 C min -1 in nitrogen atmosphere.X-ray Diffraction. X-ray powder diffraction patterns were collected by using a Philips X'pert PRO automatic diffractometer operating at 40 kV and 40 mA, in theta-theta configuration, secondary monochromator with Cu Kα radiation (λ = 1.5418 Å) and a PIXcel solid-state detector (active length in 2θ = 3.347 ).Table I.Composition of the Different Prepared WBPUU Systems and Molecular Weight of the Different Reagents Data were collected from 5 to 75 2θ (step size 0.026 and time per step = 80 s) at room temperature. A fixed divergence and antiscattering slit giving a constant volume of sample illumination were used.

Figure 2 .

 2 Figure 2. TEM images of different WBPUU systems. ×2500 magnification. Evolution of particle size and structure with the PEG content.

Figure 3 .

 3 Figure 3. (a) ATR-FTIR spectra of PCL100 and PCL80PEG20 (up) and (b) 1 H MNR spectra of PCL100 and PCL80PEG20 (down). [Color figure can be viewed at wileyonlinelibrary.com]

Figure 5 .

 5 Figure5. Analyzing the obtained results, PCL-based PUU showed one degradation step around 300 C, whereas WBPUUs containing PEG also showed a second degradation step around 375 C. Comparing with the TGA and DTG curves of neat PCL and PEG, a higher degradation temperature of PEG compared with PCL is observed; so, the second degradation appeared in PCL/PEG containing WBPUUs seems to correspond to the PEG. The obtained behavior does not correspond with the aforementioned general behavior, since the steps assigned to urea and urethane linkages and PCL polyol are overlapped in a single degradation process. Thermal stability of this WBPUU films is an important parameter in order to determine the potential applications of this material; all systems were stable until 250 C where the first degradation process happens; however, the endothermic transition observed at 70-80 C in the DSC thermogram, for all systems, limits clearly the operational stability below this temperature. The prepared materials could be reuse by reprocessing the films above their melting temperature.

Figure 4 .

 4 Figure 4. DSC thermograms of synthesized WPUU films and pure polyols. Inset: X-ray diffractograms of WBPUUs and pure polyols. [Color figure can be viewed at wileyonlinelibrary.com]

Figure 5 .

 5 Figure 5. Variation of the weight (up) and the derivated weight (down) of the WBPUU as a function of the temperature. Weight loss of polyols (inside up) and derivated weight of polyols (inside down) have been included for comparative purposes. [Color figure can be viewed at wileyonlinelibrary.com]

Figure 6 .Figure 7 .

 67 Figure 6. Storage and loss modulus (left) and tanδ (right) as a function of temperature for PCL/PEG PUU systems. Frequency = 1 rad s -1 and scanning rate = 2 C min -1 . [Color figure can be viewed at wileyonlinelibrary.com]

Figure 8 .

 8 Figure 8. AFM 3D height reconstruction and phase images of PCL100, PCL90PEG10, and PCL80PEG20 WBPUUs. [Color figure can be viewed at wileyonlinelibrary.com]

Table II .

 II Characterization of the WBPUU Dispersions

	Sample	Particle size (nm)	Polydispersity	Zeta potential (mV)	Solid content (wt %)

PCL100 86 AE 1 0.05 AE 0.02 -44.2 AE 0.3 25.02 PCL95PEG5 97 AE 0.5 0.16 AE 0.01 -48.7 AE 0.9 23.25 PCL90PEG10 115 AE 2 0.26 AE 0.03 -41.6 AE 0.2 25.18 PCL85PEG15 105 AE 5 0.23 AE 0.03 -44.1 AE 0.9 24.50 PCL80PEG20 112 AE 15 0.13 AE 0.05 -42.7 AE 0.6 26.20

Table III .

 III Weight-Average Molecular Weight and Dispersity of Synthesized WBPUU with Different Ratios of PEG/PCL and WCAs of the Films

		M w	Dispersity	Contact
	Sample	(g mol -1 )	ð M w = M n )	angle ( )
	PCL100	35,240	1.3	90 AE 3
	PCL95PEG5	25,795	1.5	83 AE 3
	PCL90PEG10	24,314	1.7	78 AE 3
	PCL85PEG15	23,196	1.7	76 AE 2
	PCL80PEG20	21,813	1.8	72 AE 4

Table IV .

 IV Calculated Thermal Transitions of WBPU Films

	Sample	T g SS ( C)	T m HS ( C)	ΔH m HS (J g -1 )
	PCL100	-55.9	79.5	16.5
	PCL95PEG5	-56.8	78.0	15.8
	PCL90PEG10	-56.5	74.2	6.1
	PCL85PEG15	-55.6	73.5	5.5
	PCL80PEG20	-55.1	72.8	3.4

Table V .

 V Mechanical Properties of WBPUU Films

		Young	Stress at	Strain at
		modulus	break	break
	Sample	(E) (MPa)	(σ b ) (MPa)	(ε b ) (%)
	PCL100	21 AE 2	2 9AE 8	842 AE 102
	PCL95PEG5	19 AE 2	2 5AE 3	1063 AE 114
	PCL90PEG10	14 AE 2	2 4AE 8	1080 AE 41
	PCL85PEG15	14 AE 2	1 5AE 5	1106 AE 122
	PCL80PEG20	14 AE 2	1 2AE 5	1312 AE 84

J. APPL. POLYM. SCI. 2020, DOI: 10.1002/APP.48847

ACKNOWLEDGMENTS

Financial support from the University of the Basque Country (UPV/EHU) (GIU18-216) and Spanish Ministry of Economy and Competitiveness (MINECO) (MAT2016-76294R) are gratefully acknowledged. J.V. wishes to acknowledge both the University of Pau and Pays de l'Adour and the UPV/EHU for his PhD grant. Moreover, technical support provided by "Macrobehaviour-Mesostructure-Nanotechnology" and "Microscopy: Polymer characterization" SGIker units from the UPV/EHU are also gratefully acknowledged.