N Cazy

P.-B Wieber

P Robuffo Giordano

F Chaumette
email: francois.chaumette@inria.fr

Visual Servoing Using Model Predictive Control to Assist Multiple Trajectory Tracking

Nicolas Cazy, Pierre-Brice Wieber, Paolo Robuffo Giordano, and Franc ¸ois Chaumette

Abstract-We propose in this paper a new active perception scheme based on Model Predictive Control under constraints for generating a sequence of visual servoing tasks. The proposed control scheme is used to compute the motion of a camera whose task is to successively observe a set of robots for measuring their position and improving the accuracy of their localization. This method is based on the prediction of an uncertainty model (due to actuation and measurement noise) for determining which robot has to be observed by the camera. Simulation results are presented for validating the approach.

I. INTRODUCTION

Visual servoing is a very effective method for controling the robot motion for realizing tasks of all sorts using feedback from visual sensors [START_REF] Chaumette | Visual servo control, part i: Basic approaches[END_REF]. However, an important constraint of using cameras is their limited field of view. This constraint is usually mitigated by imposing that the target features are always kept within the field of view. Avoiding that visual features leave the camera field of view is taken into account in [START_REF] Morel | Dynamic sensor planning in visual servoing[END_REF]. Various works on visual trajectory planning that take into account occlusions and feature loss avoidance have also been realized [START_REF] Marchand | Dynamic sensor planning in visual servoing[END_REF]- [START_REF] Kazemi | Randomized kinodynamic planning for robust visual servoing[END_REF].

However, strictly imposing the visibility of all targets is not always possible nor desirable. One way to perform visual servoing when the necessary visual information is missing is to simply predict how this information would evolve when no measurement is available [START_REF] Folio | Dealing with visual features loss during a vision-based task for a mobile robot[END_REF], [START_REF] Cazy | Pose Error Correction For Visual Features Prediction[END_REF]. This solution is limited, however, since the mismatch between the prediction model and the real measurements is bound to increase with time. In this respect, we propose in this paper to monitor this discrepancy for guaranteeing that it stays between predefined bounds. This requires ensuring that the necessary visual information can be collected regularly, when needed. For this, we consider a classical eye-in-hand setup where the camera view can be modified at will. We model the drifts between real and modeled positions and predict their evolution over a future time window. We set bounds on these drifts, and ensure that they will not be exceeded thanks to new camera measurements, using a Model Predictive Control scheme. This control strategy has been already used for visual servoing handling constraints for different applications in [START_REF] Murao | Predictive visual feedback control with eye-in-hand system via stabilizing receding horizon approach[END_REF]- [START_REF] Assa | Robust model predictive control for visual servoing[END_REF].

As an example, we propose to consider one camera carried by a UAV hovering over ground at a constant height. This UAV is subject to a drift between its real and its modeled position. We consider that the only way to correct the UAV/camera position consists in observing a landmark on the ground whose position is known. We also consider a set of mobile ground robots which have to follow some planned trajectories. These trajectories are far from each other and from the landmark so that the robots and the landmark cannot be seen by the flying camera at the same time. The UAV will then have to decide when to look at which robot and when to look at the landmark for making sure that the visual information, necessary for the ground robots to realize their task, is regularly collected.

A link can be established between our method and the selftriggered control in the sense that the quantity of information sent by the camera to the ground robots is intentionally reduced [START_REF] Fiter | Tutorial on arbitrary and state-dependent sampling[END_REF]. Indeed at each sampling instant the knowledge of the UAV and of the robots dynamics allows to predict the camera velocity without any measurement. Furthermore, our proposed method can be considered as an instance of active perception [START_REF] Bajcsy | Active perception[END_REF]. As discussed in [START_REF]Artificial intelligence and information-control systems of robots[END_REF], the proposed control scheme has to decide online the control inputs and sensor measurement strategies in order to minimize the uncertainty in the ground robot localization. Multiple works have already been conducted in this sense: for instance, in the context of object search, a mobile robot generates its path thanks to a web-based knowledge which delivers a customized execution plan in [START_REF] Riazuelo | Roboearth web-enabled and knowledge-based active perception[END_REF]. A robot path is determined on current knowledge of the object location encoded by a probability distribution computed during the search process in [START_REF] Shubina | Visual search for an object in a 3d environment using a mobile robot[END_REF]. Object recognition is obtained by predicting the optimal viewpoints in [START_REF] Browatzki | Active object recognition on a humanoid robot[END_REF], [START_REF] Le | Active perception: Interactive manipulation for improving object detection[END_REF]. Finally, a grasping method of deformable object by a robot arm using depth information is proposed in [START_REF] Ribas | Active perception of deformable objects using 3d cameras[END_REF]. In this paper, we exploit a model of the uncertainty evolution of the ground robots and UAV position to determine the sequence of camera visual servoing tasks to correct model drifts one after the other. The principle of active perception is thus focused on the control of the camera such that mobile ground robots and UAV correct their drift.

Section II starts by describing the tasks assigned to mobile ground robots, as well as the model errors which can be corrected by camera measures. Section III reviews the considered Model Predictive Control scheme which is used to realize the desired camera task. Simulation results are presented in Sect. IV. Finally, Sect. V concludes the paper and discusses about future works.

II. MODELING

In this paper, we consider N r mobile robots that have to follow some desired trajectories on the ground, one camera embedded on a UAV stabilized at a constant height, and a fixed landmark on the ground whose position is known in the world frame. The robot and camera are modeled as kinematic systems with velocity inputs. We also assume that each real position drifts from its respective model because of non-idealities such as actuation noise. This noise represents the fact that the robots and the UAV/camera do not react perfectly to their control schemes due to, e.g., modeling errors or delays. The UAV and camera are modeled as simple kinematic systems in order to focus on the the characteristics of the model predictive control scheme presented in the Section III. Future extensions of this work will consider more realistic dynamical models for the considered robots.

In our settings, the only way to rectify these drifts is to correct the modeled positions by measuring the robots (for the robot models) and landmark (for the camera model) positions. The problem is that, in general, only one robot or the landmark can be seen by the camera at once. Moreover, we choose to force the camera to stay focused on one robot or on the landmark as long as the other models do not need to be corrected. This is meant to minimize the camera total displacement. To determine which model should be corrected, the uncertainty of each position is propagated in order to quantify the amount of drifts. According to these uncertainties, the camera controller is able to manage the priority of which model must be corrected.

A. Representation

We consider that the camera and the robots have 2 DOFs each, namely their Cartesian coordinates on a plane. More precisely, we set c = c x , c y and r i = r ix , r iy respectively as camera and robot i (∀i = 1, ..., N r ) coordinates in a world frame (in red and orange on Fig. 1). We set also m = m x , m y as the motionless landmark used for the camera model correction, whose coordinates in the world frame are supposed known (in black in Fig. 1).

We set also c m and r im as the modeled coordinates of the camera and the robots (in blue on Fig. 1). These models are introduced for two reasons: the first one lies in the structure of the employed model predictive control scheme (see Section III) in wich future positions of the robots and the camera are determined using a model. The second one lies in the fact that, when the robots and the landmark are outside the camera field of view, no measurement is available to determine precisely the robot and camera position. Thus, using a model allows to cope with the lack of direct measurements and approximate the unmeasurable positions. Obviously, when a measurement becomes available, the corresponding model can be corrected accordingly.

B. Dynamics

The dynamics of camera and robots is assumed to evolve according to this simple kinematic model:

ċ = u c + n c ṙi = u i + n i ; (1)
where:

• the velocities applied to the camera allow it to move parallel to the ground: u c = u cx , u cy . • the velocities applied to each robot allow them to move on the ground: u i = u ix , u iy . • n c and n i are additive Gaussian noises applied to the camera and robots velocities:

n ∼ N (0, σ 2 ), (2)
with σ 2 being the variance. They define the error between the applied and the actual velocity.

As discussed in Sect. II-A, the modeled positions of camera and robots are the only information always available. Following (1), the dynamics of the modeled coordinates are defined in a discrete time form at each instant t k as:

c m (t k+1 ) = c m (t k ) + τ u c (t k ) r im (t k+1 ) = r im (t k ) + τ u i (t k ) (3)
with τ being the sampling time. From (3), the modeled coordinates can be predicted over a prediction horizon N as:

-→ c m = Ac m (t k ) + B -→ u c --→ r 1m = Ar im (t k ) + B -→ u i (4)
where :

• the sets of the camera and robot positions are:

-→ c m = c m (t k+1 ) , . . . , c m (t k+N ) ∈ R 2N ×1 --→ r im = r im (t k+1 ) , . . . , r im (t k+N ) ∈ R 2N ×1 , (5)
• the sets of the camera and robot inputs are:

-→ u c = u c (t k ) , . . . , u c (t k+N -1 ) ∈ R 2N ×1 -→ u i = u i (t k ) , . . . , u i (t k+N -1 ) ∈ R 2N ×1 , ( 6
) • and A =        1 0 0 1 . . . . . . 1 0 0 1        ∈ R 2N ×2 , B =          τ 0 0 0 . . . 0 0 τ 0 0 . . . 0 τ 0 τ 0 . . . 0 0 τ 0 τ . . . 0 . . . . . . . . . . . . . . . . . . 0 τ 0 τ . . . τ          ∈ R 2N ×2N . (7)
This dynamic over a future horizon is used for the model predictive control scheme (see Section III).

We now describe how the velocities applied to the robots are computed to allow them to follow their desired trajectory.

C. Robot control

The desired trajectories to be followed by the robots are represented by a time-varying desired r * i (t) (in green on Fig. 1). The robots are controlled to track their desired trajectories by applying the simple control scheme:

u i = k(r * i -r im ) (8)
with a scalar gain k > 0. It therefore appears that the drift between the real and modeled robot coordinates will also induce a drift between the real and the desired robot position.

We see now how to correct the modeled position thanks to camera measurements.

D. Models correction

The camera coordinates in the world frame are given by:

s c = c -m, (9)
while the robot coordinates in the camera frame are:

s i = r i -c. (10)
However because of image processing errors, camera measurements are subject to an additive noise n s with same form as (2). This implies a slight error for each measurement sc and si as

sc = s c + n s si = s i + n s . ( 11
)
These coordinates are available as measurement only when the landmark and the robots appear in the camera field of view delimited by the visibility range s min = s minx , s miny and s max = s maxx , s maxy (orange lines in Fig. 1). A measurement allows obtaining either the camera model coordinates c m or one robot model coordinates r im . Thus, using ( 9), [START_REF] Murao | Predictive visual feedback control with eye-in-hand system via stabilizing receding horizon approach[END_REF], and [START_REF] Sauvée | Image based visual servoing through nonlinear model predictive control[END_REF], we can write:

c m ← sc + m if s min sc s max r im ← si + c m if s min si s max (12)
The camera model drift can then be corrected by the landmark observation, while the robots and therefore also their trajectories are corrected by robots observation.

In this section, we discussed the modeling about camera and robots position, and highlighted the importance of obtaining a camera measurement for correcting the camera and robot models (and, thus, improving the robot tracking performance). The main remaining issue is to develop an appropriate control scheme for the camera moving to the robots or to the landmark to correct the position models, which is the goal of the next section.

III. MODEL PREDICTIVE CONTROL SCHEME

In this section we design a model predictive control strategy for allowing the camera to move towards either the landmark or one of the robots. Our aim is to force the camera to switch from one target to another one as a function of the evolution of the position uncertainties. These uncertainties are computed from a model characterizing how the drifts evolve over time. This strategy can be considered as a particular instance of active perception, since the camera motion must be optimized online in order to gather information about the scene.

The use of this approach is justified by the fact that the camera cannot move instantaneously from one point to another. It is therefore necessary to anticipate the camera and robot motion exploiting the evolution of position uncertainties over a prediction horizon. In order to develop this control strategy, we need to define the overall system dynamics (which has already been done in Sect. II-B), a suitable cost function able to capture our control objective, and the constraints imposed to the system. A proper definition of these constraints provides the desired behavior of the camera.

A. Cost function

The cost function to be minimized is given by:

min -→ uc∈K J( -→ u c ) ( 13
)
where K is the constraint domain described in the next section III-B, and

J( -→ u c ) = ( --→ s cm - --→ s * cm ) T ( --→ s cm - --→ s * cm ) + ( --→ s im - --→ s * im ) T ( --→ s im - --→ s * im ) , (14)
where --→ s cm and --→ s im are the sets of the corresponding models of -→ s c and -→ s i defined as:

s cm = c m -m s im = r im -c m (15)
Moreover, --→ s * cm and --→ s * im are the sets of the desired position of the landmark and the robot models in the camera frame along the prediction horizon. To obtain the landmark and robots in the center of the camera field of view, we set s * cm and s * im always null. This cost function has been designed so that the camera keeps at equal distance from the landmark and the robots. However, this distance will be strongly disturbed by the constraints [START_REF] Browatzki | Active object recognition on a humanoid robot[END_REF]. Namely, these constraints will force the camera to move towards the landmark or one robot (as it will be explained in Sect. III-B.2.b).

B. Constraints

A great advantage of model predictive control is the possibility to handle constraints in a principled way. We will use this characteristic to influence the camera motion.

1) Camera inputs constraints: First, constraints on the control input are included in the domain K as:

u - c u c u + c (16)
to ensure the camera velocity will not exceed its physical limitations, with u - c = (u - cx , u - cy ) and u + c = (u + cx , u + cy ). Obviously, these constraints are applied for all the set of inputs -→ u c . 2) Relative positions constraints: Secondly, the propagation of the positions uncertainty over a future time is modeled. This propagation is used to select which model needs to be corrected. Then, to force the camera to move to the selected model, constraints on the camera position relative to the landmark and to the robot positions are included in the domain K.

a) Uncertainties: As introduced in Sect. II-B, the velocity applied to camera and robots is subject to noise in [START_REF] Chaumette | Visual servo control, part i: Basic approaches[END_REF], which causes a drift between the real and the modeled position. Knowing the variance of that noise, it is possible to model the drift propagation over time. For this, we introduce the position uncertainty p j where j = 0, ..., N r , and j = 0 represents the camera.

Two situations are involved to determine the uncertainty propagation. Firstly, when a model is corrected thanks to an observation by the camera, a minimum value can be assigned to its uncertainty:

p j (t k ) = σ 2 s if s min sj (t k ) s max , (17)
which corresponds to the variance σ 2 s of the measurement error [START_REF] Sauvée | Image based visual servoing through nonlinear model predictive control[END_REF]. Secondly when a model is not corrected, the propagation of the uncertainty, exlploiting the dynamics (1), is defined as [START_REF] Thrun | Probabilistic robotics[END_REF]:

p j (t k+1 ) = p j (t k ) + τ 2 σ 2 j . (18)
Propagating over the future horizon N , we obtain:

p j (t k+N ) = σ 2 s if s min sj (t k ) s max p j (t k+N ) = p j (t k ) + N τ 2 σ 2 j otherwise . (19)
By predicting the position uncertainty over the prediction horizon, we can define which model has to be corrected. For this, a condition C j is associated to each uncertainty as:

C j ⇔ p j (t k+N ) p maxj
and p j (t k+N ) -p maxj > p l (t k+N ) -p max l or p j (t k+N ) < p maxj and p j (t k+N ) < p l (t k+N )

∀l = j, ∀j = 0, ..., N r (20)
The uncertainty threshold p maxj , which corresponds to the uncertainty value to not exceeded for allowing the correction of model j will be described in Sect. III-B.2.c. More precisely:

• If the uncertainty p j is larger than p maxj over the prediction horizon and if the difference between p j and p maxj is the largest among all the other uncertainties, the condition is held. This ensures that the model whose uncertainty exceeds the most the threshold over the prediction horizon has to be corrected. • Or if all uncertainties are lower than their threshold, and if the uncertainty p j is lower than all the other ones over the prediction horizon, the condition is held. This ensures that either the landmark or one of the robots is always corrected even when no uncertainty exceeds its threshold over the prediction horizon. This choice minimizes the camera displacement since it will remain focused on the same target as long as no other target needs to be corrected. The prediction of uncertainties allows the control strategy to determine which model should be corrected by the observations of the camera. We now describe how to inject this information in the model predictive control.

b) Tolerances: Tolerances represent the set of distances -→ δ j between the limits of the camera field of view and the modeled coordinates of the landmark/robot in the camera frame. The constraints that they define are added to the domain K:

s min - - → δ j --→ s jm s max + - → δ j , ∀j = 0, ..., N r . ( 21
)
We set δ -and δ + as respectively a low and a high tolerance value. They are assigned to the tolerances -→ δ j following the conditions defined in [START_REF] Shubina | Visual search for an object in a 3d environment using a mobile robot[END_REF]:

- → δ j = δ -if C j - → δ j = δ + otherwise . ( 22
)
The idea consists in having a low value δ -for moving the camera to the model that must be corrected. For instance on the Figure 2a, the tolerance δ 1 is low, the camera moves then to the first robot. However, even if δ -is low, we cannot guarantee that the real position will be in the camera field of view at the end of the camera motion because of the drifts from the models. This is why the value δ -is set to a negative value to allow the robot or the landmark to be in the camera field of view despite the drifts (see Figure 2a). Obviously, the camera can move to only one robot or to the landmark at a time. This is why a high value of tolerance δ + is assigned to the other models for ensuring that the system tolerates that they keep far from the camera field of view. Thus, on Figure 2b, while the tolerance assigned to the camera δ 0 is low, the tolerance assigned to the robot δ 1 is high, which allows the camera to move to the landmark and to keep far from the robot. The constraints defined by ( 21) allow the control scheme to force the camera to move towards the landmark or one robot when it is needed. δ -and δ + are assigned according to the conditions in [START_REF] Shubina | Visual search for an object in a 3d environment using a mobile robot[END_REF] depending on the evolution of uncertainties p j and on the uncertainty threshold p maxj that is described now. c) Uncertainty threshold: As described above, δ -has to be negative to allow the real robot or the landmark to be in the camera field of view despite the drifts from the models. However, whatever the value chosen for δ -the drift distance between the real and the modeled position will exceed |δ -| after several iterations. If that would occur, the real position would be outside the camera field of view and the model would not be corrected. The solution consists to correct the model before the distance |δ -| is reached by the drift.

For this, we define N min j as the number of iterations required by the drift distance to reach |δ -| from a null value. Thus, by propagating the uncertainty p j from the minimum value defined by [START_REF] Bajcsy | Active perception[END_REF] over N min j iterations we obtain the uncertainty threshold , namely the uncertainty reached when the drift distance attains |δ -| from a null value. Finally, using this threshold in [START_REF] Shubina | Visual search for an object in a 3d environment using a mobile robot[END_REF], we ensure that the drift will be corrected before reaching |δ -|. Using [START_REF] Riazuelo | Roboearth web-enabled and knowledge-based active perception[END_REF], the uncertainty threshold p maxj is given by:

p maxj = σ 2 s + N min j τ 2 σ 2 j . ( 23
)
To understand how we compute N min j , we firstly introduce the drift distances of the camera and the robots:

e 0 = ||c -c m || e i = ||r i -r im || ∀i = 1, . . . , N r , ( 24
)
with e 0 being the distance of the camera from its model, and e i the distance between the robot i and its model. Because both drift ditances of the camera and the robot i increase together, it appears that the distance |δ -| can be reached by the sum of these two drifts after a minimal number of iterations. As an example illustrated in Figure 3, we considere the worst case which can be considered. Because δ 1 = δ -, the camera represented in orange moves to the robot in red while this one follows its trajectory represented in green. We can see the drifts between their real and their model positions, and that the sum e 0 +e 1 reachs the distance |δ -| after N min 1 iterations. The exact values of e 0 and e i cannot be exactly computed, so we need to further explore the considered model of noise (2) which represents the drift evolution. As well known, it follows the normal distribution whose probability density establishes that 99.7% of noise values are distributed between -3σ j and +3σ j . The drift distance can then be confidently modeled by the propagation of 3σ j at each iteration. From this, the corresponding uncertainty is propagated as 9σ 2 j at each iteration. N min j is then the ration between the square of the limit distance |δ -| 2 and the uncertainty 9σ 2 j . Finally, two different N min j are computed. The first N min i is linked to the robot i. As we saw at the beginning of this paragraph, the distance |δ -| is reached by the sum of drift distances of the camera and the robot i. The corresponding minimal number of iterations can then be quantified as:

N min i = |δ -| 2 9τ 2 (σ 2 c + σ 2 i ) . (25)
Finally, N min i is injected in [START_REF] Ribas | Active perception of deformable objects using 3d cameras[END_REF] to compute the uncertainty threshold of the robot i.

The second N min 0 is linked to the camera. The goal is to ensure that all models of the robots can be corrected despite the camera drift. This means that we must ensure that the model of the camera is corrected as much as the model of the robot which has the largest drift. Therefore N min 0 is equal to the N min i with the lowest value:

N min 0 = min(N min i ) (26)
N min 0 can then be injected in [START_REF] Ribas | Active perception of deformable objects using 3d cameras[END_REF] to compute the uncertainty threshold of the camera. We described in this part how to compute the uncertainty thresholds p maxj in order to use it in the conditions [START_REF] Shubina | Visual search for an object in a 3d environment using a mobile robot[END_REF]. We now describe how to compute properly the value of the prediction horizon N to ensure that a satisfactory behavior will always be obtained. Fig. 3: The robot r 1 in red follows the trajectory represented in green. The camera c and its field of view are in orange. At t = t k , both models are corrected. At t = t k+N min1 , the robot is in the camera field of view despite the sum of errors caused by robot and camera drifts e 0 + e 1 .

C. Prediction Horizon

To guarantee that the models can be corrected despite errors introduced by noises, we must ensure that no uncertainty exceeds its corresponding threshold. This can be completed by a suitable computation of the prediction horizon N .

The camera cannot move instantaneously from a position to another. The prediction horizon must thus be greater than or equal to the number of iterations required to complete the longest path taken by the camera according to the minimum of its maximum speed:

N d max τ u cmin (27)
where

u cmin = min(|u - cx |, |u - cy |, |u + cx |, |u + cy |)
to ensure that the lower camera speed is taken into account, and d max is the longest path taken by the camera to reach robots and landmark. It can be seen here that the achievement of the proposed control scheme does not depend on the number of robots N r , but rather on d max whose value depends on the considered initial configurations (an example is presented in Section IV-A.3).

We can then use N computed in (27) to determine if the desired behavior can be realized without any loss of information. Indeed, the drift distance [START_REF] Thrun | Probabilistic robotics[END_REF] could exceed the distance defined by |δ -| before the camera has time to reach the position. We can then validate the system settings if:

N min(N min j ). (28)

IV. SIMULATION RESULTS

Simulations results are presented in this section. All coordinates described in the following are expressed in meters.

A. Parameters 1) Initial configurations: The camera is initially located at c(t 0 ) = 0.7 , 0.6 , the landmark at m = 0.5 , 0.5 and two robots at r 1 (t 0 ) = 0.5 , 0.7 and r 2 (t 0 ) = 1 , 0.7 . We set also the model of the positions as being perfeclty estimated at the beginning of the simulation: c m (t 0 ) = c(t 0 ), r 1m (t 0 ) = r 1 (t 0 ) and r 2m (t 0 ) = r 2 (t 0 ). Uncertainties are at their minimum value at the initialization p j (t 0 ) = σ 2 s . The range of visibility is delimited by s min = -0.08 , -0.08 and s max = 0.08 , 0.08 . Concerning the noise added to the robots and the camera motion, the averages are null µ c = µ 1 = µ 2 = 0, and the standard deviations are σ c = 0.07, σ 1 = 0.09 and σ 2 = 0.08. The noise added to the measures is defined by µ s = 0 and σ s = 0.005. The maximum value for the tolerance is set to δ + = 2, while the minimum value is δ -= -0.08 wich corresponds to the size of the field of view from the center of the image plane. As for the camera control, the minimum and maximum values that can be reached are u - c = -2m/s and u + c = +2m/s. And for the robots control, the gain is set at k = 1. The total simulation time is divided in 375 steps with sampling time of τ = 0.04s.

2) Desired trajectories: The robot desired trajectories are

       -→ r * 1 = Ar * 1 (t k ) + BE 0.2 cos(t k ) 0.2 sin(t k ) -→ r * 2 = Ar * 2 (t k ) + BE -0.2 cos(t k ) -0.2 sin(t k ) . ( 29
)
with r * 1 (t 0 ) = r 1 (t 0 ), r * 2 (t 0 ) = r 2 (t 0 ) and

E =        cos(τ ) -sin(τ ) sin(τ ) cos(τ ) . . . . . . cos(N τ ) -sin(N τ ) sin(N τ ) cos(N τ )        ∈ R 2N ×2 (30)
to obtain two circular paths repeated indefinitely. The radii of the circles are R 1 = R 2 = 0.2 and their centers are C 1 = 0.5 , 0.9 and C 2 = 1 , 0.5 .

3) Prediction Horizon: To compute the prediction horizon, we first determine the longest path taken by the camera to correct the models. With the considered configurations, the longest path can be defined as:

d max = d m + d r (31)
with d r the farthest distance between the two robots on their trajectory:

d r = R 1 + R 2 + (C 1x -C 2x ) 2 + (C 1y -C 2y ) 2 (32)
and d m the distance between the landmark and the farthest robot from the landmark on its trajectory:

d m = R 2 + (C 2x -m x ) 2 + (C 2y -m y ) 2 (33)
Combining ( 27) and (31) we can then set N = 22. The condition (28) is then held, which means that the correction of the model is guaranteed.

B. Small prediction horizon

We first consider our control strategy with a very small prediction horizon N = 1 (see the first part of the video submitted as supplementary material). The numerical method chosen for minimizing J under the constraints K is available in the function COBYLA (Constrained Optimization BY Linear Approximation) [START_REF] Powell | Direct search algorithms for optimization calculations[END_REF] from the C++ library NLOPT [START_REF] Johnson | The nlopt nonlinear-optimization package[END_REF]. This function is not the most efficient method to solve our optimization problem, but nevertheless allows to obtain simulation results which approach strongly the desired camera behavior. In this first simulation, the computation time required is approximately 1 ms for each function COBYLA call.

Figures 4a and4b show the evolution of real s x , s y and desired s *

x and s * y robot coordinates in the camera frame, while Figure 4c represents the evolution of the landmark coordinates. The camera field of view is depicted in orange while the evolution of the tolerances is in magenta (for the first robot), cyan (for the second robot) and black (for the landmark). We can thus visualize when the models have been corrected, that is when the blue and red curves are between the two orange lines at the same time. Finally, Figure 4d shows the evolution of uncertainties together with the uncertainty thresholds.

It can be observed at t = 0 that the uncertainties do not exceed their thresholds. However, the one related to the camera is smaller than the other ones on the prediction horizon. Thanks to the conditions (20), δ -is assigned to the landmark tolerance while δ + is assigned to the robots tolerance. Moreover, we can see on Figure 4c that s cx (t 0 ) and s cy (t 0 ) are not located in the interval described by s min -δ 0 and s max +δ 0 . The camera must therefore move to the landmark to satisfy the constraint [START_REF] Browatzki | Active object recognition on a humanoid robot[END_REF]. The model of the camera is then corrected by the observation of the landmark.

In the same way, the camera must move to the first robot at t = t 1 , when the prediction of the uncertainty p 1 (t k+N ) reaches p max1 . We can see that p 1 exceeds p max1 while the camera moves to the first robot. However, the camera is still able to recover the robot in its field of view. Indeed, the drift distance of the first robot and the camera is not large enough for exceed the distance described by |δ -|.

This limit is however exceeded for the second robot at t = t 2 , the minimum tolerance value δ -is fixed to the second robot tolerance, while δ + is fixed to the landmark and the first robot until the end of the simulation. The robots are not assisted by the camera observations anymore and diverge from their desired trajectories. Thus, the blue and red curves diverge from green curves on Figures 4a and4b.

The solution consists in properly computing the prediction horizon following the method described in Section III-C.

C. Large prediction horizon

We now apply our method with a prediction horizon set to N = 22 following the computation presented in Section IV-A.3 (see the last part of the video submitted as supplementary material). The computation time required is obviously greater than in the previous case. To be consistent with the sampling time τ , and for real-time issue, we have limited the resolution time of (13) to 40 ms. This is possible with the NLOPT library but may introduce a loss of accuracy in the solution.

As we can see on Figures 5a, 5b, 5c, the large prediction horizon allows the camera to switch from a target to another much earlier than in the previous simulation, resulting in an increase of the tolerances modification frequency. It causes a greater number of corrections achieved by the camera to satisfy the constraint [START_REF] Browatzki | Active object recognition on a humanoid robot[END_REF].

It can also be observed Figure 5d that the uncertainties never exceed their threshold. The drift distance is never large enough for exceed the distance described by |δ -|. The camera can then always retrieve the robots in its field of view, resulting in the success of the trajectory tracking of the robots throughout the simulation thanks to the proper horizon prediction.

V. CONCLUSION

We introduced in this paper a new camera control scheme to achieve a succession of visual servoing tasks by following the principle of active perception. This method is based on the model predictive control to anticipate the model errors over a future time horizon and, thanks to constraints, force the camera to move for correcting these errors. This method has been used to assist multiple robots subjected to noises to perform a trajectory tracking task. With the reported simulation results, we showed the effectiveness of our method and the importance of the prediction horizon in the parameters settings.

We applied this method in a 2D configuration. Future works will consider this method applied in more realistic applications. This will involve considering more complex models for the UAV and robotics dynamics, as well as a more realistic propagation of the state estimation uncertainty. Morover, we will also consider different possible numerical solvers for solving the proposed minimization problem in order to meet real-time constraints.

Fig. 1 :

 1 Fig. 1: Camera with limited field of view monitoring the trajectory of two robots on the ground (top view).

Fig. 2 :

 2 Fig. 2: Tolerance principle: The camera field of view is represented by orange dotted lines. (a) δ 1 represented in magenta is negative. The camera c retrieves the robot r 1 in its field of view despite the drift from the model r 1m .(b) δ 0 is negative, and δ 1 is high. The camera has moved to the landmark and its model is corrected, while the robot is allowed to stay far away from the field of view.

Fig. 4 :

 4 Fig. 4: Small prediction horizon: (a) and (b) show the desired and real coordinates of robots in the camera image plane, with smin-δi and smax + δi represented in magenta and cyan. (c) depicts the real coordinates of the landmark in the camera image plane, with smin -δ0 and smax + δ0 represented in black. (d) represents the uncertainties and their thresholds. At t = t2, the camera is not able to observe the second robot in its field of view because of the small prediction horizon which does not anticipate the model errors.

Fig. 5 :

 5 Fig. 5: Large prediction horizon: Compared to Fig. 4, the tolerance modification frequency increases thanks to the large horizon prediction, resulting by a greater number of model correction to achieve by the camera, which allows to assist indefinitely the robots trajectory tracking.

ACKNOWLEDGMENTS

This work was partly funded by the Oseo Romeo 2 Project.


Vo Thu

An Nguyen

Marie-Claire De Pauw-Gillet

Olivier Sandre

Mario Gauthier

Biocompatible Polyion Complex Micelles Synthesized from Arborescent Polymers

Keywords: Polyion Complex Micelle, Arborescent Polymer, Double-Hydrophilic Block Copolymer, ATRP, Core-Shell Structure, Cytotoxicity, Cell Internalization

Water-dispersible polyion complex (PIC) micelles were prepared by the self-assembly of an arborescent polystyrene-graft-poly(2-vinylpyridine) copolymer (denoted G0PS-g-P2VP or G1) serving as core and a poly(acrylic acid)-block-poly(2hydroxyethyl acrylate) (PAA-b-PHEA) doublehydrophilic block copolymer (DHBC) forming a shell. Varying the density of hydrophilic polymer chains in the stabilizing layer provided control over the size and structure of the entities obtained, from large flocculated species to stable isolated PIC micelles with diameters ranging from 42 to 67 nm. The hydrodynamic radius (determined from dynamic light scattering measurements), and the weight-average molar mass (M̅ w ) and radius of gyration of the scatterers (extracted from static multi-angle light scattering data) evidenced the formation of either isolated or aggregated PIC micelles depending on the self-assembly conditions used (pH, concentration and mixing molar ratio f). Changes in the morphology of the arborescent copolymer after complexation were observed by atomic force microscopy (AFM) imaging. In particular, by varying the force applied with the AFM tip on the samples, the core-shell structure of the PIC micelles was clearly evidenced. The PIC micelles displayed no significant cytotoxicity towards mouse fibroblast L929 cells, a standard cell line recommended for toxicity assays, due to the good biocompatibility of the hydrophilic PAA-b-PHEA shell. In spite of a negative residual zeta potential due to an excess of negative charges, fluorescently labeled PIC* micelles were successful internalized by L929 cells, as confirmed by laser scanning confocal microscopy (LSCM) and transmission electron microscopy (TEM).

INTRODUCTION

In biomedical applications, colloidal stability is of critical importance to ensure the safety and efficacy of nanoscale products, 1 since long blood circulation times are needed both for contrast agents and nanomedicines, while particles larger than 5 μm can cause capillary blockage and embolism. 2 The modification of the surface of nanoparticles (NPs) by self-assembly or complexation with a copolymer is a technique commonly applied to the delivery of DNA (gene therapy) and oligonucleotides (anti-sense therapy), 3 dendrimer porphyrin photosensitizers (photodynamic therapy), 4 drugs 5 and other types of hydrophobic nanomaterials, 6 by protecting (shielding) specific functions within the micellar core. Van der Waals (VDW) interactions are considered the main driving force leading to particle dispersion instability. However these can be counterbalanced by steric repulsions arising from a polymer coating, for example through adsorbed or grafted chains. [7][8] When the magnitude of the attraction force between two colloidal particles at contact is greater than the thermal energy (k B T) flocculation occurs, as seen for large cores stabilized by a low M̅ w polymer. 7 Increasing the M̅ w of the stabilizing polymer efficiently reduces the magnitude of the VDW forces by limiting the minimal approach distance between the particles. Qualitatively, the polymer layer thickness must exceed the VDW effective attraction distance of 5 nm to impart stability to the dispersed phase. 7 Since electrostatic stabilization is very sensitive to the ionic strength and the dielectric constant of the dispersion medium, polymeric stabilization (also referred to as steric stabilization), or a combination of steric and electrostatic (electrosteric) stabilization were preferred in this work, because the objective was to disperse the micelles in cell culture media. Considering that the grafted P2VP units of the arborescent G1 or G0PS-g-P2VP copolymer used in this investigation can be protonated, a polymer stabilizing shell bound through polyionic interactions appears promising, since it offers stronger interactions than physical adsorption and is much more versatile than covalent bonding.

This document is the Accepted Manuscript version of a Published Work that appeared in final form in Langmuir, 2016, 32 (50), 13482-13492, after peer review and technical editing by the publisher. Copyright © American Chemical Society. To access the final edited and published work see http://pubs.acs.org/doi/abs/10.1021/acs.langmuir.6b03683

2

The use of electrostatic interactions between two linear and oppositely charged polymers to form PIC micelles was first reported in the mid-1990s and immediately drew considerable attention. [9][10][11] The main driving force for polyion complexation is a combination of Coulombic attraction and an entropy gain through the release of counterions. 9-11 More precisely, polyion complex formation is ascribed to a decrease in total electrostatic free energy, due to a change in translational entropy for the small counterions and to the total electrostatic (or Born) energy.

12 Hydrophobic interactions and hydrogen bonding are considered additional contributions to the polyion complex formation process.

Hydrogen bonding was sometimes cited as the driving force for the complexation of poly(2-vinylpyridine) (P2VP) or poly(4-vinylpyridine) (P4VP) homopolymers with poly(acrylic acid) (PAA) or poly(methacrylic acid) (PMAA) in ethanol/water (1:1) solutions. 13 Intermolecular hydrogen bonding was also mentioned in the complexation of P4VP/P2VP-containing block copolymers (BCs) with PAA in DMF, 14 with preformed PS-b-PAA micelles in ethanol, 15 as well as with triblock copolymers of poly(ethylene oxide) monomethyl ether methacrylate and acrylic acid, (PEO 11 MA) 63 -b-PAA 657 -b-(PEO 11 MA) 63 in DMF. 16 By changing the solvent, tuning the chain length and adjusting the stoichiometry ratio of the complex, the extent of hydrogen bonding between PAA and poly(3-caprolactone)-block-poly(2-vinyl pyridine) in THF/ethanol could be controlled. 17 The solvent was found to have a strong impact on the nature of the driving force for the complexation between P4VP and PS-b-PAA: Weak hydrogen bonding was observed in ethanol, while strong polyionic interactions were reported in ethanol/water (1:1) solutions.

18

Polyion complexation in an aqueous environment is not straightforward for the weak base P2VP, since the pH affects the degree of ionization of weak polyelectrolytes before complexation.

19

Thus PIC micelles obtained by the complexation of P2VP-b-PEO with the strong polyanion sodium poly(4styrenesulfonate) in water acidified to pH 5 exhibited reversible dissociation at elevated pH and reassembly upon decreasing the pH. 20 The formation of PIC micelles upon mixing protonated P2VP-b-PEO with PMAA-b-PEO could only be achieved within a limited pH range (2 < pH < 6.1) to avoid the self-assembly into pure (single polymer) micelles of either PMAA-b-PEO at low pH or P2VP-b-PEO at high pH. 21 Interesting phase behavior and micellization phenomena (e.g. heteroarm star-like micelles, irregular aggregates…) were also observed at various pH in the self-complexation of a P2VP-b-PMMAb-PAA block terpolymer in aqueous dilute solutions.

22

Wang et al. 23 further reported the penetration of poly(Nisopropylacrylamide-co-4-vinylpyridine) statistical copolymer chains dissolved in an HCl solution into rigid preformed core-shell PS-co-PMAA microspheres.

To the best of our knowledge, most of the studies in the literature were performed with polyvinylpyridine (PVP) homopolymers or block copolymers, but not with complex polymer architectures containing PVP blocks as side-chains. Therefore this is the first report on the complexation of poly(acrylic acid)-b-poly(2hydroxyethyl acrylate) (PAA-b-PHEA) with an arborescent graft copolymer (G0PS-g-P2VP) in water to produce well-defined (non-aggregated) PIC micellar species. [24][START_REF] Fan | Recent Development of Unimolecular Micelles as Functional Materials and Applications[END_REF][START_REF] Ambade | Dendrimeric Micelles for Controlled Drug Release and Targeted Delivery[END_REF][START_REF] Wang | Magnetic and Dendritic Catalysts[END_REF][START_REF] Lukowiak | Dendritic Core-Shell Systems as Soft Drug Delivery Nanocarriers[END_REF] Water was used as the complexation solvent to obtain PIC micelles in one step, instead of relying upon micellization in organic solvents followed by transfer to aqueous media by heating or stepwise dialysis.

18,22

The current investigation aimed to characterize the PIC micelles in their dispersed form before examining their structure in the dry state, in contrast to most studies mentioned above. The goal of the PIC process was to anchor PAA-b-PHEA onto the surface of the G1 copolymer, such that the PHEA tails would form a swollen shell increasing the separation distance between the particles to the extent that VDW attractions would become negligible (> 5 nm, as observed for many colloidal systems). 7 The micellar structures thus obtained were expected to be colloidally stable in water at pH 7, and most importantly biocompatible due to the shell of PHEA, a highly biocompatible material. [START_REF] Lin | Preparation, Characterization, and Release Behavior of Aspirin-Loaded Poly(2-hydroxyethyl acrylate)/Silica Hydrogels[END_REF][START_REF] Hoogenboom | Nitroxide-Mediated Copolymerization of 2-Hydroxyethyl Acrylate and 2-Hydroxypropyl Acrylate: Copolymerization Kinetics and Thermoresponsive Properties[END_REF] The ability of PAA-b-PHEA DHBCs (with different block lengths) to stabilize the hydrophobic G1 substrate was evaluated using dynamic light scattering (DLS) and static multi-angle laser light scattering (MALLS) measurements, while the PIC micelle internal structure was further analyzed by atomic force microscopy (AFM). Cytotoxicity and cell internalization studies are presented in the last section, to demonstrate the usefulness of such PIC micelles in a biological context.

EXPERIMENTAL SECTION

Synthesis of polymers. The G1 or G0PS-g-P2VP copolymer (M̅ n = 1.1×10 Preparation of Polyion Complex Micelles. Complexation of the G1 arborescent copolymer with various amounts of different PAA-b-PHEA DHBCs was carried out in aqueous solutions. Hereafter the parameter f = CO 2 H/N will be used to quantify the molar ratio between the CO 2 H groups (from the PAA segment of the DHBCs) and the N moieties (2VP units of G1). The 3 following procedure describes the complexation of G1 with PAA 27 -b-PHEA 260 for f = 0.5. In a 10-mL vial, G1 (1.7 mg, 14.7 µmol of N) was completely dissolved in 300 µL of aqueous HCl solution at pH 1.4 by sonication (37 PIC Micelles in Water. G0PS-g-P2VP is insoluble in aqueous media at neutral pH, while PAA-b-PHEA is soluble. No micelles nor aggregates are formed when mixing them directly at pH 7. To maximize the interactions between the two weak polyelectrolytes PAA and P2VP, it is crucial to ionize these components before the complexation, 19 by protonating a portion of the amine groups of P2VP (pK a = 5) 19 into NH + , and ionizing some of the CO 2 H moieties of PAA (pK a = 6.5) [START_REF] Choi | Influence of the Degree of Ionization on Weak Polyelectrolyte Multilayer Assembly[END_REF] into CO 2 -. It should be easier to form stable electrostatic bonds between polycations and polyanions within a pH range where a majority of both the acid and basic moieties exist in their ionic forms, 19 but this was difficult to achieve for our system. Figure SI-5 illustrates the challenge in finding a pH range where the majority of both components exist in their ionic forms.

Complexation is unfavorable at either too low (acidic) or too high (basic) pH values, but pH 4.7 was selected as a compromise solution such that ca. 67% of the 2VP moieties would be protonated, while 1.6% of the carboxylic groups would exist in their anionic form according to the classical Henderson-Hasselbalch law of ionization for weak acids and bases. The interactions between PAA and P2VP may be much stronger at that pH in practice, since the apparent pK a value of PAA was found by Burke and Barrett [START_REF] Burke | Acid-Base Equilibria of Weak Polyelectrolytes in Multilayer Thin Films[END_REF] to shift downwards by up to 4 pH units when incorporated into multilayer assemblies, which would result in larger acid and base activities.

Intermolecular hydrogen bonding between neutral Hdonor CO 2 H groups in the PAA segments and neutral Hacceptor N groups in P2VP has been claimed to be the main driving force for complexations performed in organic solvents. 14,17,[START_REF] Zhao | Acid Terminated PS: not Polyion Micelle-like Particles Formed by Carboxylic Acid-Terminated Polystyrene and Poly(4-vinyl pyridine) in Chloroform/Methanol Mixed Solution[END_REF][START_REF] Xiong | Temperature-Responsive Multilayered Micelles Formed from the Complexation of PNIPAM-b-P4VP Block-Copolymer and PS-b-PAA Core-Shell Micelles[END_REF] In aqueous environments at pH 4.7, multiple intermolecular hydrogen bonding interactions may also assist at least transiently the electrostatic interactions in the complexation process. However at pH 7, these hydrogen bonds should be perturbed due to increased ionization of the CO 2 H groups, allowing the electrostatic interactions to play a dominant role in the polyion complexation process. [START_REF] Mori | Intelligent Colloidal Hybrids via Reversible pH-Induced Complexation of Polyelectrolyte and Silica Nanoparticles[END_REF] At pH 7 ca. 76% of the CO 2 H functionalities should exist in their ionic form, while much less (i.e. < 1%) of the N moieties may stay protonated, which should result in negatively charged PIC micelles at every f investigated. This was confirmed with zeta potential measurements (Figure 1). Electrostatic repulsions between these negatively charged PIC micelles, as indicated by their large zeta potential (ZP) values (Figure 1), may contribute significantly to colloidal stability of the particles, in addition to steric stabilization by the neutral PHEA chains.

Dynamic Light Scattering (DLS) Measurements. The PIC process was optimized in terms of its ability to yield exclusively isolated (non-aggregated) micellar structures, i.e. incorporating only one G1 substrate molecule within their core. To this end, DLS measurements at 90 o were used to generate numberand intensity-weighted size distributions and the corresponding average hydrodynamic diameters for most sample compositions investigated. The G1 substrate had an intensity-average hydrodynamic diameter 𝐷 h I = 29 ± 4 nm, with an extremely low polydispersity index (PDI = 0.04) at pH 4, which is much larger than the theoretical hard-core diameter (D G1 = 14.6 nm calculated from the bulk densities of PS and P2VP of 1.05 and 1.17 g/cm, 3,37 respectively, and the molar mass of the G1 polymer) for a collapsed PS-P2VP hydrophobic sphere at pH 7. The coiled dimensions of the PHEA segments (l PHEA ) in the different DHBCs, ranging from 7.1 to 12.2 nm, suggests that the absolute This document is the Accepted Manuscript version of a Published Work that appeared in final form in Langmuir, 2016, 32 (50), 13482-13492, after peer review and technical editing by the publisher. Copyright © American Chemical Society. To access the final edited and published work see http://pubs.acs.org/doi/abs/10.1021/acs.langmuir.6b03683 4 maximum hydrodynamic diameter of isolated PIC micelles (derived from a single G1 molecule), denoted thereafter D max , should be in a range of ca. 29-40 nm, with a narrow and unimodal size distribution, in analogy to the G1 substrate. The thickness of the protecting layer

𝛿 n = 1 2 [𝐷 h(PIC) n -𝐷 𝐺1 ]
can be calculated for an isolated micelle, where D G1 ≈ 14.6 nm for a collapsed spherical PSg-P2VP core. This information was used to determine the nature of the core-shell nanoparticles obtained, i.e. whether they were built around a single G1 polymer core (isolated micelles) or aggregated (multi-core) species.

In all cases where an intensity-weighted unimodal size distribution was obtained by DLS analysis, similar results were also obtained for number-and volumeweighted distribution curves, but with peaks in a lower size range (Figures SI-6 and SI-7). The intensity-weighted z-average 𝐷 h I was obtained by cumulants analysis of the autocorrelation function for the scattered intensity (see ESI S3.1). This value is shifted towards larger sizes as compared to the number-average radius 𝐷 h n , because it is dominated by the larger components of the size distribution.

38-39

Since only unimodal distributions were obtained in the current investigation, the use of cumulants analysis was justified and the number-weighted mean diameter 𝐷 h n can be used to describe the size of the PICs without bias towards larger sizes. The 𝐷 h n values obtained, reported together with the 𝐷 h I values in Table 1, therefore provide more realistic PIC dimensions than the z-average 𝐷 h I for comparison with other number-weighted measurements (e.g. AFM analysis).

Table 1. Hydrodynamic diameter and polydispersity index of G1@PAA-b-PHEA PIC micelles in water at 90°, pH 7 and 25 °C for various f ratios. The optimal f values for each DHBC are bolded. -𝐷 𝐺1 ] calculated for an isolated micelle, where D G1 ≈ 14.6 nm for a collapsed spherical PSg-P2VP core. The coiled dimensions of PHEA blocks indicated in parentheses (l PHEA ) were used to estimate the maximum theoretical diameter of an isolated core-shell assembly for each copolymer.

f 9 6 3 1 0.5 0.25 0.125 G1@PAA 27 -b- PHEA 56 (l PHEA ≈ 7.3 nm, D max ≈ 29 nm) 𝐷 h I /𝐷 h n (
G1@PAA 27 -b-PHEA 56 and G1@PAA 13 -b-PHEA 50 (l PHEA ≈ 7.1-7.3 nm, D max ≈ 29 nm). These two systems had similar characteristics and thus will be discussed concomitantly. The f = 1 ratio failed to yield a denseenough polymeric shell to stabilize the G1 molecules, resulting in > 1.2 µm diameter agglomerates. For f = 3, 6 and 9, stable PIC micelle dispersions were obtained with decreased 𝐷 h n values approaching the theoretical D max . Increasing the f ratio provided more DHBC chains to the stabilizing shell and increased the CO 2 -charge density surrounding the G1 molecules, leading to stronger electrosteric repulsions, as well as smaller and more stable particles due to their enhanced colloidal stability. The reduction in the PAA chain length from 27 to 13 units, whilst maintaining a comparable length for the PHEA block, resulted in smaller particle sizes and much narrower size distributions (for f = 3, 6, 9). Indeed when PAA 27 -b-PHEA 56 and PAA 13 -b-PHEA 50 are used at the same f ratio, the latter corresponds to twice as many chains (providing a higher density of stabilizing segments) for the same number of CO 2 H functionalities. However even for the highest ratio (f = 9), the micelles seem to be aggregated since: i) the 𝐷 h n = 44-36 nm is much larger than D max ≈ 29 nm; ii) the thickness 𝛿 n ≈ 15-This document is the Accepted Manuscript version of a Published Work that appeared in final form in Langmuir, 2016, 32 (50), 13482-13492, after peer review and technical editing by the publisher. Copyright © American Chemical Society. To access the final edited and published work see http://pubs.acs.org/doi/abs/10.1021/acs.langmuir.6b03683 5 11 nm is larger than l PHEA ≈ 7.3-7.1 nm. The very negative ZP of the PIC micelles (ca. -29 mV, Figure 1) also confirms the excess of CO 2 -at pH 7 in all cases, and suggests an important contribution of electrostatic repulsions to PIC micelle stability. In most cases, however, even a combination of electrostatic and steric stabilization was insufficient to produce isolated PIC micelles. The optimal composition of these systems was determined to be for f = 9, which is relatively high. G1@PAA 13 -b-PHEA 150 (l PHEA ≈ 10.1 nm, D max ≈ 35 nm) and G1@PAA 27 -b-PHEA 260 (l PHEA ≈ 12.2 nm, D max ≈ 39 nm). Good stabilization was observed for a ratio f = 3, yielding PIC micelles with 𝐷 h n = 32 nm and 37 nm (< D max ), respectively, and relatively narrow size distributions (PDI = 0.18). The seemingly nonaggregated nature of these micelles highlights the advantages of these DHBCs. The thicker PHEA shell (shell thickness 𝛿 n ≈ 9 and 11 nm, exceeding the 5 nm effective distance of VDW forces) provides enhanced steric stability for the PIC micelles, which favors the formation of isolated PIC species even though the contribution from electrostatic stabilization is fairly small (ZP = -13 and -19 mV, Figure 1).

Stabilization was also investigated for f =1, yielding G1@PAA 13 -b-PHEA 150 PIC micelles with 𝐷 h n = 29 nm (PDI = 0.14, ZP = -15 mV), and for G1@PAA 27 -b-PHEA 260 𝐷 h n = 34 nm (PDI = 0.13, ZP = -17 mV). These PIC micelles are still likely non-aggregated, with a shell thickness 𝛿 n ≈ 7-10 nm; for the same f ratio, PAA 27 -b-PHEA 260 increased the layer thickness by ca. 2-3 nm, in agreement with the larger dimensions of the PHEA chains. Reducing the f ratio from 3 to 1 decreased the density of DHBC in the micelles 3-fold, such that steric stabilization due to the long polymer chains presumably dominated colloidal stability. These results show that by introducing more HEA units into the polymeric stabilizer, the colloidal stability of the PIC micelles changed from electrosteric (a combination of electrostatic repulsion and steric stabilization, when using short PHEA chains) to predominantly steric stabilization (when employing longer PHEA blocks). Upon reducing the f ratio further to 0.5, 0.25 and 0.125, the decreased density of DHBC chains in the PIC assemblies led to flocculation for the G1@PAA 13 -b-PHEA 150 system, suggesting that the optimal f for this DHBC is f = 1. The influence of steric stabilization imparted by longer chains is also evident as G1@PAA 27 -b-PHEA 260 f = 0.5 had 𝐷 h n = 26 nm, fairly close to the size of the collapsed G1 core (D G1 = 14.6 nm, 𝛿 n ≈ 6 nm) and a narrow size distribution (PDI = 0.08), thus confirming their non-aggregated nature. The G1@PAA 27b-PHEA 260 f = 0.5 PIC micelles are the smallest colloidally stable micelles obtained. The ca. 6 nm shell thickness gives a closest distance of approach between This document is the Accepted Manuscript version of a Published Work that appeared in final form in Langmuir, 2016, 32 (50), 13482-13492, after peer review and technical editing by the publisher. Copyright © American Chemical Society. To access the final edited and published work see http://pubs.acs.org/doi/abs/10.1021/acs.langmuir.6b03683 6 two cores of 12 nm, still exceeding the 5 nm minimum effective distance 7 and further confirming the superior stabilizing ability of this copolymer. Further reduction in the f ratio leads to fewer stabilizing chains per micelle. For f = 0.25 and 0.125, the DHBC content was insufficient for stabilization of the G1 copolymer in its isolated form, which led to the formation of larger, aggregated micellar species. The most efficient copolymer tested for the preparation of isolated PIC micelles was therefore PAA 27 -b-PHEA 260 , as it maintained a sufficient thickness for the protecting shell while allowing a sufficient density of chains to cover the G1 substrate. For these copolymers, the optimal f was about 0.5-1, i.e. close to charge stoichiometry.

Multi-Angle Static and Dynamic Light Scattering. Static MALLS measurements were performed on selected PIC micelle samples over a concentration range of 0.2-1 mg⋅mL -1 and at 25 angles ranging from 30-150°, to be extrapolated to zero concentration and angle by the Zimm technique (Figures SI-8 to SI-10) to determine their weight-average absolute molecular weight (M̅ w ), radius of gyration (R g ), and second virial coefficient (A 2 ).

The hydrodynamic radius (R h ) of these samples was also determined at 25 different angles by 2 nd order cumulants analysis of the dynamic light scattering (DLS) data at a concentration of 1 mg⋅mL -1 , and extrapolation to zero angle as discussed in ESI S3.3 (Fig. SI-11). The results obtained by both techniques are summarized in Table 2.

Isolated vs. Aggregated Micelles. The nature of the PIC micelles may be investigated by comparing the experimental weight-average molecular weight M̅ w(Exp) obtained from static multi-angle laser light scattering (MALLS) data analysis with the molecular weight of the arborescent substrate (M̅ w(G1) = 1.1×10 6 g⋅mol -1 ) and the theoretical molecular weight based on the amounts of each component used in the complex formation, M̅ w(Theor) = M̅ w(G1) ×wt (G1+BC) /wt (G1) where wt (G1+BC) is the total weight of PIC micelles recovered after dialysis. The formation of isolated micelles can be confirmed if M̅ w(Exp) is in a range comparable to M̅ w(G1) , while for M̅ w(Exp) >> M̅ w(Theor) the formation of aggregates is unambiguously demonstrated. The non-aggregated nature of PIC micelles prepared from the DHBC with long PHEA segments, G1@PAA 13 -b-PHEA 150 with f = 3 and 1, and G1@PAA 27 -b-PHEA 260 with f = 0.5 was previously concluded from the DLS measurements at 90°. This is also confirmed by the MALLS data since M̅ w(Exp) < 2×M̅ w(G1) with M̅ w(G1) = 1.1×10 6 g⋅mol -1 . This is probably also true for G1@PAA 27 -b-PHEA 260 f = 3 and f = 1, for which M̅ w(Exp) < 3×M̅ w(G1) and non-aggregated character was likewise concluded from the DLS measurements when comparing the 𝐷 h n of the arborescent substrate and the PIC micelles. This assumption allows the calculation of the average number of chains shielding a G1 core in the micelles. For the five samples deemed to be non-aggregated, the number of DHBC chains per micelle varied from 27 PAA 13 -b-PHEA 150 surrounding one G1 core for f = 1, to 68 molecules for G1@PAA 27 -b-PHEA 260 with f = 3. The number of DHBC molecules per micelle increased as the amount of copolymer (f ratio) used in their preparation was increased, which seems reasonable. It is also larger for the PAA 27 -b-PHEA 260 sample series in comparison to the PAA 13 -b-PHEA 150 samples, in spite of the longer PHEA 260 segment increasing the DHBC bulkiness. This highlights the dominating influence of the anchoring PAA segment length in determining the number of copolymer chains incorporated in the PIC micelles.

This document is the Accepted Manuscript version of a Published Work that appeared in final form in Langmuir, 2016, 32 (50), 13482-13492, after peer review and technical editing by the publisher. Copyright © American Chemical Society. To access the final edited and published work see http://pubs.acs.org/doi/abs/10.1021/acs.langmuir.6b03683 7 Since the M̅ w(Exp) determined for PIC micelles G1@PAA 13 -b-PHEA 150 with f = 0.5 and 0.25, and G1@PAA 27 -b-PHEA 260 with f = 0.25 was larger than M̅ w(Theor.) , the micelles clearly flocculated due to the lack of stabilizer within the PIC shell. Contrary to our expectations, the G1@PAA 13 -b-PHEA 50 and G1@PAA 27 -b-PHEA 56 systems with f = 9, thought to be aggregated on the basis of the DLS measurements, were found to have M̅ w(Exp) < 2×M̅ w(G1) . The expected structure of these two assemblies will be further discussed on the basis of their ρ-parameter below.

ρ-Parameter. The ratio of the gyration and hydrodynamic radii R g /R h , also known as the ρparameter, is structure-sensitive and can provide information on the morphology of PIC micelles. For example, ρ = 0.775 is expected for rigid spheres of uniform density, while ρ = 1.7-2.0 is typically observed for randomly coiled polymer chains. [START_REF] Schmidt | Translational Diffusion and Hydrodynamic Radius of Unperturbed Flexible Chains[END_REF] However, for particles combining a dense core and partly coiled, less dense chain segments forming the shell, the concentration of the mass within their core can lead to a small radius of gyration, and ρ-values below the rigid sphere "limit" are thus often observed, while ρ-values close to 0.775 more likely reflect a core-shell structure with a non-homogenous mass density. Thus ρparameters below the rigid sphere limit (0.58 < ρ < 0.61)

were reported for lightly and moderately cross-linked PBMA microgels in THF, [START_REF] Kunz | Dynamic light scattering from spherical particles[END_REF] and 0.61 < ρ < 0.68 for spherical PS microgels measured in toluene. [START_REF] Antonietti | Microgels: Model Polymers for the Crosslinked State[END_REF] Micelles with PS segregated within the micellar core, surrounded by less dense PMAA segments in a dioxane : water 80 : 20 v/v solvent mixture had ρ-values ranging from 0.37 to 0.6. [START_REF] Qin | Polystyrene-Poly(methacrylic acid) Block Copolymer Micelles[END_REF] It is also possible for a core-shell structure to have ρ > 0.775, as reported for CdS@PS-b-PMMA hybrid inorganic-organic core-shell NPs with ρ = 1.3 in toluene (where both PMMA and PS chain segments were swollen), and with ρ = 1.1 in acetone (where the PS chains were collapsed).

44

The DLS data used in the ρ-parameter analysis were extrapolated to zero angle at a concentration of 1 mg⋅mL -1 (ESI S3.3) to determine the hydrodynamic radius (R h ).

The differences observed by extrapolation in comparison to the 90° data were relatively small in most cases (11% on average). Concentration-dependent DLS measurements also revealed that the influence of concentration (0.2-1 mg⋅mL -1 ) on the results was negligible, with ca. 1% variation among the samples. Consequently, only the data obtained at 1 mg⋅mL -1 were considered in the multi-angle DLS analysis. Since a collapsed core-loose stabilizing shell morphology was postulated earlier for our PIC micelles, the ρ-parameter should reflect the relative masses of the G1 core and shell components. Thus for G1@PAA 27 -b-PHEA 56 with f = 9, the value ρ = 0.93 obtained could suggest a core-shell structure. However the unexpectedly large 𝐷 h n = 44 nm (Table 1) and extrapolated D h = 57 nm (2×R h in Table 2) vs. D max ≈ 29 nm suggest an aggregation pattern corresponding to a mixture of isolated and aggregated species. The M̅ w(Exp) of these assemblies, being less than 2×M̅ w(G1) , most likely results from a dominant population of isolated species in comparison to aggregated species, resulting in an average hydrodynamic diameter larger than that expected for isolated micellar species. The situation is different for G1@PAA 13 -b-PHEA 50 f = 9, with a very large ρ = 1.97 and M̅ w(Exp) = 1.4×10 6 g⋅mol -1 , slightly larger than M̅ w(Exp) = 1.3×10 6 g⋅mol -1 in the previous case, corresponding to more repulsive interactions (larger A 2 ) resulting from a denser neutral polymer shell, decreasing the effective hydrodynamic diameter.

The non-aggregated PIC micelles G1@PAA 13 -b-PHEA 150 with f = 3 and 1 have ρ = 1.46 and 1.13, respectively, consistent is with a core-shell structure. These values are indeed well above the rigid dense sphere value, which evidences the large contribution from the mass of swollen block copolymer chains in the shell surrounding the collapsed arborescent core. The protecting shell was likely less dense for f = 1 than for f = 3, as evidenced by the somewhat lower M̅ w(Exp) . This also led to smaller R g (27.7 vs. 35.8 nm) and ρ (1.13 vs. 1.46)

values for f = 1, in agreement with the former interpretation of the results. Among the R g values obtained for the G1@PAA 27 -b-PHEA 260 system, only the values for f = 3 and 0.25 are considered reliable since the other two samples had R g < λ/20 ~ 15 nm, i.e., dimensions in a range outside the Guinier approximation domain, leading to large errors.

On that basis, ρ = 0.8 obtained for the PIC micelles with f = 3 is considered consistent with a dense core-swollen shell morphology. It is tempting to come to the same conclusion for f = 1 and 0.5, in spite of the large errors expected on the R g values. Furthermore, the nonaggregated PIC micelles in the G1@PAA 27 -b-PHEA 260 series with f = 3, 1 and 0.5 exhibit a larger ρ-value for f = 3. This could be due to a more extended conformation for the copolymer chains in the shell for f = 3, resulting from increased steric crowding effect in this sample which had the highest molecular weight among all the non-aggregated samples (M̅ w(Exp) = 3.2×10 6 g⋅mol -1 ).

The aggregates produced by the PIC micelles G1@PAA 27 -b-PHEA 260 with f = 0.25 were characterized by ρ = 0.57, suggesting the formation of a dense collapsed core with a relatively small number of G1 molecules (M̅ w(Exp) = 9.7×10 6 g⋅mol -1 ). Similarly, ρ = 0.89 obtained for the aggregates G1@PAA 13 -b-PHEA 150 f = 0.5, would correspond to a dense core formed by a larger number of G1 molecules (M̅ w(Exp) = 3.9×10 7 g⋅mol -1 ), surrounded by a loose shell of PHEA chains. This low ρ-parameter value could also be due to the presence of small clusters composed of many PIC micelles as suggested by Qin et al. [START_REF] Qin | Polystyrene-Poly(methacrylic acid) Block Copolymer Micelles[END_REF] The ρ = 1.56 value obtained for G1@PAA 13 -b-PHEA 150 micelles with f = 0.25, apart from corresponding to even larger aggregates of G1 molecules (M̅ w(Exp) = 17×10 7 g⋅mol -1 ), could also correspond to the formation of linear cluster structures, given that ρ is significantly larger than in the previous case. 8

Second Virial Coefficient. The second virial coefficient (A 2 ) of polymers in solution obtained from MALLS data is a measure of solvent quality or solventpolymer interactions. All the measured A 2 values were positive, reflecting favorable interactions between the -OH groups in the PHEA segments of the PIC micelles with the solvent (water). However the range of values observed (from 0.2×10 -5 to 19.3×10 -5 cm 3 ⋅mol⋅g -2

) is indicative of variations in the solvent-polymer interactions. It can be noted that: 1) Values in the highend were observed mostly for systems with f > 1, regardless of the specific DHBC used, thus reflecting the importance of interactions between the -OH groups in the PHEA chains and water. 2) Lower A 2 values were observed for lower f ratios among the series of PIC micelles derived from the same DHBC type. This is also consistent with the first statement. 3) Comparison between the G1@PAA 13 -b-PHEA 150 and G1@PAA 27 -b-PHEA 260 PIC micelles at the same f ratio produces the same result: The A 2 values for the former are smaller than for the latter. This is again related to differences in the number of -OH groups per PIC micelle. Taking f = 3 as an example, the G1@PAA 13 -b-PHEA 150 micelle had ca. 5,000 -OH groups provided by 33 DHBC side chains; G1@PAA 27 -b-PHEA 260 carried ca. 17,500 -OH groups on 68 DHBC side chains.

The lowest A 2 values were observed for the aggregated systems as expected: The largest PIC micelle G1@PAA 13b-PHEA 150 with f = 0.25 (M̅ w = 170×10 6 g⋅mol -1 ) had A 2 = 0.2×10 -5 cm 3 ⋅mol⋅g -2

, and so on. There is no obvious correlation between A 2 and the M̅ w of the aggregated PIC micelles however, presumably because the micelles have a heterogeneous chemical composition between the hydrophobic G1 core and the hydrophilic PAA-PHEA components. Considering the dominating contribution from -OH/water interactions to A 2 , a correlation was nevertheless found between the number of -OH groups in the molecules and the A 2 values for non-aggregated systems with f > 1 (Figure 2a).

pH Responsiveness of the PIC Micelles. The ZP of the PIC micelles G1@PAA 27 -b-PHEA 260 for f = 0.5 was pH-dependent. A large negative ZP (ca. -27 mV) was measured at pH ≥ 7 (Figure 2b), due to the presence of the CO 2 -functionalities. The ZP of the PIC micelles remained fairly constant under basic conditions, indicating complete ionization of the CO 2 H groups beyond pH 7. The increase in ZP in the acidic pH range is ascribed to protonation of the carboxylate groups, yielding the less polar CO 2 H functionalities, and of the amine groups in the G1 core to yield hydrophilic NH + derivatives. The isoelectric point of the PIC micelles was around pH 4. The corresponding pH-dependent variations in 𝐷 h I and 𝐷 h n of the PIC micelles were similar (Figure 2c), but the discussion will focus on the trends observed in 𝐷 h n . At pH 7.4 the number-weighted hydrodynamic diameter was 28.5 nm. Slight swelling of the PIC micelles to ca. 33 nm at pH 8-9.5 was likely caused by electrostatic repulsion between the CO 2 -groups, even at higher ionic strengths after the addition of 0.1 M NaOH. The hydrodynamic diameter almost doubled as the pH was decreased from 7.4 to 5.8, and tripled at pH to 2, indicating the formation of aggregates. At the isoelectric point (pH 4) the PIC micelles formed small clusters, but no PIC micelle dissociation was observed within the pH range tested. The variations in ZP and size were reversible over the range of pH ≥ 7 (isolated structures), while between 3.5 ≤ pH ≤ 6.5 small clusters formed that persisted even if the pH was subsequently brought back to 7.

Atomic Force Microscopy. The bare G1 substrate, deposited from acidic solutions (pH 4), exhibited illdefined morphologies because the arborescent copolymer flattened extensively on mica (visible more clearly in the phase image of Figure 3, column 1), presumably due to strong electrostatic interactions between the protonated G1 and the negatively charged mica. Both the molecular diameter measured on the AFM image (ca. 43 nm, larger than the 𝐷 h I = 29 nm in acidic solutions) and the height (ca. 1.9 nm) obtained by section analysis of the height image confirm strong flattening of the molecules. It is noteworthy that the PIC micelles obtained after complexation appeared as uniformly distributed, almost spherical particles with a narrow size distribution on the mica surface. The spherical morphology of the PIC micelles, shown in the 3D images and in section analysis of the images (Figure 3, column 2), are attributed in part to electrostatic This document is the Accepted Manuscript version of a Published Work that appeared in final form in Langmuir, 2016, 32 (50), 13482-13492, after peer review and technical editing by the publisher. Copyright © American Chemical Society. To access the final edited and published work see http://pubs.acs.org/doi/abs/10.1021/acs.langmuir.6b03683 9 repulsions between the PIC micelles and mica, both negatively charged. Section analysis reveals a particle diameter for the solid PIC micelles of 27 nm (close to 𝐷 h n = 26 nm measured in the aqueous dispersions), and a particle height of ca. 2.3 nm.

The phase lag of the cantilever oscillations with respect to the applied force is sensitive to variations in composition, adhesion, friction, viscoelasticity, as well as other factors. [START_REF] Schmitz | Phase Imaging as an Extension to Tapping Mode AFM for the Identification of Material Properties on Humidity-Sensitive Surfaces[END_REF] Since the pioneering work of Leclère et al., [START_REF] Leclère | Microdomain Morphology Analysis of Block Copolymers by Atomic Force Microscopy with Phase Detection Imaging[END_REF][START_REF] Leclère | Direct Observation of Microdomain Morphology in "All-Acrylic" Thermoplastic Elastomers Synthesized via Living Radical Polymerization[END_REF] darker domains of AFM phase images have been assigned to "soft" materials in several reports, [START_REF] Xu | Dynamics of Hydrated Polyurethane Biomaterials: Surface This document is the Accepted Manuscript version of a Published Work that appeared in final form in Langmuir[END_REF][START_REF] Raghavan | Mapping Polymer Heterogeneity Using Atomic Force Microscopy Phase Imaging and Nanoscale Indentation[END_REF] whilst the bright regions were ascribed to stiffer domains. 49 The surface of PIC micelles changed dramatically in comparison to the G1 substrate, from a "stiff" G0PS-g-P2VP matrix appearing in bright color to a dark "soft" coating layer. This is interpreted as a transition from a glassy PS-g-P2VP core to the acrylic hydrophilic PHEA phase of lower T g , also potentially plasticized by adsorbed water. The size, size distribution and uniformity of the PIC particles assessed by AFM analysis provide strong evidence for the success of the polyion complexation process using DHBC PAA 27 -b-PHEA 260 .

Figure 3. 3D images (row 1), height (row 2), phase AFM images (row 3) and section analysis (row 4) for the G1 substrate (column 1), G1@PAA 27 -b-PHEA 260 with f = 0.5 at set-points ratios of 0.96 (column 2), 0.92 (column 3), and 0.86 (column 4). The scale of all the images is 400 × 400 nm 2 .

In Tapping Mode™, the oscillating force applied during scanning is related to the ratio of the set-point amplitude to the free-oscillation amplitude of the cantilever, usually defined as the set-point ratio. [START_REF] Eaton | Atomic Force Microscopy[END_REF] For set-point ratios close to but less than 1, a low force is applied on the surface whilst at smaller set-point ratios, the AFM tip interacts more strongly with the sample. 49 By tuning the amplitude set-point and thus varying the interaction forces between the AFM tip and the particles, the morphology of PIC micelle can be probed. The PIC micelles (Figure 3) appear as spherical particles surrounded by a layer of "soft" PHEA (in darker color) at a set-point ratio of 0.96 (column 2). A stronger force applied at a set-point ratio of 0.92 (column 3) reveals the This document is the Accepted Manuscript version of a Published Work that appeared in final form in Langmuir, 2016, 32 (50), 13482-13492, after peer review and technical editing by the publisher. Copyright © American Chemical Society. To access the final edited and published work see http://pubs.acs.org/doi/abs/10.1021/acs.langmuir.6b03683 10 "hard" PS core (bright color) inside each PIC micelle, with a surrounding softer ring (darker color), thus confirming the core-shell morphology of the micelles. This structure was further confirmed by the "mountainlike" topology of PIC micelles (shown in the 3D images and the section analyses). When the set-point ratio was further decreased to 0.86 (column 4), at which point the PIC micelles appeared in a "sunny side up" egg-like morphology on the height AFM images, only the rigid PS component could be detected in the phase image. It may thus be concluded beyond any doubt that AFM analysis confirmed the formation of a DHBC stabilizing layer on the G1 arborescent substrate, as deduced from the DLS and MALLS measurements.

Cytotoxicity Assessment.

Due to the hydrophobicity of PS and P2VP at physiological pH, these polymers would be most likely isolated within the core of the PIC micelle, thus shifting the need for biocompatibility to the protecting shell, namely PAA and PHEA in our case. PAA has been used as an adhesive and surface-protective agent in dentistry, in eye drops, suppositories, capsules, wart creams, and in ophthalmic gels. [START_REF] Prati | Hydrostatic Intrapulpal Pressure and Bond Strength of Bonding Systems[END_REF][START_REF] Seldon | Silastic with Polyacrylic Acid Filler: Swelling Properties, Biocompatibility and Potential Use in Cochlear Implants[END_REF] PHEA has biocompatibility including low cytotoxicity and good cell compatibility, [START_REF] Lin | Preparation, Characterization, and Release Behavior of Aspirin-Loaded Poly(2-hydroxyethyl acrylate)/Silica Hydrogels[END_REF][START_REF] Hoogenboom | Nitroxide-Mediated Copolymerization of 2-Hydroxyethyl Acrylate and 2-Hydroxypropyl Acrylate: Copolymerization Kinetics and Thermoresponsive Properties[END_REF] similarly to the more widespread poly(2-hydroxyethyl methacrylate). PHEA-based materials have been used in controlled release drug delivery [START_REF] Arun | In Vitro Drug Release Studies of 2-Hydroxyethyl Acrylate or 2-Hydroxypropyl Methacrylate-4-{(1E,4E)-5-[4-(acryloyloxy)phenyl]-3oxopenta-1,4-dienyl}phenyl Acrylate Copolymer Beads[END_REF] and in cell culture media without significant cytotoxicity. [START_REF] Martínez-Ramos | Differentiation of Postnatal Neural Stem Cells into Glia and Functional Neurons on Laminin-Coated Polymeric Substrates[END_REF] Within the concentration range tested (from 0.2 to 10 mg⋅mL -1 ), none of the PIC micelles displayed cytotoxicity on the L929 cells (Figure 4a Even though they had a net negative charge, the PIC* micelles G1@PAA* 27 -b-PHEA 260 f = 0.5 were successfully internalized in the L929 cells, as evidenced by the green fluorescence found primarily in the cytoplasm at the periphery of the nuclear membrane, and probably in vesicles (Figure 4c to f). This observation is in good agreement with the residence of NPs in endosomes or lysosomes reported by other groups.

55-56

The ease of internalization of the NPs is attributed to their small size (𝐷 h I = 42 nm, PDI = 0.08). The PIC* micelles were clearly visualized as individual green This document is the Accepted Manuscript version of a Published Work that appeared in final form in Langmuir, 2016, 32 (50), 13482-13492, after peer review and technical editing by the publisher. Copyright © American Chemical Society. To access the final edited and published work see http://pubs.acs.org/doi/abs/10.1021/acs.langmuir.6b03683 11 fluorescent dots below optical resolution (∼300 nm), thereby evidencing their non-aggregation within the cytoplasm of cells and their integrity: The PHEA shell may prevent degradation induced by proteases and other enzymes existing in the low pH environment of late endosomes and lysosomes. The cells contained variable numbers of PIC micelles. The 3D construction from a 50 CLSM image stack (Figure 4g) clearly shows the location of the PIC micelles inside the cytoplasm, although some also stayed at the periphery of the nuclear membrane.

CONCLUSIONS

PAA-b-PHEA DHBCs with four different block lengths were synthesized by ATRP, and for the first time employed as polymeric stabilizers for P2VP-containing arborescent substrate cores through polyion complexation in water. Control over the size and structure of the PIC micelles, from large flocculated species to stable isolated (non-aggregated) entities, was accomplished by tuning the density of the polymer stabilizing layer, either by varying the DHBC stoichiometry or the length of the PAA anchoring block, and most notably by varying the length of the nonionic PHEA block. Slightly better stability was achieved by decreasing the PAA block length from 27 to 13 units at constant length of the PHEA block, which presumably led to increased coverage by the DHBC on the surface of the micelles. Short PHEA tails were insufficiently bulky to impart good colloidal stability, while most effective stabilization was achieved when the length of the nonionic segments was increased to 260 units, even for the sub-stoichiometric complexation ratio f = 0.5. Evidence for isolated vs. aggregated structures formation was obtained by: i) comparing the hydrodynamic diameter of the micelles with the dimensions calculated for the collapsed core and ii) obtaining molar mass (M̅ w ) and gyration-to-hydrodynamic radii ratio (ρ-parameter) values from static and dynamic light scattering measurements. The second virial coefficient A 2 obtained by MALLS also highlighted the dominating role of -OH/water interactions. The success of the complexation process and the core-shell morphology of PIC micelles were confirmed by imaging of the soft shell of PHEA surrounding the stiff G1 core through variations in the amplitude set-point in Tapping Mode™ AFM.

None of the PIC micelles incorporating one or multiple G1 arborescent core(s) and a shell of PAA-b-PHEA (tested at concentrations of up to 10 mg⋅mL -1 ) had significant cytotoxicity for L929 cells after 48 h of incubation, in spite of their different coating densities and sizes. The good colloidal stability and noncytotoxicity of the PIC micelles are ascribed to their good biocompatibility and hydrophilicity, and to the neutrality of the PHEA shell. In spite of being negatively charged, fluorescently labeled PIC* micelles were successfully internalized through the L929 cell membrane to reach the cytoplasmic compartment and perinuclear region. The response of PIC micelles to pH variations could be useful for applications such as the preparation of metallic or metal oxide NPs. A forthcoming article will describe the preparation of magnetic PIC micelles, making use of their G0PS-g-P2VP core as chelator for iron oxide precursors, and as templating medium for the synthesis of inorganic nanoparticles. The PIC process using PAA-b-PHEA to stabilize hydrophobic P2VP-containing arborescent copolymers can be applied to different P2VP copolymers, or to more complex arborescent materials (e.g. hybrid organic-inorganic systems).

ASSOCIATED CONTENT Supporting Information

The Supporting Information is available free of charge on the ACS Publications website http://pubs.acs.org/ at DOI: 10.1021/acs.langmuir.6b03683 Detailed procedures for the synthesis and characterization of the double-hydrophilic block copolymers and the PIC micelles. Methodology for biocompatibility and cell internalization assessment (PDF).

Figure 1 .

 1 Figure 1. Intensity-(top) and number-weighted hydrodynamic diameters (bottom) for four series of PIC micelles. The first data point in each series is disconnected from the trend line because it is off-scale. The zeta potential values for the samples are provided next to each data point.

Figure 2 .

 2 Figure 2. a) Dependence of A 2 on the number of -OH groups/PIC micelle. b) Zeta potential and c) Hydrodynamic diameter of PIC micelles G1@PAA 27 -b-PHEA 260 with f = 0.5 at different pH.

  , b), despite their different sizes (𝐷 h I = 42 and 87 nm for G1@PAA 27 -b-PHEA 260 f = 0.5 and 0.25; 𝐷 h I = 46 and 104 nm for G1@PAA 13 -b-PHEA 150 f = 1 and 0.5). Even after a 48-h incubation period, the L929 cells were well-spread in the wells and some were still undergoing mitosis, indicating the proliferation capability of the cells after exposure to the NPs. The good cell viability is attributed to the high hydrophilicity, biocompatibility and electrostatic neutrality of the PHEA shell.

Figure 4

 4 Figure 4. a) and b) Cytotoxicity profiles for PIC micelles G1@PAA-b-PHEA at various f ratios with the L929 cells determined via MTS assay at different concentrations after 48 h of incubation. Cell viability is expressed as the mean value and the standard deviation from three independent experiments, each with four replicates per condition, relatively to untreated cells (100% control); c) to f) confocal laser scanning microscopy images; and g) 3D construction and xz, yz sections for L929 cells treated with PIC* micelles (2 mg⋅mL -1 , 24 h incubation). Cell Internalization Study. A preliminary qualitative cell internalization study by confocal laser scanning microscopy (CLSM) used the same cell line as the cytotoxicity assessment. The cytoskeleton F-actin was labeled with phalloidin (red), while the nuclei were labeled with DAPI (blue), as shown in Figure SI-12 (for L929 cells not treated with PIC micelles), and in Figure 4c to f (for L929 cells treated with the PIC* micelles). The PIC* micelles labeled with the fluorescent probe fluoresceinamine were visualized as green fluorescent species.

RESULTS AND DISCUSSION Synthesis of the PAA-b-PHEA DHBCs and Fluorescent Labeling. The determination

			kHz,
	30 min, Elmasonic™ Ultrasonic Cleaner SH075EL). A pH
	7 aqueous solution of PAA 27 -b-PHEA 260 (8.5 mg, 5 mg⋅mL -1 , 7.35 µmol of CO 2 H) was quickly added. The
	mixture was stirred on a magnetic stirrer for 1 h before
	the pH was adjusted to 4.7 with a 1 M NaOH solution,
	and stirring was continued for 1 h. The pH was further
	adjusted to 7 with 0.1 M NaOH and the solution was
	stirred for 30 min. The solution was then dialyzed
	(50,000 MWCO Spectra/Por	® 7 regenerated cellulose
	bag) against Milli-Q water (5 L) for 24 h before it was
	collected and stored at 4	o C. The preparation of the
	fluorescently labeled PIC micelles (hereinafter referred
	to as PIC* micelles) was performed in the dark by the same procedure, but with a mixture of 5% PAA 27 * -b-
	PHEA 260 (7.0% labeled) and 95 mol% of (non-labeled)
	PAA 27 -b-PHEA 260 . Characterization of the PIC micelles
	by DLS, MALLS and AFM is described in ESI S3.
	Biocompatibility Assessment. The materials used
	for the cytotoxicity and cell internalization
	investigations of PIC micelles in L929 cells are described
	in ESI S4. These murine fibroblasts are recommended in
	the ISO10993-1: 2009 procedure "Biological evaluation of
	medical devices".
			of the
	number-average degree of polymerization of the
	polymers is described in ESI S1.2. The apparent molar mass dispersity of copolymers Ð SEC = M̅ w •M̅ n -1 ,
	determined by size-exclusion chromatography (ESI S1.2),
	was as follows: PAA 13 -b-PHEA 50 (Ð SEC = 1.15), PAA 13 -b-
	PHEA 150 (Ð SEC = 1.23), PAA 27 -b-PHEA 56 (Ð SEC = 1.08),
	PAA 27 -b-PHEA 260 (Ð SEC = 1.16). The number of
	fluoresceinamine (FA) molecules conjugated per PAA* 27 -
	b-PHEA 260 molecule (containing 27 CO 2 H groups) was
	1.88 ± 0.15, corresponding to 7.0 mol% labeling, which
	was sufficient to image the micelles in fluorescence
	microscopy whilst not changing their colloidal
	properties (hydrodynamic diameter and zeta potential;
	ESI S2.2).

Table 2 .

 2 Data derived from multi-angle static and dynamic light scattering measurements on PIC micelles. Italicized values indicate aggregated micelles due to insufficient coating and weak repulsion (low A 2 values).

	G1@PAA x -b-PHEA y M̅ w (10 6 g⋅mol	-1 )	#BC /PIC	R g (nm)	R h (nm)	PDI	ρ = R g /R h	-5 cm 3 ⋅ A 2 (10 mol⋅g -2 )
		f	Theor.	Exp.
	27-56	9	18.7		1.3	22	26.7	28.7	0.30	0.93	9.5
	13-50	9	32.5		1.4	54	45.3	23	0.16	1.97	12.1
	13-150	3	33.2		1.7	33	35.8	24.6	0.18	1.46	12.3
		1	14.0		1.6	27	27.7	24.5	0.23	1.13	1.8
		0.5	7.2		39	-	49.6	55.5	0.02	0.89	0.3
		0.25	3.5		170	-	124	79.6	0.08	1.56	0.2
	27-260	3	32.8		3.2	68	23.9	30	0.18	0.80	19.3
		1	10.5		2.8	51	12.2	29.8	0.20	0.41	1.9
		0.5	6.6		2.1	32	9.9	21.3	0.22	0.46	3.3
		0.25	2.9		9.7	-	17.4	30.3	0.15	0.57	1.7

  This document is the Accepted Manuscript version of a Published Work that appeared in final form in Langmuir, 2016, 32(50), 13482-13492, after peer review and technical editing by the publisher. Copyright © American Chemical Society. To access the final edited and published work see http://pubs.acs.org/doi/abs/10.1021/acs.langmuir.6b03683

ACKNOWLEDGEMENTS

We thank DE. Emmanuel Ibarboure (LCPO), DE. Patrick Garrigue (ENSCBP), and Pr. Neso Sojic (ENSCBP) for assistance with AFM, DE. Nicolas Guidolin (LCPO) for assistance with MALLS, and Dr. Sandra Ormenese (Imaging and flow cytometry platform -GIGA-R) for her help with confocal laser scanning microscopy.

(24) One of the referees expressed strong objection to using "PIC micelle" terminology to describe the selfassembled structures obtained in the present investigation, insisting that they rather be described as "polymer complex colloids", given the high molecular weight of the arborescent polymers used to prepare them. We insist on using this particular nomenclature This document is the Accepted Manuscript version of a Published Work that appeared in final form in Langmuir, 2016, 32 (50), 13482-13492, after peer review and technical editing by the publisher. Copyright © American Chemical Society. To access the final edited and published work see http://pubs.acs.org/doi/abs/10.1021/acs.langmuir.6b03683 13 however, considering that 1) under optimal conditions, the self-assembled structures incorporate one single arborescent polymer molecule with well-defined characteristics (narrow molecular weight distribution, controlled branching density, etc.), as opposed to a solid particle, 2) coupling of the double-hydrophilic block copolymer with the hydrophobic arborescent substrate is achieved via electrostatic interactions, and 3) the amphiphilic character of the assemblies is indeed analogous to other micellar species such as those obtained by the self-assembly of amphiphilic block copolymers, as well as amphiphilic dendritic structures. The use of the term "micelles" to describe different types of dendritic amphiphiles (the closest analogues to the systems being investigated) is indeed widespread in the literature, as demonstrated by the following four reviews on these types of amphiphilic constructs.