Aude Bolopion
email: bolopion@isir.fr

Barthélemy Cagneau
email: cagneau@isir.fr

Stéphane Régnier
email: regnier@isir.fr

2D micro teleoperation with force feedback

est destinée au dépôt

I. INTRODUCTION

During the last decades, the importance of micro and nano robotic systems has increased dramatically with the development of MEMS and micro assembly. However, handling of the components (objects of micrometer or nanometer sizes) remains a major problem [START_REF] Sitti | Microscale and nanoscale robotics systems [grand challenges of robotics[END_REF]. Several manipulation tools exist, including micromanipulators with capacitive or piezoresistive sensors, micro tweezers and cantilevers. Among them, Scanning Probe Microscopes based manipulation provide the highest resolution for real time force sensing.

To perform manipulation tasks at micro or nanoscales, a major problem is the lack of real-time visual feedback. To overcome this difficulty, virtual reality (VR) is widely used [START_REF] Ferreira | Virtual reality and haptics for nanorobotics[END_REF]. The environment with which the operator interacts can be purely virtual (for teaching purposes, for example) [START_REF] Kim | Task-based and stable telenanomanipulation in a nanoscale virtual environment[END_REF], or based on periodic measurements made during an experiment [START_REF] Li | Development of augmented reality system for AFM-based nanomanipulation[END_REF]. Haptic feedback would also appear to be an interesting solution, since it adds the sense of touch during the manipulation. When based on direct measurements performed during the manipulation, it allows the user to receive information about the objects being manipulated [START_REF] Tian | 3D nano forces sensing for an AFM based nanomanipulator[END_REF]. It can also be used in addition to a VR system [START_REF] Ammi | Robotic assisted micromanipulation system using virtual fixtures and metaphors[END_REF]. In this case, metaphors can be derived from the simulation, and be rendered as virtual guides to the operator. So far, manipulations with force feedback based on direct force measurement mainly consist in feeling or indenting objects [START_REF] Sitti | Teleoperated touch feedback from the surfaces at the nanoscale: modeling and experiments[END_REF] and characterizing them [START_REF] Guthold | Investigation and modification of molecular structures with the nanomanipulator[END_REF]. These operations are the first steps towards more complex tasks.

In this paper, we propose 2D telemanipulation with force feedback. We use a cantilever to manipulate objects, and measure the deformations of the beam using a laser reflected onto a photodiode. Using these measurements and mechanical properties of the probe, we determine the force applied on it by the object. This information is used to derive comprehensible indications about the manipulation that will be given to the user. Haptic feedback is chosen as the means to provide the user with the derived information. This approach is successfully demonstrated by microsphere rolling experiments.

This paper is organized as follows: In section II, forces available from the measurement of a cantilever's deformations are detailed. They will be used in section III to derive comprehensible forces that will be sent to the user to help him or her perform rolling manipulations. Results are presented in section V using the material and the control scheme described in section IV.

II. 2D FORCE MEASUREMENT FOR HAPTIC FEEDBACK

The manipulation platform is composed of a cantilever with no tip. As explained in detail in [START_REF] Haliyo | Controlled rolling of micro objects for autonomous micro manipulation[END_REF], this kind of probe can be used to manipulate objects. Their deformations are measured using a laser, reflected onto a photodiode (figure 1). Knowing the stiffnesses of the beams, forces applied on the probes are determined from the deformation measurements. Calibration methods of the cantilevers' spring constants can be found in particular in [START_REF] Langlois | Spring constant calibration of atomic force microscopy cantilevers with a piezosensor transfer standard[END_REF] or [START_REF] Cain | Force calibration in lateral force microscopy[END_REF].

A. Measurement of the cantilever's deformations

The photodiode is composed of four different parts (T l , T r , B l , B r ), as shown on figure 1. Two output voltages are defined, u v and u h , that are computed as:

u v = (T l + T r ) -(B l + B r ) u h = (T l + B l ) -(T r + B r ) (1)
They correspond respectively to a displacement of the laser beam's impact along the y and z axes. To estimate the deformations of the cantilever, the vertical and horizontal sensitivities of the photodiode (S v and S h ) need to be calibrated. Since the laser beam is focused in point E situated in the middle line and at the extremity of the probe (see figure 1), only the cantilever's deformations at point E can be measured. The deflection δ(L, 0), and angle of rotation φ(L, 0) are measured:

δ(L, 0) = S v u v (2) φ(L, 0) = S h u h (3)

B. Relation between the cantilever's deformations and the forces

The AFM cantilever is a simple beam the length, width and thickness of which will be noted respectively L, w, and t. All the parameters related to the cantilever are summarized in table I. Numerical values are given for a silicon beam, and the dimensions are those of the cantilever used in section V for the experiments. As a force F z (x p , y p ) is applied on the beam at point P (x p , y p ) along the vertical direction (z axis), the cantilever will experience two different deformations, deflection and rotation. In the following paragraphs, we will derive relations between the applied force and the deformations of the probe.

1) Deformations along the x axis:

The force F z (x p , y p ) bends the beam. Considering a clamp-free cantilever, the deflection of the probe δ(x, y p ) is given by:

δ(x, y p ) = F z (x p , y p ) EI y x x 2 p 2 - x 3 p 6 for x p ≤ x ≤ L (4)
As seen in section II-A, this bending can be measured using equation [START_REF] Ferreira | Virtual reality and haptics for nanorobotics[END_REF]. Therefore, the force F z (x p , y p ) can be estimated using this measurement:

F z (x p , y p ) = k z (x p )δ(x p , y p ) (5)
where k z (x p ) is the stiffness of the cantilever at point x p :

k z (x p ) = Ewt 3 4x 3 p = L x p 3 k z (L) (6)
The normal stiffness at the extremity of the cantilever k z (L) is calibrated. According to (4):

δ(x p , y p ) = 2x p 3L -x p δ(L, y p ) (7)
Combining equations ( 6) and ( 7) leads to:

F z (x p , y p ) = 2L 3 x 2 p (3L -x p ) k z (L)δ(L, y p ) (8)
2) Deformations along the y axis: Deformation is derived considering a cantilever along the y axis (free at its extremity), and the hypothesis that the problem is monodimensional. The force F z (x p , y p ) applied in the vertical direction produces a rotation φ(x p , y p ) of the beam:

φ(x p , y p ) = F z (x p , y p ) 2EI x y 2 p + φ(x p , 0) (9)
This angle can be used to compute the corresponding moment M (x p , y p ):

M (x p , y p ) = k t φ (x p )φ(x p , y p ) (10)
where k t φ (x p ) = Gwt 3 3xp is the torsional spring constant. According to [START_REF] Haliyo | Controlled rolling of micro objects for autonomous micro manipulation[END_REF], this leads to:

M (x p , y p ) = k t φ (x p ) F z (x p , y p ) 2EI x y 2 p + φ(x p , 0) (11)
The relations between the force applied on the cantilever and the measure of its deformations presented in this section will be used in the following paragraphs to derive comprehensible information about the interaction forces, to help the user perform a given manipulation task.

III. AN EXAMPLE OF 2D MANIPULATION WITH FORCE

FEEDBACK

The use of haptic feedback greatly improves the dexterity of an operator while performing a manipulation task in the microworld [START_REF] Millet | Improving perception and understanding of nanoscale phenomena using haptics and visual analogy[END_REF]. This is especially true when the visual feedback is limited, which is the case at microscales since optical microscopes provide only two dimensional images. A rolling experiment has been chosen to demonstrate the use of force feedback for micro manipulation. This is a promising way to manipulate objects (spheres, carbon nanotubes) at this scale. It can also be used to release objects attached to a cantilever [START_REF] Haliyo | Controlled rolling of micro objects for autonomous micro manipulation[END_REF], or to perform patterning on a substrate [START_REF] Tafazzoli | Force-controlled microcontact printing using microassembled particle templates[END_REF]. It has to be noted that other strategies of manipulation could have been chosen, as well as different object sizes. Rolling is only one possible application to demonstrate how to render interaction forces to the operator using haptic feedback.

A. Longitudinal rolling

In [START_REF] Haliyo | Controlled rolling of micro objects for autonomous micro manipulation[END_REF], the authors state that to make objects roll without sliding, a force similar to the pull-off force must be applied on them by the cantilever. Since the cantilever is not in the plane (x, O, y), but tilted by several degrees (see figure 1) the deflection at the extremity of the cantilever will be kept constant to apply this force. The position of the origin of the cantilever will therefore be servoed to maintain it at the desired value.

The initial position of the sphere under the cantilever is noted x p (0). This point, as well as the position of the cantilever along the x axis x c (j) at time step j, are presumed to be known (see paragraph IV-A). Since rolling without sliding is performed, the current position of the sphere x p at time j is estimated from:

x p (j) = x p (0) + x c (j) -x c (0) 2 (12)
For longitudinal rolling, the sphere is initially positioned at the extremity of the probe, in its middle line (point E (x max , 0) on figure 1). The position along the y axis is kept constant and is such that the cantilever experiences no torsion: φ(L, 0) = 0. The cantilever is then moved along the x axis, to make the sphere roll. To provide the user with information about the position of the object being manipulated a force F x e will be sent to him or her. This force is derived from the normal force F z applied by the sphere on the cantilever. However, since δ(L, y p ) is not available, we chose to send to the user a force F x e computed as:

F x e (x p , y p ) = 2L 3 x 2 p (3L -x p ) k z (L)δ(L, 0) (13)
where δ(L, 0) is measured using the photodiode (see ( 2)).

The profile of the force is plotted on figure 2 (as stated above, the position of the origin of the cantilever is servoed so that δ(L, 0) is constant). Position of the sphere along the x axis (µm) Profile of the force F x e (µN ) Fig. 2. Profile of the force that will be sent to the user along the x axis This force is minimal at the extremity of the cantilever. So that this point represents a stable equilibrium position for the user, an offset will be used, in order to render a force equal to zero when the sphere is at the extremity of the cantilever.

B. Lateral rolling

For lateral rolling, the position along the x axis is kept constant, while the cantilever is moved along the y axis so that the sphere rolls. While performing rolling tasks, it is important to keep the sphere in the middle line of the cantilever, so that it does not loose contact on a lateral side of the beam. To help the user maintain the sphere on the middle line, a force F y e is sent to him or her via the haptic interface. It will correspond to the torsion force that will tend to keep the cantilever horizontal:

M (x p , y p ) y p (14)
However, according to [START_REF] Cain | Force calibration in lateral force microscopy[END_REF], the variation of this force is not easily interpretable. Therefore, we chose to linearize the expression:

F y e (x p , y p ) = M (L, 0) w 2 = k t φ (L) w 2 φ(L, 0) (15)
where φ(L, 0) is the measure of the rotation given by the photodiode output (see equation ( 3)). The variations of the force F y e are plotted on figure 3. It can be seen that depending on the sign of φ(L, 0), this force is either positive or negative. The higher φ(L, 0) is, the higher the magnitude of the force. If sent back to the user through a haptic interface, this force will indeed tend to keep the sphere on the middle line of the cantilever (which is the point where the torsion is nil). Since the angle increases the further the force F z (x p , y p ) is situated from the middle line, and F y e increases as the angle increases, it can also help to estimate the position of the sphere (x p , y p ) under the cantilever.

C. 2D force feedback rolling operation

As can be seen from ( 13) and (15), the forces F x e and F y e sent to the user are not coupled. Therefore, 2D rolling experiments can be carried out using the same force feedback rendered on two different axes of the haptic interface.

IV. EXPERIMENTAL SETUP

A. Platform

The manipulation platform used to perform the experiments described in section III is depicted on figure 4(a).

The vision system is composed of an optical microscope with a 20× lens. The initial contact point between the sphere and the probe has been determined using this vision system.

To position the cantilever accurately with respect to the scene, three nanotranslators are available to manipulate objects along the x, y and z directions. The range (resp. resolution) provided by this system is equal to 50µm (resp. 0.1nm) for the x and y directions, 12µm (resp. 1.83nm) The available haptic device is a Virtuose manufactured by Haption1 . This arm, represented on figure 4(b), allows 3 degrees of freedom of force feedback. For ease of manipulation, and to avoid damaging the master device, forces sent back to the user are saturated to 6N .

B. Control scheme

The choice of the control scheme used to connect the haptic device to the slave device is of primary importance for the quality of the rendering. [START_REF] Bolopion | Tuning the gains of haptic couplings to improve force feedback stability in nanorobotics[END_REF] provides a detailed analysis of two different couplings. It concludes that among these two control schemes the Force-Position (FP) is the one that leads to the best results for performing manipulation tasks at micro or nanoscales. It is depicted on figure 5. Scalar variables are used, since the two forces F x e and F y e are not coupled (see equations ( 13) and (15)), and will be rendered on two different axes of the haptic interface. The parameter i is for x or y, depending on the considered axis. The operator applies a force F i op on the haptic device handle, the velocity of which is V i m . This velocity will be used in the coupling to compute the desired velocity of the nanotranslator (V i s ). The actual velocity of the cantilever with respect to the substrate is V i c . F i e represents either F x e or F y e computed as shown in equations ( 13) and (15). The force fed back to the user F i m is computed using this variable.

V i m H i m (s) F i op F i m C i (s)(B i f p , K i f p ) G i n A i d 1/A i f V i c V i s H i s (s)
For each axis, the haptic device's continuous time transfer function H i m (s) has been identified as being composed of inertia M i v and damping B i v . The slave's H i s (s) is a second order transfer function, with two time constants τ i 1 and τ i 2 :

H i m (s) = 1 B i v + M i v s
(16)

H i s (s) = 1 (1 + τ i 1 s)(1 + τ i 2 s) (17)
Due to scale differences between micro and macro worlds, scaling factors have to be used to enhance the forces fed back to the user (by a factor A i f ), and scale down his or her motions (by a factor A i d ). For stability reasons, two controllers are added. The first one is C i (s), a proportionalintegral (PI) controller:

C i (s) = B i f p + K i f p s ( 18
)
It is used to compute the forces reflected to the user. The integral term determines the stiffness of the feeling, while the proportional term adds viscosity into the system and stabilizes it. The second controller is G i n , a proportional (P) term, used to control the slave device.

Tuning of this coupling is important for the quality of the force rendering. A i f and A i d are determined by force amplification and precision of manipulation requirements. The range of admissible values for the gains of the (PI) and (P) controllers is derived taking into account the stability and transparency conditions.

In the following section the control scheme's parameters will be tuned according to table II [START_REF] Bolopion | Tuning the gains of haptic couplings to improve force feedback stability in nanorobotics[END_REF]. T e (the sampling period) is equal to 5ms for our system.

Gn high for non-contact transparency

A f A d K f p Te 2 +A d B f p V. EXPERIMENTAL RESULTS
The strategy described in section III will be evaluated with rolling experiments using glass spheres with a radius of 25µm.

A. Longitudinal rolling

In this section, a rolling experiment along the x axis of the cantilever is presented. The vertical position of the probe is servoed in order to maintain deflection at its extremity constant (δ(L, 0) = 0.7µm , which corresponds to a force of 1.7µN when the sphere is at point E).

The controller gain coefficients are chosen according to table II. A x d = 50 • 10 3 is such that the displacement of the sphere is precise enough for the experiment. A x f is chosen so that the force F x m is on the order of a few newtons so that it can be easily felt by the user. In the case of longitudinal rolling, the force will be amplified by a factor 25•10 6 . K x f p = 100N.m -1 and B x f p = 0.1N.s.m -1 are chosen to allow a stiff feeling, while avoiding oscillations by adding some viscosity. G x n is computed using the values of the other gains. During the experiment, the user makes the sphere roll from the point x = x max = L (extremity of the cantilever) to a point closer to the origin of the cantilever (noted x min , and then back to a point situated between x min and x max , noted x int (for x intermediate). The user makes small oscillations by moving the sphere back and forth around this position x int .

The results are plotted on figure 6. The position of the sphere along the x axis, as well as the forces sent to the operator F x m are represented.

A x f = 25 • 10 6 , A x d = 50 • 10 3 , K x f p = 100N.m -1 , B x f p = 0.1N.s.m -1 , G x n = 825m.N -1 .s -1
According to the velocity scaling factor A x d used, a displacement of 1cm in the haptic workspace corresponds to a displacement of 0.2µm for the cantilever. During this experiment, the probe is moved by around 40µm, which implies a displacement of 2m of the haptic handle. Due to the limited workspace of the master device (around 35cm for the x axis), a function to reset the position of the haptic device without moving the probe is used. The steps that can be seen on the plot of the position along the x axis are due to these instants when the user resets the position of the haptic device (and therefore keeps the position of the sphere constant). As a consequence, the force sent back to the user is constant during this operation.

As shown by the results on figure 6, and as predicted in section III-A (figure 2), the force sent to the user increases as the sphere gets further from the extremity of the cantilever. For x = x max , the force is nil. The operator has to make an effort to move the sphere further from this equilibrium position. The force felt by the user also helps him or her to evaluate how far the sphere is from point x = x max , as the magnitude of the force increases.

B. Lateral rolling

As in the case of longitudinal rolling, the probe's vertical deflection is kept constant by servoing the vertical position of the origin of the cantilever. The x position is also kept constant.

The velocity scaling factor is the same as that used for lateral rolling (the desired precision of the manipulation is the same for both axes). Due to the expression of the force F y e that has been chosen (moment transformed as a force), its magnitude is around 10 times higher than F x e . Since the same magnitude as F x m is required for F y m (so that it can be felt by the user and rendered through the haptic interface), the force scaling factor has been decreased by a factor 10. Concerning the PI controller, the same integral gain as for the x axis is used. However, the proportional gain has to be increased, such that B y f p = 1.5N.s.m -1 . This is due to the variation between positive and negative forces that arise when the sphere moves from the left to the right side of the cantilever, and vice-versa. To avoid oscillations, viscosity must be added. As for lateral rolling, G y n is computed according to the value of the other gains. Due to the difference between the gains used for the x and y axes, the proportional gain of the y axis is smaller than that for the x axis.

The user manipulates the cantilever along the y axis. Starting from the middle line of the cantilever (y = 0), he or she moves the sphere towards negative y (phase 1). During phase 2, he or she moves back to positive y. In phase 3, he or she performs oscillations with the sphere, moving back and forth between positive and negative y.

The user feels the force F y m , computed from F y e (see equation ( 15)). Results of both forces and positions along the y axis are presented on figure 7. The same velocity scaling factor as in section V-B is used (A y d = 50 • 10 3 ). For the same reasons of limited haptic workspace (maximum 64cm for the y axis), the reset function is used, and produces the steps that can be seen on the plots of both positions and forces.

As stated in section III-B, the user feels a force that guide him or her to bring the cantilever into an equilibrium position (no torsion). The forces sent back allow an untrained user to feel distinctly on which side of the cantilever the sphere is positioned. Moreover, it has been verified experimentally that the system remains stable even if the user creates small oscillations (see plots at around t = 22s).

C. 2D Rolling

This experiment consists in combining both lateral and longitudinal rolling, to perform a 2D rolling task with force feedback. This is possible since the signals used are independent: there is no coupling between the F x e and F y e forces computed. Therefore they can be rendered to the user using two axes of the haptic device.

As for lateral and longitudinal rolling the deflection δ(L, 0) is constant. The movement performed by the operator in the plane (xOy) is depicted on figure 8. The shape of the movement is roughly circular, but noticeable differences have been made to better distinguish the corresponding forces sent to the user. The movement has been repeated three times in order to verify that the forces felt by the operator are constant for a given position. The gains used are the same as those given in sections V-A and V-B. Contrary to plots on figures 6 and 7, the positions along the x and y axes are not stepped. Since the displacements of the cantilever were small enough, it was not useful to use the haptic device's reset function.

The forces sent back to the user while performing this experiment are represented on figure 9, where the lateral force is plotted against the longitudinal force. Since the user chose to perform these circles without reaching the extremity of the cantilever, the force along the x axis is never nil. For a given position, the operator feels the same force, as demonstrated by the three attempts and highlighted by the characteristic points noted P 1 , P 2 and P 3 . This helps the operator to estimate the position of the sphere under the cantilever. During these experiments, the forces provide the user with comprehensible information, since he or she has to make an effort to get further from the defined equilibrium position (point E). This is possible thanks to the appropriate choice of the forces sent to the user, and a control scheme able to render these forces sufficiently amplified (so that the operator can feel them), while remaining stable. Using the same setup, other strategies to provide force information during a manipulation task can be considered.

VI. CONCLUSION

In this paper, micromanipulation with force feedback using tipless cantilevers has been demonstrated. Forces that help the user bring objects to a desired position were derived from raw measurements of the cantilever's deformations, and the mechanical properties of the beams. These indications were provided to the user through a haptic interface, using a force position control scheme. The structure of this coupling, as well as the tuning of the gains, allowed for stable and transparent manipulation.

This approach has been validated using rolling experiments on 25 micrometer radius spheres. As shown experimentally for both longitudinal and lateral rolling, attractive forces indeed helped the user since he or she had to make an effort to move the sphere further from a defined equilibrium position. 2D rolling has also been demonstrated. Based on the magnitude and the direction of the forces felt, the user was also able to estimate the position the sphere under the cantilever.

This work was the first step toward more complex teleoperations. An analysis to determine the most relevant profile of the forces sent to the user should be carried out. The possibility of using other strategies to manipulate objects via a haptic interface should also be studied.

Fig. 1 .

 1 Fig. 1. Principle of force measurement using a laser

Fig. 3 .

 3 Fig.3. Profile of the force that will be sent to the user along the y axis

Fig. 4 .

 4 Fig. 4. Experimental setup

Fig. 5 .

 5 Fig.5. Force-Position control scheme used to control the cantilever and provide force feedback. The i parameter is for the variable x or y, depending on the considered axis

Fig. 6 .

 6 Fig. 6. Positions and forces during the longitudinal rolling experiment. Parameters used: Ax f = 25 • 10 6 , A x d = 50 • 10 3 , K x f p = 100N.m -1 , B x f p = 0.1N.s.m -1 , G x n = 825m.N -1 .s -1

Fig. 7 .

 7 Fig. 7. Positions and forces during the lateral rolling experiment. Parameters used: A y f = 2 • 10 6 , A y d = 50 • 10 3 , K y f p = 100N.m -1 , B y f p = 1.5N.s.m -1 , G y n = 19.8m.N -1 .s -1

2 1 P 2 P 3 Fig. 8 .

 1238 Fig. 8. Positions of the sphere on the plane (xOy) during the 2D rolling experiment

1 P 2 P 3 Fig. 9 .

 1239 Fig. 9. Forces felt by the user on the x and y axes of the haptic interface during the 2D rolling experiment

TABLE I VARIABLES

 I DESCRIBING THE AFM PROBE, AND RELATED NUMERICAL VALUES (FOR A SILICON CANTILEVER). DIMENSIONS GIVEN ARE THE ONES OF THE CANTILEVER USED IN SECTION V)

		Signification	Numerical values
	L, t, w kz, kt φ E, G	length, thickness and width of the cantilever normal stiffness and torsional spring of the cantilever resp. Young's modulus and the	L = 220µm, t = 2.9µm, w = 27µm kz(L) = 2.4N.m -1 , kt φ (L) = 64.0 • 10 -9 N.m E E = 150GP a, G = 2(1+ν)
	ν	shear modulus Poisson's ratio	ν = 0.17
	Ix, Iy	second moments of area about the x (resp. y) axis	Ix = tw 3 12 , Iy = t 3 w 12

TABLE II VALID

 II RANGE OF VALUES FOR THE GAINS OF THE (PI) AND (P)

			CONTROLLERS.
		Minimum value	Maximum value
	K f p	high for a stiff contact feeling	4Mv T 2 e non-contact transparency , low for
	B f p	K f p Te 2	-Bv	low for non-contact transparency

http://www.haption.com

VII. ACKNOWLEDGMENT

This work was supported by the French National Agency of Research, through the PACMAN project.


Aude Bolopion

Barthélemy Cagneau

D Sinan Haliyo

Stéphane Régnier

Tuning the gains of haptic couplings to improve force feedback stability in nanorobotics

This paper deals with the problem of bilateral haptic control in nanorobotics. At this scale, a human operator cannot interact directly with objects. He needs special tools manipulated through robotic systems. Therefore, force feedback devices are the only solution to provide him a sense of touch. However, the quality of the rendering strongly influences his ability to perform a given task. Stability is the main requirement that the system must fulfil to be usable. As the choice of the controller and its tuning are critical issues, a general method to tune the parameters of two haptic controllers is presented. A theoretical study is carried out and the methodology is validated with an experiment composed of several phases with high dynamic phenomena. Intrinsic limitations of the two controllers are also pointed out.

I. INTRODUCTION

The importance of microscale and nanoscale robotic systems is growing and new application fields are emerging. From mini-invasive surgery to ultra-high density storage devices and sensor networks for environmental monitoring, micro and nano technologies are becoming part of our every day life [START_REF] Sitti | Microscale and nanoscale robotics systems [grand challenges of robotics[END_REF]. However, at this scale, manipulation of objects is still a challenge.

At the submicrometer scale, it is necessary to overcome a critical problem before performing any task: human operators cannot interact directly with the environment. Consequently, special tools are needed to allow operations and to provide visual and tactile feedback. However, the resolution of optical microscopes is limited. Scanning Probe Microscopes (SPM) provide greater accuracy, but it is not possible to use them to capture images and manipulate objects simultaneously. Scanning Electron Microscopes (SEM) do not provide realtime images. That is why haptic coupling appears to be an interesting solution to establish a link between these two worlds, ( [START_REF] Hollis | Toward a tele-nanorobotic manipulation system with atomic scale force feedback and motion resolution[END_REF], [START_REF] Hatamura | Direct coupling system between nanometer world and human world[END_REF]). Indeed, even elementary tasks lithography or indentation, or characterization of objects' mechanical properties cannot be performed without visual or haptic feedback [START_REF] Ferreira | Virtual reality and haptics for nanorobotics[END_REF], [START_REF] Sitti | Teleoperated touch feedback from the surfaces at the nanoscale: modeling and experiments[END_REF], [START_REF] Guthold | Controlled manipulation of molecular samples with the nanomanipulator[END_REF].

Based on dimensional analysis, [START_REF] Goldfarb | Dimensional analysis and selective distortion in scaled bilateral telemanipulation[END_REF] presents a method to select the coefficients that are used to scale forces and positions for haptic coupling. However, practical issues, like limited workspace and range of forces that can be felt by a human operator while manipulating a haptic device (typically from 0.1 to 10N ), will prevent using this kind of reasoning.

Other works on coupling issues include [START_REF] Kim | Task-based and stable telenanomanipulation in a nanoscale virtual environment[END_REF], which also proposes a method to derive scaling factors based on Llewelyn's criteria. A passivity controller is implemented to assure {bolopion,cagneau,haliyo,regnier}@isir.fr stability. However, it has been tested only on virtual environments and no method to tune the gains of the controller is provided. [START_REF] Onal | A scaled bilateral control system for experimental 1-D teleoperated nanomanipulation applications[END_REF] also presents very interesting results with the cancellation of the cantilever's stiffness on the rendered forces and more detailed conclusions about the passivity controller.

In this paper, we will present two different control schemes. To tune the gains of the controllers, an original method that can be adapted to most of the usual systems is used. It considers simple inequalities between the parameters of the controllers so that they can be chosen according to simple relations. Stability is the main requirement that will be considered in deriving these relations. Compared to [START_REF] Venture | Force-feedback micromanipulation with unconditionally stable coupling[END_REF], the resolution of the force sensor has been increased, allowing users to feel forces in the range of nanonewton (instead of micronewton). Morevoer, we do present rules to tune the gains, and experimental results for cantilevers of different stiffnesses.

This paper is organized as follows: in section II, we present the experiments and the experimental setup that we will use to validate the control schemes; then, the most intuitive control scheme, the Direct Force Feedback is presented in section III. We will show its limitations in our context. Section IV introduces another control scheme, the Force-Position control scheme, and an original method for choosing the controllers' gains. Throughout this paper, experimental results will be presented to underscore our conclusions.

II. EXPERIMENTAL SETUP

A. Coupling validation's protocol

To compare the performances of the different couplings and the influence of the controllers' gains, we will perform a basic one-dimensional experiment. It consists of approaching the substrate with the probe, applying a force and then retracting. The well-known plot representing the force applied on the cantilever as a function of the distance from the substrate is given in figure 1(a). The corresponding plot of force applied on the cantilever as a function of time is given in figure 1(b). This is what the operator should ideally feel. Two discontinuous phenomena (referenced as A and E on the plots) are respectively representative of the instant when the cantilever is attracted by the substrate (pull-in) and when the contact is broken (pull-off). Two kinds of forces are present: the very first ones (B) and the last ones (D) are attractive (just after the pull-in and just before the pull-off), while the ones in between are repulsive (C). Since attractive forces and discontinuous phenomena are present at nanoscales, the coupling will have to be designed and tuned taking these characteristics into consideration. Indeed, it has to remain stable and the user must be able to feel these phenomena. As the control scheme used depends on the master and slave devices, the experimental setup has to be detailed before performing any analysis.

B. Experimental platform

The experimental platform used to validate the coupling is depicted on figure 3. It is composed of a nanotranslator which can move along the vertical direction. The force applied on the cantilever is measured with a laser. The deflection is measured using a laser beam focused on the cantilever. The beam is reflected onto a photodiode. The normal force is computed as:

F = k S V (1)
where k and S, the stiffness of the cantilever and the sensitivity of the photodiode, are calibrated [START_REF] Xie | Calibration of lateral force measurements in atomic force microscopy[END_REF]. V is the output voltage of the photodiode. As highlighted in equation ( 1), the force depends on k. Therefore, the coupling will be influenced by this factor. However, the proposed method must be adapted for cantilevers of any stiffness.

The available haptic device is a Virtuose manufactured by Haption 1 . This arm, represented on figure 2, allows 3 degrees of freedom of force feedback.

F op F v V p V v V v V n F e F e

Operator

Master device Coupling Slave device Environment • F e : force applied by the substrate on the cantilever • F op : force applied by the operator on the master arm The set of variables that corresponds to the outputs is:

• V p : velocity of the cantilever in the operational space • V v : velocity of the master arm in the operational space F v is the force fed back to the user, through the haptic interface, and V n is the desired nanotranslator velocity. The subsystems' inputs and outputs are the only assumptions we make in designing the control scheme. However, they are not really restrictive since many teleoperation systems presented in literature fulfill these requirements.

III. DIRECT FORCE FEEDBACK

Direct Force Feedback (DFF) control scheme seems to be the most intuitive formulation for providing amplified forces to the operator. It is depicted on figure 5. The blocks and Master (Virtuose) and slave (nanotranslator) continuoustime transfer functions are modeled as follows:

V v V (s) F op F v 1/A d A f V p V n N (s) F e + +
V (s) = [(B v + M v s)] -1 N (s) = [(1 + τ 1 s)(1 + τ 2 s)] -1 (2)
where s is the Laplace variable. Numerical values of the characteristic parameters have been identified as:

M v = 0.4kg ; B v = 0.1N.s.m -1 τ 1 = 1.35 × 10 -3 s ; τ 2 = 0.57 × 10 -3 s (3)
where M v and B v are respectively the inertia and damping of the Virtuose. τ 1 and τ 2 are time constants of the nanotranslator. A f and A d are respectively the force and velocity scaling factors. These coefficients are necessary to adjust the scale difference between macro-and nano-worlds.

The main advantage of this control scheme is that it requires only these two parameters to achieve DFF control. On the other hand, the system's stability will entirely depend on these coefficients.

A. Routh stability criterion

The slave device interacts with a remote environment, which will influence the stability of the coupling. Therefore, it has to be considered while performing stability analysis. Two approaches can be used. The first is to consider a method that does not require modeling the environment (e.g., methods based on passivity study). However, it is more conservative, and not useful in pointing out limitations as a function of the cantilever' and the environment's characteristics. The second approach is to model the environment. This is what we will do to derive limitations of the control scheme for our particular application. The environment will be modeled with respect to assumption [START_REF] Sitti | Microscale and nanoscale robotics systems [grand challenges of robotics[END_REF].

Assumption 1: The slave device is linked with its environment through a spring of stiffness k c (the cantilever). The contact between the cantilever and the substrate is modeled as a spring k s . These two serial springs are linked such that the equivalent spring k eq is:

1 k eq = 1 k s + 1 k c (4)
Considering this assumption and the control scheme depicted on figure 5, it is possible to determine the transfer function between F op and V v . Since the system is linear time invariant (LTI), the Routh-Hurwitz criterion can be applied to the transfer function. A necessary and sufficient condition of stability is derived. It leads to the following inequality:

R = A f A d ≤ γ k eq = R max (5)
where γ =

Bv(τ1+τ2)[M 2 v +MvBv(τ1+τ2)+B 2 v τ1τ2] [Mv(τ1+τ2)+Bvτ1τ2] 2
As γ only depends on the system's parameters, for a given environment, equation [START_REF] Sitti | Teleoperated touch feedback from the surfaces at the nanoscale: modeling and experiments[END_REF] shows that the system's stability only depends on the ratio R = A f A d . The system's stability is assured by selecting R ≤ R max with respect to equation [START_REF] Sitti | Teleoperated touch feedback from the surfaces at the nanoscale: modeling and experiments[END_REF]. R max will also determine the closed loop performances in terms of force/velocity scaling. However, it is obvious that the ratio R max , inversely proportional to k eq , is decreasing sharply when environmental stiffness increases. The next subsection will provide a compromise between stability and the values of the scaling factors through experimental examples.

For stability issues, the worst case is when equivalent stiffness is the highest. Using Hertz's theory [START_REF] Maugis | Contact, adhesion and rupture of elastic solids[END_REF], it can be shown that it corresponds to k eq = k c . Therefore, in the following sections we will use this approximation.

B. Experimental results

We used a cantilever with k c = 2.4N.m -1 so that:

R max = 21.7 (6)
The experiment described in section II-A is performed with the following values:

A d = 0.05 • 10 6 A f = 0.12 • 10 6 ⇒ R = 2.4 < R max (7)
A d is chosen according to the master and slave motion ranges. The value A d = 0.05 • 10 6 is such that the master motion range, which is 25cm, corresponds to a displacement of 5µm for the slave. This gives a good compromise between ease of manipulation and precision. This value will be kept constant for all experiments, to allow a better comparison between them. In this first experiment, A f is chosen so that the Routh criterion given in equation ( 5) is satisfied.

Stability will be studied with respect to this factor in next paragraphs. The results we have obtained on figure 6 confirm Approach-retract curve for a cantilever of stiffness kc = 2.4N.m -1 , using A d = 0.05 • 10 6 , A f = 0.12 • 10 6 . R=2.4. our predictions. Indeed, this figure represents the magnitude of the forces sent back to the user. During the first stage (C), the operator applies forces on the substrate. The maximum value of the forces applied is 0.30N . While retracting, he must counterbalance the forces resulting from adhesion effects (D). A force equal to -0.13N is necessary to release the cantilever from the substrate (E).

Even if the system remains stable as expected, force feedback does not allow an operator to distinguish the different stages described on section II-A (attraction, repulsion, pulloff, etc.). Therefore, it might be interesting to increase A f in order to provide better force feedback to the user. The pull-in phenomenon is not visible on the plot. Indeed, to avoid time consuming operations, we chose A d = 0.05 • 10 6 . With this value, the velocity of the nanotranslator is too high compared to the dynamics of the pull-in effect, and so it could not be reflected to the user.

During the second experiment we performed, the value of A d remained the same, but A f was increased, so that A f = 6.0 • 10 6 . With these values:

R = 120 > R max (8)
Equation [START_REF] Kim | Task-based and stable telenanomanipulation in a nanoscale virtual environment[END_REF] shows that the Routh criterion is no longer satisfied. The system is thus predicted to be unstable. The results we have obtained are plotted on figure 7. Remarkably, the forces have been amplified compared to the forces of the Approach-retract curve for a cantilever of stiffness kc = 2.4N.m -1 , using A d = 0.05 • 10 6 , A f = 6 • 10 6 . R=120. figure 6. It is then easier for the user to detect repulsive and attractive forces (C, D) and the high variation of the forces due to the pull-off (E). This result shows that force reflection has been improved by modifying the value of A f . However, the stability of the system is affected. When the cantilever establishes the contact with the substrate, it creates high amplitude oscillations in the system (O). These oscillations are very disturbing for the user who must compensate for these perturbations. He has to act like a damper to absorb the amount of energy responsible for instability.

IV. FORCE-POSITION CONTROL

All studies on this coupling will be conducted using a discrete time variable z. Indeed, contrary to the DFF control scheme, FP involves numerical integrations which make the system very sensitive to the sampling period.

A. Control scheme design

It has been shown in section III that the DFF scheme presents limitations in terms of stability, depending on the desired scaling factors. To improve that issue, two controllers are added in the FP control scheme detailed on figure 8. G n and C(z) are respectively proportional (P) and proportionalintegral (PI) controllers. Only P and PI controllers have been used. Indeed, their simple form highlights the influence of each gain on the system transparency and stability. A good understanding of the interactions between gain values and system performances is necessary to later design more complex control schemes. On-line gains tuning based on the rules that will be derived in next paragraphs will be studied in future works. The star superscript is for discrete parameters. C(z) has been discretized using Tustin's approximation. This leads to:

V * v V (z) F * op F * v C(z)(Bfp, Kfp) Gn Ad 1/Af V * p V * n N (z)
C(z) = B f p + K f p T e (z + 1) 2(z -1) (9)
In the following, V (z) (resp. N (z)) will refer to the discrete time transfer function corresponding to V (s) (resp. N (s)).

They are such that:

V (z) = 1 B v 1 -δ z -δ with: δ = e -Bv Te Mv ( 10
)
N (z) = 1 - τ 1 τ 1 -τ 2 z -1 z -e -Te τ 1 - τ 2 τ 2 -τ 1 z -1 z -e -Te τ 2 (11)
Compared to the DFF control scheme, the force fed back to the user, F v , is computed with the PI controller. The integral gain K f p and the proportional gain B f p can be used to modify the stiffness and damping of the rendered force. The gain G n is used to compute the desired velocity of the slave device.

B. Determination of the control scheme parameters 1) Scaling factors:

As in the first control scheme, A f and A d depend on the user's requirements. Indeed, one might want a precise positioning, and/or important force feedback, according to the manipulation tasks. As in section III, we will choose A d = 0.05 • 10 6 , A f will be such that the forces sent back to the user are high enough for an untrained user to distinguish between the different phenomena encountered during the experiment.

2) Proportional controller G n : Problems of stability can be due to control schemes but also to numerical computation. Indeed, the force F * v at time k + 1 determined using information of positions and velocities at time k (see figure 8):

F * v (k + 1) = B f p ∆V (k) + K f p ∆X(k) (12)
V * n (k + 1) = G n F * e (k) - 1 A f F * v (k) (13)
where: 12) and ( 13) and the fact that the position X * n is computed using Tustin's discretization, when the tip is well above the substrate (no force applied on it, i.e., F * e = 0), F * v is given by:

∆V (k) = A d V * n (k) -V * v (k) ∆X(k) = A d X * n (k) -X * v (k) Considering (
F * v (k + 1) = λ 1 F * v (k -1) + λ 2 V * v (k) + λ 3 K f p A d + λ 4 K f p (14)
where:

λ 1 = -G n B f p A d + K f p A d Te 2 A f λ 2 = -B f p + K f p T e 2 λ 3 = X * n (k -1) + V * n (k -1)T e 2 λ 4 = -X * v (k -1) + V * v (k -1)T e 2
Avoiding numerical instabilities leads to an upper boundary on G n , a necessary condition for stability:

|λ 1 | < 1 ⇔ G n < A f A d K f p Te 2 + A d B f p (15)
However, for transparency reasons, G n has to be as high as possible. Otherwise, it leads to a tracking error on the slave side and thus to viscosity on the force fed back to the user.

3) Proportional integral controller B f p and K f p : In [START_REF] Gil | Stability analysis of a 1 DOF haptic interface using the Routh-Hurwitz criterion[END_REF] a relation between B f p , K f p and T e is derived to assure the stability of the system while in contact with an infinitely stiff environment. The same methodology applied to our system will be used here. As for the DFF control scheme, the system considered is LTI. The Tustin equation is used for the discretization. The discrete time transfer function is:

V * v F * op = V (z) 1 + V (z)C(z) (16)
After applying the bilinear transformation z = 2+wTe 2-wTe , the resulting characteristic equation becomes:

b 2 w 2 + b 1 w + b 0 = 0 (17)
where:

b 2 = 4B v (1 + δ) -4(1 -δ)B f p (18) b 1 = (1 -δ)(4B v + 4B f p -2K f p T e ) (19)
b 0 = 2K f p T e (1 -δ) (20)
The Routh-Hurwitz criterion is achieved if and only if b 0 , b 1 and b 2 have the same sign. Since < 1, (20) is always positive. Therefore, the system will be stable if and only if b 1 > 0 and b 2 > 0, which implies:

B v + B f p > K f p T e 2 (21)
B v 1 + δ 1 -δ > B f p (22)
(21) is the same condition as that found in [START_REF] Gil | Stability analysis of a 1 DOF haptic interface using the Routh-Hurwitz criterion[END_REF], and states that the stiffness of the coupling is bounded by the damping of the haptic interface and that added by the coupling. Moreover, if the sampling period increases, K f p must decrease for the same amount of damping to guarantee stability. Using the first order Taylor development of x → exp(x) in the neighborhood of 0 ( BvTe Mv → 0 since T e → 0), δ ≈ 1 -BvTe Mv . ( 22) can be approximated by:

2M v T e > B v + B f p (23)
This highlights the fact that the maximum damping (and therefore, according to (21), the maximum stiffness) admissible is limited by the inertia of the master arm, and is inversely proportional to the sampling period T e . This is a convincing argument of the importance of the sampling period for stable haptic feedbacks.

C. Experimental results

This subsection presents experimental results for the FP control scheme. To validate the methodology described above, we will use the same parameters as for the DFF control scheme. Taking the same cantilever as in section III-B (k c = 2.4N.m -1 ), we require the same scaling factors (A d = 0.05 • 10 6 , A f = 6 • 10 6 ).

on conclusions of section IV-B, the following considerations have to be taken into account to choose the gains:

• A f and A d are determined by the user's requirements • K f p has to be high to allow a stiff contact feeling.

However, according to ( 21) and ( 23), it can not be higher than

4M v T 2 e
• B f p to be low to allow for transparent feeling when no forces are applied on the cantilever. However, according to (21), it has to be higher than

K f p Te 2 -B v .
The trade off between K f p and B f p greatly depends on the application. It is necessary to bear in mind the upper and lower boundaries but the values that match his requirements can be freely chosen.

• knowing all the other parameters, G n is given by (15) For our application, an admissible set of gains is:

B f p = 2.0 N.s.m -1 , K f p = 100.0 N.m -1 , G n = 47.5 m.N -1 .s -1 .
It is necessary to verify experimentally that the method we described and the gains we chose are adapted to the stated problem. The experimental results we have obtained are plotted on figure 9. They must be compared to those on figure 7. It is obvious that the system remains stable during this experiment, unlike the DFF control scheme. The small oscillations are induced by the virtual coupling. However, due to the limited bandwith of the haptic interface, this phenomenon is not disturbing for manipulations. As expected, the user is able to feel the pull-off phenomenon with a peak of amplitude 1N . Consequently, the method described through this paper allows the controller presented in section IV-A to be tuned efficiently. To prove the robustness of our approach with respect to environment's stiffness, we performed the same experiment using different cantilevers. The results are presented on figure 10. We required the same velocity-scaling factor for the three probes. The force amplification was chosen so that the user could clearly feel the contact (about 5N are fed back via the haptic device), for a cantilever's deflection of less than 5µm. This lead to A f = 200.10 6 for the cantilever of stiffness k c = 0.05N.m -1 , and A f = 0.2.10 6 for that of stiffness k c = 48N.m -1 . Other gains were chosen using the methodology described in section IV-B.

Plots on figure 10 show that the operator was able to feel the contact in a stable way for cantilevers of different stiffnesses (from 0.05 up to 48N.m -1 ). It should be noted that

k c = 0.05 N.m -1 k c = 2.4 N.m -1 k c = 48 N.m -1
Fig. 10.

Approach-retract curve for cantilevers of stiffnesses kc = {0.05, 2.4, 48}N.m -1 (K f p = {100, 100, 100}N.m -1 , B f p = {1.2, 2.0, 1.5}N.s.m -1 , Gn = {2329, 47.5, 1.98}m.N -1 .s -1 ).

these experiments have been performed in a non-controlled environment. Conditions of humidity and temperature may have changed between the experiments. However, the pulloff is indeed greater for cantilevers of low stiffness. The FP control scheme seems adapted to the application described in section II-A, in particular since the system is stable, even when in contact with the substrate. Moreover, the methodology to tune the gains is validated through experiments. Indeed, it allowed us to feel forces of 10nN (pull-off for the cantilever of stiffness k c = 0.05 N.m -1 ). However, for the same reasons as in paragraph III-B, the pull-in phenomenon is not visible on the plot.

V. CONCLUSION

Two different control schemes designed for nanorobotic applications have been presented in this paper. The Direct Force Feedback scheme suffers from a trade-off between force amplification and workspace limitation. To overcome this problem, the Force-Position control scheme is introduced. Compared to the DFF control scheme, damping is added. It improves stability but degrades transparency. Using the FP control scheme, we were able to feel forces of 10nN . Moreover, stable contact for cantilevers of stiffnesses from 0.05N.m -1 to 48N.m -1 is demonstrated. Since performances of haptic couplings greatly depend on the tuning of the gains, the choice of these values is a critical issue. Therefore, an original method is applied to tune the gains of these two controllers. Intrinsic limitations of the couplings are exposed theoretically and experimentally.

In future work, we expect to use force feedback to develop original assistance for more complex telemanipulations. Two main tasks need to be performed to reach this goal. The first is to determine how to use the photodiode's outputs to retrieve more information than is presented in this paper (i.e., contact or non-contact). The second is to find new sources of information, such as modifications of objectinduced vibrations of excited cantilevers.

VI. ACKNOWLEDGMENT

This work was supported by the French National Agency of Research, through the PACMAN project.

Fig. 1 .

 1 Fig. 1. Approach-retract curve

Fig. 2 .

 2 Fig. 2. Haptic interface

Fig. 4 .Figure 4

 44 Fig. 4. Power flow between the subsystems

Fig. 5 .

 5 Fig. 5. Control scheme for DFF.

  Fig.6.Approach-retract curve for a cantilever of stiffness kc = 2.4N.m -1 , using A d = 0.05 • 10 6 , A f = 0.12 • 10 6 . R=2.4.

  Fig. 7.Approach-retract curve for a cantilever of stiffness kc = 2.4N.m -1 , using A d = 0.05 • 10 6 , A f = 6 • 10 6 . R=120.

Fig. 8 .

 8 Fig.8. Force-Position control scheme

Fig. 9 .

 9 Fig. 9.Approach-retract curve for a cantilever of stiffness kc = 2.4N.m -1 , using A d = 0.05 • 10 6 , A f = 6 • 10 6 . FP control scheme.

http://www.haption.com/


Mariela Aguilera Sammaritano

Mauro González Vera

Tatiane Pablo Marcelo Cometto
email: pablocometto@gmail.com

Nicola Tejero

Glauco F Bauerfeldt

Abdelwahid S Mellouki

Tatiane Nicola Tejero

Temperature dependence of rate coefficients for the gas phase reaction of OH with 3-chloropropene. A theoretical and experimental study

Keywords: allyl chloride, OH radical, gas phase reaction, PLP-LIF, theoretical calculations investigation, formal analysis, visualization, conceptualization, investigation, formal analysis, writing software, resources, formal analysis

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction.

3-Chloropropene is an important intermediate in the petrochemical industry that is used to synthesize epichlorohydrin and glycerin, allyl compounds such as allyl alcohol, allyl amines, allyl esters, and polyesters [START_REF]Ullmann`s Fine Chemicals[END_REF] . Its derivatives are found in varnishes, plastics, adhesives, perfumes, pharmaceuticals, and insecticides [START_REF]Health and Environmental Effects Profile for Allyl Chloride[END_REF] . U.S. Environmental Protection Agency (EPA) considers 3-Chloropropene to be a possible human carcinogen [START_REF]Integrated Risk Information System (IRIS) on Allyl Chloride[END_REF] and that individuals may be exposed through breathing contaminated air [START_REF]Health and Environmental Effects Profile for Allyl Chloride[END_REF] .

Its atmospheric chemistry has been investigated by various groups, Albaladejo et al. [START_REF] Albaladejo | [END_REF] have concluded that the gas phase reaction with OH radical constitutes its main atmospheric removal process in the troposphere. Grosjean 5 has proposed that the reaction releases Cl atoms and leads to carbonyl compounds as reported by Zhang et al. 6 and Tuazon et al. 7 To the best of our knowledge, the reaction of OH with 3-chloropropene has been investigated by a limited number of groups. Edney et al. 8,9 and Tuazon et al. 7 used the relative rate method to determine the rate coefficient value at room temperature in N 2 as bath gas. Martinez et al. 10 have estimated k 298K by the relationship between the rate coefficients of the reactions of the OH and NO 3 radical with a haloprenes series. The only absolute measurement was performed by Albaladejo et al. [START_REF] Albaladejo | [END_REF] in the temperature range 228 -388 K, using the pulsed laser photolysis-laser induced fluorescence technique (PLP-LIF) with He as carrier gas.

OH + CH 2 =CHCH 2 Cl  products k [1] k refers to the second order rate coefficient for the global reaction (that will be under scripted as k -cVTST , when is theoretically calculated in this work). The temperature dependence of k shows slight negative activation energy. This behavior is widely observed in the gas phase reaction of OH radical with unsaturated volatile organic compounds (UVOCs) [11][12][13] . In general, the negative dependence of the rate coefficients with the temperature has been justified by a mechanism in which the reversible formation of a pre-barrier complex takes place in the OH radical electrophylic addition to the double bond, followed by an irreversible reaction step through a saddle point, to products. [14][15][16] . In particular, Zhang et al. 6 have computationally investigated the gas phase reaction of OH radical with 3-Chloropropene to products at 100 Torr and between 200-600 K temperature range. They have considered two addition and four hydrogen abstraction pathways. They have employed a variational transition state model and multichannel RRKM theory to calculate the temperature and pressure dependence of the rate constants.

In the present work we have reinvestigated the temperature dependence of k using a PLP-LIF apparatus, in order to provide new data for critical revisions of kinetic parameters. In addition, we have conducted a computational study paying special attention to the OH addition channels. To the best of our knowledge, this is the first study that uses a canonical Variational Transition State Theory (cVTST) over four addition and one H atom abstraction pathways to calculate the temperature dependence of k.

Methodology

Experimental

The experiments were conducted using the absolute method of PLP-LIF to determine the rate coefficients for the reaction [1] in the temperature range 253-371 K at around 100 Torr.

The PLP-LIF experimental system used is located at the CNRS (Orléans, France). The details of the experimental setup and methodology used have been presented previously 17 . Therefore, the description below is limited to the necessary features to understand the current experiments. premixed with helium in a 10 L glass bulb to form a 0.2-0.4% mixture at total pressure of about 1000 Torr. The gas mixture, the photolytic precursor (H 2 O 2 ), and the bath gas (He) were flowed through the cell with a linear velocity in the range 3-10 cm s -1 and the residence time in the cell reaction zone is estimated less than 0.1 s. Each photolysis/probe sequence interrogated a fresh gas mixture, and reaction products did not build up in the cell. The 3-chloropropene concentrations were calculated from their mass flow rates, the temperature and the pressure in the reaction cell. All flow rates

were measured with mass flow meters calibrated by measuring the rate of pressure increase in a known volume. The pressure in the cell was measured with a capacitance manometer connected at the entrance of the cell.

Rate coefficients were determined under pseudo-first-order conditions with the concentration of 3-Chloropropene in excess over that of OH radicals

([CH 2 =CHCH 2 Cl] 0 >10 [OH] 0 )
. The rate of disappearance of OH radical followed a simple exponential rate law:

[OH] t = [OH] 0 e -k't where k' = k [CH 2 =CHCH 2 Cl] + k' 0
Where k represents the second order rate coefficient for the global reaction [1] and k' 0 is the first-order rate coefficient for OH removal in the absence of 3chloropropene due to the diffusion of OH radicals out of the detection zone (which have a rate coefficient of 30 s -1 in the set up) and the reaction between OH radicals and their precursor (H 2 O 2 ).

Experiments were carried out in the temperature range 253-371 K and at total pressure between 101 and 107 Torr of helium. The experimental conditions are summarized in Table 1.

Materials.

The helium carrier gas (Alpha Gas UHP certified to > 99.9995%). The 50 wt% H 2 O 2 solution, from Prolabo, was concentrated by bubbling helium through the solution to remove water for several days prior to use and constantly during the experiment. 3-Chloropropene Aldrich (99%), was degassed by repeated freeze-pump-thaw cycles and purified by vacuum distillation before use.

Theoretical Calculations

Taking into account the experimental and theoretical results of Tuazon et al. 7 and Zhang et al. 6 , we have considered in this work that at temperatures lower than 600 K the rate limiting step of the reaction [1] is determined by a mechanism composed by OH addition to double bond and the β-H atom abstraction parallel reactions, according to the following general scheme:

CH 2 =CHCH 2 Cl + OH → σ-PC (k 1 ) [2] σ-PC → CH 2 =CHCH 2 Cl + OH (k-1 ) [3] σ-PC → I c (k 2 (c)) [4] σ-PC → I t (k 2 (t)) [5] CH 2 =CHCH 2 Cl + OH → σ-PC β (k 3 ) [6] σ-PC β → CH 2 =CHCH 2 Cl + OH (k-3 ) [7] σ-PC β → I abs + H 2 O (k 4 ) [8]
where, σ-PC is the pre-barrier complex, I c and I t are the products of the OH addition to the central and terminal carbon atoms, σ-PC β is the pre-barrier complex and I d is the product of the β-H atom abstraction, and, in parentheses, the k i (being i= 1, 2, 3, 4, -1 and -3) are the rate coefficients corresponding to each reaction step. k 4 is the rate coefficient for the H abstraction channel.

Reaction [1] (involving the elementary reactions [2-8] and the further splitting explained in the discussion section) has been described at Density Functional Theory (DFT) level, [START_REF] Levine | Quantum Chemistry[END_REF] adopting the M06-2X functional [START_REF] Becke | [END_REF] along with the aug-cc-pVTZ basis sets 20

. Geometry optimizations aiming to the location of all minima and saddle points have been performed and the characterization of these stationary points, after the converged geometry optimization calculations, has been done by analysis of the vibrational frequencies, calculated at the same level. Also, in order to guarantee that the optimized structure corresponds to a global or local minimum, relaxed scans along one or more dihedral angles have been calculated. This procedure has been adopted because the potential energy surface is expected to show a high density of minima, due to the combination of some internal rotations, differing one from the other by a few kcal mol -1 .

Theoretical calculations have been performed with the GAUSSIAN09 program packages 21 . The stationary points can be summarized as reactants, pre-barrier complex, saddle point and products. The minimum energy paths connecting the prebarrier complex and product and passing through the saddle point have been calculated using the intrinsic reaction path (IRC) method 22 . For the description of the minimum energy path connecting the pre-barrier complex and the reactants, scan calculations have been performed by partially optimizing the geometries along a path of increasing C-OH interatomic distances. Thermochemical quantities have been calculated using conventional statistical thermodynamics relations, assuming the harmonic oscillator, rigid rotor and ideal gas models 23 . In this work, we have also Canonical Variational Transition State rate coefficients have been calculated using the model described elsewhere 24 . Briefly, the minimum energy path, described by the total energy as a function of the reaction coordinate (s) and suggested from IRC or scan calculations, is transformed into a vibrationally adiabatic potential energy curve by including the zero-point energy corrections of the non-stationary points that define the reaction path. A further transformation into a Gibbs free energy curve, G(s,T), at each temperature, is possible with the inclusion of the enthalpy and entropic correction terms provided from the thermodynamics formulations mentioned above. A polynomial of third or fifth order is fitted to the Gibbs free energy curves and the resulting G(s,T)

functions are analytically maximized to obtain the s values that correspond to the location of the generalized transition state at each temperature. Molecular properties (vibrational frequencies, moments of inertia and critical energy) of the generalized transition states are interpolated and applied as the input quantities for the conventional Eyring equation 25 over the temperature range 200-400 K.Tunneling coefficients were calculated using the Wigner's expression [START_REF] Cramer | Essentials of Computational Chemistry Theories and Models[END_REF] .

Since formation of the prebarrier complexes are barrierless reaction paths, conventional transition state method does not apply for the prediction of the rate coefficients. Moreover, entropy effects are important for the best evaluation of rate coefficients of reaction that proceed through relatively small barrier heights. For these reasons, the adoption of the canonical variational method for predicting rate coefficients is crucial.

Results and discussion

Experimental

The absolute rate coefficients were measured in the temperature and pressure ranges 253 -371 K and 101-104 Torr (of helium) under pseudo-first-order conditions in which the concentration of the 3-Chloropropene was at least 10 times than OH radicals' concentration. The OH decays were found to be exponential over at between 2.5 and 3 lifetimes while k' 0 and k' were in the ranges 212-617 s -1 and 552-2875 s -1 , respectively.

The variation of the photolysis fluence (7.7-21 mJ cm -2 ) had no effect on the determined rate coefficient indicating that there was no noticeable contribution of photofragments to the OH consumption. The CH 2 =CHCH 2 Cl samples were purified to better than 99% and hence loss of OH radicals by reaction with impurities is expected to be insignificant. 1, related to experimental conditions.

The rate coefficient, taken as the average of all values obtained at 298 ± 2 K was k = (1.69 ± 0.12) x 10 -11 cm 3 molecule -1 s -1

The errors quoted for k is 2σ from the linear least-squares fit to the data points and do not include systematic errors. This value is in good agreement with those obtained by other authors 4, 6, 7-9 using different techniques, which are presented in Table 2. There is no significant difference between the k value, obtained in this work at 298 K and a pressure of 100 Torr, and the value of (1.64±0.18)x10 -11 cm 3 molecule -1 s - 1 obtained in similar experimental conditions by Albaladejo et al. [START_REF] Albaladejo | [END_REF] , and the values of (1.66±0.23)x10 -11 (1.70±0.70)x10 -11 , (1.96±0.32)x10 -11 and (1.69±0.07)x10 -11 , (in units of cm 3 molecule -1 s -1 ), obtained by Albaladejo et al. [START_REF] Albaladejo | [END_REF] , Edney et al. 8,9 and Tuazon et al. 7 , respectively, at 298 K and 1 atm. Thus, our results confirm that at 100 Torr k is in its high-pressure limit. All these values are presented in the inset of the Figure 2. Table 2 presents theoretical and experimental rate coefficients temperature dependence for the global reaction [1] determined previously by other authors [START_REF] Albaladejo | [END_REF]6 . The results show negative temperature dependence for k in agreement with the Albaladejo et al. [START_REF] Albaladejo | [END_REF] work. This fact is consistent with the hypothesis that the addition mechanism is the main reaction pathway in the considered temperature range.

Theoretical Calculations

We As far as we know, this work provides the first description of these OH addition pathways for the reaction [1].

On the other hand, in the CH 2 =CHCH 2 Cl-C s rotamer the chlorine atom lies in the same plane defined by the three C atoms, as can be observed in Summarizing, as a consequence of the presence of the σ-PC down and the σ-PC up , the reactions [2]- [5] are splitted, and the ractions 6-8 are reformulated as follows:

CH 2 =CHCH 2 Cl-C 1 + OH → σ-PC down (k 1-down ) [2a] σ-PC down → CH 2 =CHCH 2 Cl-C 1 + OH (k-1-down ) [3a] σ-PC down → I c-down (k 2c-down ) [4a] σ-PC down → I t-down (k 2t-down ) [5a] CH 2 =CHCH 2 Cl-C s + OH → σ-PC up (k 1-up ) [2b] σ-PC up → CH 2 =CHCH 2 Cl-C s + OH (k-1-up ) [3b] σ-PC up → I c-up (k 2c-up ) [4b] σ-PC up → I t-up (k 2t-up ) [5b] CH 2 =CHCH 2 Cl-C 1 + OH → σ-PC β (k 3 ) [6c] σ-PC β → CH 2 =CHCH 2 Cl-C 1 + OH (k-3 ) [7c] σ-PC β → I abs + H 2 O (k 4 ) [8c]
Regarding on the β-H atom abstraction channel, in this work only the CH 2 =CHCH 2 Cl-C 1 rotamer was considered because it is the most stable one and presents two specular structures. A σ-PC β connecting reagents to the I d via the TS β-H was found, which presents two energetically degenerated stereoisomers. The molecular structures are, also, shown in the Figures 4e-f. The CH 2 ClCH=CH 2 -C 1 rotamer belongs to the C 1 symmetry group, thus in the β position the two H atoms are not equivalent. The TS β-H was found only when the OH attack was on the H atom located below the plane of the three carbon atoms. At the considered temperature the abstraction pathways are expected to be minority 6 . Then, in order to study alternative reactions channels to those presented by Zhang et al. 6 in this paper we have just focused beta H abstraction.

The molecular properties (including optimized geometries and frequencies) for all the stationary points, shown in Figure 3, obtained at the CCSD(T)/ aug-cc-PVTZ// M06-2X/ aug-cc-PVTZ level, are given as Supporting Information.

The Table 3 summarizes the relative energy (with respect to the isolated Reactants), expressed in kcal mol -1 , for the stationary points along the described reaction pathways. At the M06-2X/aug-cc-PVTZ level the S 2 values were lower than 0.78, then, regarding to the spin contamination, the wave function and electronic energies are not affected by states of higher multiplicity. No calculations with different basis set were carried out because Zhang et al. 6 have obtained ΔE values comparable to the obtained in this work for the species σ-PC up , TS c-up , TS t-up , I c-up and I t-up , at the CCSD(T)/cc-PVTZ//M06-2x/6-311++G(d,p)) level. Also, they have not found dependency of the basis sets in the results.

A description of the potential energy surface for the four addition reactions channels and the β -H atom abstraction pathway is provided in the Figure 5 (and Table 3) and as can be observed, the pre-complexes present ΔE ranging between -3.30 and -4.09 kcal mol -1 . The TS c-down , TS t-down , TS c-up and TS t-up present negative ΔE ranging between -1.07 and -1.80 kcal mol -1 , i.e., below reactants. TS β-H has a ΔE of 0.19 kcal mol -1 , over the reactants.

Starting from each saddle point, IRC calculations were performed for the description of the reaction profiles leading to the formation of the chlorinated hydroxy radicals resulting from the addition of the hydroxyl radical to either the central or the terminal carbon atoms, from upward and downward attacks: I c-down , I t-down , I c-up and I t-up .

On the side, in the β-H atom abstraction channel, the IRC confirmed the reaction path from σ -PC β to the I d radical.

For the OH addition to the double bond reactions, π pre-barrier complexes have been proposed in previous studies [14][15][16]25 , in which the OH radical moiety lies nearly perpendicular to the C=C plane. Such pre complexes were not found in the present work at the level of theory used.

Regarding the reactions pathways, the rate coefficients were calculated for the association step and dissociation and addition reactions of the PC, represented in the reactions 2a-5a, 2b-5b and 6c-8c.

The canonical variational transition state method was adopted using the theoretical molecular properties calculated at M062X level with both and aug-cc-pVTZ basis sets. For the prediction of the global rate coefficients, the mechanism described by the chemical steps 2a-5a, 2b-5b (addition channels) and Eq. 1

CH 2 =CHCH 2 Cl-C 1 + OH → -PC  (6c); -PC  → CH 2 =CHCH 2 Cl-C 1 + OH (7c); -PC  →I  (8c)
Eq. 2

Eq. 3

Eq. 4

The factor 2 in the Eq. 1-3 accounts for the degeneracy of the 3 reaction paths (addition -up and down-and abstraction) due to stereochemistry. Tunneling corrections do not apply for the barrierless addition but can slightly increase k 2c-down , k2t-down , k 2c-up , k 2t-up and k 4 values. For the hydrogen abstraction channel, the tunneling coefficients ranged from 2.46 to 1.36, as the temperature increases from 200 to 400 K, whereas a range from 1.41 to 1.06 was observed for the Wigner coefficients related to the addition channels. The Table 4 summarizes the results for k down, k up, k abs and k -cVTST (in this work, predicted using the cVTST method) between 200 and 400 K. These values were plotted in the Figure 2 along with the Arrhenius plot obtained in this work and experimental and theoretical rate coefficient values in previous literature.

The obtained Arrhenius equationis:

k cVTST = (1.54 ± 0.08) x 10 -12 exp [(532 ± 18) / T], 275 -400 K
where k cVTST is the expression obtained by fitting the rate coefficients between 275 -400 K. Linearity of the classical Arrhenius expression is lost with the extension of the temperature range from 200 -400 K and the rate coefficients are better fitted by:

k cVTST = (3.01 ± 1.21) x 10 -19 [T (2.27 ± 0.06) ] exp [(1284 ± 16) / T], 200 -400 K
The quoted uncertainties in the equations above were obtained by evaluation he standard errors of regression, at 95% of confidence level.

In agreement with the previous literature concerning the OH addition to unsaturated compounds, addition is the dominant channel at this range of temperatures and shows negative temperature dependence. As is shown in the Table 4 and Figure 2, the H abstraction pathway present positive activation energy and the k abs values are, at least, one order of magnitude lower than the values of k down and k up , at the same temperature. Then, other less important abstraction channels studied by Zhang et al. 6 can be neglected in the k -cVTST calculation.

The k -cVTST value at 298 K of 9.13 x 10 -12 cm 3 molecule -1 s -1 reach the 54% and the 59% of the values (1.69 ± 0.12) x 10 -11 and 1.56 x 10 -11 (in unit of cm 3 molecule -1 s -1 ), experimentally obtained in this work and calculated by Zhang et al. 6 , respectively.

The rate coefficients predicted from the canonical variational transition state method, employed in this work, are underestimated by factors ranging from 1.4 -2.0 (200 -400 K). In particular, the room temperature rate coefficient is underestimated by a factor of 1.9. Calculated rate coefficients are also lower than those predicted by Zhang and coworkers, who adopted a microcanonical method, based on the RRKM model, for the calculation of the rate expression. A comparison among canonical and microcanonical rate coefficients for OH reactions with unsaturated compounds can be found elsewhere 15,16 .

Although the rate coefficients have been underestimated with respect the experimental and literature values, the predicted global activation energy agrees very well with the experimental value (-1.06 kcal mol -1 , in comparison with -1.07 kcal.mol -1 , respectively, calculated from 275 -400 K). The ratio between the calculated and the experimental activation energies between 275 and 400 k is 0.99, which represent a good agreement taking into account the two different employed methods. Our calculations show negative temperature dependence for k , in accordance with the experimental results of Albaladejo et al. [START_REF] Albaladejo | [END_REF] This behavior is also predicted by theoretical calculations by Zhang et al. 6 At the same range, the activation energy calculated for the rate coefficients reported by Zhang and coworkers is -1.48 kcal mol -1 .

The deviation of the cVTST rate coefficients from the experimental values mostly arise from the entropic term in the Eyring expression, which is governed by the values of the lowest vibrational frequencies. Considering the errors in the theoretical prediction of all these parameters, the agreement between predicted and experimental rate coefficients must be considered very good.

Conclusions

There is only one previous study of the rate coefficient temperature dependence which is not enough for recommendation in critical revisions, then, this work provides new valuable experimental data of kinetic parameters of the title reaction.

As far as we know, this is the first study that uses a cVTST to obtain the temperature dependence of the k of the reaction [1] considering four additions and one H atom abstraction pathways.

The ratio between the calculated and the experimental activation energies indicates a good agreement between experimental and theoretical results, then, although the calculation of cVTST may underestimate the values of rate coefficients, considering its simplicity, it could be used as an alternative method to describe the temperature dependence for the reactions of haloalkenes with the OH radical.

The negative temperature dependence of k exp and k -cVTST (the last one obtained considering the proposed global mechanism described in the reactions 2a-5b and 6c-8c) could explain the experimental results, and supports the hypothesis that the OH radical addition is the main reaction mechanism for the studied global reaction, being the downward addition the most important reaction channel in the considered temperature range. To the best of our knowledge these not negligible reaction channels have been considered for the first time in this work in order to calculate the temperature rate constant dependence for the global reaction.

As a perspective of the ongoing research, a future study on the effects of the DFT method and basis sets over the computational capability to found the π or σ pre-complexes could be studied. Also, a further study on the impact of the kind of precomplexes on the cVTST rate coefficient calculations is previewed to be carried out. a The values were calculated using the expression k= 2.64 x 10 -14 x T 0.57 x exp (934.47/T) for the high-pressure limit and the 200-500K temperature range. Ref. 6 the k values were theoretically calculated.

5 - 3

 53 OH radicals were generated by photolysis of H 2 O 2 at λ= 248 nm (KrF excimer laser). The concentration of OH radicals was monitored at various reaction times ranging from 10 μs to 10 ms, by pulsed laser induced fluorescence (LIF). A frequencydoubled dye laser pumped by a Nd:YAG laser was used to excite the OH radicals at λ= 282 nm, and fluorescence from OH radicals was detected by a photomultiplier tube fitted with a 309 nm narrow bandpass filter. The output pulse from the photomultiplier was integrated for a preset period by a gated charge integrator. Typically, the fluorescence signal resulting from 100 probe laser shots was measured for 10 to 15 different delay times and averaged to generate OH concentration-time profiles over 2.lifetimes. H 2 O 2 was introduced into the reaction cell by passing a small flow of helium through a glass bubbler containing a solution of H 2 O 2 . 3-chloropropene was

  carried out single point calculations at the CCSD(T)/aug-cc-PVTZ level based on the M06-2X/ aug-cc-PVTZ optimized geometries, in order to improve the electronic energies. The Zero Point Energy (ZPE) corrections at the CCSD(T)/cc-aug-PVTZ were assumed the same corrections obtained from the vibrational frequencies calculated at the M06-2X/cc-aug-PVTZ level, then, unless otherwise specified, the geometric parameters and energies used in the following discussion is CCSD(T)/cc-aug-PVTZ// M06-2X/aug-cc-PVTZ + ZPE level. All open-shell stationary points showed T1 diagnosis values lower than 0.04 and no multiconfigurational character should be expected.

Figure 1 .

 1 Figure 1.a. shows an example of pseudo-first-order plots for the OH radical signal decay and in Fig 2.b., as an example, a plot of (k'-k' 0 ) as a function of CH 2 =CHCH 2 Cl concentration at different temperatures, is presented. The k values obtained in this

Figure 2

 2 Figure 2 show the measured rate coefficients for the studied reaction plotted as a function of the reciprocal of temperature along with k obtained in this work. A leastsquares analysis of the ln k versus 1/T plot leads to the following expression for the temperature dependence of k, in the temperature range 253-371 K (in units of cm 3 molecule -1 s -1 ): k = (2.82 ± 0.38) x 10 -12 exp {(540 ± 42) / T} Uncertainties are 2 σ lnA and 2 σ E/R for A and E/R, respectively.

  begin our discussion by investigating the CH 2 =CHCH 2 Cl potential energy surface aiming to explore the possible rotamers of the reactant and the location of the minimum energy geometry. Therefore, scan calculations over the dihedral angles were performed at the M06-2X/aug-cc-pVTZlevel. Possible rotamers can arise from the rotation of the -CH 2 Cl group. Three minima were found, one of them correspond to the structure CH 2 =CHCH 2 Cl-C s (rotamer belonging to the C s symmetry group), shown in the Figure 3, the other minima are mirror images belonging to the C 1 symmetry group, and have the structure shown in the same Figure labeled as CH 2 =CHCH 2 Cl-C 1 .CH 2 =CHCH 2 Cl-Cs lies 0.49 kcal mol -1 above the CH 2 =CHCH 2 Cl-C 1 rotamer and the barrier height separating them is 2.47 kcal mol-1 , as predicted at the M06-2X/aug-cc-pVTZ level. Reactions of each rotamer were treated individually. After the location of the possible rotamers, from the scan calculations, the energy was corrected at the CCSD(T)/aug-cc-pVTZ level.The finding of the CH 2 =CHCH 2 Cl-C s and CH 2 =CHCH 2 Cl-C 1 rotamers suggests that the OH radical can attack in two different ways. The first one is from the upside of the plane defined by the three C atoms, i.e., from the same side where the Cl atom is located, hereafter referred as the upward attack. The second way is given if the OH attack is from the opposite side of that plane, thus, a downward attack is proposed. For the CH 2 =CHCH 2 Cl-C 1 rotamers only the downward attack leads to a sigma pre-barrier complex (σ-PC down ) connected by TS c-down and TS t-down to I c-down and I t-down , whose geometries are also shown in Figure3. The subscripted c-down and t-down denote the downward addition pathways to the central and the terminal carbon atoms of the double bound of the CH 2 =CHCH 2 Cl-C 1 , respectively. In the Figure4a-b, the geometries of 2 stereoisomers of the σ-PC down (energetically degenerated) are shown.

Figure 3 .

 3 Figure4c-d. Then, two parallel reaction pathways (one for each stereo-isomer) for the upward OH addition have been considered. Along the upward reaction channel, we have found molecular structures similar to the CR1, TS1, TS2, IM1 and IM2 proposed by Zhang et al.6 for the OH addition to CH 2 ClCH=CH 2 .

  was assumed. The steady state is assumed for each PC in this mechanism (ie. d[-PC down ]/dt = d[-PC up ]/dt = d[-PC  ]/dt = 0), and following expressions are obtained for k down , k up and k abs and k cVTST (which is equal to the sum k down + k up + k abs ):

Figure 1 :

 1 Figure 1: a. Example of pseudo-first-orden plots for the OH radical signal decay at 298 K and four CH 2 ClCH=CH 2 concentrations. b. Examples of plots of the rate of disappearance of the OH radical (k'-k' 0 ) vs. CH 2 ClCH=CH 2 concentration at different temperatures.

Figure 2 .

 2 Figure 2. Plot of k 1 as function of 1/T for the OH reaction with CH 2 ClCH=CH 2 in the temperature range 263-369 K, compared with previous results in literature.

Figure 3 .

 3 Figure 3. Geometries of the stationary points found along the studied reaction pathways

Figure 4 :

 4 Figure 4: Geometries obtained for the 2 stereoisomers of σ-PC down (a-b), σ-PC up (b-c) and σ -PC β (e-f).

Figure 5 .

 5 Figure 5. Relative energy of the stationary points of the studied reaction pathways.

Table 1 :

 1 Summary of experimental conditions and results for measurements of the rate coefficient of the gas phase reaction between OH and CH 2 ClCH=CCH 2 .

	a molecule cm -3
	b cm 3 molecule -1 s -1
	♣ Laser Pulse Fluency. mJ cm -2

Table 2 :

 2 Summary of previous experimental and theoretical values obtained for k global in the high-pressure limit between 253 K and 371 K.

	Temperature	k*	k*	k*	k*	k*	k* a
	(K)	Ref. 4	Ref. 8	Ref. 9	Ref. 7	Ref.10	Ref.6
	228	2.64±0.40					3.51
	253	2.41±0.32					2.49
	273	2.05±0.28					1.98
	296	1.66±0.23					1.59
	298	1.64±0.18 1.70±0.70 1.96±0.32 1.69±0.07 1.53±0.40	1.56
	323						1.28
	328	1.48±0.25					1.24
	347						1.10
	370						0.960
	371						0.955
	388	1.18±0.20					0.877

* All the values are in units of cm 3 molecule -1 s -1 and were multiplied by 1x10

11

.

Table 3 :

 3 Summary of the relative energy (with respect to the isolated Reactants), expressed in kcal mol -1 , obtained in this work along with comparable values from Ref. 6.

		ΔE + ZPE	ΔE + ZPE*	ΔE +
	Species	(M06-2X/aug-	(CCSD(T)/aug-cc-	ZPE a Ref.
		cc-PVTZ)	PVTZ)	6.
	Addition
	CH 2 ClCH=CH 2 -C t + OH	0.00	0.00
	σ-PC down	-3,30	-1,90
	TS c-down	-1,28	-1,58
	TS t-down	-1,07	-1,13
	I c-down	-30,44	-27,44
	I t-down	-31,16	-27,51
	CH 2 ClCH=CH 2 -C c +OH	0.00	0.00	0.00
	σ-PC up	-4,09	-2,49	-3.56
	TS c-up	-1,80	-2,05	-1.96
	TS t-up	-1,15	-1,28	-1.03
	I c-up	-31,84	-28,76	-29.91
	I t-up	-31,86	-27,43	-29.86
	β-H atom abstraction
	CH 2 ClCH=CH 2 + OH	0.00	0.00
	σ -PC β	-3,48	-2,41
	TS β-H	0,189	0,64
	I d + H 2 O	-33,38	-31,61
	* Zero Point Energy corrections were calculated from the vibrational frequencies obtained at
	the M06-2X/cc-aug-PVTZ level.

a (CCSD(T)/cc-PVTZ//M06-2x/6-311++G(d,p))

Table 4 :

 4 Summary of conditions and results for cVTST calculations of the rate coefficients for the pathways down. up and abstraction and the global reaction.

	Temperature (K)	k down	k up	k abs	k -cVTST
	200	1.01x10 -11	1.98x10 -11	6.87x10 -13	3.06x10 -11
	225	7.28x10 -12	1.17x10 -11	7.36x10 -13	1.97x10 -11
	250	5.65x10 -12	7.67x10 -12	7.82x10 -13	1.41x10 -11
	275	4.62x10 -12	5.50x10 -12	8.23x10 -13	1.09x10 -11
	298	4.00x10 -12	4.27x10 -12	8.62x10 -13	9.13x10 -12
	300	3.96x10 -12	4.19x10 -12	8.67x10 -13	9.01x10 -12
	325	3.50x10 -12	3.39x10 -12	9.06x10 -13	7.80x10 -12
	350	3.17x10 -12	2.84x10 -12	9.52x10 -13	6.96x10 -12
	375	2.93x10 -12	2.47x10 -12	9.89x10 -13	6.39x10 -12
	400	2.73x10 -12	2.18x10 -12	1.04x10 -12	5.95x10 -12

* All the values are in units of cm 3 molecule -1 s -1 .

Acknowledgments

We thank CONICET, ANPCyT-FONCyT (PICT2016-0698) from Argentina, CNPq from Brazil, and CNRS and ECOS from France and the EU for financial support of this research and for doctoral and posdoctoral fellowships.