Véronique Billat

Nicolas J-B Brunel

Thomas Carbillet
email: thomas.carbillet@gmail.com

Stéphane Labbé

Adeline Samson

Humans are able to self-paced constant running accelerations until exhaustion

Keywords: Acceleration, Exercise, Running, Self-paced, Performance, Ornstein-Uhlenbeck

Although it has been experimentally reported that speed variations is the optimal way of optimizing his pace for achieving a given distance in a minimal time, we still do not know what the optimal speed variations (i.e accelerations) are. At first, we have to check the hypothesis that human is able to accurately self-pacing its acceleration and this even in a state of fatigue during exhaustive self-pacing ramp runs. For that purpose, 3 males and 2 females middle-aged, recreational runners ran, in random order, three exhaustive acceleration trials. We instructed the five runners to perform three self-paced acceleration trials based on three acceleration intensity levels: "soft", "medium" and "hard". We chose a descriptive modeling approach to analyse the behaviour of the runners. Once we knew that the runners were able to perceive three acceleration intensity levels, we proposed a meanreverting process (Ornstein-Uhlenbeck) to describe those accelerations: dat = -θ(at -a)dt+ σdWt where a is the mean acceleration, at is the measured acceleration at each time interval

Introduction

Endurance running is considered to have played a major role in human evolution. Humans have developed the ability to fine-tune their running speed in order to run for several days and still catch their fastest prey [START_REF] Bramble | Endurance running and the evolution of homo[END_REF]. Indeed, it has been reported that speed variation is the ideal way of optimizing its pace for achieving a given distance in a minimal time [START_REF] Foster | Effect of pacing strategy on cycle time trial performance[END_REF][START_REF] Foster | Pacing strategy and athletic performance[END_REF][START_REF] Koning | Regulation of pacing strategy during athletic competition[END_REF][START_REF] Billat | Effect of free versus constant pace on performance and oxygen kinetics in running[END_REF][START_REF] Bishop | The influence of pacing strategy on vo2 and supramaximal kayak performance[END_REF][START_REF] Sandals | Influence of pacing strategy on oxygen uptake during treadmill middle-distance running[END_REF][START_REF] Tucker | An analysis of pacing strategies during men's world-record performances in track athletics[END_REF][START_REF] Tucker | The physiological regulation of pacing strategy during exercise: a critical review[END_REF]. Hence, the rate at which the velocity of a body changes with time requires positive and negative accelerations and then strength variations according to Newton's second law of motion.

However, there is a direct relationship between force impulse and running acceleration [START_REF] Hunter | Relationships between ground reaction force impulse and kinematics of sprint-running acceleration[END_REF] and the minimum-jerk model [START_REF] Flash | The coordination of arm movements: an experimentally confirmed mathematical model[END_REF] predicts that to save energy and optimize performance, running must be as smooth as possible and variations in acceleration must be close to 0 m.s -3 . Furthermore, there is a clearly non-random speed fluctuation during self-paced exercise [START_REF] Tucker | Non random fluctuations in power output during self paced exercise[END_REF][START_REF] Noakes | From catastrophe to complexity: a novel model of integrative central neural regulation of effort and fatigue during exercise in humans[END_REF][START_REF] Abbiss | Describing and understanding pacing strategies during athletic competition[END_REF][START_REF] Koning | Regulation of pacing strategy during athletic competition[END_REF]. It has been suggested that speed variability operates through a feed forward mechanism [START_REF] Ansley | Anticipatory pacing strategies during supra-maximal exercise lasting more than 30 s[END_REF] and we know that most of the races are stochastic when the velocity is not imposed [START_REF] Billat | Detection of changes in the fractal scaling of heart rate and speed in a marathon race[END_REF]. Despite the large published body of work on pacing strategy and speed control, this is the first study to have examined acceleration control during running. We know that the fastest runners on middle and long-distance run with light speed variations [START_REF] Billat | Effect of free versus constant pace on performance and oxygen kinetics in running[END_REF][START_REF] Ely | Effect of ambient temperature on marathon pacing is dependent on runner ability[END_REF][START_REF] Haney | A description of variability of pacing in marathon distance running[END_REF]. Now that sensors make the recording of acceleration data easy, we would like to check if humans are able to integrate instructions in acceleration. The purpose is to take part in a runner's training and race in future studies. Given the fact that we are looking for a stochastic process to describe the ability for humans to maintain a constant acceleration and the hypothesis that self-paced accelerations lightly fluctuate around a constant value, we thought about the Ornstein-Uhlenbeck process [START_REF] Uhlenbeck | On the theory of the brownian motion[END_REF]. Such process is described as a stochastic process with a mean-reverting property. The use of Ornstein-Uhlenbeck process is present in fields such as mathematical finance [START_REF] Schobel | Stochastic volatility with an ornstein-uhlenbeck process: An extension[END_REF] (volatilities of asset prices and dynamics of interest rates) and biological processes [START_REF] Ricciardi | The ornstein-uhlenbeck process as a model for neuronal activity[END_REF]. Before going deeper in the effective mechanism of speed variation control during exhaustive exercise, we have to verify human's ability to maintain constant acceleration in a conscious way until exhaustion. The present study tests the hypothesis whereby humans are able to maintain a constant acceleration in a self-paced trial with no external cues regardless of the speed and the magnitude of acceleration and fatigue. For that purpose, we study more specifically the following two points:

(1) The ability for humans to maintain three different intensity levels of constant acceleration when so instructed. Those three intensities are described as "soft", "medium" and "hard" which are the instructions we gave to the runners for respectively a slow, a medium and a hard constant acceleration.

(2) The characterization of the acceleration data by introducing a new interpretation using a mean-reversing process (Ornstein-Uhlenbeck).

The paper is organized as follows. Section 2 describes the experimental protocol and presents the mathematical model. Section 3 compares the three acceleration intensities for each runner to see if we notice a genuine difference and gives an estimation of the parameters of the Ornstein-Uhlenbeck model for characterizing human accelerations. Finally, section 4 and 5 conclude the paper.

Methods

Subjects

The study population is composed of three males and two females recreational runners (aged 38 ± 3 yrs., total running distance per week: 36.1 ± 4.3 km; body weight: 66.9 ± 12.4 kg and height 171.1 ± 6.7 cm). All subjects were first informed of the risks and constraints associated with the protocol and gave their written, informed consent to participation. The present study conformed to the precepts of the Declaration of Helsinki and all procedures were approved by the local investigational review board (Saint Louis Hospital, Paris, France). On a physiological aspect, we cannot compare openly males and females. Given this fact plus the sample size of the study, we choose to focus on a descriptive modeling approach to analyse the runner behaviours.

No statistical tests will be applied.

Experimental Design

Subjects ran alone and performed three types of track sessions until exhaustion, in random order and with a one-hour interval between sessions: the first, second and third track tests were selfpaced acceleration trials at respectively slow, medium and high accelerations. We characterize those accelerations as "soft", "medium" and "hard". Velocity and acceleration were measured with a GPS-enabled accelerometer of 10 and 100 Hz the Minimax from Catapult Sports (Pty Ltd, Victoria, Australia). The GPS and accelerometer signals were sampled at 5 and 50 Hz, respectively, and averaged per second. The difference between the real distance (track) and the recorded distance (GPS) was less than 1% and 0.92% over 800 m and 1500 m, respectively. This result agrees with previous GPS studies for maximal efforts run by humans or horses [START_REF] Barbero-Alvarez | The validity and reliability of a global positioning satellite system device to assess speed and repeated sprint ability (rsa) in athletes[END_REF][START_REF] Larsson | The use of dgps and simultaneous metabolic measurements during orienteering[END_REF].

Acceleration trials

In the self-paced acceleration trial protocol, the runners performed three freely paced acceleration sets in which they were asked to maintain constant acceleration by progressively increasing their speed until exhaustion. The trials were run at three constant acceleration values, based on ratings of perceived acceleration ("soft", "medium" and "hard"), in random order. In all of the the selfpaced acceleration trials, the runners started at an initial velocity between 2.2 and 3.05 m.s -1 and then increased their velocity at three different, constant accelerations ("soft", "medium" and "hard" intensities). There was a one-hour interval between acceleration trials. In the self-paced acceleration trials set, the runners were not provided with any external information other than the distance covered. All tests were performed between 3pm and 6pm on wind-free, spring days (< 2 m.s -1 according to an anemometer, the Windwatch from Alba, Silva, Sweden) with a temperature of 20 • C, as in a previous study of the energetics of middle-distance running [START_REF] Billat | Energetics of middle-distance running performances in male and female junior using track measurements[END_REF].

Mathematical Model

We know that most of the races are stochastic when the velocity is not imposed [START_REF] Billat | Detection of changes in the fractal scaling of heart rate and speed in a marathon race[END_REF]. Our hypothesis analyzing the measured data was that humans can maintain a constant acceleration with little variations around a mean value. Therefore, we are characterizing the human acceleration on self-paced exercises with three different instruction intensities ("soft", "medium" and "hard") with a Ornstein-Uhlenbeck process. From the model to the simulation, this subsection explains the method used for characterizing the runners self-paced acceleration when so instructed.

The Ornstein-Uhlenbeck process

To assess the subjects' ability to maintain self-paced, constant acceleration, we check that the equation for constant-acceleration motion is satisfied. If the acceleration of the runner (a) is constant then the runner's speed at time t, denoted v(t), is a linear function of time v(t) = v(0) + at, where v(0) = v 0 is the initial speed. Here, v 0 is within the range of 2.2 to 3.05 m.s -1 . As a consequence, the distance d(t) covered at time t has a closed form obtained directly by integration as d(t) = 0.5at 2 + v 0 .t if we consider that d(0) = 0. Hence, the theoretical relationship between distance and time is a polynomial of second order. As mentioned in the introduction, in order to save energy and optimize performance, running must be as smooth as possible (variation in acceleration must be close to 0 m.s -3 ) during self-paced exercise. In this study, we want to see if the following analogy is possible: self-paced acceleration is as smooth as possible during self-paced exercise. Let us denote a(t) the acceleration at time t. The objective of the paper is to propose a reasonable stochastic model for a(t). Assuming that each runner has an internal variability and an autocorrect ability, we define a(t) as the solution of:

da(t) = -θ[a(t) -a]dt + σdW (t) (1)
The above equation is the definition of a mean-reverting process also called Ornstein Uhlenbeck process. a(t) (m.s -2 ) is the acceleration of the runner at time t (s). a (m.s -2 ) is the mean acceleration. We distinguish two human factors, θ depicting the autocorrect ability of the runner (that is also the speed reversion parameter) and σ the human variability (that is also the short term standard deviation). The stochastic nature of the model is brought by the Brownian motion

W (t).
A solution in terms of integral of a mean-reverting process can be written as the following equation [START_REF] Gillespie | Exact numerical simulation of the ornstein-uhlenbeck process and its integral[END_REF]:

a(t) = a(0)e -θt + a(1 -e -θt ) + σe -θt t 0 e θs dW (s) (2)
The three parameters a, θ and σ are unknown. The objective is to estimate them from the data and to validate a posteriori the fit obtained with model (1). From equation ( 2), we would like to highlight the fact that unlike the beginning of the current subsection (2.4.1), a(t) is not consistent anymore. In addition a(t) and v(t) are not linear in time. The next subsection focuses on the estimation of the parameters a, θ and σ from the data.

Parameter estimation

Parameter estimation from discrete observation a(i∆) for i = 0, .., n can be performed by optimizing a least square criteria. Indeed, we have

a(t) = e -θ∆t a(t -∆ t ) + (1 -e -θ∆t )a + σ 1 -e -2θ∆t 2θ (W (t) -W (t -∆ t )) (3)
Let us introduce a new parametrization: α = (1 -e -θ∆t )a, β = e -θ∆t and γ = σ 1-e -2θ∆ t 2θ

The least square criteria is then defined as ( α, β) = argmin

(α,β) n i=1 (a(i∆) -βa((i -1)∆) -α) 2 and γ2 = 1 n n i=1 (a(i∆) -βa((i -1)∆) -α) 2 .
The estimators correspond to the maximum likelihood estimators. They are consistent, unbiased and asymptotically normal. The solution of α and β are explicit. The Fisher information matrix can be computed explicitly, that corresponds to the asymptotic variance denoted ˆ .

To obtain the estimation of the original parameters, we use the following transformation:

φ(α, β, γ) = ( α 1-β , -ln(β) ∆ , γ ( -2ln(β) ∆(1-β 2 )
)). The estimator of (a, θ, σ) are then defined as (â, θ, σ) = φ(α, β, γ). By the delta method, we can prove that their are asymptotically normal: (â, θ, σ)

L → N ((a * , θ * , σ * ), J φ ˆ J φ )
where (a * , θ * , σ * ) are the true (unknown) parameters and J φ is the Jacobian of φ. This allows to compute confidence interval for (a, θ, σ).

Results

Comparison of the three different acceleration exercises

In order to know if humans are able to perceive different intensity levels of acceleration, we instructed the five runners to perform three self-paced constant acceleration trials (see subsection 2.3). The result of the three trials for the first male runner is shown in Figure 1.

On each plot, the velocity corresponding to "hard" intensity has the highest slope, the "medium" intensity has the medium slope and the "soft" intensity acceleration trial has the smallest one.

Results of the four other runners can be found in the appendix A, Figures 4,5,6,7. Table 1 presents the distance and time at exhaustion for each runner and each trial. Moreover the average ratios of the "Soft" to "Medium" instructions on "Distance" and "Time" are respectively 1.65 and 2.22 and the average ratios of the "Medium" to "Hard" instructions on "Distance" and "Time" are respectively 1.81 and 2.32. Tables 2 and3

Characterizing the human faculty to maintain a constant acceleration

As shown before, we can consider that humans are able to perceive different intensity levels of acceleration. The objective is how to characterize those accelerations. To do so, the Ornstein-Uhlenbeck model ( 3) is applied.

The goal is to show that it is possible to characterize the human acceleration. We would like to give an estimation and interpretation of the parameters of the Ornstein-Uhlenbeck model (â, θ, σ) and their confidence interval. By applying the method described in subsection 2.4.2, we have estimated â, θ and σ and the confidence interval for â. Results of the estimated parameters and confidence intervals for â are shown in Table 4.

We observe the following two points from Table 4:

1. The mean autocorrect abilities ( θ) of the runners are steady with a slight drop in the "Hard" trial. This means that the runners are less able to correct their mistakes during the "Hard" trials.

2. The human variability (σ) is steady too with a slight rise in the "Hard" trial. This could be an answer to the drop of autocorrect ability. This means that the runner variability tends toward increasing during "Hard" trial. In order to evaluate the validity of the mean-reverting process applied to acceleration, the goodness-of-fit of the model has been tested in three steps:

1. Checking the ability of the Ornstein-Uhlenbeck model to distinguish three distinct mean accelerations.

2. Checking for non autocorrelation of residuals with the Durbin-Watson test.

3. Testing for stationarity with the Augmented Dickey-Fuller (ADF) Test.

Checking the ability of the Ornstein-Uhlenbeck model to distinguish three distinct mean accelerations

Let us now highlight the ability of our model to distinguish the three intensities of acceleration.

In order to do so, we use the delta method to compute a confidence interval for each estimated acceleration â. The Figure 2 However, because of the low sample size, this conclusion is purely descriptive. A statistical test will be applied on the future on a larger dataset. The Figure 3 shows the result of one simulation for the first male runner. We can distinguish We can quickly observe that the confidence interval is consistent with the reality of our data and that most of the recorded acceleration data seem to be inside the confidence interval. Results

for the four other runners can be found in appendix B, Figures 8 to 11.

To confirm that a significant amount of measured acceleration are inside the confidence boundaries, Table 6 shows that the values of percentages of measured data found in the 95% confidence boundaries are above 92%. Thus the Ornstein-Uhlenbeck model described in subsection 2.4.2 seems to be consistent and could be used to describe the reality.

To go further, it would be interesting to perform self-paced acceleration following the same protocol with first, more runners in order to check if there is a trend in the ratios, second, splitting the runners into two categories (men and women) to check if there is a gender effect on those ratios.

Testing for non autocorrelation of residuals

In this subsection, we want to detect the presence of autocorrelation among residuals. In our case, the residuals is a vector defined by the difference between our measured data a(t) and the estimated acceleration parameter from the Ornstein-Uhlenbeck model (â). We perform the Durbin-Watson statistic on the following linear regression:

a(t) -a(t -∆ t ) = θ(a(t) -â)∆ t + η(t), ( 4
)
where η(t) is the vector of residuals. If the model is correct, the residuals are independent and their distribution follows a normal distribution described as follows: N (0, σ2 ∆t)

In order to perform the Durbin-Watson statistic, we define the null hypothesis (H 0 ) as follows:

the residuals of the linear regression (4) are uncorrelated. The alternative hypothesis (H 1 ) is that there is autocorrelation among the residuals. In order to test H 0 against H 1 , we used the function dwtest1 from Matlab R . The results of the test are shown in Table 7.

The range of p -value is included between 0.08 and 0.98. Given those results, we cannot reject the null hypothesis stating that the residuals are uncorrelated.

Testing for stationarity

In this subsection, we test for no unit root and wether the process of acceleration is stationary or not to insure that the estimated mean accelerations neither grow nor decrease over time. We define the null Hypothesis (H 0 ) as follows: the unit root is present in the measured acceleration time series. The alternative hypothesis (H 1 ) is defined as follows: the accelerations time series is trend-stationary. In order to test H 0 against H 1 , we used the function adf test 2 from Matlab R .

The results of the test are shown in Table 8. Given those results, we reject the null hypothesis stating that the simulation of acceleration is non-stationary.

Discussion

Our results showed that recreational runners are able to control their acceleration until exhaustion at three significantly different accelerations values perceived to be "soft", "medium" and "hard". Indeed, our data showed that runners are able to apply distinct, subjective acceleration values when so instructed and maintain constant acceleration until exhaustion, regardless of the acceleration value. In addition, our results showed that self-paced acceleration fitted the following Ornstein-Uhlenbeck process da t = -θ(a t -a)dt + σdW t where a is the mean acceleration, θ is the human ability for autocorrection, σ the human variation and a t the measured acceleration.

Indeed, a fine-grain analysis of the data shows that accelerations are obtained by the application of brief and frequent corrections designed to reach a mean target value a, and that there is no drift in the accelerations. Moreover, the fact that there is no unit-root in the acceleration time series shows that possible strong deviance from the mean value a are rapidly corrected and vanishes by mean-reverting corrections. Further studies are now required for characterizing the deceleration of a recreational runner.

Given the fact that this is the first study having examine the human ability to maintain self-paced acceleration, we are going to discuss the two following points:

1. The self-paced acceleration range values compared with current use of acceleration protocols in laboratory.

2. The possible mechanisms allowing humans to control its body acceleration in accordance with prior results obtained on manual task.

The self-paced acceleration range values compared with current use of acceleration protocols in laboratory

The range of these perceived acceleration values is in accordance with observed values in middledistance running [START_REF] Billat | Differential modeling of anaerobic and aerobic metabolism in the 800-m and 1,500-m run[END_REF]. Furthermore, these freely chosen accelerations also correspond to the imposed acceleration values frequently used in treadmill ramp protocols for determining VO2max [START_REF] Myers | exercise protocols for clinical and cardiopulmonary exercise testing[END_REF][START_REF] Porszasz | A treadmill ramp protocol using simultaneous changes in speed and grade[END_REF]. In ramp protocols, the work rate is ramped up as a continuous increase and then a continuous acceleration. Given that the linearity of the oxygen uptake response is a major discriminating cardiovascular feature for assessing exercise intolerance, it is important to be sure that the work rate profile is linear and then the acceleration is constant. That is why the tests are currently performed on a treadmill. Even though manufacturers have developed a range of technologies for enabling ramp tests (e.g. controlled cycle ergometer), some subjects have difficulty walking and running on a treadmill and then reaching their maximum VO2 and speed at VO2max. It has been argued that the treadmill use induces a higher maximum metabolic rate and requires a mode of exercises that more closely approximates some activities of daily living [START_REF] Balke | An experimental study of physical fitness of air force personnel[END_REF][START_REF] Froelicher | Exercise and the Heart[END_REF]. In a much easier protocol, the present study shows that male and female middle-aged, recreational runners were able to self-monitor acceleration and thus reproduced an outdoor ramp protocol. Indeed, the present results showed that it is possible to apply a self-paced, ramp-like running protocol on a real running track. Regardless of the acceleration intensity level ("soft", "medium" or "hard"), the protocol uses a continuous change in speed and brings the subjects to exhaustion in approximately 2 minutes and 30 seconds to 20 minutes; this meets the criteria for clinical exercise testing issues by the relevant international organizations (American College of Sports Medicine 2000, American Thoracic Society 2003, [START_REF] Myers | Individualized ramp treadmill. observations on a new protocol[END_REF][START_REF] Gibbons | Guidelines for exercise testing: a report of the american college of cardiology/ american heart association task force on practice guidelines[END_REF][START_REF] Will | Exercise testing: improving performance with a ramped bruce protocol[END_REF]).

The possible mechanisms allowing humans to control its body acceleration in accordance with prior results obtained on manual task

Here we showed that human are accurately able to accelerate smoothly and constantly, with acceleration vs. time values close to 0 m.s -2 (even near to exhaustion). However, the question of how acceleration is controlled remains to be determined. There is still no studies having worked on the ability of self-pace its acceleration in locomotion. Therefore we can only discuss versus prior studies on arm and hand movement in catching tasks. In that tasks, neurophysiologists have demonstrated that the human brain imposes (in a top-down fashion) its rules of interpretation on sensory data. It transforms the perceived world according to rules of symmetry and stability and kinematic laws derived from principles of maximum smoothness [START_REF] Berthoz | Neurobiology of "Umwelt[END_REF]. They hypothesized that the first possibility was that humans can perceive distance very accurately [START_REF] Mossio | Traveled distances: new insights into the role of optic flow[END_REF]. It has been proved that humans possess efficient mechanisms for building an abstract representation of distances, which can be transferred from one perceptual context to another [START_REF] Mossio | Traveled distances: new insights into the role of optic flow[END_REF].

This paper seemed to show that whatever the variation in velocity, from "soft" acceleration intensity level at 0.0026 m.s -2 to ten times that at 0.0253 m.s -2 in the "hard" acceleration intensity level in self-paced acceleration trial, the acceleration was well controlled and steady, with a human ability to autocorrect and a minimal jerk value. The concept of "jerk-cost" has been validated for running and walking stride smoothness [START_REF] Hreljac | Stride smoothness evaluation of runners and other athletes[END_REF].

Conclusion

To the best of our knowledge, a human's ability to maintain constant acceleration in a conscious way until exhaustion has not yet been investigated. We therefore decided to test the hypothesis whereby humans are able to maintain a constant acceleration, regardless of the velocity and the magnitude of acceleration. Indeed, despite the large, published body of work on pacing strategy and speed control (especially concerning feedback vs. feed forward power output control and central vs. peripheral mechanisms), this is the first study to have examined acceleration control during running.

The main findings of this research are as follows:

1. The measured data seemed to show that runners are able to apply at least three distinct and subjective acceleration intensities when so instructed.

2. The measured data seemed to show that the runners tend towards reducing the time and distance to exhaustion between the "soft" and the "medium" instruction as well as between the "medium" and "soft" instructions.

3. The simulation using the Ornstein-Uhlenbeck process seemed to show that accelerations are obtained by the application of brief and frequent corrections designed to reach a mean target value a.

4. The simulation showed that the runners tend towards increasing the mean acceleration between the "soft" and the "medium" instruction as well as between the "medium" and "soft" instructions.

Our results show that runners are able to control their acceleration until exhaustion at three significantly different accelerations values perceived to be "soft", "medium" and "hard". According to Newton's Second Law, humans could control their force of motion as it is directly proportional to the acceleration opening the door on future study related to human physiology and human energetic. We can think of two research opportunities go further. One would be to study the deceleration and see if humans are also able to control it. Once we successfully describe both acceleration and deceleration with mathematical models, we could think about alternating acceleration and deceleration phases to see how runners would react and study the sequence. The other would be gathering data from many runners, both males and females, in order to establish a predictive model for human acceleration. This could be done using Stochastic Differential Mixed-Effects Equations. VS. Time (s) -Comparison of the three acceleration intensity instructions: "soft" (blue curve), "medium" (red curve) and "hard" (yellow curve)

Appendices

Figure 1 -

 1 Figure 1 -Male Runner 1 -Top chart: Velocity (m/s) VS. Time (s); Bottom chart: Acceleration(m/s 2 ) VS. Time (s) -Comparison of the three acceleration intensity instructions: "soft" (blue curve), "medium" (red curve) and "hard" (yellow curve)

  display those results. Note that we do not apply any statistical test to determine the significativity of the three regimes because of the low sample size (3 males and 2 females). This highlights the fact that humans are able to feel three different intensities of acceleration and divide distance and time to exhaustion from one intensity to another.The fact that the three regimes are clearly distinct on the plot strengthens the idea that humans are able to maintain at least three different intensities of acceleration.

  Figures 12,[START_REF] Noakes | From catastrophe to complexity: a novel model of integrative central neural regulation of effort and fatigue during exercise in humans[END_REF][START_REF] Abbiss | Describing and understanding pacing strategies during athletic competition[END_REF][START_REF] Koning | Regulation of pacing strategy during athletic competition[END_REF] for the four other runners show that the confidence intervals do not overlap.

Figure 2 -

 2 Figure 2 -Male Runner 1 -Comparison of the three estimated mean accelerations

3

  information per chart with two curves and one filled region on each chart. The black curve (dotted) represents the measured data (i.e. measured accelerations), the red one (solid) represents one random simulation of the Ornstein-Uhlenbeck model (equation (3) subsection 2.4) and the filled region stands for the 95% confidence interval.

Figure 3 -

 3 Figure 3 -Male Runner 1 -From top to bottom: Soft, Medium and Hard Acceleration Intensities -Measured data VS. Ornstein-Uhlenbeck model

  0e-03 < 1.0e-03 < 1.0e-03 < 1.0e-03 < 1.0e-03 Medium < 1.0e-03 < 1.0e-03 < 1.0e-03 < 1.0e-03 < 1.0e-03 Hard < 1.0e-03 < 1.0e-03 < 1.0e-03 < 1.0e-03 < 1.0e-03

Figure 4 -

 4 Figure 4 -Male Runner 2 -Top chart: Velocity (m/s) VS. Time (s); Bottom chart: Acceleration(m/s 2 )

Figure 5 -Figure 6 -Figure 7 -Figure 8 - 22 Figure 9 -Figure 10 -Figure 11 -

 56782291011 Figure 5 -Male Runner 3 -Top chart: Velocity (m/s) VS. Time (s); Bottom chart: Acceleration(m/s 2 ) VS. Time (s) -Comparison of the three acceleration intensity instructions: "soft" (blue curve), "medium" (red curve) and "hard" (yellow curve)

Figure 12 -

 12 Figure 12 -Male Runner 2 -Comparison of the three estimated mean accelerations

Table 1 -

 1 The three different intensity levels are distinct for each runner. The times to exhaustion are in the range 96 s (1 minute and 36 seconds) to 1 204 s (approximately 20 minutes). The

	average time to exhaustion for "soft" intensities is 677.6 seconds (approximately 11 minutes and
	6 seconds), for "medium" intensities is 368.6 seconds (6 minutes and 26 seconds) and for "hard"
	intensities is 169.8 seconds (2 minutes and 49 seconds).

Distance and Time at exhaustion for each runner and for each acceleration trial intensity:

"soft", "medium" and "hard" Male 1 Male 2 Male 3 Average Males Female 1 Female 2 Average Females

Table 2

 2

	-Ratios of distances to exhaustion for "soft" vs. "medium" and "medium" vs. "hard" acceleration
	intensities for each runner

  Male 1 Male 2 Male 3 Average Males Female 1 Female 2 Average Females

	sof t medium	1.38	2.02	2.13	1.84	1.81	1.69	1.75
	medium hard	2.07	3.58	2.94	2.86	1.48	1.54	1.51

Table 3 -

 3 Ratios of times to exhaustion for "soft" vs. "medium" and "medium" vs. "hard" acceleration intensities for each runner

Table 4 -

 4 The next step consists in checking if our model could explain the reality. By its very nature, an Ornstein-Uhlenbeck process is Gaussian. This property allows us to compute the quantiles exactly. Once such confidence boundaries are computed, we check the percentage of the measured acceleration data present inside.

	Ratios of estimated acceleration for "soft" vs. "medium" and "medium" vs. "hard" acceleration
	intensities in Table 5 seem to confirm the hypothesis that humans are able to distinct at least 3
	intensities of acceleration.

Mean acceleration (â), autocorrect ability ( θ) and human variability (σ) estimated for each runner and for each acceleration trial intensity: "soft", "medium" and "hard" and confidence interval (C.I) for â Male 1 Male 2 Male 3 Female 1 Female 2 Average

Table 6 -

 6 Results of percentages of measured data found in the 95% confidence interval for accelerationfor each trial and each runner.

			Male 1(%) Male 2(%) Male 3(%) Female 1(%) Female 2(%)
	Acceleration	Soft	94.42	95.43	94.02	97.51	97.89
		Medium	95.24	97.14	94.33	94.29	95.63
		Hard	99.32	97.60	96.00	92.89	95.45

Table 7 -

 7 p -values of Durbin-Watson statistic to check for autocorrelation among residuals for each

	runner and for each acceleration trial intensity ("soft", "medium" and "hard")

Table 8 -

 8 p -value of Augmented Dickey-Fuller static to check for stationarity for each runner and for

	each acceleration trial intensity ("soft", "medium" and "hard")

See the documentation on dwtest: https://fr.mathworks.com/help/stats/dwtest.html

See the documentation on adf test: https://fr.mathworks.com/help/econ/adftest.html

Figure 15 -Female Runner 2 -Comparison of the three estimated mean accelerations


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Jack Harrowfield

Structure-Directing Effects of Counterions in Uranyl Ion Complexes with Long-Chain Aliphatic α α α α,ω ω ω ω-Dicarboxylates: 1D to Polycatenated 3D Species

)2(C7)2(HC7)2Ni(cyclam)]⋅2H2O (8) due to axial coordination of the nickel(II) centre, hydrogen bonding mediated by water molecules generating a three-dimensional (3D) net. [(UO2)2K2(C7)3(H2O)]⋅0.5H2O (9) contains convoluted uranyl dicarboxylate 2D subunits which generate a 3D framework through 2D  3D parallel polycatenation similar to that previously found in [NH4]2[(UO2)2(C7)3]⋅2H2O; further linking of these subunits is provided by bonding of the potassium cations to carboxylate and uranyl oxido groups. The solid state emission spectra of complexes 1-6 and 9 display maxima positions typical of hexacoordinated uranyl carboxylate complexes, but uranyl luminescence is quenched in 7. A solid-state photoluminescence quantum yield of 11.5% has been measured for complex 1, while those for compounds 3-6 and 9 are in the range of 2.0-3.5%.

INTRODUCTION

The aliphatic α,ω-dicarboxylic acids HOOC-(CH2)n-2-COOH (H2Cn) in their deprotonated carboxylate form constitute a family of potentially ditopic ligands of variable length and flexibility, in which the relative orientation of the two binding sites can be modulated over a large range. Fully extended chains provide divergent metal ion binding sites which, for large n values, can widely separate metal centres and favour polymer formation, while curved or buckled conformations of the chains may provide more or less convergent ligands compatible with closer locations of the metal centres and, possibly, closed oligomer formation. A consequent problem is that of finding structure-directing species enabling control of the ligand conformation, although unpredictability is obviously a characteristic of such flexible systems.

In uranyl ion coordination chemistry, and in particular in the investigation of uranyl-organic coordination polymers, [START_REF] Wang | Extended Structures and Physicochemical Properties of Uranyl-Organic Compounds[END_REF][START_REF] Andrews | Uranyl Bearing Hybrid Materials: Synthesis, Speciation, and Solid-State Structures[END_REF][START_REF] Loiseau | The Crystal Chemistry of Uranium Carboxylates[END_REF][START_REF] Su | MOFs of Uranium and the Actinides[END_REF][START_REF] Thuéry | Recent Advances in Structural Studies of Heterometallic Uranyl-Containing Coordination Polymers and Polynuclear Closed Species[END_REF] no such control is known in the case of the first two aliphatic dicarboxylates, oxalate (n = 2) and malonate (n = 3), as these ligands appear to favour simple chelation involving both carboxylate groups, often associated with bridging and polymer formation, and it is yet to be systematically demonstrated whether or not it is possible in the case of the next two members of the series, succinate (n = 4) and glutarate (n = 5), for which 15 and 20 examples, respectively, are reported in the Cambridge Structural Database (CSD, Version 5.39), [START_REF] Groom | The Cambridge Structural Database[END_REF] although these are now ligands clearly favouring polymer formation. Uranyl complexes of dicarboxylates with longer chains (n = 5-10) have been investigated by Cahill's group, with a particular emphasis on their association with molecules such as 4,4ʹ-bipyridine, 1,2-bis(4-pyridyl)ethane and trans-1,2-bis(4-pyridyl)ethylene, in a deliberate effort to use these large molecules to influence the form of uranyl-dicarboxylate coordination polymers. [START_REF] Borkowski | Crystal Engineering with the Uranyl Cation I. Aliphatic Carboxylate Coordination Polymers: Synthesis, Crystal Structures, and Fluorescent Properties[END_REF][START_REF] Borkowski | Crystal Engineering with the Uranyl Cation II. Mixed Aliphatic Carboxylate/Aromatic Pyridyl Coordination Polymers: Synthesis, Crystal Structures, and Sensitized Luminescence[END_REF][START_REF] Kerr | Crystal Engineering with the Uranyl Cation III. Mixed Aliphatic Dicarboxylate/Aromatic Dipyridyl Coordination Polymers: Synthesis, Structures, and Speciation[END_REF] These bipyridyl species assume the various roles of coligands, counterions, and structure-directing species, depending on their length compared to that of the dicarboxylates. Other diverse species, either neutral such as cucurbiturils, [START_REF] Thuéry | Uranyl Ion Complexation by Aliphatic Dicarboxylic Acids in the Presence of Cucurbiturils as Additional Ligands or Structure-Directing Agents[END_REF] 2,2ʹ-bipyridine (bipy) or 1,10-phenanthroline (phen), [START_REF] Thuéry | Anchoring Flexible Uranyl Dicarboxylate Chains through Stacking Interactions of Ancillary Ligands on Chiral U(VI) Centres[END_REF] or cationic such as NH4 + , 12 H2NMe2 + , [START_REF] Thuéry | The Crystalline α,ω-Dicarboxylate Metal Complex with the Longest Aliphatic Chain to Date: Uranyl 1,15-Pentadecanedioate[END_REF] [M(L)x] n+ , with M = Mn, Fe, Co, Ni, Cu, Ag, L = bipy or phen, x = 2 or 3 and n = 1 or 2, [START_REF] Thuéry | Counterion-Induced Variations in the Dimensionality and Topology of Uranyl Pimelate Complexes[END_REF][START_REF] Thuéry | A New Form of Triple-Stranded Helicate Found in Uranyl Complexes of Aliphatic α,ω-Dicarboxylates[END_REF][START_REF] Thuéry | From Helicates to Borromean Links: Chain Length Effect in Uranyl Ion Complexes of Aliphatic α,ω-Dicarboxylates[END_REF][START_REF] Thuéry | Uranyl Ion Complexes with Long-Chain Aliphatic α,ω-Dicarboxylates and 3d-Block Metal Counterions[END_REF][START_REF] Thuéry | Ag I and Pb II as Additional Assembling Cations in Uranyl Coordination Polymers and Frameworks[END_REF] [Co(en)3] 3+ , with en = ethylenediamine, [START_REF] Thuéry | Counterion-Induced Variations in the Dimensionality and Topology of Uranyl Pimelate Complexes[END_REF] or Pb 2+ , [START_REF] Thuéry | Ag I and Pb II as Additional Assembling Cations in Uranyl Coordination Polymers and Frameworks[END_REF] have also been found to act as structure-directing agents in uranyl ion complexes with Cn 2-(n = 6-10, 12, 13 and 15), whether they be neutral co-crystallized species, coligands, counterions or additional metal groups included in the polymer. A nice example of control of the uranyl complex geometry through choice of aliphatic chain length and counterion is provided by the isolation of discrete, triple-stranded binuclear helicates [M(bipy)3][(UO2)2(C9)3] and [M(phen)3][(UO2)2(C12)3] (M = Mn, Co, Ni), [START_REF] Thuéry | A New Form of Triple-Stranded Helicate Found in Uranyl Complexes of Aliphatic α,ω-Dicarboxylates[END_REF] or of polymeric [Mn(phen)3][(UO2)2(C13)3] species displaying Borromean-type entanglement. [START_REF] Thuéry | From Helicates to Borromean Links: Chain Length Effect in Uranyl Ion Complexes of Aliphatic α,ω-Dicarboxylates[END_REF] Following our recent use of PPh4 + and PPh3Me + , [START_REF] Thuéry | Uranyl-Organic Coordination Polymers with trans-1,2-, trans-1,4-, and cis-1,4-Cyclohexanedicarboxylates: Effects of Bulky PPh4 + and PPh3Me + Counterions[END_REF][START_REF] Thuéry | Counterion-Controlled Formation of an Octanuclear Uranyl Cage with cis-1,2-Cyclohexanedicarboxylate Ligands[END_REF][START_REF] Thuéry | Closed Uranyl-Dicarboxylate Oligomers: A Tetranuclear Metallatricycle with Uranyl Bridgeheads and 1,3-Adamantanediacetate Linkers[END_REF] [Ni(cyclam)] 2+ and [Ni(R,S-Me6cyclam)] 2+ (cyclam = 1,4,8,11-tetraazacyclotetradecane and R,S-Me6cyclam (meso isomer) = 7(R),14(S)-5,5,7,12,12,14hexamethylcyclam) [START_REF] Thuéry | Ni(cyclam)] 2+ and [Ni(R,S-Me6cyclam)] 2+ as Linkers or Counterions In Uranyl-Organic Species with cis-and trans-1,2-Cyclohexanedicarboxylate Ligands[END_REF] and alkali metal ion complexes of crown ethers [START_REF] Thuéry | Crown Ethers and Their Alkali Metal Ion Complexes as Assembler Groups in Uranyl-Organic Coordination Polymers with cis-1,3-, cis-1,2-, and trans-1,2-Cyclohexanedicarboxylates[END_REF] as bulky counterions in the synthesis of uranyl-containing coordination polymers with conformationally restricted 1,3adamantanediacetate and diverse members of the cyclohexanedicarboxylate family, we have attempted to use these or similar counterions along with conformationally labile and flexible Cn 2-α,ω-dicarboxylates, and the first results are reported here. Nine complexes including C6 2- to C9 2-and either PPh4 + , PPh3Me + , [Ni(cyclam)], 2+ [Cu(R,S-Me6cyclam)] 2+ or simple K + ions as counterions to anionic uranyl polymers, have been obtained and characterized by their crystal structure and, in most cases, their emission spectrum and luminescence quantum yield in the solid state. While most of them crystallize as one-or two-dimensional (1D or 2D) coordination polymers, one is a polycatenated three-dimensional (3D) framework, while another provides an example of hydrogen bonded interpenetration. Network entanglements in uranyl chemistry are now well documented, [START_REF] Thuéry | Structural Consequences of 1,4-Cyclohexanedicarboxylate Cis/Trans Isomerism in Uranyl Ion Complexes: From Molecular Species to 2D and 3D Entangled Nets[END_REF] and several examples have been reported recently. [START_REF] Thuéry | Structural Consequences of 1,4-Cyclohexanedicarboxylate Cis/Trans Isomerism in Uranyl Ion Complexes: From Molecular Species to 2D and 3D Entangled Nets[END_REF][START_REF] Liu | Interpenetrated Uranyl-Organic Frameworks with bor and pts Topology: Structure, Spectroscopy, and Computation[END_REF][START_REF] Wang | Large-Pore Layered Networks, Polycatenated Frameworks, and Three-Dimensional Frameworks of Uranyl Tri(biphenyl)amine/Tri(phenyl)amine Tricarboxylate: Solvent-/Ligand-Dependent Dual Regulation[END_REF][START_REF] Zhang | Semirigid Tripodal Ligand Based Uranyl Coordination Polymer Isomers Featuring 2D Honeycomb Nets[END_REF] It is notable that one of the first cases of polycatenation in a uranyl ion complex was found with C6 2-as ligand, [START_REF] Borkowski | Crystal Engineering with the Uranyl Cation II. Mixed Aliphatic Carboxylate/Aromatic Pyridyl Coordination Polymers: Synthesis, Crystal Structures, and Sensitized Luminescence[END_REF] other cases having later been reported involving C7 2-and C13 2-. [START_REF] Thuéry | Counterion-Induced Variations in the Dimensionality and Topology of Uranyl Pimelate Complexes[END_REF][START_REF] Thuéry | From Helicates to Borromean Links: Chain Length Effect in Uranyl Ion Complexes of Aliphatic α,ω-Dicarboxylates[END_REF] EXPERIMENTAL SECTION Syntheses. Caution! Uranium is a radioactive and chemically toxic element, and uranium-containing samples must be handled with suitable care and protection. UO2(NO3)2•6H2O (depleted uranium, R. P. Normapur, 99%) was purchased from Prolabo, and the carboxylic acids were from Aldrich. R,S-Me6cyclam (meso isomer of 7(R),14(S)-5,5,7,12,12,14-hexamethyl-1,4,8,11-tetraazacyclotetradecane) was prepared as described in the literature, [START_REF] Tait | Inorganic Syntheses[END_REF] and [Ni(cyclam)(NO3)2] as described in previous work. [START_REF] Thuéry | Ni(cyclam)] 2+ and [Ni(R,S-Me6cyclam)] 2+ as Linkers or Counterions In Uranyl-Organic Species with cis-and trans-1,2-Cyclohexanedicarboxylate Ligands[END_REF] Elemental analyses were performed by MEDAC Ltd. at Chobham, UK. For all syntheses of uranyl ion complexes, the mixtures in demineralized water/organic solvent were placed in 10 mL tightly closed glass vessels and heated at 140 °C under autogenous pressure.

[Cu(R,S-Me6cyclam)(NO3)2] diastereoisomers. A solution of Cu(NO3)2•3H2O (0.48 g, 2.0 mmol) in methanol (5 mL) was added to a stirred solution of R,S-Me6cyclam (0.58 g, 2.03 mmol) in methanol (5 mL), the initially partly gelatinous precipitate rapidly transforming into a partly soluble, violet, crystalline solid. Precipitation was improved by the addition of diethylether (30 mL) to give 0.57 g of violet crystals of N(R,S,R,S)-[Cu(R,S-Me6cyclam)(NO3)2] (61% yield) but the filtrate retained quite a deep blue-violet colour and the additional 30 mL of diethylether used to wash the crystals on the filter did not seem to cause further deposition, nor did addition of dichloromethane (50 mL). Evaporation of this filtrate/wash to dryness gave a blue residue which dissolved readily in water to provide large blue crystals of N(R,R,R,R;S,S,S,S)-[Cu(R,S-Me6cyclam)(NO3)2] (0.27 g, 35% yield) on evaporation at room temperature. The identification of the diastereoisomers was based on that of the analogous perchlorates [START_REF] Clay | An X-Ray Structural and Thermodynamic Investigation of the Blue and Red Forms of (C-meso-5[END_REF] and confirmed for the N(R,S,R,S) isomer by the present structural work.

[PPh4][UO2(C6)(NO3)] (1). H2C6 (15 mg, 0.10 mmol), UO2(NO3)2•6H2O (35 mg, 0.07 mmol), and PPh4Br (42 mg, 0.10 mmol) were dissolved in water (0.5 mL). Crystals of complex 1 were obtained within one week (36 mg, 63% yield based on U). Anal. Calcd for C30H28NO9PU: C, 44.18; H, 3.46; N, 1.72. Found: C, 43.95; H, 3.36; N, 1.79%.

[PPh4][UO2(C8)(NO3)] (2). H2C8 (18 mg, 0.10 mmol), UO2(NO3)2•6H2O (35 mg, 0.07 mmol), and PPh4Br (42 mg, 0.10 mmol) were dissolved in water (0.5 mL). Only a few crystals of complex 2 were obtained within two weeks, and the yield could not be determined.

[PPh3Me][UO2(C7)(HC7)] (3). H2C7 (16 mg, 0.10 mmol), UO2(NO3)2•6H2O (35 mg, 0.07 mmol), and PPh3MeBr (36 mg, 0.10 mmol) were dissolved in water (0.7 mL) and DMF (0.2 mL). Crystals of complex 3 were obtained within two months (13 mg, 30% yield based on the acid). Anal. Calcd for C33H39O10PU: C, 45.84; H, 4.55; N. Found: C, 45.55; H, 4.25%.

[PPh4][H2NMe2][(UO2)2(C7)3] (4). H2C7 (16 mg, 0.10 mmol), UO2(NO3)2•6H2O (35 mg, 0.07 mmol), and PPh4Br (42 mg, 0.10 mmol) were dissolved in water (0.5 mL) and DMF (0.2 mL). Crystals of complex 4 were obtained within one week (23 mg, 49% yield based on the acid). Anal. Calcd for C47H58NO16PU2: C, 40.32; H, 4.18; N, 1.00. Found: C, 40.25; H, 4.11; N, 1.04%.

[PPh3Me]2[(UO2)2(C8)3] (5). H2C8 (18 mg, 0.10 mmol), UO2(NO3)2•6H2O (35 mg, 0.07 mmol), and PPh3MeBr (36 mg, 0.10 mmol) were dissolved in water (0.8 mL) and DMF (0.2 mL). Crystals of complex 5 were obtained within two months (32 mg, 60% yield based on the acid). Anal. Calcd for C62H72O16P2U2: C, 46.22; H, 4.50. Found: C, 46.11; H, 4.42%.

[PPh3Me]2[(UO2)2(C9)3]⋅2H2O (6). H2C9 (19 mg, 0.10 mmol), UO2(NO3)2•6H2O (35 mg, 0.07 mmol), and PPh3MeBr (36 mg, 0.10 mmol) were dissolved in water (0.6 mL) and DMF (0.2 mL). Crystals of complex 6 were obtained within one month (24 mg, 43% yield based on the acid). Anal. Calcd for C65H82O18P2U2: C, 46.22; H, 4.89. Found: C, 46.31; H, 4.65%.

[Cu(R,S-Me6cyclam)][(UO2)2(C7)2(C2O4)]⋅4H2O (7). H2C7 (16 mg, 0.10 mmol), UO2(NO3)2•6H2O (35 mg, 0.07 mmol), and N(R,S,R,S)-[Cu(R,S-Me6cyclam)(NO3)2] (24 mg, 0.05 mmol) were dissolved in water (0.8 mL) and acetonitrile (0.2 mL). Crystals of complex 7 were obtained within one week (16 mg, 34% yield based on U). Anal. Calcd for C32H64CuN4O20U2: C, 28.17; H, 4.73; N, 4.11. Found: C, 28.03; H, 4.47; N, 4.10%.

[(UO2)2(C7)2(HC7)2Ni(cyclam)]⋅2H2O (8). H2C7 (16 mg, 0.10 mmol), UO2(NO3)2•6H2O (35 mg, 0.07 mmol), and [Ni(cyclam)(NO3)2] (20 mg, 0.05 mmol) were dissolved in water (0.4 mL). Only a few crystals of complex 8 were obtained within two weeks, and the yield could not be determined.

[(UO2)2K2(C7)3(H2O)]⋅0.5H2O (9). H2C7 (16 mg, 0.10 mmol), UO2(NO3)2•6H2O (35 mg, 0.07 mmol), and KNO3 (10 mg, 0.10 mmol) were dissolved in water (0.6 mL) and acetonitrile (0.2 mL). Crystals of complex 9 were obtained overnight (10 mg, 27% yield based on the acid).

Anal. Calcd for C21H33K2O17.5U2: C, 22.53; H, 2.97. Found: C, 23.17; H, 2.84%.

Crystallography. The data were collected on a Nonius Kappa-CCD area detector diffractometer 29 using graphite-monochromated Mo Kα radiation (λ = 0.71073 Å). The crystals were introduced into glass capillaries with a protective coating of Paratone-N oil (Hampton Research). The unit cell parameters were determined from ten frames, then refined on all data.

The data (combinations of ϕand ω-scans with a minimum redundancy of 4 for 90% of the reflections) were processed with HKL2000. [START_REF] Otwinowski | Processing of X-Ray Diffraction Data Collected in Oscillation Mode[END_REF] Absorption effects were corrected empirically with the program SCALEPACK. [START_REF] Otwinowski | Processing of X-Ray Diffraction Data Collected in Oscillation Mode[END_REF] The structures were solved by intrinsic phasing with SHELXT, [START_REF] Sheldrick | SHELXT -Integrated Space-Group and Crystal-Structure Determination[END_REF] expanded by subsequent difference Fourier synthesis and refined by full-matrix least-squares on F 2 with SHELXL-2014. [START_REF] Sheldrick | Crystal Structure Refinement with SHELXL[END_REF] All non-hydrogen atoms were refined with anisotropic displacement parameters. The hydrogen atoms bound to oxygen and nitrogen atoms were retrieved from difference Fourier maps when possible (see details below), and the carbon-bound hydrogen atoms were introduced at calculated positions. All hydrogen atoms were treated as riding atoms with an isotropic displacement parameter equal to 1.2 times that of the parent atom (1.5 for CH3, with optimized geometry). Crystal data and structure refinement parameters are given in Table 1. The molecular plots were drawn with ORTEP-3, [START_REF] Farrugia | WinGX and ORTEP for Windows: an Update[END_REF] and the polyhedral representations with VESTA. [START_REF] Momma | VESTA: a Three-Dimensional Visualization System for Electronic and Structural Analysis[END_REF] The topological analyses were conducted with TOPOS. 35 Special details, when present, are as follows.

Complex 4. The aliphatic chains are partly disordered over two positions which were refined with occupancy parameters constrained to sum to unity and restraints for bond lengths and angles. Complex 6. Extensive disorder affects two aliphatic chains. One of them, disordered over two positions related by symmetry, was refined as a complete molecule with 0.5 occupancy, and two atoms of the other were refined over two positions with occupancy parameters constrained to sum to unity. Many restraints on bond lengths and displacement parameters had to be applied, and a damping factor was used in the refinement due to the instability of atoms O3 to O6, located very near their images by symmetry due to disorder. The solvent water molecule is disordered over two positions and its hydrogen atoms were not found.

Twinning was detected with TwinRotMat (PLATON [START_REF] Spek | Structure Validation in Chemical Crystallography[END_REF] ), and was taken into account during the refinement.

Complex 7. The highest residual electron density peaks are located near uranium atoms, probably as a result of imperfect absorption corrections.

Complex 8. Two atoms of one aliphatic chain are disordered over two positions which were refined with occupancy parameters constrained to sum to unity. Restraints on bond lengths, angles and/or displacement parameters were applied for atoms in the disordered part, and also in the cyclam molecule which is probably affected by unresolved disorder. The highest residual electron density peaks are located near the disordered chain.

Complex 9. The solvent water molecule was given half-occupancy in order to retain an acceptable displacement parameter, and its hydrogen atoms were not found. The value of the refined Flack parameter was -0.006 (11). Luminescence Measurements. Emission spectra were recorded on solid samples using a Horiba-Jobin-Yvon IBH FL-322 Fluorolog 3 spectrometer equipped with a 450 W xenon arc lamp, double-grating excitation and emission monochromator (2.1 nm mm -1 of dispersion; 1200 grooves mm -1 ) and a TBX-04 single photon-counting detector. The powdered compounds were put into a quartz tube and pressed to the wall of the tube, and the measurements were performed using the right angle mode. An excitation wavelength of 420 nm, a commonly used point although only part of a broad manifold, was used in all cases and the emission was monitored between 450 and 650 nm. The quantum yield measurements were performed by using an absolute photoluminescence quantum yield spectrometer Hamamatsu Quantaurus C11347 and exciting the sample between 300 and 400 nm.

RESULTS AND DISCUSSION

Synthesis. Crystals of complexes 1-9 were grown under purely hydrothermal (1, 2 and 8) or solvo-hydrothermal conditions, with the organic cosolvent being N,N-dimethylformamide (3-6) or acetonitrile (7 and 9), at a temperature of 140 °C. Systematic attempts have been made for all possible diacid/counterion/solvent combinations, and the results reported herein correspond to the experiments which succeeded in giving single crystals of suitable quality for structure determination. The crystals obtained were deposited directly from the pressurised and heated reaction mixtures and not as a result of subsequent cooling. The uranium/ligand ratio was 7:10 in all cases, so as to favour the formation of an anionic species, and the expected ratio of 2:3 is retained in complexes 4-6 and 9, while it becomes 1:1 in 1, 2 and 7, and 1:2 in 3 and 8 (with one ligand monoprotonated). Apart from the countercations introduced during the synthesis, other charged moieties are present in some cases: nitrate ions are retained in 1 and 2, H2NMe2 + cations are present in 4, and oxalate ligands in 7. Dimethylammonium cations result from DMF hydrolysis, as very frequently observed in solvo-hydrothermal syntheses. [START_REF] Thuéry | Modulation of the Structure and Properties of Uranyl Ion Coordination Polymers Derived from 1,3,5-Benzenetriacetate by Incorporation of Ag(I) or Pb(II)[END_REF] In situ oxalate formation under hydrothermal conditions is also very frequent, [START_REF] Thuéry | Structural Variations in the Uranyl/4,4ʹ-Biphenyldicarboxylate System. Rare Examples of 2D → 3D Polycatenated Uranyl-Organic Networks[END_REF] and some in-depth studies of the mechanisms involved in particular cases have been reported. [START_REF] Andrews | In Situ Oxalate Formation during Hydrothermal Synthesis of Uranyl Hybrid Materials[END_REF][START_REF] Knope | Investigation of in Situ Oxalate Formation from 2,3-Pyrazinedicarboxylate under Hydrothermal Conditions Using Nuclear Magnetic Resonance Spectroscopy[END_REF] In the present experiments, it was found to be present only in the product obtained from the solution to which N(R,S,R,S)-[Cu(R,S-Me6cyclam)(NO3)2] had been added and thus it may have arisen from oxidative decomposition of the macrocycle. polymer provides evidence for both some CH⋅⋅⋅C(aromatic) ("CH⋅⋅⋅π") and more abundant CH⋅⋅⋅O hydrogen bonding [START_REF] Taylor | Crystallographic Evidence for the Existence of C-H⋅⋅⋅O, C-H⋅⋅⋅N, and C-H⋅⋅⋅Cl Hydrogen Bonds[END_REF][START_REF] Desiraju | The C-H⋅⋅⋅O Hydrogen Bond: Structural Implications and Supramolecular Design[END_REF] interactions beyond dispersion. Reciprocal CH⋅⋅⋅C(aromatic) interactions between cations with a P⋅⋅⋅P separation of 6.9044(13) Å may be considered to define a cation pair involved in a "phenyl embrace", [START_REF] Scudder | Crystal Supramolecular Motifs. Two-and Three-Dimensional Networks of Ph4P + Cations Engaged in Sixfold Phenyl Embraces[END_REF] 19) Å], and one of the C8 2-ligands is centrosymmetric and in its fully extended ttttt conformation, while the other is the curved gtgtt conformer. A 1D polymer is formed here also, but one in which doubly-bridged uranyl dimers are connected to one another by the extended, divergent ligands. A similar connectivity was previously found in a complex with C6 2- including diprotonated 1,2-bis(4-pyridyl)ethane, [START_REF] Borkowski | Crystal Engineering with the Uranyl Cation II. Mixed Aliphatic Carboxylate/Aromatic Pyridyl Coordination Polymers: Synthesis, Crystal Structures, and Sensitized Luminescence[END_REF] and in a complex with C10 2-including cucurbit [6]uril molecules. [START_REF] Thuéry | Uranyl Ion Complexation by Aliphatic Dicarboxylic Acids in the Presence of Cucurbiturils as Additional Ligands or Structure-Directing Agents[END_REF] The chains in 5 run along either [2 1 0] or [2 ī 0], alternate sheets Only one complex with the C9 In contrast to complexes 1-5, a two-dimensional (2D) assembly parallel to (0 1 0) is formed here, which has the point (Schläfli) symbol {6 3 } and the honeycomb hcb topology. This geometry has previously been found with C10 2-, C12 2-, and C13 2-, [START_REF] Thuéry | From Helicates to Borromean Links: Chain Length Effect in Uranyl Ion Complexes of Aliphatic α,ω-Dicarboxylates[END_REF][START_REF] Thuéry | Uranyl Ion Complexes with Long-Chain Aliphatic α,ω-Dicarboxylates and 3d-Block Metal Counterions[END_REF] with in all cases very distorted hexagonal cells, far from planar, as could be expected for such long and flexible ligands. The present case is no exception, the layers being deeply corrugated, particularly when viewed down the [1 0 ī] axis, and they are associated into pairs, with slight interdigitation visible along the same axis (Figure 6c). These pairs are then stacked with, here also, slight interdigitation apparent when the packing is viewed down the a axis (Figure 6d) or the c axis.

Crystal

As in complex 5, no parallel-displaced π-stacking interaction between cations is evident but The second compound including a 3d-block cation complex is dicarboxylate ligand is bridging bidentate at both ends, while the singly deprotonated one, which is partly disordered, is chelating through the carboxylate group, the carboxylic group being uncoordinated as in complex 3. Both ligands have irregular conformations, the fully deprotonated one being the gtgt conformer which is essentially a divergent connector. Disorder in the HC7 -ligands appears to involve the tgtg and tgtt conformers. Planar ribbon-like chains running along the a axis are formed, which contain doubly bridged uranyl dimers and pendent carboxylic acid groups on both edges. These ribbons are further connected to one another by the Ni(cyclam) 2+ moieties so as to form a 2D assembly parallel to (0 1 1), the ribbons being tilted with respect to the layer plane. The network is binodal (dicarboxylate ligand and uranyl cations), the nickel(II) cations being simple links, and it has the point symbol {4 2 .6 3 .8}{4 2 .6} and the V2O5 topological type. Due to the tilting of the ribbons within the layers, the carboxylic groups are directed toward the interlayer space; however, those pertaining to different layers do not face exactly each other so as to form double hydrogen bonds as in 3, but each of them is bound instead to a lattice water molecule which is itself linked to one carboxylate oxygen atom and to a carboxylic group of the adjacent layer (Figure 9), thus forming a water-mediated 3D network. The carboxylic groups and water molecules form a ring corresponding to the graph set descriptor 46,47 R4 4 (12), while larger R4 4 (24) rings are built by two water molecules and two HC7 -ligands. The amine atom N2 is hydrogen bonded to the bridging carboxylate atom O3 [N2⋅⋅⋅O3 2.917(6) Å, N2-H⋅⋅⋅O3 152°], a feature previously found in other carboxylate complexes including this counterion, [START_REF] Thuéry | Ni(cyclam)] 2+ and [Ni(R,S-Me6cyclam)] 2+ as Linkers or Counterions In Uranyl-Organic Species with cis-and trans-1,2-Cyclohexanedicarboxylate Ligands[END_REF] and the other amine group is bound to the lattice water molecule.

[(UO2)2(C7)2(HC7)2Ni(cyclam)]⋅2H2O (
The last compound in the present series, [(UO2)2K2(C7)3(H2O)]⋅0.5H2O ( 9), contains a counterion quite different from those in complexes 1-8, since it is a single alkali metal cation.

Although uranyl ion complexes with Cn 2-dicarboxylate ligands incorporating alkali or alkaline-earth metal cations have been reported for n = 4 (succinate), [START_REF] Novikov | Potassium and Magnesium Succinatouranilates -Synthesis and Crystal Structure[END_REF] and n = 5 (glutarate), 49- 52 for example (with formation of hydrogen bonded interpenetrated frameworks in one case 50 planar array is thus connected to three potassium cations, thus making a tetranuclear, trigonal building unit. The UO6K3 unit is nearly planar in the case of U1 (root mean square (rms) deviation 0.13 Å), but, in the case of U2, the potassium cation K2 involved in oxido bonding is displaced by 2.076(8) Å from the plane defined by the other 9 atoms (rms 0.30 Å). Overall, the coordination polymer formed is 3D, but it is worth examining the assembly formed by uranyl ions and dicarboxylate ligands alone. The latter is a uninodal 2D network parallel to (0 1 0) and with the point symbol {8 2 .10}, similar to that found in [NH4]2[(UO2)2(C7)3]⋅2H2O, which crystallizes in the same space group as 9, and with quite similar unit cell parameters. [START_REF] Thuéry | Counterion-Induced Variations in the Dimensionality and Topology of Uranyl Pimelate Complexes[END_REF] These networks are very convoluted, being built of central chains running parallel to one another and surrounded on either side by two sets of chains running in the transverse direction (Figure 11).

In both compounds, each network is entangled with its two neighbours along the b axis, resulting in 2D  3D parallel polycatenation. Uranyl and C7 2-ligands are thus sufficient for framework formation, and K + cations are further linkers between the individual 2D subunits, a role analogous to that of the ammonium cations in the former compound, which are hydrogen bonded to carboxylate oxygen atoms; this may be viewed as a consequence of the "pseudometal" behaviour of NH4 + , with properties akin to those of K + , [START_REF] Whiteside | Is Electronegativity a Useful Descriptor for the Pseudo-Alkali Metal NH4 ?[END_REF] as previously noted in the case of other uranyl carboxylate complexes. [START_REF] Thuéry | Tetrahedral and Cuboidal Clusters in Complexes of Uranyl and Alkali or Alkaline-Earth Metal Ions with rac-and (1R,2R)-trans-1,2-Cyclohexanedicarboxylate[END_REF] It is notable that the thick layers, with significant extension in the third dimension, found in 9 give rise to polycatenation, while the flatter hydrogen bonded layers in 3 result in interpenetration only (with no increase in dimensionality). With a KPI of 0.75, the framework has no significant free space. Luminescence properties. Emission spectra under excitation at 420 nm, a value suitable for the uranyl chromophore, [START_REF] Knope | Uranyl Sensitization of Samarium(III) Luminescence in a Two-Dimensional Coordination Polymer[END_REF] were recorded for complexes 1-7 and 9 in the solid state (Figure 12) (A spectrum could not be obtained for complex 8 due to the contamination of the small quantity of crystals with some amorphous material). Except for complex 7 for which complete quenching of uranyl luminescence is observed, as is frequent when d-block transition metal cations are present, probably due to the latter providing a nonradiative relaxation pathway, [START_REF] Kerr | Postsynthetic Rearrangement/Metalation as a Route to Bimetallic Uranyl Coordination Polymers: Syntheses, Structures, and Luminescence[END_REF] all the other spectra display the usual series of peaks associated with the vibronic progression corresponding to the S11 → S00 and S10 → S0ν (ν = 0-4) electronic transitions. Å in 3] in the present series. Another parameter that should be taken into account is the distance between the luminescent uranyl center and the electron-withdrawing phosphonium ions, which are potential photoluminescence quenchers, in complexes 1 and 3-6. In this regard, complex 1, which has the highest PLQY of the series, also has a minimum U⋅⋅⋅P separation of 6.8382(8) Å which is longer than any of those in complexes 3-6 [6.0169(7)-6.6330(4) Å], although the difference is rather small. However, it is no doubt an oversimplification to consider general correlations between quantum yields and such simple parameters.

CONCLUSIONS

Adding to our previous work on anionic uranyl ion complexes with α,ω-dicarboxylates using countercations of varying bulkiness and offering different possibilities of interactions with these anions (apart from dominating Coulombic forces), we have reported here the synthesis, crystal structure and, in most cases, luminescence properties of nine new complexes. In these compounds, crystallized under solvo-hydrothermal conditions, the Cn 2-ligands are of medium length, with n = 6-9, and three families of counterions are represented: PPh4 + /PPh3Me + , macrocylic ligand complexes of 3d-block metal cations, and, in one case, a cation, K + , labile with respect to its coordination sphere and flexible in its coordination geometry. Amongst other factors, there is systematic variation here in the accessibility toward lone pair donors of the centre on which the positive charge is formally located, P + in the phosphonium cations being inaccessible, Ni II and Cu II in their tetra-azamacrocycle complexes being capable of accepting up to two extra donors in a specific stereochemistry, and K I being completely accessible.

Overall, one-dimensional coordination polymers are the most frequent assemblies formed, being found in five complexes with PPh4 + /PPh3Me + (1-5) and in the complex including [Cu(R,S-Me6cyclam)] 2+ (7). The prevalence of chains in this series is in contrast to the higher level of aggregation (and sometimes uranyl oligomerization) found in neutral uranyl complexes with Cn 2-ligands having similar n values. [START_REF] Borkowski | Crystal Engineering with the Uranyl Cation I. Aliphatic Carboxylate Coordination Polymers: Synthesis, Crystal Structures, and Fluorescent Properties[END_REF][START_REF] Borkowski | Topological Evolution in Uranyl Dicarboxylates: Synthesis and Structures of One-Dimensional UO2(C6H8O4)(H2O)2 and Three-Dimensional UO2(C6H8O4)[END_REF] The separating effect of bulky counterions, which promotes formation of discrete, polynuclear species in other cases, [START_REF] Thuéry | A New Form of Triple-Stranded Helicate Found in Uranyl Complexes of Aliphatic α,ω-Dicarboxylates[END_REF][START_REF] Thuéry | Counterion-Controlled Formation of an Octanuclear Uranyl Cage with cis-1,2-Cyclohexanedicarboxylate Ligands[END_REF][START_REF] Thuéry | Closed Uranyl-Dicarboxylate Oligomers: A Tetranuclear Metallatricycle with Uranyl Bridgeheads and 1,3-Adamantanediacetate Linkers[END_REF] may be at play here, although it must be noted that some of these other cases involve ligands with significant degrees of conformational restriction. Notwithstanding the common low dimensionality in the present instances, there is some structural variety, with simple zigzag chains including terminal nitrate coligands in 1 and 2, curved ribbons with lateral uncomplexed carboxylic acid groups giving threefold parallel 2D interpenetrated hydrogen bonded networks in 3, ladderlike chains in 4 and 7, and a chain displaying an alternation of simple and double links in 5. A heavily corrugated honeycomb 2D assembly is found in complex 6, with C9 2-ligands and PPh3Me + counterions.

An increase in dimensionality also occurs when the counterions become part of the polymer itself, which occurs in the 2D assembly with C7 2-ligands and [Ni(cyclam)] 2+ axially bound to two carboxylate groups (complex 8) and in the 3D framework obtained from C7 2-and potassium cations (9). The latter has the particularity of containing uranyl carboxylate subunits displaying 2D  3D parallel polycatenation.

Except when one carboxylic acid group retains its proton and is uncoordinated, the Cn 2- ligands in the present series are all bis-chelating toward uranyl, with additional bridging of two potassium cations in complex 9, except in complex 8, in which the bridging bidentate µ2κ 1 O:κ 2 O' mode is observed (with two uranyl or one uranyl and one nickel(II) cations as coordinated species). Thus, the variations in the geometry of the assemblies formed can be seen as reflecting both the specific nature of the non-uranyl cation and the flexibility of the aliphatic chains which can adopt a large range of conformations with variable curvature. Given the expectation that rotational barriers to conformer interconversion in polymethylene chains should be small, [START_REF] Eliel | Stereochemistry of Organic Compounds[END_REF] probably smaller than CH⋅⋅⋅O interaction energies [START_REF] Desiraju | The C-H⋅⋅⋅O Hydrogen Bond: Structural Implications and Supramolecular Design[END_REF] and certainly smaller than coordinate bond energies, priority as structure-directing influences should be given to the various interactions of the non-uranyl cations with the UO6 entities formed by carboxylate binding. The sometimes marked differences in the interactions of similar species, such as seen between PPh4 + and PPh3Me + or [Cu(R,S-Me6cyclam)] 2+ and [Ni(cyclam)] 2+ in the present work, let alone differences in the pattern of interactions of a particular cation in different complexes, [START_REF] Thuéry | Counterion-Induced Variations in the Dimensionality and Topology of Uranyl Pimelate Complexes[END_REF][START_REF] Thuéry | Uranyl-Organic Coordination Polymers with trans-1,2-, trans-1,4-, and cis-1,4-Cyclohexanedicarboxylates: Effects of Bulky PPh4 + and PPh3Me + Counterions[END_REF][START_REF] Thuéry | Ni(cyclam)] 2+ and [Ni(R,S-Me6cyclam)] 2+ as Linkers or Counterions In Uranyl-Organic Species with cis-and trans-1,2-Cyclohexanedicarboxylate Ligands[END_REF] is, however, an indication that the predictability of cation influences is low.

Nevertheless, it is notable that [Ni(cyclam)] 2+ counterions, with their ability to bind to two axial donors, appear here also as suitable linkers between essentially planar uranyl-containing subunits. [START_REF] Thuéry | Ni(cyclam)] 2+ and [Ni(R,S-Me6cyclam)] 2+ as Linkers or Counterions In Uranyl-Organic Species with cis-and trans-1,2-Cyclohexanedicarboxylate Ligands[END_REF] In addition to the desirability of a structure containing potential reaction cavities, an important property in regard to any possible application of uranyl ion coordination polymers is their solid state PLQY. While phosphonium countercations appear to have an advantage over transition metal species in that they do not lead to complete luminescence quenching, the PLQY values obtained in the several systems we have investigated are all relatively low and the structural factors which need to be optimized to obtain higher values remain somewhat obscure.

Replacement of the phenyl groups on phosphorus with large saturated substituents, however, is a strategy which the present results indicate may be worthy of pursuit.

ASSOCIATED CONTENT Accession Codes

CCDC 1869722-1869730 contain the supplementary crystallographic data for this paper. These data can be obtained free of charge via www.ccdc.cam.ac.uk/data_request/cif, or by emailing data_request@ccdc.cam.ac.uk, or by contacting The Cambridge Crystallographic Data Centre, 12 Union Road, Cambridge CB2 1EZ, UK; fax: +44 1223 336033.

  along [0 1 0] in 1 and [3 0 1] in 2, with distances between adjacent uranium atoms of 10.4060(3) Å in 1 and 13.7353(5) Å in 2. The well-ordered aliphatic chains in 1 adopt a gauche-transgauche (gtg) conformation giving the ligand an S shape, with the two nearly parallel CH2COO - groups being separated by a central C-C bond nearly perpendicular to them, and thus being in two planes offset from one another, while the equally well-ordered chains of the nearly fully extended tgttt conformation in 2 are kinked so that, here also, the two nearly parallel carboxylate groups are offset with respect to one another. The anionic chains are organized into sheets, parallel to (1 0 0) in 1 and to (0 1 0) in 2, between which the countercations are located. Similar 1D polymers have previously been found in [Ag(bipy)2]2[UO2(C7)(NO3)]2,[START_REF] Thuéry | Counterion-Induced Variations in the Dimensionality and Topology of Uranyl Pimelate Complexes[END_REF] while linear chains with all nitrates on the same side are present in another form of the same complex.[START_REF] Thuéry | Ag I and Pb II as Additional Assembling Cations in Uranyl Coordination Polymers and Frameworks[END_REF]

Figure 1 .Figure 2 .

 12 Figure 1. (a) View of compound 1. Displacement ellipsoids are drawn at the 50% probability level. Symmetry codes: i = 1 -x, y + 1/2, 3/2 -z; j = 1 -x, y -1/2, 3/2 -z. (b) View of the packing with uranium coordination polyhedra colored yellow. Hydrogen atoms are omitted in both views.

Figure 3 .

 3 Figure 3. (a) View of compound 3. Displacement ellipsoids are drawn at the 50% probability level. Carbon-bound hydrogen atoms are omitted. Symmetry codes: i = x, 3/2 -y, z -1/2; j = x, 3/2 -y, z + 1/2. (b) Packing with chains viewed side-on and hydrogen atoms omitted. (c) and (d) Two views of the 1D polymer, down the a and c axes, respectively. (e) View of one three-fold interpenetrated hydrogen bonded layer with the 1D coordination polymers viewed end-on and hydrogen bonds shown as dotted lines (top), and simplified view in the same orientation with the three networks in red, blue and green and hydrogen bonds shown as pale blue rods (bottom). (f) Simplified view of the hydrogen bonded entanglement down the a axis.

Figure 4 .

 4 Figure 4. (a) View of compound 4. Displacement ellipsoids are drawn at the 50% probability level. Carbon-bound hydrogen atoms are omitted and the hydrogen bond is shown as a dashed line. Symmetry codes: i = x, y -1, z; j = 3/2 -x, -y, z; k = x, y + 1, z. (b) View of the 1D ladder-like chain. (c) Packing with chains viewed side-on. (d) Packing with chains viewed end-on. Only one position of the disordered atoms is shown in all views.

Figure 5 .

 5 Figure 5. (a) View of compound 5. Displacement ellipsoids are drawn at the 50% probability level. Symmetry codes: i = 1 -x, 1 -y, 1 -z; j = -x -1, 2 -y, 1 -z. (b) View of the 1D coordination polymer. (c) View of the packing. Hydrogen atoms are omitted in all views.

Figure 6 .Figure 7 .

 67 Figure 6. (a) View of compound 6. Displacement ellipsoids are drawn at the 40% probability level. Only one position of the disordered atoms is represented. Symmetry codes: i = 1 -x, y, 3/2 -z; j = 2 -x, y, 1/2 -z; k = 2x, y, 3/2 -z. (b) View of the honeycomb 2D network. (c) Interdigitation in the bilayers. (d) Packing with layers viewed edge-on. Both positions of the disordered aliphatic chains are shown and counterions are omitted in the last three views. Solvent molecules and hydrogen atoms are omitted in all views.

Figure 8 .

 8 Figure 8. (a) View of compound 8. Displacement ellipsoids are drawn at the 30% probability level. Carbon-bound hydrogen atoms are omitted and the hydrogen bonds are shown as dashed lines. Symmetry codes: i = 2 -x, 2 -y, 1 -z; j = x -1, y, z; k = x + 1, y, z; l = 2 -x, 1 -y, 2 -z. (b) View of the 2D network with uranium coordination polyhedra colored yellow and those of nickel(II) green. (c) Packing with layers viewed edge-on. Only one position of the disordered atoms is represented in all three views. (d) Simplified view of the 2D network (yellow, uranium nodes; green, nickel(II) links; blue, ligand nodes); same orientation as in (b).

Figure 9 .

 9 Figure 9. Hydrogen bonding network involving HC7 -ligands and water molecules in compound 8. Carbon-bound hydrogen atoms are omitted and the hydrogen bonds are shown as dashed lines. Symmetry codes: j = x -1, y, z; m = 1 -x, -y, 2 -z; n = -x, -y, 2 -z.

Figure 10 .

 10 Figure 10. (a) View of compound 9. Displacement ellipsoids are drawn at the 40% probability level. Symmetry codes: i = 3/2 -x, -y, z -1/2; j = x -1, y, z; k = x -1/2, 1/2 -y, 1 -z; l = 1 -x, y + 1/2, 3/2 -z; m = 1 -x, y -1/2, 3/2 -z; n = 2 -x, y -1/2, 3/2 -z; o = x + 1/2, 3/2 -y, 1 -z; p = x, y + 1, z; q = 3/2 -x, 1 -y, z -1/2; r = x -1/2, 3/2 -y, 1 -z; s = 3/2 -x, -y, z + 1/2; t = x + 1, y, z; u = x + 1/2, 1/2 -y, 1 -z; v = x, y -1, z; w = 3/2 -x, 1 -y, z + 1/2; x = 2 -x, y + 1/2, 3/2 -z.(b) and (c) Two views of the 3D framework with uranium coordination polyhedra colored yellow and potassium atoms shown as blue spheres. Solvent molecules and hydrogen atoms are omitted in all views.

Figure 11 .

 11 Figure 11. (a) and (b) Nodal representation of the uranyl-based 2D subunit in compound 9 (yellow, uranium; blue, centroid of dicarboxylate ligand). (c) Four of the polycatenated 2D arrays viewed edge-on.

Table 1 . Crystal Data and Structure Refinement Details

 1

		1	2	3	4	5
	chemical formula	C30H28NO9PU	C32H32NO9PU	C33H39O10PU	C47H58NO16PU2	C62H72O16P2U2
	M (g mol -1 )	815.53	843.58	864.64	1399.97	1611.19
	cryst syst	monoclinic	monoclinic	monoclinic	orthorhombic	monoclinic
	space group	P21/c	P21/n	P21/c	Pcca	P21/c
	a (Å)	15.0644(4)	9.2845(3)	14.1580(5)	29.4157(8)	9.4938(2)
	b (Å)	14.7060(4)	27.5528(13)	15.2308(5)	11.6583(3)	14.6791(4)
	c (Å) α (deg) β (deg) γ (deg)	14.2217(4) 90 104.4913(17) 90	13.0534(7) 90 110.848(3) 90	14.9688(3) 90 90.051(2) 90	14.3357(2) 90 90 90	21.9542(7) 90 100.437(2) 90
	V (Å 3 )	3050.40(15)	3120.6(3)	3227.84(17)	4916.2(2)	3008.93(14)
	Z	4	4	4	4	2
	T (K)	100(2)	100(2)	100(2)	100(2)	100(2)
	reflns collcd	96650	69451	84565	65010	140334
	indep reflns	5758	5884	6119	4656	5706
	obsd reflns [I > 2σ(I)]	5250	4741	5584	4122	5208
	Rint	0.029	0.056	0.030	0.020	0.029
	params refined	379	397	407	343	371
	R1	0.024	0.029	0.021	0.027	0.021
	wR2	0.062	0.063	0.052	0.068	0.052
	S ∆ρmin (e Å -3 ) ∆ρmax (e Å -3 )	1.043 -1.42 0.53	0.973 -1.14 1.67	1.033 -0.95 1.02	1.072 -1.19 0.97	1.051 -1.08 1.65
		6	7		8	9
	chemical formula M (g mol -1 )	C65H82O18P2U2 1689.30	C32H64CuN4O20U2 1364.47	C38H70N4NiO22U2 1469.75	C21H33K2O17.5U2 1119.73
	cryst syst	monoclinic	triclinic	triclinic		orthorhombic
	space group	C2/c	Pī	Pī		P212121
	a (Å)	16.3553(10)	11.6530(7)	10.3680(5)		9.5448(4)
	b (Å)	22.4056(16)	14.6666(8)	11.0593(6)		17.5103(8)
	c (Å) α (deg) β (deg) γ (deg)	18.3831(16) 90 100.475(5) 90	19.1956(8) 88.220(3) 83.038(3) 85.811(2)	12.4935(4) 64.222(3) 84.722(3) 72.292(2)		19.6040(10) 90 90 90
	V (Å 3 )	6624.2(9)	3247.0(3)	1227.51(10)	3276.5(3)
	Z	4	3	1		4
	T (K)	100(2)	100(2)	150(2)		100(2)
	reflns collcd	139310	173420	66617		73671
	indep reflns	6271	12308	4660		6214
	obsd reflns [I > 2σ(I)]	4638	10379	4419		5633
	Rint	0.063	0.053	0.047		0.027
	params refined	465	808	323		389
	R1	0.058	0.042	0.029		0.031
	wR2	0.128	0.096	0.074		0.085
	S ∆ρmin (e Å -3 ) ∆ρmax (e Å -3 )	1.152 -1.75 1.16	1.112 -3.06 5.39	1.212 -2.31 2.47		1.025 -1.58 1.05

  2-azelate ligand has been obtained in the present work,

	[PPh3Me]2[(UO2)2(C9)3]⋅2H2O (6). Here also, the unique uranyl ion is chelated by three
	carboxylate groups [U-O(oxido) 1.773(4) and 1.775(4) Å, U-O(carboxylate) 2.444(11)-

2.481(

14

) Å] (Figure

6

). Two carboxylate ligands have twofold rotation symmetry and the other is disordered around a twofold rotation axis (see Experimental Section); the ligand containing O7 and O8 is in its fully extended tttttt conformation, that containing the disordered atoms O3 to O6, although of irregular conformation, is divergent, but the third, including O9 and O10, which, like the second, has some gauche links, is bent at both ends so that the two nearly parallel carboxylate groups are pointing in the same direction. These different conformations result in quite different separations between uranyl centres

[15.3647(12)

, 11.7801(9) and 8.7447(8) Å].

  ), no example has been reported with larger n values. The asymmetric unit in 9, which crystallizes in the Sohncke group P212121, contains two uranyl cations, both chelated by three carboxylate

	groups [U-O(oxido) 1.763(8)-1.774(9) Å, U-O(carboxylate) 2.443(7)-2.508(7) Å], and two
	potassium cations in different environments (Figure 10). Atom K1 is bound to six carboxylate
	oxygen atoms from six different ligands [K-O 2.673(8)-2.738(8) Å], its environment being of
	irregular geometry, and atom K2 is bound to six carboxylate oxygen atoms [K-O 2.677(9)-
	3.095(8) Å] and one water molecule [2.690(10) Å]. K2 is also possibly bound to the uranyl
	oxido atom O3, with a distance of 3.462(9) Å which is longer than the usual values of ∼2.6-2.8
	Å; 22 however, this interaction appears to be significant from examination of the Hirshfeld
	surface of the potassium cation. The three inequivalent dicarboxylate ligands present adopt tttg,
	tggt and gttg conformations, linking uranyl ion centres with U⋅⋅⋅U separations of 10.8006(7),
	10.8374(6) and 9.5448

  [START_REF] Brachmann | Study of Uranyl(VI) Malonate Complexation by Time Resolved Laser-Induced Fluorescence Spectroscopy (TRLFS)[END_REF]

four main peaks are at 481-483, 501-504, 522-525, and 542-549 nm, these positions being typical of uranyl carboxylate complexes with six equatorial donors. [START_REF] Thuéry | Structural Consequences of 1,4-Cyclohexanedicarboxylate Cis/Trans Isomerism in Uranyl Ion Complexes: From Molecular Species to 2D and 3D Entangled Nets[END_REF] Figure 12. Emission spectra of compounds 1-7 and 9 in the solid state at room temperature, under excitation at a wavelength of 420 nm. All spectra are normalized except for that of complex 7.

Solid-state photoluminescence quantum yields (PLQYs) have been measured in the cases for which a sufficient quantity of pure complex could be isolated. The values found are 11.5 (1), 3.0 (3), 2.5 (4), 3.0 (5), 2.0 (6), and 3.5% (9), with a standard deviation of ca. ±2.5%.

All complexes thus feature a relatively low PLQY except for complex 1, which is 4 to 5 times more emissive (for comparison, the PLQY of uranyl nitrate hexahydrate measured under the same conditions is 24%). The last five values, for complexes 3-6 and 9, are smaller than those previously measured in similar conditions for other uranyl carboxylate complexes, 6% for

AUTHOR INFORMATION

Corresponding Authors

*E-mail: pierre.thuery@cea.fr. (P.T.) *E-mail: harrowfield@unistra.fr. (J.H.) ORCID Pierre Thuéry: 0000-0003-1683-570X Youssef Atoini: 0000-0003-4851-3713 Jack Harrowfield: 0000-0003-4005-740X

Notes

The authors declare no competing financial interest.

For Table of Contents Use Only

Structure-Directing Effects of Counterions in Uranyl Ion

Complexes with