Mediterranean countries provide a test case for examining the effectiveness of foreign capital in promoting economic growth. Focusing on the supply side of the economy, the econometric model answers two questions, using available panel data from 1960 to 1996: -does foreign aid have a positive impact on growth of per capita income? -does aid substitute or complete domestic savings? We find three significant results: -aid efficiency is indirect, it is dependent on the way it will be transmitted to other exogenous variables, specially to savings and to FDI and thus, on the efforts of the Mediterranean countries to change in-depth their productive structure; -aid must be differentiated, in its contents as well as in its objectives depending on the development model of the beneficiary country; -aid must be more regular. Aid allocation in south of the Mediterranean is often chaotic and caused by geopolitical logic.

Introduction

The subject of this paper is to answer two questions :

-does foreign aid have a positive impact on growth of per capita income ?

-does aid substitute or complete domestic savings ?

Econometric study of aid has always met with difficulties and hasn't produced very good results [START_REF] Islam | Foreign Aid and Economic Growth : an Economic Study of Bangladesh[END_REF], but new works allow to clarify the link between aid and growth in a lot of developing countries. They generally take into account a wide sample of poor or very poor economies. But, numerous middle income countries can't finance their growth either, in particular the countries south of the Mediterranean, and, in this case, the results of the econometric tests are probably quite different.

Our sample includes the three North African countries, Egypt, Syria, Jordan, Israel, Cyprus and Turkey. The tests use panel data on the long period . The first area is made up of the three Maghrebi countries, the second area of Egypt, Syria and Jordan and, the last one of Cyprus, Israel and Turkey. This cut-out follows the rationality of aid allocated to the southern Mediterranean economies.

All the variables were subjected to unit root tests (or, Dick and Fuller's tests) to avoid artificial regression problem. They are generally stationary once they were differentiated. Estimation methodology by Ordinary Least Squares (OLS) respect temporal behaviour of the variables. The T-ratio linked to each coefficient are indicated into brackets. Finally, we have taken into account dynamics, considering endogenous lagged variable like an exogenous in order to reveal some steadiness in growth rate and foreign capital flows.

The initial model

We test the effective contribution of official assistance towards development of the southern countries in two different ways:

-first, the direct effect of aid on growth (ODA) i on per capita income growth; -secondly, the indirect effect of aid on development through the other exogenous variables;

This analysis based on " expansion methodology " allows to show aid-growth link fluctuate according to its economic context ; i.e. its saving rate, FDI and exports (Casetti, 1991). The parameters of the initial model are redefined to insert ODA in the first equation ii :

(1) GGDPC = a0 + a1 S + a2 FDI + a3 POP + a4 X with GGDPC, the Gross Domestic Product per capita growth, S, the saving rate, FDI, the FDI income ratio, POP, the population growth rate and X, the export growth rate ;

(2) a0 = b00 + b10 ODA a1 = b01 + b11 ODA a2 = b02 + b12 ODA a4 = b04 + b14 ODA where ODA represents the aid income ratio ;

(3) GGDPC = b00 + b10 ODA + a1 S + a2 FDI + a3 POP + a4 X ;

(4) GGDPC = a0 + b01 S + b11 S.ODA + b02 FDI + b12 FDI.ODA + a3 POP + b04 X + b14 X.ODA The first equation results from neo-classic theory on growth in an open economy within domestic savings and foreign aid finance capital accumulation necessary to reach the expected growth.

The initial model to test is :

GGDPCt =  +  GGDPCt-1 +  St, t-1 +  POPt, t-1 +  FDIt, t-1 +  Xt, t-1
The results in chart 1 show that all the exogenous variables initially defined have a positive impact on growth. FDI contribution towards growth is essentially due to their settlement in primary and manufacturing sectors in the North African countries. The case of exports is similar ; the positive export growth relationship comes from the substantial openness of most of the southern countries and from the exports of gas and oil products. On the other hand, the coefficient tying per capita income growth and savings in the second area, masks capital flies in Jordan where saving rate is more often than not negative on the period.

Chart n°1 : initial growth model (* : gives the T-ratios)

Furthermore, the demographic variable is assumed curbing growth in the countries south of the Mediterranean. However, reality seems to be more different. Most of these countries have achieved substantial efforts as far as birth control is concerned, but, they are not so obvious in the Maghrebi countries. One can think that a positive coefficient linked to the POP variable is not always a wholesome growth guarantee, as far as population growth curbs capital labour substitution and, consequently, technological renewal.

We now integrate aid direct impact on per capita income growth in order to see if capital flows change the previous results and if they have a positive contribution on development. While some economists suppose that growth rate increases with investment rate, itself being improved with foreign aid, others show that investment increase is not sufficient to reach a high level of growth and aid can lead a rise of aggregated consumption as well as investment.

The aid direct effect

GGDPCt =  +  GGDPCt-1 +  St, t-1 +  POPt, t-1 +  FDIt, t-1 +  ODAt,t-1 +  Xt, t-1
with ODA : ODA share in GDP (in %).

The results in chart 2 show that :

-in the case of the whole sample as well as of the different areas, aid has a significantly negative impact on per capita income growth on the long period. It could be explained by the fact that foreign capital flows finance emergency projects and short term requirements or, they are used in sectors with weak spillover effects on economy ; -on the other hand, domestic savings and exchange openness have always a positive impact on development, domestic capacities and trade being more efficient than Official Development Assistance. This result reinforces the " Trade-No Aid " theory ;

-finally, if one distinguish ODA from FDI, aid can have a lagged effect on growth whereas FDI have a short term one. In fact, Assistance is used to finance infrastructure projects with a weak immediate impact because of their strong capital output ratio. Direct investment, on the other hand, is directly tied to technology transfers. The extent the southern countries take advantage of this growth potential depends on their political stability and/or their economic system.

If we only consider aid direct effect on growth, as for the most part of the studies, the results with panel data show that there is an opposite relationship between aid and per capita income growth. Does the insertion of aid indirect effect change the previous outcomes ? Chart n°2 : Aid direct effect on growth (* : gives the T-ratios)

The aid indirect effect

Aid is supposed to have a positive impact on growth through other exogenous variables. The equation to test is :

GGDPCt =  +  GGDPCt-1 +  St,t-1 +  S.ODAt, t-1 +  POPt,t-1 +  FDIt,t-1 +  FDI.ODAt, t-1 +  Xt,t-1 +  X.ODAt, t-1

Chart n°3 : Aid indirect effect on growth (* : gives the T-ratios)

When interactions between ODA and other explanatory variables are taken into account, we note that :

-savings, FDI and exports always have a significant direct effect on per capita income growth ; -the coefficient linked to the interaction between aid and savings is generally positive. Two explanations can be stated : aid is either allocated to higher saving rate countries, which seems the case of countries in areas 1 and 3 or completes domestic savings. It thus reinforces its significant positive contribution to development. One note that in the case of North African countries, aid has an immediate positive impact on domestic accumulation but it becomes negative in the medium and long term : there is either a substitution effect between these variables, or savings, too much concentrated on a single sector, the gas and oil one, have no effects on the productive structure as a whole ; -finally, aid is either allocated to higher FDI rate countries, or, it takes the place of foreign private investments. If we compare FDI and aid development, we opt for the first explanation iii .

Conclusion

At the end of this study, it isn't possible to say that aid is efficient or not, but we can conclude that :

-aid efficiency is indirect, it is dependent on the way it will be transmitted to other exogenous variables, specially to savings and to FDI and thus, on the efforts of the Mediterranean countries to change in-depth their productive structure ; -aid must be differentiated, in its contents as well as in its objectives depending on the development model of the beneficiary country. Instead of aiming at a structural adjustment logic, aid must tend to reinforce the density of the productive system and to bring it towards people consumption needs ; -aid must be more regular. Aid allocation in south of the Mediterranean is often chaotic and caused by geopolitical logic (Teboul and Moustier, 2000). This lack of rationality probably leads to unstable econometric relationships on time. TRUMBULL, W.N. and WALL, H.J. (1994)

n°1 : initial growth model (* : gives the T-ratios)

  , Estimating Aid Allocation Criteria with Panel Data, The Economic Journal, 104, 876-882. WHITE, H. (1992), The Macroeconomic Impact of development Aid : a Critical Survey, The Journal of Development Studies, 28, 163-240.

		Chart Cst GGDPCt-1	S		FDI	X	POP
					t	t-1		t	t-1	t	t-1	t	t-1
	Panel	0.016		0.104	0.439	0.157	1.092	0.049	R2 :
		(3.64)*	(1.84)	(5.25)	(1.88)	(2.80)	(2.34)	0.199
									DW :
									2.01
	Area 1	0.011						1.418	0.076	3.749	R2 :
		(1.88)						(2.72)	(2.46)	(4.23)	0.262
									DW :
									2.18
	Area 2	0.012		0.333	0.436				R2 :
		(1.76)		(3.57)	(4.58)				0.245
									DW :
									2.17
	Area 3	0.025			0.677				0.107	R2 :
		(2.91)			(3.00)				(2.27)	0.256
									DW :
									1.82
		Cst	GGDPCt-	ODA		S	FDI	X	POP
				1
					T	t-1	t		t-1	t	t-1	t	t-1 t t-1
	Panel 0.015		0.097	-0.427		0.395	0.142	1.057	0.061	R2 : 0.218
		(3.36)		(1.72)	(-2.74)		(4.69)	(1.71)	(2.73	(2.86)	DW : 2.01
		*							)
	Area	0.011			-0.898				1.384	0.052	R2 : 0.176
	1	(1.78)			(-2.53)				(2.51	(1.63)	DW : 2.35
									)
	Area	0.012		0.321	-0.275		0.403	R2 : 0.265
	2	(1.73)		(3.45)	(-1.68)		(4.18)	DW : 2.14
	Area	0.024			-1.226	0.703	0.117	R2 : 0.280
	3	(2.70)			(-1.75)	(3.07)	(2.41)	DW : 1.87

Chart n°2 : Aid direct effect on growth (* : gives the T-ratios)

Chart n°3 : Aid indirect effect on growth

		Cst	GGDPCt-1	S		SA	FDI		FDIA		X			EA	POP
				t	t-1	t	t-1	t	t-1	t	t-1	t	t-1	t	t-1	t	t-1
	Panel	0.021		0.352		7.448		0.962		-25.81		0.046
	R2 : 0.232	(4.99)*		(4.65)		(4.60)		(2.59)		(-2.03)		(2.27)
	DW : 1.86
	Area 1	0.010				36.91	-24.04	1.454				0.077
	R2 : 0.347	(1.85)				(4.43)	(-3.21)	(2.95)				(2.69)
	DW : 1.89
	Area 2	0.017	0.201	0.397		5.472				-22.69
	R2 : 0.321	(2.45)	(2.06)	(4.32)		(3.18)				(-1.69)
	DW : 2.06
	Area 3	0.025		0.656			24.65					0.117
	R2 : 0.273	(2.89)		(2.86)			(1.44)					(2.40)
	DW : 1.91
	Notes

(* : gives the T-ratios)


S Tanashkin

M N Chernodub

V A Goy

A V Molochkov

A S Tanashkin

Casimir boundaries, monopoles, and deconfinement transition in 3+1 dimensional compact electrodynamics

Casimir boundaries, monopoles, and deconfinement transition in 3+1 dimensional compact electrodynamics

I. INTRODUCTION

Physical bodies affect fluctuations of quantum fields and modify dispersion relations of quantum fluctuations in the vacuum around them. This phenomenon is the essence of the Casimir effect [1], which predicts that the energy of vacuum fluctuations is modified by the presence of physical bodies [START_REF] Bordag | Advances in the Casimir effect[END_REF][START_REF] Milton | The Casimir effect: physical manifestations of zero-point energy[END_REF]. Moreover, the shift in the energy of virtual particles has a fundamental physical consequence as the Casimir effect results in a tiny force between neutral objects [START_REF] Casimir | [END_REF] that can be detected experimentally [5][6][7]. The Casimir effect is one of the demonstrations of the importance of vacuum fluctuations and the physical significance of the mysterious vacuum energy.

Most straightforwardly, the Casimir effect reveals itself in non-interacting field theories. However, even in the absence of interactions, the shift in vacuum energy sets a complex analytical problem apart from a few simplest geometries of physical bodies. Therefore, the Casimir effect is usually studied using, for example, the analytical proximity force approximation [8], or utilizing numerical tools [9] that include the world line approaches [10] and first-principle methods of lattice gauge theory [11][12][13][14][15][16][17][18].

In the presence of (self-) interactions of fields, for example, in quantum electrodynamics (QED), the calculations of the Casimir interaction become much more involved. However, in the experimentally relevant cases, the effect of interactions of fundamental fields is very small due to the weakness of the QED coupling constant. The electron-photon coupling affects the Casimir-Polder force in the second-order of the perturbation theory, thus making this contribution undetectable with current experimental techniques [19].

In strongly coupled field theories, interactions can substantially change the magnitude of the Casimir force and, unexpectedly, modify the vacuum structure of the theories themselves. In theories with dynamical matter fields, the presence of reflective boundaries can affect vacuum condensates and generate new (and modify exist-ing) phase transitions such as the chiral phase transition in the four-fermion effective field theory [20][21][22]. Interactions can also change the sign of the Casimir-Polder force in fermionic systems with condensates [23], as well as in the CP N -1 model [24] (see also [25]).

The Casimir geometry can also affect non-perturbative phenomena associated with the gauge (vector) fields. For example, vacua of compact Abelian U(1) gauge theory and non-Abelian SU(2) gauge theory in two spatial dimensions lose the confining property at sufficiently small separations between, respectively, perfectly metallic and chromometallic plates [17,18]. In compact Abelian theory, the deconfining transition is associated with the binding transition of the Abelian monopoles, which emerges in the monopoles plasma due to the presence of the boundaries [17].

We study the effect produced by two closely-spaced, perfectly-conducting boundaries on the phase structure of 3+1 dimensional compact U(1) gauge theory. The study may be relevant to the MIT bag model, which treats the hadron as a (spherical) deconfinement region separated from the confining exterior by a (reflective) wall [26,27]. Thermodynamically, the wall is supported from the collapse by a Casimir pressure of this finitevolume system [28][29][START_REF] Horowitz | 50th International Symposium on Multiparticle Dynamics[END_REF]. We also investigate the simplest statistical properties of the Abelian monopoles in the space between the plates since these topological objects are responsible for the confinement of electric charges and mass gap generation in the compact U(1) gauge theory [START_REF] Stack | [END_REF]32].

The structure of this paper is as follows. In Section II we describe the lattice model, the monopoles, and the definition of the perfectly metallic Casimir boundary conditions. Then, in Section III we show how the Casimir plates affect the monopoles. Next, we use the monopole properties to determine the phase diagram of the model. The deconfinement order parameter in the inter-plate space are discussed in Section IV. The last section is devoted to our conclusions.

II. COMPACT ELECTRODYNAMICS ON THE LATTICE, CASIMIR PLATES AND MONOPOLES

A. The model

We study a 3+1 dimensional compact U(1) gauge theory in lattice regularization suitable for numerical simulations. The calculations in thermal equilibrium are performed, after Wick rotation, in four Euclidean spacetime dimensions. Below we briefly discuss the model, topological defects, and the Casimir boundary conditions, following the discussion in 2+1 dimensional case [15,16] closely.

The compact U(1) gauge model describes the dynamics of the lattice gauge (photon) field θ x,µ ∈ [-π, +π) which is defined on the elementary links l = {x, µ} set by its starting point x and the direction µ. In the continuum limit, a → 0, the lattice field θ xµ = aA µ (x) is related to the continuum gauge field A µ (x) and the lattice spacing a. The lattice analogue of the field-strength tensor

F µν ≡ ∂ µ A ν -∂ ν A µ is played by the plaquette angle, θ Px,µν = θ x,µ + θ x+μ,ν -θ x+ν,µ -θ x,ν , (1)
constructed from the link fields (link angles) θ x,µ . Each plaquette P ≡ P x,µν is set by the position x of one of its corners and by two vectors of the plaquette plane, µ < ν, with the axes labeled by the indices µ, ν = 1, . . . , 4. The indices µ = 1, 2, 3 correspond to spatial directions while µ = 4 marks the imaginary Euclidean time.

In the continuum limit, the plaquette angle (1) reduces to its continuum analogue θ Px,µν = a 2 F µν (x) + O(a 4 ) for small (perturbative) fluctuations of the photon field. In addition to the perturbative fluctuations, the model also possesses the topological configurations of the gauge fields, the Abelian monopoles, which correspond to large variations of the lattice gauge field θ x,µ ∼ 1. These configurations are singular in the continuum limit.

The action of the lattice model,

S[θ] = β P (1 -cos θ P ) , (2)
is given by the sum over all elementary lattice plaquettes P . For configurations without monopoles, the lattice action (2) becomes the standard photon action if one associates the lattice coupling constant β = 4/e 2 with the electric charge e. In the presence of the Abelian monopole singularities, the continuum action becomes more complicated as it includes singular Dirac sheets attached to the worldlines of the Abelian monopoles. The continuum formulation of the compact QED has been briefly discussed in one of our previous papers [16]. The model (2) is also called the "compact" model because the Abelian gauge group of the theory corresponds to a compact manifold, S 1 . The action is invariant under discrete shifts of the plaquette variable, θ P → θ P + 2πn P with an integer n P ∈ Z. This invariance implies that two lattice field strengths θ P and θ ′ P = θ P + 2πn with n ∈ Z are physically equivalent to each other, thus reducing the physical gauge group to a circle, R/Z ∼ S 1 . The same symmetry is also applied to the gauge field itself, θ x,µ → θ x,µ + 2πk x,µ with k x,µ ∈ Z.

B. Magnetic monopoles and electric confinement

The compactness of the model naturally leads to the appearance of singular configurations of the gauge field, the Abelian monopoles. In the continuum limit, the mentioned 2π shifts, which leave invariant the lattice action (2), correspond to the physically unobservable displacements of the singular Dirac sheets (i.e., the worldlines of the Dirac strings attached to the Abelian monopoles). The end-points of the open Dirac strings correspond to the trajectories of the Abelian monopoles, which are physical, gauge-invariant topological defects.

The monopoles are particle-like objects in the (3+1)dimensional compact electrodynamics. On the lattice, the monopole current j x,µ can be determined via a finitedifference divergence of the physical part

θP = θ P + 2πk P ∈ [-π, π) , k P ∈ Z , (3)
of the lattice field-strength tensor θ P . The monopole trajectory corresponds to a collection of three-dimensional cubes C x,µ which contain a nonzero magnetic charge, j x,µ ̸ = 0:

j x,µ = 1 2π P ∈∂Cx,µ θP ∈ Z . (4)
Here the sum goes over all elementary sides P of the cube C x,µ , and the index µ specifies the local direction of the monopole current. The index µ is normal to the three axes that form the 3d cube C x,µ . For example, if j x,4 ̸ = 0, then the corresponding 3-cube is a spatial cube which contains a static segment of the monopole trajectory.

In the continuum limit, the sum in Eq. ( 4) reduces to the divergence of the magnetic field, thus signaling violation of the Bianchi identities for singular field configurations, ϵ µναβ ∂ ν F αβ ̸ = 0. The monopole trajectory (4) forms a closed loop defined at the dual hypercubic lattice (for a review, see Ref. [START_REF] Chernodub | NATO Advanced Study Institute on Confinement, Duality and Nonperturbative Aspects of QCD[END_REF]).

It is convenient to define the global monopole density,

ρ = 1 Vol 4 x,µ |j x,µ | , (5)
where the sum is performed over the volume Vol 4 of the four-dimensional hypercubic lattice. We will also calculate the density (5) and the related quantities in restricted volumes (in between the Casimir plates).

In lattice gauge theories, the Abelian monopoles have been intensively probed for their possible role in the phenomenon of the charge confinement, which is, presumably, closely related to the color confinement in non-Abelian gauge theories such as QCD [START_REF] Chernodub | NATO Advanced Study Institute on Confinement, Duality and Nonperturbative Aspects of QCD[END_REF][START_REF] Mandelstam | [END_REF][35]. The monopole condensation in an Abelian gauge theory leads to linear confinement of the electric charges because the monopole condensate squeezes the electric flux emanating from the electric charges into a thin electric tube. The tube plays the role of a confining string. Since the string is a linear object with a constant energy density σ per string length, the increasing distance R between the particle-antiparticle pair leads to a linearly rising potential V (R) ≃ σR at large distances. The dimensionful parameter σ has the sense of string tension.

This confining mechanism is similar (and dual) to the formation of the Abrikosov vortices in superconductors, where the electrically charged condensate of electron Cooper pairs squeezes the magnetic flux into thin vortices. If a monopole-anti-monopole pair were immersed into the superconductor, it would be confined due to the appearance of the Abrikosov vortex stretched between the constituents of the pair. The mechanism of the charge confinement based on the monopole condensation is often called the dual superconductor mechanism [START_REF] Mandelstam | [END_REF]35].

The dual superconductor mechanism was shown to work in the four-dimensional compact electrodynamics, which possesses the straightforward and unambiguous definition (4) of the Abelian monopoles [36]. The onset of the monopole condensation, related to percolation of monopole trajectories [37], is a well-defined phenomenon in this model. More complicated, non-Abelian Yang-Mills theories were also shown to possess the dual superconductivity phenomenon in low-temperature, confining phase [37][38][39][40][41][42].

C. Casimir plates for compact gauge fields

In (3+1) dimensions, the Casimir problem is defined, in general, for three-dimensional physical materials possessing two-dimensional surfaces. If the surfaces are made of an ideal metal, then two tangential (to the surface at each point) components of the electric field and a normal component of the magnetic field vanish. These boundary conditions can be written in a covariant form:

ε µνλσ F νλ (x)v σ (x) = 0 , µ = 1, . . . 4 , (6)
where

v µ (x) = ε µνλσ d 3 ξ ∂ xν ∂ξ 1 ∂ xλ ∂ξ 2 ∂ xσ ∂ξ 3 δ (4) x -x( ⃗ ξ ) , (7)
is the dual 3-volume element of the world sheet of the surface. The latter is described by the vector function xµ = xµ ( ⃗ ξ ) parameterized by the three dimensional vector ⃗ ξ = (ξ 1 , ξ 2 , ξ 3 ). In our paper, we consider two static flat plates normal to the x 1 axis set at the positions x 1 = l 1 and x 1 = l 2 as shown in Fig. 1. For each plate, the local volume element of the corresponding world volume (7) is given by the following formula where the parameter a labels the surfaces. To derive the above formula, one can use the parameterization of the ath surface as follows: xµ a = (l a , ξ 1 , ξ 2 , ξ 3 ). Consequently, the covariant conditions (6), reduce to the three conditions that include the normal component of the magnetic field and two tangential components of the (Euclidean) electric field, respectively:

ν µ (x) = δ µ,1 δ(x 1 -l a ) , a = 1, 2 , (8)
B 1 ≡ F 23 (x) x1=la = 0 , (9)
E 2 ≡ F 24 (x) x1=la = 0 , (10)
E 3 ≡ F 34 (x) x1=la = 0 . (11)
The definition of the lattice field strength tensor (1) and its physical part (3), imply that the lattice Casimir conditions (11) read as follows [15,16]:

cos θ x,µν x1=la = 1, (µ, ν) = (23, 24, 34) , (12)
for all possible (x 2 , x 3 , x 4 ) and fixed x 1 = l a with a = 1, 2.

The simplest way to implement the boundary condition (12) in the path integral formalism is to add a set of Lagrange multipliers to the standard action (2):

S ε [θ] = P β P (ε) cos θ P , (13)
To this end, we modify the lattice plaquette couplings β P → β P (ε). The inhomogeneous coupling

β P (ε) = β 1 + (ε -1) δ P,V . (14)
is a function of the dielectric permittivity ε of the Casimir plates. Here we used the notation V to denote the collection of plaquettes P x,µν belonging to the world volume of the plates. The plates are effectively absent if ε = 1 while in the limit ε → ∞, the components of the physical lattice field-strength tensor (3) vanish at the word-volume of the plates and, consequently, the lattice condition ( 12) is satisfied. In this limit, the plates become perfectly metallic [START_REF] Bordag | Advances in the Casimir effect[END_REF].

We perform our simulations on a 24 4 lattice corresponding to a zero-temperature model. To generate and update gauge field configurations, we used the Monte-Carlo heatbath algorithm [START_REF] Gattringer | Quantum chromodynamics on the lattice[END_REF][START_REF] Boudreau | Applied Computational Physics[END_REF]. For each point, set by gauge coupling constant β and distance between plates R, we generated 7.5 × 10 5 trajectories. The first 10 5 configurations are omitted to achieve thermalization.

III. MONOPOLES AND CASIMIR PLATES

A. Monopole density in the absence of the plates It is well known that in the strong coupling region of the theory, β ≲ 1, the monopole trajectory forms a dense percolating cluster [37]. This cluster corresponds to the monopole condensate, which, according to the dual superconducting scenario, produces the confinement of oppositely charged electric test particles. The property of percolation implies that any of two points in space have a nonzero probability of being connected by a monopole trajectory. Furthermore, the infrared nature of the monopole condensate implies that at infinitely large separation between the points, the nonzero percolation probability stays constant as the distance between the points increases.

In the weak coupling regime, β ≳ 1, the percolation cluster breaks down, thus signaling that the monopole condensate disappears. The confinement property is, consequently, lost. The confining stronglycoupled regime and the weakly coupled deconfining phase are separated by the first-order phase transition [START_REF] Arnold | [END_REF]46]. While we do not study the percolation properties of the monopole cluster in our paper, the transition point can be deduced from the behavior of the much simpler quantity, the monopole density (5).

Before performing calculations of Casimir effects, we consider case of the homogeneous lattice in the absence of metallic plates. In Fig. 2, we show the monopole density ρ, the corresponding susceptibility,

χ ρ = ⟨ρ 2 ⟩ -⟨ρ⟩ 2 , ( 15
)
and the Binder cummulant (with ρ = ρ),

B ρ = ⟨ρ 4 ⟩ ⟨ρ 2 ⟩ 2 -3 , (16)
as the functions of the lattice coupling β. The quantities ( 15) and ( 16) characterize the fluctuations of the monopole density ρ.

The monopole density diminishes with the increase of the coupling constant β. The position of the deconfining phase transition,

β c = 1.010(1), (no plates) , (17)
is well visible from the discontinuity in the monopole density, Fig. 2(a). The discontinuity -which is pertinent to the first-order phase transition -coincides with high

B. Monopole properties with Casimir plates

Perfectly conducting Casimir plates are introduced via the inhomogeneous coupling ( 14) which serves as the Lagrange multiplier that reduces physical fluctuations of the gauge field at the plates. In practice, we take the sufficiently large value of the dielectric constant, ε = 10 3 , which corresponds to the asymptotically large coupling constant at the plates, β P → ∞. The coupling constant in bulk (outside and inside the plates) is fixed to take a homogeneous value β. We consider the separations R ≡ |l 1 -l 2 | = 1, . . . , 8 between the plates. We immediately notice that closely separated plates affect the monopoles between them. The effect is readily visible in the examples of the typical monopole configurations shown in Fig. 3 for two values of the bulk coupling constant β. The plates tend to diminish the monopole density in the volume between them compared to the monopole density outside the plates. The suppression effect is enhanced for larger values of the coupling constant β (at weaker coupling), as one can see from comparisons of Figs. 3(a) and(b). The suppression of the monopole density suggests that the confining property should be weakened in between the plates, and therefore the confinement-deconfinement phase transition should occur at stronger values of the coupling constant (smaller β's). This observation will be confirmed below.

The ratio of the monopole density in between the plates ρ ins and the monopole density at the same β in the absence of the plates, ρ np all , are shown in Fig. 4 for various separations R. All couplings β shown in the plot, the shrinking plates lead to the diminishing monopole density. At weaker coupling (larger β), the relative monopole density is affected stronger than at stronger coupling (smaller β). The relative monopole density, ρ ins /ρ np ins , has an inflection point at certain R = R * at fixed β. This point moves towards smaller values of R as the lattice coupling β decreases. The latter fact indicates that the model may have a β-dependent transition which moves towards smaller R as the coupling β gets larger.

In Fig. 5 we show the monopole density in between the plates ρ ins , its susceptibility, and the corresponding Binder cumulant for three values of the inter-plate distance R. One can immediately make a few qualitative observations from these figures.

Firstly, we notice that all these quantities behave similarly to the case without plates (Fig. 2) implying affinity of these transitions. Secondly, we notice that for any fixed R, the monopole density, its susceptibility, and the Binder cumulant experience the singularities at the same value of the coupling constant β, highlighting the presence of a genuine thermodynamic instability. Thirdly, the positions of these singularities, β c = β c (R), shift towards the strong coupling region as the distance between R the plates diminishes. In other words, the closer the plates, the weaker the monopole component of the vacuum.

The dependence of the critical coupling β c of the interplate distance R is shown in Fig. 6. As we discussed above, the smaller R the smaller β c . The dependence can be fitted by the function

β fit c (R) = β ∞ c -α exp -(R 2 /R 2 0 ) ν , (18)
where the best fit parameter β c = 1.0071 (16) gives us the critical coupling at the infinitely separated plates. This value is close to the transition point in the absence of the plates (17), highlighting the self-consistency of our approach. The other best fit parameters in Eq. ( 18) are as follows: α = 3.7(6), R 0 = 0.28 (7) and ν = 0.257 (16).

The best fit function is shown in Fig. 6 by the solid line. The critical coupling (18) separates the confinement phase (β < β c ) and the deconfinement phase (β > β c ). The phase transition line is a rising function of the inter-plate distance R. The critical coupling vanishes, β fit c (R c ) = 0, at the critical distance R c = 0.47 (7). Formally, at the separations smaller than critical value, R < R c , the theory cannot reside in the confining phase.

Since the inter-plate separation is a positive integer number in the discretized theory, R = 1, 2, . . . , this asymptotic critical value cannot be reached on the lattice.

The loss of the confinement property of the vacuum between the metallic plates in 3+1 dimensions, revealed by Fig. 6, can be understood using the analogy with the similar effect in the 2+1 dimensional theory [16]. Namely, the monopoles and anti-monopoles interact with each other via the long-range massless photon exchange. In the absence of the plates, this interaction, in 3+1 dimensions, decays as |x| -2 as the 4-dimensional distance x between the objects increases. In the presence of the plates, the system experiences the dimensional reduction from four-to three-dimensional spacetime. In the latter, the interaction between the (anti-)monopoles strengthens and decays slower, as |x| -1 . These two factors lead to the breaking of the infrared monopole clusters into smaller clusters and, consequently, to the disappearance of the monopole condensate.

In the 2+1 dimensional model, the same effect leads to the pairing of monopoles and anti-monopoles into the magnetically neutral monopole pairs (the lowerdimensional counterparts of the small clusters) and to the decay of the Coulomb monopole gas (the lowerdimensional analogue of the monopole condensate). The neutral pairs (the small clusters) cannot support the confinement and confining property is lost between the sufficiently close plates [16,17].

IV. DECONFINEMENT ORDER PARAMETER

We determined the nature of confinement and deconfinement phases in the whole phase diagram of Fig. 6 using the simple fact that these phases at finite separation R are smoothly connected to the known phases in the R → ∞ limit. In this section, we quantify this assertion by calculating the deconfinement order parameter, the expectation of the Polyakov loop, in between the plates.

Usually, the Polyakov loop is determined at a finite temperature where the extension of the lattice in the imaginary time direction is finite. The same order parameter can also be used on the zero-temperature lattice with a finite extension in the temporal direction, N T .

In the Abelian gauge theory, the Polyakov loop P x at the spatial space point x is given by a cyclic product of temporal link variables:

P x = N T -1 x4=0 e iθx,x 4 ;µ=4 . ( 19
)
Due to the property of the temporal cyclicity, this quantity does not depend on the time slice where it is defined. The expectation value of this gauge-invariant quantity, P = ⟨P x ⟩ serves as a deconfinement order parameter: at an infinite-volume lattice, P ̸ = 0 in the deconfinement phase and P = 0 in the confinement phase. At finite lattice (as in our case), the expectation value of the Polyakov loop is nonzero in both phases, being small (large) in the confinement (deconfinement) phase.

We compute the expectation value of the modulus of the operator (19) averaged over a set of lattice points in a fixed timeslice of the 3-dimensional volume V 3 :

|P | = 1 V 3 x∈V3 P x . (20)
The expectation value of the Polyakov order parameter (20) in the absence of the plate is shown in Fig. 7. We show the region around the phase transition point (17), where the change in the behavior is well seen.

The order parameter evaluated in the space between the plates is presented in Fig. 8. We show this quantity, as a function of the inter-plate separation R, for the same set of β values as used in Fig. 4 for the inter-plate monopole density. The shrinking plates induce the deconfining phase resulting in the increase of the Polyakov loop (Fig. 8) in agreement with the diminishing monopole density (shown in Fig. 4). The effect appears to work at all values of the coupling constant β. The similar tendency is seen in Fig. 9 which shows the same quantity vs. β at a set of fixed inter-plate separations R.

These results support the phase diagram of Fig. 6.

V. CONCLUSIONS

Using the first-principle numerical simulations, we show that the vacuum of confining gauge theory, the compact U(1) gauge model in 3+1 dimensions, is affected by closely spaced perfectly conducting parallel plates. The non-perturbative Casimir effect modifies the vacuum structure and leads to the Casimir-induced deconfining phase transition in between the plates. Our main result, the phase diagram in the plane "coupling constant"-"distance between the plates" is given in Fig. 6.

1 Figure 1 .

 11 Figure 1. The geometric setup of two parallel Casimir plates.

Figure 2 .

 2 Figure 2. (a) Monopole density ρ, (b) its susceptibility and (c) the Binder cummulant (16) for O = ρ vs lattice coupling β in the absence of plates. The vertical line marks the position of the phase transition calculated from these observables.

Figure 3 .

 3 Figure 3. Typical examples of monopole configurations in (a) the confining phase (β = 0.8) and (b) the deconfining phase (β = 0.9) for the plates separated by the distance R = 3. The monopoles and anti-monopoles are represented by the red and blue dots, respectively. The plates, positioned vertically in the middle of the lattice, are not shown.

Figure 4 .

 4 Figure 4. The ratio ρins/ρ np ins of the monopole density ρins inside the Casimir plates to the monopole density in the absence of the plates, ρ np ins , vs. the inter-plate separation R for a fixed set of the lattice coupling β.

Figure 5 .

 5 Figure 5. The monopole density (the left panel), its susceptibility (the middle panel) and the Binder cumulant (the right panel) plotted as the function of β for three separations between the plates (from top to bottom): R = 2, 4, 8. The corresponding critical coupling constants, βc = βc(R), are shown in the insets.

Figure 6 .

 6 Figure 6. The phase diagram of the vacuum of the compact U(1) gauge theory in between the perfectly metallic plates separated by the distance R. The solid line represents the best fit (18) of the critical coupling βc which corresponds to the confinement-deconfinement phase transition at the interplate distance R. The R → ∞ limit is shown by the dashed horizontal line.

Figure 7 .

 7 Figure 7. The Polyakov loop as the function of lattice coupling β in the absence of plates.

Figure 8 .

 8 Figure 8. The expectation value of the Polyakov loop in the space between the Casimir plates at the separation R at a set of fixed coupling constants β.

Figure 9 .

 9 Figure 9. The Polyakov loop inside the plates vs β at fixed R.

ACKNOWLEDGMENTS

The numerical calculations were performed in 2021 on computing cluster Vostok-1 of Far Eastern Federal Uni-versity. The work of VAG, AVM, and AST was supported by Grant No. 0657-2020-0015 of the Ministry of Science and Higher Education of Russia.


Durgesh Salunkhe
email: durgesh.salunkhe@ls2n.fr

Jose Capco
email: jose.capco@uibk.ac.at

Damien Chablat
email: damien.chablat@ls2n.fr

Philippe Wenger
email: philippe.wenger@ls2n.fr

Geometry based analysis of 3R serial robots

Cuspidal robots can travel from one inverse kinematic solution (IKS) to another without meeting a singularity. This property can be analyzed by understanding the inverse kinematic model (IKM) as well as the singularities in the joint space and in the workspace. In this article, we revisit the geometrical interpretation of the IKM with conics. The conditions of getting different conics and their implication on singularities are discussed and the observations regarding the nature of the conics are presented. Further, a sufficient condition for a 3R robot to be binary (i.e. with up to 2 IKS) as well as quaternary (i.e. with up to 4 IKS) is put forth by analyzing the geometrical interpretation of the IKM. The possibility to derive a necessary and sufficient condition is presented too.

Introduction

Cuspidal robots can travel from one inverse kinematic solution (IKS) to another without encountering a singularity. For these robots, posture identification is very difficult [START_REF] Wenger | A New General Formalism for the Kinematic Analysis of All Nonredundant Manipulators[END_REF], which makes the task of trajectory planning more challenging [START_REF] Wenger | Uniqueness Domains and Regions of Feasible Paths for Cuspidal Manipulators[END_REF]. It is known that a 3R robot can have at most four IKS, and it is generally preferred to choose a robot geometry that maximizes the size of regions with four IKS. This allows an end-user to choose IKS from different regions to counter the collision issues in the workspace. Robots that have 4 IKS regions in their workspace are referred to as quaternary robots, while robots that have at most 2 IKS are referred to as binary robots [START_REF] Wenger | A DH-parameter based condition for 3R orthogonal manipulators to have 4 distinct inverse kinematic solutions[END_REF]. It is im-portant to note that though quaternary robots have their advantages, they can be cuspidal too, while on the other hand binary robots cannot be cuspidal [START_REF] Wenger | A New General Formalism for the Kinematic Analysis of All Nonredundant Manipulators[END_REF]. A particular class of 3R robots, orthogonal 3R robots, have been studied extensively. These robots have three mutually orthogonal joint axes. A D-H parameter based condition for an orthogonal 3R robot to be quaternary was provided by algebraic analysis in [START_REF] Wenger | A DH-parameter based condition for 3R orthogonal manipulators to have 4 distinct inverse kinematic solutions[END_REF]. An extension of such an analysis to non-orthogonal 3R robots is more challenging, and no conditions for binary or quaternary non-orthogonal robots have been reported yet. Recently, the cuspidality of generic 3R robots was analyzed by using a geometric interpretation of the inverse kinematic model (IKM) [START_REF] Salunkhe | Necessary and sufficient condition for a generic 3r serial manipulator to be cuspidal[END_REF]. This work studied the intersection of a conic with a unit circle and derived important observations regarding the existence of reduced aspects as well as the necessary condition for cuspidality in generic 3R robots. The presented work reports few more properties of the conic and its implication on the maximum number of IKS in the workspace. The classification presented provides a simple and clear geometric interpretation for the condition of binary and quaternary robots. This work can be extended to have a necessary and sufficient condition for a generic 3R robot to be quaternary. The following work is divided into three sections: Section 2 revisits the geometrical interpretation of the inverse kinematics of 3R robots and singularities. This section also discusses the classification of 3R robots based on the geometry of the IKM as well as discusses the implication of the same in joint space and workspace. Section 3 shows a case of binary and quaternary robot by analyzing the geometrical properties of the IKM. Section 4 concludes the work by discussing the implications of the contribution and addressing a few pointers to future work.

Inverse kinematic model

Let x = (𝑥, 𝑦, 𝑧) be the vector of coordinates of the robot's end effector in the workspace W ⊂ R 3 at a configuration q = (𝜃 1 , 𝜃 2 , 𝜃 3 ) in the joint space J = 𝑆 1 × 𝑆 1 × 𝑆 1 . The mapping between J and W, denoted by 𝑓 : J → W, defines the direct kinematics: x = 𝑓 (q), x ∈ W, q ∈ J . The elements in the preimage 𝑓 -1 (x) are the IKS of x. A robot configuration associated with an IKS is called a posture.

Solving the inverse kinematics of 3R serial robots was first reported in [START_REF] Pieper | The Kinematics of Manipulators Under Computer Control[END_REF] where it was noted that the solutions correspond to the intersection of a conic with a circle in 𝑐 3 𝑠 3 -plane, where 𝑐 3 and 𝑠 3 denote cos 𝜃 3 and sin 𝜃 3 , respectively. Using the classical D-H parameters to describe the geometry of the robot (see [START_REF] Salunkhe | Necessary and sufficient condition for a generic 3r serial manipulator to be cuspidal[END_REF]), the solution to the IKM is revisited briefly. Let, 𝑅 = 𝜌 2 +𝑧 2 , where 𝜌 2 = 𝑥 2 + 𝑦 2 = 𝑔(𝜃 2 , 𝜃 3 ). The terms 𝑅 and 𝑧 can be written as where 𝐹 𝑖 = 𝑔 𝑖 (𝜃 3 ), for 𝑖 = 1, .., 4. Upon rearrangement, we obtain the general equation of a conic in 𝑐 3 𝑠 3 -plane as given in [START_REF] Wenger | A New General Formalism for the Kinematic Analysis of All Nonredundant Manipulators[END_REF].

𝑅 = (𝐹 1 cos 𝜃 2 + 𝐹 2 sin 𝜃 2 ) 2𝑎 1 + 𝐹 3 𝑧 = (𝐹 1 sin 𝜃 2 -𝐹 2 cos 𝜃 2 ) sin 𝛼 1 + 𝐹 4 Axis 𝑖 -1 Axis 𝑖 Axis 𝑖 + 1 𝜃𝑖 Link 𝑖 -1 Link 𝑖 𝑎𝑖 𝑑𝑖 𝛼𝑖-1 𝑧𝑖-1 𝑦𝑖-1 𝑥𝑖-1 𝑧𝑖 𝑦𝑖 𝑥𝑖
𝐴 𝑥 𝑥 𝑐 2 3 + 2𝐴 𝑥 𝑦 𝑐 3 𝑠 3 + 𝐴 𝑦𝑦 𝑠 2 3 + 2𝐵 𝑥 𝑐 3 + 2𝐵 𝑦 𝑠 3 + 𝐶 = 0 (1)
The coefficients of the conic are skipped for brevity, but they are functions of the D-H parameters and of, (𝑅, 𝑧) as shown in [START_REF] Wenger | Uniqueness Domains and Regions of Feasible Paths for Cuspidal Manipulators[END_REF],

𝐴 𝑥 𝑥 , 𝐴 𝑥 𝑦 , 𝐴 𝑦 𝑦 = 𝑓 1 (𝑎 1 , 𝑎 2 , 𝑎 3 , 𝑑 2 , 𝛼 1 , 𝛼 2 ) 𝐵 𝑥 , 𝐵 𝑦 , 𝐶 = 𝑓 2 (𝑎 1 , 𝑎 2 , 𝑎 3 , 𝑑 2 , 𝑑 3 , 𝛼 1 , 𝛼 2 , 𝑅, 𝑧) (2)
The inverse kinematic solutions are defined by the intersection points between the conic (1) and the unit circle 𝑐 2 3 + 𝑠 2 3 = 1 in 𝑐 3 𝑠 3 -plane. This conic can be a hyperbola, parabola or an ellipse depending on the D-H parameters. An example of each one is shown in Fig. 2.

Performing the tangent half-angle substitution, 𝑡 = tan 𝜃 3 2 , we get a quartic inverse kinematic polynomial 𝑀 (𝑡) = 𝑎𝑡 4 + 𝑏𝑡 3 + 𝑐𝑡 2 + 𝑑𝑡 + 𝑒 similar to the one mentioned in [START_REF] Kohli | Workspace Analysis of Mechanical Manipulators Using Polynomial Discriminants[END_REF]. The coefficients of 𝑀 (𝑡) are functions of the D-H parameters and of 𝑅 and 𝑧. The solutions to the polynomial equation, 𝑀 (𝑡) = 0, are the intersection points between the conic and the circle and are labeled as m 𝜓 , where 𝜓 ∈ {𝑖, 𝑗, 𝑘, 𝑙} in the

𝑐 3 𝑠 3 -plane. m 𝑗 m 𝑘 m 𝑙 m 𝑖 𝑠 3 𝑐 3 𝑠 3 𝑐 3 m 𝑖 m 𝑗 𝑠 3 𝑐 3 m 𝑖 m 𝑗
𝑑 = [0, 1, 0], 𝑎 = [1, 2, 3 2 ], 𝛼 = [ 𝜋 2 , 𝜋 6 , 0], (𝜌, 𝑧) = (2.46, 0.15) (center): 𝑑 = [0, 1, 0], 𝑎 = [1, 2, 3 2 ], 𝛼 = [ 𝜋 3 , 𝜋 2 , 0], (𝜌, 𝑧) = (2.33, -0.26) (right): 𝑑 = [0, 1, 0], 𝑎 = [1, 2, 3 2 ], 𝛼 = [ 𝜋 6 , 𝜋 2 , 0], (𝜌, 𝑧) = (2.4, 0.6).

Singularities

It has been reported in [START_REF] Kohli | Workspace Analysis of Mechanical Manipulators Using Polynomial Discriminants[END_REF] that the singularities in the workspace correspond to all the points where the root multiplicity of 𝑀 (𝑡) = 0 is greater than or equal to 2. Their geometric interpretation in the conic is two intersection points, m 𝑖 , m 𝑗 , merging together at a tangent point between the conic and the circle.

The nonsingular change of posture in cuspidal robots relates to two intersection points interchanging their position, with at least one of them not meeting any other intersection point [START_REF] Salunkhe | Necessary and sufficient condition for a generic 3r serial manipulator to be cuspidal[END_REF]. This interpretation is also helpful to understand why binary robots are compulsorily noncuspidal. It has been shown in [START_REF] Salunkhe | Necessary and sufficient condition for a generic 3r serial manipulator to be cuspidal[END_REF] that the orientation of the conic remains constant. Thus, if we have only two intersection points, then they cannot interchange their position without meeting each other at a tangent point.

Degenerate conic

The nature of the conic depends on the sign of the determinant of N, where N is the Hessian of the conic. The degeneracy of a conic is given by det(D) = 0, where D is the Hessian of the quadratic form. We know that the hyperbola (det(N) < 0) degenerates into two intersecting lines, while the ellipse (det(N) > 0) degenerates to a point. The degenerate case of a parabola is of particular interest as it degenerates to two parallel lines, and they can be distinct or coincident. The presented work discusses the case when a parabola degenerates to two coincident lines, resulting in two multiple roots. This case is important to analyze as this is a special degeneracy case, and it is expected to show a special property in the workspace too. Following are the conditions for a parabola to degenerate into two coincident lines:

         det(N) = 0 det(D) = 0 𝐵 2 𝑥 + 𝐵 2 𝑦 -( 𝐴 𝑥 𝑥 + 𝐴 𝑦𝑦 ) 𝐶 = 0 (3)
Solving det(N) = 0 for 𝑑 2 yields:

𝑑 2 = ± √ (𝑎 1 + 𝑎 2 ) (𝑎 1 -𝑎 2 ) (𝑠𝑎 1 -𝑠𝑎 2 ) (𝑠𝑎 1 + 𝑠𝑎 2 ) 𝑠𝑎 1 𝑠𝑎 2 (4)
We conclude that the parameters 𝑎 3 and 𝑑 3 do not play any role to define a parabola. Upon substituting any value from (4) into det(D) = 0 and solving for 𝑅, we obtain the same solution. Solving the last equation in (3) for 𝑧, the solutions 𝑅 and 𝑧 take the following form, provided that 𝑐𝑎 2 ≠ 0 and

𝑠𝑎 1 ≠ 𝑠𝑎 2 : 𝑅 = 𝑓 (𝑧) 𝑠𝑎 2 1 -𝑠𝑎 2 2 𝑧 = 𝑐𝑎 1 𝑐𝑎 2 (𝑑 3 𝑠𝑎 1 𝑠𝑎 2 + 𝑑 2 𝑐𝑎 2 ) (5)
From ( 5), it is interesting to note that a robot such that (𝑐𝑎 2 ≠ 0, 𝑠𝑎 1 ≠ 𝑠𝑎 2 ) and corresponding to a parabola will always have a point in the workspace (𝑅, 𝑧) such that its geometric interpretation is a parabola degenerating into a coincident line. When 𝑐𝑎 2 = 0 (resp. 𝑠𝑎 1 = 𝑠𝑎 2 ), 𝑧 (resp. 𝑅) is indeterminate. A point satisfying ( 5) is a tangency point between two loci of critical values, as shown in Fig. 3.

-2 -1 0 1 2 -2 -1 0 1 2 𝑐 3 𝑠 3 -𝜋 -𝜋 2 0 𝜋 2 𝜋 -𝜋 -𝜋 2 0 𝜋 2 𝜋 𝜃 2 𝜃 3 0 1 2 3 4 5

Special classes of robots

In this section, we present a sufficient condition for a 3R robot to be binary and quaternary, respectively. The motivation for the search of binary robots comes from the well-known property that two circles have at most two distinct intersections. Section 3.1 discusses the neighbourhood of such binary robots. We claim that the parameters corresponding to all the ellipses that are in a sufficiently small neighbourhood of the parameters corresponding to a circle, also result in binary robots. Section 3.2 discusses a condition for a hyperbola to compulsorily have four intersections with the unit circle in the 𝑐 3 𝑠 3 -plane, thus resulting in a quaternary robot.

Binary robots

We now consider a robot such that its associated conic is an ellipse. We prove that if the ellipse is almost a circle, then many of these robots are binary. We consider a Lemma in Geometry whose proof is straightforward and thus not presented in the text:

Lemma 1 Consider the unit circle 𝑆 1 and a number, 𝑒 ∈ (0, 1)

(i)
There is an ellipse C with eccentricity 𝑒 such that #C ∩ 𝑆 1 = 4 (ii) As 𝑒 → 0 the ellipses with eccentricity 𝑒 with property (i) will have centers that approach the origin (center of 𝑆 1 ) and have minor and major semiaxes that approach length 1.

We now make an important remark that was not emphasized in [START_REF] Salunkhe | Necessary and sufficient condition for a generic 3r serial manipulator to be cuspidal[END_REF]: given a robot, the eccentricity of its associated conic is fixed. Indeed, the eccentricity is only dependent on the entries of N (see [START_REF] Salunkhe | Necessary and sufficient condition for a generic 3r serial manipulator to be cuspidal[END_REF][START_REF] Pieper | The Kinematics of Manipulators Under Computer Control[END_REF]), and these entries only depend on the D-H parameters of the robot and not on the position of the end-effector, as shown in [START_REF] Wenger | Uniqueness Domains and Regions of Feasible Paths for Cuspidal Manipulators[END_REF]. By combining this observation along with the lemma, we can prove the following theorem:

Theorem 1 There are infinitely many binary robots whose associated conic is an ellipse (that is not a circle).

Proof We claim that it suffices to have one binary generic robot with this condition. The associated ellipse for such a robot will never degenerate, and so the minor and major axes must achieve their (non-zero) minimum values.

From previous Lemma, these lengths must lie in an interval 𝐼 ⊂ R centered at 1 (radius of an ellipse) for the ellipse to intersect the unit circle four times and for a fixed sufficiently small eccentricity 𝑒. If a generic binary robot is given with an associated ellipse of eccentricity 𝑒, then the minimum major (or minor) axes (recall the axes' length now depends on the end-effector position) is outside 𝐼. This minimum value is not in the boundary of 𝐼 and is continuously dependent on the D-H parameters. One parameter that does not affect the eccentricity but does affect the minimum major/minor axes' length is 𝑑 3 . So we may perturb 𝑑 3 within a small interval of 𝑑 3 of the given binary robot and still obtain a binary robot.

To conclude the proof, we give an example of a binary robot whose associated conic is an ellipse. The D-H parameters of this robot are:

𝑎 = [- 1503 1879 , -1, -1] 𝑑 = [0, 0, 3] 𝛼 = [-2.21, -𝜋 2 , 0] -2 -1 0 1 2 -2 -1 0 1 2 𝑠 3 𝑐 3 -𝜋 -𝜋 2 0 𝜋 2 𝜋 -𝜋 -𝜋 2 0 𝜋 2 𝜋 𝜃 2 𝜃 3 0 1 2 3 4 -4 -2 0 2 4 𝜌 = √ 𝑥 2 + 𝑦 2

𝑧

Fig. 4: An example of a binary robot's representation in 𝑐 3 𝑠 3 -𝑝𝑙𝑎𝑛𝑒, the joint space and the workspace.

Robot parameters: 𝑑 = [0, 0, 3], 𝑎 = [- 1503 1879 , -1, -1], 𝛼 = [-2.21, -𝜋 2 , 0].
As the eccentricity of the associated ellipse approaches 0 (the ellipse becomes more like a circle) we will get more such binary robots. However, we will always be able to find a robot that is quaternery for such an eccentricity (as long as it is not 0). This is a conjecture that we aim to prove in the future.

Quaternary robot

In this subsection, we discuss the case of a 3R robot such that the hyperbola degenerates and the center of the conic is inside the circle. It is straightforward to argue why such a robot is compulsorily quaternary. If the intersecting lines have their intersection point inside a circle, then each line will intersect the circle twice, thus yielding four intersection points in total. If 𝑐 𝑥 , 𝑐 𝑦 is the center of the conic in the 𝑐 3 𝑠 3 -plane, then the sufficient condition for a quaternary robot is:

           det(N) < 0 det(Q) = 0 √ 𝑐 2 𝑥 + 𝑐 2 𝑦 < 1 (6)
To illustrate the simplicity of the derivation of the sufficient condition, we present a case of orthogonal 3R robots (see an example in 5).

det(N) = - 𝑎 4 3 𝑑 2 2 𝑎 2 1 det(Q) = 𝑎 4 3 𝑑 2 2 (𝑑 2 3 -𝑧 2 ) 𝑎 2 1 𝑐 2 𝑥 + 𝑐 2 𝑦 = 𝑎𝑅 2 + 𝑏𝑅 + 𝑐 4 𝑑 2 2 𝑎 2 3 (7)
In ( 7), 𝑎 and 𝑏 are functions of the D-H parameters only and are not expressed fully for brevity. It is clear from (7) that an orthogonal 3R robot always corresponds to a hyperbola in 𝑐 3 𝑠 3 -plane. The condition for degeneracy depends only on 𝑑 3 and 𝑧 while the condition for the center of the conic to lie inside the circle is a quadratic in 𝑅. It is important to note that the degeneracy depends only on 𝑧 and not on 𝑅 and thus, for 𝑧 = 𝑑 3 , the conic is always degenerate. This property also leads to some interesting observations about the hyperbolas corresponding to an orthogonal 3R robot but are not discussed here for lack of space.

Conclusions and future work

In this work, we have revisited the geometric interpretation of the inverse kinematic model of 3R robots. The interpretation of singularities as well as the nonsingular change of posture have been briefly introduced. The special case of a parabola degenerating to two coincident lines is presented along with its interpretation in the workspace. The work also presented a sufficient condition for a 3R serial chain to be binary (ellipse case) (section 3.1) or quaternary (hyperbola case) (section 3.2) by using geometric observations. The advantages of the geometry based analysis is that the conditions for binary or quaternary robots can be extended to more generic cases of 3R robots without resorting to complex algebraic derivations. In future work, we aim to present a necessary and sufficient condition for a generic 3R robot to be binary or quaternary. This will allow the designer to include the condition while optimizing for a workspace with 4 IKS. We will also prove a few conjectures on binary and quaternary robots whose associated conic is an ellipse.

Fig. 1 :

 1 Fig. 1: D-H convention used and a schematic of a 3R serial robot

Fig. 2 :

 2 Fig. 2: The conic and unit circle in 𝑐 3 𝑠 3 -plane for robots with different 𝛼 parameters and at different points. Robot parameters (left): 𝑑= [0, 1, 0], 𝑎 = [1, 2, 3 2 ], 𝛼 = [ 𝜋 2 , 𝜋 6 , 0], (𝜌, 𝑧) = (2.46, 0.15) (center): 𝑑 = [0, 1, 0], 𝑎 = [1, 2, 3 2 ], 𝛼 = [ 𝜋 3 , 𝜋 2 , 0], (𝜌, 𝑧) = (2.33, -0.26) (right): 𝑑 = [0, 1, 0], 𝑎 = [1, 2, 3 2 ], 𝛼 = [ 𝜋 6 , 𝜋 2, 0], (𝜌, 𝑧) = (2.4, 0.6).

𝑦 2 𝑧Fig. 3 :

 23 Fig. 3: Degenerate parabola case: joint space (center) and workspace (right). The point shown in red is associated with two coincident lines in the 𝑐 3 𝑠 3 -plane (left). Robot parameters: 𝑑 = [0, 2.828, 0.5], 𝑎 = [1, 2, 3 2 ], 𝛼 = [ 𝜋 6 , 𝜋 3 , 0], (𝜌, 𝑧) = (1.471, 3.315).

Fig. 5 :

 5 Fig.5: An example of an orthogonal quaternary robot corresponding to an hyperbola.Robot parameters : 𝑑= [0, 1, 2], 𝑎 = [1, 2, 3], 𝛼 = [ -𝜋 2 , 𝜋 2 , 0], (𝜌, 𝑧) = (2, 2).

Acknowledgements

The authors are supported by the joint French and Austrian ECARP project: ANR-19-CE48-0015, FWF I4452-N. The authors also thank Christoforos Spartalis for his contribution in the initial stages of the work.


Rania Harrabi

Dr Thomas Halbritter

Dr Fabien Aussenac

Dr Ons Dakhlaoui

Dr Johan Van Tol

Krishna K Damodaran

Dr Daniel Lee

Dr Subhradip Paul

Dr Sabine Hediger

Dr Frederic Mentink-Vigier

] Snorri
email: snorrisi@hi.is[c]

Th Sigurdsson

Gaël De Paëpe
email: gael.depaepe@cea.fr

] R Harrabi

Prof. K K Damodaran

Prof. S. Th Sigurdsson

Highly efficient polarizing agents for MAS-DNP of proton-dense molecular solids

Keywords: Dynamic Nuclear Polarization, Nuclear Magnetic Resonance, Polarizing Agents, MAS-DNP, Pharmaceuticals, Biomolecules

Introduction

Solid-state Nuclear Magnetic Resonance (ssNMR) with Magic Angle Spinning (MAS) is a powerful and versatile spectroscopy for studying structure and dynamics at the atomic level. [1] Despite its strengths, ssNMR often suffers from a lack of sensitivity when detecting nuclei with low gyromagnetic ratio, low isotopic abundance, or carrying a nuclear quadrupole moment. These limitations can be overcome through the use of hyperpolarization techniques, such as high field Dynamic Nuclear Polarization under MAS (MAS-DNP) [2][3][4] , which has emerged as a powerful approach to perform previously impossible NMR experiments for a wide range of applications from materials to life science. [5][6][7][8] The MAS-DNP amplification effect is based on the use of special paramagnetic centers, called Polarizing Agents (PAs), which act as a polarization source under high frequency microwave irradiation to hyperpolarize nuclear spins. [4,9,10] These developments at high magnetic field, pioneered by the Griffin group at MIT, have required the development of dedicated instrumentation such as high-power high-frequency gyrotron microwave sources, low temperature MAS-DNP probes, as well as efficient PAs. [4,9,11] This last research axis witnessed an important milestone in the early 2000s with the introduction of chemically-bound bis-nitroxides [12] that can provide a substantial improvement in Cross Effect (CE) MAS-DNP efficiency. Since then, considerable efforts have been devoted to further improvements. Today, the most used biradicals for CE MAS-DNP are AMUPol [13] in aqueous solutions and TEKPol [14] in organic solvents, owing to their good efficiency and commercial availability. Although these radicals provide relatively high enhancement factors (𝜀 𝑂𝑛/𝑂𝑓𝑓 ) at 9.4 T and 100 K, [13] they exhibit a marked depolarization effect (𝜀 𝑑𝑒𝑝𝑜 ), [15] and a relatively long nuclear spin hyperpolarization build-up time (𝑇 B ), and thus overall, a limited sensitivity gain. In addition, the CE MAS-DNP efficiency [16,17] of these biradicals tends to decrease significantly with increasing magnetic field (>9.4 T) and MAS frequency (>10 kHz). [18] To alleviate these limitations, hetero-biradicals were recently introduced, in which a carbon-based radical is linked to a nitroxide. [18] The first example of an efficient hetero-biradical for water-based applications was TEMTriPol-I, [18] shown to perform well at high magnetic field [18,19] while having a limited depolarization effect and requiring less microwave power. [19] The hetero-structure was recently extended to the NaTriPol family by functionalizing the TEMTriPol-I linker, which limits the propensity of the radical to aggregate. [20] Furthermore, BDPA-nitroxide mixed biradicals, known as the HyTEK family, were also introduced for organic solvent-based applications. [21,22] Advanced numerical simulations can help guide the design of new PAs, [23] as illustrated recently with a family of bis-nitroxides, dubbed "AsymPol". [23] The efficiency of this family partly arises from the sizable dipolar and exchange interactions between the two electron spins. This property limits the depolarization effect, [15] generates fast hyperpolarization build-ups and overall yields high MAS-DNP enhanced NMR sensitivities. [23] Despite advances in the design of new PAs, hyperpolarization of proton-dense systems is still a challenge. In other words, the impressive improvements of MAS-DNP, qualified on partially deuterated model systems, do not immediately translate in proton-dense systems such as organic powders and biomolecular systems. Here, we show that this problem is a consequence of the inappropriate assessment of the polarizing agents' effectiveness and that it hinders the design of improved PAs. We illustrate this by comparing a new PA, cAsymPol-POK, to several known PAs while varying the number of protons to be polarized per biradical. The new PA, cAsymPol-POK, in which the geminal methyl groups of the pyrrolinoxyl radical of AsymPol-POK [23] are substituted with cyclohexyl groups, offers exceptional efficiency for proton-dense systems. Its performance is illustrated at high magnetic fields (9.4 to 18.8 T) and high MAS frequencies (10 to 40 kHz) for microcrystalline drugs and a 12.76 kDa protein. Its hyperpolarization ability enables high polarization levels for systems with short intrinsic bulk proton longitudinal relaxation times 𝑇 1,𝑛 caused for example by the presence of methyl groups. The characteristics of cAsymPol-POK are demonstrated by MAS-DNP simulations, combined with Electron Paramagnetic Resonance (EPR), Density Functional Theory (DFT) and Molecular Dynamics (MD) calculations. They highlight the contribution of the two major conformers in AsymPol-POK and cAsymPol-POK, for which the variations of the electron-electron (e-e) dipolar coupling, the exchange interaction (J), as well as the relative orientation of the g-tensors are determined. A novel MAS-DNP simulation model also rationalizes their hyperpolarization capability for media that are difficult to hyperpolarize. Experiments performed with cAsymPol-POK have yielded timesaving up to a factor of 9, compared with AMUPol. This unlocks the acquisition, in only few hours, of short and long range 2D 13 C- 13 C and 15 N- 13 C correlation experiments at Natural isotopic Abundance (NA) of microcrystalline solids, here pharmaceutical drugs: the antibiotic ampicillin and the anti-inflammatory drug indomethacin. We also report excellent sensitivity on the 12.76 kDa U-13 C, 15 N labeled sedimented LecA protein and present 2D spectra at 40 kHz MAS frequency recorded within minutes.

Results and Discussion

cAsymPol-POK: A new member of the AsymPol family. The "AsymPol family" has recently been introduced as an efficient family of PAs. [23] The design of the chemical linker was guided by computational simulations and resulted in the generation of sizeable e-e couplings. For instance, AsymPol-POK, the best water-soluble derivative reported in the previous study, is composed of a short tether with a conjugated carbon-carbon double bond in the 5-membered ring to improve the rigidity and provide a favorable relative orientation, by avoiding parallel nitroxide orientations. In addition, the two methyl groups of the piperadine-radical were replaced with spirocyclohexanolyl groups functionalized with phosphate moieties. To further tune the characteristics of AsymPol-POK, the geminal methyl groups of the pyrrolinoxyl radical were substituted by cyclohexyl groups in cAsymPol-POK. The removal of the methyl groups from the TEMPO moieties has already been investigated in the case of bCTbK, [14] TEKPOL, [14] AMUPol, [13] bcTol, [24] bcTol-M, [24,25] cyolyl-TOTAPOL, [13] and the TinyPol family. [26] The introduction of charged water-soluble moieties with AsymPol-POK (Figure 1) was crucial in its development and resulted not only in highly soluble PAs, but also prevented aggregation even at high concentration. In addition, increased molecular weight of biradicals can lengthen electron spin relaxation times, [27][28][29] which is beneficial for CE MAS-DNP efficiency. [30] In the case of cAsymPol-POK (Figure 1), first introduced here, the substitution of the geminal methyl groups in the pyrrolinoxyl radical by cyclohexyl-groups further improves the DNP efficiency of the AsymPol family. Synthesis. The synthesis of cAsymPol-POK (Scheme 1) is similar to that of AsymPol-POK. [23] First, TBDMS-protected spirocyclohexanolyl-amino-TEMPO (1) was conjugated to spirocyclohexyl-3-carboxypyrrolinoxyl nitroxide (2), followed by removal of TBDMS protecting groups with TBAF to give compound 3. Phosphitylation and deprotection of 3 gave cAsymPol-POK.

EPR and MAS-DNP simulations for AsymPol-POK and cAsymPol-POK.

In a previous study, we attempted to characterize the geometry and the magnetic properties of AsymPol-POK, but the resulting structure enabled only a partial match of the high field EPR spectrum and high values of exchange interaction were not accounted for. [23] To better determine AsymPol-POK and cAsymPol-POK geometries, the EPR fit was carried out starting from the predicted geometries given by DFT and MD calculations. DFT and MD calculations reveal that two conformers of both AsymPol-POK and cAsymPol-POK must be considered to fit the multi-frequency EPR data, yielding an excellent agreement between experiments and simulations, as shown in Figure 2 (b). These conformers have similar relative energies and are thus approximately equally populated at low temperature. However, they lead to a different predicted exchange interaction and relative orientation of the nitroxides (see SI, Table S4). The DFT calculations also revealed that steric hindrance forces one of the spirocyclohexyls on the 5-membered ring to take a "closed conformation". [31] This fact was confirmed by X-ray crystallography (see SI, Figure S10). Conformer #1 has an exchange interaction Ja,b of ~120 MHz and ~100 MHz for conformer #2. These values are significantly higher than in our previous assessment. [23] Conformers #1 and #2 account for 55/45% and 60/40% for AsymPol-POK and cAsymPol-POK, respectively. Single-crystal X-ray analysis revealed only one conformer that is an "average" of conformer #1 and #2, and thus those conformers may only exist in glass forming matrices (see SI, Figure S10). Importantly, the numerical simulations predict that conformer #2 consistently performs better than conformer #1 but both conformers provide similar hyperpolarization build-up times (𝑇 𝐵 ). Thus, the introduction of the spirocyclohexyl rings on the pyrrolinoxyl radical in cAsymPol-POK has a negligible impact on the biradical geometry (i.e., cAsymPol-POK vs AsymPol-POK). The geometrical and magnetic parameters extracted from the EPR fit were also used to compute MAS-DNP field sweep profiles.

As seen in Figure 2 (c), the simulations match well all the experimental data recorded at 9.4 T. Assessing MAS-DNP hyperpolarization performance: the returned sensitivity. The MAS-DNP efficiency of a PA should not be solely judged by measuring the enhancement factor 𝜀 𝑂𝑛/𝑂𝑓𝑓 , i.e. the ratio between the NMR signal intensities with and without microwave irradiation. [32] This ratio does not consider the time it takes to build-up the 1 H hyperpolarization (𝑇 𝐵 ) or depolarization ( 𝜀 𝑑𝑒𝑝𝑜 ) / quenching factors that depend on the structure of the PA. [15,[32][33][34] Such correction factors should be evaluated to properly quantify the DNP gain compared to Boltzmann equilibrium, defined as 𝜀 𝐵 = 𝜀 𝑂𝑛/𝑂𝑓𝑓 × 𝜀 𝑑𝑒𝑝𝑜 . [15] One can then define 𝜀 𝐵 √𝑇 𝐵 ⁄ as a measure of the relative sensitivity gain from DNP. [32] However, the precise measurement of 𝜀 𝑑𝑒𝑝𝑜 , although very informative, is time-consuming and sensitive to the background signal of the probe. Instead, it is more convenient to directly measure the returned sensitivity. [3,35] It is defined as the hyperpolarized NMR signal intensity, detected in presence of microwave irradiation, per unit square root of time, for a given PA concentration and sample amount. This quantification method provides a fair approach to compare the performance of different PAs and can be extended to applications where the polarization build-up curves are not mono-exponential. [24,25] Effect of the proton density on the DNP enhancement factor 𝜺 𝑶𝒏/𝑶𝒇𝒇 . We tested the impact of the proton concentration by performing experiments in glass-forming matrices composed of glycerol and water. A high glycerol/water ratio (6:4 v/v) ensures a good and repeatable glass formation, necessary to conduct MAS-DNP experiments, and therefore kept constant. Two proton concentrations were tested, using either d8-glycerol/D2O/H2O (60:30:10 v/v, [ 1 H] = 11 M), or glycerol/H2O (60:40 v/v, [ 1 H] = 110 M) as a DNP matrix. The first matrix is referred to as the, standard, deuterated DNP matrix and the second one as the protonated DNP matrix. Hyperpolarizing the protonated matrix is challenging since there are x10 more 1 H nuclei per biradical, and the 1 H 𝑇 1.𝑛 is also reduced from ~70 s to 30 s. [36,37] As shown in Table 1, the highest 𝜀 𝑂𝑛/𝑂𝑓𝑓 value for AMUPol, measured at 8 kHz MAS and 14.1 T, is obtained in the standard deuterated matrix with 𝜀 𝑂𝑛/𝑂𝑓𝑓 ≈ 160 . Nevertheless, the depolarization factor is significant too, reaching approximately 0.5 at 14.1 T (see SI, Table S3), impacting the absolute polarization gain 𝜀 𝐵 ≈ 80. This situation is very different for the radicals of the AsymPol family, that exhibit a small depolarization effect under the same experimental conditions of 0.78 and 0.77 for cAsymPol-POK and AsymPol-POK, respectively (see SI, Table S3). This limited depolarization, also predicted from the simulations (see SI, Table S5 andS6), is one of the important characteristics of the AsymPol family. [23] Table 1. Experimental performance of 10 mM AMUPol, AsymPol-POK and cAsymPol-POK in the deuterated DNP matrix at 14.4 T, and 8 kHz MAS using a 3.2 mm rotor. The sensitivity was measured as 𝐼 𝑜𝑛 (𝑇 𝐵 )/√𝑇 𝐵 with 𝐼 𝑜𝑛 (𝑇 𝐵 ) the signal intensity measured for a same number of scans, the same sample amount, and an inter-scan delay set to 1.3 × 𝑇 𝐵 . The sensitivity is normalized to the highest value.

𝑇 𝐵 [ values increase with the proton concentration of the matrix for both AsymPol-POK and cAsymPol-POK. A proper quantification of 𝜀 𝐵 and 𝜀 𝑑𝑒𝑝𝑜 may explain these observations. However, in the fully protonated matrix, depolarization measurements are even more challenging due to radiation damping, which we were not able to quantify. In the following, we therefore used the returned sensitivity instead.

Effect of the proton concentration on the DNP build-up time.

As shown in Table 1, the 𝑇 𝐵 is similar for both AsymPols (~ 2 s) and much shorter than for AMUPol (4.8 s) at 10 mM PA concentration, at 14.1 T, and 8 kHz MAS. This feature is a direct consequence of the stronger e-e couplings (dipolar and exchange interaction) present in the AsymPols compared to AMUPol. [23] Interestingly, the build-up time constants are unchanged for the AsymPols when going from 11 M to 110 M 1 H concentration (see Table 2). This is quite remarkable and proves the ability of these PAs to efficiently hyperpolarize a much larger number of protons. This is not the case for AMUPol, which requires a longer build-up time for 110 M proton concentration, as was also observed at 9.4 T. [38] It is important to stress that a fast nuclear polarization buildup is essential to maximize the overall NMR sensitivity, and that the measured build-up time constants can be reproduced by MAS-DNP simulations (see SI, table S5 andS6). The limits of using a deuterated matrix for benchmarking and implications for the design of improved PAs. Beyond the MAS-DNP enhancement factor ( 𝜀 𝑂𝑛/𝑂𝑓𝑓 ) , and build-up time constant (𝑇 𝐵 ) , Table 1-2 also show the experimental DNP-enhanced NMR sensitivity at 14.1 T for both DNP matrices using the three radicals. Interestingly, the AsymPols yield a much higher sensitivity than AMUPol in both matrices, especially for the fully protonated one. This critical result highlights the importance of using sensitivity measurements to evaluate and compare radical efficiencies. It also underscores the importance of using densely protonated matrices when the goal is to assess the efficiency of PAs for hyperpolarizing proton-rich samples. At 14.1 T and under moderate MAS frequency, AsymPol-POK and cAsymPol-POK yield a similar sensitivity, which is x1.4 and x2.5 higher than AMUPol for a DNP matrix with proton concentration of 11 M and 110 M, respectively. This translates into an impressive x6 gain in timesaving for the fully protonated matrix. As shown below, the performance contrast between the AsymPols, in particular cAsymPol-POK, and AMUPol is even more striking at higher magnetic fields and faster spinning frequencies and/or when applied to organic microcrystals. The challenge of polarizing large, protonated microcrystals with DNP. The sensitivity offered by MAS-DNP has enabled numerous experiments that were previously unobtainable within reasonable time, such as structural studies of powdered organic 1 (blue) and three-bonds (yellow) correlations. Spinning side bands, signals from the carrier frequency and signals from the glycerol solvent are indicated with red asterisks samples at NA. [7,[39][40][41] 2D 1 H-X heteronuclear dipolar correlation spectra can be used to assign 13 C or 15 N resonances [42] but they are not always resolved due to the low 1 H spectral resolution at the usually available MAS frequencies. 13 C-13 C or 13 C- 15 N correlation experiments at NA offer better resolution but are very challenging, even when using MAS-DNP, due to the very low natural isotopic abundance of 13 C (1.1%) and 15 N (0.4%). [7,35,39- 41,43,44] Previous work has mostly been performed either on microcrystalline samples with sufficiently long bulk 𝑇 1,𝑛 , [7,40,41] to allow large enhancements, or on nano-assemblies where one of the dimensions is in the nanometer scale, such as protein fibrils, [39] cyclo-FF nanotubes, [43] cellulose nanofibrils [45] or whiskers, [35] all of which are favorable for DNP experiments. However, the DNP efficiency decreases strongly with increasing particle size and/or in the presence of a short (few seconds) 1 H 𝑇 1,𝑛 relaxation time (before doping), since the magnetization needs to diffuse from the surface of the particle to its core, while competing against returning to Boltzmann equilibrium. [7] As such, organic microcrystals (micrometer size in all dimensions), in particular bearing methyl groups, are challenging targets and useful for testing new PAs. [38,46] cAsymPol-POK can efficiently hyperpolarize large microcrystals with short nuclear 𝑻 𝟏,𝒏 relaxation time constant. AMUPol, AsymPol-POK and cAsymPol-POK were used to polarize microcrystals of the antibiotic ampicillin and the anti-inflammatory drug indomethacin, that possess a 1 H 𝑇 1,𝑛 of 80 s and 3 s, respectively, at 9.4 T and 100 K. Both powders are sparingly soluble in water-based solvents and contain a relatively high proton concentration, ~80 M for ampicillin and ~58 M for indomethacin. SEM images of both powders confirm the presence of large micrometer-size crystals, especially in the case of indomethacin (see SI, Figure S5). Several PA concentrations were tested, from 10 to 80 mM (Figure 4 and SI Figure S6) at 9.4 T. Interestingly, the sensitivity is optimal at around 40 mM radical solution for all three PAs. These results correlate with those reported by Thureau et al., [47] where higher PA concentrations were used to suppress the solvent signals and improve the overall sensitivity. Figure 3 reports the returned sensitivity at the optimal recycle delay time 𝑇 𝑜𝑝𝑡 (see SI, DNP build-up analysis) for the three radicals at 40 mM. It shows that the AsymPol PAs clearly outperform AMUPol in terms of sensitivity, with impressive short build-up times both at 9.4 T and at 14.1 T (Figure 3 and SI Figure S7). In the case of ampicillin, the sensitivity is ~90 s -1/2 for cAsymPol-POK at 40 mM (see Figure 3a), which is about x2.7 higher than with AMUPol at 40 mM, corresponding to a factor of > 7 in timesaving. Moreover, the corresponding optimal recycle delay time 𝑇 𝑜𝑝𝑡 (Figure 3b) is much shorter for AsymPol-POK (2.8 s) and cAsymPol-POK (3.5 s), compared to 10.4 s for AMUPol (see SI, Tables S1 andS2 for the fitting of the build-up curves with a bi-exponential function, and calculation of 𝑇 𝑜𝑝𝑡 ). Similar conclusions can be drawn for indomethacin, which has a much shorter 1 H 𝑇 1,n . The sensitivity is, here too, optimal for cAsymPol-POK with a value of ~36 s -1/2 , proving almost one order of magnitude in timesaving as compared to AMUPol. This result is remarkable, considering the much shorter 1 H 𝑇 1,n of only 3 s for indomethacin.

Unlocking NA 13 C-13 C and 15 N-13 C 2D dipolar correlation spectroscopy of microcrystals with cAsymPol-POK. The unprecedented sensitivity gain provided by 40 mM cAsymPol-POK enabled the acquisition of 13 C- 13 C double quantum-single quantum (DQ-SQ) and 15 N-13 C dipolar correlation experiments for both ampicillin and indomethacin samples at NA in only a few hours (see Figure 4). It is worth noting that the signal-to-noise ratios of all 2D data presented in Figure 4 are excellent, with corresponding experimental times in the range of 3 to 5 h. The through-space 13 C-13 C and 15 N- 13 C correlation spectra were recorded using the dipolar recoupling sequence SR26 [48] combined with StiC phase shifts [43] and zf-TEDOR, [49] respectively. Since dipolar truncation is statistically negligible in the case of NA samples, the mixing time was fixed to 4 ms to favor short throughspace polarization transfer in both cases (equivalent to one or two bond lengths). Using the dataset in Figure 4, we were able to assign nearly all 13 C and 15 N resonances of both ampicillin and indomethacin. Correlations of the two methyl groups of ampicillin (carbon 9 and 10) and one of the methyl of indomethacin (carbon 8) could not be detected, which is often the case for measurements at 100 K. AsymPol-POK and cAsymPol-POK are efficient PAs at high magnetic field and fast MAS: Application to a proton-dense biomolecular system. Additional experiments were conducted with a faster MAS frequency of 40 kHz at both 9.4 and 18.8 T (1.3 mm rotors). As expected, the AsymPol biradicals return large DNP enhancements and relatively short build-up times. A quantitative comparison between AMUPol and cAsymPol-POK was conducted on a deuterated frozen solution at 40 kHz MAS (Table 3). It shows a x2.4 and x3.1 higher sensitivity compared to AMUPol, corresponding to a factor of ~5,8 and ~10 in timesaving at 9.4 and 18.8 T, respectively. This confirms the widening gap between cAsymPol-POK and AMUPol at increased magnetic fields, which is very promising for applications at fast and ultrafast MAS. This substantial improved sensitivity from cAsymPol-POK is accompanied with, once again, a short build-up time. This latter is ~40% shorter for cAsymPol-POK with respect to AMUPol at both 9.4 and 18.8 T, and 40 kHz MAS spinning. We note that high DNP enhancements were reported for the TinyPol family [26] at 18.8 T and 40 kHz MAS on a deuterated matrix. However, these enhancements come at the cost of long build-up times of 13 s or more, compared to only 3.6 s in the case of cAsymPol-POK. The robustness of cAsymPol-POK at fast MAS and high magnetic fields can be directly attributed to the presence of a large exchange interaction between the electron spins, which is one of the main characteristics of the AsymPol family, confirmed by simulations at these regimes (See SI Table S4).

Finally, the performance of cAsymPol-POK at fast MAS was evaluated on a 12.76 kDa U-13 C, 15 N labeled LecA protein. Such protein sample has a very dense proton content and a short 𝑇 1,n of ~ 7 s, mainly due to the presence of numerous methyl groups. At 40 kHz MAS and using 32 mM cAsymPol-POK, a large enhancement factor of 𝜀 𝑂𝑛/𝑂𝑓𝑓 = 130 with an impressively short 𝑇 B of 0.5 s were obtained on the LecA sample (Figure 5a). Interestingly, the returned enhancement factor is similar to the one obtained for a deuterated frozen solution (Table 3), highlighting the effectiveness of cAsymPol-POK to polarize targets that are proton-rich and have relatively short 1 H 𝑇 1.𝑛 . Figure 5b shows a 2D 13 C-13 C SQ-SQ DARR spectrum with an excellent signal-to-noise ratio, acquired at 40 kHz MAS with less than 1 mg of protein in 19 min of experimental time. [50] Such impressive DNP signal enhancements are particularly important for emerging biomolecular applications, in particular if they can be combined with Targeted DNP and Selective DNP approaches. [1]

Conclusion

In this work, we provide a rational for the limited DNP efficiency of PAs developed thus far for NMR studies of proton-dense molecular solids. Part of the problem originates from the fact that common PAs have been benchmarked and selected based on their performance for solutes in matrices with low proton density. Such deuterated matrices also have a long 1 H 𝑇 1,𝑛 and are thus relatively easy to hyperpolarize, while also being prone to amplifying depolarization effects present with some PAs. This triggered the report of large DNP enhancement factors, which do not always correlate with large DNP-NMR sensitivity. Keeping this in mind, we introduced here a new polarizing agent, cAsymPol-POK, which provides outstanding efficiency for proton-dense molecular systems at high magnetic fields (up to 18.8 T) and fast MAS frequencies (up to 40 kHz). This new biradical is compatible with challenging biomolecular applications and unlocks the rapid acquisition of 13 C- 13 C and 15 N- 13 C correlation experiments on pharmaceutical drugs at natural isotopic abundance, which are key experiments for structure determination. The performance of cAsymPol-POK is rationalized by MAS-DNP simulations, combined with EPR, DFT and MD calculations. They that the introduction of the spirocyclohexyl groups does not significantly change the linker geometry, nor the e-e spin couplings involved. Thus, the improved performance of cAsymPol-POK relative to AsymPol-POK is attributed to the replacement of methyl groups with cyclohexyl groups, which tends to lengthen both the electron and the core protons relaxation times leading to better CE-DNP efficiency.

Figure 1 .

 1 Figure 1. Chemical structure of AsymPol-POK and cAsymPol-POK.

Scheme 1 .

 1 Scheme 1. Synthesis of cAsymPol-POK. BCDP: bis(2-cyanoethyl)-N,Ndiisopropyl-phosphoramidite, BTT: 5-benzylthio-1H-tetrazole.

Figure 2 .

 2 Figure 2. (a) DFT calculated structure of cAsymPol-OH viewed from the side, showing conformer #1 on the left and conformer #2 on the right. (b) Concatenated X-band, 240 GHz EPR spectra and (c) MAS-DNP field profile at 9.4 T for AsymPol-POK (black) and cAsymPol-POK (blue). The corresponding simulated EPR spectra and MAS-DNP field profiles are given as red dotted lines. For the MAS-DNP field profiles, experimental points are reported as circles for AsymPol-POK and squares for cAsymPol-POK, full for experiments and open for simulations.

Figure 3 .

 3 Figure 3. Experimental performance of (a)-(b) ampicillin and (c)-(d) indomethacin powders impregnated with 10 mM (grey) and 40 mM (blue) AMUPol, AsymPol-POK and cAsymPol-POK in the standard deuterated DNP matrix, d8-glycerol/D2O/H2O (60:30:10 v/v), at 105 K, 9.4 T and 8 kHz MAS. The plots show the signal-to-noise ratio per square root of the experimental time (𝑆/𝑁) √𝑡 ⁄ in (a) and (c), and the optimal build-up time constant 𝑇 𝑜𝑝𝑡 in (b) and (d) for ampicillin and indomethacin, respectively.

Figure 5 .

 5 Figure 5. (a)1 H-13 C CP-MAS spectra with (red) and without (black) µwave irradiation of U-13 C,15 N LecA, leading to an enhancement factor 𝜀 𝑂𝑛/𝑂𝑓𝑓 of 130. (b) DNP-enhanced 2D SQ-SQ13 C-13 C correlation spectrum using the DARR

Table 2 .

 2 Experimental performance of 10 mM AMUPol, AsymPol-POK and cAsymPol-POK in the protonated DNP matrix at 14.4 T, and 8 kHz MAS using a 3.2 mm rotor. The sensitivity was measured as 𝐼 𝑜𝑛 (𝑇 𝐵 )/√𝑇 𝐵 with 𝐼 𝑜𝑛 (𝑇 𝐵 ) the signal intensity measured for a same number of scans, the same sample amount, and an inter-scan delay set to 1.3 × 𝑇 𝐵 . The sensitivity is normalized to the highest value.Interestingly, the values for the DNP enhancement factors 𝜀 𝑂𝑛/𝑂𝑓𝑓 are very different when measuring the fully protonated matrix and even more so, they do not follow the same trend for AMUPol and the AsymPols at 8 kHz MAS and 14.1 T. The DNP enhancement factor of AMUPol is strongly reduced from ~160 to ~120 (Table1-2), whereas it is increased for the AsymPol-POKs, from ~110 to ~130 for AsymPol-POK and from ~110 to ~150 for cAsymPol-POK. A decrease in 𝜀 𝑂𝑛/𝑂𝑓𝑓 , as observed for AMUPol, is expected as the 1 H bulk 𝑇 1,𝑛 (undoped) is shorter for the fully protonated matrix (cf. 70 vs 30 s). It is therefore surprising that the 𝜀 𝑂𝑛/𝑂𝑓𝑓

		𝑠]	𝜀 𝑜𝑛/𝑜𝑓𝑓	Sensitivity
	AMUPol	4.8	160	0.7
	AsymPol-POK	1.8	110	1
	cAsymPol-POK	1.9	110	1

Table 3 .

 3 Experimental results of 10 mM cAsymPol-POK and AMUPol in a deuterated DNP matrix at magnetic field strengths of 9.4 and 18.8 T strength and high MAS frequency of 40 kHz. Note that the AMUPol reference sample was provided by the TGIR-RMN-THC FR3050 CNRS facility in Lyon.

		9.4 T, 1.3 mm rotor		18.8 T, 1.3 mm rotor
		at 40 kHz			at 40 kHz
		Norm. Sensitivity	𝑇 𝐵 [s]	𝜀 On/Off	Norm. Sensitivity	𝑇 𝐵 [s]	𝜀 On/Off
	AMUPol	0.42	4.8	330	0.32	6	40
	cAsymPol-POK 1	2.5	140	1	3.6	50

Acknowledgements

This work was supported by the French National Research Agency (CBH-EUR-GS and ARCANE ANR-17-EURE-0003, Glyco@Alps ANR-15-IDEX-02, and ANR-16-CE11-0030-03) and the European Research Council Grant ERC-CoG-2015 No. 682895 to G. D. P. Part of this work, carried out on the Platform for Nanocharacterisation (PFNC), was supported by the "Recherches Technologiques de Base" program of the French National Research Agency (ANR). This work was supported by the Icelandic Research Fund, grant No. 173727, and the University of Iceland Research Fund (S.Th.S). T.H. thanks the Deutsche Forschungsgemeinschaft (DFG) for a postdoctoral fellowship (414196920). The National High Magnetic Field Laboratory is supported by the NSF (DMR-1157490 and DMR-1644779) and by the State of Florida. The 14.1 T DNP system at NHMFL is funded in part by NIH S10 OD018519 (magnet and console), NSF CHE-1229170 (gyrotron), and NIH P41 GM122698. FMV thanks Shimon Vega for fruitful discussions. Financial support from the TGIR-RMN-THC Fr3050 CNRS for conducting DNP experiments at high magnetic fields (18.8 T) is gratefully acknowledged.

Conflict of interest

The authors declare no conflict of interest