Nabile Boussaïd

Marco Caponigro

Thomas Chambrion

An approximate controllability result with continuous spectrum : the Morse potential with dipolar interaction *

This note presents a positive approximate controllability result for a bilinear quantum system modeling a chemical bond. The main difficulties are due to the presence of a continuous spectrum part for the uncontrolled Hamiltonian (modeled by a Morse potential) and the unboundedness of the interaction potential (dipolar interaction). The proof is based on averaging technique and spectral analysis.

1 Problem Specification.

Physical motivation

In this note, we consider the dynamics of a molecular bond, such as a H-H bond in a dihydrogen molecule (Figure 1), modeled by a bilinear Schrödinger equation (1.1) i 2m

∂ψ ∂t = [(-∆ + V (x)) + u(t)W (x)] ψ(x, t)
where = h 2π is the reduced Planck constant, m is the reduced mass of the system, x > 0 is the length of the bond between the atomic nuclei, ∆ is the Laplacian (with Dirichlet boundary conditions) on (0, +∞), V : (0, +∞) → R is the potential of the free dynamics (without external interaction) and W : (0, +∞) → R is the interaction potential, modeling the influence of an electric field (e.g., a laser) on the molecule. The function u : [0, +∞) → R is the intensity of the external field. The wave function ψ ∈ L 2 ((0, +∞), C) determines the quantum state of the system. For instance, the probability for the molecular bond to have length in [l 0 , l 1 ] at time t is l1 l0 |ψ(x, t)| 2 dx. In this note, we focus on the case where V :

x → D e (1 -e -a(x-xe) ) 2 is the celebrated Morse potential [START_REF] Morse | Diatomic Molecules According to the Wave Mechanics[END_REF]. With an abuse of notation we identify the function V with the multiplicative operator ψ → V ψ in L 2 ((0, +∞), C). The constants D e and a represent, respectively, the depth and the width of the potential well, the length x e is the equilibrium length of the bond. For a dihydrogen molecule in standard conditions, D e ≈38292 cm -1 , a≈1.4426 and x e ≈7.741410 -11 m (see [START_REF] Hulburt | Potential energy functions for diatomic molecules[END_REF]). The Morse potential is nowadays essentially used for academic purposes and it has been replaced in practical applications by more realistic potentials, see for instance, [13, page 65] and [START_REF] Le Roy | Accurate analytic potentials for Li2(X 1 Σ + g ) and Li2(A 1 Σ + u ) from 2 to 90[END_REF]. It has the advantage of a complete explicit analytic description of the eigenstates of the free Hamiltonian. Indeed, the operator -∆ + V admits a finite number of simple eigenvalues

λ j = hν 0 j + 1 2 - hν 0 j + 1 2 2 4D e
where ν 0 = a 2π 2D e /m and j = 1, . . . , 2D e /(hν 0 ) -1. The associated eigenfunctions φ j may be expressed in terms of Laguerre polynomials. A direct computations shows that, for every j = k, (1.2) φ j , xφ k = 0.

In the present analysis we consider as control operator W (x) = x. We consider controls u : [0, T ] → [0, +∞) which are piecewise constant. To define the solution of (1.1) associated with an admissible control we consider, for every constant u ≥ 0, the operator -∆ + V + uW which is the generator of a unitary group on L 2 ((0, +∞), C), hence one can define the solution of (1.1) for every initial condition. By concatenation, one defines the solution t → Υ u t,0 ψ 0 of (1.1) for every initial condition ψ 0 , for every piecewise constant function u with value in [0, +∞).

Control problem

An important question for applications is the existence and the efficient design of controls u steering the wave function ψ from a given initial state ψ 0 to a given desirable target ψ f , with a particular focus on the transfer from one energy level to another.

This question has been tackled by many authors, using different approaches, in the last decades. The linear test [START_REF] Beauchard | Local controllability of a 1-D Schrödinger equation[END_REF][START_REF] Beauchard | Controllability of a quantum particle in a moving potential well[END_REF][START_REF] Beauchard | Local controllability of 1D linear and nonlinear Schrödinger equations with bilinear control[END_REF], Lyapunov approach [START_REF] Mirrahimi | Lyapunov control of bilinear Schrödinger equations[END_REF][START_REF] Beauchard | Practical stabilization of a quantum particle in a one-dimensional infinite square potential well[END_REF][START_REF]Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications[END_REF][START_REF] Nersesyan | Growth of Sobolev norms and controllability of the Schrödinger equation[END_REF][START_REF] Beauchard | Semi-global weak stabilization of bilinear Schrödinger equations[END_REF] were used to give a precise description of the attainable set of toy models of (1.1) where x belongs to a compact set (and -∆ + V thus has a discrete spectrum). Very few works have considered the actual form of (1.1), including the continuous part of the spectrum of -∆ + V . Let us cite [START_REF] Mirrahimi | Lyapunov control of a particle in a finite quantum potential well[END_REF] (Lyapunov techniques) and [START_REF] Chambrion | Periodic excitations of bilinear quantum systems[END_REF] (averaging techniques). In both cases, the function W , and hence the associated multiplication operator in L 2 ((0, +∞), C), was assumed to be bounded.

Results of approximate simultaneous controllability in a more general framework (in the sense of Section 2 below) using piecewise constant controls have been firstly introduced in [START_REF] Chambrion | Controllability of the discrete-spectrum Schrödinger equation driven by an external field[END_REF] and then refined following works (see, for instance, [START_REF] Boscain | A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule[END_REF] and [START_REF] Boscain | Multi-input Schrödinger equation: Controllability, tracking, and application to the quantum angular momentum[END_REF]).

Contribution

Combining averaging techniques and spectral theory, we prove the simultaneous approximate controllability of (1.1) in the space spanned by the two first eigenvectors. More precisely we have the following result. To the best of our knowledge, this result is the first positive approximate controllability result of a bilinear quantum dynamics when the interaction potential W is unbounded and the spectrum of the free Hamiltonian -∆ + V has a continuous part.

Content

In Section 2, we introduce our assumptions in an abstract framework that will be used as the "black box" requirements in our application. The idea is that the interested reader can include a variety of other examples based on this template. Thus this framework includes the dynamics (1.1) with W : x → x. In Section 3, we recall some averaging results for bilinear quantum systems when there exists a Hilbert basis made of eigenvectors of the free Hamiltonian. In Section 4, we first use spectral analysis in Section 4.1 to study the discrete spectrum of approximations of the dynamics (1.1). Then, we use the results of Section 3 in Section 4.2 to prove Proposition 1.1. From these assumptions we deduce that for every u ≥ 0, i(A + uB) is bounded from below. Moreover for every u > 0, the form domain Q(iA + uiB) of iA + uiB is included in the form domain Q(iuB) of iB and iA + uiB ≥ iuB in the form sense.

For every u > 0, A + uB is a well-defined skewadjoint operator and its form domain is Q(A) ∩ Q(B): the intersections of the form domains of A and B, see [START_REF] Reed | Methods of modern mathematical physics[END_REF]Theorem VIII.15], as the associated quadratic form is a closed positive quadratic form. Since the form domain is invariant by the unitary group e t(A+uB) , by interpolation see [1, Section 2.8], the space Q(A)∩Q(B) is invariant by e t(A+uB) for any u > 0.

We denote by φ 0 1 , φ 0 2 , . . . , φ 0 n an orthonormal family of eigenvectors of A associated with the family of eigenvalues {iλ 1 , iλ 2 , . . . , iλ n }, and with (φ u j ) j∈N a Hilbert basis of H made of eigenvectors of A + uB associated with the family of eigenvalues (-iλ u j ) j∈N .

Proposition 2.1. The dynamics (1.1) fits within the frame of Assumption 1 with

1. A = i(-∆ + V ) with domain H 2 (0, +∞) ∩ H 1 0 (0, +∞) (Dirichlet boundary conditions at 0) ; 2. B = -iW , where W (x) = x, with its maximal domain {ψ ∈ L 2 (0, +∞), Bψ ∈ L 2 (0, +∞)} ; 3. the constants - 16 D e 2 -56 h ν 0 D e + 41 h 2 ν 0 2 -4 D e + h ν 0 2 h ν 0 and 16 D e 2 -56 h ν 0 D e + 41 h 2 ν 0 2 + 4 D e -h ν 0 2 h ν 0
are not integers larger than 3 and smaller than 2D e /(hν 0 ) , and

56 h ν 0 D e -41 h 2 ν 0 2 < 16D 2 e .
Proof. Assumptions 1.1 and 1.2 are immediate as V -D e is a Kato-Rellich perturbation of -∆ + D e . Assumptions 1.3 is straightforward.

To check the first part of Assumption 1.4, one computes φ 1 , Bφ 2 , which is not zero (see [11, Eq. ( 21)]).

To check the second part of Assumption 1.4, one computes

2λ 2 -λ 1 = 56 h ν 0 D e -41 h 2 ν 0 2
16 D e and one solves 2λ 2 -λ 1 = λ j for j ∈ N. The solutions (if any) are

j 1 = - 16 D e 2 -56 h ν 0 D e + 41 h 2 ν 0 2 -4 D e + h ν 0 2 h ν 0 and j 2 = 16 D e 2 -56 h ν 0 D e + 41 h 2 ν 0 2 + 4 D e -h ν 0 2 h ν 0 .
The non-degeneracy hypothesis is verified as soon as both j 1 and j 2 are not integers larger than 3 and smaller than 2D e /(hν 0 ) , and 2λ 2 -λ 1 < 0. These conditions are verified for the H-H molecule (λ

1 = D e -103.2 kcal/mol, λ 2 = D e -91.4 kcal/mol, hence 2λ 2 -λ 1 = D e -79.6 kcal/mol thus 2λ 2 -λ 1 < D e and λ 3 = D e -80.3 kcal/mol < 2λ 2 -λ 1 < λ 4 = D e -69.8 kcal/mol).
The proof of Assumption 5 follows from [22, Theorem XIII.64.(iv)]. Indeed the unit ball associated to the form -∆ + V + uW is included in H 1 0 (0, ∞) and in the unit ball form the norm u → √ xu which provides the localization needed for the Rellich-Kondrachov theorem. The fact that the eigenvectors are in the domain of B comes from the exponential decay of eigenvectors [START_REF] Hislop | Exponential decay of two-body eigenfunctions: a review[END_REF]Theorem 4.1]. Then since H 2 (0, +∞) is the maximal domain of the Laplacian it follows that the eigenvectors are in H 2 (0, +∞). The boundary condition is contained in the form domain definition.

Let Φ = (φ j ) j∈N be a Hilbert basis of H contained in the domains of A and B (such a basis exists by Assumption 1.5). For every N in N, denote by L Φ N the linear space spanned by φ 1 , φ 2 , . . . , φ N and by π Φ N : H → L Φ N the associated orthogonal projection. We define the compression of A (resp. B) in basis Φ at order N as the linear operator

A φ N = π Φ N A L Φ N (resp. B φ N = π Φ N B L Φ N ).

Definition of propagators

In general, Assumption 1 is not enough to define the solutions of (2.3) for usual functional classes such as continuous or locally integrable functions u. We restrict ourselves to the much smaller class of piecewise constant functions:

Definition 1. A function u : [0, +∞) → R is piece- wise constant if there exists an increasing unbounded sequence (t i ) i∈N such that u |(ti,ti+1
) is constant (with value denoted u i ) for every i ∈ N.

Let (A, B) satisfy Assumption 1. For every piecewise constant function u : [0, +∞) → [0, +∞), we define the propagator t ∈ [0, +∞) → Υ u t of (2.3) by concatenation

Υ u t = e (t-t l )(A+u l B) • • • • • e t1(A+u0B)
for t belonging to [t l , t l+1 ). For every s, t ≥ 0, we define also

Υ u t,s = Υ u t • (Υ u s ) -1 . Notice that if u > 0 then Υ u t leaves invariant Q(A) ∩ Q(B).
Using Assumption 1.5, we can prove the strong continuity of the map u → Υ u t for any fixed t: For any ψ and φ in Q(A) ∩ Q(B) we have for any positive piecewise constant functions u and v ψ,

Υ u t φ -ψ, Υ v t φ = t 0 ψ, Υ u s (u(s) -v(s))BΥ v s φ ds
which provides the continuity of u → ψ, Υ u t φ in the L 1 norm whenever we consider positive piecewise constant functions. For φ and ψ in H we conclude by density and thus obtain the weak continuity. The general case of non-negative piecewise constant functions follows from the strong convergence of e t(A+δB) to e tA as δ → 0 as A+δB tends to A in the strong resolvent sense. For this we have to use a symmetrized version of the resolvent expansion, see [START_REF] Jensen | A unified approach to resolvent expansions at thresholds[END_REF]. Finally, the strong continuity is obtained by the fact that the operators are unitary.

3 Averaging techniques in quantum control Notice that under the assumptions of Proposition 3.1, A has no essential spectrum.

General principle

For finite dimensional systems, averaging methods consist in replacing oscillating dynamics ẏ = f (t, y) by its average ż = f z where f (z) := lim 1 T T 0 f (t, z)dt. When the dynamics f is sufficiently regular, the solutions y and z of ẏ = f (t, y) and ż = f (z) have similar behaviors (difference of the order of ) for time less than 1/ . Averaging theory has grown to a whole theory in itself, we refer to [START_REF] Sanders | Averaging methods in nonlinear dynamical systems[END_REF] for an introduction. In quantum mechanics, averaging theory has been extensively used (under the name of "Rotating Wave Approximation") since the 60's, for finite dimensional systems. Let us briefly recall the core ideas of the method.

Up to the change of variable y = e -tA x, the system ẋ = A+u(t)Bx may be rewritten as ẏ = u(t)e -tA Be tA y. If A is diagonal with diagonal entries (iλ j ) 1≤j≤N and B has entries (b jk ) 1≤j,k≤N , then u(t)e -tA Be tA has entries (b jk u(t)e (λj -λ k )t ) 1≤j,k≤N . Choosing u small enough and T = 2π/(λ 2 -λ 1 ) periodic such that

(3.4) T 0 b jk u(t)e (λj -λ k )t dt = 0, 2 ≤ j, k ≤ N,
then the averaged system behaves like a rotation in the plane spanned by the first two eigenvectors of A. The non-degeneracy condition 1.4 is indeed a sufficient condition to ensure the existence of a periodic control u satisfying the moment vanishing hypothesis (3.4).

A simple yet wrong proof of Proposition 3.1

Extending the finite dimensional procedure described in Section 3.2 to the infinite dimensional setting of Proposition 3.1 is not completely straightforward. Our proof is inspired by [9, Theorem 1]. Let us assume for a while that (2.3) admits strong solutions for control u of class C 1 , that the propagator u → Υ u is continuous with respect to the L 1 norm and that some Duhamellike formula is available. Projecting (2.3) on the N -dimensional space L Φ N , we get

π Φ N d dt ψ = A Φ N π Φ N Aπ Φ N ψ + u(t) B Φ N π Φ N Bπ Φ N ψ +u(t)π Φ N B(1 -π Φ N )ψ. (3.5)
Let X u,(N ) be the propagator of the finite dimensional dynamics x = A Φ N x + u(t)B Φ N x. Using Duhamel formula, we deduce relation between the infinite dimensional propagator Υ u and X u,(N ) which is

π Φ N Υ u t ψ 0 = X u,(N ) t,0 ψ 0 + t 0 u(s)X u,(N ) t,s B(1 -π Φ N )Υ u s ψ 0 ds
Projecting again on L 2 and recalling that π Φ 2 and π Φ N do commute we have

π Φ 2 Υ u t ψ 0 = π Φ 2 X u,(N ) t,0 ψ 0 + t 0 u(s)π Φ 2 X u,(N ) t,s B(1 -π Φ N )Υ u s ψ 0 ds = π Φ 2 X u,(N ) t,0 ψ 0 + t 0 u(s)X u,(N ) t,s π Φ 2 B(1 -π Φ N )Υ u s ψ 0 ds (3.6) + t 0 u(s)[X u,(N ) t,s , π Φ 2 ]B(1 -π Φ N )Υ u s ψ 0 ds.
We choose now to use a control u = u * /n with n large enough, where u * is a continuous function satisfying the vanishing moments condition (3.4). Then, by [START_REF] Chambrion | Periodic excitations of bilinear quantum systems[END_REF], there exists T * such that X u,(N ) nT * ,0 exchange φ 1 and φ 2 , up to an arbitrary small error.

Notice, and this is crucial for our purpose, that the L 1 norm of u on [0, nT * ] is uniformly bounded in n (since it converges to the L 1 norm of u * on [0, T * ]). As a consequence, the last two terms of (3.6) tend to zero as N and n tend to infinity. Indeed

t 0 u(s)X u,(N ) t,s π Φ 2 B(1 -π Φ N )Υ u s ψ 0 ds ≤ π Φ 2 B(1 -π Φ N ) nT * 0 |u(s)|ds (3.7) tends to zero as N tends to infinity (uniformly in n) since π Φ 2 B(1 -π Φ N ) = (1 -π Φ N )Bπ Φ 2 tends to zero. While t 0 u(s)[X u,(N ) t,s , π Φ 2 ]B(1 -π Φ N )Υ u s ψ 0 ds ≤ sup s,t [X u,(N ) t,s , π Φ 2 ] B(1 -π Φ N ) nT * 0 |u(s)|ds (3.8)
tends to zero as n tends to infinity (for any fixed N ) by finite dimensional averaging.

In conclusion, π Φ 2 Υ u nT * ψ 0 tends to π Φ 2 X u * /n,(N ) nT * ,0 ψ 0 as n tends to infinity which is the expected result if u n = u * for every n. If u n = u * but (u n ) n tends to u * in L 1 sense, the conclusion follows from the continuity of the propagators.

Two major flaws

The sketch of proof presented in Section 3.3 is not correct for the following reasons.

The first reason is that the only way to guarantee the vanishing moments condition (3.4) for any N is to use u * in a sine form, for instance u * (t) = 1 + sin((λ 2 -λ 1 )t), however in the framework of Assumption 1 the propagators are defined only for piecewise constant functions (Section 2.2).

The second flaw in the proof of Section 3.3 is that in general, even for piecewise constant control function, (2.3) does not admit strong solutions, only a very weak notion of solutions (where the test vector are taken in the space spanned by the eigenvectors of A). In particular, there is no hope, in general, to prove a Duhamel-like formula for this dynamics.

M † = φ 1 , Bφ 1 φ 1 , Bφ 2 i 2 φ 2 , Bφ 1 i 2 φ 2 , Bφ 2
and the function v * as the reciprocal bijection of t → t + 1-cos((λ2-λ1)t) λ2-λ1 . To begin with, we recall a finite dimensional convergence estimate that will be instrumental in the following: Proposition 3.2. For any N in N, there exists K N > 0 such that, for every n in N,

π Φ 2 X u∞ n ,(N ) t,s π Φ 2 -exp (v * (t) -v * (s))M † ≤ K N n v * t n -v * s n . ( 3
.9) Proposition 3.2 follows from a straightforward computation. A proof and the explicit expression of K N can be found in [9, Eq. ( 12)].

We now come back to the proof of Proposition 3.1. Let ε > 0. There exists

N in N such that π Φ 2 B(1 - π N ) < ε K .
By assumption, there exists n 0 in N such that for n ≥ n 0 we have that u n -u ∞ L 1 ([0,T ]) < ε/(K + 1)/ B Φ N ). Finally, from (3.9), there exists n 1 in N such that, for every n ≥ n 1 , for every s, t in [0, nT * ],

π Φ 2 X u∞ n ,(N ) t,s π Φ 2 -exp (v * (t) -v * (s))M † ≤ ε K and [X u∞ n ,(N ) (t,s) , π Φ 2 ] < ε/K. Let n > max n 0 , n 1 , 2 √ 2K N T * ε .
There exists a finite sequence 0 = t 0 < t 1 < . . . < t p = nT * such that u n is constant on each interval (t i , t i+1 ), i = 0, ..., p -1. Then

π Φ 2 Υ un n nT * ,0 π Φ 2 -exp(KM † )π Φ 2 ≤ π Φ 2 Υ un n nT * ,0 -X un n ,(N ) nT * ,0 π Φ 2 + π Φ 2 X un n ,(N ) nT * ,0 -X u∞ n ,(N ) nT * ,0 π Φ 2 + π Φ 2 X u∞ n ,(N ) nT * ,0 π Φ 2 -exp(KM † )π Φ 2 (3.10)
We give below an estimate of each of the three terms appearing in the RHS of (3.10). The first term is the most difficult to handle. We first notice that, on each interval (t i , t i+1 ), i < p, the solution of (2.3)

with u = u n /n such that ψ(t i ) ∈ span(φ 0 1 , φ 0 2 ) is such that, for any t in (t i , t i+1 ), π Φ 2 ψ(t) = π Φ 2 X un n ,(N ) t,ti ψ(t i ) + ti+1 ti u n (s) n [π Φ 2 , X un n ,(N ) t,s ]B(1 -π N )ψ(s)ds + ti+1 ti u n (s) n X un n ,(N ) t,s π Φ 2 B(1 -π N )ψ(s)ds. (3.11)
The proof of (3.11) follows exactly the computation (3.6) with the difference that since u n is constant on (t i , t i+1 ) and span(φ 0 1 , φ 0 2 ) ⊂ D(A + un n B), ψ is a strong solution of (2.3) and the Duhamel formula is now valid.

A straightforward computation gives, for every

s, t ≤ nT * : [π Φ 2 , X un n ,(N ) r,s ] ≤ [π Φ 2 , X u∞ n ,(N ) r,s ] + 2 X u∞ n ,(N ) r,s -X un n ,(N ) r,s ≤ ε K + B Φ N 1 n s r |u ∞ -u n |(t)dt ≤ 2 ε K . (3.12)
As a consequence of (3.11) and (3.12), for every t in (t i , t i+1 ),

π Φ 2 Υ un n t,ti π Φ 2 -π Φ 2 X un n ,(N ) t,ti π Φ 2 ≤ ti+1 ti u n (s) n ds × sup r,s∈(ti,ti+1) [π Φ 2 , X un n ,(N ) r,s ] + π Φ 2 B(1 -π N ) ≤ 3ε K ti+1 ti |u n (s)| ds. (3.13) One has π Φ 2 X un n ,(N ) t,ti π Φ 2 = π Φ 2 exp(v * ( t i+1 n ) -v * ( t i n ))M † ) +π Φ 2 (X u∞ n ,(N ) t,ti -exp(v * ( t i+1 n ) -v * ( t i n ))M † ))π Φ 2 +π Φ 2 (X un n ,(N ) t,ti -X u∞ n ,(N ) t,ti )π Φ 2 . hence π Φ 2 X un n ,(N ) t,ti π Φ 2 ≥ 1 - B Φ N n ti+1 ti |u n -u ∞ |(s)ds + 2K N |t i+1 -t i | n 2
and, since 1

-(1 -u 2 ) ≤ √ 2u for u in [0, 1], (1 -π Φ 2 )X un n ,(N ) t,ti π Φ 2 ≤ √ 2 B Φ N n ti+1 ti |u n -u ∞ |(s)ds + 2K N |t i+1 -t i | n 2 .
Since Υ un n t,ti π Φ 2 = 1, using Pythagoras Lemma and the property 1

-(1 -u) 4 ≤ 4u for u in [0, 1], one gets (1 -π Φ 2 )Υ un n ti+1,ti π Φ 2 = 1 -π Φ 2 Υ un n ti+1,ti π Φ 2 2 ≤ 1 -π Φ 2 X un n ,(N ) t,ti π Φ 2 - 3ε nK ti+1 ti |u n (s)| ds 2 ≤ 1 -π Φ 2 X un n ,(N ) t,ti π Φ 2 - 3ε nK ti+1 ti |u n (s)| ds 4 ≤ 4 n B Φ N ti+1 ti |u n -u ∞ | + 2 K N n |t i+1 -t i |+ + 3ε K ti+1 ti |u n | Noticing that π Φ 2 Υ un n nT * ,0 π Φ 2 -π Φ 2 X un n ,(N ) nT * ,0 π Φ 2 ≤ π Φ 2 (Υ un n tp,tp-1 -X un n ,(N ) tp,tp-1 )Υ un n tp-1,0 π Φ 2 + π Φ 2 X un n ,(N ) tp,tp-1 (Υ un n tp-1,0 -X un n ,(N ) tp-1,0 )π Φ 2 ≤ π Φ 2 (Υ un n tp,tp-1 -X un n ,(N ) tp,tp-1 )π Φ 2 + π Φ 2 (Υ un n tp,tp-1 -X un n ,(N ) tp,tp-1 )(1 -π Φ 2 ) + π Φ 2 X un n ,(N ) tp,tp-1 (Υ un n tp-1,0 -X un n ,(N ) tp-1,0 )π Φ 2 ≤ π Φ 2 (Υ un n tp,tp-1 -X un n ,(N ) tp,tp-1 )π Φ 2 + π Φ 2 (Υ un n tp,tp-1 -X un n ,(N ) tp,tp-1 )(1 -π Φ 2 ) + π Φ 2 X un n ,(N ) tp,tp-1 (1 -π Φ 2 )(Υ un n tp-1,0 -X un n ,(N ) tp-1,0 )π Φ 2 + π Φ 2 X un n ,(N ) tp,tp-1 π Φ 2 (Υ un n tp-1,0 -X un n ,(N ) tp-1,0 )π Φ 2 ≤ π Φ 2 (Υ un n tp,tp-1 -X un n ,(N ) tp,tp-1 )π Φ 2 + π Φ 2 (Υ un n tp,tp-1 -X un n ,(N ) tp,tp-1 )(1 -π Φ 2 ) + π Φ 2 X un n ,(N ) tp,tp-1 (1 -π Φ 2 ) + π Φ 2 (Υ un n tp-1,0 -X un n ,(N ) tp-1,0 )π Φ 2 ,
one gets after an easy induction

π Φ 2 Υ un n nT * ,0 π Φ 2 -π Φ 2 X un n ,(N ) nT * ,0 π Φ 2 ≤ ε(32 + 4 √ 2).
The last two terms of (3.10) are easier to treat, since

π Φ 2 X un n ,(N ) nT * ,0 -X u∞ n ,(N ) nT * ,0 π Φ 2 ≤ ε B Φ N B Φ N = ε
and, by definition of n 1 , In our example,

π Φ 2 Υ u∞ n nT * ,0 π Φ 2 -exp(KM † )π Φ 2 ≤ ε. Finally, π Φ 2 Υ un n nT * ,0 π Φ 2 -exp(KM † )π Φ 2 ≤ ε(34 + 4 √ 2
Q(A) = H 1 0 ((0, +∞)) et Q(A + B) = H 1 0 ((0, +∞)) ∩ Q( √ x).
Notice that the second domain is everywhere dense in the first one for the topology of H 1 ((0, +∞)). Moreover, the exponential decay at infinity of the eigenvectors (see Proposition 2.1) ensures that the eigenvectors of A and A + B belong to the domains of A and A + B.

In the sequel N := 2D e /(hν 0 ) -1 is the number of eigenvalues of A and L Φ N the linear space spanned by (φ 0 j ) j∈N a Hilbert basis of H made of eigenvectors of A. Proof. Since ∈ (0, +∞) → λ n (A + B) is nondecreasing, for every > 0, we got λ n (A+ B) ≥ λ n (A).

Consider now the compression

A Φ n + B Φ n and λ Φ k,n its k-the eigenvalue. It is larger than λ k (A + B) (see [22, Theorem XIII.3]) . Then, since B Φ n is finite here for k ≤ n λ Φ k,n ≤ λ k (A) + B Φ n and thus λ k (A + B) ≤ λ k (A) + B Φ n . Hence lim →0 λ k (A + B) = λ k (A).
We can even impose the convergence of the eigenvectors.

Proposition 4.2. For j an integer smaller than N the family (φ j ) j∈N,j≤N of eigenvectors of A + B associated to (λ j (A + B)) j∈N,j≤N can be chosen, if is small enough, such that φ j → φ 0 j as → 0.

Proof. Recall that we write λ n for λ n (A + B).

Let us build (up to normalization) φ j as φ 0 j + η( ) with η( ) orthogonal to φ 0 j smooth and exponentially decaying. The equation (A + B)φ j = λ j φ j leads to λ j + φ 0 j , B(φ 0 j + η( )) = λ j Aη( ) + π ⊥ j B(φ 0 j + η( )) = λ j η( ) where π ⊥ j is the orthogonal projector onto {φ 0 j } ⊥ . The second equation solves as

η( ) = -A + π ⊥ j B -λ j -1 π ⊥ j Bφ 0 j .
As long as j ≤ N the invertibility of the operators follows from the convergence of the corresponding discrete part of the spectrum exactly as in Proposition 4.1. Notice that plugging the above solution into the first equation leads to fixed point problem that can be solved by means of the implicit function theorem using (1.2). This provides an alternative proof to Proposition 4.1.

Conclusions

This note presents an example of positive approximate controllability result for a bilinear quantum dynamics when the free Hamiltonian has a continuous part. The main ingredients of the proof were the min-max principle, for the spectrum perturbation, and the averaging techniques, to obtain controllability. Unlike most of the recent controllability results for conservative quantum dynamics obtained with geometric techniques, the method used for this particular example heavily depends on the fact that the spectrum of the free Hamiltonian is bounded from below. In particular, the method presented in this note cannot be adapted to the control of bilinear Dirac dynamics.

Nonetheless the provided "black box", applied here in a one dimensional framework with dipolar interaction, can be used in a variety of examples that we think to be of some useful physical interest.

Proposition 1 . 1 .

 11 Let W : x → x. There exists a sequence of piecewise constant functions u n : [0, T n ] → [0, +∞) such that the two sequences (| φ 2 , Υ un Tn,0 φ 1 |) n∈N and (| φ 1 , Υ un Tn,0 φ 2 |) n∈N tend to one as n tends to +∞.

3. 1

 1 Result The aim of this Section is to prove the following Proposition 3.1. Let (A, B) satisfy Assumptions 1.1, 1.2, and 1.4. Assume that there exists a Hilbert basis (φ j ) j∈N of H made of eigenvectors of A contained in the domain of B. Let T = 2π/|λ 2 -λ 1 | and u n be a sequence of positive T -periodic piecewise constant functions converging in L 1 ([0, T ]) to u ∞ : t → 1 + sin((λ 2λ 1 )t). Then there exists T * such that the two sequences (| φ 2 , Υ un/n nT * ,0 φ 1 |) n∈N and (| φ 1 , Υ un/n nT * ,0 φ 2 |) n∈N tend to one as n tends to infinity.

3. 5

 5 Proof of Proposition 3.1 For the ease of notation, we introduce T * = π | φ1,Bφ2 | (the charateristic time appearing in Proposition 3.1) which happens to be equal to K = T * 0 |u ∞ | (the L 1 norm of u ∞ [0, T * ]). We also introduce the 2 × 2 matrix

Theorem 4 . 1 .

 41 If A is a self-adjoint operator on H bounded from below, A ≥ cI in the form sense for some c ∈ R. Define λ n (A) = sup φ1,...,φn-1 inf ψ∈[φ1,...,φn-1] ⊥ ψ∈Q(A), ψ =1 (ψ, Aψ) . Then, for each n, either (a) there are n eigenvalues (counting multiplicity) below the bottom of the essential spectrum, and λ n (A) is the n-th eigenvalue counting multiplicity; or (b) λ n (A) is the bottom of the essential spectrum, i.e., λ n (A) = inf{λ : σ ess (A)} and in this case λ n (A) = . . . = λ n+k (A) = . . . an there are at most n -1 eigenvalues below λ n (A). Proof. See [22, Theorem XIII.1]. In [22, Theorem XIII.2] the domain Q(A) is considered in the form sense.

Proposition 4 . 1 .

 41 Let (A, B) satisfy Assumption 1. Then, for every n ≤ N , the eigenvalue λ n (A + B) of A + B tends to λ n (A) as tends to zero.

4. 2

 2 Proof of Proposition 1.1 Since λ 1 (A + B) tends to λ 1 (A) and λ 2 (A + B) tends to λ 2 (A) as tends to zero (Proposition 4.1), (1, 2) is a weakly non degenerate transition of (A + B, B) for small enough. Then, by Proposition 3.1, defining u : t → + sin((λ 1 -λ 2 )t), the two sequences (| φ 2 , Υ u/n nT * ,0 φ 1 |) n∈N and (| φ 1 , Υ u/n nT * ,0 φ 2 |) n∈N tend to one. Conclusion follows from the convergence of φ 1 and φ 2 to φ 0 1 and φ 0 2 respectively as tends to zero (Proposition 4.2).

* Supported by European Research Council ERC StG 2009 "GeCoMethods", contract number 239748.


Mahian

S Zeinali Heris

M A Razbani

P Estellé

O Mahian

Rheological behavior of zinc-oxide nanolubricants

Keywords: nanolubricant, ZnO nanoparticles, viscosity, yield stress, aggregates

  L'archive ouverte pluridisciplinaire

Introduction

It is well understood that adding nanoadditives like metal dichalcogenides to common lubricants can enhance their tribological properties [1]. In fact, the nanolubricant approach is used to overcome the disadvantages of conventional anti-wear and friction reduction additives related to their need of chemical reactions with substrates and the resulting induction period to produce the tribofilm on sliding surfaces [2]. These nanoengineered lubricants can also be classified as a new kind of fluid, namely nanofluid [START_REF] Choi | Enhancing thermal conductivity of fluids with nanoparticles[END_REF]. Nanofluids are stable suspensions of particles with less than 100 nm in diameter dispersed in base fluids (BFs) which are commonly oils and heat transfer mediums like water and ethylene glycol. The most beneficial characteristics of nanofluids are their enhanced thermal properties in comparison to the BF as reviewed recently by Kleinstreuer et al. [START_REF] Kleinstreuer | [END_REF]. Because one of the main responsibilities of lubricants is heat transfer from heat loaded components, utilizing of nanoparticles as additives in lubricants becomes even more interesting.

One of the most crucial characteristics of lubricants is their rheological properties which are tuned to work in specific conditions. Adding nanoparticles can change these properties like increasing the viscosity of the lubricant that should not be compromised [5]. Regarding the fact that heat transfer characteristics of nanofluid and the tribological properties of nanolubricants have been under focus by researchers more than their rheological properties, addressing these properties in the intersection of nanofluid and lubrication has been discussed previously [6][7][8][9][10][11][12][13][14]. However, when it comes to mineral based lubricants there are few researches in this field [6,[8][9][10]13]. In this work a ZnO nanoparticle based lubricant was prepared and its rheological properties measured and analysed. To the best of our knowledge, there is no published research that investigated rheological properties of a mineral based lubricant with ZnO nanoparticles as additive.

Nanolubricants are mainly prepared dispersing synthesized nanoparticles in common non-polar solvents and mineral oils to provide a stable and transparent solution [15]. Most of the nanolubricants has been made from synthetic based lubricants and have been characterized for their anti-wear and anti-friction properties. For example, Hernandezbattez et al. [16] investigated the tribological behaviour of ZnO nanoparticles as an additive to PAO6 without addressing rheological phenomena.

From a rheological point of view, nanofluids can be divided into Newtonian [17][18][19] and non-Newtonian materials [20][21][22]. Adding nanoparticles to a BF can also change its rheological properties. Determining whether a nanofluid behaves is not straightforward. Some researchers reported that nanofluid behaviour can change from non-Newtonian to Newtonian following test conditions [6,14,23,24]. The main parameters which govern the rheological properties of nanofluids are temperature [7,25], nanoparticle loading [11,12], (3) shear rate [6,14,26,27], aggregation of nanoparticles [8,27,28,29] and morphology of nanoparticles [5,21,30,31].

One of the challenges in producing nanofluids is the aggregation of nanoparticles.

There are some theoretical frameworks which deal with this issue in aqueous solutions like extended DLVO theory [32] which describes the stability of nanofluids. The stability of nanofluid becomes more challenging when nanoparticles are dispersed in a non-aqueous BF [33]. The effects of aggregation on the rheological properties of nanofluids were addressed in a lot of researches [18,27,[34][35][36][37]. As explained in [14,18], the presence of aggregates contributed to non-Newtonian behaviour. However, in the case of mineral based nanolubricants, few reports addressed the effects of aggregation on the rheological properties of nanofluid [8,11].

Chou et al. [6] dispersed nanodiamond particles in commercial oil and investigated tribological and rheological properties of nanofluid. They showed that nanofluid behaviour is consistent with Bingham plastic fluid. Among the most recent studies, Kole et al. [8] investigated the effect of aggregation on the viscosity of copper oxidegear oil nanofluids. They reported presence of aggregated CuO nanoparticles in the fluid, with average cluster size ~7 times the primary diameter of CuO nanoparticles.

The rheological properties of the nanofluid followed Newtonian and shear thinning model at low and high volume fraction of nanoparticles respectively. Aggregation of nanoparticles in nanofluids can also induce yield stress, which is the required shear stress for a fluid to start to flow. It was shown that the yield stress increased significantly under conditions that were favourable for the formation of complex net-like aggregate structures, such as high solid volume fractions, pH values near the iso-electric point, and high ionic strengths [38]. In addition, all of the researchers showed that with an increasing nanoparticles volume fraction, the yield stress increases [6,37,[39][40][41].

In this work, nanolubricant was prepared dispersing ZnO nanoparticles in commercial mineral based turbine oil. A full characterization of rheological properties of the resulting nanolubricant is performed. The effects of volume fraction of nanoparticles and temperature are discussed. Appropriate rheological models based on contemporary theories are proposed to correlate and predict the experimental data.

Materials and methods

Nanolubricant

Rod-like zinc oxide nanoparticles used in this study were supplied by TECNAN Navarrean Nanoproduction Technology. The ZnO nanoparticles are quite spherical but not necessarily uniform in morphology and can be in the form of aggregates. Behran Turbine SP-1 32 turbine oil was supplied by Behran Oil Company. It is a mineral based oil lubricant and was used as BF for preparing nanofluids. The overwhelming majority of compressors are best served by premium grade turbine oils with ISO viscosities 32 or 46 [START_REF] Bloch | Practical lubrication for industrial facilities[END_REF]. According to manufacturer's specifications, the nanoparticles are mainly elongated with an average particle size of 25nm, with a purity of 99.98%, a density of 5.6 g/cm 3 and a specific surface area of 35 to 50m 2 /g. The density of the Behran Turbine SP-1 32 oil at 15.6°C is 0.87 g/cm 3 . Its viscosity is 35cSt at 40°C, 5.5cSt at 100°C, and its viscosity index is 104.

Nanolubricants using ZnO nanoparticles with 0.1, 0.5, 1.0, 2.0, 3.0, and 4.0% by weight were formulated, which correspond nearly to 0.01, 0.07, 0.1, 0.3, 0.4, and 0.6% by volume at ambient temperature. Sample preparation was carried out at ambient temperature using a very sensitive mass balance with an accuracy of 0.1 mg. Intensive ultrasonication (Misonix Incorporated, Model XL2020) was performed continuously for 2 h after mixing with BF to possibly de-agglomerate the nanoparticles. Finally, the suspension was homogenized for 1h by magnetic force agitation. The prepared nanofluids do not display any visual sedimentation of ZnO nanoparticles even after keeping the fluids stationary for one day.

Rheological measurements

Rheological behaviour of the nanolubricants and BF was measured with Brookfield programmable viscometer (LVDV-III Ultra) connected to a PC to record the data.

WiseCircu temperature controlled bath which can vary the fluid temperature between -20 and 100°C and control to ±0.1 °C was used to stabilize the sample temperature. The viscometer drives a spindle immersed in nanofluid sample. Due to rotation of the spindle, a viscous drag of the fluid against the spindle is developed, which is measured by the deflection of the calibrated spring. Model LVDV-III Ultra viscometer has a viscosity measurement range between 1.3 and 30,000 cP. Spindle SC-18 was used in this study and was calibrated by using Brookfield viscosity standard fluids. The maximum uncertainty was found to be 1.9%. The viscometer contains a sample chamber where the fluid is tested. The temperature inside the sample chamber is carefully monitored using an RTD temperature sensor during the rheological measurements. The spindle type and speed combinations produce satisfactory results when the applied torque is between 10% and 100%; therefore spindle types and speeds are selected in such a way that the imposed torque values lie in this prescribed range. A wide range of spindle speeds is available in this Viscometer (0-250 RPM). All the measurements were performed under steady state conditions. Tests were done at four different temperatures 0, 20, 40, and 60.

Results and discussion

Rheological behaviour of turbine oil

The shear flow behaviour of the turbine oil measured as a function of shear rate between 0 and 60°C is shown in Fig. 1. It can be seen that over the measured temperatures and shear rate ranges, the shear stress linearly increases with shear rate.

The shear rate range is increased when the temperature increase. Regression of shear flow data can be well fitted by the Bingham fluid model as:

γ η τ τ β & + = (1)
where is the shear stress and is yield stress both in dyn/cm 2 , is the plastic viscosity in Poise (P) and is the shear rate in (s -1 ).

Fitting rheological parameters are presented in Table 1. It is clear that the viscosity of turbine oil increased with decreasing temperature as shown in both Table 1 and Fig. 1.

A similar trend is obviously reported for the yield stress of turbine oil.. One of the basic equations that can describe viscosity-temperature relation of mineral based lubricants is Walther-Function which is defined by equation ( 2).

T n b W log )) log(log( - = + = ε ν (2)
in which, ν is the dynamic viscosity (mm 2 /s), ε is equal to 0.7 for the dynamic viscosity higher than 2mm 2 /s, T is the temperature in K, and b and n are constant coefficients which are determined based on the oil type. Based on the experimental data presented in Table 1, equation (2) leads to the following equation with a regression coefficient of 0.99:

T W log 0.03) ± (3.64 0.07) ± (9.29 )) 7 . 0 log(log( - = + = ν (3)

Rheological behaviour of nanolubricants

Shear stress was measured at different shear rates for all the nanolubricant samples. As for the mineral oil, the shear flow curves of the nanolubricants were linear in form.

Then, the Bingham model is also used to describe the shear flow behaviour of nanolubricants. Regression results based on the Bingham model are presented in Table 2. This table shows that the plastic viscosity of nanolubricants mainly increases with volume fraction and decreases when the temperature is increased.

To study the effects of nanoparticles on the plastic viscosity of nanolubricants, relative viscosity is introduced, and writes as follows:

0 0 η η η η - = i (4)
where is the plastic viscosity of nanolubricant and is the plastic viscosity of the mineral oil.

The relative viscosity versus volume fraction is depicted for 20°C in Fig. 2. A second degree polynomial is used to fit the data. Regression coefficient (R 2 ) is almost equal to unity. A same trend was followed by all nanolubricant samples. The results of fitting versus volume fraction with a polynomial of the second degree are presented in Table 3. It is observed in particular that the coefficients of the polynomial are quite identical for all temperatures, indicating that the viscosity of nanolubricants seems to be slightly influenced by temperature change, in particular at low volume fraction as discussed later.

More rigorously, a semi-empirical relationship for the relative shear viscosity covering a broad range of particle volume fraction was obtained by Krieger and Dougherty [START_REF] Krieger | [END_REF]:

p p φ η φ φ η η ] [ 0 ) 1 ( - - = (5)
is the critical particle packing fraction, and is approximately 0.605. [η] is the intrinsic viscosity and equals 2.5 for quite spherical particles. Assuming that hydrodynamic forces are not powerful enough to break the particle aggregation down to individual particles, but increase the volume fraction into an effective volume fraction , equation ( 4) can be modified as follows to take into account aggregates influence [17,18,27]. where is the critical packing fraction of aggregates, is the radius of the aggregate, and is the number of particles contained in an aggregate. Because aggregates are not growing in a simple order, it is assumed that density of packing decrease with moving along the radius of aggregate. The number of particles in an aggregate also takes the following form [17,18]:

p p a φ η φ φ η η ] [ 0 ) 1 ( - - = (6)
D a a a a n       = (8)
Inserting equation ( 8) into (7) gives a φ as follows:

D a a a a - = 3 ) ( φ φ (9)
Combining equations ( 9) and ( 6

-               - = a a a φ η η (11)
From equations ( 3) and ( 11), the relative viscosity of nanolubricants versus volume fraction is plotted in Fig. 3 for / equals to 1, 3, 3.6, and 4. The case of / 3.6 is obtained from fitting the experimental results with the model. In this way, an approximation of aggregation size is obtained from analysing rheological data.

This aggregate size is about 90 nm. On the other hand, for particle volume fraction less than 4% which is the case here, the Krieger and Dougherty model reduces to the Batchelor equation [44]:

2 0 2 . 6 5 . 2 1 φ φ η η + + = (12)
So, replacing by , and replacing equation ( 9) into (12), one obtain:

2 3 3 0 ) ( 2 . 6 ) ( 5 . 2 1       + + = - - D a D a a a a a φ φ η η (13)
After substituting the value of parameters, the relative viscosity finally becomes:

2 127 . 134 628 . 11 1 0 φ φ η η η + = - = i (14)
If one compares equation ( 14) with the results that are shown in Table 3, it reveals that Bachelor equation can well predict the rheological behaviour of nanolubricants if volume fraction is well replaced by the modified volume fraction.

As stated above, rheological properties of nanolubricant and turbine oil were affected by temperature. Fig. 4 shows that all the nanolubricants follow Walther function with no detectable change in the parameters. This means that the temperature effect seems to be mainly governed by the mineral oil characteristics. To investigate the effect of temperature more precisely, versus temperature is plotted in Fig. 5. For low volume fraction of nanoparticle, is almost constant but as the volume fraction grows, starts to fall gradually with increasing temperature. This behaviour can be possibly attributed to the changes in / and D with temperature. With regard to equation ( 11) decreasing / will decrease . One of the factors which may play a role in the size of the aggregates is Brownian movement. The Einstein-Stokes equation can be used to relate Brownian movement [START_REF] Einstein | Investigations on the Theory of the Brownian Movement[END_REF]:

η π a kTt x 3 2 = ( 15
)
in which a is the diameter of particles, k is the Boltzmann constant, is the viscosity, T is the temperature, t is the time, and x is the distance.

Considering equation (15), it is clear that with increasing the temperature, Brownian movement increases. Consequently, increasing Brownian movement increases inter molecular interaction between aggregates especially at high nanoparticle volume fraction which can decrease / . Finally, the relative yield stress of nanolubricants is introduced as follows,

        - = 0 0 β β β β τ τ τ τ (16)
where 0 β τ is the yield stress of BF at the corresponding temperature. Variation β τ versus volume fraction at different temperatures is depicted in Fig. 6. First of all, Fig. 6 shows that the relative yield stress of nanolubricants slowly increases for volume fraction lower than 0.3%. For this volume fraction range, the relative yield stress is quite constant for the tested temperature under the experimental and fitting uncertainties. This indicates the slight influence of temperature for this concentration range. For higher volume fraction up to 0.6%, the relative yield stress strongly depends on both volume fraction and temperature. So, it is observed that the higher the volume fraction and temperature, the higher the relative yield stress of nanolubricants. As mentioned above, increasing the temperature increases the Brownian motion of the particles. Particularly, at high concentrations of nanoparticles the preventing role of nanoparticles on clotting will be diminished and then the yield stress increases.

Conclusion

The rheological behaviour of turbine oil with ZnO nanoparticles were investigated Temperature had direct effect on the viscosity of base fluid and nanofluid but nearly the same pattern followed by both of them. Interestingly, at high volume fraction of nanoparticles and high temperatures, the effect of temperature was intensified in such a way that increasing the temperature would weaken viscosity magnifying effect of nanoparticles on nanolubricants. Inversely, the yield stress of nanolubricants is increased with both temperature and volume fraction.

  equations, D is a measure of how the density of the packing varies along the radius. For nanofluid containing quite spherical particles, D is about 1.8. By substituting as 2.5 and D as 1.8, equation(10)

  experimentally as a function of volume fraction and temperature. It was shown that this kind of dispersion behaves like Bingham fluid. To study the effect of the nanoparticle volume fraction and temperature on the viscosity of nanolubricant, Krieger and Dougherty equation was used with modified effective volume fraction. The results showed that nanoparticles are packed within the base fluid in aggregates with diameter about 3.6 times of a single particle. Replacing the modified volume fraction with the correct aggregation size into the Bachelor equation well fitted the experimental results within the volume fraction range investigated. This can be helpful to predict nanolubricant viscosity for other concentrations and commercial oils under the presence of nanoparticles aggregates.

Figure 1 .Figure 2 .Figure 3 .Figure 4 .Figure 5 .Figure 6 .

 123456 Figure 1. Shear flow curves for turbine oil

Figure 1 .Figure 2 .Figure 3 .

 123 Figure 1. Shear flow curves for turbine oil

Figure 4 .Figure 5 .

 45 Figure 4. Viscosity versus temperature for nanolubricants

Table Captions Table 1 .

 Captions1 Regression results for turbine oil based on Bingham fluid model

Table 2 .

 2 Regression results for nanolubricants based on Bingham fluid model

Table 3 .

 3 Regression of as a function of

Table 1 .

 1 Regression results for turbine oil based on Bingham fluid model

	T(°C)	η (cP)	β (dyn/cm 2 ) τ	R 2
	0.0	316.6±0.9	0.67±0.07	0.9998
	20.0	79.68±0.06	0.50±0.02	0.9999
	40.0	29.17±0.04	0.42±0.03	0.9997
	60.0	13.75±0.03	0.27±0.02	0.9999

Table 2 .

 2 Regression results for nanolubricants based on Bingham fluid model

		Weight
	T(°C)	Percent	0.1%	0.5%	1.0%	2.0%	3.0%	4.0%
		Parameter
		η (cP)	317.2±0.4	319.5±0.4	322.5±0.9	328.6±0.8	335±2	342±3
	0.0	β τ (dyn/cm 2 )	0.7±0.03	0.81±.03	0.88±0.06	0.95±0.04	2.1±0.1	3.9±0.2
		R 2	0.9999	0.9997	0.9999	0.9999	0.9999	0.9999
		η (cP)	79.8±0.2	80.4±0.1	81.1±0.11	82.7±0.2	84.4±0.6	86±2
	20.0	β τ (dyn/cm 2 )	0.45±0.07	0.57±0.03	0.6±0.03	1.28±0.06	3.0±0.1	5.5±0.3
		R 2	0.9999	0.9999	0.9996	0.9999	0.9995	0.9999
		η (cP)	29.22±0.08	29.4±0.1	29.7±0.1	30.2±0.1	30.8±0.2	31.4±0.4
	40.0	β τ (dyn/cm 2 )	0.32±0.06	0.34±0.08	0.4±0.1	0.62±0.06	2.37±0.09	4.2±0.1
		R 2	0.9999	0.9999	0.9997	0.9999	0.9999	0.9999
		η (cP)	13.8±0.02	13.86±0.03	13.98±0.02	14.22±0.02	14.5±0.04	14.7±0.1
	60.0	β τ (dyn/cm 2 )	0.38±0.04	0.43±0.05	0.41±0.03	0.5±0.3	2.41±0.05	4.43±0.1
		R 2	0.9999	0.9999	0.9999	0.9999	0.9997	0.9999

Table 3

 3

	. Regression of	as a function of
		T(°C)	A	B	R²
		0.0	130.5	11.65	0.98
		20.0	135.7	11.62	0.99
		40.0	133.0	11.64	0.98
		60.0	137.2	11.61	0.99