Thomas Risler

Jacques Prost

Frank Julicher

Frank Jülicher

Universal Critical Behavior of Noisy Coupled Oscillators

Keywords: numbers: 64.60.Ht, 05.45.Xt, 64.60.Ak

HAL is

The collective behavior of many interacting elements generally leads to transitions and critical points in the large-scale and long-time properties of complex systems. This is well understood in the study of systems at thermodynamic equilibrium [START_REF] Zinn-Justin | Quantum Field Theory and Critical Phenomena[END_REF][START_REF] Brézin | Phase Transitions and Critical Phenomena[END_REF][START_REF] Hohenberg | [END_REF]. Non-equilibrium critical behaviors have been studied in a number of systems [4][START_REF] Schmittmann | Phase Transitions and Critical Phenomena[END_REF][START_REF] Täuber | [END_REF] but remain a serious challenge. An important example for criticality far from thermodynamic equilibrium is the behavior of coupled oscillators in the vicinity of a continuous homogeneous oscillatory instability or supercritical Hopf bifurcation. Such instabilities are important in many physical, chemical and biological systems [START_REF] Strogatz | Nonlinear Dynamics and Chaos[END_REF][START_REF] Bergé | Order within chaos[END_REF].

In this Letter, we apply concepts of the theory of dynamic critical points to study the generic properties of systems of coupled oscillators in the thermodynamic limit. In particular, we discuss linear response and twopoint correlation functions defined for the oscillator ensemble. Since the system is far from a thermodynamic equilibrium, the fluctuation-dissipation (FD) relation between correlation and response functions in equilibrium systems is broken. We show that a Hopf bifurcation represents a non-equilibrium critical point and study the universal behaviors characterizing its approach from the nonoscillating state. We apply field theoretic renormalization group (RG) methods and develop an RG procedure which is appropriate for the case of a spontaneously oscillating system. This RG is performed in an oscillating reference frame with a scale-dependent oscillation frequency. The RG fixed points characterize the universal critical properties of locally coupled oscillators. We find that at the critical point of a Hopf bifurcation, an FD relation is formally satisfied if the system is described within the oscillating reference frame. In terms of physical variables, the FD relation is strongly broken but a relationship between correlation and response functions appears. Even though our calculations are performed in a d = 4 -dimensional space, we suggest that the main features of our results apply to Hopf bifurcations in general.

The generic behavior of an oscillator in the vicinity of a supercritical Hopf bifurcation can be described by a dynamic equation for a complex variable Z, which characterizes the phase and amplitude of the oscillations [START_REF] Strogatz | Nonlinear Dynamics and Chaos[END_REF]. This variable can be chosen such that its real part is, to linear order, related to a physical observable, e.g., the displacement X(t) generated by a mechanical oscillator: X(t) = Re(Z(t)) + nonlinear terms. In the presence of a periodic stimulus force F (t) = F e -iωt with a frequency ω close to the oscillation frequency at the bifurcation ω 0 , the generic dynamics obeys [START_REF] Camalet | [END_REF]

∂ t Z = -(r + iω 0 )Z -(u + iu a ) |Z| 2 Z + Λ -1 e iθ F (t). (1)
For F = 0 and r > 0, the static state Z = 0 is stable. The system undergoes a Hopf bifurcation at r = 0 and exhibits spontaneous oscillations for r < 0. The nonlinear term characterized by the coefficients u and u a stabilizes the oscillation amplitude for u > 0. The external stimulus appears linearly in this equation and couples in general with a phase shift θ. In the case of a mechanical oscillator, the coefficient Λ has units of a friction. From the point of view of statistical physics, the Hopf bifurcation is a critical point and Eq. (1) characterizes the corresponding mean field theory. Indeed, at r = 0 and ω = ω 0 and in terms of the amplitude X of the limit cycle Z(t) = Xe -iωt , the system exhibits a power-law response

| X| | F | 1/δ where δ = 3 is a mean field critical exponent. For frequency differences |ω -ω 0 | Λ -2/3 | F | 2/3 |u+iu a | 1/3 , the response becomes linear with | X| Λ -1 | F |/|ω -ω 0 |.
In the presence of fluctuations, the critical point of an individual oscillator is concealed in the same way as finite-size effects destroy a phase transition in equilibrium thermodynamics. However, a true critical point can exist in a thermodynamic limit where many oscillators, distributed on a lattice in a d-dimensional space, are coupled by nearest-neighbor interactions. The combined system undergoes a dynamic phase transition at which all oscillators synchronize and an order parameter, which char-acterizes the global phase and amplitude of oscillators, becomes nonzero. Since at the critical point the correlation length diverges and oscillators become synchronized over large distances, a discrete model can on large scales be described by a continuum field theory which characterizes the universal features of the critical point [START_REF] Zinn-Justin | Quantum Field Theory and Critical Phenomena[END_REF]10]. In the case of locally coupled oscillators, this field theory is given by the complex Ginzburg-Landau equation [START_REF] Newell | Instability of Continuous Systems[END_REF] with fluctuations

∂ t Z = -(r + iω 0 )Z + (c + ic a )∆Z -(u + iu a ) |Z| 2 Z +Λ -1 e iθ F + η. (2)
Here, Z(x, t) becomes a complex field defined at positions x in a d-dimensional space and ∆ denotes the Laplace operator. The coefficients c and c a characterize the local coupling of oscillators and the effects of fluctuations are described via a complex random forcing term η(x, t). For a vanishing external field F (x, t) and in the absence of fluctuations, Eq. ( 2) is invariant with respect to phase changes of the oscillations Z → Ze iφ . As far as long-time and long-wavelength properties are concerned, η can be chosen Gaussian and white with the correlations η(x, t)η(x , t ) = 0 and η(x, t)η * (x , t ) = 4Dδ d (x -x )δ(t -t ), which respect phase invariance.

The linear response function χ αβ and the two-point autocorrelation function C αβ of this field theory are defined by

ψ α (x, t) = d d x dt χ αβ (x -x , t -t )F β (x , t ) + O(|F | 2 ) and C αβ (x -x , t -t ) = ψ α (x, t)ψ β (x , t ) c .
Here we have expressed Z = ψ 1 +iψ 2 and F = F 1 +iF 2 by their real and imaginary parts, and ... c denotes a connected correlation function. Because of phase invariance, these functions obey symmetry relations, e.g., C 11 = C 22 , C 21 = -C 12 . In the following, we focus for simplicity on the elements C ≡ C 11 and χ ≡ χ 11 , which characterize the correlation and response of the observable X.

It is convenient to eliminate the frequency ω 0 from Eq. ( 2) by a time-dependent variable transformation Y ≡ e iω0t Z, H ≡ e iω0t Λ -1 e iθ F and ζ ≡ e iω0t η. This leads to the amplitude equation

∂ t Y = -rY +(c+ic a )∆Y -(u+iu a ) |Y | 2 Y +H +ζ, (3)
where the noise ζ has the same correlators as η. For the particular case c a = 0 and u a = 0, Eq. ( 3) becomes identical to the model A dynamics of a real Ginzburg-Landau field theory with an O(2) symmetry of the order parameter [START_REF] Hohenberg | [END_REF]. The critical behavior of this theory at thermodynamic equilibrium has been extensively studied [12]. This leads, in this particular case, to a formal analogy between an equilibrium phase transition and a Hopf bifurcation. The correlation and response functions C αβ and χ αβ can here be obtained from those of the equilibrium field theory by using the time-dependent variable transformation given above. Since the theory at thermodynamic equilibrium obeys an FD relation, a generic relation between the correlation and response functions C αβ and χ αβ appears. This special case provides a further example of an equilibrium universality class found in a nonequilibrium dynamics with nonconserved order parameter [START_REF] Täuber | [END_REF]. It is the case, e.g., for the model A dynamics of the real Ginzburg-Landau theory with a Z 2 symmetry [13], even when the symmetry is broken by the nonequilibrium perturbations [14], and for some of its generalizations to the O(n) symmetry [15]. This raises the question to know whether the equilibrium universality class also characterizes the general case where u a and c a are finite. Dimensional analysis reveals that for d > 4 mean field theory applies. In this case,

χ mf (q, ω) = 1 2Λ e iθ R -i(ω -Ω 0 ) + e -iθ R -i(ω + Ω 0 ) C mf (q, ω) = D R 2 + (ω -Ω 0 ) 2 + D R 2 + (ω + Ω 0 ) 2 , ( 4
)
where R = r + cq 2 , Ω 0 = ω 0 + c a q 2 and where q and ω are wave vector and angular frequency, respectively.

For d < 4, mean field theory breaks down. We apply RG methods using an expansion near the upper critical dimension (d = 4 -) [16]. Defining two real fields φ α by Y = φ 1 + iφ 2 , Eq. ( 3) reads

∂ t φ α = -R αβ φ β -U αβ φ β φ γ φ γ + H α + ζ α , (5)
where

H = H 1 + iH 2 , R αβ = (r -c∆)δ αβ -c a ∆ ε αβ , and
U αβ = uδ αβ + u a ε αβ , with 21 = -12 = 1 and ij = 0 for i = j.
We introduce the Martin-Siggia-Rose response field φα [17] and apply the Janssen-De Dominicis formalism [18] to write a generating functional with action

S φα , φ α = d d x dt D φα φα -φα [∂ t φ α + R αβ φ β ] -U αβ φα φ β φ γ φ γ . (6)
Using a Callan-Symanzik RG scheme [START_REF] Zinn-Justin | Quantum Field Theory and Critical Phenomena[END_REF][START_REF] Brézin | Phase Transitions and Critical Phenomena[END_REF], we define the renormalized theory such that its effective action is of the form [START_REF] Täuber | [END_REF]. This requires to introduce a phase shift δθ and a frequency shift δω 0 between the bare fields (φ 0 α , φ0 α ) and the renormalized fields (φ α , φα ):

φ 0 α (x, t 0 ) = Ω αβ (-δω 0 t) Z 1/2 φ Z ω φ β (x, t) φ0 α (x, t 0 ) = Ω αβ (-δθ -δω 0 t) Z 1/2 φ Z ω φβ (x, t). (7)
Here we have introduced Z-factors for the renormalization of the fields and the time (t 0 = Z -1 ω t), and Ω αβ (θ) denotes the rotation matrix by an angle θ in two dimensions. We furthermore introduce dimensionless coupling constants g and g a and a scale factor µ by u = µ (4π) -/2 g and u a = µ (4π) -

χ αβ (x, t 0 ) = Λ -1 Ω ασ (θ -ω 0 t) (Z φ Z φ) 1/2 Z 2 ω γ σβ (x, t) C αβ (x, t 0 ) = Ω ασ (-ω 0 t) Z φ Z 2 ω G σβ (x, t). (8)
The dependence of the renormalized parameters g, g a and c a on µ defines three β functions. Writing g = (g, g a , c a ), we have β ( g, ) = µ(∂ µ g) 0 , where β = (β, β a , β c ) and (∂ µ ) 0 denotes differentiation with fixed u 0 , u 0 a and c 0 a . To one-loop order in perturbation theory (see Fig. 1 for examples of Feynman diagrams of the theory), only ω 0 , r, g, and g a are renormalized. The two nontrivial β functions are given by:

β a -g a -D c a 1 + c 2 a (g 2 -g 2 a + 2gg a c a ) -6gg a β -g -D g 2 a -g 2 -2gg a c a 1 + c 2 a -4g 2 , (9)
where D = 4D/(4π) 2 . The RG fixed point corresponds to the values g * of the parameters g for which the three β functions are simultaneously zero. Since β c = 0 to oneloop order, one condition is lacking to fully determine g * . Choosing c a as a parameter, we obtain g * /5D and g * a c a /5D. In order to determine completely the fixed point g * , we need to go to two-loop order in perturbation theory. To this order, all parameters (apart from c and D), the fields and the time are renormalized and explicit expressions for all the Wilson's functions and Z factors of the theory can be obtained.

Only one fixed point exists that describes the universality class of Hopf bifurcations. It obeys c * a = 0, g * a = 0 and is infrared-stable. This fixed point is formally equivalent to the one of the real Ginzburg-Landau theory with O(2) symmetry. As a consequence, we find ν 1/2 + /10, η 2 /50 and z 2 + 2 (6 ln(4/3) -1) /50, which are the corresponding equilibrium critical exponents. Here, ν denotes the exponent characterizing the divergence of the correlation length ξ, z is the dynamic exponent and η denotes the exponent characterizing the field renormalization [START_REF] Zinn-Justin | Quantum Field Theory and Critical Phenomena[END_REF][START_REF] Hohenberg | [END_REF].

The RG flow in the vicinity of the fixed point, however, is defined here in a larger parameter space as the one corresponding to the O(2) dynamic model. Furthermore, the effective theory discussed here is expressed in an oscillating reference frame with scale-dependent frequency and phase. Therefore the correlation and response functions C αβ and χ αβ differ from those of the equilibrium model and additional universal exponents appear. We can derive generic expressions for these functions using the RG flow of all parameters in the vicinity of the critical point and employing a matching procedure [20]. For example, we find for qξ 1 and for stimulation at the effective oscillation frequency ω eff 0 :

χ(q, ω = ω eff 0 ) 1 q 2-η 1 2Λ eff e iθ(q) 1 + iγ(q) , (10)
where we have introduced the functions θ(q) θ eff + α eff q ω1 + β eff q ω2 and γ(q) γ eff q ω2 , and nonuniversal effective quantities denoted by the index "eff". The universal exponents ω 1 /5 and ω 2 2 /50 here are characteristic for a Hopf bifurcation. Similarly, we find expressions for the correlation function:

C(q, ω = ω eff 0 ) 1 q z+2-η D eff 1 + γ(q) 2 , (11)
and for the frequency dependence of the homogeneous mode q = 0 in the regime (ω -ω eff 0 )ξ z 1 [19]. Because of the formal analogy of the RG fixed point discussed here with the one of an equilibrium field theory, an FD relation appears exactly at the critical point and relates the functions G αβ and γ αβ . Since the physical correlation and response functions C αβ and χ αβ can be determined from G αβ and γ αβ using the scale-dependent variable transformations of Eq. [START_REF] Bergé | Order within chaos[END_REF]

Here, χ αβ = χ αβ + iχ αβ has been separated in its real and imaginary parts. The relation ( 12) is asymptotically satisfied in the long-time and wavelength limits at the critical point.

The physical correlation and response functions C αβ and χ αβ , however, do not obey the equilibrium FD relation. The degree of this violation can be characterized by a frequency-dependent effective temperature T eff :

T eff (ω) T = ω 2k B T C 11 (ω, q = 0) χ 11 (ω, q = 0) . (13)
Here, k B denotes the Boltzmann constant and T is the temperature of the system. We find that T eff /T ∼ (ω -ω eff 0 ) -σ diverges at the critical point with a universal exponent σ. For the particular case c a = 0 and u a = 0, σ = 1, while otherwise σ 1 -/5 to first order in . The power-law divergence of T eff as a function of frequency reveals a violent breaking of the FD relation. This divergence at the oscillation frequency has been experimentally observed on a single active oscillating system [START_REF] Martin | Proc. Natl. Acad. Sci. U.S.A[END_REF].

We have shown that the critical point in d = 4 -dimensions of locally coupled oscillators is formally related to the equilibrium phase transition in the XY model. From this analogy it follows that on the oscillating side of the Hopf bifurcation, the system of coupled oscillators exhibits long range phase order and coherent oscillations for d > 2. We can speculate how our results are modified in lower dimensions d. In analogy with the equilibrium XY model, we expect the phase order of the oscillations to vanish for d < 2, and to be quasi-long range exactly at the lower critical dimension d = 2. In the last case, spectral peaks on the oscillating side of the Hopf bifurcation are expected to exhibit power-law tails with nonuniversal exponents. If the formal analogy with the equilibrium critical point found here in d = 4 -persists in d = 2, we would expect to see features of the Kosterlitz-Thouless universality class [START_REF] Kosterlitz | [END_REF] in systems of coupled oscillators in this dimension.

Critical oscillators are ideally suited for nonlinear signal detection and amplification. Indeed, close to the critical frequency, the linear response function exhibits divergent behaviors, indicative of a high sensitivity of the system. It has been suggested that the ear of vertebrates uses critical oscillations of mechanosensitive hair cells for the detection of weak sounds and that the properties of the critical point provide the basis to explain the observed compressive nonlinear response to mechanical stimuli and to frequency selectivity in the ear [START_REF] Camalet | [END_REF]23]. The correlation and response functions of single mechanosensory hair bundles have been determined experimentally [START_REF] Martin | Proc. Natl. Acad. Sci. U.S.A[END_REF]. These single cell experiments detected vibrations at the scale of tens of nanometers. There, the Hopf bifurcation was concealed by finite-size effects but its signature could be observed. In the cochlea of mammals, powerlaw responses over several orders of magnitude have been seen [24]. This suggests that in such systems a large number of oscillating degrees of freedom operate collectively and bring the system closer to true criticality.

The critical oscillations discussed here can in principle be realized in artificial systems. Nanotechnology aims to build functional units on the submicrometer scale. Large arrays of nanorotators or oscillators on patterned substrates coupled to their neighbors by elastic or viscous effects would provide a two-dimensional realization of our field theory. This could permit in the future experimental studies of the critical phenomena discussed here.

  /2 g a . The bare and renormalized quantities are now related depending on µ [25]. We define the correlation and response functions G αβ = φ α φ β c and γ αβ = φ α φβ c . They are related to the physical observables C αβ and χ αβ via [26]

FIG. 1 :

 1 FIG. 1: (a) Graphic representation of the propagators G αβ and γ αβ and of the vertex U αβ δ γδ . (b) Examples of Feynman diagrams of the theory to one-and two-loop orders.

  , a relation between correlation and response functions appears: cos θ eff χ 11 -sin θ eff χ 12 = 1 2Λ eff D eff ωC 11 + iω eff 0 C 12 cos θ eff χ 12 + sin θ eff χ 11 = 1 2Λ eff D eff ω eff 0 C 11 + iωC 12 .

We thank Edouard Brézin, Erwin Frey and Kay Wiese for useful discussions.


Claude Martin

(Re)-Discovering parents and parenting in France: What really is new?

Keywords: Family policy, parenting, parenting support, policy framing, social policy, welfare state restructuring

Re)discovering parents and parenting in France. What really is new? ",

Introduction

As in many other European and non-European countries, an explicit parenting support policy made its way onto the French political agenda during the early 1990s: a 'turn to parenting' with its trail of new terms, instruments, and institutions. In France, some decision makers now consider this parenting support policy (politique de soutien à la parentalité) as a new, but still financially marginal, 'pillar of French family policy'. It responds to a growing political demand for a solution capable of guaranteeing the success of the parental educational mission or, at the very least, of helping to avoid its failure along with the consequences that such a failure can have for the entire community.

One may well wonder, however, whether this policy represents a real innovation or merely the revival of a longstanding tradition of 'policing families'. In spite of this well-known tradition that first emerged in the 18 th century and has been thoroughly documented by historians, many international experts argue that we are currently at a turning point (Daly 2013a) and engaging in a new 'parenting culture' combining moral panic with acute risk consciousness [START_REF] Furedi | Paranoid Parenting. Why Ignoring the Experts May Be Best for your Child[END_REF][START_REF] Lee | Parenting Culture Studies[END_REF]. One of the main issues is thus the question of change. Although parenting support does indeed concern many European and non-European countries and follows a mainstream of ideas, instruments, and issues, we argue that the French configuration has certain specificities that need to be addressed not only in order to estimate the commonalities but also to avoid making hasty generalizations.

My article 1 is organized in two main steps: after preliminary considerations on my perspective and main arguments, I first contrast the current international turn to parenting with the way it has occurred in the French context (policy process and arguments). In a second section, I return to the question of the revival of a very longstanding tradition of policing the family in France, before discussing the cultural and socio-political backgrounds of this parenting support policy. This enables me to clarify what really is new about this policy and what corresponds to structural elements of policy framing.

Policy framing

The slow policy process that has led to current French parenting support policy is simultaneously the consequence of a political will to structure and manage a myriad of grassroots initiatives since the early 1990s (local policy practices) and the outcome of an ideological battle [START_REF] Martin | Gouverner les nouveaux risques sociaux. Le cas des politiques en direction de l'enfance et de la parentalité dans les Etats-providence Européens[END_REF]. This battle brings into conflict fundamental values and generates controversies pitting universalism against targeting, parental empowering against parental control, offering support to parents via services versus re-educating them through counselling and behavioural training, and local and community actions versus national regulatory actions. A fiercely fought battle of ideas around parenthood and parenting issues is thus taking place behind the scenes -a battle that is mixing old and new ideas while reactivating norms and stereotypes that are deeply rooted in the French social history of the private and public spheres and the respective roles of mothers, fathers, and public institutions.

1 This article is based on a collective research project funded by four national research agencies (France, England, Germany, and the Netherlands): the PolChi research (see http://www.uni-goettingen.de/en/213091.html ). In France, our material is based on a systematic analysis of official reports at national and international level as well as interviews with 20 experts and high-ranking civil servants involved in this policy domain in France and 20 professionals in charge of implementing the policy at local level (see [START_REF] Martin | Gouverner les nouveaux risques sociaux. Le cas des politiques en direction de l'enfance et de la parentalité dans les Etats-providence Européens[END_REF]. We want to thank our reviewers and Ilona Ostner for their suggestions and comments on a previous version of this paper.

Following Donald A. Schön and Martin Rein's (1994) 'frame reflection' perspective, I propose to map the policy controversies in terms of field of positions and conflicts by presenting the arguments of a selection of the actors engaging in this 'policy framing' in the French context. In their seminal book, Schön and Rein: assert that the parties to policy controversies see issues, policies, and policy situations in different and conflicting ways that embody different systems of belief and related prescriptions for action, often crystallized in generative metaphors. These frames determine what counts as a fact and how one makes the normative leap from facts to prescriptions for action . . . What is needed [they argue] is empirical study of policy controversies in actual policy practice. (Schön/Rein 1994: xviii) Using Schön and Rein's recommendations, I suggest going beyond the first rungs of a 'ladder of reflection', namely, the policy practices (first rung) and the policy rules (second rung: laws, entitlements, resource allocations), and exploring higher levels: 'the particular positions and accompanying arguments held by advocates and opponents in policy debates and struggles' or the 'beliefs and values held by particular institutions and interest groups', and even the 'broadly shared beliefs, values and perspectives familiar to the member of a societal culture' -what they call 'metacultural frames'. (idem) Among these frames, I wish to stress the reactivation of old and deep-rooted oppositions and controversies -not only between left-and right-wing parties but also between Republican and Catholic milieus -concerning family, the private life domain, and the challenge of secularization. In the French case, one of these meta-cultural frames concerning welfare refers to a central conflict between State and Church. This conflict was absolutely crucial in the early decades of the Third Republic (at the beginning of the formation of the welfare state), but also, as Manow and Palier argue, explains the 'premature vanishing of a Christian Democratic party in 1967' (Manow/Palier 2009: 171), after playing a central role immediately following World War II, particularly in the field of family policy. This conflict is being reactivated nowadays and represents a deep structure for the current controversies over family issues. I argue that something is still at work concerning the respective roles of State and civil society when considering the family and parental issues, something that follows up the initial structural fight during the Third Republic that led to the pivotal Loi 1905 that separated Church and the State and provoked the guerrilla scolaire. As Manow and Palier recall:

The church and its followers fought with much intransigence, not only for the preservation and public subsidization of private, Catholic schools, but also against a stronger role of the state in social policy. In the church's view, a state playing providence not only undermined individual thrift and prudence but also threatened to destroy the smallest and holiest unit of social solidarity, the family. (Manow/Palier 2009: 153)

The turn to parenting

An international turn to parenting

According to analyses over the past 25 years, it seems clear that parenting is becoming a new issue on an international level, as well as one that is commonly admitted to. The first common indicator of a change is certainly the adoption and spread of new terms. This is currently the case in English, with the verb to 'parent' and the noun 'parenting', as well as in French with the noun parentalité (and parentage in Québécois). These terms are not substitutes for previous and well-established other terms such as family, parenthood, or kinship. Indeed, the emergence of these neologisms in the two languages is relatively recent and has been gradually integrated, over the past two decades or so, into the lexicon of public decision makers, politicians, media, and professionals working in the field of childhood and the family, as well as in the health education and disease prevention sector.

Ellie Lee underlines this popularity in the introduction to the book she co-edited on Parenting Culture Studies and indicates that the number of books on parenting more than doubled between 1980 and 2000 (Lee et al. 2014: 5). To give another example of this success, a basic consultation of the SAGE database using parenting as keyword provides access to almost 21,000 published articles, 15,600 of which (almost 75%) have been published since January 2000 and mainly in the following disciplines: psychology, public health, sociology, youth studies, and behavioural sciences.2 Last but not least, in the French context, we can mention the explosion of the word match frequency for the term parentalité in the media, as noted by Julien [START_REF] Damon | Les programmes performants de parentalité[END_REF].

Although the French expression parentalité integrates the meanings of both parenthood and parenting (leading to a certain amount of misunderstanding), both neologisms -parenting and parentalité -indicate a new focus on the role of parents regardless of gender: both mothers and fathers. For Mary [START_REF] Daly | Parenting support: Another gender-related policy illusion in Europe?[END_REF], the apparent gender neutrality of the term 'parent' with regard to the respective roles of mother and father is not so much a proactive strategy in defence of the idea that the parental role is (or should be) less and less gendered (more gender neutral), as a way of denying this gender issue in parenting concerns (gender blindness). Other experts argue that this 'neutral' term could also refer to the extension to fathers of the normative messages and prescriptions traditionally addressed to mothers alone.

Be that as it may, as Ellie Lee argues: the message to mothers (and also fathers) is that the health, welfare and success (or lack of it) of their children can be directly attributed to the decisions they make about matters like feeding their children; 'parenting', parents are told, is both the hardest and most important job in the world. Tomorrow depends on it. (Lee et al. 2014: 2) Without returning to the emergence and roots of these notions of parenting and parentalité themselves and their anchorage in various theoretical and disciplinary backgrounds (see Martin 2012a), it should be noted that these terms aim to focus on parents and on parental practices and their impact on children. As New Labour aptly formulated it, parenting is 'what parents do rather than what they are' (cited in Lewis 2012: 102). One could add: 'what they should do'. The nature of parenting support is thus to 'support' and 'educate' parents in their childrearing role (Daly/Bray 2015: 634;and in this Special Issue); that is, to socialize these primary actors of socialization.

Out of the debates and institutional reforms concerning childhood and private issues over recent decades, a second argument aimed at identifying change has emerged: a new backdrop. One crucial impetus that must be mentioned is the 1989 signing of the International Convention on the Rights of the Child (CRC) -in other words, the recognition that children have specific interests that must be guaranteed. Another contextual element is the impact of the Belgian Dutroux affair on public opinion in 1996. Many other family tragedies have occurred since then, demonstrating the need to protect children from potential harm from parents (and other adults). Since the beginning of the 1990s, the general concept of a 'risk society' has developed dramatically. This also affects family and private life, particularly when youth delinquency is presented as a result of parenting (ir)responsibility. Although these issues are highly controversial among experts and political actors, they impose a double-edged risk in case of the failure of the parent-child relationship: the child is either a victim or a threat as she or he grows older, yet the parent is always to blame.

A third level of change refers to family policy reforms in different countries3 -and more precisely the development of a parenting support policy as such. As [START_REF] Daly | Parenting support in England: The bedding down of a new policy[END_REF] argue, parenting support seems better established in England than in many other European countries following a turning point initiated by New Labour between 1997 and 2010. Jane [START_REF] Lewis | From Sure Start to Children's Centres: An analysis of policy change in English early years programmes[END_REF] has undertaken a detailed analysis of this process. Over time, this parenting policy has shifted away from mainly targeting those parents whose children and adolescents display anti-social behaviour problems towards a territorially organized universal policy. A 'commissioner' is charged with its local implementation, combined with recourse to evidence-based programmes. Despite the limitations of this investment, which presupposed recourse to costly, commodified programmes (for the training of contributors), this policy was continued by the coalition government that succeeded New Labour in 2010 -even though this government was extremely concerned about reducing public spending.

According to Mary Daly and Rachel Bray, the nature of the Labour government's concern with poverty and inequality among children as well as the availability of a number of evidence-based programmes and a fascination at the time with their supposed efficacy explain why parenting support grew so quickly in England [START_REF] Daly | Parenting support in England: The bedding down of a new policy[END_REF].

An analogous trend is readable in other countries. For example, starting from the longstanding experience of public child healthcare centres, mainly oriented towards public health and prevention issues (vaccinations, weight monitoring, physical development, as well as motor and language skills), the Netherlands turned, with the new Youth Act in 2005, towards the prevention of negative childrearing practices with increasingly systematic recourse to evidence-based programmes for parents, thereby demonstrating this strong concern for proof and validation (particularly at the Dutch Centres for Youth andFamily, 2008-2011). This change went hand in hand with a coming together of policy on youth and the family [START_REF] Knijn | Parenting support in the Dutch "Participation society[END_REF]and in this Special Issue). The process was slower in Germany, where it was not until 2010-2011 that family policy turned to this type of programme, mainly by pursuing a similar, public health approach via providing federal funding for hitherto only random measures such as the Familienhebammen (a sort of United Kingdom-style family-nurse partnership) [START_REF] Ostner | Investing in children, monitoring parents: Parenting support in the changing German welfare state[END_REF]. This primacy of public health is also discernible in choices made in Sweden from 2009-2010, with the same recourse to standardized programmes -even though enthusiasm for it has waned somewhat in this country in recent years [START_REF] Lundqvist | Parenting support in Sweden : New Policies in old settings[END_REF].

In this process of defining national parenting support policies, it is important to add the incentive role played by European institutions. Even though these policies are a matter for individual member states, it is undeniable that the EU has facilitated the circulation of ideas and methods, thus contributing to the definition of orientations and suggesting best practices. Three publications can be mentioned on this issue in the course of the 2000s. First, the publication by a committee of experts on childhood and the family under the auspices of the Council of Europe in 2006 entitled: Parenting in contemporary Europe: a positive approach (Daly 2007). With its recommendation prioritizing positive parenting, the Council of Europe opened a new chapter -that is, 'parental behaviour guaranteeing that the interests of the child will be fulfilled' that, for the first time, unambiguously prioritized the relationship between parents and children as an object of intervention.

In 2007, the report of the European Network of National Observatories on Childhood (ChildOnEurope 2007), entitled Survey on the Role of Parents and the Support from the Governments in the EU, followed the same orientation, promoting socio-educational parent support in the form of advice and counselling services. In 2012, a report ordered by the European Commission from RAND Europe, 4 entitled Parenting Support Policy Brief (European Commission 2012), finally offered a general synthesis on the issue by placing it within its historical context and defining its principles and philosophy prior to describing national experiences that highlight best practices. The RAND report also argues for the necessary shift towards a social investment state, with explicit references to the works of Anthony Giddens, Gøsta Esping-Andersen, and James [START_REF] Heckman | Invest in the very Young[END_REF] who was awarded the Nobel Memorial Prize in Economics for his idea of 'capitalizing later in life' thanks to an investment in early childhood to avoid future expenditure (European Commission 2012: 7ff.).

The French turn to parenting

The policy that took shape in France during the 1990s seems to have taken a similar -yet original -trajectory (Hamel et al. 2012;[START_REF] Martin | Gouverner les nouveaux risques sociaux. Le cas des politiques en direction de l'enfance et de la parentalité dans les Etats-providence Européens[END_REF]Martin , 2015a)). As in other European countries, it mixes a variety of logics and objectives, blending new types of intervention, professionals, and techniques with pre-existing ones. It also combines local initiatives on the ground with new official national institutions to supervise and regulate them. Overall, this policy appears to be mainly the recognition by public authorities, in the late 1990s, of a myriad of practical grassroots initiatives at local level and ultimately (and this seems to be a formal turning point) the creation of a new national institution: the Comité national de soutien à la parentalité [National parenting support committee]. Created in 2010, this committee is currently the official governance body for parenting policy as such in France.

Nevertheless, initiatives aimed at supporting families in their educational role already have a long history in France. Our interviewees in the PolChi project have identified some key moments in this framing. Though not going as far back as the creation of the école des parents in 1930, one of the experts we interviewed for the PolChi research presented the beginning as follows:

Concerning the key dates, it is important to go back to the free nursery at the Sorbonne University in 1968 -it was in a way the first parental nursery, and one of the first initiatives to accompany the family transformations of the 1970s. It arose out of civil society -not even from the associative network. For me, the first movement to support parentalité, the first parenting support, came from civil society and nongovernmental organizations. Next, we had the beginnings of the Maison Verte initiated by Françoise Dolto in 1979; that is, new childcare places to welcome both children and parents together, as well as the introduction of family mediation at the beginning of the 1980s, imported from North America . . . All these initiatives came from civil society and were recognized by public authorities as being promising. (Expert on family, childhood, and parenting issues, Polchi interview 5 ) Another interviewee, this time in charge of parenting support policy at the CNAF (Caisse Nationale des Allocations Familiales), which is the main social security institution in charge of family policy in France, distinguishes four key moments and sequences in the definition of this new policy. From her point of view, the first step was taken at the end of the 1990s with the creation of the REAAP network (Réseau d'Écoute, d'Accueil et d'Accompagnement des Parents) to coordinate and organize a myriad of local initiatives by associations and nongovernmental organizations.

1998/1999 was really the first step, with the role of the Child and Family Institute6 and the creation of the REAAP. At the beginning, the main idea was to support the initiatives that were emerging at the time in various associations and localities, and to coordinate them via a network. The state sought to federate these initiatives and answer the associations' demand for better recognition. The Délégation interministérielle à la famille was in charge of the coordination. (Responsable pôle jeunesse et parentalité, CNAF)

For this actor, four other key steps remain discernible. A second step was taken with the creation of another coordination body addressing family mediation: the Conseil national consultatif de la médiation familiale that facilitated the creation of a national degree in 2003 (diplôme d'Etat de médiateur familial), a process of training centre accreditation -and ultimately a service delivery funded by the CNAF. The objective was clearly to organize the provision of this service across the entire nation. Our interviewee also identified a third step in the wake of an official (and fairly critical) report by the Cour des Comptes in 2009 on the relatively low efficacy of this emerging parenting policy. The CNAF thus decided to vote on a new Convention d'Objectif et de Gestion [planning and management agreement] for the 2009-2012 period to once again reinforce and improve how the sector operated. However, the funding level remained relatively low (a 40% increase from 53 to 75 million Euros per year). The fourth step, according to this high-ranking civil servant, was the creation of the new national body called Conseil National de Soutien à la Parentalité.

This really was a structuring phase, bringing points of view together, working on the definition of parentalité and the types of intervention included in parenting support.

(Responsable pôle jeunesse et parentalité, CNAF)

For this interviewee, the fifth step is the publication of another official report, this time by the Inspection Générale des Affaires Sociales (IGAS) recommending another strong impulse in favour of this parenting support policy -and, in particular, the doubling of CNAF funding, which has indeed been implemented (the budget rose from €75 million to €150 million) [START_REF] Jacquey-Vazquez | Evaluation de la politique de soutien à la parentalité (MAP -volet1), Tome I et II. Rapport[END_REF].

The general trend is thus a progressive structuring, recognition, and reinforcement of this policy. Yet it remains a relatively modest investment, accounting for just 0.2% of the overall budget devoted to family policy and allowances (exclusive of housing). Moreover, of the €150 million per year, central government invests just €18 million; local authorities contribute some €50 million, and the social security system (the CNAF) provides the remainder. Although the state remains a relatively weak partner in financial terms, in comparison to the CNAF, its plays a crucial role in terms of governance and incentives towards local and national stakeholders.

Nevertheless, this role of the state as guide and catalyst is not as unambiguous as it appears to be. The political agenda and political changes in government play a crucial role in renewing the arguments and priorities, the discourse around this parenting issue, and the measures to be favoured. From this perspective, important gaps separate the discourse and priorities of Ségolène Royal (Socialist minister in charge of the family between 2000 and 2002) from those of the new Right-wing minister in charge Christian Jacob (between 2002 and 2004), as well as from the arguments of Nadine Morano (another Right-wing minister between 2008 and 2010) and Dominique Bertinotti (Socialist minister in charge of the family between 2012 and 2014). These repeated shifts between Right-and Left-wing majorities play unquestionably a role in the rotation of arguments, 'systems of belief, and related prescriptions for action', to use Schön and Rein's terms.

The moralizing, punitive, and security-related discourse and orientation that dominated the 2002 Presidential campaign on the Right-wing is a good example of the ideological fight that took place prior to the Right-wing government's comeback. The issue of security -and juvenile delinquency in particular -became absolutely central. In the report on Parentalité I submitted to the Haut conseil de la population et de la famille in 2003 [START_REF] Martin | La parentalité en questions[END_REF], I mapped this slide towards a security-focused political landscape. This ideological turn provoked a certain number of responses -such as the publication of a decree introducing a 'parental responsibility contract' in the event of problems being caused by a pupil at school or significant absence from school and the establishment of 'parental responsibility courses' in 2006; the creation of the Conseil pour les droits et devoirs des familles [Council for the rights and responsibilities of families] within municipal councils in 2007; and the publication of a decree introducing the suspension (or even cancellation) of family allowances for parents of children repeatedly absent from school (at least 4 half days per month) or whose absence was unjustified in January 2011 [START_REF] Martin | The invention of a parenting policy in the French context: Elements for a policy tracing[END_REF].

With the political turn and the return of a Socialist presidency and government in 2012, this structural opposition was revived. As the Minister for the Family (between May 2012 and March 2014) argued in one of our interviews:

For the previous (Right-wing) government, parenting policy was geared towards the stigmatization of families. To put it somewhat crudely, this was because the parents were not good parents and so, the sanctions road was the preferred option -we even went as far as the notorious 'décret Ciotti' -that advocated the withdrawal of family benefit payments when children's school attendance was poor. When I arrived, I considered that we had to turn this problem on its head by saying: it is not up to us to stigmatize families but rather, on the contrary, to admit that at some point in the process of educating their children, they may come up against questions, difficulties, and problems -and the issue is to discover how we -that is, we the state, we the institutions, we the local authorities -might try to offer responses. (Minister in Charge of the Family, PolChi interview) These eruptions of political debate over the definition of parenting support policy clearly have multiple consequences affecting how measures are defined and implemented as well as how family and childhood professionals are expected -or intended -to carry out their roles. Gérard Neyrand places considerable emphasis on this opposition between the two models of support and control:

By focusing on the relationship to the child, the social management of the family ends up being caught between two competing logics of intervention in which the desire to support parents is set against the desire to control them. On the one hand, there is the idea of joint responsibility for bringing up children and its watchwords of participation, working together, targeted prevention . . . and on the other hand, the denunciation of parental abdication of responsibility, parenting courses, and the idea of getting back on the straight and narrow, and of systematic prevention. 7 (Neyrand 2011: 11) On this particular issue, the French national configuration clearly joins up with that in the United Kingdom during the same period in its sense of punitive accountability of parents who are incompetent and therefore guilty of their offspring's poor behaviour (the parenting contracts of 2005 in England and the 2006 contrats de responsabilité parentale in France; for more details see [START_REF] Martin | La parentalité en questions[END_REF]).

Yet in comparison with the United Kingdom, one major difference persists: in France, there has been very little recourse to the evidence-based programmes that were in force in the United Kingdom and some other countries, mainly due to resistance to this behavioural orientation among childhood and family professionals. In the professionals' discourse, the universal objective has priority over targeting and formulating prescriptions. Nobody wonders whether these parents are at Triple P level 3 or 5. To me, that's just crazy. (Family Mediator, PolChi interview)

The evidence-based programmes in France -people will not stand for it. At REAAP or CAF level, it would not be accepted. It goes against our approach to parents. We are not here to say: 'you should do this or that' . . . We are not experts . . . This type of expert who gives orders does exist, but it's neither our position nor our role. (Professional in charge of a REAAP in a Caisse d'allocations familiales, PolChi interview)

One of the best examples of this resistance is the strong professional mobilization against the publication of an Inserm summary report (Inserm 2005) on the knowledge acquired on early childhood troubles and their links to high-risk behaviours in adolescence. This report, 7 . Author's translation. which offered to survey the results and evidence from international research, argues that a link has been established between the behaviour of under-3s or under-5s and future risks. This evidence was however immediately strongly rejected by many professionals and experts (psychoanalysts, paediatricians, psychiatrists, neuropsychiatrists, psychologists, etc.) arguing that such types of knowledge were just giving rise to a " carrière de deviance" (carrier of deviance), by labelling children at a very early stage and stigmatizing their behaviour. This movement (collectif 'Pas de zéro de conduite pour les enfants de trois ans') argued in a petition: 'By medicalizing to the extreme phenomena of educational, psychological, or social order, the INSERM expertise maintains confusion between social malaise and mental suffering, or even hereditary illness'. 8 In spite of this resistance, implementation of evidencebased programmes is also emerging in France via public health and health education channels, but it remains highly controversial.

Cultural and meta-cultural frames in France Policing families: a longstanding tradition

To what extent might we consider this French turn to parenting since the mid-1990s to really be a new phenomenon? In order to identify any change in recent developments, it is important to bear in mind one longstanding tradition of analysis: what Jacques Donzelot called La police des familles, 'policing the family'. Beyond the legal issue and the significance of the laws governing and framing the family, the challenge is clearly normative in the sense used by Michel Foucault in his analysis of biopower:9 'moving from a simple opposition between "obedience" and "disobedience" to a game of "distributions" around a norm' and 'shifting the perspective from direct coercion to regulation' (Darmon 1999: 5).

From such a perspective, normativity with regard to the parental role is far from being a new issue. Indeed it is deeply buried in the history of our contemporary societies. We could even trace it as far back as Jean-Jacques Rousseau's famous treatise on education Emile ou de l'éducation [Emile or On Education], first published in 1762. The terms of this reflection (which intensified during the 18 th century) on the status of the child and the purpose of education laid down the basis not only for distinguishing between instruction and education but also for a new division of roles between family and state. Right from the outset, this normative production and this 'advice' to parents on education came up against the barrier of the family institution itself, and above all, in fact, against paternal authority -with fathers, in many cultures, considered heads of the family and guarantors of compliance with the rules within their own small 'community'. Although Rousseau recommended advances in favour of public education, he did not call paternal authority into question. On the contrary, he contributed to re-legitimizing its power -including at the expense of mothers. Almost two hundred years were to pass before this power was removed in France ('parental authority' reform in 1970).

The construction of the 'welfare state' (education, health, and social) in the course of the 19 th and 20 th centuries has continued to modify relationships between the private and public domains, shifting this privacy barrier and intensifying this normative work (in particular for protecting children and mothers) (see [START_REF] Castel | Les métamorphoses de la question sociale. Une chronique du salariat[END_REF][START_REF] Commaille | Les enjeux politiques de la famille[END_REF][START_REF] Donzelot | La police des familles[END_REF][START_REF] Joseph | Disciplines à domicile. L'édification de la famille[END_REF][START_REF] Lenoir | Généalogie de la morale familiale[END_REF]. In addition, by extending the scope of its action, the state has gradually relieved the family of several of its functions. Alongside this, it has, however, created the epicentre of what was gradually to become known as 'social' by developing responses to compensate for disabilities and support certain citizens experiencing weaknesses and vulnerabilities. Mothers are in the front line here (especially those belonging to the working class) as special targets for recommendations and other measures aimed at control and management from the 18 th century to the present day -particularly through public health interventions [START_REF] Boltanski | Prime éducation et morale de classe[END_REF][START_REF] Garcia | Mères sous influence. De la cause des femmes à la cause des enfants[END_REF][START_REF] Gojard | Le métier de mère[END_REF].

Indeed, the definition of the 'job of parenting' is an outdated notion that has been pursued over more than a century by a multitude of experts anxious to dispense advice and recommendations to parents 'in distress' -as well as by public decision makers denouncing the risks to which parents from the 'dangerous classes' would expose children perceived as being 'in danger'. Yet have not the very terms of this definition of the problem changed?

Advice and recommendations aimed at parents have also, for more than a century now, represented a real market that relies particularly on the feeling shared by parents that their task is difficult and that many obstacles block the path to their child's successful socialization and life. Many psychologists, paediatricians, psychiatrists, psychoanalysts -and more broadly all those who might be qualified alongside Robert Castel as 'therapists for normal people', using 'medical-psychological techniques' [START_REF] Castel | Le psychanalysme[END_REF][START_REF] Castel | La gestion des risques. De l'anti-psychiatrie à l'après-psychanalyse[END_REF] -operate within the niche of this growing demand for expertise and advice to parents, devoting a more or less substantial share of their professional practice to it. 10Most of the knowledge accumulated over the course of the 20 th century in the field of psychology, paediatrics, child and adolescent psychiatry, and psychoanalysis has thus been devoted to understanding, defining, and directing this parental role in order to come up with the best possible conditions for the guidance and socialization of the child (see Neyrand 2000). These experts in the parental role are now at work in the multiple links of a sprawling and mediatized market -a phenomenon accurately spotted by Robert Castel as long ago as the late 1970s. 11 This is the reason why, to identify what has really changed in the last 25 years, I suggest picking up on the socio-historical lineage of Isaac Joseph and Philippe Fritsch in Disciplines à domicile, L'édification de la famille (1977), Jacques Donzelot in La police des familles (1977), or Luc [START_REF] Boltanski | Prime éducation et morale de classe[END_REF] study of the rules of childrearing and household teaching in the late 19 th and early 20 th centuries in Prime éducation et morale de classes. These works (that were influenced in part by the work of Michel Foucault 12 ) quickly took stock of the impressive observation project represented by this interaction between the state, public authorities, social and healthcare professionals, and the 'entrepreneurs of family morality' on the one hand and the familial sphere and its actors on the other.

A quick recap of the main orientations in these works reveals their proximity to the current phenomenon. To take into account his perspective in La police des familles, Donzelot, for example, insisted on his determination to decode the modern passage from a model in which families were subjected to coercion into a far subtler mechanism comprising overinvestment in the role of the family 'by making it the pre-condition for each member's fulfilment ' (1977/2005: 5), and also, in the eyes of many, the guilty party in the event of a failed socialization process:

To describe the reform movement that constituted the modern family, we have thought of it in the same way as the passage from 'government of families to government by the family'. Government of families: this is the family of the Ancien Régime, a political subject, capable of using its members as instruments, deciding upon their fate, accountable for the behaviour of their members before the royal power, yet likely in return to draw upon it to impose its order on recalcitrant members. Government by the family: this time, the family is no longer the policy subject in its own history. Rather, it becomes the object of a policy. Its members are no longer expected to make alliance strategies or manage affiliations, because the family is now a means for each member to achieve their own fulfilment, each person being able to justify the deficit in their own fulfilment and blame it on the family, on condition that they have the support of a judge, social worker, or therapist who will help them to identify the source of their difficulties in the failings of their family (past or present) and to free themselves in one way or another. [START_REF] Donzelot | La police des familles[END_REF](Donzelot /2005: 6) : 6) Is not this phenomenon completely analogous when it comes to the parental role? Is it possible to establish a parallel between [START_REF] Donzelot | La police des familles[END_REF] and [START_REF] Furedi | Paranoid Parenting. Why Ignoring the Experts May Be Best for your Child[END_REF] hypothesis on 11 'The discourse put out by fans of Family Planning or the 'Ecole des parents', spokespersons on radio shows specializing in family and conjugal advice, women's magazines, and 'society' sections in magazines and weekly publications places responsibility for the ultimate reality of the family squarely on its members' ability to intensify their relationships and use psychology to regulate them (Castel 1981: 185). 12 We could add the work of Norbert Elias as a source of inspiration in this lineage of interactions between individuals and society [START_REF] Elias | La société des individus[END_REF].

parental determinism? It is just as easy to connect the contemporary parenting support issue to the questioning developed by Isaac Joseph and Philippe Fritsch (1977) when explaining their project in Disciplines à domicile, that is to say: working out the lineage of the normalization of intra-familial relationships, and more specifically of educational relationships since the end of the 18 th century. Our hypothesis is that this normalization owes less to the overall subordination of the family to the logic of state apparatus and its role in reproducing social relationships than to the import to its field and in its practice of disparate disciplinary tactics that originated in school, prison, and hospital environments or the field of social assistance. (p. 19) For these two writers if, in the 19 th century, we shifted from being a society of law to become a society of norms, as Michel Foucault says, this has been via a series of dramatized figures, the logic of which is laid bare by a body of specialists in the order of knowledge. (p. 22; author's translation)

The question is thus to understand the current way to extend and reconfigure this normative work and this tradition of analysis. In so doing, one could certainly focus more precisely on the novelty of this emerging policy towards parents' practices and distinguish which part of this novelty is context-related and which part is linked directly to interventions and practices.

Frames behind the novelty of parenting support:

Summing up my exploration, I propose three key trends. First, new policies and institutions are undeniably emerging in different countries during a single period (late 1990s-early 2000s) using a new and similar terminology, common arguments, and issues but also generating a new market in programmes and instruments. But, second, this emergence has been framed by pre-existing ideological and professional fights, by political battles in the course of political change and campaigns, as we argued for the French case. And third, the turn to parenting is also rooted in a long tradition that probably varies according to countries, but is being reactivated during the process by multiple comebacks and old ghosts from meta-cultural frames. Because of this ideological battle, the observable change over the past two decades is structured by very classic oppositions, most of which are implicit. This is clearly the case in France, where normative production and institutions around family issues are particularly active. We therefore suggest a line of analysis referring to some national specificities that may have to do with (among others) the aforementioned State and Church conflict but also with a particular political party system shared by other southern European countries.

A common 'parenting culture'?

In the United Kingdom, the current parenting support policy is analysed as a new trend concomitant to or initiated by New Labour's Third Way that has its roots in older Christian Socialist thinking and also Poor Law traditions (see [START_REF] Daly | Shifts in family policy in the UK[END_REF]. Following different authorsin particular Frank Furedi (2008) -Ellie Lee, Jennie Bristow, Charlotte Faircloth, and Jan Macvarish recognize that a 'parenting culture' has been developing for a long time, and that its 'basis lies in the working through of the separation out of "the family" from the wider economy and society' (Lee et al. 2014: 7). Yet the novelty now refers to the 'explicit focus on the parent and their behaviour' (op. cit.: 9), as well as to 'parental determinism' and the necessity of targeting the 'parenting practices of those who claim welfare benefits' (ibid), in short: parenting as a social problem.

We can be sure that 'parenting' is not a neutral term to describe what parents do as they raise their children. Rather, the transformation of the noun 'parent' into the verb 'parenting' has taken place through a sociocultural process centring on the belief that 'parenting' is a highly important and problematic sphere of social life; indeed, 'parenting' is almost always discussed as a social problem and in some way blamed for social ills. (Lee et al. 2014: 9) Val [START_REF] Gillies | Perspectives on Parenting Responsibility: Contextualizing Values and Practices[END_REF] speaks of the rise of parenting in the United Kingdom driven by the 'Third Way'. She promotes the view that a profound cultural change affecting the role of parent, due to a focusing of the attention of public authorities on the act of 'parenting', gave rise to the idea of thinking of parenting as a competence likely to give rise to learning, leading to training actions, and necessitating a process of professionalization.

Parents have always been held responsible for the behaviour and development of their children but recent years have seen a cultural shift in the way childrearing is conceptualized and targeted by policymakers. In the past, intimate family relationships tended to be viewed as personal, private, and outside the remit of state intervention . . . Parenting is no longer accepted as merely an interpersonal bond characterized by love and care. Instead it has been re-framed as a job requiring particular skills and expertise which must be taught by formally qualified professionals. (Gillies 2008: 95-96) One could bring together the United Kingdom and French situation, because this idea of thinking about parenting as a job requiring skills and training employs precisely the terms used by a movement named l'école des parents, created in 1930 (and still in existence today), whose objective was, at the very beginning: 'to teach parents to educate and instruct one another so that their children become imbued with future social and moral values' (quoted in Donzelot 1977: 181). But such an assumed affinity could also miss a central point. This French movement (largely inspired by Catholicism and psychoanalysts) had a precise historical context: defending family and parents against intrusion by the state and its agents (teachers and public health professionals) in private matters. Whereas in the United Kingdom, the family has been a private matter -with the exception of 'dangerous' poor classes, the child in France has been constructed as 'the nation's child', against which Catholic and other groups turned and fought around the issue: 'to whom the child belongs?'. As one of the European countries in which the family has been considered an affaire d'Etat since the very beginning of the 20 th century, France thus has an important specificity, namely the struggle between two forms of familialism: State familialism versus Church familialism [START_REF] Lenoir | Généalogie de la morale familiale[END_REF]. The current parenting support policy provides a perfect battlefield on which to expand these arguments by opposing Republican and religious positions.

For the moment, as a high-ranking civil servant we interviewed in the PolChi project stated, Republican argumentation is still clearly dominant, even when it takes into account the fact that some citizens have more needs and demands than others. But in this radical movement:

We remain faithful to the ideal of Republican universality, but we are careful to ensure we support more people having higher needs, in a sort of proportioned universalism. (High-ranking civil servant, author of an official report on parenting support, Polchi interview)

State versus Catholic church: a metacultural frame

France offers many historic examples of this structural opposition between the Catholic church and the République. But it is important to underline that this fight is still ongoing. One of the latest expressions of this opposition was La manif pour tous, a social movement contesting both the adoption of mariage pour tous (same-sex marriage) and the Socialist government's initiative aimed at raising children's awareness of gender equality issues at school (l'ABCD de l'égalité). 13 This conservative movement condemns a hypothetical 'gender theory' at the same time as it defends 'natural' sex differences and the necessity of distinguishing between mother and father. They only have a problem with an (apparently gender neutral) parenting policy where they believe it leaves the door open to gender confusion.

To understand this drastic opposition, it is useful to recall the crucial arguments concerning political conflicts and coalitions and their specific combination in many southern European countries, including France, that are linked to the fight between the State and the Catholic church, between a Republican elite and Catholic movements [START_REF] Martin | Southern Welfare States: Configuration of the Welfare Balance between State and the Family[END_REF]). This opposition is absolutely central to understanding the configuration of the welfare state in these countries, particularly where family issues are concerned.

Following van [START_REF] Van | Social Capitalism. A Study of Christian Democracy and the Welfare State[END_REF] work on Christian democracy and the link between religion and the welfare state, Kees van Kersbergen and Philip [START_REF] Manow | A Conservative Welfare State Regime without Christian Democracy? The French État-Providence, 1880-1960[END_REF] paid special attention to the crucial role of the electoral and political party systems. Their aim was to explain the difference between Social Democrats and Christian Democrats when looking at the role of religion in the framing of western welfare states. They reveal two different coalitions in each configuration:

The Social Democratic generous welfare states, which we find in the Nordic countries, have been the result of a coalition between Social Democratic parties and parties of agrarian defence (Red-Green coalition). One important precondition for this coalition has been the absence of a strong religious cleavage in the Scandinavian countries. On Europe's continent, in turn, we find welfare states that are the product of a coalition between Social and Christian Democracy (Red-Black coalition). This is due to the fact that the second cleavage represented in the party systems of continental Europe, besides the dominant left-right or labour-capital cleavage, has been the religious cleavage, a cleavage inherited from the State-Church conflicts in the wake of the national revolution in which liberal states' elites challenged the church in its former domains such as education or poor relief. (van Kersbergen /Manow 2009: 22) In a recent paper, Philip Manow goes further and argues:

The fundamental character of the political conflict reveals the explanatory limits of an argument based solely on socio-economic analysis. In these conflicts, religion becomes relevant, first in the explanation of the totalitarian episodes of the southern countriessince religion renders coalitions between workers and peasants impossible and thereby fosters the fascist path. These totalitarian episodes are then, second, an important explanatory factor for the persistence of political polarization in the postwar period, inter alia manifested in the fragmentation of the left in their party systems. (Manow 2015: 37) For this author, this combined religious/political factor is crucial in each mono-confessional southern country, because it simultaneously explains the radicalization to both right and left of the political arena: reactionary and anti-republican Catholicism since the 19 th century, the rift between a reformist and a radical left wing, and the violent clash between radical secularism and radical religiosity, between a clerical right and an anticlerical left.

The split between communist and social-democratic parties in the countries of southern Europe (Italy, Spain and Portugal, but also France) . . . is closely related to the decidedly anti-republican position held by the Catholic church in the mono-denominational Catholic countries of Europe's South, since the deep divide between a sharp anticlerical labor movement and pious farmers under close tutelage of the church left the political left without allies for a reformist strategy. (Manow 2015: 33).

These religious and political factors offer a much better explanation than economic ones: the differences between Nordic, continental, and southern countries, all of them concerned with late industrialization. Where a political alliance was possible between smallholding farmers and workers in the (Protestant) northern countries, it was clearly rejected 'due to noneconomic reasons' (Manow 2015: 40) in the Catholic south, marked by the violence of the confrontation between pious rural classes and the Marxist workers' movement. These conflicting ideologies are undoubtedly still at work in the framing of parenting support policy; and in France, they take the form of contrasting options: supporting (rather than controlling or condemning) families, and choosing universalist and egalitarian (rather than targeted and punitive) measures.

Concluding remarks: A turn brought about by a new intergenerational challenge

Though the processes of constructing a parenting issue do resemble one another from here to there and from then to now in Europe -for instance, in the act of backing measures and policy with arguments and 'scientific proof' or the use of a universal purpose to cover a targeted approach and differentiated social treatment -in short, of euphemistically addressing the question of social class and inequalities -it does seem that the context in which this construction is produced also has its specificities, because of the changes that have occurred in terms of not only familial practice and lifestyles but also the state's ability to intervene to guide these transformations (Martin 2012b).

We can thus, for example, ask whether today's parenting support may allow us to update Robert [START_REF] Castel | La gestion des risques. De l'anti-psychiatrie à l'après-psychanalyse[END_REF] diagnosis in terms of the management of risk. Back in the late 1970s, Castel had observed the increased use of the psychological 'human potential intensification' techniques that were typical of the humanist psychology of that time. He wrote that it was no longer a matter of 'healing a disease, fixing a dysfunction, or remedying a deficiency, but rather of working on each individual's personal and relational capital in order to intensify and improve its performance' (Castel 2011: 11). In this way, Castel identified 'a reorganization of social policy and interventions by the social state ("the active social state") in the sense of activation of the individual' (Castel 2011: 12). Today, it seems we have moved on to an additional stage: to one in which the problem is no longer just the production of the "working adult" via psychological optimization, but rather that of focusing attention on the role played by those very adults who have become parents in their function as agents of socialization -within a context in which the levers of social advancement appear in total collapse.

In this sense, it could be argued that parenting support serves to complete personal development (even, to some extent, standing in for it), because demand has moved from the ambient hedonism and individualism of the post-May 1968 generation towards the uncertainties of the present time, particularly with regard to the future of new generations [START_REF] Castel | La montée des incertitudes. Travail, protections[END_REF], and, as a consequence, from adult-centred to child-centred policy. The contemporary focus on the parenting issue defines novel priorities because of the rising pertinence of the question of children's rights and public powerlessness in the face of the difficulties encountered by new generations -children, adolescents, and young adults -that, in turn, echo the supposed threats posed by antisocial behaviour.

The popularity of this keyword is such that a new journal entitled Parenting, Science and Practice was created in 2001, defining its field as: 'Parenting: Science and Practice strives to promote the exchange of empirical findings, theoretical perspectives, and methodological approaches from all disciplines that help to define and advance theory, research, and practice in parenting, caregiving, and childrearing broadly construed . . . The journal brings parenting to science and science to parenting', see http://www.tandfonline.com/action/journalInformation?show=aimsScope&journalCode=hpar20

For a development of these national cases, see the Special Issue 'Parenting support in European countries' edited by Mary Daly in Social Policy and Society, 14(4), 2015. See also[START_REF] Boddy | International perspectives on parenting support non-English language sources[END_REF]; Richter and Andresen (2012); and[START_REF] Ramaekers | The claims of parenting. Reasons, responsibility and society[END_REF].

An international think-tank of experts founded in California in 1945 and issuing opinions on a great many research and development issues.

Translation of all interviews by the author.

This Institute (Institut de l'Enfance et de la Famille) was created in 1984 and then merged in 1997 with the Centre international de l'Enfance created in 1947 to become the Centre International de l'enfance et de la famille.

. See http://www.pasde0deconduite.org/appel/

'An important consequence of the development of biopower is the growing space occupied by the norm set at the expense of the law's legal system'(Foucault 1976: 189).

For example, we might mention certain very popular authors advising parents throughout the 20 th century such as the behaviourist John Watson in the 1920s and 1930s, Benjamin Spock in the 1940s and 1950s, Thomas Brazelton and Penelope Leach in the 1970s, or John Rosemond in the 1990s. For a historical analysis of these experts and messages, see Ann[START_REF] Hulbert | Raising America: Experts, Parents, and a Century of Advice About Children[END_REF].

See http://www.cndp.fr/ABCD-de-l-egalite/accueil.html.


G Prabhudesai

I Bihi

F Zoueshtiagh

J Jose

M Baudoin

Nonspherical armoured bubble vibration †

In this paper, we study the dynamics of cylindrical armoured bubbles excited by mechanical vibrations. A step by step transition from cylindrical to spherical shape is reported as the intensity of the vibration is increased, leading to a reduction of the bubble surface and a dissemination of the excess particles. We demonstrate through energy balance that nonspherical armoured bubbles constitute a metastable state. The vibration instills the activation energy necessary for the bubble to return to its least energetic stable state: a spherical armoured bubble. At this point, particle desorption can only be achieved through higher amplitude of excitation required to overcome capillary retention forces. Nonspherical armoured bubbles open perspectives for tailored localized particle dissemination with limited excitation power.

Introduction

Armoured bubbles, i.e. bubbles coated with a dense layer of partially wetting particles, exhibit fascinating properties such as increased stability toward dissolution 1,2 or the ability to sustain non-spherical shapes 3 . These bubbles can be massively produced by bulk emulsification techniques [4][START_REF] Binks | Colloidal particles at liquid interfaces[END_REF][START_REF] Fujii | [END_REF] or by bubble injection in a suspension of particles 7 , but with a limited control over the bubbles properties. Recently microfluidic techniques have been considered to produce some calibrated armoured bubbles of tailored shapes, sizes and composition 8 . Some of these techniques simply rely on the natural inclination of particles to be adsorbed at air/liquid interfaces 9,10 , while others exploit fancy chemicalor temperature-mediated processes 11,12 or the deep modification of liquid/air interface dynamics in particle covered capillary tubes 13,14 . In parallel, much effort has been devoted to the characterization of armoured bubbles properties, and in particular their stability toward dissolution 1,2 , their lifetime when exposed to surfactant 15 or their mechanical strength when submitted to ambient overpressure 16 . Surprisingly, less effort has been devoted to the study of the response of armoured bubbles to mechanical vibrations [START_REF] Poulichet | Proc. Nat. Ac. Sci[END_REF][START_REF] Poulichet | [END_REF] , while bubbles are known to be outstanding resonators [START_REF] Strutt | Lord Rayleigh[END_REF][20][21][22][START_REF] Brennen | Cavitation and bubble dynamics[END_REF] , with high quality factors and remarkably low resonance frequencies 20 . Recently, Poulichet et al. [START_REF] Poulichet | Proc. Nat. Ac. Sci[END_REF][START_REF] Poulichet | [END_REF] investigated the acoustically driven oscillation of particle-coated bubbles. They showed that these oscillations can lead to the tailored expulsion of the particles from the surface at sufficient acoustic power. To the best our knowledge, the response of their nonspherical counterpart to mechanical vibrations has not been reported so far in the literature.

In this paper, we characterize the dynamics of cylindrical armoured bubbles excited with vertical vibrations produced by a vibration exciter. We show that the vibration leads to a step by step transition of the bubble shape from cylindrical to spherical as the amplitude of the vibration is increased. This shape evolution leads to a decrease of the surface of the armoured bubble and thus a release of excess particles in the liquid. This evolution is shown to be highly dependent on the excitation frequency: lower excitation amplitude is required near the Minnaert bubble resonance to trigger this transition. We demonstrate through energy balance that nonspherical armoured bubbles constitute a metastable state. The bubble vibration enables the release of the mechanical constraint between the particles which maintains the bubble shape and thus lets it evolve toward the least energetic stable state: a spherical armoured bubble. This transition does not require to overcome capillary retention forces and thus enables particle dissemination with reduced excitation power. Once the armoured bubble is spherical, massive particle desorption is only observed when the kinetic energy instilled from the surface vibration overcomes the capillary retention forces. The required power in this case is thus larger than for nonspherical bubble and should increase as the particle dimensions are decreased as demonstrated through dimensional analysis.

Materials and methods

Cylindrical armoured bubbles of tailored radius R c = 0.49 ± 0.01 mm and length L c = 8.5 ± 0.5 mm are produced with the method introduced in ref. 13: First, glass capillary tubes of radius R t = 501 µm are cleaned by successive sonication in acetone and alcohol. Then the tubes are treated with piranha solution (sulfuric acid + hydrogen peroxide) to clean organic residues off the glass. They are then kept in DI Water in a sealed recipient before use. Prior to the particle dispersion, they are dried for 1 hour in an oven at 120 • C. Second, Rilsan (Polyamide 11) particles of mean radius R p = 15 ± 1 µm are scattered into the tube by gently blowing them with an air jet. Third, a DI water liquid finger is pushed at constant flow rate (Q = 0.2 ml/h) with a syringe pump inside the tube leading to the formation of a long cylindrical armoured bubble through the process described in ref. 13.

Finally the bubble is extracted from the tube and injected inside a rectangular plexiglas closed chamber of height 20 mm, length 75 mm and width 55 mm filled with DI Water, by applying a larger flow rate of Q = 5 ml/h. The bubbles are cut at the appropriate length by twisting the capillary sideways while keeping it horizontal. This cuts the bubble on the open end of the capillary. The evolution of the bubble shape is recorded with a Hamamatsu C9300 high resolution camera mounted on an Olympus SZX7 macroscope, while the fast bubble oscillations are captured with a Photron SA3 high speed camera. Finally images are treated with ImageJ software and the evolution of the bubble shape is quantified by using the roundness shape descriptor R o = 4S/πL2 m , with S the apparent surface of the bubble (in the pictures), and L m the length of the bubble major axis. This shape descriptor gives a measure of the deviation of the bubble from a circular shape, with R o = 1 as the bubble becomes perfectly spherical.

Experimental results

Shape evolution

The vertical vibration of the cylindrical armoured bubbles at a given amplitude A (0.1 µm< A < 50 µm) and frequency f e (0.5kHz< f e < 4kHz) leads to an isovolume evolution of its shape through a reduction of its surface and a release of excess particles in the liquid (see Fig. 2 to 4 and Movie S1). When the power is turned on, the bubble shape evolves at first over characteristic times < 1min and then evolves slowly toward an asymptotic shape, which depends on the excitation amplitude A and the excitation frequency f e (see Fig. 5). As the amplitude of excitation is increased gradually, this asymptotic shape is more and more spherical, eventually leading to a spherical armoured bubble (Ro → 1) (see Fig. 2 and4). This transition is nevertheless not regular as a function of the excitation amplitude A: Under a frequency-dependent threshold, the bubble keeps its cylindrical shape (see Fig. 5 and6). Above this threshold, the bubble shape evolves at first weakly as the amplitude of the excitation is increased and then undergoes a sharp transition toward a spherical shape for a critical amplitude ≈ A c , defined here as the amplitude necessary for the bubble to achieve a roundness of 0.4 (see Fig. 6).

Minnaert resonance frequency

This critical amplitude A c highly depends on the excitation frequency f e (see Fig. 7) and decreases by several orders of magnitude close to the Minnaert resonance frequency of the bubble

f M = 3.26 × R -1 s , with R s = (3/4π V ) -1/3
the radius of a spherical bubble of same volume V = πR 2 c L c as the initial cylindrical ar-moured bubble. Indeed we verified that the bubble undergoes a isovolume shape transformation (less than 4% of volume change have been measured between the initial and final state, including errors in the estimation of the volume from the 2D pictures of the bubble). The precedent formula gives a value of the Minnaert resonance frequency of f M = 2.8 ± 0.1 kHz.

It is interesting to note that two extremely different behaviors are observed around the Minnaert frequency: At 3 kHz the transition from cylindrical to spherical shape is achieved with the weakest excitation amplitude. The evolution of the bubble shape starts at very low excitation amplitude (A = 0.1 µm A c ) and then goes on progressively when the amplitude is increased, until a spherical armoured bubble is obtained (see Fig. 4 and Fig. 6). At 2.5 kHz, no shape evolution is observed until the critical amplitude A c is reached. Then, when A = A c the bubble directly evolves into a particle-free spherical bubble with a release of all surrounding particles in the liquid (see Fig. 3, Fig. 6 and movie S2).

Discussion

Cylindrical bubbles: a metastable state

Nonspherical armoured bubbles constitute a metastable state. This can be shown by computing the interfacial energy required for a quasi-static and isovolume transformation of a spherical armoured bubble into a nonspherical one. Such a transformation necessarily requires an increase in the particles-covered interface of the bubble since the spherical shape minimizes the bubble surface. We can compute the energy ∆E, which is necessary for the migration of a particle from the liquid to an extended air-liquid interface (see Fig. 8). In the calculation below we assume that the particles are hydrophilic (with a liquid-air-particle contact angle 0 ≤ θ p ≤ π/2), spherical and that their radius R p is much smaller than the radius of curvature of the bubble.

The energy ∆E is simply the difference between the interfacial energy E b of the particle integrated at the air-liquid interface and the interfacial energy E a of the particle lying in the liquid: The interfacial energy E a in configuration (a) is:

∆E = E b -E a (1
E a = γ LP A a LP (2)
with γ LP the interfacial energy per unit area of the liquid-particle interface and A LP = 4 π R 2 p the liquid-particle interface in the configuration (a). The interfacial energy E b in the configuration (b) is the sum of the gas-particle (GP), liquid-particle (LP) and gasliquid (GL) interfacial energies:

E b = γ GP A b GP + γ LP A b LP + γ GL A b GL (3)
with γ and A the interfacial energies per unit area and the areas of the corresponding interfaces. Since the particles intersect the liquid-air interface with a contact angle θ P (to minimize interfacial energy), we deduce from simple geometric considerations that:

A b GP = 2πR 2 p (1 -cos θ P ) (4)
A b LP = 2πR 2 p (1 + cos θ P ) (5)
Then, due to their shape and arrangement, the particles cover only a fraction φ of the gas-liquid interface, called the specific surface area. Thus, the gas-liquid interface that is necessary to incorporate a new particle is πR 2 P /φ , with φ < 1. If we subtract the surface occupied by the particle itself, we obtain:

A b GL = πR 2 P /φ -πR 2 P sin 2 θ P . (6)
If we now combine equations 1 to 6, we obtain:

∆E = πR 2 p [2γ GP (1 -cos θ P ) + 2γ LP (1 + cos θ P ) +γ GL 1/φ -sin 2 (θ P ) -γ LP 4 π R 2 p
From the definition of the contact angle cos θ P = (γ GPγ LP )/γ GL , we finally obtain:

∆E = γ GL πR 2 P 1 φ -(1 -cos θ P ) 2 (7)
Since 0 ≤ (1cos θ P ) 2 ≤ 1 (for hydrophilic particles) and 1/φ > 1, we have * :

∆E > 0
This energy is positive indicating that the quasi-static transformation of a spherical bubble into a nonspherical one costs some energy. In other words a nonspherical bubble is not a stable state and would return to its least energetic state, i.e. a spherical armoured bubble in the absence of other constraint. Nevertheless, as discussed initially in ref. 3, jamming between the particles prevents this evolution and maintain the bubble in a metastable state.

In the present experiments, the bubble vibration instills the activation energy required to induce this transition. Indeed, during the expansion phase of the bubble oscillation, the particles are no more in contact, and are thus freed from their interparticle mechanical constraint. In addition, the kinetic energy instilled to the particles by the vibration of the surface contribute to the release of the particles. This enables the bubble shape reconfiguration into a spherical armoured bubble.

Particle expulsion from spherical armoured bubbles

Spherical armoured bubbles on the other hand constitute a stable state. Indeed, a bubble without particles is a minimum surface energy configuration. Then, if a particle lying in the liquid migrates on the already existing liquid-air interface, the energy necessary for this migration ∆E * is simply [START_REF] Binks | [END_REF] :

∆E * = ∆E - πR 2 P φ γ GL = -γ GL πR 2 P (1 -cos θ P ) 2 < 0
which is negative, indicating that the minimization of surface energy leads to an adsorption of the particles which come into contact with the air-liquid interface. This principle was used in ref.

9,10,16 to create armoured bubbles. Thus the desorption of particles lying at the surface of a spherical armoured bubble requires to overcome capillary retention forces as noticed by Poulichet and Garbin [START_REF] Poulichet | Proc. Nat. Ac. Sci[END_REF][START_REF] Poulichet | [END_REF] . These authors showed experimentally that localized particle desorption occurs for weak oscillations of spherical armoured bubbles but that the massive release of particles in the liquid requires higher amplitudes of vibrations.

We can estimate the kinetic energy that is necessary to overcome the capillary barrier, by simply equating E c = 1 2 m P V 2 p and |∆E * |, with m p = 4/3πρ p R 3 p the mass of the particles, ρ p their density, V p = 2π f e ∆R s their velocity at the oscillating interface, and ∆R s the radial variation of the bubble.

We performed experiments on individual spherical armoured bubbles of radius R s = 1.25 mm covered with Rilsan particles of radius R p ≈ 15µm, density ρ p ≈ kg m -3 , contact angle θ P ≈ 70 • 13 , lying in water with surface tension γ GL ≈ 70 mN m -1 (see e.g. movie S3). Massive particle detachment was observed when the bubble oscillation magnitude reaches ∆R s /R s ≈ 10% of the initial radius of the bubble. This leads to E c /|∆E * | ≈ 0.4. For the experiments performed at 2.5kHz (see e.g. Movie S2), the ratio E c /|∆E * | measured when massive desorption of the particles occurs is approximatively equal to 2. These experiments are thus consistant with this criterion.

It is also interesting to note that this simple criterion is in qualitative agreement with the data provided by Poulichet et al. [START_REF] Poulichet | [END_REF] . Indeed, in their experiments, the latex particle size and density are respectively R p = 250 nm and ρ p = 1040 kg m -3 , the air-liquidparticle contact angle is θ p ≈ 45 o 17 , the frequency of excitation is f e = 40 kHz, the surface tension is γ GL ≈ 70 mN m -1 and the amplitude of oscillation required for the particles dispersion is typically 50 µm. This leads to a ratio E c /|∆E| required for particle dispersion of ≈ 3.

Analysis of the dynamics observed near the Minnaert resonance frequency

The analysis detailed in the two previous subsection provides insightful elements, which help in understanding the two different behaviors observed at 2.5 kHz and 3 kHz, near the Minnaert resonance frequency of the spherical bubble. When the bubble is excited at 2.5 kHz, no shape modification is observed when A < A c and then brutally, all particles are ejected from the interface when A = A c = 1 µm. While at 3 kHz, the shape evolution starts at low amplitude A = 0.2 µm<< A c and reaches a spherical armoured bubble shape at A = 1 µm. This suggests that 2.5 kHz is closer to the resonance frequency of the spherical bubble and that 3 kHz is closer to the resonance frequency of the cylindrical armoured bubble. Indeed, at 2.5 kHz, the bubble response is weak (since the excitation differs from the cylindrical armoured bubble resonance frequency) until the vibration is sufficient to induce a shape transformation. As the shape becomes more and more spherical, the bubble amplitude increases up to an amplitude that is sufficient to overcome capillary retention forces, resulting in the release of all interfacial particles in the liquid. At 3 kHz, the bubble starts responding at low amplitude of excitation (since the excitation frequency is close to the cylindrical armoured bubble resonance frequency), but as the bubble shape evolves, the excitation frequency differs from the resonance frequency, requiring more and more power to enable a shape evolution of the bubble. When the spherical shape is reached, the excitation amplitude of A = 1 µm is not sufficient to overcome the capillary retention forces and thus the bubble keeps its armour.

Nonspherical bubbles: powerfull vectors for particles transport and dissemination with reduced activation power

All the experimental and theoretical results provided in this paper indicate that particles dissemenation can be achieved with less power with nonspherical than with spherical armoured bubble. The dimensional analysis below suggests that this tendency should increase as the size of the bubble and particles are decreased.

Let's consider a spherical bubble of radius R s covered with particles of radius R p excited at its resonance frequency f M (for maximum efficiency) oscillating at an amplitude of oscillation ∆ Rs = ∆R s /R s sufficient to disseminate the particles. The equality E c = |∆E * | leads to:

1/2 ρ p × (4/3πR p ) 3 × R s ∆ Rs × 3.26 R -1 s 2 = γ GL πR 2 P (1-cos θ P ) 2
If the density and wetting properties of the particles are kept constant and only the size of the bubble and particles are changed, one obtain:

∆ Rs ∝ R -1 p
The bubble amplitude of oscillation required for particles dissemination does not seem to depend on the bubble size but is nevertheless inversely proportional to the size of the particles. This means that for smaller particles, a larger relative amplitude of oscillation is required to untrap the particles located at the surface of a spherical armoured bubble. While for a cylindrical bubble, the same relative amplitude of oscillations would enable the release of particles since there is no need to overcome capillary retention forces, but only to suppress the contact between the particles. This might be essential for medical applications. Indeed, contrast agents currently used in the body have a typical size of a few microns and active material of typically a few nanometers are generally required for efficient dissemination in the body. It might thus become difficult to overcome capillary retention forces at this scale.

Conclusion and perspectives

In this paper, the response of nonspherical armoured bubbles to mechanical vibrations was studied experimentally and analyzed theoretically. In particular, it was shown that a nonspherical armoured bubble constitute a metastable state, which evolves toward its minimum energy state, i.e. a spherical armoured bubble, as its surface is vibrated. This shape evolution leads to a reduction of the bubble interface and consequently a release of the excess particles. Once the spherical shape is reached, the massive release of particles from the interface requires that the kinetic energy instilled to the particles by the vibration of the bubble surface exceeds the desorption energy. Scaling laws indicate that the release of particles from spherical armoured bubbles becomes extremely challenging as the size of the bubbles is decreased. Thus nonspherical armoured bubbles appear as a tremendous alternative for controlled particle dissemination at small scales with limited input power. Applications can be envisioned in medicine for in-vivo active drug delivery through the use of robust acoustical methods precedently developed for ultrasound contrast agents.

Fig. 1

 1 Fig. 1 Scheme of the experimental setup

Fig. 2

 2 Fig. 2 Evolution of a cylindrical armoured bubble of length L c = 8.5 mm and radius R c = 0.5 mm vibrated vertically with a vibration exciter at frequency f e = 1 kHz and at an amplitude A indicated at the top of each picture. Each picture is taken after two minutes of excitation.

Fig. 3

 3 Fig. 3 Evolution of a cylindrical armoured bubble of length L c = 8.5 mm and radius R c = 0.5 mm vibrated vertically with a vibration exciter at frequency f e = 2.5 kHz and at an amplitude A indicated at the top of each picture. Each picture is taken after two minutes of excitation. In the final picture, all the particles initially at the surface of the bubble have been disseminated in the liquid.

Fig. 4

 4 Fig. 4 Evolution of a cylindrical armoured bubble of length L c = 8.5 mm and radius R c = 0.5 mm vibrated vertically with a vibration exciter at frequency f e = 3 kHz and at an amplitude A indicated at the top of each picture. Each picture is taken after two minutes of excitation.

Fig. 5

 5 Fig.5 Transient evolution of the roundness Ro of a cylindrical armoured bubble excited at 1 kHz as function of time (in min). The squares show the evolution of the bubble roundness when the amplitude A is increased step by step every 10 min from A = 9.7 µm to A = 51.7 µm. The circles show the evolution when the amplitude is directly set up to 51.7 µm.

Fig. 6

 6 Fig.6 Evolution of the Roundness as a function of the excitation amplitude for different excitation frequencies f e . For each frequency, the roundness is measured sequentially after 2 minutes of excitation at each amplitude.

Fig. 7

 7 Fig. 7 Critical excitation amplitude A c required for the bubble to reach a roundness R o = 0.4 as a function of the excitation frequency f e . Two experiments have been carried out for each frequency.

Fig. 8

 8 Fig.8 Sketch illustrating the migration of a particle lying in the liquid to an extended meniscus.

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Acknowledgements

This work was supported by Agence Nationale de la Recherche (Grant No. ANR-12-BS09-0021-01).