Nicolas Letalleur

Franck Plouraboué

Marc Prat

F Plouraboue

Ḿ Prat

Average Flow Model of Rough Surface Lubrication:

Flow Factors for Sinusoidal Surfaces

Introduction

The impact of surface roughness on lubrication is an important problem for many applications, albeit quite a difficult one. From a geometrical point of view only, it requires the computation of the flow over a time dependent fluid domain resulting from the motion and deformation of the moving solid surfaces. The more complex the surface ͑e.g., random, multi scaled, tridimensional rough-ness͒, the more difficult it is to understand the interplay between the surfaces topography and the observed macroscopic pressure, flux, and shear. Additional physical effects, such as cavitation, piezoviscosity or compressibility, among others, may contribute to the complexity of the problem but will be ignored in the analysis since, for simplicity, as far as up-scaling is concerned, they are generally discarded ͓1,2͔.

There is nevertheless an important body of literature that have been interested in such up-scaling description of sliding surfaces lubrication since the first developments of Christensen ͓3͔ and Chow et Cheng ͓4͔. Different techniques have been used to derive macroscopic equations coupling the macroscopic flux, pressure and shear stress, with some ''flow factors'' that are related to the underlying surfaces geometry. Patir and Cheng ͓1,2͔ were the first to propose a heuristic model for general roughness patterns. Tripp ͓5͔ derived the correct tensorial form of the average flow model using a stochastic approach, while Bayada et al. ͓6,7͔ addressed the problem within the framework of the homogenization theory for spatially periodic structures and more recently volume averaging techniques ͓8͔ were used for macroscopisation of the flow between lubricated sliding surfaces with solid contacts ͓9͔.

The literature thus provides some averaged macroscopic model for lubricated sliding surfaces propounding a systematic method for the flow factors computation. The latter reference ͓9͔ has nevertheless pointed out that flow factors must be computed combining spatial and time average when the two surfaces are intercorrelated. Such correlation between solid surfaces occurs naturally during rolling process where roughness is transferred from the steel roll to the workpiece ͓10͔ as well as in any process where deformations conform the two pieces one another.

In the present paper, this precise issue is further analyzed considering a specific geometry for the sliding surfaces. Simple unidirectional sinusoidal wavy surface ͑Fig. 1͒ are chosen for careful investigation. Such one-dimensional anisotropic surface roughness, are relevant, in the context of laminating processes ͓10͔ or in those of magnetic recording disk ͓11-13͔ where surfaces presenting one-dimensional like streaks are commonly found. This problem is of interest from at least four points of view: ͑1͒ it is, to a great extent, amenable to analytical developments, and thus it could be interesting to estimate the impact of some coarse description of surface roughness on flux, pressure and shear without any numerical computations, ͑2͒ the flow factor behaviors near contact can be obtained analytically, ͑3͒ the obtained results can be useful for testing numerical computation procedures, in particular for close contact lubrication studies, for which we give an explicit analytical treatment ͑4͒ it sheds light on the influence of inter-correlation between surfaces. In particular, we show that the concept of composite roughness does not apply where two surfaces are inter-correlated. Obviously, considering sinusoidal roughness as model roughness is not a novelty in the lubrication literature, ͓14 -18͔. However, to the best of our knowledge, the complete hereby presented analytical developments have not been reported previously.

As in ͓9͔ and ͓19͔, the analysis is restricted to non-deformable surfaces, isoviscous Newtonian and incompressible fluids. Cavitation, if any, is ignored as well as temperature variations. The local slopes being small, the flow at the roughness scale is described by the Reynolds ͑lubrication͒ equation. The flow is generated by the relative surfaces motion and the macroscopic pressure gradient.

Geometry and Kinematics

Figure 1 illustrates the geometry of the problem. y ͑resp. x͒ is the coordinate associated with ͑resp. perpendicular to͒ the streak direction. The top surface-defined as surface number 2-is moving at velocity U 2 and the bottom one ͑surface 1͒ at velocity U 1 . Note that U 2 and U 1 are not necessarily collinear, and that there have zero vertical components. The characteristic wavelength of sinusoidal surfaces is the same on each surface. The surface amplitude is noted a and b respectively for surfaces 1 and 2. h is the local aperture and h m is the mean distance between the two sur- faces. h m can be defined as h m ϭaϩbϩ␦ where ␦ is the minimum local aperture between the two surfaces ͑Fig. 2͒. In order to study the flow factor behaviors near contact, we define the dimensionless variable ⑀ as ⑀ϭ␦/ where is the composite r.m.s roughness classically defined as ϭͱ 1 2 ϩ 2 2 ͑where 1 and 2 are the r.m.s roughness of each surface, which gives here 1 2 ϭa 2 /2 and 2 2 ϭb 2 /2͒. When surfaces touch one another, then ⑀ϭ0 ͑i.e., h m ϭaϩb͒. In this limit, permanent contact occurs only when one of the two amplitudes a or b is zero. The time and space evolutions of surfaces height h i , iϭ1,2, are given by

h i ͑ x,t ͒ϭh io ϩͱ2 i sin ͩ 2 ͑ xϪU ix t ͒ϩ i ͪ
where h io is the mean planes of surfaces i related to the distance between both surfaces mean plane h m ϭh 2o Ϫh 1o , and i is the surface phase at initial time t o . The aperture field h(x,t) then reads

h͑x,t ͒ϭh m ϩb sin ͩ 2 ͑ xϪU 2x t ͒ϩ 2 ͪ Ϫa sin ͩ 2 ͑ xϪU 1x t ͒ϩ 1 ͪ
Defining the following dimensionless quantities:

H ⑀ ͑ X,t ͒ϭ h͑x,t ͒ h m ; Xϭ 2x ;
(1)

A ⑀ ϭ a h m ϭ a aϩbϩ⑀ ; B ⑀ ϭ b h m
leads to express the local aperture in dimensionless form as

H ⑀ ͑ X,t ͒ϭ1ϩB ⑀ sin͑XϪ 2 tϩ 2 ͒ϪA ⑀ sin͑XϪ 1 tϩ 1 ͒ (2)
where surface iϭ1,2 pulsations are given by i ϭ2U ix /. According to Eq. ͑2͒, H ⑀ (X,t) can be regarded as the sum of two harmonic oscillators of pulsation 1 and 2 , respectively. Classical developments ͑Appendix A͒ lead to express H ⑀ (X,t) as

H ⑀ ͑ X,t ͒ϭ1ϩC ⑀ ͑ t ͒sin͑ Xϩ ⑀ ͑ t ͒͒ (3)
with

C ⑀ ͑ t ͒ϭͱA ⑀ 2 ϩB ⑀ 2 Ϫ2A ⑀ B ⑀ cos͓͑ 1 Ϫ 2 ͒tϩ 2 Ϫ 1 ͔ (4)
and

⑀ ͑ t ͒ϭarctan ͫ B ⑀ sin͑ 2 Ϫ 2 t ͒ϪA ⑀ sin͑ 1 Ϫ 1 t ͒ B ⑀ cos͑ 2 Ϫ 2 t ͒ϪA ⑀ cos͑ 1 Ϫ 1 t ͒ ͬ (5)

Flow Factors Definition

As shown in ͓9͔, the unit flow averaged over the spatial period can be expressed through a macroscopic Reynolds equation:

ٌ•qϭ0 (6) ͗q͘ϭϪ h m 3 12 •ٌ͗p͘ϩ ͗h͘ 2 ͑ U 2 ϩU 1 ͒ϩ 2 s •͑U 1 ϪU 2 ͒
Where, in the case of mono-dimensional surface roughness, flow factor tensors , and s are diagonal and given by:

ϭ ͩ x 0 0 y ͪ ϭ ͩ 1 h m 3 ͗h Ϫ3 ͘ 0 0 ͗h 3 ͘ h m 3 ͪ , (7) s ϭ ͩ sx 0 0 sy ͪ ϭ ͩ 1 ͗h Ϫ3 ͘ ͳ h ˜1ϩh ˜2 h 3 ʹ h m 0 0 0 ͪ (8)
While the averaged shear takes the form:

͗͘ϭ 2 ͑ f Ϯ f s ͒•͑ U 2 ϪU 1 ͒Ϯ h m 2 f p •ٌ͗p͘ (9)
With diagonal shear flow factors of the form:

f ϭ f Iϭh m ͗h Ϫ1 ͘I (10) f s ϭ ͩ 3h m ͫͳ h ˜1ϩh ˜2 h 3 ʹ ͗h Ϫ2 ͘ ͗h Ϫ3 ͘ Ϫ ͳ h ˜1ϩh ˜2 h 2 ʹͬ 0 0 0 ͪ
where I is the identity tensor and 11) In these equations h ˜i is the height variation of surface iϭ1,2 around its mean plane, i.e., h i ϭ͗h i ͘ϩh ˜i . ͗h i ͘ denotes the super- ficial surface average of h i as defined in ͓9͔ and ͑12͒. Obviously, ͗h i ͘ϭh io .

f p ϭ ͩ f px 0 0 f py ͪ ϭ ͩ ͗h Ϫ2 ͘ h m ͗h Ϫ3 ͘ 0 0 ͗h͘ h m ϭ1 ͪ (
As pointed out in ͓9͔, Eqs. ͑6͒-͑11͒ are in complete agreement with the Patir and Cheng approach in this special case where the off-diagonal coefficient of the flow factor tensors are equal to zero. Being non-dimensional, the flow factors can be more appropriately expressed using dimensionless quantities H ⑀ , H ˜1⑀ , and H ˜2⑀ ͑note that H ⑀ ϭ1ϩH ˜2⑀ ϪH ˜1⑀ ͒. The expressions are similar to the above tensorial expressions, changing h and h ˜i by H ⑀ , H ˜i⑀ , for iϭ1,2 respectively, and the composite r.m.s roughness by its following dimensionless counterpart ⌺ϭ/h m ͑which is a function of ⑀͒. As all geometric quantities are spatially periodic, of period in the x direction, and independent of y, the superficial average ͗͘ of any function can be expressed in terms of the dimensionless coordinate X as ͗͑X,t͒͘ϭ

1 2 ͵ 0 2 ͑X,t͒dX (12)
The derivation of flow factors involves evaluating ''Sommerfeld'' integrals described in Appendix A. These integrals are defined if and only if C ⑀ (t)Ͻ1. It can be easily shown that this condition is verified for ⑀Ͼ0 ͑non-contact condition͒. This leads to express the flow factors as Equations ͑3͒-͑5͒ show that H ⑀ (X,t) is harmonic only for the trivial case where 1 ϭ 2 , i.e., U 2x ϭU 1x ͑we recall that the wavelength of sinusoidal surfaces is assumed to be the same on each surface͒.

x ͑ t ͒ϭ 2͑1ϪC ⑀ ͑ t ͒ 2 ͒ 5/2 ͑ 2ϩC ⑀ ͑ t ͒ 2 ͒ y ͑ t ͒ϭ1ϩ 3 2 C ⑀ ͑ t ͒ 2 sx ͑ t ͒ϭ 3ͱ2͑A ⑀ 2 ϪB ⑀ 2 ͒ ͑ 2ϩC ⑀ ͑ t ͒ 2 ͒ͱA ⑀ 2 ϩB ⑀ 2 (13) f ͑ t ͒ϭ 1 ͱ1ϪC ⑀ ͑ t ͒ 2 f sx ͑ t ͒ϭ 3͑A ⑀ 2 ϪB ⑀ 2 ͒ ͑ 2ϩC ⑀ ͑ t ͒ 2 ͒ͱ1ϪC ⑀ ͑ t ͒ 2 f px ͑ t ͒ϭ 2͑1ϪC ⑀ ͑ t ͒ 2 ͒ ͑ 2ϩC ⑀ ͑ t ͒ 2 ͒
Expressions ͑13͒ show that flow factors depend on: ͑1͒ both surfaces amplitudes A ⑀ and B ⑀ ͑2͒ time through the resulting amplitude C ⑀ (t) which is a periodic function of period Tϭ/(U 2x ϪU 1x ). Thus, in general, the above flow factors are time periodic. It can be also seen that the flow factors x , y , f , f p are always positive, while the sign of the flow factors s , f s depends on the difference between surfaces amplitude.

At this stage, it is interesting to clearly distinguish two types of situation depending on the inter-correlation between the top and bottom surfaces. The intercorrelation function can be defined here as

C 12 ͑ l,t ͒ϭ 1 ͵ 0 ͑ h 1 ͑ x,t ͒Ϫ͗h 1 ͑ x,t ͒͘ ͒͑h 2 ͑ xϩl,t ͒Ϫ͗h 2 ͑ x,t ͒͘ ͒dx (14)
which leads to

C 12 ͑ l,t ͒ϭ 1 2 cos ͩ 2 ͑ U 2x ϪU 1x ͒tϪ 2l ϩ 1 Ϫ 2 ͪ (15)
Hence, the two aforementioned types of situations can be described as:

1. The case where there is no correlation between the two surfaces (C 12 (l,t)ϭ0). In this case one can consider that one of the two surfaces is smooth ͑for instance the top surface, 2 ϭ0͒, taking advantage on the surfaces roughness decorrelation. Then, in the referential frame of the moving rough surface, there is a permanent contact point in the limit ⑀ϭ0. Flow factors obtained in ͑13͒ become time independent. It is interesting to note that, in this case, the flow factors ͑as given by Eqs. ͑7͒, ͑8͒, and ͑9͒͒ can be expressed in terms of spatial averages involving only the local aperture h ͑since here h ˜2ϭ0 and hϭ͗h͘Ϫh ˜1͒:

⌽ sx ϭ 1 ͩ͗ h͘Ϫ ͗h Ϫ2 ͘ ͗h Ϫ3 ͘ ͪ ⌽ f sx ϭ3h m ͩͳ h Ϫ1 Ϫ ͗h Ϫ2 ͘ 2 ͗h Ϫ3 ͘ ͪ (16)
where the symbol ⌽ is used here ͑instead of ͒ for clearly stressing that the above equations are valid only for the smooth/rough case and do not apply to the rough/rough case. For this first case, the spatial average performed above is sufficient to determine the average behavior, for which C ⑀ (t)ϭC ⑀ ϭA ⑀ . 2. The case where there is a correlation between the upper and lower surfaces (C 12 (l,t) 0). Such correlation occurs naturally from elasto-plastic deformation during rolling process ͑see for example ͓10͔͒. Here, we examine the case where both surfaces are rough ͑A ⑀ 0 and B ⑀ 0͒ leading to a sinusoidal inter-correlation function ͑15͒. As can be seen from Eq. ͑15͒, the two surfaces are perfectly correlated when they are in phase and perfectly uncorrelated when out of phase. On the contrary to the first case, contact is intermittent in the limit ⑀ϭ0. The relative motion of the two surfaces induces time variations of the local apertures and slopes. Under these circumstances, a time average must follow the spatial average for obtaining the average behaviors ͑that also could be obtained by ensemble averaging over all relative positions of two surfaces in the x di-rection͒. This is consistent with the fact that the time average of the inter-correlation function of Eq. ͑15͒ is zero.

Therefore, these two types of situation are associated with two types of averaging. The first one, i.e., the spatial averaging, is classical, see for instance ͓2,20,21͔ and references therein. The second type-i.e., spatio-temporal average-has not been considered to our knowledge, for flow factors computation. Discarding the spatio-temporal flow factor dependence is equivalent to choose a surface inter-correlation function equal to zero. Time averaging aspect of the problem has been, however, pointed out by Elrod ͓22͔.

Flow Factors Computation

4.1 Smooth-Rough Stationary Case. One of the two surfaces is smooth. Let us assume that this is the top surface, i.e., bϭB ⑀ ϭ0, then C ⑀ ϭA ⑀ and flow factors trivially reads from ͑13͒

x ϭ 2͑1ϪA ⑀ 2 ͒ 5/2 ͑ 2ϩA ⑀ 2 ͒ y ϭ1ϩ 3 2 A ⑀ 2 sx ϭ 3ͱ2A ⑀ ͑ 2ϩA ⑀ 2 ͒ (17) f ϭ 1 ͱ1ϪA ⑀ 2 f sx ϭ 3A ⑀ 2 ͑ 2ϩA ⑀ 2 ͒ͱ1ϪA ⑀ 2 f px ϭ 2͑1ϪA ⑀ 2 ͒ ͑ 2ϩA ⑀ 2 ͒
One first notes that these flow factors have the expected behaviors for large gaps between surfaces, i.e., x , y , f , f px →1 and sx , f sx →0 ͑since A ⑀ →0 when ⑀→ϱ͒. Let us now consider their behaviors near contact, i.e., when ⑀→0. According to the definition of A ⑀ ͑Eq. ͑1͒͒, lim ⑀→0 A ⑀ ϭ1Ϫ⑀/ͱ2. This leads to the following flow factors behaviors near contact, when ⑀→0

x → 2 3

͑ ͱ2⑀͒ 5/2 y → 5 2 Ϫ 3 ͱ2 ⑀ sx →ͱ2Ϫ⑀ (18) f →͑ͱ2⑀ ͒ Ϫ1/2 f sx →͑ͱ2⑀ ͒ Ϫ1/2 f px → 2ͱ2 3 ⑀
As shown in Appendix B, these behaviors ͑except that of y ͒ can be obtained from the flow-factors definition itself using saddle point method, avoiding their explicit computation ͑17͒. Noting that ͗h͘ϭaϩ␦ (␦ϭ⑀) and ϭa/ͱ2, one obtains in terms of average unit flow when ⑀→0 goes to zero, for obvious geometrical reasons. The 5/2 exponent dependence reflects the specific parabolic structure of the aperture field near contact, and can be obtained from simple saddle point argument ͑Appendix B͒. This behavior is generic for symmetrical aperture around the minimum in the x direction but will be different for non symmetrical ones. Flux in the y direction is simply that expected from a plane geometry. Let us now turn to the macroscopic shear:

͗q x ␦ ͘→U 1x aϩU 2x ␦Ϫ 2ͱ2a 9 ␦ 5/2 ͗ץp͘ ץx (19) ͗q y ␦ ͘→ U 1y ϩU 2y 2 ͑ aϩ␦ ͒Ϫ a 3 12 ͩ 5 2 Ϫ3 ␦ a ͪ ͗ץp͘
͗ x ␦ ͑ h 2 ͒͘→͑ U 2x ϪU 1x ͒ 1 ͱ2a ␦ Ϫ1/2 ϩ 2 3 ␦ ͗ץp͘ ץx ͗ x ␦ ͑ h 1 ͒͘→Ϫ 2 3 ␦ ͗ץp͘ ץx (20) ͗ y ␦ ͑ h 2 ͒͘→͑ U 2y ϪU 1y ͒ 1 ͱ2a ␦ Ϫ1/2 ϩ a 2 ͗ץp͘ ץy ͗ y ␦ ͑ h 1 ͒͘→͑ U 2y ϪU 1y ͒ 1 ͱ2a ␦ Ϫ1/2 Ϫ a 2
͗ץp͘ ץy

As expected, the main macroscopic shear comes from the Couette contribution. The obtained behaviors display a rather drastic divergence as the minimum aperture ␦ decreases. Here again, the Ϫ1/2 exponent is geometry dependent, related to the aperture symmetry around the contact point. It is worth noting that the y direction shows a symmetrical shear behavior ͑i.e., same behavior on surfaces 1 and 2͒, while on the x direction the shear is fully asymmetric: the flat surface experiences a major shear, while the sinusoidal one is almost not sheared at all. This is nothing but a lubrication effect of the sinusoidal surface in the vicinity of which a fluid layer moving at velocity U 1x annihilates the Couette shear.

4.2 Rough-Rough Unstationary Case. In this case, A ⑀ 0 and B ⑀ 0. Time averaging ͑13͒ over one period T is not straightforward owing to elliptic integrals of the first kind (K 1 ) and third kind (K 3 ) for Couette shear stress flow factors ͑see Appendix A͒. The final expressions read

x ϭ 4͑1Ϫ͑B ⑀ ϪA ⑀ ͒ 2 ͒ 5/2 ͑2ϩ͑B ⑀ ϪA ⑀ ͒ 2 ͒ W͑k 2 ,p 2 ͒ y ϭ1ϩ 3 2 ͑ A ⑀ 2 ϩB ⑀ 2 ͒ sx ϭ 3ͱ2͑A ⑀ 2 ϪB ⑀ 2 ͒ ͱ͑A ⑀ 2 ϩB ⑀ 2 ͒͑2ϩ͑ B ⑀ ϪA ⑀ ͒ 2 ͒͑2ϩ͑ B ⑀ ϩA ⑀ ͒ 2 ͒ f ϭ 2 ͱ1Ϫ͑B ⑀ ϪA ⑀ ͒ 2 K 1 ͑ k 2 ͒ (21) f sx ϭ 6͑A ⑀ 2 ϪB ⑀ 2 ͒ ͑2ϩ͑B ⑀ ϪA ⑀ ͒ 2 ͒ͱ1Ϫ͑ B ⑀ ϪA ⑀ ͒ 2 K 3 ͑ k 2 ,p 2 ͒ f px ϭ 6 ͱ͑2ϩ͑B ⑀ ϪA ⑀ ͒ 2 ͒͑2ϩ͑ B ⑀ ϩA ⑀ ͒ 2 ͒ Ϫ2
where W is a definite and positive integral defined by

W͑k 2 ,p 2 ͒ϭ ͵ 0 /2 1Ϫk sin 2 v 5/2 dv ͑ 1ϩp 2 sin 2 v ͒ , k 2 ϭ 4A ⑀ B ⑀ 1Ϫ͑B ⑀ ϪA ⑀ ͒ 2 , p 2 ϭ 4A ⑀ B ⑀ 2ϩ͑B ⑀ ϪA ⑀ ͒ 2
These results show that the flow factors x , y , and f px , associated with the average pressure gradient are always positive as represented in Figs. 3 and4. The fluid can flow both in the direction perpendicular to streaks ͑i.e., along x͒ and in the direction parallel to streaks ͑i.e., along y͒ for the contact is only intermittent. As before, the shear stress flow factors sx and f sx depend on the difference between the surface amplitudes. It can be shown that the elliptic integrals of first kind K 1 (k 2 ) and third kind K 3 (k 2 ,p 2 ) diverge when the two surfaces are near contact, i.e., when ⑀→0 and k 2 →1Ϫ⑀/2(1/aϩ1/b). As a result the flow factors behave asymptotically as ⑀→0

x → 32 These results display a drastic difference from the stationary case ͑18͒. First as already observed on results ͑21͒, flow factors x and f px converge to some positive constant which depends on the surfaces amplitude in a non trivial way, on the contrary to ͑18͒ where they tends to zero. These result simply indicate that even in the contact limit the aperture instationarity is a sufficient mechanism to produce flow and shear through pressure gradient in both x and y direction. Secondly, shear flow factors f and f sx diverge in the vicinity of contact, as observed in ͑18͒. Nevertheless, this divergence is quantitatively very different, being algebraic in ͑18͒ and logarithmic in ͑22͒. Such behavior can not be predicted using a saddle point approximation due to his non algebraic nature. It is nevertheless a result specific to the chosen geometry.

͑ ab͒ 3/2 ͑ aϩb ͒ 3 ͭ p ͩ 1ϩ 1 p 2 ͪ 2 arctan͑ p ͒Ϫ 5 6 Ϫ 1 p 2 ͮ y →1ϩ 3 2 ͩ a 2 ϩb 2 ͑ aϩb ͒ 2 ͪ sx → ͱ6͑a 2 Ϫb 2 ͒ ͱ͑a 2 ϩb 2 ͓͒2͑ aϩb ͒ 2 ϩ͑aϪb ͒ 2 ͔ f →Ϫ ͑ bϩa ͒ 2ͱab ln ͫ ⑀ 32 ͩ 1 a ϩ 1 b ͪͬ (22) f sx →Ϫ ͑ aϪb ͒ 2ͱab ln ͫ ⑀ 2 5 ͩ 1 a ϩ 1 b ͪͬ f px → 2ͱ3͑aϩb ͒ ͱ͓2͑aϩb͒ 2 ϩ͑aϪb ͒ 2 ͔ Ϫ2
As a result, the average stress on surfaces in x direction, which is given, in the ⑀→0 limit by

͗ x ␦ ͑ h 2 ͒͘→ ͑ U 1x ϪU 2x ͒ ͑aϩb ͒ ͱ a b ln ͫ ␦ 2 5 ͩ 1 a ϩ 1 b ͪͬ ϩ ͑ aϩb ͒ 2 f px ͗ץp͘ ץx (23) ͗ x ␦ ͑ h 1 ͒͘→ ͑ U 1x ϪU 2x ͒ ͑aϩb ͒ ͱ a b ln ͫ ␦ 2 5 ͩ 1 a ϩ 1 b ͪͬ Ϫ ͑ aϩb ͒ 2 f px ͗ץp͘ ץx
also diverges logarithmically. Nevertheless, on the contrary to previous results ͑20͒ the Couette contribution to the shear in the x direction is now symmetrical on both surfaces, reflecting the geometry symmetry.

Regarding the average shear stress in the y direction, its behavior when ⑀→0 reads

͗ y ␦ ͑ h 2 ͒͘→ ͑ U 1y ϪU 2y ͒ ͑aϩb ͒ ͱ b a ln ͫ ␦ 2 5 ͩ 1 a ϩ 1 b ͪͬ ϩ ͑ aϩb ͒ 2 ͗ץp͘ ץy (24) ͗ y ␦ ͑ h 1 ͒͘→ ͑ U 1y ϪU 2y ͒ ͑aϩb ͒ ͱ a b ln ͫ ␦ 2 5 ͩ 1 a ϩ 1 b ͪͬ Ϫ ͑ aϩb ͒ 2 ͗ץp͘ ץy

Discussion

The previously computed flow factors are sketched in Figs. 3-7 in order to investigate the composite roughness approximation in the context of sinusoidal surfaces. The flow factors displayed in these figures have been carefully compared with a direct numerical integration of expression Eqs. ͑7͒-͑11͒. Such numerical integration is in full agreement with the analytical results presented in the previous section. Both smooth/rough and rough/rough configuration are chosen with the same composite roughness leading to a rough surface amplitude ͱ2a in the smooth/rough case and aϭb in the rough/rough one.

For this particular choice, the Couette flow factors of the unstationary case are equal to zero ͑ sx ϭ0 and f sx ϭ0͒. The asymptotic behaviors near contact deduced from the saddle point approximation ͑Appendix B͒ are also represented. These figures illustrate the following points:

• As shown in Figs. 3 and4, the''Poiseuille'' flow factors x and f px tend to zero near contact in the stationary case ͑smooth/rough͒, because, in this limit, contact permanently clogs the flow. Then, the fluid cannot migrate from one valley to the another whatever the pressure gradient is in the x direction. It is interesting to observe the quality and range of validity of the saddle point approximation in the close contact region shown in the insets of Figs. 3 and4. This domain gives the region for which macroscopic behavior is dominated by the local structure of the aperture field near contact. On the contrary, in the unstationary case ͑rough/rough͒, flow factors x and f px tend towards non zero constants in the contact limit. A fluid flow in the x direction driven by an average pressure gradient is then possible owing to the intermittent nature of contact.

• The shear flow factor sx depends on the difference between the standard deviations of each surface. Hence, it can be either positive or negative. Interestingly, sx is zero at contact in the particular case where the surfaces are strictly identical ͑Fig. 5͒. Roughness contribution to mean flow in the x direction due to the surfaces relative sliding is then canceling out.

We recall from ͑6͒ that the mean flow due to sliding in the x direction is made up of two components. The first one is the Couette mean flow (U 2x ϩU 1x )͗h͘/2. The second one, (U 1x ϪU 2x ) s /2, is associated with the additional drag due to roughness. In the stationary case ͑smooth/rough͒ when the top surface is fixed and the rough bottom one is moving at velocity U 1x , the net mean flow due to sliding is U 1x ͗h͘.

This exactly corresponds to the volume of dragged valleys moving at the bottom surface velocity. This illustrates the coupled influence of geometry and kinematics on the mean flow due to sliding. • In the stationary case, Couette shear stress flow factors f and f sx ͑Figs. 6 and 7͒ diverge as ⑀ Ϫ1/2 near contact. f is always positive whereas the sign of f sx depends on the considered configuration. f sx Ͼ0 when the top surface is smooth and f sx Ͻ0 otherwise. This induces significant differences on the asymptotic behaviors of mean shear stress on each surface. We recall that the top surface experiences a mean shear stress proportional to f ϩ f sx while proportional to f Ϫ f sx for the bottom one. Thus, in the stationary case, the rough surface experiences a mean shear stress diverging near contact as ⑀ Ϫ1/2 , while the mean shear stress tends to zero on the smooth surface. Obviously, this is only true in an average sense. Locally, the shear stress value can be very high. As previously noted, this singular behavior is related to the symmetry of the aperture geometry around the contact point. • In the unstationary configuration, the flow factors f and f sx diverge logarithmically near contact ͑Figs. 6 and 7͒. The log-log plot in the inset displays the comparison between the algebraic and logarithmic divergence. f sx depends however on the surface amplitude difference, and is equal to zero when surfaces amplitude are identical. The ''Couette'' mean shear stress experienced by each surface is positive and diverges logarithmically. Hence, here, the unsteadiness of the aperture field leads to a striking modification of shear stress behavior compared to the smooth-rough case. The local squeeze effects induced by the time evolution of the aperture field modify continuously the flow structure. As a result, pressure generation may occur in a divergent. This effect cannot occur in the steady smooth-rough case.

Lastly, when two rough surfaces are not intercorrelated, the flow factors s and f s , of each surface, can be expressed in terms of the corresponding flow factors of the smooth/rough case ͓2,5͔, 25) 2 2

s ϭ 1 2 Ϫ 2 2 2 ⌽ s ͩ h m ͪ f s ϭ 1 2 Ϫ 2 2 2 ⌽ f sͩ h m ͪ (

Conclusions

Newtonian lubricant flow between two unidirectional wavy sinusoidal surfaces in sliding motion has been studied. Analytic expressions have been derived for flow factors describing the roughness impact on the macroscopic flow and shear. Two cases have been distinguished. In the first one, accounting for decorrelated surfaces, the aperture field is stationary and one of the two surfaces is smooth. Flows factor are then stationary and computed from simple spatial averaging. On the contrary, in the second case accounting for correlated surfaces ͑rough/rough case͒, the spatially averaged flow factors are time periodic. A temporal average leads to significant differences in the flow factor expressions. The obtained differences, can be regarded as resulting from the surface inter-correlation. Accordingly, the rough/rough situation investigated in the present paper is an example for which the composite roughness concept does not apply.

It is clearly expected that the observed drastic inter-correlation influence is enhanced by the deterministic nature of the chosen geometry as well as the considered roughness one-dimensional character. Nevertheless, for one-dimensional anisotropic surfaces, averaged stochastic geometries will share common features with our analysis.

Nomenclature

a, b ϭ bottom surface and top surface amplitudes A ⑀ ϭ a/h m B ⑀ ϭ b/h m C 12 (l,t) ϭ inter-correlation function between the two surfaces h(x,t) ϭ aperture field between the two surfaces h ˜i ϭ spatial fluctuation of the surface i h m ϭh 2o Ϫh 1o ϭaϩbϩ␦ ϭ distance between surfaces mean plane which coincides with the mean distance between the two surfaces ͗h i ͘ϭh io ϭ mean plane of surface i with i ϭ1,2 where is the composite roughness as defined previously ( ϭͱ 1 ϩ 2 ) and ⌽ s , ⌽ fs correspond to the smooth/rough case. Clearly, these relations do no apply to the surfaces considered in the present paper ͑as can be easily checked analytically͒.In our case, the inter-correlation function is different from zero when both surfaces are rough and the composite roughness approximation associated with Eq. ͑25͒ is not valid.

H ⑀ ϭ h/h m I ϭ identity tensor k 2 ϭ 4A ⑀ B ⑀ /1Ϫ(B ⑀ ϪA ⑀ ) 2 K 1 ϭ elliptic integral of the first kind K 3 ϭ elliptic integral of the third kind q ϭ unit flow q ␦ ϭ unit flow near contact p 2 ϭ 4A ⑀ B ⑀ /2ϩ(B ⑀ ϪA ⑀ ) 2 t ϭ
H ⑀ ͑ X,t ͒ϭ1ϩB ⑀ sin͑XϪ 2 tϩ 2 ͒ϪA ⑀ sin͑XϪ 1 tϩ 1 ͒
H ⑀ (X,t) can be expressed in terms of sin(X) and cos(X) as

H ⑀ ͑ X,t ͒ϭ1ϩsin͑ X ͓͒B ⑀ cos ⌽ 2 ϪA ⑀ cos ⌽ 1 ͔ ϩcos͑X ͓͒B ⑀ sin ⌽ 2 ϪA ⑀ sin ⌽ 1 ͔ where ⌽ i ϭϪ i •tϩ i (iϭ1,2).
Then, defining the angle by

tan ϭ B ⑀ sin ⌽ 2 ϪA ⑀ sin ⌽ 1 B ⑀ cos ⌽ 2 ϪA ⑀ cos ⌽ 1 allows one to express H ⑀ (X,t) as H ⑀ ͑ X,t ͒ϭ1ϩ ͓B ⑀ cos ⌽ 2 ϪA ⑀ cos ⌽ 1 ͔ cos ͕sin X cos ϩcos X sin ͖ which gives H ⑀ ͑ X,t ͒ϭ1ϩ ͓B ⑀ cos ⌽ 2 ϪA ⑀ cos ⌽ 1 ͔ cos sin͑Xϩ ͒
As cos 2 ϭ1/(1ϩtan 2 ) one finally obtains

H ⑀ ͑ X,t ͒ϭ1ϩͱA ⑀ 2 ϩB ⑀ 2 Ϫ2A ⑀ B ⑀ cos͓⌽ 2 Ϫ⌽ 1 ͔ sin͑Xϩ ͒ Sommerfeld Integral.
As can be seen from Eqs. ͑7͒-͑11͒, the computation of flow factors necessitate the evaluation of the spatial averages ͗H ⑀ 3 ͘, ͗H ⑀ Ϫn ͘, nϭ2,3, and ͗(H ˜1⑀ ϩH ˜2⑀ )/H ⑀ n ͘, n ϭ2,3. These averages reads,

͗H ⑀ 3 ͘ϭ 1 2 ͵ 0 2 ͑ 1ϩC ⑀ ͑ t ͒sin͑ xϩ ⑀ ͑ t ͒͒ 3 dx, ͗H ⑀ Ϫn ͘ϭ 1 2 ͵ 0 2 1 ͑ 1ϩC ⑀ ͑ t ͒sin͑ xϩ ⑀ ͑ t ͒͒ n dx, ͳ H ˜1⑀ ϩH ˜2⑀ H ⑀ n ʹ ϭ 1 2 ͵ 0 2 B ⑀ sin͑xϩ⌽ 2 ͒ϩA ⑀ sin͑xϩ⌽ 1 ͒ ͑ 1ϩC ⑀ ͑ t ͒sin͑ xϩ ⑀ ͑ t ͒͒ n . 2 2 S ͑ m,n ͒ ϭ 1 2 ͵ 0 2 cos m ͑ 1ϩC cos ͒ n d
These integrals ͑referred to as Sommerfeld integrals, see for instance ͓17͔͒ can be evaluated analytically. The most useful formulas for our problem are the cases where m can take integer values 0 or 1 and nϭ1,2,3.

S ͑ 0,1͒ ϭ 1 ͱ1ϪC 2 S ͑ 0,2͒ ϭ 1 ͑ 1ϪC 2 ͒ 3/2 S ͑ 0,3͒ ϭ 2ϩC 2 2͑1ϪC 2 ͒ 5/2 S ͑ 1,2͒ ϭ ϪC ͑ 1ϪC 2 ͒ 3/2 S ͑ 1,3͒ ϭ Ϫ3C 2͑1ϪC 2 ͒ 5/2
This leads to the expressions of flow factors given by Eq. ͑11͒.

One can refer to ͓23͔ for more details Elliptic Integrals. Let us first consider the definition of Elliptic integral of the first kind:

K 1 ͑ k 2 ͒ϭ ͵ 0 /2 dv ͱ1Ϫk 2 sin 2 v ϭ 2 F ͩ 1 2 ; 1 2 ;1;k 2 ͪ
where F is the hyper-geometric function as described in ͓23͔. In general, this integral must be evaluated numerically ͑see ͓24͔͒.

Again, an analytical evaluation can be performed near contact ͑i.e., when k 2 →1͒. This yields lim

k 2 →1 K 1 ͑ k 2 ͒ϭϪ 1 2 ln ͩ 1Ϫk 2 2 4 ͪ
Let us now turn to Elliptic integral of the third kind:

K 3 ͑ k 2 ,p 2 ͒ϭ ͵ 0 /2 dv ͑ 1ϩ p 2 sin 2 v ͒ͱ1Ϫk 2 sin 2 v
Again, in the general case, this integral must be evaluated numerically, ͓24͔.

Its asymptotic evaluation when k 2 →1 can be derived from that of K 1 (k 2 ) ͓23͔ since lim

k 2 →1 K 3 ͑ k 2 ,p 2 ͒ϭ 1 1ϩ p 2 lim k 2 →1 K 1 ͑ k 2 ͒

Appendix B

Saddle Point Estimate of Flow Factors Near Contact. Let us first turn our attention to the stationary case. The evaluation of flow factors ͑except y , sy , f sy , and f py which are trivial ones͒ involves integrals of the form ͗h Ϫn ͘ϭ1/͐h Ϫn dx. Near contact, i.e., when ⑀→0., these integrals are dominated by the contribution associated with the lowest values of h. For this reason, it seems quite reasonable to seek an estimate of these integrals by the saddle point method ͓25͔. Noting f ϭh Ϫn , this amounts to estimating ͗ f ͘ as ͱ2/ f (0) 3/2 /ͱϪ f Љ(0) ͑in a coor- dinate system where hϭ⑀ at xϭ0͒. With hϭ⑀ϩ2a(1 Ϫcos(2/x)), this leads to the following estimates as ⑀→0:

x →ͱ3ͱ2⑀ 5/2 sx →ͱ2 ͩ 1Ϫ ͱ3 ͱ2 ⑀ ͪ f → 1 ͱ2 ⑀ Ϫ1/2 f sx → 3͑2Ϫͱ3 ͒ 2ͱ2 ⑀ Ϫ1/2 f px →ͱ 3 2 ⑀
Scalings with ⑀ are in full agreement with the results of the direct calculus ͑18͒. The prefactors are correctly estimated for x , sx , f px while not accurate for f and f sx . Such approximation can be applied to the unstationary case, but fails. The direct computation provides the explanation, as logarithmic divergence are beyond the scope of this rough estimate.

Recalling that

C ⑀ ͑t͒ϭͱA ⑀ ϩB ⑀ Ϫ2A ⑀ B ⑀ cos͓͑ 1 Ϫ 2 ͒tϩ 2 Ϫ 1 ͔
The evaluation of the first integral is trivial. The second and third integrals are a little more difficult. They can be expressed in terms of integrals of the form

Fig. 1

 1 Fig. 1 Sketch of the situation studied

Fig. 2

 2 Fig. 2 Geometry in the plane yÄ0

ץy

  From Eq. ͑19͒, it is interesting to note that the average Couette flow near contact in the x direction is made up of two contributions. The first one is the flow induced by the roughness corresponding to a fluid volume of mean thickness a transported at velocity U 1x . The second one is the flow induced by the top smooth surface carrying a small fluid volume of mean thickness ␦ at velocity U 2x . It is expected that the Poiseuille contribution to flow in the x direction, cancels out as the minimum aperture ␦

Fig. 3

 3 Fig. 3 Comparison of pressure flow factors x calculated for stationary case "smoothÕrough, solid line… and unstationary one "roughÕrough, dotted line…. The dashed line corresponds to the saddle point estimate near contact for the smoothÕrough case. The inset shows the log-log plot of x near contact with same conventions.

Fig. 5

 5 Fig. 5 Comparison of shear stress flow factors sx calculated for the stationary case "smoothÕrough, solid line… and the unstationary one "roughÕrough, dotted line…. Same conventions as Fig. 3 have been used.

Fig. 7

 7 Fig. 7 Comparison of shear stress flow factors fsx calculated for the stationary and the unstationary one. Same conventions as Fig. 3 have been used.

  ϭ phase of surface i at tϭt 0 , f p ϭ Poiseuille flow factors tensors s , f , f s ϭ Couette flow factors tensors ⌽ sx ϭ sx ϭ when surface 2 is smooth ⌽ f sx ϭ f sx ϭ when surface 2 is smooth ϭ wave length of sinusoidal surface ϭ fluid dynamic viscosity 1 ϭ ͱ a 2 /2, 2 ϭͱb 2 /2 ϭ bottom surface, and top surface

	r.m.s roughness
	ϭ ͱ 1 2 ϩ 2 2 ϭ composite r.m.s roughness ⌺ ϭ /h m
	ϭ macroscopic tangential shear
	stress
	␦ ϭ macroscopic tangential shear
	stress near contact
	͗ ͘ ϭ spatial average
	Appendix A
	Derivation of Eq. "3…. Starting from Eq. ͑2͒,

i

Acknowledgments

This work was supported by the Research project contract ͑CPR͒ ''Mise en forme des mate ´riaux: Contact outil-me ´tallubrifiant'' between CNRS ͑SCA͒, Irsid ͑Usinor Group͒, Pe ´chiney center de recherche de Voreppe, Paris Sud Orsay University ͑LMS͒, Colle `ge de France ͑PMC͒, ECL ͑LTDS͒, INPT͑IMFT͒, INSA de Lyon ͑LMC͒, ENSMP͑CEMEF͒. The authors thank P. Montmitonnet for his remarks and suggestions.


Hadhami Mejbri

Mariem Mahfoudh
email: mariem.mahfoudh@gmail.comgermain.forestier@uha.fr

Germain Forestier

Deep learning-based sentiment analysis for predicting financial movements

Keywords: Sentiment Analysis, Stock movement prediction, Opinion Mining, Automatic Natural Language Processing (NLP), Deep Learning, Machine Learning, Time series

Sentiment analysis is a computational study of opinions, feelings, emotions, ratings and attitudes towards entities such as products, services, organizations, individuals, issues, events, subjects and their attributes. Our research is used to predict stock market movements, aims to improve the accuracy of polarity of comments, in order to accurately predict financial movements. This by creating a dictionary of emojis that contains the emoji as keys and its meanings as values. We used the dictionary at the preprocessing level to keep the meanings of emojis because they carry a lot of emotions that help us to clearly specify the polarity of the comments. We have also created a list of stopwords related to the financial field to properly clean our database. Time series and linking sequences of data is very important to properly predict stock market movements. We have therefore chosen to work with the Long short-term memory (LSTM) model. Next, we came up with two models: the first model to predict stock market movements using investor sentiment analysis of Amazon stock which gives us 93% accuracy. The second model is used to predict financial movements through historical Amazon prices. We extracted the database we used for the sentiment analysis from Twitter as the Twitter comments are up to date. As for the historical prices of Amazon stock we extracted from the most famous trading platform YahooFinance.

Introduction

Sentiment analysis have been used in diverse fields such as health, finance, sports, politics, hospitality, and consumer behavior. We are interested in our work to the financial field, more specifically in online trading platforms.

Online trading is simply buying and selling financial securities through on-Line trading platforms or mobile trading apps, to make money between buying and reselling, and vice versa. Intervening on the financial markets represents risks that can lead to financial losses. For example, market risk due to price instability linked to general economic and market fluctuations, liquidity risk and the difficulty of finding a counterparty (to sell a financial instrument at a reasonable price at a given time), etc. [START_REF] Hugo | what are the different types of fianacial risks[END_REF], so to minimize the risk of loss, we help through our proposed model to create a computer system that helps investors make the right decision in order to have a good trading experience in the financial markets. There are several areas of trading like commodity trading, currency trading and stock trading. Statistic shows that 90% of traders do not make money when trading the stocks [START_REF] Gillham | Trading the Stock Market -Why Most Traders Fail[END_REF], so we chose stock trading in the research and particularly we chose to predict the financial movements of Amazon.

Our research is divided into three main parts: the first part consists of analyzing investor sentiment towards Amazon. The reviews are taken from the official twitter page of Amazon. The second part is to predict the financial movement of Amazon stocks using sentiment analysis. Finally, the third part focuses on the use of historical prices extracted from the most famous trading site: YahooFinance.

Our article is composed of three parts. The first part is the related works in which we presented some works related to sentiment analysis in the financial field whereas in the second part, we presented our proposed approach and in the third part we discussed the results of our research.

Related work

Kordonis et al. [START_REF] Kordonis | Stock Price Forecasting via Sentiment Analysis on Twitter[END_REF] predicted how the market would subsequently behave via sentiment analysis on a set of tweets over the past few days, so they developed a system that collects past tweets, drafts, and examines the usefulness of various machine learning techniques such as Naive Bayes Bernoulli classification and Support Vector Machine (SVM), to provide positive or negative sentiment on the tweet corpus. The results still show that changes in public sentiment can affect the stock market.

Kalyani et al. [START_REF] Kalyani Joshi | stock trend prediction using news sentiment analysis[END_REF] predicted market trends, and for this they used a polarity detection algorithm to label news articles. For this algorithm, a dictionary-based approach was used. Positive and negative word dictionaries are created using general and finance-specific sentiment words. They created a dictionary for stopword removal that also includes finance-specific stopwords. Based on this data, they implemented three classification models which they tested in different test scenarios. Then, after comparing their results, Random Forest performed very well for all test cases ranging from 88% to 92% accuracy. The accuracy tracked by SVM is also considerable around 86%. The performance of the Naive Bayes algorithm is around 83%.

There are some work on Deep Learning architecture for sentiment analysis. Nti et al. [START_REF] Xu | IEEE International Conference on Big Data (Big Data), F.: Stock Prediction using Deep Learning and Sentiment Analysis[END_REF] investigated the application of the attention-based deep neural net-work LSTM in predicting future stock market movements. They also built an aggregate stock dataset, and an individual dataset, including stock history data, financial tweets sentiment, and technical indicators on the US stock market. The experiment investigates the temporal sensitivity of financial tweet sentiment and methods for calculating collective sentiment. The researchers also experimented with conventional LSTM and attention-based LSTM for performance comparison. The results prove that financial tweets published from the close to the open of the market have more predictive power on the movement of stocks the next day.

Liu et al. [START_REF] Liu | Recurrent convolutional neural kernel model for stock price movement prediction[END_REF] proposed a model named RCNK (Recurrent Convolutional Neural Kernel) for the prediction of stock price movement. In order to improve the prediction accuracy of the model and the cumulative returns of the trading simulation, the RCNK model has been optimized in three aspects: data collection, textual data processing, and the model classifier.

Researchers have proposed several models that improve the results of predicting financial movements with several techniques.

There is researchers who analyze the temporal system (when are the periods of time which represent well the prices of stocks, and give better predictions ), on the other hand there is researchers who have worked well to improve the results of sentiment analysis towards stocks. In our approach we tried to improve on both, we used "Open" and "High" respectively, as the training dataset, The "Open" column, which represents the stock's opening price each day and the "High" column, which represents the highest stock price reached each day. Furthermore, we created a dictionary of emojis and a list of financial stopwords to improve the stock sentiment analysis result (in our research we analyzed amazon stocks).

Proposed approach

To improve the trading experience, it is important to know the opinion of the investors on the share price as well as on the stocks. The objective of our work is to identify the courses of action that will be descending and those that will be ascending. That's why we offer sentiment analysis of tweets to help investors choose when and which stock to buy or sell in order to increase their earnings. In the other hand we predict the stock market movements using the historical prices extracted from the most famous trading site "Yahoo finance". The purpose of these two methods is to improve the prediction through comparing the results of the sentiment analysis and the historical prices methods.

Figure 1 illustrates the steps and techniques of our proposed sentiment analysis approach which includes seven main components:

1. Data collection. Collection of tweets from the Amazon page and the date of each. Extract the historical prices of Amazon stocks from the Yahoo finance trading site and the date of each. 2. Data pre-processing. it consists to clean the data and extract the relevant information in order to have an accurate classification of the sentiments.

Data collection

We created a Developer twitter account to have an API that allows us to extract tweets from the official Amazon page and the date of each one. In our experiment, we used tweets which were posted from 2021-02-12 to 2021-08-17. We examined the tweets to see if there is any relation between the future stock price and users sentiment. In other words, we want to see if we can predict a future stock price based on the current sentiment of many users in order to improve the prediction using the historical prices of the stock, so we have loaded AMAZON's past stock price data. From the most famous trading site Yahoo-Finance, we have selected the values of the first and second columns ("Open" and "High" respectively) as training dataset. The "Open" column represents the stock's opening price each day and the "High" column represents the highest stock price reached each day.

We used historical stock prices from Amazon from 2018/08/01 to 2021/07/29 to train our lstm model, and prices from August 2021 to test our model. We used Textblob library to labbel our tweets.

Pre-processing

1. Date standardization. Transfer the dates of each comment to the date format necessary for the prediction of financial movements over time (2021-02-12T17:22:53Z to 2021-02-12 17:22:53) . 2. Cleaning emojis. Before starting the known natural language processing techniques (punctuation removal, stopword removal, tokenization, stemming and TfIdf), We noticed that our database contains a lot of emojis with a lot of feelings and which must be removed if we removes punctuation, and for this reason we decided to create a dictionary of emojis. We have chosen to extract the twitter emojis with their meanings from the French website fr.piliapp.com in order to have a dictionary that contains all the emojis with their meanings (1665 emojis) in this form 'emoji':'meaning' (the emoji as keys and the meaning as value) the dictionary we created is in French and since the tweets in our database are in English, we used the translate library to translate it into English. After creating our data dictionary, we created a function clean text round1 which uses the re library and the string library to replace emojis with its meaning and remove punctuations in order to apply known NLP techniques to our data. 3. Tokenization. Tokenization is a method of dividing a piece of text into smaller units called tokens. Here the tokens can be words, characters or sub-words. Therefore, tokenization can be roughly divided into 3 types: word, character, and subword (n-gram characters) tokenization. In our research we need the sentiments of the words so we applicated word tokenization. 4. Remove stop words. Stop words are words that don't add much meaning to a sentence and don't convey any emotion so they don't indicate any valuable information about the sentiment of a sentence. They can be safely ignored without sacrificing the meaning of the sentence. For example, words like the, he, have etc. We used the nltk library to import English stopwords to remove unnecessary words from our database. And like our research done in financial databases, we created a new Word stop list related to the financial domain (new stopword), and removed all the financial stop words in our list from our database. And to be sure that our database is well cleaned, we used wordcloud to display the words, repeat the most and add the superfluous words to our list of stopwords. 5. Feature extraction. Machines cannot understand characters and words. So when it comes to textual data, we need to represent it in numbers for it to be understood by the machine. A simple text could be converted into features with various techniques such as Bag of Words (BOW), tf-idf wish is very often used for text features, there is also another class called TfidfVectorizer that combines all the options of CountVectorizer and TfidfTransformer in a single model, so in our research we used the TfidfVectorizer technique.

Sentiment Analysis and Prediction

In our research, we used time series and linking data sequences which are very important to predict stock market movements well. So we have chosen to work with the LSTM model which uses a technique allowing it to process information over extended time intervals. The LSTM technique learns to store information over extended time intervals via recurrent back-propagation is time consuming mainly due to insufficient and decreasing error feedback [START_REF] Hochreiter | Long Short-Term Memory[END_REF].

Results and discussion

In this section, we will explain our results curves. At first, we have presented the result of our expected stock price using historical prices and actual stock prices in the picture 2. Then, we have presented the opinions of Internet users over time as shown in the Figure 3, 4, 5, 6 (sentiments on the vertical axis and time on the horizontal axis polarities) Although the exact prices of the predicted Fig. 2. Prediction of financial movements with the LSTM model using historical prices prices are not always close to the actual price, our model always indicates general trends, such as upwards or downwards. This research tells us that LSTMs can be effective in time series forecasting. This prediction can be improved by merging the two prediction methods used in our research: sentiment analysis of tweets and historical prices.

We evaluated our model to improve it based on what we learn to evaluate it, and whether our model is useful, whether we need to train our model on more data to improve its performance, and whether we should we include more features? The three main parameters used to evaluate a classification model are: accuracy, precision and recall. The best result we got using LSTM model is 93% accuracy. Using sentiment analysis, we found that tweets with date 2021/08/03 are positive tweets, which affects stock prices which therefore increased, which is the case as shown in Figure 2 of actual stock prices. On the day of 2021/08/04 the comments are increasingly negative throughout the day, which clearly affects the prices of stocks which will therefore go down. But as Figure 5 shows at the end of the day 2021/08/04 and the beginning of the day 2021/08/05, the feedback becomes positive, which will affect the stock prices, and what is coming rise, as indicated by the curve of actual stock prices from the figure 2. And since August 6, 2021 until 16 august 2021, the comments are increasingly negative, and at the same time, we find that the prices of actual Amazon stocks are decreasing considerably. Our research is inspired by the works [START_REF] Kordonis | Stock Price Forecasting via Sentiment Analysis on Twitter[END_REF], [START_REF] Kalyani Joshi | stock trend prediction using news sentiment analysis[END_REF] and [START_REF] Xu | IEEE International Conference on Big Data (Big Data), F.: Stock Prediction using Deep Learning and Sentiment Analysis[END_REF]. Such as the work [START_REF] Kordonis | Stock Price Forecasting via Sentiment Analysis on Twitter[END_REF] the researchers created a dictionary of emojis which indicates the polarity of the emojis which carries a feeling. Which makes us think about creating a dictionary of emojis which carries the meaning of the emojis and we have put all the emojis without exception. In order to attract all the feelings of our twitters and find out if there is any sarcasm. We also adopted the idea of [START_REF] Kalyani Joshi | stock trend prediction using news sentiment analysis[END_REF] to create a list of financial stopwords wish improve sentiment analysis results. And finally, the work [START_REF] Kordonis | Stock Price Forecasting via Sentiment Analysis on Twitter[END_REF] that analyzes the aspect of time on the prediction results, that guided us to try different stock prices on the same day and choose to use the Open and High columns as our training data set. The Open column represents the stock's opening price each day and the High column represents the highest stock price reached each day. As the table 1 indicate our model give the best accuracy.

Conclusion

The stock market is often volatile and changes abruptly due to economic conditions, the political situation and major events for the country. To improve the trading experience: it is important to know the opinion of investors on the share price as well as on stocks, and thanks to social networks now, we can know the opinions of Internet users. In our research, we used Amazon page tweets to do Amazon customer sentiments analysis to identify which courses of action are going to be bottom-up. Our task consists in two main steps : in the first step, we cleaned our database by creating a dictionary of emojis that replaces emojis with their meanings, and we created a list of stopwords related to the financial field, then we applied the TF / IDF, to represent the words numerically. Time series and linking data sequences are very important for predicting stock market movements well. And for that we then chose to use several models of the "LSTM" classifier and a model of the "decision tree" classifier. After a comparison between these models, the results of our proposed approach, the LSTM classifier that demonstrates the best reliability demonstrates a performance of 93%. The second part of our proposed approach is to use our investor sentiment analysis Yahoo-Finance, Twitter

They have created a dictionary of emojis that indicates the polarity of emojis that carry sentiment Naive Bayes 80%

[2] Yahoo-Finance, Reauters

They have created a dictionary for stopword removal which also includes finance-specific stopwords.

Random Forest 92% [START_REF] Xu | IEEE International Conference on Big Data (Big Data), F.: Stock Prediction using Deep Learning and Sentiment Analysis[END_REF] Yahoo-Finance, Twitter

The experiment investigates the time sensitivity of financial tweet sentiment and methods for calculating collective sentiment.

LSTM 62% [START_REF] Liu | Recurrent convolutional neural kernel model for stock price movement prediction[END_REF] guba eastmoney

The different from previous studies is that The researchers treated the text data as sequential data and they used the RCNK model to train sentiment embeddings with the temporal features.

Recurrent convolutional neural kernel 66.62%

Our work

Yahoo-Finance, Twitter

We created a dictionary of emojis that contains the emoji as keys and its meanings as values, this dictionary we used at the preprocessing level to keep the meanings of emojis , we have also created a list of stopwords related to the financial field to properly clean our database.

LSTM 93%

to predict market movements of Amazon stocks. As for the other method, we used to predict the stock market movement, the method using historical prices extracted from the most famous trading site Yahoo Finance for this prediction we used LSTM which gives us good results as showing Figure 2. The perspectives of our work consist in changing our database to make it dynamic in real time, and in uniting the results of the sentiment analysis with the results of historical prices, in order to develop and exploit a computer system allowing to invest in the financial markets automatically (without human intervention). We also aim to solve the problem of ironic sentences using our dictionary of emojis because an opposition between the emotion of the emoji and the emotion of the text usually constitutes sarcasm.

Fig. 1 . 6 .

 16 Fig. 1. Summary of our proposed approach

Fig. 3 .

 3 Fig. 3. Prediction of financial movements with the LSTM model using sentiment analysis on August 3, 2021.

Fig. 4 .

 4 Fig. 4. Prediction of financial movements with the LSTM model using sentiment analysis on August 4, 2021.

Fig. 5 .

 5 Fig. 5. Prediction of financial movements with the LSTM model using sentiment analysis on August 4 and 5, 2021.

Fig. 6 .

 6 Fig. 6. Prediction of financial movements with the LSTM model using sentiment analysis on August 6, 2021.

Fig. 7 .

 7 Fig. 7. Prediction of financial movements with the LSTM model using sentiment analysis on August 7 and 8, 2021.

Table 1 .

 1 Comparison of our work with previous works

	Works Database The technique used	Algorithm	Accuracy
	[3]


Renaud Baillou

Jonas Ranft
email: jonas.ranft@ens.psl.eu

Non-equilibrium cluster-cluster aggregation in the presence of anchoring sites

Non-equilibrium cluster-cluster aggregation of particles diffusing in or at the cell membrane has been hypothesized to lead to domains of finite size in different biological contexts such as lipid rafts, cell adhesion complexes, or postsynaptic domains in neurons. In this scenario, the desorption of particles balances a continuous flux to the membrane, imposing a cut-off on possible aggregate sizes and giving rise to a stationary size distribution. Here, we investigate the case of non-equilibrium cluster-cluster aggregation in two dimensions where diffusing particles and/or clusters remain fixed in space at specific anchoring sites, which should be particularly relevant for synapses but may also be present in other biological or physical systems. Using an effective mean-field description of the concentration field around anchored clusters, we derive an expression for their average size as a function of parameters such as the anchoring site density. We furthermore propose and solve appropriate rate equations that allow us to predict the size distributions of both diffusing and fixed clusters. We confirm our results with particle-based simulations, and discuss potential implications for biological and physical systems.

I. INTRODUCTION

The aggregation of particles into larger structures is observed not only in inactive physical [1? , 2] but also in biological systems, where it plays an important role e.g. in the formation of larger protein assemblies in cells. Such protein clusters typically form in or at cellular membranes; examples range from the clustering of chemotactic receptors in bacteria [3,4] to the aggregation of cellcell adhesion proteins in epithelia into dense clusters [5]. More generally, aggregation processes have been hypothesized to govern the spatio-temporal organization of domains in cellular membranes e.g. in the context of lipid raft formation [6,7] or the maturation of the Golgi apparatus [8]. Such instances of 'healthy' aggregation are not to be confounded with the abnormal protein aggregation often linked to neurodegenerative diseases [9], which has also received theoretical attention [10,11].

In the context of diffusion-limited growth, aggregates (domains) grow by the addition of individual particles or by fusion with other clusters (domain coalescence). Whereas the former typically occurs at the molecular scale in supersaturated solutions and is described by Lifshitz-Slyozov-Wagner theory [12], the latter case of cluster-cluster aggregation or domain coalescence is more relevant for (typically larger) strongly interacting particles with small or vanishing nucleation barrier and has been described by Smoluchowski coagulation equations [1, 5-7, 11? ]. Submonolayer molecular beam epitaxy, during which monolayer islands form due to the diffusion and aggregation of deposited adatoms on the crystal surface, is another growth scenario that has been described using rate equations for the concentrations of islands of a given size [2,13]. In the absence of a specific mechanism for growth arrest, eventually only one large component will remain and grow until the depletion of available particles. On the contrary, biological systems are generally out of equilibrium and can use energy to maintain a stationary state with finite aggregate or domain sizes. For example, arrested liquid-liquid phase separation (LLPS) has become a very active field of research since the discovery of its importance in the formation of P-granules in the C. elegans oocyte [14][15][16]. In the context of LLPS, liquid droplets of the minority phase are limited in size by chemical reactions that are driven out of equilibrium by externally imposed gradients. In the context of particle aggregation or domain coalescence in cellular membranes, the recycling of individual particles, clusters or membrane domains, i.e. their removal from the surface, can serve to maintain a stationary distribution of sizes of diffusing clusters or domains despite continuous fusion and aggregation [5][6][7].

Recently, a similar mechanism has been hypothesized to underly the formation of postsynaptic domains (PSDs) at inhibitory synapses [17] (but see [18]). In neurons, synaptic transmission depends on the concentration of transmembrane neurotransmitter receptors at PSDs due to transient interactions with scaffold proteins that form larger domains by homotypic interactions; both excitatory and inhibitory PSDs have been shown to be highly dynamic structures subject to continuous turnover of their constituent particles [19,20], raising the question of how domains of a well-defined size may be maintained. In ref. [17], the authors characterized the stationary size distributions that arise from the interplay of the (possibly size-dependent) diffusion of particles and clusters, aggregation, and the turnover of individual particles. Here, we will follow up on this earlier work by extending the analysis to include the existence of anchoring sites that can locally pin clusters at specific positions at the surface and hinder their diffusion. At synapses, transsynaptic adhesion complexes link the presynaptic bouton to the Sketch of diffusion-aggregation dynamics with turnover in the presence of anchoring sites. (a) Particles and clusters that are not attached to anchoring sites can diffuse freely as indicated by the thin black arrows. Clusters at anchoring sites do not diffuse. Individual particles are subject to turnover ("monomer recycling scheme") as indicated by the blue arrows. (b) Corresponding mean-field picture with a smooth concentration field c(r, t) (gray shading) outside of fixed domains (red).

z C x R h o F R s d H g 4 1 e u N G R + X T t K 4 = " > A A A C 3 H i c b Z J L a 9 w w E M e 1 7 i t 1 X 0 l 7 z E V 0 K W w u i 1 1 C m m N o L z k m k E 0 C t h t k e b w r o o e R 5 J T F + N Z b 6 C 3 J N + q H 6 L f p e N e F 7 G 4 H J P 2 Z 3 w w a z S i v p H A + i v 4 M g i d P n z 1 / s f U y f P X 6 z d t 3 2 z v v z 5 2 p L Y c J N 9 L Y y 5 w 5 k E L D x A s v 4 b K y w F Q u 4 S K / / t b x i x u w T h h 9 5 u c V Z I p N t S g F Z x 5 d p 9 H V 9 j A a R w u j m y L u x Z D 0 d n K 1 M / i d F o b X C r T n k j m X x F H l s 4 Z Z L 7 i E N k x r B x X j 1 2 w K C U r N F L i s W V T a 0 k / o K W h p L C 7 t 6 c L 7 O K N h y r m 5 y j F S M T 9 z 6 6 x z / o 8 l t S 8 P s 0 b o q v a g + f K i s p b U G 9 o 9 m x b C A v d y j o J x K 7 B W y m f M M u 6 x O W G Y a v j B j V J M F 0 2 q b I s b X m F V u 0 o K k I i q 7 q 1 M I r P w m I 6 Q S S j 9 q K X r a A + R F d O Z 3 9 t k S Z + W b K L s X 1 q G b L W U 3 L V J n O F p Z N F 1 z M h m G L d r 9 U q 7 D J J M T y V 0 A a l d y D b E w c f r Y 9 4 U 5 5 / H 8 c F 4 / 3 R / e P S 1 / w J b Z J d 8 J C M S k y / k i B y T E z I h n A C 5 J X f k P v g e / A x u g 1 / L 0 G D Q 5 3 w g K x Y 8 / A X k N O r t < / l a t e x i t > 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " O N n a 7 H M O d G / 4 H F K P A G 5 l y 0 l / 1 O Q = " > A A A C 3 n i c b Z L N b t Q w E M e 9 4 a u E r x a O X C x W S N v L K k F V 4 V j B h W M R b F s p i R b H m W S t j u 3 I d l q t o l y 5 V d y A F + I h e B u c 3 S B 1 d x n J 9 l / z m 5 H H M 8 5 r F N Z F 0 Z 9 R c O f u v f s P 9 h 6 G j x 4 / e f p s / + D 5 m d W N 4 T D j G r W 5 y J k F F A p m T j i E i 9 o A k z n C e X 7 5 o e f n V 2 C s 0 O q L W 9 a Q S V Y p U Q r O n H d 9 5 v N o v j + O p t H K 6 K 6 I B z E m g 5 3 O D 0 a / 0 0 L z R o J y H J m 1 S R z V L m u Z c Y I j d G H a W K g Z v 2 Q V J F 4 q J s F m 7 a r W j r 7 2 n o K W 2 v i l H F
PSD [21] and can thus be expected to act as such anchoring sites in the postsynaptic membrane. Beyond the specific example of PSDs, our theory generalizes earlier approaches to describe two-dimensional diffusion-limited aggregation with turnover to the case where surface impurities or interactions with cytoplasmic or extracellular structures locally impedes the diffusion of protein clusters or membrane domains, with the aim to predict the stationary size distribution of fixed and diffusing domains as a function of parameters.

This article is organized as follows. In Sec. II, we first derive a self-consistent equation for the radius of the fixed domains based on the solution of the diffusion equation for particles outside the domains. While this approach provides insight into the scaling of the domain size with anchoring site density, it does not account for the distribution of sizes of domains. In Sec. III, we then present the results for those stationary size distributions obtained by rate and master equations for diffusing and fixed domains, respectively. In Sec. IV, we finally compare our results to particle-based Brownian dynamics simulations and discuss the role of spatial order in the distribution of anchoring sites, before we conclude with a short summary discussion of our results in Sec. V.

II. MEAN-FIELD DESCRIPTION BASED ON THE DIFFUSION EQUATION

In a first approach, we describe the system as follows, see sketch of Fig. 1. Diffusing particles outside of anchored domains are captured by a concentration field c(r, t), r ∈ R 2 , which obeys the diffusion equation

∂ t c(r, t) = D∆c(r, t) -kc(r, t) + J (1)
between fixed domains. Here, D is the diffusion constant, k is the recycling (or turnover) rate of individual particles, and J is a flux that re-injects recycled particles into the system. Diffusion and turnover define a characteristic length scale λ = D/k, and the recycling constants k and J define an equilibrium (average) concentration c 0 = J/k. Upon encounter with a fixed domain, diffusing particles are adsorbed and added to that domain. Each domain i thus acts as a local sink for c(r) with a flux B i that depends on its radius R i . In principle, this is described by absorbing boundary conditions for c(r, t) at each domain boundary. However, for simplicity we consider point-like sinks located at anchoring sites r j , j ̸ = i, when describing the concentration field around a specific domain i, assuming that the typical extension of the domains is small compared to the distance between domains. In this limit, the concentration field obeys the equation

∂ t c = D∆c -kc + J - j̸ =i B j δ(r -r j ) . (2)
The size N i of each domain i expressed as the number of constituent particles obeys

d dt N i = B i -kN i , (3)
where B i is the flux of adsorbed particles and kN i is the particle loss due to turnover within the domain.

We can now try and coarse-grain Eq. ( 2) with respect to the (random) positions of the individual domains when all anchored domains are considered to be at steady state and of the same size N , i.e. B i = kN i = kN . Stationarity furthermore implies a constant concentration field c(r, t) = c(r). Introducing the density n of anchoring sites and assuming radial symmetry around the specific domain that we consider, Eq. ( 2) then becomes

0 = D∆c(r) -k(c(r) -c 0 ) -kN n , (4)
which needs to be solved with the absorbing boundary condition c(R) = 0 at the domain edge r = R. One then readily obtains

c(r) = (c 0 -N n) 1 - K 0 (r/λ) K 0 (R/λ) , (5)
where K α are the modified Bessel functions of the second kind.

The radius R = N/(πρ) is linked to the domain size via the domain surface density ρ. The domain size N itself follows self-consistently from the flux equilibrium condition dN/dt = 0 (Eq. ( 3)), namely

kN = 2πRD∂ r c(r)| R . (6)
Here, the lateral influx is given by the diffusive particle flux j = -D∇c(r) integrated along the domain circum- ference. With Eq. ( 5), this condition leads to the following implicit equation for R:

R λ = 2 c 0 ρ -π R λ 2 nλ 2 K 1 (R/λ) K 0 (R/λ) . (7)
The resulting domain size N = πR 2 ρ can thus be numerically determined as a function of the parameters of the system. For fixed c 0 , ρ, and λ, we show the predicted variation of N with the density of anchoring sites n in Fig. 2a. The concentration field around anchored domains for different values of n is shown in Fig. 2b. When the density n of anchoring sites increases, an increasing fraction of available particles remains trapped in the anchored clusters. We can obtain the mass fraction of anchored particles directly from the size N of fixed domains via N n/c 0 , which we plot in Fig. 2c as a function of n/c 0 . Note that a priori N changes with n in a nontrivial fashion (cf. Eq. ( 7), Fig. 2a); however, when almost all particles are trapped at anchoring sites, ≈ c /n which we also on Fig. 2a comparison. we can ask how the domain size N changes with particle turnover rate k for a fixed value of n, or more generally with c 0 , D, and k at a fixed relative value n/c 0 . This is shown in Fig. 2d. Note that while in the dilute limit (n = 0) the domains can become arbitrarily large, their size can become at most c 0 /n (in the meanfield regime) when n > 0, which corresponds to limiting case when all particles are trapped at anchoring sites.

So far, we have only considered aggregation at the fixed anchoring sites but completely ignored possible effects due to aggregation and cluster formation of diffusing particles outside of the anchored domains. Assuming a vanishing nucleation barrier for cluster formation of particles coming into contact, one however cannot exclude that particles diffuse as entire clusters in the free surface. In order to take into account the potentially non-negligible size of diffusing clusters, we introduce a typical radius R typ and a typical diffusion constant D typ of diffusing clusters and modify the above Eqs. ( 4) and (6) accordingly, i.e.,

0 = D typ ∆c(r) -k(c(r) -c 0 ) -kN n , (8)
kR 2 ρ = 2R eff D typ ∂ r c(r)| R eff , (9)
where R eff = R + R typ is the effective radius at which anchored and diffusing clusters fuse. With λ = D typ /k, the implicit equation for R (Eq. ( 7)) then becomes

R 2 λR eff = 2 c 0 ρ -π R λ 2 n λ2 K 1 (R eff / λ) K 0 (R eff / λ) . ( 10
)
To be able to make an educated guess about R typ and D typ if not to outright calculate them, we need a theory that allows to predict the size-distribution of diffusing clusters, which will be the subject of the next section.

III. CLUSTER SIZE DISTRIBUTIONS OBTAINED FROM RATE EQUATIONS

In an earlier work, some of us have shown that in the dilute limit n = 0, Smoluchowski rate equations capture extremely well the size distribution of diffusing clusters that one obtains in full particle-based simulations of Brownian dynamics with particle aggregation and turnover, in which a stationary size distribution is observed after a characteristic time 1/k [17]. In the same spirit, we propose here rate equations that describe the evolution of the (spatially averaged) concentrations c m (t) of diffusing clusters of size m, where we introduce an additional term that accounts for the fusion with anchored domains. In particular, this approach allows us to take into account a possible size dependence of the diffusion constant of diffusing clusters, which we assume to be

D m = D 0 m -σ (11)
in a scale-free way such that no additional length scale is introduced in the system. Once we have obtained the size distribution of diffusing clusters, we can use a master equation approach to calculate the size distribution of fixed clusters, providing us with a full description of the system that we set out to study.

In the presence of anchoring sites at a density n, the rate equations now read

d dt c m = -kmc m + k(m + 1)c m+1 + Jδ m1 -K j (D j + D m )c j c m + K j<m (D j + D m-j )c j c m-j -KD m c m n , ( 12
)
where K is a kinetic coefficient (see e.g. ref. [22] for a discussion). In the dilute limit n = 0 and for σ = 0, an analytical solution for the stationary solution c * m can be found [17]. For n > 0 and/or σ > 0 one can determine the c * m numerically by integration of Eq. ( 12) until a stationary state is reached. It follows from the above equation that the normalized cluster concentrations c * m /c 0 only depend on three dimensionless parameters,

c * m = c 0 f n (c 0 λ ′2 , σ, n/c 0 ) , (13)
where we introduced the characteristic length scale λ ′ = KD 0 /k reminiscent of λ introduced in the previous section.

While the stationary cluster size distribution depends on n via the last term of Eq. ( 12), it does not depend on the size N of the fixed domains. To obtain N , we can proceed along two lines. Let us first invoke average particle number balance on the surface. The total concentration of particles c tot is given by the weighted sum of concentrations c m plus the surface density of anchored particles which is given by N n. At the same time, the balance of desorption and adsorption to the surface requires kc tot = J, or c tot = c 0 . One thus obtains an average size for the anchored domains

N = (c 0 - ∞ m=0 mc * m )/n ( 14
)
in the stationary state. Note that with Eq. ( 13) it follows that N only depends on the same three dimensionless parameters as c * m /c 0 . Second, we can use a previously developed master equation approach to compute the distribution of sizes for fixed clusters that fluctuate in size due to the stochastic aggregation with diffusing clusters and recycling [22]. For the sake of completeness we repeat here the governing equation for probabilities p l (t) for a domain to be of size l,

d dt p l = -klp l + k(l + 1)p l+1 -p l K j D j c j + K j<l D l-j c l-j p j (15)
for l > 0, where the first two terms describe size changes due to the desorption of individual particles and the last two terms size changes due to fusion with diffusing aggregates. Note that strictly speaking the p l a b c d (c,d) The mean-field theory of Sec. II is shown for comparison with dots ("naïve" theory, Eq. ( 7)) and crosses ("effective" theory, Eq. ( 10)). For the rate equations, K = 2 was used.

for l ≥ 1 do not define a probability distribution, as (d/dt) l≥1 p l = -kp 1 and l = 0 is an absorbing state. However, one can define a quasi-stationary solution pl such that p l (t) = pl e -νt on long times, with ν ≪ k and l≥1 pl = 1. The pl can be obtained in a semi-analytical way from a recursion relation [22] or via the relaxation of an initial p l (0) towards the (quasi-)stationary state by integrating Eq. ( 15) numerically. Before solving for the pl , we can again make a scaling argument using Eq. ( 13) and find they also depend only on c 0 λ ′2 , σ, and n/c 0 .

Assuming now that each anchoring site indeed harbors a fixed domain, the absolute concentrations of fixed domains of size l are then simply given by ĉl = np l .

(

) 16
The total density of clusters of size m follows as c tot m = c * m + ĉm . Finally, following this approach the average size of anchored domains is found to be

N = l l pl , (17)
which happens to give basically the same result as Eq. ( 14) when evaluated.

The results for the size distributions of diffusing and anchored clusters for different values of the anchoring site density n and size-depedence exponent σ of the diffusion constant are shown in Fig. 3a,b. In the dilute limit n = 0, a strong dependence of the size distribution c * m of diffusing clusters on σ has been described [17]. For σ = 0, one can furthermore theoretically predict in this dilute limit that the size distribution follows a power law according to c * m ∼ m -1 until a typical size

M = m m 2 c * m m mc * m ( 18
)
that scales as M dil 0 ≈ c 0 λ ′2 . In the more general case of σ > 0 in the dilute limit, the power law no longer holds, while the typical size of diffusing clusters follows the more general scaling

M dil σ ≈ (c 0 λ ′2 ) 1 1+σ
. This picture changes in the presence of a finite n > 0. Notably, the total concentration c tot m starts to deviate from a power law for σ = 0 (Fig. 3a), and differences between the size distributions of diffusing clusters c * m for different σ become less pronounced with increasing n (compare Fig. 3a,b). For increasing n, the size distributions of diffusing and anchored clusters overlap less and less, which manifests itself by an increasingly prominent trough in the total concentrations c tot m that separates both types of clusters. The average size N of the fixed domains as a function of n as predicted by the rate-equation approach is shown in Fig. 3c. We first observe that in line with the results of the previous section, N decreases with the density of anchoring sites n. Moreover, the difference between the average size for different values of σ does indeed decrease with n, as one can expect from the fact that it does only depend on the concentrations of diffusing clusters c * n , which become increasingly similar. With increasing n, the mass fraction nN/c 0 of anchored domains does increase despite decreasing N , as shown in Fig. 3d. Consequentially, as more and more mass is concentrated in the anchored domains, their average size must decrease as N ≈ c 0 /n for large n, see Eq. ( 14) (Fig. 3c).

Before we turn to particle-based simulations in the next section, we can already ask how well the "effective" theory of the previous section (Eq. ( 10)) accounts for the average size N of anchored domains when taking into account a typical radius R typ = ⟨⟨r m ⟩⟩ and diffusion constant D typ = ⟨⟨D m ⟩⟩ of diffusing clusters, where we defined ⟨⟨a m ⟩⟩ = m ma m c * m / m mc * m to be the mass-weighted average over diffusing clusters and r m = m/(πρ). The comparison between the "naïve" theory (Eq. ( 7)) with D = D 0 , the "effective" theory with R typ and D typ informed from the rate-equation approach, and the result for N from the rate-equation approach itself is also shown in Fig. 3. Perhaps not surprisingly, the corrections based on R typ and D typ almost entirely account for the discrepancy between the "naïve" theory and the result from the rate-equation approach.

IV. PARTICLE-BASED SIMULATIONS

To validate our theory, we implemented particle-based simulations following the model of refs. [17,22]. In brief, c 0 L 2 particles were distributed on a square simulation box of lateral dimension L with periodic boundary conditions. Upon diffusive encounter, particles fuse and form larger clusters, which can also diffuse. Here, we assumed that particles within clusters rearrange quickly and considered disc-like clusters the radius of which is related to the number of constituent particles according to R = N/(πρ), where ρ is the particle density within clusters, but note that we have shown in earlier work that the resulting cluster size statistics is almost identical in the absence of any particle rearrangement [17]. Individual particles (monomers or constituent particles of larger clusters) are removed from the surface with rate k and randomly injected at the surface to keep the total particle number constant. In addition, nL 2 anchoring sites are distributed on the surface either randomly or forming a square lattice. While particles and clusters in principle diffuse with a size-dependent diffusion constant D m , clusters that are in contact with these anchoring sites do not diffuse. Example snapshots of the simulations are shown in Fig. 4(a,c). We compare the simulation results for the observed size distributions of diffusing and anchored domains with the corresponding predictions from the rateequation approach of Sec. III in Fig. 4(b,d). (Note that the predictions are obtained without any free adjustable parameter as we chose the kinetic coefficient K = 1.81 according to the calibration in the absence of anchoring sites of ref. [22].) The comparison of the average (typical) size of fixed and diffusing clusters as a function of anchoring site density for both random and ordered cases with the theoretical predictions is shown in Fig. 4(e).

In the disordered case, the excellent agreement between the size distributions for specific parameter choices [Fig. 4(b)] and the dependence of average sizes on anchoring site density [Fig. 4(e)] allows us to conclude that our theory provides a quantitatively precise account of the aggregation dynamics and the ensuing non-equilibrium stationary state. As biological systems often display disorder, as can be seen for example in the random distribution of inhibitory PSDs on the dendritic membrane, our theory can thus be considered as quantitatively relevant beyond the general qualitative agreement between simulation and theory independent of spatial order. Interestingly, the agreement becomes less exact in the case of orderly distributed anchoring sites [Fig. 4(d,e)], in which case fixed clusters tend to be slightly larger then predicted by our theory, and diffusing clusters conversely slightly smaller. It seems to make intuitively sense that the disordered case is better described by the mean-field rate equations for the concentrations of domains of a given size, assuming that potential spatial correlations between neighboring clusters effectively self-average and cancel each other in this case. While it is not entirely clear why spatial correlations in the ordered case should give rise to an increased average size of the anchored domains, one may speculate that the regular tiling of space tends to suppress the growth of larger diffusing clusters and thus leads to the accumulation of a larger mass fraction of particles in fixed clusters.

V. CONCLUSION

To summarize, we developed a framework to predict the size distribution of diffusing and fixed domains that form by irreversible aggregation of particles combined with particle turnover in the presence of anchoring sites. In doing so, we followed three separate approaches that provided increasing insight into the statistics of the resulting non-equilibrium stationary state. First, by considering the concentration field of diffusing particles around a stationary domain, we derived an implicit, selfconsistent equation for the domain radius. The calculation predicts a decrease of the domain size with the density of anchoring sites and a concomitant increase in the fraction of particles at fixed domains (Sec. II, Fig. 2). We then combined Smoluchowski rate equations for the freely diffusing clusters with a Master equation for the stochastic size dynamics of fixed clusters to predict more precisely the shapes of the distribution of sizes of both types of cluster (Sec. III, Fig. 3). Notably, the results obtained using the diffusion-equation approach capture rather well the average size of fixed domains predicted by the rate-and Master equation approach [Fig. 3(e,f)]. Given that both approaches are quite different insofar as the first relies on the spatial gradient in the concentration field whereas the second discards spatial variations completely and focuses on the kinetics of aggregation instead, this is in and of itself indicative of the validity of our theory. Finally, we used particle-based simulations to assess the quantitative agreement between our mean-field theory and the aggregation dynamics in space (Sec. IV). Here, we found that a random distribution of anchoring sites leads to a non-equilibrium stationary state the size statistics of which are very well captured by our theory [Fig. 4(b,e)]. In the (biologically less relevant) case of ordered anchoring sites, the agreement becomes less precise, although our theory still captures qualitatively the dependence of the size distributions and average sizes on anchoring site density [Fig. 4(d,e)].

Our results indicate that the presence of anchoring sites modifies the cluster size statistics in several ways: (i) Diffusing and fixed clusters show distinct characteristic size distributions, the typical (average) sizes of which increasingly differ with increasing cluster site density; (ii) fixed clusters have a well-defined average size for arbitrary anchoring site density n whereas (depending on the diffusion size-dependence exponent σ) diffusing clusters may show a power-law dependence until a cut-off size imposed by turnover at vanishing n, with large sizes becoming increasingly suppressed with increasing n. As a corollary, even if the diffusion constant does not depend on cluster size (σ = 0), the size distribution of diffusing and fixed clusters combined would show a deviation from a power-law scaling at finite anchoring site density n. Such a deviation is therefore not alone sufficient to indicate that diffusion is indeed size-dependent, as would be the case in the absence of any anchoring.

We note that the results we have obtained here for the case of a finite density of anchoring sites are somewhat similar to the finite-size effects discussed in the context of membrane nanodomains in vesicles [23], where the finite amount of aggregating particles leads to the separation of clusters into one "giant" cluster and small clusters when the system size becomes smaller or comparable to the typical cluster size in the infinite system. While the authors considered two distinct recycling schemes (monomer vs. cluster recycling), they did not investigate the effects of a possible size-dependence of diffusion and used a simplified Master equation for the size evolution of the single large cluster; their study can thus be considered complementary to our work. In addition, our particle-based simulations reveal the subtle impor-tance of spatial correlations in the resulting size statistics, which is a property of the extended system, i.e. containing much more particles than the typical cluster size. It would be interesting to see whether analytical approaches beyond pure mean-field description may shed additional light onto these kinds of effects (see e.g. [24,25] for a discussion of the interaction between domains).

Another interesting consequence of the clear distinction between diffusing and fixed clusters is that the conceptual "boundary" between (actively suppressed) Ostwald ripening on the one hand [24,26] and domain or droplet coalescence on the other becomes somewhat blurred, as mass becomes increasingly concentrated in the larger fixed clusters which are replenished by diffusing small clusters.

Here, we restricted ourselves to the statistics of the non-equilibrium stationary state that can arise in systems where aggregation is combined with turnover [6,23]. Recently, the dynamics of the cluster size distribution for size-dependent recycling has been analytically studied using a continuum version of Smoluchowski coagulation equations, including results for the transient dynamics of the average cluster size for given initial conditions [7]. It would be worthwhile to see whether these approaches may be extended to the case with size-dependent diffusion coefficient and a finite density of anchoring sites. Given the quantitative agreement between our stationary solution and the simulations, we would expect that the time evolution of the cluster size distribution from non-stationary initial conditions would equally well be captured using the time-dependent (numerical) solutions of the rate equations and Master equation, although admittedly this does not amount to an analytical theory. This work has been inspired by the observation that inhibitory neurotransmitter receptors and scaffold proteins can spontaneously organize into surface domains of a well-defined size even in non-neuronal cells [27,28], which lead to the hypothesis that these domains form according to the aggregation process with turnover described here [17]. A recent study assessed whether the proposed non-equilibrium recycling scheme could explain the observed receptor and scaffold kinetics at synapses upon the suppression of lateral receptor fluxes at the extrasynaptic membrane, and concluded that direct recruitment of scaffold proteins to synaptic sites cannot be neglected [18]. Given the ubiquity of protein diffusion and aggregation in membranes, and the potential sites for protein anchoring at the submembrane structures such as the cytoskeleton, transmembrane or extracellular structures, we are confident that the results presented here remain nevertheless relevant for a large number of biological systems.

  arXiv:2308.15162v1 [cond-mat.soft] 29 Aug 2023 t e x i t s h a 1 _ b a s e 6 4 = " N l

0 Figure 1 .

 01 Figure 1. Sketch of diffusion-aggregation dynamics with turnover in the presence of anchoring sites. (a) Particles and clusters that are not attached to anchoring sites can diffuse freely as indicated by the thin black arrows. Clusters at anchoring sites do not diffuse. Individual particles are subject to turnover ("monomer recycling scheme") as indicated by the blue arrows. (b) Corresponding mean-field picture with a smooth concentration field c(r, t) (gray shading) outside of fixed domains (red).

Figure 2 .

 2 Figure 2. Mean-field theory for a single domain based on the diffusion equation. (a) Variation of the size of anchored domains with anchoring site density n/c0. The asymptote N = c0/n when all mass is concentrated in fixed clusters is also shown, see text. (b) Concentration field c(r) outside of the anchored domains for different values of nλ 2 . (c) Mass fraction of anchored domains as a function of anchoring site density n/c0. (d) Size of anchored domains as a function of c0λ 2for fixed values of n/c0. For all plots, ρ/c0 = 7000. Cluster radii R (and equivalently cluster sizes N ) were obtained from numerically solving Eq. (7).

Figure 3 .

 3 Figure 3. Characterization of diffusing and anchored domains. (a,b) Size distributions for different values of n/c0, for (a) σ = 0 and (b) σ = 1. Solid lines: ĉm, dotted lines: c * m , dashed-dotted lines: c tot m ; colors indicate the value of n/c0, see legend in panel (b). (c) Average (typical) size of the anchored (diffusing) clusters as a function of n/c0 for different values of σ. (d) Relative mass fraction of anchored domains as a function of n/c0 for different values of σ.(c,d) The mean-field theory of Sec. II is shown for comparison with dots ("naïve" theory, Eq. (7)) and crosses ("effective" theory, Eq. (10)). For the rate equations, K = 2 was used.

Figure 4 .

 4 Figure 4. Particle-based simulations. (a) Snapshot of a simulation with 144 randomly distributed anchoring sites in a square box of side length L = 2000. Lengths are in units of single particle diameter a, see refs. [17, 22]; cluster radii are not to scale but increased five-fold for better visibility. Clusters at anchoring sites are colored in red. (b) Size distributions of anchored and diffusing clusters for different values of n/c0, comparing simulations with randomly distributed anchoring sites to the rate-equation-based theory. Note that the theory has no free parameter. (c,d) Same as (a,b) but for simulations with regularly distributed anchoring sites. (e) Average size N of anchored clusters (solid lines) and typical size Mof diffusing clusters (dotted lines) as a function of n/c0 as predicted by theory (blue, large dots) and obtained by simulations with randomly (green, crosses) or regularly (brown, small dots) distributed anchoring sites. For all simulations, parameters were identical to case II of ref.[22] (c0/ρ = 9•10 -4 , c0D0/k = 45, σ = 0.5), with a simulation box size L = 2000 and for a duration T = 5 • 10 7 .

Acknowledgments. The authors would like to thank Vincent Hakim and Benjamin Lindner for comments on an earlier version of this manuscript.