M Vynnycky
email: michael.vynnycky@ul.ie

J Lacaze

On the modelling of joint formation in dissolutive brazing processes

Keywords: Asymptotic analysis, Dissolution, Wetting . Vynnycl-y (181

In this paper, earlier dissolutive wetting models describing the dynamics of an axisymmetric alloy drop spreading on pure metal substrate are extended to describe reactive wetting and subsequent joint formation in brazing processes. A two-dimensional time-dependent problem is formulated, and the model equations are nondimensionalized, revealing the possibilities for asymptotic mode! reduction. Whilst the numerical solution of the time-dependent problem, which contains two moving contact lines and would not in general be amenable to lubrication theory, is relegated to future work, the steady-state problem is analyzed in detail. The analysis offers an arguably more transparent alternative to an earlier energy minimization approach for finding the location of the meniscus, which ultimately constitutes the joint. The results of the present mode! are found to compare favourably to those of earlier experimental and theoretical work.

Introduction

In the controlled atmosphere brazing (CAB) process, separate parts are assembled by being submitted to a thermal cycle during which a cladding consisting of a low melting temperature alloy becomes liquid and forms the necessary joints by capillarity. After a time allocated for joint formation, the assembly is cooled so that the joints solidify. For manufacturing aluminium heat exchangers, the parts to be assembled are plates and fins made of an essentially binary Al-Mn alloy. Sorne of these parts are clad with an essentially binary Al-Si alloy that has a liquidus 4�50 K below the solidus of the Al-Mn alloy. Proper joint formation requires that enough liquid is made available to form the joints, but not too much as this may lead to unexpected deformations of the assemblies and other potential defects. Accordingly, the brazing cycles are designed with a maximum temperature close to the clad alloy liquidus, slightly lower to minimize deformations or slightly higher if more liquid is needed.

Since the very beginning of development of the CAB process, the materials and process parameters have been tested using an inverted tee-joint configuration, wherein a plate stands vertically on a horizontal plate [START_REF] Terrill | Diffusion of silicon in aluminum brazing sheet[END_REF][START_REF] Woods | Flow of aluminum dip brazing filler metals[END_REF][START_REF] Zhao | Analysis of fin-tube joints in a compact heat exchanger[END_REF]. A theoretical prediction of the brazed joint shape to be formed during manufacturing can be performed using a joint shape modelling advocated in [START_REF] Sekulic | Prediction of the fillet mass and topology of aluminum brazed joints[END_REF]. Recent studies of some relatively simple brazed joint configurations [START_REF] Sekulic | Molten aluminum equilibrium membrane formed during controlled atmosphere brazing[END_REF], and more complex two-dimensional (2D) and three-dimensional (3D) geometries [START_REF] Zellmer | Numerical modelling and experimental verification of the formation of 2D and 3D brazed joints[END_REF][START_REF] Sekulic | Influence of joint topology on the formation of brazed joints[END_REF], have convincingly demonstrated that a minimization of potential energy of the molten metal free surface at the onset of cladding solidification may efficiently be utilized for joint topology predictions. A summary of these approaches is given in [START_REF] Sekulic | Modeling of the sequence of phenomena in brazing[END_REF].

Nevertheless, the modelling of brazing should be amenable to an even more fundamental approach, such as that which has been applied to reactive wetting in metal-metal systems [START_REF] Warren | Modeling reactive wetting[END_REF][START_REF] Su | Modeling dissolution and spreading of Bi-Sn alloy drops on a Bi substrate[END_REF][START_REF] Villanueva | Multicomponent and multiphase modeling and simulation of reactive wetting[END_REF][START_REF] Villanueva | Effect of phase change and solute diffusion on spreading on a dissolving substrate[END_REF][START_REF] Villanueva | A diffuse-interface model of reactive wetting with intermetallic formation[END_REF]. Thus, rather than appealing to energy minimization so as to find, for example, the final equilibrium meniscus shape, it should be possible to formulate time-dependent conservation equations that describe how the final state is reached. One reason this approach has not been adopted for brazing is that it has been proven difficult to obtain solutions for real parameters in solder systems [START_REF] Sekulic | Molten aluminum equilibrium membrane formed during controlled atmosphere brazing[END_REF][START_REF] Braun | Lubrication theory for reactive spreading of a thin drop[END_REF]. Hence, one of the aims of this paper is to apply the concepts previously used for reactive wetting to the modelling of microscale phenomena in brazing. Moreover, this approach should serve to complement the energy minimization approach: for example, it is already a well-accepted concept in physics that the shape that a fluid meniscus adopts is the same as that required to minimize the energy of the system [START_REF] Racz | A general statement of the problem and description of a proposed method of calculation for some meniscus problems in materials processing[END_REF].

To fix ideas, we focus on the inverted tee-joint configuration considered in [START_REF] Sekulic | Molten aluminum equilibrium membrane formed during controlled atmosphere brazing[END_REF], as it is not only the simplest one to consider geometrically, but there are also experimental data to compare with. The structure of the paper is as follows. In Sect. 2, we motivate the modelling approach to be used. In Sect. 3, we formulate the problem mathematically, and in Sect. 4, we nondimensionalize the equations. The main body of the analysis is in Sect. 5, which considers the time-dependent equations from a qualitative perspective and the solution of the steady-state equations. Results are given in Sect. 6, and conclusions are drawn in Sect. 7. Finally, auxiliary results are given in four appendices.

Modelling considerations

Compared to the modelling of reactive spreading of an axisymmetric droplet of a molten metallic alloy on a metal substrate [START_REF] Warren | Modeling reactive wetting[END_REF][START_REF] Su | Modeling dissolution and spreading of Bi-Sn alloy drops on a Bi substrate[END_REF][START_REF] Villanueva | Multicomponent and multiphase modeling and simulation of reactive wetting[END_REF][START_REF] Villanueva | Effect of phase change and solute diffusion on spreading on a dissolving substrate[END_REF][START_REF] Villanueva | A diffuse-interface model of reactive wetting with intermetallic formation[END_REF], the modelling of brazing is substantially more complex, since the clad, which will ultimately flow and cause the dissolution of the core, is initially solid. Moreover, once the heating of the core-clad assembly starts, solid-state diffusion of the silicon in the clad occurs; also, it may be the case that heating is not isothermal, with different parts of the clad and the core reaching a certain temperature at different times. In particular, this will mean that some parts of the clad will reach the solidus temperature sooner than others; thus, these parts would in principle be available for flow sooner than others. However, if there is no solid-state diffusion from the clad to the core and if the final holding temperature is lower than the liquidus temperature corresponding to the initial composition of Si in the clad, then not all of the clad will melt; on the other hand, if there is solid-state diffusion, then even this holding temperature might not be high enough to melt the clad, and an even higher temperature would be required to do so. Furthermore, once melting has begun, capillary forces can be expected to disrupt the initial planar configuration of the clad, leading to the formation of a moving meniscus, at which point coupled two-phase momentum, i.e. heat and solute transfers will be occurring. A more complete description of these stages can be found in [START_REF] Lacaze | Numerical simulation of brazing aluminium alloys with Al-Si alloys[END_REF].

Therefore, in order to avoid getting bogged down in these details, and to be able to focus on reactive spreading, we will simply assume that the clad is initially molten and at a uniform temperature that is greater than the liquidus temperature for the clad; furthermore, we will also assume that the core is at the same uniform temperature as the clad. In fact, these assumptions should not affect what will be the main body of the results in this paper, i.e. the determination at steady state of the joint shape and of the dissolved region.

Mode) formulation

Goveming equations

Assume all of the clad, of composition co, has reached the temperature T, where T > Tliq(co) and occupies 0 � x � wo, 0 � y � ho, as shown in Fig. 1, which depicts the right-hand half of an inverted tee joint. Here, Î! i q denotes the liquidus curve in the Al-Si phase diagram and is a fonction of the Si concentration, c, as shown in Fig. 2a; however, in what follows, we approximate Tliq by Î! i q = Tmme, (3.1) where Tm denotes the melting temperature of pure aluminium and-mis the slope of the liquidus curve, which is now assumed straight, as is shown in Fig. 2b. Thus, T > Tliq(co) implies that the clad is assumed to be completely molten initially. Note also that we implicitly assume that the horizontal core and clad are initially in contact, so that the clad immediately proceeds to wet the core; clearly, in practice, this contact may be limited or sporadic along the length of the joint, leading to possible variations along the axis perpendicular to the x-y plane that we consider. Thereafter, the molten clad begins to move and at time t, it will occupy O � x � x 1 (t), 0 � y � h (x, t), as shown in Fig. 3. At the same time, the core which occupies x < 0 and y < 0 will dissolve, with the limit of the where p and µ, are, respectively, the density and viscosity of molten clad; these will be assumed to be constant. In addition, u and v are the velocity components in the xand y-directions respectively, pis the pressure and gis the gravitational acceleration. Moreover, conservation of solute is given by f (x, t) � y � h (x, t), (3.5) where c is the solute concentration and D is the diffusion coefficient for Si. Also, by conservation of mass for the clad, we must have

{ XI (f) l o h (x, t) dx = woho.

Bound ary conditions

At y = h (x, t) , we have zero tangential stress, given by ( av au )

( au av ) { 2 } 2µ, ---hx + µ, -+ -1 -h = 0,
ay ax ay ax

X (3.6) (3.7)
and a normal stress condition, given by

p a -p + 2μ 1 + h 2 x ∂u ∂ x h 2 x + ∂v ∂ y - ∂u ∂ y + ∂v ∂ x h 2 x = σ fs K, ( 3.8)
where σ fs denotes the interfacial tension between the molten clad and the surroundings, p a is the pressure of the surroundings and K is the curvature of the meniscus, given by

K = h x x 1 + h 2 x 3/2 .
(3.9)

In addition to (3.7) and (3.8), which are called dynamic conditions, there is a kinematic condition for the motion of the meniscus, given by

v = h t + uh x . (3.10)
Also, there is no normal flux of solute at the meniscus, and hence

n h • ∇c = 0, (3.11)
where n h is the unit normal vector to y = h (x, t), and is given by

n h = 1 1 + h 2 x (-h x , 1) . ( 3.12)
At the dissolution front, y = f (x, t) , we will have

v = f t + u f x , (3.13) u + v f x = 0, (3.14) c = c E (T ) + Γ κ, (3.15) c 1 + f 2 x ∂ f ∂t = (cu -D∇c) • n f , ( 3.16)
where u = (u, v) , n f is the unit normal vector to y = f (x, t) , and is given by

n f = 1 1 + f 2 x (-f x , 1) , ( 3.17)
and κ denotes the curvature and is given by

κ = f x x 1 + f 2 x 3/2 . (3.18)
Note that - (3.15) expresses local equilibrium at the liquid-solid interface including the Gibbs-Thomson effect [START_REF] Warren | Modeling reactive wetting[END_REF][START_REF] Su | Modeling dissolution and spreading of Bi-Sn alloy drops on a Bi substrate[END_REF], with c E (T ) as the equilibrium concentration of Si at the brazing temperature, T, as given by the phase diagram and Γ as the normalized liquid-solid surface energy or solute capillary length, which is assumed to be constant; -in (3.16), we have neglected any solid-state diffusion in the core; -applying the Reynolds transport theorem [START_REF] Leal | Advanced transport phenomena: fluid mechanics and convective transport processes[END_REF] to Eq. (3.5), we find, with the help of (3.11) and (3.16) that

Ω(t)
c (x, y, t) dy dx = w 0 h 0 c 0 , (3.19) where Ω denotes the liquid region, i.e. solute is conserved.

Furthermore, conditions are required at the contact lines, which are at (x 1 (t) , 0) and (0, y 1 (t)) . It seems reasonable to generalize the condition derived by Warren et al. [START_REF] Warren | Modeling reactive wetting[END_REF], and later used by Su et al. [START_REF] Su | Modeling dissolution and spreading of Bi-Sn alloy drops on a Bi substrate[END_REF] and Singler et al. [START_REF] Singler | Modeling and experiments in dissolutive wetting: a review[END_REF], for the contact line of an axisymmetric droplet during reactive spreading on a horizontal substrate. Warren et al. [START_REF] Warren | Modeling reactive wetting[END_REF] assume that the contact line remains at the initial level of the substrate and that the total angle between the meniscus and the liquid-solid interface, θ L , remains constant; in the present context, and with reference to Fig. 3,

θ b,x 1 + θ t,x 1 = θ L , θ b,y 1 + θ t,y 1 = θ L . (3.20)
However, since there is a discontinuity in the slope between the liquid surface and the liquid-solid interface, there will arise an additional restriction from the requirement that the liquid concentration must be well-defined and differentiable at the triple junction. The resulting expressions for the velocities of the contact lines, ẋ1 and ẏ1 , are

ẋ1 = -D tan θ b,x 1 + tan θ t,x 1 c tan θ b,x 1 ∂c ∂ x (x 1 (t),0) = -DΓ tan θ L c tan θ b,x 1 ∂κ ∂ x (x 1 (t),0) , (3.21) ẏ1 = -D tan θ b,y 1 + tan θ t,y 1 c tan θ b,y 1 ∂c ∂ y (0,y 1 (t)) = -DΓ tan θ L c tan θ b,y 1 ∂κ ∂ y (0,y 1 (t)) , (3.22)
where the dots denote differentiation with respect to time.

In the model, θ L is a prescribed constant-and has to be determined experimentally-but θ b,x 1 , θ t,x 1 , θ b,y 1 and θ t,y 1 are all functions of time and must all be determined. Equation (3.20) can then be written in terms of f and h as

h x (0, t) -f x (0, t) 1 + h x (0, t) f x (0, t) = tan θ L , (3.23) h x (x 1 , t) -f x (x 1 , t) 1 + h x (x 1 , t) f x (x 1 , t) = -tan θ L , ( 3.24)
and further conditions are

f (0, t) = h (0, t) , (3.25) f (x 1 , t) = 0, (3.26) h (x 1 , t) = 0. (3.27)
Equations (3.25)-(3.27) are perhaps worthy of further explanation, as they are not obvious. In general, one would expect f (x 1 , t) = h (x 1 , t) , so that both the x-and y-locations of the contact are not known and have to be determined. However, Warren et al. [START_REF] Warren | Modeling reactive wetting[END_REF]take f (x 1 , t) = 0, h (x 1 , t) = 0, which at least leads to a well-posed mathematical problem. On the other hand, Villanueva et al. [START_REF] Villanueva | Multicomponent and multiphase modeling and simulation of reactive wetting[END_REF] treat a similar problem using phase-field methods and obtain an elevation of the contact line relative to the y = 0 level, which is apparently qualitatively similar to the experimental observations of Saiz et al. [START_REF] Saiz | Reactive spreading: adsorption, ridging and compound formation[END_REF][START_REF] Saiz | Ridging effects on wetting and spreading of liquids on solids[END_REF]; nevertheless, the elevation is small compared to the maximum height of the droplet, which suggests that (3.26) and (3.27) can be a suitable approximation. Thus, having adopted this approximation at the horizontal boundary, we may adopt it at the vertical one also, giving Eq. (3.25). Note that although it is written as one equation, it may be thought of as two if we write

f -1 (y 1 (t) , t) = 0, h -1 (y 1 (t) , t) = 0, (3.28)
where f -1 and h -1 denote the inverse functions of f and h, respectively.

The initial conditions at t = 0 should be

u = 0, v = 0, c = c 0 for 0 < x < w 0 , 0 < y < h 0 , (3.29) h -1 (y, 0) = w 0 , 0 ≤ y ≤ h 0 , h (x, 0) = h 0 , 0 ≤ x ≤ w 0 , (3.30) f -1 (y, 0) = 0, 0 ≤ y ≤ h 0 , f (x, 0) = 0, 0 ≤ x ≤ w 0 . (3.31)
The last of these implies that none of the cores has dissolved.

Nondimensionalization

The relevant length scale that should appear is the capillary length scale for the meniscus, l = σ f s /ρg. Thus, we nondimensionalize with

X = x l , Y = y l , U = u [u] , V = v [u] , P = p -p a √ ρgσ f s , τ = t [t] , H = h l , F = f l ,
where [u] is a velocity scale to be determined and [t] = l/ [u]; in fact, it turns out to be appropriate to take [u] = D/l, whence [t] = l 2 /D, although we discuss the selection of [u] in Appendix A. Also, for the time being, we set

X 1 = x 1 l , Y 1 = y 1 l . (4.1)
We obtain, from (3.2)-(3.5),

∂U ∂ X + ∂ V ∂Y = 0, (4.2)
Re ∂U ∂τ + U ∂U ∂ X + V ∂U ∂Y = - 1 Ca ∂ P ∂ X + ∂ 2 U ∂ X 2 + ∂ 2 U ∂Y 2 , (4.3) Re ∂ V ∂τ + U ∂ V ∂ X + V ∂ V ∂Y = - 1 Ca ∂ P ∂Y + ∂ 2 V ∂ X 2 + ∂ 2 V ∂Y 2 - 1 Ca , ( 4.4
)

∂c ∂τ + U ∂c ∂ X + V ∂c ∂Y = ∂ 2 c ∂ X 2 + ∂ 2 c ∂Y 2 , ( 4.5)
where

Ca = μ [u] σ f s , Re = ρ [u] l μ ; (4.6)
these are, respectively, the capillary and Reynolds numbers. Also, conservation of mass for the clad gives

X 1 (τ ) 0 H (X, τ ) dX = W 0 H 0 , ( 4.7)
where W 0 = w 0 /l, H 0 = h 0 /l, whereas conservation of solute for the liquid region gives

X 1 (τ ) 0 H (X,τ ) F(X,τ ) c (X, Y, τ ) dY dX = W 0 H 0 c 0 . ( 4
2 ∂ V ∂Y - ∂U ∂ X H X + ∂U ∂Y + ∂ V ∂ X 1 -H 2 X = 0, (4.9)
-P + 2Ca 1 + H 2 X ∂U ∂ X H 2 X + ∂ V ∂Y - ∂U ∂Y + ∂ V ∂ X H 2 X = H X X 1 + H 2 X 3/2 , (4.10) V = H τ + U H X , ( 4.11
)

n H • ∇c = 0, (4.12)
respectively, where

n H = 1 1 + H 2 X (-H X , 1) . (4.13) Equations (3.13)-(3.16) give, at Y = F (X, τ ) , V = F τ + U F X , (4.14) U + V F X = 0, (4.15) c = c E (T ) + δ F X X 1 + F 2 X 3/2 , (4.16) c 1 + F 2 X ∂ F ∂τ = n F • (cU -∇c) , ( 4.17)
respectively, where U = (U, V ) , δ = Γ /l and

n F = 1 1 + F 2 X (-F X , 1) . (4.18)
At the contact lines, (3.21) and (3.22) become, respectively,

Ẋ1 = - tan θ b,x 1 + tan θ t,x 1 c tan θ b,x 1 ∂c ∂ X (X 1 (τ ),0) = - δ tan θ L c tan θ b,x 1 F X X 1 + F 2 X 3/2 X (X 1 (τ ),0) , (4.19) Ẏ1 = - tan θ b,y 1 + tan θ t,y 1 c tan θ b,y 1 ∂c ∂Y (0,Y 1 (τ )) = - δ tan θ L cF X tan θ b,y 1 F X X 1 + F 2 X 3/2 X (0,Y 1 (τ )) , ( 4
H X (0, τ ) -F X (0, τ ) 1 + H X (0, τ ) F X (0, τ ) = tan θ L , (4.21) H X (X 1 , τ ) -F X (X 1 , τ ) 1 + H X (X 1 , τ ) F X (X 1 , τ ) = -tan θ L , (4.22) F (0, τ ) = H (0, τ ) , (4.23) F (X 1 , τ ) = 0, (4.24) H (X 1 , τ ) = 0. (4.25)
Lastly, the initial conditions at τ = 0 are

U = 0, V = 0, c = c 0 for 0 < X < W 0 , 0 < Y < H 0 , (4.26) H -1 (Y, 0) = W 0 , 0 ≤ Y ≤ H 0 , H (X, 0) = H 0 , 0 ≤ X ≤ W 0 , (4.27) F -1 (Y, 0) = 0, 0 ≤ Y ≤ H 0 , F (X, 0) = 0, 0 ≤ X ≤ W 0 . (4.28)

Analysis

In this section, we consider first in Sect. 5.1 the qualitative behaviour of the transient solution; then, in Sect. 5.2, we compute steady-state solutions.

Transient behaviour

Using the parameters in Table 1, we find that l ≈ 5 mm, which will give

Ca = μD lσ f s ∼ 10 -10 , Re = ρ D μ ∼ 10 -3 , δ ∼ 10 -7 .
We observe now that Ca, Re 1, and we can consider the equations at leading order in Ca. Equations (4.3) and (4.4) reduce to just

∂ P ∂ X = 0, (5.1)
∂ P ∂Y = -1, (5.2)
giving

P = P * (τ ) -Y, ( 5.3)
where P * is a function of τ to be determined. The only other place in the problem formulation where Ca appears is in (4.10), which, on using (5.3) gives, at leading order in Ca,

H X X 1 + H 2 X 3/2 = H -P * (τ ) .
(5.4)

Although the governing equation for H appears to decouple from the rest of the formulation, this is not so significant for the transient problem, since (5.4) requires boundary conditions at X = 0 and X = X 1 (τ ) which will depend on the solutions for U, V and c, to which we turn next. We may as well now set P = (P * (τ ) -Y ) + Ca P 1 , so that (4.2)-(4.5) become

∂U ∂ X + ∂ V ∂Y = 0, (5.5
)

Re ∂U ∂τ + U ∂U ∂ X + V ∂U ∂Y = - ∂ P 1 ∂ X + ∂ 2 U ∂ X 2 + ∂ 2 U ∂Y 2 , ( 5.6
)

Re ∂ V ∂τ + U ∂ V ∂ X + V ∂ V ∂Y = - ∂ P 1 ∂Y + ∂ 2 V ∂ X 2 + ∂ 2 V ∂Y 2 , ( 5.7
)

∂c ∂τ + U ∂c ∂ X + V ∂c ∂Y = ∂ 2 c ∂ X 2 + ∂ 2 c ∂Y 2 . ( 5.8)
The boundary and initial conditions, (4.9)-(4.28), are unchanged, apart from (4.10), which becomes

-P 1 + 2 1 + H 2 X ∂U ∂ X H 2 X + ∂ V ∂Y - ∂U ∂Y + ∂ V ∂ X H 2 X = 0.
(5.9)

Note that since δ 1, we can expect from (4. [START_REF] Saiz | Reactive spreading: adsorption, ridging and compound formation[END_REF]) and (4.20) that X ˙ 1 ≈ 0, Y ˙1 ≈ 0, respectively, and then that the problem for U, V, P 1 would have the trivial solution, but for the fact that F τ = 0atY = F (X,τ) ; thus, the movement of the reaction front drives the velocity field in the melt via the boundary condition at Y = F (X,τ) . Whilst this is true for intermediate times, prior to the steady state which we shall consider shortly, there is an additional complication concerning (4. [START_REF] Saiz | Reactive spreading: adsorption, ridging and compound formation[END_REF]) and (4.20). Since θ b,x 1 and θ b,y 1 are initially zero, it is clear that the simplification that X ˙ 1 ≈ 0, Y ˙1 ≈ 0 will only become valid once θ b,x 1 ,θ b,y 1 δ; further analysis would be required to determine a time scale for when this happens, and we do not consider it any further here. Moreover, since X ˙ 1 and Y ˙1 are initially very large, i.e. when θ b,x 1 ,θ b,y 1 δ, this stage is often identified in droplet spreading situations as that of the initial fast hydrodynamic spreading of the liquid, during which very little dissolution takes place [START_REF] Warren | Modeling reactive wetting[END_REF]. Another interesting point is that once X ˙ 1 ≈ 0, Y ˙1 ≈ 0, the two contact lines stop moving; thereafter, the meniscus adjusts its profile in order that (4.21) and (4.22) are satisfied, but its endpoints are fixed.

Recalling that l 2 /D was determined in Sect. 4 as the timescale for the dissolution problem, we can now attempt to infer the order of magnitude of the time required to reach steady state. Using the parameters from Table 1, l 2 /D would correspond to around 25,000 s, i.e. around 7 h. Nevertheless, it needs to be emphasized that this value may be a severe overestimation. For example, if the meniscus has equal extents of O (L) in x-and y-directions, then we should have L 2 ∼ w 0 h 0 , and the relevant time scale will be L 2 /D. With L ∼ (w 0 h 0 ) 1/2 ∼ 900 µm, we would obtain L ∼ 750 s, which compares well with the length of about 5 minutes of the dwell time during an optimized brazing cycle, e.g. see Fig. 3 in [START_REF] Sekulic | Prediction of the fillet mass and topology of aluminum brazed joints[END_REF]. In practice, a full numerical simulation would be necessary to determine when the transient model reaches a steady state.

Steady state

Having identified how the transient problem as a whole operates qualitatively, we proceed to determine the steady state quantitatively. First, we note that, at steady state, U, V, P 1 ≡ 0 and c is constant, so that (5.5)-(5.8) are satisfied automatically, as are boundary conditions (4.9), (4.11), (4.12), (4.14), (4.15), (4.17), (4. [START_REF] Saiz | Reactive spreading: adsorption, ridging and compound formation[END_REF]) and (4.20). We recall that boundary condition (4.10) resulted in Eq. (5.4), so that if we now set C 1 = -P * (∞) , where C 1 is a constant whose value will need to determine, we will have

H X X 1 + H 2 X 3/2 = H + C 1 .
(5.10) Also, boundary condition (4.16) gives

δ F X X 1 + F 2 X 3/2 = C * 2 -c E (T ) , ( 5.11)
where C * 2 is the constant steady-state value of c, which is as yet unknown; note here that even though δ 1, whereas c E (T ) is nominally O (1) , the term on the right-hand side of (5. (5.12)

C * 2 X * 0 (H (X ) -F (X )) dX = c 0 A, ( 5.13)
H (0) = Y * , (5.14) H X (0) -F X (0) 1 + H X (0) F X (0) = tan θ L , ( 5.15)
H (X * ) = 0, (5.16)

F (X * ) = 0, (5.17) H X (X * ) -F X (X * ) 1 + H X (X * ) F X (X * ) = -tan θ L , ( 5.18)
where

A = ρgw 0 h 0 σ f s , (5.19) also, X * = X 1 (∞) , Y * = Y 1 (∞).
The equations can be simplified further by recalling that δ 1, so that

C * 2 = c E (T ) + O(δ). Setting C * 2 = c E (T ) + δC 2 + O δ 2 , ( 5.20)
we see that (5.11) becomes

F X X 1 + F 2 X 3/2 = C 2 , ( 5.21)
whereas (5.13) simplifies to

X * 0 F (X ) dX = c E (T ) -c 0 c E (T ) A. (5.22)
At this stage, we have

H X X 1 + H 2 X 3/2 = H + C 1 , (5.23) F X X 1 + F 2 X 3/2 = C 2 , ( 5.24)
subject to

H (0) = F (0) , (5.25) H X (0) -F X (0) 1 + H X (0) F X (0) = tan θ L , (5.26)
H (X * ) = 0, (5.27)

F (X * ) = 0, (5.28) H X (X * ) -F X (X * ) 1 + H X (X * ) F X (X * ) = -tan θ L , (5.29) X * 0 H (X ) dX = A, (5.30) X * 0 F (X ) dX = c E (T ) -c 0 c E (T ) A.
(5.31)

We observe that (5.24) implies that the curvature of the curve describing the dissolved zone is constant, and it is tempting to infer that F will describe the arc of a circle in the core from (0, Y * ) to (X * , 0) . However, it will be shown that this cannot in general be the case. The easiest way to see this is to consider what happens when

ε := (c 0 -c E (T )) /c E (T )
1, although this is relegated to Appendix B; in addition, in Appendix C, it is shown that, even in the general case when ε ∼ O (1) , there cannot be a circular arc from (0, Y * ) to (X * , 0) . Instead, the only resolution appears to be that there are two circular arcs, one starting at (0, Y * ) and the other at (X * , 0) , both of which pass through (0,0) and have the same radius. Thence, we append We consider the circles (5.33)

F (0) = 0 (5.
(5.34)

(X -X + ) 2 + (Y -Y + ) 2 = R 2 , (X -X -) 2 + (Y -Y -) 2 = R 2 ,

X. X

where X+, Y+, X_, Y_ and R must be determined; we will later set Y= F. Equation (5.33) will represent a circle of radius Rand having its centre at (X + , Y+) that passes through (0,0) and (0,Y.), whereas Eq. ( 5.34) will represent a circle of radius R and having its centre at (X_, f_) that passes through (0,0) and (X*, 0); this is summarized schematically in Fig. 4. From (5.25),(5.28) and (5.32) twice, we must have

xi+ cr. -r+> 2 = R 2 , ex. -x_) 2 + r.: = R 2 ,
Xi+ Y;= R 2 , X:.+ Y.:= R 2 , respectively; note that (5.32) is used twice since both circular arcs must satisfy it. We quickly see that Y;= (Y* -f+) 2 , X:.= (X* -X_) 2 , and hence

x _ = x ., where

tan θ + = Y + X + , tan θ -= X - Y - . (5.44) So, R 2 (θ + + θ -) -X + Y + -X -Y -= εA, (5.45) i.e. R 2 tan -1 Y * 4R 2 -Y 2 * 1/2 + tan -1 4R 2 -X 2 * 1/2 X * - Y * 2 R 2 - Y 2 * 4 1/2 - X * 2 R 2 - X 2 * 4 1/2 = εA, (5.46)
which is a relation for R in terms of X * and Y * . We use (5.33) and ( 5.34) to simplify (5.26) and (5.29). First, we see from differentiating (5.33) that

(X -X + ) + (Y -Y + ) dY dX = 0, (5.47)
whence, on setting Y = F (X ) , we obtain

F X (0) = 2X + Y * .
(5.48) From (5.34),

(X -X -) + (Y -Y -) dY dX = 0, (5.49)
whence, similarly,

F X (X * ) = X * 2Y - .
(5.50)

Now, we can re-arrange (5.26) and (5.29) to obtain

H X (0) = F X (0) + tan θ L 1 -F X (0) tan θ L , (5.51) H X (X * ) = F X (X * ) -tan θ L 1 + F X (X * ) tan θ L , (5.52)
respectively, and hence

H X (0) = 2X + + Y * tan θ L Y * -2X + tan θ L , (5.53) H X (X * ) = X * -2Y -tan θ L 2Y -+ X * tan θ L , ( 5.54)
where

X + = R 2 - Y 2 * 4 1/2 , Y -= R 2 - X 2 * 4 1/2 , (5.55)
and with R given by the solution to Eq. (5.46). A further interesting observation in relation to earlier work concerns the function H, rather than F. Terrill [START_REF] Terrill | Diffusion of silicon in aluminum brazing sheet[END_REF] approximated the meniscus of a tee joint with a quarter of a circle. This would mean neglecting the term in H on the right-hand side of (5.23), leaving us with an equation that is qualitatively similar to (5.24), with the end result being just a meniscus having constant curvature. However, it is clear from the analysis that the term in H is a leading-order term in the model, and should therefore not be neglected.

Model summary and numerical solution

The model equations have been reduced to just one nonlinear second-order ordinary differential equation, (5.23), subject to (5.27), (5.30), (5.46), (5.53) and (5.54), with the unknowns being H ,R, X * and C 1 ; there are thus five constraints for five unknowns, with the latter being R, X * , C 1 and the two constants of integration from (5.23). Once, these are determined, X + , Y + , Y * and C 2 can be found.

Although it is a computationally inexpensive task to solve these equations, some care is necessary in finding a suitable initial guess for H ,R, X * and C 1 ; a poor guess leads to the square root of negative numbers in (5.46). Thereafter, once a solution is found for one (θ L , ε)-combination, it can be used as the initial guess for finding solutions for other combinations by parameter stepping. Moreover, although what we have is a boundary-value problem, the location of one of the boundaries, X = X * , is not known beforehand. It is therefore convenient to use boundary immobilization, and this is done by introducing the variables X and Ĥ , which are given by X = X * X , H = X * Ĥ , (5.56) leading to, for 0 ≤ X ≤ 1,

Ĥ X X 1 + Ĥ 2 X 3/2 = X * X * Ĥ + C 1 , (5.57) subject to Ĥ (1) = 0, (5.58) X 2 * X * 0 Ĥ (X ) dX = A, (5.59) Ĥ X (0) = 2X + + Y * tan θ L Y * -2X + tan θ L , (5.60) Ĥ X (1) = X * -2Y -tan θ L 2Y -+ X * tan θ L , (5.61)
and (5.46), where Y * = X * Ĥ (0) . These equations were solved using the finite element software Comsol Multiphysics, typically using a few hundred elements to discretize the interval 0 ≤ X ≤ 1. It is straightforward to ensure that the results are mesh-independent, and we do not document those calculations here.

To validate the model, data parallelling the experimental details given by Sekulic [START_REF] Sekulic | Molten aluminum equilibrium membrane formed during controlled atmosphere brazing[END_REF] were considered. The core alloy consists of AA3003 alloy which is mainly an Al -Mn alloy with 1. E-1.5 wt% Mn and little Fe and Si. The clad alloy is a AA4343 alloy with 8 wt% Si, 0.8 wt% Fe, 0.25 wt%Cu, 0.10 wt% Mn and 0.20 wt% Zn (in wt.%), and has a liquidus of 607 .8 °C, as calculated using the TCAL2 database [23]. The thickness of the clad was wo = 200µ,m.

Heating to the brazing upper temperature and cooling from it were both performed at a rate higher than 50 °C/min, which would be large enough that solid-state diffusion of silicon does not affect the amount of clad available [START_REF] Lacaze | Numerical simulation of brazing aluminium alloys with Al-Si alloys[END_REF].

Although Sekulic [5] indicates an upper brazing temperature of 600 °C for the experiment he reported, he implicitly assumed in his approach that ail the clad had melted. lndeed, brazing for 3003/4343 assemblies may be performed from 600 to 615 °C [START_REF] Gao | Residual clad formation and aluminum brazed joint topology prediction[END_REF], i.e. from slightly below to slightly above the clad liquidus temperature.

In the present modelling approach, the clad alloy is considered as a binary Al -Si alloy, the liquidus temperature of which is estimated as Tmmco, where the values for Tm and mare as given in Table 1. As for co, we note that if we take 8 wt% Si, as indicated above, this would imply a value lower than that of c E at 600 °C, which is 8.8 wt% Si; this would give s < O. On the other hand, if we sum up the non-Al compositions given above and assign this sum to be co, then we obtain 9.4 wt% Si, giving s > O. Since the mode! is only self-consistent ifs ::: 0, we will therefore take co = 9.4 wt%.

At this stage, we still need to prescribe ho and 0L. To calculate ho, we use Fig. 3(b) and(c) from [START_REF] Sekulic | Molten aluminum equilibrium membrane formed during controlled atmosphere brazing[END_REF], which indicate the cross-sectional area of the joint to be 7.060x 10-7 m 2 and 6.423x 10-7 m 2 ; using these and the value of wo given above leads to ho = 3.53 x 10-3 m and 3.21 x 10-3 m, respectively. As for 0L, this is a quantity that does not explicitly appear at ail in [START_REF] Sekulic | Molten aluminum equilibrium membrane formed during controlled atmosphere brazing[END_REF], where it is argued that the fact that a good brazed joint indicates a very small inclination angle of the joint surface close to the mating surfaces implies that an assumption of a close-to-zero contact angle between the molten metal and the solid substrate at the onset of solidification is plausible indeed.

Here, we will keep 0L as a parameter, the value of which we will vary, thereby seeing what effect it has on the solution.

Moving to the results, Figs. 5 and 6 compare the experimental results from Fig. 3(b) and (c), respectively, in [START_REF] Sekulic | Molten aluminum equilibrium membrane formed during controlled atmosphere brazing[END_REF] with our mode! results for 0L = 0.05, 0.07 and 0.09, corresponding to 2.86 °, 4.01 ° and 5.16 °, respectively; for these, s = 0.06. From this, it is notable the all three mode! curves agree reasonably well with the experimental data, although the height of the meniscus is somewhat underpredicted towards x = 0 in Fig. 5; this was also a feature of the mode! results in [START_REF] Sekulic | Molten aluminum equilibrium membrane formed during controlled atmosphere brazing[END_REF]. Also of note is the fact that the predicted profile is insensitive to 0 L as its value is increased from 0.05 to 0.09; consequently, even though we may not have a way to determine 0L, it is not much of consequence.

This may lead us to believe that we should simply set 0L = O. However, an interesting theoretical result that can even be obtained analytically in this case is that the contact line can extend to infinity in the zero-dissolution limit, x[mm] s = 0; this is explained in detail in Appendix O. Moreover, we also found that, in our model, it became no longer possible to obtain converged solutions for values of 0 L lower than around 0.044; this was in spite of the fact that we use the best available initial guesses for the solution at each value of 0 L , This behaviour is no doubt due to the fact that s > 0; there was no problem at ail to find solutions in the limitas 0 L ➔ 0 fors = 0, and indeed there is even an analytical solution in this case, as shown by Eq. (D.12) in Appendix O.

Although the above results might suggest that the solution is more or less independent of 0 L, this is not the case, and its value can affect the meniscus profile. To see this, consider Fig. 7 wherein values have been computed as 0 L = 0.05, 0.45, and 0.85, corresponding, respectively, to 5.16 °, 25.78 ° and 48.70 °, fors= 0.06, and the same cross-sectional area as for Fig. 6; hence, as 0 L is increased, the profile becomes almost linear, and hence far-removed from the experimental observations. Lastly, we consider how varying the value of s affects the solution. Figure 8 shows the profiles for f and h fors = 0.44, 0.74 and 1.04, corresponding to temperatures of 614.8 °C, 622.6 °C and 628.l °C, respectively; 0L = 0.85, corresponding to 48.70 °; and a cross-sectional area of 7.060x 10-7 m 2 . These values of s and 0L have been chosen arbitrarily, and are sufficiently different from the values used for Figs. 5 and 6 so as to illustrate that the model works for a wide range of parameters. In this case, we see that, even though we have chosen a high value of 0 L , the profiles for h are qualitatively reasonable, and do not resemble tihe linear profile in Fig. 7. This is of course in part due to the fact that there is now a significant dissolved region, with the /-profile being responsible for a greater share of 0L than the h-profile.

In this paper, we have considered a framework for the mathematical modelling of reactive wetting during brazing; more specifically, a simplified 2D inverted tee-joint geometry consisting of an unclad horizontal plate and a vertical plate with cladding was considered. The key points are as follows: l. Once the clad is molten, the problem bears some similarity to those of the reactive wetting that occurs during the spread of a molten drop let, as bas been considered by others [START_REF] Warren | Modeling reactive wetting[END_REF][START_REF] Su | Modeling dissolution and spreading of Bi-Sn alloy drops on a Bi substrate[END_REF][START_REF] Singler | Modeling and experiments in dissolutive wetting: a review[END_REF]. 2. Although it is a challenging mathematical problem to determine the lime evolution of the molten clad meniscus and the dissolution front that forms, it is much easier to determine the theoretical steady state, and this was done here; the standout plot for this is Fig. 8, which shows the steady-state profiles for the meniscus and dissolved region for three different brazing temperatures, assuming a prescribed value for the angle, 0L, between the meniscus and the dissolution front. This constitutes the maximum possible spatial extent of the dissolved zone. 3. Moreover, there exists a simple approximate relation between the area of the dissolved zone (Adiss ), the initial amount of clad (woho), the equilibrium concentration of Si at the brazing temperature (cE (T)) and the initial composition of the clad, co :

Adiss = woho ( c o T -1) . CE ( ) (7.l)
Possible directions for further modelling work would involve the following:

-computation of the time evolution of the meniscus and the dissolved region towards a steady state. The problem of a T-joint is considerably more complicated than that of the spreading droplet because there are two contact lines to consider and the use of so-called "lubrication theory" that is highlighted in [START_REF] Warren | Modeling reactive wetting[END_REF][START_REF] Su | Modeling dissolution and spreading of Bi-Sn alloy drops on a Bi substrate[END_REF][START_REF] Singler | Modeling and experiments in dissolutive wetting: a review[END_REF] is no longer strictly speaking valid since the aspect ratio of the molten region is not small. Another possibility, although computationally more challenging, is the use of a phase-field approach [l l-13], which involves the use of a particular type of diffuse-interface model that is based on the free energy as a function of state variables; this bas the benefit, at least, that the force singularity which arises in the classical model of moving contact lines, first pointed out by Huh and Scriven [START_REF] Huh | Hydrodynamic model of steady movement of a solid/liquid/fluid contact line[END_REF], is no longer present due to mass transfer across the interface [START_REF] Seppecher | Moving contact lines in the Cahn-Hilliard theory[END_REF]. In this context, we can note that the model we have presented here would indeed lead to such a singularity, although it could be alleviated by introducing a slip coefficient into the no-slip boundary condition, as bas previously been done by many others, e.g. [START_REF] Ehrhard | Nonisothermal spreading of liquid-drops on horizontal plates[END_REF]. Yet another possibility is to use a sharp-interface method via an Arbitrary .. ,. , where Xo, Yo and R are to be determined; then we would need ,,,,.

/ /\• 1/' /, \ i I \ i I \� .. , •� 11 -�� � \ '.},� \. \ -�).
��-•-•••••••• •• •••••••• • / 5 ..... _, _ ---- 10 X [m]
X5+(Y*-Yo) 2 =R 2 , (X*-Xo) 2 +YJ=R 2 . Also, since dY (X -Xo) + (Y -Yo) -= 0,
-5 '------'----'-----'------'----' -5 where

θ = π 2 + tan -1 Y * -Y 0 X 0 tan -1 X * -X 0 Y 0 ; (C.10)
here, (C.9) comes from Eq. (5.31), i.e. the conservation of solute condition. Numerical experimentation to determine f and h indicated that there are not only great difficulties in finding a numerical solution, but also difficulties in finding a good initial guess; it is necessary to find this because the governing equations are nonlinear. Moreover, it is not even known if a solution exists. Thus, it turns out to be more worthwhile to consider what would be the properties of such a solution, if it exists. We know that we must have

H X (0) = X 0 + (Y * -Y 0 ) tan θ L Y * -Y 0 -X 0 tan θ L , (C.11) H X (X * ) = X * -X 0 -Y 0 tan θ L Y 0 + (X * -X 0 ) tan θ L , (C.12)
which we can rewrite as ' ' • 0 0 ',, 0.5 is ever zero or not. Here, the idea is that we solve (5.10), subject to (5.12), (5. where a and f3 are the angles that the curve Y = H (X) makes with the Y-and X-axes, respectively; note that this computation is independent of 0L, and that we merely sweep over ail values of a and /3, where O < a, f3 < n/2.

X 0 + (Y * -Y 0 ) tan θ L = (Y * -Y 0 -X 0 tan θ L ) H X (0) , (C.13) X * -X 0 -Y 0 tan θ L = (Y 0 + (X * -X 0 ) tan θ L )
Each computation generates values for X ,.. and Y ,.. , which are then used in (C.25). If it is found that </> is never zero, then this is an indication that X o and Yo cannot be found, and hence the sought-after circle does not exist. Numerical experimentation suggests that this indeed is the case. As an example, Fig. 11 which leads to

H ∼ 1 2 C 1 (X * -X ) 2 .
(D.10) However, note that if C 1 < 0, which is clearly possible from (D.7) if A > 1, then H < 0, which would be physically unrealistic. On the other hand, if C 1 = 0, i.e. A = 1, we have, instead of (D.9), H 2 X ∼ H 2 , (D.11)

which would give H ∼ e ±X . It is clear that only the behaviour with the minus sign is of relevance here. Taking this one, we see that an alternative to (D.10) is the possibility that the contact line extends to infinity with H ∼ e -X as X → ∞. In fact, for this case, there is an analytical solution for the meniscus given by [30-32]

X = √ 2 -ln 1 + √ 2 -4 -H 2 + ln 2 + √ 4 -H 2 H . (D.12)
Finally, it is worth noting that, from the data in Table 1 and the values for h 0 and w 0 used in this paper, A ≈ 0.02, indicating that the relevant behaviour is that given by Eq. (D.10).

Fig. 1 Fig. 2 a

 12 Fig. 1 Initial configuration for an inverted tee joint

0 H

 0 11) is critical to what follows, and is therefore not neglected. Equations (5.10) and (5.11) are subject to (4.7), (4.8) and (4.21)-(4.25), which are now, respectively, X * (X ) dX = A,

32 )

 32 to Eqs. (5.25)-(5.31) and proceed by considering the case ε ∼ O (1).

Fig. 4

 4 Fig. 4 Schematic for determining the extent of the dissolved region by finding the centres and radii of two circles

2 5

 25 but with X+, y_ and R still unknown. A further relation cornes from(5.31), which gives (

Fig. 5 Fig. 7 6 •

 576 Fig.5 Comparison of experimental data from Fig. 3(b) in [5] with the meniscus profile, h, computed using the current mode) with 0L = 0.05, 0.01, 0.09 and e = 0.06. Note that the curvesfor 0L = 0.05, 0.01, 0.09 are literally on top of each other

Fig. 8 f

 8 Fig. 8 f and h (0L = 0.85, corresponding to 48.70 °, and e = 0.44, 0.74, 1.04), obtained not using the assumption that e « 1

  .......... '\� ''�•!;!•�•-�.--.--.. ,,,� '-•.;:-: -------0 ' • ':._• ::

Fig. 10 f

 10 Fig. 10 f (0L = 0.1, T = 595 °C, 615 °C, 625 °C), using the assumption that e « 1

  Fx (0) = --, Fx (X*)= ---. . ;:.., ...... . ... -:;.::: ........ -. .... . _ ----•--• --,.

= 1 +

 1 H X (X * ) , H X (0) tan θ L , (C.16) b = H X (0)tan θ L , (C.17) c = H X (X * ) tan θ L -1, (C.18) d = -H X (X * )tan θ L , (C.19) e = Y * (H X (0)tan θ L ) , (C.20) f = X * (H X (X * ) tan θ L -1) , 2Y * Y 0 = Y 2 * -X 2 * + 2X * X 0 , (C.23) from (C.8) we have 2Y * (a fce) = Y 2 * -X 2 * (adbc) + 2X * (deb f ) ,

Fig. 11 <

 11 Fig. 11 </> vs. oe for 0L = 0, 1r/4, 1r/3 (/3 = 1r/3)

  [START_REF] Lacaze | Numerical simulation of brazing aluminium alloys with Al-Si alloys[END_REF] andHx (0) = -tan(n/2-a) , Hx (X ,.. )= -tan {3,

5 ) 2 + C 1 H

 521 shows</> vs. a for 0L = 0, n /4, n /2, for f3 = n /3, indicating that </> is never zero, no matter what value of 0L is tried. Similar graphs were also obtained for other values of f3. Appendix D: 8L = 0, s = 0 When s = 0, we have F = 0, and Eqs. (5.23)-(5.31) reduce to just Hxx Hx (X*)= -tan0L, fo x * H (X) dX = A. Multiplying (D.1) by H X , integrating with respect to X and applying (D.3) and (D.4) cos θ L . (D.6) Alternatively, integrating (D.1) with respect to X and applying (D.2), (D.4) and (D.5) gives cos θ Lsin θ L = A + C 1 X * (D.7)Now, consider the behaviour of H near X = X * when θ L = 0. Since H, H X 1, we have, from (D.6),

Table 1

 1 Model parameters

	Parameter	Symbol	Value	Unit	References
	Diffusion coefficient	D	10 -9	m 2 s -1	[9]
	Gravitational acceleration	g	9.81	m s -2	-
	Initial clad height	h 0	1.5×10 -4	m	[ 5]
	Liquidus slope	m	6.80	wt% • C -1	-
	Al melting temperature	T m	660.3	• C	-
	Initial clad width	w 0	0.005	m	[5]
	Solute capillary length	Γ	5×10 -10	m	[ 9]
	Clad viscosity	μ	0.001	Pa s	[21]
	Clad density	ρ	2650	kg m -3	[21]
	Surface tension	σ fs	0.65	N m -1	[22]

Acknowledgements

The first author acknowledges the award of a visiting researcher grant from FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) [Grant Number 2018/07643-8]. Both authors would like to thank the anonymous referees for their constructive comments.

Lagrangian Eulerian (ALE) formulation; interestingly, ALE also appears to have been used by others [START_REF] Singler | Modeling and experiments in dissolutive wetting: a review[END_REF][START_REF] Su | The development of computational models for studying wetting, evaporation and thermal transport for electronics packaging applications[END_REF]. There do exist even simpler models based around the Noyes-Whitney equation [START_REF] Yang | Evolution of the interfacial shape in dissolutive wetting: coupling of wetting and dissolution[END_REF].

-consideration of how the model differs if the brazing temperature T is such that T < T mmc 0 . In this case, the clad is only partly molten, and therefore not all of it is available for flow. Hence, there will be a mushy zone and a Darcy-law term needs to be added to the Navier-Stokes equation. Moreover, it is evident that, although ε > 0 when all of the clad is molten, we will now have ε < 0. When ε > 0, it is clear that it gives a characteristic scale for the extent of the dissolved region; on the other hand, if ε < 0, the interpretation is far less straightforward, since not all of the clad will have melted.

Appendix A: Determining the velocity scale, [u]

In addition to [u] = D/l, another possibility is to take [u] = DΓ /l 2 . In this case, we have

and (4. [START_REF] Saiz | Reactive spreading: adsorption, ridging and compound formation[END_REF]) and (4.20) become

respectively. Also, (4.5) and (4.17) become

respectively. It is now convenient to refer to Table 1, which contains the model parameters. Now, since δ 1, (A.4) and (A.5) reduce to

i.e. this is already the steady state. Also, from (A.1), we would obtain [t] ∼ 10 10 s, and it is clear that we must have taken the wrong choice for [u] .

Appendix B: ε 1

In this case, the F and H problems decouple in an interesting way. From (5.31), and assuming that A ∼ O (1) , we now expect that

Considering the first case, if F = ε F, with the expectation that F is an O (1) function, implies that (5.24) becomes, at leading order in ε,

where C 2 = C2 /ε, with C2 an O (1) constant, whereas (5.28) and (5.29) reduce to

respectively. For the second case, we set X = ε X , so that (5.24) becomes

with (5.26) reducing to

Note that, at this stage, (B.2) and (B.5) are consistent with each other, and the implication is that the radius of curvature of F, which is in effect 1/C 2 , is small. Also, we can note that the problem for H has now decoupled from that for F. More specifically, we have, for H, Eq. (5.23) subject to

Each of these problems is clearly missing a boundary condition, and hence two more conditions would be required. One would hope for continuity of F and its derivative, so that the two problems would "join up", in the sense that F and dF/dX as computed by the two problems should be continuous at some value of X that would have to be determined. However, it is impossible to achieve this, whilst at the same time ensuring that the curvature of F is constant. Thus, the only apparent possibility is that

and hence that

which abandons the requirement that dF/dX is continuous. In this case, dissolution does not start at all from the point (0,0). Note that (B.15) will mean that F is multivalued at X = 0, and indeed for X < 0.

The remaining problem for F is (B.11), subject to (B.12) and (B. [START_REF] Lacaze | Numerical simulation of brazing aluminium alloys with Al-Si alloys[END_REF]), and we obtain

The remaining problem for F is (B.13), subject to (B.14) and (B.15). To solve, it is easier to rewrite the problem as

subject to

which leads to the solution A preliminary result for f and h is given in Fig. 9 using 0L = 0.1 (5.71 °) and T = 615 °C, corresponding to s = 0.445. Note that the sum of the area in y < 0 and x < 0 should be 0.445 that in x, y,> O.

Furthermore, Fig. 10 shows the extent of the dissolution zone for T = 595 °C, 615 °C and 625 °C. Note that the curve for T = 595 °C is literally on top of the xand y-axes. For this s. « 1 analysis, these curves are calculated using the results from the O (1) problem for H; thus, X * and Y* are the same for ail of these curves.

Appendix C: A one-circle dissolution zone fore « 1?

It was seen in Appendix B that if s « 1, then it is not possible that a single circular arc will join the two contact lines. However, it is not yet clear that this is the case ifs. is larger, as will be the case when the difference between co and CE (T) is large, or CE (T) is sufficiently small. Thus, we now consider this situation.

Suppose that such a circular arc exists and that it is part of the circle (X -Xo) 2 + (Y -Y 0 ) 2 = R 2 , (C.l )


Olivier Ferret
email: [ferret@limsi.fr

Brigitte Grau
email: grau]@limsi.fr

A Topic Segmentation of Texts based on Semantic Domains

Thematic analysis is essential for many Natural Language Processing (NLP) applications, such as text summarization or information extraction. It is a two-dimensional process that has both to delimit the thematic segments of a text and to identify the topic of each of them. The system we present possesses these two characteristics. Based on the use of semantic domains, it is able to structure narrative texts into adjacent thematic segments, this segmentation operating at the paragraph level, and to identify the topic they are about. Moreover, semantic domains, that are topic representations made of words, are automatically learned, which allows us to apply our system on a wide range of texts in varied domains.

INTRODUCTION

Thematic analysis covers different kinds of problems. Text segmentation consists of delimiting parts of texts related to different topics. Topic structuring makes the underlying thematic structure of a text explicit while topic identification associates topic representations to parts of texts. Depending on the kind of texts to be analyzed and the problem to be solved, methods require background knowledge or not. Scientific or technical texts may be segmented by examining how words are distributed in the texts [7] [11]. Within this paradigm, Salton [START_REF] Salton | Automatic Text Decomposition Using Text Segments and Text Themes[END_REF] proposed a method to structure a text based on a graph of links between segments. However, when dealing with narratives, as newspaper articles, such methods are inappropriate and require some domain knowledge.

Approaches for analyzing narratives can roughly be categorized in two classes, knowledge-based approaches and word-based approaches. Knowledge-based systems [START_REF] Grosz | Attention, Intentions and the Structure of Discourse[END_REF] lead to a precise decomposition and identification of discourse topics by using in-depth understanding processes and a knowledge base about situations (their characters, their events chronologically ordered, their consequences). As such bases do not exist, apart possibly for small domains, word-based approaches have been developed [START_REF] Hearst | TextTiling: Segmenting Text into Multi-paragraph Subtopic Passages[END_REF] [8] [START_REF] Ferret | How to thematically segment texts by using lexical cohesion?[END_REF] allowing to overcome this limitation and process texts regardless of the topic. The purpose of these systems is to segment texts, but not to recognize topics. Improving such methods requires going towards in-depth understanding and thus using knowledge about topics, but without the former limitation.

The thematic analysis we propose relies on specific semantic domains, automatically learned from texts. These topic representations, described by sets of words, allowed us to develop a segmentation process at the paragraph level. We detail in this paper the implementation of this process and we show its performances on a classical evaluation task for topic segmentation algorithms.

OVERVIEW OF THE SEGAPSITH SYSTEM

SEGAPSITH has two main components: a module that learns semantic domains from texts and a topic segmentation module that relies on these domains. Semantic domains are topic representations made of weighted words. They are built by incrementally aggregating a large number of similar text segments. These segments are delimited by a rough segmentation module called SEGCOHLEX [START_REF] Ferret | How to thematically segment texts by using lexical cohesion?[END_REF]. Figure 1 shows the overall architecture of the system. The segmentation modules operate on pre-processed texts in order to keep only their lemmatized content words. In accordance with works on discourse segmentation as [START_REF] Grosz | Attention, Intentions and the Structure of Discourse[END_REF], the SEGAPSITH one processes texts linearly and detects topic shifts without delaying its decision, i.e., by only taking into account the data extracted from the part of text already analyzed. A window that delimits the current focus of the analysis is moved over each text to be segmented. A topic context is associated to this window. It is made up of the semantic domains that are the most activated by the content of this window. The current segment is also given a topic context. This context results from the fusion of the contexts associated to the focus window when this window was in the segment space. A topic shift is then detected when the context of the focus window and the context of the current segment are not similar any more for several successive positions of the focus window. This process not only locates topic shifts but by identifying topics of segments, also links non adjacent segments related to the same subject.

SEGCOHLEX

SEMANTIC DOMAIN LEARNING

As the focus of this paper is the segmentation component of the SEGAPSITH system, we give here only a brief overview of the learning semantic domain module. This module is described more precisely in [START_REF] Ferret | A Thematic Segmentation Procedure for Extracting Semantic Domains from Texts[END_REF]. It builds topic representations incrementally; these representations are made of weighted words, from discourse segments delimited by SEGCOHLEX [START_REF] Ferret | How to thematically segment texts by using lexical cohesion?[END_REF]. The module works without any a priori classification or hand-coded pieces of knowledge. Processed texts are typically newspaper articles. They are pre-processed to only keep their lemmatized content words (adjectives, single or compound nouns and verbs).

The topic segmentation implemented by SEGCOHLEX is based on a large collocation network, built from 24 months of the French newspaper Le Monde, where a link between two words aims at capturing semantic and pragmatic relations between them. The strength of such a link is evaluated by the mutual information between its two words. The segmentation process relies on these links for computing a cohesion value for each position of a text. It assumes that a discourse segment is a part of text whose words refer to the same topic, that is, words are strongly linked to each other in the collocation network and yield a high cohesion value. On the contrary, low cohesion values indicate topics shifts. After delimiting segments by an automatic analysis of the graph resulting from the cohesion computation, only highly cohesive segments, named Thematic Units (TUs), are kept to learn topic representations. This segmentation method entails a text to be decomposed in small TUs, whose size is equivalent to a paragraph. To enrich the particular description given by a text, we add to TUs those words of the collocation network that are particularly linked to the words found in the corresponding segment. Learning a complete description of a topic consists of merging all successive points of view, i.e. similar TUs, into a single memorized thematic unit, called a semantic domain. Each aggregation of a new TU increases the system's knowledge about one topic by reinforcing recurrent words and adding new ones. Weights on words represent the importance of each word relative to the topic and is computed from the number of occurrences of these words in the TUs. This method leads SEGAPSITH to learn specific topic representations as opposed to [START_REF] Lin | Robust Automated Topic Identification[END_REF] for example whose method builds general topic descriptions as for economy, sport, etc.

We have applied the learning module of SEGAPSITH on one month (May 1994) of AFP newswires, corresponding to 7823 TUs. The learning stage has produced 1024 semantic domains. Table 1 shows an example of a domain about justice that gathers 69 TUs. The segmentation module of SEGAPSITH only works with the most reliable of these domains. Thus, we have selected those whose number of aggregations is superior to 4. Moreover, we have selected in these 193 domains words whose weight is superior to 0.1, since below this limit a lot of words only represent noise.

THE TOPIC SEGMENTATION

The segmentation algorithm of SEGAPSITH locates topic shifts by detecting when a significant difference is found between the set of semantic domains selected for each position of a text and the set of domains associated to the segment that is currently active at this time. The first set of domains defines the window context and the second set the segment context.

Topic contexts

A topic context aims at characterizing the entity it is associated to from the thematic point of view and is represented by a vector of weighted semantic domains. The weight of a domain expresses the importance of this domain with regard to the other domains of the vector. A context contains several domains because domains are rather specific. Thus, having several domains that are close to each other allows the system to cover a larger thematic field. Secondly, SEGAPSITH handles representations made of words, whose meaning may be ambiguous and refer to different topics. By putting several domains into a context, we cope with this ambiguity without having to choose explicitly one interpretation.

Building the topic context of the focus window

The topic context of the focus window is built first, by activating the semantic domains that are available from the words of the focus window and then, by selecting the most activated of these domains. The activation value of a semantic domain is given by:

) ( ) , ( ) ( j j i j i w nbOcc w dom ght w dom activ ⋅ = ∑ (1)
where the first factor is the weight of the word w j in the domain dom i (cf. [START_REF] Ferret | A Thematic Segmentation Procedure for Extracting Semantic Domains from Texts[END_REF] for more details) and the second one is the number of occurrences of w j in the focus window. After this activation step, the context of the focus window is set by selecting the N th first semantic domains according to their activation value. Their weight in this context is equal to their activation value. N is the fixed size of all the contexts.

Building the topic context of a segment

The topic context of a segment contains the semantic domains that were the most activated when the focus window was moving in the segment space. This is achieved by combining the contexts associated to each position of the focus window inside the segment (see Figure 2). This fusion is done incrementally: the context of each new position of a segment is combined with the current context of the segment. First, the semantic domains of both contexts are joined. Then, their weight is revalued according to this formula:

) , , ( ) ( )
⋅ + ⋅ = + β α (4)
with Cw, the context of the window, Cs, the context of the segment and wght(dom i ,Cx,t), the weight of the domain dom i in the context Cx for the position t. The results we present in the next sections are obtained with α(t)=1 and β(t)=1. These functions are a solution halfway between a fast and a slow evolution of the context of segments. The context of a segment has to be stable because if it follows too narrowly the thematic evolution given by the context of the window, topic shifts cannot be detected. However, it must also adapt itself to small variations in the way a topic is expressed when progressing in the text in order not to detect false topic shifts. After weight revaluation, the joined domains are sorted in decreasing order of weight and finally, the N th first of them are selected for building the new version of the segment context.

Evaluating the similarity of two contexts

In order to determine if the content of the focus window is thematically coherent or not with the segment that is currently active, the topic context of the window is compared to the topic context of the segment. This comparison is achieved by a similarity measure taking into account the following four factors:

1. The significance of the domains shared by the two contexts with regard to the window domains in terms of weight; 2. The significance of the domains shared by the two contexts with regard to the segment domains in terms of weight; 3. The significance of the number of domains shared by the two contexts with regard to the size of contexts. This ensures that a high similarity is not found with only a few domains in common having a very high weight; 4. The difference of order among the domains shared by the two contexts. This difference is given by:

p N Cs dom rank Cw dom rank Cs Cw rankDiff p c c c ⋅ - - = ∑ = ) 1 ( ) , ( ) , ( ) , ( 1 (5)
with p, the number of common domains, dom c , one of these common domains and rank(dom c ,Cx), the rank of this domain in the context Cx. In this factor, the sum of the rank differences of domains in the two contexts is normalized by an upper bound assuming that the difference of rank is maximal (N-1) for each common domain.

These factors are combined in a geometric mean [START_REF] Grosz | Attention, Intentions and the Structure of Discourse[END_REF]. The first two factors are gathered in the first term of the global product. The term p/N corresponds to the third factor and the last term is the complement of the fourth factor, as two contexts are more similar if they share domains in the same order. The values of this similarity measure are in the interval [0,1] since the values of each of its four components are also in the same interval. Two contexts are considered as similar if the value of the similarity measure is above a fixed threshold. In all the experiments we present here, this threshold was set to 0.5.

Topic shift detection

The basic algorithm that detects topic shifts is the following: for each position of a text, if the value of the similarity measure between the topic context of the focus window and the topic context of the current segment is lower than a fixed threshold, a topic shift is assumed and a new segment is opened. Otherwise, the active segment is extended up to the current position.

This basic algorithm assumes that the transition phase between two segments is punctual. It actually must be more complex because of the lack of precision of SEGAPSITH. This imprecision makes it necessary to set a short delay before deciding that the active segment really ends and similarly, before deciding that a new segment with a stable topic begins. Hence, the algorithm for detecting topic shifts distinguishes four states:

1. The NewTopicDetection state. This state takes place when a new segment is going to be opened. This opening will be confirmed provided that the content of the focus window context stays mainly the same for several positions. Moreover, the core of the segment context is defined when the topic segmenter is in the NewTopicDetection state; 2. The InTopic state, which is active when the focus window is inside a segment with a stable topic; 3. The EndTopicDetection state. This state is active when the focus window is inside a segment but a difference between the context of the focus window and the context of the current segment suggests that this segment could end soon. As for 2, this difference has to be confirmed for several positions before a change of state is decided; 4. The OutOfTopic state. This state occurs between two segments. Most of the time, the segmentation algorithm stays in this state no longer than 1 or 2 positions but when the semantic domains that should be related to the current topic of the text are not available, this number of positions may be equal to the size of a segment.

The segmentation algorithm follows the transitions of the automaton of the Figure 3 according to three parameters:

1. the current state of the algorithm; 2. the similarity between the context of the focus window and the context of the current segment: Sim or non Sim; 3. the number of successive positions of the focus window for which the current state stays the same: confirmNb, which must be above the T confirm threshold for going away from the states NewTopicDetection and EndTopicDetection. The processing of a segment starts with the OutOfTopic state, after the end of the previous segment or at the beginning of the text. As soon as the set of semantic domains of the focus window is stable enough between two successive positions, the topic segmenter enters into the NewTopicDetection state. The InTopic state can then be reached only if the same stability of the window context is found for the next confirmNb-1 positions. Otherwise, the segmenter assumes that it is a false alarm and returns to the OutOf-Topic state. The detection of the end of a segment is symmetrical to the detection of its beginning. The segmenter goes into the End-TopicDetection state as soon as the content of the window context begins to change significantly between two successive positions but the transition towards the OutOfTopic state is done only if this change is confirmed for the next confirmNb-1 next positions.

This algorithm is completed by two specific mechanisms. First several segments of a text may refer to the same topic, which is necessary to detect for making the structure of a text explicit. Hence, when the topic segmenter goes from the NewTopicDetection state to the InTopic state, it first checks whether the current context of the new segment is similar, according to [START_REF] Grosz | Attention, Intentions and the Structure of Discourse[END_REF], to one of the contexts of the previous segments. If such a similarity is found, the new segment is linked to the corresponding segment and it takes the context of this one as its own context. It assumes that the new segment continues to develop a previous topic.

The second mechanism is related to the OutOfTopic state. When the topic segmenter stays too long in this state (this time is defined as 10 positions of the focus window in our experiments), it assumes that the topic of the current part of text is not represented among the available domains and it creates a new segment with an unknown topic that covers all the concerned positions. Of course, this mechanism cannot separate several connected segments of this kind but it allows the system to segment texts without having all the topic representations that should be necessary.

EXPERIMENTS

Qualitative results and discussion

A first qualitative test of the segmentation method was done with a small set of texts and without a formal protocol as in [START_REF] Passonneau | Discourse Segmentation by Human and Automated Means[END_REF]. We have tested several range of values for the different parameters of the method and have found that for the kind of texts as the one given in Figure 4, the best results are obtained with a size of 19 words for the focus window and a value of 3 positions for the confirmNb parameter. Furthermore, results are rather stable around these values. Figure 4 shows the value of the similarity measure between the context of the focus window and the context of the current segment for each position of the given text. The two topic shifts, from the Miss Universe topic to the terrorism topic and then the return to the Miss Universe topic, are clearly detected through significant falls of the similarity values (positions 62-63 for the first et 89 to 91 for the second; these shifts are marked in bold in the text). On the other hand, the method misses the last topic shift (from the Miss Universe topic to the demonstration topic) because it is expressed very shortly and not in a very specific way.

<ST> An 18 year old Indian model, Sushmita Sen, caused a surprise on Sunday in Manilla when winning the Miss Universe 1994 title ahead of two South-American beauties, Miss Colombia, Carolina Gomez Correa, and above all Miss Venezuela, Minorka Mercado, who appeared as favorite in the competition. The young Indian, a brown beauty, hazel eyed and 1.75 meters tall, is the first candidate of her country to win this title. She succeeds to Miss Porto Rico, Dayanara Torres, 22, who gave her her crown in front of a television hearing estimated to six hundred million people all over the world. Among the six finalists also appeared Miss United States, Frances Louis Parker, Miss Philippines, Charlene Gonzales, and Miss Slovak Republic, Silvia Lakatosova. The new miss was chosen among a group of ten finalists that also included the representatives of Italy, Greece, Sweden and Switzerland. </ST> <ST> A few hours before the ceremony, a man was killed by the explosion of an appliance he carried, at about one kilometer from the Congress Center where the beauty competition was being held, in front of the Manilla bay. The police was not immediately able to establish if this incident was in relation with this competition. On Thursday, a weak-power craft bomb had exploded in a garbage can of the congress center without any damages. </ST> <ST> The new Miss Universe, who won more than 150,000 Dollars in different prizes, declared that she intended to do theater, publicity or writing. However, her most cherished wish, she assured, was to meet Mother Teresa because she was "a perfect example of a person totally devoted, unselfish and completely involved". </ST> <ST> During the election, about a hundred feminists demonstrated pacifically in front of the Congress Center to denounce the competition, stating that it promoted sexual tourism in Philippines. </ST> AFP newswire, translated from French (may 1994) -The <ST> tags delimit the segments resulting from a human judgment The analysis of this example also illustrates two important characteristics of our method. As it makes use of an explicit representation of topics, it allows us to recognize that two disconnected segments are related to the same topic, as it is done here for the segments 1 and 3 about the Miss Universe topic.

Our method also segments texts without having an exact representation of the topics of the texts. Thus, the newswire above was segmented without having a semantic domain related to beauty competitions. This topic was represented here by only one of its dimensions, competition, through a set of domains about sport competitions. More generally, as a context is a set of domains, a topic representation can be dynamically built by associating several domains related to different dimensions of this topic.

Topic shift threshold

Quantitative evaluation

In order to have a more objective evaluation, we applied our method to the "classical" task of discovering boundaries between concatenated texts. As we are interested in segmenting texts at the paragraph level, our evaluation was performed with short texts, precisely 49 texts from Le Monde newspaper of 133 words long on average. Results are shown in Table 2. As in [START_REF] Hearst | TextTiling: Segmenting Text into Multi-paragraph Subtopic Passages[END_REF], boundaries found by the method are sorted in decreasing order of their probability to be document breaks (we rely for this on the similarity between contexts). For the first N b boundaries, N t is the number of boundaries that match with document breaks. Precision is given by N t / N b and recall, by N t / D, with D the number of document breaks. The f-measure is the harmonic mean of precision and recall. The match between a boundary and a document break was accepted if the boundary was not further than 9 words (after pre-processing). Globally, our results are comparable to the other studies in the field although a significant comparison is not easy to do. In [START_REF] Hearst | TextTiling: Segmenting Text into Multi-paragraph Subtopic Passages[END_REF], Hearst reports a similar evaluation but with much larger texts (average length of 16 paragraphs). For 44 texts, she gets 0.95 as precision, 0.59 as recall and 0.728 as f-measure (comparison must be done with the last line of Table 2). However, a direct comparison is not completely relevant as Hearst's method cannot be applied to texts as smaller as ours. The work in [START_REF] Bigi | Detecting topic shifts using a cache memory[END_REF] is more similar to ours. Although its evaluation method is slightly different, its results can be compared to ours: in the best case, 0.75 as precision, 0.80 as recall and 0.774 as f-measure. The differences can be explained by the nature of topics: [START_REF] Bigi | Detecting topic shifts using a cache memory[END_REF] focuses on a small set of very general topics (such as business, politics) while we focus on a large set of specific topics. As a consequence, our recall is better -our topic representations are closer to the topics of the texts -while [START_REF] Bigi | Detecting topic shifts using a cache memory[END_REF] shows a better precision -we are likely to have more noise. However, it is important to note that our precision does not decrease as more document bounds are found.

RELATED WORKS

Our method is a synthesis between methods that rely on lexical cohesion for segmenting texts and use word recurrence [START_REF] Hearst | TextTiling: Segmenting Text into Multi-paragraph Subtopic Passages[END_REF] or a lexical network [START_REF] Kozima | Text Segmentation Based on Similarity between Words[END_REF], and methods that rely on explicit topic representations, as [START_REF] Bigi | Detecting topic shifts using a cache memory[END_REF] or studies done in the Topic Detection and Tracking (TDT) framework [START_REF] Fiscus | NIST's 1998 Topic Detection and Tracking Evaluation (TDT2)[END_REF]. As these last methods, ours makes use of explicit topic representations but it exploits them with the same tools as [START_REF] Hearst | TextTiling: Segmenting Text into Multi-paragraph Subtopic Passages[END_REF] or [START_REF] Kozima | Text Segmentation Based on Similarity between Words[END_REF] and not with the probabilistic approach generally found in TDT or in [START_REF] Beeferman | Statistical Models for Text Segmentation[END_REF]. In [START_REF] Bigi | Detecting topic shifts using a cache memory[END_REF], topics are very general. On the contrary, they are very specific in [START_REF] Fiscus | NIST's 1998 Topic Detection and Tracking Evaluation (TDT2)[END_REF] and often comparable to events. Domains in SEGAPSITH are halfway between these two extremes: they aim at describing specific topics but not events.

Studies done in the TDT framework also differ from ours in the delay for deciding if a topic shift occurs. They take a decision after a deferral period going from 100 up to 10000 words while this parameter is equal to only 3 content words in our method.

Having topic representations clearly allows us to work at a finer grain than methods based on lexical cohesion. But on a large scale, it also requires to automatically build these representations, preferably in an unsupervised way. This problem is tackled to some extent in the Detection task of the TDT evaluation but not in the segmentation one. On the contrary, SEGAPSITH includes a module that learns in an unsupervised way the topic representations that support its segmentation module. Moreover, its association with SEGCOHLEX allows our global system to progressively go from a segmentation module based on lexical cohesion to a segmentation module based on topic representations.

CONCLUSION

The segmentation module of SEGAPSITH implements a topic segmentation of texts at the paragraph level. Using semantic knowledge about domains provides a mean to identify these topics. SEGAPSITH was also designed to recognize links between non adjacent segments. Furthermore, it is able to manage lack of knowledge about some domains. The evaluation of this system gives good results on narrative texts. However, we envisage to structure domains hierarchically and to take into account this hierarchy for going further in the thematic structuring of texts.

Figure 1 .

 1 Figure 1. The architecture of the SEGAPSITH system

Figure 2 .

 2 Figure 2. Building of the context segment

Figure 3 .

 3 Figure 3. The automaton for topic shift detection

Figure 4 .

 4 Figure 4. A text and its context similarity graph (for the French text)

Table 1 .

 1 The most representative words of a domain about justice

	words	occ.	weight
	examining judge	58	0.501
	police custody	50	0.442
	public property	46	0.428
	indictment	49	0.421
	to imprison	45	0.417
	court of criminal appeal	47	0.412
	receiving stolen goods	42	0.397
	to presume	45	0.382
	criminal investigation department	42	0.381
	fraud	42	0.381

Table 2 .

 2 Results of the evaluation

	N b	N t	recall	precision	f-measure
	10	7	0.146	0.636	0.237
	20	11	0.229	0.524	0.319
	30	18	0.375	0.581	0.456
	40	24	0.5	0.585	0.539
	50	28	0.583	0.549	0.566
	60	33	0.688	0.541	0.606
	70	38	0.792	0.535	0.639
	77	43	0.896	0.551	0.683