Michel Bellet

Finite Element Analysis of Compressible Viscoplasticity Using a Three

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Finite element analysis of compressible viscoplasticity using a three-field formulation: Application to metal powder hot compaction

Michel Bellet

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Michel Bellet. Finite element analysis of compressible viscoplasticity using a three-field formulation: Application to metal powder hot compaction. Computer Methods in Applied Mechanics and Engineering, 1999, 175 (1-2), pp.Pages 19-40. [START_REF] Coupez | A mesh improvement method for 3D automatic remeshing[END_REF].1016/S0045-7825(98)00317-X. hal-00574862

INTRODUCTION

Compaction of powdered materials has experienced great development in many industries for production of various parts. The reasons for manufacturing a product from powder are either economic or technical. They can be divided into three classes: ability to elaborate materials which are difficult or even impossible to melt; achievement of particular structures or properties (porosity, fine grains, isotropy, purity); simplification of manufacturing routes, providing near net shape components and raw material savings.

Densification of metal powders is obtained by compression and/or sintering. A typical process is outlined as follows:

-Mixing of powder with some binder-lubricant.

-Cold compaction step. This can be achieved either by punch compaction or by cold isostatic pressing, typically up to 75 to 85% of full density. -Dewaxing (binder removal) and optional sintering under specific conditions, to increase particle bonding. -Hot compaction step: hot forging or hot isostatic pressing, in order to achieve full density.

Mathematical models of such a manufacturing process can be useful since they can provide the production engineers with indication of possible underfill or final porosity, evolution of density distribution and material flow throughout the process, tooling load estimates, press size requirements, etc. They can also allow accurate and inexpensive parametric studies of process variables.

The present work is essentially focused on the application of the finite element method (FEM) to the analysis of hot forging of cold pre-compacted preforms. Hence we adress here large viscoplastic deformations through generalized Newtonian fluid flow models, including compressibility effects and coupling with heat transfer. Several authors have already contributed to this research field. Among them, let us quote Im and Kobayashi [START_REF] Im | Analysis of axisymmetric forging of porous materials by the finite element method[END_REF] who developed a FEM formulation for powdered metal forging an implemented it into the bulk forming computation code DEFORM ® , for treating two-dimensional axisymmetric and plane strain problems. The material is considered as a compressible rigid-viscoplastic continuous medium. The authors have used a penalty-like formulation. Such an approach penalizes the terms connected with volumetric strain in both the porous and the fully dense regions of the preform. The only unknown of the discretized problem is the nodal velocity field. A similar approach has been developed by Barata Marques and Martins [START_REF] Barata Marques | Finite element simulation of powder metal forming[END_REF] who have clearly shown that some terms of the finite element tangent stiffness matrix tends to become infinite as the material approaches the dense state. A cut-off value of the relative density is then used in order to limit the value of those terms and transform them into classical penalty terms enforcing the incompressibility constraint, as in flow formulation. In order to avoid locking, they have used a reduced Gauss integration. However, as pointed out by Jinka and Lewis [START_REF] Jinka | Finite element simulation of hot isostatic pressing of metal powders[END_REF], such a formulation yields a wrong estimation of pressure, and consequently stresses in the compacted material. These authors preferred using a mixed velocity-pressure formulation and applied it successfully to the two-dimensional analysis of hot isostatic pressing.

All these works were limited to two-dimensional analysis of compaction processes. The first three-dimensional FEM simulation of powder forming has been developed by Chenot et al. [START_REF] Chenot | Finite element simulation of metal powder forming[END_REF], but using a similar penalty-like flow formulation as [START_REF] Im | Analysis of axisymmetric forging of porous materials by the finite element method[END_REF][START_REF] Barata Marques | Finite element simulation of powder metal forming[END_REF] and 8-node linear hexahedral elements. Following this preliminary attempt to model metal compaction in three-dimensions, the objective of the present study is to set out a new three-dimensional mixed formulation, which would be easily implemented in FORGE3 ® , a three-dimensional finite element code initially developed for non steadystate large transformations of pure viscoplastic material [START_REF] Chenot | 3-D finite element simulation of the forging process: scientific developments and industrial applications[END_REF]. In FORGE3 ® , the non linear equilibrium equations are solved for the primitive variables velocity and pressure using tetrahedral elements of P1+/P1 type. This permits the use of an automatic remeshing procedure [START_REF] Coupez | A mesh improvement method for 3D automatic remeshing[END_REF] and iterative solvers such as the preconditioned general minimum residual method. Such solvers give rise to efficient parallelization [START_REF] Coupez | From a direct solver to a parallel iterative solver in 3-D forming simulation[END_REF].

Recently, elastic-viscoplastic constitutive equations have been implemented in the code [START_REF] Aliaga | 3D numerical simulation of thermo-elasto-visco-plastic behaviour using stabilized mixed F.E. formulation: application to heat treatment[END_REF]. As part of these developments, it is required that the aimed compressible formulation for hot powder compaction should be cast in the same type of element in order to be consistent with the general software environment.

Hence, in the present paper, we shall first recall the governing equations of compressible viscoplastic continuous media. The finite element formulation of the mechanical problem is discussed, regarding particularly the choice of primitive variables. We justify the choice of a three-field formulation and give details about its finite element discretization using tetrahedral elements. The heat transfer problem and its coupling with the mechanical one are also presented. This new formulation is implemented in the computation code FORGE3 ® and is validated by comparison with analytical solutions derived in the case of uniaxial free compression test. Finally, an example of application to the hot forging of a connecting rod preform is presented.

GOVERNING EQUATIONS

Material Constitutive Equations

The powdered metal is considered as an average continuous medium. It is characterised by its local relative density r ρ , defined as the ratio of the apparent specific mass by the specific mass of the dense metal. Assuming material isotropy, the theoretical concepts of plasticity can be extended to such an average medium, as proposed by Green [START_REF] Green | A plasticity theory for porous solids[END_REF], by defining the equivalent stress as:

2 / 1 2 ) (Tr : 2 3       + = σ σ s s f c (1)
where σ σ σ σ is the Cauchy stress tensor, s is the deviatoric stress tensor and, using the Einstein summation convention,

ij ij s s = s s : ii σ = σ Tr ( 2
)
Coefficients c and f are functions of the relative density r ρ . They verify c(1) = 1 and f(1) = 0 (fig. 1), so that the classical definition of the von Mises equivalent stress is obtained for the dense material.

Assuming an associated plasticity flow rule, it can be shown [START_REF] Green | A plasticity theory for porous solids[END_REF] that the corresponding expression of the equivalent plastic strain rate is:

2 / 1 2 ) (Tr 9 2 9 1 : 3 2 1                 - + ρ = ε pl pl pl r c f c ε ε ε ) (Tr 9 1 : 3 2 1         + ρ = pl pl pl r f c ε e e & & & (3)
where pl ε & is the plastic strain rate tensor and pl e & its deviatoric part. In addition, the volumic power of plastic deformation is given by:

ε σ ρ = ε σ = & & & r pl ij ij pl ε σ : (4)
The assumption of associated flow rule may be criticized in the field of cold metal compaction, due to the important contribution of grain rearrangement in the global plastic deformation of the medium. However, during hot powder forging, the powdered metal has already been cold pre-compacted, so that it exhibits a rather large initial relative density (typically 0.75 to 0.85), preventing from significant grain rearrangement. In this context, the normality rule holds and the first extension of the above compressible plasticity concepts to compressible viscoplasticity has been proposed by Abouaf et al. [START_REF] Abouaf | Finite element simulation of hot isostatic pressing of metal powders[END_REF]. In this case, a relationship between the equivalent stress and the equivalent strain rate is substituted for the notion of instantaneous plasticity criterion. If the dense metal obeys a viscoplastic flow rule of the type:

) , ( T g σ = ε & (5)
where T is the temperature and the equivalent stress and strain rate are defined in the von Mises sense, then the average continuous medium representing the porous metal obeys the extended form:

) , ( T g r σ = ε ρ & (6)
with σ and ε & defined according to (1) and ( 3). Here we shall assume that the dense material obeys the purely viscoplastic Norton-Hoff law:

m K / 1 3 3 1         σ = ε & ( 7
)
where K is the metal consistency and m the strain rate sensitivity coefficient, both being temperature dependent. Hence, the one-dimensional constitutive equation of the porous metal can be written, according to (6):

m r K / 1 3 3 1         σ = ε ρ & or ( ) m r K ε ρ = σ & 3 3 (8)
In order to derive the three-dimensional constitutive equation, the viscoplastic potential ϕ is introduced:

( ) 1 3 1 + ε ρ + = ϕ m r m K & (9)
Then, the stress tensor is obtained as follows: [START_REF] Coupez | A mesh improvement method for 3D automatic remeshing[END_REF] where I denotes the identity tensor. Strain hardening and temperature softening effects can be taken into account, by use of the following law: [START_REF] Coupez | From a direct solver to a parallel iterative solver in 3-D forming simulation[END_REF] where β is the temperature softening coefficient, 0 ε and n are the strain hardening coefficients, 0 K is a material parameter and ε is the viscoplastic equivalent strain defined by:

ε ε σ & & & & ∂ ε ∂ ε ∂ ϕ ∂ = ∂ ϕ ∂ = ( )                 - + ε ρ = - I ε ε ) (Tr 9 2 9 1 3 2 3 3 1 & & & c f c K m r ( )         + ε ρ = - I ε e ) (Tr 9 1 3 2 3 3 1 & & & f c K m r
      β ε + ε = T K K n exp ) ( 0 0
∫ ε = ε t du u 0 ) ( & (12)

Friction Model

Regarding hot forming of dense metals, sliding contact along tooling surfaces is often modelled using a viscoplastic friction law. The tangential shear stress vector τ is then related to the relative velocity g v by the following power law:

g p g f K v v n n σn σn τ 1 ) . ( - α - = - = (13)
where α and f p are friction coefficients, n is the normal vector at the tool/workpiece interface, and g v is the relative velocity between the metal and the die, whose velocity is denoted die v :

n n v v v v v ) ). (( die die g - - - = (14)
In the case of hot forming of porous metals, this suggests using the same friction model, but with friction coefficients α and f p possibly dependent on the relative density of the material. Hence, the friction shear stress vector τ also derives from the following potential:

1 1 + + α = ϕ f p g f f p K v and v τ ∂ ϕ ∂ - = f (15)

Heat Transfer

Since the forming is performed under hot environment, the temperature evolution has to be modelled during the process. However, the thermal conductivity of porous media is known to decrease with porosity. In the literature, different models have been proposed: see the review of Cheng and Vachon [START_REF] Cheng | A technique for predicting the thermal conductivity of suspensions, emulsions and porous materials[END_REF]. These authors found, by comparing theoretical predictive models with experimental measurements, that convection and radiation which occur in the pore spaces can be neglected for small pore size and low or intermediate temperature. Despite the rather high temperature encountered in powder hot forging, the same assumption will be done in the present work, as a first approach. The thermal conductivity is then given by:

dense r k k ρ = ( 16
)
where dense k is the thermal conductivity of the dense metal.

The energy balance equation due to heat transfer includes Fourier standard conduction law:

ε σ & : ) .( r T k dt dT c p + ∇ ∇ = ρ ( 17
)
where ρ is the specific mass of the porous metal (

r dense ρ ρ = ρ
), p c is the specific heat of the metal (independent on r ρ ), dT/dt is the material or total derivative of temperature with respect to time, k is the thermal conductivity (equ. 16), r is the fraction of the deformation energy which is really transformed into heat. In the sequel, r will be assumed equal to 1.

The boundary Ω ∂ of the processed material is basically subjected to two different types of thermal boundary conditions, depending whether contact with die is established or not. The boundary Ω ∂ will then be split into a partition

s c Ω ∂ ∪ Ω ∂ = Ω ∂
, where

c Ω ∂
is the part of the boundary contacting dies whereas

s Ω ∂
is the free surface. The following boundary conditions apply on these regions.

• Convection and radiation on

s Ω ∂ : ) ( ) ( . 4 4 ext r r ext cv T T T T h T k - σ ε + - = ∇ - n (18)
where ext T is the external temperature, cv h the convection coefficient, r ε the material emissivity, r σ the Stefan-Boltzmann constant and n the outward normal vector. When convection and radiation occur simultaneously equation ( 18) can be expressed:

) ( . ext cr T T h T k - = ∇ - n (19)
) )( ( with

2 2 ext ext r r cv cr T T T T h h + + σ ε + = (20)
In the context of non steady state time incremental computations, this permits a linearisation of the radiation law by using the value of temperature which comes from the previous time increment in [START_REF] Im | Analysis of axisymmetric forging of porous materials by the finite element method[END_REF].

• Conduction and additional friction flux on c Ω ∂ :

T T h T k φ - - = ∇ - ) ( .n (21)
where cd h is the heat transfer coefficient between die and workpiece, die T the die surface temperature, and f φ the inward friction flux corresponding to the splitting of the friction surfacic heat source g v τ. , between die and workpiece. According to a quadratic approximation of the temperature profile in the normal direction [START_REF] Soyris | Modélisation tridimensionnelle du couplage thermique en forgeage à chaud (Three-dimensional modelling of thermomechanical coupling in hot forging)[END_REF], the expression of f φ is:

1 + α + = φ f p g die f K b b b v (22)
in which b is the effusivity (

p c k b ρ =
) and die b the effusivity of the die material.

Mass conservation

The evolution of the relative density is governed by the mass conservation equation:

0 Tr = ρ + ρ ε & r r dt d ( 23
)
where dt d r / ρ is the material or total derivative in time of the relative density.

FINITE ELEMENT RESOLUTION OF THE MECHANICAL PROBLEM

Problem Statement

At time t, the mechanical equations can be summarised as follows, neglecting gravity and inertia forces:

0 . = ∇ σ on Ω ( )         + ε ρ = - I ε e σ ) (Tr 9 1 3 2 3 3 1 & & & f c K m r on Ω d T σn = on s Ω ∂ (24)
g p g f K v v τ 1 - α - = on c Ω ∂ 0 ). ( = - n v v die on c Ω ∂
It should be noted that, in practice, the module of prescribed stress vector d T on the "free" surface is equal to the atmospheric pressure and can be neglected with respect to internal stresses. However, it will be kept in the equations for generality.

Let us define the following spaces:

{ } 3 2 2 1 )) ( ( ) ( ) ( Ω ∈ ∇ Ω ∈ = Ω L q L q H (25) { } c die H Ω ∂ = - Ω ∈ = ϑ on 0 ). ( )) ( ( 3 1 n v v v (26) { } c H Ω ∂ = Ω ∈ = ϑ on 0 . )) ( ( 3 1 0 n v v (27)
ϑ is the space of "kinematically admissible" velocity fields and 0 ϑ is the space of "zero kinematically admissible" velocity fields. The virtual power principle states that the solution velocity field ϑ ∈ v should fulfill the following condition:

0 * . * . * : * 0 = - - ϑ ∈ ∀ ∫ ∫ ∫ Ω ∂ Ω ∂ Ω s c dS dS dV d v T v τ ε σ v & ( 28
)
This is equivalent to say that the solution velocity field, v, should minimise the following functional Φ on the space of kinematically admissible velocity fields ϑ :

∫ ∫ ∫ Ω ∂ Ω ∂ Ω - ϕ + ϕ = Φ s c dS dS dV d f v T v . ) ( (29)
A more detailed expression of the functional Φ is:

  ⌡ ⌠                 - + + = Φ Ω + + dV c f c m K m m 2 / ) 1 ( 2 1 ) (Tr 9 2 9 1 : 3 2 1 ) 3 ( ) ( ε ε ε v & & & ∫ Ω ∂ Ω ∂ + -  ⌡ ⌠ + α + s c f dS dS p K d p g f v T v . 1 1 (30)

Choice of a Formulation

A flow formulation with the velocity field as unknown can be straightforwardly derived from this functional [START_REF] Chenot | Finite element simulation of metal powder forming[END_REF]. However, as already said, its main drawback is that it yields erroneous pressure and stresses values and can exhibit spurious velocity fields or become inaccurate when some regions of the domain Ω reach a fully dense state. Since in hot powder forging, the part is supposed to be fully densified at the end of the process, a mixed formulation passing the Brezzi-Babuska compatibility condition [START_REF] Brezzi | On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers[END_REF][START_REF] Oden | Finite Element: Mathematical Aspects[END_REF] has to be developed. Two approaches of a mixed two-field formulation are possible: the first one is the velocity-pressure (

H p , v
) formulation and the second one is the velocity-volumetric strain rate ( θ & , v ) formulation.

• Velocity-pressure formulation

The velocity-pressure formulation consists in solving the equilibrium equation (28) under the constraint of the volumetric equation, issued from (10):

( ) ε σ & & Tr 3 3 Tr 3 1 1 - ε ρ - = - = m r H f K p (31)
Consequently, the mixed problem can be written:

), ( ) , ( Find 2 Ω × ϑ ∈ L p H v ), ( *) *, ( such that 2 0 Ω × ϑ ∈ ∀ L p v ( ) ∫      = ε ρ + = - - - Ω - Ω ∂ Ω ∂ Ω Ω ∫ ∫ ∫ ∫ 0 Tr ) 3 ( 3 * 0 * . * . * Tr * : ) ( 1 dV K fp p dS dS dV p dV m r H d H s c ε v T v τ ε ε v s
where *

ε & is the virtual strain rate tensor associated with any virtual velocity field * v . Because of the non-linearity of the constraint equation (32b), it can be seen that the derived discrete velocity-pressure formulation would result in a non symmetric tangent stiffness matrix. This is the case in the similar mixed velocity-pressure approach developed by Jinka and Lewis [START_REF] Jinka | Finite element simulation of hot isostatic pressing of metal powders[END_REF].

• Velocity-volumetric strain rate formulation

The additional constraint to the equilibrium equation ( 28) is now given by

ε & & Tr = θ
, and the corresponding mixed formulation is the following.

), ( ) , ( Find 2 Ω × ϑ ∈ θ L & v ), ( *) *, ( such that 2 0 Ω × ϑ ∈ θ ∀ L & v      = θ - θ = - - θ ∫ ∫ ∫ ∫ Ω Ω ∂ Ω ∂ Ω 0 ) (Tr * 0 * . * . * : ) , ( dV dS dS dV s c ε v T v τ ε v σ (33)
Once again, the tangent stiffness matrix is found non symmetric. It should be noted, however, that when the volumetric strain rate θ & is chosen constant per element, using for example P1/P0 quadrangles or P2/P0 triangles in two-dimensional problems, θ & can be eliminated from (33b) at the element level [START_REF] Hans Raj | Simulation of industrial forging of axisymmetrical parts[END_REF][START_REF] Fourment | Hot powder forging[END_REF], its value being injected in the equilibrium equation (33a). This equation can then be solved for the velocity field only, using again a non symmetric tangent stiffness matrix. In the frame of three-dimensional problems, using linear tetrahedra such as in FORGE3 ® , such a formulation cannot be used and the global resolution for v and θ & should be carried out with a non symmetric matrix.

Non-symmetry might be more or less acceptable when using iterative solvers, although it implies somewhat more storage and computational effort per iteration. However, it has been preferred to develop a three-field formulation. As explained hereafter, it results in a symmetric tangent stiffness matrix and permits a minimal storage requirement as compared to previous formulations.

Three-Field Formulation

The three-field formulation consists in introducing an additional variable in order to obtain a symmetric tangent stiffness matrix. This variable must permit the formulation of the problem with a Lagrangian function. As the relative density directly depends on the volumetric strain rate ( 23), the volumetric strain rate θ & is the variable to be added. Firstly, Φ is rewritten as a function of v and θ & , on

) ( 2 Ω × ϑ L :   ⌡ ⌠         θ         - + + = θ Φ Ω + + dV c f c m K m m 2 / ) 1 ( 2 1 9 2 9 1 : 3 2 1 ) 3 ( ) , ( ~& & & & ε ε v ∫ Ω ∂ Ω ∂ + -  ⌡ ⌠ + α + s c f dS dS p K d p g f v T v . 1 1 (34) Secondly, a weak form of the constraint ε & & Tr = θ is written: ∫ 0 )) ( Tr ( * ) ( * 2 = - θ Ω ∈ ∀ Ω dV p L p v ε & & (35) Finally, the following Lagrangian ) , , ( p L θ & v is defined on 2 2 )) ( ( Ω × ϑ L
, in which p is a field of Lagrange multipliers of the constraint:

∫ Ω - θ + θ Φ = θ dV p p L )) ( Tr ( ) , ( ) , , ( v ε v v (36)
The stationarity of the Lagrangian L with respect to p (i.e. 0 / = ∂ ∂ p L

) ensures the fulfilment, in a weak form, of the constraint relation

ε & & Tr = θ . Once this is achieved, ) , ( ~θ& v Φ can be considered equal to ) (v Φ . Again the stationarity of the functional ) (v Φ
with respect to v leads to the principle of virtual work (see [START_REF] Chenot | Finite element simulation of metal powder forming[END_REF] for a detailed demonstration).

Moreover, it can be shown [START_REF] Fourment | Hot powder forging[END_REF] that the relation between the effective hydrostatic pressure H p and the Lagrange multiplier p is:

( ) ε & & Tr 3 3 2 1 - ε ρ + = m r H c K p p (37)
When the material approaches the dense state, then 0 Tr → ε & and H p p → : the Lagrange multiplier p, is in this case nothing but the hydrostatic pressure.

The additional condition of non-penetration of forming tools (the solution velocity field v must be kinematically admissible in the sense of ( 26)) can be enforced by different methods, among which the penalty method which is used in software FORGE3 ® [START_REF] Chenot | Recent progresses in finite element simulation of the forging process[END_REF][START_REF] Chenot | 3-D finite element simulation of the forging process: scientific developments and industrial applications[END_REF][START_REF] Fourment | An implicit contact algorithm for the 3D simulation of the forging process[END_REF]. This method consists in adding a complementary term to the Lagrangian L, which will finally be written on

2 2 3 1 )) ( ( )) ( ( Ω × Ω L H : ∫ ∫ Ω ∂ Ω - χ + - θ + θ Φ = θ c dS dV p p L die p 2 2 1 ). ( )) ( Tr ( ) , ( ) , , ( n v v v ε v v
χ is a large positive constant and denotes the Mac Cauley's bracketing (i.e. 2 / ) (

x x x + =
).

Finite Element Discretization

In powder forging, the material undergoes large deformations. Remeshing is required in order to avoid severely distortion of the finite element mesh. As reliable threedimensional automatic remeshing procedures are based on linear tetrahedra, this kind of element is recommended here. In order to satisfy the Brezzi-Babuska compatibility condition, a three-field version of the P1+/P1 element [START_REF] Arnold | A stable finite element for Stokes equation[END_REF][START_REF] Fortin | Experiments with several elements for viscous incompressible flows[END_REF] has been developed and is presented hereafter. This choice has been fixed following a previous investigation of P2/P1 10-noded element [START_REF] Jinka | A new three-dimensional finite element model for the simulation of powder forging processes: Application to hot forming of P/M connecting rod[END_REF], but which proved too computationally expensive.

In the P1+/P1 element, the velocity field is quasi-linear continuous, including additional degrees of freedom at the centre of gravity of the element (bubble formulation). It can be decomposed in two parts: a linear one, resulting from linear interpolation between the four apex velocities, and a complementary one issued from interpolation of velocity correction degrees of freedom defined at the centre of the element:

) ( ) ( ) ( b l b k k N N v v b V v + = + = (39)
In (39), k N denotes the linear interpolation function associated to apex k, k V is the velocity vector at node k and the summation is extended to the four apexes (k = 1, 4) of the tetrahedron. Vector b is the vector of velocity corrections at element centre. The value of the "bubble" interpolation function

) (b
N is 1 at the centre and 0 on the element boundary (i.e. the four facets). This function is defined separately on each of the four sub-tetrahedra (the three nodes of each facet, plus the central node), so that the velocity field v is linear on each sub-tetrahedron (fig. 2).

On another hand, the field of Lagrange multipliers p and the volumetric strain rate field θ & are linear continuous. Inside each finite element, we have:

k k P N p = k k N Θ θ & & = (40)
both summation being extended to the four apexes (k = 1, 4) of the tetrahedron. Let us denote V the global vector of nodal velocities (vector of 3Nbnoe components), B the global vector of central velocity corrections (3Nbelt components), P the global vector of nodal Lagrange multipliers (Nbnoe components) and Θ & the global vector of nodal volumetric strain rates (Nbnoe components). The previous interpolations (39-40) are extended to the whole finite element mesh, by writing:

q b q n n N N B V v ) ( + = n n P N p = n n N Θ & & = θ ( 41
)
where index n varies from 1 to Nbnoe (total number of apexes of tetrahedra) whereas index q varies from 1 to Nbelt (total number of elements). In (41) the interpolation functions have been extended to the whole mesh, ) (

x k N (respectively ) ( ) ( x b q N
) having a zero value when x is outside the elements the node k belongs to (respectively outside element q). The discrete finite element Lagrangian is written as ) , , , (

P Θ B V & L
. Its stationarity condition provides the following system of non linear equations to be solved:

           = ∂ ∂ = ∂ ∂ = ∂ ∂ = ∂ ∂ 0 0 0 0 P Θ B V L L L L & (42)
The discretized equations of this system are expressed hereunder:

Nbnoe k , 1 3 , 1 = ∀ = λ ∀ ∫ ∫ Ω ∂ λ Ω λ Ω λ - λ - -  ⌡ ⌠ ε ρ = ∂ ∂ s dS N T dV P N dV c K V L n d n m m n m r n ) Tr( : ) 3 ( 2 1 B B ε & & 0 ) ( , , 1 = - χ - α + ∫ ∫ Ω ∂ λ µ µ µ Ω ∂ λ - c c f dS N n n v V N dS N v K n die m m p n g p g v (43) Nbelt q , 1 3 , 1 = ∀ = λ ∀ 0 ) Tr( : ) 3 ( 2 ) ( ) ( 1 = -  ⌡ ⌠ ε ρ = ∂ ∂ ∫ Ω λ Ω λ - λ dV P N dV c K L b q m m b q m r q B B ε B & & (44) 0 9 2 9 1 ) 3 ( 3 , 1 1 =  ⌡ ⌠       +         - ε ρ = ∂ ∂ = ∀ Ω - dV P N N c f K N L Nbnoe n m m m m m r n n ( ) [ ] ∫ 0 ) Tr( ) Tr( , 1 ) ( = + - = ∂ ∂ = ∀ Ω λ λ λ λ dV B V N N P L Nbnoe n q b q m m m m n n B B Θ & (46)
In the previous equations, repeated indices m and q varies from 1 to Nbnoe and from 1 to Nbelt respectively; B and ) (b B are the discrete differential operators defined by:

n n V λ λ ∂ ∂ = ε & B or, in components,         δ ∂ ∂ + δ ∂ ∂ = ∂ ∂ = λ λ λ λ j i n i j n n ij n ij x N x N V ε 2 1 & B ( 47
) q b q B λ λ ∂ ∂ = ε & ) ( B or, in components,         δ ∂ ∂ + δ ∂ ∂ = ∂ ∂ = λ λ λ λ j i b q i j b q q ij b q ij x N x N B ε ) ( ) ( ) ( 2 1 & B ( 48
)
It should be noted that the integrals in (43-46) have to be computed on the subtetrahedra when they involve the "bubble" interpolation function

) (b

N

or its spatial derivative.

Let us use the following global notation for the set of non linear equations (42):

0 = ∂ ∂ Y / L with ) , , , ( P Θ B V Y & =
. It is solved using a Newton-Raphson method. At each iteration of the method, the equations are linearised and a correction ) , , , (

P Θ B V Y δ δ δ δ δ & =
is computed to improve the current estimate Y:

) ( ) ( 2 2 Y Y Y Y Y ∂ ∂ - =       ∂ ∂ L L δ (49)
This is worth noting that, according to the definition of the bubble interpolation function, the stationarity equations (44), which correspond to the derivation of L with respect to the additional velocity degrees of freedom B, can be written on each element in a decoupled manner. Consequently, the Newton-Raphson correction B δ can be eliminated at element level by writing, in each element e:

) ( ) ( e e e L L Y B Y Y B Y ∂ ∂ - =             ∂ ∂ ∂ ∂ δ (50)
While assembling the tangent stiffness matrix

&

and to report this value in other equations. Since there are about four to five times more tetrahedra than nodes (apexes) in a three-dimensional tetrahedral mesh, this elimination proves quite efficient. After elimination of these degrees of freedom, the system to be solved can be written as:

) ' ( ' ) ' ( Y G Y Y H - = δ (51) where ) , , ( ' P Θ V Y δ δ δ δ & =
. In practice, the vector ' Y is partitioned into Nbnoe blocks, each block containing the four primary nodal unknowns.

This linear system is solved with a general minimum residual method [START_REF] Saad | Practical use of some krylov subspace methods for solving indefinite and nonsymmetric linear systems[END_REF][START_REF] Saad | GMRES: a generalized minimum residual algorithm for solving nonsymmetric linear systems[END_REF], as in the standard incompressible viscoplastic formulation in FORGE3 ® , except that in the present case, it has been extended to the solution of a system with five unknowns per node (instead of four). A block-diagonal preconditioning is used and is designed as follows. After elimination of additional velocity degrees of freedom, the matrix H of the system (51) has the following structure:

( ) ( ) ( )           pp T p T vp p T v vp v vv H H H H H H H H H θ θ θθ θ θ (52)
The block diagonal preconditioning matrix C, which is required to be definite positive and to approach the inverse of H is then composed of Nbnoe diagonal blocks It can be noted in particular that such a block-diagonal preconditioning can remedy the ill-conditioning issued from the penalty method for contact. As a matter of fact, a nodal penalty method is used in FORGE3 ® : the last integral in equation ( 38) is approached by a summation on nodes with application of repulsive nodal forces if nodes tend to penetrate the tooling [START_REF] Fourment | An implicit contact algorithm for the 3D simulation of the forging process[END_REF]. Hence, such a method only produce diagonal terms in the tangent stiffness matrix. The resulting poor conditioning can be counterbalanced by the block-diagonal preconditioner and does not affect the convergence rate of the iterative solver.

An isoparametric continuous four node interpolation is used for the discretization of temperature, the finite element mesh being the same as for the mechanical resolution (except central nodes, not used here). In each tetrahedral element, we have:

k k T N T = (54)
the summation being extended to the four apexes (k = 1, 4) of the tetrahedron The use of this discretization in the variational form of equation ( 17), according to the Galerkin finite element method, leads to the following classical set of non linear differential equations in which T is the global vector of nodal temperature (Nbnoe components).

F KT T C = + dt d ( 55
)
C is the heat capacity matrix, K the conductivity matrix and F the thermal loading vector. It should be noted that using a moving mesh formulation, dT/dt is here nothing but the partial derivative in time of the vector of nodal temperatures, since the nodes are convected with the material flow (see the scheme for configuration updating in section 5).

The expressions of matrices C and K and of vector F are as follows:

∫ Ω ρ = dV N N c C k n p nk (56) ∫ ∫ ∫ Ω ∂ Ω ∂ Ω + + ∇ ∇ = c s dS N N h dS N N h dV N N k K k n cd k n cr k n nk (57) ∫ ∫ ∫ Ω ∂ Ω ∂ Ω φ + + + = c s dS N T h dS N T h dV N F n f die cd n ext cr n n ) ( : ε σ & (58)
where n and k vary between 1 and Nbnoe.

The first order differential equation ( 55) is solved using a second order three level (two step) finite difference scheme [START_REF] Hogge | A comparison of two and three level integration schemes for non linear heat conduction[END_REF]. Given two successive time steps 1 t ∆ and 2 t ∆ (not necessarily of same size), the nodal temperature vector T and its time derivative dT/dt are expressed as:

2 1 3 2 1 t t t t t ∆ + ∆ - α + α + α = T T T T (59) 2 1 2 1 ) 1 ( t g t g dt d t t t t t t ∆ - + ∆ - - = ∆ + ∆ - T T T T T ( 60
)
where g and , , 3 2 1 α α α are numerical coefficients providing different integration schemes. In fact, such schemes are second order accurate on condition that [START_REF] Hogge | A comparison of two and three level integration schemes for non linear heat conduction[END_REF]:

        ∆ ∆ + - ∆ ∆ α - + α - = α 2 1 2 1 1 1 2 1 2 ) 2 1 ( ) 1 ( t t g t t ( 61
)
and

        ∆ ∆ + + ∆ ∆ - α = α 2 1 2 1 1 3 1 2 ) 2 1 ( t t g t t ( 62
)
It can be noted that 1

3 2 1 = α + α + α
and that each scheme is determined by only two parameters, for instance 1 α and g. Moreover, such schemes are unconditionally stable (no limitation on ∆t) on condition that [START_REF] Hogge | A comparison of two and three level integration schemes for non linear heat conduction[END_REF]:

) 1 ( 2 1 and 2 1 1 g g - ≥ α ≥ ( 63
)
As regards matrices C, K and F, equation ( 59) is not used because it would result in an iterative resolution of (55). This is avoided by using a linearisation technique, suggested by Zlamal [START_REF]Finite element for non linear parabolic equations[END_REF] and shown to be second order accurate [START_REF] Soyris | Modélisation tridimensionnelle du couplage thermique en forgeage à chaud (Three-dimensional modelling of thermomechanical coupling in hot forging)[END_REF], each non linear matrix W (C, K or F) being expressed as:

      - ∆ ∆ + α + α + α = ∆ - ∆ - ) ( 1 1 1 2 3 2 1 t t t t t t t t t W W W W W W (64)
which yields, after injection of (61) and (62), the following simple expression:

t t t t t g t t g W W W               ∆ ∆ + + +               ∆ ∆ + - = ∆ - 1 2 1 2 1 2 2 1 1 2 2 1 1 (65)
Finally, the injection of the time discretization defined by equations (59-60) in equation ( 55) results in a linear system

B AT = ∆ + 2 t t
(66) to be solved for the unknown nodal temperature vector 2 t t ∆ +

T

. In practice, we use the two-step implicit scheme ( 2 3 , 0

1 = = α g ).
It can be noted that this scheme is different from the one-step backward Euler implicit scheme (the expression of the time derivative is different, which yields to second order accuracy). For the first time increment, a one-step Crank-Nicolson scheme is used ( 1 , 2 1 , 0

3 2 1 = = α = α = α g ).
When there is a large temperature difference between the workpiece and the forging dies, space and time numerical oscillations are observed, whatever the integration scheme used. To overcome this numerical problem, a classical remedy is to approach C by its diagonal lumped form L C :

∫ Ω ρ δ = dV N c C m p mn L mn (67)
where index m is not summed. However, this approximation is not always efficient enough, more particularly using three-dimensional tetrahedral elements. The size of the thermal time step is then chosen with a different value from that of the mechanical time step, so that it remains greater than the critical value c t ∆ issued from the Courant limit [START_REF] Comini | Stability characteristics of time integration schemes for finite element solutions of conduction-type problems[END_REF]:

k l c t p c 32 ∆ ρ = ∆ ( 68
)
where l is a reference length which is computed as the distance between neighbouring nodes along the normal direction to the workpiece/die interface.

THERMO-MECHANICAL COUPLING AND CONFIGURATION UPDATING

There is a thermomechanical coupling between mechanical equations [START_REF] Saad | Practical use of some krylov subspace methods for solving indefinite and nonsymmetric linear systems[END_REF], and heat transfer equations [START_REF] Green | A plasticity theory for porous solids[END_REF][START_REF] Hans Raj | Simulation of industrial forging of axisymmetrical parts[END_REF][START_REF] Hogge | A comparison of two and three level integration schemes for non linear heat conduction[END_REF][START_REF] Im | Analysis of axisymmetric forging of porous materials by the finite element method[END_REF][START_REF] Jinka | A new three-dimensional finite element model for the simulation of powder forging processes: Application to hot forming of P/M connecting rod[END_REF][START_REF] Jinka | Finite element simulation of hot isostatic pressing of metal powders[END_REF]. However, the material flow does not strongly depends on temperature, so there is no need to solve these problems simultaneously at each time increment [t, t+∆t]. A staggered coupling algorithm is preferred. At any time t, the temperature field, t T is assumed to be known. It allows the calculation of the consistency of the material [START_REF] Coupez | From a direct solver to a parallel iterative solver in 3-D forming simulation[END_REF] and then the computation of the flow. Once the velocity field is known, at time t, the internal source of heat due to deformation (4) and to friction [START_REF] Jinka | Finite element simulation of hot isostatic pressing of metal powders[END_REF] are calculated, allowing the computation of the temperature field at time t+∆t.

The configuration is then updated, according to a Euler explicit scheme. Denoting X the global vector of nodal coordinates, we have:

t t t t tV X X ∆ + = ∆ + (69)
The same scheme is applied to the mass conservation equation to express the evolution of nodal relative density. Let R denote the vector of nodal relative densities (components

Nbnoe n R n , 1 , = ). ) 1 ( ) exp( t n t n t n t n t t n t R t R R Θ Θ & & ∆ - ≈ ∆ - = ∆ + (70)
A strict application of (70) would yield relative density values slightly greater than 1 as the material becomes fully dense. To prevent this, the upper limit of 1 is applied.

More generally, the previous procedures have been implemented in the software FORGE3 ® . In addition, an automatic remeshing procedure is triggered when necessary in order to avoid mesh degeneration. In case of remeshing, a transportation of nodal temperature and relative density values from old to new mesh is carried out by means of direct interpolation. The complete incremental resolution scheme is given at figure 3.

VALIDATION

In order to validate the three-field formulation, the reference solution of a uniaxial free compression test (fig. 4) has been established. It is possible to exhibit a closed form solution, on condition that coefficients c and f are kept constant. Under this assumption, the following solution has been derived (details are given in appendix):

c f f r r h h + -       ρ = ρ 3 0 0 (71) m m h V f c K               + - = σ + 2 1 3 ( 72
)
in which 0 h and h denote the initial and current height of the compacted specimen, 0 r ρ and r ρ are the initial and current relative density, V is the compaction prescribed velocity (which is not necessarily constant). The other coefficients are those used in the above presentation.

The values that have been used for the comparison are: 0 h = 120 mm, V = 3 mm.s -1 , 0 ρ = 0.80, K = 100 MPa.s m , m = 0.25. Coefficients c and f are kept constant (c = 6.463, f = 0.45) until the full densified state is reached. Then the corresponding values c = 1 and f = 0 are used, as they are associated with dense state. In such conditions, a uniform dense state is reached for h = 38.3 mm (height reduction ∆h = 81.7 mm). The agreement between the finite element solution and the analytic one is excellent for both relative density and stress values (fig. 5). This is observed before full densification, as well as after, when the deformation becomes purely viscoplastic and incompressible. The discontinuity of the stress evolution is clearly a consequence of the abrupt change in the values of coefficients c and f at full densification. The error on the axial stress (respectively the relative density) is less than 0.13% (respectively 0.40%) up to the end of the test, corresponding to a cylinder height of 20 mm.

APPLICATION

The finite element model has been applied to the simulation of the hot compaction step of a connecting rod preform, made of aluminium alloy, in order to test the ability to simulate a complex three-dimensional compaction process. The preform is normally obtained by a cold compaction step. Since this first step has not been simulated, the thickness of the preform has been taken uniform along the compaction direction z (fig. 6). For the same reason, its initial relative density distribution has been assumed uniform, equal to 0.80. Figure 6 shows the initial volumic mesh of the preform (initially rather coarse) as well as the surfacic meshes of the tooling components in initial position. The tooling is composed of a lateral die, two cylindrical dies at both ends of the rod and a punch which has a prescribed vertical velocity. For symmetry reason, only one quarter of the part is effectively computed. The process and material parameters used in the computation come from previous experimental work [START_REF] Grazzini | Etude expérimentale du comportement rhéologique de milieux granulaires à constituants plastiquement déformables: comparaison de poudres de plasticine et d'alliages d'aluminium[END_REF] and are given in table 1.

Figure 7 illustrates the progress of the forming. It can be seen that the preform is first deformed in its central I-beam region. Then the deformation extends to the small end (piston end) and to the big end (crankshaft end). This figure also reveals the efficiency of the remeshing procedure, resulting in a finer discretization in regions of small curvature radii.

As expected with such an initial preform geometry, densification occurs first in the central region (fig. 8), and then extends to the piston end and the crankshaft end. This can be seen on figure 9, on which distributions of relative density have been plotted in five transverse sections along the connecting rod, at an intermediate stage of compaction. The compaction of the centre of the arm of the rod is associated to some de-densification near the zone contacting the punch. The dilatation (or de-densification) affects the free surface zone near the punch edge. A similar phenomenon occurs in the big end region where the outer zone is first compacted: there is a rapid densification of the material under the punch and some de-densified material beside, near the inner radius.

Figure 10 shows the configuration reached near the end of the compaction process. At this stage, the preform has been almost completely compacted. The last region to be densified is located at the crankshaft end. The de-densification phenomena are significant in this region, the relative density being even noticeably lower than the initial value of 0.80. Such a non uniform compaction could be avoided by defining a preform of non uniform thickness or/and relative density (this is actually the case in real production).

Finally, figure 11 shows the temperature distribution at the end of the process. The competition between heat loss at tool contact and heat dissipation by plastic deformation and friction results in a maximum temperature drop of about 60°C. It can be seen that some regions near both ends of the rod suffer from significant temperature inhomogeneities which could induce deformations and residual stresses after cooling. The complete simulation took 80 increments, 26 remeshing operations and about 15 hours (CPU time) on IBM-Risc6000-43T, the maximum number of nodes being about 5000. Finally, the relative variation of the material mass is only 0.038% at the end of the computation.

CONCLUSION

A new three-field formulation (velocity, volumetric strain rate and pressure) has been developed for an accurate simulation of the hot forging of metal powder. The finite element discretization is based on the new linear tetrahedral P1+/P1/P1 element. It has been implemented in the computation code FORGE3 ® , whose iterative solver has been adapted to the resolution of non linear systems with five nodal unknowns. The finite element resolution of the three-dimensional mechanical problem has been coupled in a staggered manner with the heat transfer resolution, providing an efficient tool for the simulation of hot powder compaction processes. This has been illustrated by treating the hot forging of a connecting rod preform. In the future, such a formulation should be extended to the modelling of cold compaction, in order to simulate the global powder forming process. (A4)

The application of (A4) to (A1a) yields:

zz r rr c f σ - σ ε ρ = ε = ε θθ ) 2 ( 2 & & & zz r zz f c σ + σ ε ρ = ε ) ( & & (A5)
Relation (A5b), combined with ( 8) and (A2) permits to obtain the analytical expression of the axial stress (72):

m m zz h V f c K               + - = σ + 2 1 3 (A6)
As regards the relative density evolution, it is obtained in the following way. The summation of the three strain rate components (A5), combined with (A1a) yields the expression of volumetric strain rate:

f c f h V + - = = ∇ = θ 3 Tr . ε v & & (A7)
Integration of (A7) can be achieved easily under the assumption of constant coefficients c and f during compaction. Denoting 0 V and V the initial and current volume of the specimen respectively, we have:

0 0 0 0 ln 3 3 ) ( V V ln h h f c f dt h V f c f du u t t + =  ⌡ ⌠ + - = θ = ∫ & (A8)
which yields finally relation (71):

c f f r r h h + -       ρ = ρ 3 0 0 (A9)
It should be noted that equation ( 72) is not submitted to the restrictive condition of constant coefficients c and f, and that (71) and (72) hold for time dependent compression velocity V.

CaptionsFig. 1 :

 1 Fig. 1: Variation of coefficients c and f versus the relative density r ρ (schematic).

Fig. 2 :

 2 Fig. 2: P1+/P1/P1 tetrahedral element.

Fig. 3 :

 3 Fig. 3: Incremental resolution scheme.

Fig. 4 :

 4 Fig. 4: Uniaxial free compression test.

Fig. 5 :

 5 Fig. 5: Comparison between finite element results and analytical solution. Plots of relative density and axial stress vs height reduction.

Fig. 6 :

 6 Fig.6: Hot compaction step of a connecting rod preform made of aluminium alloy powder. Initial position of the volumic mesh of the preform and of the surfacic meshes of the tooling.

Fig. 7 :Fig. 8 :Fig. 9 :

 789 Fig. 7: Successive deformed configurations of the preform at 10, 20, 26 and 35% height reduction.

Fig. 10 :Fig. 11 :

 1011 Fig. 10: Distribution of relative density near the end of forming.

Figure 1 :

 1 Figure 1: Variation of coefficients c and f versus the relative density r ρ (schematic).

Figure 2 :

 2 Figure 2: P1+/P1/P1 tetrahedral element.

Table 1 :

 1 process and material parameters used in the simulation

& & & & 2 / 1 2

f die cd

& & & (32)

& & & & &

& & & &

& & & & (38)where p

& & &(45)

FINITE ELEMENT RESOLUTION OF THE HEAT TRANSFER PROBLEM

ACKNOWLEDGEMENT

The present study has been conducted in the frame of a European Brite/Euram project : BRE2 CT 92-0220 with RENAULT and PEAK as industrial partners.

APPENDIX

The uniaxial compression of a compressible viscoplastic axisymmetric specimen is considered. The contact along tool surfaces is supposed perfectly sliding (no friction). The deformation is then homogeneous throughout the specimen. Denoting V the modulus of the compression velocity, the components of the strain rate tensor are:

The stress tensor is diagonal, the only non zero component being zz σ . Then the equivalent stress is, according to (1):

Let us express now the deviatoric part and the volumetric part of the flow rule [START_REF] Coupez | A mesh improvement method for 3D automatic remeshing[END_REF]. This yields respectively:

The flow rule [START_REF] Coupez | A mesh improvement method for 3D automatic remeshing[END_REF] can then be inverted, using a combination of (A3) and (8):

• Mechanical equilibrium resolution

• Eventual automatic remeshing • If remeshing, transport from old to new mesh:

• Go to next increment Figure 3: Incremental resolution scheme.


Marianne Astruc

Patrick Navard

A flow-induced phase inversion in immiscible polymer blends containing a liquid-crystalline polymer studied by in situ optical microscopy

The phase inversion from a morphology of hydroxypropylcellulose in water ͑HPC50%͒ droplets in polydimethylsiloxane ͑PDMS͒ matrix to a morphology of PDMS droplets in HPC50% matrix can be induced by a change of shear rate, due to a viscosity ratio inversion. Such a process passing through four different transient morphological stages was studied by optical microscopy in a transparent shear device. In a certain concentration region, at a fixe shear rate, after sheets of PDMS were formed, the transition ''hesitates'' between phase inversion and refine starting morphology. The influenc of PDMS concentration, shear rate, elasticity, and phase dimension on the fina morphology was investigated and compared with different models given in the literature. The influenc of concentration and shear rate on the duration of the transient flo was also studied. We propose a simplifie model of breakage of filament to explain a part of our results.

I. INTRODUCTION

The increasing application of polymer blends for the elaboration of new materials is motivating the general interest in the study of morphology development in immiscible polymer blends, since morphology is one of the major factors controlling their fina properties. For two-phase immiscible polymer blends, different morphologies such as droplet, fibrilla or co-continuous structures can be found. Several parameters are important in determining the fina or steady-state morphology: composition of the blend, shear rate, viscosity and elasticity of the two phases, interfacial tension, and time of mixing. Among the many morphological phenomena that occur during preparation and processing of polymer blends, the study and prediction of the point of phase inversion at which co-continuity is observed has attracted a lot of attention. All the published studies are based on semiempirical models. Some authors have related the point of phase inversion with the rheological properties of the components. Avgeropoulos et al. ͑1976͒ suggested that the point of co-continuity is reached when the torque ratio of the components in an internal mixer is equal to the component volume fraction ratio. Later, Paul and Barlow ͑1980͒ proposed a similar formulation using the viscosity ratio instead of the torque ratio. Miles and Zurek ͑1988͒ proposed the following generalized expression:

⌽ A B ͑␥ ˙͒ ⌽ B A ͑␥ ˙͒ ͭ Ͼ 1: phase A is continuous ϭ 1: co-continuity Ͻ 1: phase B is continuous , ͑1͒
where ⌽ A and ⌽ B are the volume fractions of the phases A and B, and A and B their viscosities at the applied shear rate ␥ ˙. Several other models predicting the point of phase inversion are based on the filamen instability concept. Metelkin and Blekht ͑1984͒ considered the dispersed phase to be that with the shorter breakage time of filaments The critical volume fraction for the phase inversion was calculated to be

⌽ B ϭ 1 1ϩF͑͒ ͑2͒
where F͑͒ ϭ 1ϩ2.25 log ϩ1.81͑log ͒ 2 , ͑3͒

with being the viscosity ratio of the blend components at the blending shear rate. Luciani et al. ͑1993͒ made the assumption that there is a two-fibrilla network structure at the co-continuity point. They supposed that the threads have the same length and stability at the phase inversion point, so that

ͩ ⌽ B ⌽ A ͪ 0.5 ϭ A ⍀ ͩ B A ͪ B ⍀ ͩ A B ͪ ͑4͒
where ⍀͑͒ is a function tabulated by Chappelear ͑1964͒.

Utracki ͑1991͒ suggested a model that can be used to predict the point of phase inversion for blends with a viscosity ratio far from unity. It is based on the dependence of the viscosity on the volume fraction of monodispersed hard spheres in the matrix, as proposed by Krieger and Dougherty ͑1959͒. The point of co-continuity is where the viscosity of polymer A ͑at the volume fraction ⌽ A ) dispersed in polymer B equals the viscosity of polymer B ͑at the volume fraction ⌽ B ) in polymer A. At this point, the ratio is

ϭ ͫ ͑⌽ m Ϫ⌽ B ͒ ͑⌽ m Ϫ⌽ A ͒ ͬ ͓͔⌽ m , ͑5͒
where ͓͔ is an intrinsic viscosity and ⌽ m ϭ 1Ϫ⌽ c is the maximum packing volume fraction. ⌽ c is the critical volume fraction for percolation, equal to 0.156 for threedimensional spherical dispersions ͓Chapoy ͑1986͔͒. An arbitrary value of ͓͔ ϭ 1.9 was given by Utracki ͑1991͒, based on published data on the points of phase inversion of thermoplastics and rubbers. The model of Avgeropoulos predicts satisfactorily the point of phase inversion for torque ratios close to unity. The model of Miles and Zurek ͑1͒ and the model of Metelkin and Blekht ͑2͒ were found to overestimate the composition at which phase inversion occurs if the viscosity ratio departs from unity while the model of Utracki ͑5͒ was found to agree with some experimental results for blends having a viscosity ratio far from unity ͓Utracki ͑1991͒, Mekhilef and Verhoogt ͑1996͒, Mekhilef and Verhoogt ͑1997͔͒. The agreement of these models with experiments is strongly dependent on the nature of the blend under consideration, underlining the imperfection of these models. The failure of the models may be due to the fact that the elasticity of each component polymer or the duration of the blending process are not taken into account. More recently, other models where the elasticity or the interfacial tension are considered have been established. Because the more elastic phase will tend to encapsulate the less elastic one and form the matrix, Bourry and Favis ͑1998͒ proposed the following expression for the point of co-continuity:

⌽ A ⌽ B ϭ G B Ј G A Ј • G A Љ G B Љ , ͑6͒
where G i Ј and G i Љ are, respectively, the storage and the loss modulus of the component i.

This expression seems to give better results than expressions ͑1͒, ͑2͒, and ͑5͒ when the elasticities of the two components differ strongly. By considering a co-continuous structure as a dense packing of randomly oriented rodlike particles of the minor phase and taking into account the microrheological requirements for the stability of such rods, Willemse et al. ͑1998͒ established an equation predicting the range of compositions within which fully co-continuous structures can exist. The equation relates the critical volume fraction of the minor phase at complete co-continuity ⌽ d,cc to the matrix viscosity m , interfacial tension , shear rate ␥ ˙and phase dimension R 0 :

1

⌽ d,cc ϭ 1.38ϩ0.0213 ͩ m ␥ R 0 ͪ 4.2
.

͑7͒

An absolute prediction based on this model is not straightforward because the phase dimension has to be measured ͑or even worse, has to be define for a real blend͒. The Flow-induced phase inversion occurs when the original morphology of polymer A dispersed in the matrix of polymer B is transformed under the action of a flo into polymer B dispersed in polymer A. Clearly, this must involve crossing the point of co-continuity. Such crossings have been observed after blend extrusion ͓Shih ͑1992͔͒ and studied in batch mixers ͓Shih ͑1995͒, Ratnagiri and Scott ͑1996͔͒. In some cases, the observations could be explained by differences in the melting temperature of the components ͑i.e., phase A supposed to be the matrix is molten at a higher temperature than phase B, which gives solid A in molten B, and which inverses when A melts͒. But it appears clearly that in most cases, a variation of viscosity or elasticity during the blending process plays a major role ͓Ratnagiri and Scott ͑1996͒, Ratnagiri and Scott ͑1997͔͒.

The main problem is that observations after extractions do not allow us to clearly know how co-continuity is reached during the flo and how the morphologies change around that point. This lack of knowledge makes it difficul to build a comprehensive model. To our knowledge, no in situ experiment on this effect has been reported.

Using the torque or viscosity ratio concept, one may understand qualitatively how a phase inversion can be induced by the flow By varying the shear rate, the viscosity ratio may vary. If this variation is large enough, it could induce a phase inversion ͓Eqs. ͑1͒, ͑4͒ or ͑5͒, depending on the model͔. Bouilloux et al. ͑1997͒ studied a phase inversion induced by the variation of viscosity ratio with time of mixing. By quenching the blend at various stages of mixing, they always observed the co-continuous structure. They could not reach the inverted morphology. Willemse et al. ͑1999a͒ assumed that sheets of the initial dispersed phase could be the precursors of the co-continuous morphology. Depending on the stability of the network of ligaments in the mixer after sheet breakup, the co-continuous morphology could be either a transient or a stable morphology. The formation and the stability of these kinds of sheets under a simple shear flo in viscoelastic blends was firs reported by Sundararaj et al. ͑1995͒. More recently, Lazo and Scott ͑1999͒ confirme with extraction experiments on PS/PE blends that phase inversion in steady-shear flo pass through sheet formation. Sheet formation that transforms into threads or drops seems to be an important factor in polymer blend deformation ͓Willemse et al. ͑1999b͔͒.

The objectives of the present work are to use rheo-optical tools to visualize a shearinduced phase inversion in an immiscible polymer blend, to study how it occurs, and to compare this to the theories that are available. For reasons that will be detailed later, we used a blend of PDMS and a solution of hydroxypropylcellulose in water ͑HPC50%͒. One of the difficultie in studying phase inversion and how it occurs is that the point of phase inversion is classically reached by increasing concentration, which implies changing samples. The advantage of the phase inversion observed here is that the cocontinuous morphology should be a transient state that takes place between two dropletmatrix morphologies, in the same sample. This opens the way to study two related phenomena, i.e., the mechanisms controlling both the co-continuity and the phase inversion.

II. EXPERIMENTAL SETUP AND PREPARATION PROCEDURES

A. Rheo-optical tools

Flow-induced morphology observations were performed on a rheo-optical system comprising three main parts:

͑i͒

A Linkam CSS 450 shearing system. It is composed of two transparent rotating parallel plates permitting the observation of the sample at 7.5 mm from the center axis of rotation. The shearing system is monitored by a PC using the LINKSYS 1.41 software. ͑ii͒

An optical system composed of an optical microscope Leitz Metallux 3 with a lighting system, a charge-coupled device camera, and a monitor. ͑iii͒ A video data acquisition system ͑video recorder and time code generator͒.

Experimental difficulties

From a practical point of view, the study of phase inversion mechanisms with rheooptical tools is not simple. The formation of co-continuous structures in polymer blends requires a relatively high concentration of both components. But the observation of morphology in concentrated blends is difficult concentrated blends are highly scattering and so turbidity limits observation of their morphology by optical microscopy. That is why it was chosen to shear thin layers ͑a gap between 50 and 100 m͒. We are aware of the fact that this restriction could influenc the morphologies. But this is the only way to perform in situ observations with such highly concentrated systems. In the experiments special case was taken to ensure that the observations are not significantl modifie by the small dimensions of the gap. Furthermore, due to the complex morphologies that were formed, it is difficul to identify each phase and to recognize which polymer is where. A way to overcome this is to label one of the polymers. By choosing an isotropic component and a birefringent one, it is possible to distinguish the two phases between crossed polarizers. Unfortunately, our blends are liquid at room temperature and do not allow us to carry out extraction experiments after the shear experiments.

B. Selected materials

Materials with the required rheological and optical properties were selected. All experiments were performed on blends of PDMS from Rho ˆne-Poulenc ͑Rhodorsil 47V200000, M W ϭ 149 350) and a solution of hydroxypropylcellulose diluted by 50 wt % in water ͑HPC50%͒. The HPC is produced by Aqualon ͑Klucel L, M W ϭ 100 000). These components are liquid at room temperature. To avoid problems of phase separation that have been reported for the solution of 50% of HPC in water ͓Guido and Grizzuti ͑1995͔͒, the experiments were performed at 18 °C. Under these conditions, the solution of HPC is in a liquid-crystalline phase. It is birefringent, whereas the PDMS is not. It is thus possible to distinguish the two components by using depolarized light microscopy. We take a literature value of the interfacial tension equal to 12.2 mN/m as reported by Tsakalos ͓Tsakalos ͑1995͔͒ as obtained for 50 wt % HPC ͑Klucel L͒ in a PDMS ͑Rhodorsil͒ matrix.

Rheology

The steady shear viscosity measurements of the pure components were carried out at 18 °C with a controlled stress Stresstech Rheologica rheometer, in parallel plate geometry ͑1 mm gap and 25 mm diameter͒. Shear rates from 0.1 to 20 s Ϫ1 were investigated. The measurements performed on the HPC solution were made after sufficientl large deformations to reach the steady-state conditions in lyotropic LCPs ͓Moldenaers ͑1987͔͒. Viscosity data of the two components are presented in Fig. 1͑a͒. The zero shear viscosity of the PDMS is 240 Pa s compared with 480 Pa s at 0.1 s Ϫ1 for HPC50%. Below 8 s Ϫ1 , the PDMS is Newtonian while the HPC50% solution presents a strong shear thinning behavior. The viscosity curves cross at a shear rate of 0.8 s Ϫ1 . Below this shear rate, the PDMS is the least viscous component while above it is the most viscous. Over the shear rate range under investigation ͑0.1-20 s Ϫ1 ͒, the viscosity ratio of the components can easily be changed by varying the shear rate. Different viscosity ratios depending on the shear rate are specifie in Table I. By changing the shear rate from 0.2 to 10 s Ϫ1 , the viscosity ratio varies from 1.7 to 0.4.

First normal stress difference N 1 data are plotted in Fig. 1͑b͒. Below 12 s Ϫ1 , the solution of HPC50% has the highest elasticity. This is confirme by the measurements of the storage and loss moduli carried out at a stress of 100 Pa. At low frequencies, the solution of HPC50% has the highest elasticity while the loss modulus is the lowest ͑Fig. 2͒. The storage moduli cross at 11 rad s Ϫ1 and the loss moduli at 0.30 rad s Ϫ1 , close to the values found in steady-state experiments. Even if the Cox-Merz rule is not strictly verifie for the HPC solution, these results show that dynamics measurements allow us to compare the relative elasticity of both components, especially for the higher shear rates that cannot be reached in continuous measurements.

C. Preparation of the blends

Preparation of the solutions of HPC was made following the method described by Tsakalos ͑1995͒. To avoid possible diffusion of water in PDMS that could change the HPC concentration in the HPC50% component, the PDMS was saturated with water before blending ͓Tsakalos ͑1995͔͒.

In order to study phase a inversion induced by a change in the viscosity ratio, it was necessary to prepare blends for which the PDMS is the continuous phase at rest. During this blending procedure, special care was taken not to cross the point of phase inversion. This means that the samples had to be submitted to very low shear rates ͑under 2 s Ϫ1 ͒. The blends were hand mixed after cooling in a refrigerator. Suitable amounts of the components were weighed and mixed very slowly for ϳ 10 min with a spatula until a homogeneous creamy appearance was obtained.

For the preparation, two opposite points have to be balanced. On one side, the blends have to be mixed during sufficientl long time and with sufficientl high speed of rotation to give a good homogenization and stability. On the other hand, for high concentrations of the dispersed phase, the point of phase inversion is easily crossed under these conditions. A phase separation is observed after a few hours and it is also difficul to reproduce the same initial morphology for different samples with the same concentrations. Blends of 70, 67, 63, 60, and 55 vol. % of PDMS were prepared by this procedure. They are, respectively, referred to PDMS70, PDMS67, PDMS63, PDMS60, and PDMS55. We were not able to prepare blends with a lower concentration of PDMS where PDMS was the continuous phase at rest.

D. Experimental conditions

Startup experiments in the Linkam shearing system were carried out at different shear rates ͑0.8-30 s Ϫ1 ͒ and concentrations ͑55%-70% of PDMS͒. The indicated shear rates correspond to the shear rates applied at the location of observation in plane-plane ge- ometry, given the thickness of the sample and speed of rotation. All the experiments were done at Ϯ8% of the indicated shear rates. These variations are associated with the radial gradient of the shear rate throughout the width of visualization. First, the transient flo for blends with concentration of PDMS varying from 55% to 70% was studied at a startup of 10 s Ϫ1 . Second, the transient flo of PDMS63 was studied under different startup experiments with shear rates varying from 0.8 to 30 s Ϫ1 . The lower limit of the range of concentrations is limited by the difficultie of preparation described in the preceding section ͑phase inversion may occur during the blending procedure͒. The highest value of shear rate that can be used is 30 s Ϫ1 , above which the blend is expulsed from the shearing device after few minutes of shearing.

In Sec. III, the transient flo observed for the blends at a shear rate of 10 s Ϫ1 will be presented in detail. These results illustrate the typical mechanisms occurring during a well-controlled but highly specifi flo induced phase inversion.

III. RESULTS AND DISCUSSION

A. Observed morphologies

At-rest morphology

After carefully and slowly fillin the Linkam shearing system with a small amount of blend and adjusting the gap, the sample was allowed to relax and to reach a temperature of 18 °C for 30 min. For the concentrations of PDMS from 55% to 70%, similar morphologies were observed ͓Fig. 3͑a͔͒; droplets of HPC50% of diameter under 100 m are dispersed in the matrix of PDMS. Some droplets contain small subinclusions of PDMS. Favis and Chalifoux ͑1988͒ and Bourry and Favis ͑1998͒ reported the presence of similar encapsulations in blends of polypropylene/polycarbonate and in blends of polyethylene/ polystyrene around the point of phase inversion. In our blends, we attribute the presence of these encapsulations in the at-rest morphology to the fact that some local phase inversion may have occurred during the blending preparation.

Transient morphologies (example of shear at 10 s ؊1 )

Submitted to a simple shear flo at a shear rate of 10 s Ϫ1 , the blend is in a region where a phase inversion is expected to take place. Several different transient morphologies were observed before the fina steady-state equilibrium morphology was reached. Four different stages in the mechanism leading to the fina equilibrium can be distinguished. These four stages occurred in all the observed blends and can be considered as the general route that the PDMS/HPC50% blends take toward phase inversion. First stage (t 0 → t 0 ϩ10 s at 10 s Ϫ1 ͒: bi-fibrilla morphology. During the firs 10 s of shearing at 10 sϪ1, the inclusions of HPC50% and the subinclusions of PDMS are deformed and stretched along the flo direction ͓Fig. 3͑b͔͒. Soon after, it is difficul to tell which phase is the continuous one. The PDMS and the HPC50% phases appear like long filaments and the morphology can be described as a bifibrilla structure ͓Fig. 3͑c͔͒.

Second stage (t 0 ϩ10 s → t 0 ϩ60 s at 10 s Ϫ1 ͒: emergence of sheets of PDMS. The bifibrilla morphology is not stable. The shorter filament of PDMS begin to break ͓Fig. 3͑d͔͒ while at the same time, sheets of PDMS layered in the flo direction form ͓Fig. 3͑e͔͒. These domains of PDMS contain filament and droplets of HPC50% with small subinclusions of PDMS. They are dispersed in a continuous phase of HPC50% also containing small droplets stemming from the shorter filament of PDMS ͓Fig. 3͑f͔͒.

Third stage (t 0 ϩ60 s → t 0 ϩ600 s at 10 s Ϫ1 ͒: stripe morphology. For a longer shearing time, the sheets of PDMS continue to stretch along the flo direction ͓Fig. 4͑a͔͒ until they form stripes: continuous phases of HPC50% with small droplets of PDMS dispersed alternately with continuous phases of PDMS containing inclusions of HPC50% with subinclusions of PDMS ͓Fig. 4͑b͔͒. In fact, we can consider that by this stage, a phase inversion has already occurred in the stripes of HPC50%, between the stripes of PDMS.

Fourth stage (t 0 ϩ600 s → t 0 ϩ10 000 s at 10 s Ϫ1 ͒: homogenization of the morpholo- gy. After the stripes have been formed, the morphology evolves extremely slowly. Different mechanisms occur independently in different zones in the blend. ͑i͒ In the stripes where the PDMS is the continuous phase, we observe long filament of HPC50% constituting a kind of relatively unstable network in which branches are continuously forming and breaking. Some of these filament break to form smaller droplets. These small inclusions coalesce to reform filaments Inside the inclusions of HPC50%, the subinclusions of PDMS are also subject to the mechanisms of breakup and coalescence. As the inclusions of HPC50% form filaments we also observe small droplets of PDMS coalescing to form filaments In droplets of HPC50%, one or two encapsulated drops of PDMS can generally be observed. These mechanisms are shown schematically in Fig. 5͑a͒.

͑ii͒ In the stripes where the HPC50% is continuous, filament and inclusions of PDMS break to form small droplets. Although it is not easy to measure it by optical microscopy, their size distribution seems to be very narrow compared with the size distribution of the inclusions of HPC50% in the stripes of PDMS. This could be due to an absence of coalescence of PDMS droplets in these stripes.

͑iii͒ At the boundary between PDMS and HPC50% stripes, several complex mechanisms compete.

Droplets of PDMS dispersed in the matrix of HPC50% can coalesce with the stripes of PDMS and contribute, in this way, to the broadening of the PDMS stripes.

The existence of a filamen of HPC50% at the edge of a stripe of PDMS can lead to two opposite mechanisms.

The layer of PDMS between the filamen of HPC50% and the stripe of HPC50% can also constitute a filament Depending on the breakage time of this filamen ͑whether it is shorter or longer than that of HPC50%͒, the stripe of PDMS will broader or become thinner. In the case of the development of Rayleigh instabilities in the thread of PDMS, the drops created will disperse in the stripe of HPC50%. The filamen of HPC50% joins the continuous phase of HPC50% and the subinclusions of PDMS encapsulated in this filamen are thus dispersed in the stripe of HPC50% too. This mechanism leads to the broadening of the stripes of HPC50% and so to the phase inversion. It is schematized in Fig. 5͑b͒. If the breakage time of the filamen of HPC50% is shorter than that of the filamen formed by the layer of PDMS, most of the droplets of HPC50% created will contain subinclusions of PDMS. This mechanism leads to the broadening of the stripes of PDMS and to the refinement of the initial morphology. This mechanism is schematized in Fig. 5͑c͒.

After such long shearing times, two different equilibrium morphologies can thus be obtained. In some cases, due to the thinning of the stripes of PDMS, the local phase inversion propagates into the whole sample ͓Figs. 4͑c͒ and 4͑d͔͒. Consequently, a general flow-induce phase inversion occurs. In other cases, the stripes of HPC50% thin and disappear. The flow-induce phase inversion does not take place. As can be seen in Fig. 4͑e͒, the resulting fina morphology is in this case a continuous phase of PDMS with filament and droplets of HPC50% that themselves contain subinclusions of PDMS.

We did not note any influenc of the radial gradient of the shear rate over the width of visualization during these experiments. Indeed the formation and the broadening of the different stripes did not occur preferentially at one side of the window. Some experiments were performed at higher gaps ͑up to 500 m͒ to check the weak influenc of the gap. We observed with less sharpness the morphologies of the blends but their evolution was similar as for small gaps ͑50 or 100 m͒. This was confirme once the shear was stopped at different stages, and the gap was slowly decreased up to the point where it was possible to observe the morphology distinctly. This is not direct in situ experiments, but interrupted ones.

Evolution of the final morphology after cessation of flow

Once the fina morphology was reached, the shear was stopped. For the blends where the phase inversion did not occur, a morphology similar to the initial morphology before shearing, but much finer develops a few minutes after cessation of flow The PDMS is the continuous phase, and the dispersed phase is composed of droplets of HPC50% that encapsulate between one and three droplets of PDMS. This morphology is similar to the morphology of high-impact polystyrene.

If the flo is stopped after a phase inversion, a completely opposite morphology is observed. Immediately after stopping, the shear-deformed droplets of PDMS relax into spherical drops. The drops of PDMS coalesce but remain in the dispersed phase. In order to confir the stability of the inverted morphology ͑blends with HPC50% continuous and PDMS dispersed͒ and the nonreversibility of the phase inversion, these blends were resheared 2 min after cessation of flo at 0.2 s Ϫ1 , a shear rate where PDMS should be the continuous phase. But even after 15 h of shear, the HPC50% was still the continuous phase as shown in Fig. 4͑f͒.

B. Influence of polymer concentration and shear rate on the final steadystate morphology

At 10 s Ϫ1 , whether a phase inversion is induced or not depends on concentration. As shown in Fig. 6, the occurrence of phase inversions observed during the tests at a given concentration increases with the concentration of HPC50%. With 60% of PDMS, the phase inversion is always observed. With 70% of PDMS, a phase inversion is never induced. Between these two limits, there is a range of concentrations where, for the same blend, we obtain in some cases the inverted morphology after shearing and in other cases not. The HPC50% viscosity being very sensitive to temperature, small variations could modify the viscosity ratio. Furthermore because the initial morphologies are not well controlled for the reasons discussed above, small differences in the at-rest morphologies ͑size of the inclusions, volume of PDMS encapsulated or small variations of concentra-tions͒ could influenc the fina morphology after shearing at 10 s Ϫ1 near the region of phase inversion. All these parameter variations could be sufficien to shift the system into or out of the region of phase inversion.

As co-continuity was observed for a range of concentrations instead of a single point of phase inversion, the critical concentration for flow-induce phase inversion cannot be determined by one point. There is rather a range of concentration for the blend to ''hesitate'' between both morphologies. Nevertheless, without extraction experiments we cannot confir that the fina steady-state morphology is passing through a transient cocontinuous morphology.

Comparison with models based on viscosity ratio and elasticity ratio

Our observations can be compared with the different models presented in the literature. From Eqs. ͑1͒, ͑2͒, and ͑5͒, the viscosity ratio necessary to have HPC50% as the continuous phase can be calculated for each concentration. The minimal shear rate to apply for having a continuous phase of HPC50% can be inferred from the rheological data in steady state ͓see Fig. 1͑a͔͒. But since in the absence of interfacial slip the stresses are continuous across the interface while the deformation rates are not, the models have in common the fact that they consider only the viscosity ratio should be used at constant stress as reported by Utracki ͑1991͒. Calculations based on constant stress of one of the two phases give similar results as made at constant shear rate for the models of Miles and Zurek ͓Eq. ͑1͔͒ and Metelkin and Blekht ͓Eq. ͑2͔͒ ͑see Table II͒. Data calculated with the model of Utracki ͓Eq. ͑5͔͒ depend on the way the viscosity ratio is calculated and cannot be compared with our experimental results. In the case of Bourry and Favis ͓Eq. ͑6͔͒, the necessity to know GЈ and GЉ at constant shear stress ͑that we did not measure͒ does not allow us to perform such a comparison. At 10 s Ϫ1 , PDMS and HPC50% have about the same elasticity ͑see Table I͒, which strongly simplifie evaluation of the influenc of viscosity and elasticity on the point of phase inversion. Nevertheless, the models of Miles and Zurek ͓Eq. ͑1͔͒ and Metelkin and Blekht ͓Eq. ͑2͔͒ have limits. For shear rates above 7 s Ϫ1 , they predict a phase inversion on all of our blends. However, refinemen is observed for some blends with 63% of PDMS or more.

For shear rates below unity, the elasticity ratio and viscosity ratio are greater than unity. They should have opposite effects on the morphology. In order to test the influenc of elasticity on the fina morphology, a blend of PDMS63 was sheared at 0.8 s Ϫ1 ͑viscosity ratio ϭ 1) for 30 h ͑i.e., 86 400 strain units͒. After a few hours, the stripe morphology appeared and no more evolution of the morphology was observed during the following 20 h of shearing. We did not get a phase inversion, as predicted by Eqs. ͑1͒ and ͑2͒ that nevertheless do not take into account the elasticity ratio.

Between 10 and 30 s Ϫ1 , the elasticity ratio changes with the shear rate ͑from 1 to 0.5͒ while the viscosity ratio remains at a constant value of 0.4. By increasing the shear rate and so decreasing the elasticity ratio, elasticity should limit the phase inversion. On the contrary, we observed with PDMS63 blends either phase inversion or refinemen with even a slight increase of the rate of occurrence of phase inversion ͑Fig. 7͒. a Fitted at 10 s Ϫ1 with our experimental results, using R 0 ϭ 23 m.

FLOW-INDUCED PHASE INVERSION

The elasticity ratio does not seem to influenc the concentration of shear-induced phase inversion for elasticity ratios between 0.5 and 10. But we have clear experimental limitations for exploring higher elasticity differences.

Comparison with the model of Willemse et al.

Although the model of Willemse et al. ͓Eq. ͑7͔͒ presents the disadvantage of having the phase dimension as an unknown parameter, it has the advantage of taking into account more mixing parameters such as viscosity of the major phase, shear rate, and interfacial tension, which are well define in our experiments. Contrary to the other models, this model does not allow us to predict a point of phase inversion but a minimal concentration so that the initial dispersed phase becomes a continuous phase. Equation ͑7͒ was used to predict the effects of concentration or shear rate by using the phase dimension R 0 as a f t parameter.

At 10 s Ϫ1 , the minimal concentration to have HPC50% as a continuous phase is experimentally known: phase inversion is observed with a minimal concentration of HPC50% of 33% ͑blend PDMS67͒. Knowing the shear rate, concentration of HPC50% (⌽ HPC50%,cc ), matrix viscosity and interfacial tension R 0 can be calculated with Eq. ͑7͒. This is justifie because the phase dimension R 0 only changes during the initial stage of mixing. By supposing that the phase dimension is slightly influence by concentration on this concentration range, the critical shear rates for phase inversion can be calculated for the different concentrations. The results are presented in Table II ͑with R 0HPC50% ϭ 23 m for PDMS67͒.

Ghodgaonkar and Sundararaj ͑1996͒ proposed a formulation to predict D, the dispersed phase drop diameter in viscoelastic blends

D ϭ 2 m ␥ ˙Ϫ2͑G d ЈϪG m Ј ͒ , ͑8͒
with G d Ј and G m Ј the storage moduli of the dispersed and matrix phases. If we calculate D with HPC50% as the dispersed phase and PDMS as the matrix, we do not fin major differences in the range of shear rates calculated with Eq. ͑7͒ ͑see Fig. 8͒. This can justify the use of a constant phase dimension R 0 in Table II. Because of the qualitative deter- mination of ⌽ HPC50%,cc and R 0 , these results do not allow us to determine quantitative values but to highlight a trend: the minimal shear rate to apply for having HPC50% as the continuous phase increases with concentration of PDMS. By reversing the role of PDMS and HPC50% in Eq. ͑7͒, the apparent stability of the inverted morphology ͑blends with HPC50% continuous and PDMS dispersed͒ and the nonreversibility of the phase inversion at low shear rates can be understood. We calculated at 0.2, 0.8, and 10 s Ϫ1 the minimal phase dimension R 0 of PDMS necessary to return as the continuous phase. With 67% of PDMS, R 0 of PDMS should be larger than 263 m at 0.2 s Ϫ1 , 108 m at 0.8 s Ϫ1 , and 46 m at 10 s Ϫ1 to return as the continuous phase. This is much larger than what is experimentally observed at 0.2 s Ϫ1 ͓Fig. 4͑f͔͒ and 10 s Ϫ1 ͓Fig. 4͑d͔͒ or calculated with Eq. ͑8͒ ͑see Fig. 8͒.

C. Phase change dynamics: Time necessary to reach the steady-state morphology

The existing models give no information about the duration of the flo before the occurrence of phase inversion. Lazo and Scott ͑1999͒ made the assumption that the relevant parameter in the kinetics of phase inversion of PS/PE blends is the rate of hole nucleation in the forming sheets. But the sheets are formed with the least viscous component. They are so more easily stretched than if they were formed with the most viscous component. Phase inversion occurs after 1000-3500 strain units. In our case, the sheets of PDMS seem to be more stable. Our observations have shown that the time necessary to reach the inverted morphology is very long. The mechanisms leading to the fina steady-state morphology seem to occur at the edge of the sheets and not on their whole surface as proposed by this hole nucleation theory. Furthermore, it does not explain the case of refinement This mechanism is certainly present but not prevalent in our blends.

We can attribute a characteristic time to the different identifie transient stages of the flow For each type of blend, measurements were carried out for at least four or fiv samples. We defin t 0 as the start of shearing, t 1 as the time when the large domains of PDMS emerge ͑equivalent to the beginning of the second stage of morphology as described in the preceding paragraph͒, t 2 as the beginning of the stripe morphology ͑equivalent to the beginning of the fourth stage of morphology͒ and t 3 as the time necessary to reach the steady-state morphology ͑equivalent to the end of the fourth state of morphology͒. The boundary between the different stages is not very sharp, especially between the second and third stage. Actually, formation of the large domains of PDMS can continue while others are already being stretched to form stripes. Furthermore, it is difficul to determine the end of the last stage for the blends for which the inverted morphology does not occur. By contrast, for the blends that invert their morphology, we can easily determine the time t 3 when the last filamen of PDMS breaks up in the matrix of HPC50%. The results are given in Tables III andIV. It is clear that most of the time necessary to reach the steady-state morphology ͑refinemen or phase inversion͒ is devoted to the t 3 process. Compared to t 2 and t 3 , the time necessary to obtain the bifibrilla morphology is very short, being nevertheless a few hundred strain units. The time t 2 necessary for obtaining the stripe morphology is much larger, and the corresponding deformation is a few thousands strain units. The limiting process is the interplay between coalescence and breakup that occurs during the fourth stage. It is also during this stage that the system will either refin its morphology or phase invert. The description of the phase inversion must thus take into account this stage. The other stages can be ignored as a firs approximation. We will now go into more details on the influenc of concentration and shear rate on the transition to the fina morphology. We recorded large discrepancies ͑up to 20%͒ in characteristic times between samples at the same concentration. These discrepancies can be explained by the different initial morphologies. The size of the inclusions of HPC50%, and the size and especially the volume of encapsulated droplets of PDMS can be slightly different from one sample to another. The presence and the total volume of encapsulations of PDMS in the sample probably influence the duration of the mechanism of phase inversion. For samples of the same concentration, the phase inversion occurs earlier in blends containing a larger volume of encapsulated PDMS. Considering the experimental variability due to the different initial morphologies and the difficult of identifying t 1 and t 2 , it is difficul to measure very precisely the influenc of concentration on t 1 and t 2 . On the other hand, its influenc on t 3 can easily be measured.

The results for t 1 , t 2 , and t 3 are reported in Table III. Only t 3 is plotted in Fig. 6. It is important to observe how, in term of t 3 , concentration has opposite effects on phase inversion and refinement Below a concentration of 60%, only phase inversion occurs. Increasing the concentration increases the time taken for phase inversion. The time taken follows the reverse trend, decreasing with increasing concentration. At 63%, refinement are sometimes obtained. At 70%, we observe only refinemen as the steady-state morphology.

If we fi the two sets of points in Fig. 6 with straight lines, it is interesting to note that the two slopes have similar absolute values. This may indicate that these phenomena are similarly proportional to concentration, in this range of concentration. This makes sense if we recall that both steady-state morphologies are reached through the same mechanism of breaking fiber as schematized in Fig. 9. By making the assumption that the relevant parameter is t b , the time of breakage of the filament at the boundary of the stripes, t 3 will be the number of filament which must be broken multiplied by the time of breakage of one filament

t 3 ϭ n HPC50% •t bHPC50% Ͻ n PDMS •t bPDMS in the case of refinemen ͑9͒ t 3 ϭ n PDMS •t bPDMS Ͻ n HPC50% •t bHPC50% in the case of phase inversion,
where n HPC50% and n PDMS are, respectively, the average number of filament of HPC50% per stripe of PDMS and the average number of filament of PDMS per stripe of HPC50% and t bHPC50% and t bPDMS , respectively, the time of breakage of one filamen of HPC50% and one filamen of PDMS. Tsakalos et al. ͑1998͒ estimated t b , the time necessary for a drop to break through the development of capillary instabilities in blends of PDMS/HPC50%

t b ϭ 1 ␥ ˙•ͩ D 0 d b ͑␥ ˙͒ͪ 2 , ͑10͒ with
d b ϰ 1 ␥ ˙, ͑11͒
where D 0 is the initial diameter of the drop before shearing and d b (␥ ˙) the thread diameter necessary for capillary instabilities leading to breakup. By supposing that

n HPC50% ϰ ͑ 1Ϫ⌽ PDMS ͒ ͑12͒ n PDMS ϰ ⌽ PDMS
and that d b is independent of concentration ͓Tsakalos et al. ͑1998͔͒:

t 3 ϰ ͑1Ϫ⌽ PDMS ͒•A͑ ␥ ˙͒ in the case of refinemen ͑13͒ t 3 ϰ ⌽ PDMS •B͑␥ ˙͒ in the case of phase inversion,
where A(␥ ˙) and B(␥ ˙) are the terms describing dependence on the shear rate. At a given shear rate, the time to obtain a given phase as the continuous phase should be proportional to the concentration of this phase. This is what we observe in Fig. 6.

Influence of the shear rate

PDMS63 blends were sheared at various shear rates above 10 s Ϫ1 , i.e., 15, 20, and 30 s Ϫ1 . As shown in Table I, the viscosity ratio is the same at these shear rates, but not the elasticity ratio. Increasing the shear rate decreases G HPC50% Ј /G PDMS Ј by a factor 2 at 30 s Ϫ1 ͑compared to G HPC50% Ј /G PDMS Ј ϭ 1 at 10 s Ϫ1 ͒. Figure 7 shows that the amount of deformation necessary to reach equilibrium is an apparently nonmonotonically function of the shear rate. Deformation versus shear rate for refinemen and phase inversion both increase with shear rate, but not at the same rate. Once more, we see here that since the same mechanisms are responsible for both morphologies, external parameters have the same influenc on both morphologies. The question remains whether to consider the plateau of deformation between 10 and 20 s Ϫ1 for refinemen and between 20 and 30 s Ϫ1 for phase inversion as real, or as being simply due to the difficult of measuring when the steady state is really reached. The only relevant parameters acting in the process are breakup and coalescence, which are monotonically dependent on the shear rate. In addition, the viscosity ratio is constant and the elasticity ratio varies monotonically between 10 and 30 s Ϫ1 .

If we consider the breakage of filament as the relevant process, it is found from Eqs. ͑10͒ and ͑11͒ that t b is proportional to ␥ ˙. Also, at a given concentration, the deformation for having one phase to be the continuous one should be deformation ϭ ␥ ˙•t 3 ͭ ϰ ␥ ˙2•C͑1Ϫ⌽ PDMS ͒ in the case of refinemen ϰ ␥ ˙2•D͑⌽ PDMS ͒ in the case of phase inversion , ͑14͒

where C(1Ϫ⌽ PDMS ) and D(⌽ PDMS ) regroup terms dependent on concentration. At a given concentration of 63 vol %, this is not what we observe experimentally ͑Fig. 7͒. This demonstrates the limits of this simplifie model. The equations for the time necessary for a drop to break ͓Eq. ͑10͔͒ and the critical diameter of the filamen for breakup ͓Eq. ͑11͔͒ proposed by Tsakalos et al. ͑1998͒ have been established for isolated droplets and so do not take into account the influenc of neighboring threads nor the influenc of coalescence. Nevertheless, for highly concentrated blends, the surrounding drops and threads probably have an effect on the stability on the filaments Furthermore, elasticity of the components has not been considered. But the decrease of G HPC50% Ј /G PDMS Ј with increasing shear rate certainly influence d b and t b . Another question is which D 0 to consider in Eq. ͑10͒, especially in the case of t bPDMS where PDMS is the continuous phase at rest.

IV. CONCLUSIONS

By varying the shear rate applied to blends of PDMS/HPC50%, a flow-induce phase inversion can be observed and studied with rheo-optical microscopy. These observations show that phase inversion occurs after the blend passes through several different transient morphologies. A bifibrilla network structure appears during the transient flow as proposed by Luciani et al. ͑1993͒. But the subsequent transient stripe morphologies that follow the formation of sheets of PDMS demonstrate that the mechanisms leading to the inverted morphology are more complex. The blends must be subjected to larger deformations ͑strain units above 70 000͒ than it was previously reported in a blend where the phase to disperse is the minor and least viscous component ͓Lazo and Scott ͑1999͔͒. In our system, the phase to disperse is the major and most viscous component. Fixing concentration and viscosity ratio, the amount of deformation necessary to reach the phase inversion is not constant and seems to depend on shear rate. This suggests that one of the controlling parameters is the amount of deformation needed to break a filament which is known to be shear rate dependent. At a fixe shear rate of 10 s Ϫ1 , we did not observe a single clear concentration above which phase inversion would not occur. On the contrary, we rather observed a range of concentration for which either phase inversion or refine ment occurs. Again, if the rupture of the filament' mechanism is important, its dependence on the elasticity of the matrix and the filamen must be taken into account.

  authors have experimentally shown that ⌽ d,cc is slightly influence by m ͓Willemse et al. ͑1998͔͒ whereas an increase of leads to an increase of ⌽ d,cc and a decrease of the range of co-continuity ͓Willemse et al. ͑1999a͔͒.

FIG. 1 .

 1 FIG. 1. Shear viscosity ͑a͒ and N 1 ͑b͒ of PDMS and HPC50% measured at 18 °C in a plate-plate geometry.

FIG. 2 .

 2 FIG. 2.Storage and loss modulus data of PDMS and HPC50% measured at 18 °C in a cone and plate geometry.

FIG. 3 .

 3 FIG. 3. The different morphologies of the blends PDMS/HPC50% observed between crossed polarizers in the Linkam device at 18 °C. The PDMS appears black and the HPC50% bright. ͑a͒ At-rest morphology of the blend PDMS60. ͑b͒ and ͑c͒ Morphology of the blend PDMS67 during the firs stage of shearing at 10 s Ϫ1 . ͑d͒ and ͑e͒ Morphology of the blend PDMS60 and ͑f͒ the blend PDMS67 during the second stage of shearing at 10 s Ϫ1 .

FIG. 4 .

 4 FIG. 4.In the blend PDMS60: ͑a͒ the large domains of PDMS are stretched along the flo direction; ͑b͒ stripes of continuous phases of PDMS and continuous phases of HPC50% are formed; ͑c͒ the stripes of PDMS become thinner. Filaments originating from these stripes are dispersed in the continuous phase of HPC50%; ͑d͒ The stripes of PDMS have completely disappeared and the phase inversion has propagated through the whole sample. We can observe a continuous phase of HPC50% with droplets of PDMS inside. This is case ͑b͒ of Fig.5. ͑e͒ In PDMS67: at the end of the fourth stage, the stripes of local phase inversion have disappeared: the flow-induce phase inversion does not occur and the fina morphology consists of droplets of HPC50% in the continuous phase of PDMS, as shearing. The difference is the much smaller size of the HPC50% droplets ͓see Fig.3͑a͔͒. This is case ͑c͒ of Fig.5. ͑f͒ PDMS67 sheared 15 h at 0.2 s Ϫ1 after phase inversion.

FIG. 5 .

 5 FIG. 5. ͑a͒ Schematic representation of the network of threads of HPC50% in the stripes where the PDMS is the continuous phase observed during the fourth stage. ͑b͒ Schematic representation of the mechanisms observed at the edge of the stripes during the fourth stage when a f lament of HPC50% is close to the edge: the edge of PDMS breaks up and the stripe of HPC50% enlarges. ͑c͒ The f lament of HPC50% breaks up and the stripe of HPC50% thins.

FIG. 6 .

 6 FIG. 6. Blends sheared at 10 s Ϫ1 . Occurrence of phase inversion ͑*͒ vs concentration of PDMS. Duration of the transient flo t 3 vs concentration of PDMS in case of phase inversion ͑᭡͒ and refinemen ͑͒. The lines ͑ ͒ and ͑-͒ are linear fit of the experimental points.

FIG. 7 .

 7 FIG. 7.Deformation in strain units necessary to reach either the refine ͑--͒ or the phase inverted morphology ͑--᭡--͒ and frequency of occurrence ͑*͒ of phase inversion vs shear rate for the blend PDMS63.

FIG. 8 .

 8 FIG. 8.Radius of the dispersed phase versus shear rate calculated with Eq. ͑8͒ proposed by Ghodgaonkar and Sundararaj ͑1996͒.

FIG. 9 .

 9 FIG. 9. Schematized mechanisms of breakage of one filamen occurring at the boundary of the stripes that leads to the propagation of one phase.

TABLE I .

 I Viscosity ratio and storage modulus ratio of the HPC50% and the PDMS at 18 °C calculated at constant shear rate.

	Shear rate
	(1/s)	HPC50% / PDMS	G HPC50% Ј	/G PDMS Ј a
	1.5ϫ10 Ϫ3	7.3	-
	0.2	1.7	40
	0.5	1.3	11
	0.8	1	9.8
	2.5	0.7	3.1
	10	0.4	1
	15	0.4	0.8
	20	0.4	0.7
	30	0.4 b	0.5

a Assuming that the Cox-Merz rule is verified b Value extrapolated from Fig. 1͑a͒.

TABLE II .

 II Minimal shear rate to apply to have HPC50% as the continuous phase determined with the models of Miles and Zurek ͓Eq. ͑1͔͒, Metelkin and Blekht ͓Eq. ͑2͔͒, Utracki ͓Eq. ͑5͔͒, and Willemse et al. ͓Eq. ͑7͔͒ for blends of different concentrations ͓͑i͒ calculated at constant shear rate; ͑ii͒ calculated at constant shear stress of HPC50%; ͑iii͒ calculated at constant shear stress of PDMS͔.

			Shear rate ͑s Ϫ1 ͒ calculated
			with the different equations
	Blend	͓Eq. ͑1͔͒	͓Eq. ͑2͔͒	͓Eq. ͑5͔͒	͓Eq. ͑7͔͒ a
	PDMS55
	͑i͒	1.8	1.7	3.1
	͑ii͒	1.8	1.7	2.9	7.5
	͑iii͒	1.5	1.2	1.8
	PDMS60
	͑i͒	2.2	1.7	7
	͑ii͒	2.2	1.7	5.5	7.8
	͑iii͒	1.6	1.2	2.1
	PDMS63
	͑i͒	3.1	1.7	No
	͑ii͒	2.9	1.7	9	8.5
	͑iii͒	1.8	1.2	2.5
	PDMS67
	͑i͒	4.6	1.8	No
	͑ii͒	4	1.8	13	10
	͑iii͒	2	1.5	2.8
	PDMS70
	͑i͒	7	1.8	No
	͑ii͒	5.5	1.8	30	14
	͑iii͒	2.1	1.5	31

TABLE III .

 III Characteristic times of the three identifie steps to reach the fina steady-state morphology, for blends of different concentrations sheared at 10 s Ϫ1 .

	Final
	steady-state		t 1	t 2	t 3	␥ ˙t3
	morphology	Blend	͑s͒	͑s͒	͑s͒	͑strain units͒
	Phase inversion	PDMS60	23	310	7000	70 000
		PDMS63	47	160	7600	76 000
		PDMS67	60	260	10 400	104 000
	Refinemen	PDMS63	54	180	4300	43 000
		PDMS67	330	910	5700	57 000
		PDMS70	23	61	1100	11 000

TABLE IV .

 IV Characteristic times and deformation necessary to reach the steady-state morphology at different shear rates, for the blend PDMS63.

	1.
	Final
	steady-state morphology	␥ ͑s Ϫ1 ͒	t 1 ͑s͒	␥ ˙t1	t 2 ͑s͒	␥ ˙t2	t 3 ͑s͒	␥ ˙t3
	Phase inversion	10	47	470	160	1600	7600	76 000
		15	32	480	90	1350	4900	73 500
		20	11	220	440	8800	7900	158 000
		30	23	690	100	3000	5500	165 000
	Refinemen	10	53	530	140	1400	7300	73 000
		15	24	360	116	1700	3200	48 000
		20	35	700	120	2400	3800	76 000
		30	29	870	88	2640	2700	81 000

Influence of the concentration at a shear rate of 10 s ؊1

FLOW-INDUCED PHASE INVERSION

FLOW-INDUCED PHASE INVERSION

FLOW-INDUCED PHASE INVERSION