Pierre Guillot
email: guillot@math.unistra.fr

THE GROTHENDIECK-TEICHMÜLLER GROUP OF PSL(2, q)

We show that the Grothendieck-Teichmüller group of PSL(2, q), or more precisely the group GT 1 (PSL(2, q)) as previously defined by the author, is the product of an elementary abelian 2-group and several copies of the dihedral group of order 8. Moreover, when q is even, we show that it is trivial.

We explain how it follows that the moduli field of any "dessin d'enfant" whose monodromy group is PSL(2, q) has derived length ≤ 3.

This paper can serve as an introduction to the general results on the Grothendieck-Teichmüller group of finite groups obtained by the author.

Introduction & Statement of results

In [Gui], we have introduced the Grothendieck-Teichmüller group of a finite group G, denoted GT (G). Motivation for the study of this group stems from the theory of dessins d'enfants. Recall that a dessin is essentially a bipartite graph embedded on a compact, oriented surface (without boundary), and that the absolute Galois group Gal(Q/Q) acts on (isomorphism classes of) dessins. As explained in loc. cit., there is an action of GT (G) on those dessins whose monodromy group is G, and the Galois action on the same objects factors via a map Gal(Q/Q) -→ GT (G).

Motivation for the study of all groups GT (G), for all groups G, is increased by the fact that the combined map

Gal(Q/Q) -→ GT := lim G GT (G) is injective.
The group GT (G) possesses a normal subgroup GT 1 (G), which is such that the quotient GT (G)/GT 1 (G) is abelian. It follows that the commutator subgroup of Gal(Q/Q) maps into GT 1 (G), and injects into the inverse limit GT 1 formed by these as G varies. There is little mystery left in GT (G)/GT 1 (G) (see [Gui] again), and the challenge is in the computation of GT 1 (G).

In this paper we treat the case of G = PSL(2, q). We obtain the following result.

Theorem 1.1 -The group GT 1 (PSL(2, 2 s )) is trivial for all s ≥ 1.

The group GT 1 (PSL(2, q)), when q is odd, is isomorphic to a product

C n1 2 × D n2 8 .
Here D 8 is the dihedral group of order 8. Note that this result was observed experimentally for small values of q in [Gui].

This theorem depends crucially on the work of MacBeath in [START_REF] Macbeath | Generators of the linear fractional groups, Number Theory[END_REF], which classifies the triples (x, y, z) in PSL(2, q) in various ways. Indeed, we feel that the group GT 1 (PSL(2, q)) encapsulates part of this information neatly.

Let us give an application to dessins d'enfants. The first part of the next theorem was implicit in [Gui], and indeed it hardly deserves a proof once the statement is properly explained. However, it seems worth spelling it out for emphasis.

Theorem 1.2 -Let G be a finite group. There exists a number field K, Galois over Q, such that Gal(K/Q) is a subgroup of GT (G), and containing the moduli field of any dessin whose monodromy group is G.

For example, suppose that X is a dessin whose monodromy group is PSL(2, q). If q is even, then the moduli field of X is an abelian extension of Q. If q is odd, then the Galois closure F of the moduli field F of X is such that Gal( F /Q) has derived length ≤ 3.

A word of explanation. First, when Γ is a group we write Γ ′ for the derived (commutator) subgroup, and we say that Γ has derived length ≤ 3 when Γ ′′′ is trivial. Also, the moduli field of a dessin is the extension F of Q such that Gal(Q/F ) is the stabilizer of the isomorphism class of X under the Galois action. Note that if we can write down explicit equations for X with coefficients in the number field L, then certainly the moduli field F is a subfield of L. While there are subtle counterexamples of dessins for which there are no equations over F , it is still intuitively helpful to think of F as the smallest field over which the dessin is defined.

For example in [START_REF]An elementary approach to dessins d'enfants and the Grothendieck-Teichmüller group[END_REF], Example 4.6 and Example 4.13, we have examined a certain dessin X (a planar tree), whose monodromy group is the simple group of order 168, that is PSL(2, 7) (or PSL(3, 2), as it is written in loc. cit.). We found explicit equations with coefficients in a field of the form Q(α) with the minimal polynomial of α having degree 4 (though not all details are provided); if L is the Galois closure of Q(α), then Gal(L/Q) is a subgroup of S 4 , which has derived length 3, confirming the prediction. However, there is even an easier way to see that the moduli field is very simple: there are only two dessins in the Galois orbit of X, so the moduli field is in fact a quadratic extension of Q.

It is an open problem to explicitly exhibit a dessin such that Gal( F /Q) is nonabelian.

The examples treated in this paper are a less technical illustration of the ideas discussed in [Gui], and may serve as an introduction to the latter. Note that, motivation and background aside, it is not necessary to be familiar with [Gui] in order to follow the arguments we present, leading to the computation of GT 1 (PSL(2, q)).

Definitions

We take a definition of GT 1 (G) which is only suitable when G is non-abelian and simple, such as G = PSL(2, q) ; see [Gui] for the more general definition.

So let G be such a finite group, and let T denote the set of triples (x, y, z) ∈ G 3 such that xyz = 1 and x, y, z = G. Further, we let T /G denote the set of orbits in T under simultaneous conjugation by an element of G. We write [x, y, z] for the class of (x, y, z). (In [Gui] we write P instead of T , thinking of these elements as pairs (x, y).)

There is a free action of Out(G), the group of outer automorphisms of G, on T /G. Moreover, there is also an action of S 3 , the symmetric group of degree 3. This is essentially a permutation of the coordinates, but to be more precise, one usually introduces the permutation θ of T /G defined by θ • [x, y, z] = [y, x, z x ], and the permutation δ defined by δ•[x, y, z] = [z, y, x y ]. These are both well-defined, and square to the identity operation of T /G. There is a homomorphism S 3 → S(T /G), where S(T /G) is the symmetric group of the set T /G, mapping (12) to θ and (13) to δ.

The two actions described commute, and together define an action of

H := Out(G) × S 3 on T /G. Let us write [x, y, z] ≡ [x ′ , y ′ , z ′ ] when
x is a conjugate of x ′ , while y is a conjugate of y ′ , and z is a conjugate of z ′ . This is an equivalence relation on T /G.

The group GT 1 (G) is defined, in this context, to be the subgroup of the symmetric group S(T /G) comprised by those permutations ϕ which:

• commute with the action of H; in other words, if h ∈ H, t ∈ T /G then ϕ(h• t) = h • ϕ(t). • are compatible with ≡; that is, t ≡ t ′ implies ϕ(t) ≡ ϕ(t ′ ), if t, t ′ ∈ T /G.
(Somewhat arbitrarily, we write h • t for the action of h ∈ H, and ϕ(t) for the action of ϕ ∈ GT 1 (G), in order to set the elements of GT 1 (G) apart.)

Characteristic two

We start by assuming that q is a power of 2, so that PSL(2, q) = SL(2, q). Following MacBeath [START_REF] Macbeath | Generators of the linear fractional groups, Number Theory[END_REF], we partition the set of triples (x, y, z) of elements of SL(2, q) satisfying xyz = 1 into the subsets E(a, b, c), where a, b, c ∈ F q , by requiring (x, y, z) ∈ E(a, b, c) when Tr(x) = a, Tr(y) = b, Tr(z) = c (here Tr is the trace).

Since elements of GT 1 (SL(2, q)) are assumed to be compatible with the relation ≡, the following observation is trivially true.

Lemma 3.1 -Suppose (x, y, z) ∈ E(a, b, c), with x, y, z = SL(2, q), let ϕ ∈ GT 1 (SL(2, q)), and suppose that x ′ , y ′ , z ′ satisfy

ϕ([x, y, z]) = [x ′ , y ′ , z ′ ] . Then (x ′ , y ′ , z ′ ) ∈ E(a, b, c).
Note that SL(2, q) acts on E(a, b, c) by simultaneous conjugation. The crucial point is this: Proposition 3.2 (after MacBeath) -When the set E(a, b, c) contains a triple (x, y, z) such that x, y, z = SL(2, q), it consists of just one conjugacy class.

Proof. In [START_REF] Macbeath | Generators of the linear fractional groups, Number Theory[END_REF], the triples (a, b, c) are divided into the "singular" ones and the "non-singular" ones ; also, the type of (x, y, z) is the type of (Tr(x), Tr(y), Tr(z)) by definition. Theorem 2 asserts that when (x, y, z) is singular, the group x, y, z is "affine", and in particular it is not all of SL(2, q). Our hypothesis guarantees thus that (a, b, c) is non-singular.

We may then apply (ii) of Theorem 3 in loc. cit., giving the result.

Corollary 3.3 -The group GT 1 (SL(2, q)) is trivial.

Proof. Let ϕ ∈ GT 1 (SL(2, q)). Any t ∈ T /G is of the form t = [x, y, z] with (x, y, z) ∈ E(a, b, c) for some a, b, c, and x, y, z = SL(2, q) by definition. The Lemma applies, showing that ϕ(t) = [x ′ , y ′ , z ′ ] with (x ′ , y ′ , z ′ ) ∈ E(a, b, c), while the Proposition proves that all triples in E(a, b, c) are in fact conjugate. As a result ϕ(t) = t.

Odd characteristics

Now we assume that q = p s is a power of the odd prime p, and we turn to the description of GT 1 (G) where G = PSL(2, q). 4.1. Sets of triples. As in the previous section, we define E(a, b, c) to be the set of triples (x, y, z) ∈ SL(2, q) 3 such that xyz = 1 and with Tr(x) = a, Tr(y) = b, Tr(z) = c. We also define E(a, b, c) to be the subset of E(a, b, c), which may well be empty, of triples generating SL(2, q) (or equivalently, whose images generate G). Finally, we write PE(a, b, c) for the image of E(a, b, c) in G 3 .

Lemma 4.1 -The notation behaves as follows.

(1) If PE(a, b, c) and PE(a ′ , b ′ , c ′ ) are not disjoint, then they are equal, and (a ′ , b ′ , c ′ ) = (±a, ±b, ±c) for some choices of signs.

(2) We have

PE(a, b, c) = PE(-a, -b, c) = PE(-a, b, -c) = PE(a, -b, -c) .
In other words, the set PE(a, b, c) is not altered when an even number of signs are introduced.

(3) When abc = 0, all choices of signs give the same set PE(±a, ±b, ±c). (4) When abc = 0, the sets PE(a, b, c) and PE(a, b, -c) are disjoint.

Proof. (1) An element (g, h, k) ∈ PE(a, b, c) is of the form (x, y, z), where x, y, z ∈ SL(2, q) and the bar denotes the morphism to G, where the traces of these elements are a, b, c respectively. If (g, h, k) also belongs to PE(a ′ , b ′ , c ′ ), given that the possible lifts of g, h, k are ±x, ±y, ±z respectively, we see that 3)-(4) (since these properties imply that PE(a, b, c) and PE(±a, ±b, ±c) are either equal or disjoint).

a ′ = ±a, b ′ = ±b, c ′ = ±c. The fact that PE(a, b, c) = PE(a ′ , b ′ , c ′ ) will follow from (2)-(
(2) If (x, y, z) ∈ E(a, b, c), then (-x, -y, z) ∈ E(-a, -b, c), and these two triples map to the same element in G 3 . This shows that an element of PE(a, b, c) also belongs to PE(-a, -b, c), and conversely. The other arguments are similar.

(3) If abc = 0, then one of a, b, c is 0, say a = 0, so that a = -a. We are thus free to change the sign of a, and an even number of other signs, which gives the result.

(4) If x ′ = ±x, and Tr(x ′ ) = Tr(x) = 0, then x ′ = x. We see thus that, whenever two triples (x, y, z) ∈ E(a, b, c) and (x ′ , y ′ , z ′ ) ∈ E(a, b, -c) map to the same element of G 3 , we must have x ′ = x and y ′ = y, so that z ′ = z since xyz = 1 = x ′ y ′ z ′ . This is a contradiction since the traces of z and z ′ are c = 0 and -c. As a result, PE(a, b, c) and PE(a, b, -c) are disjoint in this case.

Example 4.2 -Trying the example of PSL(2, 5), one finds that PE(0, 2, 3) is nonempty, showing that the case abc = 0 does occur non-trivially. The set PE(2, 2, 4) is also non-empty, as is PE(2, 2, -4), so the case abc = 0 occurs and states here the disjointness of non-empty sets. However, PE(1, 2, 4) is non-empty, but PE(1, 2, -4) is empty, an instance where (4) still holds, but in a degenerate way.

We define finally

T (a, b, c) = signs PE(±a, ±b, ±c) = PE(a, b, c) ∪ PE(a, b, -c) .
This is a subset of T , and T (a, b, c)/G is a subset of T /G. As (a, b, c) varies, the subsets T (a, b, c)/G are disjoint, and constitute an initial partition of T /G. Lemma 4.3 -The subset T (a, b, c)/G is stable under the action of GT 1 (G).

Proof. Suppose ϕ ∈ GT 1 (G), and ϕ([g, h, k]) = [g ′ , h ′ , k ′ ], with g, h, k, g ′ , h ′ , k ′ ∈ G.
Since ϕ is compatible with ≡ by definition, we see that g ′ is conjugate to g within G; writing g = x for x ∈ SL(2, q), and similary g ′ = x ′ , we conclude that x ′ is a conjugate of ±x, so Tr(x ′ ) = ± Tr(x). Similar considerations apply to h and h ′ , and to k and k ′ .

We conclude that if (g, h, k) ∈ PE(a, b, c), then (g ′ , h ′ , k ′ ) ∈ PE(±a, ±b, ±c), as we wanted.

1 0 0 -1 ∈ GL(2, q) SL(2, q) .
One verifies that α is not inner (below we recall the description of Out(G)). Moreover, since conjugate matrices have the same trace, we see that the action of α on the triples in T also preserves the sets PE(a, b, c).

Proposition 4.5 (after MacBeath) -When PE(a, b, c) is non-empty, it is made of precisely two conjugacy classes, which are exchanged by α.

Proof. First we argue as in Proposition 3.2, relying on (i) of Theorem 3 in [START_REF] Macbeath | Generators of the linear fractional groups, Number Theory[END_REF].

The conclusion is that when E(a, b, c) is non-empty, that is when E(a, b, c) contains a triple generating SL(2, q), then E(a, b, c) consists of two conjugacy classes exactly. If (x, y, z) ∈ E(a, b, c), then (α(x), α(y), α(z)) cannot be in the conjugacy class of (x, y, z), lest we should conclude that α is inner (here we view α as an automorphism of SL(2, q), rather than G). However (α(x), α(y), α(z)) ∈ E(a, b, c), showing that E(a, b, c) intersects both conjugacy classes in E(a, b, c), and that E(a, b, c) = E(a, b, c).

When α is viewed as an automorphism of G, it is still non-inner. So the same reasoning applies, showing that there are triples in PE(a, b, c) which are not conjugate to one another, and more precisely that (g, h, k) and (α(g), α(h), α(k)) are never conjugate. The Proposition has been proved.

The cardinality of PE(a, b, c)/G is thus 2, when it is not 0; and T (a, b, c)/G contains 2 or 4 elements (or 0). These sets are unions of orbits of α (recall that Out(G) acts freely on T /G).

The action of H. Recall that we write

H = Out(G) × S 3 . According to [Wil09], Theorem 3.2, when G = PSL(2, p s ) with p odd, we have Out(G) = α × Gal(F p s /F p ) ∼ = C 2 × C s .
Here α is as above, and the Galois group acts on matrix entries in the obvious way. In particular, note that α is central in H. Now suppose that (a, b, c) is a fixed triple, and let H 0 denote the subgroup of H leaving the subset T (a, b, c)/G stable, assuming the latter is non-empty. Note that α ∈ H 0 .

Lemma 4.6 -The permutation group induced by H 0 on the set T (a, b, c)/G is isomorphic to either C 2 , or C 2 2 , or D 8 . The same can be said of the centralizer of this permutation group in the symmetric group S(T (a, b, c)/G).

Proof. If T (a, b, c)/G has only 2 elements, there is nothing to prove, so we turn to the alternative, namely, we assume that this set has 4 elements. These are freely permuted by α, which has order 2, so they may be numbered 1, 2, 3, 4 in such a way that α acts as (12)(34).

The centraliser of α in S 4 is isomorphic to D 8 , generated, say, by (12) and (13)(24). Since α is central in H, we have a map H 0 -→ D 8 , and the first part of the Lemma is about its image. The non-trivial subgroups of D 8 are all of the form indicated, except for the presence of cyclic groups of order 4.

So we assume that

h = α i σπ ∈ α × Gal(F q /F p ) × S 3 = H
belongs to H 0 and acts as a 4-cycle on T (a, b, c)/G, and work towards a contradiction. First, we may replace h by αh if necessary, and assume that i = 0, that is h = σπ. The element π ∈ S 3 has order 1, 2 or 3; if it has order 3, we replace h by h 3 = σ 3 π 3 = σ 3 and we are reduced to the case when π = 1. So we assume that the order of π divides 2.

Elements of order 4 in D 8 , when squared, give the non-trivial central element, here (12)(34). Thus h 2 = σ 2 acts as α does. However, this is a contradiction, since α and σ belong to Out(G), which acts freely on T /G, while α = σ 2 does not hold.

This proves the first part. For the second part, since α ∈ H 0 , we note that the centralizer in question must centralize (12)(34), so it is a subgroup of the D 8 under consideration. The centralizer, in D 8 , of a subgroup which is not cyclic of order 4 is again not cyclic of order 4, as is readily checked. 4.4. The partition of T /G. We now let

X(a, b, c) = h∈H h • T (a, b, c)/G .
As a, b, c vary, the subsets X(a, b, c) provide a partition of T /G. Note that, given the description of H (and Out(G)), we certainly have, for any h ∈ H,

h • T (a, b, c)/G = T (a ′ , b ′ , c ′ )/G for some a ′ , b ′ , c ′ .
Lemma 4.7 -Let GT 1 (G) abc be the permutation group on X(a, b, c), consisting of those permutations commuting with the action of H, and compatible with the relation ≡. Then GT 1 (G) is the direct product of the various groups GT 1 (G) abc .

Proof. This is a completely general fact: when T /G is partitioned into subsets which are stable under the action of H, and which are unions of equivalence classes for ≡, then GT 1 (G) splits as a corresponding direct product, as one sees from the definition. Now suppose a, b, c are fixed, and resume the notation H 0 from the previous section.

Lemma 4.8 -The permutation group GT 1 (G) abc is isomorphic to one of {1}, C 2 , C 2 2 , or D 8 . Proof. Since the action of GT 1 (G) abc commutes with that of H, it is determined by its restriction to T (a, b, c)/G. In other words, the map GT 1 (G) abc → S(T (a, b, c)/G), which is well-defined since T (a, b, c)/G is stable under GT 1 (G), is injective.

The image Γ of that map is a permutation group which commutes with the action of H 0 , and so by Lemma 4.6 it is a subgroup of either C 2 , C 2 2 or D 8 . Thus it remains to prove that Γ is not cyclic of order 4, which potentially could happen when the centralizer C of H 0 is isomorphic to D 8 . Indeed, suppose Γ contains a 4-cycle. We infer that GT 1 (G) acts transitively on T (a, b, c)/G. It follows that the equivalence relation ≡, preserved by GT 1 (G), is trivial, in the sense that it has just one class in this set: all the triples in T (a, b, c) are "coordinate-wise conjugate". Thus the same can be said of ≡ on all the translates h • T (a, b, c)/G, easily. As a result, these translates are precisely the equivalence classes of ≡ on X(a, b, c) (see Remark 4.4).

However, let us now consider the action of the full centralizer C ∼ = D 8 , extended to all of X(a, b, c) by requiring commutation with the action of H. Given the description of the classes of ≡, it is clear that C is compatible with this equivalence relation. We conclude that GT 1 (G) abc contains a copy of D 8 , and in particular it is not cyclic of order 4.

The last two lemmas establish that, as announced: Theorem 4.9 -When q is a power of an odd prime, there exist integers n 1 , n 2 such that GT 1 (PSL(2, q)) ∼ = C n1 2 × D n2 8 .

In [Gui], explicit examples have been computed (with the help of the GAP software). We found the following table. The first line is in accordance with the isomorphism PSL(2, 5) ∼ = PSL(2, 4).

Application to dessins

We will conclude the paper with a proof of Theorem 1.1. Recall that Gal(Q/Q) acts on the isomorphism classes of dessins, and that the action on those dessins with monodromy group G factors via a certain map

λ G : Gal(Q/Q) -→ GT (G) .
If K is that field such that Gal(Q/K) = ker(λ G ), then K/Q is Galois and Gal(K/Q) is identified with a subgroup of GT (G).

The moduli field of the dessin X is that field F such that Gal(Q/F ) is the subgroup of elements stabilizing X (up to isomorphism). This subgroup contains ker(λ G ) if the monodromy group of X is G, so that F ⊂ K. This proves the first part of the Theorem. Now we specialize to G = PSL(2, q). If q is even, then GT 1 (G) = 1, so that GT (G) is abelian (since the commutators belong to GT 1 (G)). In this case K/Q is an abelian extension of Q, as is F/Q in the notation above.

When q is odd, we can at least state that GT 1 (G) is of derived length ≤ 2. As a result, the derived length of GT (G) is ≤ 3. The same can be said of Gal(K/Q) and of Gal( F /Q), where F ⊂ K is the Galois closure of F .

  Remark 4.4. Similar arguments show that T (a, b, c)/G is a union of equivalence classes for ≡. 4.2. Number of conjugacy classes of triples. The action of G on T by (simultaneous) conjugation restricts to an action on each set PE(a, b, c), clearly. Moreover, let us introduce the automorphism α of G induced by conjugation by


Cedric Quinard

Jiu Peng Song

Thierry Barriere
email: thierry.barriere@univ-fcomte.fr

Jean C Gelin

Elaboration of PIM feedstocks with 316L fine stainless steel powders for the processing of micro-components

Elaboration of PIM feedstocks with 316L fine stainless

steel powders for the processing of mico-components

Introduction

Manufacturing of metallic micro-components has rapidly increased in the recent years due to the increasing demands from various applications, such as sensors, medical and energy devices [START_REF] Federzoni | Micro-channelled heat exchanger manufactured by ceramic injection moulding and co-firing[END_REF]. Powder injection moulding (PIM) is now used to produce micro-parts since 15 years [START_REF] Rota | Wear resistant tools for reproduction technologies produced by micro powder metallurgy[END_REF], but until now there is no commercial available feedstock based on powders with particles size smaller than 6 μm [START_REF] Barriere | Experimental and numerical investigations on the properties and quality of parts produced by MIM[END_REF]. More and more investigations are currently focused on the manufacturing of micro-components with very fine stainless steel or other metallic powders [START_REF] Wang | Effects of Mold Dimensions on Rheological of Feedstock in Micro Powder Injection Molding[END_REF]. On the other hand, the final mechanical properties and resulting microstructures of the sintered microcomponents made by micro-powder injection moulding (μPIM) is becoming another important issue as well as the dimensional accuracy.

μPIM which combines the advantages of plastic micro-injection moulding and powder metallurgy is a competitive net-shape forming technology for manufacturing three dimensional micro-components in large batch [START_REF] German | Status of PIM Research and Development[END_REF][START_REF] Piotter | Micro PIM moves into the zone of industrial possibility[END_REF][START_REF] Kukla | Micro injection moulding[END_REF]. The basic steps of μPIM process include mixing, injection moulding, debinding and sintering. Mixing step aims to form the feedstock for injection moulding by mixing the metallic or ceramic powders with thermoplastic binders. Injection moulding stage leads to the green parts by filling the mould cavities with powder/binder mixture. The debinding step is used to remove the binder in the injection moulded parts without shape distortion by thermal, solvent or catalytic methods. In the subsequent sintering step, the debinded components are heated at a temperature just below the melting point of the main metallic constituent in order to obtain the required final density by bonding the powder particles together through solid state diffusion mechanisms.

The dimensional accuracy of the final components, as well as the physical and mechanical properties are mainly determined in the sintering stage. Sintering process for PIM parts significantly differs from the die-pressed sintering as it is only associated to solid state diffusion process, leading to large shrinkage (10-15%) and fast sintering kinetics due to the large porosity after debinding and generally smaller particle sizes. Uneven shrinkage may be caused by gravity, friction between the component and the sintering support, small heterogeneity of the moulded parts due to segregation phenomena occurring during the mixing or injection moulding steps. Experimental and analyses of studies on the sintering processes have been carried out for more than 50 years. Models based on different physical approaches have been established for the modelling of sintering process based on simplifications of the complete physical processes [START_REF] German | Computer modelling of sintering processes[END_REF]. Based on particle geometries for two or several particles, micro-structural based models focus on the analysis of neck growth between powder particles, pore geometries and shrinkage, according to different mass transport mechanisms [START_REF] Coble | Sintering crystalline solids. I. Intermediate and final state diffusion models[END_REF]. The mesoscopic models, such as the Monte Carlo one, are based on a representative cell of the powder medium that is used to get the evolution of the microstructure [START_REF] Braginsky | Numerical simulation of solid state sintering[END_REF]. The phenomenological models, based on continuum solid mechanics according to density variations mainly use finite element method accounting macroscopic sintering physical models to predict the dimensional changes of the sintered components during the sintering stage [START_REF] Olevsky | Theory of sintering: from discrete to continuum[END_REF]. The previous works of the authors are related to the development of a sintering model and numerical simulations based on continuum mechanics to predict the shrinkages and distortions of the parts during and after sintering. The physical and material parameters used in the sintering models have been identified by experiments carried out in a dilatometer in real sintering conditions.

In the proposed analysis, the feedstock for μPIM has been developed on the basis of a fine 316L stainless steel powders, with particle size d 50 equal to 3.4 μm. The viscous behaviour of this feedstock during mixing, injection moulding, debinding and sintering steps has been investigated. The paper reports the shrinkage occurring during bending tests where the sintering process that has been carried out in a dilatometer under different sintering kinetics. Then finite element simulations have been used to predict the shrinkage of the sintered parts based on the phenomenological sintering model and physical parameters determined through the experiments in dilatometer.

Experimental investigations

Material characterization

316L stainless steel powders that are widely used in PIM, exhibits excellent mechanical and corrosion properties, as well as biocompatibility. The fine 316L stainless steel powders that are used for μPIM have generally a particle size d 50 equal to 3.4 μm. In the present analysis, the 316L stainless steel powders (Osprey® Sandvik, UK) that are used exhibits a particle size distribution as follows: d 10 = 1.8 μm, d 50 = 3.4 μm and d 90 = 6.0 μm. The pycnometer density of the such powders is equal to 7.9 g cm -3 . The chemical compositions for these stainless steel powders by weight are respectively 17.4% Cr, 10.9% Ni, 2.5% Mo, 1.2% Mn, Si, C, P and S b 1%. The SEM images show that the powders have spherical shapes that are well adapted for PIM due to the appropriate flowability (Fig. 1).

Several formulations based on wax binder systems as the primary binder, secondary binder and surfactant have been proposed for PIM feedstock [START_REF] German | Status of PIM Research and Development[END_REF]. The primary polymeric binders such as polypropylene (PP) and polyethylene (PE) are used to maintain the component shape after the injection moulding and debinding stages. The secondary binder such as paraffin wax (PW) is used to decrease the feedstock viscosity and consequently to increase the replication ability of the feedstock. The surfactants such as stearic acid (SA) or oleic acid (OA) are suitable to increase powder wetting. The characteristics and compositions of the binder systems which are suitable for thermal debinding are given in Table 1 [START_REF] Liu | Mixing and characterisation of 316L stainless steel feedstock for micro powder injection molding[END_REF][START_REF] Tay | Characterisation of micro gears produced by micro powder injection moulding[END_REF]. Some of these formulations for the wax-based binder systems have been tested with powder loadings (powders volume fraction) equal to 60 in vol.% [START_REF] German | Status of PIM Research and Development[END_REF]. Considering the facility to mix the feedstock and to get a mixture with excellent homogeneity [START_REF] Quinard | Development and properties identification of 316L stainless steel feedstock for PIM and μPIM[END_REF], three binder formulations have been chosen for μPIM in this study (Table 2).

Experimental procedures

The first experiment consists to choose the well appropriate binder formulation from the three ones related in Table 2 in order to get a well adapted feedstock for μPIM with the fine 316L stainless steel powders. A torque rheometer with two blade screws (Plastograph®, Brabender, Germany) has been used to mix the powder and binder. The powder loading was chosen as 60 vol. %. The mixing temperature was 160 °C. The screw rotated at 30 rpm during 30 min. Finally, the feedstock has been granulated in small pellets for easy injection moulding.

After the determination of the appropriate binder formulation, the critical powder loading was investigated using a torque rheometer. The method consists to progressively increase the solid volume loading during the test, with a fixed binder volume [START_REF] Barreiros | Role of Nanostructured Layers on Powders during Sintering of Low Carbon Austenitic Stainless Steel[END_REF]. The powder loading varies from 50 to 66 vol.% with increasing step equal to 2 vol.% for each level by adding the powders content. The principle of this method consists to incrementally add powders content up to a maximum value corresponding to the significant increasing of the mixing torque vs. powder loading. The test presents the advantage to provide the way determine the critical powder loading by using just one testing cycle. But the entire heating cycle take around 225 min (test including 15 powder loading levels), and the intensive shearing can lead to the decomposition of the stearic acid. Consequently, a mixing temperature was chosen as low as possible (160 °C) to reduce this effect. Formulation 2, based on PP, PW and SA, was used to determine the critical powder volume loading.

A two-plate mould with micro-cavities corresponding to tensile and bending test specimens has been designed (Fig. 2a). The cross section area for the cavity gates (type 3 and 4) is 0.5 mm 2 . These cavities have been obtained by electrical discharge machining (EDM) and high speed machining (HSM). The developed feedstock used for injection moulding consists of fine 316L stainless steel powders with particle size d 50 equal to 3.4 μm and binder system corresponding to the formulation 2 related in Table 2. The powder loading of the feedstock corresponds to 60 vol.%. Micro-components has been injected in using a hydraulic injection moulding equipment (Arburg® Allrounder 220S). A special screw for the MIM feedstock injection, with a diameter equal to 15 mm and a length equal to 600 mm, has been used for the different injection moulding tests. The process parameters for the injection moulding stage are the following: injection temperature equal to 190 °C, mould temperature equal to 65 °C, maximum injection pressure equal to 100 MPa, flow rate equal to 20 cm 3 s -1 associated to a screw speed equal to 10 m min -1 . The total injected volume corresponds to 2 cm 3 . The microstructures of the mixed and injected components have been studied by SEM observation. The stainless steel powders exhibited an appropriate homogeneity where the stainless steel powders are uniformly distributed within the binder, see Fig. 3. The debinding is performed to eliminate the binder after injection moulding. After debinding, only the primary binder remains with the powders that ensure a minimum strength of the component [START_REF] German | Wick debinding distortion of injection molded powder compacts[END_REF]. The thermal debinding was selected associated to its simplicity, safety and environment protection. The debinding was realized in a thermal debinding oven at 220 °C in argon atmosphere. The debinding temperature cycle consist in two stages: (1) heating from ambient temperature to 130 °C in 2 h and (2) heating from 130 °C to 220 °C in 24 h.

µm

The sintering was carried in a high temperature furnace under vacuum conditions. The heating rate for the sintering cycles varies from 5 to 20 °C min -1 . Two holding stages have been used: (1) 600 °C during 30 min and (2) 1360 °C during 120 min. The furnace vacuum was 10 -3 mbar and alumina plates were used for sintering supports. The sintered parts were cooled in furnace from sintering temperature to the ambient one with a cooling rate corresponding to 10 °C min -1 .

Results

The mixing torque curves for the feedstock with a fixed powder loading equal to 60 vol.% corresponding to the 3 binder formulations are related in Table 2 and are plotted in Fig. 4a viscosities of the three formulations are related in Fig. 4b, formulation 2 (PP + PW + SA) exhibits the smallest shear viscosity among them. So the binder associated to formulation 2 is more appropriate than formulation 1 and 3 for mixing with the fine 316L stainless steel powders to elaborate the μPIM feedstock with a proper flowability.

The formulation 2 corresponding to a binder based on PP, PW and SA, has been used for the determination of the critical powder loading. In Fig. 5, one can clearly observe that there are three obvious stages of the mixing torque corresponding to the increasing of powder content loading. In the first stage, the mixing torque keeps at a low value and increases slightly, as the powder loading ranges from 50 to 60 vol.%. In the second stage, the powder is added until the powder loading reaches 66 in vol.%, and the third stage corresponds to the abrupt increase of the mixing torque. The resulting feedstock cannot be used for injection moulding due to high viscosity. So the critical powder loading for the related feedstock system is in the range from 64 to 66 vol.%. Generally the optimal powder loading should be about 2 to 5 vol.% lower than the critical one [START_REF] German | Status of PIM Research and Development[END_REF]. In this study, the powder loading used for the injection moulding tests is chosen as 60 vol.%.

The micro-components have been moulded without defects such as jetting and dead zone, as shown in Fig. 2b. The thickness of the parts is 1 mm. The micro parts corresponding to bending test specimens and cylindrical ones are used for the sintering experiments in order to determine the material parameters related to the sintering model that will be described in the following section. The debinded parts are homothetic to the moulded ones, but a small shrinkage around 1% for the large tensile test specimen is observed (Fig. 6). The sintered parts are carefully measured in order to determine the shrinkage. The average shrinkage with reference to the mould dimensions are between 11.5% and 15.0%, depending on the heating rates used in the thermal cycles during sintering. In a first approximation and considering an isotropic shrinkage, the average relative density d of the sintered parts can be calculated from the average shrinkage using the following expression [START_REF] Song | Experimental and numerical analysis on sintering behaviours of injection moulded components in 316L stainless steel powder[END_REF]:

d = Φ = 1-δ ð Þ 3 ð1Þ
where Φ is the powder loading and δ is the shrinkage after sintering. The relative density increases generally as the heating rate increases during sintering, see Fig. 7. The relative density varies from 92% to 98% vs. heating rates. A higher final relative density corresponding to 98% has been obtained by using a heating rate equal to 10 °C min -1 .

Physical modelling and numerical simulation of the sintering stage

Continuum sintering model

The macroscopic shrinkage and distortion of the resulting parts during and after sintering can be analyzed in using continuum mechanics conservation laws coupled with a thermo-elasto-viscoplastic constitutive laws [START_REF] Olevsky | Theory of sintering: from discrete to continuum[END_REF]. The strain rate consists in three parts: elastic strain rate associated to the elastic deformations εe , thermal strain rate associated to thermal deformations ˙εth , viscoplastic strain rate associated to viscoplastic deformations εvp , and can be written:

ε = εe + εth + εvp ð2Þ
The stress rate σ can be expressed vs. ˙εe , ˙εth and ˙εvp as following:

σ = D e ˙εe = D e ε- εth - ˙εvp ð3Þ
where D e is the isotropic elasticity tensor. The thermal strain rate εth is limited to thermal expansion as expressed in Eq. ( 4):

εth = αΔ ṪI d ð4Þ
where α is the coefficient of thermal expansion, ΔT is the temperature increment and I d is the second order identity tensor. The viscoplastic 0 0,5 strain rate ˙εvp , corresponding to the sintering mechanisms by solid state diffusion, is determined through the expression below [START_REF] Olevsky | Theory of sintering: from discrete to continuum[END_REF]:

εvp = devσ 2G + σ m -σ s 3K I d ð5Þ
where devσ is the deviatoric stress tensor, G is the shear viscosity modulus, σ m = tr(σ) /3 mean stress, σ s is the sintering stress. K is the bulk viscosity modulus and I d is the second order identity tensor. The shear and bulk viscosity modules are expressed through Eqs. ( 6) and ( 7) issue from [START_REF] Bordia | On constrained sintering-I Constitutive model for a sintering body[END_REF]:

G = η z 2 1 + ν p ð6Þ
K = η z 3 1-2ν p ð7Þ
where η z is the uniaxial viscosity and ν p is the Poisson's ratio of the porous material. These coefficients can be determined through experiments. The uniaxial viscosity η z e could be measured in using bending tests during sintering with the following expression [START_REF] Lee | Bending creep test to measure the viscosity of porous materials during sintering[END_REF]:

η e z = 1 δ FL 3 s 4bh 3 ð8Þ
where δ is the measured deflection rate of the bending specimen, L s is the span length of the bending tests, h and b are respectively the thickness and the width of specimen before deformation and F is the axial load applied perpendicularly to the specimen (Fig. 8). The Poisson's ratio of the sintering material can be expressed through the following expression [START_REF] Bordia | On constrained sintering-I Constitutive model for a sintering body[END_REF]:

ν p 1 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi d 3-2d r ð9Þ
where d is the relative density that could increase from 0.6 to 1. The sintering stress is a function of relative density and particles size [START_REF] Peterson | Constitutive behavior of WC-Co materials with different grain size sintered under load[END_REF] expressed as:

σ s = Bd C ð10Þ
where B and C are the physical parameters to be determined from the experiments in dilatometer.

Determination of the constitutive parameters

The identification of the physical parameters entering in the constitutive and phenomenological relationship as previously mentioned is realized through an inverse method based on free or loaded tests and bending tests under fixed load during sintering in dilatometer [START_REF] Song | Experimental and numerical analysis on sintering behaviours of injection moulded components in 316L stainless steel powder[END_REF]. The dilatometric measurements have been carried out at high temperature in an horizontal dilatometer (Netzsch® DIL 402C). This equipment proceeds under primary vacuum and a normal load equal to 15 cN is applied on the specimen. The span distance for the bending tests is 10 mm. The axial deflection and shrinkage measurements were conducted with different heating rates equal to 5, 10 and 15 °C min -1 . The temperature of these tests are in the range from 25 °C to 1360 °C. The holding times at 1360 °C are 120 min. The results are related in Figs. 9 and10.

In the sintering model used in the analysis, uniaxial viscosity η p and sintering stress σ s are the parameters that should be identified. The bending and shrinkage tests in dilatometer during sintering are used to determine both parameters respectively [START_REF] Song | Experimental and numerical analysis on sintering behaviours of injection moulded components in 316L stainless steel powder[END_REF]. The obtained uniaxial viscosity for the heating and holding periods during sintering with different heating rates are related in Fig. 11. An inverse identification method is used to determine the sintering stress. This method is based on the minimization of the following objective function:

minG x ð Þ G x ð Þ = ∑ n i = 1 jλ e T i ; x ð Þ-λ m T i ; x ð Þj 2 x = B; C ½ 8 > > > < > > > : ð11Þ
where G(x) is the objective function, λ e is the shrinkage issued from dilatometric tests, λ m is the shrinkage calculated with the sintering model, n corresponds to the number of the points on the shrinkage curves recorded by the dilatometer, B and C are the physical parameters to be identified. Based on the determined uniaxial viscosity and shrinkage tests in dilatometer, an optimization algorithm [START_REF] Lagarias | Convergence properties of the Nelder-Mead simplex method in low dimensions[END_REF] is used to solve the minimisation problem [START_REF] Olevsky | Theory of sintering: from discrete to continuum[END_REF]. The optimization is divided into two stages by a critical relative density 80%. The determined sintering stress σ s expressed as a function of relative density (Eq. ( 10)) is shown in Fig. 12.

Numerical simulations of the sintering stage

The numerical simulations based on the above sintering model have been implemented in the Abaqus® finite element software, through the user subroutine UMAT. The initial geometries of components used for the simulation consist of a bending test specimen with a length equal to 25 mm, a width equal to 5.5 mm and a thickness equal to 1 mm, and a tensile test specimen with a length equal to 10 mm, a section width at central position equal to 2 mm and a thickness equal to 0.9 mm. The powder loading of the green parts is 60 vol.%. The sintering cycles, used to perform the simulations correspond to heating rates equal to 5, 10 and 15 °C min -1 and holding temperature equal to 1360 °C during 120 min. The sintering simulations have been realized considering an alumina support plate, with a friction coefficient equal to 0.5, the gravity effect is also considered [START_REF] Song | Experimental and numerical analysis on sintering behaviours of injection moulded components in 316L stainless steel powder[END_REF].

The simulations of the sintering stage have been carried out for components corresponding to the 316L stainless steel powders with d 50 particle size equal to 3.4 μm. The shrinkage of the specimens after sintering stage, corresponding to a heating rate equal to 5 °C min -1 are obtained by simulation, are shown in Fig. 13a andb. The results of all the simulations indicate that the shrinkage as well as the relative density increase with the heating rate (Table 3). The maximum of the isotropic shrinkage and final relative density have been obtained for a thermal kinetics corresponding to 15 °C min -1 .

A comparison between simulations and experiments considering shrinkage and relative density is related in Table 3. The values of relative density are lower than the experimental ones. The maximum discrepancy is less than 10%. Nevertheless, the difference between simulations and experiments can be reduced by considering heterogeneous powders volume fractions after injection moulding stage or near proper friction coefficient carefully measured from experiments.

Shrinkage, %

Sintering stress, MPa

Conclusions

This study demonstrates the possibility to realize metallic microcomponents by micro-powder injection moulding (μPIM). It has been shown that the fine metallic powders can be properly mixed with polymers to make suitable feedstocks for micro-injection moulding. The micro-components exhibit very small shrinkage after injection and debinding. However, the components exhibit inhomogeneous shrinkage after sintering ranging from 11.5% to 15%, depending on the part size and sintering kinetics. The continuum sintering model based on thermo-elasto-viscoplastic constitutive law could provide component shrinkage, which is helpful to determine the die cavity geometry that has to be used in injection moulding. The parameters used in the sintering model could be obtained from bending and shrinkage tests carried out in a dilatometer. In order to improve the accuracy of the simulation results, the powder inhomogeneity in the moulded parts has to be accounted as well as friction coefficient between the parts and sintering supports that should be properly defined.

Fig. 1 .

 1 Fig. 1. Photography of 316L stainless steel powders with particle size d 50 equal to 3.4 μm, used in the subsequent tests.

Fig. 2 .

 2 Fig. 2. (a) Die cavity comprising two bending test specimens (type 1 and 3) and two tensile test specimens (type 2 and 4) with a thickness equal to 1 mm; (b) microcomponents after injection moulding based on 316L stainless steel powders (d 50 equal to 3.4 μm).

Fig. 3 .Fig. 4 .

 34 Fig. 3. SEM pictures of feedstock after injection stage, corresponding to the formulation 2 (powder loading equal to 60% in vol., 316L stainless steel powders, d 50 equal to 3.4 μm) (mixing temperature: 160 °C, mixing rotation speed: 30 rpm and mixing time: 30 min.).

Fig. 6 .Fig. 7 .

 67 Fig. 6. Injected, debinded and sintered specimens in 316L stainless steel powders (d 50 equal to 3.4 μm), with heating rate equal to 10 °C min -1 and then constant temperature equal to 1360 °C during 120 min.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Description of simply supported beam model for beam-bending tests.

Fig. 10 .Fig. 11 .

 1011 Fig. 10. Shrinkage variation vs. temperature and time during the sintering stages for the components in 316L stainless steel (d 50 equal to 3.4 μm).

Fig. 12 .

 12 Fig. 12. Evolution of sintering stress during heating and holding stages for 316L stainless steel components, with particle size d 50 equal to 3.4 μm.

Fig. 13 .

 13 Fig. 13. Shrinkage of the specimens after sintering stage corresponding to a heating rate equal to 5 °C min -1 : a) beam bending specimen (initial length: 23 mm); b) tensile specimen (initial length: 10 mm).

Table 1

 1 Chemical composition and physical properties of the binder.

	Designation	Type	Melting temperature, °C Density, g cm -3	Decomposition, °C
	Polyethylene (LDPE) Primary	130	0.91	400
	Polypropylene (PP)	Primary	140	0.90	400
	Paraffin wax (PW)	Secondary	60	0.91	300
	Stearic acid (SA)	Surfactant	70	0.94	395
	Oleic acid (OA)	Surfactant	17	0.91	350

Table 2 Polymers

 2

		PP, %	PE, %	PW, %	SA, %	OA, %
	1	0	4 0	0	6 0	0
	2	4 0	0	5 5	5	0
	3	9 4	0	0	0	6

volume fraction used for the feedstocks. Formulation (vol.%).

Table 3

 3 Comparison of relative density obtained from experiments and simulation at different temperature rates.

	Kinetic	Density obtained		Discrepancy
	(°C/min)	From simulation (%)	From experiments (%)	(%)
	5	86.22	92.70	7.00
	10	89.85	99.00	9.24
	15	92.87	95.70	2.96


The mobility of academic staff to South Africa is expected to benefit higher education institutions through teaching-research collaboration and capacity building. However, South African institutions do not always have the adequate organisational processes to facilitate host and international staff collaboration. Drawing on individual interviews with 16 lecturers from 12 different countries, all of th m t ch g S uth Afr c u v r ty, th rt cl ly l ctur r ' p rc v d c tr but to their host university and the challenges they encountered. Recommendations revolve around the development of appropriate induction programmes and faculty forums to promote cross-cultural collaboration and the cross-fertilisation of ideas.

 as international lecturers are confronted with a highly racialized context that values the contributions of black South African and sees foreigners

Introduction

One objective of the Continental Education Strategy for Africa (CESA) over the period 2016-2025 is to facilitate Pan-African mobility and academic integration through regional cooperation on teaching and research. This policy objective is consistent with the global rationale for higher education internationalisation. This internationalisation agenda aims to support the development of teaching and research quality in partnering universities, in addition to exposing students to international perspectives [START_REF] Egron-Polak | Internationalization of higher education: Growing expectations, fundamental values[END_REF]. In Africa in particular, the expected gains of internationalisation pertain to a strengthened capacity for knowledge production and opportunities for staff mobility [START_REF] Egron-Polak | Internationalization of higher education: Growing expectations, fundamental values[END_REF]. These internationalisation gains depend however on the opportunities for scholars to cross borders and contribute to their host institutions through equal collaboration with colleagues on site. The risk of unfair distribution of gains in the international brain game is a current source of worry among university leaders in Africa [START_REF] Egron-Polak | Internationalization of higher education: Growing expectations, fundamental values[END_REF]. Not only Western scholars, nor Western institutions with greater revenue streams, should benefit from international collaborations.

International academic staff members represent a valuable resource to universities competing for talent as they bring with them different sets of expertise and an inter-national network [START_REF] Kim | Transnational academic mobility and the academic profession[END_REF]. In this internationalised context, higher education institutions within the Southern African Development Community (SADC) are encouraged to increase international partnerships with other universities. This is promoted by organisations like the Southern African Regional Universities Association (SARUA) that approach student and academic mobility as an opportunity for capacity building of leadership, research, and teaching that has the potential of increasing attractiveness of the region [START_REF] Kotecha | Engaging universities in the regional integration project in 520 Southern Africa[END_REF]. As academic exchanges bring about cultural diversity in higher education institution, these contexts are perceived as more productive in terms of the production of new ideas. This academic mobility will be facilitated by a credit-equivalent systemlike the SADC Qualification Framework currently in implementationand quality assurance strategies within the Southern African region.

as a potential threat to indigenous African cultures. [START_REF] Peberdy | Rooted in racism: The origins of the aliens control act[END_REF] identified an identitarian movement within South African society as result of the apartheid regime. This is exacerbated by debates on the decolonisation and the Africanisation of the curriculum in South African universities [START_REF] Le Grange | Decolonising the university curriculum[END_REF]Vorster & Quinn, 2017). Academic mobility also takes place in the context of the transformation of South African of universities, which sometimes takes the form of recruitment preferences that favour historically disadvantaged people in order to achieve a more balanced representation of society in terms of race, gender, and class, especially at managerial levels. This definition of transformation is characterised by Du Pr z, S mm d , d V rh f (2016) 'tr f rm t thr ugh r dr g qu ty' (p. 6). Th r article engages with current discussions around the transformation of higher education in the context of the afore-mentioned decolonisation and Africanisation of knowledge, but also in the broader context of the internationalisation of universities in South Africa [START_REF] Du Preez | Rethinking and researching transformation in higher education: A meta-study of South African trends[END_REF]. Du Preez et l. ( 2016) r c mm d d "gr t r tt t b g v t th t r t l t d b t f rm g th higher education context in order to broaden the discourse on transformation and encourage the r th k g th r f" (p. 6). Th curr t p p r c tr but t th r th k g pr c by xpl r g h w the internationalisation of university academic staff may contribute to the transformation of university processes related to teaching and learning. As the multi-faceted notion of transformation raises qu t r l t d t u v r t ' tructur , qu ty, ucc ful cc , p d g g c l pr ct c , d curriculumcurriculum content that should depart from a Eurocentric focus, according to Heleta (2016)the process of internationalisation in South Africa could contribute to a transformative movement towards social justice and appreciation of the epistemologies of the South through the development of inclusive pedagogical and research practices [START_REF] Cross | Emerging concept of internationalisation in South African higher education: Conversations on local and global exposure at the university of the witwatersrand (wits)[END_REF][START_REF] Maringe | Creating opportunities for a socially just pedagogy: The imperatives of transformation in post-colonial HE spaces[END_REF].

Within this broader context of internationalisation, a more particular regionalisation can also favour the development of distinctive and collaborative research programmes within Southern Africa. Academic cross-cultural forums can provide enabling spaces to produce relevant research for African contexts, embracing African indigenous epistemological frameworks, and addressing African socio-economic realities (Sehoole & de W t, 2014). Th y c uld m t g t th r k f v r mph g 'W t r ' p t m l g c l ppr ch Africa [START_REF] Egron-Polak | Internationalization of higher education: Growing expectations, fundamental values[END_REF]. Representing 25% of the African population, the 15 member states of the SADC region share similarities, such as a wide ethnic and language diversity, a colonial past, and common constraints (limited resources, brain drain). In 2012, it included 109 public universities, 526 technical colleges, and a fast-growing pool of private institutions [START_REF] Wilson-Strydom | A profile of higher education in Southern Africa[END_REF]. Confronted with specific challenges on the continent, Southern African researchers can work collaboratively towards developing appropriate research and policy programmes. For example, African partnerships can enable researcher-practitioners to design locally relevant research strategies as well as regionally relevant curricula in accordance with the current emphasize on indigenous frames of references in South Africa [START_REF] Le Grange | Decolonising the university curriculum[END_REF][START_REF] Sehoole | The regionalisation, internationalisation, and globalisation of African higher education[END_REF].

However, as each SADC country presents specific cultural and administrative contexts, 95 including distinctive educational cultures and a myriad of languages (beyond the colonial English, French, or Portuguese languages), moving from one country to another requires lecturers to learn and adjust to different teaching philosophies and practices [START_REF] Williams | Boundary crossing and the professional learning of teacher educators in new international contexts[END_REF]. Crossing borders brings about disturbing experiences that can be conceptualised in culture shock models [START_REF] Dongfeng | Culture shock and its implications for cross-cultural training and culture teaching[END_REF][START_REF] Kim | Intercultural personhood: Globalization and a way of being[END_REF], which describe the difficulties of the integration and socialisation processes of international academic scholars in new sociocultural contexts [START_REF] Kim | Intercultural personhood: Globalization and a way of being[END_REF][START_REF] Volet | Cultural transitions in higher education: Individual adaptation, transformation and engagement[END_REF]. Assessing the advantages of academic staff mobility to South Africa necessitates a review of the scope of their contributions to their host institutions and the challenges they need to overcome to unleash their full potential in a new sociocultural environment. This article focuses on transnational faculty members who teach in one South African university; more precisely, it draws on data collected from 16 international lecturers and 12 different nationalities teaching in one university in Johannesburg. This case study aims to provide an in-depth illustration of the perceived contributions of international lecturers to their host institution. Two research questions were devised to guide the study:

• How did international faculty members perceive their contribution(s) to their host institution?

• Which challenges did they encounter in this specific context? Aft r r v w g th curr t l t r tur c d m c m b l ty d th v lu f f r g l ctur r ' fl w from an institutional perspective, I will present the conceptual framework and the methodology before h ghl ght g th f d g d r c mm d t . F c l t t g t r t l l ctur r ' t gr t t and collaboration within the host university could help them contribute more extensively to their institution.

Contributions of transnational lecturers to their institutions: a literature review

Building on the current literature on academic mobility and partnerships (Fellesson hlck, 2017;[START_REF] Ishengoma | Strengthening higher education space in Africa through North-South partnerships and links: Myths and realities from Tanzan publ c u v r t . C mp r t v d I t r t l duc t duc t C mp r t I t r t l[END_REF][START_REF] Kot | The perceived benefits of international partnerships in Africa: A case study of two public universities in Tanzania and the democratic republic of Congo[END_REF], this literature review is organised around the expected gains and challenges attached to academic staff mobility. Issues and benefits are considered from an institutional perspective.

A growing body of cross-boundary literature explores the conditions for successful teaching experiences needed for reducing mismatched expectations between foreign teachers and their host institutions [START_REF] Smith | Exploring flying faculty teaching experiences: Motivations, challenges and opportunities[END_REF][START_REF] Volet | Cultural transitions in higher education: Individual adaptation, transformation and engagement[END_REF]. A large share of the literature promotes the mutual learning of faculty from diverse cultural back-grounds [START_REF] Bovill | Transnational approaches to teaching and learning in higher education: Challenges and possible guiding principles[END_REF][START_REF] Lai | Teacher agency and professional learning in cross-cultural teaching contexts: Accounts of Chinese teachers from international schools in Hong Kong[END_REF][START_REF] Mizzi | Bridging borders: Toward a pedagogy of preparedness for visiting faculty[END_REF]. The various collaboration opportunities between host and international lecturers are framed by the institutional culture, structure, and processes of the host university that together delimit the scope of their interaction opportunities [START_REF] Lai | Teacher agency and professional learning in cross-cultural teaching contexts: Accounts of Chinese teachers from international schools in Hong Kong[END_REF][START_REF] Shagrir | Collaborating with colleagues for the sake of academic and professional development in higher education[END_REF]. [START_REF] Lai | Teacher agency and professional learning in cross-cultural teaching contexts: Accounts of Chinese teachers from international schools in Hong Kong[END_REF] interviewed 14 Chinese language teachers from 13 Western/international schools in Hong Kong where English was the medium of instruction. They shed light on the sense of powerlessness perceived by these m r ty t ch r th y r p rt d 'l m t d g cy t flu c th r W t r c ll gu ' d m t m b c f d r t d ( t l., p. 18). In the study by Lai et al., the participants mentioned only very few opportunities to interact with their Western colleagues through cross-disciplinary forums. In addition to addressing the limited opportunities for cross-cultural collaboration, Lai et al. highlighted the importance of considering the power relationships between staff members and of finding ways to move away from a hierarchical relationship between Northern and Southern faculty members.

A lack of individual self-confidence combined with the social roles imposed on Chinese language t ch r ' cultur l g t r th r th p d g g c l r curr cul r xp rt ' xpl d th l ck f influence of Chinese teachers on their Western colleagues (Lai et al., 2016, p. 19). Therefore, the authors warned against the risk of local teachers undermining their own position. They recommended developing opportunities for collaboration between equally positioned teachers, and encouraged Chinese instructors to develop their agency while enabling other faculty members to learn from their perspectives -r c mm d t th t l g d w th Fr t' (2012) dv c cy for the development of t ch r ' -positional leadership and collaboration. 2017) noted that research collaboration between the North and the South could be more beneficial to Northern researchers than their Southern counterparts. They explained that funding streams from the North tended t cr t f l g f ub rd t S uth r r rch r (F ll hlck, 2017), a relationship that has been denounced as unequal [START_REF] Altbach | The internationalization of higher education: Motivations and realities[END_REF]. Hence, distorted power relationships between Northern and Southern teachers, based on unequal access to resources and historical inequalities, should be addressed by the administration of receiving universities.

g th c f h r rch c ll b r t b tw T z , z mb qu , d Sw d , F ll d hlck (
These illustrations of power dynamics in international contexts suggest that equal cross-cultural collaboration does not happen as a matter of course and should be supported by the culture, structure, and processes of host institutions [START_REF] Brooks | Exploring opportunities and challenges of cross-cultural and international teaching, research, and service for higher education faculty members[END_REF]. Spencer-Oatey (2013) highlighted that power inequalities, a lack of openness to new thinking, and differences in the understanding of time could impede cross-cultural collaboration. However, when effectively managed, cross-cultural academic partnerships have been praised for contributing to the professional growth of all faculty members (Dunn & Wallace, 2008;[START_REF] Volet | Cultural transitions in higher education: Individual adaptation, transformation and engagement[END_REF]. In the literature, the positive outcomes of cross-cultural collaboration include mutual learning in relation to 170 research and teaching, selfreflection on previously taken-for-granted assumptions about cultural differences, a greater awareness f d ff r t p t m l g , d th d v l pm t f l ctur r ' t rcultur l t v ty (K m, 2008;Montgomery & Bourassa-Dansereau, n.d.;[START_REF] Volet | Cultural transitions in higher education: Individual adaptation, transformation and engagement[END_REF][START_REF] Williams | Boundary crossing and the professional learning of teacher educators in new international contexts[END_REF]. Hence, academic staff mobility can contribute to the co-construction of ideas as international lecturers bring with them increased diversity of knowledge, competencies, and ways of doinga promising opportunity for universities within South Africa. However, for this promise t b fulf ll d g d r qu r bu ld g th d v r ty f l ctur r ' xp r c ( t l., 2016) d funds of knowledge [START_REF] Moll | Funds of knowledge for teaching: Using a qualitative approach to connect homes and classrooms[END_REF].

Conceptual framework

The conceptual framework of this study drew on the theories of boundary crossing [START_REF] Akkerman | Boundary crossing and boundary objects[END_REF][START_REF] Williams | Boundary crossing and the professional learning of teacher educators in new international contexts[END_REF] and professional learning communities [START_REF] Stoll | Professional learning communities: A review of the literature[END_REF][START_REF] Wenger | Communities of practice: A brief introduction[END_REF]. The concept of boundary crossing is used to consider the changes encountered by lecturers who move from one country to another, thereby experiencing new national and institutional cultures. Crossing geographical and sociocultural boundaries expose lecturers to different practices and socially accepted frames of reference, which may create disturbing experiences but could also bring about identity repositioning [START_REF] Kim | Intercultural personhood: Globalization and a way of being[END_REF][START_REF] Williams | Boundary crossing and the professional learning of teacher educators in new international contexts[END_REF]. In this study, The theoretical literature on professional learning communities allows envisioning multicultural teams of lecturers as an opportunity for learning through socialisation activities and the exchange of ideas, support, and best practices [START_REF] Vescio | A review of research on the impact of professional learning communities on teaching practice and student learning[END_REF][START_REF] Wenger | Communities of practice: A brief introduction[END_REF]. The notion of the crossfertilisation of ideas was important to frame this research as it encompasses the positive outcomes of collaboration between different-but-equal individuals with a common purpose. According to the l t r tur , t tut wh ch pr m t f culty m mb r ' ct v v lv m t nable their lecturers to pool their funds of knowledge, spur mutual learning, and enhance their collective skillsets [START_REF] Lai | Teacher agency and professional learning in cross-cultural teaching contexts: Accounts of Chinese teachers from international schools in Hong Kong[END_REF][START_REF] Moll | Funds of knowledge for teaching: Using a qualitative approach to connect homes and classrooms[END_REF][START_REF] Tam | The role of a professional learning community in teacher change: A perspective from beliefs and practices[END_REF].

Methodology

This paper builds on a qualitative study of 16 international lecturers with 12 nationalities (15 male and 1 female lecturers) in one South African university. The participants primarily came to South Africa for personal, economic, or political reasons, but also out of research interests. Some of them came for their studies without planning to remain in South Africa but ended up staying permanently. They held permanent job positions and had been teaching in the university from a few months to 12 years in humanities and sciences departments. At the time of the interviews, however, the majority of participants had been teaching for two to four years in Johannesburg. Six of them originated from Southern African countries, namely from the Democratic Republic of Congo (DRC), Malawi and Zimbabwe. Others came from other countries in Africa (3 participants from Cameroun and Ivory Coast), five different countries in Europe (five participants), Asia (one participant), and North America (one participant). All participants were volunteers and had been teaching in the university for at least 10 months. The 10-month selection criterion was arbitrarily chosen to include a wide range of international lecturers while ensuring they had been working in the university long enough to be able to reflect on their onsite contributions and challenges. All participants had been teaching in another country before coming to South Africa, which could have facilitated their contributions to their new teaching institution but could also explain experiences of culture shock as they would have formed specific expectations based on their past experiences.

After receiving ethical clearance from the institution, I invited international lecturers by m l d bt d th r c t ( g d c t f rm ). p d g p rt c p t ' pr f r c , I conducted the semi-directed interviews face-to-face or online [START_REF] Meho | E-mail interviewing in qualitative research: A methodological discussion[END_REF][START_REF] Rubin | Qualitative interviewing: The art of hearing data[END_REF]. The possibility to respond by email was offered to minimize the time burden of being interviewed face-to-face [START_REF] Meho | E-mail interviewing in qualitative research: A methodological discussion[END_REF]. Only four participants chose the online interview method and the content of their responses was similar to the face-to-face interviews, even though they provided shorter answers.

The interview protocol was reviewed by an international scholar to improve the relevance and accuracy of the questions. The interviews (20 to 50 minutes) were transcribed and hand-coded to reveal prominent themes related to the research questions [START_REF] Creswell | Educational research: Planning, conducting, and evaluating quantitative and qualitative research[END_REF][START_REF] Lodico | Methods in educational research: From theory to practice[END_REF]. Codes were refined and re-organised throughout the data analysis to encompass p rt c p t ' r ch d d v r xp r c . Th c d r l t d t p d g g c l pr ct c , p rt c p t ' com-petencies, and challenging experiences. Each participant was able to review the verbatim quotes from their interview used in the research output to add any clarifications they deemed necessary. This procedure helped to enhance the validity of the results presented below.

The university that forms the context of the study is a large university in South Africa accommodating students from poor and historically disadvantaged backgrounds. Even though the university was developing a strategy for the internationalisation of its academic staff and student population at the time of the study, there was no clear policy to attract or retain international lecturers on site.

The findings presented below describe the multiple channels through which transnational lecturers thought they contributed to their host institution. These results are limited in scope as they rely on the experiences of only 16 international lecturers. However, they provide new insights into the contributions of international lecturers to their host university despite encountered challenges.

Findings: the contribution of international lecturers to their South African institution

Th tudy l c t d th p rt c p t ' p rc pt f th r c tr but t th r S uth Afr c t tut in terms of research, teaching, and professional development. Despite individual challenges to settle in, the transnational lecturers perceived that these contributions drew on their distinctive experiences, which could inform the university development and facilitate the inclusion of multicultural students in class. Four themes were prominent to describe how lecturers enriched their faculty or the university as wh l . I th f ll w g ct , p ud ym r u d thr ugh ut t pr t ct p rt c p t ' anonymity.

Contribution 1: different experiences and pedagogical practices

Lecturers perceived that they brought different pedagogical experiences to the university, introduced fresh perspectives to administration, and contributed to benchmarking of practices before implementation of new programmes and qualifications. Eli explained that international lecturers could bring multi-site, cross-country experiences that might be useful in times of curriculum transformation: At that point, you want to benchmark with what is happening overseas. That is where I come in. [The university] will then benefit because it is in transition; they want to change and offer new qualifications and now they have a bunch of people who have done the same thing.

Drawing on his own experience, Hans explained that he introduced new research practices to students as it was possible in his country of r g t t t tud t ' r rch pr j ct v ry rly th c ur f th r tud , wh ch w l c mm S uth Afr c : 'S , m tud t c m t m t t rt th r w pr j ct.' rt c p t also reported their ability to be culturally sensitive and to effectively accommodate local and international students in class, as presented in the next subsection.

Contribution 2: multicultural perspectives unfolded in teaching practices

Participants reported that their multicultural experiences were useful in accommodating students from different cultural backgrounds. Ahmed explained: I have colleagues around the world and we work together, so it is a kind of a United Nation experience and that translates to how I accommodate everybody. I have students from Kenya, Zimbabwe, Zambia, DRC, so I am able to relate to each person on a different level, including South Africans. [It is] like a multidimensional tool in order to accommodate those stud t ' d ff r c . Lecturers acknowledged they came from countries with a different racial history than South Africa. This difference in historical heritage appeared in the way they interpreted their daily interactions with colleagues and students from diverse cultural backgrounds. As international academic staff were not impacted by the apartheid regime as were their Black and White South African colleagues, they felt they did not share the same frames of reference. Therefore, participants reported that their relationships with colleagues and students from different ethnic groups c uld b m r r l x d. uk f lt th t 'th r a room for international white lecturers [in South Africa]: they can easily step aside and deal with race issues in much less emotional and with much m r d t c th wh t S uth Afr c .' ctur r xpl d th t th r cr -cultural exposure could help them accommodate students from various national and cultural backgrounds, but they also felt surprised by the racial frame of references commonly used in South African universities. This example shows that international lecturers experienced a new sociocultural environment and approached it with their distinctive frames of references. They could reflect, learn and adjust to this new environment but also bring about different practices.

Contribution 3: different research interests and awareness of world issues

The transnational lecturers highlighted their awareness of topics or contexts that go beyond South Africa. In addition to being informed about what is happening in the rest f th w rld, th y l br g b ut d t ct v r rch t r t . Rudy xpl d h br ught 'd v r ty in terms of res rch'. Th r ult c uld b c mp r d t Sh gr r' (2017) f d g c ll b r t , where lecturers reported a preference for research collaboration (carrying out, writing, and presenting academic research) as it permits the cross-fertilisation of ideas.

H

xpl d, 'I m t ly f cu d S uth Afr c . I h v cl c ll b r t w th m y c u tr .' ch l rgu d th t h br ught d ff r t 'w rldv w, fr h p r p ct v '. Th comparative perspective enabled lecturers to reflect on the South African specificities, encouraging South African students to be proud of their country. For example, Luke explained:

There is a lack of knowledge about the uniqueness of SA for [his discipline]. Indeed, the [disciplinary] inventory and resources of this country are unique in the world, and students have no conscience about it. They are not proud of their country and one challenge is to make them realise the chance they have to study [his discipline] in SA.

Finally, lecturers highlighted their language competences in a South African context with 11 official languages.

Contribution 4: language competences

The international respondents finally mentioned their language competences as a benefit for their institution and colleagues. Ahmed, from an African francophone country, illustrated his contribution. 'F r x mpl , f w g t d g c r r m cr t c R publ c f C g , m t f th c ll gu w ll be lost (. . .) and so having an international colleague like me who is able to navigate those worlds has been u ful.' rt c p t h ghl ght d th t th d v r ty f th r pr f l c tr but d t th internationalisation agenda of the university. However, these contributions were only possible after lecturers overcame initial adaptation issues that are presented in the next paragraphs.

Difficulties in unleashing lecturers' full potential

Participants acknowledged that their productive participation in the faculty was not without hurdles. They stressed difficulties in comprehending the South African culture, deeply marked by postapartheid racial inequalities and a racial interpretive framework. They also experienced ambiguity w th r g rd t th r t r t l pr f l . l l b r t d: ' y xp r c th t th r qu t l t of insecurity. In South Africa, th r l v d h t r l t h p w th p pl fr m v r .' l' xp r c ch d th mb v l t r l t h p th t S uth Afr c c ty h d v l p d w th foreigners from poorer Southern African countries [START_REF] Peberdy | Rooted in racism: The origins of the aliens control act[END_REF] or from Western countries as a result of colonisation and the apartheid regime. Some lecturers also mentioned they were surprised, and sometimes unprepared, to meet the demand for student-centred pedagogy promoted in the university, considering they were coming from countries that tended to use lecturer-centred approaches. This difference in pedagogical cultures was observed between faculty members coming from French-speaking and English-speaking countries within Africa, the former being more lecturer-centred than the latter due to their different higher education heritage. Ella explained:

The South African approach is mainly student-driven and for [her French-speaking country of origin], it is the opposite, it is teacher-driven. Student-driven means that we tend to find out what the students want or what they will be happy with and then, we follow that. (. . .), in [her country of origin], the lecturer is the one who knows what is good for you and you must follow. So, it took me a while to actually understand that.

Being exposed to students with different learning needs and new institutional expectations with regard to educational objectives, lecturers reflected on their teaching roles and questioned their ways of doing. Eli departed from viewing higher education as an elitist enterprise as he t rt d t ch g tud t fr m d dv t g d b ckgr u d . H xpl d: 'S m th g h ft d in me when I came to the realisation that these students were under-prepared (. . .) I started questioning the way I was presenting my subj ct.' rt c p t ' m x d xp r c f lly ugg t d th t t w p bl t b tt r tr duc t r t l lecturers to the institutional and national cultures, enabling them to meet the wide learning needs of diverse students in South Africa and prepare them for the national debates around the decolonisation and the Africanisation of the curriculum. Induction programmes, faculty participative forums, and inclusive leadership might help transnational lecturers to thrive in their new environment by feeling valued for their distinctive competencies, epistemologies, and pedagogical practices, as argued in the last subsections.

Discussion

Conceptual contributions of the paper

The above results illustrate and confirm the relevance of boundary crossing theory [START_REF] Akkerman | Boundary crossing and boundary objects[END_REF][START_REF] Williams | Boundary crossing and the professional learning of teacher educators in new international contexts[END_REF] in the South African context. More precisely, it highlights how lecturers felt puzzled by the culture and requirements of their host institution. Lecturers did not expect the racial interpretive framework that is pervasive 380 in the university as a consequence of the p rth d l g cy. Th y xp r c d t d cr t c l c d t th t r typ c l wh 'w rk g b rd rl d ' ( rr, 2012). Th xp r c r ult d t k g m t p b ck t r c sider their l ctur r ' r l a country with wide inequalities. For example, some African participants originating from French-speaking countries felt unprepared for the student-centred pedagogical approach required to accommodate students from poor economic backgrounds.

The findings also illustrate the notion of cross-fertilisation or cross-pollination of ideas in the South African context, as an international professional learning community (Stoll t l., 2006) c uld dr w l ctur r ' d ff r t xp r ences, pedagogical practices, research interests, and linguistic competences. Building an international academic staff community would require the university to set effective processes to move from a community of differences to a community of learning, which is the main objective of the practical recommendations below.

Implications for academic developers

This study has important implications for cross-cultural collaboration in a context of 395 increasing academic mobility. As the university under study was developing an internationalisation policy at the time, these results had the potential to be helpful for effectively activating the benefits of accommodating a growing body of international academic staff. It became clear from the results that host and international lecturers should be invited to collaborate in faculty meetings devoted to brainstorming and benchmarking different teaching ideas. Organising regular meetings around teaching and research could enable both host and international staff to collaborate and confront their different perspectives (National Comprehensive Center for Teacher Quality, 2010;[START_REF] Shagrir | Collaborating with colleagues for the sake of academic and professional development in higher education[END_REF][START_REF] Steinert | Faculty development: From workshops to communities of practice (AMEE guide no. 33)[END_REF] ). This would enable researchers to contribute to the development of Southern African societies by mixing different-but-equally-valued epistemologies. For example, drawing on the South African Ubuntu philosophy [START_REF] Waghid | Reconceptualizing ubuntu as inclusion in African higher education: Towards equalization of voice[END_REF] does not mean that other foreign epistemologies should be discarded but that a discussion between different approaches could contribute to the d v l pm t f clu v d v t v d . Th r f r , l ctur r ' m b l ty t S uth Afr c m y continue developing and disseminating culturally and regionally relevant knowledge through inclusive forums in their host institutions.

Limitations

This case study drew on the experiences of only 16 lecturers in one South African university. Hence, other lecturers could have reported different experiences or distinctive contributions to their host institution. International lecturers who participated in the research came from 12 different countries. The study was therefore limited in scope and breadth, but nevertheless illustrated some of the contributions and adaptation issues reported in the literature and demonstrated their salience in the South African context. The findings highlighted the necessity of supporting international lecturers through appropriate induction programmes and participative forums.

Conclusion

As African leaders have identified the rest of Africa as their top priority to develop international partnerships [START_REF] Egron-Polak | Internationalization of higher education: Growing expectations, fundamental values[END_REF], it is crucial to review the advantages of academic staff mobility in the context of the transformation and internationalisation of the South African higher education system [START_REF] Du Preez | Rethinking and researching transformation in higher education: A meta-study of South African trends[END_REF][START_REF] Maringe | Teaching large classes in an increasingly internationalising higher education environment: Pedagogical, quality and equity issues[END_REF]. In this research, I investigated tr t l l ctur r ' xp r c S uth Afr c u v r ty t l c t th r p rc v d c tr but to their new institution. This research study explored how international faculty members could c tr but t th r S uth Afr c t tut th t l c t xt f u v r t ' tr f rm t , tud t ' d m d f r th d c l t d Afr c t f th curr culum (H l t , 2016; Le Gr g , 2016), d c ty' upp rt f r c lly ju t p d g g ( r g , 2017). R v lv g r u d four themes, the 16 participants stressed their ability to bring fresh perspectives to their host institution through capitalising on their own experiences abroad, using their intercultural sensitivity to better accommodate their students, and bringing about different language competences and research t r t . It h w d h w l ctur r ' cultur l d v r ty p t t l urc f r th cr t f d nd should be better valued in a context of occasional distrust against non-South African citizens [START_REF] Peberdy | Rooted in racism: The origins of the aliens control act[END_REF].

As a result, the administration of the university should encourage the contributions of international lecturers through the organisation of participative forums to enhance the cross-fertilisation of ideas. The findings of this qualitative research indicate that induction sessions could benefit international faculty members by accelerating their socialisation and acculturation processes in the university, but also in better valuing their contributions in this post-apartheid context. Peer support, organised at the t tut l l v l, c uld l cc mp y l ctur r ' lf-reflection process while teaching in 'b rd rl d '. Th r c mm d t ch t l.' (2016) ugg t t crut th t tut ' cultur , tructur , d c p c ty bu ld g pr c dur rd r t f c l t t h t d international academic staff collaboration. They also encourage South African institutions to support international staff adaptation to their new sociocultural environment.

  the notion of culture refers to soci lly cc pt d b h v ur d umpt th t flu c l ctur r ' teaching philosophies and practices. The adjectives transnational and international 190 are used interchangeably to qualify the lecturers who have crossed borders in order to teach in South Afr c , cc rd c w th K m d ck ' (2010) u f th t rm tr t l m b l ty the movement of academic staff across countries. Their research is important for investigating underxpl r d ph m , clud g 'th tur f t r t l c d m c ' xp r c th r h t institutions and countries and the broader impact of academic mobility on styles of scholarship and t ll ctu l tr d t ' (K m ck , 2010, p. 27).

  Patrick explained: When interacting with a White person at [the university], I cannot pick up racist overtones immediately. I can pick them if they are clearly hatred, but if they are subtle, you won't me picking them up. I would assume these are normal human errors, while my Black colleagues, th y mm d t ly p ck up r c t v rt . A d b c u I d 't p ck th m up, th cl h b tw m d my Bl ck c ll gu , b c u th y y, 'Y u f v ur th m'. I u u lly y, 'N I d 't, I d 't y pr bl m h r . It t r c m, t mply hum rr r'. S , th t' th main challenge I picked. Because of that, I decided in the very first years to be very quiet and that was problematic. I was quiet on controversial issues, very quiet, people debating racist u . A d I mply d, 'N , I h v c mm t' d p pl d d t l k t.

  . Staff mentoring and pairing could also help transnational lecturers receive peer support in order to facilitate their acculturation process(Darling-405 Hammond, Wei, Andree, Richardson, & Orphanos, 2009).Aligned with the SADC Protocol on Education and Training, SARUA recommended academic collaboration within the Southern African region to support leadership skill development and education quality[START_REF] Kotecha | Engaging universities in the regional integration project in 520 Southern Africa[END_REF]. Cross-cultural collaboration will enable research teams to address locally relevant issues, while 410 being inspired by international comparative perspectives.

					As
	S h l	d d W t (2014) t pul t d: 'African scholars are challenged to produce knowledge that is
	relevant to their own development context and to explore avenues by which to disseminate this
	k wl dg by d t tud t	d ch l r w rldw d ' (p. 235). Afr c p rt r h p c 'pr duc
	tangible solutions to Southern African problems 415 that draw on both local as well as global systems
	f k wl dg	d r gr u d d S uth r Afr c	xp r c , rm	d v lu ' (K t ch , 2011, p.
	16