Daniel Gonçalves

3-colorable planar graphs have an intersection segment representation using 3 slopes

Keywords:

In his PhD Thesis E.R. Scheinerman conjectured that planar graphs are intersection graphs of segments in the plane. This conjecture was proved with two different approaches. In the case of 3-colorable planar graphs E.R. Scheinerman conjectured that it is possible to restrict the set of slopes used by the segments to only 3 slopes. Here we prove this conjecture by using an approach introduced by S. Felsner to deal with contact representations of planar graphs with homothetic triangles.

Introduction

In this paper, we consider intersection representations for planar graphs. A segment representation of a graph G maps every vertex v ∈ V (G) to a segment v of the plane so that two segments u and v intersect if and only if uv ∈ E(G). Although this graph family is simply defined, it is not easy to manipulate. Actually, even if this class of graphs is small (there are less than 2 O(n log n) such graphs with n vertices [START_REF] Pach | Crossing Patterns of Segments[END_REF]) a segment representation may be long to encode (in the representations of some of these graphs the endpoints of the segments need at least 2 √ n bits to be coded [START_REF] Kratochvíl | Intersection Graphs of Segments[END_REF]). There are also interesting open problems concerning this class of graphs. For example, we know that deciding whether a graph G admits a segment representation is NP-hard, indeed it is ∃R-complete [START_REF] Kratochvíl | String graphs. II. Recognizing String Graphs is NP-hard[END_REF] but it is still open whether this problem belongs to NP or not. Here we focus on segment representations for planar graphs.

In his PhD Thesis, E.R. Scheinerman [START_REF] Scheinerman | Intersection classes and multiple intersection parameters of graphs[END_REF] conjectured that every planar graph has a segment representation. This conjecture attracted a lot of attention. H. de Fraysseix and P. Ossona de Mendez [START_REF] De Fraysseix | Representations by Contact and Intersection of Segments[END_REF] proved it for a large family of planar graphs, the planar graphs having a 4-coloring in which every induced cycle of length 4 uses at most 3 colors. In particular, this implies the conjecture for 3-colorable planar graphs. Then J. Chalopin and the author finally proved this conjecture [START_REF] Chalopin | Every planar graph is the intersection graph of segments in the plane: extended abstract[END_REF]. Recently, a much simpler proof was provided by the author, L. Isenmann, and C. Pennarun [START_REF] Gonçalves | Planar Graphs as L-intersection or L-contact graphs[END_REF]. Here we focus on segment representations of planar graphs with further restrictions.

In his PhD Thesis, E.R. Scheinerman [START_REF] Scheinerman | Intersection classes and multiple intersection parameters of graphs[END_REF] proved that every outerplanar graph has a segment representation where only 3 slopes are used, and where parallel segments do not intersect. Let us call such a representation a 3-slopes segment representation. This result led E.R. Scheinerman conjecture [START_REF] Scheinerman | Private communication to D. West[END_REF] (see also [START_REF] De Fraysseix | Representations by Contact and Intersection of Segments[END_REF]) that such representation exists for every 3-colorable planar graph. Later, several groups proved a related result on bipartite planar graphs [START_REF] Czyzowicz | A simple proof of the representation of bipartite planar graphs as the contact graphs of orthogonal straight line segments[END_REF][START_REF] De Fraysseix | Representation of planar graphs by segments. Intuitive geometry[END_REF][START_REF] Hartman | On grid intersection graphs[END_REF]. They proved that every bipartite planar graph has a 2-slopes segment representation, with the extra property that segments do not cross each other. Let us call such a representation a 2-slopes contact segment representation. More recently de Castro et al. [START_REF] Castro | Triangle-free planar graphs as segment intersection graphs[END_REF] considered a particular class of 3-colorable planar graphs. They proved that every triangle-free planar graph has a 3-slopes contact segment representation. Such a contact segment representation cannot be asked for any 3-colorable planar graph. Indeed, up to isomorphism, the octahedron has only one 3slopes contact segment representation depicted in Figure 1, and one can easily check that this representation does not extend to the (3-colorable) graph obtained after gluing a copy of an octahedron in each of its faces. However, we will use 3-slopes contact segment representations in the proof of our main result.

Theorem 1. Every 3-colored planar graph has a 3-slopes segment representation such that parallel segments correspond to the color classes. As every 3-colored planar graph is the induced subgraph of some 3-colored triangulation we only consider the case of triangulations in the following. In Section 2 we review some basic definitions. Section 3 is devoted to the so-called triangular contact schemes. It is shown that every 3-colorable triangulation admits such a scheme. Then, those schemes are used in Section 4 to build 3-slopes segment representations. Finally, we conclude with some remarks on 4-slopes segment representations.

Terminology

A triangulation is a plane graph where every face has size three. Unless stated otherwise, in this paper triangulations are simple, that means without loops nor multiple edges. A triangulation T , simple or not, is Eulerian if every vertex has even degree. It is folklore that these triangulations are the 3-colorable triangulations. Actually these triangulations are uniquely 3-colorable (up to color permutation). Hence their vertex set V (T ) is canonically partitioned into three independent sets A, B and C. In the following we will denote the vertices of these sets respectively a i with 0 ≤ i < |A|, b j with 0 ≤ j < |B|, and c k with 0 ≤ k < |C|. In such a triangulation T any face is incident to one vertex a i , one vertex b j , and one vertex c k , and these vertices appear in this order either clockwisely or counterclockwisely. In the following, the vertices of the outerface are always denoted a 0 , b 0 and c 0 , and they appear clockwisely in this order around T . As the orders of two adjacent faces are opposite, the dual graph of T is bipartite. Given an Eulerian triangulation T with face set F (T ), let us denote by F 1 (T ) and F 2 (T ) (or simply F 1 and F 2 if it is clear from the context) the face sets partitioning F (T ), such that no two adjacent faces belong to the same set, and such that F 2 (T ) contains the outer-face. Note that by construction for any face f ∈ F 1 (T ) (resp. f ∈ F 2 (T )) its vertices a i , b j and c k appear in clockwise (resp. counterclockwise) order around f . Note that the vertices a 0 , b 0 and c 0 appear in clockwise order around T , but in counterclockwise order w.r.t. the outer face. Let n = |V (T )|. As T is a triangulation, by Euler's formula it has 2n -4 faces. Hence, as T 's dual is bipartite and 3-regular,

|F 1 (T )| = |F 2 (T )| = n -2.
In the following we build 3-slopes segment representations. The 3 slopes used are expected to be distinct, but apart from that the exact 3 slopes considered do not matter. Indeed, for any two triples of slopes, (s 1 , s 2 , s 3 ) and (s 1 , s 2 , s 3 ), there exists an affine map of the plane turning any 3-slopes segment representation using slopes (s 1 , s 2 , s 3 ) into a 3-slopes segment representation using slopes (s 1 , s 2 , s 3 ). We denote -→ a , -→ b , and -→ c the vectors corresponding to slopes of the sets A, B , and C respectively. The magnitude of these vectors is chosen such that -

→ a + - → b + - → c = - → 0 .

TC-representations and TC-schemes

We begin with the definition of particular 3-slopes contact representations illustrated in Figure 2.

Definition 1. A Triangular 3-slopes segment Contact representation (TC-representation for short) is a 3-slopes contact segment representation using the same slopes as -→ a , -→ b , and -→ c , and where:

-Three segments a 0 , b 0 , and c 0 , form a triangle which contains all the other segments. -Every inner region is a triangle, whose each side is contained in a segment of the representation.

-Two parallel segments intersect on at most one point, their endpoint. Definition 2. Let the plane graph M (R) induced by a TC-representation R be the map whose vertices correspond to the segments of the representation, and where two vertices are adjacent if and only if the corresponding segments form a corner of one of the inner triangles. The orders of the neighbors around a vertex v correspond to the order of the segments around v. Note that the plane graph induced by a TC-representation has several properties. For example, two parallel segments correspond to non-adjacent vertices. The slopes hence define a 3-coloring of the graph. Note also that the dual graph of M (R) is bipartite. Indeed such a map has two types of faces, one set contains the (triangular) faces corresponding to the inner regions of the TC-representation, and the other set contains the outerface and the faces corresponding to intersection points. Let us denote the latter faces degenerate faces, and note that those faces have size three or six. A size six face (a i , b j , c k , a i , b j , c k ) comes from the intersection point of six segments, and as those six segments go in distinct directions they do not intersect elsewhere, so this cycle has no chord in M (R). Finally note that going clockwise in any inner region one successively follows α -→ a , α -→ b , and then α -→ c , for some not necessarily positive value α. Definition 3. A TC-representation R is a TC-scheme of an Eulerian triangulation T if M (R) is a subgraph of T with the same outer-face as T and such that the vertices and edges of V (T ) \ V (M (R)) lie inside degenerate faces of M (R) (see Figure 3).

Actually as in M (R), the inner faces around any vertex alternate among degenerate and non-degenerate. This implies that every edge of M (R) bounds a non-degenerate face, and a face that is degenerate or that is the outerface. We thus have the following.

Remark 1. A TC-representation R is a TC-scheme of T if and only if the non-degenerate faces of M (R) and its outerface are faces of T .

The main ingredient in the proof of Theorem 1 is the following.

Theorem 2. Every Eulerian triangulation T has a TC-scheme, and this scheme is unique.

To prove this theorem we proceed by the following steps. We first model TCschemes of T by means of a system of linear equations and we sketch out why this linear system always has a unique solution.

The linear system model

In a TC-representation all the triangles are homothetic. Let us define the size of a triangle as its relative size with respect to the outer-triangle. We may require that the outer-triangle has size 1, the triangles with a corner on the left have positive sizes, while the triangles with a corner on the right have negative sizes. The variables of our linear system correspond to the sizes of the triangular regions. So for each face f ∈ F 1 we have a variable x f . Informally, the value of x f will prescribe the size and shape of the corresponding triangle in a TC-representation. If x f < 0, x f = 0, or if x f > 0 the corresponding triangle has a corner on the right, is missing, or has a corner on the left, respectively.

Let us denote by F 1 (v) the subset of faces of F 1 incident to v. As the outer triangle has size 1 and contains the other triangles, the faces in F 1 (a 0 ) should have non-negative sizes, and they should sum up to 1 (see Figure 4, left). We hence consider the following constraint.

f ∈F1(a0)

x f = 1 (a 0 )
We add no constraint about the sign of these sizes. Note that similar constraints hold for b 0 and c 0 .

f ∈F1(b0)

x f = 1 (b 0 ) f ∈F1(c0) x f = 1 (c 0 )
Similarly, around an inner segment of a TC-representation all the triangles on one side have same size sign, which is opposite to the other side. Furthermore, by summing all these sizes one should obtain 0 (see Figure 4, right). Hence, for any inner-vertex u we consider the following constraint.

f ∈F1(u)

x f = 0 (u) + 0.33 + 0.17 In the following, Equation (a j ) will refer to Equation (u) where vertex u is replaced by a j . Note that every face f ∈ Let us define the set V = V \ {b 0 , c 0 } of size n -2. Finding a solution to L is equivalent to finding a vector S ∈ R F1 (that is a vector indexed by elements of F 1 ) such that M S = I, where M ∈ R V ×F1 (a square matrix indexed by elements of V × F 1 ) and I ∈ R V are defined by

M (x i , f ) = 1 if f ∈ F 1 (x i ) 0 otherwise. I(x i ) = 1 if x i = a 0 0 otherwise.
Given some bijective mappings g

V : [1, . . . , n -2] -→ V and g F1 : [1, . . . , n - 2] -→ F 1 , one can index the elements of M by pairs (i, j) ∈ [1, . . . , n-2]×[1, . . . , n- 2]
, and thus define the determinant of M . By the following lemma, L has a solution vector S, and this solution is unique.

Lemma 1. The matrix M is non-degenerate, i.e. det(M ) = 0.
The full proof of this lemma is inspired by the work of Felsner [START_REF] Felsner | Triangle Contact Representations[END_REF] on contact representations with homothetic triangles. The proof is not provided in this extended abstract but the main idea is to consider the bipartite graph T M with independent sets V and F 1 such that v ∈ V and f ∈ F 1 are adjacent if and only if v and f are incident in T . Note that M is the biadjacency matrix of T M . From the embedding of T one can easily embed T M in such a way that all the inner faces have size 6, and such that a 0 is on the outerboundary.

Note that every perfect matching of T M (if any) corresponds to a permutation σ on [1, . . . , n -2] (we say σ belongs to the permutation group S n-2 ) defined by σ(g -1 F1 (f )) = g -1 V (v), for any edge vf of the perfect matching. If the obtained permutation is even we call such perfect matching positive, otherwise it is negative. From the Leibniz formula for the determinant,

det(M ) = σ∈Sn-2 sgn(σ) i∈[1,...,n-2] M (g V (σ(i)), g F1 (i))
, one can see that det(M ) counts the number of positive perfect matchings of T M minus its number of negative perfect matchings.

Claim. The graph T M admits at least one perfect matching.

Given a graph G and a perfect matching M of G, an alternating cycle C is a cycle of G with edges alternating between M and E(G) \ M . Note that replacing in M the edges of M ∩ C by the edges of C \ M yields another perfect matching. We call such operation a cycle exchange. It is folklore that the set of perfect matchings of a graph are linked by cycle exchanges. Actually, for T M one can restrict itself to cycles of length six.

Claim. All the perfect matchings of T M are linked by 6-cycle exchanges. This implies that the perfect matchings of T M are either all positive, or all negative. Thus det(M ) = 0. The following lemma (not proved in this extended abstract) allows us to conclude the proof of Theorem 2.

Lemma 2. Every Eulerian triangulation T admits a TC-scheme R that corresponds to the solution of its linear system L.

3-slopes segment representations

In this section we use Theorem 2 to prove the main theorem of the article, Theorem 1. As already mentioned it is sufficient to prove it for Eulerian triangulations. Given an Eulerian triangulation T , let us denote a 1 , b 1 and c 1 the vertices forming a face with vertices b 0 and c 0 , with a 0 and c 0 , and with a 0 and b 0 , respectively. Theorem 1 follows from the following technical proposition. Proposition 1. For every > 0, every Eulerian triangulation T admits a 3-slopes segment representations R such that:

-The segments a 0 , b 0 , and c 0 form a triangle ∆ of size 1 (its sides are obtained by following -→ a , -→ b , and -→ c ).

-Every segment is contained in the hexagon centered on ∆, obtained by successively following

(1-) - → a , -2 - → c , (1-) - → b , -2 - → a , ( 1
-) -→ c , and -2 -→ b (see Figure 5). Given such representation R, we define the shape of R as the triplet (s a , s b , s c ) of sizes in R (w.r.t. the outer-triangle) of the triangles corresponding to a 1 b 0 c 0 , a 0 b 1 c 0 , a 0 b 0 c 1 , respectively. Note that if is chosen sufficiently small, that is for < 1, as the vertices a 1 , b 1 , and c 1 have neighbors that are inner vertices, a 1 , b 1 , and c 1 intersect ∆, and we have s a > 0, s b > 0, and s c > 0.

Proof. We proceed by induction as we assume that the proposition holds for any Eulerian triangulation with less vertices. The initial case of this induction, when |V (T )| = 3 clearly holds.

Given an Eulerian triangulation T with at least four vertices, we consider a TCscheme R of T (given by Theorem 2), and by successively resolving degenerate points (i.e. intersection points of at least three segments) from left to right, we eventually reach the sought representation. Here resolving means that the segments of a 3-degenerate point (resp. a 6-degenerate point) are moved to form a triangle (resp. a polygon) inside which we are going to draw a 3-slopes representation of the graph corresponding to this degenerate face of M (R), this is possible by using the induction on this smaller graph. The degenerate points of R are resolved from left to right. This means that at a given stage of this process there is a vertical line (parallel with -→ b ) V such that on its left there is no intersection point of three or more segments. This implies that on the left of V the representation handles some small perturbations: one can slightly move the segments without changing the intersections.

Let V be the leftmost vertical line containing degenerate points. We resolve those degenerate points by slightly moving segments on the left of or on V, while maintaining the right side of the representation unchanged. We consider different cases according to the degenerate points on V.

If V contains a 3-degenerate point p in the interior of a (vertical) segment b j and at the end of two segments a i and c k lying on the left of V, the situation is rather simple. Move these segments a little to the left and slightly prolong them to intersect b j (see Figure 6). As there is no degenerate point on the left of V these moves can be done while maintaining the existing intersections and avoiding new intersections. If a i b j c k is If V contains a double 3-degenerate point p in the interior of a (vertical) segment b j , the situation is similar to the previous one. Move the segments on the left of V as depicted in Figure 7. If the new triangle is not a face of T , we add a representation inside. We are now left with a simple 3-degenerate point at p. This corresponds to the following case.

If V contains a 3-degenerate point p in the interior of a (vertical) segment b j and at the end of two segments, a i and c k , lying on the right of V, one can move b j slightly to the right or slightly to the left and resolve these points without changing the right part of the representation. The choice of moving b j to the right or to the left is explained in the next paragraph, but we can assume this move to be arbitrarily small. Whatever the direction b j is moved one has to prolong a i and c k to have all the intersections, between these segments or with b j (see Figure 8). Note that in order to preserve the representation on the right of V the segments a i and c k are not moved, they are only prolonged around p. Again, if a i b j c k is not a face of T , we draw a representation inside the newly formed triangle. Note that if b j moves to the right, the triangle bordered by a i , b j and c k has negative size, but it suffices to apply a homothety with negative ratio to obtain a representation that can be drawn inside.

Consider now the degenerate points at the end of a (vertical) segment b j of V. Let b 1 , b 2 , . . . , b t be a maximal sequence of segments on V such that b j and b j+1 intersect on a point. We are going to move these segments alternatively to the right and to the left, for example the segments with even index are moved to the left while the ones with odd index are moved to the right. The exact magnitude of these moves will be set later, but first note that the 3-degenerate points in the interior of the segments b j with 1 ≤ j ≤ t can be dealt if the move of b j is sufficiently small (see previous cases). Consider the intersection point p between b j and b j+1 . The case of b 1 and b t 's end is similar and it is not detailed here. If there is a segment a i going through p. It is shown in Figure 9 how to resolve these two overlapped 3-degenerate points, in order to create two triangles, where one can add a small representation if needed. The case where there is a segment c k going through p is similar. Fig. 10. From left to right : A 6-degenerate point on V. Resolution if there is no chord in bjacbj+1a c with the shape of R . Resolution if none of bjc, ca , or a bj is a chord, with the shape of R . Resolution if ac and ca are chords, with the shape of R2.

Assume now that six segments intersect at p. Let b j be the one below p, and let a, c, b j+1 , a , and c be the other ones around p clockwisely. Let us assume wlog that b j has to move to the left, while b j+1 has to move to the right. The degenerate face corresponding to p is bounded by these six vertices and there are several cases according to whether there are chords among them in T (see Figure 10).

If there is no chord inside the cycle b j acb j+1 a c we consider the subgraph of T induced by the vertices on and inside this cycle, add we add the edges ab j+1 , b j+1 c , and ac outside the cycle, and we denote by T the obtained simple Eulerian triangulation. By the induction we know that T admits a 3-slope segment representation R , and let (s a , s b , s c ) be the shape of R . We resolve the point by moving the segments as depicted in Figure 10, and the magnitude of each of these moves is prescribed by the shape (s a , s b , s c ) in order to allow us to copy R inside the triangle formed by a, b j+1 , and c . Then we shorten a, b j+1 , and c to avoid intersections among them. Actually, the case where none of ab j+1 , b j+1 c , or ac is a chord is similar.

If none of b j c, ca , or a b j is a chord of b j acb j+1 a c we proceed similarly. The only difference is that we add the edge b j c, ca , or a b j outside b j acb j+1 a c to obtain T , and that we have to perform a homothety with negative ratio to include R .

Finally, if there are two opposite chords on b j acb j+1 a c , say ac and ca , we consider two triangulations. Let T 1 be the one inside the cycle c b j a and let T 2 be the one obtained from the interior of the 5-cycle acb j+1 a c by adding the edges ab j+1 and b j+1 c . By the induction we know that T 1 and T 2 admit 3-slopes segment representations R 1 , and R 2 , and let (s a , s b , s c ) be the shape of R 2 . We resolve the point by moving the segments as depicted in Figure 10, and the magnitude of each of these moves, except for b j , is prescribed by the shape (s a , s b , s c ) in order to allow us to copy R 2 inside the triangle formed by a, b j+1 , and c . Then we shorten a, and b j+1 to avoid the intersections corresponding to ab j+1 and b j+1 c . The segment b j is moved sufficiently to the left to avoid the interior of the triangle containing R 2 . Then R 1 is drawn inside the triangle bordered by b j , a and c . This is possible because R 2 does not intersect this triangle.

Finally note that the moves of b j and b j+1 are opposite but of proportional magnitudes (up to some constant depending on the shapes (s a , s b , s c ) of R or R 2 ). So it is clear that we can simultaneously move all the segments b j on V. This concludes the proof of the lemma.

Conclusion

D. West [START_REF] West | Open problems[END_REF] and H. de Fraysseix and P. Ossona de Mendez [START_REF] De Fraysseix | Representations by Contact and Intersection of Segments[END_REF] independently ask for a generalization of Scheinerman's Conjecture.

Conjecture 1. Planar graphs that are k-colorable admit a k-slopes segment representation.

The case k = 1 is trivial. We have seen that the case k = 2 holds with 2-slopes contact representations. We have seen that the case k = 3 also holds. For the final case k = 4 we would like to apply the same approach as here. This means that we would like to go through a TC-representation of the considered triangulation, and then resolve its degenerate points. There are at least two obstacles for this approach. The first one is to find an order on the degenerate points to resolve them. The left to right approach does not seem sufficient here. The second one is connected to degenerate points p such that going clockwisely around p we successively cross segments a, b, c, d, and a again. In that case its is impossible, restricting ourselves to small pertubations to create a region bordered by a, b, c, and d in the clockwise or counterclockwise order. To avoid this issue we want 4-colorings (i.e. 4-slopes assignements) of planar graphs with particular properties.

Conjecture 2. Planar graphs admit a {1, 2, 3, 4}-coloring such that ther is no induced C 4 colored with colors 1, 2, 3 and 4 in clockwise order.

Examples show that one cannot extend this condition to non-induced 4-cycles. A positive answer to this conjecture would imply that simple signed planar graphs have chromatic number at most 4, positively answering a conjecture of E. Máčajová, A. Raspaud, and M. Škoviera [START_REF] Máčajová | The Chromatic Number of a Signed Graph[END_REF].

Fig. 1 .

 1 Fig. 1. The octahedron and a 3-slopes contact representation. It is unique, up to vertex automorphism, up to scaling, and once the slopes are set.

Fig. 2 .

 2 Fig. 2. (left) Vectors -→ a , -→ b , and -→ c . (middle) A TC-representation with various types of intersection points. (right) Its induced graph, where gray faces are particular degenerate faces. One has size six, and there are two faces of size three that correspond to the same intersection point.

Fig. 3 .

 3 Fig. 3. From left to right. A TC-representation R; its induced map M (R), where gray faces are the degenerate faces; and two triangulations having R as TC-scheme.

Fig. 4 .

 4 Fig. 4. (left) The size of the triangles around a0. (right) The size of the triangles around some inner vertex bi.

F 1

 1 is incident to exactly one vertex of A, one vertex of B, and one vertex of C. Hence by summing Equations (a 0 ), (a 1 ),. . . ,(a |A| ), one obtains that f ∈F1 x f = 1. The same holds with Equations (b 0 ), (b 1 ),. . . ,(b |B| ), or with Equations (c 0 ), (c 1 ),. . . ,(c |C| ). Equations (b 0 ) and (c 0 ) are hence implied by the others and thus we do not need to consider them anymore. Let us denote by L the obtained system of n -2 linear equations on |F 1 | = n -2 variables.

Fig. 5 .

 5 Fig. 5. (left) A 3-slopes segment representation inside an hexagon. (right) A scheme representing its shape.

Fig. 6 .

 6 Fig. 6. (left) A 3-degenerate point on V (middle) Small perturbation of R (right) The addition of a representation inside the new triangle.

Fig. 7 .

 7 Fig. 7. (left) A double 3-degenerate point on V (right) Small perturbation of R.

Fig. 8 .

 8 Fig. 8. (left) A 3-degenerate point on V (middle) Slightly moving b j to the right (right) Slightly moving b j to the left.

Fig. 9 .

 9 Fig. 9. (left) A double 3-degenerate point on V (middle) & (right) Small moves that resolve this point.

Acknowledgements. The author is thankful to Marc de Visme for fruitful discussions on this topic, and to Pascal Ochem for bringing [15] to his attention.

This research is partially supported by the ANR GATO, under contract ANR-16-CE40-0009.


Marian Cors

Oliver Wrede

Anne-Caroline Genix

Dario Anselmetti

Julian Oberdisse

Thomas Hellweg

Core-shell microgel based surface coatings with linear thermo-response

We study the swelling and shrinking behavior of core-shell microgels adsorbed on silicon wafers. In these systems, the core is made of cross-linked poly(N-isopropylmethacrylamide) and the shell consists of cross-linked poly(N-n-propylacrylamide). In suspension, these particles exhibit an extended linear swelling behavior in the temperature interval between the lower critical solution temperatures of the two polymers. Using ellipsometry and AFM, we show that this linear response is also observed in the adsorbed state.

Introduction

Acrylamide-based microgels are colloidal particles comprising a gel network of solvated polymer chains. [START_REF] Pelton | Temperature-sensitive aqueous microgels[END_REF] As a function of external stimuli, and in particular temperature, these particles show a reversible shrinking and swelling behavior granting them the name "smart microgels". [START_REF] Nayak | Soft Nanotechnology with Soft Nanoparticles[END_REF][START_REF] Richtering | Gel architectures and their complexity[END_REF][START_REF] Ballauff | Smart" nanoparticles: Preparation, characterization and applications[END_REF][START_REF] Dietsch | Soft Nanotechnologyfrom Colloid Physics to Nanostructured Functional Materials[END_REF] The most prominent examples are microgels based on poly(N-isopropylacrylamide) (pNIPAM) [START_REF] Saunders | Microgel particles as model colloids: theory, properties, and applications[END_REF][START_REF] Geisel | New Insight into Microgel-Stabilized Emulsions Using Transmission X-ray Microscopy: Nonuniform Deformation and Arrangement of Microgels at Liquid Interfaces[END_REF] , which have a volume phase transition temperature of ca. 32-33 °C in water, corresponding to the lower critical solution temperature (LCST) in the phase diagram. [START_REF] Murray | The preparation, characterisation and applications of colloidal microgels[END_REF] However, also other monomers lead to the formation of microgels with thermo-responsive behavior, with various phase transition temperatures. [START_REF] Berndt | Temperature-Sensitive Core-Shell Microgel Particles with Dense Shell[END_REF][START_REF] Balaceanu | Heterogeneous Morphology of Random Copolymer Microgels as Reflected in Temperature induced Volume Transition and ¹H High-Resolution Transverse Relaxation NMR[END_REF][START_REF] Wedel | Non NIPAM Based Smart Microgels: Systematic Variation of the Volume Phase Transition Temperature by Copolymerization[END_REF][START_REF] Wedel | Role of Anionic Surfactants in the Synthesis of Smart Microgels Based on Different Acrylamides[END_REF][START_REF] Wedel | Smart Homopolymer Microgels: Influence of the Monomer Structure on the Particle Properties[END_REF][START_REF] Wu | Behavior of Temperature-Responsive Copolymer Microgels at the Oil/Water Interface[END_REF] By copolymerizing them statistically, the phase transition temperature can be tuned. [START_REF] Wedel | Non NIPAM Based Smart Microgels: Systematic Variation of the Volume Phase Transition Temperature by Copolymerization[END_REF][START_REF] Hoare | Dimensionless plot analysis: A new way to analyze functionalized microgels[END_REF][START_REF] Lally | Microgels containing methacrylic acid: effects of composition on pH-triggered swelling and gelation behaviours[END_REF][START_REF] Kim | The volume transition in thermosensitive core-shell latex particles containing charged groups[END_REF][START_REF] Keerl | Temperature Sensitive Copolymer Microgels with Nanophase Separated Structure[END_REF][START_REF] Snowden | Colloidal copolymer microgels of Nisopropylacrylamide and acrylic acid: pH, ionic strength and temperature effects[END_REF][START_REF] Hertle | Thermoresponsive copolymer microgels[END_REF] Moreover, by designing complex particle architectures [START_REF] Ballauff | Smart" nanoparticles: Preparation, characterization and applications[END_REF][START_REF] Berndt | Doubly Temperature Sensitive Core-Shell Microgels[END_REF][START_REF] Lu | Thermoresponsive Core-Shell Particles as Carriers for Ag Nanoparticles: Modulating the Catalytic Activity by a Phase Transition in Networks[END_REF][START_REF] Duracher | Adsorption of Modified HIV-1 Capsid p24 Protien onto Thermosensitive and Cationic Core-Shell Poly(styrene)-Poly(N-isopropylacrylamaide) Particles[END_REF][START_REF] Hatto | Novel microgel-particle colloids: the detailed characterisation of the layer structure and chain topology of silica:poly(NIPAM) core-shell particles[END_REF][START_REF] Hellweg | Responsive Core-Shell Microgels: Synthesis, Characterization, and possible Applications[END_REF][START_REF] Zeiser | Linearly thermoresponsive core-shell microgels: Towards a new class of nanoactuators[END_REF][START_REF] Dubbert | How Hollow Are Thermoresponsive Hollow Nanogels?[END_REF][START_REF] Dubbert | Core-shell-shell and hollow double-shell microgels with advanced temperature responsiveness[END_REF][START_REF] Karg | Well defined hybrid PNIPAM core-shell microgels: Size variation of the silica nanoparticle core[END_REF][START_REF] Königer | Thermophoresis of thermoresponsive polystyrene-poly(N-isopropylacrylamide) core-shell particles[END_REF] , i.e. microgel particles with compartments of different monomeric composition or composition gradients, particle properties are expected to be varied, and stimuli-response controlled. Furthermore, also the properties of single adsorbed microgels and of microgel coatings have been studied. [START_REF] Schmidt | Thermoresponsive surfaces by spincoating of PNIPAM-co-PAA microgels. A combined AFM and ellipsometry study[END_REF][START_REF] Spears | Dynamic materials from microgel multilayers[END_REF][START_REF] Vikulina | Temperature-induced molecular transport through polymer multilayers coated with PNIPAM microgels[END_REF][START_REF] Islam | Poly (N -isopropylacrylamide) microgel-based etalons and etalon arrays for determining the molecular weight of polymers in solution[END_REF][START_REF] Nerapusri | Absorption of cetylpyridinium chloride into poly(N-isopropylacrylamide)-based microgel particles, in dispersion and as surface-deposited monolayers[END_REF][START_REF] Howard | Swelling and collapse of an adsorbed pH-responsive film-forming microgel measured by optical reflectometry and QCM[END_REF][START_REF] Höfl | Volume phase transition of "smart" microgels in bulk solution and adsorbed at an interface: A combined AFM, dynamic light, and small angle neutron scattering study[END_REF][START_REF] Herman | Polyelectrolyte exchange and diffusion in microgel multilayer thin films[END_REF][START_REF] Clarke | Core/Shell Microgels Decouple the pH and Temperature Responsivities of Microgel Films[END_REF] A core-shell architecture, where the core and the shell have a different phase transition temperature is particularly promising. Zeiser et al. have recently shown that for core-shell particles with a shell of smaller LCST than the core, a qualitatively different bulk swelling behavior is present, including a linear swelling over a broad range of temperatures. [START_REF] Zeiser | Linearly thermoresponsive core-shell microgels: Towards a new class of nanoactuators[END_REF] Beside the scientific interest in this volume phase transition, growing attention arises from potential applications of these materials. Prominent examples are their use as surface coatings for smart cultivation of vertebrate cells and as etalons. [START_REF] Islam | Poly (N -isopropylacrylamide) microgel-based etalons and etalon arrays for determining the molecular weight of polymers in solution[END_REF][START_REF] Schmidt | Adhesion and Mechanical Properties of PNIPAM Microgel Films and Their Potential Use as Switchable Cell Culture Substrates[END_REF][START_REF] Uhlig | Patterned Thermoresponsive Microgel Coatings for Noninvasive Processing of Adherent Cells[END_REF][START_REF] Smiley-Wiens | Solvent exchange kinetics in poly(N-isopropylacrylamide) microgel-based etalons[END_REF][START_REF] Sorrell | Glucose sensitive poly (N-isopropylacrylamide) microgel based etalons[END_REF] In this context, using different particle architectures might be advantageous, as it would allow making use of progressive swelling/shrinking as a function of temperature, as opposed to a sudden collapse. However, it remains to be shown that core-shell microgels made of poly(N-isopropylmethacrylamide) (pNIPMAM) and poly(N-n-propylacrylamide) (pNNPAM) still exhibit a linear swelling response in the adsorbed state. Hence, in this article, we focus on coreshell microgels made of a large pNIPMAM core having a high LCST of 44°C, surrounded by a shell of pNNPAM having an inferior LCST of 21°C, similar in architecture to those described by Zeiser et al [START_REF] Zeiser | Linearly thermoresponsive core-shell microgels: Towards a new class of nanoactuators[END_REF] .

The magnitude of the response of these core-shell microgels is mainly controlled by the cross-linker content of the core if the shell composition is chosen to be constant. Therefore, in this work we systematically vary the core crosslinker content (CCC) of the pNIPMAM cores. Both the swelling properties of the cores and of the core-shell microgel particles are studied in suspension by photon correlation spectroscopy (PCS) first. Then the particles are deposited on a silicon wafer by spin-coating, and the thickness of the layer and individual height profiles are studied by AFM and ellipsometry in the dry state, followed by an investigation of the swelling properties when the layers are covered with water.

Experimental Section

Synthesis: The core monomer N-isopropylmethacrylamide (NIPMAM, purity > 97 %), the initiator ammonium peroxodisulfate (APS, >98 %), the crosslinker N,N'-methylenebisacrylamide (BIS, 99 %), the ionic surfactant sodium dodecyl sulfate (SDS, >99 %) were purchased from Sigma-Aldrich and used as received. The shell monomer N-n-propylacrylamide was synthesized following a classical Schotten-Baumann reaction described by Hirano et al. [START_REF] Hirano | Hydrogen-bondassisted syndiotactic-specific radical polymerizations ofN-alkylacrylamides: The effect of theNsubstituents on the stereospecificities and unusual large hysteresis in the phase-transition behavior of aqueous solution of syndiotactic poly(N-n-propylacrylamide)[END_REF] , using acryloyl chloride (97 %, Aldrich, USA), triethylamine (99 %, Grüssing, Germany), propylamine (99 %, Fluka, USA) and methylene chloride (p.

A.) as solvent. For the synthesis of pNIPMAM cores with different crosslinker contents, a 250 mL threenecked flask under gentle nitrogen flow with a mechanical stirrer on the top opening is used. This flask was placed in a heating oil bath with magnetic stirrer. The two side openings were equipped with a septum with a cannula as nitrogen inlet and a reflux condenser with bubble counter, respectively. The crosslinker contents, SDS concentration and radical starter specified as mol% with respect to the amount of monomer are listed in Table 1 for all samples. The crosslinker was dissolved in water (140 mL) first and subsequently heated to 70 °C. Its concentration in the core is given as CCC (core crosslinker content) throughout this article. After purging the solution for 1 h with nitrogen the monomer followed by surfactant was added as powder and after 5 minutes the reaction was initiated by injecting 1 mL of an aqueous solution of the radical starter. After rinsing the neck of the flask and initiation the volume was 150 mL. The total reaction took 5 h, followed by continued stirring at room temperature overnight, and purification. The purification was done at room temperature by centrifugation at 48,384 g for 45 minutes, decantation of the supernatant and redispersion of the precipitate with water. The redispersion was dome with a vortexer (MS 3 basic, IKA-Werke GmbH & CO. KG, Staufen, Germany) for typically 1 minute at maximum speed. This procedure was repeated five times. Shell synthesis was done with a similar protocol, also at high temperature and thus in the collapsed state, in the same setup with a 100 mL flask, by precipitating NNPAM on previously synthesized and purified pNIPMAM cores resulting in a pNIPMAM-core pNNPAM-shell microgel system. The total suspension volume was 50 mL. The shell had always the same nominal composition. The amount of crosslinker BIS was 1.90 mol% with respect to the NNPAM feed and the added amount of the core was 0.15 wt% with respect to the solvent. The concentration of SDS was 1.1 mmol per Liter (2.75 mol%). The core-shell particles were again purified by the same purification procedure as the core particles. The resulting molecular architecture is depicted in Scheme 1. Photon correlation spectroscopy (PCS): PCS experiments were performed using two different setups.

Angular dependent PCS experiments were done using a classical goniometer setup (ALV GmbH, Langen, Germany) equipped with an Ar + -ion laser (Stabilite 2017, Spectra Physics, Santa Clara, USA) emitting at 514.5 nm. The sample was placed in a decalin matching bath kept at constant temperature using a thermostat (Haake DC50, Thermo Fisher Scientific, Waltham, USA). The scattered light was analyzed using a multiple τ hardware correlator (ALV 5000 Fast, ALV GmbH, Langen, Germany). The obtained correlation functions were analyzed using numerical Laplace inversion by means of CONTIN [START_REF] Provencher | A constrained regularization method for inverting data represented by linear algebraic or integral equations[END_REF][START_REF] Provencher | Contin: a general purpose constrained regularization program for inverting noisy linear algebraic and integral equations[END_REF] . This method provides the intensity-weighted relaxation rate Γ of the observed dynamics in the sample. For pure diffusional modes the dispersion Γ ~ q 2 is expected and the diffusion coefficient can be obtained from the corresponding plot. The expected dispersion is found for both the core and for the core-shell particles. Hence, the temperature-dependent measurements were performed at a fixed angle of = 60 ° in a partly home-made setup. The PCS-setup had a He-Ne laser (HNL210L-EC, 632.8 nm, Thorlabs, Newton, USA) as light source, a multiple tau digital correlator (ALV-6010, ALV GmbH, Langen, Germany) and a detector (SO-SIPD, ALV GmbH, Langen, Germany). The sample was placed in a decalin matching bath. For temperature control a refrigerated bath (Haake C25P, Thermo Fisher Scientific, Waltham, USA) with a controller (Phoenix II, Thermo Fisher Scientific, Waltham, USA) was used.

Spin coating: Microgel monolayers on silicon wafers (Sigert Wafer, diameter = 50.8 mm, Aachen, Germany) were obtained by spin coating (LabSpin6, SÜSS micro Tec, Garching, Germany), after plasma cleaning (Zepto B, Diener Electronics, Ebhausen, Germany) with an O2-plasma for 15 min at 100 W. The calculated arithmetical mean deviation roughness as well as the root mean square roughness was below 1nm. For microgel layers with a high surface coverage first, a dilute purified microgel dispersion (0.5 wt%) was put on the whole silicon wafer for 45 min, followed by spinning for 300 s at a rotation speed of 1000 rpm. For well-separated microgel particles the same procedure was used without waiting 45 min before spinning the wafer.

Ellipsometry: For ellipsometry measurements a single-wavelength ellipsometer (SE 400adv, SENTECH Instruments GmbH, Berlin, Germany) was used. The method of ellipsometry was described elsewhere. [START_REF] Bonn | Molecular Layering on a Fluid Substrate[END_REF][START_REF] Beaglehole | Ellipsometric Study of the Surface of Simple Liquids[END_REF] Temperature control and liquid cell are custom-made products from SENTECH. The accessible temperature range was 15 °C to 55 °C. The measurements and modeling were done with the software "SE400advanced" from SENTECH. For modeling, a first guess of the microgel layer thickness was used based on the layer thickness determined by AFM measurements, and the initial refractive index was estimated based on literature to 1.35. [START_REF] Schmidt | Thermoresponsive surfaces by spincoating of PNIPAM-co-PAA microgels. A combined AFM and ellipsometry study[END_REF] Ellipsometry was already used to characterize the swelling behavior of adsorbed microgels in our previous work from Schmidt et al. [START_REF] Schmidt | Thermoresponsive surfaces by spincoating of PNIPAM-co-PAA microgels. A combined AFM and ellipsometry study[END_REF] and in literature. with the open source data evaluation program for scanning probe microscopy Gwyddion. [START_REF] Nečas | Gwyddion: an open-source software for SPM data analysis[END_REF] After background correction we extracted the profile of at least 20 microgels and calculated the mean profile.

Determination of the amount of water of dried microgels:

The water content of dried microgels can be determined by Karl-Fischer titration. [START_REF] Gawlitza | Immobilization of water-soluble HRP within poly-N-isopropylacrylamide microgel particles for use in organic media[END_REF] The measurements were done using a Karl-Fischer coulometer

(

Results and Discussion

After microgel synthesis, the particles have been characterized in aqueous suspension first. Then they have been deposited on silicon wafers by spin-coating, and their swelling properties in the dry and wet state have been investigated by ellipsometry. Moreover, AFM has been used to measure the height profile of individual dry microgel particles.

Synthesis and properties in suspension

The microgel cores were produced following the synthesis protocol outlined in Section 2 and the parameters are summarized in Table 1. The key parameter is the core crosslinker content (CCC), which allows fine tuning of the global swelling properties of the core-shell particles in suspension. [START_REF] Zeiser | Linearly thermoresponsive core-shell microgels: Towards a new class of nanoactuators[END_REF] We have first checked that the cores alone have the expected swelling properties of pNIPMAM, see Supporting

Information Figure S1 for details. Then, we have measured the hydrodynamic radius RH as a function of temperature of the series of core-shell microgels by PCS. These results are shown in Figure 1. They agree nicely with previous findings by Zeiser et al [START_REF] Zeiser | Linearly thermoresponsive core-shell microgels: Towards a new class of nanoactuators[END_REF] and thereby validate the synthesis. The original swelling curves shown in Figure 1a display the hydrodynamic radius RH as a function of temperature, for microgel suspensions of various CCC as given in the legend. Three swelling regimes are found. Below T = 20 °C, the entire particles are swollen, and the total hydrodynamic radius is typically greater than 200 nm. Note that due to the different formulations with the various crosslinker concentrations, the total swollen sizes of the core-shell microgels are not identical. At the VPTT of pNNPAM (23 °C), the system starts to collapse, resulting in a size transition step. The region of interest in the intermediate temperature range, between 23 °C and 44 °C, shows a swelling behavior where RH changes linearly with temperature. Instead of some average transition temperature with a steep collapse, the swelling extends thus over the entire temperature range. At high temperature, finally, both core and shell are collapsed. A second size transition step at the VPTT of pNIPMAM (44 °C) does not occur because the particle size is already compressed to the range of the collapsed core (compare Figure S1).

As the microgels do not have the same size from the start, it is instructive to normalize the size by the collapsed radius RH min taken as the high-temperature average (T = 45 -50 °C). The result, RH/RH min , is shown in Figure 1b, where the high temperature regime now superimposes by construction. This representation illustrates that the swelling is higher for the samples with lower CCC. The slope in the linear regime connecting the collapsed and the swollen state can therefore be tuned by the core crosslinker content.

A zoom of the linear region is given in the inset of Figure 1b for different crosslinker concentrations.

The relative value of the slope is found to decrease from 2.25 %/°C to 1.22 %/°C, with reasonable accuracy (± 0.06 %). These values are in line with the total relative decrease of about 30 % in radius, over a temperature range of some 15 degrees. The change in slope corresponds to a diminished sensitivity of swelling to temperature by a factor of 1.8, when the CCC is changed from 5 to 15 mol%.

Finally, the swelling is sometimes expressed in terms of volume change which is given by (RH/RH min ) [START_REF] Richtering | Gel architectures and their complexity[END_REF] .

This leads to a qualitatively similar figure as Figure 1b, with the linear domain slightly curved, and the low-temperature values giving the maximum volumetric swelling. In our case, the maximum swelling varies from about six to eight with CCC.

Spin coating and characterization of dry surfaces

Well-separated core-shell microgel particles have been deposited by spin coating on silicon wafers, and studied by AFM. The height profiles of individual particles in the dry state can be determined by image treatment of the AFM results, and averaged over several particles, typically 20. In the SI (Figure S3), sample measurements are shown for illustration. In Figure 2a, the average height profiles for microgels with different CCC are shown. One should pay attention to the different X-and Z-scales: all microgel particles are squeezed and adopt a rather flat, pancake-like shape. With increasing CCC, the particles become less flat: an increase in CCC by a factor of three from 5 to 15 mol% results in an increase in height by a factor of three as well, but even particles with the highest degree of crosslinking have an aspect ratio of ca. four. Figure 2b shows a schematic drawing of an adsorbed microgel particle. The total volume of these pancakes can be determined by integration, i.e. adding up the volume of hollow cylinders delimited by the height profile. If this volume is taken as the microgel volume (with water) then the total polymer volume (without water) can be determined by subtraction of the volume of the water present in the "dry" sample, determined by Karl-Fischer-Titration. Then the water contents can be determined by comparing the polymer volume (without water) to the swollen (low-T) volume in water estimated from PCS. The calculated volumes are given in Table 2. From the ratio of the swollen to the dry volume, a water content of ca. 75-95 % in the swollen state can be deduced. This is approximately four times higher compared to the water content of dried microgels (Φ H2O,dry) of about 20 % residual water. The water content in the dried state was determined by Karl-Fischer-Titration. As a next step, successful deposition of microgel particles with a high surface coverage was obtained by spin coating on silicon wafers, following the protocol given in Section 2. The optimization of system parameters, and in particular the introduction of a waiting time of 45 minutes before spinning, as well as the appropriate concentration, were essential to promote the formation of homogeneous layers. In Figure 3, an AFM height profile scan over a lateral size of 5 µm of a dry microgel particle layer at room temperature (22±1 °C) is presented. It shows that the core-shell particles cover the surface, forming a dense monolayer. The mean height of an area of 25 µm 2 was determined and is given in Table 3. To verify homogeneity of the surface coating, different magnitudes up to an image with 50 µm lateral size are shown in the SI (Figure S2). Given that the geometry in Figure 3 is not a laterally homogeneous flat layer due to the density fluctuations caused by the array of squeezed spherical microgel particles and the presence of holes, the average thickness of the dry layers determined by AFM has been confirmed by ellipsometry. The ellipsometry measurements yielded also the refractive index of the microgel particles. Both thickness measurements are compared in Table 3. The general tendency of increase in thickness with CCC is well captured by both methods, and the results agree at least semi-quantitatively. The difference in thickness determined by AFM and ellipsometry may be due to surface roughness. While AFM measures the detailed topography of the microgel surface, and average height can be easily computed, single wavelength ellipsometry averages over the layer. Then surface roughness like the one due to the individual microgel particles contributes, and may result in an inaccuracy in thickness. 52

Comparison of swelling in water at surfaces and in bulk suspension

The thicknesses in water of the core-shell microgel layers obtained by spin coating on silicon wafers have also been assessed with ellipsometry. The raw results are plotted as a function of temperature in Figure 4a, for three different CCC as given in the legend. At low temperature, the total thicknesses are a bit smaller than twice the hydrodynamic radii in suspension (cf. Figure 1a), which can be understood from the probable fuzzy nature of the particle surface, not dense enough to be picked up by ellipsometry.

Moreover, particles are probably slightly deformed upon contact with the wafer. The ellipsometric thickness is found to be about 70% to 85% of the hydrodynamic diameter, which seems to be compatible with such a scenario. Furthermore, as the initial microgel particles differ in size due to the varying compositions, their thicknesses d is normalized to the unswollen, high-temperature (45 °C -50 °C) thickness d min in Figure 4b, in analogy to Figure 1b.

Summary and Conclusion

In the present contribution, we have prepared surface coatings using core-shell microgels with core and shell made of two distinct polymers with a significant contrast in LCST (ΔLCST = 23 °C). The shell has the inferior LCST, thus it shrinks first as temperature is increased, and compresses the core.

Inversely, starting from the high synthesis temperature and decreasing T, the core tends to swell first, but is hindered by the still collapsed shell, which leads to a peculiar -linear -swelling behavior. The slope of the evolution of the radius with temperature has been successfully correlated with the crosslinker density of the core: softer, less crosslinked cores have a stronger response. We have then deposited the microgel particles by spin-coating. The key result of this article is that the same particles respond also in a linear manner in size to temperature changes when they are adsorbed to a hard surface.

In particular, we have shown by combinations of PCS, ellipsometry and AFM, that an extended linear behavior very similar to the one of the respective suspended particles persists, with a convincing overlap of the linear regions when expressed as relative swelling. Moreover, the average shape of individually adsorbed dry microgel particles has been measured by AFM. The total particle height could again be correlated with the core crosslinker density, the weaker microgels being flatter. The CCC is thus again shown to be the most important system parameter. Moreover, a volume analysis using the AFM height profiles allows an estimation of the average dry volume, and thus of the water content in suspension, in both the swollen and the collapsed state.

Such layers with a linear change in thickness might allow to tune, e. g., the interaction with vertebrate cells. Such a behavior might be of special interest in the context of stem cell differentiation since it can be expected that the Young modulus of the layers changes linearly and can be adjusted by the correct choice of the temperature. Moreover, such layers are certainly advantageous in microgel based etalons [START_REF] Zhang | Lipase-modified pH-responsive microgel-based optical device for triglyceride sensing[END_REF][START_REF] Sorrell | Color Tunable Poly(N-Isopropylacrylamide)-co-Acrylic Acid Microgel-Au Hybrid Assemblies[END_REF] . As a promising perspective, one may explore the system where the LCST contrast is inverted. As the temperature is increased in such an inverse system, the core made of pNNPAM shrinks first (above 22°C), while the shell made of pNIPMAM is still in good solvent conditions. Such as system is expected to show completely different swelling curves, as well as modified mechanical properties.

For Table of Contents Only:

Scheme 1 :

 1 Scheme 1: Core-shell architecture obtained by precipitating and cross-linking of NNPAM (red) on the pNIPMAM core (black). The collapsed core particles are used as seeds in the precipitation polymerization of pNNPAM. The shell synthesis is done at 70 °C, which is largely above both LCSTs of the two polymers.

49

 49

AFM:

  AFM measurements were performed on a FlexAFM (Nanosurf AG, Liestal, Switzerland) with an Easyscan 2 controller (Nanosurf AG, Liestal, Switzerland). To avoid background vibrations, the AFM was placed on a sample stage (Flex, Nanosurf AG) on top of a marble plate which lies on tennis balls. All measurements where performed in tapping ("Phase Contrast") mode. We used standard silicon cantilevers with 30 nm aluminum reflex coating from BudgetSensors® (Tap300Al-G, BudgetSensors®, Sofia, Bulgaria). The cantilevers had a force constant of 40 N/m and a resonance frequency of approximately 290 kHz. The scan rate was between 1 Hz and 0.33 Hz with 256 or 512 points per line.The setpoint was 55 % of the maximal deflection and the free vibration amplitude was approximately 200 mV.The thickness of the dry microgel monolayer was determined as the average height over an area of 25 µm 2 using the software of the AFM (Nanosurf Easyscan 2, Nanosurf AG, Liestal, Swiss). For investigation of the individual height profiles of single microgels, we analyzed the recorded raw data

Figure 1 :

 1 Figure 1: Swelling curves of core-shell microgels as a function of temperature with different core crosslinker contents (CCC) (5.0 mol% CCC (□), 7.5 mol% CCC (○), 10.0 mol% CCC (△), 12.5 mol% CCC (▽) and 15.0 mol% CCC (◇)). a) Hydrodynamic radius RH. The linear swelling zone is indicated by the vertical dashes. An example of the error of PCS data estimated to be 5% is indicated. b) Same data normalized with respect to the high-temperature average (T = 45 C° -50 °C) of the hydrodynamic radius RH min . Inset: Zoom of the linear region.

Figure 2 :

 2 Figure 2: a) Average height profiles for dry core-shell microgels determined from AFM height scans, for different core crosslinker contents as indicated in the legend. b) Schematic drawing of a microgel in suspension and an adsorbed microgel.

Figure 3 :

 3 Figure 3: Zoom of an AFM height scan of a monolayer of dried core-shell microgels with 5 mol% core crosslinker content. The lateral image size is 5 µm and the average height is 30.8 nm.

Figure 4 :

 4 Figure 4: Ellipsometric measurements in water of the a) layer thickness d of core-shell microgels deposited on a silicon wafer, for different core crosslinker contents as indicated in the legend, as a function of temperature. In b),

Figure 5 :

 5 Figure 5: Swelling curves of hydrodynamic radius RH or ellipsometric thickness d normalized to their hightemperature average (45 °C -50 °C) of core-shell microgel particles in bulk (PCS, RH/RH min ) and on surfaces (ellipsometry, d/d min ), for different core crosslinker contents a) 5 mol% CCC b) 10 mol% CCC c) 15 mol% CCC).

Table 1 :

 1

	pNIPMAM-core crosslinker (BIS) content. Constant amounts of pNIPMAM (12.4 mmol,
	1.5628 g ± 0.0004 g), SDS (3 mol%, 1.0730 g ± 0.0010 g) and radical starter (3.3 mol%, 0.0924 g ± 0.0001 g)
	were used.
	BIS / mol%	BIS / g
	5.00	0.0947
	7.50	0.1421

  Methanol AnalaR NORMAPUR® ACS, Reag. Ph. Eur. zur Analyse, VWR International, Radnor, USA) resulting in a solution of 1 wt%. The measurements was done with a sample volume of 100 µL. The water content of the solvent was measured and subtracted.

684 KF Coulometer, Metrohm AG, Herisau, Swiss). The titration solution was CombiCoulomat fritless (Meck KGaA, Darmstadt, Germany). The microgels were dried for 24 h at 70 °C in a drying oven (U30, Memmert GmbH + Co. KG, Schwabach, W -Germany) and after storing them for 24 h at ambient they were dissolved in methanol (

Table 2 :

 2 Volume of dry core-shell microgels estimated with the average height profiles determined by AFM (Figure3), comparison to volumes in suspension (PCS), estimated water fractions in suspension, in the swollen state (ΦH2O,sw) and measured water contents in the dried state (ΦH2O,dry) by Karl-Fischer-Titration for different core crosslinker contents (CCC).

	Core crosslinker content / mol% Vdry. / nm 3 Vcollapsed / nm 3 Vswollen / nm 3 ΦH2O,dry ΦH2O, sw
	5	1.19•10 7	5.13•10 6	3.88•10 7	18 %	74 %
	7.5	3.61•10 6	6.71•10 6	4.46•10 7	17 %	93 %
	10	4.84•10 6	8.58•10 6	5.44•10 7	22 %	93 %

Table 3 :

 3 Thickness of dried layers of core-shell microgel particles deposited on a wafer, measured by AFM and ellipsometry. The thickness was deduced by the mean thickness of an area of 25 µm 2 for AFM and ca. 1 mm 2 for ellipsometry. The refractive index was determined by ellipsometry. The error of thicknesses determined by AFM is estimated to be 5 % based on multiple measurements.

	Core crosslinker	Thickness AFM	Thickness by	Refractive index
	content (mol%)	(nm)	ellipsometry (nm)	by ellipsometry
	5	30.8	36.0±0.8	1.48±0.02
	7.5	46.8	41.1±0.9	1.51±0.02
	10	59.6	51.8±1.2	1.50±0.02
	12.5	71.8	56.0±1.2	1.49±0.02
	15	74.5	64.0±1.4	1.46±0.01

Acknowledgements:

The authors are thankful for support by the joint ANR/DFG CoreShellGel project, grant ANR-14-CE35-0008-01 of the French Agence Nationale de la Recherche, and grant HE2995/5-1 by Deutsche Forschungsgemeinschaft. We also thank Dagmar Wiechoczek for the help with the Karl-Fischer titration.

Supporting Information

PCS measurements of used PNIPMAM-cores, AFM images with different magnitudes and a sample for height profile extraction.


Malé Kital

Romain Courault

Marianne Cohen

Present and future distribution modeling of Pinus sylvestris and Betula pubescens within the

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Stanislaw Schukajlow
email: schukajlow@uni-muenster.de

Chair Inés M A Gómez-Chacón

Çiğdem Haser
email: cigdem.haser@utu.fi

Peter Liljedahl
email: liljedahl@sfu.ca

Karen Skilling
email: karen.skilling@kcl.ac.uk

Hanna Viitala
email: hanna.viitala.edu@gmail.com

Introduction to the work of TWG 8: Affect and the teaching and learning of mathematics

Introduction

In this chapter, we would like to introduce discussions about the affective constructs and their relation to teaching and learning of mathematics. Before the conference, we agreed to change the title of TWG from "Affect and mathematical thinking" to "Affect and teaching and learning of mathematics". The new title of the group better reflects the broad scope of research that we welcome in our group. All types of affective constructs that emerges in relation to both learning and teaching of mathematics could be represented and discussed in the working group during the conference.

The number of submissions and participants confirmed the interest of mathematics education researchers in affect. In total 26 papers and 3 posters were submitted to our group, with 24 papers and 2 posters were accepted for presentation, and 23 papers and 2 posters accepted for publication in these conference proceedings. Our group included presentations from Canada, Israel and different European countries. Participation of researchers from 15 countries (including 13 newcomers) reflected the spirit of inclusion that is traditional for this group that has been discussing affect in mathematics education in recent years.

We started our work with an introductory, ice-breaking activity and a reflection of the work done in previous conferences. This year we decided to considerably increase the time for discussions of each paper and structured the paper and poster presentations and discussions in a new way. After a short presentation (7 min) and clarifying questions (3 min) of papers that we scheduled within each session, authors of presentations discussed their research (30 -40 min) with other group members in small groups. Participants could decide which paper or poster they would like to join for discussion. We assigned contributions to the sessions in groupings related to: affect; emotions; identity; beliefs; and motivation or attitudes/engagement. At the end of each session, we shared our thoughts about these topics with the whole group. In the final session, we discussed new perspectives on affect and related constructs that emerged during our work.

Reflection on prior research in affect

Peter Liljedahl presented a comprehensive overview of the work that has been done during the past CERME conferences. Çiğdem Haser and Stanislaw Schukajlow picked up and elaborated on this overview during the presentation of the report about the work of the TWG8. The starting point for systematic research on affect in mathematics education is usually a review paper by [START_REF] Mcleod | Research on affect in mathematics education: A reconceptualization[END_REF]. In his seminal work, the author introduced an influential taxonomy of affect. According to this taxonomy, affective constructs can be assigned to different points on the line that ranges between beliefs through attitudes to emotions. Beliefs were proposed to be most stable, less affective and most cognitive, whereas emotions were noted as less stable, most affective and less cognitive.

In the last decades, researchers underlined repeatedly that the complexity of affect requires elaborating the taxonomy proposed by McLeod. For example, [START_REF] Hannula | Exploring new dimensions of mathematics-related affect: embodied and social theories[END_REF] analyzed research in mathematics education and suggested a need to distinguish between three dimensions as common foundations for a theoretical framework. These included: cognitive, motivational and affective dimensions; unstable states and stable traits; and social, psychological or physiological nature of affect.

Recently, object of the affect (affect about life, learning, problem solving or strategy use), subject (teacher or students) or valence (positive or negative) were suggested to be important for research on affect [START_REF] Schukajlow | Emotions and motivation in mathematics education: theoretical considerations and empirical contributions[END_REF]. Further, the theoretical approach (acquisitionist or participationist) play an important role. Affective constructs that were explicitly anchored in the taxonomy by McLeod were investigated and new dimensions were proposed. For example, teacher beliefs about the nature of mathematics and their relation to learning can be assigned to beliefs about nature of mathematics, beliefs about mathematics teaching and beliefs about mathematics learning [START_REF] Beswick | Teachers' beliefs about school mathematics and mathematicians' mathematics and their relationship to practice[END_REF]. Teacher beliefs build a so-called belief system that start its development in the school years, changes in university years and is shaped based on the practical experience encountered by in-service teachers.

An important point raised in prior discussions related to the definitions used for characterization of affective constructs. As affective constructs are seen as complex phenomena the definitions of particular constructs often overlap with other affective constructs and are criticized as being circular. For example, emotions are defined as phenomena that included cognitive, affective, motivational, physiological and expressive parts. Affect and motivation are used in this definition thus for characterization and grounding of emotions.

Finally, reciprocal relationships between affective constructs and students' achievement is acknowledged and are crucial for understanding the role of the affect in teaching and learning. Most of the studies in the TWG8 targeted affect and implied problems with affective constructs as a reason for difficulties in teaching and learning. However, another perspective is also valuable, whereby students' or teachers' affective problems while learning or teaching might result from problematic achievements in the past. [START_REF] Liljedahl | Mathematics Education Graduate Students' Thoughts About Becoming Researchers[END_REF].

Engagement in students' interactive storytelling and its impact on attitudes were explored in the contribution by Pierry et al. Novotna analyzed private supplementary tutoring and its relation to students' attitudes. Different levels of engagement (high and low) were distinguished in students with the same achievement level in the study by Skilling. Haataja et al. demonstrated that gazes at teachers' and at students' faces differed in smalland whole-group instructions. Kourty analyzed cognitive, behavioral and affective aspects of engagement in inquiry-based learning.

In posters, Hansen discussed collaborative processes while solving problems and Barton introduced a study on self-made tutorials.

Evolution of the TWG

The overall quality of the submitted contributions increased and this resulted in acceptance of nearly all papers and posters for presentation at the conference. Contributions investigated affective measures of different subjects: primary and secondary school students, pre-and in-service teachers and doctoral students. Most of the researchers applied qualitative analysis and performed case studies, in order to answer their research questions. Thus, the distribution of submissions followed the general trend toward the dominating role of qualitative studies in research on affect [START_REF] Schukajlow | Emotions and motivation in mathematics education: theoretical considerations and empirical contributions[END_REF]. One reason for this trend might be the efforts of researchers to capture the whole complexity of the affective phenomena and analyze it in relation to cognitive and strategical behavior. In our discussions about methodological issues, we pointed out that even researchers who applied qualitative methods should be aware that they can get only one "slice" of phenomena, even though it is usually more rich than in quantitative studies. On the other hand, researchers who applied quantitative analysis often pretend to produce objective and valid results. In this type of analysis, limitations concerning operationalization and generalizing should be carefully addressed.

The choice of research method should be derived not by general preferences of what is the "right" or "wrong" type of analysis, but by research questions that the researcher is interested in.

A new trend in the group was the appearance of several papers on identity. In their studies, researchers reported on how they assess identity and tried to understand this complex phenomenon. Assessment and theoretical background seem to be closely related to each other in research on identity. New theoretical approaches, such as theory of mindsets and theory of identity appeared in the papers this year. Further, we had again contributions that referred to psychological theories such as control-value theory of achievements emotions, theory of self-concept and other theories. Most of the studies investigated students affective state reflecting researchers' interest in what is going on during learning. The new object of affective construct, emotions and perceptions regarding strategies used during problem solving, appeared for the first time in our group. Another trend is an investigation of the affect in learning environments that included programming as part of the treatment.

During our general discussion, Liljedahl summarized by using the scheme for assessment categories in research on affect (Table 1

Figueiredo

  Figueiredo and Guimarães carried out a study on learning styles and their relation to performance. Van Hoeve et al. investigated students' mindset. An analysis of questionnaires for students' selfconcept of mathematics in school and university was presented in the contribution by Rach et al. Westerhout et al. analyzed the effect of the project about game programming. The development and evaluation of the general and domain-specific scales for self-concept and interest were analyzed by Sproesser et al. First-person vicarious experiences was found to be crucial for the change of beliefs by Rouleau et al. Pantziara et al. presented a study on validity of the scale about teachers' epistemic beliefs. Wiik and Vos reported on the study about reasons of choosing advanced mathematics courses in secondary school. Using metaphors for exploring pre-service teachers' beliefs was analyzed by Haser. Liljedahl presented an analysis of master and doctoral students' beliefs about research[START_REF] Liljedahl | Mathematics Education Graduate Students' Thoughts About Becoming Researchers[END_REF].

Table 1 : Assessment categories in research on affect

 1 ). Studies can address individuals or social variables (subject) within social or individual context. For example, if researchers investigated beliefs of individuals during group work, the subject is the individual and context is social.

	Subject
	social	individual

Contributions in the TWG8

In this section, we would like to present the papers that we included in these proceeding. Gómez-Chacón and Barbero investigated students' perception of backwards strategy in problem solving. Forbes et al. analyzed students' mindsets and its relation to self-confidence in programming. A low achiever's mathematical thinking was analyzed by Viitala in a case study. Alcantara et al. explored students' failure experience and identified emotions hopeless, shame, frustration and fear. Relation between anxiety about generating drawings and students' gender, strategical and cognitive factors was analyzed by Schukajlow et al. Lake reported on the study that investigated how data about emotional state can be used during reflection on teachers practice.

An instrument for measurement of students' identity was developed and applied by OReilly at al. Narratives on student identity within a social-cultural environment were explored by Ben-Dor and Heyd-Metzuyanim. Howard et al. applied thematic topic analysis for research on students' identity and its relation to mathematics in higher education.