Zezhou Liu

Helen Minsky

Costantino Creton

Matteo Ciccotti

Chung-Yuen Hui

Mechanics of zero degree peel test on a tape -effects of large deformation, material nonlinearity, and finite bond length

Keywords:

Introduction

Many of us have experienced hanging a poster on a wall or a picture on a refrigerator with tape. Our intuition tells us that if the tape is too short, it is not going to hold. Interestingly, this mode of failure is rarely studied. Instead, most measurements on the mechanical strength of adhesive tape use a high angle peel test, e.g. the tape is peeled at an angle of 90 degrees. The mode of failure has been well studied [START_REF] Wei | Interface strength, work of adhesion and plasticity in the peel test[END_REF][START_REF] Kaelble | Theory and analysis of peel adhesion: Mechanisms and mechanics[END_REF][START_REF] Kendall | Thin-film peeling-the elastic term[END_REF][START_REF] Creton | Fracture and adhesion of soft materials: a review[END_REF][START_REF] Villey | In-situ measurement of the large strain response of the fibrillar debonding region during the steady peeling of pressure sensitive adhesives[END_REF][START_REF] Kaelble | Peel adhesion: Micro-fracture mechanics of interfacial unbonding of polymers[END_REF]: the adhesive fails by nucleation and growth of cavities directly ahead of the peel front; the growth of these cavities creates tiny fibrils bridging the adhesive backing and the substrate [START_REF] Creton | Fracture and adhesion of soft materials: a review[END_REF][START_REF] Villey | In-situ measurement of the large strain response of the fibrillar debonding region during the steady peeling of pressure sensitive adhesives[END_REF][START_REF] Kaelble | Peel adhesion: Micro-fracture mechanics of interfacial unbonding of polymers[END_REF] and failure occurs as these fibrils at the interface break. The same mode of failure can be observed in a probe tack test [START_REF] Creton | Micromechanics of flat-probe adhesion tests of soft viscoelastic polymer films[END_REF][START_REF] Lakrout | Influence of molecular features on the tackiness of acrylic polymer melts[END_REF][START_REF] Zosel | The effect of fibrilation on the tack of pressure sensitive adhesives[END_REF][START_REF] Lakrout | Direct observation of Cavitation and fibrillation in a probe tack experiment on model acrylic pressuresensitive-adhesives[END_REF][START_REF] Crosby | Deformation and failure modes of adhesively bonded elastic layers[END_REF][START_REF] Shull | Axisymmetric adhesion tests of soft materials[END_REF][START_REF] Lin | A detailed elastic analysis of the flat punch (tack) test for pressure-sensitive adhesives[END_REF], where a flat cylindrical steel punch is brought into intimate contact with a thin adhesive film under a slight compressive force. This is followed by applying a tensile force to pull the punch off from the adhesive. In both tests, the adhesive is strongly confined, resulting in a very high triaxial state of stress which promotes cavitation [START_REF] Lin | A detailed elastic analysis of the flat punch (tack) test for pressure-sensitive adhesives[END_REF][START_REF] Gent | Fracture mechanics and cavitation in rubber-like solids[END_REF]. This is not the case with the lap shear geometry, here our intuition suggests that the adhesive mainly deforms by shear. Hence, one may expect a different mode of failure. To the best of our knowledge, this mode of failure has not been studied in detail.

The lap shear geometry is a special case of a peel test where the peel force is aligned with the peel direction. In the literature, this is called the zero degree peel test. Here we remind the reader that the usual adhesive tape consists of a thin stiff backing (≈20 µm thick) bonded to an adhesive layer of similar thickness. The Young's modulus of the backing is on the order of several GPa [START_REF] Villey | In-situ measurement of the large strain response of the fibrillar debonding region during the steady peeling of pressure sensitive adhesives[END_REF], while the adhesive has a much lower modulus, in the kPa range. The mechanics of the peel test is well studied [START_REF] Kaelble | Theory and analysis of peel adhesion: Bond stresses and distributions[END_REF][START_REF] Gent | Adhesion of viscoelastic materials to rigid substrates. II. Tensile strength of adhesive joints[END_REF][START_REF] Gent | Adhesion of viscoelastic materials to rigid substrates[END_REF][START_REF] Zosel | Shear strength of pressure sensitive adhesives and its correlation to mechanical properties[END_REF][START_REF] Ponce | Effect of friction on the peeling test at zero-degrees[END_REF][START_REF] Kinloch | Adhesion and Adhesives: Science and Technology[END_REF][START_REF] Collino | Detachment of compliant films adhered to stiff substrates via van der Waals interactions: role of frictional sliding during peeling[END_REF][START_REF] Mojdehi | Friction of extensible strips: An extended shear lag model with experimental evaluation[END_REF][START_REF] Sosson | Investigation of shear failure mechanisms of pressure-sensitive adhesives[END_REF][START_REF] Xia | Adhesion of heterogeneous thin films-I: Elastic heterogeneity[END_REF][START_REF] Cohen | Competing failure modes in finite adhesive pads[END_REF][START_REF] Bartlett | Designing bio-inspired adhesives for shear loading: From simple structures to complex patterns[END_REF]. Kendall's seminal work [START_REF] Kendall | Thin-film peeling-the elastic term[END_REF][START_REF] Kendall | The adhesion and surface energy of elastic solids[END_REF] determined the energy release rate in the steady peeling of an linearly elastic tape off a rigid substrate at arbitrary peel angles. More details on the complex state of stress at the peel front have been analyzed in detail by Thouless et al. [START_REF] Thouless | A parametric study of the peel test[END_REF] using a linear elastic fracture mechanics approach [START_REF] Yang | Mixed-mode fracture analyses of plasticallydeforming adhesive joints[END_REF][START_REF] Suo | Interface crack between two elastic layers[END_REF]. However, these works assume an infinitely thin adhesive layer, which is not the case for adhesive tapes. The key contribution in adhesive peeling is due to Kaelble [START_REF] Kaelble | Theory and analysis of peel adhesion: Bond stresses and distributions[END_REF] who accounts for the presence of a finite adhesive layer. The peel angle in Kaelble's theory is arbitrary, so the zero degree peel test is a special case of his analysis. Although Kaelble's analysis gives valuable insight into the mechanics of the zero degree peel test, it has several limitations. His analysis assumes that the adhesive is linearly elastic, that is, the strains are small and stress is directly proportional to the strain. This is certainly not the case for most common adhesives. Indeed, the shear strain in the adhesive layer can easily exceed 500% and 1000% strain is not uncommon [START_REF] Chopin | Nonlinear viscoelastic modeling of adhesive failure for polyacrylate pressure-sensitive adhesives[END_REF][START_REF] Bellamine | Design of nanostructured waterborne adhesives with improved shear resistance[END_REF][START_REF] Lindner | Adhesive and rheological properties of lightly crosslinked model acrylic networks[END_REF][START_REF] Tanguy | Quantitative analysis of the debonding structure of soft adhesives[END_REF]. In this regime, the small strain assumption breaks down. In addition, the assumption of material linearity also breaks down: the shear strain is no longer directly proportional to the shear stress. Indeed, most adhesives exhibit complex nonlinear stress-strain behavior. Roughly, for small strains, the stress-strain curve is linear; at larger strains, the adhesive softens, as indicated by a decrease in tangent modulus; finally, at high strains, the adhesive hardens [START_REF] Boyce | Constitutive models of rubber elasticity: A review[END_REF]. In this work, we use a nonlinear material model which captures this feature. Yet another limitation involves local stress concentration: the complex 3D state of stress at the peel front cannot be well represented by a one-dimensional model. Finally, Kaelble assumes the bond length (the length of tape that is sticking to the substrate) is infinite [START_REF] Kaelble | Theory and analysis of peel adhesion: Bond stresses and distributions[END_REF], where the more interesting situation involves a finite bond length. Our analysis also introduces a feature absent from Kaelble's classical theory: we treat the peel front as an interface crack and determine the energy release rate of this crack. Our energy release rate expression is valid for arbitrary large deformation and for a wide class of hyperelastic material strain energy density functions.

The geometry considered in this work differs from the traditional lap joint test where the joint length is relatively short in comparison with relevant dimensions. Most lap joint analyses assume an adhesive layer of zero thickness [START_REF] Yang | Mixed-mode fracture analyses of plasticallydeforming adhesive joints[END_REF][START_REF] Kendall | Crack propagation in lap shear joints[END_REF][START_REF] Kafkalidis | The effects of geometry and material properties on the fracture of single lap-shear joints[END_REF]. In some studies, the adhesive layer is modeled as a cohesive zone [START_REF] Yang | Mixed-mode fracture analyses of plasticallydeforming adhesive joints[END_REF][START_REF] Kafkalidis | The effects of geometry and material properties on the fracture of single lap-shear joints[END_REF]. Here we explicitly account for the finite thickness of the adhesive layer and we focus on the effect of large deformation and material nonlinearity of this layer.

In a previous work [START_REF] Hui | Mechanics of an adhesive tape in a zero degree peel test: effect of large deformation and material nonlinearity[END_REF] we studied the large deformation mechanics of an adhesive tape subjected to zero-degree peel. In that study we focus on the situation where the bond length is infinite. Here we consider the effect of finite bond length on the mechanics of zero degree peel. It turns out a finite bond length introduces many interesting effects which are absent from the case of an infinitely long tape. For example, we shall demonstrate that the stress state at the free edge of the bond tape is highly triaxial and can nucleate cavities.

The organization of the paper is as follows: in Section 2 we study the strain and stress distribution in the adhesive layer using a nonlinear shear-lag model. We also present an exact expression for the energy release rate which is valid for large deformation and nonlinear materials. We compare our results with the linear theory established by Kaelble [START_REF] Kaelble | Theory and analysis of peel adhesion: Bond stresses and distributions[END_REF]. Our analytic results are validated by a nonlinear FE model. In Section 3 we consider the stress and strain near the edges, where the shearlag model breaks down. Summary and discussion are given in Section 4.

Theoretical analysis

Statement of problem and geometry

The geometry is shown in Fig. 1(a). The bond length is denoted by L and the initial thickness of the adhesive layer is denoted by h. The backing layer has thickness h b which is on the same order as h. In the following we assume that the backing layer is linear elastic, with plane strain Young's modulus E * b . As the width of the tape is much greater than its thickness, we assume plane strain deformation where the out-of-plane displacement u 3 is zero and the in-plane displacements u α (α = 1, 2) depend only on the in-plane coordinates (x 1 , x 2 ). Another assumption is that the hanging weight is uniformly distributed across its width, therefore the appropriate force measured in a plane strain analysis, denoted by F , is defined as the weight divided by the width of the tape and has units of force per unit length. This force F is applied horizontally on one end of the tape (note F is negative in our coordinate system). Since the substrate is rigid, it can be extended pass the origin in the negative x 1 direction, so the point x 1 = x 2 = 0 can be considered as the tip of an interface crack (see Fig. 1(b)). After load is applied, the material point at (x 1 , x 2 ) moves to (y 1 , y 2 ) with respect to the same coordinate system, thus, y α = x α + u α .

Special case of a linear elastic adhesive -small strains

To position this work in relation to previous works and to motivate the idea, we first consider a situation with a linear elastic adhesive layer with shear modulus µ; this assumption is valid provided that the deformation of the adhesive layer or the applied force F is small. A straightforward calculation based on the shear-lag model [START_REF] Kaelble | Theory and analysis of peel adhesion: Bond stresses and distributions[END_REF][START_REF] Kaelble | Theory and analysis of peel adhesion: Adhesive thickness effects[END_REF] shows that shear strain in the adhesive layer is (see the Supplementary Information (SI) for details)

γ = F µl LT cosh ( L -η ) sinh L
, where L = L/l LT , η = x 1 /l LT .

(1a)

The tension in the backing layer τ B 11 and the shear stress in the adhesive layer τ 12 are given by

τ B 11 = - E * b h sinh L F µl LT sinh ( L -η ) , η ∈ [ 0, L ] Backing layer, (1b)
τ 12 = F l LT sinh L cosh ( L -η ) , η ∈ [ 0, L ] , Adhesive layer (1c)
respectively. Note in the shear-lag model, the backing layer is subjected to pure in-plane tension and because the strains are assumed to be small, the only non-vanishing stress in the adhesive layer is the shear stress τ 12 .

Equations (1a)-(1c) contain a very important length scale which is the load transfer length l LT . It is

l LT = √ E * b hh b /µ. (1d)
This is the length of the adhesive tape that carries most of the load -more precisely, it is the characteristic decay distance of the shear stress along the tape. For a typical tape, the shear modulus of the adhesive is on the order of tens of kPa, and the plane strain Young's modulus of the backing layer is on the order of GPa. The thicknesses of the adhesive and backing layers are on the order of tens of microns. Taking h = h b = 20 microns, E * b ≈ 3 GPa and µ = 10 kPa [START_REF] Villey | In-situ measurement of the large strain response of the fibrillar debonding region during the steady peeling of pressure sensitive adhesives[END_REF], the load transfer length is about 10 mm.

Extension to large deformation and nonlinear stress-strain behavior

The results in Section 2 are based on two simplifying assumptions: (1) the strains are small in the adhesive layer so we can avoid finite strain kinematics; (2) the adhesive is linearly elastic. Both assumptions are unrealistic in practical applications where the strains in the adhesive can reach 1000% [START_REF] Chopin | Nonlinear viscoelastic modeling of adhesive failure for polyacrylate pressure-sensitive adhesives[END_REF][START_REF] Bellamine | Design of nanostructured waterborne adhesives with improved shear resistance[END_REF][START_REF] Lindner | Adhesive and rheological properties of lightly crosslinked model acrylic networks[END_REF][START_REF] Tanguy | Quantitative analysis of the debonding structure of soft adhesives[END_REF]. Here we explore how removal of these two assumptions affects the results above. In our previous work [START_REF] Hui | Mechanics of an adhesive tape in a zero degree peel test: effect of large deformation and material nonlinearity[END_REF], we assumed the bond length is infinite and modeled the nonlinear elastic behavior of the adhesive using a three-term Yeoh solid. We showed that the stress and deformation in the adhesive layer can be accurately predicted Since the equation governing γ was obtained in our previous work [START_REF] Hui | Mechanics of an adhesive tape in a zero degree peel test: effect of large deformation and material nonlinearity[END_REF], we state it here without derivation, leaving some of the details in the SI. Briefly, Eq. ( 2) below expresses the equilibrium of a material element in the backing layer, that is, the net tension on this element must be balanced by the shear stress τ (γ ) exerted by the adhesive layer to maintain equilibrium. Due to material nonlinearity, τ (γ ) is a nonlinear function of the shear strain γ

d 2 γ dη 2 = τ (γ ) , (2)
where η = x 1 /l LT is the normalized distance from the crack tip and τ (γ ) ≡ τ (γ ) /µ is the normalized shear stress and µ is the small strain shear modulus in the Yeoh model. The strain energy density function W of the Yeoh solid is

W = C 1 (I 1 -3) + C 2 (I 1 -3) 2 + C 3 (I 1 -3) 3 (3a)
where the C k 's are material parameters with units of stress, I 1 is the trace of the right Cauchy-Green tensor. For incompressible materials, C 1 = µ/2. Softening is controlled by setting C 2 < 0 and strain hardening is controlled by C 3 > 0. In our previous work, we have shown that the normalized shear stress in Eq. ( 2) is related to the shear strain by

τ = f (γ ) γ , f (γ ) = ( 1 + 4C 2 µ γ 2 + 6C 3 µ γ 4
) .

(3b)

and

I 1 -3 ≈ γ 2 (3c)
The dimensionless function f (γ ) in Eq. (3b) represents strain hardening behavior. For example, C 2 = C 3 = 0 corresponds to a neo-Hookean solid. It is interesting to note the shear strain distribution is governed by the same equation whether or not the adhesive is linear elastic or neo-Hookean. However, it must be noted that the stress state in the adhesive is highly dependent on whether the material is linear or nonlinear, as revealed in Section 2.6. The boundary conditions are

γ | η=0 = γ 0 , dγ dη ⏐ ⏐ ⏐ ⏐ η=L/l LT = 0. (4a,b)
Here γ 0 denotes the maximum shear strain which occurs at x 1 = 0. The last condition in (4b) states that the tension in the backing layer at x 1 = L is zero. The maximum shear strain γ 0 is determined by the peel force F which is the integral of the shear stress along the adhesive/backing interface, i.e.,

F = µl LT ∫ L/l LT 0 f (γ ) γ dη.
(

) 5
The shear strain distribution γ is obtained by solving equation [START_REF] Kaelble | Theory and analysis of peel adhesion: Mechanisms and mechanics[END_REF] subjected to the boundary conditions (4a,b). It is (see the SI for details)

η = - ∫ γ γ 0 dw √ ( w 2 -(γ min ) 2 + c 2 2 ( w 4 -(γ min ) 4 ) + c 3 w 6 -(γ min ) 6 3
) ,

where γ min denote the minimum shear strain at x 1 = L or η = L/l LT , w is a dummy integration variable, and

c 2 = 4C 2 /µ, c 3 = 6C 3 /µ. (6b)
Eq. ( 5) motivates us to introduce a normalized peel force F ≡ F /µl LT . The peel force can be evaluated exactly (see the SI); it is

F = γ min √ ρ 2 -1 + c 2 2 (γ min ) 2 ( ρ 4 -1 ) + c 3 3 (γ min ) 4 ( ρ 6 -1 )
,

where ρ = γ 0 /γ min [START_REF] Creton | Micromechanics of flat-probe adhesion tests of soft viscoelastic polymer films[END_REF] Note that both γ 0 , γ min are unknowns which depend on the peel force and the bond length. To determine these relationships, we evaluate Eq. (6a) at η = L/l LT ; after some simple algebra, it reduces to the following equation relating bond length and the maximum and minimum shear strains, i.e.,

L ≡ L l LT = ∫ ρ 1 dw √ w 2 -1 √ 1 + c 2 (γ min ) 2 2 ( w 2 + 1 ) + c 3 (γ min ) 4 3 ( w 4 + w 2 + 1 ) . (8)
Eqs. ( 7) and ( 8) give two equations to determine γ 0 and γ min as a function of F = F /µl LT and L = L/l LT . Specifically, these two equations imply that these two strains depend only on (for the sake of simplifying notations, we have not indicated their dependence on c 2 and c 3 )

γ 0 = f 0 ( F , L ) , γ min = f min ( F , L ) . (9a,b)
Unfortunately, there is no closed form solution for the integrals (6a) and ( 8) unless c 2 = c 3 = 0. In the SI, we show how to numerically solve for γ 0 and γ min as a function of F and L and to determine the shear strain distribution in the adhesive layer. For the linear elastic case, these strains are directly proportional to the peel force and from (1a), the maximum and minimum shear strains are:

γ 0 = F coth L µl LT , γ min = F µl LT sinh L , ρ = γ 0 γ min = cosh L.
(10a,b)

Numerical results

The minimum shear strain γ min is plotted against the normalized peel force F for different normalized bond lengths L in Fig. 2. The special case of L = ∞ where it vanishes is shown as circles. Unless otherwise stated, we use the same parameters c 2 =-0.0948 and c 3 = 0.00996 from our previous study [START_REF] Hui | Mechanics of an adhesive tape in a zero degree peel test: effect of large deformation and material nonlinearity[END_REF] in all the figures below. The special case of small strain linear elasticity (SSLE) (solid lines) are also plotted as a comparison. In the linear theory, γ min and γ 0 are directly proportional to F (see Eqs. (10a,b)). Fig. 2(a) shows that as long as F > -1, there is no significant differences between linear and nonlinear theory.

However, for larger forces, the linear theory overpredicts γ min considerably. The reason is that the nonlinear material strain hardens significantly; therefore, for large force, the nonlinear material deforms much less. Indeed, due to strain hardening, γ min increases very slowly with the applied force for F < -5. This result is also illustrated in Fig. 2 is approximately independent of the peel force, i.e.,

γ min ≈ f min ( L ) , F ≪ -1 (11)
Fig. 3 plots the maximum shear strain γ 0 versus the normalized force F for three normalized bond lengths. We also plotted the prediction of the linear theory for comparison. Note in particular, the special case of L = ∞ (cross symbols) sets a lower bound for the peel force (for the same γ 0 ). As expected, for small F , there is no significant difference between the linear and nonlinear theory. The effect of strain softening occurs at peel forces roughly between -4 to -2, after which strain hardening takes over and the increase in maximum shear stress is much lower than that predicted by the linear theory. An interesting result is that, γ 0 versus F curves for different normalized bond lengths lie on top of each other for large F . Mathematically, this result is:

γ 0 ≈ f 0 ( F ) , F ≪ -1 (12)
Thus, decreasing the bond length in the high force regime has only small effect on the maximum shear strain -a surprising result.

The shear strain distribution is plotted in Fig. 4 as the horizontal axis.

Energy release rate

The energy release rate Γ of the interface crack in Fig. 1(b) can be computed using the path independent J integral in the undeformed reference configuration [START_REF] Rice | A path independent integral and the approximate analysis of strain concentration by notches and cracks[END_REF][START_REF] Budiansky | Conservation laws and energy-release rates[END_REF], which is

J = ∫ ω Wdy -P αβ N β u α,1 ds, ω = ω 1 + ω 2 + ω 3 + ω 4 , α, β = 1, 2, (13)
where P αβ , N β are the components of nominal stress tensor and unit normal in the reference configuration respectively. Also u α,1 is defined as ∂u α /∂x 1 . For the path ω chosen in Fig. 1(b), the integral is zero on ω 1 and ω 3 . On ω 2 the traction is zero, so P αβ N β u α,1 is zero, thus only the first term contributes on ω 2 and it is

∫ ω 2 Wdy = ∫ htape 0 W (x 1 = L, x 2 ) dx 2 , (14)
where h tape = h b + h. Note that W cannot be evaluated exactly on this path, since the corners can have weak singularities. The integral on path ω 4 can be readily evaluated, since the adhesive on this path is practically stress free and all the loads are carried by the much stiffer backing layer, which is linear elastic. Hence

∫ ω 4 Wdy -P αβ N β u α,1 ds = (F /h b ) 2 2E * b h b . (15)
Putting all this together, the energy release rate Γ = J is

Γ = F 2 2E * b h b + ∫ h b +h 0 W (x 1 = L, x 2 ) dx 2 (16)
Physical consideration suggests that W (x 1 = L, x 2 ) must increase with decreasing L since the same load has to be distributed over a smaller bond area. Since dL = -dc as the crack grows by dc, Eq. ( 16) implies that the energy release rate Γ increases with crack growth for any finite bond length L. Note that the energy release rate given by Eq. ( 16) is valid for any nonlinear elastic solid and arbitrary large deformation of the adhesive layer.

Special case of a linear elastic adhesive

The special case of a linear elastic adhesive can be readily determined using Eq. (1a)-(1c). The strain energy density of the adhesive at x 1 = L can be estimated using the shear-lag model,

W (x = L) = 1 2 τ 12 γ ⏐ ⏐ ⏐ ⏐ x=L = 1 2µ ( F l LT sinh (L/l LT ) ) 2 . ( 17
)
The energy release rate equation ( 16) is

Γ = F 2 2E * b h b + h 2µ ( F l LT sinh (L/l LT ) ) 2 = F 2 cot h 2 (L/l LT ) 2E * b h b = Γ ∞ cot h 2 (L/l LT ) . (18)
The first term in Eq. ( 18) is independent of bond length, while the second term in [START_REF] Zosel | Shear strength of pressure sensitive adhesives and its correlation to mechanical properties[END_REF] vanishes as L goes to infinity, hence the

first term, Γ ∞ ≡ F 2 2E * b h b
is the energy release rate for the case of infinite bond length. Fig. 5(a) plots the normalized energy release rate Γ /Γ ∞ versus L.

Energy release rate for nonlinear adhesive and large deformation

We evaluate the energy release rate by Eq. ( 16) using the results in Section 2.3, specifically equations (3a) and (3c). The key is to compute the strain energy density W in the adhesive layer at x 1 = L. In the nonlinear shear-lag model, the energy density function W at the bond end is related to the shear strain there by

W (x 1 = L) = µ 2 γ 2 min [ 1 + c 2 γ 2 min 2 + c 3 γ 4 min 3 ] . (19)
Note that the strain energy density increases rapidly with the minimum shear strain γ min at x 1 = L. The energy release rate based on Eqs. ( 16) and ( 19) is

Γ = F 2 2E * b h b + µh 2 γ 2 min [ 1 + c 2 γ 2 min 2 + c 3 γ 4 min 3
] .

(

) 20
Note that if we set c 2 = c 3 = 0 in Eq. ( 20), we recover the linear theory energy release rate given by Eq. ( 18). Here we emphasized that γ min is a nonlinear function of F and L, hence the energy release rate is no longer proportional to F 2 . Consistent with our previous notation, we decompose the energy release rate into two parts,

Γ = Γ ∞ + Γ L , Γ L ≡ µh 2 γ 2 min [ 1 + c 2 γ 2 min 2 + c 3 γ 4 min 3 ] . (21)
The first term in [START_REF] Collino | Detachment of compliant films adhered to stiff substrates via van der Waals interactions: role of frictional sliding during peeling[END_REF],

Γ ∞ = F 2 /2E * b h b is independent of the bond
length L whereas the second term Γ L is very sensitive to it.

The analysis above provides several interesting results:

(1) The effect of adhesive modulus only comes in through the load transfer length; for long bond length, the energy release rate is insensitive to the adhesive thickness and its elastic properties.

(2) Griffith criterion states that crack growth occurs when

Γ = G c , (22)
where G c is the critical interfacial fracture energy which we assumed to be a material constant, independent of crack length. Eq. ( 20) tells us when the bond length is long in comparison with the load transfer length, the second term is small since γ min → 0, hence versus bond length for a linear elastic adhesive. Note that critical normalized peel force goes to zero as bond length become small. Fig. 5b will be discussed in the following section.

Γ ≈ F 2 2E * b h b . (23)
For this case, the critical peel force F ∞ c needed to initiate crack growth is given by

F ∞ c = √ 2E * b h b G c . (24a)
The superscript ∞ indicates that this result is valid for long bond lengths. Eq. (24a) suggests it is advantageous to have stiff and thick backing layer as it increases the peel force without changing the adhesion. In other words, one can increase the peel force of a weak interface by geometry. This conclusion has been justified by experiments [START_REF] Ponce | Effect of friction on the peeling test at zero-degrees[END_REF][START_REF] Bartlett | Designing bio-inspired adhesives for shear loading: From simple structures to complex patterns[END_REF][START_REF] King | Optimizing adhesive design by understanding compliance[END_REF][START_REF] Bartlett | High capacity easy release adhesives from renewable materials[END_REF]. These experiments showed that the critical force is proportional to 1/ √ C where C is the compliance of the system, and C ∼ 1/E * b h b for the zero degree peel test [START_REF] Ponce | Effect of friction on the peeling test at zero-degrees[END_REF]. Note that the only place where the adhesive properties appear is the critical fracture energy G c .

The critical peel force F c drops dramatically as the bond length decreases (see Fig. 5(b)). Indeed, Eq. ( 18) implies that the fracture condition ( 22) is

F c = F ∞ c tanh 2 (L/l LT ) . ( 24b
)
Since tanh 2 (L/l LT ) ≈ (L/l LT ) 2 for small L/l LT , the critical peel force to grow the crack decreases rapidly with decreasing bond length.

(3) A short bond length destabilizes crack growth. Note that the first term of Eq. ( 20), F 2 /2E * b h b , does not change with bond length or adhesive properties whereas the second term does. In our system, extension of the interface crack by ∆c > 0 is exactly -∆L, that is, the bond length decreases by the amount of crack extension. Thus, the second term in Eq. ( 20) controls the change of energy release rate as the crack extends. Note this term increases with increasing crack length (or decreasing L). If the interfacial fracture energy is a material constant, then crack growth is stable if

∂Γ ∂c ⏐ ⏐ ⏐ ⏐ F = - ∂Γ ∂L ⏐ ⏐ ⏐ ⏐ F < 0. ( 25
)
In our case ∂Γ ∂c ⏐ ⏐ F > 0, so the condition equation ( 25) can never be satisfied as long as the critical energy is a constant independent of the amount of crack extension; the best one can manage is neutral stability ∂Γ ∂c ⏐ ⏐ F = 0, which occurs at long bond lengths, i.e., L/l LT ≫ 1.

In Fig. 6, we plot the normalized energy release rate

Γ Γ ∞ = 1 + Γ L Γ ∞ (26)
versus L for four different values of normalized peel force F (dashed lines). We also plotted the prediction of the linear theory by Eq. ( 18) in the same figure (solid line). For linear theory, Γ /Γ ∞ is independent of peel force -that is, the energy release rate scales as the square of the peel force. Surprisingly, the normalized energy release rate Γ /Γ ∞ predicted by the nonlinear theory at a given bond length decreases with the peel force: the linear theory dramatically overestimates the energy release rate at large peel force. A striking example is to consider large peel force (e.g. F = -10); according to Fig. 6 (purple dashed line), Γ is approximately independent of the bond length for L ≥ 1/2 and is given by

Γ ≈ Γ ∞ = F 2 /2E * b h b .
However, the energy release rate based on the linear theory (see Eq. ( 18)) predicts that energy release rate increases rapidly with decreasing bond length for 1/2 ≤ L ≤ 2. This surprising result is due to the fact that the linear theory overestimates γ min in this regime (see Fig. 2) and hence overestimates the energy release rate (see Eq. ( 20)). Finally, it is necessary to point out that the shear-lag model may not be accurate at x 1 = L because of a stress concentration at the free edge (due to edge effects). Thus, the results in this section should be checked against FE result in the next section.

Stresses in the adhesive layer

As shown in our previous work [START_REF] Hui | Mechanics of an adhesive tape in a zero degree peel test: effect of large deformation and material nonlinearity[END_REF], an important distinction between the linear and nonlinear theory is that the stress state in the adhesive layer is entirely different. In the linear theory, the stress state is one dimensional, that is, the only non-vanishing stress component is the shear stress. However, under large deformation, the stress state in the adhesive is three dimensional. Indeed, the true normal stress τ 11 in the adhesive layer is not zero and is in fact much greater than τ 12 for large peel force. The inplane stresses in the adhesive layer are completely determined by the shear strain γ distribution and the hardening function f, τ 11 = µγ 2 f (γ ) , τ 12 = µγ f (γ ) , and τ 22 ≈ 0.

(27)

Note for small shear strains (small applied force), the transverse stress τ 11 is insignificant compare to the shear stress, consistent with the linear theory, where it is assumed to be zero.

Finite element analysis

Finite element method (FEM) results

In the following, we use a nonlinear FE model to check the results of the shear-lag model in the previous section and to study edge effects. These FE calculations account for nonlinear material behavior using the three term Yeoh's material behavior and large strain kinematics. All simulations are carried out using ABAQUS and plane strain deformation is assumed (as in the shear-lag model). Since the FE model is similar to our previous work, relevant details are given in the SI. Fig. 7 plots the maximum shear strain γ 0 and minimum shear strain γ min in the adhesive layer versus normalized peel force F for different normalized bond length L. The blue, green and red colors indicate L = 1.0, 2.0 and 5.0. The nonlinear shearlag solution of γ 0 and γ min are plotted as solid and dashed lines respectively. FEM results are represented as symbols for comparison. Fig. 7 shows excellent agreement between predictions of nonlinear shear-model and FEM results.

Energy release rate and stress distribution in adhesive layer

ABAQUS can also evaluate the energy release rate and these energy release rates (symbols) are compared with the energy release rates obtained using the nonlinear shear-lag model given by Eq. (20) (dashed lines) in Fig. 8. The agreement between our analytic model and FEM is excellent. This justifies our results in Section 2, in particular, Eq. ( 20). Fig. 9 compares the true stress distributions τ 12 , τ 11 and τ 22 in the adhesive layer obtained by our FEM with our nonlinear model [START_REF] Kendall | The adhesion and surface energy of elastic solids[END_REF] for F = -10. All stresses are normalized by the maximum stress. The normalized position r is defined by r ≡ x 1 /L. The nonlinear shear-lag model is able to capture accurately the true stress distribution accurately as long as ( 27) is used to describe the stress state of the adhesive layer and the edges at x 1 = 0, L are avoided. In the next section, we focus on the stress state near these edges, where the shear-lag model is expected to break down.

Edge effects

The FEM results in Fig. 7 to 9 show that the nonlinear model accurately captures the stress and strain in the adhesive layer as long as one avoids the edges at x 1 = 0 and L. This is expected, since the edge at x 1 = 0 is the tip of an interface crack and the other a sharp 90 • corner. Since one of the key assumptions of the shear-lag model is that the stress and strain is independent of the vertical coordinate, this model cannot possibly capture the true state of stress at these locations. Indeed, the stress distributions can be significantly different along the adhesive/substrate and adhesive/backing interfaces near these edges. In the following, we use our FEM model to study these edge stress states. All figures below use the coordinates in the deformed configuration. Specifically, the true stresses are plotted against the deformed coordinates y α . An illustration of the difference between the reference coordinate (undeformed configuration, where a material point is denoted by its positionx α ) and deformed coordinates (where the material point x α moves to y α ) is shown in the bottom of Fig. 11(a). Note, on the adhesive/substrate interface, there is no difference between deformed and undeformed coordinates, very high and tensile near the crack tip, this is due to the large normal stress parallel to the interface. It must be noted that here we assume the interface is perfectly bonded whereas in reality, local slip or debonding will relax these extremely high stresses near the crack tip. As expected, the stress concentration is much less severe along the adhesive/backing interface. Note, however, the hydrostatic stress here is still very large and tensile.

Figs. 12 and 13 plot the normalized stress components along the adhesive/substrate and adhesive/backing interfaces near the right edge corner at x 1 = L respectively. As shown in the bottom of Fig. 13, we define the distances measuring from the right corner along the adhesive/substrate and adhesive/backing interfaces by m 1 and n 1 , and these distances are normalized by the adhesive thickness h, i.e., m 1 = m 1 /h and n 1 = n 1 /h. These figures show that the stresses on both interfaces increase rapidly as the bond length decreases beyond one load transfer length. Further, the stress state at this edge along the adhesive-back interface is triaxial with τ 22 , τ 11 , τ 33 all positive; therefore, cavitation is likely to occur at this edge. However, for bond length longer than 2 times the load transfer length, these stresses are small, and debonding or cavitation here is less likely.

Summary and discussion

In zero degree peel, the dominant deformation in the adhesive layer is shear. Except at distances very close to the peel front and the free edge, finite element analysis has shown that the shear strain and the stresses in the adhesive layer is well predicted by our analytic model in Section 2. Here we recall that in large deformation, even simple shear can induce normal stresses via the Poynting effect [START_REF] Poynting | On pressure perpendicular to the shear planes in finite pure shears, and on the lengthening of loaded wires when twisted[END_REF]. Thus, even though the dominant deformation mode in the adhesive layer is shear, the stress state there is not one dimensional. Indeed, at large shear strains, the adhesive is subjected to a normal stress τ 11 (in the direction of the peel force) that is much larger than the shear stress τ 12 , as noted by [START_REF] Kendall | The adhesion and surface energy of elastic solids[END_REF].

This, together with the out of plane stress due to the plane strain confinement, raise the possibility of cavitation inside the adhesive layer despite the fact that the stress normal to the adhesive layer τ 22 is zero.

Due to strain hardening, the minimum shear strain γ min at the free end of the tape depends primarily on the bond length and is quite insensitive to the peel force, as long as the peel force is large. This result significantly affects the energy release rate Γ , which consists of two parts: the first is the energy release rate of an infinite long tape, Γ ∞ which is independent of the mechanical property of the adhesive and the second part depends on γ min which is insensitive to the peel force in the nonlinear high force regime. As a result, Γ /Γ ∞ increases much slower with decreasing bond length, which means that the driving force for unstable crack growth is reduced significantly in comparison with linear theory. It is interesting to note that for long tapes (i.e., L ≫ l LT ) γ min is practically zero, this means that the force to drive the crack is controlled by the stiffness of backing layer and is independent of the mechanical behavior of adhesive. For a stiff backing, the force to pull the tape off can be very high. Thus, even though the force is supported by a small segment of the tape, the part of the tape that is not supporting shear actually plays an important role in protecting the tape as it significantly reduces the energy release rate. This is one reason to use a longer tape in the hanging weight experiment. The maximum shear strain is found to increase much less rapidly with the applied force in the nonlinear model. Furthermore, in the high force regime it is insensitive to the bond length. However, it should be aware that the maximum shear strain evaluated using the shear-lag model is most likely a lower bound because of the high stress concentration and 3D effect near the peel front. Here, our FE results indicate the normal stress τ 22 on the adhesive/substrate interface close to the peel front is highly compressive, so cavitation, if exist, is expected to occur at a small distance away from the peel front, where the normal stress is tensile. In contrast, the normal stress τ 22 on both the adhesive/substrate and the adhesive/backing interfaces close to the free edge are found to be tensile. For short bond lengths, the stress state on the interfaces near the free edge can be very high, and cavitation here is a definite possibility.

A limitation of our model is that the adhesive is assumed to be elastic while in reality they are nonlinear viscoelastic. The main difficulty with extending our analysis to account for viscoelastic effect is the absence of an accurate 3D large deformation viscoelastic model for adhesives.

Finally, although this study is motivated by understanding how adhesive tapes work under predominantly shear loading, our results have much wider generality and can be applied to study the mechanics of other composite systems with similar geometry. For example, the energy release rate expression and the stresses in the adhesive layer as well as the solution of the nonlinear shear-lag model, are valid for a wide class of hyperelastic strain energy density functions W, as shown in the supplementary information.

Fig. 1 .

 1 Fig. 1. Schematics of a zero degree peel test. (a) L is the finite bond length. The adhesive and backing thickness are denoted by h and h b , respectively. The applied force F is usually provided by a hanging weight per unit width. (b) The overhanging adhesive surface up to the peel front (x 1 < 0, x 2 = 0) can be seen as an interface crack. The path ω = ω 1 + ω 2 + ω 3 + ω 4 used in the J integral is illustrated by black arrows (see Section 2.5).

Fig. 2 .

 2 Fig. 2. (a) Minimum shear strain γ min versus the normalized force F for three normalized bond lengths L. The dashed and solid lines represent SSLE and NL solutions, respectively. The limiting case of L = ∞ is plotted as circles. (b) Minimum shear strain γ min versus the normalized bond lengths L for different normalized force F . by a nonlinear shear-lag model. Here we use the same nonlinear shear-lag model to study the finite bond length problem. The average shear strain in the adhesive layer is γ = u b /h, where u b is the horizontal displacement in the backing layer.

  (b), where γ min is plotted against normalized bond length for large normalized forces. In contrast to the linear theory, for large applied force, γ min approximately depends only on the bond length in the high force regime; the curves of γ min versus L practically lies on top of each other. Mathematically, in this regime the minimum shear strain

Fig. 3 .

 3 Fig. 3. Maximum shear strain γ 0 versus the normalized force F for three normalized bond lengths L. The dashed and solid lines represent SSLE and NL solutions, respectively. The limiting case of L = ∞ is plotted as cross symbols.

  figure, we use a normalized position r, defined by r ≡ x 1 /L = η/L,

Fig. 4 .

 4 Fig. 4. Shear strain profiles in the adhesive layer. (a) L = 1, with different normalized applied peel force; (b) large peel force F = -10, with different normalized bond lengths.

Fig. 5 .

 5 Fig. 5. (a) Normalized energy release rate versus bond length for a linear elastic adhesive. Note energy release rate increases rapidly as bond length decreases beyond the load transfer length l LT . For bond length longer than the load transfer length, the energy release rate Γ becomes independent of bond length and reaches a steady state value of Γ ∞ = F 2 /2E * b h b . (b) Critical normalized peel force F c /F ∞ c

Fig. 6 .

 6 Fig. 6. Normalized energy release rate Γ /Γ ∞ versus normalized bond length L for different normalized peel force F . The predictions of SSLE and NL are plotted as the solid line and dashed lines, respectively. Note normalized energy release rate is independent of the peel force in the linear theory.

Fig. 7 .

 7 Fig. 7. Maximum shear strain γ 0 and minimum shear strain γ min versus normalized peel force F for different normalized bond length L (blue, green and red colors indicate L = 1.0, 2.0 and 5.0). γ 0 and γ min of nonlinear shear-lag model are plotted as solid and dashed lines respectively. FEM results are represented as symbols.

Fig. 8 .

 8 Fig. 8. Normalized energy release rate Γ /Γ ∞ versus normalized bond length L for different normalized peel force F . The blue, green, red and purple colors correspond to F = -1.0, -2.0, -5.0 and -10.0. The FEM results are represented by symbols. SSLE and NL solution are plotted as the solid line and dashed lines, respectively.

Fig. 9 .

 9 Fig. 9. Comparison of true stress distributions in the adhesive layer obtained using nonlinear shear-lag model (solid lines) and finite element models (symbols) for four different bond lengths. The normalized applied force is F = -10. (a) τ 22 ; (b) τ 11 /τ max 11 ; and (c) τ 12 /τ max 12 . The maximum stress τ max 11 for different bond lengths L = 0.5, 1.0, 2.0, 5.0 are 351, 312, 292 and 285, respectively; and the maximum stress τ max 12 for these different bond lengths are -59, -53, -50 and -49, respectively.

Fig. 10 .

 10 Fig. 10. Stresses near the crack tip along the adhesive/substrate interface for different normalized bond length L (blue, green, red and purple colors indicate L = 0.5, 1.0, 2.0 and 5.0). The applied normalized force F = -10. The black dashed lines indicate FEM results are singular at the tip. (a) τ 22 ; (b) τ 11 ; (c) τ 12 ; and (d) p.i.e., x α = y α , as the substrate is rigid and we assume perfect bonding.Figs. 10 and 11 plot the normalized stress components along the adhesive/substrate and adhesive/backing interfaces near the crack tip respectively. Here we used different normalized coordinates to reflect our focus near the crack tip. The coordinates

Fig. 11 .

 11 Fig. 11. Stresses near the crack tip along the adhesive/backing interface for different normalized bond length L (blue, green, red and purple colors indicate L = 0.5, 1.0, 2.0 and 5.0). The applied normalized force is F = -10. (a) τ 22 ; (b) τ 11 ; (c) τ 12 ; and (d) p.

Fig. 12 .

 12 Fig. 12. Stresses near the corner along the adhesive/substrate interface for different normalized bond length L (blue, green red and purple colors indicate L = 0.5, 1.0, 2.0 and 5.0). The applied normalized force F = -10. The black dashed lines indicate FEM results are singular at the corner. (a) τ 22 ; (b) τ 11 ; (c) τ 12 ; and (d) p.

Fig. 13 .

 13 Fig. 13. Stresses near the corner along the adhesive/backing interface for different normalized bond length L (blue, green red and purple colors indicate L = 0.5, 1.0, 2.0 and 5.0). The applied normalized force F = -10. The black dashed lines indicate FEM results are singular at s = 0 (1) τ 22 ; (2) τ 11 ; (3) τ 12 ; and (4) p.

Acknowledgments

C.Y. Hui acknowledges National Science Foundation (NSF), USA MoMS program under grant number 1903308. H. K. Minsky is supported by the 3M Company, USA and Centre national de la recherché scientifique, France. We are grateful to the reviewers for their helpful comments.

Appendix A. Supplementary data

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.eml.2019.100518.


Julien Es Sayed

Dr Cédric Lorthioir

Dr Philippe Banet

Patrick Perrin
email: patrick.perrin@espci.fr

Dr Nicolas Sanson
email: nicolas.sanson@espci.fr

Philippe Lorthioir

PatrickP Banet

Sanson

Dr J Essayed

Prof P Perrin

Reversible Assembly of Microgels by Metallo-Supramolecular Chemistry

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

Introduction

Assembly of colloidal particles is aparamount challenge. Understanding the collective behavior of colloidal particles paves the way to unravelling the crystal nucleation and growth mechanisms by assimilating one particle to abig atom, or to the elaboration of so-called colloidal molecules or supracolloids. [1] To this end, the fine-tuning of particle assembly first requires precise control over the potentials acting on it. Besides,the particle assembly can be directed by designing geometrically and/or chemically anisotropic building blocks leading to so-called patchy particles. [2] Recently, research has been oriented towards assembly of deformable and versatile colloids named microgels,which are extensively studied as soft colloidal particles made of as olvent swollen network of crosslinked polymer chains. [3] Their chemical nature,which can be adapted on-demand to introduce specific functionalities,and their soft hairy structure,allowing them to come into contact without irreversible aggregation, make microgels outstanding candidates to form colloidal assemblies. [4] Theversatility of building block microgel particles can be enhanced by coupling it with supramolecular chemistry, paving the way to the design of selective or reversible colloidal assemblies.F or instance,host-guest supramolecular chemistry has been widely used to selectively assemble microgels functionalized by b-cyclodextrins as host with microgels bearing either adamantane,a zobenzene or ferrocene moieties as guests. [5] Another type of supramolecular interactions,w hich appears to be well-suited to perform supramolecular assemblies of microgels,i sm etal-ligand coordination chemistry. [6] This chemistry has been widely used and studied in the field of polymer chemistry to design metallo-supramolecular polymers with well-defined architectures. [7] However,m etal-ligand chemistry remains scarcely used in the field of colloidal assembly. [8] Fori nstance,G erth et al. connected silica particles by the intermediate of terpyridine-functionalized PEO chains adsorbed onto their surface.T hey showed that the gelation of these suspensions can be triggered by adding cobalt(II) cations. [8b] Recently, Lee et al. investigated the rheological properties of macrogels resulting from the gelation of thermoresponsive microgels functionalized with terpyridine moieties through the addition of different metal ions. [8c] To the best of our knowledge,t he elaboration of reversible microgel assemblies using metallosupramolecular chemistry has not yet been reported.

Hereafter, we describe as traightforward assemblefreeze-disassemble route to design bulk assemblies of thermoresponsive microgels in aqueous dispersions (Scheme 1). Our strategy is first based on the synthesis of well-defined terpyridine-functionalized thermoresponsive microgels used as building blocks.T hen, the control over the assembly of PNiPAM microgels involves two essential prerequisites: 1) the terpyridine ligands (Tpy) need to be accessible when adding the metal cation to establish metallo-supramolecular connections between microgels;2 )the probability of contact between microgels needs to be sufficiently high to develop short-range inter-particle supramolecular links as the metal cation is introduced in the aqueous dispersion. Microgels can be considered, at least at first glance,asclassical hard colloids, meaning that their proximity in solution is controlled by abalance between attractive and repulsive potentials and by the concentration of the dispersion. Consequently,t he average distance between microgels can certainly be adjusted by varying the volume fraction (f)a nd by modulating both the repulsive interactions by adding salt and the short-range attractive depletion potential originating by adding depletant molecules in solution. Then, the [M II Tpy 2 ] 2+ complexes formed upon addition of the metal cation (Fe II or Co II )a ct as supramolecular links between microgels to freeze the formed assemblies.W ea lso take advantage of the supramolecular nature of the [M II Tpy 2 ] 2+ links to use oxidation as an original stimulus to break up the [M II Tpy 2 ] 2+ complexes and disassemble the connected microgels into individual microgel building blocks (Scheme 1). This work reports on ap romising way to generate new stimuli-responsive microscale and macroscale materials using microgels as functionalized building blocks.Although our work mainly focuses on the 3D assemblies,w es how that our assemble-freezedisassemble concept can readily be applied to develop 1D and 2D materials using microgel building blocks.

Results and Discussion

Microgel Building Blocks

Te rpyridine-functionalized and thermoresponsive microgels (MG-Tpy) were synthesized in ao ne-step synthesis in water by surfactant-free dispersion copolymerization of Nisopropylacrylamide (NiPAM) as main monomer and poly-(ethylene glycol) monomethacrylate end-modified by at erpyridine group (PEGMA-Tpy) as comonomer (0.25 mol %vs. NiPAM) in presence of ac hemical crosslinker, N,N'-methylenebisacrylamide (MBA, 2mol %v s. NiPAM;S cheme 2). Labeled microgels with fluorescein (MG-TpyF) or rhodamine B( MG-TpyR) monomers have also been synthesized for visualization by confocal laser scanning microscopy (CLSM; Supporting Information, Table S1). Before using MG-Tpy microgels as building blocks for colloidal assemblies,t he chemical composition, thermal behavior,c olloidal stability, and morphology of the microgels were investigated using dynamic light scattering (DLS), nuclear magnetic resonance (NMR), scanning electron microscopy (SEM), UV/Vis spectroscopy,and zetametry.Our results show that microgels with terpyridine groups have been obtained in aone-step synthesis. Their behavior in water is similar to that of the pure NiPAM microgels (Figure 1a,b). Owing to the possible protonation of terpyridine groups in acidic conditions,M G-Tpy microgels Scheme 1. General assemble-freeze-disassembles trategy for the design of reversible microgel assemblies by metallo-supramolecular chemistry using MG-Tpy microgels as building blocks. Starting from well-dispersed microgels, the assemblies are dictated by adjusting both the volume fraction and the interaction potentials between them. Then, the formation of metallo-supramolecularcomplexes, [M II Tpy 2 ] 2+ , enables the freezing of the assembledm icrogels. Finally,the disassembly of the formed microgel assemblies can be triggered by the oxidation of the metal cation trapped in the bis-terpyridine complex leading to individualmicrogels.

Scheme 2. One-step synthesis of terpyridine-modified thermoresponsive microgels, MG-Tpy.Microgels are used as building blocks for complex assemblies induced by metallo-supramolecularc hemistry.The synthesis of the PEGMA-Tpy macromonomer is described in the SupportingInformation,F igures S1, S2. were characterized both at pH 3a nd pH 6. [9] Irrespective of pH, the MG-Tpy microgels exhibit ad ecrease of the hydrodynamic radius throughout the polymer phase transition with avolume phase transition temperature (VPTT) around 33 8 8C whereas the value of the electrophoretic mobility increases owing to the presence of cationic initiator.S uch features are characteristics of thermoresponsive pure NiPAM microgels (Figure 1a,b;S upporting Information, Figure S3). Thep resence of terpyridine groups in the microgel network and its quantification were established by UV/Vis spectroscopy,with absorption bands at 280 and 323 nm being characteristic of the terpyridine groups (Figure 1c), and 1 HNMR spectroscopy (Supporting Information, Figure S4 and Table S2). Both methods are in excellent agreement as they give as imilar amount of terpyridine groups incorporated in the microgel network evaluated at 2.6.10 À5 mol per gram of microgel. From SEM observations,m icrogels exhibit as pherical shape with al ow polydispersity index (PDI < 0.1;F igure 1d). The localization of the terpyridine groups into the microgel network is an important question, as microgel connection requires their accessibility.T op robe the distribution of the Tpy functions, 1 HNMR spectra of MG-Tpy microgels in D 2 O were recorded at different equilibrium temperature between 1a nd 57 8 8C( Figure 1e). Thep eaks related either to the isopropyl CH of the NiPAM units at 3.9 ppm or to the backbone CH/CH 2 protons between 1.4 and 2.4 ppm broaden and correspondingly,t heir intensity significantly decreases with temperature.T his is attributed to ad ecrease in the mobility of the copolymer chain segments with the microgel deswelling above VPTT.H owever,t he protons of the terpyridine-functionalized PEG side chains at 3.5 ppm still yield an observable 1 HNMR peak indicating that some of the PEG repeat units (EG) are mobile at temperature higher than VPTT despite the collapse of the microgel. Thetemperature dependence of the fractions of mobile NiPAM (f NiPAM )a nd EG (f EG )units determined at pH 6(Figure 1f)indicates that 70 %ofthe incorporated PEGMA-Tpy groups remain mobile at 45 8 8Cw hereas f NiPAM is close to zero above 38 8 8C. The protonation of terpyridine groups at pH 3does not affect the fraction of PEGMA-Tpy mobile units above the VPTT (Supporting Information, Figure S5). As ac onsequence,w e can conclude that the major part of the Tpy ligands are located at the periphery of the microgels and available for further supramolecular interactions.

Supramolecular-Driven Microgel Assembly

Using microgels in contact with each other is certainly one of the most straightforward ways to establish that metalligand complex formation can indeed connect MG-Tpy microgels.C onsequently,w ep ut into contact compacted dry powder of microgels (around 50 mg) with 100 mLo f1 0 À3 m FeCl 2 •4 H 2 O, CoCl 2 or CaCl 2 respectively (Figure 2a). The system is maintained at rest for 2h at 25 8 8C, at ime long enough to allow the diffusion of cations inside the microgels powder.T hen, 4mLofwater are added. Thecations Fe II and Co II are chosen because both are readily and spontaneously chelated by terpyridine ligands to form highly stable thermodynamic bis-terpyridine complexes in water (association constant b 2 > 10 16 ). [6,10] After adding cations solution (Fe II and Co II ), the respective pink and beige colors characteristic of the formed complexes were observed (Figure 2a). The structure of macrogels is maintained over time without significant release of microgels,i nc ontrast to the control experiment performed Ca 2+ at the same concentration where microgels spontaneously disperse in water. Remarkably,t hese supramolecular macrogels exhibit at emperatureinduced deswelling above the VPTT of the microgels,that is, 33 8 8C( Supporting Information, Figure S6). This swelling/ deswelling behavior is fully reversible and comparable to the one of dispersed microgels.From this result, we conclude that as upramolecular assembly of MG-Tpy microgels can be formed by introducing Fe II or Co II ,l eading to am acrogel of microgels.

Considering the concept of colloidal molecule evocated in the introduction, af iner control over size of the MG-Tpy microgel assemblies is necessary.F igure 2b-f illustrates the influence of the key physico-chemical parameters,such as the microgel volume fraction (f), the ionic strength, and the effect of depletant molecules,n ecessary to induce the assembly of the microgels prior to the freezing of the structure using supramolecular links as depicted in Scheme 2. By centrifuging microgel dispersions at agiven mass fraction we were able to prepare al arge range of volume fraction of MG-Tpy microgel dispersions from near close-packing (f = 0.56) to highly diluted (f = 0.014). Similarly to the experiments conducted on the dry powder, we added at T = 25 8 8C either Fe II or Co II as complexing cations of Tpy units.A control sample was also prepared by adding Ca 2+ at the same ionic strength. To further ascertain the formation of supramolecular assemblies,s amples were diluted to f = 0.014 and left under mild stirring overnight at 25 8 8C. Then, dynamic light scattering (DLS) measurements and confocal microscopy observations were performed to probe the aggregation state of the microgels.T he effect of the nature of the metal cation on the supramolecular connections of microgels with Fe II and Co II in water as af unction of the volume fraction is summarized in Figure 2b.The connection was probed in pure water (pH 6) at 25 8 8Cf or volume fractions, f,r anging from 0.21 to 0.56. Irrespective of f,a ll microgel dispersions with added Ca 2+ do not contain microgel aggregates,e xhibit ah omogeneous aspect and can flow under their own weight even at f = 0.56 (Supporting Information, Figure S7). In contrast, at an identical volume fraction the samples in which Fe II or Co II were added exhibit agel-like behavior.Moreover, as in the case of macrogels,the addition of Fe II or Co II cations induced the appearance of either the pink ([Fe II Tpy 2 ] 2+ )o r beige ([Co II Tpy 2 ] 2+ )c olor associated to the presence of the characteristic absorption bands observed by UV/Vis spectroscopy performed immediately after dilution (Supporting Information, Figure S8). [6,7] These macroscopic observations are first hints supporting the formation of supramolecularcontrolled assemblies of microgels.F igure 2b shows the size of the formed assemblies given in terms of equivalent hydrodynamic spheres (cumulant analysis) for each system one night after dilution (f = 0.014). In the case of Ca 2+ (not chelated by Tpy units) the average hydrodynamic radius is similar to that of the bare MG-Tpy microgel whatever the volume fraction. In the case of iron(II), we clearly identify the formation of aggregates of MG-Tpy microgels characterized by an increase of the average hydrodynamic radius concomitant to an increase of the polydispersity index (PDI) for volume fractions f ! 0.35. We can thus confirm that the microgel assemblies are metallo-supramolecularly frozen by the formation of [Fe II Tpy 2 ] 2+ complexes.However, in the case of Co II ,t he values of the measured average hydrodynamic radius are similar to the ones obtained for individual microgels in the range of investigated volume fraction. This result seems to be inconsistent with the macroscopic observations performed just after dilution of the samples.A ctually,t he disassembly of microgels observed in the case of Co II may result from the dynamics of the formed complexes.I ndeed, even though both Fe II and Co II exhibit av ery strong affinity with terpyridine units (b 2 > 10 16 ), they highly differ by their kinetics of dissociation of the bis-complexes.The cobalt-based complex, [Co II Tpy 2 ] 2+ ,d isplays am ore pronounced lability compared to the iron-based one,[ Fe II Tpy 2 ] 2+ (k À2 ,Co II = 4.10 À2 min À1 and k À2 ,Fe II = 10 À5 min À1 ). [10,11] This means that the characteristic dissociation frequency of the cobalt(II) biscomplexes is higher than that of the iron(II) based ones.Thus, after dilution of the samples to f = 0.014, dissociation events between connected microgels lead to the redispersion of microgels.These results do not necessarily imply that iron(II) induces more connections between microgels than cobalt(II); it rather means that iron(II) based complexes are much more stable over time.Finally,kinetically frozen MG-Tpy microgel assemblies induced by [Fe II Tpy 2 ] 2+ supramolecular complexes can instantly form in pure water at 25 8 8Cw hen the microgel volume fraction is higher than f = 0.35 (see kinetics experiments in the Supporting Information, Figure S9). In what follows,t he physico-chemical parameters controlling MG-Tpy microgel assemblies are only investigated in the presence of Fe II cations.

Thed ecrease of the electrostatic repulsions between microgels allows increasing the probability of contact at agiven volume fraction. Figure 2c shows the evolution of the average hydrodynamic radius of the formed assemblies supramolecularly linked with iron(II) at 25 8 8Cf or av olume fraction between f = 0.07 and f = 0.56 in pure water and in 10 À3 m NaCl. It is worth noting that the ionic strength does not affect the colloidal stability of the dispersed MG-Tpy microgels.O nce again, an increase of both the average hydrodynamic radius and PDI of the microgel aggregates was observed after addition of Fe II in the solution. However,t his increase occurs from avolume fraction of f = 0.14 lower than that observed in pure water (f > 0.35). TheC LSM images presented in Figure 2f (upper right) confirms the formation of assemblies of MG-TpyR and MG-TpyF microgels at f = 0.14 in 10 À3 m NaCl solution. It is thus possible to decrease very significantly the extent of electrostatic repulsions by addition of salt, hence increasing the probability of contact between microgels at lower volume fractions than in pure water even if the charge density of microgels is quite low in these conditions (Figure 1b). Then, the decrease of electrostatic repulsions enables the formation of metallo-supramolecular frozen assemblies of MG-Tpy microgels in water at 25 8 8Ca bove ac ritical volume fraction f = 0.14. Below this value,the presence of salt is not sufficient to make microgels come close enough to build aggregates as iron(II) is added.

To aggregate microgels at al ower volume fraction (f = 0.014) before adding iron(II), an attractive depletion potential between them was introduced by adding dextran chains (M w = 100.000 gmol À1 )i naMG-Tpy dispersion. [12] Addition of dextran chains at 3wt% to aM G-Tpy dispersion in pure water at 25 8 8Cs uccessfully enables to form aggregate microgels (Figure 2d left). However, they disassemble upon dilution in water to form again am icrogel solution. The addition of iron(II) to ad ispersion of depletion-driven microgel aggregates enables to freeze the aggregates by metallo-supramolecular links and maintain the structure upon dilution (Figure 2d right and Figure 2f down right). Lastly,w es how that ac ombination of depletion and partial screening of repulsive electrostatic repulsions followed by subsequent addition of iron(II) enables to perform metallosupramolecular assemblies of MG-Tpy at both low volume fraction f = 0.014 and low concentration of depletant dextran molecules (0.5 wt %; Figure 2e). Therefore,d epletion-induced attraction between microgels appears as an efficient pathway to favor contact between them at low volume fraction.

Responsive Disassembly of Microgels

We took advantage of the supramolecular nature of the links to open the metal-ligand complexes in response to an external stimulus with the aim of dissociating the microgel assemblies.C ompetitor molecules or pH are among the stimuli commonly used to cleave the terpyridine-based supramolecular complexes incorporated in macromolecular systems. [13] Here,westudied the possibility to oxidize iron(II) embedded between terpyridine units into iron(III) using potassium persulfate (KPS) as an original route to dissociate the metal-terpyridine complex and consequently to break microgel assemblies (Scheme 2). Indeed, it is reported that iron(III)/ bis-terpyridine complexes are not kinetically stable and exhibit av ery high lability so that microgel aggregates disassembly can be expected. [14] Oxidation of Fe II by KPS is possible because the standard redox potential of the [Fe III Tpy 2 ] 3+ /[Fe II Tpy 2 ] 2+ couple (1.02 Vv s. standard hydrogen electrode (SHE);Supporting Information, Figure S10) is lower than that of the S 2 O 8 2À /SO 4 2À couple (2.08 Vvs. SHE). Figure 3p resents UV/Vis and DLS experiments performed on microgels aggregates samples prior and after addition of KPS.T he absorption band at 567 nm characteristic of the [Fe II Tpy 2 ] 2+ complex at the molecular scale disappears after adding the oxidant molecules (Figure 3a). Accordingly,atthe microgel scale,D LS measurements show that the microgel aggregates break into individual microgels (Figure 3b)a fter 3h (Supporting Information, Figure S11). Finally,w ed emonstrated the reversibility of the assembly-disassembly process by successively adding iron(II) to freeze the microgel assemblies and KPS to cleave the metal-terpyridine complex and release microgels in solution. Thereversibility was shown by confocal images (Figure 3c)a nd supported by dynamic light scattering experiments performed at each stage (Supporting Information, Figure S12).

Toward New Supramolecular Materials

Thesystematic study of the microgel assemblies paves the way to the development of novel 1D,2D, or 3D supramolecular organic materials in which microgels are the building blocks.O nt he one hand, ac oncentrated dispersion of MG-Tpy microgels (f = 0.5) can be extruded in an iron(II) aqueous solution to form af ilament (Figure 4a;S upporting Information, Movie S1). After extrusion, the microgels filament was instantaneously frozen by the formation of [Fe II Tpy 2 ] 2+ supramolecular links between close-packed microgels as outlined by the appearance of the pink color. The macrostructure resists to mechanical shear induced by as tirring bar.O nt he contrary,a ne xtrusion realized with an identical MG-Tpy microgels dispersion in pure water led to the complete dispersion of the microgels (Supporting Information, Movie S2). As presented before with micro-assemblies of microgels,t he disassembly of this macro-structure may be triggered by addition of KPS (Supporting Information, Figure S13). By taking advantage of the spontaneous adsorption of NiPAM-based MG-Tpy at air-water interface, air bubbles stabilized by the microgels are formed (Fig- ure 4b). [15] Thec rosslinking of the thin film occurs by depositing the bubble on an iron(II) aqueous solution (Figure 4b,l eft).

Indeed, iron(II) ions can diffuse through the film inducing metallo-supramolecular links between microgels in close contact at the interface.Amechanical stress of the crosslinked microgel thin film clearly demonstrates the formation of amicrogel skin (Figure 4b,right;Supporting Information, Movie S3) contrary to ac lassical bubble stabilized by MG-Tpy microgels in absence of iron(II) (Supporting information, Movie S4).

Conclusion

By combining colloidal assembly and supramolecular chemistry,w eh ave demonstrated as trategy to design reversible metallo-supramolecular assemblies of thermoresponsive microgels in water. To this end, af ine-tuning of the building block microgels is achieved by introducing accessible terpyridine-based supramolecular ligands into the microgel network. Next, we showed the possibility to prepare frozen macro-assemblies such as macrogels or aggregates of microgels maintained by the formation of metallo-supramolecular links involving the metal coordination complex. Thephysicochemical parameters (microgel volume fraction, ionic strength, metal cations,a nd depletant molecules) have been screened to define rules for supramolecular assembly of microgels.L astly,w ee mphasize on the possibility to induce ar esponsive disassembly of the connected microgels by oxidation of the metallo-supramolecular links,[ Fe II Tpy 2 ] 2+ , between connected microgels.A saproof-of-concept, the elaboration of new supramolecular materials resulting from microgels as building blocks has been explored, leading to the formation of injectable macrogels or supramolecular crosslinked membranes.T his assemble-freeze-disassemble strategy can serve as astarting point for future research owing to the versatility of both the microgels and the metallosupramolecular chemistry.

Figure 1 .

 1 Figure 1. Characterization of terpyridine-modified thermoresponsive NiPAM microgels MG-Tpy.a)Hydrodynamic radius as afunction of temperature at pH 6and pH 3. b) Electrophoretic mobility as afunction of temperature at pH 6and pH 3. c) Absorbanceofsynthesized MG-Tpymicrogels (DMSO). d) SEM image of MG-Tpy microgels (scale bar:5mm). e) Temperature dependence of 1 HNMR spectra of MG-Tpy in D 2 OatpH6.f)T emperaturedependence of the fraction of mobile NiPAM and PEGMA-Tpy units, f NiPAM and f EG ,a sdetermined at pH 6. The fractions were calculated using the area of the NMR peak assigned to either the isopropyl CH protons of the NiPAM units or the CH 2 CH 2 Oprotons of the PEGMA-Tpy units normalizedbyt he area value measured at T = 1 8 8C.

Figure 2 .

 2 Figure 2. Assembly of MG-Tpy microgels induced by metallo-supramolecularchemistry at 25 8 8C. a) Formationo fmacrogel of microgelsc onnected by supramolecular links, [Fe II Tpy 2 ] 2+ and [Co II Tpy 2 ] 2+ .Nomacro-structure was observed in the case of anon-chelatedc ation (Ca 2+ ). b),c) Evolution of the average hydrodynamic radius, R h ,ofthe MG-Tpy microgel aggregates as afunction of the microgel volume fraction:b)Effect of the nature of the added cations and c) effect of the ionic strength. Labels denote the values of the polydispersity indexes, PDI, determined by cumulant analysis. Green dashed lines correspond to the average hydrodynamic radius of bare MG-Tpy microgels. d) Effect of depletant molecules (Dextran) on microgel assembly at f = 0.014 with 3wt% of dextran in pure water at 25 8 8Cand evolution of the average hydrodynamic radius upon dilution without or with previous addition of iron(II). e) Combined effect of depletantmolecules (dextran) and ionic strength on the supramolecular assemblyo fMG-Tpy microgels (f = 0.014) at 25 8 8C. f) CLSM pictures of bare (MG-TpyR + MG-TpyF microgels) (left) and supramolecularassembled MG-TpyR and MG-TpyF microgels with iron(II) at f = 0.14 in H 2 O, 10 À3 m NaCl (up right) and at f = 0.014, H 2 O, 10 À3 m NaCl, 1.5 wt % dextran (down right). Scale bar:10mm.

Figure 3 .

 3 Figure 3. Responsived isassembly of microgels aggregates (f = 0.14 in H 2 O, 10 À3 m NaCl). a) UV/Visible spectra of assembled and disassembled microgel dispersion. b) Intensity size distribution of bare MG-Tpy microgel dispersions, assembled MG-Tpy microgel dispersions with Fe II and disassembled MG-Tpy microgel dispersions upon addition of KPS. c) Reversibility of the assembly-disassembly process shown by confocal microscopyo fover two cycles. Scale bar:5mm.

Figure 4 .

 4 Figure 4. Macroscopic observations of the MG-Tpy microgel aggregation (f = 0.5) induced by metallo-supramolecularchemistry at 25 8 8C with Fe 2+ .a)Microgel filamento btained by extrusion of amicrogel dispersion into aF e 2+ solution (see also the SupportingInformation, Movie S1). b) Left:Amicrogel bubble was deposited onto athin layer of Fe 2+ solution;r ight:after migration of Fe 2+ ions into the microgel membrane,apipette was used to mechanically stress the membrane (Supporting Information, Movie S3).

Acknowledgements

Thea uthors gratefully acknowledge the financial support of CNRS,E SPCI, and the PhD school of Sorbonne UniversitØ (ED 397, Sorbonne UniversitØ)f or the PhD fellowship funding of J. Es Sayed. Theauthors thank Christophe Meyer for his help in organic chemistry,Bruno Bresson for technical advice on SEM experiments,N adge Pantoustier and Philippe Devilliers for fruitful discussion. Thea uthors thank the microscopy platform of Institut Pierre Gilles de Gennes for technical advice on CLSM.

Conflict of interest

Theauthors declare no conflict of interest.

Keywords: colloid assembly •microgels •responsive materials • supramolecular chemistry •t erpyridine complex