Effect of Uniformly Applied Force and Molecular Characteristics of a

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Effect of uniformly applied force and molecular characteristics of a polymer chain on its adhesion to graphene substrates Sunil Kumar, Sudip K Pattanayek, Gerald Pereira, and Sanat Mohanty Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.5b04028 • Publication Date (Web): 26 Feb 2016 Downloaded from http://pubs.acs.org on March 1, 2016

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Effect of uniformly applied force and molecular characteristics of a polymer chain on its adhesion to graphene substrates Sunil Kumar,† Sudip K. Pattanayek,∗,† Gerald G. Pereira,‡ and Sanat Mohanty† Department of Chemical Engineering Indian Institute of Technology, New Delhi, 110016 India, and CSIRO Mathematics, Informatics & Statistics Private Bag 33, Clayton South, 3169, Australia E-mail: [email protected] Phone: +91 11 26591018. Fax: +91 11 26581120

∗

To whom correspondence should be addressed Department of Chemical Engineering Indian Institute of Technology, New Delhi, 110016 India ‡ CSIRO Mathematics, Informatics & Statistics Private Bag 33, Clayton South, 3169, Australia †

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Abstract The force-induced desorption of a polymer chain from a graphene substrate is studied with Molecular Dynamics (MD). A critical force needs to be exceeded before detachment of the polymer from the substrate. It is found for a chain to exhibit good adhesive properties the chain configuration should consist of fibrils - elongated, aligned sections of polymers and cavities which dissipate the applied energy. A fibrillation index is defined to quantify the quality of fibrils. We focus on the molecular properties of the polymer chain which can lead to large amounts of fibrillation and find both strong attraction between polymer and substrate and good solvency conditions are important conditions for this. We also vary the stiffness of the chain and find for less stiff chains a plateau in the stress-strain curve gives rise to good adhesion but for very stiff chains, there is limited elongation of the chain but the chain can still exhibit good fibrillation by a lamella-like rearrangement. Finally, it is found the detachment time, t of a polymer from the adsorbed substrate is inversely proportional to force, F , i.e. t ∝ F −γ where the exponent γ depends on solvent quality, polymer-substrate attraction and chain stiffness.


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Introduction An understanding of the behavior of polymers near a hard, flat substrate in the presence of an externally applied force is vital in the development of high performance polymeric systems such as Pressure Sensitive Adhesives (PSA) 1,2 and nano-composites. 3,4 The development of such applications require a detailed understanding of the mechanisms involved in the dissipation of stress energy due to various nano-substrate-polymer interactions. A number of experimental 5–17 and simulation studies 18–33 have been reported to understand the structure and dynamics of polymers under the application of stresses, especially close to a substrate. Experimentally, researchers have studied the effect of a normal force applied to the polymer chain. (Here the normal force is in the perpendicular direction to the flat substrate on which the polymer chain resides.) They found that the dissipation of energy is dependent on factors such as the interaction between polymer chains, the polymer-substrate energy and the various morphological structures which evolve as the polymer chain is pulled off the substrate. Creton and his group 5,31 have studied the response of polymers with different cohesive energies at constant applied velocity. They pointed out that as the substrate force decreases, the polymer undergoes an adhesive failure in that it is cleanly detached from the substrate. However, as the substrate interactions become more favorable between the polymer and the substrate, the polymer is stretched before forming fibrils and subsequently being cleanly displaced. The processes of fibrillation and formation of cavities are important pathways to dissipation of applied energy and thus impact the properties of polymer adhesion. Lenhart and Cole 34 have studied the adhesive properties of polymer gels which are know to behave similarly to PSA. They showed that the maximum pull-off force increased with cross-link density which was associated with an increase in energy required for cavitation. They also performed tack adhesion tests and found the best performed gels were those that exhibit fibrillation and extension, while the worst performed were gels which exhibited a non-dissipative failure. In general, in all the gels they studied an increased value of practical 1


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work of adhesion resulted due to the process of cavitation and fibrillation. Experiments with laser optical tweezers and/or atomic force microscope (AFM) have been carried out to study the effect of mechanical forces on individual bio-macromolecules such as polysaccharides, proteins and DNA. 8–14,35 In conjunction with these experiments, theoretical and simulations studies have been implemented to study the application of a normal force at one end of a single, adsorbed polymer chain. 19–27 These studies are primarily concerned with the various configurations the polymer chain may adopt and the force required to pull the chain off the substrate. Recently Raos and Sluckin, 28 using molecular dynamics methods, applied the same force to all the beads of a single polymer chain to investigate the frictional properties of the chain. The substrate chosen was chemically heterogeneous and the chain followed a stick-slip motion at low pulling force and sliding motion at a high pulling force. However, note that since the author’s were concerned with frictional/lubrication properties the external force was only applied in a direction parallel to the substrate. Very recently Fechine et al. 36 have used both experimental and simulation methods (MD) to understand graphene-polymer adhesion. In their MD simulations, they used many short chains (up to 50 repeat units) and pulled the graphene surface away from the polymer film. They found higher temperatures (heating) resulted in better adhesion and larger pull-off forces. The adhesion was found to be hysteretic, depending on both polymer treatment (temperature) and also chemical composition of the polymer film. While there have previously been experimental and theoretical (continuum) models of polymer adhesion, an understanding of the molecular origins and contributions to adhesion are still unclear. In this work we therefore are motivated to build a molecular model of polymer adhesion. A computational model of a polymer adhesive system may consist of a large number of entangled polymer chains which are adsorbed between flat substrates. As the substrates move apart, a large fraction of the adsorbed polymer beads will feel a normal force which will tend to pull them off the adsorbed substrate. Computationally it is extremely challenging to model a large number of entangled chains and so we limit ourselves

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in this study to one long chain. However, application of a normal force to only one end of the chain is not a realistic model of the adhesive, where the applied force is spread over many beads. Thus, in line with the study of Raos and Sluckin, 28 we propose a more realistic model would be to apply the same force to all beads in the chain. To the best of our knowledge, no study has been performed where all beads of the polymer chain are pulled with an equal amount of force and which we argue is one of the simplest model for polymer adhesives. In this study we apply a force on a single polymer chain which is adsorbed randomly on a graphene substrate using molecular dynamics simulations. The force is applied to all beads on the chain and both parallel and perpendicular to the substrate in separate simulations. However, we are primarily interested here in the adhesive properties rather than any frictional or lubrication effects and so application of the perpendicular force is more important. The choice of polymer (polyethylene) allows simulations of simple, straight chains that interact primarily through van der Waals potentials with the substrate. A graphene substrate is chosen because it results in a well-defined, regular substrate structure. We note that graphite based nanoparticles have also been in use for adhesive formulations. 37 The behavior of polyethylene on graphene substrates is perhaps the simplest of systems that allows us to focus on the role of substrate interactions with the polymer and polymer-polymer interactions (both of which are tunable through the simulation parameters) and their roles in debonding mechanisms and the strength of the bond.

Simulation details We implement constant temperature molecular dynamics simulations 38,39 in this study. The procedure adopted can be found in our recent publications 40,41 based on a simulation box (X, Y, Z) of dimension (±362˚ A, ±362˚ A, ±362˚ A). The substrate consists of a square graphene A) which is made up of a network of sp2 hybridized carbon atoms, sheet (edge length = 100˚ which are orientated hexagonally and defined by two lattice translation vectors a1 and a2 .

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The magnitude of these vectors are the same (i.e. 2.46˚ A) and this corresponds to an in-plane nearest neighbor distance of 1.4˚ A. 42 The total number of beads present in the graphene sheet is 3936. The co-ordinates of the beads are generated by using VMD (Visual Molecular Dynamics). 43 The graphene sheet was kept stationary at the center in the X-Y plane of the simulation box during the simulation (see Figure S1a). Next, a randomly generated polymer chain containing N connected beads, where each bead corresponds to a united CH2 atom of mass m, is placed near the graphene substrate. The following interaction potentials act among various beads present within the system: (a) Lennard-Jones potential, ELJ , (b) bonded interactions potential, Ebond , (c) harmonic cosine bond angle potential, Eangle and (d) torsional angle potential, Etorsion . We implement a non-bonded interaction potential among all the nonbonded beads of the polymer (P) and in between the polymer and graphene beads (S). The non-bonded potential between any two polymer beads and between a bead of the polymer and a bead of graphene substrate is assumed to follow the Lennard-Jones potential given by ELJ (rij ) E ij (rij ) ≡ = 4ϵij ϵP P

!

1 1 − 6 12 rij rij

"

,

(1)

where rij denotes a dimensionless distance between two beads i and j and ϵij denotes a dimensionless interaction energy which is either ϵP P or ϵP S depending on the type of bead corresponding to i and j. All unscaled distances are made dimensionless by dividing them by the equilibrium polymer-polymer bead separation distance σP P , which corresponds to the separation at which the polymer-polymer LJ potential is zero. The LJ potential between polymer beads on the same chain is only calculated for beads which are separated by four or more beads. In addition, the cut-off distance between two beads for the LJ interaction is taken to be 2.5σP P . σP S is the distance at which the polymer-substrate potential is zero. The bonded (stretching) interaction potential between two bonded beads i and i + 1

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of the polymer chain is given by 1 EBonded = k s (ri,i+1 − req )2 , ϵP P 2

(2)

where the dimensionless bond vector joining ith and i+1th beads of the polymer is ri,i+1 . The equilibrium separation between two adjacent beads is denoted by req . k s is the dimensionless spring constant of the harmonic potential defined as k s = ks /ϵP P σP2 P . The harmonic (cosine) bond angle potential is given by EBondangle 1 = k θ (cos θi − cos θ0 )2 , ϵP P 2

(3)

where θi is the bond angles between two successive bond vectors, ri−1,i = ri−1 − ri and ri,i+1 = ri − ri+1 , θ0 is the equilibrium bond angle between adjacent bonds and k θ is the dimensionless bond angle constant. The torsional angle potential given by Etorsion 1 = k ϕ (1 + cos 3ϕ), ϵP P 2

(4)

where ϕ is the torsion angles between three successive bond vectors, ri−1,i = ri−1 − ri , ri,i+1 = ri − ri+1 and ri+1,i+2 = ri+1 − ri+2 while k ϕ is the dimensionless torsional constant. All beads of the polymer chain are subjected to the external force (Fext ), whenever required, in addition to the above interaction potentials. The equation of motion of a polymer bead due to the external forces and various interaction potential can be written as:

m

d 2 ri = Fi = −∇i E + Fext dt2

(5)

where E is the sum of all potential energies. We make all quantities dimensionless, i.e. time in terms of τ (≡

# 2

mσP2 P /ϵP P ), so that t = t/τ , energy E = E/ϵP P and lengths r = r/σP P .

The atomic potential parameters between substrate atoms and polymer atoms used here is

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the DREIDING 44 potential. The equation of motion of a bead of mass m due to the forces arising from the sum of the various potentials, E can be integrated up to any time according to the velocity Verlet 45 algorithm.

Parameters The polymer chain length, N chosen for all simulations in this study is 1000. The parameters required to define the Lennard-Jones potential among non-bonded beads are ϵP P , ϵP S , σP P , and σP S . The equilibrium separation between two adjacent beads, req , and the spring constant of the harmonic potential, ks , are required to define bonded interaction potentials. The harmonic cosine bond angle potential requires a rotational constant, kθ , and equilibrium angle, θ0 . The torsional angle potential requires a dimensionless torsional constant, kϕ . The values of all these parameters 44 are shown in Table S1 and lead to a chain which is effectively inextensible.

Parameters studied To quantify the change in the chain conformation during the period a force is applied to it, we measure the chain’s radius of gyration, Rg 46 and the fibrillation index. These parameters help us to characterize the polymer chain conformation during the desorption process. To study the average size of a polymer chain, we measured all three components of Rg defined 1 1 N 1 2 2 2 2 as ⟨Rgx ⟩ = ΣN Σj=1 (yi − Ycm )2 and ⟨Rgz ⟩ = ΣN (zi − Zcm )2 . j=1 (xi − Xcm ) , ⟨Rgy ⟩ = N N N j=1 Here xi , yi , and zi are the components of position vector of the ith beads and Xcm , Ycm , and Zcm are the components of position vector of the center of mass of the polymer chain. On application of a force, we expect that a higher load can be taken by the adhesive when there is formation of fibrils. Fibrils are bundles of highly oriented polymer segments which are aligned along the applied force direction. The fibrils are located a small distance from the substrate and extend a (comparatively) long distance. To design the best materials leading to fibril formation, we need to quantify the proportion of fibrils and the quality of 6


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the fibrils which are formed. Qualitatively, fibrillation is dependent on the following two factors: (a) the degree of order in the polymer structure in the applied force direction, which is characterized by order parameter, Sp and defined below; (b) the formation of cavities in between fibrils, which are characterized by the exposed area of fibrils, Af . During fibril formation, the polymeric material can dissipate the applied energy by the formation of one (or more) cavities within the polymer structure leading to the creation of exposed surface area around the fibrils. The fibrils, themselves, are ordered, stretched polymer sections. We define the fibrillation index (F I) based on these two variables as

FI =

Af (Sp + 1) , Am

(6)

where Af is the total exposed area of all fibrils and Am is the maximum possible exposed area of all the beads, which leads to the Af area. Am can be thought of as a normalization constant and makes F I dimensionless.Am is determined from the solvent accessible area (SASA) 47 of a fully stretched PE chain by using VMD software. 43 The calculated SASA of the pulled section of polymer, which exist as fibril gives the value of Af . Sp denotes the bond order parameter of the polymer in the applied force direction and is defined as 41

Sp =

2 1 −2 3 cos αj − 1 ΣN ⟨ ⟩, N − 2 j=1 2

(7)

where αj is the angle between a sub-bond vector of the polymer chain and a unit vector parallel to the external applied force. A sub-bond vector joins the mid-point of two consecutive bonds.

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Polymer simulations External force — Fzext An external force, Fzext , is applied to all beads of an initially adsorbed polymer chain. The polymer chain adsorption process is discussed in supplementary material (see Figure S1 and S2). The properties of the chain are described by the chosen parameters ϵP P = 0.3968, ϵP S = 0.30, kϕ = 2.0 kcal/mol. The equilibrated chain lies on the graphene substrate and the applied force is perpendicular to this substrate. Figure 1 shows snapshots of the polymer chain as it is pulled off the graphene substrate at a series of times after application of the force, Fzext . The various magnitudes of the used force are Fzext = 7, 14, 21 and 70pN . For application of forces greater than 14pN the chain eventually detaches from the substrate. We have calculated the detachment time of a chain which we define as the detachment of all but a single adsorbed bead. We propose that the force and detachment time will be related by γ Fzext t = const.

(8)

The exponent γ may vary depending on characteristics of the polymer (ϵP P , ϵP S , kϕ ). Although Eq.8 is empirical there is some theoretical basis for this equation. As has been discussed in works by Williams and Kauzlarich 48 and Kovalchick et al. 49 when peeling an adhesive tape of width b from a substrate one can write the applied force, F in term of adhesion energy per unit area Gc as F (1 − cos ψ) = Gc , b

(9)

an equation initially derived by Rivlin in 1944. Here ψ is the peel angle (i.e., angle the tape makes with substrate). From a variety of experiments 48,49 one finds the adhesion energy is

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related to the peel velocity in a power law manner (with exponent z), i.e.

Gc =

$ %z

x t

,

(10)

where x is the distance along the tape that has peeled off the substrate in time t. In our work we consider the detachment time of a chain. For given chain parameters (i.e., chain length, interactions, etc) the detachment distance x is roughly constant, so that combining Eqs. 9 and 10 for peel angle of 90 degrees gives Eq. 8 if we define γ ≡ z −1 . It is found that the γ value for the present case is 1.6. We discuss the variation of γ with ϵP P and ϵP S below (see Section: Detachment process dependence on ϵPS and ϵPP ). In all the cases where the polymer detaches from the substrate, the polymer backbone is found to increasingly align along the applied force direction. This can be deduced from the Sp values for the chain, given in Table 1. (Recall, an Sp value of 1 indicates bond vectors are parallel to the substrate normal while an Sp value of 0 indicates a random distribution of angles.) The highest Sp value corresponds to the chain with the medium force (21pN ) just before detachment (at 30ps) with a value of 0.8. The other two cases have much lower Sp values (of 0.61 for a force of 14pN and 0.66 for 70pN ). Not only does the polymer backbone align with the substrate normal they also tend to be co-planar just before detachment. This is indicated by the torsion angle distribution - a torsion angle of 180◦ (trans configuration) indicates three consecutive beads are co-planar. The torsion angle distribution, at various times during detachment are given in Figure 2. For the small force case, Fzext = 14pN , the maximum number of torsion angles at 180◦ (trans configuration) is around 750. For the medium force this increases to 950, while for the largest force it is around 800. Note, a large force pulls beads rapidly off the substrate and bond vectors have insufficient time to straighten while in the smallest force case even though many bonds straighten when pulled off the substrate, the long detachment time (>60ps) allows beads furthest from the substrate to reorganise into a small compact blob (see Fig.

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Top view Side view Time (ps)

0

25

50

75

100

40

50

60

25

30

6

7

(a)Fzext = 7pN Top view

Side view

Time (ps)

0

25

(b)Fzext = 14pN Top view

Side view

Time (ps)

0

15

20

(c)Fzext = 21pN Top view

Side view

Time (ps)

0

3

5

(d) Fzext = 70pN

Figure 1: Snapshots of polymer chain during the period it is being pulled perpendicular to the substrate (z-direction) by different values of external force,Fzext in pN at T = 300K (a) Fzext = 7, (b) Fzext = 14, (c) Fzext = 21 and (d) Fzext = 70. Other parameters are ϵP P = 0.3968, ϵP S = 0.30, kϕ = 2.0 kcal/mol.

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60ps 40ps 30ps 0ps 500

0 0

60

120

(Degree)

(a)

180

1000

1000

6ps 4ps 2ps 0ps

30ps 15ps 0ps 500

0 0

60

120

180

(Degree)

(b)

Number of torsional angle ( )

1000


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500

0 0

60

120

180

(Degree) (c)

Figure 2: Number distribution of torsional angle (ϕ) during the period the chain is pulled in the z-direction at constant applied force Fzext (in pN ) (a) Fzext = 14 (b) Fzext = 21 and (c) Fzext = 70pN . Other parameters are the same as indicated in Figure 1. 1). A co-planar strand allows subsequent strands to also lie co-planar and adjacent to it, leading to a lamella-like structure. By doing this, the polymer chain maximizes the number of polymer-polymer contacts and hence minimizes the polymer-polymer energy. A large Sp value and a large number of torsion angles close to 180◦ indicates much of the applied force has been transferred to rearranging the chain structure. On the other hand, if the chain structure is not rearranged, the applied force is transferred to pulling beads off the substrate. An internal rearrangement of the chain structure will therefore be more beneficial to a polymer chain’s adhesive capacity. It is found that for large values of Fzext small gaps tend to separate polymer strands which are pulled off the substrate giving the overall appearance of cavities in the polymer structure. On the other hand, for lower values of Fzext the polymer detaches slowly either as a single strand or a double strand but with a very little cavity formation. The structural response of the polymer chain to a high external force (Fzext ) resembles a fibrillated polymer which we believe is essential for the chain’s enhanced tackyness. Due to this fibril formation, the polymer dissipates energy through the formation of new surfaces. During the new surface creation, the polymer-polymer bead contact interaction energy decreases, which can be captured through determination of EP P . The fibril formation depends on 11


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the configuration of polymer and also the polymer-surface contacts. We give a quantitative measure of this through the fibrillation index, F I which was defined in Eq. (6). These structures will now be analyzed with respect to their radius of gyration, non bonded polymerpolymer contact energy, gradient of total energy and fibrillation index, F I. We note that while the single chain system may not entirely replicate the experimental system containing many chains, it will give insights into the role of fibrillation and its effects on adhesion. Table 1: Value of order parameter for different simulation conditions. Note at the lowest force of 14 pN and 60ps the chain comes off as a single strand (see Fig. 1) and so the FI index is ill-defined at this time. Fzext (pN) 70 70 70 70 70 21 21 21 14 14

Time (ps) 3 4 5 6 7 20 25 30 40 50

Sp

FI

0.17 0.41 0.51 0.60 0.66 0.23 0.62 0.80 0.30 0.61

0.82 0.97 1.03 1.07 1.03 0.58 0.66 0.85 0.88 0.98

Chain statistics during detachment The variation of the z-component of the radius of gyration, Rgz with time gives an indication of the deformation of the polymer chain. Prior to the application of an external force the radius of gyration is nearly zero. After detachment of the polymer from the substrate, the effect of the applied force on the polymer is negligible as the polymer attempts to form a globule-like structure, which again leads to a (equilibrium) small Rgz . Figure 3a shows the variation of Rgz with time after the application of the external force, Fzext . This variation of Rgz with simulation time strongly depends on the magnitude of Fzext . The data points are

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an average of five different runs using the same set of parameters. At Fzext = 7pN , (which is not shown to keep clarity in the Figure), the value of Rgz remains constant (the polymer remains adsorbed) as such a low value of Fzext is unable to cause a significant change in the polymer configuration. At Fzext = 14 and 21pN , the value of Rgz increases due to extension of the polymer chain away from the graphene substrate and it subsequently decreases when the chain completely detaches from the substrate. At the highest value of Fzext = 70pN , the polymer beads only remain on the substrate for a short time. During this period, many sections of the polymer chain align perpendicular to the substrate (along the force direction) and subsequently the polymer detaches from the substrate. Due to the small contact time, a small change in Rgz of the polymer chain is observed. -1000

150

Fzext

Fzext

70pN 21pN 14pN

100

Epp (Kcal/mol)

70pN 21pN 14pN

Rgz (Å)

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50

0

-1500

-2000 0

25

50

0

25

50

Time (ps)

Time (ps)

Figure 3: Effect of externally applied force Fzext (pN ) on (a) z-component of radius of gyration and (b) polymer-polymer contact energy. The variation of EP P with time (shown in Figure 3b) indicates that after initial rearrangement of the polymer there is a local decrease in EP P values. As beads are detached from the substrate, there is an increase in polymer-polymer contacts which is due to nearby strands of the polymer chain tending to align resulting from an increase in trans conformations of the polymer chain. However, as more beads which are far apart along the chain detach from the substrate, new polymer surface is created which results in cavities in the chain structure. At this point EP P begins to increase again, leaving a local minima in the 13


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EP P versus time curves. Eventually, the chain detaches entirely, at which point the chain forms a globule and EP P then decreases monotonically. Each of the cases, thus has at least one minima followed by a maxima - for the highest force of 70 pN this occurs very early on (by about 3ps) while for the 21pN case it takes about 12ps. For the lowest force of 14pN, a series of minima and maxima appear until approximately 31ps at which point the chain detaches entirely from the substrate. Figure 3 thus demonstrates that as the polymer is pulled off the substrate nearby beads tends to align and clump together while distant strands leave cavities in the polymer structure. The large cavities expose polymer to the solvent, resulting in an increase of polymer-polymer energy before the chain entirely detaches from the substrate. We have determined the fibrillation index, F I, of the detaching polymer chain using Eq. 6. This requires the bond order parameter, Sp , (see Eq. 7), Af and Am at different times for the stretched chain. The variation of Af /Am , Sp and the calculated F I at the different applied forces are plotted in Figure 4. The values of Af depends on the surface area of the fibrils while Am is a constant quantity. Sp increases with pulling time due to favorable trans conformations of the polymer chain. The curve of F I passes through the maxima for the applied force 70pN. In general, the time required to attain the maximum F I decreases with increasing applied force. A small value of the maximum fibrillation index indicates a smaller cavity within the formed fibril or no fibril formation. Thus higher values of fibrillation (typically F I > 1) indicate the formation of fibrils with cavities. This value of F I is justified from the values of Sp and Af /Am . A Sp value of 0.5 is indicative of good alignment of the polymer along the applied force direction. Assuming a hexagonal crosssection for a stiff polymer, 50 if three chain segments align parallel to each other, one third of each strand area will be covered. Thus a typical value of Af /Am is 2/3. Thus using Eq. 6, we calculate typically good values of F I to be 1 or more. The substrate force gives an indication of the response of a polymer chain during the detachment process. We define the substrate force as the gradient of the non-bonded

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80

1

(

70pN 21pN 14pN

70pN 21pN 14pN

05

m

o

o

40

0 0

25

0

-0 5

50

0

Time (ps)

25

50

Time (ps)

(a)

(b)

F

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

70pN 21pN 14pN

0 0

Time (ps) (c)

Figure 4: Effect of externally applied force Fzext (pN ) on various parameters at different time (a) Af /Am (b) Sp and (c) F I. interaction potential energy between polymer chain and the graphene substrate beads. This is calculated using the energy in Eq. (1), i.e., FS =⟨Fi ⟩=⟨−∇i Eij (rij )⟩. Supplementry Figure S3 shows the average substrate force profile as the chain is pulled away at different values of the external force. It is found that the x and y components of the substrate force are zero throughout the application of the externally applied force and only the z-component of the substrate force varies with time. For the smallest external force of 14pN the substrate force has oscillations in its profile. Such a profile has been observed previously when single (adjacent) beads of a polymer chain are being sequentially detached from a substrate and has been referred to before as a saw-tooth profile. 11,23 The profile for the medium force (21pN ) also has a fluctuating profile but here it is because distant beads on the chain are 15


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being detached (and also possibly re-attached) as the polymer is pulled from the substrate. Finally for the largest force (70pN ) the profile is smooth with a steep valley, reaching a maximum absolute value of 70pN and is back to zero within 5ps. The initial rapid decrease occurs when all beads are almost simultaneously rapidly detached from the substrate. The substrate force then smoothly returns to zero as the chain moves further away from the substrate. The reason for the difference in the substrate force profiles (shown in supplementry Figure S3) can be attributed to the rate of breakage of polymer-substrate contact with the application of the external force. Supplementry Figure S4 compares the rate of breakage of polymer bead-substrate contacts with time for two different applied forces — Fzext = 70 and 21pN . The rate of breakage of polymer-substrate contacts goes through a maximum for both cases. However, the curve is smooth for Fzext = 70pN but fluctuates for Fzext = 21pN . In the later case, after the detachment of a polymer bead from the substrate, the stretching of subsequent bonds allow the polymer to fluctuate in the space just above the substrate. Eventually the external force pulls the chain far enough away from the substrate and whole chain detaches. It is well known that the addition of small amounts of solvent may change the properties of pressure sensitive adhesives. The addition of solvent changes properties of the polymer. In simulations, solvent quality can be changed through ϵP P and substrate strength through ϵP S . Another important chain property is stiffness or kϕ . Below we discuss the effect of variation of these parameters on fibrillation.

Detachment process dependence on ϵPS and ϵPP We note that Lopez et al. 5 have conducted experiments on pressure sensitive adhesives and discussed the process of fibrillation with respect to the failure of the polymeric materials at a constant applied force. The appearance of a plateau region in the probe tack experiments, at a constant applied probe velocity, was deemed to be a characteristic of cavitation and 16


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350

1.6

ɛPS (kcal/mol)

0.5 175

FI

Stress (pN/Å2)

0.3

0.8

PS

(kcal/mol) 0.3 0.5

0 0

20

40

0

60

0

Strain

5

10

15

Time (ps)

(a) 1000

(b) 1.5

ɛPP (kcal/mol) 0.5968 0.3984

100

1

0.1984 FI

Stress (pN/Å2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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10

PP

0.5

(kcal/mol) 0.5958 0.3968 0.1984

1 0

10

20

30

0

40

0

4

Strain

8

Time (ps)

(c)

(d)

Figure 5: Variation of stress as polymer chains of different ϵP S and ϵP P are pulled with external forces Fzext = 70pN (a) variation in stress with strain at different values of ϵP S (b)variation of F I with time at different values of ϵP S (c) variation in stress with strain at different values of ϵP P (d)variation of F I with time at different values of ϵP P . fibrillation. We measure the substrate force and the corresponding strain for the applied force (which corresponds to experimental probe velocities) as shown in Figure 5 at different values of ϵP P and ϵP S . The stress is calculated by dividing the substrate force by area at the base of the fibril, Ab , which is calculated using the SASA method. 47 For ϵP S = 0.3 the extension is small, which corresponds to smooth removal of the material from the substrate, but for ϵP S = 0.5 a long plateau is observed (between 20 to 40 seconds) in the stress-strain curve. The long plateau in the stress-strain curves corresponds to the larger fibrillation indices as shown in Figure 5b. In Figure 5 c and d, where ϵP P is varied at constant ϵP S , one sees that 17


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

the largest ϵP P corresponds to a quick and smooth removal of material from the substrate. The other two ϵP P values (0.3984 and 0.1984) yield a much longer extension of the chain and corresponding larger fibrillation indices. The experimental work of Lenhart and Cole 34 indicates an increase in adhesive energy of polymer gels with an increase in good solvency conditions. We have also run simulations for three different values of the external force Fzext = 14, 35 and 49pN and at various values of ϵP S , ϵP P as listed in Table S2. We fixed the value of kϕ to 2kcal/mol. We determine the extension of the chain just before complete detachment (i.e. detachment of the last adsorbed bead) and the detachment time. These are plotted against the externally applied force and shown in Figure 6. The following trends can be extracted from these plots: • At fixed ϵP S and applied force the extension of polymer decreases with increasing ϵP P • The larger the polymer-substrate energy (ϵP S ), the longer the polymer sticks to the substrate and consequently the larger the polymer extension • At a higher applied force, the polymer extension is smaller as the polymer has less time to respond to the applied force • At a fixed value of ϵP S , with a lower value of ϵP P the chain remains attached to the substrate for a longer time. This indicates that increasing the solvency (to good solvent conditions) leads to the adsorbed chain becoming more extended which delays detachment of the polymer from the substrate • On the other hand, for poor solvent conditions the polymer beads attempt to reduce solvent contact and hence make as globular a conformation as possible. On application of a force, the chain would rather be detached (and globular) than extended. Hence, the polymer detaches more rapidly as the solvent conditions are made poorer. The external force versus detachment time curves are fitted with the equation Fγzext t = 18


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Page 21 of 32

1000 etac em t (Å)

100 PS

PS

Time needed for detachment (t)

PS = 0.5

= 0.3

st e

100 PS

= 0.1

Exte sio

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

10

= 0.5 = 0.637 =0.633 =0.584 =1.652 =1.623 =1.664

10 PS

= 0.3

1

14

28

42

56

70

84

14

140

Fzext (pN)

Fzext (pN) (a)

(b)

Figure 6: (a) Extension of polymer chain just before detachment from the substrate as a function of applied external force. (b) Detachment time as a function of applied external force. Both graphs are plotted for different ϵP S and ϵP P . All circles ϵP P =0.1984, all triangles ϵP P = 0.3968, and all diamonds ϵP P =0.5952. Blue color for ϵP S =0.1, red color for ϵP S =0.3 and black color for ϵP S =0.5. Other conditions are temperature T =300K and kϕ =2.0 kcal/Mol. C, where C is a constant. The fitted γ values are given in Figure 6 and listed in Table S2. It is found that at a fixed ϵP S the γ value is almost independent of ϵP P while it is strongly dependent on ϵP S . At ϵP S = 0.5 the γ value is around 0.60 while at ϵP S = 0.3 the γ value is around 1.65. The constant value, C, is dependent on ϵP P with a larger constant corresponding to smaller ϵP P . As a comparison, we consider these results in the context of a fully flexible chain. For the fully flexible chain (kθ = 0 and kϕ = 0) we run the simulations keeping all other parameters the same as above (see Table S1). We determine the extension of the fully flexible chain just before complete detachment and the detachment time. These quantities are plotted against the external force (see Figure 7). These results follow the same trends as detailed for the semi-flexible chain (listed above). The fully flexible chain also detaches com-

19


Langmuir

600

100

Time needed for detachment (t)

PS = 0.5 kcal/mol

350

i

t

e

tac

(Å)

PS

Ex

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 32

PS

= 0.3 kcal/mol

= 0.5 =0.572 =0.658 =0.685

PS

10

=1.133 =1.135 =1.228

= 0.3

100 14

28

42

56

70

1

84

14

140

Fzext

Fzext (pN) (a)

(b)

Figure 7: (a) Extension of fully flexible polymer chain just before detachment from the substrate as a function of applied external force. (b) Detachment time as a function of applied external force. Both graphs are plotted for different ϵP S and ϵP P . All circles ϵP P =0.1984, all triangles ϵP P =0.3968, and all diamonds ϵP P =0.5952. Red color for ϵP S =0.3 and black color for ϵP S =0.5. Other conditions are temperature T =300K and kϕ =2.0 kcal/Mol. paratively quickly in poor solvent conditions while as the solvency improves the detachment time increases. This can be seen from the γ exponents extracted from the detachment time versus force curves (as indicated in Figure 7 and listed in Table S3). The larger ϵP S value corresponds to the smaller γ exponent. Comparing Figures 6 and 7, we find that there is a quantitative difference in the size of the extension and length of detachment period. The fully flexible chains tend to be more extended for the same values of parameters ϵP S and ϵP P and applied force. In summary, it is found that the adhesive capacity of the polymer is enhanced when the polymer has a favorable interaction with the solvent in conjunction with a large polymersubstrate interaction. Fully flexible chains tend to have large extensions at detachment, due to the applied force being used to stretch the chain rather than rearrange the internal structure into lamellae or form cavities.

20


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Langmuir

Detachment time dependence on polymer twist We now vary the torsional constant, kϕ , of the polymer chain, keeping other constants fixed (i.e., ϵP P = 0.3968 and ϵP S = 0.3). We applied three different forces of Fzext = 21, 35 and 70pN . Since the results for the three different forces are qualitatively similar we focus on only one case (of Fzext = 35pN ) which are shown in Figure 8. Figure 8a shows the variation of total contact energy between polymer and substrate, EP S with time. The chain with the largest kϕ is attached to the substrate for the longest time (around 25ps) while the chain with the smallest kϕ stays attached only for about 10ps. All chains begin with roughly the same EP S value of around -4000 kcal/mol which indicates they all initially lie flat on the substrate. Figure 8b plots Rgz as a function of time. The chain with the largest variation in Rgz once again is the one with the largest kϕ value of 5.0. As the torsional constant decreases to 0.5 kcal/mol, Rgz varies to a much smaller degree. This indicates that the chain with the larger kϕ yield a fibrillated structure during application of the external force (e.g., compare Figure 8b with Figure 3a). Figure 8c shows why the chains with the larger torsional constant result in a fibrillated structure and hence a long detachment time. Initially, the chain with the lowest EP P value corresponds to the largest kϕ value. This indicates the beads along the chain tend to align and hence maximize their polymer-polymer contacts. On the other hand, the chain with the smallest kϕ value tends to have a much more random configuration on the substrate and consequently has fewer polymer-polymer contacts. On application of the external force, the chain with the smallest kϕ value quickly detaches from the substrate since it readily forms a globular structure in solvent (and hence easily increases the number of polymer-polymer contacts). Conversely, since the chain with the larger kϕ values are initially highly aligned on the substrate they would like to remain in this configuration. (Notice the local minima in the kϕ = 5 and 2 kcal/mol curves in Figure 8c while the kϕ = 0.5 kcal/mol curve shows a monotonic decrease.) Hence these chains resists detachment from the substrate, on application of an external force. Moreover, when beads do detach from the substrate, a fibrillated structure forms which prolongs the detachment 21


Langmuir

(Å)

150

-2000

al/mol) 5 2 0

100

z

cal/mol)

0

al/mol) E

50

5 2 5

-4000 0

10

20

0 0

0

10

0

Time (ps)

Time (ps) (a) 0 cal/mol)

20

(b) al/mol) 5 2

-2000

E

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 32

-4000 0

10

20 Time (ps) (c)

Figure 8: Effect of variation of torsional constant (kϕ in kcal/mol) during the period the polymer is pulled perpendicular to the substrate (z-axis) by a fixed external force of Fzext = 35pN and ϵP P = 0.3968 kcal/mol. (a) Substrate-polymer energy profile EP S as a function of time, (b) Rgz as a function of time and (c) Polymer-polymer contact energy EP P as a function of time. time of the entire chain. For these three kϕ values we have also extracted the detachment exponent γ. From our data, we estimate γ = 1.3, 1.14 and 1.05 for kϕ = 0.5, 2 and 5 respectively. It should be noted that a longer detachment time has been also been seen for fully flexible chains (previous subsection). However, the longer detachment time for fully flexible chains corresponded to a very large chain extension, whereas in this case (chains with larger torsion constant) it corresponds to a realignment of the chain. If an adhesive was between two substrates, it would therefore mean in the former case (fully flexible chains) the substrates would move apart while in the later case (chain with larger torsion constant) 22


Page 25 of 32

it would mean the substrates would be still quite close. Figure 9 plots the variation of stress with strain (9a) and the fibrillation index, F I, with time (9 b) for polymer chains with different kϕ values. The stress-strain curve shows that the stiffest chain has a small extension (only up to 10-15%). However, the same chain is in contact with the surface for a long time (up to around 7ps) and achieves a high FI value of over one. Thus the stiff chains do not have the long stress-strain plateau, as the chains do not extend significantly. However, they still display good adhesive capacity as indicated by the large FI and the long time the chain remains in contact with the substrate. Although the least stiff chain can extend beyond 20% strain, its FI value is comparatively small (around 0.65-0.75) and thus keeps in contact with the substrate for a shorter time (less than 6ps). The larger FI values (and larger detachment times) for the stiffer chains (kϕ = 2, 5) can be attributed to more bonds in the trans configuration which leads to more polymer-polymer contacts. 15

50

( cal/mol) 5 2

1

05 25

F

Stress (pN/Å2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

05

( cal/mol) 5 2 05

0

0 0

20

0

40

4 Time (ps)

(a)

(b)

Figure 9: (a) Variation of stress as polymer chains of different kϕ are pulled with external force of Fzext = 70pN . (b) Variation of F I with time as polymer chains of different kϕ are pulled with external force of Fzext = 70pN . In summary, chains with larger torsional constants are initially highly aligned on the substrate and subsequently yield fibrillated structures resulting in longer detachment

23


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times but still relatively small extension of the chain. On the other hand, chains with small torsional constants have a much more random configuration which results in a smooth removal from the substrate and hence a rapid detachment.

Conclusions In this study we have sought to understand the adhesive capacity of a polymer at the molecular level. Although a simple MD model has been used, where the same force is applied to each bead in the chain, the simulations have yielded some important insights. To achieve good adhesion, the externally applied force needs to be dissipated through the polymer rather than being used to pull beads of the substrate. It has been observed that the applied energy can be dissipated in two main ways — firstly, the formation of cavities between polymer fibrils which produces new surface area and secondly a rearranged polymer structure which tends to align segments of the chain into lamella layers. To quantify these qualitative observations we defined a measure called the fibrillation index (FI). In our simulations we observed a long plateau in the stress-strain curves which had been previously identified (through experiments 5,31 ) as a characteristic of enhanced tackyness. We calculated the FI values for these polymer chains and found they also had a large FI index. Thus we propose a large FI value is a quantitative indication of adhesion capacity. A series of simulations were carried out varying polymer-polymer, polymer-substrate and torsion constant of the chain. Observations from these simulations as well as calculations of the FI values lead to the following conclusions: • For good adhesion, a large polymer-substrate energy together with a small polymerpolymer energy (i.e. good solvent conditions) is required. This leads to the formation of fibrils and cavities • For chains with large torsional constants, a stress-strain plateau is not observed, but 24


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Langmuir

the chain can still display good adhesion. This is because while the chains do not stretch, they still remain in contact with the substrate and dissipate energy through chain rearrangement to an ordered polymer structure. • The detachment time, on application of an external force, is inversely proportional γ to the force. In fact we propose the relationship Fext t = Constant. The exponent γ

depends on the polymer-polymer, polymer-substrate and chain torsion energies. The stiffness of the polymer, the relative energy of interaction between the polymer moiety and between the polymer and substrate, all affect the energy dissipated during the process of stress transfer. While these ideas were presented and developed for macroscopic polymer systems by Creton et al. 5,31 this study shows that the mechanisms of stress dissipation hold true for single polymers in the vicinity of a nano-substrate. The above analysis also implies that the presence of nano-substrates affect the nature of polymer response to stresses and thus will affect polymer failure. Choice of the polymer, the nano-substrate (and how it interacts with the polymer) affects the mode of failure for both PSAs and composites. The presence of many polymers will further affect the fibrillated structure and formation of cavities and it is envisaged this study can be extended to more complex systems. In addition, further analysis of the nano-substrate structure and its impact on the polymer configuration and stress dissipation will also help provide greater understanding of design principles for development of strength and toughness of polymer formulations or of designed failure in polymer systems.

Supporting Information Available This material is available free of charge via the Internet at http://pubs.acs.org/.

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(43) Humphrey, W.; Dalke, A.; Schulten, K. VMD: visual molecular dynamics. Journal of molecular graphics 1996, 14, 33–38. (44) Mayo, S. L.; Olafson, B. D.; Goddard, W. A. DREIDING: a generic force field for molecular simulations. Journal of Physical Chemistry 1990, 94, 8897–8909. (45) Verlet, L. Computer” experiments” on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Physical review 1967, 159, 98. (46) Dhondi, S.; Pereira, G. G.; Hendy, S. C. Effect of Molecular Weight on the Capillary Absorption of Polymer Droplets. Langmuir 2012, 28, 10256–10265. (47) Lee, B.; Richards, F. M. The interpretation of protein structures: estimation of static accessibility. Journal of molecular biology 1971, 55, 379–IN4. (48) Williams, J. A.; Kauzlarich, J. J. The influence of peel angle on the mechanics of peeling flexible adherends with arbitrary load–extension characteristics. Tribology International 2006, 38, 951–958. (49) Kovalchick, C.; Molinari, A.; Ravichandran, G. Rate dependent adhesion energy and nonsteady peeling of inextensible tapes. Journal of Applied Mechanics 2014, 81, 041016. (50) Pereira, G. G.; Williams, D. R. Toroidal globules of DNA and other semi-flexible polymers: Jumps in radius caused by hexagoanl packing. Europhysical Letters 2000, 50, 559–564.

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For Table of Contents use only Effect of uniformly applied force and molecular characteristics of a polymer chain on its adhesion to graphene substrates

21pN

70pN

ati

Fzext = 14pN pN

x (F )

Sunil Kumar, Sudip K. Pattanayek*, Gerald G. Pereira and Sanat Mohanty

F

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70pN 21pN 14pN

0 0

Time (ps) Time = 40ps

Time = 20ps

Time = 4ps

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Effect of Uniformly Applied Force and Molecular Characteristics of a

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