Wetting characteristics of polymer adhesives with different chain bending stiffness

Abstract

Polymer adhesives are widely used in daily applications and in industry owing to their flexibility and overall non-toxicity, particularly in interfacial adhesion. The spreading of polymer adhesives on adherend is one of the essential considerations for the interfacial adhesion of polymer adhesives, which is strongly related to their wetting behaviors. While relationships between polymer microstructure and adhesion have been investigated in previous studies, it remains challenging to unveil the effect of polymer microstructure on wettability. To address this issue, here we utilize coarse-grained molecular dynamics (CGMD) simulations to systematically elucidate how the wettability of a polymer adhesive droplet on a surface depends on bending stiffness. The wetting dynamics and the contact angle are studied to show the evolution of morphology of droplets during the wetting process. The results indicate the wettability is weakened by the increase of bending stiffness of polymer chain. Detailed thermodynamic property analysis is further conducted, revealing that the adhesion between the polymer droplet and substrate deteriorates due to the decline of wettability. Interestingly, we observe such deterioration becomes more significant by both increasing the temperature and decreasing the bending stiffness. Our study sheds light on the dependence of chain bending stiffness and temperature on the wetting behavior of polymer adhesive droplets, and offers insights, which, upon experimental validation can then be used for the design of adhesives or hydrogels.

Keywords

Polymer adhesives coarse-grained molecular dynamics contact angle bending stiffness

Introduction

Polymer adhesives play a progressively vital role in various industries, from daily used memo notes to medically used tissue adhesives.¹ Recently developed strong and fast tissue adhesives or hydrogels^2,3 have potential applications of tissue repairing and wound dressing. Many interesting studies focus on the influence of polymer microstructure on adhesion properties^4-7; however, it remains challenging to reveal the effect of polymer microstructure on wettability. The wetting of a polymer adhesive droplet on a solid surface plays a critical role in describing the characteristics of polymer adhesives in an extensive range of technical applications. For coating and painting technology, the adsorption of polymers on the surface will determine the quality of the painting and the stability of the coating. The information about spreading rate and equilibrium conditions is important for attaining better adhesion performance. Controlling the wettability of polymer adhesives can enhance the production of adhesives, especially for those coating on release films. Good wettability makes the adhesives spread quickly and evenly on the release films and reduces the probability of generation of bubbles and hence makes the adhesives more homogeneous between the two release films.^8-10 On the other hand, exploring the influencing factors of wettability can provide guidance for designing antifouling or self-cleaning surfaces.¹¹ For these applications, it is crucial to understand the fundamental mechanisms about wetting behaviors of polymer adhesives at the molecular level.

It is a long-standing goal to unravel the relationship between macroscopic properties and microstructure, and molecular simulation provides a viable means to derive macroscopic properties from molecular structures and atomic interaction. Over the years, molecular simulation has become an almost irreplaceable tool for physics, chemistry, and materials science. Molecular simulation provides a viable means to conduct computer experiments under perfectly controlled conditions and has been applied to study the wetting behavior of droplets. However, most previous works devote their efforts to water droplets or molten metal droplets.^12-16 Although some researches pay attention to the wetting behaviors of polymers,^17-19 the effect of microstructure of polymer chains on the wetting behaviors is still undiscovered. At the atomic scale, the microstructure dictates the morphology of the polymer chain. Sinan et al. demonstrated that there is a connection between different morphologies of polymer chains and macro-properties.⁷

Although the wettability of a droplet on a solid surface involves numerous factors, such as chemical composition, chemical reactions at the interface, roughness and flatness of the solid substrate, impurities of liquid, and applied external field (e.g., electric field),²⁰ we make following assumptions in order to emphasize the influence of polymer structure. In this regard, we use a homogeneous model and the contact between a droplet and a substrate is perfect. The influence of generation of microbubbles during the wetting process is not considered. Although the wettability is affected by surface flatness and roughness, we use a flat coarse-grained substrate to perform simulations. As for the effects of electric or magnetic field, they are beyond our research scope in this article.

We start the investigation from the calculation of glass transition temperature of polymer adhesives with different bending stiffness, and the temperature for the simulations is determined to be higher than the glass transition temperature. Then coarse-grained molecular dynamics (CGMD) simulations are carried out to catch the time history of contact radius and precursor foot during the wetting process. The contact angles at equilibrium status are calculated by circle fitting method. The thermodynamic analysis is performed to study the influence of wettability controlled by different chain bending stiffness on the interfacial adhesion. Our purpose is to gain a better understanding of the effects of microstructure on the wettability of polymer adhesives and give rudimentary suggestions in designing of high-performance adhesives based on our predictions.

Simulation details

The polymer chains are represented by the Kremer–Grest model,²¹ which is well known for capturing the properties of bulk polymers. Within our Kremer–Grest-based model, a cluster of interconnected spherical beads with mass of m form a polymer chain. The nonbonded interactions within and between polymer chains are illustrated by a Lennard-Jones (LJ) potential

U_{LJ} (r_{i j}) = 4 ε [{(\frac{σ}{r_{i j}})}^{12} - {(\frac{σ}{r_{i j}})}^{6}], r_{i j} < r_{c}

(1)

All other parameters are deduced and based on these three fundamental reduced Lennard-Jones units ( $ε$ , $σ$ , and m are the reduced units of energy, length, and mass) with the Boltzmann constant set as $k_{B} =$ 1. The cutoff radius is set as $r_{c} = 2.5 σ$ in all types of interactions for enhancing the calculation efficiency. Within the Lennard-Jones potential, $σ$ is the radius where the interaction is zero between two adjacent beads, and the parameter $ε$ equals the strength of interaction and is our unit of energy.

For bonded beads, the Kremer–Grest model uses finite extensible nonlinear elastic (FENE) bond to connect each bead. The FENE bond potential can be expressed as

U_{bond} (r_{i j}) = - 0.5 k_{fene} R_{0}^{2} \ln [1 - (\frac{r_{i j}}{R_{0}})]

(2)

which can effectively confine the maximum of bond length to

R_{0} = 1.5 σ

. The bond strength is set to

k_{fene} = 30.0 ε \cdot σ^{- 2}

. Since real polymer chains have a certain degree of rigidity, the determination of first bond angle of a polymer chain will constrain the rest of orientation of bonds,²² and this will decrease the flexibility of polymer chain. However, it is hard to control the bending stiffness by only adopting Kremer–Grest model as its theoretical form is oversimplified. Therefore, we use chain bending stiffness to describe the flexibility of a polymer chain, and the energy induced by the curling of polymer chain is defined as:

E_{B} = \frac{κ}{2} \int_{0}^{L} {(\partial T / \partial s)}^{2} d s

, where L is the length when polymer chain is in the undeformed state, T is the unit tangent vector of a CGMD bead,

κ

is the bending rigidity which constrains the flexibility of a polymer chain, and

\partial T / \partial s

embodies the ratio of local bending.^23,24 To achieve the purpose of making several polymer adhesive droplets with different microstructure, the angle potential is derived for tuning the bending stiffness of the polymer chain directly.

The bending energy of this model is given as

\begin{matrix} E_{B} = \frac{κ}{2} \int_{0}^{L} {(\frac{\partial T}{\partial s})}^{2} d s \end{matrix}

(3)

and the discretized version of equation (3) can be illustrated as follows

\begin{array}{l} \approx \frac{κ}{2} R_{balance} \sum_{i} {(\frac{T_{i + 1} - T_{i}}{R_{balance}})}^{2} \\ = \frac{κ}{2 R_{balance}} \sum_{i} (T_{i + 1}^{2} + T_{i}^{2} - 2 \cos α) \\ = \frac{κ}{R_{balance}} \sum_{i} (1 - \cos α) \\ \begin{matrix} = \frac{κ}{R_{balance}} \sum_{i} (1 - \cos (π - θ)) \end{matrix} \end{array}

(4)

where

α

is the complementary angle between the bond vectors,

θ

is the bond angle depicted in Figure 1(a), and

R_{balance}

(

R_{balance} = 0.97 σ

in FENE bond potential) is the equilibrium bond length of the polymer chain. The angle potential is then defined as

\begin{matrix} U_{angle} (θ) = K [1 - \cos (θ - θ_{0})] (0 < θ < 2 π) \end{matrix}

(5)

where

θ_{0} = π

K = κ / R_{balance}

, and K is linearly proportional to bending stiffness. The following equation describes the total energy of the polymer system

\begin{matrix} U_{total} = U_{bond} + U_{LJ} + U_{angle} + U_{kinetic} \end{matrix}

(6)

Figure 1.

(a) Illustration of relationship between $θ$ and $α$ . (b) The model of polymer adhesives. (c) Hemisphere model of polymer adhesive used in coarse-grained molecular dynamics simulation. (d) Initial configuration of polymer adhesive droplet consisting of 35000 monomers and residing on the FCC (1 1 1) surface.

We choose microcanonical ensemble (NVE) to update position and velocity, and Langevin thermostat is combined to make energy transfer available for the whole wetting system. This combination also could be regarded as a canonical ensemble (NVT). The damping term²⁵ of Langevin thermostat in all the simulations is set as $10 τ^{- 1}$ .

To explore the wetting behavior of polymer adhesives, we divide the procedure of generating hemispherical polymer adhesive droplets into two steps. The first step is to generate randomly distributed polymer networks. In this step, 1000 coarse-grained polymer chains are placed in a $58.5 σ \times 58.5 σ \times 58.5 σ$ periodic simulation box as shown in Figure 1(b). Every polymer chain contains 200 beads which is good for avoiding the vaporization of polymer monomers at high temperature. In order to adjust the position of overlapping beads in the initial configuration, a soft repulsive potential with well depth of $ε$ and cutoff radius of $2^{1 / 6} σ$ is applied first and the simulation runs for 1 million steps in Langevin thermo bath at the temperature of $T = 1.0 ε \cdot k_{B}^{- 1}$ . Then the soft pair potential is turned off and the LJ pair potential is turned on and the simulation runs for 1 million steps in Langevin thermo bath at the same temperature. The last step is to cut off a hemisphere model with radius of $26 σ$ from this cubic model, and all beads and bonds outside the hemisphere region will be deleted directly in this process.

The hemisphere is balanced in NVT ensemble for 1 million steps to adjust the position of overlapped beads. Then a substrate composed of four layers of (1 1 1) surface of FCC lattice is constructed. The interaction of substrate atoms is set as $5 ε$ and the mass of the substrate atom is set as $2 m$ . We combine the hemisphere and the substrate by putting the hemisphere at the center of the substrate and the potential between droplet and substrate is set as $2 ε$ . The distance between the bottom of the hemisphere and the top of the substrate is set as $1.5 σ$ , which is within the attraction zone of LJ potential. The hemisphere shown in Figure 1(d) would contact the substrate immediately once the simulation begins. The equations of motion are solved by the Verlet algorithm implanted in LAMMPS²⁶ using a timestep of $Δ t = 0.009 τ$ , where $τ = σ {(m / ε)}^{1 / 2}$ , and the visualizations are performed by OVITO.²⁷

Results and discussions

Determination of glass transition temperature

The wetting phenomenon of polymer adhesives can happen when the experimental temperature is higher than glass transition temperature. It is reported that chain bending stiffness or angle potential may influence the glass transition temperature of polymers.^28,29 We use the method proposed by Tsige and Taylor³⁰ to acquire the glass transition temperatures through calculating the changes of mean-squared displacement (MSD). For models with different chain bending stiffness, the time-related MSD used for describing the movements of monomers at different temperatures are calculated.

The MSD data are fitted linearly by two lines with different slopes; then the glassy-state region and the rubbery-state region could be recognized clearly. The glass transition temperature $T_{g}$ is obtained from the intersection point of these two lines as shown in (Supplementary Material Figure S1). Figure 2 is the prediction of glass transition temperature of polymer adhesives with different bending stiffness.

Figure 2.

$T_{g}$ prediction for polymer adhesives with different bending stiffness via the calculation of mean-squared displacement at different temperatures.

In order to determine the influence of temperature, we have conducted two sets of simulation for comparison. For droplets with bending stiffness of

κ = 2.0 σ \cdot ε

and

κ = 10.0 σ \cdot ε

, the simulation temperature is set as

1.0 ε \cdot k_{B}^{- 1}

. For droplets with bending stiffness of

κ = 10.0 σ \cdot ε

and

κ = 50.0 σ \cdot ε

, the simulation temperature is set as

1.5 ε \cdot k_{B}^{- 1}

. Table 1 shows the properties of four different systems.

Table 1.

Properties of the different wetting system.

System name	Strength of bond angle $K$	Bending stiffness $κ$	Temperature T
B2.0-T1.0	$2.06 ε$	$2.0 σ \cdot ε$	$1.0 ε \cdot k_{B}^{- 1}$
B10.0-T1.0	$10.3 ε$	$10.0 σ \cdot ε$	$1.0 ε \cdot k_{B}^{- 1}$
B10.0-T1.5	$10.3 ε$	$10.0 σ \cdot ε$	$1.5 ε \cdot k_{B}^{- 1}$
B50.0-T1.5	$51.5 ε$	$50.0 σ \cdot ε$	$1.5 ε \cdot k_{B}^{- 1}$

B and T represent bending stiffness and temperature, respectively

The dynamic analysis of wetting process for different systems

The progressive wetting process for B2.0-T1.0 and B50.0-T1.5 are presented in Figure 3 for comparison. To further compare the wetting process of different systems, we conduct a statistical analysis on the wetting behaviors. Figure 4(a) and 4(b) show how contact radius and precursor foot of droplets change with time. The precursor foot first surges rapidly and keeps enlarging at steady rate while the droplet body spreads for a short time and then becomes stabilized. In addition, the circle fitting method^31-33 is applied for fitting the time-dependent contact angles of droplets with different bending stiffness. At first, the polar axis is placed at the center of droplet along the Z direction, then simulation box is divided evenly by cylindrical surface into bins with $Δ r = 0.5 σ$ and $Δ z = 0.5 σ$ . The density distribution could be determined by averaging the number of polymer beads in each bin from 10 isolated wetting tests for each wetting system. Figure 4(c)–(f) show the contours of density distribution computed from the spatial variation of the local density. And it would be convenient to distinguish the vacuum part (density equal to 0) and the contour of droplet (navy-blue line). Therefore, the contact angles are fitted by a tangent line at the edge of droplet and vacuum part.

Figure 3.

The wetting process of polymer adhesive droplets on the coarse-grained substrate with (a) B2.0-T1.0 and (b) B50.0-T1.5.

Figure 4.

(a)–(b) Change of polymer adhesive droplets with different bending stiffness on substrate during the wetting process, in which cr represents the time histories of contact radius, and pf represents the time histories of precursor foot. (c)–(f) Density contours of each droplet at equilibrium status with different bending stiffness.

The balanced contact angles of different systems show that higher bending stiffness leads to larger contact angle and hence decreases the wettability. For the droplet with bending stiffness of $10.0 σ \cdot ε$ , increasing temperature will decrease the contact angle. Besides, from Figure 4(f) we notice that the internal density of polymer adhesives with higher bending stiffness is relatively low, which means a high probability of generation of cavities inside the polymer adhesives with high bending stiffness. For the application of polymer adhesives in drug transportation,³⁴ polymer adhesives with more inside cavities may be an appropriate choice.

Figure S2 in supplementary material shows the distribution of coarse-grained beads of different system along the Z direction. The results show that polymer adhesive droplets with higher bending stiffness tend to have a smaller value at the first peak, reflecting the negative effect of the increase of bending stiffness on the distribution of coarse-grained beads at the interface between droplets and substrates. Hence, the coarse-grained beads in droplets with higher bending stiffness cannot interact with the substrate effectively; thus, the higher bending stiffness leads to weaker wettability which is identical to the contact angle analysis.

Thermodynamic analysis of wetting behavior

Thermodynamic analysis has been used to explain the wetting behavior of polymer adhesives. For instance, the free energy perturbation (FEP) method³⁵ and potential of mean force (PMF) method³⁶ are applied to compute adhesion free energy and adhesion energy, respectively. In the process of calculating the free energy of adhesion, the interaction between polymer adhesive droplet and substrate is gradually declined from $2 ε$ to 0. To change the interaction between the droplet and substrate, a coupling parameter $(λ)$ is employed and tuned from 1 to 0 in 20 steps to calculate the free energy of adhesion.¹⁶ $λ = 1$ is the state that the substrate is fully wetted whereas $λ = 0$ indicates the state that the polymer adhesive droplet has no interaction with the substrate. The free energy is obtained from the free energy difference between the initial and the perturbed state and can be illustrated as the following equation

\begin{matrix} Δ_{0}^{1} A = - k_{B} T \sum_{i = 0}^{n - 1} \ln 〈 exp (\frac{U_{(λ_{i + 1})} - U_{(λ_{i})}}{k_{B} T}) 〉 \end{matrix}

(7)

where

n

is 20 in this calculation.

To further explore the influence of temperature, we divide adhesion free energy into entropic $Δ S$ and energetic $Δ U$ contributions according to the thermodynamic formula $Δ A = Δ U - T Δ S$ , where $Δ U$ is the sum of internal energy calculated by accumulating the potential energy change between the polymer adhesive droplet and substrate during the FEP process.¹³ Hence, the change in entropy, $Δ S$ , can be obtained from the thermodynamic formula. Previous studies have explained higher free energy enhances the wettability of droplets while higher entropy reduces it.^37,38 As can be seen from Figure 5, increasing the bending stiffness will largely reduce the internal energy, then the free energy between droplet and substrate will be reduced, therefore the wettability of polymer adhesives decreases. As the temperature increases, the system possesses more internal energy which is beneficial to wettability while entropy is harmful to it by comparing B10.0-T1.0 and B10.0-T1.5, which is similar to the result from previous work.³⁸

Figure 5.

Distributions of free energy (pure orange), internal energy (green with dense oblique lines), and entropy (purple with sparse oblique lines) of the wetting systems.

Generally, the calculation of adhesion energy through PMF pulling tests gives the essential knowledge about the magnitude of adhesion at the interface between droplets and substrates. In particular, for polymer adhesive droplets, since polymer chains could untangle, slip, or deform under external influences, the adhesion behavior could be dependent on the loading mode.³⁹ To solve the above issues regarding interfacial adhesion behavior, here we implement two different separation modes for characterization of the adhesion energy at the interface: Mode I (under normal force) and Mode II (under shear force). The pulling force is vertical to the substrate for Mode I, and for Mode II we test the shear force along X axis and Y axis to see whether the PMF is sensitive to the direction of shear force or not.

The adhesion energy of the polymer adhesive droplets can be calculated through PMF by dragging the polymer adhesive droplet away from the liquid–solid interface. The PMF method for adhesion energy calculation is realized by steered molecular dynamics (SMD)⁴⁰ according to Jarzynski equality. In the SMD simulations, one end of virtual stiff spring with a spring constant of $k = 50 ε \cdot σ^{- 2}$ is fixed at the centroid of the polymer adhesive droplet and another end is pulled by a constant force with a pulling velocity of $v = σ \cdot τ^{- 1}$ . Previous work has testified that the pulling rate has minimal effect on the results of PMF calculations.⁷ The elastic potential energy produced by the deformation of virtual spring is expressed as follows

\begin{matrix} U_{spring} = \frac{1}{2} k {v t - (\vec{r_{t}} - \vec{r_{0}}) \cdot \vec{n}}^{2} \end{matrix}

where

\vec{n}

is the pulling direction perpendicular (Mode I) or parallel (Mode II) to the substrate,

\vec{r_{t}}

and

\vec{r_{0}}

represent the current and initial position of the centroid of polymer adhesive droplets. Consequently, the total energy consumed to make separation between polymer adhesive droplet and substrate is calculated by the following formula

W = \int_{r_{0}}^{r_{f}} \nabla U_{spring} \cdot d \vec{r}

(9)

where

r_{f}

is the position of droplet’s centroid when separation is happening. Finally, PMF is calculated using the Jarzynski equation

\begin{matrix} PMF = - \frac{1}{k_{B} T} \log 〈 e^{- W / k_{B} T} 〉 \end{matrix}

(10)

where

W

is the sampled work from different trajectories during the pulling process, and

〈 \dots 〉

denotes the ensemble average.

In Figure 6(a), we present a schematic view describing how each of these two modes is realized. The PMF curves have an upward growing trend for both Mode I and Mode II until reaching a constant value in all systems when separation between droplets and substrate occurs. The results also confirm that the maximum value of PMF decreases with the increase of chain bending stiffness at the same temperature, illustrating a decrease of the adhesion energy as the chain bending stiffness increases. In addition, it takes a long distance for PMF curves to reach the constant value. The reason for this phenomenon may be that a large amount of strain energy is stored in the wetted droplet from the wetting process.⁶ Interestingly, the trend for these two modes is basically same by comparing B2.0-T1.0 and B10.0-T1.0, B10.0-T1.5 and B50.0-T1.5. It can be seen that separation occurs fast for droplets with higher bending stiffness, which means for polymer adhesives with undesired wettability will lead to disastrous adhesion. From $PM F_{∥}$ , we find that the adhesion energy is not sensitive to the shear direction, which also testifies the spreading for homogeneous polymer adhesives on the perfect substrate is isotropic. Also, $PM F_{∥}$ is almost threefold larger than $PM F_{⊥}$ , which is due to the continuous interaction between droplets and the substrate during the process of separation.

Figure 6.

Mode I (a) Schematic illustration of a wetted droplet subjected to a normal force. (b) PMF for droplets with different bending stiffness moved away from the initial position in Z direction. (c) Side view of CGMD simulation snapshots during the separation process under normal force. Mode II (a) Schematic illustration of a wetted droplet subjected to a shear force. (b) PMF for droplets with different bending stiffness moved away from initial position in X/Y direction. (c) Top view of CGMD simulation snapshots during the separation process under shear force. Abbreviation: CGMD = coarse-grained molecular dynamics, PMF = potential of mean force.

Conclusions

In summary, indepth understanding of the interaction between polymer adhesive droplets and substrates is of crucial importance for developing high-performance polymer adhesives. To elucidate the influence of microstructure of polymer adhesives on the wettability, we have systematically performed coarse-grained molecular dynamics simulations to study the wettability of polymer adhesive droplets with different bending stiffness. We first determine the glass transition temperature, then we conduct a series of calculations and find out that polymer adhesive droplets with higher bending stiffness exhibit weaker wettability. The spreading rate of precursor foot with higher bending stiffness is relatively low at the interface, which leads to a low spreading rate of droplet body before the droplet reaches a steady state. The contact angle at equilibrium status increases as the bending stiffness increases, therefore the wettability decreases. The FEP analysis confirms that the wettability decreases as the bending stiffness increases which coincides with the dynamic analysis. In addition, a good wetting process gives a tough adhesion at the interface from the PMF analysis. These results which upon experimental validation should provide insights for optimal designing of adhesives or hydrogels.

Supplemental Material

sj-pdf-1-hip-10.1177_09540083211035016 – Supplemental Material for Wetting characteristics of polymer adhesives with different chain bending stiffness

Supplemental Material, sj-pdf-1-hip-10.1177_09540083211035016 for Wetting characteristics of polymer adhesives with different chain bending stiffness by Wenhao Sha, Jimin Fu and Fenglin Guo in High Performance Polymers

Footnotes

Acknowledgments

We are grateful to Mr Anran Wei for helpful discussions. The computations in this work were run on the π 2.0 cluster supported by the Center for High Performance Computing at Shanghai Jiao Tong University.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Grant No.11972226).

ORCID iD

Wenhao Sha

Supplemental material

Supplemental material for this article is available online.

Appendix

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