Letter Cite This: ACS Macro Lett. 2018, 7, 116−121
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Surface Stress and Surface Tension in Polymeric Networks Heyi Liang, Zhen Cao, Zilu Wang, and Andrey V. Dobrynin* Department of Polymer Science, University of Akron, Akron, Ohio 44325, United States S Supporting Information *
ABSTRACT: Understanding of how surface properties could change upon deformation is of paramount importance for controlling adhesion, friction, and lubrication of soft polymeric materials (i.e., networks and gels). Here, we use a combination of the theoretical calculations and coarse-grained molecular dynamics simulations to study surface stress dependence on deformation in films made of soft and rigid polymeric networks. Simulations have shown that films of polymeric networks could demonstrate surface properties of both polymer melts and elastic solids depending on their deformation. In particular, at small film deformations the film surface stress ϒ is equal to the surface tension obtained at zero film strains, γ0, and surface properties of networks are similar to those of polymer melts. The surface stress begins to show a strain dependence when the film deformation ratio λ approaches its maximum possible value λmax corresponding to fully stretched network strands without bond deformations. In the entire film deformation range the normalized surface stress ϒ(λ)/γ0 is a universal function of the ratio λ/λmax. Analysis of the simulation data at large film deformations points out that the significant increase in the surface stress can be ascribed to the onset of the bond deformation. In this deformation regime network films behave as elastic solids. required to stretch a surface, and surface free energy γ. For liquids, ∂γ/∂a = 0, the surface stress ϒ and the surface free energy γ are the same. For elastic solids, the surface stress could be larger or smaller than γ.23 While the majority of the research over the years has been focused on understanding the nature of the surface stress and surface elastic constants for solid materials (see for review refs 2, 5, and 16), the recent studies of deformation of thin polymeric films and soft substrates have highlighted the importance of the surface elastic response in interfacial mechanics of polymeric networks.10,13,17,24−27 In polymeric systems this is particularly important since polymer networks and gels are viscoelastic materials that could demonstrate properties of both liquids and solids depending on the time and length scales. Therefore, in such systems it should be possible to establish how the surface stress and surface free energy change with the cross-linking density to elucidate crossover between liquid-like and solid-like behavior in different film deformation regimes. To elucidate interfacial properties of polymeric systems under the applied stress we have developed a model of deformation of thin films of polymeric networks and use it to analyze results of coarse-grained molecular dynamics simulations of the network films to obtain dependence of surface stress on film deformation. We begin our discussion with derivation of a model of deformation of thin films made of polymeric networks. We consider a uniaxial compression of an incompressible polymeric film with initial area A0 and thickness h0 made by cross-linking
M
aterials’ surface properties play an important role in controlling mechanics of deformation of nano- and microscale size objects.1−13 Their macroscopic manifestations are seen in the size dependence of the apparent elastic constants (shear or Young’s modulus) obtained from analyzing data for deformation of nanorods, thin plates, and films14−21 and for wrinkling of thin polymeric films at the surface of a liquid.7,10 The physical origin of this dependence is an increase of the surface area upon an object deformation resulting in an increase of the surface contribution to the total free energy of the system. In materials, creation of a new surface area could either proceed by increasing the number of molecules (atoms) in the surface layer by keeping the area, a, per molecule constant or by keeping the number of molecules constant and increasing the area per molecule. The first case corresponds to liquids where redistribution of molecules between bulk and interface can easily occur. Since the structure of the surface layer does not change, the surface free energy per unit area γ remains constant. The second case describes deformation of the elastic solids where creation of a new area requires increase of interatom spacing. This deformation leads to changes in the structure of the surface layer manifested in nonzero values of ∂γ/∂a ≠ 0. Thus, in a general case two processes could happen simultaneously, and the reversible work associated with creation of a new surface area is determined by the surface stress ϒ=γ+a
∂γ ∂a
(1)
which accounts for variations of both area per molecule (or atom) and number of molecules. This equation is a simplified version of the Shuttleworth equation22 that relates a tensor of the surface stress, describing in-plane forces per unit length © XXXX American Chemical Society
Received: October 13, 2017 Accepted: January 2, 2018
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DOI: 10.1021/acsmacrolett.7b00812 ACS Macro Lett. 2018, 7, 116−121
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Figure 1. Illustration of the uniaxial compression (biaxial extension) of a section of polymeric film. The film is periodic in x and y directions.
To test the applicability of eq 5 to deformation of the network films we performed coarse-grained molecular dynamics simulations of biaxial film stretching using the LAMMPS simulation package.32 Polymer chains making up a network are modeled as bead−spring chains composed of N = 32 beads with diameter σ interacting through truncated shifted LennardJones (LJ) potential. The bonds connecting monomers into chains and cross-linking bonds connecting chains together are described by the combination of the FENE and truncated shifted LJ potentials.33 Network films are prepared by randomly cross-linking the films of polymer melts with FENE bonds. To control the film bulk modulus we have changed the number of cross-links per chain, ncr. The simulation details are summarized in the Supporting Information. A macroscopic approach to calculation of the bulk and surface properties of a film is based on analysis of the stress− strain curves of a film deformation. In simulations the corresponding value of the true stress, σfilm, obtained from biaxial film stretching is calculated using components of the pressure tensor Pij as follows
polymer chains. Under uniaxial compression, as shown in Figure 1, the thickness of the film decreases by a factor λ < 1, h = h0λ, and the area of the film increases as A0/λ. Note that for incompressible materials, like polymeric networks, the uniaxial compression can be directly related to a biaxial film extension.28 The total Helmholtz free energy of the deformed film with surface free energy per unit area of the film γ(λ) is a sum of the surface free energy and elastic energy contributions F (λ ) ≅
−1 ⎡ I (λ ) ⎛ 2γ(λ)A 0 β I (λ ) ⎞ ⎤ + GA 0h0⎢ 1 + β −1⎜1 − 1 ⎟ ⎥ ⎝ 3 ⎠ ⎥⎦ λ ⎢⎣ 6
(2)
where G is the structural shear modulus; I1(λ) = λ + 2/λ is the first deformation invariant; and β = ⟨R2in⟩/R2max is the network strand extension ratio defined as a ratio of the mean square end-to-end distance of the network strands in the as-prepared state ⟨R2in⟩ to the square of the size of the fully extended strands R2max (see for details refs 29−31). The true stress, σfilm(λ), generated in a film upon compression is equal to 2
⎞ ∂F(λ) 2 ⎛ ∂γ(λ) ⎜λ = − γ(λ)⎟ ⎠ ∂λ λh0 ⎝ ∂λ −2 ⎤ ⎡ ⎛ βI (λ) ⎞ G + (λ 2 − λ−1)⎢1 + 2⎜1 − 1 ⎟ ⎥ ⎝ 3 ⎢⎣ 3 ⎠ ⎥⎦
σfilm(λ) = V 0−1λ
|σfilm| =
∂γ(λ) ∂λ
⎤ ⎡ 1 ⎢⎣Pzz(z) − 2 (Pxx(z) + Pyy(z))⎥⎦dz z
Lz
∫−L
(6)
where h = λh0 is the thickness of deformed film and integration is performed over a simulation box with size 2Lz along the zdirection. Figure 2 illustrates a typical pressure difference and monomer density profile across thickness of the deformed film.
(3)
Equation 3 generalizes the expression for the stress in networks undergoing uniaxial deformations29−31 to the case of network films with finite thickness. The first term in the rhs of this equation is associated with surface contribution to the film stress. Therefore, we can define a surface stress due to deformation of the surface layer of the network film as ϒ(λ) = γ(λ) − λ
1 h
(4)
This equation is an analogue of the Shuttleworth’s equation modified for deformation of films of polymeric networks (incompressible elastic materials with Poisson ratio equal to 0.5). The second term in the rhs of eq 3 corresponds to the bulk stress. For the data analysis it is convenient to rewrite eq 3 in the following form 2ϒ(λ) |σfilm(λ)| = + |σbulk(λ)| λh 0
Figure 2. Monomer density profile (blue solid line) and pressure 1 difference profile,PN (z) − PT(z) = Pzz(z) − 2 (Pxx(z) + Pyy(z)) (red solid line), across film thickness for deformed film (ε = 0.093) with average number of cross-links per chain ncr = 13.54 and initial thickness h0 = 79.6σ. The red vertical dash lines show positions of the peaks in the pressure difference profile. These lines are considered as boundaries for calculation of the film thickness. The average value of the bulk stress is calculated by averaging of the pressure difference between two green vertical dashed lines.
(5)
where |σbulk(λ)| is the absolute value of the bulk stress which in the framework of our network deformation model is given by the second term in the rhs of eq 3. It follows from this equation that as the thickness of the film h0 → ∞ the film stress approaches the bulk stress contribution. 117
DOI: 10.1021/acsmacrolett.7b00812 ACS Macro Lett. 2018, 7, 116−121
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obtain the film apparent modulus, Gapp = G0 + 4γ2/3h0, and surface tension at zero strain, γ0. These values for apparent shear modulus, plotted as a function of 1/h0 in Figure 4, do not
As one can see from this figure, changes in the monomer density and pressure difference occur within a surface layer of the finite thickness. In a macroscopic approach the properties of the surface layer are assigned to a dividing surface separating macroscopic (bulk) phases. The location of this surface could be either determined from the monomer density or from the pressure difference profile.34 Here we use a distance between two peaks in the pressure difference profile (see Figure 2) to obtain the film thickness h (or h0 for undeformed films). Discussion of how selection of the film boundary could influence the results is presented below with corresponding plots shown in the Supporting Information. In the limit of small film deformations, ε = 1 − λ ≪ 1, it is possible to approximate the surface free energy by a quadratic function of strain, γ ≈ γ0 + γ2ε2 and rewrite eq 5 as follows λh0|σfilm(λ)| ⎛ 4 γ2 ⎞ 3h0ε ≈ ⎜G0 + + γ0 ⎟ 2 3 h0 ⎠ 2 ⎝
(7) Figure 4. Dependence of the apparent shear modulus in the films of polymeric networks on the inverse film thickness for network films with average number of cross-links per chain ncr = 6.25 (purple circles), 10.42 (blue triangles), 13.54 (green left triangles), 16.67 (yellow right triangles), and 19.79 (red squares). Dashed lines show average values of the apparent modulus.
G
where G0 = 3 [1 + 2(1 − β)−2 ] is the shear modulus of the film at small deformations, λ → 1. For soft networks of flexible chains, for which the extension ratio β ≪ 1, the shear modulus at small deformations G0 is equal to the network structural modulus, G. By plotting λh0|σfilm(λ)|/2 as a function of 3h0ε/2, we can obtain apparent shear modulus Gapp = G0 + 4γ2/3h0 from the slope and the surface tension γ0 from the intercept. We refer to γ0 as a surface tension to highlight its relation with that of liquid interfaces for which there is no strain dependence of surface properties in the entire interval of film deformations. It is important to point out that rearrangement of terms in the form of eq 7 makes results for Gapp and γ0 to be independent of the choice of boundaries for calculation of the film thickness. This is due to the fact that the value λh0|σfilm(λ)| is equal to the integral across the simulation box (see eq 6), and εh0 = −Δh is a change in the film thickness upon deformation which does not depend on a particular selection of the film boundary as long as the film deforms affinely. Thus, the particular value of the initial film thickness h0 is only required for separation of the bulk and surface contributions to the apparent shear modulus, Gapp. Figure 3 presents our results for dependence of the film’s true stress on deformation. By fitting the data sets to eq 7, we
show any dependence on the film thickness. This means that the value of the parameter γ2 in the expansion of the surface free energy as a function of the film strain is equal to zero with simulation accuracy. Therefore, for all our networks in the limit of small deformations the surface free energy is strain independent, and surface stress is equal to the surface tension, ϒ = γ = γ0 (see eq 4). This result is not sensitive to selection of the interface location. For example, moving a dividing surface into equimolar location34 does not change the value of the apparent modulus and its dependence on h0 as shown in Supporting Information. We can directly obtain a value of the surface stress ϒ from integration of the pressure difference (see Figure 2) across the simulation box in accordance with eqs 5 and 6 ϒ=
1⎡ ⎢ 2⎣
⎤ ⎡ 1 ⎢⎣Pzz(z) − 2 (Pxx(z) + Pyy(z))⎥⎦dz z ⎤ − |σbulk|h⎥ ⎦ +Lz
∫−L
(8)
The factor 1/2 in eq 8 accounts for two film surfaces. The last term in the square brackets in eq 8 eliminates the bulk stress contribution arising from stretching of the network strands. It immediately follows from eq 8 that the final result for surface stress obtained in the framework of this approach will depend on selection of the film boundary used for calculation of a film thickness h. Figure 5 shows calculation results for which the peak positions in the pressure difference are used as the boundaries of the film bulk (red vertical dashed lines in Figure 2). These data (see Figure 5) confirm that for polymeric networks contribution of the strand deformation into the surface properties of the network films is negligible, and surface stress is equal to the surface tension, ϒ = γ0. However, a shift in the interface (boundary) location by δh will lead to a correction to the surface stress, δϒ = |σbulk|δh. This correction could be positive or negative depending on the interface shift with respect to its location determined from the
Figure 3. Dependence of the reduced true stress on normalized strain in network films with average numbers of cross-links per chain ncr and initial film thicknesses h0: 6.25 and 101.5 σ (purple circles); 10.42 and 100.5 σ (blue triangles); 13.54 and 99.8 σ (green left triangles); 16.67 and 99.1 σ (yellow right triangles); and 19.79 and 98.5 σ (red squares). 118
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deformation range the surface properties of the network films are similar to those of polymer melts. However, one should expect that with increasing the strands’ extension the transversal to the direction of strand deformation fluctuations should be hindered. This in turn should result in suppression of the ability of the network strands to redistribute monomers to cover the surface area increase without additional strand deformations. Such a strand ability will be significantly diminished when the film deformation ratio λ approaches the maximum possible deformation ratio of the network strands λmax without bond deformations. Therefore, at large film deformations such as λ > λmax the creation of a new surface area will require deformation of individual bonds since strands have exhausted the ability for redistribution of monomers just through changes in their conformations. This corresponds to a crossover to a solid-like behavior of polymer networks. To confirm this qualitative picture we have performed simulations of the large film deformations. Figure 6a shows results for the normalized surface stress, ϒ(ε)/γ0, obtained from the integration method (see eq 8) for network films undergoing biaxial stretching covering both linear and nonlinear deformation regimes. As follows from this figure the data for strongly cross-linked networks corresponding to the large number of cross-links per chain (ncr = 16.7) begin to deviate from the plateau value first. Such networks have the shortest network strands between cross-links and therefore reach the fully extended strand limit at smaller values of the strains, ε ≈ εmax = 1 − λmax. To further highlight the importance of the λmax in Figure 6b we have rescaled the film deformation ratio λ by its value λmax. The maximum elongation ratio of network strands, λmax, is calculated from the strand extension ratio β as λmax = β−1/2.29,30 The values of the parameter β are obtained from fitting the stress−deformation curve calculated from the bulk part of the film stress (see Supporting Information). All data sets have collapsed into one universal curve as a function of ratio λ/λmax. It follows from this figure that as this ratio approaches 0.9 the ability of network strands to redistribute monomers to cover the increase of the surface area diminishes, forcing deformation of the network strands belonging to the interfacial region and leading to generation of the additional stress at the interface. The increase of the surface stress correlates with the increase in the bond elastic energy (see inset
Figure 5. Dependence of the surface stress on strain for films of polymer networks with average numbers of cross-links per chain ncr and initial film thicknesses h0: 6.25 and 101.5 σ (purple circles); 10.42 and 100.5 σ (blue triangles); 13.54 and 99.8 σ (green left triangles); 16.67 and 99.1 σ (yellow right triangles); and 19.79 and 98.5 σ (red squares). The dashed lines correspond to the surface tension γ0 obtained from the intercept in Figure 3.
peak positions. For example, shifting the film boundary to equimolar location determined from the film density profile (see blue line in Figure 2) will lead to a decrease of the surface stress by the amount δϒ ≈ −|σbulk|0.8σ. This is due to the fact that shifting of the interface results in an increase in the number of stress supporting strands in the bulk of the film at the expense of those at the interface region. This effect becomes more pronounced with increasing magnitude of the bulk stress as illustrated in the Supporting Information. Therefore, the presence of the nonzero stress in a film bulk eliminates the freedom in selection of the interface (boundary) location. A true location of the interface which provides identical results for macroscopic and microscopic analysis of the film deformation data is one obtained from locations of the two peaks in the pressure difference across the film thickness. The behavior of surface properties of network films at small deformations (see Figures 4 and 5) points out that network strands could redistribute monomers from outside the interface region without additional strand deformation to keep the number of monomers per unit area unchanged. In this
Figure 6. (a) Dependence of the normalized surface stress on strain for large deformations. (b) Universal dependence of the normalized surface stress on the ratio λ/λmax. Inset: relative change of the normalized surface stress as a function of the relative change of the bond energy. Eb(λ) is the bond energy in the film at different deformation ratios, and Eb,0 is the bond energy in the undeformed film. Data are shown for films of polymer networks with average numbers of cross-links per chain ncr and initial film thicknesses h0: 6.25 and 60.3σ (purple circles); 10.42 and 59.6σ (blue triangles); 13.54 and 59.2σ (green left triangles); 16.67 and 58.9σ (yellow right triangles). 119
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ACS Macro Letters in Figure 6b), pointing out that the changes in the bond length provide dominant contribution to significant variations in the surface stress. Our results for equality of surface stress and surface tension in a wide range of network deformations are consistent with the molecular dynamics simulations35−40 and experimental data41 for contact mechanics of nano- and microscale particles with soft elastic substrates. However, our finding appears to be in contradiction with the conclusions reached in refs 13, 27, and 42. In these papers the values of the surface stress were calculated from balance of the force triangle at the triple-phase contact line. In such calculations, one obtains a value of the line force, assuring equilibrium of the contact line. Note that this force is not necessarily associated with the surface contribution alone. For elastic materials there is an additional contribution coming from stresses generated outside the surface or interface region. The universality of the surface deformation profile near the triple phase contact line for droplets of different sizes cannot be used as a justification of the pure interface origin of the line forces. The universal surface profile points out that close to the contact line the interface deformation is a solution of the 2D rather than 3D elastocapillary problem. Therefore, a value of the surface stress obtained from the force balance analysis might not provide a true surface stress in the Shuttleworth sense22 since it is “contaminated” by a bulk stress contribution. A better name for a surface stress determined from analysis of the substrate deformation in a complex geometry of the contact should be an apparent surface stress. Furthermore, our results for soft film deformations disagree with conclusions reached in a recently published paper43 which reports that there is a strong strain effect on the surface stress even in the case of soft silicone gels with Young’s modulus E = 5.6 kPa. In particular, it was reported that there is about a 50% increase in the surface stress at strain ε ≈ 0.1. Our simulations show no effect of the film deformation on magnitude of the surface stress in this deformation interval (see Figures 5 and 6). In conclusion, our coarse-grained molecular dynamics simulations have shown that in rigid and soft polymeric networks the surface stress is a universal function of the ratio λ/ λmax and does not depend on the network deformations at small deformations. The direct experimental verification of our results for surface stress dependence on the film deformation could be done by implementing the bubble blowing technique developed by McKenna’s group for analysis of the viscoelastic response of the ultrathin polymeric films.24 We hope our work will serve as inspiration for such experiments.
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Andrey V. Dobrynin: 0000-0002-6484-7409 Funding
National Science Foundation (DMR-1624569). Notes
The authors declare no competing financial interest.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsmacrolett.7b00812. Simulation details: system set up, film deformation simulations (PDF)
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REFERENCES
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Heyi Liang: 0000-0002-8308-3547 Zhen Cao: 0000-0001-5499-3130 Zilu Wang: 0000-0002-5957-8064 120
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