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Interface Components: Nanoparticles, Colloids, Emulsions, Surfactants, Proteins, Polymers
Surface Stresses and a Force Balance at a Contact Line Heyi Liang, Zhen Cao, Zilu Wang, and Andrey V. Dobrynin Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b01680 • Publication Date (Web): 30 May 2018 Downloaded from http://pubs.acs.org on May 30, 2018
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Surface Stresses and a Force Balance at a Contact Line Heyi Liang, Zhen Cao, Zilu Wang, and Andrey V. Dobrynin* Department of Polymer Science, University of Akron, Akron, Ohio 44325, USA
ABSRACT: Results of the coarse-grained molecular dynamics simulations are used to show that the force balance analysis at the triple phase contact line has to include a quartet of forces – three surface tensions (surface free energies) and an elastic force per unit length. In the case of the contact line formed by a droplet on an elastic substrate an elastic force is due to substrate deformation generated by formation of the wetting ridge. The magnitude of this force fel is proportional to the product of the ridge height h and substrate shear modulus G. Similar elastic line force should be included in the force analysis at the triple phase contact line of a solid particle in contact with an elastic substrate. For this contact problem elastic force obtained from contact angles and surface tensions is a sum of the elastic forces acting from the side of a solid particle and an elastic substrate. By considering only three line forces acting at the triple phase contact line one implicitly accounts the bulk stress contribution as a part of the resultant surface stresses. This “contamination” of the surface properties by a bulk contribution could lead to unphysically large values of the surface stresses in soft materials.
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INTRODUCTION The equilibrium surface properties of the solids and liquids can be determined from the line force equilibrium at the triple phase contact line.1-4 In the case of the solid substrates the equilibrium contact angle θ1 of a droplet (see Figure 1a) satisfies the Young’s equation1 which represents in plane balance of the line forces associated with the surface free energies per unit area (surface tensions) of the liquid, γ L , substrate, γ S , and substrate/ liquid γ SL interfaces. For a droplet placed on a liquid substrate at equilibrium a vector sum of the corresponding line forces must be equal to zero. This leads to a well-known Neumann’s triangle conditions2 corresponding to balance of the vertical and horizontal components of the surface tensions at a triple phase contact line (see Figure 1b). It is worth pointing out that the Young’s and Neumann’s equations are simple visualizations of the conditions determining an equilibrium shape of a droplet corresponding to the minimum of the system free energy attained at a constant droplet volume.36
Figure 1: Line forces acting at a triple phase contact line. (a) Droplet on solid substrate. Force balance conditions: (Young’s law). Note that the vertical component of a droplet surface tension γ sin should be balanced by the elastic force fel (not shown) due to substrate deformation. See text for details. (b) Droplet on liquid substrate. Neumann’s triangle conditions: and
.
The situation becomes more complex when a droplet is in contact with a deformable elastic substrate when both capillary and elastic forces determine the equilibrium shapes of a droplet and a substrate.7-28 It was argued recently that a substrate deformation could be taken into account by substituting the surface stresses instead of surface tensions (surface free energies) to account for the local balance of capillary and elastic forces at the triple phase contact line17,21,27 (see Figure 2a). The surface stress balance approach is attractively simple since it requires only measurements of the corresponding contact angles. The justification for this approach provided in refs17,
21
is based on the following
arguments: (i) the universality of the shape of a wetting ridge close to the triple phase contact line and its similarity to that of a liquid substrate; and (ii) at short length scales (smaller than the elastocapillary 2 ACS Paragon Plus Environment
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length) the surface forces dominate bulk elasticity controlling a local substrate deformation. Extension of this approach was also used to calculate surface modulus of soft gels.27,29
Figure 2: A droplet on soft (gel-like) substrate. (a) Line force balance conditions for surface stresses: and . (b) Line force balance conditions
with
elastic
force:
and .
Unfortunately, these arguments cannot be used as a justification of the pure surface origin of the surface stresses obtained from a force balance. The universality of the ridge profile means that close to the triple phase contact line a contact line curvature can be neglected and problem of the ridge shape becomes a solution of the 2D elastocapillarity problem. It is important to point out that a ridge will be present even if one considers a deformation of an elastic film by a line force without accounting for capillary forces acting at the film surface. The solution of this problem for a ridge height as a function of the magnitude of the applied force and substrate modulus G was used to explain the force balance perpendicular to the solid surface direction (see Figure 1a). In particular, a line force γ L sin θ1 induces a stress field manifested in deformation of the substrate in the vicinity of the triple line, producing a wetting ridge with height h on the order of h ∝ γ L / G .9-12 Thus, in a general case of an elastic substrate the force balance analysis at the triple phase contact line should include a quartet of forces - three surface tensions (free energies) and an elastic force, acting at a contact line from the side of the elastic substrate (see Figure 2b). One can think of sum of surfaces tensions as a net line force acting at the triple phase contact line which action is balanced by the elastic force. By considering only three surface stresses, one implicitly accounts the elastic bulk force contribution as a part of the resultant surface stresses. RESULTS AND DISCUSSION To prove that a quartet of forces should be used to determine equilibrium at triple phase contact line we present analysis of the simulation data of the elastic (gel-like) substrate deformation induced by 3 ACS Paragon Plus Environment
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polymeric droplets and solid particles of different sizes. In simulations the corresponding surface tensions (free energies) are determined from additional set of simulations of the flat surfaces and interfaces and therefore can be independently used to confirm a force balance condition. Furthermore, in the case of substrate deformation by a droplet we verify a relationship between an elastic force, a ridge height and a substrate shear modulus.
Figure 3: Simulation snapshots of (a) a polymeric droplet, (b) a solid spherical nanoparticle and (c) a solid cylindrical nanoparticle in contact with a soft elastic (gel-like) substrate. In our force balance analysis we used results of the coarse-grained molecular dynamics simulations of polymeric droplets24 and solid nanoparticles30 in contact with soft elastic substrates (see Figure 3). Polymeric droplets with initial radii between Rd = 10.92σ and 34.62σ were made of beadspring chains31 of the Lennard-Jones (LJ) beads with diameter σ and the degree of polymerization N = 32. The connectivity of beads into polymer chains was maintained by the FENE bonds. Solid cylindrical and spherical nanoparticles with radii between Rp = 9.8 σ and 31.4 σ consisted of the LJ-beads with diameter
σ arranged in a hexagonal closed-packed (HCP) lattice and connected by the FENE bonds. The parameters of the FENE bond potential are selected in such a way to preclude particle deformations effectively making them rigid. Elastic (gel-like) substrates were made by randomly cross-linking beadspring chains by the FENE bonds. The number of cross-links per chain was varied to cover a range of the substrate shear modulus between 0.024 kBT/σ3 and 0.833 kBT/σ3 (where kB is the Boltzmann constant and T is the absolute temperature). The pairwise interactions between beads belonging to polymer chains, nanoparticle and elastic substrates were modeled by the truncated-shifted LJ-potentials which strength is characterized by the LJ-interaction parameter εLJ. Simulation details and explicit values of the interaction parameter can be found in the original publications24,30 and are summarized in the Supporting Information.
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We first discuss the effect of the substrate deformation induced by a polymeric droplet. The analysis of the droplet and substrate profiles indicates that close to the triple phase line profiles produced by droplets of different sizes can be superimposed as shown in Figures 4. It also follows from these figures that the shape profiles for small droplets with sizes Rd=10.92σ and 15.81σ begin to deviate from
Figure 4: Superimposed droplet and surface profiles at triple phase contact line for droplets of different initial sizes Rd = 10.92σ (black squares), Rd = 15.81σ (red circles), Rd = 19.78σ (blue triangles), Rd = 25.74σ (green triangles), Rd = 30.64σ (pink triangles), and Rd = 34.62σ (dark blue triangles) having value of the LJ-interaction parameter between polymer-substrate pairs εLJ = 1.20kBT on substrates with values of shear modulus: G = 0.0kBT/σ3 (a), G = 0.162kBT/σ3 (b), and G = 0.833kBT/σ3 (c). The red lines show direction of the elastic force.
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Figure 5: (a) Dependence of the magnitude of the elastic force on the substrate shear modulus. The elastic force is obtained from analysis of the force balance at the triple phase contact line for polymeric droplets with different values of the LJ -interaction parameters for droplet-substrate pairs: εLJ=1.2 kBT (blue triangles), εLJ=0.75 kBT (red circles), and εLJ=0.4 kBT (black squares). The shaded areas of corresponding colors demonstrate uncertainties in calculations of the elastic force. (b) Vertical z-component of the elastic force as a function of Gh for largest droplets with initial size Rd = 34.62 σ. (c) Dependence of the normalized z-component of the elastic force on the dimensionless parameter
. (d) Dependence of the normalized difference between surface
stress and surface tension of the substrate
on the dimensionless parameter
.
The values of the surface stress for these plots were obtained from Neumann’s triangles conditions shown in Figure 2a. In panels (b-d) notations are the same as in panel (a). the universal shape curve indicating a finite size effect. Therefore, in our calculations of the mutual orientation of the line forces we use the profiles produced by the largest droplets. For a liquid substrate (G = 0.0 kBT/σ3) the corresponding surface tensions balance to zero. In the case of the elastic substrates we first establish that the local substrate deformations induced by droplets are not sufficient to cause 6 ACS Paragon Plus Environment
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renormalization of the substrate surface tension. In particular, our analysis shows that the local strain near the contact line is ε < 0.04 for all the simulations performed. For such range of the substrate deformations the surface stress is equal to surface tension as was shown in ref 32, where we have studied effect of the elastic film deformation on its surface properties. Therefore, for the force balance at the triple phase contact line we can use surface tensions obtained for undeformed substrates. In this case, a balance of a net line tension force requires an elastic force which direction is shown by the red lines in the middle and right panels. Furthermore, with increasing substrate shear modulus the direction of the elastic force moves closer to the vertical direction as one would expect based on the force balance for solid substrates. The magnitude of the elastic force monotonically increases with substrate rigidity (see Figure 5a). Figure 5b confirms correlation between the vertical component (along z-axis) of the elastic force , , substrate shear modulus G and ridge height h which follows from the general expression for the pulling force
f el , z ≈ 2π Gh / ln ( H / rcut ) (where H is substrate thickness and rcut ≈ ξ is a cutoff distance on the order of the interface thickness).9-12 To make connections with experiments21 it is convenient to replot the data in Figures 5a and b in terms of dimensionless parameters. Figure 5c shows these data for a normalized z-component of the elastic force f el , z / γ L sin θ1 as a function of the dimensionless parameter Gh / γ L . The parameter
Gh / γ L is proportional to a ratio of the restoring elastic bulk force acting per unit length of the ridge with height h to the surface tension of the droplet. In the limit of solid substrates this ratio is on the order of unity.9-12 In recent experiments21 on silicon-glycerol systems, glycerol droplets with surface tension
γ L = 46 ± 4 mN / m were placed on silicon substrates with shear modulus G ≈ 1.0 kPa and the ridge height h was measured to be on the order of 8 µm. This gives the value of the parameter Gh / γ L to be equal to 0.17. As follows from Figure 5c this range of parameters is covered by our simulations and analysis of the experimental data in ref 21 should explicitly take into account an elastic bulk force acting at the triple phase contact line. In conclusion of this section, we would like to point out that presented above analysis proves that there is an elastic force generated by the substrate deformation induced by the ridge formation. Therefore, by implicitly accounting this elastic force as a part of the surface stresses (see Figure 2a) one can obtain unphysically large values of the surface stresses in experiments21 and in simulations as illustrated in Figure 5d. In particular, it was reported in ref 21 that the surface stress for silicon gels calculated from the force balance (see Figure 2a) is equal to 31 ± 5 and 28 ± 2 mN/m for silicon-glycerol and siliconfluorinated oil systems respectively which is significantly higher than that for liquid silicon, 21 mN/m. Furthermore, these values of the surface stresses are inconsistent with the small substrate deformations, ε 7 ACS Paragon Plus Environment
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~ 0.1, estimated from the surface profile at the triple phase contact line. For such small deformations the network strands of the substrates with the shear modulus G ≈ 1.0 kPa remain close to their ideal (Gaussian) conformations and surface properties of the weakly deformed networks are similar to those of a polymer melt (see ref 32). Similar conclusions can be reached about the surface stress values shown in Figure 5d. These data are inconsistent with those obtained in molecular dynamics simulations of the network films of similar rigidity undergoing a biaxial stretching.32
Figure 6: (a) Line forces acting at a triple phase contact line formed by a solid spherical (or cylindrical) nanoparticle in contact with a soft gel-like substrate. (b) Expression of the force balance in terms of the work of adhesion. The identification of different contributions to the elastic force becomes difficult when one considers a force balance at a triple phase contact line when two phases are elastic solids. This is the case of the solid nanoparticles in contact with an elastic substrate. In this case due to the fact that the direction of line force corresponding to surface energy of the particle, , and surface tension of the particle/substrate interface, , are aligned with the tangent line to the particle surface passing through the triple phase contact point, thus the force balance can be reduced to analysis of only three forces (see Figures 6). As shown in the right panel (see Figure 6b) these forces can be expressed as substrate surface tension γS, difference between work of adhesion ( ) and substrate surface tension, W -
γS, and elastic force, fel. It is worth pointing out that the elastic force is a sum of the elastic forces generated by deformation of the substrate and solid particle caused by the capillary forces. Figures 7a and b show results of our calculations of the elastic force as a function of the substrate shear modulus for cylindrical (a) and spherical (b) particles. It follows from these figures that the magnitude of the elastic force increases with increasing the substrate modulus. Also as evident from these figures there is a particle size dependence of the elastic force magnitude. This size effect weakens as a particle radius increases. It should also be noted that even though the magnitudes of the elastic forces appear to be similar for different particle sizes, their orientations could be different (see Figure S1 in SI). 8 ACS Paragon Plus Environment
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Figure 7: Dependence of the magnitude of the elastic force on the substrate shear modulus. The elastic force is obtained from analysis of the force balance at the triple phase contact line for (a) cylindrical and (b) spherical nanoparticles of varied sizes. In panel (a), Rp = 9.8 σ (red squares), 14.3 σ (orange circles), 17.9 σ (green triangles), 23.3 σ (cyan inverted triangles), 27.7 σ (blue left triangles), 31.4 σ (purple right triangles). In panel (b), Rp = 9.6 σ (red squares), 14.8 σ (orange circles), 17.8 σ (green triangles), 22.9 σ (cyan inverted triangles), 27.3 σ (blue left triangles), 31.4 σ (purple right triangles). CONCLUSIONS In conclusion, we have shown that in the case of the elastic substrates deformed by liquid droplets, the contact line force balance analysis should include an elastic force generated by the substrate deformation in the vicinity of the triple phase contact line. By accounting for this force, one eliminates the bulk stress contribution into surface stresses. However, the presence of this force poses a serious obstacle in evaluation of the surface properties of elastic materials. If the elastic force is taken into account and only surface tension of the liquid is known there are not enough equations to independently determine surface tensions and elastic force which magnitude and orientation is unknown (see figure caption to Figure 2). Thus application of the force balance analysis at the triple phase contact line requires independent measurements of the surface properties of the elastic substrate. By considering only three forces, one sidesteps this issue by implicitly accounting the bulk stress as a part of the resultant surface stresses. This bulk stress contribution results in unexpectedly large surface stresses reported for soft materials.21 Note that in soft polymeric networks one can achieve a renormalization of the network surface properties only in the limit of nonlinear substrate deformations.32 It is also important to point out that utilization of the surface stress data for interpretation of the results for the substrate deformation by solid particles33 can lead to a double counting of the substrate deformation effects. This raises a question about usefulness of the force balance approach at a triple phase contact line to determine surface properties of soft elastic materials.
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A direct approach to evaluation of the surface properties of the elastic materials can be based on bubble blowing technique34 developed by the McKenna’s group for analysis of the viscoelastic response of the ultrathin polymeric films. The biaxial deformation of the film allows for a straight forward separation of the bulk and surface stress contributions into the film deformation32. One can also use a macroscopic angle analysis for a drop on elastic substrates to extract information about surface tension and surface stresses35. These direct measurements35 are in agreement with the results of the computer simulations32 for thin film deformation showing that for soft substrates the surface stress is equal to the surface tension for incompressible films with the Poisson ratio 0.5 in a wide deformation range. ASSOCIATED CONTENT Supporting Information Simulation details and data sets used for figures. AUTHOR INFORMATION Corresponding Author *Email:
[email protected] Funding Sources National Science Foundation (DMR-1624569). Notes The authors declare no competing financial interest.
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