Article pubs.acs.org/Langmuir
Shear-Induced Detachment of Polystyrene Beads from SAM-Coated Surfaces Kwun Lun Cho,† Axel Rosenhahn,‡ Richard Thelen,§ Michael Grunze,†,∥ Matthew Lobban,⊥ Markus Leopold Karahka,⊥ and H. Jürgen Kreuzer*,⊥ †
Institute for Functional Interfaces, Karlsruhe Institute of Technology, Kaiserstraße 12, 76131 Karlsruhe, Germany Analytical ChemistryBiointerfaces, Ruhr-University Bochum, 44801 Bochum, Germany § Institute of Microstructure Technology, Karlsruhe Institute of Technology, Hermann-von-Helmholtz-Platz 1, 76344, Eggenstein-Leopoldshafen, Germany ∥ Applied Physical Chemistry, University of Heidelberg, D-69120 Heidelberg, Germany ⊥ Department of Physics and Atmospheric Science, Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada ‡
S Supporting Information *
ABSTRACT: In this work we experimentally and theoretically analyze the detachment of microscopic polystyrene beads from different self-assembled monolayer (SAM) surfaces in a shear flow in order to develop a mechanistic model for the removal of cells from surfaces. The detachment of the beads from the surface is treated as a thermally activated process applying an Arrhenius Ansatz to determine the activation barrier and attempt frequency of the rate determing step in bead removal. The statistical analysis of the experimental shear detachment data obtained in phosphate-buffered saline buffer results in an activation energy around 20 kJ/mol, which is orders of magnitude lower than the adhesion energy measured by atomic force microscopy (AFM). The same order of magnitude for the adhesion energy measured by AFM is derived from ab initio calculations of the van der Waals interaction energy between the polystyrene beads and the SAM-covered gold surface. We conclude that the rate determing step for detachment of the beads is the initiation of rolling on the surface (overcoming static friction) and not physical detachment, i.e., lifting the particle off the surface.
I. INTRODUCTION Adhesion of biological species, biota, ranging from cells to multicellular organisms on surfaces is a critical process in their life cycle and essential for their proliferation and survival. The physical and chemical nature of the interface between biota and the substrate are key to understanding the biological behavior of cells and bacteria on surfaces. However, understanding the detailed mechanisms and kinetics on how cells and organisms settle and detach from surfaces is equally important, and not only for biomedical applications,1 but in particular for the prevention of detrimental and unwanted biofouling in the ambient or marine environment.2−5 Analysis of the kinetic processes leading to attachment and removal of the organism is complicated due to analytical challenges encountered in in vivo and in situ measurements which required novel tracking methods6 to obtain a fourthdimensional (space and time) plot of the exploration and settlement process. Eukaryotic cell migration on surfaces involves both rolling7 and migration by forming and releasing focal adhesion contacts.8 The mechanism by which they can be removed from a surface by shear flow may either involve rolling and then lift-off, or for a spread cell a ”peel-off ” mechanism where a propagating crack is formed at the leading edge exposed to shear.5,9,10 In this work we focus on model experiments designed to facilitate a quantitative understanding of the mechanism(s) by © 2015 American Chemical Society
which cells and unicellular organisms are removed from a surface by shear flow. Central to this work is our discussion about how the energetics and kinetics of detachment are influenced by the magnitude of the adhesion energy of the cell to the substrate. To analyze how a cell is removed from a surface, microfluidic shear force assay measurements have advantages compared to atomic force microscopy,11 quartz crystal microbalance,12,13 and centrifugation.14 The use of microfluidic channels allows for both the imaging of a large population of particles adherent on a substrate and the application of a well-defined laminar shear flow that can be controlled precisely with small incremental increases across several orders of magnitude.5,15 Further,the observation of many species simultaneously simplifies and accelerates a statistical analysis of the detachment experiments. Using the setup previously described by us,5 we investigate the mechanism of detachment of polystyrene particles from self-assembled monolayers (SAMs) in response to different shearing rates. SAMs provide ideal surfaces to model interfacial bonding interactions due to their well-defined surface properties16 and have been used extensively in fundamental adhesion studies as well as for various biomedical applications.17 In our Received: June 29, 2015 Revised: August 22, 2015 Published: September 24, 2015 11105
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optical properties described by the Cauchy model on top of the underlying gold layer using WVASE (Woollam). The values reported are an average of three measurements on different spots on the sample. D. Contact Angle. Sessile water drop contact angle measurements were performed using a G1 goniometer (Krüss, Hamburg, Germany) in ambient conditions (T = 20 °C) . Each measurement reported is an average of five measurements along the sample surface. E. AFM Adhesion Force Measurements. A Bioscope catalyst atomic force microscope (Bruker, Karlsruhe, Germany) mounted on an Nikon Ti-E inverted microscope (Nikon Instruments, Melville, NY, USA) was used for the adhesion measurements between polystyrene spheres and the SAM-modified surfaces. For the experiments a sphere of 4.5 μm was mounted onto a tipless cantilever (spring constant, 0.235 N/m) using a two-component epoxide glue. An optical interferometer was used to confirm the attachment of the sphere onto the cantilever and the lack of glue on the scanning side of the sphere. Prior to force measurements, the samples’ surfaces were cleaned by immersion in ethanol for 10 min followed by rinsing and drying under a stream of nitrogen. The AFM measurements were done as the shear force experiments in PBS buffer solution at pH 7. Buffer was injected into the fluid cell slowly to avoid bubble formation. The fluid was allowed to equilibrate for 5 min under ambient conditions (T = 20 °C) before force measurements. Adhesion force was calculated from the retraction curves. For each sample, at least 20 force−displacement profiles, all at different locations within a 5 × 5 μm2 grid on the surface, were averaged to obtain the adhesion force. Measurements performed in air were conducted using the same procedure but with a stiffer cantilever (0.75 N/m) F. Microfluidic Flow Measurements. The detachment of the polystyrene microspheres was measured by the hydrodynamic shear force technique.21 A Poiseuille model was used to describe the shear stress τ generated on the surfaces of the microfluidic channel in relation to fluid flow Q, fluid viscosity μ, channel height h, and channel width w:22
experiments, after allowing the particles to sediment onto the surfaces, their detachment process was quantified by exposure to shear stresses up to 2700 dyn/cm2. Particle detachment is shown to be dependent on the rate of increase in applied shear stress, similar to surface science thermal desorption experiments in which the activation energy of desorption from surfaces is measured.18,19 The probability of polystyrene particles adherent on the surface as a function of shear stress is analyzed using an Arrhenius Ansatz to determine the activation energy and frequency factor of detachment.
II. EXPERIMENTAL SECTION A. Materials. 1-Hexanethiol (C6-OH), dodecanthiol (C11-CH3), 11-hydroxy-1-undecanthiol (C11-OH), absolute ethanol and pH 7.4 phosphate buffered saline (PBS) was purchased from Sigma-Aldrich. 1-Hexadecanethiol (C16-OH) and 11-(tridecafluorooctyloxy)-1-undecanthiol (FUDT) were purchased from Prochimia Surfaces (Sopot, Poland), and carboxylate functionalized polystyrene microspheres (Polybead carboxylate microspheres, 4.5 μm diameter, 2.5% (w/v) aqueous suspension, 4.99 × 108 particles/mL) was obtained from Polysciences Inc. (Eppelheim, Germany), respectively. All chemicals were used as received unless otherwise stated. Gold substrates were manufactured by George Albert (PVD coatings, Silz, Germany), with Nexterion B glass slides (Schott, Mainz, Germany). Before thermal vapor deposition of polycrystalline gold (30 nm, 99.99% purity), a 5 nm titanium layer was deposited as an adhesion promoter. The deposition was carried out in high vacuum (2 × 10−7 mbar) with a deposition rate of 0.5 nm/s. The roughness of the gold substrates was around 1 nm (RMS) as measured by AFM. The gold substrates were stored under an argon atmosphere until use. B. Preparation of SAMs. Prior to SAM functionalization the gold substrates were placed under UV light (ozone generating 150 W mercury vapor lamp, model TQ150, Heraeus GmbH, Hanau, Germany) for 2 h and subsequently sonicated for 3 min in ethanol to remove organic contaminants. The cleaned gold substrates were immersed in the respective thiol solution (1 mmol) for 24 h to facilitate monolayer assembly. After 24 h, the modified gold substrates were rinsed with ethanol and sonicated in ethanol for 3 min. Subsequently the modified substrates were dried under a nitrogen stream and kept under argon until use. All SAMmodified substrates were characterized using contact angle goniometry and spectral ellipsometry. Table 1 compiles the static contact angles, the measured ellipsometric thickness, and the theoretical thickness expected for a
τ = 6Qμ/(h2w)
Table 1. Static Contact Angle and Ellipsometric Thickness of the SAM-Modified Gold Substratesa surface gold C6-OH C11-OH C16-OH C11-CH3 FUDT
contact angle (deg) 60 35 35 32 105 112
± ± ± ± ± ±
3 3 4 4 5 2
thickness (Å)
theoretical thickness (Å)
± ± ± ± ±
10.1 15.6 21.1 15.6 25.5
4 13 21 15 16
0.4 0.3 0.5 0.8 0.8
(1)
The microfluidic channel assembly (see5 for a detailed description) used consists of a PDMS channel sealed by an upper glass slide and the sample substrate on the bottom. Two holes were drilled into the glass slide for inlet and outlet fluid flow. The PDMS ring was made by cast molding from a polished micromachined brass mold and cured at 65 °C for 8 h to form channels with dimensions approximately 150 μm high, 1500 μm wide, and 2500 μm long. From the channel dimensions and a maximum volumetric flow rate (Q) of 80 mL/min, the maximum Reynolds number is calculated to be 9̃ 00, far below the critical Reynolds number for turbulent flow to begin. Therefore, the flow regime for the entire experiment can be considered laminar. For the current experiment several replicate rings were made and interchanged between samples. The exact channel dimensions used to calculate shear stress and the channel walls were measured before each experiment to ensure accuracy. The microfluidic channel assembly was mounted onto a microscope (Nikon 90i) housed inside an incubator at 37 °C. Fluid flow was supplied by a pressurized flask filled with PBS buffer (pH 7), and the flow rate was regulated by a syringe pump (PI, Physics Instruments GmbH & Co. Karlsruhe, Germany) in incremental steps determined by computer software (Mikromove). Before bead injection in an experiment the channels were flushed with PBS (pH 7) buffer. Prior to the shear flow assay the polystyrene microsphere were washed 3 times. In a typical washing step, the polystyrene beads (35 μL) were dispersed in PBS buffer solution (1 mL) using a vortex mixer and centrifuged (3.5 min, 13000 rpm) to form a white pellet with the supernatant discarded. The particles were then resuspended in PBS buffer (1 mL). A suspension of polystyrene microspheres (1.8 × 107 particles/mL, ∼1000 particles/microscope image) in PBS buffer was injected into the channel. After an incubation period of 5 min to facilitate sedimentation, the syringe pump was activated to induce detachment. The pump speed was increased incrementally at predetermined exponential rates (α) to generate shear stresses from
a
The theoretical thickness represents the thickness expected for a close packed monolayer oriented at 30° to the surface normal.
SAM consisting of alkanethiol molecules in the all-trans conformation and tilted by 30° with respect to the surface normal.20 The theoretical thickness exceeds in all cases the ellipsometric thickness, indicating that the SAMs are partially disordered and the molecules are not homogeneously in an all-trans conformation. C. Ellipsometry. Thickness of the self-assembled monolayer was determined using a fixed angle M-44 spectral ellipsometer (J. A. Woollam Co. Inc., Lincoln, NE, USA) with an operating wavelength between 280 and 800 nm. The film was modeled as a single layer with 11106
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between beads and substrate. The errors shown in Figure 1, calculated as the standard error from three replicate measurements, are representative for all detachment experiments done in this work. As shown in Figure 1, with increasing SAM thickness, an overall reduction in the shear force profile to lower forces was observed, thus indicating a reduced barrier for detachment. Obviously the interaction with the substrate depends on the thickness of the alkane layer, which is indicative of a significant interaction through van der Waals forces. However, there must also be a substantial electrostatic component as inferred from the pH dependence of the detachment curves as shown in Figure 2. As expected, below the pKa of the carboxylate group (pKa ∼ 5.5), the carboxylate group becomes deprotonated, resulting in an increase in negative surface charge and electrostatic interaction, leading to a significant increase in the force required for detachment. We note that besides van der Waals and electrostatic interactions also hydration forces and direct interactions will be part of the total adhesion energy, as discussed in detail in the work by Pertsin and Grunze.23 The largest uncertainty in the data are at the very beginning and the end of the detachment curves, where the first or last beads detach from the surface, respectively. In thermal desorption experiments the leading and trailing edges of the desorption traces are often used to derive the energetic parameters of the desorption process, but this is clearly not feasible in our case here due to the scatter in the data.18,19 Hence, in this work we have to resort to a complete fit of the detachment curves in order to extract the activation energy and preexponential factor of detachment (see Theory).
(2)
Particle detachment was recorded using a 10× objective via digital time lapse microscopy. Particles were counted using computer software (Nikon elements Ar, Nikon), resulting in a plot of the fraction of microspheres still attached versus applied shear stress. A number of different systems, as summarized in Table 2, were studied to establish trends and to identify the effects of film thickness and chemistry.
III. EXPERIMENTAL RESULTS In subsequent text we show representative detachment curves for the series of measurements done on hydroxyl-terminated aliphatic SAMs of different thickness (Figure 1) and for C16-OH as a function of pH (Figure 2) to demonstrate the presence of electrostatic interaction
Table 2. Minimum Shear Stress (τmin), Minimum H Hydrodynamic Force (FH min) and Torque (Tmin) Calculated Using Faxen’S Law for the Hydrodynamic Drag of a Particle near a Channel Wall, At the Point of Detachment surface gold
C6-OH
C11-OH
C16-OH
C11-CH3
FUDT
C11-OH, 3 μm
C16-OH, pH 3
shear ramp rate (s−1)
τmin (dyn/cm2)
FHmin (nN)
THmin (nN·μm)
0.0154 0.0462 0.0837 0.1153 0.1284 0.0154 0.0462 0.0837 0.1153 0.1284 0.0154 0.0462 0.0837 0.1153 0.1284 0.0154 0.0462 0.0837 0.1153 0.1284 0.0154 0.0462 0.0837 0.1153 0.1284 0.0462 0.0837 0.1153 0.1284 0.0154 0.0462 0.0837 0.1153 0.1284 0.0154 0.0462 0.0837 0.1153 0.1284
100 55 95 120 130 32 36 40 64 90 18 20 22 24 25 5 6 7 10 13 120 90 140 170 200 30 228 280 300 0.1 0.3 0.9 2 3 35 40 60 70 75
0.15 0.085 0.14 0.18 0.19 0.049 0.055 0.061 0.098 0.14 0.027 0.031 0.033 0.035 0.038 0.0075 0.0092 0.011 0.015 0.020 0.18 0.14 0.21 0.26 0.31 0.061 0.46 0.57 0.61 0.00024 0.00078 0.0025 0.0030 0.0071 0.053 0.060 0.089 0.10 0.11
1.5 0.84 1.4 1.8 1.9 0.47 0.53 0.59 0.94 1.3 0.27 0.30 0.33 0.35 0.38 0.072 0.088 0.10 0.15 0.19 1.8 1.3 2.1 2.5 3.0 0.60 4.5 5.6 6.0 0.0023 0.0077 0.025 0.029 0.070 0.52 0.60 0.89 1.0 1.1
IV. THEORY In our analysis of the experimental data we treat the detachment of objects (cells, beads, and so on) from a solid surface as a bond-breaking mechanism.24 Of importance for a successful description of detachment is the fact that the process exhibits a dependence on the shear ramp, i.e., the rate of the increase of the applied shear stress. We first develop the model itself and then give the details on how to analyze data for the final interpretation of the detachment process. Bond breaking by an external force or shear stress τ(t) is a thermally activated process. We introduce the probability, P(t), that the object is still attached at time t for which we write an Arrhenius rate equation dP = −A exp[−β ΔV (τ )]P dt
(3)
A reattachment process can be included easily but does not seem to be important. In (3) ΔV(τ) is the activation energy or energy difference between the free energy minimum of the bonds to the surface and the barrier to be overcome in bond breaking under the influence of an applied shear stress, see Figure 3. The prefactor A contains information about the change in entropy when the object, initially fixed on the surface, is free to move along or away from the surface. In the typical force ramp mode, the force is increased linearly in time with a force loading rate α.25 However, in the present experiment the shear stress increases exponentially with time as shown in eq 2. Eliminating t in favor of τ we rewrite the Arrhenius rate eq 3 as dP dP dτ dP = = ατ dt dτ dt dτ
so that dP A exp[−β ΔV (τ )]P =− dτ ατ
(4)
which can be solved to give 11107
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Figure 1. Detachment profiles of 4.5 μm particles on hydroxylated SAM surfaces of different thickness at five different shear ramp rates. The errors are calculated as the standard error from three replicate measurements for each ramp rate for each system.
Figure 2. Detachment of 4.5 μm beads at pH 7.4 and pH 3 from a hexadecanethiol (C16-OH) surface. The errors are calculated as the standard error from three replicate measurements for each ramp rate for each system.
⎡ A P(τ ) = exp⎢ − ⎣ α
∫τ
τ
0
⎤ 1 exp[−β ΔV (τ′)] dτ′⎥ ⎦ τ′
breaking too long to break earlier. Likewise the distribution gets narrower and as a consequence higher to preserve its normalization. In addition of course, more strongly bonded objects will break at higher shear stresses. These two effects can be untangled with this model. To go further analytically, we need to specify the activation barrier ΔV(τ) that must be overcome to detach the bead from its binding site. This has been discussed earlier, and we take over the simple end result, namely, that a good approximation of the dependence of the activation barrier on shear stress (force) is given by24
(5)
We obtain the distribution of the detachment stresses by taking the derivative of −P(τ) with respect to τ. Its maximum gives the most probable bond-breaking shear stress and is obtained by equating the second derivative of (5) to zero, which yields dβ ΔV dτ
=− τb
A 1 exp[−β ΔV (τb)] − τbα τb
(6)
ΔV ≃ V0(1 − τ /τmax )2
Likewise, we calculate the width of the breaking shear stress distribution by setting the third derivative equal to zero. Note that for larger rates of the shear stress ramp the maximum shifts to higher shear stresses, simply because it takes the bond
(7)
where τ max is the maximum possible shear stress which makes the activation barrier disappear. V0 is the depth of the holding potential in which the bead is trapped. One finds that 11108
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ensemble of Ns adsorption sites for which we introduce the probabilities Pi(t) subject to (3) for i = 1...Ns with different ΔVi and Ai. The average probability that the bead is still attached at time t is
P ̅ (t ) =
γV0 2as
(12)
For each experimental trial (j,κ) we find the best parameter set (A,V0,τ max) by first minimizing
where γ is the inverse range of the holding potential and as is the area of contact of the object with the substrate. For a linear force/stress ramp the rate equation can be solved explicitly in terms of error functions. However, for an exponential ramp we are confined to do the integration numerically; i.e.,
Sjκ =
1 Nκ
Nκ
∑ (P(τi) − Pijκ)2
(13)
i=1
In a second step we average the probability values from the Ntκ experimental trials for a given experimental condition κ and minimize the statistical deviations defined by
2 ⎧ ⎡ ⎤ ⎫ τ 1 ⎛ ⎪ ⎪ τ′ ⎞ A P(τ ) = exp⎨− exp⎢− V0⎜1 − ⎟ /kBT ⎥ dτ′⎬ , ⎪ ⎪ ⎢⎣ ⎥ τ0 τ ′ α τ ⎝ ⎠ max ⎦ ⎭ ⎩ τ0 ≤ τ ≤ τmax
1 Sκ = Nκ
∫
⎡ 1 ∑ ⎢⎢P(τi) − Ntκ i=1 ⎣ Nκ
⎤2 ∑ Pijκ ⎥⎥ ⎦ j=1 Ntκ
(14)
The result is the optimal parameter set (A,V0,τ max) for a given experimental condition κ for all trials. An estimate of the uncertainty is then obtained by analyzing the distribution around the minimum.
τmax < τ (9)
τ0 is the initial stress when the exponential increase was started. For τ ≤ τmax, the preceding first line, bead detachment is an activated process in which the remaining barrier (7) has to be overcome. Once τ > τ max the process changes which is best understood by rewriting the second line as P(t ) = P(tm) exp( −A(t − tm))
(11)
i
{τ0κ , αk , {τiκ ,{Piκj} Nj =tκ1}iN=κ1}κN=c 1
(8)
⎛ τ ⎞−A / α = P(τmax )⎜ , ⎟ ⎝ τmax ⎠
∑ Pi(t )
which is also subject to the Arrhenius rate eq 3. Experiments were done under Nc different conditions c specified by {τ0κ,ακ}Nκ=1 . Nκ denotes the number of (τ, P) data points (measurements) per trial for the κth experimental condition, and Ntκ denotes the number of experimental trial runs done for the κth experimental condition. The collection of all data is then
Figure 3. Morse potential in bond breaking as a function of the fragment separation with and without an external force applied, where ΔV is the activation barrier height.
τmax =
1 Ns
VI. RESULTS OF THE DATA ANALYSIS To establish the data analysis, we give details for a particular system, namely, carboxyl-terminated 4.5 μm diameter beads on a SAM of C16-OH, S−(CH2)16−OH. There were a total of three data sets available for four shear stress ramps α = 0.0154 s−1, 0.0462 s−1, 0.0837 s−1, and 0.1153 s−1. In Figure 4 we show the data, averaged over the three data sets, together with the best theoretical fit. To get an idea of the quality of the data and the fit, we show in Figure 5 the binding energy V0/kBT for all of the data sets, with error bars included. The average is V0 = (7.5 ± 0.6)kBT ≈ 188 ± 15 meV or 18.1 ± 1.4 kJ/mol. For all systems studied we found that the attempt frequency A rises linearly with the ramp speed α. For this data set we find the average value of A/α = 1.52 ± 0.12. The third parameter in our model, the maximum shear stress τ max, also rises but the increases are quite different for different systems. This is not a system-specific parameter but controlled by the experimental shear rate α: the greater α is, the greater τ max is (Figure 6). Before we continue with the physical interpretation of the extracted parameters A/α, V0/kBT, and τ max we show the results for all of the systems we studied in a summary Table 3. The extracted values for the parameters were averaged over the results from several trials over each of the five different ramp rates α. We should note that the average RMS value of the statistical analysis for all data sets is of the order 10−2.
(10)
This is a simple decay process controlled by the rate constant A. For slow shear ramps α ≪ A the total removal of beads is achieved with stresses τ < τ max so that only the first line of (9) is needed. However, as the ramp is increased, i.e., if the characteristic time scale of the ramp α −1 approaches or surpasses the removal time A−1, not all of the beads have been removed with stress τ < τ max. We will show in our data analysis that in such nonequilibrium situations the parameters A and τ max increase with α. The model has three parameters: the depth of the holding potential V0, the maximum shear stress τ max, and the prefactor A. Note that τ max itself is related to the range of the holding potential γ and the contact area as, both of which can be estimated. These parameters will be obtained on the basis of our data analysis.
V. METHODOLOGY OF THE DATA ANALYSIS To develop an algorithm for the data analysis, we need to generalize the theory as developed so far to account for an 11109
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VII. DISCUSSION The physical model for our data analysis contains four parameters: the depth of the holding potential or the binding energy V0 of the polystyrene beads on the SAM surface, the range of the holding potential γ, the prefactor or rate constant A, and the contact area as of the bead with the SAM. The statistical analysis produced numerical values for V0/kBT, A, and τ max. γ and as appear together with V0 in τmax = γV0/2as. The motivation for this experiment of shear-stress-induced detachment of objects such as beads (in order to develop a mechanistic model for the removal of cells from surfaces) was to measure its adhesion strength; i.e., the hypothesis was that the bead would be detached from the surface. For this bead experiment we showed earlier that the maximum activation barrier, i.e., the holding potential is actually quite weak. Indeed, a surface binding energy of 200 meV is typical for heavier rare gases on metals. One is therefore tempted to assume that these beads are attached to the surface mainly by van der Waals forces although other forces such as solvation forces or electrostatic forces between surface charges on the bead and on the SAM will also contribute. Although it seems at first glance implausible that a massive polystyrene sphere would be bound so weakly by van der Waals forces we need to check out this possibility in detail. Fortunately there are spectroscopic data available for polystyrene beads26−28 so that we can calculate the dispersion force for a bead in front of a wall from first principles29 as outlined in the Appendix, provided in the Supporting Information. The van der Waals potential for a bead in front of a wall is30
Figure 4. Attachment probability P(τ) for varying shear ramps α (averaged over three runs each) as a function of increasing shear stress τ for COO-coated 4.5 μm beads on a SAM of 1-hexadecanethiol (C16OH).
Vb = −A
rb ℏω r =− ̅ b 6h 8π h
(15)
where rb is the radius of the bead, h is the thickness of the layer between the bead and the metal, i.e., the SAM, and ℏω is the characteristic energy calculated from first principles.29 In a first calculation we treat the SAM gap of about 20 Å in height as vacuum and get a van der Waals energy for a 4.5 μm bead to be 130 eV. Next we assume a layer of water of the same thickness and find a reduction to 20 eV. A similar reduction by about an order of magnitude was found in the Hamaker constant for a system of two polystyrene sheets, namely, from 70.9 to 7.7 zJ. This is still a factor of 100 too large compared to what our analysis of the shear-stress experiments produced. Changing the layer from water to water-plus-alkane leads to a further reduction by perhaps another factor of 2, which is still not enough. Although these calculations of the van der Waals interaction are reliable, we looked for an experimental confirmation. We have therefore done AFM experiments to measure the force to rip a bead off the surface along the direction of the surface normal. The results are listed in Table 4, together with the respective energies which, from (16), are given by Vb = Fh; see Table 4. They decrease, as expected, with increasing SAM thickness. Although the magnitude of these energies is close to what we predicted theoretically, the decrease like h−1 is not seen. One source of a slower decrease of the interaction with increasing SAM thickness could be the fact that the bead can depress the thicker SAM easier so that the SAM thickness would effectively be reduced. We can also look for additional interaction mechanisms. First we consider repulsive forces. They would be largest for thin SAMs and decay for thicker ones, i.e., lead to an even faster decrease in the overall binding energy with SAM thickness. On the other hand,
Figure 5. Binding energy V0/kBT for all experimental runs for the system in Figure 4 where experiment ID means α (trial no.).
Figure 6. Averages of the maximum shear stress τmax as a function of the shear stress ramp for the system in Figure 4.
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Langmuir Table 3. Extracted Parameters A/α, V0/kBT, and τ
(for Different α in Table 2) for Various Systemsa
A/α
system 4.5 μm, FUDT 4.5 μm, C6-OH 3 μm, C11-OH 4.5 μm, C11-OH 4.5 μm, C16-OH 4.5 μm, C16-OH, pH 3 4.5 μm, C11-CH3 4.5 μm, gold a
max
1.43 2.70 0.98 1.60 1.52 1.62 2.00 1.53
± ± ± ± ± ± ± ±
0.30 0.25 0.12 0.11 0.12 0.17 0.21 0.22
V0/kBT
τmax(dyn/cm2) (α = 0.0154, 0.0462, 0.0837, 0.1153, 0.1284 s−1)
± ± ± ± ± ± ± ±
1673 ± 551 (NA, 142, 1178, 1825, NA) 244 ± 55 (NA, 158, 225, 269, 324) 6 ± 5 (0.3, NA, 1.4, NA, 12.3) 56 ± 10 (40, 47, 56, 69, 69) 39 ± 10 (23, 29, 39, 48, 51) 155 ± 35 (NA, 122, 149, 175, 187) 1119 ± 426 (NA, 489, 955, 1378, 1430) 406 ± 129 (233, 269, 399, 533, 543)
8.1 8.5 8.1 9.2 7.5 8.7 7.5 7.6
1.4 0.7 1.3 1.1 0.6 0.6 0.6 0.9
NA means that the data for that particular rate was unavailable or could not be fitted.
Table 4. Adhesion Force of a 4.5 μm Bead on the Surfaces, Measured by AFM sample
gold
C6-OH
C11-OH
C16-OH
C11-CH3
FUDT
thickness (Å) force (nN) Vb (eV)
0 65.4 ± 4.2
4 60.5 ± 2.7 150 ± 7.0
13 25 ± 2.0 170 ± 5.0
21 8.7 ± 0.6 120 ± 1.5
39.2 ± 1.6
14.3 ± 1.2
so that the correction is about an enhancement of Stokes law by a factor of 3. For a velocity of roughly 5 mm/s we get a drag force of 0.1 nN. If we set V0 = Fd, we get the range over which this force acts to start the beads movement to be of the order of a few angstroms, i.e., essentially a contact force as one expects for dry friction.
weak covalent bonding and electrostatic interactions might lead to a slower decrease. The inescapable conclusion from this discussion of van der Waals forces is that they are much larger by about 2 orders of magnitude than the energies we extracted from the shear stress experiments. Indeed, what one observes is that the bead is removed from its initial attachment site. There are therefore two options, one that the bead is removed from the surface akin to desorption or that it rolls along the surface. According to the binding energies we have extracted from the experiment and explained by van der Waals forces either scenario is still possible: detachment and rolling, or detachment and lift-off from the surface. The end result is that the bead is no longer attached no matter where it goes. If we accept the fact that the bead remains in contact and moves along the surface, the obvious mechanism is a rolling motion. As the shear force increases, a few contacts at the edge facing the shear (trailing edge of the bead) break and the bead rolls, instantly re-establishing similar contacts at the front edge and hence re-establishing the energy balance. Thus, the onset of rolling is an example of dry static friction whereby the externally applied force is lower than the frictional adhesion force between the particle and the surface.31,32 This is in good agreement with theoretical models and traditional systems, where the force required to initiate rolling for a surface attached sphere is significantly lower than that to initiate detachment via sliding or lift-off.33,34 This picture not only agrees with the small activation barrier but also with the fact that both the activation energy and the prefactor are approximately the same for all systems studied. With hindsight this is not too surprising as it is known from the shear-induced detachment of cells that they are peeled-off and not pulled-off.9 Another point in favor of static dry friction is the following: Stokes drag law of a sphere in a Poiseuille flow has to be modified when the sphere is close to a surface. In 1924 Faxén21,35−37 got an approximate formula for the drag force parallel to the surface, namely,
VIII. CONCLUSIONS A lesson learned from these experiments and our statistical analysis is that characterizing bead detachment by noting the stress at which half the beads have been removed (i.e., the maximum in the integrated detachment curve)as done in our previous experimental work5has no relation to the activation energy of detachment. It can only be a measure of the relative difference in the effect of surface chemistry on the detachment process. Clearly one needs a statistical data analysis to extract physically meaningful and important parameters which we have done here on the basis of a bond-breaking model. Our data convincingly show that detachment occurs via a rolling motion, where the activation barrier to rolling is due to dry static friction. One complication in the analysis and interpretation of the current experiments on shear-induced detachment of beads turns outwith hindsightto be that the shear stress ramps were too fast, meaning one was working in a kinetic regime where nonequilibrium effects away from the linear regime dominated. At present, however, controlled lower shear stress ramps cannot be generated in our experimental setup. As we pointed out earlier shear-induced detachment of cells is not a roll-off but a peel-off mechanism. In addition, one needs to account for the phenomenon of ”catch-bonds”, where the lifetime of the adhered substance (such as cells) increases with an increase of external stress (shear flow in this case) rather than diminishes. This increase goes through a maximum after which it is reduced as expected.38,39 Such phenomena we intend to address in future work.
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6πηaU⃗
FD⃗ = γU⃗ = 1−
9 16
3
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−
45 256
4
( ha )
−
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ASSOCIATED CONTENT
S Supporting Information *
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( ha )
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.5b02321. Appendix providing details of van der Waals interactions in bead adsorption and detachment (PDF)
(16)
For our geometry the distance of the center of the sphere from the wall h is very close to the radius of the sphere rb (or a) 11111
DOI: 10.1021/acs.langmuir.5b02321 Langmuir 2015, 31, 11105−11112
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AUTHOR INFORMATION
Notes
The authors declare no competing financial interest.
ACKNOWLEDGMENTS The work at the University of Heidelberg and the Institute for Functional Interfaces at KIT was supported by grants from the Office of Naval Research Washington, D.C. (Grant Nos. N00014-12-1-0498 and N00014-15-12324). This work was partly also carried out at the Karlsruhe Nano Micro Facility, a Helmholtz Research Infrastructure at Karlsruhe Institute of Technology (KIT). Initial support in setting up the experiments by Dr. Maria Alles is gratefully acknowledged. The work at Dalhousie University was supported by grants from the Natural Sciences and Engineering Council of Canada and from the Office of Naval Research, Washington, D.C. (Grant No. N00014120497).
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DOI: 10.1021/acs.langmuir.5b02321 Langmuir 2015, 31, 11105−11112
