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Langmuir 2007, 23, 9704-9712
Forced Detachment of Immersed Elastic Rubber Beads H. Ge´rardin, A. Burdeau, A. Buguin,* and F. Brochard-Wyart Institut Curie, Centre de recherche, CNRS UMR 168 - UniVersite´ Pierre et Marie Curie, 11 rue Pierre et Marie Curie, 75231 Paris Cedex 05 - France ReceiVed March 14, 2007. In Final Form: July 5, 2007 We investigate the strength of adhesion and the dynamics of detachment of elastic beads (Young’s modulus E ≈ 1 MPa) adhering to a horizontal solid surface in a viscous liquid. The beads are initially compressed on the surface. Their unbinding is imposed by fast vertical stretching (above a certain threshold value). The decrease in the contact radius is monitored by interferential microscopy. We find that the dynamics of detachment involves three steps: (i) fast elastic decompression, (ii) slow adhesive detachment, and (iii) catastrophic rupture. They can be interpreted by a transfer of the Johnson Kendall Roberts (JKR) energy toward viscous losses in the liquid wedge, near the rubber/ solid/liquid (R/S/L) contact line.
I. Introduction The adhesion of soft objects onto wet substrates is important in many fields, from driving on a wet road to cell/cell adhesion. Here we are concerned with separation processes on a model system: a rubber bead surrounded by a viscous liquid, adhering to a silanized glass substrate. Our experiments start with a bead pressed onto the solid, achieving an equilibrium contact radius ai that is well described by classical JKR analysis.1,2 The bead is suddenly stretched by imposing a vertical displacement ∆ to the top. Above a certain threshold |∆| > ∆/c , the contact will ultimately be lost. We monitor the decreasing contact radius as a function of time, a(t), by interference microscopy. The experimental procedure is illustrated in Figure 1, where we specify the notations used in this article. Many previous teams have studied the rupture of a rubber/ solid contact in air,33 with the main object being ultrasoft beads.4 In these systems, mechanical energy is dissipated by viscous processes in the weak rubber,5 which depends critically on the presence of free chains or long side chains. Furthermore, fibrillation and cavitation processes contribute and are complex.6,7 In our case, with an ambient viscous liquid and “hard” rubber beads (E ≈ 1 MPa), we expect simpler regimes controlled only by friction inside the liquid. Previous work in our group was focused on the buildup of a contact condition under prescribed squeezing. In air, the spreading of a rubber bead (studied by Shanahan8) results from the transfer of mechanical energy into dissipation in the rubber. In a liquid, we found that upon fast squeezing we deal with elastohydrodynamics: the sphere is flattened, a liquid film is trapped at the interface, and the contact is achieved by dewetting.9 When we impose slow squeezing, the bead is only weakly deformed, touches (1) Johnson, K. L. Contact Mechanics; Cambridge University Press: Cambridge, U.K., 1987. (2) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Sect. A 1971, 324, 301-313. (3) de Gennes, P.-G. C. R. Acad. Sci. Paris 1988, 307, 1949-1953. (4) Vondracek, P.; Vallat, M.-F.; Schultz, J. J. Adhes. 1991, 35, 105-113. (5) Barquins, M.; Maugis, D. J. Adhes. 1981, 13, 53-65. (6) Poivet, S.; Nallet, F.; Gay, C.; Fabre, P. Europhys. Lett. 2003, 62, 244250. (7) Creton, C.; Fabre, P. In Adhesion Science and Engineering; Dilliard, D. A. and Pocius, A. V. Eds.; Elsevier: Amsterdam, 2002; Vol. 1, pp 535-575. (8) Michel, F.; Shanahan, M. E. R. C. R. Acad. Sci. Paris, Ser. II 1990, 310, 17-20. (9) Martin, P.; Brochard-Wyart, F. Phys. ReV. Lett. 1998, 80, 3296-3299.
the solid at the center, and spreads with a quasi-static shape.10 The data can be accounted for by the viscous dissipation of mechanical energy in the fluid (dissipation in the rubber being negligible with hard rubber beads). Our aim is to study the inverse process of detachment in the limit ruled by elastohydrodynamics only. The experimental procedure is explained in section II. We determine the elastic modulus via a classical JKR test (performed at fixed load). An overview of JKR theory and the relation between the contact radius and the displacement a(∆) under static conditions is given in section III. Section IV shows the data on unbinding. One technical point is relevant here: JKR is usually operated at a prescribed load, whereas in our detachment experiments we work at prescribed displacement ∆. We measure the decreasing contact radius a(t) under stretching: we vary the initial condition ai, the stretching amplitude ∆, and the liquid viscosity η. We have found three successive regimes during separation: (i) fast elastic decompression, (ii) slow detachment dominated by adhesive forces, and (iii) catastrophic rupture. In section V, we interpret the three regimes by a dynamic model incorporating JKR mechanics and viscous flow in the liquid. II. Materials and Methods To make small rubber lenses, we deposit droplets from the liquid reaction mixture onto a silanized glass slide. We use a commercial elastomer (Dow Corning Corp.) prepared by cross linking endfunctionalized poly(dimethylsiloxane) (PDMS). It is supplied in two liquid parts (Sylgard 184 A & B) comprising a vinyl end-capped oligomeric PDMS (∼250 monomers), a cross linker, and a catalyst for the cross-linking reaction. A 10:1 mixture (w/w) of the two liquid parts is prepared. Cross linking is obtained by curing the reaction mixture at 65 °C for 4 h. We get spherical caps (lens) with radii of curvature R on the order of 2 mm. The elastic modulus E measured by JKR tests in air (as explained below) is E ) (0.94 ( 0.02) MPa. As solid substrates, we use microscope glass slides chemically modified to obtain low surface energies. The slides are silanized by grafting a monolayer of n-octadecyltrichlorosilane onto the surface following a standard procedure.11 (10) Verneuil, E.; Clain, J.; Buguin, A.; Brochard-Wyart, F. Eur. Phys. J. E 2003, 10, 345-353. (11) Brzoska, J.-B.; Shahidazeh, N.; Rondelez, F. Nature 1992, 360, 719721.
10.1021/la700757x CCC: $37.00 © 2007 American Chemical Society Published on Web 08/11/2007
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Figure 1. Description of the detachment experiments. (a) In the initial state, we impose the displacement ∆0 > 0 to press the bead against the slide. (The contact radius is then ai.) (b) At time t ) 0, we impose the displacement ∆1 < 0 to stretch the bead and (c) follow the evolution of the contact radius with time. If |∆| > ∆/c , then detachment will occur to reach the final state (d). Displacements are considered to be positive if the bead is pressed (displacements oriented along the z axis) and negative if the bead is stretched.
Figure 2. Schematic of the experimental setup. Detachment experiments have been made by using a hard cantilever (a microscope glass slide) and controlling the displacement in the z direction with the piezo controller. The observation of the contact is made by RICM, and a typical interferogram obtained for a PDMS bead in air (not in contact) is shown on the right. For JKR tests, the bead is placed on a soft cantilever (stiffness k ≈ 10 N‚m-1) whose deflection is detected by a laser beam. Using the micromanipulator, the bead is pressed against the glass slide, and the establishment of the relation between the contact radius and the force leads to values for the elastic modulus and adhesion energy. JKR tests were made only in air to determine the elastic modulus. The liquids are fluorinated silicone oils (Hu¨ls-Petrarch Corp., used as received) that are immiscible with PDMS. The oil’s viscosity η, measured using a classical rheometer, varies between 0.1 and 1 Pa‚s. The experimental setup is shown in Figure 2. The silanized glass slide is placed on the mobile stage of the microscope, and the rubber bead is fixed on a glass slide set to a three axis micromanipulator (Narishige). A piezosystem (Jena) allows vertical displacement monitored by a computer. We follow the lens’s normal approach along the axis of a 20× objective of an inverted microscope (Zeiss, Axiovert 135) using, in its simplest form, a technique known as reflection contrast interferential microscopy (RICM). The monochromatic illumination is given by a 100 W HBO lamp and interference filters (wavelength λ ) 546 nm). Interfaces are monitored with a classical video camera (COHU). Because the distances between glass and rubber are micrometric or smaller, we can see fringes of equal thickness due to interference between rays reflected at the liquid/glass and at the liquid/rubber interfaces: two successive bright fringes are separated by a thickness variation of λ/2n ≈ 200 nm, where n ) 1.38 is the liquid refractive index. The adhesion energy W at the S/L/R interface is measured from the profile of droplets trapped at the PDMS/solid interface. The height H and the contact radius r of these droplets are related by W/E ) (π/6)H2/r. We obtain W ) (4.8 ( 0.1) mJ‚m-2.12,13 JKR tests were made by using a homemade setup whose principle is analogous to a “macro AFM” (Figure 2), which allows us to (12) Martin, P.; Silberzan, P.; Brochard-Wyart, F. Langmuir 1997, 13, 49104914. (13) Sneddon, I. N. Proc. R. Soc. London, Sect. A 1946, 187, 229-260.
Figure 3. Plot of a(∆) obtained by using the JKR theory at fixed displacement. ac and ∆c are the critical values of the contact radius and displacement at fixed load, respectively. Working at fixed displacement ∆ allows us to reach lower values of the contact radius and displacement: detachment occurs at A, a ) a/c ) (ac/32/3), and ∆ ) -a/c ) -32/3∆c. Table 1. Experimental and Theoretical (Obtained by Using Experimental Values of E, W, and R) Values of the Threshold Deformation ∆/c for Detachment in Several Fluids for Different Beads of Radius R, with the Error Corresponding to the Difference between Experimental and Theoretical Values of ∆/c silicon oil medium
water
W (mJ m-2) R (mm) ∆/c , exp ∆/c , th error (%)
34 1.90 1.48 ( 0.02 1.80 ( 0.07 18
silicon oil
silicon oil
(η ) 0.10 Pa.s) (η ) 0.68 Pa.s) (η ) 12.8 Pa.s) 4.8 2.37 0.36 ( 0.02 0.52 ( 0.02 31
4.8 2.33 0.54 ( 0.02 0.52 ( 0.02 4
4.8 2.23 0.40 ( 0.02 0.49 ( 0.02 22
measure normal forces from 1 µN to 0.1 mN. The “tip” is the PDMS bead and is placed on the extremity of a glass cantilever (dimensions of 26 × 1 × 0.1 mm3, stiffness k ≈ 10 N‚m-1), whose deflection is measured using a laser detection system: a laser beam is reflected from the top of the cantilever into a photodiode, and the difference signal on it gives a measurement of the cantilever bending. A micromanipulator allows the vertical displacement to press the bead on the substrate, which enables the step-by-step loading of the bead by a small increase in the vertical displacement ∆z of the system (∆z ) 2.5 µm). In our JKR tests, we have waited 1 min after each step to measure the normal force and the contact radius.
III. JKR Test Johnson, Kendall, and Roberts2 constructed a standard test to measure weak adhesion such as van der Waals energies. When a bead is pressed on a plane solid surface, the contact area is a function of the load f applied to the bead. The JKR analysis is an extension of Hertz elastic theory14 and includes adhesion (14) Hertz, H. Miscellaneous papers, McMillan and Co.: London, 1896.
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Figure 4. Typical detachment sequence. The experiment was conducted under conditions of ai ) 3.0a/c , |∆| ) 1.2∆/c , and liquid viscosity η ) 0.68 Pa‚s. The time between images is ∆t ) 6 s.
The minimization of G with respect to δ2 leads to
3 f ) Kaδ2 2
(2)
The free enthalpy becomes
G)
f2 a2 1 Ka5 - fR2 - πa2W 2 15 R R 3Ka
(3)
The JKR equation corresponds to the minimum in G(a). For W ) 0, dG/da ) 0 must obey the Hertz law f ) Ka3/R, which imposes R2 ) 1/3. For W * 0, dG/da ) 0 leads to
f2 Ka4 2 fa )0 2πWa + 3R2 3 R 3Ka2
Figure 5. Typical evolution of the contact radius a with time. The detachment is made when |∆| ) ∆/c and the initial contact radius ai ) 6.3 a/c , and the liquid viscosity is η ) 0.68 Pa‚s. As specified in the text, we can define three regimes with crossover values aI and aII.
features. It explains why, even at f ) 0, the contact size is finite (radius a ) a0). The JKR equation involves the load f (f > 0 for squeezing, f < 0 for stretching) and the adhesion energy W. It can be deduced from the free enthalpy of the squeezed bead that
G)
(
)
a2 1 Ka5 3 + δ2 - πa2W + Kaδ22 f R 2 15 R2 R 4
(1)
where K is proportional to the Young’s modulus E (K ) (16/9)E for an incompressible elastomer). The first enthalpy term is the elastic energy to flatten a sphere of radius R (E(δ1/a)2a3, δ1 ) R2 a2/R ≈ a2/R). The second term is the work of the load -f∆, where ∆ ) δ1 + δ2 is the sum of a squeezing term δ1 and a stretching term δ2 associated with the deformation of the bound bead (elastic modulus (3/2)Ka) induced by f. The third term is the surface energy gain, and the last one is the elastic energy of the stretched bound bead.
(4)
By setting X ) Ka3/R, eq 4 can be written as (X - f)2 ) 6πWRX. The equilibrium radius a(f) deduced from eq 4 is given by the classical JKR equation
Ka3 ) f + 3πWR + x(3πWR)2 + 6πWRf R
(5)
For f ) 0, a03 ) 6πWR2/K. When the force f is negative, the bead is stretched from the surface, and for f ) fc ) -(3/2π)WR, contact is lost. (fc is the rupture force at fixed load.) The radius at separation is ac3 ) (3π/2)WR2/K, and the displacement is ∆ ) -∆c ) -ac2/3R. In a usual JKR test, we obtain W and K by plotting a3/2/R versus f/a3/2. Indeed, we can write eq 5 as a3/2/R ) (1/K)f/a3/2 + x6πW/K. The slope and the contact radius at zero force lead to K and W. For a PDMS bead pressed against a silanized glass slide in air, we obtained E ) (0.94 ( 0.02) MPa and W ) (47 ( 3) mJ‚m-2. In our detachment experiments, we do not impose the load but the displacement ∆. To derive the contact radius a versus ∆, we have to work with the corresponding potential F ) G + f∆:
F)
(
3 a2 K a5 2 πa W + ∆ Ka 15 R2 4 3R
)
2
(6)
Detachment of Immersed Elastic Rubber Beads
Figure 6. Effect of the final displacement ∆ on the dynamics of detachment. (a) Experiments performed with ai ) 4.0 a/c , ∆/c ) 0.56 m, and η ) 0.68 Pa‚s. The representation of a versus (t/tdet) in part b allows us to see the disappearance of the first regime when |∆| increases. This shows that aI depends on the value of ∆.
The minimization of F ((∂F/∂a)∆ ) 0) leads to ∆ ) a2/R x8πaW/K. The plot of a(∆) is shown in Figure 3. Contact is thus lost for d∆/da ) 0, f ) f/c ) (5/9)fc, a ) a/c ) ac/32/3, and ∆ ) -∆/c ) -32/3∆c. f/c is the rupture force at fixed displacement.15 Experimental measurements of ∆/c , performed in air or in various liquid, are listed in Table 1. ∆/c corresponds to the displacement for which the dynamics of detachment becomes extremely slow. One should point out that the strain energy release rates derived from eqs 3 and 6 are the same (1/2πa(∂G/∂a)f ) 1/2πa(∂F/∂a)∆).
IV. Dynamics of Detachment Experiments are performed as follows. We first put the bead on the slide linked to the micromanipulator and the silanized glass slide on the microscope. Using the micromanipulator, the bead is positioned on the center of the microscope objective. Its radius is deduced from the interference fringes due to reflections on the bead and slide interfaces, which are characterized by good contrast in air. (15) Maugis, D. Contact, Adhesion, and Rupture of Elastic Solids; Springer: Berlin, 2000.
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Figure 7. Fits obtained for the slope of the second regime m2 and the time of detachment tdet. Data are issued from Figure 6. (a) Evolution of m2 vs ∆. The graph represents (m2/θ0) versus |∆|3/2, where θ0 is the initial angle between the bead and the substrate (cf. Figure 1). (b) Evolution of (tdetθ0/ai) versus ((|∆|/∆/c ))3/2 - 1, leading to tdet ≈ (((|∆|/∆/c ))3/2 - 1)-0.99.
Once the bead is centered, we introduce silicon oil between the rubber and the slide. The bead is then pressed onto the slide with a displacement ∆0 using the piezo controller, and the sphere is flattened. Initially, an intercalated film of oil is trapped at the interfaces. After a few minutes, the film is removed by drainage and dewetting.9 Once the contact is established between the bead and the slide, the contact radius is ai. We suddenly stretch the bead by imposing a negative displacement ∆1 such that the total deformation ∆ ) ∆0 + ∆1 becomes negative. If |∆| > ∆/c , then the bead will detach from the substrate (Figure 4), and we follow the radius decrease with time. After each detachment, an evaluation of the zero altitude of the bead is made by moving the sphere near the contact until no movement of the fringes can be detected. A typical radius evolution, evaluated by drawing circles on each image, is represented in Figure 5. We clearly see three regimes in the dynamics of detachment: (i) a fast decrease of a(t) for t < tI and a > aI corresponding to the decompression of the bead, (ii) a slow decrease where the contact radius decreases linearly (a(t) ≈ -m2t, where m2 is the slope at the inflection point), and (iii) a final ultrafast rupture for t > tII. We also notice on this curve that the second process is slow, thus it dominates the overall time of detachment tdet.
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Figure 9. Effect of ai on the dynamics of detachment. Data are issued from the graph plotted in Figure 8b. We have plotted (ai/tdet) versus ∆0, and only the black points were used for the linear fit because they correspond to higher values of ai.
Figure 8. Effect of the initial contact radius ai on the dynamics of detachment. Experiments were conducted with a/c ) 14 m, |∆| ) 6.1∆/c , and η ) 0.85 Pa‚s. Curves obtained in part a are plotted in reduced time units in part b to extract the data showing the decompression regime clearly.
We have studied the unbinding dynamics as a function of displacement ∆, liquid viscosity η, and initial contact radius ai. (i) Influence of ∆. As shown in Figure 5, when |∆| approaches ∆/c , the dynamics is strongly slowed down. In Figure 6, we have varied |∆|/∆/c from 1.4 to 5.7, with the other parameters being fixed. When |∆| decreases, the time of detachment tdet increases, and the slope m2 of the intermediary regime decreases. Figure 6b also shows that the first regime becomes less and less visible as |∆| increases. When |∆| ) 3.2 µm, the detachment curve is almost linear. The dependence of tdet and m2 on |∆| is shown in Figure 7. (ii) Influence of ai (via changing ∆0). We show in Figure 8 the evolution of a(t) for different values of ai at fixed ∆. The first regime of decompression is very sensitive to ai. As shown in Figure 8b, if we start with large value of ai, then we clearly find the three regimes, but if the bead is not compressed enough (ai < aI ≈ 55 µm under these conditions), then the first regime disappears. As shown in Figure 9, we have found a linear variation between ai/tdet and ∆0. (iii) Influence of viscosity. We conducted several experiments with different beads (of similar contact radii) and different viscosities (η varies from 0.1 to 1 Pa‚s), with ai/a/c and ∆/∆/c
Figure 10. Effect of viscosity on detachment. Experiments are conducted with different beads of similar contact radii, ai ) 4.3a/c and |∆| ) 2.4 ∆/c . The time of detachment and the slope m2 have been plotted versus η, and we have found tdet ≈ η1.0 and m2 ≈ η-1.0 (results not shown).
being fixed. As shown in Figure 10, we found that tdet increases linearly with the viscosity and that the slope m2 decreases as the inverse of the viscosity. The dynamics of detachment is slowed down when η increases. This shows that dissipation in the liquid dominates the process and that the PDMS lens is fully elastic. We can conclude from these observations that, depending on the ai values, the detachment curves can show three or two regimes: when ai eaI, the first regime disappears and the second one is truncated, and aI depends on the value of ∆. Moreover, the second regime is the longest one and fixes the time of detachment: tdet ≈ (m2)-1. tdet diverges when ∆ tends to ∆/c .
V. Interpretation Equation of Motion for the Contact Line. We shall now write down a dynamic equation for the S/R/L triple line. The driving force f⊥ acting on the line per unit length is derived from the JKR energy. If the unbinding is processed at constant displacement ∆, then f⊥ is derived from the potential F : f⊥ )
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We model the dynamic angle θ by
θ ) 1
∆0 δ(t) + 2 ai a
where 1 and 2 are numerical constants of order unity. The first term is the angle of the truncated sphere before stretching. The second term is the increase in the contact angle induced by stretching δ(t) ) |∆1| - (a2i - a2)/3R, which decreases from δinitial ) |∆1| to δterminal ) |∆|. We notice that the plot of θ(t) is almost stationary because δ(t)/a is minimal for a2 ) 3R|∆| (except in the final “catastrophic regime” where θ increases up to π/2). In this discussion, we approximate θ by a constant value θ ) θ0 ) (1∆0 + 2|∆1|)/ai. The equation of motion becomes
Figure 11. Detachment force versus contact radius in adimensional units plotted for ∆ ˜ ) -1.5. a˜ m corresponds to the minimum in the function a˜ m ) x|∆ ˜ |.
-(1/2)πa ∂F/∂a. Using eq 6 for F, we obtain
f⊥ ) -
[
3
2
2E ∆ 2E a 4E a∆ + -W 3π R2 3π a 3π R
]
f⊥ ˜2 3 a˜ 3 9 ∆ )- + ∆ ˜ a˜ + 1 W 16 16 a˜ 8
(7)
(8)
˜ ) (∆/∆/c ). Notice that f⊥ is related to where a˜ ) (a/a/c ) and ∆ the energy release rate usually named G for Griffith (f⊥ ) G W).5 A plot of |f˜⊥| versus a˜ is shown in Figure 11. |f˜⊥|(a˜ ) decreases up to a minimal value of |f˜⊥| ) |∆ ˜ |3/2 - 1 for a˜ m ) x|∆ ˜ |. • If ∆ ˜ > -1, then the force becomes positive for a˜ ) a˜ m. The contact relaxes to a finite size corresponding to ˜f⊥ ) 0. • If ∆ ˜ < -1, then the force is always negative, and the contact is ultimately lost. We describe now the detachment in this regime. To derive the line motion equation, we balance the driving force f⊥ by a resisting friction force. For a rubber bead in air, the balance force is5 f⊥ ) G - W ) Wφ(aTa˘ ), where φ is a dissipative function that depends upon the fracture velocity a˘ and the WLF factor shift aT. Taking φ ≈ a˘ n (n ) 0.6), one can derive the dynamics of detachment in air and fit the experiments for detachment at imposed separation velocity ∆˙ .5 For a bead immersed in a liquid, the resisting friction force is associated with the retracting liquid wedge characterized by a dynamic angle θ.19 The dissipation in the moving liquid wedge has been calculated by de Gennes for the spreading of liquid droplets.16,17 This leads to
a˘ fv ) 12η L θ where L is a logarithm factor that describes the dissipation in a wedge and the numerical factor 12 corresponds to solid boundary conditions on both S/L and L/R interfaces. (16) de Gennes, P.-G.; Brochard-Wyart, F.; Que´re´, D. Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, WaVes; Springer: New York, 2003. (17) de Gennes, P.-G. ReV. Mod. Phys. 1985, 57, 827-863. (18) Ge´rardin, H. Ph.D. Thesis, Universite´ Paris 6, 2006.
(9)
By dividing this equation by W and introducing the dimensionless variables a˜ , ∆ ˜ (defined after eq 8), and a˜ ) da˜ /dt˜ ) τ da˜ /dt (τ is the characteristic time of detachment), we obtain
a˜ ) -
or, in adimensional units,
˜f ⊥ )
2E a3 2E ∆2 4E a∆ a˘ + +W 12η L ) θ0 3π R2 3π a 3π R
˜2 3 9 ∆ a˜ 3 + ∆ ˜ a˜ + 1 16 16 a˜ 8
(10)
with τ ) 12ηLa/c /Wθ0. Evolution of a(t). Equation 9 shows three regimes. At large contact radius (region I), the first term in eq 10 is dominant: this corresponds to the decompression of the bead. For intermediate values, the force |f⊥| goes through a minimum and is nearly constant. This second regime corresponds to the adhesive rupture. Finally, as a tends toward zero, the second term in eq 10 dominates and creates a catastrophic rupture. (a) Fast elastic decompression (large contact radius): a gaI For large values of a˜ (a), the first term in the force is dominant, and the equation of motion (eq 10) becomes a˜ ) - a˜ 3/16, which leads to
1 1 1 ) ˜t a˜ (t) a˜ i2 8 2
θ0Et t 1 1 ) (11) - 2) /2 a (t) ai 8τac 9ηLπR2 2
(b) Slow detachment dominated by adhesive forces: aII ea eaI The force |f˜⊥(a˜ )| is minimal when a˜ ) a˜ m ) x|∆ ˜ | and ˜f⊥ (a˜ m) ) 1 - |∆ ˜ |3/2 (Figure 11). This minimum corresponds to the inflection point of the curve a˜ (t˜). The force is nearly constant for a broad range of contact radius values around a˜ m (Figure 11), and eq 10 leads to a˜ ) 1 - |∆ ˜ |3/2. We then have
˜2 a˜ (t) ) (1 - |∆ ˜ |3/2) ˜t + C
( ( ))
a(t) ) -m2t + C2 ) 1 -
|∆| ∆/c
3/2
Wθ0t + C2 (12) 12ηL
The second regime is the longest one. We then expect to have for the detachment time (in adimensional units) ˜tdet ≈ ˜tII - ˜tI ) (a˜ I - a˜ II/|∆ ˜ |3/2 - 1). (19) The profile is approximated by a wedge of angle θ. In fact, the profile at the contact line becomes sharp : z ≈ xxh0 (h0 ) W/E). z˘ ) θ leads to xM ) h0/θ2 (∼10 nm). The viscous dissipation in this microscopic region ∫ η(a˘ /z)2 zdx gives a contribution ηa˘ xxM/h0 ≈ ηa˘ /θ which scales like the “macroscopic” friction form.
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Figure 12. (a) Numerical resolution of eq 10 for a˜ i ) 6 and ∆ ˜ ) -1.05. M corresponds to the inflection point. Functions corresponding
to eqs 11-13 are plotted with this curve: (I) a˜ (t˜) ) x(a˜ i/1+(a˜ i2˜t/8)), (II) a˜ (t˜) ) a˜ m + (1 - ˜|∆|3/2)(t˜ - ˜tm), and (III) a˜ (t˜) ) x(9/8)∆ ˜ 2(t˜det-t˜). (b) ˜tdet versus a˜ i for different values of ∆ ˜ . The initial slope varies linearly with |∆ ˜ |3/2 - 1 and the plateau values as (|∆ ˜ |3/2 - 1)1/2 (insets), as predicted by eqs 18 and 17.
(c) Catastrophic rupture: a eaII As a˜ f 0, the elastic modulus of the bound lens falls to zero, and ˜f⊥ diverges (f˜⊥ ) -(9/16)∆ ˜ 2/a˜ ) a˜ ). Thus we have
9 2 a˜ 2(t) ) - ∆ ˜3 ˜ ˜t + C 8 Eθ0∆2 a (t) ) -m3t + C3 ) t + C3 (13) 9πηL 2
We have considered the angle θ to be constant during the whole process, but just before detachment occurs, θ increases strongly. we propose that during the last step of the detachment the angle can be approximated as θ ≈ |∆|/a. Whereas ∆/a e (∆0 + |∆1|)/ai, we can consider θ to be constant, but as soon as |∆|/a g (∆0 + |∆1|)/ai, we have to take its variations into account. Then, when a e aθ ) |∆|/(∆0 + |∆1|)ai, eq 13 must be modified as da/dt ) -(2E/3π)(θ/12ηL)(∆2/a) ) -(E/18πηL)(|∆|3/a2), which implies a3(t) ) -E|∆|3/6πηL t + C4. However, this last step is too fast to be followed with accuracy (when a tends to zero), and the correction is not very relevant. Crossover and Prediction of the Time of Detachment. To derive crossover values a˜ I and a˜ II, we develop f⊥ around its
maximum (parabolic approximation), and eq 10 becomes
3 1/2 a˜ ) 1 - |∆ ˜ |3/2 - |∆ ˜ | (a˜ - |∆ ˜ |1/2)2 4
(14)
The solution of this equation is
arctan u ) -
|∆ ˜ |3/2 - 1 (t˜ - ˜t 0) R
(15)
with u ) (a˜ - |∆ ˜ |1/2/R), R ) x4/3(|∆ ˜ |3/2-1)/|∆ ˜ |1/2, and ˜t0 being a constant of interpolation (u(t˜ ) ˜t0) ) 0). For u , 1, arctan u ≈ u, and the solution of eq 15 is linear and is given by eq 12. For u . 1, arctan u ≈ π/2 - 1/u, and we obtain
1 1 3 ˜ |t˜ ) x|∆ 4 a˜ i - x|∆ ˜ | a˜ - x|∆ ˜|
Detachment of Immersed Elastic Rubber Beads
Langmuir, Vol. 23, No. 19, 2007 9711
Figure 13. Analysis of the three regimes. (a) Detachment curve (|∆| ) 2.0 ∆/c , ai ) 4.0a/c , and η ) 0.68 Pa‚s) and fits obtained for the three regimes. (b) Fast elastic decompression. (c) Slow adhesive detachment. (d) Catastrophic rupture.
The crossover between these two limits can be defined as arctan u ) π/4, and then we deduce the crossover values a˜ I and a˜ II
a˜ I,II ) |∆ ˜ |1/2 ( R ) |∆ ˜ |1/2 (
x
VI. Comparison with Experimental Data
˜ | - 1) 4(|∆ 3 |∆ ˜ |1/2 3/2
(16)
When a˜ i > a˜ I, the time of detachment is then
˜t det ≈
a˜ I - a˜ II |∆ ˜|
3/2
-1
)
(|∆ ˜|
3/2
1 - 1)1/2|∆ ˜ |1/4
(17)
When a˜ i becomes smaller than a˜ I, the decompression regime disappears, and the second regime is cut down. We then have
tdet ≈
ai 3/2
|∆ ˜|
12ηL - 1 Wθ0
We have also determined that the thermodynamic slowdown for |∆| f ∆/c is well described by eqs 18 and 17 in the two limits of large and small initial compression of the bead (Figure 12b).
(18)
We have solved eq 10 numerically by varying ∆ ˜ and the initial conditions (a˜ i). The results are shown in Figure 12. We can clearly define three regimes for the numerical solution plotted in Figure 12a; eqs 11-13 are plotted on this solution and confirm our interpretation of the three regimes.
Three Regimes. Figure 13 presents a comparaison of eqs 11-13 and experimental fits obtained for a detachment curve a(t), for which the three regimes were clearly observed. These data correspond to high initial compression of the bead. (a) Fast elastic decompression (Figure 13.b): In the first regime, we expect the plot of 1/a2 versus time to be linear, and we obtained good agreement. We predict the slope to be θ0E/9ηLπR2. Values of E, η, and R are known, and we can assume that θ0 ≈ ∆0 + |∆1|/ai ≈ 5.5 × 10-2. Identification with the predicted slope leads to L ) 3. (b) Slow detachment dominated by adhesive forces (Figure 13c): In the second regime, a(t) + (1 - (|∆|/∆/c )3/2)Wθ0t/12ηL + C2. In practice, a(t) is linear, and the identification of the slope with the theoretical value leads to L ) 4. (c) Catastrophic rupture (Figure 13d): For the third regime, we have plotted a versus tdet - t; the power fit leads to a ≈ (tdet - t)0.53, which is close to the expected value (0.5). The plot of a2(t) leads to L ) 8. The values of L are very close in the first two regimes, but the higher value in the catastrophic regime may be related to our assumption of constant wedge angle θ ) θ0: an increase in θ leads to a higher L value.
9712 Langmuir, Vol. 23, No. 19, 2007
Influence of ai. The initial contact radius ai measures the initial compression of the bead and also fixes the initial contact angle. Large ai implies that the three regimes are observed and that θ0 is larger, which means that the characteristic time τ (eq 10) is smaller. It explains why the role of ai is subtle. To extract the curves showing clearly the decompression regime, we have plotted in Figure 8b the contact radius versus reduced time (t/tdet) for high values of ai. We see an initial fast decompression for all of these curves, showing that ai > aI. The first regime predicts that the initial slope of a(t) will be given by eq 11, which leads to a˘ (t ) 0) ) -θ0E/18ηLπR2)(ai)3 ≈ -ai2(∆0 + |∆1|). The plot of (|a˘ (t ) 0)|/ai2 versus ∆0 is well fitted by a straight line (data not shown here, see ref 18), which confirms our analysis. In this limit of large ai, the time of detachment is given by eq 17. The graph tdet(ai) must reflect the variations of τ(ai). We have τ ∝ 1/θ0 ≈ ai/(∆0 + |∆1|) and tdet ≈ (ai/(∆0 + |∆1|). We have verified that the plot of ai/tdet versus ∆0 for the data presented in Figure 8b is well fit by a straight line (Figure 9). Influence of ∆. ∆ had no effect on the slope of the first regime, which is dominated by the initial compression. For the second regime, we predict a(t) ≈ -m2t and m2 ) (|∆ ˜ |3/2 - 1)Wθ0/12ηL. We have plotted in Figure 7a m2/θ0 versus |∆|3/2. We find a linear relationship as expected theoretically from eq 12. Moreover, the interpolation of the linear fit must lead to ∆/c . We have obtained ∆/c ) (0.53 ( 0.18) µm, in good agreement with the experimental value of (0.56 ( 0.02) µm. An identification of the slope of m2(|∆|3/2) with the value calculated from eq 12 leads to L ) 4, which corroborates the value found previously for this regime. For the third regime (eq 13), we expect m3 ≈ E∆2θ0/9πηL. We have plotted m3/θ0 versus ∆2, and the linear fit leads to L ) 8. We can also predict the influence of ∆ on the detachment time tdet. Most of the data for a(t) presented in Figure 6 concern ai
Ge´ rardin et al.
< aI, which means that the decompression regime is not present. The initial slope must then be given by eq 12 and is (|a˘ (t ) 0)|/θ0 ≈ ˜|∆|3/2 - 1, which was confirmed (data not shown). The time of detachment will be given by eq 18: tdet ≈ a˜ i/(|˜∆|3/2 - 1)τ. We have plotted tdetθ0/a2i versus |∆ ˜ |3/2 -1 in Figure 7b, and we find a linear variation with a good agreement.
VII. Conclusions We have focused on soft beads adhering via van der Waals forces to a glass plate. The unbinding of a bead involves three steps: (i) fast decompression, (ii) slow adhesive unbinding, and (iii) catastrophic rupture. The overall time span of detachment is controlled by the second step and diverges when the imposed vertical displacement reaches a certain critical value ∆/c . We find that the time of detachment scales as (ai/(|∆|/∆/c )3/2 - 1), where ai is the initial size of the contact and ∆ is the displacement. Our results may be useful in a broad range of applications, from car tires separating from a wet road to the motility of living cells. When driving on a wet road at uniform velocity V, the rubber asperities have to attach and detach continuously. We have previously studied the dynamics of attachment, and we have now begun to understand the inverse process. Comparing the detachment time to the passage time allows us to predict energy losses by elastic deformations versus the velocity V. To extend our studies to cellular biophysics, we have grafted specific proteins on the bead and on the substrate. The main difference is that the elastic energy is dissipated upon rupture of the binding bridges. This work will be published in a forthcoming article. Acknowledgment. We thank P. Nassoy and P.-G. de Gennes for stimulating discussions and K. Guevorkian for carefully reading the manuscript. LA700757X
