Ultrathin Hydrated Dextran Films Grafted on Glass: Preparation and

4866

Langmuir 1996, 12, 4866-4876

Ultrathin Hydrated Dextran Films Grafted on Glass: Preparation and Characterization of Structural, Viscous, and Elastic Properties by Quantitative Microinterferometry Martin Ku¨hner* and Erich Sackmann Physik Department E22, Biophysics Laboratory, Technische Universita¨ t Mu¨ nchen, James Franck Strasse, D85747 Garching, Germany Received March 25, 1996. In Final Form: July 2, 1996X We report the deposition and partial anchoring of ultrathin dextran films on glass surfaces and the characterization of their structural, viscous, and elastic properties by using latex beads as colloidal probes. The average distance between bead and surface was measured to an accuracy of (0.2 nm by application of reflection interference contrast microscopy (RICM) and by application of an improved theory of image formation of this microinterferometric technique. The fluctuation of the height h(t) of the beads above the glass surface with time t was analyzed in terms of the theory of Brownian motion of particles in a potential V(h). The interaction potential V(h) of the bead was obtained by analyzing the height probability distribution of a bead in terms of the Boltzmann distribution law. It is shown that V(h) is determined essentially by the gravitational attraction and the polymer-induced disjoining pressure. The latter could be best interpreted in terms of a simple spring model, whereas the classical scaling theories of polymer-induced forces failed. An elastic constant of 300 N/m2 was obtained. An apparent viscosity of the dextran film was determined by the evaluation of the damping constant of the height correlation function 〈h(t) h(0)〉. Applying the Brinkmann model of flow in porous media, the damping constant is further used to evaluate a hydrodynamic decay length of the hydrodynamic field in the dextran film.

Introduction The biofunctionalization of solids by supported membranes suffers from a severe drawback: the high density of defects (local density fluctuations or highly curved domains), which are mainly introduced by the surface roughness of the solid. They reduce the electrical resistivity of the lipid bilayers compared to those of black lipid membranes by orders of magnitude.1 They form strongly attractive centers for proteins and thus impede the application of supported lipid-protein films for proteinprotein binding studies. A major aim is therefore to deposit self-healing membranes on solids exhibiting the same intrinsic lateral tension as free lipid bilayers. The most convenient and promising way to achieve this goal is to separate the membrane and the solid substrate by ultrathin polymer cushions. These must be soft and highly hydrated in order to provide a natural environment for proteins penetrating the bilayer. On the other hand the polymer film must be stabilized by local anchoring in order to prevent the collapse of the lipid-polymer composite film due to Van der Waals attraction between the bilayer and the solid.2 In the present work we have explored the usefulness of dextran films as polymer cushions on glass surfaces, focusing on the preparation and characterization of the dextran films. The anchoring was achieved by deposition of silane layers onto glass slides containing a controllable density of epoxy groups to which hydroxyl groups of the dextran were coupled. The principle of the anchorage is similar to that used for the deposition of dextran layers on the gold-covered surfaces of the surface plasmon resonance sensor of Pharmacia.3 X Abstract published in Advance ACS Abstracts, September 1, 1996.

(1) Stelzle, M.; Weismu¨ller, G.; Sackmann, E. J. Phys. Chem. 1993, 97, 2974-2981. (2) Elender, G.; Sackmann, E. J. Phys. II 1994, 4, 455-479. (3) Lo¨fa˚s, S.; Johnson, B. J. Chem. Soc., Chem. Commun. 1990, 15261528.

S0743-7463(96)00282-X CCC: $12.00

In this paper we present measurements of the structural, viscous, and elastic properties of the dextran films by using latex beads as colloidal probes. The thickness of the polymer film was determined by measuring the distance of the beads from the solid surface by quantitative reflection interference contrast microscopy (RICM). The theory of RICM image formation was improved in order to increase the accuracy of the thickness measurements. By analyzing the distance fluctuations of the bead (in terms of the theory of Brownian motion of a particle in a potential), the (repulsive) interaction potential between the bead and the polymer film was determined and the polymer-induced repulsive forces were measured. An apparent viscosity of the dextran film was evaluated. An alternative interpretation of the damping in terms of the Brinkmann model of hydrodynamic flow in porous media is presented. Materials and Methods Sample preparation. Glass cover slides were cleaned by successive ultrasonification first in a 2 vol % aqueous solution of Hellmanex (Hellma GmbH, Mu¨lheim, Germany), second in Millipore filtered water (Millipore Milli-Q-System, Molsheim, France), and third once again in Millipore filtered water for 15 min, respectively. Between each sonification step the glass slides were rinsed about 10 times with Millipore filtered water. Thereafter the glass slides were dried for 1 h at 75 °C. Glass slides, cleaned in this way, are hydrophilic. They are called hydrophilic glass slides in the following. Hydrophilic glass slides were silanized with octadecyltrichlorosilane (OTS, Aldrich) by incubation for 30 s in a solution of 0.2 vol % OTS, 20 vol % chloroform, and about 80 vol % n-hexadecane. After incubation the glass slides were rinsed with chloroform and dried for 1 h at 75 °C. The resulting glass slides are hydrophobic. They are called OTS-silanized glass slides (OTS glass slides) in the following. To couple dextran to glass slides covalently, the procedure of Elam et al.4 was slightly modified: Hydrophilic glass slides were (4) Elam, J. H.; Nyren, H.; Stenberg, M. J. Biomed. Mater. Res. 1984, 18, 953-959.

© 1996 American Chemical Society

Ultrathin Hydrated Dextran Films Grafted on Glass

Langmuir, Vol. 12, No. 20, 1996 4867

Figure 1. Latex bead (radius R ) 14.9 µm) hovering over a dextranized GOPTS glass slide in a 100 mmol aqueous NaCl solution. (a) (+) Measured interaction potential of the bead V. h is the height of the bead above the glass slide. (s) Best fit assuming the effective gravitational force and the modified Hertz model. Optimized parameters: K ) 1400 N/m2, Wad ) 5.2 × 10-8 J/m2, L ) 111.0 nm. (b) (+) Measured autocorrelation function of the height fluctuations. τ is the time. (‚‚‚) Best fit assuming an exponential decay. Optimized parameters: 〈h2r (0)〉 ) 4.7 nm2, τr ) 0.13 s, see eq 4.

Figure 2. Latex bead (radius R ) 10.8 µm) adhering to an OTS glass slide in a 100 mmol aqueous NaCl solution. (a) (+) Measured interaction potential of the bead V. h is the height of the bead above the glass slide. (s) Best fit obtained by the modified Hertz theory and the gravitational force. Optimized parameters: K ) 143 000 N/m2, Wad ) 3.5 × 10-7 J/m2. (b) (+) Measured autocorrelation function of the height fluctuations. τ is the time. The relaxation time is clearly below the time resolution of the video frequency, since for τ > 0 the autocorrelation is not significantly different from zero.

silanized with (3-(glycidyloxy)propyl)trimethoxysilane (GOPTS, Aldrich Chemie GmbH, Steinheim, Germany) by incubation for 5 min in a 0.2 vol % solution of GOPTS in 2-propanol. This solution was prepared 5 min before incubation. After incubation the glass slides were dried for 1 h at 75 °C. The substrates were once again rinsed with 2-propanol and then dried for 30 min at room temperature (T ≈ 24 °C). Glass slides, silanized in this way, are hydrophobic. They are called GOPTS-silanized glass slides (GOPTS glass slides) in the following. GOPTS-silanized glass slides were dextranized by incubation of the substrates for 24 h at room temperature (T ≈ 24 °C) in an aqueous solution of dextran (MW ) 500k, randomly branched, Pharmacia Biotech Europe GmbH, Freiburg, Germany, concentration 30 g of dextran in 100 mL of Millipore filtered water). The dextran solution was prepared 6 h prior to substrate incubation. Prior to experiments freshly dextranized glass slides were washed in Millipore filtered water for 1 week, while shaking the glass slides on an incubator table and occasionally exchanging the water. Glass slides, prepared in this way, are hydrophilic. They are called dextranized GOPTS glass slides in the following. The success of each of the preparation steps can easily be checked by the different wettability of the different substrates with water. The chemistry involved in the covalent coupling of dextran can be read in standard organic chemistry textbooks, and only one point should be mentioned. If the epoxy group of the silane does not react with the hydroxyl groups of the dextran, it can react with water molecules (of the washing bath). This passivation of the epoxy group leads to a more hydrophilic substrate. As shown by Elender et al.5 the dextran layer of the dextranized GOPTS glass slides is covalently bound to the glass slide with an ellipsometric thickness of 0.8 nm for the dried dextran film. Reflection Interference Contrast Microscopy. In the experiments described here, reflection interference contrast microscopy (RICM) was used to determine the height of small polystyrene latex beads (Polysciences Inc., Warrington, PA) above glass slides. This height can be evaluated by analyzing the interference pattern (see Figure 5 in the Appendix) of the light

which is reflected from the glass slide/solution interface and the light which is reflected from the solution/latex bead interface (see Figure 6a in the Appendix). For that purpose one has to fit the experimental pattern with the theoretically calculated pattern of a bead, leaving the height of the bead as an adjustable parameter. In the past, RICM patterns were evaluated by two theories: a “simple theory” and a more refined “finite aperture theory”.6,7 The simple theory assumes that the illuminating light has a zero aperture angle and that the light is reflected at interfaces which are approximated to be parallel to the glass slide. The finite aperture theory takes into account a nonzero aperture angle of illumination. But as in the simple theory, it is assumed that the light is reflected at interfaces, which are approximated to be parallel to the glass slide. In our evaluation of the RICM patterns we have used a third approach. This approach takes into account the finite aperture angle of illumination and detection and the reflection at curved interfaces. This new theory of RICM image formation is described in the Appendix. With this new theory, the height of the bead above the glass slide can be measured with an accuracy of (4 nm and the difference between two heights is exact to (0.2 nm. RICM image formation theory is further complicated if an additional silane or polymer layer is adsorbed onto the glass slide. Thus more than two reflecting interfaces have to be taken into account. Luckily the refraction index, n, of silane layers is normally between 1.45 and 1.5,8 while the refraction index of glass is about 1.52 and that of the aqueous solution is about 1.33.8 Therefore, the difference in refraction indices is smaller at the silane/glass interface than at the silane/water interface. Since for zero angle of incidence the reflection coefficient is proportional to the square of the difference between the refraction indices, the reflection at the silane/water interface dominates over that at the glass/silane interface and the silane layer can be approximated as an additional glass layer. Moreover, if the adsorbed polymer layer swells very well in the presence of water, there is almost no difference in refraction indices between water (n ≈ 1.33) and the swollen polymer. As shown in the Discussion,

(5) Elender, G.; Ku¨hner, M.; Sackmann, E. Biosens. Bioelectron. 1996, 11, 565-577.

(6) Ra¨dler, J.; Sackmann, E. J. Phys. II 1993, 3, 727-748. (7) Gingell, D.; Todd, I. Biophys. J. 1979, 26, 507-526. (8) Weast, R. C. Handbook of Chemistry and Physics, 51st ed.; The Chemical Rubber Co. (CRC): Cleveland, OH, 1971.

4868 Langmuir, Vol. 12, No. 20, 1996

Ku¨ hner and Sackmann

the refraction index of the swollen dextran layer is about 1.332. Therefore the reflectivity at the dextran/water interface is small compared to that at the surface of the latex bead (n ≈ 1.55).8 The hardware setup for the RICM microscope was already described earlier.9 RICM images were recorded using a SuperVHS-video recorder (AG7350, Panasonic). Image processing was done on a Macintosh Quadra 950 with the help of a time base corrector (FA310, For-A, Tokyo), a video board (Perceptics, Knoxville, TN), and the Image software (Wayne Rasband, N.I.H.). The evaluation of the interference pattern was done on a Power Macintosh 6100 using the Igor software (Wavemetrics, Lake Oswego, OR). Measurement and Evaluation Procedure. Two kinds of height measurements were performed, single height measurements and height fluctuation measurements. For a single height measurement, about 20 RICM images of the bead were digitized from the videotape and the picture with the apparently smallest height was selected. For this picture a single interference pattern along a line passing through the center of the bead was recorded and analyzed. For the fluctuation measurements a computer program controlled the digitization of the RICM images, found the center of the bead, and recorded the interference pattern. This procedure was performed automatically for all the RICM images of the fluctuation sequence. The single interference pattern or the interference patterns of the fluctuation sequence were then automatically fitted to the theoretical description of the pattern to determine the height of the bead. Potential and Damping of the Latex Bead Fluctuation. The thermally driven motion of a latex bead in the potential V(h) of the glass slide surface can be described by a Langevin equation for the height h of the bead above the glass slide:

∂2hr

m

2

∂t

∂hr ∂V + ) fstoch ∂t ∂hr

(1)

+γ

Here hr(t) ) h(t) - 〈h〉 is the height relative to the most probable height 〈h〉 at the minimum of the potential V, t is the time, m is the mass of the bead, γ is the frictional coefficient, and fstoch is the fluctuating force of the thermal environment. The frictional coefficient at the interface may be described by the Reynolds formula10

Fr R2 γ) ) 6πηeff ∂h/∂t h

(2)

where ηeff denotes the viscosity of the medium at the glass surface and R denotes the radius of the bead. This viscosity is actually an effective viscosity, since it may include the effect of the pure water solution and the effect of molecules adsorbed at the surface. These effects will be treated more closely in the Discussion. The Langevin equation cannot be solved in the form stated above considering all dependencies on the height h. Therefore some approximations are made, which are experimentally (see Results) reasonable: (1) The Reynolds number

Re ) R

∂hr g 10 nm g F/ηw ≈ 10 µm‚ ‚1.05 3/0.01 ) ∂t 0.03 s cm‚s cm 4 × 10-10

is much smaller than 1. Thus the movement is overdamped and the inertia term can be neglected. (2) For small height fluctuations the potential V can be harmonically approximated

V)

∂2V ∂h2r

|

h2r ) V′′h2r

(3)

hr)0

(3) The frictional coefficient is assumed to be equal to its value at the average height 〈h〉. With these simplifications the Langevin equation can be treated analytically in order to calculate the autocorrelation function of the height fluctuations 〈hr(τ) hr(0)〉. (9) Ra¨dler, J.; Sackmann, E. Langmuir 1992, 8, 848-853. (10) Reynolds, O. Trans. R. Soc. London 1886, 177, 157.

By assuming that the damping of the motion can be described by a single correlation time τr, this autocorrelation function is a simple exponential function:

(∆h)2(τ) t 〈hr(τ) hr(0)〉‚e-τ/τr

(4)

The correlation time is given by

τr )

6πηeffR2 γ ) V′′ 〈h〉V′′

(5)

while the amplitude can be calculated using the equipartition theorem and is

〈h2r (0)〉 )

kBT V′′

(6)

Here kB is Boltzmann’s constant and T is the absolute temperature. Experimentally the autocorrelation function can be evaluated from the time sequence of the height fluctuations by

(∆h)2(τ) ) 〈[h(τ+t) - 〈h〉]‚[h(t) - 〈h〉]〉t

(7)

where 〈 〉t denotes the time averaging over the time sequence. The most probable height 〈h〉 can be approximated by 〈h〉 ) 〈h〉t because of the harmonic approximation of the potential. Since the radius of the bead can be measured by phase contrast microscopy, the temperature T is known, and 〈h〉 can be calculated from the fluctuation sequence, one can evaluate the elastic constant of the potential (V′′) and the effective viscosity. The height dependence of the potential V can be evaluated more precisely, because at thermal equilibrium the relative probability density p(h) of the heights is related to the potential by the Boltzmann distribution law:11

p(h) ) p0e-V(h)/kT

(8)

so that

( )

V(h) p(h) ) -ln ∝ - ln(p(h)) kT p0

(9)

In analogy to the autocorrelation function this probability density p(h) can be determined from the time sequence of the height fluctuations. In our experiments the time sequence of the heights normally consisted of about 800 values.

Results We have studied the Brownian motion of latex beads above two kinds of substrates: GOPTS- and OTS-silanized glass slides. For each kind of the substrates about 10 beads were examined. Figure 1a shows an example of the potential V(h) of a latex bead, hovering over a dextranized GOPTS glass slide in a 100 mmol NaCl solution at T ) 30 °C. Figure 1b shows the corresponding autocorrelation function (∆h)2(τ). Figure 2a shows an example of the potential of a latex bead, sitting on an OTS-silanized glass slide again in a 100 mmol NaCl solution at T ) 30 °C. Figure 2b shows the corresponding autocorrelation function. While there is no doubt about the experimental significance of the fluctuations of the beads on the dextranized substrate, the fluctuations of the latex bead on the OTS substrate are close to the theoretical height resolution by RICM, which is (0.2 nm (see Appendix). Thus, some care has to be taken about the potential of the latex beads on OTS. Nevertheless, one can clearly distinguish the two substrates. The height of the beads, the width of the potential, and the relaxation time of the autocorrelation function for the OTS-silanized (11) Prieve, D. C.; Bike, S. G.; Frej, N. A. Faraday Discuss. Chem. Soc. 1990, 90, 209.


Langmuir, Vol. 12, No. 20, 1996 4869 Table 1: Typical Height of Latex Beads above Different Substrates in NaCl Solutions as Measured by Single Height Measurements (See Measurement and Evaluation Procedure)a NaCl conc of the aqueous solution/mmol

single height over dextranized GOPTS glass slide/nm

single height over GOPTS glass slide/nm

2 100

103.0 ( 3.3 106.2 ( 5.5

33.8 ( 7.0

a

The values are averages over about 10 different samples, where each latex bead had a radius of about 12 µm. Table 2: Parameters of the Exponential Fits to the Autocorrelation Function of Height Fluctuations of Latex Beadsa

Figure 3. (a) Geometry of a latex bead as assumed for the microscopic approach. (b) Deformation of the bead and the dextran layer as assumed by the Hertz theory.

〈h〉/nm V′′/(J/m2) ηeff/(N‚s/m2) ηdex/(N‚s/m2) ξh(〈h〉)/nm

95.5 ( 2.2 (1.4 ( 0.4) × 10-3 (5.0 ( 2.6) × 10-3 1.7 3.4

a The latex beads are hovering over a dextranized GOPTS glass slide in a 100 mmol aqueous NaCl solution. Also shown are the resulting values for the parameters of the “viscosity model” (apparent viscosity ηdex) and of the “two-fluid model” (hydrodynamic correlation length ξh, see text). The values are averages over about 10 different samples.

2 to 0.5 mmol.9 Table 1 further shows the result of single height measurements of latex beads over a GOPTSsilanized glass slide in a 100 mmol NaCl solution at T ) 30 °C. The decrease in height compared to that of the dextranized GOPTS substrate clearly indicates the influence of the dextran layer. Table 2 shows the effective viscosity and the elastic constant of the potential as determined from the autocorrelation functions. One can clearly see that the viscosity is about a factor of 5 higher than that observed for the aqueous bulk solution. The latter can be approximated by the viscosity of pure water (ηw ) 0.001 N‚s/m2).8 Discussion

Figure 4. (+) Measured interaction potential of a polystyrene latex bead of radius R ) 14.9 µm. The bead is hovering over a dextranized glass slide in a 100 mmol NaCl solution. h is the height of the bead above the glass slide. Also the best fits of the data points are shown, as obtained by assuming the effective gravitational and the different models of repulsive interactions. (a) Electrostatic forces. In fit 1 (‚‚‚) the Debye length is held constant to the theoretically expected value. In fit 2 (s) the Debye length is optimized to κ(1) ) 0.14 ( 0.01/nm. Fit 1 is unacceptable, because of the very large deviations. Fit 2 is discarded, because the fitted value for the Debye length is too far away from the theoretically expected value, so that it cannot be explained by experimental inaccuracies or by the inadequacy of the DLVO theory. (b) (s) Simple spring model. Optimized parameters: E ) 81 N/m2, dextran layer thickness L ) 110.1 nm); (‚‚‚) scaling theory of adsorbed polymers according to deGennes.16

slide are much smaller than the corresponding values of the dextranized sample. Table 1 shows the results of single height measurements of latex beads over a dextranized GOPTS glass slide in a solution of two different NaCl concentrations at T ) 30 °C. No significant difference in height is observed by changing the concentration from 2 to 100 mmol. In contrast, for latex beads over hydrophilic (not dextranized, not silanized) glass slides an increase of about a factor of 1.8 is observed, when the concentration is reduced from

In the following we try to identify the major forces which determine the interaction potential V(h) of the latex beads over dextranized glass slides. Thereafter we discuss the damping of the motion of the beads in that potential. The interaction potential of the latex bead can be described by two different approaches, a microscopic and a macroscopic approach. Both approaches assume that the latex bead has a molecularly smooth surface and that only forces between the latex bead and the dextran layer are relevant. Forces between the glass slide or the GOPTS layer and the latex bead are ignored for two reasons: Firstly, the distance between these surfaces and the latex bead (≈100 nm) is larger than the typical theoretical interaction range (30 nm) of the molecular forces (see below). Secondly, the experiments show that the single height (see Measurement and Evaluation Procedure) of the latex beads over dextranized GOPTS glass slides is three times higher than the heights over GOPTS-silanized glass slides (see Table 1). This indicates that especially the repulsive part of the potential on dextran cannot be explained by the interaction between the silane layer or the glass slide but is mainly due to the dextran layer. Microscopic Approach. The microscopic approach assumes that the latex bead is not deformed during its motion in the potential and that the potential is determined by intermolecular forces like electrostatic, Van der Waals, hydration, or steric polymer forces. This situation is schematically shown in Figure 3a. The potential of the bead can be calculated from the energy per unit area f(d)

4870 Langmuir, Vol. 12, No. 20, 1996


of two parallel plates, separated a distance d, by integrating f(d) along the surface of the bead and using the socalled Derejaguin approximation:12

∫0 f(d(x))2πx dx xc

V(h) )

(10)

Here x is the lateral distance from the center of the plane and xc is a cutoff parameter, which depends on the range of the interaction. Further d(x) is the local distance between the glass slide and the bead in Figure 3a. This distance cannot be smaller than a minimal dmin due to short range repulsion forces (steric or hydration forces), and d(x) can be approximated for x , R as

d(x) )

{

dmin ≈ 0.3 nm, in the contact area of the bead and the dextran: |x| e x2R(L - h + dmin) x2 h + R - xR2 - x2 - L ≈ h + - L, 2R otherwise (11)

Here R is the radius of the bead and L is the thickness of the undeformed dextran layer. For all but nonsteric polymer forces xc can be chosen to be infinite. Only for the steric polymer forces, which are relevant when the dextran layer is deformed, is xc(s) ) [2R(L - h + dmin)]1/2 finite. The analytical form of f(d) or V(h) depends on the type of molecular interaction. Hydration Forces. The energy per unit area of two parallel plates is given by Marcelja and Radic:13

f(d) ) f0λhyde-d/λhyd

(12)

The relaxation length λhyd ≈ 0.3 nm is so small that the hydration force falls off more rapidly than any other intermolecular force. Thus, it is expected that the influence of the hydration force can be accounted for by assuming a finite distance dmin ≈ λhyd. Electrostatic Forces (DLVO Theory). The energy per unit area of two parallel plates with surface potentials ψ1 and ψ2 is given according to Verwey and Overbeck14 by

( ) ( )

eψ1 eψ2 -κd n tanh e f(d) ) 64kT tanh κ 4kT 4kT κ ) x2ne2/0kT

(13)

separated by a medium with index 3 is given by15

f(d) ) -

{

( ) ( )

eψ1 eψ2 nR tanh 4kT 4kT κ2

{1 + κ(L - h + dmin)}e-κdmin, h - dmin e L (deformed dextran layer) -κ(h-L) e , h - dmin > L (undeformed dextran layer)

(15)

The Hamaker constant is approximated by

A ) Aγ)0e-2κd + Aγ>0 Aγ)0 ) Aγ>0 )

3‚2πpγe 8x2

3 1 - 3 2 - 3 kT 4 1 + 3 2 + 3

× (n21 - n23)(n22 - n23)

x(n21 + n23)(n22 + n23){x(n21 + n23) + x(n22 + n23)}

(16)

where i are the dielectric constants and ni are the refraction indices at the main absorption frequency γe. This approximation is only valid for distances d smaller than 10 nm. For larger separations, f decays faster with d(1/d3). Using eq 16, it will be shown that Van der Waals forces are not relevant in the interaction potential V(h) of the bead over dextranized GOPTS glass slides. Thus the Van der Waals contribution to the potential V of the bead is omitted here. Steric Polymer Forces. Depending on the different features of the polymer layer, several theories predict different analytical expressions for the energy per area of a unidirectionally compressed polymer layer. Many of these theories make assumptions which are not fulfilled here. For example, the models of polymers forming mushroomlike or brushlike configurations assume that the linear polymer is anchored to the surface with one end only. This is not the case for our dextran layers, since the dextran polymer is randomly branched and several grafting sites may be possible. Thus the scaling theory of absorbed, linear polymers on surfaces may be considered: This theory assumes a semidilute solution of linear polymers between two parallel plates. Furthermore, the polymer adsorbs at one of the plates and the correlation length of the adsorbed polymer is equal to the distance from this plate. At the other plate no adsorption is considered. Applying the result of deGennes,16 who deals with the case of polymer adsorption on both plates, the energy per unit area is given by the osmotic pressure:

Here n is the concentration of the NaCl solution and 1/κ is the Debye length. Accordingly the potential is given by

V(h) ) 128πkT tanh

A 12πd2

f(d) )

c kT 2 d2

(17)

where c is a factor of order one. Thus,

V(h) )

(14)

Van der Waals Forces. The energy per surface area between two parallel plates indexed by 1 and 2 and (12) Safran, S. A. Statistical Thermodynamics of Surfaces, Interfaces and Membranes; Addison-Wesley Publishing Company: Reading, MA, 1994. (13) Marcelja, S.; Radic, N. Chem. Phys. Lett. 1976, 42, 129. (14) Verwey, E. J.; Overbeck, J. Th. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948.

{

[

cπkT 0,

]

R R - , h - dmin e L h - dmin L h - dmin > L

(18)

A more simple model, denoted as the simple spring model, treats the dextran layer as a spring, which is compressed in the vertical direction (z axis). This model starts from the Hook equation uzz ) p/E for the relation between the strain in the normal direction uzz and the disjoining pressure p in the case of a homogeneous compression17 of (15) Israelachvili, J. N. Intermolecular & Surface Forces; Academic Press: London, 1994. (16) deGennes, P. G. Adv. Colloid Interface Sci. 1987, 27, 189-209. (17) Landau, L. D.; Lifschitz, E. M. Theory of Elasticity; Pergamon Press: London, 1980; Vol. 7.


Langmuir, Vol. 12, No. 20, 1996 4871

the dextran layer along the z axis. Here E is the elastic modulus. From this relation one can calculate the total deformation u ) pL/E by simple integration along the z axis. Thus the energy per unit area is given by

f(d) )

∫0u)L-dp(u′) du′ ) 21 EL(L - d)2

(19)

and the potential of the latex beads is calculated as

{

(

)

h - dmin 3 π 2 , h - dmin e L EL R 1 V(h) ) 3 L h - dmin > L 0,

(20)

Macroscopic Approach. The macroscopic approach assumes that the latex bead and the dextran layer have smooth surfaces and that both are deformed elastically. This situation is shown in Figure 3b. Analytically the deformation of macroscopic bodies under an external force can be described by the Hertz theory.17,18 This theory has been extended to include specific adhesion forces by Johnson et al.19 and Derjaguin et al.20 Here only a simple modification of the Hertz theory is used. This modified theory differs from the original Hertz theory by considering an additional term, which describes the adhesion energy in the contact area. In this approach no minimal distance dmin is taken into account. Contribution of the Elastic Forces to the Potential of the Latex Bead. According to the Hertz theory the external force F which deforms two bodies is given by

F(u) ) a3K/R

(21)

Herein 2

K)

1 - σi 4 , ki ) πEi 3π(k1 + k2) R1R2 , a ) xuR R1 + R2

R)

(22)

with K the effective elastic module, ki the effective elastic constant, Ei the elastic moduli, σi the Poisson ratio, and Ri the radius of the undeformed body i. Since the dextran layer is flat, R reduces to the radius of the latex bead. Furthermore a is the radius of the contact area between the two bodies. The potential of the beads is calculated as

V(h) )

{

∫0u)L-hF(u′) du′ ) 52KR3(L R- h)

5/2

0,

, heL h>L (23)

Contribution of Adhesive Forces to the Potential of the Latex Bead. The adhesion energy is given by the adhesion energy per unit area times the contact area obtained by the Hertz theory:

V(h) )

{

-πWadR(L - h), h e L 0, h>L

(24)

Gravitational Force. The last relevant force of the potential is the gravitational force: (18) Hertz, H. J. Reine Angew. Math. 1881, 92, 156-171. (19) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324, 301-313. (20) Derjaguin, B. V.; Muller, V. M.; Toporov, Yu. P. J. Colloid Interface Sci. 1975, 53, 314-325.

V(h) )

4π - FWasser)gR3h (F 3 latex

(25)

with the different densities Fi and the gravitational acceleration g ()9.81 m2/s). Comparison of the Contributions of the Different Forces. In the following we try to decide which of the forces described above are relevant in the experiments described here. If not explicitly stated otherwise, we assume a bead radius which is determined by phase contrast microscopy, a specific density Flatex ) 1.05 g/cm3 8 of the latex beads, and a specific density FWasser ) 1.00 g/cm3 of the solution. The assumption of slightly higher latex bead densities, as suggested by Ra¨dler and Sackmann,9 does not change the interpretations substantially. Judging from their chemical formulas polystyrene latex beads and dextran molecules are expected to be uncharged in the pH range of our experiments (pH 5-6). But it has been reported that polystyrene latex beads are negatively charged in an aqueous solution.9 Thus electrostatic forces cannot be ignored a priori in the system of a latex bead over a dextranized glass slide. However, as the following two experimental results, show electrostatic forces are not responsible for the repulsive part of the potential. (1) Figure 4a shows the best fits to the experimental potential assuming repulsive electrostatic forces and the effective gravitational force. In fit 1 the inverse Debye length is fixed to the theoretical value κ ) 1.02/nm calculated from eq 13. Clearly the deviation of the fit from the measured potential is unacceptably high. In fit 2 the inverse Debye length is optimized to κ(1) ) 0.14 ( 0.01/nm. This value is too low compared to the theoretical value predicted by eq 13. It cannot be explained by experimental insufficiencies or limitations of the DLVO theory. (2) According to Table 1 no increase in height of the latex beads is observed when the NaCl concentration is decreased. In contrast, a significant increase is observed for latex beads hovering over hydrophilic glass slides.9 Thus the dextran layer is not or at most very weakly charged, and in the experiments described here electrostatic forces can be ignored. To decide whether Van der Waals forces are relevant for the repulsive part of the potential, we determine the sign of the Hamaker constant in the three-layer system of the dextran layer, the solution, and the latex bead. Similar to the case for other polymers the dielectric constant of the dextran molecules is expected to be much smaller than the dielectric constant of water (r ) 81). Furthermore, the dielectric constant of polystyrene is r ) 2.5521 and thus is also smaller than that of water. Therefore, eq 16 yields a not negative value for Aγ)0. Since the refraction index of the dextran layer is between that of the pure solution (n ) 1.33) and that of the pure dry dextran (approximated to be n ) 1.56, the value of dry sucrose8), the same holds for Aγ>0 (see eq 16). Thus the total Hamaker constant A is not negative and Van der Waals forces cannot account for the repulsive part of the potential. The repulsion is therefore expected to be solely determined by steric polymer forces. Figure 4b shows fits to the experimental potential assuming different models of steric polymer forces and the effective gravitational force. The data have been fitted according to both the simple spring theory and the scaling theory for adsorbed polymers. Obviously only the simple spring theory can describe the potential adequately, whereas the scaling theory for adsorbed polymers fails. (21) Brandrup, J.; Immergut, E. H. Polymer Handbook, 3rd ed.; John Wiley & Sons: New York, 1989.

4872 Langmuir, Vol. 12, No. 20, 1996


Table 3: Parameters Obtained by fitting to the Potential of Latex Beads Hovering over a Dextranized Glass Slide in a 100 mmol NaCl Solutiona thickness of the dextran layer/nm E/(N/m2)

101.6 ( 5.2 300 ( 160

Table 5: Effective Elasticity (k) and Elastic Modulus (E) for the Polystyrene Latex Beads and the Dextran Layer as Given by the Hertz Theory ki/(m2/N) polystyrene latex bead

4.7 ×

dextran layer assuming an undeformed latex bead dextran layer assuming a deformed latex bead

1.02 ×

10-6

a

The fits take account of the simple spring model and the effective gravitational force. The values are averages over about 10 different samples. Table 4: Fitted Parameters of the Potential of Latex Beads above Different Substrates as Obtained by the Modified Hertz Model and the Effective Gravitational Forcea dextranized GOPTS glass slide K/(N/m2) Wad/(J/m2) thickness of the dextran layer/nm a

4160 ( 2240 (2.5 ( 5.0) × 10-8 101.6 ( 5.0

OTS glass slide 90000 ( 750000 (4.3 ( 1.2) × 10-7

Values are averages over about 10 different latex beads.

This failing of the adsorbed polymer model has two reasons: Firstly, this theories assumes a high compression of the polymer layer. This is not true in our case, since the undeformed thickness of the dextran layer is about 100 nm, whereas the deformation is smaller than about 10 nm. Secondly, this theory holds for linear polymer chains and is thus not adequate for the randomly branched dextran molecule. In contrast, the simple spring model is more suitable for small deformations and does not explicitly take into account the special features of the polymer molecules. Table 3 shows the results of the fits to all the experimental potentials. Since the simple spring theory can well reproduce the repulsive part of the potential, elastic polymer forces and the effective gravitational force are thought to be dominant in the interaction of latex beads and the dextran layer. Whereas electrostatic and Van der Waals forces do not contribute appreciably to the repulsive part of the interaction potential, they may contribute to the attractive part. But our fits to the experimental data do not exhibit statistically significant attractive electrostatic or Van der Waals forces. A remark should be made about the existence of socalled bridging forces.16,22 These forces may arise when dextran molecules which are grafted to the glass slide additionally adsorb onto the latex bead. Since only the simple spring theory can explain the repulsive part of the potential, one has to consider only bridging forces in the framework of this model. Luckily, in the simple spring model, the heights of the latex bead above the glass surface are smaller than the thickness of the dextran layer (see Figure 4b). Because an attractive bridging force is only relevant when the height of the bead is higher than the thickness of the undeformed dextran layer, the bridging forces are not expected to affect the potential in our situation. It should be noted that the measured interaction potential can also be explained by the modified Hertz theory. Figure 1a shows the best fit of the modified Hertz model to the experimental potential of a latex bead over a dextranized glass slide, and Figure 2a shows the best fit to the experimental potential of a latex bead over a OTS-silanized glass slide. Obviously the Hertz theory describes the situation very well. In Table 4 the averaged, fitted parameters of all experiments are shown. Using eq 22, one can first evaluate the apparent elastic modulus of the latex bead from the potential on the OTS(22) Ji, H.; Hone, D.; Pincus, P. A.; Rossi, G. Macromolecules 1990, 23, 698-707.

a

10-4

9.7 × 10-5

σia

Ei/(N/m2)

0 0.33 0 0.5 0 0.5

67500 60150 3120 2340 3270 2450

σi is the Poisson ratio of the latex beads or the dextran layer.

silanized glass slide and then the elastic modulus of the dextran layer from the potential on the dextranized glass slide. Table 5 shows the resulting values. From Table 5 one can conclude the following: (1) The elastic modulus of the latex beads is 5 orders of magnitude smaller than the value 3 × 109 N/m2 given in the polymer literature.21 This difference of the elastic modulus may be explained by a surface region whose polymer (chain) density, and thus the elastic modulus, is smaller than that in the bulk of the bead. AFM experiments on the yield strength of polystyrene latex beads in air by Schaefer et al.23 support this point of view. Schaefer et al. found that the yield strength of the polystyrene latex beads is about 3 orders of magnitude smaller than the yield strength of the bulk amorphous polymer. Schaefer et al. explained their difference by microscopic asperities of the bead. These asperities cause the density of the polymer at the bead surface to be smaller than that of the bulk of the polymer. Thus our possible explanation is consistent with the explanation of Schaefer et al. The interpretation of the difference in elastic modulus is further complicated by the surface roughness of the glass slide, which is reported to be about 2 nm24 and thus comparable to the magnitude of the fluctuations of the latex bead on OTS-silanized glass. Since the modified Hertz theory does not take into account the roughness of the glass surface, the application of the theory may lead to an apparent difference between the surface and the bulk values of the elastic modulus. However, quite another possible explanation for the difference in the measured elastic modulus is that the fluctuations over OTS-silanized glass slides are actually an artifact of the finite resolution of the height determination and that the elastic modulus at the surface is actually the same as that in the bulk. At present we are not able to decide between the different explanations mentioned above. At least, the elastic modulus of ≈6 × 104 N/m2 is a lower bound of the true value of the elastic modulus. For the determination of the structural and elastic properties of the dextran layer it is not essential to decide between the different explanations: Evaluating the elastic modulus of the dextran layer according to each explanation leads to almost the same result. (2) The elastic modulus of the dextran layer is by a factor of 20 smaller than the elasticity of the latex bead. Therefore the assumption of an incompressible latex bead in the microscopic approach is justified. The elastic modulus obtained for the dextran layer by application of the microscopic model (spring theory) is by a factor of 20 smaller than the value obtained by evaluating the data in terms of the macroscopic model (modified Hertz theory). This cannot be attributed to the neglect of the transverse concentration in the framework of our simple (23) Schaefer, D. M.; Carpenter, M.; Reifenberger, M.; Demejo, L. P.; Rimai, D. S. J. Adhesion Sci. Technol. 1994, 8, 197-210. (24) Karrasch, S.; Dolder, M.; Schabert, F.; Ramsden, J.; Engel, A. Biophys. J. 1993, 65, 2437-2446.


Langmuir, Vol. 12, No. 20, 1996 4873

spring theory, since the incorporation of that fact would lead to an even greater difference of the elastic moduli.17 The difference is more likely a consequence of the different assumptions of both theories: The macroscopic modified Hertz theory approximates the dextran layer as a half space and ignores the finite thickness of the dextran layer. This can be seen from eq 23, showing that the potential is only determined by the deformation (L - h) of the dextran layer. In contrast, the microscopic spring model explicitly takes into account the finite thickness of the dextran layer. This can be seen from eq 20, showing that both the magnitude of deformation and the thickness of the dextran layer determine the potential. Since the thickness of the dextran layer is about equal to the size of one hydrated dextran molecule, the microscopic point of view is regarded to be more reliable. At this point a remark about the degree of swelling of the dextran layer should be made: From ellipsometry it is known that the thickness of the dry dextran layer is about 0.8 nm.5 If the dextran layer is homogeneous, this thickness should correspond to a monolayer of dextran. In contrast the thickness of the swollen dextran layer is about 100 nm. Thus the dextran layer swells about a factor of 100 in an aqueous solution. Assuming a density of F ) 1.56 g/cm3 (density value of sucrose8) for dry dextran and a molecular weight of mr ≈ 171 (half of the value for the mass of sucrose8) for one sugar monomer unit, one obtains a monomer density of ≈0.057/nm3 for the swollen dextran layer. Since the density of the water molecules is ≈32/nm3 (as evaluated from the density of water and the molecular weight of water), the mass fraction of water in the dextran layer is 99%, which corresponds to a density of the dextran layer of F ≈ 1.004 g/cm3. This density differs only slightly from the density of water or of the solution (FWasser ) 1.00 g/cm3). Additionally the difference of the refraction index between water and the dextran layer is negligible. This is explained in the following. The refraction index nh of the swollen dextran layer can be estimated according to the Garnet equation25,26

n2h - n2w

n2t - n2w ) Φ n2h + 2n2w n2t + 2n2w

(26)

Here nt denotes the refraction index of dry dextran, which is approximated by the refraction index of sucrose nt ) 1.54.8 nw ) 1.33 is the refraction index of water. Φ ) 1/100 is the volume fraction of the dextran. Using these values, a refraction index nh ) 1.332 for the swollen dextran layer is calculated. Concerning its optical properties, the dextran layer is therefore indistinguishable from water or solution and the RICM image is not influenced by reflections at the interface between the swollen dextran layer and the water. In the remaining part of the section we discuss the enhanced damping of the latex beads as shown by the high effective viscosity of the media surrounding the bead. The enhanced damping is attributed to the additional friction of the bead, caused by the immersion of the bead into the dextran layer (see discussion of the potential). Since the hydrodynamic problem of the motion in such a system cannot be solved exactly, the following approximation is made: The frictional force Fr of the bead consists of two additive terms.

Fr ) Fr,w + Fr,dex

∂h R2 Fr,w ) 6πηw ∂t h

(28)

In the region where the bead touches the dextran layer, the frictional force is determined by two contributions: the direct friction due to the polymer surface and the friction caused by the water of the dextran layer. Since the dextran is covalently bound to the glass slide, the polymer molecules can only glide locally. Thus only elastic but not viscous forces are exerted on the bead. Therefore the friction is mainly attributed to the friction of the water in the dextran layer. The motion of the water within the dextran layer is itself determined by the friction between water and polymer. To evaluate this frictional force, two models are proposed: In the first model (“viscosity model”) the water of the dextran layer is treated as a medium with an apparent viscosity ηdex. Solving the Navier Stokes equation and integrating the resulting hydrodynamic pressure over the part of the surface of the bead which is immersed in the dextran layer leads to

Fr,dex ) 6πηdex

2

(L -L h) ∂h∂t Rh 2

(29)

(This equation is obtained by modifying the calculation of Kim and Karrila27). Here once again L is the thickness of the undeformed dextran layer. In the second model (“two-fluid model”28) the dextran layer is treated as a porous medium, through which water can flow. Such a system can be described by the Brinkmann equation29

ηw∇2vj -

ηw

(vj - u j˘ ) - ∇P ) 0

ξh(h)

(30)

Here vj denotes the velocity of the hydrodynamic flow field. ηw is the viscosity of bulk water. P is the hydrodynamic j˘ is its pressure. u j is the shear field of the polymer, u derivative with respect to time, and ξh is the phenomenological decay length of the hydrodynamic field. Since the simple spring model, which describes the potential best, assumes an homogeneous dextran layer, ξh is proportional to the inverse of the polymer density. Thus ξh ∞ h. The polymer is covalently fixed to the glass slide j˘ | holds and u j˘ can be and does not flow. Therefore |vj | . |u neglected in eq 30. With these approximations the Brinkmann equation can be solved. Integrating the resulting pressure over that part of the surface of the bead which is immersed into the dextran layer leads to

( )(

1 h ∆Fr,dex ) 6πηw 12 ξ(h)

2

1-

)

h ∂h R2 L ∂t h

(31)

(by modifying the calculation of Fredrickson and Pincus30). The total frictional force can be characterized by an effective viscosity ηeff according to eq 2. This effective

(27)

Fr,w is the frictional force exerted by the bulk solution. (25) Garnet, M. Philos. Trans. 1904, A203, 385-420. (26) Garnet, M. Philos. Trans. 1906, A205, 237-288.

Fr,dex is the frictional force exerted by the dextran layer. In the region far above the dextran layer the hydrodynamic flow field around the bead is similar to the field around a bead hovering over a bare, hydrophilic glass slide. Thus the friction of the bead in the solution can be approximated by (see also eq 2)

(27) Kim, S.; Karrila, S. J. Microhydrodynamics: Principles and Selected Applications; Butterworth-Heinemann: Boston, 1991. (28) Sens, P.; Marques, C. M.; Joanny, J. F. Macromolecules 1994, 27, 3812-3820. (29) Brinkmann, H. C. Appl. Sci. Res. 1947, A1, 27. (30) Fredrickson, G. H.; Pincus, P. Langmuir 1991, 7, 786-795.

4874 Langmuir, Vol. 12, No. 20, 1996

viscosity ηeff is measured by our experiments. Combining eqs 2, 27, 28, and 29 or 31, the effective viscosity ηeff can be related to the parameters of the two models mentioned above:

{[

ηeff )

[

)] ) ( )]

ηdex L - h 2 , for the “viscosity model” ηw L 1 h 2 h 1, for the “two fluid model” ηw 1 + 12 ξh(h) L (32)

ηw 1 +

(

(

Table 2 shows the apparent viscosity ηdex according to the viscosity model and the hydrodynamic decay length ξh obtained by the two-fluid model. The apparent viscosity is by a factor of 1700 higher than the viscosity of water and by a factor of 1400 higher than the viscosity of a 2% (w/w) solution of sucrose (0.0012 Ns/m2 31). It should be noted that the density of a 2% (w/w) solution of sucrose is slightly higher than the density of the swollen dextran layer. The large difference of ηdex with respect to the sucrose solution indicates the influence of the covalent anchoring of the dextran molecules to the GOPTS glass slides. This coupling leads to a connected, porous dextran network which exhibits a high friction to streaming water. The hydrodynamic decay length is about equal to the mean distance of 2.6 nm between two dextran monomer molecules (calculated from the monomer density mentioned above). The damping of the latex beads on OTS is completely different. The relaxation time is smaller than the integration time of video microscopy. This indicates that the latex bead adheres to the OTS glass slide and that the damping is determined by elastic and not by viscous forces. (Large V′′ values dominate over all other parameters in eq 5.) Appendix: Nonlocal Theory of RICM Image Formation In the following we describe an improved method to analyze quantitatively the interference pattern generated by a latex bead located over the surface of a glass slide (Figure 5). The image formation is illustrated in Figure 6a. Focusing on the surface of the glass slide, the intensity measured at position X is the time average of the interference between light reflected at the glass surface (field amplitude at position X, E01) and light reflected from the latex bead (field amplitude at position X, E12):

I(x) ∝ 〈E2〉t

∫ ∫E12 dΩ12)2〉t ∝ 〈∫E01E′* 01 dΩ01 dΩ′01 + ∫E12E′* 12 dΩ12 dΩ′12 + ∫(E01E*12 + E*01E12) dΩ01 dΩ′12〉t (A.1) ∝ 〈( E01 dΩ01 +

where

Ωij ) (ϑij,φij) denotes the polar angles of incidence at the position X. In order to perform the time averaging 〈 〉t, one has to consider the coherence function of the light source in a rigorous theory. However, since this function is not known explicitly, the so-called parallel illumination approxima(31) Lide, D. R. Handbook of Chemistry and Physics, 75th ed.; CRC Press, Inc.: Boca Raton, FL, 1994.


tion is made, which assumes that only parallel illuminating light rays are coherent.7 Additionally, two other points have to be considered. Firstly, the phase difference is conserved from the position X until the light rays reach the camera. Secondly, the coherence length, typically 30 µm for the high-pressure mercury lamp32 used in our experiments, is longer than the typical distances between the glass slide and the latex bead, which are smaller than 5 µm in our experiments. Thus the phase difference between the reflected rays is maintained. This phase difference can be evaluated from the local geometry, and the intensity can be approximated by (see Figure 6a)

∫ ∫E12E′*12δ(ϑ12-ϑ′12)δ(φ12-φ′12) dΩ12 dΩ′12 + ∫(E′01E*12 + E′*01E12)δ(ϑ′01-fϑ(ϑ12,φ12))δ(φ′01-

I(x) ∝ { E01E′* 01δ(ϑ01-ϑ′01)δ(φ01-φ′01) dΩ01 dΩ′01 +

fφ(ϑ12,φ12)) dΩ′01 dΩ12} (A.2)

In this equation δ( ) is the delta function. The functions fφ(φ12,ϑ12) ) φ01 and fϑ(φ12,ϑ12) ) ϑ01 correlate the angles of incidence of the coherent rays, reflected from the glass (φ01,ϑ01) and from the latex bead (φ12,ϑ12, see Figure 6a). Unfortunately, these functions cannot be evaluated analytically for arbitrary surfaces. To evaluate the integral, the following additional simpliciations are introduced: Firstly, the reflection coefficients (rij) for finite angles of incidence are approximated by their value for an angle of incidence of zero. Secondly, the amplitude of the illumination (E0) is independent of the angle of incidence and of the position X. Thirdly, the amplitude is treated as a scalar and not as a vector, corresponding to the assumption that all Ei are polarized in the same direction. Fourthly, the light source is monochromatic. With this simplification one obtains (see Figure 6a)

E01(Ω01) )

{

r01E0, 0 e ϑ01 e RIA 0, otherwise

{

E12(Ω12,x) ) 2 Θ(RIA - fϑ(φ12,ϑ12))(1 - r01 )r12E0e-ik∆(Ω12,x), 0 e ϑ12 e RDA 0, otherwise

∆(Ω12,x) ) n1(BA + AX - CX)

(A.3)

In these equations Θ(R) is the Heaviside function, k ) 2π/λ is the wavenumber, and λ is the wavelength of the illuminating light. ni is the refraction index of the medium i. Furthermore, RDA is the aperture angle of detection, and RIA is the aperture angle of illumination, as evaluated in the medium of the solution. The Heaviside function Θ(R) ensures that the reflected ray only exists when the corresponding illumination ray exists too. BA, AX, CX are the distances between the points depicted in Figure 6a. This figure is only two dimensional in order to simplify the representation. Therefore, it appears that the optical path difference ∆(Ω12,x) could be evaluated analytically. However, this is not possible for the threedimensional case. The same holds for the functions fφ and fϑ, as already mentioned. Insertion of eq A.3 into eq A.2 leads to (32) Reynolds, G. O.; DeVelis, J. B.; Parrent, J. B.; Thompson, B. J. The New Physical Optics Notebook: Tutorials in Fourier Optics; SPIEsOptical Engineering Press: Washington, DC, 1990.


Langmuir, Vol. 12, No. 20, 1996 4875

Figure 5. RICM Pattern (+) of a latex bead which adheres to an OTS-silanized glass slide. The pattern is recorded along a line through the contact point of the bead with the glass slide. Also shown are the best fits obtained by the nonlocal theory (s), the simple theory (‚‚‚), and the finite aperture theory (- - -), respectively.

( )

∫

RIA 2π,RDA (r01E0)2 + 0,0 Θ(RIA 2 2 2 fϑ(φ12,ϑ12))[((1 - r01 )r12E0)2 + 2r01(1 - r01 )r12E02 × cos(kδ(Ω12,x)] dΩ12 (A.4)

I(x) ∝ 4π sin2

This integral of the nonlocal theory can only be evaluated numerically. In contrast to the nonlocal theory, in a simple theory only an optical path difference of twice the local height times the refraction index (2n1h, see Figure 6b) is considered. Thus the intensity can be expressed as 2 )r12E0)2 + I(x) ∝ (r01E0)2 + ((1 - r01 2 2r01(1 - r01 )r12E02 cos(2kn1h(x)) (A.5)

In the framework of the finite aperture theory6,7 the optical path difference of a single illuminating beam is approximated by 2n1(BA - CX), where the distances BA and CX are measured between the points indicated in Figure 6c. The contributions of the different rays are integrated over the illumination aperture. Thus the intensity can be expressed as

( )

RIA I(x) ∝ 4π sin 2 2

{

2

(r01E0) + ((1 -

(

2 r01 )r12E0)2

( )) ( ) ( )))

sin 2kh(x) sin2

2 )r12E20 2r01(1 - r01

RIA 2

RIA 2kh(x) sin 2 2

(

(

+

cos 2kh(x) 1 - sin2

RIA 2

×

}

(A.6)

To test the validity of the nonlocal theory and to compare it with the simple theory and the finite aperture theory, respectively, the RICM picture of a latex bead adhering to an OTS glass slide is analyzed in the following. Figure

Figure 6. Local geometry of a latex bead hovering above a glass slide. The figure shows the interfering rays of the RICM according to the nonlocal theory (a), the simple theory (b), and the finite aperture theory (c). For simplification only the twodimensional geometry is shown.

5 shows the interference pattern along a line drawn through the contact point. Additionally shown are the best fits according to the simple, the finite aperture, and the nonlocal theory, respectively. In these fits the refraction indices of the different media and the aperture angles of illumination and of detection are held constant. Their values are given in the literature8 or were obtained by separate independent measurements. The fitted parameters, the radius of the bead and the height of the bead above the glass slide, are given in Table 6. A closer inspection of Figure 5 and Table 6 shows the following: Firstly, the fit according to nonlocal theory has the smallest deviations from the experimental interference pattern. Secondly, the radius of the bead as determined by phase contrast microscopy agrees best with the value obtained by the nonlocal theory. Thirdly, the value of the height obtained by the nonlocal theory is closer to zero than the corresponding values of the other theories. The deviation from zero in the case of the nonlocal theory may

4876 Langmuir, Vol. 12, No. 20, 1996

be attributed either to the inaccuracy of focusing at the glass surface, which causes an experimental error of the height measurement of (4 nm, or to the surface roughness of the substrate, which is typically in the nanometer range.24,33 As shown in the Discussion and as can be seen from Figure 2a, the deformation of the bead can be neglected compared to the errors mentioned above. Thus the nonlocal theory describes the interference pattern more adequately than the other theories. Nevertheless, there remain significant deviations between the nonlocal fit and the experimental pattern. These deviations are due to the approximations introduced to calculate the intensity. The most severe assumption is that the reflection coefficients do not depend on the angle of incidence. As can be seen from Table 6 the values of the absolute height obtained by the nonlocal theory and the finite aperture theory are different. However, simulations have shown that the difference of two absolute heights, obtained by the nonlocal theory, agrees to within 0.1 nm with the difference obtained by the finite aperture theory, if the difference is smaller than 10 nm. For the determination of the height fluctuations around a reference value, one can therefore apply the following simplified procedure. The height fluctuations are determined by the analytic expression of the finite aperture theory, whereas the reference height is determined by (33) Radmacher, M. Thesis, Technische Universita¨t Mu¨nchen, 1993.

Ku¨ hner and Sackmann Table 6: Parameters of the Best Fits to the RICM Interference Pattern of the Latex Bead Shown in Figure 5a

theory

optimized parameters of the fits phase contrast radius of the height of the radius of the bead/nm bead/nm bead/nm

simple 12080 ( 100 finite aperture 11160 ( 80 nonlocal 10720 ( 70

13.3 ( 3.4 13.7 ( 2.5 3.8 ( 1.7

10830 ( 500

a The fits are evaluated according to the nonlocal theory, the simple theory, and the finite aperture theory.

the numerical solution of the nonlocal theory. Finally, the difference of the values from the nonlocal and the finite aperture theories can be calculated for the reference height, and the values of all the heights obtained by the finite aperture theory can be corrected. The statistical accuracy of the differences in height, which is relevant for the significance of the fluctuations, can be estimated as follows: A realistic noise is superimposed on a simulated interference pattern given by the nonlocal theory, and the noisy pattern is evaluated by application of the finite aperture theory. This procedure is repeated several times. The probability distribution of the fitted heights in Gaussian and has a standard deviation of 0.2 nm. Thus the accuracy of the relative heights is about (0.2 nm. LA960282+

Ultrathin Hydrated Dextran Films Grafted on Glass: Preparation and

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