Langmuir 2007, 23, 10779-10787
10779
Nanomechanical Properties of Mechanical Double-Layers: A Novel Semiempirical Analysis B. Cappella* and D. Silbernagl Federal Institute for Material Research and Testing (BAM), VI.21, Unter den Eichen 87, 12205 Berlin, Germany ReceiVed April 27, 2007. In Final Form: July 23, 2007 Force-displacement curves have been acquired with a commercial atomic force microscope on a thin film of poly(n-butyl methacrylate) on glass substrates. The film thickness is nonuniform, ranging in the measured area from 0 to 30 nm, and gives the possibility to survey the so-called “mechanical double-layer” topic, i.e., the influence of the substrate on the mechanical properties of the film in dependence of the film thickness. The stiffness and the deformation for each force-distance curve were determined and related to the film thickness. We were able to estimate the resolution of the film thickness that can be achieved by means of force-distance curves. By exploiting the data acquired in the present and in a previous experiment, a novel semiempirical approach to describe the mechanical properties of a mechanical double-layer is introduced. The mathematical model, with which deformation-force curves can be described, permits to calculate the Young’s moduli of film and substrate in agreement with literature values and to determine the film thickness in agreement with the topography.
Introduction Investigation of mechanical properties is one of the most instructive and straightforward tool in material characterization, since it gives information about the structure and the applicability of a material.1 Methods to examine the mechanical properties in a macro scale are well-known and well in use, e.g., thermal mechanical analysis (TMA) and dynamic mechanical analysis (DMA).1,2 But in recent years material characterization in the nano scale gained in importance. On one hand applications have been increasingly minimized, and materials are actually implemented in this scale.3-5 On the other hand highly specialized materials are solely obtainable as colloids or thin films.6-8 Therefore, measuring techniques have to be developed in order to characterize materials in the same scale as they are deployed or available. In order to increase the resolution, atomic force microscopy (AFM)9,10 and nanoindenter are applied to this task. The basic approach for measuring mechanical properties of bulk materials consists in performing a deformation or penetration of the sample through the application of a certain load by means of a probe (indentation). Knowing the probe position relative to the sample, the sample deformation can be determined as a function of the applied force. Through the deformation and the use of an adequate continuum elastic theory, the mechanical properties of the probed volume can be determined. The probed volume is the section of the sample in which the force field is spread. The extent of the force field depends on the applied load and on the probe-sample contact area. Enlarging the contact (1) Elias, H.-G. Makromoleku¨le [Macromolecules], 6th ed.; Wiley-VCH: Weinheim, 2002. (2) Domininghaus, H. Polymers and Their Properties, 5th ed.; Springer: Berlin, Heidelberg, New York, 1998. (3) Holme, I. Surf. Coat. Int. B 2006, 89, 343. (4) Simunkova, S.; Blahova O.; Stepanek I. J. Materials Processing Technology 2003, 133, 189. (5) Xu, S.; Shi, Y.; Kim, S. Nanotechnology 2006, 17, 4497. (6) Decher, G., Schlenoff, J. B., Eds. Multilayer Thin Films, 1st ed.; WileyVCH: Weinheim, 2003. (7) Valette, L.; Pascault, J.; Magny, B. Macromol. Mater. Eng. 2003, 288, 642. (8) Burgert, I. Am. J. Bot. 2006, 93, 1391. (9) Cappella, B.; Dietler, G. Surf. Sci. Rep. 1999, 34, 1. (10) Butt, H. J.; Cappella, B.; Kappl, M. Surf. Sci. Rep. 2005, 59, 1.
area by choosing a larger probe reduces the pressure on the sample. The probed sample depth is reduced, but the width is increased, due to the larger contact area. Hence, the depth resolution is enhanced at the expense of the lateral resolution. The reasonable way of reducing the probed volume is to reduce the applied load by using a probe with a rather small contact area. Since an AFM allows applying forces in the range of 1 nN to 100 µN and uses probes with a radius in the magnitude of 10 nm, its merit is the probing of a comparatively small volume of the bulk material, a spatial resolution that is not reached by any other method. Nonetheless, in some cases, even such a resolution is not sufficient. One of these cases is the study of thin films (film thickness below 100 nm), which are usually deposited on a substrate with considerably different mechanical properties, forming a mechanical double-layer. The force field exerted during the measurement may spread beyond the film, so that the measured volume eventually includes the substrate. In the case of a compliant film on a stiff substrate, experience shows that the substrate influences the measurement if deformations are larger than 10% of the film thickness.11 This criterion is merely a rule of thumb, and the value of 10% cannot be assessed with precision. In the case of a stiff film on a compliant substrate no guideline, derived from experience, is even established. Unfortunately there is no established theory to analyze the experimental data obtained from mechanical double-layers. Existing elastic continuum theories can be applied to the analysis of deformation-load curves only if the sample is homogeneous. They do not hold for inhomogeneous samples, such as mechanical double-layers. To solve this problem either a further reduction of the probed volume has to be achieved, enhancing the depth resolution, or a theory has to be developed which allows extracting the mechanical properties of the film (and eventually of the substrate) from the experimental data obtained on a mechanical doublelayer. A further reduction of the probed volume can be achieved by reducing the applied load and hence the deformation, but due to the depth resolution of an AFM the measurement would yield (11) Domke, J.; Radmacher, M. Langmuir 1998, 14, 3320.
10.1021/la701234q CCC: $37.00 © 2007 American Chemical Society Published on Web 09/07/2007
10780 Langmuir, Vol. 23, No. 21, 2007
data with hardly any significance. Since the influence of the substrate cannot be avoided, it has to be quantified, so that the mechanical properties of the film can be obtained, at least indirectly. For this purpose the sample is considered as a mechanical system, whose mechanical properties are composed of the properties of the single materials. Further, it is reasonable to assume that the composition is depending on the film thickness. To this issue some semiempirical approaches and equations have been developed,12,13 but in a former publication we could show that those approaches are flawed and cannot be considered as a global solution for this problem.14 For the analysis of our present experiment we have extended the fit previously used for the description of plastic deformations15,16 to the analysis of the deformation-load curves on mechanical double-layers. Fitting equations have been applied to experimental data acquired on a sample with a nonuniform film thickness, in order to quantify the effect of the film thickness. Materials and Methods Sample Preparation. Poly(n-butyl methacrylate) (PnBMA) was chosen as material to form polymeric films; commercially available glass slides (Menzel-Glaeser, Germany) were found to be an adequate substrate material. The PnBMA granulate was purchased from Scientific Polymer Products (Ontario, NY) and was used without further purification. The molecular weight, the polydispersity index and the glass transition temperature given by the supplier are Mw ) 319 000 g/mol, Mw/Mn e 2.58, and Tg ) 22 °C. PnBMA permits formation of thin (10-100 nm) and ultrathin (1-10 nm) films by spin coating from a solution of PnBMA in toluene14 and meets the most important requirements of this experiment: 1. A low elastic modulus is desired to make the mechanical properties of the film and the substrate as different, and therefore distinguishable, as possible. To this aim, a polymer in the glassy state (T < Tg) would be too stiff. On the other hand a polymer in the viscoelastic or rubbery state (T > Tg) would be likely to undergo plastic deformations even at small loads, which is highly undesired in this experiment since exclusively the elastic deformations are examined. Due to its glass transition temperature (22 °C), at roomtemperature PnBMA the polymer is in the transition region and has therefore the desired low elastic modulus without undergoing plastic deformations at very low loads. 2. Since the glass-transition region of PnBMA is rather broad (∆T ≈ 70 K), changes of Young’s modulus of PnBMA due to slight changes in temperature ((3 K) can be neglected. 3. The mechanical behavior of the bulk PnBMA is well-known and was examined in detail by means of force-displacement curves in our previous experiments.14-17 For the sample preparation PnBMA was dissolved in toluene with a concentration of c ) 0.001 g/mL. The glass slides were cleaned and rinsed with toluene. A polymer film was spincoated from 100 µL of the PnBMA solution on top of the glass slide with 2000 rpm for 1 min.18 An ensuing annealing step was not required since the glass transition temperature of PnBMA is close to room temperature. To ensure obtaining equilibrated and solvent-free films, the sample was dried for 5 days under ambient conditions. Due to the very low concentration of the initial solution and the rather high angular velocity during the spin-coating process, the film slightly dewetted the substrate upon drying and relaxing. In order to increase the clean glass surface, the film was carefully scratched on several spots. (12) Tsukruk, V. V.; Sidorenko, A.; Gorbunov, V. V.; Chizhik, S. A. Langmuir 2001, 17, 6715. (13) Doerner, M. F.; Nix, W. D. J. Mater. Res. 1986, 1, 601. (14) Cappella, B.; Silbernagl, D. Thin Solid Films 2007. Manuscript submitted. (15) Cappella, B.; Kaliappan, S. K.; Sturm, H. Macromolecules 2005, 38, 1874. (16) Cappella, B.; Stark, W. J. Colloid Interface Sci. 2006, 296, 507. (17) Cappella, B.; Kaliappan, S. K. Macromolecules 2006, 39, 9243. (18) Schubert, D. W.; Dunkel, T. Mater. Res. InnoVations 2003, 7, 314.
Cappella and Silbernagl Techniques. Atomic force microscopy (AFM) measurements have been performed with an MFP-3D microscope (Asylum Research, Santa Barbara, CA). For all measurements the AFM was equipped with a Pointprobe NCL cantilever (Nanosensor, Wetzlar-Blankenfeld, Germany) with elastic constant kc ) 44 N/m and a Si tip with a radius R of approximately 15 nm. Measurements have been performed in two different modes: the topography of the sample was scanned in tapping mode, whereas the mechanical properties were examined by means of force-distance curves9,10 in force-volume mode. The vital part of an AFM is the cantilever on which the tip is placed. The tip is the actual probe which interacts with the sample surface through attractive and repulsive forces, pushing or drawing the tip toward or away from the sample, respectively. These movements result in the deflection δ of the cantilever. In order to measure the cantilever deflection δ a laser beam is focused on the back of the cantilever and reflected onto the center of a photodiode consisting of four quadrants. The laser spot on the photodiode generates a current Isum ) Itop + Ibottom, where Itop (Ibottom) is the current of the top (bottom) quadrant. When the cantilever is deflected, the spot reflected on the photodiode shifts out of the center, which leads to a change of Itop and Ibottom, Isum staying constant. In order to correlate the deflection with the change of generated current a conversion factor, the sensitivity Ω given by Ωδ ) (Itop - Ibottom)/ (Itop + Ibottom), is needed. The determination of Ω requires the measurement of a deflection-current curve on a sample, for which the ratio between the piezo displacement and the cantilever deflection is known. The mechanical properties were examined, as mentioned above, by means of force-distance curves. In order to obtain force-distance curves the sample is moved in z direction, perpendicular to the sample surface, toward and away from the tip by means of a piezotransducer. According to the two movements the acquired curves comprise two moieties, the approach and withdrawal part. In the following solely the approach curve is discussed. The approach part of force-distance curves can be roughly divided into three parts: (1) the zero line, recorded at large tip-sample distances, where no interaction is present; (2) the jump to contact, occurring when at a certain tip-sample distance the gradient of the attractive forces exceeds the elastic constant of the cantilever, so that the tip snaps abruptly onto the sample surface; (3) the contact line, where the tip is pushed against the sample. The contact line gives information about the mechanical properties of the cantilever and the sample and is therefore discussed in detail in the following. In case the stiffness of the cantilever exceeds the stiffness of the sample the tip can indent the sample and cause a deformation. The deformation can be estimated as the difference of the piezo displacement Z and the deflection δ: D)Z-δ
(1)
Under the realistic assumption that the cantilever is elastic and shows the behavior of an ideal spring, the applied force F is given by the Hooke’s law: F ) -kcδ
(2)
where kc is the spring constant of the cantilever. If deformations occur, the contact line gives information about the spring constant of the sample k. Since the forces are in balance during contact, eq 2 can be rewritten as: |δ|kc ) |D|k
(3)
This is valid under the assumption that the sample is elastic and behaves like an ideal spring. This assumption does hold only for small deformations. Equations 1 and 3 yield:
Nanomechanical Properties of Mechanical Double-Layers δ)
( )
k Z ) keffZ kc + k
Langmuir, Vol. 23, No. 21, 2007 10781 (4)
which is a direct approach to estimate the effective spring constant of the sample/cantilever system keff from the contact line. Depending on the ratio of the sample and cantilever spring constants, additional approximations can be done. Equation 4 can be written as δ ) Z for k . kc and as δ = k/kc Z for k , kc. The effective spring constant keff obtained from eq 4 is a fairly good orientation guide within the measurement and can be used as an indicator for the sample stiffness, although it is obtained by rather extensive and rough assumptions. The spring constant keff is meaningful only if the sample can be described as a spring, i.e., the stiffness S ) dF/dD is a constant. This is true only for small deformations and under the condition k , kc, but in this case keff can be directly obtained from the contact line of a force-distance curve, since it is simply the slope of the contact line. If, for a given sample (usually a very stiff sample) keff is known, force-distance curves on such a sample can be used to determine the sensitivity Ω and, hence, can be exploited as a reference for the measurement of δ in nm. In the present work we have put keff ) 0.926 and D ) 0.08δ for glass, so that the Young’s modulus calculated from the deformation-deflection curves (see below) is 70 GPa. For further examination of the contact line of the force-distance curves an adequate continuum elastic theory, which specifies the relation of the applied load F and the deformation D as a function D(F), is needed. Different approaches for the measurements of homogeneous bulk materials are provided by the theories of Hertz,19 Johnson-Kendall-Roberts (JKR),20 and Derjaguin-Mu¨ller-Toporov (DMT).21 The theory of Hertz does not consider the adhesion between sample and tip, which is accounted for in the JKR and DMT theories. The adhesion can be neglected in case the applied load is much larger than the adhesion force, which is the case for this experiment. Hence, the relation between the applied force F and the deformation D can be expressed by the Hertz theory: D)
( ) F2 RK2
1/3
(5)
where
(
1 3 1 - ν2 1 - νt ) + K 4 E Et
)
2
is the reduced elastic modulus of the probed material taking the Young’s modulus of the tip Et into account, D is the deformation, F the applied load, and R the tip radius. When E , Et , the reduced modulus can be approximated in the form
(
1 3 1-ν ) K 4 E
)
1 RD 1 RD 1 1 - exp + exp ) K Ks t Kf t
[
(6)
It is evident that, within elastic continuum theories, the stiffness S ) dF/dD is not a constant but rather is a function of the force F. Equations 5 and 6 for the probe/sample interaction were developed for homogeneous samples. As pointed out before, this work deals with inhomogeneous samples and in particular with mechanical double-layers, i.e., with samples consisting of a film on a substrate. The peculiarity of such systems is that the mechanical properties of (19) Hertz, H. J. Reine Angew. Math. 1881, 92, 156. (20) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324, 301. (21) Derjaguin, B. V.; Mu¨ller, V. M.; Toporov, Yu.P. J. Colloid Interface Sci. 1975, 53, 314.
)]
(
(
)
(7)
In this equation, the composite elastic modulus is the sum of the elastic moduli of film and substrate, weighted by a factor depending on the film thickness, the deformation, and a parameter R. The parameter R is introduced as a constant for one material couple, and has to be derived experimentally. This is a first important drawback of this approach. Another important experimental drawback is that Doerner’s analysis is based on the determination of the elastic modulus through the derivation of eq 5, which is affected by large noise, and therefore difficult to interpret. In a recent work14 we could show that Doerner’s equation is not a reliable tool to describe the mechanical properties of mechanical double-layers. Another approach was proposed by Tsukruk,12,22 which does not describe the composite elastic modulus directly, rather the deformation, respectively the contact radius a, as a function of the two elastic moduli of film and substrate and of the film thickness t:
a)
2
The form of eq 5 is based on the additional assumption that the indenting part of the tip has the shape of a half sphere. This assumption permits calculating also the contact radius a in the form: a ) xRD
the two sample components, e.g., their Young’s moduli, are probed at the same time, resulting in composite mechanical properties. From experiments it is known that the composite elastic modulus changes with the thickness of the film and hence the probed elastic modulus depends on the relationship of the deformation and of the film thickness. This is due to the fact that the larger the distance between substrate and tip and the smaller the force field applied on the sample, the less the mechanical properties of the substrate affect the probed composite modulus. Until now there is no global approach to deduce back the elastic moduli of the sample components, Ef and Es, or the film thickness t from the composite elastic modulus. Even if mathematical details are not known, it is evident that at the beginning of an indentation curve on a mechanical double-layer the tip probes only the upper, compliant polymer film, and deformation and stiffness are nearly the same as would be obtained on the bulk, homogeneous polymer without substrate. By increasing the load, and hence the deformation, the tip probes more and more the underlying substrate, and the stiffness increases until, for very high loads, the same stiffness as that on the substrate is obtained. In order to describe analytically the indentation of a mechanical double-layer, an expression for the composite modulus, depending on the two elastic moduli of film and substrate and on the film thickness, is needed. Semiempirical approaches have been introduced to describe this dependency. One semiempirical equation was introduced by Doerner et al.13 in the form:
[
]
() x x x ( x ) Kf Ks
4/3
+ 0.8t
1 + 0.8t
3
3
Kf RF
Kf RF
1/4
3
2
RF Kf
(8)
In a former work14 we could show that this equation fails for Kf/Ks = 1, since in this case eq 8 yields unphysical values. Most important, this approach underestimates the influence of the substrate, and the experimental curves obtained with films of different thicknesses could not be fitted with eq 8 and the known values of Ef, Es, and t. Hence, there is still a need for a global approach to describe the resulting composite elastic modulus, depending on the elastic modulus of film and substrate and on the film thickness. A novel approach is introduced in the following.
Results and Discussion In this work the focus lies on the survey of mechanical doublelayers and on applying a novel approach to describe the (22) Kovalev, A.; Shula, H.; Lemieux, M.; Myshkin, N.; Tsukruk, V. V. J. Mater. Res. 2004, 19, 716.
10782 Langmuir, Vol. 23, No. 21, 2007
Cappella and Silbernagl
Figure 1. Map of the topography of the examined sample section. The black regions represent the uncovered surface of the glass substrate. The gray to white areas show the regions covered with PnBMA.
mechanical behavior of such layers. It is assumed that the measurement of the mechanical properties depends on the elastic moduli of substrate and film Es and Ef and on the film thickness t. In order to validate our novel approach it is necessary to test it on a wide range of at least one of these parameters. The variation of the film thickness t is chosen as a first step. Other important parameters that will be varied in future experiments are the elastic moduli of film and substrate and their ratio. In order to collect a large number of force-distance curves under identical conditions and with a large variation of film thickness, 2500 force-distance curves were taken on a 2.5 × 2.5 µm2 area of a film of PnBMA, whose film thickness ranges from clean substrate up to 30 nm. The topography of the probed section is shown in Figure 1. It becomes apparent that approximately one-third of the chosen section is uncovered glass substrate. Also, the glass areas are distributed over the whole sample section and show characteristic, easily distinguishable edges to the PnBMA film. This is necessary in this extent since measurements on glass are needed as references for height and for stiffness, as will be shown in detail below. The effective spring constant on each point of the area was evaluated from the contact line of the force-distance curves by means of eq 4. Since the force-distance curves are spatially resolved, a map of the effective elastic constant could be calculated, as can be seen in Figure 2A. The resemblance of the topography to the map of the effective elastic constant keff is striking. All glass regions (black regions in Figure 1) have the maximum keff (keff ) 0.926 ( 0.012) and correspond to the white regions in Figure 2A. Darker regions show a lower effective stiffness, and they correspond to sections covered with the polymer film, whose effective elastic constant depends on the film thickness. This is confirmed by the histogram of keff, shown in Figure 2B. The values of keff of polymer and glass can be easily distinguished. The portion of the histogram acquired on glass (gray bars) can be fitted with a Gaussian curve, and 95% of the values are in the interval keff ) 0.926 ( 0.012. Due to drift during the measurements and their different resolution, the topography in Figure 1 and the map of the effective elastic constant in Figure 2A cannot be simply superimposed in order to determine the film thickness t corresponding to each point of the effective elastic constant map. In order to make the two maps congruent the glass surfaces can be exploited, since
Figure 2. (A) Map of the effective elastic constant keff of the examined sample section, calculated from force-distance curves, following eq 4. The white regions represent the highest values of keff, i.e., the effective elastic constant of glass, whereas darker spots show a lower keff, i.e., the effective elastic constant of the polymer film, varying with the film thickness. (B) Histogram of the effective elastic constant keff, showing two distinguishable peaks. The peak at keff ) 0.926 is highlighted in gray and is superposed by the Gaussian fit, shown as a black plot.
they can be unambiguously identified as the regions with height values between -2 and 2 nm in Figure 1 and with keff values between 0.914 and 0.938 in Figure 2A. The areas on both maps (Figures 1 and 2A) showing the values for glass are brought to coincide. The result of this superposition is shown in Figure 3. In this picture the same topography as in Figure 1 is shown, with the only difference being that its resolution has been adjusted to the resolution of the effective elastic constant map. On the topography we have superimposed the contour corresponding to keff ) 0.914, i.e. the line at which the effective elastic constant goes through the value keff ) 0.914, thus indicating the edges between glass and polymer. It can be seen that the contour traces not only the largest glass regions, i.e. the approximately diagonal stripe in the top half and the bottom region, but also five smaller spots surrounded by polymer. Since now the film thickness t for each force-distance curve has been determined through the superposition, it can be plotted versus the effective elastic constant. To this purpose the film thickness has been grouped in intervals of 1 nm, and the average keff has been calculated for each interval. The resulting plot is shown in Figure 4, where the error bars are the average deviation over each interval. It can be clearly seen that the stiffness decreases with increasing film thickness t.
Nanomechanical Properties of Mechanical Double-Layers
Figure 3. Map of the topography with the same resolution as the map in Figure 2A. The contour line at keff ) 0.914, obtained from the map of keff in Figure 2A (white-black-white plot), superposes the topography. This contour line is assumed to be the edge between glass and polymer and hence can be used to bring both pictures to coincide.
Figure 4. Effective stiffness keff of each force-distance curve, assigned to the film thickness yielded by the topography, grouped in thickness intervals and averaged. The error bars show the average deviation for each film thickness interval.
The relatively large error bars associated to thicknesses larger than 2 nm show that the correlation of film thickness and keff is affected by uncertainties, which appear mostly at the edges of the glass areas. Such uncertainties are due to three phenomena. The first one is that loose polymer residues at the edges of the compact polymer film may provoke unpredictable movements of tip and cantilever, i.e., sticking or slipping, resulting in artefacts in the force-distance curves. In this case the corresponding force-distance curve, showing an extreme erratic behavior, has not been considered for the further analysis. The second reason for the anomalous curves at the edges is that a force-distance curve acquired on relatively steep edges yields a false measurement of the stiffness due to changes of the contact area between tip and sample during the indentation.9,23 The third reason is that, when a force-distance curve is acquired at the edges, a small deviation in the position of the tip may have dramatic consequences on the measurement of the effective elastic constant, since a curve that is thought to be on glass is actually on the polymer film and vice versa. In order to draw further conclusions about the thickness dependence of the mechanical properties of polymer films, it is (23) Mizes, H. A.; Loh, K.-G.; Miller, R. J. D.; Ahuja, S. K.; Grabowski, E. F. Appl. Phys. Lett. 1991, 59, 2901.
Langmuir, Vol. 23, No. 21, 2007 10783
necessary to analyze the deformation D, calculated by means of eq 1, versus the deflection δ, proportional to the applied load. Also in this case we have calculated the average of the curves acquired in film thickness intervals of 1 nm, starting at a film thickness of 2 nm up to 15 nm, yielding 14 deformationdeflection curves on polymer. All curves taken on a film thickness below 2 nm were averaged to a glass curve. In this way at least 30 curves were considered for each interval. Unfortunately, the number of force-distance curves taken on a film thickness larger than 16 nm was too small to generate a reasonable average curve. Figure 5 shows four exemplary deformation-deflection curves, i.e. the curves obtained on glass and on the 2-, 3-, and 8-nm thick polymer regions. The curves acquired on the polymer film are plotted with their average deviation. Curves whose average deviations do not overlap were assessed to be distinguishable. Hence, it can be claimed that AFM indentation is an adequate technique to distinguish the mechanical properties of a 2-nm thick film from those of a 3-nm thick film as well as from those of the substrate. Such a thickness resolution of the deformation curves, i.e. the possibility of distinguishing two films of different thicknesses via the relative deformation curves, becomes worse and worse with increasing thickness. With ultrathin polymer films, i.e. up to approximately 15 nm, small differences of thickness are sufficient to change the low-forces part of the deformation curve in a significant, measurable extent. With thicker films, measurable thickness differences become larger and larger with increasing thickness (in the range 40 nm < t < 200 nm only films with a thickness difference of about 10 nm can be distinguished). Finally, for very thick films (t > 200 nm), differences in the low-forces part of the deformation curve almost disappear, because the lowforces part is completely dominated by the mechanical properties of the film. Depending on the thickness range of interest, deeper indentations or a stiffer cantilever can be employed to distinguish thicker films via the high-forces part of the deformation curve. Further, it can be noted that the dependence of the curves on film thickness and deformation is as expected: for small loads the resulting deformation and the stiffness (inverse of the first derivative of the shown curves) are dominated by the polymer film, whereas at high loads the glass substrate gains influence on the indentation curve, so that the stiffness increases. The transition between the film- and substrate-dominated regimes depends on the film thickness t. For example, in the curve relative to the 2-nm thick regions, for δ > 35 nm, the stiffness is largely the same as that on glass, whereas in the other curves the “glass stiffness” has not been reached in the experimental range of forces. Also, the deformation and the compliance in the initial portion of the curve increase with the film thickness. The final goal of this work is to find a function D(F) for the deformation of mechanical double-layers. This function should depend on the parameters Es and Ef (elastic moduli of substrate and film) and on the film thickness t. Such a function should reduce to the Hertz equation (eq 5) in the two limits D/t f ∞ (the tip probes only the substrate) and D/t f 0 (the tip probes only the polymer film). Please note that D/t f ∞ can be achieved if either D f ∞ (the tip indents the whole polymer film) or t f 0 (there is no polymer film on top of the substrate). Also the second limiting case D/t f 0 is achieved by both conditions D f 0 and t f ∞. Hence, we know that for
D/t f ∞ D3/2 f D3/2 s )
F xRKs
10784 Langmuir, Vol. 23, No. 21, 2007
Cappella and Silbernagl
Figure 5. Averaged deformation-deflection curves for the thickness intervals 2, 3 and 8 nm together with the curve taken on glass. The deformation-deflection curves on polymer are plotted with their average deviation to show that the curves are distinguishable.
and for
whose first derivative is given by:
D/t f 0 D3/2 f Df3/2 )
F xRKf
∂D3/2 )β-R ∂δ
where Ks and Kf are the reduced elastic moduli of substrate and film, respectively. The case D/t f 0 must be considered very carefully. From a mathematical point of view, both an infinite thickness and an infinite small deformation are possible. Yet, from a physical point of view, an infinite small deformation, i.e., a deformation being much smaller than the film thickness t, may be such that it cannot be measured or it cannot be described by elastic continuum theories, i.e. theories that do not account for the discrete nature of matter. Elastic continuum theories hold approximately for deformations larger than 1 nm. Hence, if the film is thinner than 10-20 nm, measurable deformations that can be described through elastic continuum theories are not much smaller than the film thickness. This means that the mechanical properties of a 10-20-nm thick film are always influenced by the substrate. Generally speaking, the mechanical properties of compliant films which go below a critical film thickness tc cannot be measured without taking into account the effect of the substrate. This physical feature of the function D(F) or E(D) is not accounted either in the formula of Tsukruk (eq 8) nor in the formula of Doerner (eq 7), which both predict that E ) Ep for D ) 0. If t > tc, the function D3/2(F) or D3/2(δ) should present two linear regions for D/t f ∞ and D/t f 0, where the first derivative of the function is proportional to 1/Ks and to 1/Kf, respectively. In between, the first derivative of the function should decrease, as the substrate influences more and more the deformation. As in the case of plastic deformations,15 these considerations lead to the idea of describing D3/2 curves as a hyperbola in the form:
D3/2(δ) ) βδ + - xR2δ2 + 2(β - γ)δ + 2
(9)
δ+
x
(β - γ) R R
(β - γ) 2 δ +2 δ+ 2 R R R
(10)
2
All parameters of the fitting function are positive and, since the first derivative is always positive, is R < β. The first derivative is a sigmoid function with plateaus β - R for δ f (∞. The plateau values are the slopes of the two asymptotes of the hyperbola. As already explained, the slopes of the two asymptotes are inversely proportional to the Young’s moduli of glass and polymer:
kc kc 3 3 ; β + R ) (1 - ν2f ) β - R ) (1 - ν2s ) 4 4 xRE xRE s
f
Hence the two parameters R and β are known, once the Young’s moduli of polymer and substrate are known. Figure 6 shows the D3/2 curves obtained in the present measurement on regions with thickness 2, 3, 4, 5, 6, 8, 10, 12, and 15 nm (panel A) together with the curves obtained in a previous measurement14 (kc ) 37 N/m and R ) 25 nm) on films with thickness 50, 60, 90, 110, 190, and 430 nm (panel B). This second set of curves is shown in order to cover a wider range of thicknesses. It can be noted that, starting from 70-nm film thickness, the polymer film is thick enough to compensate the effect of the substrate, at least at the beginning of the indentation, so that the deformation increases with the load rather steeply for small forces (i.e. the stiffness is rather small, at least for low forces). In other words, the curves relative to thicker films are not “substrate-dominated”, like the ones on thinner films in Figure 6A; rather they are “film-dominated”. The curves between 40 and 70 nm (the first two ones in Figure 6B) are particularly interesting because they clearly show, in our experimental range
Nanomechanical Properties of Mechanical Double-Layers
Langmuir, Vol. 23, No. 21, 2007 10785
Figure 6. (A) Averaged D3/2 curves for the thickness intervals 2, 3, 4, 5, 6, 8, 10, 12, and 15 nm (markers) plotted vs the deflection δ and fitted with the hyperbola described by eq 9 (black solid line). The asymptotes of the hyperbola, describing the extreme cases of a measurement of glass and bulk polymer, are shown as well (black solid lines). (B) D3/2 curves obtained on glass and on polymer films with a film thickness of 50, 60, 90, 110, 190, and 430 nm (markers) plotted vs the deflection δ and fitted with the hyperbola described by eq 9 (black solid line). The cantilever elastic constant and the tip radius in this measurement are 37 N/m and 25 nm.
of forces, both the substrate- and the film-dominated regime, and hence the transition between the two. In both figures, the experimental curves are plotted with gray points and the fitted curves with black lines. As can be seen, all curves can be fitted very well with the hyperbola in eq 9. As mentioned above, the values for R and β are derived from the known elastic properties of glass and PnBMA and were kept the same for all fits. The value of the Young’s modulus of glass has been fixed through the sensitivity since the very beginning of the analysis and is Es ) 70 GPa for both measurements. Fixing the sensitivity Ω determines completely Es, but not Ef, that can be measured from the value of β + R. Despite several important differences in the two measurements (different samples and different cantilevers) the two sets of curves yield the same value for Ef, namely Ef ) 3.76 GPa. Moreover, this value is in very good agreement with previous results obtained in our group on bulk PnBMA.14,15,17
The parameter γ has a central meaning for the shape of the fitting function. Since the radicand in eq 9 has to be positive, it must be β - R < γ < β + R. Furthermore, γ is the value of the first derivative for δ ) 0. Hence, the mathematical condition β - R < γ < β + R assumes a physical meaning: the Young’s modulus of the sample probed at very small deformations (δ = 0) must be intermediate between the Young’s modulus of bulk polymer and the Young’s modulus of substrate. Figure 7 shows the first derivative in eq 10 in three cases: γ = β - R (4), γ = β (O), and γ = β + R (3) and the corresponding D3/2 functions. It is evident that the parameter γ is an indicator of whether the curve is dominated by the polymer or by the substrate. Since this is “tuned” by the film thickness, γ must also indicate how thick the polymer film on top of the substrate is. If γ ) β - R (the sample is only the substrate or, in other words, the film thickness t is 0), the hyperbola in eq 9 degenerates
10786 Langmuir, Vol. 23, No. 21, 2007
Cappella and Silbernagl
t)c
Figure 7. (A) First derivative of the hyperbola, eq 10, (panel B), vs the applied load F ) kcδ (black plots). The plateaus of the sigmoid at the values β + R and β - R are shown as a gray dashed lines.
into the line D3/2 ) (β - R)δ. If γ ) β + R (the sample is only the polymer or, in other words, t f ∞), the hyperbola in eq 9 degenerates into the line D3/2 ) (β + R)δ. But in this case must be δ < /R, so that the radicand in eq 9 is positive. This shows that also the parameter increases with increasing thickness. Hence, it is reasonable to assume that the film thickness t is given by
(β - R) - γ R γ - (β + R)
(11)
where c is a proportionality constant. In this way the film thickness is proportional to /R, becomes 0 on glass (because on glass γ ) β - R), and tends to infinity on bulk polymer (because on bulk polymer γ ) β + R). In order to test and verify eq 11, we have fitted each single curve obtained on the section shown in Figures 1, 2, and 3. The result is shown in Figure 8, where the circles represent the thickness of each point calculated through eq 11 vs the thickness measured from the topography (Figure 1), the black line with error bars represents the mean value of the calculated thickness with the average deviation, and the empty squares are the thickness calculated from the fit of the average curves shown in Figure 6. The agreement between the mean value of the calculated thicknesses and the thicknesses calculated from the mean curves shows that the two procedures (averaging and fitting) are interchangeable. Please note that the experiment provides far more data points for film thicknesses below 15 nm than for film thicknesses above 15 nm. Figure 8 shows the most significant section (below 15 nm) of the plot for clarity. The plot shows several significant attributes confirming that the expression of thickness in eq 11 is correct. First of all, for curves on glass the thickness calculated from the fit is 0.5 ( 1.5 nm. Additionally the thickness calculated from the fit parameters is proportional to the thickness obtained from the topography. The plot of the single points as well as the error bars of the averages shows a certain distribution of the values, due to several factors. The most important are, as already explained, the error sources at the edges of the polymer film and the uncertainty of the association of a certain point of the topography to a given force-distance curve. Nevertheless, it is possible to calculate the thickness from the fit parameters with an uncertainty of (3 nm. Another important reason for the distribution of the calculated thicknesses in such a plot is that, strictly speaking, the two quantities plotted in Figure 8 are not the same. As a matter of
Figure 8. Film thickness calculated from the fit parameters by means of eq 11 vs the film thickness obtained from the topography. The circles show the film thickness obtained from each force-distance curve, with its average curve and average deviation (black solid line). Thickness values calculated from the averaged deformation curves in Figure 6A are shown by square markers.
Nanomechanical Properties of Mechanical Double-Layers
Langmuir, Vol. 23, No. 21, 2007 10787
respondence is further confirmed through Figure 9B, showing the height profile (black line) of one column of Figure 9A (marked by arrows) and the height profile of the corresponding column obtained from the topography (gray line). The five glass intervals can be singled out with high precision in the height profile obtained from the fit, and also the height of the polymer intervals is in very good agreement with the topography and differs from the topography only in few points.
Conclusion
Figure 9. (A) Reconstructed topography: Each force-distance curve was fitted by means of eq 9 and the film thickness was calculated through eq 11. The arrows flag one column of the map that is shown in panel B. (B) Line profiles of the column marked by the arrows in panel A. The black line shows the film thickness calculated from the fit, the gray line the corresponding line profile from the topography.
fact the thickness measured by means of force-distance curves is actually the thickness of the polymer film, whereas the thickness obtained from the topography is the height difference between the single point of the topography and the substrate surface. Hence, the two quantities are the same only if the substrate surface is absolutely flat. This is not the case for common glass substrates, which have a roughness in the order of our resolution in the determination of the film thickness. This explains also why the thickness measured from the topography can assume negative values, whereas the film thickness is always positive. We have arbitrarily put the mean height of the glass surface equal to zero, and since the mean roughness is 2 nm, there are also points of the glass surface with “negative” height. On the other hand, the expression in eq 11 is always positive, because negative film thicknesses would have no physical meaning. From the plots shown in Figure 8 the proportionality constant c can be determined. It is now possible to reconstruct the topography, as shown in Figure 9A. By comparing the reconstructed topography with Figure 4, a very good agreement of the two maps can be seen. Characteristic features of the topography such as the glass spots inside the film region can be clearly identified in both maps. Also, between higher and lower polymer regions there is a good correspondence, even if not punctually precise. This cor-
A large number of force-distance curves were taken on an area of a PnBMA film, with film thickness ranging from clean substrate up to 30 nm. The large variation of film thicknesses gave us the possibility to survey the so-called “mechanical doublelayer” topic, i.e., the influence of the substrate on the mechanical properties of the film in dependence of the film thickness. A first analysis of the experimental results has been based on the calculation of a map of the effective spring constant, showing a striking resemblance with the topography: glass regions have a maximum effective spring constant, whereas polymer regions have a lower effective spring constant, depending on the thickness. Bringing the topography to coincide with the effective spring constant map, the film thickness for each force-distance curve could be determined, and the effective elastic constant could be plotted vs the thickness. Such a plot shows that the stiffness decreases with increasing film thickness. The most important step of the present work is the analysis of the deformation-deflection curves. A comparison of the curves obtained on regions with different thicknesses proves that AFM indentation is able to distinguish the mechanical properties of a 2-nm thick film from those of a 3-nm thick film and from those of the substrate. Such a 1-nm resolution can be assessed only in case of ultrathin films (up to 10 nm), and it should be kept in mind that the possibility of distinguishing two films of different thicknesses via the respective deformation curves becomes worse and worse with increasing thickness. The final goal of this work was to develop a semiempirical equation for the deformation of mechanical double-layers. Fitting the experimental curves with a hyperbola was found to be an adequate and promising model. In order to cover a wider range of thicknesses, deformation-deflection curves obtained in a previous experiment have been included in the analysis. All experimental curves considered in this work could be fitted with high precision. The first important requirement of a model of the mechanical properties of double-layers is the possibility to calculate the Young’s moduli of the constituents. Our model yields the same value of the Young’s modulus of the polymer for two different sets of indentation curves, and this measured value is in very good agreement with previous published results. Also, the second claim of such a model, namely the possibility to calculate the film thickness from indentation curves, could be fulfilled. Our expression for the film thickness in dependence of the fit parameters is confirmed by the proportionality between the thickness calculated from fit and the thickness obtained from the topography and by the successful reconstruction of the topography starting from the fit parameters via such expression. LA701234Q
