Modeling the Interactions between Atomic Force Microscope Tips and

Polymer Nanomechanics. J.K. Li , I.T.S. Li , G.C. Walker , R.M.A. Sullan , S. Zou , Y. Sun. 2012,377-404. Nano-Mechanical Properties of Modified Poly(...
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Langmuir 1998, 14, 4615-4622

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Modeling the Interactions between Atomic Force Microscope Tips and Polymeric Substrates Ekaterina Zhulina,† Gilbert C. Walker,‡ and Anna C. Balazs*,† Department of Chemical and Petroleum Engineering and Department of Chemistry, The University of Pittsburgh, Pittsburgh, Pennsylvania 15261 Received March 26, 1998

We use scaling theory to model the interactions between a layer of end-grafted polymers in a poor solvent and an atomic force microscope (AFM) tip. We focus on two distinct types of interactions as the tip is brought in contact with the polymeric substrate. In the first case, the polymers preferentially wet the surface of the tip, whereas in the second case, chains form chemical bonds with sites on the tip surface. For both cases, we analyze the morphological changes and determine the force profiles when the tip is embedded in the polymer layer and subsequently drawn away from this interface. The profiles for both cases exhibit a plateau region, where the force does not depend on the degree to which the polymer layer is stretched or deformed. This plateau region is more pronounced in the case where the chains are chemically bound to the tip. The results are in agreement with recent AFM studies on a collapsed polymer layer and help rationalize the experimental findings. The results also provide useful insight into the relationship between microscopic interactions at the polymer-tip interface and experimentally measured results.

Introduction Recent advances in atomic force microscopy (AFM) have enabled researchers to probe not only the structure and elasticity of thin polymer films but also the behavior of individual macromolecules.1-8 For example, the technique has been used to characterize the features of polymer layers adsorbed on inhomogeneous surfaces3 and determine the properties of strongly stretched strands of DNA.6,7 To properly interpret the experimental observations, it is important to understand how the interactions between the AFM tip and the polymer chains can affect the structure of the film and influence the observed force versus distance profiles. In this paper, we use scaling theory to examine the interactions between a layer of end-grafted polymers in a poor solvent and an AFM tip. We focus on two specific cases. In the first case, the interactions between the polymer and tip are governed by van der Waals forces that drive the polymer to preferentially wet the surface of the tip. In the second case, polymer-tip interactions lead to chemical bonding between the tip and chains within the grafted layer. In the latter example, we are modeling the situation where the tip has been chemically modified by attaching functional groups on this surface. When such a tip is embedded in the polymer film, reactions between * Author to whom correspondence should be addressed. † Department of Chemical and Petroleum Engineering. ‡ Department of Chemistry. (1) Frisbie, C. D.; Rosznyai, L. F.; Noy, A.; Wrighton, M. S.; Leiber, C. M. Science 1994, 265, 2071. (2) Green, J.-B.; McDermott, M. T.; Porter, M. D.; Siperko, L. M. J. Chem. Phys. 1995, 99, 10960. (3) Siqueira, D. F.; Kohler, K.; Stamm, M. Langmuir 1995, 11, 3092. (4) Stamouli, A.; Pelletier, E.; Koutsos, V.; van der Verte, E.; Hadziioannou, G. Langmuir 1996, 12, 3221. (5) Rief, M.; Oesterhalt, F.; Heymann, B.; Gaub, H. E. Science 1997, 275, 1295. (6) Shivashankar, G. V.; Libchaber, A. Appl. Phys. Lett. 1997, 71, 3727. (7) Lee, G. U.; Chrisey, R. J.; Colton, R. J. Science 1994, 266, 771. (8) Guffond, M. C.; Williams, D. R. M.; Sevick, E. M. Langmuir 1997, 13, 5691.

the functional groups and sites within the tethered chains lead to the formation of irreversible bonds between the tip and substrate. Through our scaling models, we analyze how the nature of the polymer-tip interactions (preferential wetting versus chemical bonding) affect the morphology of the system and the force versus distance profiles. As we show later, the chains undergo a novel structural rearrangement in response to the applied force. The rearrangement involves the unwinding of polymers segments from a region of high polymer density and the localization of these segments into a strand that interconnects the tip and substrate. The feeding of monomers into the strand mediates the effects of the external force. The overall results from this study provide useful insight into the relationship between microscopic interactions at the polymer-tip interface and the experimentally measured observables. The findings also add to our fundamental knowledge on the tribological interactions between surfaces and polymer-coated substrates. We begin our discussion by describing the specific model for the polymer layer. We then determine the response of the layer to the different tips, indicating the unique characteristics of each example. Model We consider flexible polymer chains that are composed of N monomers and are tethered at one end onto an impenetrable substrate. The chains are anchored at a grafting density given by 1/s, where s is the area per chain. The diameter of a monomer, a, is chosen as the unit of length. The chains are immersed in an infinite reservoir of solvent. The quality of the solvent is specified by the value of the second virial coefficient in the free energy expression for the polymer-polymer interactions, or v ) a3(θ - T)/T ) -τa3. Here, T stands for temperature. The variable θ represents the theta temperature for the system and thus, τ indicates the deviation from theta solvent conditions. For τ > 0, the surrounding solution represents a poor solvent, and for τN1/2 > 1, an individual polymer chain will collapse into a globule.

S0743-7463(98)00344-8 CCC: $15.00 © 1998 American Chemical Society Published on Web 07/17/1998

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Figure 1. The schematic in (a) shows the undeformed pinned micelle; the tip lies at a large distance H from the substrate. The schematic diagrams in (b)-(c) are for the cases where the tethered chains wet the tip; the diagrams in (d)-(e) are for the cases where the chains form chemical bonds with the tip. In (b), the pinned micelles attach to the surface of the tip and form separated legs at small H. The diagram in (c) shows that the legs aggregate into a strand at higher values of H. In the case of chemical bonding, the diagram in (d) shows that the strand is conserved for a wide range of H and the image in (e) reveals that the strand is disrupted at strong deformations.

We assume that the chains are grafted rather sparsely on the surface. Because of the solvophobic interactions, the chains aggregate into “pinned micelles”.9,10 Such micelles consist of a droplet-like core of collapsed polymers and “legs” that anchor the core to the surface (see Figure 1(a)). Because of the interaction between the polymer and substrate, the core can partially wet the surface and form a spherical cap. The contact angle between this spherical cap and the surface is given by Θ1. (Here and later, the subscript 1 refers to the grafting surface.) The value of Θ1 is determined by Young’s law:

cos Θ1 ) (γsw - γpw)/γ

(1)

where γsw, γpw, and γ are the respective surface tensions at the solvent-surface, polymer-surface, and polymersolvent interfaces. Each pinned micelle consists of f chains and occupies a total area equal to D2 ) fs. The majority of the monomers are localized in the core of the micelle. The polymer concentration (volume fraction of polymer units) within this micellar core is given by τ. The volume occupied by the core of a micelle is equal to 4πRo3/3 ) (fNa3/τ), where Ro is the radius of the undeformed (spherical) core. (Unless specified otherwise, we omit all numerical coefficients that are on the order of unity in our scaling expressions.) The structure of a pinned micelle can be analyzed in the context of the “blob” model. In this model, the micellar core is viewed as a densely packed system of blobs, whereas the stretched legs are viewed as strings of blobs.9 The size of each blob, ξ, is given by ξ ) a/τ. The surface tension at the polymer-solvent boundary is given by the number of blobs per unit area, or γ/kT ) ξ-2 ) (τ/a)2. (9) (a) Klushin, L. I. Preprint 1992; (b) Zhulina, E. B.; Birshtein, T. M.; Pryamitsyn, V. A.; Klushin, L. I. Macromolecules 1995, 28, 8612. (10) Williams, D. R. M. J. Physique II 1993, 3, 1313.

The equilibrium characteristics of the pinned micelles are determined by a balance of the surface free energy associated with the core of the micelle, and the elastic stretching of the legs. The surface contribution is given by

Fsurf/kT ) 4πRo2γg[cos(Θ1)]/f ) k1N2/3τ4/3f-1/3g[cos(Θ1)] (2) where the function g[x] ) (2 + x)1/3(1 - x)2/3/41/3 accounts for the partial wetting and spreading of the micellar core on the substrate,11 and k1 is a numerical coefficient on the order of unity. Later, we focus on the case where the contact angle between the micellar core and the surface is Θ1 ) 180° and thus, g1 ) 1. In this scenario, the polymer effectively dewets the surface. Having determined the free energy contribution from the micellar core, we now consider the terms stemming from the micellar legs. The expression for the free energy of the legs (the strings of blobs) accounts for the losses associated with the elastic stretching of these segments, and the losses associated with contact between the solvent and the surface of the legs. Both terms are of the same order of magnitude and are proportional to the average length of the leg, l ) D. Thus, we write

Fleg/kT ) k2τl ) k2τD ) k2τ(sf)1/2

(3)

where k2 is on the order of unity. By minimizing the sum of the two terms, (Fsurf + Fleg), with respect to f, we arrive at the equilibrium number f1 (11) Singh, C.; Zhulina, E. B.; Gersappe, D.; Pickett, G. T.; Balazs, A. C. Macromolecules 1996, 29, 7637.

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Langmuir, Vol. 14, No. 16, 1998 4617

of chains in a micelle,

f1 ) (2k1/3k2)6/5N4/5τ2/5/s3/5

(4)

The lateral size of such undeformed micelles is given by

D1 ) (f1s)1/2

(5)

The corresponding value for the total free energy, F ) Fleg + Fsurf, yields

F1/kT ) (5k2/2)(2k1/3k2)3/5τ6/5N2/5s1/5

core. From Figure 1(b), we see that the average length of a leg is now given by l ) (D2 + H2)1/2. Thus, we obtain the following expression for the free energy of the legs,

Fleg/kT ) k2τl ) k2τ(sf + H2)1/2

where f now refers to the number of chains in a stretched micelle on surface 2. The surface free energy per chain is given by

Fsurf/kT ) 4πRo2γg[cos(Θ2)]/f ) k1N2/3τ4/3f-1/3g2

(6)

Such pinned micelles are thermodynamically stable9 for the range of s, area per chain, given by the following inequality: s2 ) N1/2τ-1 < s < s1 ) N4/3τ2/3. At the upper boundary, s ) s1, the micelles split into separate, isolated globules (i.e., at s ) s1, f1 ) 1). At the lower boundary, s ) s2, the micelles merge into a laterally homogeneous film. In scaling terms, the formation of this film occurs when the radius of the core, R1 () Ro/g1 ) Ro) equals D1. We are now ready to consider the interactions between a substrate covered with pinned micelles and the AFM tip. We model the tip by an infinite planar surface (which we will refer to as surface 2) that is located at a fixed distance H above the substrate. In the first case we consider, the polymer partially wets the tip, forming a contact angle Θ2 < Θ1 with this surface. In the second case, the surface of the tip is chemically modified in such a way that polymer-tip interactions result in the formation of covalent links between the tip and a few polymer chains in each micelle. The rest of the polymer units dewet the tip surface. Results and Discussion Force versus Distance Profiles. Interactions between the tip and substrate can lead to structural rearrangements within the polymer micelles and morphological changes at the tip-substrate interface. These structural changes will, in turn, affect the force between the tip and substrate. We start with an analysis of the morphology and force profile for the case where the polymer preferentially wets the surface of the tip. For this scenario, we first consider the situation where the distance between the tip and substrate is relatively small, and thus, the system is only weakly deformed. Then, we examine the case where the micelle is greatly spread out on the tip and the distance between the tip and substrate is relatively large. This is the large deformation regime; here, the pinned micelles undergo further structural changes. Polymers Wet the Tip Surface. Weak Deformations of the Polymer Layer. In this case, when the distance between the tip and the polymer layer is small, preferential interactions with the tip drive the tethered chains to form droplet-like cores on the surface of the tip, rather than on the substrate. In other words, the micelles, whose ends are still tethered to surface 1, now spread on the surface of the tip (surface 2), at the cost of stretching the polymeric legs. The schematic in Figure 1(b) indicates the morphology of this system. Increasing the distance between the tip and substrate, H, leads to a corresponding increase in the lengths of the tethered legs. To analyze the consequences of this preferential wetting, we adopt the following model. We assume that at small deformations, the structure of the pinned micelles is still determined by a balance of the free energy for the legs and the surface free energy associated with the micellar

(7)

(8)

where g2 ) g[cos(Θ2)]. The term g2 describes the deformation of the micellar core on surface 2 relative to its spherical shape on surface 1. Recall that we are focusing on the case where Θ2 < Θ1, thus, g2 < 1. By minimizing the sum (Fsurf + Fleg) with respect to f, we arrive at an expression for the equilibrium number of chains in the micelle on surface 2, or f2. (Here and in the following equations, the subscript 2 refers to surface 2, the AFM tip.) At H ) 0, when the tip comes in contact with the substrate, we find

f2 ) (2k1/3k2)6/5 N4/5τ2/5g26/5/s3/5 ) f1g26/5

(9)

As indicated by eq 9, at zero distance between the tip and substrate, we find that f2 < f1. Thus, due to the interactions with the tip, the micelles contain fewer chains that in their initial state. The radius of the core at surface 2, R2 ) Ro(f2)/g2 ) Ro(f1)g2-3/5, becomes larger than that for the initial micelle, whereas the overall size,

D2 ) (f2s)1/2 ) g23/5D1

(10)

diminishes. We assume, however, that the ratio, D2/R2 ) g26/5 D1/R1, remains higher than unity when g2 is decreased. In other words, the preferential wetting of the polymer onto the tip does not lead to a merger of the micelles and the formation of a homogeneous film at surface 2. Increases in H, the distance between the tip and the substrate, lead to corresponding increases in f2 and the micelles grow in size. In particular, the cost of stretching the anchored chains can be offset by incorporating more of the grafted polymers into each micelle because this rearrangement reduces the surface tension or surface free energy per chain.11 Note that because the size of each micelle increases, the total number of pinned micelles decreases. Such lateral rearrangements in the size of the micelles are, however, expected to be quite slow because the system has to overcome energetic barriers to reach the most favorable conformation. Thus, these rearrangements might not occur within the times scales of the relevant experimental studies. Consequently, in the subsequent analysis, we assume that the number of chains per micelle, f2, remains constant during the deformation process. The corresponding value for the total free energy per chain, F ) Fleg + Fsurf, in such a micelle is determined by eqs 7-9 and can be written as

F2 ) F1[0.4(x2 + g26/5)1/2 + 0.6g23/5]

(11)

Here, we have introduced the parameter x ) H/D1, which characterizes the reduced distance between the tip and substrate. (We now retain the numerical coefficients in our equation; these will be necessary for the calculations described later.) At relatively small distances H between

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the tip and substrate, where F2 < F1, the micelles are attached to the tip and their characteristics are given by eqs 9-11. At larger distances, where F2 > F1, the micelles detach from the tip and return to their initial state, possessing features described by eqs 4-6. By setting F1 ) F2, we can determine the height Hc where the micelles will undergo a discontinuous transition or “jump” from the tip to the substrate. By applying eq 11, we arrive at the equation for the reduced critical deformation,

xc ) [(2.5 - 1.5g23/5)2 - g26/5]1/2

(12)

We can now calculate the attractive force acting between the tip and the substrate. By differentiating F2 with respect to H, and taking into account eqs 6 and 11, we obtain 6/5

Ga/kT ) k2τx/(g2

2 1/2

+x )

(13)

where G is the force per chain (or, equivalently, per area s of the substrate). At x > xc, the force G ) 0, whereas at x < xc, the value of force is given by eq 13. At the transition point x ) xc, which corresponds to the discontinuous detachment of the chains from the tip, the tip experiences the maximum force Gc ) G(xc), where xc is given by eq 12. Large Deformations of the Polymer Layer. The picture just described holds for relatively high values of g2 < 1; that is, where the polymer is not highly spread out on the tip. At lower values of g2 (greater spreading of the polymer on the tip) and larger separations between the tip and substrate, the pinned micelles undergo further structural rearrangements. As shown in Figure 1(c), the legs now start to merge near the core of the micelle and form a long bundle, or strand, adjacent to the core. Aggregation of these monomers reduces the free energy of the system because the aggregated segments are effectively shielded from the solvent. We can approximate the strand of aggregated segments by a cylinder of radius F and height hc. As shown in Figure 1(c), the nonaggregated parts of the legs form a cone of height (H - hc) and each leg is comprised of a string of blobs. The free energy per chain in such a micelle, F, is given by

F ) Fsurf + Fleg + Fcyl

(14)

The surface free energy associated with the core of the micelle, Fsurf, is still given by eq 8. The free energy of the legs, Fleg, is now modified in the following way:

Fleg/kT ) k2τ[(H - hc)2 + sf2]1/2

(15)

The last term in eq 14, Fcyl, accounts for the parts of the chains that are in the cylindrical strand. We let n be the number of monomers per chain in the cylinder. Because the polymer concentration in the strand is fixed at τ, the radius of the cylinder is related to n in the following way:

F ) a[nf2/(τhc)]1/2

(16)

The free energy per chain in the strand involves losses due to the elastic stretching of the chains,

F′str/kT ) hc2/a2n

(17)

and the surface free energy associated with the surface of the cylinder,

F′surf/kT ) 2πγhcF/f2 ) τ3/2n1/2hc1/2f2-1/2

(18)

By minimizing Fcyl ) (F′str + F′surf) with respect to n at a fixed value of hc, we arrive at an equilibrium value of n,

n ) hcf21/3/aτ

(19)

Note that by inserting eq 19 into eq 16, we find that F does not depend on hc, but remains constant in value. Equation 19 also allows us to determine the value of Fcyl

Fcyl/kT ) hcτ/af21/3

(20)

The total free energy per chain in the micelle reads

F/kT ) k2τ[(H - hc)2 + sf2]1/2 + k2hcτ/af21/3 + k1N2/3 τ4/3 f2-1/3g2 (21) where, for computational purposes, we now retain the numerical coefficients. As indicated by eq 21, an increase in hc leads to a decrease in the free energy of the legs (first term in eq 21) and an increase in the strand free energy (second term in eq 21). By minimizing F with respect to hc, we obtain the equilibrium height of the cone h ) (H - hc),

h ) D2(f22/3 - l)-1/2 ) D1/(f12/3g22/5 - g26/5)-1/2

(22)

Thus, we find that the size of the cone, h, does not depend on the distance H between the tip and the substrate, and is solely determined by the number of chains, f2, and the grafting area s. Increases in H lead to the formation of the cylindrical strand only when H exceeds the distance given by eq 22. Thus, the appearance of the aforementioned conformation for the pinned micelle is only possible when H > h. Here, the free energy per chain can be calculated through eqs 21 and 22. After some algebra, we arrive at the following expression for this free energy:

F2′ ) F1g23/5[0.4(l - g2-4/5f1-2/3)1/2 + 0.6 + 0.4xg2-1f1-1/3] (23) where F1 is given by eq 6 and x ) H/D1 is the reduced distance between the tip and substrate. By differentiating F2′ with respect to H (or equivalently, x), we can calculate the force acting between the tip and substrate. Because F2′ depends linearly on x, we arrive at a constant value of force,

G′a/kT ) k2 τ/f21/3 ) k2τg22/5/f11/3

(24)

Thus, as a result of the structural transformation within the micelles (namely, the development of the strand at H > h), the system exhibits a “plateau”-like regime in the force versus distance profile. The specific physical basis for the constant-force regime can be explained in the following manner. As H is increased, hc also increases [recall that hc ) H - h, where h remains constant (eq 22)]. As indicated by eq 19, an increase in hc necessitates an increase in n, the number of monomers in the cylindrical strand. The additional monomers are extracted from the micellar core, the reservoir of available units, and localized in the cylindrical strand. These additional units maintain the radius of the strand and effectively mediate the effects

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Langmuir, Vol. 14, No. 16, 1998 4619

Figure 2. The diagram of states plotted in the following reduced coordinates: x ) H/D1 versus g2-1. The curves marked by xc and xc′ separate the phase space into two regions. In region I, the chains form pinned micelles on the substrate and there is no contact between the tip and substrate. In region II, the pinned micelles are spread on the surface of the tip and there is an attractive force between the two surfaces. Within region II, above the dashed line, the legs aggregate into strands. Below the dashed line, the legs are separated into individual strings of blobs.

of the applied force. Put concisely, the unraveling of the micellar core gives rise to a profile where G′ does not depend on H. The break down of this plateau regime occurs when the pinned micelles detach from the tip. By equating F2′ ) F1, we can determine the critical deformation xc′ at which the micelles abruptly return to the substrate,

xc′ ) f11/3(2.5g2-2/5 - 1.5g2-1) - [f11/3 g2-2 - g2-6/5]1/2 (25) Figure 2 reveals a “phase diagram” of the system, plotted in the g2-1, x coordinate frame. The diagram is composed of two distinct regions, I and II, where the boundary xc between the regions is given by eq 12. In region I, the chains form micelles on the substrate and there is no contact between the tip and the polymer. Here, the characteristics of the micelles are described by eqs 4-6. In region II, the chains form micelles on the surface of the tip. Here, the features of the micelles are given by eqs 9-10. Below the dashed line, which is determined from the condition F2 ) F2′ and is given by xo ) (f12/3g2-2/5 g2-6/5)-1/2, the legs of the pinned micelles do not aggregate, but form separate strings of blobs [as in Figure 1(b)]. Here, the free energy of the system is given by eq 11, whereas the force acting between the tip and the substrate is described by eq 13. Above this dashed line, the segments within the legs aggregate to form a cylindrical strand. Here, increases in H lead to corresponding increases in the length of the cylinder, hc ) H - h, whereas the height of cone h formed by the nonaggregated parts of the legs stays constant. The free energy of the system is given by eq 23, whereas the force is determined by eq 24. The transition of the micelle from the tip to the substrate occurs at the critical separation xc′ given by eq 25. Figure 3 schematically depicts the force versus distance profiles for various values of g2. As shown in Figure 3, a noticeable plateau is predicted for relatively low values of g2. Here, the value of force is given by eq 24, which further indicates that G′, the force per chain, scales as

Figure 3. Schematics for force versus distance profiles for the case where the polymers wet the tip. The plots are shown for the following values of g2-1: (a) 1.01; (b) 1.04; (c) 1.08; and (d) 2.00. The dotted lines indicate the failure of the tip-polymer contact.

f1-1/3. The value of the latter exponent (-1/3, rather than 0) reflects the collective response of the chains to the deformation, namely, the formation of the cylindrical strand depicted in Figure 1(c). A constant force regime has been predicted theoretically in prior studies of stretching collapsed polymers (such as individual globules and brushes,12 and pinned micelles localized in the middle of two confining surfaces9b). These earlier studies were based on the behavior of a single leg. As a consequence, the findings showed that when the applied force is equal to the tension in the string of blobs () τ), pulling a chain out of a micelle happens with constant force. For our system of a polymer wetting the tip, the force only acts at x < xc or x < xc′. If g2 is close to unity, and the detachment of the micelles occurs at relatively small deformations, the force increases with H in a virtually linear manner according to eq 13. This behavior is due to the rather limited range of distances (small values of xc) over which the polymers are stretched or deformed. The force-distance profiles reveal a noticeable plateau only for lower values of g2 (see Figure 3), where the polymers are relatively spread out on surface 2. Here, the value of xc′ is sufficiently high to provide a range of x with a nearly constant force versus distance profile. In contrast to the studies based on the behavior of a single leg, the value of this constant force is a factor of f-1/3 less than the tension in a single string of blobs ()τ). Next, we show that the plateau region becomes even more extended for tips that undergo covalent bonding with the polymer layer. Chemical Bonding between Polymers and Tip. We now consider the behavior of chemically modified tips that can form covalent bonds with sites in the polymer layer. In particular, we assume that the reactive groups on the surface of the tip can form p irreversible links with the free ends of the grafted chains. One can envision situations where p is either relatively low or relatively high. The latter case could be realized by grafting branched or star polymers that contain functional groups at each chain end. Regardless of the value of p, we assume that the reactive groups on the tip are distributed rather sparsely on this surface. Thus, each pinned micelle only forms p , f1 links with the tip when it is embedded in the micellar core. When the tip moves away from the (12) Halperin, A.; Zhulina, E. B. Macromolecules 1991, 24, 5393; Europhys. Lett. 1991, 15, 417.

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substrate, the covalent bonds stretch the chains away from the pinned micelles, as shown in Figure 1(d). This motion gives rise to a resisting (attractive) force between the tip and substrate. Later, we analyze how the value of this force is affected by the number of irreversible bonds between the tip and the grafted chains. In effect, the number of such links is a measure of the strength of the interaction between the tip and the substrate. Figure 1(d) schematically illustrates the conformation of a pinned micelle with p chains chemically linked to the tip, which is located a distance H from the substrate. (Here, the number of links between the substrate and tip is equivalent to the number of chains bound to the tip.) With increases in H, the chemically bound chains are pulled out of the micellar core, whereas the rest of the chains stay virtually unperturbed by the deformation. As a first approximation, the total number of chains in such perturbed micelle remains constant and equals f1. (Our assumption of a constant f1 is again justified by the slow kinetics involved in the rearrangement of the collapsed polymers.) As in the case considered in the previous section, we assume that segments of the bound chains form a cylinder of radius F and length hc. Because all the linked chains are attached to the tip at a single spot, we do not have a cone of separated legs near surface 2. A cone of legs is, however, formed near surface 1. The presence of this cone is due to the lifting of the micellar core away from the substrate. Such a lifting of the core decreases the total stretching of the bound chains. Recall that we are considering polymers that dewet surface 1 and thus, the detachment and lifting of the core will not lead to increases in the surface free energy. We can now use the arguments of the previous section to analyze the conformation of such a micelle. Namely, we assume that each bound chain provides n monomers, or nb ) na2/ξ2 blobs of size ξ ) a/τ, to the cylindrical strand in Figure 1(d). The stipulation that the blobs are densely packed within the cylinder provides a relationship between F and H. Namely, F2H ) pnbξ3 ) a3pn/τ, which yields

F ) a(np/Hτ)1/2

(26)

where, as before, numerical prefactors are omitted. The free energy of such a micelle (per tethered chain) is again given by

F ) Fleg + Fsurf + Fcyl

(27)

The first term in eq 27 describes the free energy of the legs, which form a cone of height (H - hc),

Fleg/kT ) k2τ[(H - hc)2 + sf1]1/2

(28)

(which is comparable to eq 15). The second term in eq 27 accounts for the surface free energy associated with the core of the micelle (eq 2), which consists of (Nf1 - np) monomers,

Fsurf/kT ) k1(Nf1 - np)2/3τ4/3f1-1

(29)

Finally, the third term in eq 27 is due to cylindrical strand. As before, the latter contribution, Fcyl ) Fstr′ + Fsurf′, incorporates the surface free energy associated with the surface of the strand,

Fsurf′/kT ) 2πγ H F/f1 ) τ3/2n1/2hc1/2p1/2/f1

(30)

and the elastic stretching of the segments in the cylinder,

Fstr′/kT ) phc2/a2nf1

(31)

By minimizing F with respect to n at a fixed value of hc, and neglecting the smaller term arising from the surface free energy, eq 29, we obtain

n ) hcp1/3/τ

(32)

The corresponding value of Fcyl yields

Fcyl/kT ) Fsurf′/kT + Fstr′/kT ) k2hcτp2/3/f1 (33) From eqs 32 and 33, we can show that the polymer segments in a cylinder are less stretched than the segments in a micellar leg, which is formed from totally stretched strings of blobs. In particular, each segment in the cylinder consists of nb ) nτ2 ) Hτp1/3 blobs, and has a total length nbξ ) Hp1/3. Thus, the degree of stretching for such a segment, H/n ) τp-1/3, is less than the degree of stretching for the leg () τ) by a factor p1/3. The total free energy per chain yields

F/kT ) k2τ[(H - hc)2 + sf1]1/2 + k1(Nf1 - np)2/3τ4/3f1-1 + k2hcτp2/3/f1 (34) By minimizing F with respect to hc, and neglecting the contribution due to the surface free energy (second term in eq 34), we arrive at the equilibrium height of the cone, h ) (H - hc),

h ) (f1s)1/2(f12p-4/3 - 1)-1/2 ≈ D1p2/3f1-1

(35)

As in the example just shown, the height of the cone, h, formed by the individual legs, does not depend on the deformation of the micelle, provided that H > h. However, the specific value of the above h is different from that in eq 22. This specificity is because chemical bonding involves only p < f1 chains in the formation of the cylindrical strand, whereas polymer wetting of the tip drives all f2 chains to participate in a cooperative response to the deformation. The corresponding force acting between the tip and substrate is given by d(F)/dH per tethered chain. Neglecting the small contribution due to the variation in the surface free energy of the core (i.e., putting Nf1 - np ) Nf1), we obtain

Ga/kT ) d(F/kT)/dH ) k2τp2/3/f1

(36)

We thus find that the pullout of the chemically bound chains occurs at a constant value of force, independent of the extent of stretching. The pullout force G given by eq 36 acts normal to the substrate. The normal component of the elastic force within each leg () k2τ) equals k2τ sin(R), where R is the average bending angle of a leg with respect to the surface [i.e., sin(R) ) h/(h2 + D12)1/2]. Thus, eqs 35 and 36 ensure a balance of the normal forces and, correspondingly, the mechanical equilibrium of the core of the micelle. The cooperative response of the linked chains to the deformation results in a nonlinear dependence of the force G on the number of links between the tip and the substrate p. The scaling dependence G ) p2/3, is weaker than a linear dependence on p. Recall that in case of p independently stretched chains, the response would be proportional to p. Thus, our findings indicate that the bound

Model of AFM Tip-Polymeric Substrate Interaction

Langmuir, Vol. 14, No. 16, 1998 4621

chains undergo a cooperative rearrangement or adjustment in response to the applied force. In other words, at a fixed value of H, the stretching of the chain segments in the cylinder is reduced by transferring units from the core into the cylinder. This “reeling out” of segments from core to the cylinder is also the physical basis for the constant-force response to the deformation of the layer. We find the same effect in both systems, for polymers wetting the tip and polymers chemically bound to the tip. (In the first case, it is micelles on the surface of the tip that provide the monomers for the connecting strand, whereas in the second case, it is micelles localized on the substrate that yield the monomers for the chemically bound cylinder.) Although the range and magnitude of the effect will be different for the two systems, the underlying physics is, however, the same. The winding of the chain out of the core and into the cylinder also ensures that the radius of the cylinder, remains independent of H in this range of deformations. We note that eq 36 does not cover the range of very small deformations, H < ξ. Here, distortions of the core result in a linear restoring force because an undeformed micelle is found under thermodynamic equilibrium. At larger values of deformation, H < h, where the cylindrical strand is not yet formed, the response is determined by the cone of separated legs. Here, we have Fleg/kT ) k2τ(f1s + H2)1/2, and correspondingly,

Ga/kT ) k2τH/(f1s + H2)1/2

(37)

Finally, when H becomes larger than h, which is determined by eq 35, the bound chains start to form a cylinder and the force between the tip and the substrate is now given by eq 36. The latter regime persists until n becomes comparable to N, n ≈ N, or equivalently (from eq 32) until the separation between the tip and substrate reaches the critical value H ) H* ) Nτ/p1/3. Now, the majority of the units in the bound chains contribute to the cylinder [whereas (f1 - p) chains form the core]. The parts of the linked chains that constitute the cylinder are, however, less stretched than the legs of the micelle. Further increases in H > H* lead to a significant increase in the stretching free energy of the segments in the cylinder, Fstr′, according to eq 31 where n ) N. According to eq 30, the corresponding increase in Fsurf′ is relatively small. Thus, the restoring force per micelle is essentially given by d(F′str)/dH, which yields

Ga/kT ) pH/(Nf1)

(38)

per tethered chain. The force profile now reveals a linear dependence on both p and H. Here, the thickness of the cylinder decreases according to eq 26 where n ) N; that is,

F ) a(Np/Hτ)1/2

(39)

and now H > H*. Because the value of G increases with increases in H (eq 38), maintaining the mechanical equilibrium of the core requires a corresponding growth in h, the height of the cone, formed by the individual legs (or, equivalently, an increase in the bending angle R). Now, the equality of the normal forces yields

sin(R) ) pH/(f1Nτ)

(40)

Figure 4. Schematic for the force versus distance profile for the case where polymers form chemical bonds with the tip.

and the height of the cone increases as

h ) D1/[f1Nτ/(pH) - 1]-1/2

(41)

When H becomes comparable to the value H** ) Nτ, the radius of the cylinder equals F** ) (ξ2p)1/2. Correspondingly, the area per chain in the cylinder, (F**)2/p ) ξ2, and the chains in the cylinder can be viewed as totally stretched strings of blobs. In terms of our scaling model, this means that we are at the threshold of the chains being entirely pulled out of this cylinder. Indeed, the tension in the bound chains now equals that in the legs of the micelle, and retaining the cylindrical strand becomes unfavorable. As a consequence, the cylinder is expected to split into separate strings of blobs, as in the case of pinned micelles that bridge two surfaces.9b,11 In scaling terms this happens at H** ) Nτ. Here, the bound chains are totally pulled out of the cylinder and the chains in the assembly split into a micelle formed by (f1 - p) chains, tethered with an average grafting area s′ ) sf1/(f1 - p), and p independent bridging chains, which are attached to the chemical group at the tip and form strings of blobs between surfaces 1 and 2. However, because p , f1, in the scaling model, the size of the pinned micelle is only slightly affected by the pullout of the bound chains. The core of the micelle returns to the surface 1, that is, it abruptly jumps from a height h ) h(H**) to h ) 0 upon the pullout of the bound chains. Figure 1(e) shows the conformation for the latter system. Further increases in H > H** lead to the destruction of the blobs and the bound chains obey the normal Gaussian statistics of strongly stretched polymers. Here, the value of the restoring force is still given by eq 38. Finally, at H ) Hc ) aN, one expects the rupture of the chemical links between the tip and polymer, between the polymer and substrate or between the bonds within the chains. Furthermore, Ga/kT changes from its maximal value () p/f1) to zero, indicating the failure of the contact. The schematic behavior of the entire force versus distance profile is presented in Figure 4. Conclusions We investigated the interactions between a polymercoated substrate and two different AFM tips. In the first case, the polymer preferentially wets the surface of the tip, whereas in the second example, the polymer forms a chemical bond with the tip surface. The different polymer-tip interactions give rise to distinct force-distance profiles. In the case of preferential wetting, the forcedistance profile reveals the existence of an abrupt transition, which takes place at a critical tip-substrate sepa-

4622 Langmuir, Vol. 14, No. 16, 1998

ration, Hc. At smaller separations, the system exhibits an increasingly attractive force. At the critical separation, the force abruptly goes to zero, indicating the failure of the contact between the tip and substrate. A surprising result, however, is that below Hc and at small wetting angles (Θ1/Θ2 < 1), the force-distance profile displays a plateaulike region, where the force is virtually independent of the stretching or deformation of the grafted layer. In the case of chemical bonding between the tip and polymer layer, the force-distance profile is relatively complex. The profile displays a well-defined plateau region, where the attractive force between the tip and substrate is independent of the deformation of the layer. In this region, the force shows a weaker than linear dependence on the number of links between the tip and substrate, or the effective “strength” of the interaction. This regime exists for a wide range of deformations (until nearly all the units in the bound chains form a cylindrical strand between the tip and the substrate). This region is followed by a linear response regime, where the behavior of the bound, strongly stretched chains is described by Gaussian statistics. The failure of the tip-polymer contact takes place when the distance between the tip and substrate is comparable to the contour length of the tethered chains. Our findings indicate that in the case of strong wetting or chemical bonding at the polymer-tip interface, one expects a plateau regime for the force versus distance

Zhulina et al.

profiles. Our models rationalize many of the features that were recently observed in AFM studies of the hydration of fluorinated amide/urethane copolymer films.13 In these experiments, which involved the use of chemically functionalized AFM tips to probe copolymer micelles, the retraction of the AFM tips yielded force profiles where the deflection force was almost independent of the cantilever position, or where, the potential energy of the system was proportional to the tip-sample separation. Of particular interest are ongoing experiments that probe the molecular basis for the magnitude of the restoring force and the distance over which the tip-sample interaction force is constant. Future work will address these issues as the experimental data becomes more detailed. Acknowledgment. A. C. B. gratefully acknowledges support from ONR through grant N00014-91-J-1363, from the N.S.F., through grant number DMR-9709101, and from the D.O.E, through grant number DE-FG0290ER45438. G. C. W. gratefully acknowledges support from ONR through Grant N00014-96-1-0735 and a 3M Untenured Faculty Award. The authors thank Vasiliy Zhulin for assistance in preparing the figures. LA980344R (13) Akhremitchev, B.; Mohney, B. K.; Marra, K. G.; Chapman, T. M.; Walker, G. C. Langmuir, in press.