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Interaction between a Grafted Polymer Chain and an AFM Tip: Scaling Laws, Forces, and Evidence of Conformational Transition Jorge Jimenez and Raj Rajagopalan* Department of Chemical Engineering, University of Florida, Gainesville, Florida 32611-6005 Received November 13, 1997. In Final Form: March 23, 1998 The force of compression of an end-grafted polymer chain in a Θ solvent by a finite-sized object (e.g., the tip of an atomic force microscope) is studied using a new simulation technique called the contactdistribution method. This paper presents the first ever simulation of the conformational transition of a polymer chain from a “confined” state to an “escaped” state. The results corroborate the force relations that were recently obtained using scaling arguments but clearly indicate the existence of a transition region between the two states. The characteristics of the transition are affected by the size, shape, and alignment of the tip and by the temperature or quality of the solvent. We also comment on the implication of the results to the experimental confirmation of the predictions.
Introduction The deformation of polymer layers by finite-sized objects has received considerable attention in recent years.1-6 One case of special interest is the compression of grafted polymers in the so-called mushroom regime. While, from a practical perspective, this case is of importance in some biological systems,7,8 it is also of great interest from a more fundamental standpoint due to the novel and interesting conformational transitions9 that could occur in a polymer chain under compression.1-3 In this communication we use bond-fluctuation Monte Carlo simulations and a new method for determining the force due to compression to study the interaction between an endgrafted polymer chain and an AFM tip in a Θ solvent. Our objectives are to examine the accuracy of the scaling relations and the nature of the above-mentioned conformational transition and, in addition, to examine the effect of the shape, size, and alignment of the tip on the deformation and resulting force. Scaling Theory: Details of the Conformational Transition The interaction between a cylindrical tip with a flat bottom and a polymer mushroom has been studied by scaling arguments in both good1,2 and Θ solvents.3 These studies identify a conformational transition in which the chain goes from a confined state to an escaped state, giving rise to a sudden increase in the end-to-end distance of the (1) Subramanian, G.; Williams, D. R. M.; Pincus, P. A. Europhys. Lett. 1995, 29, 285. (2) Subramanian, G.; Williams, D. R. M.; Pincus, P. A. Macromolecules 1996, 29, 4045. (3) Guffond, M. C.; Williams, D. R. M.; Sevick, E. M. Langmuir 1997, 13, 5691. (4) Williams, D. R. M.; MacKintosh, F. C. J. Phys. II 1995, 5, 1407. (5) Braithwaite, G. J. C.; Howe, A.; Luckham, P. F. Langmuir 1996, 12, 4224. (6) O’Shea, S. J.; Welland, M. E.; Rayment, T. Langmuir 1993, 9, 1826. (7) Woodle, M. C.; Lasic, D. D. Biochim. Biophys. Acta 1992, 1113, 171. (8) Kuhl, T. L.; Leckband, D. E.; Lasic D. D.; Israelachvili, J. N. Biophys. J. 1994, 66, 1479. (9) As discussed in the following sections, the polymer chains undergo a conformational transition, from a “confined” state to an “escaped” state, under compression. Although this transition is referred to as a “phase” transition in earlier papers, we shall use the term “conformational” transition to avoid confusion.
chain. In this section, we summarize the essential results of Guffond, Williams, and Sevick3 for the Helmholtz potential F and the force f for subsequent comparison with our simulations. For a confined chain, the scaling laws for the Helmholtz potential and the force in a Θ solvent can be written as
Na2 Na2 Fconfined ) c1 2 , fconfined ) 2c1 3 H H
(1)
where c1 is the proportionality constant, N is the degree of polymerization, a is the monomer size, and H is the distance between the tip and the surface. For a chain that has escaped from under the tip the relations become
Rt Rt Fescaped ) 2c2 , fescaped ) 2c2 2 H H
(2)
where c2 is the proportionality constant and Rt is the radius of the cylindrical flat-bottom tip. The crossover point between the two free energies occurs at
H* )
c1 Na2 c2 2Rt
(3)
In the earlier studies,1,3 H* has been considered as the value of H at which the transition takes place. A sudden jump in the end-to-end distance and, hence, a jump in the predicted force profile led these authors to predict the existence of a first-order “phase transition”. These scaling relations are derived using order-of-magnitude arguments based on the “blob” concept of polymer chain conformations and have not been tested quantitatively so far. Moreover, although the scaling approach identifies a well-defined crossover point, it is reasonable to expect that thermal fluctuations of the system will blur the free energy crossover between the two states. It is therefore of interest to develop simulation methods to examine the scaling relations as well as the transition region between the two states. This forms the focus of the subsequent sections of this Letter.
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Langmuir, Vol. 14, No. 10, 1998 2599 Table 1. Simulation Parameters characteristics of the AFM tip
set number
chain length
shape
bottom
sizea
alignment
quality solvent
I II III IV V VI VII
100 60 60 100 100 100 100
monolith monolith monolith monolith cylinder cylinder monolith
flat flat flat flat flat hemispherical flat
Lt ) 120 Lt ) 60 Lt ) 60 Lt ) 100 Rt ) 50 Rt, Rc ) 50 Lt ) 120
centered centered skewed centered centered centered centered
Θ Θ Θ Θ Θ Θ good (athermal)
a Size given in lattice units. L is the side of the monolith tip, R is the radius of the cylindrical tip, and R is the curvature radius at t t c the bottom of the tip.
Model and Simulation Method
Results and Discussion
The simulations presented here employ the threedimensional bond-fluctuation model described in the literature.10,11 A lattice size of 200 × 200 × 200 is used with impenetrable walls at z ) 0 and z ) 200 and periodic boundary conditions in the x- and y-directions. We consider a single chain grafted to the center of the lower wall (i.e., at x ) 100, y ) 100, z ) 1) and allow the nonbonded segments to interact with each other through a square-well potential with an interaction parameter given by χ/kT ) -0.53, where k is the Boltzmann constant. This interaction gives rise to a behavior characteristic of Θ conditions.12 The shape and the size of the tip and other conditions used in the simulations are summarized in Table 1. For each set of parameters, several simulations are run for different values of H (with at least one run for each H). In each case, the system is first equilibrated and then sampled over (1-3) × 107 Monte Carlo steps to obtain the probability of the chain having contacts with the tip, from which the Helmholtz potentials and the forces can be obtained. However, obtaining the Helmholtz potentials and forces from Monte Carlo simulations is not straightforward. One method available in the literature is based on the introduction of a repulsive potential in the layer next to the boundary of the compressing object. This method, due to Dickman,13-15 is known as the repulsive-wall method and requires the calculation of the mean number of contacts of the polymer chain and the solid object for different values of the repulsive potential. The method, though useful, requires a large number of simulations and is computationally time-consuming. In order to circumvent such problems, we have developed a new method that uses not only the mean value but also the probability distribution of the number of contacts of the chain with the solid object. One can use the computed results for the probability of contacts to determine the free energy change (and hence the force) as the chain is compressed. The mathematical details of this technique, which we call the contact-distribution method, and the rationale will be presented elsewhere; here we focus on the results of the simulations and their implications to the scaling arguments for the problem outlined in the previous section. In addition, we comment on the implications of the results to the experimental determination of the conformational transition of the polymer chain and to the possibility of detecting the transition regime from experimental force profiles.
We first examine the accuracy of the scaling laws for the Helmholtz potential and the nature of the conformational transition. The conditions specified as set I in Table 1 are used for this purpose, and the results are shown in Figure 1. Figure 1a shows that the agreement between our simulations and the scaling relations, that is, eqs 1 and 2, is excellent. (The proportionality constants c1 and c2 in the equations are treated as adjustable parameters to obtain the dashed lines.) It is clear that the confirmation of the power-law dependence of the force for the confined state is unambiguous. For the escaped state only two points have been computed, but the obtained values are consistent with the expected H-2 behavior for the force. (We have not attempted to obtain more points in the “escape” region, since the chain enters a glassy state when H is about four lattice units or less. Very long computer runs are needed for obtaining results with acceptable accuracy under these conditions.) Moreover, the value of H* predicted by eq 3 from c1, c2 and Na2 obtained from the simulations (see the figure caption) is roughly about 8.2 and falls within the transition region, thus confirming the consistency of the scaling relations. The fact that the value of H* thus obtained is almost at the onset of the transition is, however, not a concern because of the uncertainty associated with the numerical estimation of c 2. Figure 1a also shows that the transition region ∆H obtained from the simulations is only a small fraction of the end-to-end distance Rend. The size of the transition region has important implications to attempts to corroborate the conformational transition experimentally, since experimental force-measurement methods typically use finite steps in H in the measurements.16,17 For the results shown in Figure 1a, ∆H is roughly equal to 10% of Rend, which places limits on the required sensitivity of the equipment or the minimum size of the chain that must be used for the transition to be accessible experimentally.18 It is also important to note that the statistical error in the force within the transition region is rather large compared to the other points. This error can be reduced further by considering a larger ensemble of simulated chains. However, for sufficiently slow compression (i.e., time scale of compression much larger than the conformational relaxation time of the chain) one would expect the existence of a well-defined transition regime rather
(10) Carmesin, I.; Kremer, K. Macromolecules 1988, 21, 2819. Deutsch, H. P.; Binder, K. J. Chem. Phys. 1991, 94, 2294. (11) Lai, P. Y.; Binder, K. J. Chem. Phys. 1991, 95, 9288. (12) Lai, P. Y.; Binder, K. J. Chem. Phys. 1992, 97, 586. (13) Dickman, R.; Hong, D. J. Chem. Phys. 1991, 95, 4650. (14) Deutsch, H. P.; Dickman, R. J. Chem. Phys. 1990, 93, 8983. (15) Dickman, R.; Anderson, P. E. J. Chem. Phys. 1993, 99, 3112.
(16) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, 1991. (17) Roters, A.; Gelbert, M.; Schimmel, J. R.; Johannsmann, D. Phys. Rev. E 1997, 56, 3256. (18) This comment also applies to lattice-based computer simulations. That is, even though we observe a transition region between the two states, the details of how the force changes within the transition region as a function of H are difficult to determine with lattice-based models. Off-lattice simulations, which permit finer intervals in H, are required for this particular task.
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Figure 1. (a) Force profile when compressing a 100-segment chain with a flat-bottom monolith (Lt ) 120). To fit the equations we use (Na2)1/2 ∼ 27 (average end-to-end distance for the unperturbed chain obtained from simulations) and “Rt” ) 60. The values of c1 and c2 obtained from fitting the simulation data to eqs 1 and 2 are 1.9 and 1.4, respectively. (b) z-component of the radius of gyration for the grafted chain. The bars above and below the points indicate the statistical errors obtained in the calculation.
than the sudden changes in the force and in the scaling behavior predicted by eqs 1-3. Figure 1b shows that the slope of the z-component of the radius of gyration Rgz with respect to H changes sign from positive to negative at the onset of escape of the chain. Previously, the variation of the end-to-end distance has been used1,3 as an indication of the escape of the chain from under the tip. However, in the case of the simulations, since the statistical errors associated with the calculation of Rgz are much smaller, we find that Rgz, rather than the end-to-end distance, is a better indicator of the point at which the escape probability becomes significant. One would expect the size, shape, and alignment of the tip (relative to the grafting point) to influence the chain conformation and the transition, and this is evident from Figure 2. This figure shows that the distance at which the chain escapes and the occurrence of the inflection point in the force profile depend on the actual size and shape of the tip and the alignment of the tip with respect to the polymer chain. The transition region can be either small or not clearly defined depending on the above consider-
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Figure 2. (a) Force profiles when compressing a 60-segment chain with an aligned and misaligned flat-bottom monolith (Lt ) 60). In the former case, the center of the monolith and the grafted end of the polymer chain [(x, y) ) (100, 100)] are aligned one over the other. In the latter, the center of the tip is located at (x, y) ) (90, 90). (b)This figure summarizes the results obtained for sets IV, V, and VI listed in Table 1. The dotted lines correspond to the statistical errors in the determination of the force.
ations (see, for example, the case of a tip with a hemispherical bottom). This implies that the typical sensitivity of the AFM might be insufficient for detecting the transition experimentally. However, as shown in Figures 1 and 2, we expect the fluctuations in the force to increase markedly within this region, which suggests that recent methods based on the analysis of the noise of the AFM tip17 might serve as better probes to detect the transition region from the force profile. Figure 3 illustrates the influence of the temperature on the force profile. The results were obtained for the same system shown in Figure 1a but at a higher temperature, that is, at good solvent conditions. One notices that at this temperature the transition region becomes smaller and almost imperceptible. This suggests that there is a critical temperature at which the thermal fluctuations of the system make a clear distinction between confined and escaped states no longer possible. This critical temperature, according to our results, is close to the good solvent condition.
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Figure 3. Force profile for a 100-segment chain and a flatbottom monolith (Lt ) 120) in a good solvent (i.e., athermal).
Although presented in the context of the interaction between a polymer mushroom and an AFM tip, the phenomenon described in this paper has broader implications, both to other force measurement techniques and particle/substrate interactions in the presence of polymers. For example, instead of an AFM, one can use evanescent wave light scattering from a particle held by an optical trap to measure particle/surface force as a function of H. This technique is much more sensitive than the AFM and is capable of detecting forces of the order of picoNewtons. The phenomenon described in the present paper is of importance in interpreting such measurements, as it is for other applications such as colloid stability and biological cellular interactions.
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Finally, we would like to emphasize that the forces considered in this paper are steric interactions between the polymer layer and the finite-sized objects. Additional interactions between the surface and the object (AFM tip or a spherical particle) can be present and can make the detection of the transition difficult. Although the specific situation discussed in this paper is more of fundamental interest than anything else, similar conformational transitions can also take place when polymer layers at higher grafting densities are compressed. This could give rise to an inflection point in the force profile and deserves attention. In a forthcoming publication, we shall present an extended report on the details of the simulation methods used here and on results that are mentioned only peripherally in this Letter. There we shall also present a discussion of interaction forces in polymer layers in the brush regime and when different objects such as a spherical particle or an AFM tip with a rough surface are used. It is also worth noting that the simulation method used in this paper is not restricted to the type of situations considered here. For example, it can be used to study interactions between particles of arbitrary shapes in the presence or absence of additives and between particles or substrates with adsorbed or grafted polymer layers, among others. Acknowledgment. We thank the Higher Education Coordinating Board of the State of Texas and the NSF Engineering Research Center for Particle Science and Technology, University of Florida, for partial support of the work presented here. We also thank one of the reviewers for pointing out an error in the original manuscript. LA971233F
