Dynamics and Friction in Submicrometer Confining Systems

Chapter 3

Dynamic Force Spectroscopy: Effect of Thermal Fluctuations on Friction and Adhesion 1

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1

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Downloaded by NORTH CAROLINA STATE UNIV on September 7, 2012 | http://pubs.acs.org Publication Date: July 17, 2004 | doi: 10.1021/bk-2004-0882.ch003

O. K. Dudko , A. E. Filippov , J. Klafter , and M. Urbakh

School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel Donetsk Institute for Physics and Engineering, 83144 Donetsk, Ukraine

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In this paper we propose a model which allows to describe the effect of thermal fluctuations on atomic scale friction and dynamical response of adhesion complexes. In the case of friction the model presents a generalization of Tomlinson model which includes the contribution of an external noise. We demonstrate that important information on friction and adhesion can be obtained from measurements of distribution functions for the maximal spring force which quantifies static friction and rupture forces. We derive relationships between equilibrium potentials and the forces measured under nonequilibrium conditions.

Experiments that probe mechanical forces on small scales provide a versatile tool for studying molecular adhesion and friction through the response to mechanical stress of single molecules or of nanoscale tips. The probing techniques include atomic force microscopy (AFM) (1,2), biomembrane force probe microscopy (3) and optical tweezers (4). Examples for processes which are investigated are friction on atomic scale (1,2), specific binding of ligandreceptor (5), protein unfolding (6), and mechanical properties of single polymer molecules such as D N A (7). In these experiments one probes forces along a reaction coordinate. Recent theoretical studies (8-10) suggest that microscopic © 2004 American Chemical Society 29 In Dynamics and Friction in Submicrometer Confining Systems; Braiman, Y., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2004.

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information on the potential and dissipative interactions, which a probe (a molecule or tip) experiences, can be obtained from dynamic force spectroscopy (DFS) by investigating the velocity dependence of the mechanical forces. Here we introduce a model to describe the dynamical response of a probe subject to an external drive in the context of DFS. The response is governed by the Langevin equation (12,13)

M x(t) = -7]x(t) -

- K(x - Vt) + T(0 .

(1)

ox Here the probe of mass M is pulled by a linker of a spring constant K connected to a support moving with a velocity V. In the context of friction the probe (tip) is pulled along the surface and U(x) is the periodic potential U(x)=U cos(27Dc/b). In this case eq 1 presents a generalization of Tomlinson model (14) which includes thermal fluctuations. In studies of adhesion the probe (molecule) is pulled in the direction perpendicular to the surface and U(x} presents the adhesion potential, rj is a dissipation constant and the effect of thermal fluctuations is given by a random force T\t), which is ^-correlated =2k Tri8(t). Below we start with a discussion of the effect of thermal fluctuations on friction and continue with the dynamical response of adhesion complexes. It is convenient to introduce the dimensionless space and time coordinates, y=l70c/b and T=tca where c^(2n/b)(Uo/M) is the frequency of the small oscillations of the tip in the minima of the periodic potential. The dynamical behavior of the system is determined by the following dimensionless parameters: a = (Q/co) is the square of the ratio of the frequency of the free oscillations of the tip Cl = ylK/M to ca i] = 7]/(Mco), O?=A /(U TJ), and V=2nVI (cob) are respectively the dimensionless dissipation constant, intensity of the noise, and stage velocity. Eq 1 describes also the response of macromolecules subject to an external drive provided for instance by optical tweezers. The observable in DFS is the spring (or frictional) force, in particular its time series and dependencies on external parameters such as driving velocity, spring constant, temperature and normal load. The main result reported in DFS is F(V), the velocity-dependent force, either maximal or time averaged. In order to understand the nature of the force and to establish relationships between the measured forces and the microscopic parameters of the system we apply two approaches: (a) direct integration of the Langevin eq 1, and (b) reconstruction of the force from the density of states, which is accumulated from the corresponding Fokker-Planck eqeation (75). In the Fokker-Planck approach the average friction force can be written as 0

B

m

t

2

2

0

In Dynamics and Friction in Submicrometer Confining Systems; Braiman, Y., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2004.

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F = \\p(x, v; V, a) (sm(2ax /b) + J]v) dxdv,

(2)


x,v where v is a tip velocity in response to V and p(x,v,V,(f) is a time averaged density of states for the driven tip in phase space {JC,V}. This equation explicitly demonstrates two contributions to the friction force: the potential given by the sin(2mc/b) term and the viscous one given by the t] v term. Figure 1 shows the velocity dependence of the time-averaged forces found through direct numerical solution of eq 1 and by using eq 2, namely the FokkerPlanck approach. The time average forces in our periodic potential provide the same information as that of an ensemble pulled over a barrier. All calculations have been done under the condition a 1

>1

0

0

0

b)

2

2

50

100

0

150

tiiMiiiiiiiH 50

y

100

150

y c)

2

2

>1

>1

0 0

0 /fMHti«::*M!fl|R^

50

100

y

150

0

50

100

150

y

Figure 4. Time evolution of the density of states distribution in the phase space. Typical stages of the time evolution of the portrait are presented: (a) the first jump of the tip from the locked state, (b) the locked state at short times, (c) intermediate stage, (d) stationary shape of the phase portrait showing mixing of locked and sliding states. All portraits are shown within the same interval of the space coordinates. Parameter values as in Figure 3.

In Dynamics and Friction in Submicrometer Confining Systems; Braiman, Y., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2004.

37 It should be noted that in order to fit the Ornstein-Uhlenbeck equation to the results of our calculations, the damping coefficient ^ ff entered the equation should be considered as a fitting parameter that is inversely proportional to the intensity of the noise. This reflects the influence of the substrate potential that traps tips leading to an increase of the effective friction i) ff. This effect is most important at low intensities of the noise. Thus, envelopes of AL(t) and 5(0 becomes steeper as a increases, and they approach asymptotic values at shorter times (see Figure 3). Dynamic response of adhesion complexes can be also described by the Langevin equation 1 and the approximate kinetic model in eq 3. Here, for illustration the adhesion potential is presented by the Morse potential, e


e

U(x) = U | [ l - exp(-26(jc - R ) / R f - 1 J .Then the distribution of the 0

c

c

maximal unbinding force and ensemble averaged inbinding force are given by eqs6,7 with F = bUjR c

,U =U ,Q

c

C

0

l/2

= 2bU

C

l/2

/(R M ).

0

Equation 7 predicts a universal scaling of

c

( ^ " " ( ^

m

a

x ) )

^

w

^

T

ln(F/7 ) which is independent of temperature. Figure 5 shows results of direct numerical calculations using Langeven equation 1 which confirm the prediction of analytical theory in eq 7. g

x10

4

1

Ol

1

11

A

A

1

x10

r

°

%

$ 5

T 3

0

V 1

ln(V/T) 133 K 213 K 293 K

o • a

0

-2

?4

ln(V/T)

2

Figure 5. Results of numerical calculations supporting the scaling behavior of the ensemble averaged unbinding according eq 7. The inset shows significantly worse scaling for the traditional description

Dynamics and Friction in Submicrometer Confining Systems

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