Investigating adsorbed polymer layer behaviour using dynamic surface force apparatuses — a review

ADVANCES IN COLLOID AND INTERFACE SCIENCE

Advances in Colloid and Interface Science 73 (1997) l-46

ELSEVIER

Investigating adsorbed polymer layer behaviour using dynamic surface force apparatuses - a review P.F. Luckham*, S. Manimaaran Department

of Chemical Engineering

and Chemical Technology, Imperial College of Science

Technology and Medicine, Prince Consort Road, London SW7 2BY, UK

Abstract The equilibrium forces encountered between polymer bearing surfaces have been investigated in great detail, at a nanoscopic scale, ever since the development of the Surface Force Apparatus (SFA). However, in doing so it has also been recognised that polymer interactions have a uniquely dynamic nature that warrants further investigation. In the last decade a number of researchers have taken this task to hand by modifying the SFA such that it attains the added capacity ofworking at a dynamic level. This review will follow the development of the SFA in its use as a dynamic tool, and also consider theoretical advances that have been made in order to interpret the results retrieved in such studies.

Contents Abstract . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . Normal motion . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . The original DSFA (low frequency) . . . Interpretation of dynamic properties . Results from the low frequency technique

. . . . . . . . . . .

. . . . . . . .

* To whom correspondence must be addressed. OOOl-8686/97/$32.00

0 1997 -

PZZ: SOOOl-8686(97)00001-8

Elsevier Science B.V. All rights reserved.

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Colloid

Interface

Sci. 73 (1997) 146

DSFA for high frequency studies ........................ The ‘new’SFA ................................... Results obtained using the ‘new’SFA ...................... Theory ......................................... Introduction .................................... Non-overlapping polymer layers ......................... Periodic compression ofweakly overlapping polymer layers .......... Compressed layers ................................ Periodic compression at high frequency ..................... Lateral motion ..................................... Introduction .................................... Experiments involving the shear technique .................. Theory ......................................... Frictional properties ............................... Brush swelling .................................. Molecular dynamics simulations ........................ Concluding remarks .................................. References .......................................

10 15 17 20 20 23 26 27 28 29 29 30 37 37 39 40 42 45

Introduction Interest in the topic of colloidal stabilisation has a strong academic and industrial basis as colloids play a pivotal role in our everyday lives. Many naturally occurring substances such as smoke, mist and clay, biological matter such as blood and milk as well as technologically important materials such as paints, ceramics and inks can be classified as colloids. In the absence of other (stabilising) materials, attractive forces that bring about aggregation are always present between colloidal particles. As aggregation may not be a desired effect, repulsive forces need to be introduced into colloidal systems for the purposes of stabilising them. Stability is achieved if the repulsive forces introduced are of sufficient magnitude for the overall inter-particle force to be repulsive or, if attractive, sufficiently weak to be overcome by thermal energy. There are in fact three distinct methods that can be employed independently or in combination for this purpose: solvent effects, electrostatic stabilisation and steric stabilisation. Of these, steric stabilisation is generally the method of greatest ease and practical value and revolves around the use of adsorbed polymer layers. These adsorbed layers act by physically preventing the close approach of colloidal particles, thereby removing any possibility of aggregation. The importance of colloidal stabilisation has dictated that the nature of the interactions that take place in such systems be probed by an ever

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increasing array of experimental techniques and theoretical treatments. Of these, one of the more recently developed techniques revolves around the use of a modified Surface Force Apparatus (SFA) 111which enables dynamic studies to be conducted at a nanoscopic scale. Interest in nano-dynamic techniques have gained momentum especially in the case of polymer based interactions, as equilibrium theories at such scales fall short of providing a complete picture. In non-polymer mediated systems, the use of equilibrium measurements for understanding behaviour are of great value, as both the electronic and ionic distributions can respond sufficiently rapidly to ensure that forces experienced are equilibrium ones. In polymer stabilised systems however, the presence of a number of distinct relaxation mechanisms that take place independently with their own characteristic relaxation times bring about interactions that are far from equilibrium - this is responsible for the viscoelastic properties exhibited by polymeric systems [2], The SFA has been modified in three ways thus far for the purposes of work at a dynamic level. In one of these, a lateral motion is applied to a surface allowing systems that are (highly) confined to be investigated with a view towards understanding their shear and frictional properties [3,4]. The two other methods revolve around the application of a normal motion, and can be used at a wide range of separations. The application of a normal oscillatory motion enables the measurement of viscoelastic properties [5]. Whereas the application of a normal load, with one of the surfaces held stationary while the other is driven towards it at a constant velocity allows the hydrodynamic forces in the system to be measured [6]. In this review all three methods will be discussed in detail especially in the context of work carried out on adsorbed polymer systems. Theoretical models that have been developed in parallel to this dynamic work will also be reviewed here.

Normal motion Introduction

The idea of using the SFA for the purpose of carrying out (normal) dynamic experiments was developed simultaneously and independently by two research groups. The work of Chan and Horn [61 involved one surface being driven towards the other, which was held stationary, allowing the rate of drainage of confined fluid to be investigated in

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relation to its increasing confinement. The experiment developed by Israelachvili 151however, involves one surface being oscillated sinusoidally with the response of the material in the gap to this dynamic load being monitored. This allows both the viscous and elastic properties of the confined system to be investigated with respect to confinement. Although both these techniques were developed primarily to gain further insight into liquid structuring adjacent to a solid surface, once established, studies were then extended to polymer melts and then finally to adsorbed polymer layers. The earliest work involving the (normal) DSFA was carried out at low frequencies (< 1 Hz) and required the use of the Fringes of Equal Chromatic Order (FECO) for the purposes of interpreting the response of the system [5]. With the advent of piezoelectric technology however studies were then extended to higher frequencies (> 3 Hz) [7,8]. This review will detail all work carried out using the aforementioned techniques for studies on adsorbed polymer systems. The more recent work of Montfort et al. [9,101 on a ‘new’ SFA developed by Tonck et al. [ll-131 will also be discussed here. The original DSFA (low frequency) The apparatuses employed for the initial dynamic experiments preserved the design and operational characteristics of the traditional SFA [14,15] (see Fig. 11, but did require additional components [5]. The requirement of high precision with respect to the measurement of motion was one easily satisfied by the use of interference fringes. However, an accurate measure of separation/amplitude of oscillation during the course of dynamic experiments was considerably more difficult to achieve and this necessitated the addition of a video camera and recorder. This configuration enabled motions to be measured to Angstrom resolution even during dynamic experiments. This technique is however limited to low frequencies (< 1 Hz) as video-recording equipment can (normally) only store a maximum of 50 frames per second. Light intensity makes it very difficult to use higher speed cameras. Interpretation

of dynamic properties

The data emerging from these dynamic experiments are converted to meaningful results by deriving an equation of motion for the lower surface. Such an analysis was performed by Montfort and Hadziioannou

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5

ilght to spectrometer A microscope

objective.

1

piezoelect r ic mica sheets movable clamp

clamp / adjusting rod

s!iff

double-

variable stiffness force, mezsurlng spring

white light

I

Fig. 1. Surface force apparatus as designed by Israelachvili [21. (Reproduced with permission from Academic Press, London, U.K).

[16], and although their paper,

The expression lower

surface

restoring

is derived which

force,

more complicated tion, F,,

there is a minor

the final expression,

error in the equation1

shown

by summing

include

an inertial

FR (= Jz3t),which expressions

and the hydrodynamic

below,

all the forces contribution,

are easily

accounting

described

takes on a similar

derived,

in

form.

that act upon the FI(= mw2>, and a as well as slightly

for the surface

force contribu-

force, FH.

In fact, the contribution of the surface forces, Fs, can be simplified to a linear function by restricting the dynamic motions to small perturbations of the distance D between the surfaces, about the mean separation, D [16]. This leads to an expression of the form: 2$(D) =

F,(D)+f . (D - 0)

(1)

1 We believe that this error arisesas a resultof usingthe expressionderivedby Chan and Horn for the hydrodynamicforceFH. In re-derivingthis expressionourselves,we foundthat a minus sign did not appearbeforethe expressionfor FH (see Eq. (2)), thus a minussign and not a plus sign appearsin front of the force gradientterm,f, in Eq. (3).

P.F. Luckham, S. ManimaaranlAdv.

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Colloid Interface Sci. 73 (1997) l-46

where f refers to the gradient of the equilibrium force at the mean separation, D. The expression for the hydrodynamic force, Fn, was first derived by Chan and Horn [6] in order to interpret the results from their drainage experiments. Although their derivation contains an error in the sign of the final expression, it is applicable at low Reynolds numbers, and in the geometry defined by the SFA. The corrected version of the hydrodynamic force for a plane-sphere geometry reads as

(2) where R is the radius of the glass formers onto which the mica is mounted, dD/dt refers to the rate at which the surfaces are being oscillated, and n is the viscosity. The final expressions for the storage, G’, and loss, G”, moduli then take on the form shown below: (3)

s

G”=-

i5

(k - mo2) 2

sin@

(4)

-

where, D is the mean separation, k is the spring constant, m is the mass of the lever, o is the frequency at which the experiments were carried out, A is the amplitude of the gap separation, A, is the amplitude of the imposed motion on the upper surface, and Q is the phase angle between the motion of the upper surface and that of the gap. In this expression the storage term, G’, can be split into two components, one of which relates to a hydrodynamic component, G,‘, and the other the equilibrium force contribution, G,, as shown below [91: G,’ =

5 3

G,=-Df

(k

6~22~

-

mo2)

(5)

(6)

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The latter term arises from the compression of the polymer chains adsorbed to the surface and can be calculated from equilibrium force distance profiles. The two hydrodynamic terms however, can only be retrieved from dynamic experiments. The data from these experiments can also be rearranged to produce a quantity known as the effective mobility, G [ll. This quantity, shown below, is retrieved by reordering the (solved) equation of motion, and is valid so long as the spring constant, K, is considerably larger than the force gradient, ‘f’ (obtained from the force distance profile), and the gap amplitude, A, is very much smaller than the gap separation, D. 6zR20

(7)

h[(A, lA)2 - 1]1’2

The effective mobility is a useful quantity to evaluate as it helps define (i) the effective position of the plane of no slip, (ii) the bulk solution viscosity to be determined form the inverse of the slope of the plot away from overlap, and (iii) is a measure of the ease with which the two surfaces can be forced together or apart - the larger the value the greater the ease. Results from the low frequency

technique

The earliest work involving adsorbed polymer layers using the low frequency technique was carried out by Israelachvili et al. [5,17]. The objective of this work was to compare the effects of confinement on the viscosity of a fluid between surfaces with or without adsorbed polymer layers. The pure fluid experiments were conducted on cyclohexane, and the polymer system investigated consisted of polystyrene (Mw 9x105) adsorbed onto mica from a bulk cyclohexane solution at 26°C (6 temperature 34.5”C) [5]. Despite the lack of emphasis on (actual) polymer behaviour this study did reveal a number of features that are of interest. As with the earlier investigations into the behaviour of pure solvents, experiments in the presence of adsorbed polymer layers revealed the system to exhibit bulk solution properties (far) away from overlap. This is illustrated by the inverse of the slope of Fig. 2 corresponding to the viscosity of the bulk solution [5]. Of greater interest though, the studies also showed the major effect of polymer adsorption to be a shift in the plane of shear, which is located at D = 0 in the case of pure liquids, to a distance D = AHaway from the surface [5,17]. This distance corresponds

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0

Distance, D t h Fig. 2. Plot of 12~2R2ulkl(A, /A)’ - 11""against distance in Cyclohexane and in PS-Cyclohexane as reported by Israelachvili 151. The fact that the x-intercept in the plot for PS-Cyclohexane shows a pronounced shift is representative of a hydrodynamic layer of thickness - 26 nm on each surface. (Reproduced with permission from Dr. Dietrich Steinkopff Verial, Darmstadt, Germany).

to what is an effectively immobilised layer on the surface, below which there is an increased resistance to solvent flow. This is commonly referred to as the hydrodynamic layer thickness 151, and this shift in the plane of shear is also illustrated by Fig. 2. Although the values reported for AH were smaller than expected, when compared with other experimental and theoretical studies [5,181, it is interesting to note that these experiments found a direct correlation between the range of the steric forces and the location of the ‘immobilised layer’, as demonstrated by the inset in Fig. 2. This suggests that a major cause for this observation must be the solvency conditions. Improved solvency conditions would increase the range of the steric forces and may also increase the hydrodynamic thickness of the adsorbed layer.

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In more recent times Klein et al. [19,20] have also carried out similar work, though in these studies there was a greater emphasis on the actual behaviour of the adsorbed polymers. End adsorbing polymers consisting of monodispersed polystyrene (PSI chains terminated with the zwitterionic group -N+(CH,),(CH&SO; (designated by PS-X) of molecular weight 1.41 and 3.75~10~ were prepared in a toluene solution under ‘good’solvent conditions for this work. As before, the results from this study showed that for surface separations greater than 2L, which refers to the range of the steric forces, the hydrodynamic forces were characterised by Newtonian flow of liquid with viscosity comparable to that of toluene 119,201(see Fig. 3). This is consistent with the findings of Israelachvili 151and suggests that, at least at these low frequencies, there is very little inter-penetration of polymer layers by the solvent velocity field, as it flows past them i.e. the plane of no-slip of the fluid is close to the edge of the polymer

1000

2000

3000

D,(i) Fig. 3. Variation of the effective mobility, G, with distance for a PS-X solution in Toluene for a range of frequencies (0.2 Hz and 1 Hz) as reported by IClein et al. [19,20]. (Reproduced with permission from American Chemical Society, Washington, USA).

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layers [19]. This is an interesting finding that will be discussed in greater detail later in this section, as it is in conflict with the theoretical view proposed by Milner 1211.Once inside the layers, the response of the system was seen to become more solid-like with increasing confinement, and the effect of solvent flow through the confined layers was found to dominate the hydrodynamic forces. In other words, any contribution to the system behaviour by the flow of free solvent was negligible when compared to that arising from flow through the polymer network [19,20].

DSFA for high frequency studies Apart from the restriction to studies at low shear rates the original version of the DSFA was also found wanting in its abilities to return accurate readings at high confinement. This arose as a consequence of the increasing difficulty in measuring a decreasing gap amplitude at high confinement. The first papers to confront these particular problems were those published by Israelachvili and co-workers [7,8] who developed a modified version of the SFA. In these papers they detailed investigations on the properties of both polymer melts and those of simple liquids at higher frequencies than had been possible prior to this. The apparatus designed for high(er) frequency studies incorporated piezoelectric technology and involved a simple modification being made to the lower surface. This results in the lower surface being attached to a piezoelectric bimorph instead of the traditional leaf spring 171.The net effect being that the lower surface now acts as a motion detector. The flexing of a bimorph in response to an applied motion results in a potential difference being developed across its surfaces, enabling the magnitude of this signal to be measured using a lock-in amplifier. A schematic representation of such an apparatus can be seen in Fig. 4. Employing such a configuration enables comparisons to be made between the input and output signals with respect to their amplitude and phase angle. These can then be re-organised so as to allow values for the moduli being sought to be retrieved using Eqs. (3)-(6). By optimising the apparatus it is possible to use piezos to measure motions as small as 0.1 A and accurate experiments can thus be conducted at frequencies where electronic equipment is at its most efficient (> 10 Hz) [7,8]. A significant advantage of this technique concerns its ability to measure signals obtained at high confinement rather more accurately than had been possible prior to this development. In fact, motions involved at high compression result in a low gap

P.F. Luckham, S. ManimaaranIAdv.

INPUT SIGNAL

Colloid Interface Sci. 73 (1997) l-46

11

PIEZOELECTRIC TUBE

CHAMBER WALL B ., (’:.:. .,:,..:.:.‘i..

BIMORPH STRIP )_

MICA SHEETS

MOVEABLE CLAMPS 5

OUTPUT SIGNAL Fig. 4. A schematic representation of the DSFA [22,231. (Reproduced from Elsevier Science Publishing Co. Inc., New York, USA).

with permission

amplitude and therefore a large lower surface amplitude, which is comparable to that of the applied motion. Consequently, a greater degree of accuracy can be obtained when surfaces are highly confined, as a large signal is easier to detect than a small one. The only detailed work to have emerged using a DSFA, at high frequencies on adsorbed polymer layers, has been carried out at Imperial College [22,23]. Two contrasting polymer systems, in terms of size, have been investigated in this study thus far. The first system chosen consisted of a (relatively) low molecular weight polymer made up of poly (12-hydroxy stearic acid) [PHSA] side chains grafted onto poly (methyl methacrylate) [PMMAI backbones (Mw - 2~10~) with decahydranaphthalene as their solvent. Polymer systems such as these have been incorporated into polymer latex particles allowing osmotic pressure and rheology measurements to be carried out on them [24-261. By way of contrast the second system investigated had a significantly higher molecular weight (2.5~10~) and consisted of poly(ethylene oxide) [PEOI side chains grafted onto poly(styrene) [PSI backbone, and used toluene as its solvent. This system possessed the advantage of being larger and therefore offering a considerably larger region of steric interaction open to investigation.

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Table 1 Response of PMMA/PHSA k 0.07 mV; * 0.10”.

at 30 Hz. Calibration signal (average): 44.37 mV; 1.05”. Error

D

Amplitude A,

Phase angle QL Gc

(nm)

(mV)

(deg)

6.1

44.19 f 0.02

8.8 10

(Pa)

(Pa) (average)

G” (Pa) (average)

1.27 Z!I0.00

0.03

0.80

0.74

44.15 f 0.03

1.30 f 0.02

0.02

1.01

0.88

0.01 43.93 z!L

1.62 + 0.02

0.015

0.51

0.51

Gn’

22.3

42.15 + 0.05

2.48 + 0.07

-0

0.34

0.17

33:‘:

41.96 f 0.04

3.22 f 0.05

0

0.09 (0.2)

0.07 (0.16)

107”’

41.32 f 0.06

10.33 I!Y0.22

0

0.15 (0.17)

0.43 (0.48)

260.12:

36.30 If:0.00

26.91 + 0.01

0

0.08 (0.08)

0.43 (0.45)

The results for the PEO/PS system and, to a lesser extent, the results from the PMIVWPHSA system reveal a favourable comparison to the results brought forward by bulk rheological studies on colloidal particles stabilised by terminally attached polymers. They support the argument put forward by Strivens, in his work on the rheology of PMMA particles stabilised by the PMMALPHSA copolymer, that, the elasticity exhibited by such systems owe their origins to hydrodynamic interactions as well as colloidal forces [261. In fact, the results obtained in these dynamic experiments suggest that, if anything, the contribution ofhydrodynamic forces, to the elasticity, at high frequencies (>lO Hz), outweigh the contribution that arises from colloidal interactions. This argument is highlighted by the observation of hydrodynamic moduli, G,’ and G”, significantly larger than the static storage modulus, Gc [22,23] (see Tables 1 and 2). In addition, the results also confirm the finding of Strivens [26] that, the storage modulus is very much smaller than the loss modulus (i.e. G” >> G’) at volume fractions which correspond to gap separations beyond polymer layer overlap, with the situation reversed (i.e. G’ > G”) where there are polymer interactions. The results obtained at large separations, away from overlap see the systems exhibit near

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Table 2 Response of PEO/PS at 25 Hz. Calibration signal (average): 26 mV; 1.41”. Error: f0.05 mV, f0.02”.

(nm)

Amplitude A, (n-iv)

Phase angle $L Gc (de& (Pa)

&I’ (Pa) (average)

45

25.74 + 0.02

1.73 f 0.03

0.15

4.57

2.56

47

25.65 + 0.01

1.80 * 0.02

0.12

3.77

1.88

51

25.81+ 0.01

1.63 + 0.02

0.10

7.46

4.00

D

G” (Pa) (average)

53.5

25.64 + 0.01

1.75 f 0.02

0.06

4.43

1.90

63

25.64 zt0.01

1.85 f 0.04

0.03

4.87

2.52

79

25.39 + 0.02

1.94 + 0.03


3.91

1.54

106.4

25.10 + 0.01

2.14 + 0.01

-0

3.59

1.32

120

24.015 0.03

2.49 f 0.07

-0

1.88

0.47

138

22.54 f 0.04

3.95 f 0.06

-0

1.12

0.38

162

22.41 f 0.06

5.90 * 0.11

-0

1.06

0.62

209.6

19.49 ?I0.08

15.48 + 0.04

-0

0.44

0.49

276

13.46 + 0.12

53.86 f 0.20

0

0.001 (0.015)

0.06 (0.14)

In the tables, the gap separations accompanied by an asterisk correspond to separations where the polymer layers no longer overlap (force distance profiles show no interactions beyond 25 nm in the case of PMMA/PHSA and 210 nm in the case of PEO/PS [22,23]. Two values are presented for each of the moduli at these separations -the ones in bold face correspond to a gap separation corrected for after subtracing the polymer layer thickness (25 nm for PMMA/PHSA, and 210 nm for PEO/PS), and those in italics correspond to a value obtained when the subtracted amount is equivalent to a hydrodynamic layer thickness suggested by Milner [21] (-60% of the layer thickness, i.e. -15 nm PMMA/PHSA and - 126 nm for PEO/PS).

bulk solution properties. This finding is further strengthened by Fig. 5 which illustrates the variation of phase angle for PEO/PS with separation [23]. The figure shows the transition of the system from a viscoelastic solid at high compression (D c 150 nm), where QL - 0, to a viscoelastic liquid at large separations (D > 210 nm), as +L + 90. Despite the encouragement offered by these results, the expressions for the hydrodynamic and static moduli raise a question with regard to the validity of a comparison between the two types of interaction [22,231. This can be illustrated by examining the expression for Gc (see Eq. (6)).

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Separation (nm) Fig. 5. Semi log plot of the variation of phase angle with separation for PEO/PS system at 30 Hz with gap separation.

In this expression, the term f/R is apparatus independent, as it is simply the gradient of the normalised force. The basic point behind the use of a concept such as the normalised force is that it allows a qualitative as well as a quantitative comparison to be made between the results of different groups - i.e. it is independent of the apparatuses used. However, in Eq. (6) the (second) division by the radius of the surfaces, R, then results in the measured moduli being apparatus specific, even for the same systems, unless the results presented by two groups are made with surfaces of equal curvature, or if the effect being reported is normalised by multiplying the measured moduli by R. In contrast, the expressions for the hydrodynamic moduli (G,'and G” - Eqs. (5) and (4)) cannot be apparatus specific, as the complex modulus, G’b, obtained by making use of the identity

where G’ = G,’ + Gc

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and, Gc = 0, away from overlap would not yield bulk solution viscosity (n”: = GVw) away from overlap as reported here, and by other workers [1,19]. This does therefore introduce an element of concern when conclusions are drawn with respect to relaxation mechanisms, for example, from the results obtained here, or for that matter with any other apparatus. Unfortunately, the lack of data away from overlap in the experiments involving PEO/PS, meant that it was not possible to confirm, or otherwise, the findings of Klein et al. [19,201 with regard to the position of the plane of no slip on a similar system (PS-X1, at lower frequencies. Results at a range of frequencies may have suggested a frequency dependence unaccounted for in the theoretical model applied to these systems; the theory derived for these experiments will be discussed at a later stage. Although the results obtained using the modified version of the traditional SFA do point to some future possibilities, it does suffer from a number of operational problems. Firstly, there is a restriction to mica surfaces that needs to be overcome if experiments are to be conducted on a wide variety of systems. More significantly, there is a slight deformation of the mica surfaces at high compression that compromises the geometric conformation of the apparatus 122,231. In the high frequency experiments carried out on the PEO/PS and PMMA/PHSA systems it was found that the mica was shown to take on a flattened appearance even in the presence of the adsorbed polymer layers. The high forces encountered at high compression can result in deformation of the mica surfaces due to the creep of the glue which is used to attach mica to the glass surfaces. Such an observation implies that the system can no longer be treated along the simple lines of a sphere approaching a plane. It was found that the flattened appearance masked the quality of the results obtained with low molecular weight polymers (PMMA/ PHSA) to such an extent that a significant amount of vital information could not be analysed correctly using the present model. Although similar observations were made in the case of the PEO/PS system, the extent to which the PEO/PS layers ‘stretched out’in solution meant that, ignoring data where the surfaces appeared flat still left a large region of strong interaction open to valid investigation 122,231. The ‘new’ SFA The ‘new’ SFA as designed by Tonck, Georges and Loubet [ll-131 operates at the same scale as the traditional apparatus [14,15] with the

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3 LC oscillators

and frequency discriminators

] I

I’

.I .

v

3 two phases lock-in analysers ji

7

;i-static Denavlor Double cantilever spring

Meniscus of liquid

f

II dynamical . . benavlor

cant ifOl ,

I

CALCULATOR

Fig. 6. Schematic representation of the ‘new’ SFA as designed by Tonck et al. [ 1 l-14. (Reproduced with permission from American Institute of Physics, New York, USA).

same geometry (that of a sphere approaching a plane), but with some significant modifications made so as to address some of the limitations of the traditional apparatus. In particular, the ‘new’ SFA attempts to be more effective in the case of dynamic experiments and also seeks to address the restriction of the traditional apparatus to mica bearing surfaces. A schematic representation of the apparatus is shown in Fig. 6. The operation of the apparatus is achieved by moving the upper, spherical, surface towards the lower, planar, one by using the expansion and vibration of the piezoelectric crystal. Generally, the sphere and plane used are metallic and the planar surface is supported by a double cantilever spring whose stiffness can be continuously adjusted. The material being investigated is introduced into the system by injecting it in between the two surfaces. The relative motions of the surfaces are then measured using a series of capacitors as shown in the figure. The capacitor C, measures the elastic deformation of the cantilever and thus the force transmitted through to the liquid plane. The second sensor C!, measures the relative displacement between the supports of the two solids and the third capacitor C measures the electrical capacitance between the sphere and the plane [ll-131.

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The apparatus described allows displacement measurements to be made to a theoretical accuracy of -lOA nm, and thus allows dynamic experiments with amplitudes as low as 5x10” nm to be investigated. This is a significant improvement (a factor of two) over the mica based apparatus. This set-up consequently allows viscosities as high as lo4 Pa s to be measured accurately; these values are generally encountered in the case of polymer melts. The use of metallic surfaces as opposed to mica also negates any possible deformation that can occur to the surfaces. It should be noted though, that the metal surfaces used have a peak to valley roughness of -1 nm unlike mica which is molecularly smooth. They do, however, suggest that this is negligible as the corrugation diameter is -50 nm. The low amplitude of the surface roughness was also thought to be negligible when compared with the thickness of the polymer layers they encountered [131.

Results obtained

using the Ztew’SFA

The results obtained from the ‘new’SFA represent the most comprehensive set of dynamic results obtained at a nanoscopic scale. Unlike the work described in the previous sections, this work has been performed on physically adsorbed layers as opposed to terminally attached layers [9,101. Figures 7-9 demonstrate, some of the results obtained by Montfort et al. 19,101using the new apparatus. The system investigated consisted of polybutadiene (Mw 3.3~10~) diluted in a highly viscous non-volatile hydrocarbon oil. At very large separations, where there are no steric interactions, the storage modulus, G’, is very much smaller than the loss modulus, G”, especially at low frequencies. At these large separations, a Maxwell-like variation [27] is observed in that, G’ varies with the applied frequency, as 02, whereas G”’exhibits a linear relationship with frequency. The gulf between the moduli therefore decreases with increasing frequency. As the surface separation is reduced, and the polymer layers interact, the storage modulus increases quite dramatically, and exhibits a plateau modulus at very low frequencies, with the values of the storage modulus being generally larger than that of ‘the loss modulus at such separations. The magnitude of the plateau modulus increases with confinement, as it is a function of the steric forces. As Fig. 9 illustrates, there is only a very small contribution from the hydrodynamic part of

18

P.F. Luckham, S. ManimaaranlAdv.

Colloid Interface Sci. 73 (1997) l-46

Fig. 7. Log plot of G’vs. fequency (0) for a variety of separations as reported by Montfort et al. [9,10]. All data except (a) obtained where there is polymer interaction, with increasing moduli for a given frequency corresponding to increasing confinement. The figure shows that a slope of two is found away from overlap, and a plateau modulus appears at low frequencies where there is polymer interaction. (Reproduced with permission from American Chemical Society, Washington, USA.)

the storage modulus to the overall storage modulus at low frequencies. The main contribution comes from a frequency independent modulus, G,. In fact, under such conditions the static term dominates the contribution to the overall complex modulus G*(o). A compressional modulus E, can also be retrieved from Go, and the values obtained were found to be in between the rubbery and glassy moduli of melts. This was taken to suggest that physically adsorbed chains have hindered motion when compressed and overlapped. The appearance of the static term with decreasing separation coincides with the onset of the steric interactions and marks the transition of the system from that of a viscoelastic liquid

P.F. Luckham,

, o ;; _, :

S. ManimaaranlAdv.

I

I

I

,;,;

_;

!

,

Colloid Interface Sci. 73 (19971 146

,‘b’ I I

-1

I,,,,,

’’: ‘;

w [ Rad.s”

]

!I

10

,

iI

19

,,,,,!

10

’

Fig. 8. Log plot for G” vs. frequency as reported by Montfort et al. [9,10]. At large separations, away from polymer interactions, the figure exhibits a slope of unity. (Reproduced with permission from American Chemical Society, Washington, USA.)

at large separations and low frequencies, to a viscoelastic solid at low separations and frequencies. With increasing frequency however, the hydrodynamic component, G,‘, of the storage modulus, G’, begins to make an increasingly significant contribution that is comparable to and then greater than the frequency independent component, G, (see Fig. 9) [9,101. It is unclear at present whether the dynamic experiments of Montfort et al. [9,10] allow a direct measurement of G’as opposed to G,‘. This is a matter of some concern relating to the apparatus specific nature of the term G,. If the measurements of Montfort et al. do allow direct measurement of G’, then the correspondence of the values obtained for Gc using Eq. (6), with the plateau modulus obtained at low frequencies in their measurements (see Fig. 9) must be fortuitous.

20

P.F. Luckham, S. ManimaaranlAdv.

Colloid Interface Sci. 73 (1997) 146

Fig. 9. Measured storage component, G’, and hydrodynamic storage component, GH’,as a function of frequency as reported by Montfort et al. [9,10]. The dashed lines stand for the hydrodynamic component. (Reproduced with permission from American Chemical Society, Washington, USA.)

Theory Introduction

At present two independent theoretical treatments have been derived to complement the experimental work that has been carried out on (normal) DSFAs 128,291. The two approaches follow a similar line, except in their choice of polymer layers and resulting concentration profile, and are thus worth considering in parallel. The two theories work on the basis of applying a two fluids model to the polymer solution. The solvent is treated as one component with the network formed by the polymer layers being the other [28,29]. By adopting the same geometric configuration encountered in the SFA (see Fig. lo), it is clear that a normal motion applied to either of the surfaces induces a pressure gradient in the radial direction, resulting in radial solvent flow. The drag of the fluid through the network creates a viscous dissipation and an elastic strain of the polymer. The description of the drag in the system revolves around the Brinkman equation [30] which was traditionally employed for the purpose of describing flow through a

P.F. Luckham, S. ManimaamnlAdv.

i h(r)

Colloid Interface Sci. 73 (1997) l-46

21

hu

I

Surface Fig. 10. Schematic representation of a sphere approaching a plane surface, each surface bearing adsorbed polymer layers.

porous medium. When extended to this case, the equation is employed such that the polymer layers are treated as the porous medium with permeability q/c2, where 5 is the local correlation length. The equation thus reads as:

qv2v - T@(c) (v

-

Ii) -

VP

=0

(8)

with the condition of incompressibility leading to:

v.v=o

(9)

In these expressions v refers to the solvent velocity field, and u describes the polymer displacement field. The first term in Eq. 8 (i.e. nV2v) refers to the viscous stress, and the last (VP) the pressure gradient. The middle term (ncz
v.

cJ+

~;c)(v-ti)=o

where o is the elastic stress tensor, which is in turn given by;

(10)

22

P.F. Luckham, S. ManimaaranlAdv.

Colloid Interface Sci. 73 (1997) l-46

E(c)[Vu + (VU)T]

0 = f

(11)

The expression for the elastic stress is obtained by taking advantage of a disappearing Poisson’s ratio. The term E(c) refers to the local Young’s modulus of the polymer solution, which in the scaling approach is proportional to the osmotic pressure 1311,leading to an expression of the form:

E(c)

-

kBT

(12)

W,W13

In this expression, h, refers to the mean height of the upper surface. An interesting feature of Eqs. (8) and (10) is that the permeability, #& is written in terms of a quantity kH, which is a length proportional to the local correlation length, 5. The definition of the local correlation length, which is a function of the polymer density profile, is crucial to what follows, and it is at this point that the approach proposed by Sens et al. [291 differs from that of Fredrickson and Pincus 1281. The approach of Fredrickson and Pincus 1281is in many ways a more basic treatment as it takes on a simplified view of the polymer density profile. The treatment is most applicable to grafted brushes and the concentration profile is approximated to that of a step-function as proposed by Alexander [32] and de Gennes 1331.In doing so it is possible to designate a fmed value to the concentration profile inside the layers that is applicable to all areas of the brush except the outer extremities and a region adjacent to the solid surface 1281. c(z) = co within the polymer layer, z 5 L (13) = 0 outside the polymer layers, z > L The local correlation length, 4, varies with concentration,

5-

c-3’4a-5’4

c(r,z), as:

(14)

where a is the monomer size. The approach of Sens et al. 1291takes on a slightly more complicated view of the layers by assuming a power law dependence with distance. This is in accordance with the de Gennes scaling picture [31] for

P.F. Luckham, 5’. ManimaaranlAdv.

Colloid Interface Sci. 73 (1997) 1-46

23

adsorbed layers. The adsorbed layer adopts the form of a self similar grid, and consequently a non-constant profile. c(z) -

(15)

,f4’3

By defining a dimensionless parameter I, which is a measure of friction, Sens et al. produce an expression for the correlation length given below: &2

=

w - 1) 22

&2 = 41 - 1) (h - z)~

if z < h/2

(16) if z > h/2

As the detailed account of the two treatments is easily accessible, in what follows a brief summary of the results will be presented alongside comments about their implications. Non-overlapping

polymer layers

In this situation the gap separation is considerably larger than the combined thickness of the adsorbed polymer layers on each surface. By assuming that there will be no deformation of the polymer network, the only forces that arise can be attributed to hydrodynamic forces as a result of fluid drag in the network. By breaking the system down into three layers - two adsorbed layers on each of the surfaces, and the intervening fluid - the equation of motion can be solved to produce an expression for the hydrodynamic force, in the case of steady compression [28,29] : (17) where LH refers to a hydrodynamic layer thickness. This expression is similar to the Reynolds equation 1341and that derived by Chan and Horn [6] (Eq. 2) except for the fact that the gap separation is reduced by 2LH. This effectively means that a layer of polymer corresponding to a thickness of & on each surface does not participate in the flow and acts as an extended surface. In the step function analysis proposed by Fredrickson and Pincus 1281the hydrodynamic layer thickness can be

24

P.F. Luckham, S. ManimaaranlAdu.

Colloid Interface Sci. 73 (1997) l-46

approximated by the brush height. This assumes little or no inter-penetration of the brush. The analysis of Sens et al. [29] however predicts the same final result though the extent of solvent flow in the layers is far more significant. Although this discrepancy can be partially attributed to the nature of the polymer-surface interaction, it may also be due to the simplifying assumption of a step function brush. Had a more realistic profile been considered, the degree of interpenetration may have been more significant in the Fredrickson and Pincus analysis. This is a point that has been investigated by Milner [21]. By using a refined self consistent field approach Milner [35] predicts a profile that assumes a near parabolic shape for polymer brushes. Small angle neutron scattering (SANS) experiments and neutron reflectometry experiments carried out by Cosgrove et al. [36,37] confirm the fact that the concentration profile does not resemble a step function and although it does not duplicate the results proposed by Milner [35], the Milner model does bear a greater resemblance. The significance of this with respect to brush hydrodynamics has also been investigated by Milner [21,38] and although this corrected profile has not been applied to the situation encountered in the SFA [19,20,22,231, Milner’s conclusions are of interest. In this analysis he points out that, although any feature that largely depends on the scaling properties of a brush are well characterised by the scaling approach, those that are affected by the structure of the brush at its outer extremities will be affected by a parabolic profile. One such property is the hydrodynamic penetration length. In his paper Milner compares the depth of penetration in the case of a simple shear flow for both the parabolic profile and the step function case [21], and his results are summarised by Fig. 11. The figure shows up a very important feature: although the velocity field in a parabolic brush does penetrate a greater depth than a step function brush, it does not penetrate the whole brush. This has important consequences to the analysis of Fredrickson and Pincus [281,in that it suggests that, although it is correct to predict the hydrodynamic force in terms of a modified Reynolds equation, the hydrodynamic penetration length, L,, may no longer be approximated by the brush height, L, in Eq. (17). It is clear from the figure that LH c L. Sens et al. [29] also investigated the effects of periodic compression at separations beyond overlap. By perturbatively solving the equations in powers of o, expressions for the loss, G”, and storage modulus were retrieved. The expressions for these moduli are shown below, and

P.F. Luckham, S. ManimaaranlAdv.

Colloid Interface Sci. 73 (1997) l-46

25

x (height)

s‘$

0.6

-

0.6

-

0.4

-

0.2

-

s z B

0.0.

0.0

’ ’ ’ ’ 1 0.2

0.4 Y

(height)

Fig. 11. Comparison of the predictions of Milner [21] for the flow penetration in a step function brush and a parabolic brush. (Reproduced with permission from American Chemical Society, Washington, USA).

suggest that the complex modulus at these separations is dominated by the loss modulus which has a value close to that of the pure solvent (i.e. G” = q), with the storage modulus as a correctional term varying as a higher power of o.

(18)

26

P.F. Luckham, S. ManimaaranlAdv.

Colloid Interface Sci. 73 (1997) l-46

In this equation z,, = T$$ /u.k,T is the Zimm relaxation time 1391 for a polymer solution with a correlation length h,. The frequency dependence of both moduli are characteristic of a Maxwell viscoelastic fluid (G” 0~o, and G’ 0~02) 1271. Periodic compression

of weakly overlapping polymer layers

The approach of Fredrickson and Pincus is not applied in this region as the concentration profile is complicated here. However, Sens et al. do consider this region and treat the system as represented by Fig. 12. This allows them to assume that in the central region the overlap is weak enough to continue being represented by a self similar grid on each surface. Once again by solving in powers of w, they produce rather (more) complicated expressions for the moduli (cf. Eq. (18)) which once again predict a Maxwell-like behaviour [27].

G”(w) = orl

l(1 + 1)2 hQ hi Cl+ 3) + o(l) R, + P(l) q

(19) 1

G’(o) = 202qz,

R; [20(l)

1 22

0

I

RF

+ 2@,(l) + Y(l)1 - h

0

Em(l) + @@)I +$ w i

(20)

The most important conclusion to draw from this is that both moduli increase sharply with increasing values of I, and that the dominant

Fig. 12. Schematic representation of radial variation of interaction. Two different regions need to be defined. In the inner region there is no pure solvent [291. (Reproduced with permission from American Chemical Society, Washington, USA.)

P.F. Luckham,

S. ManimaaranlAdv.

contribution to both expressions there is overlap.

Compressed

Colloid Interface Sci. 73 (1997) l-46

27

comes from the central region where

layers

This regime is considered by Fredrickson and Pincus 1281, with the assumption of constant surface coverage, only solvent is forced out of the gap region. In the case of steady squeezing, so long as the gap separation is considerably larger than the hydrodynamic screening length defined by Eq. (141, the hydrodynamic force is given by:

(21) This expression suggests that the force is similar to the one described by the Reynolds equation 1341except for the presence of an enhancement factor given by the term [&(h>12. This enhancement is due to the added drag of the solvent on the fixed polymer grid [281. For periodic compression in this regime the imposed strain and frequency are once again taken to be sufficiently small that the local concentration profile is instantaneously established. In doing so results showing an unusual (non-Maxwell-like) scaling behaviour with frequency for the hydrodynamic storage modulus at low frequencies, and reaching a plateau value at high frequencies are produced. G’,(R)

-

E,d3’11 (22)

G”,(Q) - $ E,Q for low frequencies

i.e. where Q -+ 0, and

16 G’H-221E0

(23) 1 clH - - 56 E,Q-r forR+w.

In these expressions Sz is a dimensionless

frequency given by

28

P.F. Luckham, S. ManimnaranlAdv.

Colloid Interface Sci. 73 (1997) l-46

111 G;,(iLjIE,

-6-

-:

-2

Inn

2

Fig. 13. Predictions for the variation of the dynamic storage modulus G,’ and dynamic loss modulus, Ga” with (dimensionless) frequency for a moderately compressed polymer as reported by Fredrickson and Pincus [281. The moduli are scaled by E,, the Youngs elastic modulus. (Reproduced with permission from American Chemical Society, Washington, USA.)

R = ozo, with 2. being the relaxation time at surface separation ho. E, corresponds to the Young’s modulus, E, at ho. Sens et al. [29] have suggested that the lack of flow penetration of the polymer layers may be responsible for the non-Maxwell-like scaling of the storage modulus of the Fredrickson and Pincus model [281. The loss modulus on the other hand varies linearly at low frequencies and then decays at high frequencies. These results of Fredrickson and Pincus are summarised by Fig. 13. Periodic compression

at high frequency

Sens et al. [29] approach this high frequency scenario by grouping the monomers that make up the chain into dynamic blobs which can be considered independent from a hydrodynamic point of view, and then solve the equation of motion by balancing the solvent friction and the elastic stress. This approach is supported by the work of de Gennes [40,41] where chain behaviour above a certain characteristic time (Zimm) is treated on an individual (independent) basis. The two fluids model used at ‘low’ frequencies is based on a polymer network, i.e. the polymer chains are grouped together and treated in terms of their collective motion. Their results with such an approach suggest that for

P.F. Luckham, S. ManimaaranIAdv.

Colloid Interface Sci. 73 (1997) l-46

29

the non-overlapping case, the mechanical behaviour of the system is equivalent to two surfaces coated with polymer, as before, except that the effective thickness of this coating has been further reduced. As for the moduli, the predicted behaviour now deviates from Maxwell treatment for both the overlapping and non-overlapping case.

Lateral motion Introduction

In the traditional SFA and the modified version discussed in the previous section, the two surfaces were restricted to motions normal to their cylindrical axis. The modifications introduced by van Alsten and Granick [3] as well as Israelachvili et al. [4] were the first to involve surface motions parallel to each other at constant separation - a shearing motion. In effect such an apparatus functions as a nanotribometer, and allows properties such as friction and lubrication to be investigated. As in the case of the original (normal) DSFA, initial experiments were conducted on simple liquids before work progressed to polymer melts and adsorbed polymer layers. Although a number of researchers [3,4,42] have worked on the design resulting in subtle differences between the various emerging apparatuses, the principle remains the same. A schematic representation of a typical apparatus can be seen in Fig. 14 and shows the upper surface now attached to two piezoelectric bimorph strips mounted symmetrically. Shear is generated by the application of a voltage to one of the strips, with the other strip acting as the force sensor detecting resistance to motion. One of the variations on this theme includes the use of a sectored piezo attached to the upper surface to induce both normal and lateral motion [42]. Two vertical leaf springs are also attached to the upper surface, and the bending of the springs is then detected using an air-gap capacitor. This configuration has been successfully adopted by Klein et al. whose work will be reviewed here as it is the most relevant to this review. More recently, Israelachvili et al. [43] have also worked on an X-Ray Surface Force Apparatus (XSFA) which can be used to both image and examine the effect of stresses on the structure of a confined medium. Although the application of shear to mica surfaces is a well established technique, only a few experiments have been carried out on

30

P.F. Luckham, S. ManimaaranlAdv.

Colloid Interface Sci. 73 (19971 l-46

SENDER CHAMBER WALL

LEAF SPRING MICA SHEETS

DIRECTION OF MOTOR TRAVEL CANTILEVER SPRING Fig. 14. A schematic representation of the shearing apparatus designed by Van Alsten and Granick [3]. (Reproduced with permission from Elsevier Science Publishing Co. Inc., New York, USA.)

polymer systems. As studies involving these systems forms the main thrust of this paper, the work of Granick et al. [31 and Israelachvili et al. [4] involving simple solutions and polymer melts will not be reviewed here. adsorbed

Experiments

involving the shear technique

The most comprehensive experiments with adsorbed polymer systems have been carried out by Klein et al. [20,42,44-461. In these studies they have simultaneously investigated both the (traditional) normal surface forces F,(D) and the shear forces F,,(D) encountered between compressed polymer bearing mica surfaces, at fixed surface separation D. By quantifying these two forces it is possible to retrieve an effective friction coefficient, p, given by the ratio F,(FI, leading to meaningful conclusions regarding the effect of polymer adsorption on system flow properties. The experiments were carried out on two distinct systems. The first incorporated the annionically polymerised polystyrene [PSI chains terminated with the zwitterionic group denoted by ‘X’ of Mw 1.4~10~ as in the previous section. This polymer was once again used in the presence of toluene which acted as a good solvent, and the polymer was believed to attach itself terminally to the surface via the active X’ group. The

P.F. Luckham,

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Colloid Interface Sci. 73 (1997) l-46

31

second system investigated however, consisted of a simple homopolymer - polystyrene [PSI of Mw 1.8~10~ in a solution of cyclopentane at or just above the 6 temperature (experiments were conducted at 22°C and 6 temperature for the PS-cyclopentane system was found to be at -19.6”C). The PS was expected to physically adsorb onto the surface under these conditions. Concentrating first on the PS-X in toluene system [20,42,44-461, the normal forces were seen to be characteristic of a swollen brush in a good solvent (see Fig. 15). When shear experiments were then conducted on this system, the polymer layers were seen to decrease the shear force considerably when compared to the case encountered in the absence of polymer under similar normal loads and sliding velocities (see Fig. 16). This corresponds to a reduction of the effective friction coefficient, p, by two to three orders of magnitude. It is also clear that the stick-slip behaviour [47,48] that characterises the trace in the case of simple liquids is less marked once polymers are attached to the surface, and was only seen to come into effect at the very highest of compressions. Stick-slip behaviour was observed in cases where the shear force

600 i Z ;E’

400

2

D,A

,/

200

61 (br=W

\;.+;..

. .. . . . . .

\ ____-____-___----

0

--I--

A--

--

.-.2:-u.-.-.--

-8

I

/

L

200

400

600

600

1000

1200

1400

DA

Fig. 15. Normal and shear forces for polymer bearing mica surfaces 1461.The inset shows the normal forces on a logarithmic scale. (Reproduced with permission from Royal Society of Chemistry, London, UK.)

32

P.F. Luckham, S. Zt4animaaranlAdu.

Colloid Interface Sci. 73 119971 146

seconds

5 Time (seconds)

5

1 10

Fig. 16. Variation of shear force with time [461. Curve (a) is obtained from two bare mica surfaces immersed in toluene, and displays the characteristic stick-slip motion. The discontinuity in (a) arises from a change in direction. Curve (b) is obtained in the presence of a PS-X brush on each of the surfaces. The inset shows curve (b) on an expanded scale. (Reproduced with permission from Royal Society of Chemistry, London, U.K.)

applied to the system exceeds the frictional force - static friction is greater than the kinetic friction. The absence of stick-slip behaviour even under moderate compressions and the reduced friction encountered in these cases is believed to be the result of the strong steric interactions that take place in between the polymer layers. These interactions enable the layers to bear a strong normal load preventing (or minimising) interpenetration, and ensuring the presence of a highly fluid film at the interface between the two layers. At the highest of compressions however (D CD,,), a stick-slip behaviour similar to that observed in the absence of polymer is observed (see inset of Fig. 17). Unlike the forces measured at higher separations (2L < D > D,), these shear forces fall in the detectable range [42,44-46] (see Fig. 17). It is believed that at this range the effect of mutual entanglement in the brushes increases, and the PS in the gap displays a ‘glassy’ behaviour. When these experiments were repeated in the case of the homopolymer PS [44,46] in a 6 solvent, the force distance profiles were characterised by bridging attractions, in the case of partial adsorption, which

P.F. Luckham, S. ManimaaranlAdv.

0.0

0.2

0.4

Colloid Interface Sci. 73 (1997) 1-46

0.6 Dl2L

0.8

1.0

33

1.2

Fig. 17. Normal and shear forces for mica surfaces bearing PS-X brushes [45]. The inset shows a plot for the shear force in the highly compressed regime. (Reproduced with permission from Macmillan Magazines Ltd., London, U.K.)

were then removed at equilibrium resulting in the characteristic monotonic repulsion. The shear forces between the PS bearing mica surfaces were seen to be largely different for a given normal load from those observed between the PS-X brushes. This is highlighted by Fig. 15 which shows the variation with time of the shear force F,,(D) in response to a displacement Ax0 applied to the top mica surface. The readings were obtained at separations, such that the normal loads being applied to both systems were similar 144,461. The shear forces required to slide the adsorbed layers past each other was once again lower than that encountered in the absence of polymer. However, as Fig. 15 demonstrates the forces involved are at least 40 times larger in the case of the PS system than that measured for the PS-X system. Figure 15 shows the similarity in range and magnitude of the normal forces for both systems, but vastly different shear forces. This can be best understood by referring to Fig. 18 illustrating the properties of the two types of layers. When brush bearing surfaces such as PS-X come to overlap and are compressed, they undergo rather limited interpenetration (an idea proposed by de Gennes [49] >. By way of contrast substantial interpenetration and entanglement is observed in the case of adsorbed layers in a 8 solvent. In addition, unlike the good solvent case there is a far lower entropic penalty for interpenetration in a 6 solvent. Consequently additional dissipation is incurred in such systems induced by entanglements

P.F. Luckham, S. ManimaaranlAdv.

34

_I ___- _. t

Colloid Interface Sci. 73 (1997) 146

d

-s-

Fig. 18. Schematic representation of (a) compressed brush and (b) compressed adsorbed layer in 0 solvent [46]. d indicates the extent of interpenetration and s the distance between anchoring sites. (Reproduced with permission from Royal Society of Chemistry, London, U.K.)

of the chains and by possible bridging interactions [44,46]. One further point worthy of mention at this stage is the repeatability of the force distance profiles both before and after the shear experiments for both systems. This implies that there is no loss of polymer from the surfaces whatever the mode of adsorption during shear. This is an important factor for consideration in the use of adsorbed polymer layers as lubricants 144,461. The studies of Klein et al. have also considered the effects of shear on the structure of the PS-X brushes formed under good solvent conditions [42]. The results for this seems to imply that at low shear velocities, no detectable change is observed in the normal forces acting between the surfaces. Above a certain critical velocity, uC,however, the situation is seen to change, and an additional repulsion, m(d), is observed. This increase was seen to occur reversibly, in that, it was seen to present itself at velocities above a certain critical value, uC,but then disappear once the velocities dropped to lower values. The resultant effect of an increased normal force with shear velocities above a critical value was observed both in the case of weakly overlapping layers (curve A in Fig. 19) and in the case of non-overlapping layers at lower velocities (curve B in Fig. 19). The origin of the additional force is thought to lie with the swelling of the polymer brushes, which comes about as a result of reduced screening between adjacent layers. Figure 20 illustrates the variation of the additional normal force as a function of separation for a given shear rate. It is interesting to note that this figure shows the additional force decreasing with increasing monomer concentration (i.e. once you get further into the polymer

P.F. Luckham,

S. Manimaaran

IAdu. Colloid Interface Sci. 73 (1997) 146

/

/

?? ? ?

??

/A

vd

$” A/’ /’

-100

.

’

I,

??-

.I

AH’

I

--\ A/ _*_-_-_-_

35

/

B Vi

-----___

;A

t& h

0

lo6 v,nm/s

2.106

Fig. 19. Variation of additional normal force as a function of sliding velocity [42]. Curve A was obtained at a separation of 95 nm and curve B at a separation of 155 nm. (Reproduced with permission from Macmillan Magazines Ltd., London, U.K.)

D (rim) Fig. 20. Variation of the normal force as a function of separation for a fixed value of the shear velocity [421. (Reproduced with permission from Macmillan Magazines Ltd., London, U.K.)

36

P.F. Luckham, S. ManimnaranlAdv.

0

200

Colloid Znterface Sci. 73 (1997) 146

400

660

Film Thickness ( A ) Fig. 21. Shear and normal forces as a function of separation for two surfaces bearing PS-PVP polymer brushes [501. (H) normal forces, (G,E) shear viscous forces. (Reproduced with permission from American Chemical Society, Washington, USA.)

layers). The observation of an increasing additional normal force beyond a certain critical distance (- 200 nm) followed by a decrease beyond a maximal point has been explained by Klein ‘et al. along the following lines. At very large separations (> 200 nm) the shear rate in the gap is sufficiently low to not lead to any significant stretching of the chains. This in turn means that there is no thickening of the layers and consequently no increase in the repulsion between the layers. At lower separations (C 200 nm) however, there is an increase in the shear rate in the gap for a given applied shear rate, and the effective shear rate in the gap continues to increase as the gap separation decreases. The immediate result of such an increase in the effective shear rate is the manifestation of an additional normal force that also increases with decreasing separation. At still higher compressions (< Donset)however, the increased concentration of chains in the gap reduces the efficiency of chain stretching in promoting repulsive interactions between the chains [42]. This eventually leads to a decrease in the measured additional force, as seen in Fig. 20. Finally, some work has also been reportedly carried out to compare the shear behaviour of brush-brush interactions with those between a brush and a bare mica surface. This work has been presented in a review by Granick et al. [501and seems to suggest greater lubrication efficiency when only one of the surfaces is coated with a PS-PVP polymer brush

P.F. Luckham,

S. ManimaaranlAdv.

lo2

ifm&

A *r

10’:

31

I

I .

-

Colloid Interface Sci. 73 (1997) 146

A

A A

%

A 0 loo:

0

A

?oOOA& ? 0

0

A

0

O0 Id 0

0 I 100

Oo

0

I 200

300

Film Thickness ( i ) Fig. 22. Shear and Normal force for a system where one surface bears the PS-PVP brush and the other is bare 1501. (Reproduced with permission from American Chemical Society, Washington, USA.)

(see Figs. 21 and 22). A possible explanation for this observation may be that there is no longer a possibility for polymer layers on opposing surfaces to interpenetrate when there is only one such layer.

Theory

At present there is no comprehensive theory that provides a complete account of the various phenomena observed during the course of shear experiments. However, a few models covering particular aspects of behaviour have been presented and these will be reviewed individually below (i) Frictional properties A basic model to interpret frictional properties during the shear of polymer brushes has been presented in a recent review paper by Klein [51]. A brief summary of this model follows below. In this model he first considers the extent, d, to which chains on opposing surfaces interpenetrate as two polymer brushes are compressed (see Fig. 18). In order to do so he assumes a parabolic brush

P.F. Luckham, S. ManimaaranlAdv.

38

Colloid Interface Sci. 73 (1997) l-46

structure for the adsorbed layers [521 as opposed to the step function picture incorporated by Fredrickson and Pincus 1281in their equivalent analysis in the case of the normal dynamic experiments. To estimate d he then considers the extent of penetration of one chain from a brush into the field of the other brush. The structure of the brushes is accounted for in this model by assigning a parabolic potential field for each brush [51]. By only accounting for moderate compressions of the brushes he then retrieves a rather weak relationship, shown below, for the variation of d with brush compression, D.

Moving to the particular case of brushes sliding past each other, he then assumes that the energy dissipated during sliding is a direct result of the friction experienced by the chains when they are dragged through the overlap region, d. The resulting shear stress for shear rates below the I&mm relaxation time is then given by [511 oS = (no. of chains/unit drag/blob) =

area)

(no. of blobs in the zone d)

Ws2>. ~z~,~(D)) (6 7~yeti, &@I

(frictional

(25)

where s is the screening length at the anchoring surface, nb d is obtained by noting that the mean volume occupied by each chain in the zone is given by (s2.d/2), so that nb,d = [(s2d/2)/{A(D)31, C&is the Alexander blob size, and the last term in Eq. (25) is the Stokes drag term for a blob of the aforementioned size moving with a velocity u,. The effective friction coefficient, peff, can thus be retrieved as it is simply the ratio of the shear force to the normal force. The Alexander-de Gennes [32,331 view of the force encountered during brush interactions is given by kBT

f(D) = 7

(pg’4 - p-3’4)

where p = 2L,/D, the compression ratio. For moderate compressions Klein then makes use of the dominance of the osmotic interactions (the first term) in the above equation, and by then substituting.the expressions for shear and normal forces retrieves an expression for the effective friction:

P.F. Luckham,

S. ManimanranlAdu.

Colloid Interface Sci. 73 (1997) l-46

peff= os /f(D) = (6n3jefiu, s’P-~‘~) Ik,T

39

(27)

This expression predicts that for a given chain size, the denser the brush (i.e. the smaller the s) the greater the effective lubrication. In addition, it also predicts a lower friction coefficient for higher (relative) compressions (large values of p> of the brush. This simply means that, in the bounds of applicability, the increase in normal forces are far greater than the increase in shear forces. It should be noted though that this prediction only applies up to moderate compressions of the brush (in line with the basic assumptions made in deriving the model), it is not applicable in the highly compressed regions where high shear forces have been encountered during the course of experiments [20,42,44-461. (ii) Brush swelling

The possibility of a polymer brush swelling under the imposition of a shear force was first examined by Rabin and Alexander [53] and then investigated in more detail by Barrat 1541.In essence the workings of the model can be illustrated by Fig. 23. The model is based on the step function concentration profile for the brushes suggested by Alexander [321 and de Gennes [33]. Each chain is then assumed to be subject to a force parallel to the surface, as shown in the Fig. 23. The effect of this shear force is thought to stretch the chains beyond their equilibrium dimension, L,, and the new configuration of the chain is described by a chain end to end vector L = (h,,h,,). In

f shear /chain

Fig. 23. Schematic representation of a sheared polymer brush 1511. (Reproduced permission from Annual Reviews Inc., California, USA.)

with

40

P.F. Luckham, S. ManimaaranlAdv.

Colloid Interface Sci. 73 (1997) 146

order to calculate L, Barrat then considers the Gibbs free energy of a chain and expresses it as

U(L) = UCL)str&&+ UCL)i,t + h[!jjl where L is the end to end vector of the stretched chain; L is the corresponding length; h,, h,, are the mean chain dimensions normal and parallel to the surface; fshearis the applied shear force; U(L) is the total is the elastic energy in a stretched chain; and brush energy; U(L)stretch U(L)i,t is the interaction energy between blobs. By then minimising the total chain energy Barrat retrieves the result [541

(29)

This result by Barrat predicts a maximum increase in chain length of up to 25% during shear experiments. The model as it stands is restricted to interpretations of interactions amongst polymer brushes and not homopolymers. As in previous section accounting for more complicated polymer layer structure should then allow more systems to be understood.

(iii) Molecular dynamics simulation (MDS)

In a study conducted by Grest [55] he extends his previous work concerned with the Molecular Dynamics Simulation (MDS) of normal force distance profiles obtained from SFA experiments. MDS runs were conducted to follow change in frictional properties as well as changes in brush thickness during the course of shear experiments. Chain molecules are simulated here using a bead spring model in which monomers interact via a Lennard-Jones potential. A purely repulsive interaction potential is then assigned to the interactions that take place between two flat plates with adsorbed polymer layers. Shear is imposed on the system by pulling the top plate with a velocity u in the x-direction. In performing the simulations the number of polymer brush chains, the temperature and the separation between the plates are held constant.

P.F. Luckham,

S. Manimaaran

300.0

IAdv. Colloid Interface Sci. 73 (1997) l-46

I

I

I A

“b

200.0-

6

\ A NO (1:

0 A

VlOOOA +e

0

0

I

I

0.0

0

0

? ?o

I A

0 0

Ao

41

I

I

A 0.06 2

-

0.04-

A A A

43 0

0

0.02 -

G?

A

0

0

O.dO5 0

0 0 0

0

o.b:T;t&

0.b2

0.025

Fig. 24. Shear stress f per unit area and mean square radius of gyration CR’+ as a function of velocity [55]. (Reproduced with permission from American Institute of Physics, New York, USA.)

Figure 24 provides an insight into the results retrieved by Grest. In the figure the dependence of the shear force, f, on sliding velocity is shown for three values of plate separation as well as the change in the radius of gyration (R,). For small values of the velocity, U,f displays a linear increase with velocity whereas little or no change was observed for R, in the same regime. The linear variation of fwith u is the type of behaviour expected from a Newtonian fluid. At higher velocities however, f no longer responds in a linear manner (i.e. non-Newtonian behaviour), and the chains can also be seen to stretch - increasing c&>~. There is also the possibility of polymer disentanglement from the surfaces in this regime. The simulation results do not however respond quite as rapidly as SFA results to decreasing compressions (see Fig. 25). In SFA experiments [42,44-46] the shear stress was seen to increase rapidly and quite dramatically at high compressions whereas the simulations show a more gradual change -one that is more in line with the normal force profiles. Interestingly though, when these results are reinterpreted in terms of a friction coefficient, l.~,along the lines of the Klein study, the predictions are seen to show a decrease in the friction coefficient with decreasing

42

P.F. Luckham, S. ManimaaranlAdv.

. 0.0

0.2

Colloid Interface Sci. 73 (1997) I46

“00

%o (A)_ D

I

I

0.4

0.6

I

0.8

D/No Fig. 25. Semilog plot of normal load P and shear force f between polymer brushes as a function of separation [551. The inset shows the results of Klein et al. [42]. (Reproduced with permission from American Institute of Physics, New York, USA.)

surface separation. The decrease in magnitude can be as large as a factor of five! This is in obvious contradiction of the SFA results and may well be a response to the model system being based on a parallel plate model, as opposed to a sphere on a flat model. The Derjaguin model [56] has been successfully used to translate normal force relationships from the two disparate geometrical systems into comparable energy calculations [57]. However, at present the relationship for shear force in the two systems remains unclear and may be partially responsible for the disagreement. It has also been pointed out that the larger range D over which fhas been investigated and the higher velocities which have been used may have also been responsible. Finally it should also be noted that at present it is unclear as to the correspondence of the two forms of data i.e. it is unclear as to which reading from an experiment corresponds to a particular theoretical reading [55].

Concluding

remarks

In comparing the theory constructed to deal with the behaviour of terminally attached layers undergoing normal dynamic motion [281 with that ofexperimental behaviour [5,17,19,20,22,23], somefavourable though rather basic analogies have been found. Namely they both show a dramatic change in behaviour upon the onset of polymer interaction, with the polymer layers acting as extended surfaces (to some extent).

P.F. Luckham, S. ManimaaranlAdv.

Colloid Interface Sci. 73 (1997) l-46

43

However, the extent to which there is solvent penetration of the layers is unclear at present and remains a rather important topic for further investigation, as it will provide insight into the actual nature of the polymer-solvent interaction. All the (successful) investigations thus far, point to little or no interpenetration of the layers [5,19,20]. The lack of concise data covering a wide range of frequencies however, has limited both the qualitative and quantitative comparisons that can be carried out. Further work will provide a clue as to the extent of applicability of the model - namely a detailed comparison with the final predictions summarised by Fig. 13. The present model proposed by Fredrikson and Pincus [28] is a rather basic one as it does not account for the flow of free polymer through the network and relies on a step function profile. Consequently it will not be surprising if only a limited correlation can be found between experiment and theory even when more experimental work has been carried out. It would interesting to note the effects on the model of a more realistic concentration profile and the incorporation of the reptation idea proposed by de Gennes 1581. Reptation is currently the best model available to simulate the movement of polymers. However, theoretical models do require experimental results to help channel attention into areas of greatest concern, and therefore it is vital that more detailed work is carried out on these systems experimentally. Moving to adsorbed layers undergoing normal motion, the theory of Sens et al. 1291has found some very favourable comparisons with the work of Montfort et al. 19,101.Namely, a Maxwell-like variation of the moduli was detected at low frequencies. However, a curious feature of the Montfort results was the deviation from Maxwell-like behaviour at higher frequencies 19,101.Although Sens et al. 1291seem to find this compatible with their high frequency predictions it is unclear whether this is strictly correct. The high frequency model is said to come in at frequencies above the Zimm relaxation time which is of order lo5 s. The frequencies at which the deviation in the Montfort results are observed are of order 10 Hz. Although it is possible that a dynamic blob model is more appropriate at these frequencies, it should also be recognised that the simplicity of the models may once again play a significant role ignoring reptation and adsorption desorption effects. In the particular case of adsorbed (as opposed to terminally attached) layers, there is a strong case for considering adsorption and desorption dynamics. An independent theoretical investigation for this has already been considered by Joanny et al. [59,60]. Incorporating these findings into the Sens

44

P.F. Luckham, S. ManimaaranlAdv.

Colloid Interface Sci. 73 (1997) 146

model may prove fruitful. Finally, it should also be recognised that at high frequencies, the internal modes of the layers may well be excited, and the semi-dilute picture may well need to be revised [9, lo]. The experiments of Klein et al. [20,42,44-461 comparing the effects of adsorbed and terminally attached layers have provided a useful insight into the possibility of using polymer layers to modify the frictional properties of a confined system. They have undoubtedly provided us with proof of the adavntages posessed by polymer bearing surfaces. However, they do not provide conclusive evidence for the preference of terminally attached layers over adsorbed layers, as the effects of solvent quality are not negligible. Until similar experiments have been carried out on terminally attached layers under poor solvent conditions and adsorbed layers under good solvent conditions it will be unjust to draw wide ranging conclusions. The theoretical models [51,54,55] reviewed thus far seem in effect to present very similar conclusions to those presented in the section dealing with normal motion. They provide the reader and the experimentalist a (theoretical) basis for understanding the reasons promoting reduced friction [20,42,44-46] and increased normal forces [421 during shear. However, as before, the relative infancy of the subject matter in question has resulted in rather basic models and also a very limited array of experimental results. At present, the systems investigated theoretically have been restricted to brush layers in good solvents, thereby neglecting the effects of solvent quality, strength of adsorption and effects of entanglement, that are quite significant in adsorbed layers. In more recent times the Atomic Force Microscope (AFM) has been successfully employed to measure equilibrium forces, and their success at doing so has renewed interest in dynamic experiments. Recent works by Overney et al. 1611 and other workers [62,631 using the normal dynamic mode, Mate et al. [641 and other groups 165-671 using the lateral mode have indicated a definite route for progress using this tool. Although some concerns have arisen regarding the applicability of models derived specifically for the SFA for use on AFMs, this may well prove a minor hurdle. The adaptability of the AFM to all manner of surfaces and its ease of usage at high frequencies (much higher than those used in the SFA), will make it a useful tool to run in parallel with the DSFA. The DSFA will remain more efficient at lower frequencies. As mentioned before progress in this field relies on a combined effort between theoreticians and experimentalists, and future experimental work is bound to pay dividends.

P.F. Luckham,

S. ManimuuranlAdv.

Colloid Interface Sci. 73 (1997) 1-G

45

Acknowledgement

We would like to thank the E.P.S.R.C. (RoPA Grant: GRK 37420) for their financial support for S.M., and Mr. G.J.C. Braithwaite for help with this work. References

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