Langmuir 1992,8, 2077-2079
2077
Comments Comment on the Interaction Forces of Polymer-Coated Surfaces In a recent paper' Costello et al. compared measurements of the interaction between mica surfaces coated with a monolayer of a comb-graft copolymer of poly(l2hydroxystearicacid) and poly(methy1methacrylate) made using the surface forces apparatus (SFA), with storage modulus2measurements made on nonaqueousdispersions (NAD)stabilizedby the same copolymer,and with osmotic pressure3 measurements made on NAD stabilized by a different batch of copolymermade by the same procedure. The comparison between the osmotic pressure and SFA measurements was made in terms of an interaction potential between flat plates derived using the Derjaguin approximation. Both measurements indicated the onset of significant interaction at a separation of about 20 nm but showed very different decay lengths, ca. 2.4 nm in the case of SFA and ca. 0.5 nm from the osmotic pressure. Costello et al. held that the SFA measurements were considerably more sensitive than the osmotic pressure measurements and concluded that the two sets of data were in tolerable agreement. The SFA and storage modulus data were compared by attempting to predict the latter from the former using an to the Zwanzig and Mountain equation4 which reads
where N = particle number concentration, dm = packing constant, n = coordination number, and R = center-tocenter distance. The force between spheres dVldR and its first derivative were obtained from the potential between flat plates derived from the SFA measurements E(H) by means of the Derjaguin approximation, viz. d'V(R) -- *a d"-"E(H)
dR'
dp-1'
It is worth noting that the procedure used to obtain the modulus from E(H) is similar to the reverse of the procedure used to obtain E(H) from the osmotic pressure measurements, the approximate relationship used being derivable from the relevant integral equation in the same way as eq 1 is derived from the Zwanzig and Mountain equation. Costello et al. found that while the form of the dependence of modulus on volume fraction was predicted correctly, the absolute magnitude was overestimated by about a decade. This is tantamount to agreement as to the decay length but disagreement as to the strength of the interaction at a given absolute separation, a shift to
shorter distances of about 7 nm being required to reconcile the results. Costello et al. tentatively suggested particlesize polydispersity as the cause of the discrepancy. However, as was noted above, the osmotic pressure and the modulus bear the same kind of relationship to E(H), whereas the osmotic pressure data indicate stronger rather than weaker interaction. Modulus data for similar NAD with different particle sizes are a~ailable.~~' It is shown below that these data give rise to potentials which are not inconsistent with the osmotic pressure and SFA results. An apparent potential can also be obtained from viscosity data. In a recent paper5the researcher proposed a scaling method for obtaining the apparent interaction potential of soft-sphere particles from the non-Newtonian viscosity. Unlike the methods used to infer a potential from the osmotic pressure or modulus, the method allows an apparent potential to be derived from measurements at a single volume fraction. In the case of the modulus (or pressure), single-point information on a derivative of the potential is obtained at any one volume fraction and variation of the volume fraction probes the distance scale. In the case of the viscosity method, the non-Newtonian viscosity tracks distance and the shear stress tracks the energy. The method is heuristic, and dimensionalanalysis apart, it lacks a firm theoretical basis. It can however be validated by confirming that the apparent potential obtained is independent of volume fraction. The method is based on the supposition that the probability of two particles approachinga center-to-center distance R is given by P(R) = exp(-V(R)/t) (3) where t is the collision energy; therefore, their effective collision diameter is (4)
this being identical to a relationship given by Barker and Henderson if t = kT. For collisions under shear, e is taken to be given by5 t
kT(1 + ~
/ T J
(5)
where 7 is the shear stress and 7,
= 7,(4,ff)
4,ff = N*R3/6
is the characteristicstress for shear thinning of hard-sphere dispersions. The potential is obtained by inversion of eq 4 as described in ref 5. The distance scale is obtained from the viscosity using a correlation for hard spheres, for example, the Krieger-Dougherty equation, where
?(7)/Tm4ium = (1 - 4eff(7)/4m)-5'"2
(1) Coatello, B. A. de L.; Luckham, P. F.; Tadros, Th. F. Langmuir 1992,8, 464.
(2) Frith, W. J.; Strivens, T. A.; Mewis, J. J . Colloid Interface Sci.
-1990. _ _,_-1. -3-9., 6.5. - -.
(3) Cairns, R. J. R.; Ottewill, R. H.; Oamond, D. J. W.; Wagstaff, 1. J . Colloid Interface Sci. 1976,54, 51. (4) Zwanzig, R.; Mountain, R. D. J . Chem. Phys. 1965, 43, 4464. (5) Buscall, R. J . Chem. SOC.,Faraday T r a m . 1991,87, 1365. (6) Evans, I. D.;Lips, A. J. Chem. Soc.,Faraday Tram. 1990,86,3413.
Apparent potentials obtained from the viscosity of NAD (7) Mewis, J. Private communication, Katholieke Universiteit Leu-
ven.
0 1992 American Chemical Society
2078 Langmuir, Vol. 8, No. 8,1992
Comments 500.0
parameter:- effective volume fraction ((pe,=1.74(p)
\SFA
'\
30
25
I
I
1
I
1 1 ,
o
0
0712 0700 0676
200.0
100.0 -_ 50.0
0 638
\ \
v!
aref = 237 nm
P
20.0 ''
kY
10.0 --
?-
5.0 ..
E
kY
2.0
i 10
\x
\ \
. 1
\
y
\
"
"
1 .o --
0.5 ..
.
00
0.2 ' 0.1 t
AAA
,
00
H=(R-Zo)/nm
Figure 2. Scaled interactionpotential (V(R)adakT;a d = 237
0
5
10 H=(R-Zo)/nm
15
20
25
Figure 1. Apparent interactionpotential derived from viscosity
for 42-nm-radius NAD particles stabilized by PHS and four different volume fractions. The parameter distinguishing the plots is the effective volume fraction calculated from the weight fraction of solid ( w ) and the intrinsicviscoaity [VI; i.e., = (2/6)[~lw.
will be compared with those obtained from modulus, pressure, and SFA below. v Note that the model assumes that the limiting low- and high-shear, relative viscosities show a hard-sphere-like dependence on effective volume fraction, with the latter defied as R%h(low)
$%h(low)
(
2a
nm)for PHS-stabilizedparticles: open circles, from modulus for 237- and 98-nm NAD;open triangles, from modulus for 146-nm NAD;dashed line labeled G, from modulus for 42-nm NAD; dashed-dotted line labeled P, from osmotic pressure; solid line labeled V, from viscosity;dashedlines labeledSFA, from surface forces apparatus, the two lines bracket the data of Costello et al. 10"
1
I '
exp'tal predicted
a=237nm
5
S
lo4
I
R
li ,42nm
i
3
)
$0
(7)
where R1""is the separation, where V(R)equals ca. 0.5 kT, a n d R s % the separation where it diverges. Hard-sphere scaling of the concentration dependence of the limiting viscosities of stericallystabilized particles has been shown elsewhere.2ve10
Results and Discussion Modulus data for NAD of four particle sizes (radius a = 42,98,145, and 237 nm) were provided by the Leuven The viscosity data (a = 42 nm) were likewise taken from the thesis of Frith2 and have been discussed in the literature? The potentials obtained from the viscosity data for four volume fractions are shown in Figure 1. The excellent superposition gives confidence in the method; note that the viscosity varied by nearly a million-fold. In order to compare the different methods,it was chosen to calculate the scaled sphereephere potential V*(R) = V(R)a,f/akT (8) with 237 nm as the reference radius (the flabplatepotential could equally well have been derived). The results are shown in Figure 2. It can be seen that the potentials derived from the modulus for three out of four of the NAD agree with t h w derived from the SFA data of Costello et al. They are also consistent with the potentials derived from the viscosity for the 42-nm particles and are not inconsistent with those derived from osmotic pressure. (8) Frith, W. J. Ph.D. Thesis, Katholieke Universiteit Leuven, 1986. (9) Mewis, J.; Frith, W. J.; Strivene, T. A.; Russel, W. B. AZCHE J. 1989,35,415. (10) DHaene, P. Ph.D. Thesis Katholieke Universiteit Leuven, in preparation.
loo
1
0.50
0.55
0.60
0.65
0.70
0.75
0.80
I
0.85
Effective volume fraction
Figure 3. Scaled moduli GaS/kTas a functionof volume fraction
q k = (2/S)[qlw: dashed linea, experimental data,a73 solid line, predictions made using eq 3 together with a fit to the potential for 42-nm particles from F w e 1 scaled with size, i.e., V(H,a) = (a + H) V(H,42)/(42 + H).
The inconsistencies are that the potential derived from the modulus for the 42-nm particles is displaced inward from the other curves, and that the viscosity and pressure indicate a hardeningof the interaction at ca. 15nm whereas the SFA show a continued exponential dependencedown to distances where there is 75% interpenetration of the adsorbed layers (i.e., penetration with compression). It should be noted that three different batches of copolymers t a b h r have been used. The PHS chains are polydisperse, and polydispersity ie likely to vary batch to batch in condensation polymers of this type. Thus, a spread in the tail is to be expected. The superficial impreeeion gained is that the modulus data for the 42-nm particles are anomalous in some way. This is also suggested by Figure 3 where the modulusdata for the four particle sizes are compared with predictions made using the potential obtained from the viscosity; the 42-nm particlea apart, the agreement is very satisfactory, given the numerical manipulations that have to be performed. It would however be remiss of the researcher not to point out a source of uncertainty in the results obtained from the viecosity,modulus, and osmotic preasure data. In these cases a value for the packing constant &, has to be assumedand the distance scale is sensitive to the choice of this value. Here, 0.7404 has been wed for the 42-nm particles (face-centered cubic packing) and 0.64
Langmuir, Vol. 8, No. 8,1992 2079
Comments
(random close-packing) for the 237-nm particles. This choice was not entirely arbitary; the contrary choice gives interaction at unrealistically small and large distances, respectively, and in the case of the 237-nm particles is inconsistent with the divergenceof the modulus and highshear viscosity. Equation 1 suggests that the quantity
been generated, and these will be analyzed using the methods described here.1° Overallthe results availabledo not appear to necessitate a conclusion of Costello et al. that the models used (to calculate the modulus from the potential) do not hold at high volume fractions. Acknowledgment. J. Mewis, Katholieke Universiteit Leuven, is thanked for providing the modulus data in tabular form.
-
should be a unique function of R given the correct choice of the packing constants, by virtue of the fact that V(R) a and R 2a. The choices used here gave the best scaling in the light of this relationship when 0.7404 was also used for the 98- and 137-nmparticles. However, given both modulus and viscosity data for a sufficiently large number of samples of differing particle size, it should be possible to eliminate entirely any ambiguity. Modulus and viscosity data for further particle sizes have recently Qc
Richard Buscall ICI Corporate Colloid Science Group, P.O.Box 11, The Heath, Runcorn, Cheshire WA? 4QE, United Kingdom Received April 17, 1992. In Final Form: May 26,1992 Registry No. (12-Hydroxystearic acid)(methyl methacrylate) (graft copolymer), 128364-06-7.
